12th International ISAAC Congress

Volume of Abstracts

12th International ISAAC Congress—

Abstracts

Edited by P. Cerejeiras and U. Kahler. Prepared and typeset using LATEX.

Departamento de MatematicaUniversidade de AveiroCampus de SantiagoP-3810-193 Aveiro, Portugal

Welcoming address

The ISAAC board, the Local Organising Committee and the Department of Mathematics at Imperial CollegeLondon, are pleased to welcome you to the 12th International ISAAC Congress in Aveiro. The 12th Inter-national ISAAC congress continues the successful series of meetings previously held in the Delaware, USA(1997), Fukuoka, Japan (1999), Berlin, Germany (2001), Toronto, Canada (2003), Catania, Italy (2005),Ankara, Turkey (2007), London, UK (2009), Moscow, Russia (2011), Krakow, Poland (2013), Macao, China(2015), and Vaxjo, Sweden (2017).

The success of such a series of congresses would not be possible without the valuable contributions of all theparticipants.

We acknowledge the financial support for this congress given by

the FCT–Fundacao para a Ciencia e a Tecnologia (FCT), through the Fundo de Apoio a ComunidadeCientıfica (FACC),

the Universidade de Aveiro (UA),

the Center for Research and Development in Mathematics and Applications (CIDMA),

and the Departamento de Matematica da Universidade de Aveiro.

Also, we wish to thank the many Portuguese companies who’s support was crucial for the success of thisevent:

Nestle Portugal, Porto de Aveiro, Real Companhia Velha, Sociedade da Agua Luso,

TAP Air Portugal, The Navigator Company, and Turismo Centro de Portugal,

ISAAC BoardMichael Reissig (Freiberg, Germany), President of the ISAACJoachim Toft (Vaxjo, Sweden), Vice President of the ISAACIrene Sabadini (Milano, Italy), Secretary and TreasurerLuigi Rodino (Torino, Italy), Former PresidentRobert P. Gilbert (Newark, Delaware, USA), Honorary PresidentHeinrich Begehr (Berlin, Germany)Okay Celebi (Istanbul, Turkey)Paula Cerejeiras (Aveiro, Portugal)Massimo Lanza de Cristoforis (Padova, Italy)Anatoly Golberg (Holon, Israel)Uwe Kahler (Aveiro, Portugal)Vladimir V. Mityushev (Cracow, Poland), WebmasterMichael Oberguggenberger (Innsbruck, Austria)Stevan Pilipovic (Novi Sad, Serbia)Tao Qian (Macao, China)Michael Ruzhansky (Ghent, Belgium / London, U.K.)Mitsuru Sugimoto (Nagoya, Japan)Ville Turunen (Aalto, Finland)Jasson Vindas (Ghent, Belgium)Jens Wirth (Stuttgart, Germany)Man Wah Wong (Toronto, Canada)

Local Organising CommitteePaula Cerejeiras (Chair)Uwe KahlerJames Kennedy (Lisboa)Milton Ferreira (Leiria)Nelson Vieira

with further support by Claudio Henriques and Fabio Henriques (technical support) as well as further studenthelpers.

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Abstracts

Plenary talks 15Afonso S. Bandeira : Statistical Estimation with Algebraic Structure: Statistical and Computational Con-

siderations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Alexander Grigor’yan : Analysis on fractal spaces and heat kernels . . . . . . . . . . . . . . . . . . . . . . . 15Karlheinz Grochenig : Totally positive functions in sampling theory and time-frequency analysis . . . . . . . 16Hakan Hedenmalm : Planar orthogonal polynomials and related point processes: random norm matrices and

arithmetic jellium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Andre Neves : Recent progress in the theory of minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . 17Roman Novikov : Multidimensional inverse scattering problem . . . . . . . . . . . . . . . . . . . . . . . . . 17Tohru Ozawa : Self-similar solutions to the derivative nonlinear Schrodinger equation . . . . . . . . . . . . 17Samuli Siltanen : Inverse Problems and the Nonlinear Fourier Transform . . . . . . . . . . . . . . . . . . . 18Durvudkhan Suragan : Hardy inequalities on homogeneous groups . . . . . . . . . . . . . . . . . . . . . . . 19

Sessions 21

S.1 Applications of dynamical systems theory in biology 21Yuanji Cheng : Backward bifurcation in SEIRS model with satuation incidence rate and treatment . . . . . 21Yan Fyodorov : Nonlinear analogue of the May-Wigner instability transition . . . . . . . . . . . . . . . . . 21Valery Gaiko : Global bifurcations of limit cycles and strange attractors in multi-parameter polynomial

dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Natali Hritonenko : Age- and size-structured models of population dynamics: optimal control and sustain-

ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Anatoli Ivanov : Two-dimensional differential delay systems as mathematical models of physiological pro-

cesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Flavia Lanzara : On the limit configuration of four species strongly competing systems . . . . . . . . . . . . 22Ana P. Lemos-Paiao : Applications of optimal control theory to cholera . . . . . . . . . . . . . . . . . . . . 22Torsten Lindstrom : Destabilization, stabilization, and multiple attractors in saturated mixotrophic environ-

ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Gulden Murzabekova : The inverse problem on tree graph with attached masses from the point of view of

neurobiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Karlygash Nurtazina : Identification algorithm for differential equation with memory in biomathematical

problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Telmo Peixe : From Lotka-Volterra systems to Polymatrix Replicators . . . . . . . . . . . . . . . . . . . . . 23Gergely Rost : Global dynamics of a new delay logistic equation arisen in cell biology . . . . . . . . . . . . 23Shigui Ruan : Bifurcations on Invariant Tori in Predator-prey Models with Seasonal Prey Harvesting . . . 24Cristiana J. Silva : Stability and optimal control of a delayed HIV/AIDS-PrEP model . . . . . . . . . . . . 24

S.2 Complex Analysis and Partial Differential Equations 24Umit Aksoy : Integral representations and some boundary value problems in Clifford analysis . . . . . . . . 24A. Okay Celebi : Boundary Value Problems in Polydomains . . . . . . . . . . . . . . . . . . . . . . . . . . 24Abdelkerim Chaabani : Well-posedness and asymptotic results of 3D Burgers equation in Gevrey class . . . 25Natalia Chinchaladze : Existence and Uniqueness Theorem for Cusped Kelvin-Voigt Elastic Plates in the

Zero Approximation of the Hierarchical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Bakur Gulua : Derivation of system of the equations of equilibrium for shallow shells and plates with voids 25Yongmin Liu : The product-type operators from logarithmic Bloch spaces to Zygmund-type spaces . . . . . . 25Grzegorz Lysik : Higher order mean value functions and polyharmonic functions . . . . . . . . . . . . . . . 25Nino Manjavidze : On the structure of generalized analytic function in the vicinity of singular point . . . . 25Rui Marreiros : On the dimension of the kernel of a singular integral operator with non-Carleman shift and

conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Mohamed Kainane Mezadek : Global existence for coupled to the structurally damped σ-evolution model . . 26Madi Muratbekov : On the existence and compactness of the resolvent of a Schrodinger operator with a

negative parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Nusrat Rajabov : To theory one class of three-dimensional integral equation with super-singular kernels by

tube domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Xiaoyin Wang : 2-D Frequency-Domain System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 26Huang Yun : Paley–Wiener-type Theorem for Analytic Functions in Tubular Domains . . . . . . . . . . . . 27

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S.3 Complex Geometry 27Ana Casimiro : Higgs bundles and Schottky representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Camilla Felisetti : Intersection cohomology of the moduli space of Higgs bundles on a smooth projective curve 27Carlos Florentino : Generating Functions for Hodge-Euler polynomials of GL(n,C)-character varieties . . . 27Emilio Franco : Cartan branes on the Hitchin system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Peter Gothen : SO(p, q)−Higgs bundles and higher Teichmuller components . . . . . . . . . . . . . . . . . 28Viktoria Heu : The Riemann-Hilbert mapping in genus two . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Norbert Hoffmann : Universal torsors over degenerating del Pezzo surfaces . . . . . . . . . . . . . . . . . . 28Angel Luis Munoz-Castaneda : Compactification of the moduli space of stable principal G-bundles over a

stable curve and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Angela Ortega : Klein coverings of genus 2 curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Ana Peon-Nieto : Mirror symmetry on some non generic loci of the Hitchin system . . . . . . . . . . . . . 29Francesco Polizzi : Surface braid groups, finite Heisenberg covers and double Kodaira fibrations . . . . . . . 29Martha Romero : On Galois group of factorized covers of curves . . . . . . . . . . . . . . . . . . . . . . . . 29Florent Schaffhauser : Higher Teichmuller spaces for orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . 29Tom Sutherland : Theta functions on moduli spaces of local systems . . . . . . . . . . . . . . . . . . . . . . 29

S.4 Complex Variables and Potential Theory 30Tugba Akyel : Some Remarks on the Uniqueness Part of Schwarz Lemma . . . . . . . . . . . . . . . . . . . 30Mohammed Alip : Stokes’ Theorem and the Cauchy - Pompeiu Formula for Polydisc . . . . . . . . . . . . 30Alberto Cialdea : Completeness theorems for systems of particular solutions of partial differential equations 30Matteo Dalla Riva : Existence, uniqueness, and regularity properties of the solutions of a nonlinear trans-

mission problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Iryna Denega : Extremal decomposition of the complex plane with free poles . . . . . . . . . . . . . . . . . . 30Grigori Giorgadze : On the irregular generalized Cauchy-Riemann equations . . . . . . . . . . . . . . . . . 31Anatoly Golberg : Nonlinear Beltrami equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Christopher Green : Harmonic measure distribution functions on spherical and toroidal surfaces . . . . . . 31Serhii Gryshchuk : Monogenic functions in 2-D commutative Complex algebras to plane orthotropy . . . . . 31Fiana Jacobzon : Linearization of holomorphic semicocycles . . . . . . . . . . . . . . . . . . . . . . . . . . 32Bogdan Klishchuk : Lower bounds for the volume of the image of a ball . . . . . . . . . . . . . . . . . . . . 32Gabriela Kohr : Approximation properties and related results for univalent mappings in higher dimensions 32Massimo Lanza de Cristoforis : An inequality for Holder continuous functions, in the wake of the work of

Carlo Miranda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Paolo Luzzini : Shape analysis of the longitudinal flow along a periodic array of cylinders . . . . . . . . . . 33Krzysztof Maciaszek : On polynomial extension property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Paolo Musolino : Converging expansions for Lipschitz self-similar perforations of a plane sector . . . . . . 33Mohamed M S Nasser : Numerical computation of conformal capacity of generalized condensers . . . . . . 33Mitja Nedic : Support of Borel measures in the plane satisfying a certain positivity condition . . . . . . . . 34Makhmud Sadybekov : Nonlocal boundary value problems for the Laplace operator in a unit ball which are

multidimensional generalizations of the Samarskii-Ionkin problem . . . . . . . . . . . . . . . . . . . . 34Vitalii Shpakivskyi : Hypercomplex method for solving linear PDEs . . . . . . . . . . . . . . . . . . . . . . 34Luis Manuel Tovar Sanchez : Weierstrass Theorem for Bicomplex Holomorphic Functions . . . . . . . . . . 35

S.5 Constructive Methods in the Theory of Composite and Porous Media 35Olaf Bar : Fast method for 2D Dirichlet problem for circular multiply connected domains . . . . . . . . . . 35Wojciech Baran : Local Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Mikhail Borsuk : The Robin problem in conical domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Marta Bryla : Conductivity of two-dimensional composites with randomly distributed elliptical inclusions:

Random elliptical inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Roman Czapla : Estimations of the effective conductivity of composites with non-circular inclusions . . . . 36Piotr Drygas : Effective elastic constants for 2D random composites . . . . . . . . . . . . . . . . . . . . . . 36Jan Guncaga : Education of Selected Notions Connected with Mathematical Analysis with Help of Visual-

ization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Pawe l Kurtyka : On quantitative assessment of In-situ and ex-situ composites structures with micro and

nano-particles reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Vladimir Mityushev : R-linear problem and its application to random composites . . . . . . . . . . . . . . . 36Wojciech Nawalaniec : Classifying and analysis of random composites using structural sums feature vector 37Kazimierz Rajchel : Trygonometric approach to the Schrodinger equation as a general case of known solu-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Roman Rosiek : Analysis of the degree of image complexity used in eye tracking research on STEM education 37Natalia Rylko : Inverse problem for spherical particles and its applications to metal matrix composites

reinforced by nano TiC particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Miroslawa Sajka : Three mental worlds of mathematics in pre-service mathematics teachers - an eye-tracking

research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Maia Svanadze : Boundary integral equation method in the theory of binary mixtures of porous viscoelastic

materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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S.6 Fixed Point Theory, Ulam Stability, and Related Applications 38Obaid Alqahtani : A New Approach to Interval Matrices and Applications . . . . . . . . . . . . . . . . . . 38Bouchra Ben Amma : Existence of Solutions to Boundary Value Problems for Intuitionistic Fuzzy Partial

Hyperbolic Functional Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Krzysztof Cieplinski : A fixed point theorem in n-Banach spaces and Ulam stability . . . . . . . . . . . . . 39Marija Cvetkovic : Impact of Perov type results on Ulam-Hyers stability . . . . . . . . . . . . . . . . . . . . 39Amar Debbouche : Mittag-Leffler stability analysis for time-fractional partial differential equations . . . . . 39Inci Erhan : Application of F -contraction mappings to integral equations on time scales . . . . . . . . . . . 39Andreea Fulga : Fixed points of discontinuous mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Tayeb Hadj Kaddour : Local and global solution for effectively damped wave equations with non linear

memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Erdal Karapinar : Recent topics on metric fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Safeer Hussain Khan : Attractive Points by a Modern Iterative Process . . . . . . . . . . . . . . . . . . . . 39Ozge Bicer Odemis : Some related fixed point theorems for multivalued mappings on two metric spaces . . . 40Taoufik Sabar : On best proximity point theory: From cyclic mappings to Tricyclic ones. . . . . . . . . . . 40Alberto Simoes : Ulam Stabilities for a Class of Higher Order Integro-Differential Equations . . . . . . . . 40Izhar Uddin : Some existence and convergence results for monotone mappings . . . . . . . . . . . . . . . . 40

S.7 Function Spaces and Applications 40Akbota Abylayeva : Weighted Hardy type inequalities with a variable upper limit . . . . . . . . . . . . . . . 41Oscar Blasco : Notes on bilinear multipliers on Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 41Stephan Dahlke : On Besov Regularity of Solutions to Nonlinear Elliptic Partial Differential Equations . . 41Oscar Dominguez : Characterizations of Besov spaces via K-functionals and ball averages . . . . . . . . . . 41Douadi Drihem : Function spaces with general weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Dorothee D. Haroske : Compact embeddings of Smoothness Morrey spaces on bounded domains . . . . . . . 41Peter Hasto : Holder continuity of quasiminimizers and ω-minimizers of functionals with generalized Orlicz

growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Alexey Karapetyants : Holomorphic Morrey, Orlicz, Grand Lebesgue and Holder spaces . . . . . . . . . . . 42Oleksiy Karlovych : Dual property of the Hardy-Littlewood maximal operator on reflexive variable Lebesgue

spaces over spaces of homogeneous type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Vakhtang Kokilashvili : Integral operators in mixed norm weighted function spaces and application . . . . . 42Alexander Meskhi : Maximal and Calderon-Zygmund operators in extrapolation Banach function lattices

and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Zdenek Mihula : Embeddings of homogeneous Sobolev spaces on the entire space . . . . . . . . . . . . . . . 42Susana Moura : Some embeddings of smoothness Morrey spaces in limiting cases . . . . . . . . . . . . . . . 42Julio Neves : Embeddings of Sobolev-type spaces into generalized Holder spaces . . . . . . . . . . . . . . . . 43Takahiro Noi : Embedding properties for weighted Besov-Morrey and Triebel-Lizorkin-Morrey spaces . . . . 43Ryskul Oinarov : Boundedness of Riemann-Liouville operator from weighted Sobolev space to weighted

Lebesgue space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Jihoon Ok : Maximal regularity for local minimizers of non-autonomous functionals . . . . . . . . . . . . . 43Lubos Pick : Moser meets Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Tao Qian : n-best approximation in reproducing kernel Hilbert spaces . . . . . . . . . . . . . . . . . . . . . 44Humberto Rafeiro : Nonstandard bounded variation spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Joel E. Restrepo : Boundedness and compactness of composition operators in holomorphic and harmonic

spaces of Holder type functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44M. Manuela Rodrigues : Time-fractional telegraph equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Natasha Samko : Embeddings of weighted local generalized Morrey spaces into Lebesgue spaces on fractal sets 44Winfried Sickel : Lizorkin-Triebel-Morrey Spaces and Differences . . . . . . . . . . . . . . . . . . . . . . . . 44Shakro Tetunashvili : On Cantor’s Λ functional and reconstruction of coefficients of multiple function series 44Tsira Tsanava : Trigonometric approximation in weighted grand variable exponent Lebesgue spaces . . . . . 45Hana Turcinova : Characterization of functions with zero traces from Sobolev spaces via the distance function

from the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

S.8 Function Spaces and their Applications to Nonlinear Evolutional Equations 45Xiuqing Chen : Global Existence and Uniqueness Analysis of Reaction-Cross-Diffusion Systems . . . . . . 45Jayson Cunanan : Inhomogeneous Strichartz estimates in some critical cases . . . . . . . . . . . . . . . . . 45Keiichi Kato : Wave packet transform and estimate of solutions to Schrodinger equations in modulation

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Tomoya Kato : Boundedness of bilinear pseudo-differential operators with symbols in an S0,0-type class . . 45Nobu Kishimoto : Ill-posedness of an NLS-type equation with derivative nonlinearity on the torus . . . . . 46Tzon-Tzer Lu : Adomian Decomposition Method for Burgers’ Equations . . . . . . . . . . . . . . . . . . . . 46Masaharu Kobayashi : Operating functions on Aqs(T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Koichi Taniguchi : Gradient estimates for heat equation in an exterior domain . . . . . . . . . . . . . . . . 46Joachim Toft : Periodic ultra-distributions and periodic elements in modulation spaces . . . . . . . . . . . . 46Naohito Tomita : Bilinear pseudo-differential operators with exotic symbols . . . . . . . . . . . . . . . . . . 46Yohei Tsutsui : A sparse bound for an integral operator with wave propagator . . . . . . . . . . . . . . . . . 46

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Xuecheng Wang : Global regularity for the 3D relativistic massive Vlasov-Maxwell system . . . . . . . . . . 47

Lifeng Zhao : Equivariant Schrodinger map from two dimensional hyperbolic spaces . . . . . . . . . . . . . 47

S.9 Generalized Functions and Applications 47Nenad Antonic : H-distributions and variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Sanja Atanasova : Characterization of wave front sets via multidimensional Stockwell transform . . . . . . 47

Christian Bargetz : Applications of topological tensor products to the theory of distributions . . . . . . . . . 47

Chikh Bouzar : On almost periodicity and almost automorphy . . . . . . . . . . . . . . . . . . . . . . . . . 48

Frederik Broucke : Absence of remainders in the Wiener-Ikehara theorem . . . . . . . . . . . . . . . . . . . 48

Soon-Yeong Chung : Fujita’s Blow-up property of Discrete Reaction-Diffusion Equations on Networks . . . 48

Antonio R. G. Garcia : A Differential Calculus in the Framework Colombeau’s Full Algebra . . . . . . . . . 48

Maximilian F. Hasler : Singularities in equations of fluid dynamics and applications to hurricanes. . . . . . 48

Natali Hritonenko : Generalized functions in the optimal control of age-structured population models . . . . 48

Jaeho Hwang : The eigenvalue problems for p-Schrodinger operators under the mixed boundary conditionson finite networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Andrzej Kaminski : The convolution of ultradistributions in the context of their supports . . . . . . . . . . 49

Sanja Konjik : Variational problems of Herglotz type with complex order fractional derivatives and lessregular Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Michael Kunzinger : Parameter-dependent linear ODE and topology of domains . . . . . . . . . . . . . . . 49

Irina Melnikova : Integro-differential equations for probabilistic characteristics of continuous and intermit-tent processes in spaces of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Svetlana Mincheva-Kaminska : Existence results concerning the convolution of ultradistributions . . . . . . 50

Akbarali Mukhammadiev : Supremum, infimum and hyperlimits of Colombeau generalized numbers . . . . 50

Marko Nedeljkov : Generalized solutions of conservation law systems containing the delta distribution . . . 50

Lenny Neyt : Asymptotic boundedness in ultradistribution spaces . . . . . . . . . . . . . . . . . . . . . . . . 50

Eduard Nigsch : The geometrization of the theory of full Colombeau algebras . . . . . . . . . . . . . . . . . 51

Michael Oberguggenberger : Propagation of singularities for generalized solutions to nonlinear wave equa-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Stevan Pilipovic : Wave fronts-new results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Bojan Prangoski : Ellipticity and the Fredholm property in the Weyl-Hormander calculus . . . . . . . . . . 51

Walter Rodrigues : Aragona Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Martin Schwarz : Stochastic Fourier Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Jailson Calado Silva : Some results on the zero set of Generalized Holomorphic functions . . . . . . . . . . 52

Diksha Tiwari : Hyperseries of Colombeau generalized numbers . . . . . . . . . . . . . . . . . . . . . . . . . 52

Todor D. Todorov : Axiomatic Approach to Colombeau Theory . . . . . . . . . . . . . . . . . . . . . . . . . 53

Daniel Velinov : f-Frequently hypercyclic C0-semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Jasson Vindas : Harmonic representations of generalized functions . . . . . . . . . . . . . . . . . . . . . . . 53

Milica Zigic : Some classes of stochastic evolution equations with Wick-type nonlinearities . . . . . . . . . . 53

S.10 Geometric & Regularity Properties of Solutions to Elliptic and Parabolic PDEs 53Virginia Agostiniani : Minkowski inequality and nonlinear potential theory - 2 . . . . . . . . . . . . . . . . 54

Angel Arroyo : Asymptotic regularity for a random walk over ellipsoids . . . . . . . . . . . . . . . . . . . . 54

Amal Attouchi : Local regularity for viscosity solutions of quasilinear parabolic equations in nondivergenceform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Diego Berti : Short-time asymptotics for solutions of the evolutionary game theoretic p-laplacian and Puccioperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Lorenzo Cavallina : On the pairs of domains that solve a two-phase overdetermined problem of Serrin-type 55

Eleonora Cinti : Some recent results in the study of fractional mean curvature flow . . . . . . . . . . . . . . 55

Matteo Cozzi : Regularity and rigidity results for nonlocal minimal graphs . . . . . . . . . . . . . . . . . . 55

Francesco Della Pietra : An optimization problem in thermal insulation . . . . . . . . . . . . . . . . . . . . 55

Marıa Angeles Garcıa-Ferrero : Strong unique continuation for higher order fractional Schrodinger equations 55

Wadade Hidemitsu : On a maximizing problem for the Moser-Trudinger type inequality with inhomogeneousconstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Phillipo Lappicy : Quasilinear parabolic equations: from Sturm attractors to Ginzburg-Landau patterns . . 56

Erik Lindgren : Extremals for Morrey’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Ilaria Lucardesi : On Blaschke-Santalo diagrams involving the torsional rigidity and the first eigenvalue ofthe Dirichlet Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Michele Marini : The sharp quantitative isocapacitary inequality . . . . . . . . . . . . . . . . . . . . . . . . 56

Lorenzo Mazzieri : Minkowski inequality and nonlinear potential theory - 1 . . . . . . . . . . . . . . . . . . 56

Michiaki Onodera : Hyperbolic solutions to Bernoulli’s free boundary problem . . . . . . . . . . . . . . . . . 57

Eero Ruosteenoja : Regularity for the normalized p-Poisson problem . . . . . . . . . . . . . . . . . . . . . . 57

Pieralberto Sicbaldi : Existence and regularity of Faber-Krahn minimizers in a Riemannian manifold . . . 57

Yachimura Toshiaki : On a thin coating problem and its related inverse problem . . . . . . . . . . . . . . . 57

Leonardo Trani : A Quantitative Weinstock Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Bozhidar Velichkov : On the logarithmic epiperimetric inequality . . . . . . . . . . . . . . . . . . . . . . . . 57

8

Abstracts

S.11 Geometries Defined by Differential Forms 57Mahir Bilen Can : The Asymptotic Nilpotent Variety of G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Alexei Kovalev : A compact G2-calibrated manifold with first Betti number b1 = 1. . . . . . . . . . . . . . . 58Jason Lotay : Ancient solutions in Lagrangian mean curvature flow . . . . . . . . . . . . . . . . . . . . . . 58Goncalo Oliveira : Generalizing holomorphic bundles to almost complex 6-manifolds . . . . . . . . . . . . . 58Tommaso Pacini : Extremal length in higher dimensions and complex systolic inequalities . . . . . . . . . . 58Vladimir Salnikov : Generalized geometry in physics and mechanics . . . . . . . . . . . . . . . . . . . . . . 59Hong Van Le : Almost formality of closed G2- and Spin(7) manifolds . . . . . . . . . . . . . . . . . . . . . 59Dimiter Vassilev : On a sub-Riemannian space associated to a Lorentzian manifold . . . . . . . . . . . . . 59Luigi Vezzoni : Geometric flows of closed forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Jordan Watts : Classifying Spaces of Diffeological Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Emily Windes : Contact Structures on G2 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

S.12 Harmonic Analysis and Partial Differential Equations 60Jonas Brinker : On Kohn-Nirenberg symbols of operators on the Heisenberg group . . . . . . . . . . . . . . 60Yonggeun Cho : Focusing energy-critical inhomogeneous NLS . . . . . . . . . . . . . . . . . . . . . . . . . . 60Simao Correia : Critical well-posedness for the modified Korteweg-de Vries equation and self-similar dynam-

ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Radouan Daher : Old and New Trends in Hausdorff Operator . . . . . . . . . . . . . . . . . . . . . . . . . 61Piero D’Ancona : Global almost radial solutions to supercritical dispersive equations outside the unit ball . 61Kazumasa Fujiwara : Remark on blow-up for the half Ginzburg-Landau-Kuramoto equation with rough

coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Noriyoshi Fukaya : Uniqueness and nondegeneracy of ground states for nonlinear Schrodinger equations

with attractive inverse-power potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Daniele Garrisi : Uniqueness of standing-waves solutions to the non-linear Schroedinger equations for com-

bined power-type non-linearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Masaru Hamano : Scattering solutions for the focusing nonlinear Schrodinger equation with a potential . . 61Gyeongha Hwang : Probabilistic well-posedness of the mass-critical NLS with radial data below Rd . . . . . 62Masahiro Ikeda : Test function method for blow-up phenomena of semilinear wave equations and their weakly

coupled systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Takahisa Inui : The Strichartz estimates for the damped wave equation and its application to a nonlinear

problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Ningan Lai : Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime . . . 62Chunhua Li : On the scattering problem for the nonlinear Schrodinger equation with a potential in 2D . . . 62Annunziata Loiudice : Asymptotic decay results for critical subelliptic equations . . . . . . . . . . . . . . . 62Tokio Matsuyama : Energy estimate for wave equation with bounded time-dependent coefficient . . . . . . . 63Hayato Miyazaki : Strong instability for standing wave solutions to the system of the quadratic NLKG . . . 63Kiyoshi Mochizuki : Spectral theory for magnetic Schrodinger . . . . . . . . . . . . . . . . . . . . . . . . . 63Ljubica Oparnica : Fractional wave equation with discontinuous coefficients . . . . . . . . . . . . . . . . . . 63Ivan Pombo : A new approach on the Calderon problem for discontinuous complex conductivities . . . . . . 63Petar Popivanov : Boundary value problem for the biharmonic operator in a ball . . . . . . . . . . . . . . . 63Peetta Kandy Ratnakumar : Local smoothing of Fourier integral operators and Hermite functions . . . . . 63David Rottensteiner : Harmonic and Anharmonic Oscillators on the Heisenberg Group . . . . . . . . . . . 64Bolys Sabibtek : Geometric Hardy inequalities in the half-space on stratified groups . . . . . . . . . . . . . 64Mohammed Elamine Sebih : On a wave equation with singular dissipation . . . . . . . . . . . . . . . . . . 64Ruben Sousa : Product formulas and convolutions for solutions of Sturm-Liouville equations . . . . . . . . 64Mitsuru Sugimoto : A trial to construct specific self-similar solutions for nonlinear wave equations . . . . . 64Tomoyuki Tanaka : Local well-posedness for higher order Benjamin-Ono type equations . . . . . . . . . . . 64Koichi Taniguchi : Endpoint Strichartz estimates for Schrodinger equation on exterior domains . . . . . . . 65Mirko Tarulli : Decay and Scattering in energy space for the solution of generalised Hartree equation . . . . 65Niyaz Tokmagambetov : Very weak solutions of wave equation for Landau Hamiltonian with irregular elec-

tromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Berikbol Torebek : Blowing-up solutions of the time-fractional dispersive partial differential equations . . . 65Igor Trushin : Inverse Scattering Problems on Quantum Graphs . . . . . . . . . . . . . . . . . . . . . . . . 65Kanat Tulenov : Optimal symmetric range for Hilbert transform and its non-commutative extensions . . . 65George Venkov : Local uniqueness of ground states for generalized Choquard model . . . . . . . . . . . . . . 66Semyon Yakubovich : Index transforms with the squares of Kelvin functions . . . . . . . . . . . . . . . . . 66Nurgissa Yessirkegenov : Hypoelliptic functional inequalities and applications . . . . . . . . . . . . . . . . . 66Yi Zhou : Global regularity for Einstein-Klein-Gordon system with U(1)× R isometry group . . . . . . . . 66

S.13 Integral Transforms and Reproducing Kernels 66Arran Fernandez : Models and classifications in fractional calculus . . . . . . . . . . . . . . . . . . . . . . . 67Rita Guerra : Paley-Wiener and Wiener’s Tauberian results for a class of oscillatory integral operators . . 67Isao Ishikawa : On the boundedness of composition operators on reproducing kernel Hilbert spaces with

analytic positive definite functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67El Aıdi Mohammed : A harmonic interpolation sequence on the real unit ball . . . . . . . . . . . . . . . . . 67

9

Abstracts

Zouhaır Mouayn : Nonlinear coherent states associated with a measure on the positive real half line . . . . 67Faical Ndairou : Distributed-order non-local optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Anabela Silva : Convolution integral equations related to Fourier sine and cosine transforms and Hermite

functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Nurlan Yerkinbayev : Noetherian Solvability of an Operator Singular Integral Equation with Carleman Shift

in Fractional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

S.14 Partial Differential Equations on Curved Spacetimes 68Andras Balogh : Computational analysis of a nonlinear wave equation with black hole embedded in an

expanding universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Dean Baskin : Radiation fields for wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Uwe Brauer : Local existence of solutions to the Euler–Poisson system, including densities without compact

support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Marcelo Ebert : About critical exponents in semi-linear de Sitter models . . . . . . . . . . . . . . . . . . . . 69Anahit Galstyan : Semilinear Klein-Gordon Equation in the Friedmann-Lamaitre-Robertson-Walker space-

time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Sergey Grigorian : Heat Flow of Isometric G2-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Fumihiko Hirosawa : Energy estimates for Klein-Gordon type equations with time dependent mass . . . . . 69Lavi Karp : Continuous dependence on the geometrical initial data for the Einstein vacuum equations . . . 70Baoping Liu : Asymptotic Behavior of nonlinear Schrodinger equations with radial data . . . . . . . . . . . 70Makoto Nakamura : Remarks on the Navier-Stokes equations in homogeneous and isotropic spacetimes . . 70Jose Natario : Solutions of the wave equation bounded at the Big Bang . . . . . . . . . . . . . . . . . . . . 70Michael Gerhard Reissig : Semilinear de Sitter model of cosmology - global existence of small data solutions 70Andras Vasy : Outgoing Fredholm theory and the limiting absorption principle for asymptotically conic spaces 70James Vickers : Quantum Field Theory on Low regularity Spacetimes . . . . . . . . . . . . . . . . . . . . . 71Seiichiro Wakabayashi : On the Cauchy problem for hyperbolic operators with triple characteristics whose

coefficients depend only on the time variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Baoxiang Wang : Navier-Stokes equation with very rough data . . . . . . . . . . . . . . . . . . . . . . . . . 71Karen Yagdjian : Properties of solutions of hyperbolic equations in the curved space-times . . . . . . . . . . 71Shiwu Yang : Global solutions of massive Maxwell-Klein-Gordon equations with large Maxwell field . . . . 71

S.15 Partial Differential Equations with Nonstandard Growth 71Gaukhar Arepova : On a multidimensional boundary value problem for a model degenerate elliptic-parabolic

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Hyeong Ohk Bae : Interior Regularity to the Steady Incompressible Shear Thinning Fluids with Non-standard

Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Nikolai V. Chemetov : Well-posedness of the Cosserat-Bingham multi-dimensional fluid equations . . . . . 72QHeung Choi : Boundary value problem involving p−Laplacian and jumping nonlinearities . . . . . . . . . 72Fernanda Cipriano : Optimal feedback control of second grade fluids . . . . . . . . . . . . . . . . . . . . . . 72Jose Carlos Matos Duque : On the numerical simulation of a parabolic equation with p-Laplacian and

memory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Jacques Giacomoni : Picone identity for variable exponents operators and applications . . . . . . . . . . . . 73Eurica Henriques : About some generalizations of the parabolic p-Laplacian and the Porous Medium Equation 73Tacksun Jung : Multiplicity results for p−Laplacian boundary value problem . . . . . . . . . . . . . . . . . 73Claudia Lederman : A free boundary problem for an inhomogeneous operator with nonstandard growth . . . 73Roger Lewandowski : About a new non linear boundary condition for the turbulent kinetic energy involved

in models of turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Pedro Marın-Rubio : On a nonlocal viscosity p-Laplacian evolutive problem. Existence and long-time behav-

ior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Cesar J. Niche : A survey of recent results on the characterization of decay of solutions to some dissipative

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Hermenegildo Borges de Oliveira : Existence results for the p(u)−Laplacian problem. . . . . . . . . . . . . 74Sergey Sazhenkov : Entropy and kinetic solutions of the genuinely-nonlinear Kolmogorov-type equation with

partial diffusion and nonlinear source term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Ugur Sert : Solvability of Nonlinear Elliptic Type Equation With Two Unrelated Non standard Growths . . 74Sergey Shmarev : Global regularity of solutions of singular parabolic equations with nonstandard growth . . 74Petra Wittbold : On a stochastic p(ω, t, x)-Laplace equation . . . . . . . . . . . . . . . . . . . . . . . . . . 75Joerg Wolf : Sufficient conditions for local regularity to the generalized Newtonian fluid with shear thinning

viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

S.16 Pseudo Differential Operators 75Viorel Catana : Two-wavelet curvelet localization operators and two-wavelet curvelet multipliers . . . . . . 75Sandro Coriasco : Pseudo-differential operators of Gevrey type and their action on classes of modulation

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Gianluca Garello : Pseudo-Differential Operators and Existence of Gabor Frames . . . . . . . . . . . . . . . 76Ivan Ivec : Continuity of linear operators on mixed-norm Lebesgue and Sobolev spaces . . . . . . . . . . . . 76

10

Abstracts

Vishvesh Kumar : C∗-algebras, H∗-algebras and trace ideals of pseudo-differential operators on locallycompact, Hausdorff and abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Daniel Lorenz : Pseudos with limited smoothness applied to strictly hyperbolic Cauchy problems with coeffi-cients low-regular in time and space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Alessandro Oliaro : Paley-Wiener Theorems of real type in ultradifferentiable spaces . . . . . . . . . . . . . 76

Christine Pfeuffer : Fredholm Property of Non-Smooth Pseudodifferential Operators . . . . . . . . . . . . . 76

Joachim Toft : Analytic pseudo-differential calculus via the Bargmann transform . . . . . . . . . . . . . . . 77

Ville Turunen : Time-frequency analysis and pseudo-differential operators on locally compact groups . . . . 77

Ivana Vojnovic : Defect distributions applied to differential equations with polynomial coefficients . . . . . . 77

Patrik Wahlberg : Strong continuity on Shubin-Sobolev spaces of semigroups for evolution equations ofquadratic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

K. L. Wong : Normality, Self-Adjointness, Spectral Invariance, Groups and Determinants of Pseudo-Differential Operators on Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

S.17 Quaternionic and Clifford Analysis 78Daniel Alpay : A general setting for functions of Fueter variables: differentiability, rational functions, Fock

module and related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Eusebio Ariza Garcia : Sufficient Conditions for Associated Operators to a Space of Harmonic-Type func-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Swanhild Bernstein : A Dirac operator in infinite dimensional analysis . . . . . . . . . . . . . . . . . . . . 78

Cinzia Bisi : On Runge Pairs and Topology of Axially Symmetric Domains. . . . . . . . . . . . . . . . . . 78

Sebastian Bock : Special classes of monogenic functions and applications in linear elastic fracture mechanics 79

Vladimir Bolotnikov : Interpolation by polynomials over division rings . . . . . . . . . . . . . . . . . . . . . 79

Dmitry Bryukhov : GASPT, Radially Holomorphic Functions, the Fueter Potential Method in an Inhomo-geneous Medium and Problems of Generalized Joukowski Transformations in R3 . . . . . . . . . . . . 79

Fabrizio Colombo : Some applications of the Spectral Theory on the S-spectrum . . . . . . . . . . . . . . . 79

Briceyda Berenice Delgado : A right inverse operator for curl + λI and applications . . . . . . . . . . . . . 79

Kamal Diki : On the Bargmann-Fock-Fueter and Bergman-Fueter integral transforms (Joint work with Prof.Krausshar and Prof. Sabadini) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Antonio Di Teodoro : Sufficient Conditions for the Fractional Vekua equation to be Associated with thefractional Cauchy-Riemann operator in the Riemann-Liouville sense . . . . . . . . . . . . . . . . . . . 80

Sirkka-Liisa Eriksson : Hyperbolic function theory and hyperbolic Brownian motion . . . . . . . . . . . . . . 80

Nelson Faustino : The discrete Cauchy-Kovalevskaya extension originating from fractional central differenceoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Minggang Fei : Real Paley-Wiener Theorem for Clifford-Fourier Transfrom . . . . . . . . . . . . . . . . . . 81

Brigitte Forster-Heinlein : THE commutative diagram of signal processing from a Clifford perspective . . . 81

Hamed Baghal Ghaffari : A Clifford Construction of Multidimensional Prolate Spheroidal Wave Functions 81

Riccardo Ghiloni : Slice regular functions in several variables . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Narciso Gomes : Compressed Sensing with Slice Monogenic Signals . . . . . . . . . . . . . . . . . . . . . . 81

Yuri Grigor’ev, Andrei Iakovlev : Radial integration operators in infinite domains and their applications inquaternion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Ali Guzman Adan : A distributional approach to integration over smooth surfaces of lower dimensionembedded in Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Jeffrey Hogan : Quaternionic splines, wavelets and prolates . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Mathias Ionescu-Tira : Time-Frequency Analysis in the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . 82

Igor Kanatchikov : On the hypercomplex Airy function in the context of quantum Yang-Mills theory . . . . 82

Yakov Krasnov : Spectral Decomposition of Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Soren Kraußhar : Applications of slice-holomorphic functions to automorphic forms and Bergman kernels . 83

Lukas Krump : Nonstable quaternionic and orthogonal complexes for k Dirac operators . . . . . . . . . . . 83

Roman Lavicka : Fischer decomposition in non-stable cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Anastasiia Legatiuk : Application of the discrete potential theory to problems of mathematical physics . . . 83

Dmitrii Legatiuk : On application of script geometry to finite element exterior calculus . . . . . . . . . . . 84

Wang Liping : Some Properties and Application of Teodorescu Operators Associated with the HelmholtzEquation and the Time-harmonic Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Maria Elena Luna-Elizarraras : On the structure of the singularities of bicomplex holomorphic functions . . 84

Peter Massopust : Clifford-Valued Cone and Hex Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Teppo Mertens : Radon-Type transforms for holomorphic functions in the Lie ball . . . . . . . . . . . . . . 84

Craig Nolder : Immersions of Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Heikki Orelma : A group classification of linear fractional partial differential equation with Laplacian on aplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Alessandro Perotti : An Almansi type decomposition for slice-regular functions on Clifford algebras . . . . . 85

Guangbin Ren : Slice Analysis of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Hu Ren : Bargmann-Radon transform for axially monogenic functions . . . . . . . . . . . . . . . . . . . . . 85

Irene Sabadini : Multivariable quaternionic functions and power series . . . . . . . . . . . . . . . . . . . . 85

Tomas Salac : Domains of monogenicity in several quaternionic variables . . . . . . . . . . . . . . . . . . . 85

11

Abstracts

Michael Shapiro : Analyticity in the sense of Hausdorff and classes of hyperholomorphic quaternionic func-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Haipan Shi : Two-sided Fourier Transform in Clifford Analysis and its Application . . . . . . . . . . . . . 86Caterina Stoppato : Zeros of slice functions over dual quaternions: theory and applications . . . . . . . . . 86Luis Manuel Tovar Sanchez : Dirichlet and Bloch Spaces in different Contexts . . . . . . . . . . . . . . . . 86Mihaela Vajiac : Ternary Regular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Wei Wang : Analysis of the k-Cauchy-Fueter complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Yonghong Xie : Important Properties of Clifford Mobius Transformation and Its Application . . . . . . . . 86Heju Yang : A new Cauchy integral formula in the complex Clifford analysis . . . . . . . . . . . . . . . . . 87

S.18 Recent Progress in Evolution Equations 87Mohammed Djaouti Abdelhamid : Modified different nonlinearities for weakly coupled systems of semilinear

effectively damped waves with different time-dependent coefficients in the dissipation terms . . . . . . 87Alessia Ascanelli : Decay of the data and regularity of the solution in Schrodinger equations . . . . . . . . . 87Halit Sevki Aslan : The influence of oscillations on energy estimates for damped wave models with time-

dependent propagation speed and dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Chiara Boiti : Regularity of linear partial differential operators with polynomial coefficients . . . . . . . . . 87Marco Cappiello : Semilinear p-evolution equations in weighted Sobolev space . . . . . . . . . . . . . . . . . 88Ruy Coimbra Charao : Existence and asymptotic properties for dissipative semilinear second order evolution

equations with fractional Laplacian operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Wenhui Chen : Weakly coupled systems of semilinear elastic waves with different damping mechanisms in 3D 88Marcello D’Abbicco : Asymptotics for a semilinear damped plate equation with time-dependent coefficients 88Tuan Anh Dao : L1 estimates for oscillating integrals and application to parabolic like semi-linear structurally

damped σ-evolution models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Andrey Faminskiy : On regularity of solutions to two-dimensional Zakharov-Kuznetsov equation . . . . . . 89Vladimir Georgiev : On Traveling Solitary Waves for nonlinear Half-Wave Equations . . . . . . . . . . . . 89Marina Ghisi : Time-dependent propagation speed vs strong damping . . . . . . . . . . . . . . . . . . . . . . 89Giovanni Girardi : L1 − L1 estimates for the strongly damped plate equation . . . . . . . . . . . . . . . . . 89Massimo Gobbino : Infinite dimensional Duffing-like evolution equations . . . . . . . . . . . . . . . . . . . 89Fumihiko Hirosawa : Some properties of time dependent propagation speed of the wave equation for the

estimates of low frequency energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Jon Johnsen : On a criterion for log-convex decay in non- selfadjoint dynamical systems . . . . . . . . . . 90Sandra Lucente : Blow-up for Quasi-Linear Wave Equations with time dependent potential . . . . . . . . . 90Cleverson da Luz : Sigma-evolution models with low regular time-dependent structural damping: effective

and non-effective dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Jorge Marques : Critical exponent for the semilinear damped wave equation with polynomial decaying prop-

agation speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Makoto Nakamura : Global solutions for the semilinear diffusion equation in the de Sitter spacetime . . . . 90Hideo Nakazawa : Some estimates of solutions of perturbed Helmholtz equations . . . . . . . . . . . . . . . 91Alessandro Palmieri : Recent progress in semilinear wave equations in the scale-invariant case . . . . . . . 91Fabio Pusateri : Some recent results on periodic waver waves . . . . . . . . . . . . . . . . . . . . . . . . . . 91Yuta Wakasugi : Lp-Lq estimates for the damped wave equation and the critical exponent for the nonlinear

problem with slowly decaying data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

S.19 Spectral Theory of Partial Differential Equations 91Giuseppina di Blasio : Some sharp spectral inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Davide Buoso : Semiclassical bounds for spectra of biharmonic operators . . . . . . . . . . . . . . . . . . . 92Maurizio Garrione : Spectral theory for beams with intermediate piers and related nonlinear evolution prob-

lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Tynysbek Kal’menov : On the structure of the spectrum of regular boundary value problems for differential

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Yulia Meshkova : On homogenization of periodic parabolic systems . . . . . . . . . . . . . . . . . . . . . . . 92Alip Mohammed : Eigenvalues of the Third Boundary Value Problem for the Heat Equation . . . . . . . . 92El Aıdi Mohammed : On the negative spectrum of generalized Hamiltonians defined on metric graphs . . . 92Kordan Ospanov : Coercive estimates for a second-order differential equation with oscillating drift . . . . . 93Marcus Waurick : Quantitative aspects for homogenisation problems in Rn . . . . . . . . . . . . . . . . . . 93Michele Zaccaron : Spectral stability for the Maxwell equations in a cavity . . . . . . . . . . . . . . . . . . . 93Yeskabylova Zhuldyz : Conditions of maximal regularity for a third-order differential equation with fast

growing intermediate coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

S.20 Theory and Applications of Boundary-domain Integral and Pseudodifferential Operators 94Tsegaye Ayele : Two-operator BDIEs for Variable-Coefficient Neumann Problem with General Data . . . . 94Tsegaye Ayele : Boundary-Domain Integral Equation systems to the Mixed BVP for Compressible Stokes

Equations with Variable Viscosity in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Lasha Baramidze : Uniform Convergence of Double Vilenkin-Fourier Series . . . . . . . . . . . . . . . . . 94Solomon Tesfaye Bekele : Two-operator Boundary-Domain Integral Equations for variable coefficient Neu-

mann boundary value problem in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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George Chkadua : Dynamical interaction problems of acoustic waves and thermo-electro-magneto elasticstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Christian Constanda : Approximate Solution for Thin Plates in an Infinite Domain . . . . . . . . . . . . . 95

Mulugeta Alemayehu Dagnaw : Boundary-Domain Integral Equation Systems to the Dirichlet BVP forStokes Equations with variable viscosity in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Mulugeta Alemayehu Dagnaw : Boundary-Domain Integral Equation Systems to the Neumann BVP for acompressible Stokes System with variable viscosity in 2D . . . . . . . . . . . . . . . . . . . . . . . . . 95

Roland Duduchava : Mixed Boundary Value Problems for an Elliptic Equation in a Lipschitz Domain . . . 95

Carlos Fresneda-Portillo : Boundary Domain Integral Equations for Diffusion Equation in Non-homogeneousMedia with Dirichlet Boundary Conditions based on a New Family of Parametrices . . . . . . . . . . . 96

Yuri Karlovich : Mellin pseudodifferential operators with non-regular symbols and their applications . . . . 96

Mirela Kohr : Layer potentials and exterior boundary problems in weighted Sobolev spaces for the Stokessystem with L∞ elliptic coefficient tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Vakhtang Kokilashvili : Solution of the Riemann boundary value problem in the case when the free termbelongs to the grand variable exponent Lebesgue space Lp(t),θ(Γ) when min

Γp(t) = 1 . . . . . . . . . . . 96

Alexander Meskhi : Singular integrals in weighted grand variable exponent Lebesgue spaces . . . . . . . . . 96

Sergey E. Mikhailov : Analysis of Boundary-Domain Integral Equations for the Variable-Viscosity Com-pressible Robin-Stokes BVP in Lp-based Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Maia Mrevlishvili : Investigation of nonclassical transmission problems of the thermo-electro-magneto elas-ticity theory for composed bodies by the integral equation method . . . . . . . . . . . . . . . . . . . . . 97

David Natroshvili : Localized boundary-domain integral equations approach with piecewise constant cut-offfunction for the heat transfer equation with a variable coefficient . . . . . . . . . . . . . . . . . . . . . 97

Julia Orlik : Rescaling and Trace Operators in Fractional Sobolev Spaces on Bounded Lipschitz Domainswith Periodic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Gvantsa Shavardenidze : On the convergence of Cesaro means of negative order of Vilenkin-Fourier series 97

Ilya Spitkovsky : On Toeplitz operators with matrix almost periodic symbols . . . . . . . . . . . . . . . . . . 98

S.21: Time-frequency Analysis and Applications 98Luis Daniel Abreu : Time-frequency analysis of signals periodic in time and in frequency . . . . . . . . . . 98

Peter Balazs : Recent Developments in Time-frequency Analysis and Applications . . . . . . . . . . . . . . 98

Brigitte Forster-Heinlein : Frame Recycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Hans G. Feichtinger : Current problems in Gabor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Nicki Holighaus : Coorbit spaces for warped time-frequency representations . . . . . . . . . . . . . . . . . . 99

Joao Nuno Prata : What is the Wigner function closest to a given square integrable function? . . . . . . . 99

Ayse Sandikci : Continuity properties of Multilinear τ -Wigner Transform . . . . . . . . . . . . . . . . . . . 99

Michael Speckbacher : Planar sets of sampling for the short-time Fourier transform - From sufficientconditions to sampling bounds on polyanalytic Fock spaces . . . . . . . . . . . . . . . . . . . . . . . . . 99

Elena Ushakova : Compactly supported linear combinations of elements of Battle–Lemarie spline waveletsystems and decompositions in weighted function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Jordy Timo Van Velthoven : Coorbit spaces associated to integrably admissible dilation groups . . . . . . . 100

Felix Voigtlaender : Invertibility of frame operators on Besov-type decomposition spaces . . . . . . . . . . . 100

Wang Yanbo : Adaptive Fourier Decomposition in Hp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 100

S.22 Wavelet theory and its Related Topics 100Yoshihiro Aihara : Deficiency of holomorphic curves for hypersurfaces and linear systems . . . . . . . . . . 100

Maria Charina : Holder regularity of anisotropic wavelets and wavelet frames: matrix approach . . . . . . . 100

Li Dengfeng : The Template-based Procedure of Biorthogonal Ternary Wavelets for Curve MultiresolutionProcessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Neil Kristofer Dizon : Optimization in the construction of nearly cardinal and nearly symmetric wavelets . 101

Kensuke Fujinoki : Two-dimensional frames for multidirectional decompositions . . . . . . . . . . . . . . . 101

Keiko Fujita : Some topics on the Gabor wavelet transformation . . . . . . . . . . . . . . . . . . . . . . . . 101

Mikhail Karapetyants : Subdivision schemes on a dyadic half-line . . . . . . . . . . . . . . . . . . . . . . . 101

Yun-Zhang Li : The time-frequency analysis on the half space . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Anastasia Zakharova : Application of frame theory to shape from defocus . . . . . . . . . . . . . . . . . . . 102

S.23 Poster Session 102Yahya Almalki : Spin Structures on p-Gonal Surfaces via Divisors . . . . . . . . . . . . . . . . . . . . . . . 102

Asmae Babaya : Analytic Improvement of Frequency Characteristic of GaN-based HEMT . . . . . . . . . . 102

David Santiago Gomez Cobos : Harmonic Analysis and the topology of Spheres . . . . . . . . . . . . . . . . 102

George Giorgobiani : On the series with an affine sum range in a Banach space . . . . . . . . . . . . . . . 102

Snezana Gordic : Colombeau generalized stochastic processes and applications . . . . . . . . . . . . . . . . . 103

Anna Kholomeeva : Exact solutions of a nonlinear equation from the theory of nonstationary processes insemiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Liudmila Novikova : Bifurcations of invariant manifolds of nonlinear operators in Banach spaces . . . . . . 103

Nicusor Minculete : Some refinements of Ostrowski’s inequality and an extension to a 2-inner product space 103

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Nico Michele Schiavone : Heat-like and wave-like behaviour of the lifespan estimates for wave equations withscale invariant damping and mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Yi Zhu : Global solutions of 3D incompressible MHD system with mixed partial dissipation and magneticdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Index 105

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15

Plenary talks

Statistical Estimation with Algebraic Structure: Statistical and Compu-tational Considerations

Afonso S. BandeiraDepartment of Mathematics and Center for Data Science, Courant Institute of Mathematical Sciences, NewYork University, New York, NY, USAbandeira@cims.nyu.edu

The massive datasets now commonplace in many fields of industry and science hold the promise of signifi-cantly changing our understanding of the world around us. Key to the fulfillment of this promise however,is the ability to perform computationally efficient statistical inference on such large datasets. This moti-vates one of the key questions in theoretical data science, to understand which statistical inference tasks areimpossible, which are possible but computationally infeasible for large datasets, and which enjoy efficientestimation procedures.In this talk, we will focus on an important class of estimation problems that are rich in algebraic structure.Motivated by problems in signal/image processing, and computer vision we will address the task of estimatinga signal, image, tri-dimensional structure/scene from measurements corrupted not only by noise but also bylatent transformations. Many such transformations can be described as a group acting on the object to berecovered. Examples include the Simulatenous Localization and Mapping (SLaM) problem in Robotics andComputer Vision, where pictures of a scene are obtained from different positions and orientations; Cryo-Electron Microscopy (Cryo-EM) imaging where projections of a molecule density are taken from unknownrotations, and several others.

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Afonso Bandeira received his Ph.D. in Mathematics from Princeton University in 2015. Currently, he isa Professor of Mathematics and Data Science at the Courant Institute. His contributions to Mathematicscover a broad range with fundamental contributions to probability, harmonic analysis, numerical analysis,mathematical signal processing, graph theory, data science, and neural networks. Among other awards hereceived a Sloan Research Fellowship in 2018 and was selected as a 2015 DARPA Riser.

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Analysis on fractal spaces and heat kernels

Alexander Grigor’yanDepartment of Mathematics, University of Bielefeld, 33501 Bielefeld, Germanygrigor@math.uni-bielefeld.de

We discuss elements of Analysis on singular metric spaces, in particular, on fractals, based on the notion ofthe heat kernel. Such spaces are characterized by two parameters: the Hausdorff dimension and the walkdimension, where the latter determines the space/time scaling for a diffusion process. We present variousapproaches to the notion of the walk dimension, including those via Besov function spaces and via Markovjump processes. We also discuss heat kernel bounds for diffusion and jump processes.

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Alexander Grigor’yan received his PhD from Moscow State University in 1982 and his Doctor of Physyco-mathematical sciences by Moscow State University in 1989. He was a visiting scholar at Harvard Universityin 1993-1994 and moved to the Imperial College London until 2005. Since then he is Professor of Mathe-matics at University of Bielefeld in Germany. He made fundamental contributions in Geometric Analysis onRiemannian manifolds, graphs, and metric spaces. This in includes analysis of partial differential equationsof elliptic and parabolic types on Riemannian manifolds, random walks on graphs, function theory and dif-fusion processes on fractal spaces. Major results are on heat kernel estimates, estimates on eigenvalues ofLaplace and Schrodinger operators and long time behavior of random walks and diffusions. Furthermore,this also resulted in fundamental contributions to the theory of stochastic processes on manifolds, graphs,

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and fractals. For his contributions he received the Whitehead prize in 1997 and the prize of the MoscowMathematical Society in 1988.

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Totally positive functions in sampling theory and time-frequency analysis

Karlheinz GrochenigFaculty of Mathematics, University of Vienna, Oscar-Morgenstern-Platz 1, 1090 Wien, Austriakarlheinz.groechenig@univie.ac.at

Totally positive functions play an important role in approximation theory and statistics. In this talk I willpresent recent new applications of totally positive functions (TPFs) in sampling theory and time-frequencyanalysis.(i) We study the sampling problem for shift-invariant spaces generated by a TPF. These spaces arise thespan of the integer shifts of a TPF and are often used as a substitute for bandlimited functions. We givea complete characterization of sampling sets for a shift-invariant space with a TPF generator of Gaussiantype in the style of Beurling.(ii) A related problem is the question of Gabor frames, i.e., the spanning properties of time-frequency shiftsof a given function. It is conjectured that the lattice shifts of a TPF generate a frame, if and only if thedensity of the lattice exceeds 1. At this time this conjecture has been proved for two important subclasses ofTPFs. For rational lattices it is true for arbitrary TPFs. So far, TPFs seem to be the only window functionsfor which the fine structure of the associated Gabor frames is tractable.(iii) Yet another question in time-frequency analysis is the existence of zeros of the Wigner distribution (orthe radar ambiguity function). So far all examples of zero-free ambiguity functions are related to TPFs, e.g.,the ambiguity function of the Gaussian is zero free.

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Karlheinz Grochenig is an Austrian Mathematician at the Faculty of Mathematics at University of Vienna.He made his Ph.D. degree sub auspiciis praesidentis at the University of Vienna in 1985. After positions at theMcMaster University and the University of Connecticut he returned to Vienna in 2006. He made fundamentalcontributions in Harmonic analysis, time-frequency analysis, wave- let theory, (non-uniform) sampling theory,pseudodifferential operators, Banach algebras and their symmetry, non-commutative harmonic analysis.Among many other awards he received the Marie-Curie Excellence Award of the European Union in 2004.

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Planar orthogonal polynomials and related point processes: randomnorm matrices and arithmetic jellium

Hakan HedenmalmDepartment of Mathematics, Royal Institute of Technology, Stockholm, Swedenhaakanh@math.kth.se

In the random normal matrix model, the correlation kernel is expressed as a sum over orthogonal polynomials.The spectra of such normal matrices typically fill a region, the “spectral droplet”. The local behaviorof the eigenvalues near the edge of the droplet has attracted attention, and it was conjectured that thebehavior is asymptotically governed by the error function kernel, for smooth boundary points. Here, weresolve this conjecture, assuming the whole boundary is smooth. The approach goes through developing anew asymptotic expansion for the orthogonal polynomials, which goes beyond the classical work of Szego,Carleman, and Suetin. This allows us to also consider ensembles where the correlation kernel has gaps in thesense of summing only over some of the orthogonal polynomials. If we sum over an arithmetic progressionwe obtain “arithmetic jellium”, a compound with orbifold structure generating a representation of SU(1, 1).This reports on joint work with Aron Wennman.

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Hakan Hedenmalm received his Ph.D. in Mathematics Uppsala University in 1985. Prior to joining theRoyal Institute of Technology, he served in the Department of Mathematics at Lund University until 2002.He made major contributions in a variety of fields including the theory of Bergman spaces and associatedreproducing kernels, Several complex variables, the theory of biharmonic operators, the theory of Dirichlet

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series, quasiconformal mappings, random matrices and polyanalytic determinantal processes. For his contri-butions he received the Wallenberg prize in 1882, the Goran Gustafsson Prize in 2002 and the Eva and LarsGarding Prize in 2015. In 1997 he was elected to the Royal Physiographic Society in Lund and in 2018 tothe Royal Norwegian Society of Sciences and Letters.

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Recent progress in the theory of minimal surfaces

Andre NevesDepartment of Mathematics, University of Chicago, IL 60637, USAaneves@math.uchicago.edu

A long standing problem in geometry, conjectured by S.-T. Yau in 1982, is that any any 3-manifold admitsan infinite number of distinct minimal surfaces. The analogous problem for geodesics on surfaces led to thediscovery of deep interactions between dynamics, topology, and analysis. The last couple of years broughtdramatic developments to Yau’s conjecture, which has now been settled due to the work of Marques-Nevesand Song.I will survey the history of the problem, its significance, and all the contributions made throughout the last30 years.

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Andre Neves received his Ph.D. in Mathematics from Stanford University in 2005. He has a Licenciaturadegree from the Instituto Superior Tecnico (1999). Prior to joining the faculty of the University of Chicagoin 2016 he was a Professor at the Imperial College. In 2012, together with Fernando Marques, he provedthat the least bended torus in space is the Clifford torus and, in doing so, solved the Willmore conjecture.For his contributions he received numerous awards, including the Philip Leverhulme Prize in 2012, the LMSWhitehead Prize in 2013, the Royal Society Wolfson Research Merit Award in 2015, a New Horizons inMathematics Prize, Oswald Veblen Prize in 2016, and a Simons Investigator Award in 2018. His currentresearch interests include Riemannian geometry, scalar curvature, Min-Max Theory, Mean Curvature Flow,and General Relativity.

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Multidimensional inverse scattering problem

Roman NovikovCNRS (UMR 7641), Centre de Mathematiques Appliquees, Ecole Polytechnique, Universite Paris-Saclay,91128 Palaiseau, Francenovikov@cmap.polytechnique.fr

We give a short review of old and recent results on the multidimensional inverse scattering problem for theSchrodinger equation. A special attention is paid to efficient reconstructions of the potential from scatteringdata which can be measured in practice. In this connection our considerations include reconstructionsfrom non-overdetermined monochromatic scattering data and formulas for phase recovering from phaselessscattering data. Potential applications include phaseless inverse X-ray scattering, acoustic tomography andtomographies using elementary particles. This talk is based, in particular, on results going back to M. Born(1926), L. Faddeev (1956, 1974), S. Manakov (1981), R.Beals, R. Coifman (1985), G. Henkin, R. Novikov(1987), and on more recent results of R. Novikov (1998 - 2019).

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Roman Novikov received his Ph.D. in Mathematics from M.V. Lomonosov Moscow State University in 1990under the supervision of S.P. Novikov and his Doctor of Physyco-mathematical sciences by the St. PetersburgBranch of V.A. Steklov Mathematical Institute in 1998. Currently, he is Directeur de recherche au CNRSat CMAP - Centre de Mathematiques Appliquees, Ecole Polytechnique, France. He made fundamentalcontributions in the theory of inverse problems, especially in the theory of inverse scattering and tomography,discrete function theory, and the theory of Riemann-Hilbert problems.

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Self-similar solutions to the derivative nonlinear Schrodinger equation

Tohru OzawaDepartment of Applied Physics, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo, Japantxozawa@waseda.jp

A class of self-similar solutions to the derivative nonlinear Schrodinger equations is studied. Especially, theasymptotics of profile functions are shown to posses a logarithmic phase correction. This logarithmic phasecorrection is obtained from the nonlinear interaction of profile functions. This is a remarkable differencefrom the pseudo-conformally invariant case, where the logarithmic correction comes from the linear part ofthe equations of the profile functions. This talk is based on my recent jointwork with Kazumasa Fujiwara(Tohoku) and Vladimir Georgiev (Pisa).

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Tohru Ozawa has been a Professor of Mathematics at Waseda University in Tokyo, Tokyo, Japan, since 2008.He graduated with honors from the Department of Physics at Waseda University in 1984. He received anM.S. degree in 1986 and Ph.D. degree in 1990 from RIMS, Kyoto University. He held a Full Professor positionfrom 1995 to 2008 at Hokkaido University in Sapporo. In 1998, he was awarded a Spring Prize from theMathematical Society of Japan for his pioneering work on the nonlinear Schrodinger equation. His researchinterests include nonlinear Schrodinger equations (scattering theory, Strichartz estimates, analyticity andsmoothing properties of solutions, derivative coupling, etc.), nonlinear hyperbolic equations (Klein-Gordonequation, Dirac equation, systems of wave equations, Strichartz estimates, nonrelativistic limit, self-similarsolutions, etc.), and function spaces (embedding theorems, interpolation inequalities, characterizations offractional Sobolev spaces over Lie groups, etc.).

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Inverse Problems and the Nonlinear Fourier Transform

Samuli SiltanenDepartment of Mathematics and Statistics, University of Helsinki, Finlandsamuli.siltanen@iki.fi

Electrical impedance tomography (EIT) is an emerging medical imaging method. It is based on probing thehuman body with harmless electric currents fed through electrodes on the skin. The voltages appearing onthe electrodes are measured, and the aim of EIT is to recover the internal distribution of electric conductivity.The resulting image can be used for diagnosing stroke or assessing the lung function of cystic fibrosis patients.The mathematical model of EIT is the inverse conductivity problem introduced by Alberto Calderon in 1980.It is a generic example of an ill-posed inverse boundary value problem, where one tries to reconstruct a PDEcoefficient from a Dirichlet-to-Neumann map. This reconstruction task is highly sensitive to modelling errorsand measurement noise, and therefore requires regularised solution.A mathematically satisfying regularisation approach is offered by a nonlinear Fourier transform, based onComplex Geometric Optics solutions introduced by John Sylvester and Gunther Uhlmann in 1987. A low-pass filter applied on the nonlinear frequency domain enables robust real-time EIT imaging, with cutofffrequency determined by the amplitude of measurement noise. This imaging method is based on solving aD-bar equation and is connected to the theory of pseudoanalytic functions.There are further interesting possibilities arising from the use of the nonlinear Fourier transform. An addedone-dimensional Fourier transform leads to singularity propagation along two-dimensional leaves, accordingto the Duistermaat-Hormander theory of complex principal type operators. This can be used in EIT imagingfor recovering boundaries between tissues and organs.Furthermore, the nonlinear Fourier transform can be used for linearising the Novikov-Veselov equation, a(2+1) dimensional generalisation of the KdV equation.Based on these examples it is safe to say that the nonlinear Fourier transform is a versatile tool applicableto very different problems. It surely holds more secrets yet to be revealed.

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Samuli Siltanen received his Ph.D. in Applied Mathematics from Helsinki University of Technology in 1999.Currently, he is Professor of Industrial Mathematics at University of Helsinki and Team leader in the FinnishCentre of Excellence in Inverse Modelling and Imaging. He made major contributions in the area of inverseproblems, especially in the conductivity problem and inverse scattering. He also focuses on practical and

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industrial implementations and is holder of several patents. Among other prizes he received the J.V. SnellmanPrize in 2018.

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Hardy inequalities on homogeneous groups

Durvudkhan SuraganDepartment of Mathematics, Nazarbayev University, Kazakhstandurvudkhan.suragan@nu.edu.kz

In this talk, we discuss Hardy inequalities and closely related topics from the point of view of Folland andStein’s homogeneous (Lie) groups. The place where Hardy inequalities and homogeneous groups meet is abeautiful area of mathematics with links to many other subjects. In short, our main idea is consistentlyworking with relations of quasi-radial derivatives and the Euler operators, so from these relations followvarious Hardy type inequalities with sharp constants on homogeneous groups. While describing the generaltheory of Hardy type inequalities in the setting of general homogeneous groups, we pay particular attentionto the special class of stratified groups. In this environment, the theory of Hardy inequalities becomes in-tricately intertwined with the properties of sub-Laplacians and subelliptic partial differential equations. Todemonstrate applications of the theory we present solutions of two previously known conjectures. Partic-ularly, we discuss the Badiale-Tarantello conjecture and the conjecture on the geometric Hardy inequalityin a half-space on the Heisenberg group with a sharp constant. The present talk is partially based on ourrecent open access book (with the same title) with Michael Ruzhansky.

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Durvudkhan Suragan received his Ph.D. in Mathematics from Al-Farabi Kazakh National University in 2013.Prior to joining Nazarbayev University in April 2018, he served in the Department of Mathematics at ImperialCollege London as a research associate and at Institute of Mathematics and Mathematical Modeling (Almaty)as a leading reseacher. His current research interests include analysis on Lie groups, applied mathematics(mathematical physics), PDE, potential theory and spectral geometry. In 2013, Suragan was awarded theKonayev prize for young scientists (in Kazakhstan) for the best work in the field of natural sciences. In2018, Suragan won the Ferran Sunyer i Balaguer Prize for his (with Professor Michael Ruzhansky, ImperialCollege London) monograph: Hardy inequalities on homogeneous groups, Progress in Mathematics, Vol.327, Birkhauser, 2019. xvi+588pp.

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19

Abstracts

20

21

Sessions

S.1 Applications of dynamical systemstheory in biology

OrganisersTorsten Lindstrom

Scope of the session: In this session we primarily accepttalks that are using dynamical systems theory in orderto analyze various models that arise in biological appli-cations. The models analyzed may be mechanisticallyformulated, fitted to data, deterministic, or stochastic.Various relations between such models that arise in dif-ferent modeling approaches and under different simpli-fying assumptions can be analyzed. Possible biologicalapplications can include ecology, epidemiology, pharma-cokinetics, evolution, physiology, pattern formation, andresource distribution, but are not limited to these topics.

—Abstracts—

Backward bifurcation in SEIRS model with satuationincidence rate and treatment

Yuanji ChengMalmo Universitet, 205 06 Malmo, SWEDENyuanji.cheng@mau.se

Abstract

———

Nonlinear analogue of the May-Wigner instability tran-sition

Yan FyodorovDepartment of Mathematics, King’s College London,London WC2R 2LS, UKyan.fyodorov@kcl.ac.uk

How many equilibria will a large complex system, mod-elled by N 1 randomly coupled autonomous nonlineardifferential equations typically have? How many of thoseequilibria are stable being local attractors of nearby tra-jectories? Such questions arise in many applications,notably in stability analysis of ecological communitiesas originally posed by Robert May in 1972. Recentlythey has been partly answered within the framework ofa model introduced and analysed in Y.V. Fyodorov, B.A.Khoruzhenko (2016), and G. Ben Arous, Y.V. Fyodorov,and B.A. Khoruzhenko (under preparation) exploitingrecent insights in the theory of large random asymmet-ric matrices. We show that with increased interactionstrength such systems generically undergo an abrupttransition from a trivial phase portrait with a single sta-ble equilibrium into a topologically non-trivial regimeof ’absolute instability’ where equilibria are on averageexponentially abundant, but typically all of them areunstable, unless the dynamics is purely gradient. Wheninteractions increase even further the stable equilibriaeventually become on average exponentially abundantunless the dynamics is purely solenoidal. We further

calculate the mean proportion of equilibria which havea fixed fraction of unstable directions.

———

Global bifurcations of limit cycles and strange attrac-tors in multi-parameter polynomial dynamical systems

Valery GaikoNational Academy of Sciences of Belarus, United Insti-tute of Informatics Problems, Surganov Str. 6, 220012Minsk, BELARUSvalery.gaiko@gmail.com

We develop new bifurcational geometric methods basedon the Wintner–Perko termination principle for theglobal bifurcation analysis of planar multi-parameterpolynomial dynamical systems. Using these methods,we present, e. g., a solution of Hilbert’s Sixteenth Prob-lem on the maximum number and distribution of limitcycles for the Kukles cubic-linear system, the generalLienard polynomial system with an arbitrary number ofsingular points, a Leslie–Gower system which models thepopulation dynamics in a real ecological or biomedicalsystem, and for a reduced Topp system which modelsthe dynamics of diabetes. Applying a similar approach,we study also three-dimensional polynomial dynamicalsystems and, in particular, complete the strange attrac-tor bifurcation scenarios in Lorenz type systems connect-ing globally the homoclinic, period-doubling, Andronov–Shilnikov, and period-halving bifurcations of limit cy-cles.

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Age- and size-structured models of population dynam-ics: optimal control and sustainability

Natali HritonenkoPrairie View A&M University, 77446 Prairie View,Texas, USAnahritonenko@pvamu.edu

Optimal control of population dynamics in forestry, fish-ery, agriculture and environmental sciences will be dis-cussed. The focus will be on arising differential and in-tegral dynamic population models with two-dimensionaldistributed controls. Such models describe a popula-tion as a deterministic dynamic system with heteroge-neous elements, whose performance depends on the cur-rent time and an additional independent variable (ageor size).Two models, the age-structured Lotka-McKendrikmodel of population dynamics and a nonlinear size-structured model of rational forest management, will beconsidered. In forestry and fishery, the link between thesize and age of population individuals is rather weakand corresponding models use the individual size ratherthan its age as a variable, which leads to size-structured(physiologically structured) models. The models havebeen employed to solve various practical problems thatinclude, but are not limited to, finding optimal harvest-ing strategies and exploring sustainable development un-der changing environmental conditions. The optimalcontrol, steady-state analysis, and bifurcation analysiswill be presented. Applied interpretation of the obtainedoutcomes will be provided.

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21

Abstracts

Two-dimensional differential delay systems as mathe-matical models of physiological processes

Anatoli IvanovPennsylvania State University, P.O. Box 264, Lehman,PA 18627, USAaivanov@psu.edu

We study dynamical properties of solutions of two-dimensional differential systems with delay which serveas mathematical models of several physiological pro-cesses in human body. One such model describes theintracellular circadian rhythm generator; it was intro-duced in 1999 by Scheper et al. in the Journal of Neuro-science. Another one describes oscillations in a glucose-insulin interaction model with time delay; its several ver-sions were introduced by different researchers who stud-ied various aspects of this physiological phenomenon.Mathematically the models can be incorporated by asystem of delay differential equations of the form

(1) x ′(t) = −αx(t) + f(x(t), y(t), x(t− τ), y(t− σ))

y ′(t) = −βy(t) + g(x(t), y(t), x(t− τ), y(t− σ)),

where nonlinearities f and g are continuous real-valuedfunctions, decay rates α, β are positive, and delays τ, σare non-negative with τ + σ > 0. Sufficient conditionsfor the existence of periodic solutions in system (1) areestablished. The nonlinearities f and g are further as-sumed to satisfy either positive or negative feedback con-dition with the overall negative feedback in the system.The stability of the unique equilibrium and oscillationof all solutions about it are studied and derived in termsof the characteristic equation of the linearized system.The instability of the equilibrium together with a one-sided boundedness of either f or g lead to the existenceof periodic solutions. The analysis of system (1) usessome of the recent results established for higher orderdifferential delay equations.

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On the limit configuration of four species strongly com-peting systems

Flavia LanzaraMathematics Department, University ”La Sapienza”,00144 Rome, ITALYlanzara@mat.uniroma1.it

According to the competitive exclusion principle (alsoknown as Gause’s law), many competing species can-not coexist under very strong competition, but whenspatial movements are permitted more than one speciescan coexist thanks to the segregation of their habitats.From a mathematical viewpoint the determination of theconfiguration of the habitat segregation is an interestingproblem which can be modelled by an optimal (in a suit-able sense) partition of a domain. In recent papers theproblem is studied modelling the interspecies competi-tion with a large interaction term in an elliptic system ofpartial differential equations inspired by classical modelsin populations dynamics.In this talk we analysed some qualitative properties ofthe limit configuration of the solutions of a reaction-diffusion system of four competing species as the com-petition rate tends to infinity. Large interaction inducesthe spatial segregation of the species and only two limit

configurations are possible: either there is a point wherefour species concur, a 4-point, or there are two pointswhere only three species concur. We characterized, fora given datum, the possible 4-point configuration bymeans of the solution of a Dirichlet problem for theLaplace equation.This is a joint work with Eugenio Montefusco (SapienzaUniversity, Rome, Italy).

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Applications of optimal control theory to cholera

Ana P. Lemos-PaiaoCIDMA, University of Aveiro, 3810-193 Aveiro, POR-TUGALanapaiao@ua.pt

Applications of optimal control theory to cholera Ab-stract: We propose and study several mathematicalmodels for the transmission dynamics of some strainsof the bacterium Vibrio cholerae, responsible for thecholera disease in humans. Control functions are addedwith the purpose to obtain different optimal controlproblems. Their study allow us to determine the bestway to curtail the spread of the disease. Such models areapplied to the cholera’s outbreak of Haiti (2010-2011)and Yemen (2017-2018).

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Destabilization, stabilization, and multiple attractors insaturated mixotrophic environments

Torsten LindstromDepartment of Mathematics, Linnaeus University, 35195Vaxjo, SWEDENTorsten.Lindstrom@lnu.se

In this talk we elucidate the dynamical consequences ofthe invasion of mixotrophs by the use of a model that is alimiting case the chemostat possessing explicit resourcedynamics modeling. The model is a hybrid of a com-petition model describing the competition for resourcesbetween an autotroph and a mixotroph and a predator-prey model describing the interaction between the au-totroph and a herbivore.Mixotrophic interactions are a strong component inharmful algae blooms, Hallenrath-Lehmann et. al.(2015) and the purpose of this paper is not predict-ing recurrent harmful algae blooms. Instead we aimat understanding the environmental conditions allow-ing for mixotrophic invasions and their dynamical con-sequences.It is possible for a mixotroph to invade both autotrophicenvironments and environments described by interac-tions between autotrophs and herbivores. The inter-action between autotrophs and herbivores might be inequilibrium or cyclic. Our first conclusion is that itis possible for an invading mixotroph to both stabilizeand destabilize autotrophs-herbivore dynamics, depend-ing on the environmental conditions and the propertiesof the invading mixotroph.Our second conclusion is that environmental conditionsallowing for multiple attractors after mixotrophic inva-sion exist. Such initial value dependent behavior may bethe consequence both after an invasion in a completelyautotrophic environment and in both cyclic and equilib-rium autotroph-herbivore environments.

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22

Abstracts

The inverse problem on tree graph with attachedmasses from the point of view of neurobiology

Gulden MurzabekovaZhenis Prospect, 62, 010008 Nur-Sultan, KAZA-KHSTANguldenmur07@gmail.com

The model is based on the practical problem of dendritictrees formed by neurons of the central nervous system.The activity of neurons is accompanied by significantvariations in electrical and physical parameters. Infor-mation is generated through an electrical impulse. Thestate of the system is described on a quantum graph [Av-donin S., Kurasov P., 2008] with differential operatorson edges and the Kirchhoff-Neumann agreement condi-tions at the vertices. Unlike the staining and visualiza-tion methods for evaluating many dendritic properties,our approach is analytically rigorous and relies on thestudy of changes in currents. In contrast to [Stephen-son E., Kojouharov H., 2018], we have given a math-ematical justification based on an inverse problem fora parabolic equation on a tree graph. We expand thetheory of neural cables by the BC method; we provesufficient conditions for the identification of a priori pa-rameters; new theorems on regularity and controllabilityin the inverse problem for a heat equation with memoryon a tree graph with attached masses in internal ver-tices. On any given branch of the dendrite (edges ei ofthe graph tree) there are sources of Ni connecting withother cells, where Ni = 0, 1, 2, . . . and Ni is a finite num-ber. The obtained results on the recovering of sourcesallow us to study the functional activity of neurons inthe process of influence of skin receptors.

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Identification algorithm for differential equation withmemory in biomathematical problems

Karlygash NurtazinaSatpayev Street, 2, 010008 Nur-Sultan, KAZAKHSTAN

nurtazina.k@gmail.com

The paper is devoted to mathematical models of sensoryphysiology which lead to different class of inverse prob-lems. Pacinian corpuscles are skin receptors that playa major role in sense of touch (Bell, 1994; Bell, 1999).A model for the membrane potential for the encapsu-lated nerve ending can be described by parabolic equa-tion with memory on tree graph with attached masses.The coefficient m(x) of this equation is linearly related tothe mechanical strain felt by the corpuscles nerve due toa local stimulus applied on the skin surface above the re-ceptor. This problem boils down to constructing an algo-rithm for identifying the coefficient of a parabolic equa-tion with memory on a tree graph with masses attachedto internal vertices. We recover the m(x) coefficientby modifying the boundary control method [Avdonin,Kurasov 2008], and develop an algorithmic approach toestimate m(x) numerically. We develop methodology of[Avdonin, Bell, 2015] to identify edge radii, lengths, andconductances for our class of graphs. It is allows us todevelop an algorithmic implementation to estimate thesequantities. Many results concerning inverse problems forPDEs on graphs dealt mostly with the hyperbolic typeequations. Our identification research involves applica-tions to parabolic equations, so our response operator

will be the Neumann-to-Dirichlet rather than Dirichlet-to-Neumann. What is more important, the matchingKirchhoff condition takes a slightly different form insome of these cases.

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From Lotka-Volterra systems to Polymatrix Replicators

Telmo PeixeULisboa, ISEG, CEMAPRE, Rua do Quelhas, 6, 1200-781 Lisbon, PORTUGALtelmopeixe@gmail.com

Consider a population divided in a finite number ofgroups, each one with a finite number of strategies,where interactions between individuals of any twogroups are allowed, including the same group. Thismodel is designated as the polymatrix game. The sys-tem of differential equations associated to a polyma-trix game, introduced recently by Alishah and Duartein 2015 and designated as polymatrix replicator, forma simple class of ordinary differential equations definedon prisms given by a product of simplexes. This class ofreplicator dynamics contains well known classes of evo-lutionary game dynamics, such as the symmetric andasymmetric replicator equations, and some replicatorequations for n-person games. As J. Hofbauer proved in1981 the replicator equation is in some sense equivalentto the Lotka-Volterra (LV) system, independently intro-duced in 1920s by A. J. Lotka and V. Volterra. The LVsystem is perhaps the most widely known system used inscientific areas as diverse as biology, physics, chemistry,and economy. In this talk I will present the definition ofpolymatrix replicator, some basic properties, and someresults about the dynamics and the inferences we canmake about the associated polymatrix game.

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Global dynamics of a new delay logistic equation arisenin cell biology

Gergely RostUniversity of Szeged, Szeged, HUNGARYrost@math.u-szeged.hu

The delayed logistic equation (also known as Hutchin-son’s equation or Wright’s equation) was originally in-troduced to explain oscillatory phenomena in ecologicaldynamics. While it motivated the development of a largenumber of mathematical tools in the study of nonlin-ear delay differential equations, it also received criticismfrom modellers because of the lack of a mechanistic bi-ological derivation and interpretation. Here we proposea new delayed logistic equation, which has clear biologi-cal underpinning coming from cell population modelling.This nonlinear differential equation includes terms withdiscrete and distributed delays. The global dynamicsis completely described, and it is proven that all feasi-ble nontrivial solutions converge to the positive equilib-rium. The main tools of the proof rely on persistencetheory, comparison principles and an L2-perturbationtechnique. Using local invariant manifolds, a unique het-eroclinic orbit is constructed that connects the unstablezero and the stable positive equilibrium, and we showthat these three complete orbits constitute the globalattractor of the system. Despite global attractivity, thedynamics is not trivial as we can observe long-lasting

23

Abstracts

transient oscillatory patterns of various shapes. We alsodiscuss the biological implications of these findings andtheir relations to other logistic type models of growthwith delays.

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Bifurcations on Invariant Tori in Predator-prey Modelswith Seasonal Prey Harvesting

Shigui RuanDepartment of Mathematics, University of Miami, CoralGables, FL 33124-4250, USAruan@math.miami.edu

We study bifurcations in predator-prey systems withseasonal prey harvesting. First, when the seasonal har-vesting reduces to constant yield, it is shown that variouskinds of bifurcations, including saddle-node bifurcation,degenerate Hopf bifurcation, and Bogdanov-Takens bi-furcation (i.e., cusp bifurcation of codimension 2), occurin the model as parameters vary. The existence of twolimit cycles and a homoclinic loop is established. Bifur-cation diagrams and phase portraits of the model arealso given by numerical simulations, which reveal farricher dynamics compared to the case without harvest-ing. Second, when harvesting is seasonal (described by aperiodic function), sufficient conditions for the existenceof an asymptotically stable periodic solution and bifur-cation of a stable periodic orbit into a stable invarianttorus of the model are given. Numerical simulations,including bifurcation diagrams, phase portraits, and at-tractors of Poincare maps, are carried out to demon-strate the existence of bifurcation of a stable periodicorbit into an invariant torus and bifurcation of a sta-ble homoclinic loop into an invariant homoclinic torus,respectively, as the amplitude of seasonal harvesting in-creases. Our study indicates that to have persistence ofthe interacting species with seasonal harvesting in theform of asymptotically stable periodic solutions or stablequasi-periodic solutions, initial species densities shouldbe located in the attraction basin of the hyperbolic sta-ble equilibrium, stable limit cycle, or stable homoclinicloop, respectively, for the model with no harvesting orwith constant-yield harvesting.

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Stability and optimal control of a delayed HIV/AIDS-PrEP model

Cristiana J. SilvaCIDMA, Department of Mathematics, University ofAveiro, 3810-193 Aveiro, PORTUGALcjoaosilva@ua.pt

Pre-exposure prophylaxis (PrEP) consists in the use ofan antiretroviral medication to prevent the acquisitionof HIV infection by uninfected individuals and has re-cently demonstrated to be highly efficacious for HIV pre-vention. In this work, we propose a delayed HIV/AIDS-PrEP model, given by a system of delayed differentialequations. We analyze the case where the implementa-tion of PrEP suffers a discrete time-delay and study theimpact of the delay on the number of new HIV infec-tions. The time delay describes mathematically barriersthat block an effective implementation of PrEP, such as,stigma, cost and adherence. The model has two equi-librium points: disease free and endemic. A local and

global stability analysis of the equilibrium points is per-formed, for any positive discrete time delay. In a finalpart, we formulate a delayed optimal control problemwith the aim to determine the PrEP implementationstrategy that minimizes the number of individuals withpre-AIDS HIV-infection, as well as the costs associatedto PrEP. The theoretical results are illustrated throughnumerical simulations.

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S.2 Complex Analysis and PartialDifferential Equations

OrganisersOkay Celebi, Sergei Rogosin

Scope of the session: This session will be the combi-nation of researchers whose interests are close to theSpecial Interest Group “Complex Analysis” and to the“Complex and functional analytic methods for differen-tial equations”.This session supports the researches in classical areasof complex analysis such as initial and boundary valueproblems for (linear and nonlinear) complex differentialequations and applications, boundary behavior of ana-lytic and generalized analytic functions, conformal map-pings, special functions in complex domains, entire andmeromorphic functions, integral equations, as well asthe modern directions in one- and several-dimensionalcomplex analysis in which functional-analytic methodsare also employed. Complex methods serve to constructexplicit solutions to linear problems in the plane withgeneralizations to higher dimensions, while functional-analytic tools are applied to treat nonlinear equationswith linear or nonlinear conditions. We also welcome thepresentations which have applications in MathematicalPhysics, Mechanics, Biology, Chemistry, Medicine, Eco-nomics etc. Applications in Fluid Dynamics, Compos-ite Materials, Porous Media, Elasticity, Visco-Elasticity,Hydro-, Aero- and Thermo-Dynamics are the most con-sidered.

—Abstracts—

Integral representations and some boundary value prob-lems in Clifford analysis

Umit AksoyAtilim University, Department of Mathematics, Incek,Golbasi, 06830 Ankara, TURKEYumit.aksoy@atilim.edu.tr

We study the Dirichlet and Neumann problems for Pois-son equation in the unit ball using integral representa-tion formulas in terms of Laplacian in the complex Clif-ford algebra Cm. By introducing integral operators inClifford analysis with their properties, we investigate theboundary value problems for higher-order Poisson equa-tion.

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Boundary Value Problems in Polydomains

A. Okay CelebiYeditepe University, Department of Mathematics, Kay-isdagi Caddesi, Atasehir, 34755 Istanbul, TURKEYacelebi@yeditepe.edu.tr

24

Abstracts

In this presentation we give a short survey of the bound-ary value problems in polydomains in the last decades.Firstly we develop an alternative method to derive in-tegral representations for functions in Cn. This unifiedmethod provides representations which are suitable tobe employed in discussions for all linear boundary valueproblems. In the rest of the article we have improvedsome results obtained for Schwarz and Dirichlet typeproblems.Subject classification: 32W10, 32W50; 31A10Keywords; Polydisc, Schwarz problem, Riquier problem,complex partial differential equations

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Well-posedness and asymptotic results of 3D Burgersequation in Gevrey class

Abdelkerim ChaabaniFaculty of mathematical, physical and natural sciences,University El Manar, Residence El jinen block 137 ap-partement 3.1 el morouj, 2097 Tunis, TUNISIAchaabanikarim89@gmail.com

We prove that the three-dimensional periodic Burgersequation has a unique global in time solution, in a crit-ical Gevrey-Sobolev space. Comparatively to Navier-Stokes equations, the main difficulty is the lack of incom-pressibility condition. In our proof of existence, we over-come the bootstrapping argument, which was a technicalstep in a precedent proof, in Sololev spaces. This makesour proof shorter and gives sense of considering Gevreyclass for a mathematical study to Burgers equation. Toprove that the unique solution is global in time, we usethe maximum principle. Energy methods, Sobolev prod-uct laws, compactness methods and Fourier analysis arethe main tools.

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Existence and Uniqueness Theorem for Cusped Kelvin-Voigt Elastic Plates in the Zero Approximation of theHierarchical Models

Natalia ChinchaladzeI.Vekua Institute of Applied Mathematics of I.Javakhishvili, Tbilisi State University, University street2, 0186 Tbilisi, Tbilisi, GEORGIAchinchaladze@gmail.com

The talk is devoted to the homogeneous Dirichlet prob-lem for the vibration problem of cusped viscoelasticKelvin-Voigt prismatic shells in case of the zero approx-imation of the hierarchical models. The classical andweak setting of the problem are formulated. The spa-cial weighted functional spaces are introduced, which arecrucial in our analysis. The coerciveness of the corre-sponding bilinear form is shown and uniqueness and ex-istence results for the variational problem are proved.We describe in detail the structure of this spaces and es-tablish their connection with weighted Sobolev spaces.

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Derivation of system of the equations of equilibrium forshallow shells and plates with voids

Bakur GuluaSokhumi State University, Iv. Javakhishvili Tbilisi StateUniversity, 61 Politkovskaya street, University st. 2.,

0186 Tbilisi, GEORGIAbak.gulua@gmail.com

In the report the three-dimensional system of equationsof equilibrium for solids with voids is considered. Fromthis system of equations, using a reduction method of I.Vekua, we receive the equilibrium equations for the shal-low shells. Further we consider the case of plates withconstant thickness in more detail. Namely, the systemsof equations corresponding to approximations N = 0and N = 1 are written down in a complex form and weexpress the general solutions of these systems throughanalytic functions of complex variable and solutions ofthe Helmholtz equation. The received general represen-tations give the opportunity to solve analytically bound-ary value problems.

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The product-type operators from logarithmic Blochspaces to Zygmund-type spaces

Yongmin LiuSchool of Mathematics and Statistics, Jiangsu NormalUniversity, 221116 Xuzhou, Jiangsu, PEOPLE’S RE-PUBLIC OF CHINAminliu@jsnu.edu.cn

The boundedness and compactness of a product-typeoperator, recently introduced by S. Stevic, A. Sharmaand R. Krishan,

Tnψ1,ψ2,ϕf(z) = ψ1(z)f (n)(ϕ(z))+ψ2(z)f (n+1)(ϕ(z)), f ∈ H(D),

from the logarithmic Bloch spaces to Zygmund-typespaces are characterized, where ψ1, ψ2 ∈ H(D), ϕ is ananalytic self-map of D and n a positive integer.

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Higher order mean value functions and polyharmonicfunctions

Grzegorz LysikJan Kochanowski University, Swietokrzyska 15, 25-406Kielce, POLANDglysik@ujk.edu.pl

We introduce integral mean value functions which areaverages of integral means over spheres/balls and overtheir images under the action of a discrete group of com-plex rotations. In the case of real analytic functions wederive higher order Pizzetti’s formulas. As applicationswe obtain a maximum principle for polyharmonic func-tions and a characterization of convergent solutions tohigher order heat type equations. Also a new Dirichlettype problem for polyharmonic functions is introducedand solved in a special case.

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On the structure of generalized analytic function in thevicinity of singular point

Nino Manjavidze3/5 Cholokashvili St., 0162 Tbilisi, GEORGIAnino.manjavidze@iliauni.edu.ge

The talk deals with the structure of the solutions ofCarleman-Vekua equations in the neighborhood of sin-gular point. The problem of the existence of a specialmajorant is studied. Some direct applications are pre-sented. Talk is partially supported by the Shota Rus-taveli National Science Foundation grant N 17-96

25

Abstracts

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On the dimension of the kernel of a singular integraloperator with non-Carleman shift and conjugation

Rui MarreirosDepartamento de Matematica, Faculdade de Cienciase Tecnologia, Universidade do Algarve, 8005-139 Faro,PORTUGALrmarrei@ualg.pt

On the Hilbert space L2(T) the singular integral op-erator with non-Carleman shift and conjugation K =P+ + (aI + AC)P− is considered, where P± are the

Cauchy projectors, A =m∑j=0

ajUj , a, aj , j = 1,m, are

continuous functions on the unit circle T, U is the shiftoperator and C is the operator of complex conjugation.An estimate for the dimension of the kernel of the opera-tor K is obtained; some particular cases are considered.

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Global existence for coupled to the structurally dampedσ-evolution model

Mohamed Kainane MezadekUniversity Hassiba Benbouali of Chlef, Department ofMathematics, Faculty of Exact Science and Informatics,02000 Chlef, Chlef, ALGERIAkainane med@yahoo.fr

We are interested in the global existence to the coupledCauchy problem for structurally damped σ-evolutionmodels:

utt(t, x) + (−∆)σu(t, x) + b(t)(−∆)δut(t, x) = |v|p,u(0, x) := u0(x), ut(0, x) := u1(x)

vtt(t, x) + (−∆)σv(t, x) + µ(−∆)δvt(t, x) = |u|p,v(0, x) := v0(x), vt(0, x) := v1(x),

for (t, x) ∈ (0,∞) × Rn, where δ ∈ [0, σ], σ > 1 andb = b(t) is a positive function.

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On the existence and compactness of the resolvent ofa Schrodinger operator with a negative parameter

Madi MuratbekovKazakh University of Economics, Finance and Inter-national Trade, 7, Zhubanov street, 010005 Astana,KAZAKHSTANmmuratbekov@kuef.kz

The talk is devoted to the questions of the existence ofthe resolvent and coercive estimates of the Schrodingeroperator with a negative parameter in the space L2(Rn),where n ≥ 2. We note that a negative Schrodinger op-erator with a negative parameter arises when studyingsingular differential operators of hyperbolic type in thespace L2(Rn+1).

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To theory one class of three-dimensional integral equa-tion with super-singular kernels by tube domain

Nusrat RajabovTajik National University, Rudaki avenue 17, 734025

Dushanbe, TAJIKISTANnusrat38@mail.ru

In this work, we investigate one class of three- dimen-sional integral equation by tube domains, are in powerbasis and lateral surface and may have supersingularity.

Let Ω denote the cylindrical domain Ω = (t, z) : a <t < b, |z| < R. Lower ground this cylinder denote byD = t = a, |z| < R and the lateral surface will be de-note by S = a < t < b, |z| = R, z = x+ iy. In domainΩ we shall consider the following integral equation

ϕ(t, z) +

∫ t

a

K1(t, τ)

(τ − a)αϕ(τ, z)dτ

+1

π

∫∫D

exp[iθ]K2(r, ρ)

((R− ρ)β)(s− z)ϕ(t, s)ds

+1

π

∫ t

a

dτ

(τ − a)α

∫∫D

K3(t, τ ; r, ρ)

(R− ρ)β(s− z)exp[iθ]ϕ(τ, s)ds

= f(t, z),

where θ = args, s = ξ + iη, ds =dξdη, ρ2 = ξ2 + η2, r2 = x2 + y2,K1(t, τ) =n∑j=1

Aj(ωαa (t) − ωαa (τ))j−1,K2(r, ρ) =

m∑l=1

Bl(ωβa (r) −

ωβa (ρ))i−1),K3(t, τ ; r, ρ) = K1(t, τ)K2(r, ρ), Aj(1 ≤j ≤ n), Bl(1 ≤ l ≤ m) -are given constants, f(t, z)-are given function, ϕ(t, z)-unknown function, ωαa (t) =[(α−1)(t−a)α−1]−1. In this work in depend of the rootsof the characteristics equations

λn +

n∑j=1

Aj(j − 1)!λn−j = 0,

and

µm +

m∑l=1

Bj(j − 1)!µn−j = 0

obtained manifold solution. In this case, when generalsolution contain arbitrary functions, stand and investi-gation different boundary value problems.

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2-D Frequency-Domain System Identification

Xiaoyin WangUniversity of Macau, Avenida da Universidade, Taipa,Macau, PEOPLE’S REPUBLIC OF CHINAyb57448@connect.umac.mo

In this article, we propose two iterative algorithms toidentify transfer functions of 2-D systems. The proposedalgorithms are modifications of the two-dimensionalAdaptive Fourier Decomposition (abbreviated as 2-DAFD) and Weak Pre-Orthogonal Adaptive Fourier De-composition (abbreviated as W-POAFD). 2-D AFD andW-POAFD are newly established adaptive representa-tion theories for multivariate functions utilizing, respec-tively, the product-TM system and the product-Szegodictionary. The proposed algorithms give rise to rationalapproximations with real coefficients to transfer func-tions. Owing to the modified maximal selection prin-ciples, the algorithms achieve a fast convergence rate

O(n−12 ) . To use 2-D AFD and W-POAFD for sys-

tem identification not only the theory is revised, butalso the practical algorithm codes are provided. Experi-mental examples show that the proposed algorithms givepromising results. The theory and algorithms studied inthis paper are valid for any n-D case, n ≥ 2.

26

Abstracts

———

Paley–Wiener-type Theorem for Analytic Functions inTubular Domains

Huang YunUniversity of Macau, Taipa, Macau, PEOPLE’S RE-PUBLIC OF CHINAyhuang12@126.com

Herein, a weighted version of the Paley–Wiener-typetheorem for analytic functions in a tubular domain overa regular cone is obtained using Hp space method.Then, the classical n-dimensional Paley–Wiener theo-rem is generalized to a case wherein 0 < p < 2 and Kis not required to be a symmetric body. Finally, a ver-sion of the edge-of-the-wedge theorem is obtained as anapplication of the weighted theorems.

———

S.3 Complex Geometry

OrganisersAlexander Schmitt

Scope of the session: The main topics will be Higgs bun-dles and, more generally, augmented principal bundles.These are objects of complex algebraic geometry withlinks to other areas, such as arithmetic and mathemati-cal physics. In the form of gauge theory, analysis plays amajor role in the investigation of these objects. In fact,the so-called Kobayashi-Hitchin correspondence relatesan algebro-geometric moduli space for these objects toa gauge theoretically defined one. The proof of the cor-respondence is highly analytical. The correspondenceand its proof played a major role, e.g., in the work ofSir Simon Donaldson on invariants of differentiable four-manifolds for which he was awarded the fields medal.

—Abstracts—

Higgs bundles and Schottky representations

Ana CasimiroFaculdade de Ciencias e Tecnologia, Universidade Novade Lisboa, Quinta da Torre, 2829-516 Caparica, POR-TUGALamc@fct.unl.pt

We relate Schottky representations to certain La-grangian subspaces of the moduli space of Higgs G-bundles (G is a connected reductive algebraic group).It is a fundamental result in the theory of Higgs bun-dles, the so-called non-abelian Hodge theorem, that byconsidering the Hitchin equations for G-Higgs bundless,one obtains a homeomorphism between the Betti space,B, and the moduli space of semistable G-Higgs bundlesover a Riemann surface X. By a remark of Baraglia-Schaposnik, when considering G-Higgs bundles over Xwith a real structure, one is naturally lead to represen-tations into G of the fundamental group of a 3-manifoldwith boundary X. These are naturally related to Schot-tky representations, as we will present in this talk. Ourapproach via Schottky representations has one advan-tage: we obtain a simple argument that shows that theBaraglia-Schaposnik brane is indeed Lagrangian with re-spect to the natural complex structure of B (coming

from the complex structure of G). More precisely, weobtain a simpler proof of the vanishing of the complexsymplectic form on the strict Schottky locus. This ajoint work with Susana Ferreira (IPL) and Carlos Flo-rentino (FCUL).

———

Intersection cohomology of the moduli space of Higgsbundles on a smooth projective curve

Camilla FelisettiUniversity of Geneva, Villa Battelle, 7 Route de Drize,1227 Carouge, Geneva, SWITZERLANDcamilla.felisetti@unige.ch

Let X be a smooth projective curve of genus g overC. The character variety MB parametrizing conjugacyclasses of representations from the fundamental groupof X into SL(2,C) is an affine irreducible singular pro-jective variety. The Non Abelian Hodge theorem statesthat there is a real analytic isomorphism between MB

and the quasi projective singular variety MDol whichparametrizes semistable Higgs bundles of rank 2 and de-gree 0 on X. During the seminar I will present a desin-gularization of these moduli spaces and I will computethe intersection cohomology of MDol using the famousDecomposition theorem by Beilinson, Bernstein, Deligneand Gabber. Moreover I will show that the mixed Hodgestructure on the intersection cohomology is pure, show-ing evidence that an analogue of the P = W conjecturemight hold for singular moduli spaces.

———

Generating Functions for Hodge-Euler polynomials ofGL(n,C)-character varieties

Carlos FlorentinoDept. Mat. Faculdade Ciencias, Univ. de Lisboa, Edf.C6, Cp. Grande, 1749-016 Lisboa, PORTUGALCaflorentino@fc.ul.pt

Given a finitely generated group F and a complex re-ductive Lie group G, the G-character variety of F ,XFG = Hom(F,G)//G, is typically a singular algebraicvariety whose geometric and topological (and sometimesarithmetic) properties can be studied via mixed Hodgestructures (MHS).The most interesting cases are when F is the funda-mental group of a Kahler manifold M , since then XFGis homeomorphic to a space of G-Higgs bundles overM . Some special classes of character varieties havetheir MHS encoded in a polynomial generalization ofthe Euler-Poincare characteristic: the Hodge-Euler, alsocalled the E-polynomial.In this seminar, concentrating in the case of the generallinear group G = GL(n,C), we present a remarkable re-lation between the E-polynomials of XFG and those ofXirrF G, the locus of irreducible representations of F into

G. We will also give an overview of known explicit com-putations of E-polynomials, as well as some conjecturesand open problems.

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Cartan branes on the Hitchin system

Emilio FrancoFaculdade de Ciencias, Departamento de Matematica,Universidade do Porto, Rua do Campo Alegre

27

Abstracts

1021/1055. 4169-007 Porto, PORTUGALemilio.franco@fc.up.pt

We study mirror symmetry on the singular locus of theHitchin system at two levels. Firstly, by covering it by(supports of) BBB-branes, corresponding to Higgs bun-dles reducing their structure group to the Levi subgroupof some parabolic subgroup P, whose conjectural dualBAA-branes we identify. Heuristically speaking, the lat-ter are given by Higgs bundles reducing their structuregroup to the unipotent radical of P. Secondly, when P isa Borel subgroup, we are able to construct a family ofhyperholomorphic bundles on the BBB-brane, and studythe variation of the dual under this choice. We give evi-dence of both families of branes being dual under mirrorsymmetry via an ad-hoc Fourier-Mukai integral functor.

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SO(p, q)−Higgs bundles and higher Teichmuller com-ponents

Peter GothenCentro de Matematica da Universidade do Porto, Fac-uldade de Ciencias da UP, Edifıcio FC1, Rua do CampoAlegre, s/n, 4169-007 Porto, PORTUGALpbgothen@fc.up.pt

Some connected components of a moduli space are mun-dane in the sense that they are distinguished only byobvious topological invariants or have no special char-acteristics. Others, such as the Hitchin component inthe moduli space of Higgs bundles, are more alluringand unusual either because they are not detected by pri-mary invariants, or because they have special geometricsignificance, or both. In this paper we describe new ex-amples of such ”exotic” components in moduli spaces ofSO(p, q)−Higgs bundles on closed Riemann surfaces or,equivalently, moduli spaces of surface group representa-tions into the Lie group SO(p, q). We also provide a com-plete count of the connected components of these modulispaces (except for SO(2, q), with q > 3). Time permit-ting, we will comment on possible generalizations. Thetalk will mainly be based on arXiv:1802.08093 which isjoint work with Marta Aparicio-Arroyo, Steven Bradlow,Brian Collier, Oscar Garcıa-Prada and Andre Oliveira.

———

The Riemann-Hilbert mapping in genus two

Viktoria HeuIRMA, 7 rue Rene Descartes, 67084 Strasbourg Cedex,Bas-Rhin, FRANCEheu@math.unistra.fr

One possible formulation of the Riemann-Hilbert prob-lem in higher genus is ask which is the vector bundle un-derlying the holomorphic connection over a curve asso-ciated to a given monodromy representation. Since themonodromy is given in terms of the topological and notthe complex structure of the curve, one may vary the lat-ter and obtains, by Riemann-Hilbert correspondence, anisomonodromic family of connections. In collaborationwith F. Loray, we obtained the following result: in themoduli space of irreducible sl2C-connections over genustwo curves, the isomonodromic foliation is transversal tothe locus of the trivial bundle and transversal to the lo-cus of flat unstable bundles. In this talk, we will presentsome applications of this result and of its proof.

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Universal torsors over degenerating del Pezzo surfaces

Norbert HoffmannMary Immaculate College, South Circular Road, V94VN26 Limerick, IRELANDnorbert.hoffmann@mic.ul.ie

Let S be a split one-parameter family of smooth delPezzo surfaces degenerating to a del Pezzo surface withADE-singularities. Let G be the reductive group givenby the root system of these singularities. We constructa G-torsor (or principal G-bundle) over S whose restric-tion to the smooth fibres is the extension of structuregroup of the universal torsor under the Neron-Severitorus introduced and studied by Colliot-Thelene andSansuc. The G-torsor is unique and infinitesimally rigid.This extends a construction of Friedman and Morgan forindividual singular del Pezzo surfaces. It is joint workwith Ulrich Derenthal.

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Compactification of the moduli space of stable principalG-bundles over a stable curve and beyond

Angel Luis Munoz-CastanedaDepartment of Mathematics, University of Leon, SPAIN

amunc@unileon.es

Given a projective manifold X, a reductive group G anda faithfull representation ρ : G→ SL(V ), A. Schmitt de-fined a singular principal G-bundle on it as a pair (F, τ)given by a torsion free sheaf and a morphism of alge-bras τ : S•(V ⊗ F )G → OX , and proved the existenceof a compact moduli space for δ-(semi)stable singularprincipal G-bundles having the space of stable principalG-bundles as an open subscheme. U. Bhosle proved theexistence of a compact moduli space for δ-(semi)stablesingular principal G-bundles over irreducible projectivecurves with at most nodes as singularities. An impor-tant feature of this moduli space is that, when the curveis smooth, it is isomorphic to the classical moduli spaceconstructed by A. Ramanathan provided the rationalparameter δ is large enough. When the base curve isreducible, we can generalize the definition of singularprincipal bundle by considering sheaves of depth one. Inthis talk, I will discuss the existence of a compact mod-uli space for δ-(semi)stable singular principal G-bundlesover nodal (possibly reducible) projective curves and itsbehavior under variations of the base curve along Mg.

———

Klein coverings of genus 2 curves

Angela OrtegaHumboldt Universitat, Institut fur Mathematik, Unterden Linden 6, D-10099 Berlin, GERMANYortega@math.hu-berlin.de

We consider etale 4 : 1 coverings of smooth genus 2curves with the monodromy group the Klein group. De-pending on the values of the Weil pairing restrictedto the group defining the covering, we distinguish theisotropic and non-isotropic case. In this talk we willdiscuss the correspondence between the non-isotropicKlein coverings and the (1, 4)-polarised abelian surface.As a consequence of this, one can show the existence

28

Abstracts

of exactly four hyperelliptic curves in a general (1, 4)-polarised abelian surface. We will also give several char-acterisations of the Klein coverings (isotropic and non-isotropic) leading to the result that the correspondingPrym maps are generically injective in both cases.This is a joint work with Pawe l Borowka.

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Mirror symmetry on some non generic loci of theHitchin system

Ana Peon-NietoUniversity of Geneva, Villa Battelle, 7 Route de Drize,1227 Carouge, Geneva, SWITZERLANDana.peon-nieto@unige.ch

Mirror symmetry for Hitchin systems has been provento be realised by a relative Fourier-Mukai transform overthe generic locus. In this talk I will explain how mirrorsymmetry operates on some natural hyperholomorphicbranes on the Hitchin system, given by fixed points bytensorisation with a torsion line bundle. Generically,they are supported over the locus of singular integralspectral curves. Many aspects of their geometry arehowever more easily understood through branes sup-ported on the most singular locus of a related Hitchinsystem. I will refer to this interplay during the exposi-tion.This is joint work with E. Franco, P. Gothen and A.Oliveira.

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Surface braid groups, finite Heisenberg covers and dou-ble Kodaira fibrations

Francesco PolizziUniversita della Calabria - Dipartimento di Matematicae Informatica, Via P. Bucci Cubo 30 B, 87036 Arcava-cata di Rende, Cosenza, ITALYpolizzi@mat.unical.it

A Kodaira fibration is a smooth, connected holomor-phic fibration f : S → B, where S is a compact com-plex surface and B is a compact complex curve, whichis not isotrivial (this means that not all its fibres arebiholomorphic to each other). Examples of such fibra-tions were originally constructed by Kodaira as a wayto show that, unlike the topological Euler characteris-tic, the signature σ of a manifold is not multiplicativefor fibre bundles. In fact, every Kodaira fibred surfaceS satisfies σ(S) > 0, whereas σ(B) = σ(F ) = 0, andso σ(S) 6= σ(B)σ(F ). On the other hand, in a clas-sical work by Chern, Hirzebruch and Serre it is provedthat the signature is multiplicative for differentiable fibrebundles in the case where the monodromy action of thefundamental group π1(B) on the rational cohomologyring H∗(F, Q) is trivial; thus, Kodaira fibrations pro-vide examples of fibre bundles for which this action isnon-trivial. In this talk, we show how to construct newexamples of double Kodaira fibrations by using finiteHeisenberg covers (i.e., Galois covers with Galois groupisomorphic to a finite Heisenberg group) of a productΣb×Σb, where Σb is a smooth projective curve of genusb ≥ 2. Each cover is obtained by providing an explicitgroup epimorphism from the pure braid group P2(Σb)to the corresponding Heisenberg group. In particular,we exhibit the first examples of surface that admits two

distinct Kodaira fibrations with base genus 2, answer-ing a stronger version of a problem from Kirby’s list inlow-dimensional topology:

Theorem. There exists an oriented 4-manifold X ofsignature 144 that can be realized as a surface bundleover a surface of genus 2 with fibre genus 325 in twodifferent ways.

This is joint work with A. Causin.

———

On Galois group of factorized covers of curves

Martha RomeroUniversidad del Cauca, Carrera 2A No. 3N-111 oficina306, 190002 Popayan, Cauca, COLOMBIAmjromero@unicauca.edu.co

Let Y ψ−−−→ X ϕ−−−→ P1 be a sequence of covers of com-pact Riemann surfaces. In this work we study and com-pletely characterize the Galois group G(ϕψ) of ϕψunder the following properties: ϕ is a simple cover ofdegree m and ψ is a Galois unramified cover of degreen with abelian Galois group of type (n1, n2, ...ns).We prove that G(ϕψ) ∼= (Zn1×Zn2×· · ·Zns)m−1oSm.Furthermore, we give a natural geometric generator sys-tem of G(ϕψ) obtained by studying the action on thecompact Riemann surface Z associated to the Galoisclosure of ϕ ψ.

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Higher Teichmuller spaces for orbifolds

Florent SchaffhauserUniversidad de Los Andes, Bogota, COLOMBIAflorent.schaffhauser@gmail.com

The Teichmuller space of a compact 2-orbifold X can bedefined as the space of faithful and discrete representa-tions of the fundamental group of X into PGL(2,R).It is a contractible space. For closed orientable sur-faces, ”Higher analogues” of the Teichmuller space are,by definition, (unions of) connected components of rep-resentation varieties of π1(X) that consist entirely of dis-crete and faithful representations. There are two knownfamilies of such spaces, namely Hitchin representationsand maximal representations, and conjectures on how tofind others. In joint work with Daniele Alessandrini andGye-Seon Lee, we show that the natural generalisationof Hitchin components to the orbifold case yield newexamples of Higher Teichmuller spaces: Hitchin repre-sentations of orbifold fundamental groups are discreteand faithful, and share many other properties of Hitchinrepresentations of surface groups. However, we also un-cover new phenomena, which are specific to the orbifoldcase.

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Theta functions on moduli spaces of local systems

Tom SutherlandUniversidade de Lisboa, Campo Grande - Edificio C6,1749-016 Lisboa, PORTUGALtasutherland@fc.ul.pt

Moduli spaces of SL(2,C)-local systems on puncturedRiemann surfaces provided examples of log Calabi-Yauvarieties, for which a generalisation of the classical thetafunctions has been developed by Gross and Siebert. We

29

Abstracts

will see an explicit computation and a modular interpre-tation of these theta functions for the moduli space oflocal systems of the four-punctured sphere, which is thetotal space of a family of affine cubic surfaces. Furtherwe will discuss the relationship with the complex geome-try of a diffeomorphic family of rational elliptic surfaceswith a singular fibre of type I∗0 , the complement of whichis the corresponding moduli space of Higgs bundles.

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S.4 Complex Variables and Potential Theory

OrganisersTahir Aliyev Azeroglu,Massimo Lanza de Cristoforis,Anatoly Golberg, Sergiy Plaksa

Scope of the session: This session is devoted to the widerange of directions of complex analysis, potential theory,their applications and related topics.

—Abstracts—

Some Remarks on the Uniqueness Part of SchwarzLemma

Tugba AkyelMaltepe University, Faculty of Engineering and NaturalSciences, Department of Computer Engineering, Istan-bul, TURKEYtugbaakyel@maltepe.edu.tr

Let f be an holomorphic function from the unit disc toitself. We generalize Schwarz Lemma at the boundaryimproving the condition on the bilogarithmic concavemajorant and provide conditions on the local behaviourof f along boundary near a finite set of the boundarypoints that requires f to be a finite Blaschke product.The basis of proofs of the main results is based on aspecial version of Phragmen-Lindelof Princible.KeyWords. Holomorphic function, Harnack inequal-ity, Bilogarithmic concave majorant, Phragmen-Lindelofprincible.

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Stokes’ Theorem and the Cauchy - Pompeiu Formulafor Polydisc

Mohammed AlipDepartment of Mathematics, College of Arts and Sci-ences, Khalifa University, Abu Dhabi, UAEmaimaiti.alifu@ku.ac.ae

The talk will present some challenges in finding theCauchy - Pompeiu formula for polydiscs. It is shownby Hormander and others that the Stokes’ Theorem isthe key in establishing the Cauchy - Pompeiu formula forgeneral functions on a disc in one dimensional complexdomain. However the Stokes’ Theorem for polydiscs isnot readily available and its proof is an open problem.For this purpose, differential forms could be applied asit was done in one variable case and in ball like domains.Questions remain for the polydisc: the Stokes’ Theoremis applicable to which boundary? To the characteristicboundary or to the entire boundary of the polydisc? Thetalk articulate on these points.

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Completeness theorems for systems of particular solu-tions of partial differential equations

Alberto CialdeaUniversity of Basilicata, viale dell’Ateneo Lucano, 10,85100 Potenza, ITALYcialdea@email.it

Roughly speaking there are two different kinds of com-pleteness theorems for systems of particular solutionsof partial differential equations. Results of a first kindshow that we can approximate in a domain a solution ofa partial differential equation by a sequence of particularsolutions of the same equation. For example, if we havea holomorphic function f of one complex variable, wemay ask when f can be approximated in some norms bypolynomials or by rational functions. The classical theo-rems of Runge and Mergelyan are the main results in thisdirection. These problems have been widely studied andextended to general elliptic partial differential equations.Results of a second kind are the so called completenesstheorems in the sense of Picone. They are much moresophisticated and related not only to a partial differen-tial equation, but also to a particular boundary valueproblem. The present talk is devoted to consider theproblem of completeness in this last sense. In particularwe shall give necessary and sufficient conditions for thecompleteness of polynomial solutions of a partial differ-ential equations of higher order related to the Dirichletproblem. We shall give also some recent results obtainedfor systems, where very little is known. All these resultshave been obtained by using potential theory.

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Existence, uniqueness, and regularity properties of thesolutions of a nonlinear transmission problem

Matteo Dalla RivaThe University of Tulsa, 74104 Tulsa, Oklahoma, USAmatteo-dallariva@utulsa.edu

We consider a nonlinear contact problem that arisesin the study of composite structures glued togetherby thermo-active materials. By an approach based onboundary integral equations and on the Schauder fixed-point theorem we can prove that the problem has solu-tions; however, such solution may not be locally uniqueand may be also very irregular. For example, we mighthave solutions that are not in Hs for any s > 1/2.The results presented are obtained in collaboration withB. Luczak (the University of Tulsa, US), G. Mishuris(Aberystwyth University, UK), R. Molinarolo (Aberyst-wyth University, UK), and P. Musolino (Universita degliStudi di Padova, Italy).

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Extremal decomposition of the complex plane with freepoles

Iryna DenegaInstitute of Mathematics of the National Academy ofSciences of Ukraine, Department of Complex Analysisand Potential Theory, 01004 Kiev-4, 3, Tereschenkivskast., UKRAINEiradenega@gmail.com

30

Abstracts

In geometric function theory of complex variable ex-tremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are re-duced to determination of the maximum of product ofinner radii on the system of non-overlapping domainssatisfying a certain conditions. We consider the well-known problem of maximum of the functional

rγ (B0, 0)

n∏k=1

r (Bk, ak) ,

where B0,...,Bn (n > 2) are pairwise disjoint domains inC, a0 = 0, |ak| = 1, k = 1, n are different points of theunit circle, γ ∈ (0, n], and r(B, a) is the inner radius ofthe domain B ⊂ C relative to a point a. This problemwas posed as an open problem in the Dubinin paper in1994. Till now, the problem has not been solved, thoughsome partial solutions are available. The proof is due toDubinin for γ = 1 and to Kuz’mina for 0 < γ < 1. Sub-sequently, Kovalev in 1996 solved this problem for n > 5under the additional assumption that the angles betweenneighbouring line segments [0, ak] do not exceed 2π/

√γ.

We obtained evaluation of the functional for all n andγ ∈ (1, n]. Theorem. Let n ∈ N, n > 2, γ ∈ (1, n].Then, for any system of different points aknk=1 of theunit circle and any mutually non-overlapping domainsBk, ak ∈ Bk ⊂ C, k = 0, n, a0 = 0, the following in-equality holds

rγ (B0, 0)

n∏k=1

r (Bk, ak) 6 n−γ2

(4

n

)n−γ.

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On the irregular generalized Cauchy-Riemann equations

Grigori GiorgadzeFaculty of Exact and Natural Sciences, Iv.JavakhishviliTbilisi State University, 0176 Tbilisi, GEORGIAgia.giorgadze@tsu.ge

This is a joint work with V.Jikia and G.Makatsaria (Sin-gular Generalized Analytic Functions J.Math. Sci. 237,30-109, 2019). We consider the generalized Cauchy-Riemann equation (Carleman-Bers-Vekua equation incomplex form)

∂w

∂z= aw + bw,

where a ∈ Llocp (C) and b ∈ Lp,2(C), p > 2. Intro-duce subclasses of the class Llocp (C), p > 2, elements

of which have∂

∂zprimitives and satisfying certain ad-

ditional asymptotic conditions. In the talk we definethese classes and prove their properties. In particular,denote by J0(C) the set of functions a ∈ Llocp (C), p > 2

for which there exists∂

∂z-primitive Q(z) satisfying the

following condition

ReQ(z) = O(1), z ∈ C

and denote by J1(C) the set of the functions a ∈ Llocp (C),

p > 2, for which there exists∂

∂zprimitive Q(z), satisfy-

ing the following condition

zn expQ(z)

= O(1), z ∈ C,

for every natural number n.

Theorem. The function a(z) of the class Llocp (C), p >

2, belongs to the class J1(C) if and only if its∂

∂z-

primitive exists and satisfies the condition

limz→∞

zk expQ(z)

= 0,

for every natural number k.

Theorem. The class J0(C) is a linear space over thefield of real numbers. Moreover, for arbitrary real p > 2the following inclusion

Lp,2(C) ⊂ J0(C)

holds.

Theorem. The following equality

J0(C) ∩ J1() = ∅

holds.

This work was supported, in part, by the Shota RustaveliNational Science Foundation under Grant No 17-96.

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Nonlinear Beltrami equation

Anatoly GolbergHolon Institute of Technology, 52 Golomb St. P.O.Box305, 5810201 Holon, ISRAELgolberga@hit.ac.il

We consider a nonlinear counterpart of the classical Bel-trami equation and study the main features of its solu-tions. It involves directional dilatations connected witha priory fixed point and a class of mappings called ringQ-homeomorphisms with respect to p-module. We alsoestablish some regularity properties of solutions to suchequation and illustrate them by several examples.

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Harmonic measure distribution functions on sphericaland toroidal surfaces

Christopher GreenMacquarie University, Sydney, NSW 2109, AUS-TRALIAchristopher.c.green@mq.edu.au

It will be shown how to generalise recent formulae forharmonic measure distribution functions, or h-functions,for multiply connected slit domains in the complex planeto two distinct compact surfaces: the sphere (genus-0)and the ring torus (genus-1).Given a domain Ω on a compact surface S, and a fixedbasepoint z0 ∈ Ω, the h-function is a piecewise smoothcontinuous function h : [0,∞) → [0, 1] which encodescertain properties of the triple (S,Ω, z0). For r > 0, thevalue of h(r) is the harmonic measure of the portion ofthe boundary ∂Ω that lies within a distance r of z0, andwhere r is measured along the surface S.Motivated by deriving analytical formulae, attention isrestricted to domains Ω exterior to a finite number ofhorizontal slits of equal latitude, and the associated h-functions determined using a combination of techniquesfrom conformal mapping and special function theory.The formulae derived hold for any finite number of slits.

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Abstracts

Monogenic functions in 2-D commutative Complex al-gebras to plane orthotropy

Serhii GryshchukKyiv-4, 3, Tereschenkivska str., 01024 Kyiv, UKRAINE

serhii.gryshchuk@gmail.com

The present research is devoted to the construction ofclasses of “analytic” functions Φ with values in two-dimensional commutative algebras over the field of com-plex numbers containing bases (e1, e2) with some alge-braic properties (in what follows, we construct men-tioned bases and the corresponding algebra in the ex-plicit form) sufficient for the real components of thesefunctions to satisfy the following equation:

(1)

(∂4

∂y4+ 2p

∂4

∂x2∂y2+

∂4

∂x4

)u(x, y) = 0,

where p > −1 (a case of elliptic type), u is a real-valuedsolution of (??), an argument (x, y) ∈ D, while the lat-ter is belonging to the Cartesian plane xOy. Eq. (1)forp 6= 1 is an equation of the stress function of somecass of orthotropic plane deformations, an appropriatealgebra is semi-simple. The case p = 1 is correspond-ing to isotropic plane deformations, Eq. (1) terns ontothe biharmonic Eq., an appropriate algebra is the bihar-monic (cf., e.g., papers of S. G. Gryshchuk, S. A. Plaksaand I. P. Mel’nichenko). A characterisation of solutionsu is done by means of components of Φ(xe1 + ye2),(x, y) ∈ D, in case when a domain under considerationDis bounded and simply connected. A solution of an ap-propriate equilibrium system in displacements is foundas a linear combinations of all real-valued componentsof monogenic functions Φ(xe1 + ye2).

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Linearization of holomorphic semicocycles

Fiana JacobzonDepartment of Mathematics, ORT Braude College P.O.Box 78, 2161002 Karmiel, ISRAELfiana@braude.ac.il

Semicocycles over semigroups of holomorphic self-mappings appear naturally in the study of non-autonomous dynamical systems in Banach spaces. Inthe theory of semicocycles over semigroups, a basic re-lation is the cohomological equivalence. Namely, twocohomological semicocycles have similar properties in-cluding continuity and differentiability, asymptotic be-havior, and so on. So that, one central goal in studyof semicocycles is to classify them up to relation of co-homology. In particular, the linearization problem isthat of determining whether a given semicocycle is coho-mologous to a constant one (that is, independent of thespace-variables). Focusing on this problem, we providesome criteria for a semicocycle to be linearizable. Theseconditions are essential even for semicocycles over lin-ear semigroups. This talk is based on a joint workwith M. Elin and G. Katriel

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Lower bounds for the volume of the image of a ball

Bogdan KlishchukInstitute of mathematics of the National Academy of

Sciences of Ukraine, Department of complex analysisand potential theory, 3 Tereschenkivska str., 01004 Kiev-4, UKRAINEkban1988@gmail.com

Let Γ be a family of curves γ in Rn, n > 2. A Borelmeasurable function ρ : Rn → [0,∞] is called admissiblefor Γ, (abbr. ρ ∈ adm Γ), if∫

γ

ρ(x) ds > 1

for any curve γ ∈ Γ. Let p ∈ (1,∞). The quantity

Mp(Γ) = infρ∈adm Γ

∫Rn

ρp(x) dm(x) .

is called p–modulus of the family Γ. For arbitrary setsE, F and G of Rn we denote by ∆(E,F,G) a set of allcontinuous curves γ : [a, b]→ Rn, that connect E and Fin G, i. e., such that γ(a) ∈ E, γ(b) ∈ F and γ(t) ∈ Gfor a < t < b. Let D be a domain in Rn, n > 2, x0 ∈ Dand d0 = dist(x0, ∂D). Set

A(x0, r1, r2) = x ∈ Rn : r1 < |x− x0| < r2 ,

Si = S(x0, ri) = x ∈ Rn : |x− x0| = ri , i = 1, 2 .

Let a function Q : D → [0,∞] be Lebesgue measurable.We say that a homeomorphism f : D → Rn is ring Q-homeomorphism with respect to p-modulus at x0 ∈ D,if the relation

Mp(∆(fS1, fS2, fD)) 6∫A

Q(x) ηp(|x− x0|) dm(x)

holds for any ring A = A(x0, r1, r2) , 0 < r1 < r2 < d0,d0 = dist(x0, ∂D), and for any measurable functionη : (r1, r2)→ [0,∞] such that

r2∫r1

η(r) dr = 1 .

Theorem. Let D be a bounded domain in Rn, n > 2,and f : D → Rn be a ring Q-homeomorphism with re-spect to p-modulus at the point x0 ∈ D for p > n. As-sume that the function Q satisfies the condition

qx0(t) 6 q0 t−α, q0 ∈ (0,∞) , α ∈ (0,∞) ,

for x0 ∈ D and almost all t ∈ (0, ε0), ε0 ∈ (0, d0),d0 = dist(x0, ∂D). Then for all r ∈ (0, ε0) the estimate

m(fB(x0, r)) > Ωn

(p− n

α+ p− n

)n(p−1)p−n

qnn−p0 r

n(α+p−n)p−n

holds true, where B(x0, r) = x ∈ Rn : |x − x0| 6 r,qx0(t) = 1

ωn−1 rn−1

∫S(x0,r)

Q(x) dA is the integral mean

value over the sphere S(x0, t) = x ∈ Rn : |x−x0| = t ,Ωn is the volume of the unit ball in Rn , ωn−1 is thesurface area of the unit sphere Sn−1 in Rn , dA is theelement of the surface area.

———

Approximation properties and related results for univa-lent mappings in higher dimensions

Gabriela KohrBabes-Bolyai University, Faculty of Mathematics and

32

Abstracts

Computer Science, 1 M. Kogalniceanu Str., 400084 Cluj-Napoca, ROMENIAmkohr@math.ubbcluj.ro

Let n ≥ 2 and let A ∈ L(Cn) be such that <〈A(z), z〉 >0, ‖z‖ = 1. Also, let S0

A(Bn) be the family of normalizedunivalent mappings on Bn which have A-parametric rep-resentation. In this talk we present a variational methodfor A-normalized univalent subordination chains on theEuclidean unit ball Bn in Cn, which allows us to obtainapproximation properties for the family S0

A(Bn) by auto-morphisms of Cn. In particular, we present approxima-tion results of starlike, convex, and spirallike mappingsby automorphisms of Cn whose restrictions to Bn havethe same geometric property. Extremal properties forthe family S0

A(Bn) will be also discussed. Finally, cer-tain questions and open problems will be mentioned.

———

An inequality for Holder continuous functions, in thewake of the work of Carlo Miranda

Massimo Lanza de CristoforisDipartimento di Matematica ‘Tullio Levi-Civita’, Uni-versita degli Studi di Padova, Via Trieste 63, 35121Padova, ITALYmldc@math.unipd.it

We plan to show an inequality for Holder continuousfunctions, which is useful to study the boundary behav-ior of layer potentials and which enables to simplify aproof of a result of Carlo Miranda.

———

Shape analysis of the longitudinal flow along a periodicarray of cylinders

Paolo LuzziniFreie Universitat Berlin, GERMANYluzzinipaolo@gmail.com

In the present talk we study the behavior of the lon-gitudinal flow along a periodic array of cylinders uponperturbations of the shape of the cross section of thecylinders and the periodicity structure, when a Newto-nian fluid is flowing at low Reynolds numbers around thecylinders. The periodicity cell is a rectangle of sides oflength l and 1/l, where l is a positive parameter, and theshape of the cross section of the cylinders is determinedby the image of a base domain through a diffeomorphismφ. We also assume that the pressure gradient is parallelto the cylinders. Under such assumptions, for each pair(l, φ), one defines the average of the longitudinal compo-nent of the flow velocity Σ[l, φ]. As the main result, weshow that the quantity Σ[l, φ] depends analytically onthe pair (l, φ), which we consider as a point in a suitableBanach space.

The results presented have been obtained in collaborationwith Paolo Musolino and Roman Pukhtaievych.

———

On polynomial extension property

Krzysztof Maciaszekul. prof. Stanis lawa Lojasiewicza 6, 30-348 Krakow,POLANDkrzysztof.maciaszek@im.uj.edu.pl

The celebrated theorem of H. Cartan states that for ananalytic variety V in a pseudoconvex domain Ω, anyholomorphic function on V extends to a holomorphicfunction on Ω. In this talk we are interested in thesesubsets V that additionally admit the (polynomial) ex-tension property, meaning that for every bounded holo-morphic function (resp. polynomial) f on V there existsa bounded holomorphic function F on Ω such that

F |V = f and supΩ|F | = sup

V|f |.

The characterization of subsets in the bidisc, that havethe polynomial extension property, were achieved byAgler and McCarthy (2003). More recently Kosinski andMcCarthy (2017) were studied one- and two-dimensionalsubsets of the tridisc. Based on the techniques fromthe latter, we have obtained the result, that an one-dimensional algebraic subset of arbitrarily dimensionalpolydisc Dn, which has the polynomial extension prop-erty, is a holomorphic retract.

———

Converging expansions for Lipschitz self-similar perfo-rations of a plane sector

Paolo MusolinoDepartment of Mathematics, via Trieste 63, 35121Padova, ITALYmusolino@math.unipd.it

In contrast with the well-known methods of matchingasymptotics and multiscale (or compound) asymptotics,the “Functional Analytic Approach” proposed by Lanzade Cristoforis allows to prove convergence of expansionsaround interior small holes of size ε for solutions of el-liptic boundary value problems. Using the method oflayer potentials, the asymptotic behavior of the solutionas ε tends to zero is described not only by asymptoticseries in powers of ε, but by convergent power series. Inthis talk we present some recent results, where we usethis method to investigate the Dirichlet problem for theLaplace operator where holes are collapsing at a polyg-onal corner of opening ω. The strategy relies on a com-bination of odd reflections and conformal mappings sothat the original problem is transformed into a similarproblem where the perforations are near the center of adisc, on potential theory on Lipschitz domains, and on adetailed analysis of the solution of the limiting problemin proximity of the corner. We show that in additionto the scale ε there appears the scale η = επ/ω whenπ/ω is irrational, the solution of the Dirichlet problemis given by convergent series in powers of these two smallparameters. The final outcome can be compared withmulti-scale expansions for which convergence does nothold in general.Based on joint work with M. Costabel, M. Dalla Riva,and M. Dauge.

———

Numerical computation of conformal capacity of gen-eralized condensers

Mohamed M S NasserDepartment of Mathematics, Statistics & Physics, QatarUniversity, Doha, QATARmms.nasser@qu.edu.qa

33

Abstracts

Consider generalized condensers of the form C =(C, E, δ) where E = Ekmk=1 is a collection of nonemptyclosed pairwise disjoint sets in C and δ = δkmk=1 is acollection of real numbers containing at least two dif-ferent numbers. We assume that G = C\ ∪mk=1 Ek is amultiply connected domain of connectivity m such that∂G = ∪mk=1∂Ek has no isolated boundary points. Theset G is called the field of the condenser C, the sets Ekare the plates of the condenser, and the δk are the levelsof the potential of the plates Ek, k = 1, 2, . . . ,m. Theconformal capacity of C, cap(C), is then given by theDirichlet integral

cap(C) =

∫∫G

|∇u|2dxdy

where u is the potential function of the condenser C,i.e., u is continuous in G, harmonic in G, and equal toδk on ∂Ek for k = 1, 2, . . . ,m.We present a boundary integral method for numericalcomputation of the capacity cap(C). The method isbased on the boundary integral equation with the gen-eralized Neumann kernel. The presented method appliesto a wide variety of generalized condenser geometry in-cluding the cases when the plates of the generalized con-denser are smooth Jordan curves, piecewise smooth Jor-dan curves, segment slits and circular slits.

The talk is based on collaborations with ProfessorMatti Vuorinen (University of Turku, Finland; vuori-nen@utu.fi).

———

Support of Borel measures in the plane satisfying a cer-tain positivity condition

Mitja NedicMatematiska Institutionen, Stockholms Universitet,Roslagsvagen 101, Kraftriket hus 6, 106 91 Stockholm,SWEDENmitja@math.su.se

Certain Borel measures in the plane appear as repre-senting parameters for classes of holomorphic functionswith non-negative real or imaginary part. Such measuresmust satisfy a positivity condition which constitutes asevere restriction within the set of Borel measures. Inthis talk, we will present how the positivity conditionthat characterizes representing measures of holomorphicfunctions of two variables with non-negative imaginarypart restricts the support of these measures. For ex-ample, we will present some subsets of the plane thatcannot contain the support of such measures. This talkis based on joint work with Annemarie Luger.

———

Nonlocal boundary value problems for the Laplace op-erator in a unit ball which are multidimensional gener-alizations of the Samarskii-Ionkin problem

Makhmud SadybekovInstitute of Mathematics and Mathematical Modeling,125 Pushkin Str., 050010 Almaty, KAZAKHSTANsadybekov@math.kz

In this talk we consider one stationary diffusion prob-lem describing the Poisson equation. The problem isconsidered in a model domain, chosen as a half disk.

Dirichlet classical boundary conditions are set on thearc of the circle. New nonlocal boundary conditions areset on the bottom base. The first condition means anequality of flows through opposite radii, and the sec-ond condition is the proportionality of distribution den-sities on these radii with a variable coefficient of pro-portionality. The uniqueness and existence of a classicalsolution to the problem are proved. An inverse prob-lem on the solution definition to the Poisson equationand its right-hand part depending only on an angularvariable are considered. As an additional condition weuse the boundary overdetermination. Inverse problemsto the Dirichlet, Neumann problems and to problemswith nonlocal conditions of the equality of flows throughthe opposite radii are considered. The well-posednessof the formulated inverse problems is proved. Someof our results can be found in [1]. The talk is basedon joint work with Aishabibi Dukenbayeva (al-FarabiKazakh National University and Institute of Mathemat-ics and Mathematical Modeling, Almaty, Kazakhstan).[1] M.A. Sadybekov and A.A. Dukenbayeva, Direct andinverse problems for the Poisson equation with equalityof flows on a part of the boundary, Complex Variablesand Elliptic Equations, 64(5) (2019), 777-791, DOI:10.1080/17476933.2018.1517340

———

Hypercomplex method for solving linear PDEs

Vitalii Shpakivskyi3, Tereshchenkivska str., 01024 Kyiv-4, UKRAINEshpakivskyi86@gmail.com

Algebraic-analytic approach to constructing solutionsfor given linear partial differential equations were inves-tigated in many papers. It involves solving two prob-lems. Problem (P 1) is to describe all the sets of vectorse1, e2, . . . , ed of commutative associative algebra, whichsatisfy the polynomial characteristic equation (or spec-ify the procedure by which they can be found). Andthe Problem (P 2) is to describe all the components ofmonogenic (i. e., continuous and differentiable in senseGateaux) functions. Earlier we got a constructive de-scription of all analytic functions with values is ob-tained in an arbitrary finite-dimensional commutativeassociative algebra over the field C. Further we statesthat it is enough to limit the study of monogenic func-tions in algebras with the basis of 1, η1, η2, . . . , ηn−1,where η1, η2, . . . , ηn−1 are nilpotents. In addition, it isshowed that in each algebra with a basis of the form1, η1, η2, . . . , ηn−1 the polynomial characteristic equa-tion has solutions. That is, the problems (P 1) and (P 2)are completely solved on the classes of commutative as-sociative algebras with the basis 1, η1, η2, . . . , ηn−1. Itis worth noting that in a finite-dimensional algebra a de-composition of monogenic functions has a finite numberof components, and therefore, it generates a finite num-ber of solutions of a given partial differential equations.In this report we propose a procedure for constructingan infinite number of families of solutions of given lin-ear differential equations with partial derivatives withconstant coefficients. We use monogenic functions thatare defined on some sequences of commutative associa-tive algebras over the field of complex numbers. Toachieve this goal, we first study the solutions of the so-called characteristic equation on a given sequence of al-gebras. Further, we investigate monogenic functions on

34

Abstracts

the sequence of algebras and study their relation withsolutions of partial deferential equations. The proposedmethod is used to construct solutions of some equationsof mathematical physics. In particular, for the three-dimensional Laplace equation and the wave equation, forthe equation of transverse oscillations of the elastic rodand the conjugate equation, a generalized biharmonicequation and the two-dimensional Helmholtz equation.We note that this method yields all analytic solutionsof the two-dimensional Laplace equation and the two-dimensional biharmonic equation (Goursat formula).

———

Weierstrass Theorem for Bicomplex Holomorphic Func-tions

Luis Manuel Tovar SanchezEscuela Superior de Fısica y Matematicas, Edificio 9,Unidad Profesional A.L.M. Zacatenco del IPN Ciudadde Mexico, Caldas 562 Depto 101 Col. Valle del Te-peyac, Mexico, MEXICOtovar@esfm.ipn.mx

How are the zeros of a bicomplex holomorphic function?So, how is the corresponding Weierstrass theorem to pre-scribe zeros and singularities of a bicomplex function?In this talk we present differences between the classi-cal Weierstrass theorem for analytic functions and thecorresponding statement for bicomplex functions.

———

S.5 Constructive Methods in the Theory ofComposite and Porous Media

OrganisersVladimir Mityushev

Scope of the session: The topic concerns application ofmathematical methods to study composite and porousmedia. In particular, it includes

• Effective properties of composites

• Homogenization theory

• Electromagnetic and optical properties of metama-terials

• Transport in random media

• Nanocomposites

• Cloaking

The theoretical methods applied in this field are usu-ally based on complex analysis, boundary value prob-lems, partial differential equations, functional equationsetc. This session will be also interesting for engineersand designers using mathematical models and computersimulations. It is plan to organize a workshop withinthe session concerning mathematical models and appliedproblems presented by engineers.

—Abstracts—

Fast method for 2D Dirichlet problem for circular mul-tiply connected domains

Olaf Barul. Podchorazych 2, 30-084 Krakow, Malopolska,POLANDbar@up.krakow.pl

In this paper we present the implementation of algo-rithm to determine the conductivity of 2D plane withnon-overlapping disk inclusions. The fast Poincare se-ries method and modified Delaunay triangulation to de-termine closest neighbours was used.

———

Local Fields

Wojciech BaranUniwersytet Pedagogiczny w Krakowie, ul. Pod-chorazych 2, 30-084 Krakow, Malopolska, POLANDwojciech.baran@up.krakow.pl

Consider a plane multiply connected domain D boundedby non-overlapping disks. Introduce the complex poten-tial u(z) = Re ϕ(z) in D where the function ϕ(z) isanalytic in D except at infinity where ϕ(z) ∼ z. Theunknown function ϕ(z) is continuously differentiable inthe closures of the considered domain. We solve approx-imately the Schwarz problem when u(z) = Re ϕ(z) isequal to an undetermined constant on every boundarycomponent of D by a method of functional equations

———

The Robin problem in conical domains

Mikhail BorsukUniversity of Warmia and Mazury in Olsztyn, 10-170Olsztyn, POLANDborsuk@uwm.edu.pl

We study the existence and behavior near a boundaryangular or conical point of weak solutions to the Robinproblem for an elliptic quasi-linear second-order equa-tion with the p− and variable p(x)-Laplacian. Co-authorof talk: Mariusz Bodzioch, University of Warmia andMazury in Olsztyn, Poland

———

Conductivity of two-dimensional composites with ran-domly distributed elliptical inclusions: Random ellipticalinclusions

Marta BrylaDepartment of Physics, Pedagogical University of Cra-cow, ul. Podchorazych 2, 30-084 Krakow, Malopolskie,POLANDmartaoleksy@gmail.com

Analytical approximate formulae for the effective con-ductivity tensor of the two dimensional composites withelliptical inclusions are derived in the form of polyno-mial approximations in concentration. New formulaeexplain the seeming contradiction between various for-mulae derived in the framework of self consistent meth-ods. Random composites with high conducting inclu-sions of two different shapes (elliptical and circle) of the

35

Abstracts

same area are compared. It is established that greaterrelative concentration of ellipses increases the effectiveconductivity.

———

Estimations of the effective conductivity of compositeswith non-circular inclusions

Roman CzaplaCracow, POLANDroman.czapla@up.krakow.pl

The closed form formula for the efective conductivityof 2D composite material with nonoverlapping ircularinclusions is known. We present a method and resultson estimations of the efective conductivity of ompositeswith non-circular inclusions by approximating their ge-ometry, i.e. by covering the considered shape with diskswhich form cluster. In addition to a general discussionof the problem of approximation, in this talk, we presentan estimation of the effective conductivity of compositematerials with triangular inclusions.

———

Effective elastic constants for 2D random composites

Piotr DrygasUniversity of Rzeszow, Rejtana 16 c str., 35–959 Rzes-zow, POLANDdrygaspi@ur.edu.pl

Basing on the MMM principle by Hashin and the theoryof homogenization we follow the method of random con-structive homogenization developed in Gluzman S., Mi-tyushev V., Nawalaniec W., (2018). Consider 2D multi-phase random composites with different radii circularinclusions located at the sites of hexagonal (triangular)lattice. The inclusions are embedded into the matrixwith different elastic properties. Plane strain elasticproblem is solved for such a composite. The effectiveshear and bulk moduli are obtained in the form of powerseries in the inclusions concentration f . The coefficientsof this series are written in analytical form, with thecoefficients expressed through the elastic constants ofcomponents. New analytical formulae for the effectiveconstants are deduced up to arbitrary O(fn) for macro-scopically isotropic composites. We derive general ana-lytical formulae for the local fields and for the effectiveconstants in 2D random composites. We consider exam-ples of simulated random media and application of thederived symbolic-numerical algorithms to them.

———

Education of Selected Notions Connected with Math-ematical Analysis with Help of Visualization

Jan GuncagaComenius University in Bratislava, Faculty of Educa-tion, Racianska 59, 81334 Bratislava, SLOVAKIAjguncaga@gmail.com

The aspect of visualization plays an important role inthe analysis teaching. The notion of the function is ba-sic notion at the lower secondary level. We will presentsome examples from the national measurements in Slo-vakia (T9). The notion of the graph of the function andhis graphical and visual representation is important at

upper secondary level and university level. If the stu-dent draws the graph of the function, then the workwith graphical representation of the function can helpto make his knowledge about notions such continuity,derivative more deep. The development of informationand communication technologies (ICT) gives the possi-bility to use different tools (for example educational soft-ware) as a supporting aspect for mathematics educationwith better understanding. We can find many suitableinspirative examples in the historical mathematical text-books and materials. We would like to show some exam-ples with the help of GeoGebra for motivational analysisteaching. This contribution is prepared with the help ofthe project VEGA 1/0079/19.

———

On quantitative assessment of In-situ and ex-situ com-posites structures with micro and nano-particles rein-forcement

Pawe l KurtykaInstitute of Technology, Pedagogical University ofKrakow, 30-084 Krakow, POLANDpkurtyka@gmail.com

The aim of the investigation is development of a quan-titative methodology for In-situ and Ex-situ compositesstructures with micro and nano-particles reinforcementassessment. The primary method is based on the an-alytical representative volume element (ARVE) theory,which was developed to determine the effective prop-erties of fiber composites. The analysis were carriedout for two types of composites manufacturing methods(in-situ, ex-situ) and two types of particles (nano-TiCand micro-SiC). The developed methodology allows tocompare composites with similar composition obtainedthrough different technological processes. It also allowsto assess the impact of process parameters on the com-posite structures. Moreover, the research methodologyis applied to structure analysis of the in-situ compos-ites.

———

R-linear problem and its application to random com-posites

Vladimir MityushevPedagogical University, ul. Podchorazych 2, 30-084Krakow, POLANDmityu@up.krakow.pl

Let Dk be mutually disjoint simply connected domainsbounded by smooth curves ∂Dk and D be the comple-ment of all closures of Dk to the torus T 2. To find afunction ϕ(z) analytic in D+ = D1 ∪ . . . ∪Dn, D− = Dand continuous in the closures of the considered domainson the torus T 2 with the following R-linear conjugationcondition

ϕ+(t) = ϕ−(t)− ρ(t)ϕ−(t), t ∈ ∂D+,

where ρ(t) is a constant on each component of ∂D+.Application of the generalized method of Schwarz yieldsa constructive algorithm to solve the problem (see S.Gluzman, V. Mityushev, W. Nawalaniec, 2017). Re-lations to the Riemann-Hilbert problem are discussed.The obtained results are applied to description of com-posites, in particular, to the constructive RVE theoryof random two-dimensional composites (see V. Mityu-shev, W. Nawalaniec, N. Rylko, 2018). We give precise

36

Abstracts

and computationally instant answers to such questionsas isotropy of structure. Applications to metamaterialsare discussed.

———

Classifying and analysis of random composites usingstructural sums feature vector

Wojciech NawalaniecPOLANDwojciech.nawalaniec@up.krakow.pl

We present the application of structural sums, mathe-matical objects originating from the computational ma-terials science, in construction of a feature space vec-tor of 2D random composites simulated by distributionsof non-overlapping disks on the plane. Construction ofthe feature vector enables the immediate application ofmachine learning tools and data analysis techniques torandom structures.

———

Trygonometric approach to the Schrodinger equationas a general case of known solutions

Kazimierz RajchelInstitute of Computer Science, Pedagogical University ofCracow, ul. Podchorazych 2, 30-084 Krakow, POLAND

kazimierz.rajchel@up.krakow.pl

General solutions of the Schrodinger equation were ob-tained by means of parametrization of the unit circleequation, related to the Riccati equation. Particular so-lutions, such as oscillator, Coulomb or Morse potentials,are given by appropriate choice of parameters. This con-cept is strictly connected with the theory of orthogonalpolynomials and hypergeometric functions. As a resultone can choose more objectively orthogonal basis in nu-merical computations, which leads to improvement ofaccuracy.

———

Analysis of the degree of image complexity used in eyetracking research on STEM education

Roman RosiekPedagogical University of Cracow, 30-084 Krakow,POLANDroman.rosiek@up.krakow.pl

The article contains a description and a proposal of usingmathematical methods to analyse the degree of imagecomplexity for eye tracking research. Complex imagesare often used as stimuli for eye tracking experimentssometimes they do not have any distinctive nor impor-tant elements , e.g. works of art, illustrations, text-books, websites. The article describes an attempt toapply mathematical methods, numerical algorithms, toanalyse and determine the degree of image complexity.An attempt was made to search for correlations betweenthe areas of images with the greatest changes of dynam-ics determined mainly by the analysis of the gradient ofluminance and chrominance of image pixels and areasof interest experimentally determined using eye track-ing methods. Research was carried out in the STEM

education by an interdisciplinary group of cognitive di-dactic at the Faculty of Mathematics, Physics and Tech-nical Sciences of the Pedagogical University of Cracow.The study was carried out using a remote stationary eyetracker.

———

Inverse problem for spherical particles and its applica-tions to metal matrix composites reinforced by nanoTiC particles

Natalia RylkoPedagogical University of Krakow, Podhorazych 2, 30-084 Krakow, Malopolska, POLANDnatalia.rylko@gmail.com

Randomly distributed non-overlapping perfectly con-ducting n balls of radius R are embedded in a conductingmatrix occupying a large ball of the normalized unit ra-dius. The potential and the normal flux are given on theboundary of the unit ball. The locations of inclusions akare not known. The perturbation term of potential in-duced by inclusions is constructed up to O(R4) by amethod of functional equations

(1) U(x) =

n∑m=1

[|x|3 x1 − |x|2am1

|x− |x|2am|3− x1 − am1

|x− am|3

].

The perturbation term (1) includes the unknown centersof inclusions in symbolic form. The inverse problem isreduced to determination of the centers ak by fitting ofthe given perturbation term on the unit sphere.

———

Three mental worlds of mathematics in pre-servicemathematics teachers - an eye-tracking research

Miroslawa SajkaPedagogical University of Cracow, ul. Podcharazych 2,30-084 Krakow, POLANDmiroslawa.sajka@up.krakow.pl

Development of mathematical thinking can be consid-ered in terms of “three mental worlds of mathematics”,according to the theoretical framework formulated byD. Tall. These are: (1) a world of (conceptual) embod-iment, (2) a world of (operational) symbolism and (3)a world of (axiomatic) formalism. These worlds shouldbe developed by every secondary school and universitystudent and a mathematics teacher should provoke theirdevelopment during mathematical classes. The empiri-cal research was designed in order to reveal some foun-dations of the process of problem solving. It will diag-nose which of the three kinds of mathematical mentalworlds was activated by the pre-service teachers in or-der to solve tasks. The presentation will be focused onunderstanding chosen notions (e.g. the notion of func-tion and its representations) by pre-service mathemat-ics teachers from this perspective. The presentation willdiscuss results of the research conducted with the use ofeye-tracking technology at the Faculty of Mathematicsand Physics and Technical Sciences of the PedagogicalUniversity of Cracow.

———

Boundary integral equation method in the theory ofbinary mixtures of porous viscoelastic materials

Maia SvanadzeTbilisi State University, I. Chavchavadze Ave., 3, 0179

37

Abstracts

Tbilisi, GEORGIAmaia.svanadze@gmail.com

In this talk the linear theory for binary mixtures ofporous viscoelastic materials is considered. The indi-vidual components of the mixture are a Kelvin-Voigtporous material and an isotropic elastic solid. The basicboundary value problems (BVPs) of steady vibrationsand quasi-static are investigated. Namely, the funda-mental solutions of the system of equations of steady vi-brations and quasi-static are constructed explicitly andtheir basic properties are established. The uniquenesstheorems for classical solutions of the internal and ex-ternal basic BVPs of steady vibrations and quasi-staticare proved. The surface and volume potentials are con-structed and their basic properties are given. The de-terminants of symbolic matrices are calculated explic-itly. The BVPs are reduced to the always solvable sin-gular integral equations for which Fredholm’s theoremsare valid. Finally, the existence theorems for classicalsolutions of the internal and external BVPs of steadyvibrations and quasi-static are proved by the boundaryintegral equation method and the theory of singular in-tegral equations.

———

S.6 Fixed Point Theory, Ulam Stability, andRelated Applications

OrganisersErdal Karapinar, Janusz Brzdek

Scope of the session: It is an indisputable fact that both“Ulam Stability” and “Metric Fixed Point Theory” areamong the most dynamic research fields of NonlinearAnalysis. Moreover, it has been clearly demonstratedthat they are closely related to each other.Ulam stability deals mainly with the following naturalissue: when is it true that an approximate solution toan equation must be close to an exact solution of theequation. It is a quite new, but rapidly growing area ofresearch with various possible applications. Motivatedby the well-known problem of S. Ulam, concerning theapproximate homomorphisms of metric groups, it hasbecome nowadays an area of investigations of approxi-mate solutions to a wide range of equations (e.g., differ-ence, differential, integral, functional) and related fixedpoint results.Due to its possible applications, Fixed Point Theory inthe metric spaces plays a key role in Nonlinear Anal-ysis. In the last fifty years, discussions on the exis-tence and uniqueness of fixed points of single and mul-tivalued operators in different kind of spaces (such asquasimetric spaces, pseudo-quasi-metric spaces, partialmetric spaces, b-metric spaces and fuzzy metric spaces,among others) has attracted the attention of numerousresearchers. The enormous potential of its applicationsto almost all quantitative sciences (such as Mathematics,Engineering, Chemistry, Biology, Economics, ComputerScience, and others) justify the present great interest inthis area. The purpose of this workshop is to bring to-gether Mathematicians, but also other researchers thatmight be interested in this topic, to present, share anddiscuss their main advances (ideas, techniques, possibleresults, proofs, etc.) in this area.

Topics in this session include, but are not limited to: sta-bility of difference, differential, functional, and integralequations, stability of inequalities and other mathemat-ical objects, hyperstability and superstability, various(direct, fixed point, invariant mean, etc.) methods forproving Ulam’s type stability results, generalized (in thesense of Aoki and Rassias, Bourgin and Gavruta) stabil-ity, stability on restricted domains and in various (met-ric, Banach, non-Archimedean, fuzzy, quasi-Banach, n-Banach, etc.) spaces, Ulam stability of operators, re-lations between Ulam’s type stability and fixed pointresults, fixed point theory in various abstract spaces,existence and uniqueness of coupled/tripled/quadrupledfixed point, coincidence point theory, existence anduniqueness of common fixed points, well-posedness offixed point results, advances on multivalued fixed pointtheorems, fixed point methods for the equilibrium prob-lems and applications, iterative methods for the fixedpoints of the nonexpansive-type mappings, Picard oper-ators on various abstract spaces, applications to variousother areas.

—Abstracts—

A New Approach to Interval Matrices and Applications

Obaid AlqahtaniDepartment of Mathematics, King Saud University,Riyadh, 11451, Saudi Arabiaobalgahtani@ksu.edu.sa

A convex combination of Intervals can be written as:

[a, b] = xα = (1− α)a+ αb : α ∈ [0, 1].

Consequently, we may adopt interval operations by ap-plying the scalar operation point-wise to the correspond-ing interval points. With the usual restriction 0 /∈ J if· = ÷. These operations are associative:

I + (J +K) = (I + J) +K,

I ∗ (J ∗K) = (I ∗ J) ∗K.These two properties, which are missing in the usual in-terval operations, will enable the extension of the usuallinear system concepts to the interval setting in a seam-less manner. The arithmetic introduced here avoids suchvague terms as ”interval extension”, ”inclusion func-tion”, determinants which we encounter in the engineer-ing literature that deal with interval linear systems. Onthe other hand, these definitions were motivated by ourattempt to arrive at a definition of interval random vari-ables and investigate the corresponding statistical prop-erties. We feel that they are the natural ones to handleinterval systems. We will enable the extension of manyresults from usual state space models to interval statespace models. This feeling is reassured by the numericalresults we obtained in a simulation examples.

———

Existence of Solutions to Boundary Value Problems forIntuitionistic Fuzzy Partial Hyperbolic Functional Dif-ferential Equations

Bouchra Ben AmmaSultan Moulay Sliman University, PO Box 523, 23000Beni Mellal, MOROCCObouchrabenamma@gmail.com

38

Abstracts

Intuitionistic partial functional differential equations arevery rare, we combine two aspects, intuitionistic fuzzymathematics and partial functional differential equa-tions to get intuitionistic fuzzy partial hyperbolic func-tional differential equations, in this paper presents somenew results on the existence and uniqueness of intu-itionistic fuzzy solutions for some classes of intuitionis-tic fuzzy partial hyperbolic functional differential equa-tions with integral boundary conditions using the Ba-nach fixed point theorem. An illustrated example forour results is given with some numerical simulation forα-cuts of the intuitionistic fuzzy solutions.

———

A fixed point theorem in n-Banach spaces and Ulamstability

Krzysztof CieplinskiAGH University of Science and Technology, Mickiewicza30, 30-059 Krakow, POLANDcieplin@agh.edu.pl

In the talk, we give a fixed point theorem for opera-tors acting on some classes of functions with values inn-Banach spaces. We also present its applications tothe Ulam stability of eigenvectors and some functionaland difference equations. The presented results were ob-tained jointly with Janusz Brzdek.

———

Impact of Perov type results on Ulam-Hyers stability

Marija CvetkovicDepartment of Mathematics, Faculty of Sciences andMathematics, University of Nis, Visegradska 33, 18000Nis, SERBIAmarijac@pmf.ni.ac.rs

Introducing a new kind of a contractive mapping whichincludes operator acting as a contractive constant givesus a possibility for a different approach on Ulam-Hyersstability of differential and operator equations. Theapplication of several different fixed point theorems ofPerov type will be presented along with some examplesand comparison with previously obtained Ulam-Hyersstability results of these classes of equations.

———

Mittag-Leffler stability analysis for time-fractional par-tial differential equations

Amar DebboucheDepartment of Mathematics, Guelma University, 24000Guelma, ALGERIE & CIDMA-Department of Mathe-matics, University of Aveiro, 3810-193 Aveiro, PORTU-GALamar debbouche@yahoo.fr

In this work, we shall investigate a class of time-fractional Keller–Segel models with boundary Dirichletconditions. We use Faedo–Galerkin method with somecompactness arguments to show the existence results ofweak solutions. Further, we establish the Mittag–Lefflerstability of these weak solutions for the indicated mod-els.

———

Application of F -contraction mappings to integralequations on time scales

Inci ErhanAtilim University, Department of Mathematics, AtilimUniversity, Kizilcasar Mahallesi, 06830 Ankara,TURKEYinci.erhan@atilim.edu.tr

In this talk we present some existence and uniquenessof fixed points of certain (φ, F -type contractions in theframe of metric like spaces. As an application of the the-oretical results we consider the existence and uniquenessof solutions of nonlinear Fredholm integral equations ofthe second kind on time scales. We also discuss particu-lar examples which demonstrate our theoretical findingsand propose directions for further studies.

———

Fixed points of discontinuous mappings

Andreea FulgaIuliu Maniu 50, 500091 Brasov, ROMANIAafulga@unitbv.ro

The goal of this study is to present and investigatesome contractive type inequalities for discontinuous self-mappings . The main results cover and extend a few ex-isting results in the corresponding literature. Further-more, we give some illustrative examples to verify theeffectiveness and strength of our derived results.

———

Local and global solution for effectively damped waveequations with non linear memory

Tayeb Hadj KaddourFaculty for Mathematics and Informatics, Oued R’hiou,48001 Boudalia Hassani, Relizane, ALGERIAhktn2000@yahoo.fr

We prove the existence of local solutions for any dataand global solution for small data for the Cauchy prob-lem of effectively damped wave equation with non linearmemory

utt −∆u+ b(t)ut =

∫ t

0

(t− s)−γ|u(s, ·)|pds

u(0, x) and ut(0, x) are given in suitably spaces.

———

Recent topics on metric fixed points

Erdal KarapinarChina Medical University, 40402, Taichung, TAIWANerdalkarapinar@gmail.com

The aim of this talk is present and discuss recent resultsand topics in the frame-work of metric fixed point the-ory. In addition, we talk possible applications of metricfixed point theory on distinct research areas.

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Attractive Points by a Modern Iterative Process

Safeer Hussain KhanDepartment of Mathematics and Physics, Qatar Univer-sity, Doha 2713, QATARsafeer@qu.edu.qa

39

Abstracts

Takahashi and Takeuchi introduced the idea of attrac-tive points. Since then some mathematicians have ap-proximated (common) attractive points by classical iter-ative processes. Classical iterative processes include Pi-card, Mann, Ishikawa iterative processes and their mul-tistep variants. A lot is still to be done in this direc-tion. The author introduced an iterative process calledPicard-Mann iterative process. This iterative processgot a good attention of researchers and many general-izations came into show. The author in another paperintroduced the ideas of common attractive points andfurther generalized mappings. He established existenceof common attractive points and approximated themthrough a generalized version of the above-mentionedPicard-Mann iterative process. Here we continue thisstudy by using a more modern iterative method to ap-proximate attractive points of such mappings. A modernthree-step iterative process may be defined as follows.Let T : C → C be a nonlinear mapping defined on aconvex closed subset C of a Hilbert space H. Define thesequence xn iteratively as follows.

xn+1 = Tyn,yn = T ((1− δn)zn + δnTzn),

zn = T ((1− µn)xn + µnTxn), n ∈ N.

This iterative process converges faster than classical it-erative processes. We prove some existence as well asconvergence results for attractive points of T using theabove iterative process. We also give a comparison ofattractive and fixed points. This will improve and gen-eralize several results including those of Khan.

———

Some related fixed point theorems for multivalued map-pings on two metric spaces

Ozge Bicer OdemisIstanbul Medipol University, The Campus of Kavacik,Department of Electronic Communication TechnologyVocational School, Istanbul, Beykoz, TURKEYozgeb89@hotmail.com

In this paper, we present some fixed point results formultivalued mappings on two complete metric spaces.First, we give a classical result which is an extension ofthe main result of Brain Fisher’s related theorem givenin 1981. After, considering the recent technique of War-dowski, we provide two related fixed point results forboth compact and closed and bounded set-valued map-pings via F-contraction type condition.

———

On best proximity point theory: From cyclic mappingsto Tricyclic ones.

Taoufik SabarHassane II University of Casablanca, faculty of scienceBen msick, Casablanca, MOROCCOsabarsaw@gmail.com

The recently introduced Tricyclic mappings are self map-pings defined on the union of three subsets A, B and Cof a metric space and maps A into B, B into C and Cinto A. Taking inspiration from the recent work by thecurrent authors. We shall discuss existence of best prox-imity points of both tricyclic contractions and tricyclicrelatively nonexpansive mappings in different subclasses

of metric spaces. First, we introduce the concept of (S)convex metric spaces, those are convex metric spaceswhose convex structure verifies an additional condition,thereby, we acquire a best proximity point theorem fortricyclic contraction mappings. Afterwards , we studythe structure of minimal sets of tricyclic relatively non-expansive mappings in the setting of Kohlenbach hyper-bolic spaces, this way we obtain an existence theoremfor such mappings.

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Ulam Stabilities for a Class of Higher Order Integro-Differential Equations

Alberto SimoesCenter of Mathematics and Applications of University ofBeira Interior (CMA-UBI), Department of Mathematics,University of Beira Interior, 6200 Covilha, AND Centerfor Research and Development in Mathematics and Ap-plications (CIDMA), Department of Mathematics, Uni-versity of Aveiro, 3810-193 Aveiro, PORTUGALasimoes@ubi.pt

We will identify sufficient conditions that allow us toguarantee different kinds of Ulam stabilities for a classof higher order integro-differential equations (within ap-propriate metric spaces). We will consider a so-calledσ-semi-Hyers-Ulam stability, which is a kind of stabil-ity somehow between the Hyers-Ulam and the Hyers-Ulam-Rassias stabilities. Conditions are obtained inview to ensure Hyers-Ulam, σ-semi-Hyers-Ulam andHyers-Ulam-Rassias stabilities for such class of integro-differential equations (which considers both cases of fi-nite and infinite intervals of integration). Fixed point ar-guments and generalizations of the Bielecki metric havea central role in the proposed method. (Joint work withL. P. Castro.)

———

Some existence and convergence results for monotonemappings

Izhar UddinJamia Millia Islamia, Department of Mathematics,110025 New Delhi, Delhi, INDIAizharuddin1@jmi.ac.in

Fixed Point Theory is one of the most exciting branchesof mathematics as it works as a bridge between pure andapplied mathematics. Banach contraction principle isone the basic and most widely used result of mathemat-ics. In 2004, Ran and Reurings generalized the Banachcontraction principle to ordered metric spaces. Later in2005, Nieto and Rodriguez used the same approach tofurther extend some more results of fixed point in par-tially ordered metric spaces and utilized them to studythe existence of solution of differential equations. Duringthis talk, we will discuss some recent existence and con-vergence results with respect to monotone nonexpansivemappings.Keywords: Fixed Point, nonexpansive mappings, Ba-nach contraction.AMS-Classifications: 47H10, 54H25.

———

40

Abstracts

S.7 Function Spaces and Applications

OrganisersAlexandre Almeida, Antonio Caetano,Stefan Samko

Scope of the session: This session aims to cover recentprogresses in the Theory of Function Spaces (includ-ing spaces of Lebesgue, Orlicz, Sobolev, Besov, Triebel-Lizorkin and Morrey-Campanato type) and their appli-cations. Various generalizations are welcome, includ-ing spaces with variable exponents and others relatedto nonstandard growth conditions. Applications mightrange from properties of operators of harmonic analysisacting in such spaces to various applications to partialdifferential equations or integral equations. Applicationsin new contexts are also welcome.

—Abstracts—

Weighted Hardy type inequalities with a variable upperlimit

Akbota AbylayevaL.N.Gumilyov Eurasian National University, SatpayevStr., 2, 010008 Astana, KAZAKHSTANabylayeva b@mail.ru

Let 0 < α < 1. The operator of the form

Kα,ϕf(x) =

ϕ(x)∫a

f(t)w(t)dt

(W (x)−W (t))(1−α), x > 0,

is considered, where the real weight functions v(x) andw(x) are locally integrable on I := (a, b), 0 ≤ a < b ≤ ∞and dW (x)

dx≡ w(x).

In this paper we derive criteria for the operator Kα,ϕ,0 < α < 1, 0 < p; q < ∞, p > 1

αto be bounded and

compact from the spaces Lp,w to the spaces Lq,v.

———

Notes on bilinear multipliers on Orlicz spaces

Oscar BlascoUniversidad de Valencia, Dr. Moliner 60, 46100 Burjas-sot, Valencia, SPAINoscar.blasco@uv.es

Let Φ1,Φ2,Φ3 be Young functions and LΦ1(R), LΦ2(R),LΦ3(R) be corresponding Orlicz spaces. We say that afunction m(ξ, η) defined on R×R is a bilinear multiplierof type (Φ1,Φ2,Φ3) if

Bm(f, g)(x) =

∫R

∫Rf(ξ)g(η)m(ξ, η)e2πi(ξ+η)xdξdη

is a bounded bilinear operator from LΦ1(R)×LΦ2(R) toLΦ3(R). We denote by BM(Φ1,Φ2,Φ3)(R) the space of allbilinear multipliers of type (Φ1,Φ2,Φ3) and investigatesome properties of such a class and, under some condi-tions on the triple (Φ1,Φ2,Φ3), give some examples ofbilinear multipliers of type (Φ1,Φ2,Φ3). We will focuson the case m(ξ, η) = M(ξ−η) and get necessary condi-tions on (Φ1,Φ2,Φ3) to get non-trivial multipliers in thisclass. In particular we recover some of the the knownresults for Lebesgue spaces.

———

On Besov Regularity of Solutions to Nonlinear EllipticPartial Differential Equations

Stephan DahlkePhilipps-Universitat Marburg, FB 12 Mathematics andComputer Sciences, Hans-Meerwein-Str., Lahnberge,35032 Marburg, Hessia, GERMANYdahlke@mathematik.uni-marburg.de

We will be concerned with the regularity of solutionsto nonlinear elliptic operator equatiions on domains ofpolygedral type. In particular, we study the smoothnessin the specific scale Bsτ,τ , 1/τ = s/d+1/p of Besov spacesThe regularity in these spaces determines the approxi-mation order that can be achieved by adaptive and othernonlinear approximation schemes. We show that for allcases under consideration the Besov regularity is highenough to justify the use of adaptive algorithms. Theproofs are performed by using general embedding theo-rems between Kondratiev spaces and Besov spaces.

———

Characterizations of Besov spaces via K-functionalsand ball averages

Oscar DominguezUniversidad Complutense de Madrid, 28040 Madrid,SPAINoscar.dominguez@ucm.es

Besov spaces occur naturally in many fields of analy-sis. In this talk, we discuss various characterizations ofBesov spaces in terms of different K-functionals. Forinstance, we present descriptions via ball averages andminimization problems for bounded variation functions(Bianchini-type norms).This is a joint work with S. Tikhonov (Barcelona).

———

Function spaces with general weights

Douadi DrihemM’sila University, 28000 M’sila , ALGERIAdouadidr@yahoo.fr

We introduce Besov and Triebel-Lizorkin spaces withgeneral smoothness. These spaces unify and generalizethe classical Besov and Triebel-Lizorkin spaces. We es-tablish the ϕ-transform characterization of these spacesin the sense of Frazier and Jawerth and we prove theirSobolev embeddings. We study complex interpolationof these function spaces by using the Calderon productmethod and we identify their duals. We also we estab-lish the smooth atomic, molecular and wavelet decom-position of these function spaces. To do these we needa generalization of some maximal inequality to the caseof general weights.

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Compact embeddings of Smoothness Morrey spaces onbounded domains

Dorothee D. HaroskeFriedrich Schiller University Jena, Institute of Mathe-matics, 07737 Jena, GERMANYdorothee.haroske@uni-jena.de

We study the compact embedding between smoothnessMorrey spaces on bounded domains and characterise its

41

Abstracts

entropy numbers. Here we discover a new phenomenonwhen the difference of smoothness parameters in thesource and target spaces is rather small compared withthe influence of the fine parameters in the Morrey set-ting. In view of some partial forerunners this was notto be expected till now. Our argument relies on waveletdecomposition techniques of the function spaces and acareful study of the related sequence space setting. Thisis joint work with Leszek Skrzypczak from Poznan.

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Holder continuity of quasiminimizers and ω-minimizersof functionals with generalized Orlicz growth

Peter HastoUniversity of Turku, FINLANDpeter.hasto@oulu.fi

I present recent results on local Holder continuityof quasiminimizers of functionals with nonstandard(Musielak-Orlicz) growth. Compared with previous re-sults, we cover more general minimizing functionals andneed fewer assumptions. We prove Harnack’s inequalityand a Morrey type estimate for quasiminimizers. Com-bining this with Ekeland’s variational principle, we ob-tain local Holder continuity for ω-minimizers. This isjoint work with Petteri Harjulehto and Mikyoung Lee.

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Holomorphic Morrey, Orlicz, Grand Lebesgue andHolder spaces

Alexey KarapetyantsSouthern Federal University, Bolshaya Sadovaya 105/42,344019 Rostov-on-Don, Rostov region, RUSSIAN FED-ERATIONkarapetyants@gmail.com

We discuss some non standard spaces of functions holo-morphic in domains on the complex plain: Orlicz, vari-able exponent Morrey, Grand/Small spaces and general-ized Holder spaces, constructed either with modulus ofcontinuity or with variable exponent. We study bound-edness of classical Bergman projection, growth of func-tions in these spaces near the boundary and some otherquestions.

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Dual property of the Hardy-Littlewood maximal oper-ator on reflexive variable Lebesgue spaces over spacesof homogeneous type

Oleksiy KarlovychUniversidade Nova de Lisboa, Faculdade de Ciencias eTecnologia, 2829-516 Caparica, PORTUGALoyk@fct.unl.pt

We show that the Hardy-Littlewood maximal operator isbounded on a reflexive variable Lebesgue space Lp(·) overa space of homogeneous type (X, d, µ) if and only if it isbounded on its dual space Lq(·), where 1/p(x) + 1/q(x)for x ∈ X. This result extends the corresponding resultof Lars Diening (2005) from the Euclidean setting of Rnto the setting of spaces of homogeneous type (X, d, µ).

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Integral operators in mixed norm weighted functionspaces and application

Vakhtang KokilashviliA. Razmadze Mathematical Institute of I. JavakhishviliTbilisi State University, 6 Tamarashvili Str., 0177 Tbil-isi, 0177 Tbilisi, GEORGIAvakhtangkokilashvili@yahoo.com

The goal of our talk is to present the boundedness crite-ria for the fundamental integral operators of HarmonicAnalysis in mixed weighted grand function spaces. Weintend to give some applications to the approximationtheory.Acknowledgement. The work was supported by ShotaRustaveli National Science

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Maximal and Calderon-Zygmund operators in extrapo-lation Banach function lattices and applications

Alexander MeskhiA. Razmadze Mathematical Institute, I. JavakhishviliTbilisi State University, 6. Tamarashvili Str., 0177 Tbil-isi, Department of Mathematics, Faculty of Informaticsand Control Systems, Georgian Technical University, 77,Kostava St., 0175, Tbilisi, GEORGIAa.meskhi@gtu.ge

We derived the boundedness of the Hardy-Littlewoodmaximal and Calderon-Zygmund operators in extrap-olation Banach function lattices. As a consequence wehave the boundedness of these operators in extrapolationspaces generated by Orlicz spaces over quasi-metric mea-sure spaces with doubling measure. These results are ap-plied to get the boundedness of the Calderon-Zygmundoperator in grand Orlicz-Zygmund spaces with Mucken-houpt weights. The investigation was carried out jointlywith V. Kokilashvili and M. Mastylo.Acknowledgement. The work was supported byShota Rustaveli National Science Foundation grant (No.FR-18-2499).

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Embeddings of homogeneous Sobolev spaces on theentire space

Zdenek MihulaCharles University, Faculty of Mathematics and Physics,Ke Karlovu 3, 121 16 Praha, CZECH REPUBLICmihulaz@karlin.mff.cuni.cz

We completely characterize the validity of the inequal-ity ‖u‖Y (Rn) ≤ C‖∇ku‖X(Rn), where X and Y arerearrangement-invariant spaces, by reducing it to a con-siderably simpler one-dimensional inequality. Further-more, we fully describe the optimal rearrangement-invariant space on either side of the inequality when thespace on the other side is fixed. We also solve the sameproblem within the environment in which the competingspaces are Orlicz spaces. A variety of examples involv-ing customary function spaces suitable for applicationsis also provided.

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Some embeddings of smoothness Morrey spaces in lim-iting cases

Susana MouraCMUC, University of Coimbra, Apartado 3008, EC

42

Abstracts

Santa Cruz, 3001-501 Coimbra, PORTUGALsmpsd@mat.uc.pt

The classical Morrey space Mu,p(Rd), 0 < p ≤ u < ∞,consists of all locally p-integrable functions f on Rd suchthat

‖f |Mu,p(Rd)‖ = supB|B|

1u− 1p

(∫B

|f(x)|p dy)1/p

is finite, where B runs over all balls in Rd. Thesespaces are considered as an extension of the scale of Lpspaces, in view of Lu(Rd) = Mu,u(Rd) → Mu,p(Rd)for any p ≤ u. Built upon these spaces Besov-Morreyspaces N s

u,p,q(Rd) and Triebel-Lizorkin-Morrey spacesEsu,p,q(Rd) attracted some attention in last years, in par-ticular in connection with Navier-Stokes equations. Inthis talk we focus on embeddings of these smoothnessMorrey spaces in the borderline case of s = n(1/u −p/u)+, and in the so-called critical case s = d/u. In thelater case we obtain embeddings in Orlicz-Morrey spacesand in generalised Morrey spaces.This is joint work with Dorothee Haroske (Jena) andLeszek Skrzypczak (Poznan).

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Embeddings of Sobolev-type spaces into generalizedHolder spaces

Julio NevesUniversity of Coimbra, PORTUGALjsn@mat.uc.pt

We present a sharp estimate for thek-modulus ofsmoothness, modelled upon a Lp-Lebesgue space, of a

function f in W kLpnn+kp

,p(Ω), where Ω is a domain with

minimally smooth boundary and finite Lebesgue mea-sure, k, n ∈ N, k < n and n

n−k < p < +∞. Thissharp estimate is used to establish necessary and suf-ficient conditions for continuous embeddings of Sobolev-type spaces into generalized Holder spaces defined bymeans of the k-modulus of smoothness.This is joint work with A. Gogatishvili and B. Opic.

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Embedding properties for weighted Besov-Morrey andTriebel-Lizorkin-Morrey spaces

Takahiro Noi1-1 Minami osawa, hachioji-city, Tokyo,192-0397 Ha-chioji, JAPANtaka.noi.hiro@gmail.com

We discuss Sobolev’s embedding theorem on weightedBesov-Morrey and Triebel-Lizorkin-Morrey spaces withappropriate weight.

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Boundedness of Riemann-Liouville operator fromweighted Sobolev space to weighted Lebesgue space

Ryskul OinarovL. N. Gumilyov Eurasian National Universiti, Satpaevstr., 2, 010008 Astana, KAZAKHSTANo ryskul@mail.ru

Work done in collaboration with Aigerim Kalybay. LetI = (0,∞), 1 < p, q < ∞, 1

p+ 1

p′ = 1 and 1q

+ 1q′ = 1.

Let υ, ρ and ω be functions nonnegative on I such thatυp, ρp, ωq, ρ−p

′and ω−q

′are locally summable on I.

Denote by W 1p (ρ, υ) ≡W 1

p (ρ, υ, I) the space of all func-tions locally absolutely continuous on I having the finitenorm

‖f‖W1p

= ‖ρf ′‖p + ‖υf‖p,where ‖ · ‖p is the standard norm of the Lebesgue spaceLp(I).Let AC(I) be the set of all locally absolutely continuousfunctions with compact supports on I.Denote by W 1

p (ρ, υ) ≡ W 1p (ρ, υ, I) the closure of the set

AC (I)∩W 1p (ρ, υ) with respect to the norm of the space

W 1p (ρ, υ).

Let Lp,υ ≡ Lp(υ, I) be the space of all measurable func-tions I with the finite norm ‖f‖p,υ ≡ ‖υf‖p.We consider the Riemann-Liouville fractional integra-tion operator Iα:

Iαf(x) =

x∫0

(x− s)α−1f(s)ds, x ∈ I. (0.1)

We establish a criterion for the boundedness ofthe Riemann-Liouville operator Iα from W 1

p (ρ, υ) toLq,(ω, I), i.e., the fulfillment of the inequality:

‖ωIαf‖q ≤ C(‖ρf ′‖p + ‖υf‖p), f ∈ W 1p (ρ, υ).

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Maximal regularity for local minimizers of non-autonomous functionals

Jihoon OkKyung Hee University, 1732, Deogyeong-daero,Giheung-gu, 17104 Yongin-si, Gyeonggi-do, KOREA

jihoonok@khu.ac.kr

We establish local C1,α-regularity for some α ∈ (0, 1)and Cα-regularity for any α ∈ (0, 1) of local minimizersof the functional

v 7→∫

Ω

ϕ(x, |Dv|) dx,

where ϕ satisfies a (p, q)-growth condition. Establishingsuch a regularity theory with sharp, general conditionshas been an open problem since the 1980s. In contrastto previous results, we formulate the continuity require-ment on ϕ in terms of a single condition for the map(x, t) 7→ φ(x, t), rather than separately in the x- andt-directions. Thus we can obtain regularity results forfunctionals without assuming that the gap between theupper and lower bounds is small, i.e. q

pneed not be close

to 1. Moreover, for ϕ(x, t) with particular structure, in-cluding p-, Orlicz-, p(x)- and double phase-growth, oursingle condition implies known, essentially optimal, reg-ularity conditions. Hence we handle regularity theoryfor the above functional in a universal way.

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Moser meets Gauss

Lubos PickKMA MFF UK, Sokolovska 83, 18675 Prague, CZECHREPUBLICpick@karlin.mff.cuni.cz

43

Abstracts

We present Moser-type estimates for Gaussian-Sobolevembeddings.

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n-best approximation in reproducing kernel Hilbertspaces

Tao QianMacau University of Science and Technology, 999078Macau, Taipa, MACAOtqian@must.edu.mo

The talk will introduce the n-best approximation modelin reproducing kernel Hilbert spaces under a ”doublingboundary vanishing condition”. The background ex-amples include Bergman and weighted Bergman andweighted Hardy spaces. In particular cases this methodgives rise to rational approximation of the underlyingspace. The method can be extended to matrix-valuedfunctions of certain hyper-complex variables.

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Nonstandard bounded variation spaces

Humberto RafeiroUnited Arab Emirates University, UAErafeiro@uaeu.ac.ae

We will discuss the notion of bounded variation spaceswith variable exponent (in the Wiener and in the Rieszsense). In particular, we will show some embedding re-sults, a Riesz representation lemma, and a characteriza-tion of global Lipschitz Nemytskii operators in the Rieszbounded variation spaces with variable exponent.

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Boundedness and compactness of composition opera-tors in holomorphic and harmonic spaces of Holder typefunctions

Joel E. RestrepoBolshaya Sadovaya, 105/42, Rostov-on-Don, 344006,RUSSIAN FEDERATIONcocojoel89@yahoo.es

We study the boundedness and compactness of compo-sition operators in the generalized Holder type space ofholomorphic functions in the unit disc with prescribedmodulus of continuity and in the variable exponent gen-eralized Holder spaces of holomorphic functions in theunit disc.

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Time-fractional telegraph equation

M. Manuela RodriguesCIDMA & University of Aveiro, 3810-193 Aveiro, POR-TUGALmrodrigues@ua.pt

In this work we obtain the first and second fundamentalsolutions (FS) of the multidimensional time-fractionalequation with Laplace operator, where the two time-fractional derivatives of orders α ∈]0, 1] and β ∈]1, 2] arein the Caputo sense. We obtain representations of theFS in terms of Hankel transform, double Mellin-Barnesintegrals, and H-functions of two variables. As an ap-plication, the FS are used to solve Cauchy problems of

Laplace type. Moreover, some considerations about itsapplications in future work and its Lp integrability willbe presented.

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Embeddings of weighted local generalized Morreyspaces into Lebesgue spaces on fractal sets

Natasha SamkoUiT The Arctic University of Norway, Narvik, NOR-WAYNatasha.G.Samko@uit.no

We study embedding of weighted local generalized Mor-rey spaces defined on a quasi-metric measure set of gen-eral nature which may be unbounded, into Lebesguespaces. In the general setting of quasi-metric measurespaces and arbitrary weights we give a sufficient condi-tion for such an embedding. In the case of radial weightsrelated to the center of local Morrey space, this generalcondition allows to obtain an effective sufficient condi-tion in terms of (fractional in general) upper Ahlforsdimension of the set X: In this case we also obtain anecessary condition with the use of (fractional in gen-eral) lower Ahlfors dimension instead of the upper one,so that we have a criterion for the embedding when thesedimensions coincide.

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Lizorkin-Triebel-Morrey Spaces and Differences

Winfried SickelInstitute of Mathematics, Friedrich Schiller UniversityJena, D-07743 Jena, Thuringia, GERMANYwinfried.sickel@uni-jena.de

In my talk I plan to speak about the characterizationof F s,τp,q (Rd) by differences. At the beginning I will re-call what is known about the characterization of theLizorkin-Triebel spaces F sp,q(R

d) itself and differences.In particular I will consider two types of characteriza-tions by differences: (a) the so-called ball mean charac-terization and (b) the Strichartz characterization. Thosecharacterizations are never valid for the maximal rangeof the parameters p, q, s and d. Usually s has to be suf-ficiently large depending on p, q and d. We will discusssome sufficient and some necessary conditions.

———

On Cantor’s Λ functional and reconstruction of coeffi-cients of multiple function series

Shakro TetunashviliGeorgian Technical University, Kostava Str., 77, 0171Tbilisi, GEORGIAstetun@hotmail.com

In our talk summable series with respect to the systemsΦ =

(ϕn(t)

)∞n=0

of finite and measurable functions de-fined on [0, 1] by positive, regular, triangular Λ matricesare considered. It is introduced the notion of Cantor’sΛ functional for Λ summable series. This notion gen-eralizes, in particular, any trigonometric integral in thesense of the reconstruction of coefficients of the series.Reconstruction of coefficients of multiple function seriesby iterated using of Cantor’s Λ functional is established.The work was fulfilled jointly with Tengiz Tetunashvili.Acknowledgement. The work was supported byShota Rustaveli National Science Foundation grant (No.DI 18-118).

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Trigonometric approximation in weighted grand variableexponent Lebesgue spaces

Tsira TsanavaFaculty of Informatics and Control Systems, GeorgianTechnical University, 77, Kostava St., 0171 Tbilisi,GEORGIAts tsanava75@yahoo.com

We intend to discuss a description of approximative sub-spaces of weighted grand variable exponent Lebesguespaces. For the functions of spaces we plan to presentthe direct and inverse theorems of trigonometric approx-imation.Acknowledgement. The work was supported byShota Rustaveli National Science Foundation grant (No.DI 18-118).

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Characterization of functions with zero traces fromSobolev spaces via the distance function from theboundary

Hana TurcinovaKMA MFF UK, Sokolovska 83, 18675 Prague, CZECHREPUBLICTurcinovaHana@seznam.cz

We characterize functions from Sobolev spaces with zerotraces by conditions involving the distance from theboundary.

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S.8 Function Spaces and their Applicationsto Nonlinear Evolutional Equations

OrganisersMitsuru Sugimoto, Baoxiang Wang

Scope of the session: The session will discuss some re-cent progress in various Banach function spaces, espe-cially arising from harmonic analysis, and their applica-tions to nonlinear evolutional equations. This session isfocusing on the following topics:Time-frequency analysisModulation spaces, Besov spacesLinear and nonlinear dispersive equationsNavier-Stokes, MHD equations

—Abstracts—

Global Existence and Uniqueness Analysis of Reaction-Cross-Diffusion Systems

Xiuqing ChenSun Yat-sen University, Math College (Zhuhai), 519082Zhuhai, Guangdong, PEOPLE’S REPUBLIC OFCHINAbuptxchen@yahoo.com

The global-in-time existence of weak and renormalizedsolutions to reaction-cross-diffusion systems for an arbi-trary number of variables in bounded domains with no-flux boundary conditions are proved. The cross-diffusion

part describes the segregation of population species andis a generalization of the Shigesada-Kawasaki-Teramotomodel. The diffusion matrix is not diagonal and gen-erally neither symmetric nor positive semi-definite, butthe system possesses a formal gradient-flow or entropystructure. The reaction part is of Lotka-Volterra type forweak solutions or includes reversible reactions of mass-action kinetics and does not obey any growth conditionfor renormalized solutions. Furthermore, we prove theuniqueness of bounded weak solutions to a special classof cross-diffusion systems, and the weak-strong unique-ness of renormalized solutions to the general reaction-cross-diffusion cases.

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Inhomogeneous Strichartz estimates in some criticalcases

Jayson CunananDepartment of Mathematics, Graduate School of Scienceand Engineering, Saitama University, 255 Shimookubo,Sakura Ward, 338-8570 Saitama, JAPANjcunanan@mail.saitama-u.ac.jp

Strong-type inhomogeneous Strichartz estimates areshown to be false for the wave equation outside the so-called acceptable region. On a critical line where theacceptability condition marginally fails, we prove substi-tute estimates with a weak-type norm in the temporalvariable. We achieve this by establishing such weak-type inhomogeneous Strichartz estimates in an abstractsetting. The application to the wave equation rests ona slightly stronger form of the standard dispersive esti-mate in terms of certain Besov spaces. This talk is basedon a joint-work with Neal Bez and Sanghyuk Lee.

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Wave packet transform and estimate of solutions toSchrodinger equations in modulation spaces

Keiichi KatoTokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, 2430432 Tokyo, Tokyo, JAPANkato@rs.tus.ac.jp

In this talk, we give a representation of solutions toSchrodinger equations with potential via wave packettransform. By using this representation, we show theestimate of solutions to Schrodinger equations with timedependent sub-quadratic potential in modulation spacesand give a construction of solutions by time slicingmethod.

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Boundedness of bilinear pseudo-differential operatorswith symbols in an S0,0-type class

Tomoya KatoGunma University, Kiryu Campus 1-5-1 Tenjin-cho, 376-8515 Kiryu city, Gunma, JAPANt.katou@gunma-u.ac.jp

In this talk, we extend the result proved by Miyachi–Tomita (2013), the L2 × L2 → h1-boundedness of bi-linear pseudo-differential operators with symbols be-longing to the Hormander class BS

−n/20,0 , in two ways.

First, we improve the target space h1 to the amal-gam space (L2, `1), which is embedded into h1 ∩ Lr,

45

Abstracts

1 < r ≤ 2, continuously. Secondly, we improve the thesymbol class BS

−n/20,0 by replacing the weight function

(1 + |ξ1|2 + |ξ2|2)−n/2 by general functions. This talkis based on a joint research with Prof. Miyachi (TokyoWoman’s Christian University) and Prof. Tomita (Os-aka University).

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Ill-posedness of an NLS-type equation with derivativenonlinearity on the torus

Nobu KishimotoRIMS, Kyoto University, Oiwakecho KitashirakawaSakyo-ku, 6068502 Kyoto, Kyoto, JAPANnobu@kurims.kyoto-u.ac.jp

We consider a nonlinear Schrodinger equation withthird-order dispersion and derivative nonlinearity, whichis regarded as a mathematical model for the photoniccrystal fiber oscillator. For the non-periodic problem,the associated Cauchy problem is known to be locallywell-posed in Sobolev spaces. We show that in theperiodic setting the derivative nonlinearity causes ill-posedness (more precisely, non-existence of local-in-timesolutions) of the Cauchy problem in Sobolev spaces andeven in Gevrey classes. This talk is based on a jountwork with Yoshio Tsutsumi (Kyoto University)

———

Adomian Decomposition Method for Burgers’ Equa-tions

Tzon-Tzer LuDepartment of Applied Mathematics, National SunYat-sen University, 80424 Kaohsiung, TAIWAN,PROVINCE OF CHINAttlu@math.nsysu.edu.tw

In this paper we like to explore the full power of Ado-mian decomposition method (ADM) to solve nonlinearproblems. ADM has the capability to obtain three dif-ferent types of solutions, namely explicit solution, ana-lytic solution and semi-analytic solution. When a closedform solution exists, ADM is possible to capture thisexplicit solution, while most of the numerical methodscan only get approximation, not exact solution. On theother hand, ADM is itself a computational method forsolution, while the method of characteristics needs assis-tance from other numerical methods to find characteris-tic curve and differential equation on it. We will demon-strate ADM to solve the prototype nonlinear PDEs, i.e.the inviscid and viscous Burgers’ equations, and con-clude the superior of ADM.

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Operating functions on Aqs(T)

Masaharu KobayashiDepartment of Mathematics, Hokkaido University, Kita10, Nishi 8, Kita-Ku, 060-0810 Sapporo, Hokkaido,JAPANm-kobayashi@math.sci.hokudai.ac.jp

In this talk, we consider operating functions on Aqs(T).This is joint work with Enji Sato (Yamagata Univer-sity).

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Gradient estimates for heat equation in an exterior do-main

Koichi TaniguchiNagoya University, Furocho, Chikusaku, 464-8602Nagoya, JAPANkoichi-t@math.nagoya-u.ac.jp

In this talk we consider gradient estimates for heat equa-tion with Dirichlet boundary condition in an exterior do-main. Our results describe the sharp time decay rates ofthe derivatives of solutions to the heat equation. As anapplication, we also consider the fractional Leibniz rulefor the Dirichlet Laplacian in the exterior domain.This talk is based on the joint work with ProfessorVladimir Georgiev (University of Pisa).

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Periodic ultra-distributions and periodic elements inmodulation spaces

Joachim ToftDepartment of Mathematics, Vejdes plats 6,7, LinnaeusUniversity, 35195 Vaxjo, Smaland, SWEDENjoachim.toft@lnu.se

In the present talk we characterize periodic elements inGevrey classes, Gelfand-Shilov distribution spaces andmodulation spaces, in terms of estimates of involvedFourier coefficients, and by estimates of their short-timeFourier transforms. We show that such spaces can becompletely characterised in terms of formal Fourier se-ries with suitable estimates on their coefficients. Forperiodic Gelfand-Shilov distributions such characterisa-tions can be found in the literature in the case when theGevrey parameter is strictly larger than 1. Our analysisis valid for all positive Gevrey parameters.As a consequence, inverse problems for diffusion equa-tions and similar equations on certain bounded domainscan be handeled.The proofs are based on new types of formulae of inde-pendent interestswhen evaluating the Fourier coefficientsand which involve short-time Fourier transforms. Thetalk is based on a joint work with E. Nabizadeh.

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Bilinear pseudo-differential operators with exotic sym-bols

Naohito TomitaDepartment of Mathematics, Osaka University, 560-0043Toyonaka, Osaka, JAPANtomita@math.sci.osaka-u.ac.jp

In this talk, we consider the boundedness of bilinearpseudo-differential operators with symbols in the bilin-ear Hormander class BSmρ,ρ, 0 ≤ ρ < 1. Our purpose isto determine the critical order m = m(ρ, p, q) to assurethe boundedness from products of Hardy spaces Hp×Hq

to Lr, 1/p+1/q = 1/r, in the full range 0 < p, q, r ≤ ∞.

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A sparse bound for an integral operator with wave prop-agator

Yohei TsutsuiDepartment of Mathematical Sciences, Faculty of Sci-ence, Shinshu University, Asahi 3-1-1, 390-8621 Mat-sumoto, Nagano, JAPANtsutsui@shinshu-u.ac.jp

46

Abstracts

The purpose of this talk is to give a sparse bound for

T`f(x) :=

∫ 2

1

∣∣∣eit√−∆ϕ`(D)f(x)∣∣∣ dt

where ` ∈ N and ϕ`(D)f := F−1[ϕ(·/2`)f ]. This opera-tor is related to the maximal Riesz means.

———

Global regularity for the 3D relativistic massive Vlasov-Maxwell system

Xuecheng WangTsinghua University, 100084 Beijing, PEOPLE’S RE-PUBLIC OF CHINAxuecheng@tsinghua.edu.cn

Given any smooth, suitably small initial data, which de-cays polynomially at infinity, we prove global regularityfor the 3D relativistic massive Vlasov-Maxwell system.In particular, neither the initial distribution for particlesnor the electromagnetic field is assumed to have a com-pact support. The compact support type assumptionwas assumed in all previous results.

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Equivariant Schrodinger map from two dimensional hy-perbolic spaces

Lifeng ZhaoUniversity of Science and Technology of China, 230026Hefei, Anhui, PEOPLE’S REPUBLIC OF CHINAzhaolf@ustc.edu.cn

We consider the equivariant Schrodinger map from H2

to S2 which converges to the north pole of S2 at the ori-gin and spatial infinity of the hyperbolic space. If theenergy of the data is less than 4?, we show that the lo-cal existence of Schrodinger map. Furthermore, if theenergy of the data sufficiently small, we prove the so-lutions are global in time. This is based on joint workwith Jiaxi Huang.

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S.9 Generalized Functions and Applications

OrganisersMichael Kunzinger,Michael Oberguggenberger, Stevan Pilipovic

Scope of the session: The session will be devoted totheory and application of generalized functions, whichcomprise, among others, distributions, ultradistribu-tions, hyperfunctions and algebras of generalized func-tions. Applications include, but are not restricted to, lin-ear and nonlinear partial differential equations, asymp-totic analysis, geometry, mathematical physics, stochas-tic processes, and harmonic analysis, both in theoreticaland numerical aspects. The session is open to contri-butions on any aspect of generalized functions and theirapplications.

—Abstracts—

H-distributions and variants

Nenad AntonicDepartment of Mathematics, Faculty of Science, Bi-jenicka c. 30, 10000 Zagreb, CROATIAnenad@math.hr

H-distributions are microlocal defect functionals (like H-measures and semiclassical/Wigner measures, microlo-cal compactness forms), the objects which determine, insome sense, the lack of strong compactness for weaklyconvergent Lp sequences.H-distributions are an extension of H-measures to theLp–Lq setting, and so far they have been success-fully applied in compactness by compensation theorywith discontinuous coefficients and to velocity averaging.For their construction, the Plancherel theorem (whichwas sufficient for H-measures) had to be replaced byHormander-Mihlin’s theorem for Fourier multipliers. Inorder to broaden their possible applicability one needsto develop some additional properties of H-distributions.In this, an appropriate variant of the Schwartz kerneltheorem is crucial: it allows to identify a bilinear formon the space of test functions with a distribution of finiteorder in both variables; in fact, being a Radon measurein the physical x space, and the distribution of finiteorder in the dual ξ space.In order to adjust H-distributions to some applications,we shall investigate different projections in the phasespace, as well as different compactifications, comparingthe constructions with that of microlocal compactnessforms.

———

Characterization of wave front sets via multidimen-sional Stockwell transform

Sanja AtanasovaSs. Cyril and Methodius University, Faculty of Elec-trical Engineering and Information Technologies, 1000Skopje, MACEDONIAksanja@feit.ukim.edu.mk

The talk is dedicated to resolution of wave front setsusing the multidimensional Stockwell transform definedwith respect to the rotation group. The main results givethe principles for directional smoothness, by providingcriteria for regular directed points using the Stockwelltransform. We work on the cases when the dimensionis 1, 2, 4 and 8, and try to generalize the results onarbitrary dimension. The talk is based on a joint workwith Katerina Saneva, Stevan Pilipovic and Bojan Pran-goski.

———

Applications of topological tensor products to the the-ory of distributions

Christian BargetzUniversity of Innsbruck, Technikerstraße 13, 6020 Inns-bruck, AUSTRIAchristian.bargetz@uibk.ac.at

The theory of topological tensor products has been animportant tool in the theory of distributions since theintroduction of theory of vector-valued distributions by

47

Abstracts

L. Schwartz in 1957. We present a number of recent re-sults in distribution theory which are shown using meth-ods from the theory of topological tensor products. Inparticular, we will present some recent results on theconvolvability and regularisation of (vector-valued) dis-tributions.

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On almost periodicity and almost automorphy

Chikh BouzarLaboratory of Mathematical Analysis and Applications,University of Oran, Oran, ALGERIAch.bouzar@gmail.com

This talk aims to give an overview on almost periodic-ity and asymptotic almost periodicity as well as almostautomorphy and asymptotic almost automorphy in theframework of classical functions, distributions, ultradis-tributions and finally generalized functions.

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Absence of remainders in the Wiener-Ikehara theorem

Frederik BrouckeGhent University, Raymond Smetstraat 32, 9170 Sint-Gillis-Waas, Oost-Vlaanderen, BELGIUMfabrouck.broucke@ugent.be

The celebrated Wiener-Ikehara theorem is a cornerstoneof Tauberian theory. It states:

Theorem. Let S be a non-decreasing function and sup-pose that

G(s) :=

∫ ∞1

S(x)x−s−1dx

converges for <s > 1 and that there exists a constant asuch that G(s)−a/(s−1) admits a continuous extensionto <s ≥ 1. Then

S(x) = ax+ o(x).

If one wants to deduce a quantitive error term forS(x) − ax, one in general needs to ask greater regu-larity of the Mellin transform G(s) of S. It is wellknown that one can get remainders if for example oneassumes analytic continuation to <s > α for some α,0 < α < 1, and certain bounds on G(s) − a/(s − 1) inthe strip α < <s < 2. In 2017, Muger raised the ques-tion if an explicit error term would follow from merelythe analytic continuation (without assuming bounds) ofG(s)−a/(s− 1). This was recently answered negativelyby G. Debruyne and J. Vindas. Their proof uses meth-ods from functional analysis such as the open mappingtheorem, and is non-constructive.In this talk, we will sketch how one can construct ex-plicit counterexamples. This talk is based on collabora-tive work with Gregory Debruyne and Jasson Vindas.

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Fujita’s Blow-up property of Discrete Reaction-Diffusion Equations on Networks

Soon-Yeong ChungDepartment of Math., Sogang University, National In-stitute for Mathematical Sciences, Seoul 04107, KOREA

sychung@nims.re.kr

The purpose of this work is to study Fujita’s blow-upproperty of the equation:

(E) :

ut = ∆ωu+ ψ(t)f(u), in S × (0,∞),

u = 0, on ∂S × (0,∞),

u(·, 0) = u0, on S,

In fact, we introduce a set Λ(ψ)(called a critical set) toprove the followings:

(i) if f ∈ Λ(ψ), then the solutions to (E) blow up infinite time for every initial data.

(ii) if f 6∈ Λ(ψ), then the solutions to (E) blow up infinite time for sufficiently large data.

(iii) if f 6∈ Λ(ψ), then the solutions to (E) exist glob-ally for small initial data.

Here,S is a finite network with boundary ∂S and weightω. (This is a joint work with Dr. J.-H.Park and M.-J.Choi)

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A Differential Calculus in the Framework Colombeau’sFull Algebra

Antonio R. G. GarciaUniversidade Federal Rural do Semi-Arido, 572, Fran-cisco Mota Avenue - Neighborhood Costa e Silva,Mossoro RN, Zip code: 59.625-900, BRAZILronaldogarcia@ufersa.edu.br

Starting from the Colombeau’s full generalized func-tions, the sharp topologies and the notion of generalizedpoints, we introduce a new kind differential calculus. Westudy generalized pointvalues, Colombeau’s differentialalgebra, holomorphic and analytic functions. We showthat the Embedding Theorem and the Open MappingTheorem hold in this framework. Moreover, we studysome applications in differential equations.

This is joint work with Cortes, Wagner., da Silva,S. Horacio and Juriaans, O. Stanley.

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Singularities in equations of fluid dynamics and appli-cations to hurricanes.

Maximilian F. HaslerUniversite des Antilles, DSI & MEMIAD, B.P. 7209,97275 Schoelcher cedex, Martinique, FRANCEmaximilian.hasler@univ-antilles.fr

We prove new theorems concerning irregular solutionsto PDE of fluid dynamics derived from Navier-Stokesequations in rotating coordinate systems. Such PDEare common in meterology, but we generalize our resultsto an arbitrary number of dimensions. Our solutions arecomposed of a regular part and singular part which is lo-calized on the border of a compact domain in space. Ourresults comprise orthogonality conditions remeniscent ofRankine-Hugoniot type jump conditions. We apply ourresults to mathematical models of hurricanes to study inparticular the dynamics near the eye of the hurricane.This also includes asymptotic expressions for the windprofile near the eye’s wall.

———

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Abstracts

Generalized functions in the optimal control of age-structured population models

Natali HritonenkoPrairie View A&M University, 77446 Prairie View,Texas, USAnahritonenko@pvamu.edu

We use generalized functions in a qualitative optimiza-tion analysis of two age-structured population modelsand discuss the applied relevance of obtained theoreticresults. Solutions from non-smooth functional spaces,including generalized functions, often appear in the op-timal control of partial differential equations but areavoided in applications. A common consensus is thatoptimization problems with state constraints have weaksolutions in wide classes with generalized functions. Thistalk demonstrates that generalized functions can arise inapplied problems without any constraints just becauseof the nature of a process under study.The first discussed problem maximizes the discountedconcave profit from harvesting in the linear age-structured Lotka-McKendrick model. Such models of-ten possess so-called bang-bang harvesting controls. Weconsider a relaxation of this optimal harvesting problemwithout upper bound on control variables and prove thatoptimal age-dependent harvesting control is expressedvia the Dirac delta-function. The results establish newlinks between bang-bang and spiked harvesting strate-gies.The second problem is the optimal distribution of in-vestments into new and old equipment under evolvingtechnology. A challenge is that solutions do not exist innormal functional classes, so, we have to employ general-ized functions to construct solutions. The optimal cap-ital and investment age-distributions involve the Diracdelta-function and its derivative. They are obtained asthe limiting case of a related model with constant elastic-ity of substitution among equipment vintages of differentages. Numeric simulation illustrates analytic outcomes.

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The eigenvalue problems for p-Schrodinger operatorsunder the mixed boundary conditions on finite networks

Jaeho HwangDepartment of Mathematics, Sogang University, Seoul04107, REPUBLIC OF KOREAhjaeho@sogang.ac.kr

In this talk, we discuss the discrete eigenvalue problemsfor the p-Schrodinger equations under the mixed bound-ary conditions defined on networks as follows:

−∆p,ωφ+ q|φ|p−2φ = λ|φ|p−2φ, in S,

µ ∂φ∂pn

+ σ|φ|p−2φ = 0, on ∂S,

where p > 1, q is a real-valued function on a networkS, and µ, σ are nonnegative functions on the boundary∂S of S, with µ(z) + σ(z) > 0, z ∈ ∂S. Here, ∆p,ω

and ∂φ∂pn

are discrete p-Laplace operator and discrete p-

normal derivative, respectively. Next, we use the aboveresult to provide the existence of a positive solution tothe discrete Poisson equation

−∆p,ωu+ q|u|p−2u = f, in S,

µ ∂u∂pn

+ σ|u|p−2u = 0, on ∂S.

This is a joint work with Soon-Yeong Chung(sychung@sogang.ac.kr).

———

The convolution of ultradistributions in the context oftheir supports

Andrzej KaminskiFaculty of Mathematics and Natural Sciences, Univer-sity of Rzeszow, Prof. Pigonia 1, 35-310 Rzeszow,POLANDakaminsk@ur.edu.pl

Various sufficient conditions for the existence and asso-ciativity of the convolution in the spaces of ultradistri-butions of Beurling and Roumieu types (as well as inthe spaces of tempered ultradistributions of both types)are given in terms of their supports. The conditionsare counterparts of the compatibility conditions of sup-ports of distributions (and of the polynomial compat-ibility of supports of tempered distributions). Severalnew particular cases are discussed and new applicationsare shown.

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Variational problems of Herglotz type with complex or-der fractional derivatives and less regular Lagrangian

Sanja KonjikDepartment of Mathematics and Informatics, Faculty ofSciences, University of Novi Sad, SERBIAsanja.konjik@dmi.uns.ac.rs

We derive optimality conditions for variational problemsof Herglotz type whose Lagrangian depends on fractionalderivatives of both real and complex order, and resolvethe case of subdomain when the lower bounds of vari-ational integral and fractional derivatives differ. More-over, we consider a problem of the Herglotz type thatcorresponds to the case when the Lagrangian dependson the fractional derivative of the action and give anexample of the problem that corresponds to the oscilla-tor with a memory. Since our assumptions on the La-grangian are weaker than in the classical theory, we ana-lyze generalized Euler-Lagrange equations by the use ofweak derivatives and the appropriate technics of distri-bution theory. Such an example is discussed in details.The talk is based on collaboration with T. M. Atanack-ovic and S. Pilipovic.

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Parameter-dependent linear ODE and topology of do-mains

Michael KunzingerUniversity of Vienna, Faculty of Mathematics, 1090 Vi-enna, AUSTRIAmichael.kunzinger@univie.ac.at

The well-known solution theory for (systems of) lin-ear ordinary differential equations undergoes significantchanges when introducing an additional real parameter.Properties like the existence of fundamental sets of so-lutions or characterizations of such systems via nonvan-ishing Wronskians are sensitive to the topological prop-erties of the underlying domain of the independent vari-able and the parameter. We give a complete character-ization of the solvability of such parameter-dependent

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systems in terms of topological properties of the domain,both classically and in the setting of Schwartz distribu-tions. This is joint work with V.M. Boyko and R.O.Popovych.

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Integro-differential equations for probabilistic charac-teristics of continuous and intermittent processes inspaces of distributions

Irina MelnikovaUral Federal University, Institute of Natural Sciencesand Mathematics, Lenina st. 51, 620083 Ekaterinburg,RUSSIAN FEDERATIONirina.melnikova@urfu.ru

The talk covers a comparison of different approaches toobtaining integro-differential (pseudo-differential) equa-tions for probabilistic characteristics of processes thatoccur under the influence of continuous and intermit-tent random perturbations.• The approach based on existence of three limits forrandom processes on time interval ∆t as ∆t→ 0. Theyare limits related to the local first an second order mo-ments of transition probability and a limit that sepa-rates continuous processes from discontinuous. Practi-cally, existence of the limits is based on statistical datafor probabilistic characteristics of random processes. Weshow the existence of these limits based on models lead-ing to stochastic equations for the processes themselves.• Semigroup approach to the Kolmogorov-Fokker-Planck and master equations. Construction of genera-tors of pseudo-differential equations based on the Levy-Khintchin formula and the techniques of the generalizedFourier transform.In the equations obtained formally there are general-ized functions such as shifted δ-functions and even theirproducts, as well as integral transforms of these func-tions. The special attention is paid to justification ofthese terms in spaces of generalized functions.

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Existence results concerning the convolution of ultra-distributions

Svetlana Mincheva-KaminskaFaculty of Mathematics and Natural Sciences, Univer-sity of Rzeszow, Prof. Pigonia 1, 35-310 Rzeszow,POLANDminczewa@ur.edu.pl

We consider and discuss various general sequential con-ditions (a) for integrability and (b) for convolvability ofultradistributions of Roumieu type in the space D′Mp,based on suitably chosen classes of approximate units. Inboth cases (a) and (b), we prove the equivalence of theconsidered conditions. Analogous results are obtainedfor tempered ultradistributions of Roumieu type in thespace S ′Mp. We also prove several existence theoremson the convolution in the spaces D′Mp and S ′Mp.

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Supremum, infimum and hyperlimits of Colombeaugeneralized numbers

Akbarali MukhammadievFaculty of Mathematics, Oskar-Morgenstern-Platz 1, A-1090 Wienakbarali.mukhammadiev@univie.ac.at

It is well-known that the notion of limit in the sharptopology of sequences of Colombeau generalized num-bers R does not generalize classical results. In fact, thering R is non-Archimedean and this topology is gener-ated by balls of infinitesimal radii. E.g. the sequence1n6→ 0 and a sequence (xn)n∈N converges if and only

if xn+1 − xn → 0. This has several deep consequences,e.g. in the study of series, analytic generalized functions,or sigma-additivity and classical limit theorems in inte-gration of generalized functions. The lacking of theseresults is also connected to the fact that R is necessar-ily not a complete ordered set, e.g. the set of all theinfinitesimals does not have neither supremum nor infi-mum.We present a solution of these problems with the intro-duction of the Robinson-Colombeau ring ρR and withthe notion of hypernatural number

ρN :=

[nε] ∈ ρR | nε ∈ N ∀ε.

The ring ρR is defined exactly as R, but where the gen-eralized number ρ = [ρε] plays the role of the classical[ε]. The net (ρε)ε∈(0,1] ↓ 0 is called a gauge. We de-fine the notion of supremum and infimum and that ofArchimedean bound, showing that the existence of theformer is equivalent to that of the latter, which is, likein the classical case, an easier and convenient concept.In this way, we can generalize all the classical theoremsfor the hyperlimit of an hypersequence (xn)n : σN→ ρR,where (σε)ε is another gauge: relations between hy-perlimits and inequalities, Cauchy criterion, monotonichypersequences, sub-hypersequences, limit superior andinferior, relations with sharp continuity of generalizedfunctions, classical examples and counter-examples, etc.The fact that 1

logn→ 0 as n → +∞, n ∈ σNσ, for a

certain gauge σ > ρ but 1logn

6→ 0 if σ = ρ shows thatthe generalization using two gauges is necessary.This is a joint work with D. Tiwari (University of Vi-enna), G. Apaaboah (Universit Grenoble Alpes) andP. Giordano (University of Vienna).

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Generalized solutions of conservation law systems con-taining the delta distribution

Marko NedeljkovDepartment for mathematics, Faculty of Sciences, Uni-versity of Novi Sad, SERBIAmarko@dmi.uns.ac.rs

There is an interesting class of conservation law systemsthat posses unbounded solutions. We will describe someof them containing the Dirac delta function as a partof a solution. There is no unified way of treating theseproblems and we will present some of them, try to bringtheir good and bad sides and compare.

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Asymptotic boundedness in ultradistribution spaces

Lenny NeytDepartment of Mathematics: Analysis, Logic and Dis-crete Mathematics, Krijgslaan 281 Building S8 3rd floor,9000 Ghent, BELGIUMlenny.neyt@ugent.be

50

Abstracts

In this talk we study the structure imposed by asymp-totic boundedness in ultradistribution spaces, which asusual is done by an analysis of its parametric behav-ior relative to some weight function. In the first partof the talk we will be considering convolution averagesand provide a full characterization. Though such struc-tural theorems have been found in the past, our ap-proach is novel as we employ the short-time Fouriertransform (STFT), which allows us to work under milderassumptions than those needed when working with theparametrix method. Because of this, the theory of theSTFT had to be extended to the context of general ul-tradistributions.In the second part of the talk, dilation is considered.By applying the previous results and certain techniquesdeveloped by Vindas and the speaker, a full charac-terization is at hand. This structure is then utilizedto consider the so-called moment asymptotic expansion(MAE) in an ultradistributional context. A generalizedfunction f is said to satisfy the MAE if there exists cer-tain complex numbers µαα∈Nd such that the followingasymptotic expansion holds

f(λx) ∼∑α∈Nd

(−1)|α|µαδ(α)(x)

α!λ|α|+d, λ→ +∞.

In this talk we consider the MAE over several spaces ofultradifferentiable functions, providing a full characteri-zation in the 1-dimensional case, whilst also consideringa uniform analog.

———

The geometrization of the theory of full Colombeaualgebras

Eduard NigschFaculty of Mathematics, University of Vienna, AUS-TRIAeduard.nigsch@univie.ac.at

A natural field of application of Colombeau algebras ofnonlinear generalized functions lies in general relativity,in particular in questions concerning metrics of low reg-ularity where classical distribution theory fails. An earlyapplication of Clarke, Vickers and Wilson (’96) in thiscontext involved the calculation of the curvature of con-ical metrics as an associated distribution. To free suchcalculations from the dependence on coordinate systemsand special mollifiers, one naturally needs to developfull Colombeau algebras which are invariant under theaction of diffeomorphisms.I will give a broad overview of the steps involved inobtaining a diffeomorphism invariant theory of nonlin-ear generalized functions and how it is extended to thecase of nonlinear generalized sections of vector bundles.Moreover, I will show how this structure accomodatesvirtually all previous variants of Colombeau algebrasand can be used to calculate the curvature of conicalmetrics.

———

Propagation of singularities for generalized solutions tononlinear wave equations

Michael OberguggenbergerUniversity of Innsbruck, Technikerstraße 13, 6020 Inns-bruck, Austriamichael.oberguggenberger@uibk.ac.at

The talk addresses propagation of singularities for non-linear wave equations in the Colombeau algebra G.While the subalgebra G∞ is not suitable for nonlinearequations, the subalgebra G0 of Colombeau generalizedfunctions all whose derivatives are of bounded type canbe used. We first show that the Cauchy problem fornonlinear wave equations of the form uε = εf(uε) canbe globally solved in G0 up to three space dimensions.Here the nonlinear function f is assumed to be smoothand polynomially bounded; the nonlinearity is small inthe sense that it is multiplied by the Colombeau param-eter ε. Then we follow the classical road laid out byMichael Reed and Jeff Rauch in the 1970s showing thatin one space dimension regularity propagates inside thelight cone emanating from a singular initial point. Bothresults are proven by means of a fixed point theorem inthe sharp topology.The presentation is joint work with Hideo Deguchi.

———

Wave fronts-new results

Stevan PilipovicDepartment of Mathematics and Informatics - Univer-sity Of Novi Sad, Novi Sad, SERBIAstevan.pilipovic@dmi.uns.ac.rs

Wave fronts are defined for the distributions, for non-quasy-analytic Gevrey type ultradistributions of orders > 1, for quasy-analytic ultradistributions of orders = 1 (analytic wave front) and for their subspaces. Twonew wave fronts: 1. for generalized functions betweenultradistributions and distributions and 2. for quasy-analytic ultradistributions of order s ∈ (1/2, 1), will bepresented. Also the short time Fourier transformationis used for a new characterization of the classical wavefront.Key words and phrases: Wave front, ultradistributions2010 Mathematics Subject Classification: 46F12,46F05,

———

Ellipticity and the Fredholm property in the Weyl-Hormander calculus

Bojan PrangoskiFaculty of Mechanical engineering - Skopje, Karposh 2b.b., 1000 Skopje, MACEDONIAbprangoski@yahoo.com

In this talk we present results concentring the Fredholmproperties and ellipticity of pseudodifferential operatorswith symbols in the Hormander classes S(M, g). Themain result is that the Fredholm property of a ΨDOacting on Sobolev spaces in the Weyl-Hormander cal-culus and the ellipticity are equivalent for geodesicallytemperate Hormander metrics whose associated Planckfunctions vanish at infinity. Additionally, we prove thatwhen the Hormander metric is geodesically temperate,and consequently the calculus is spectrally invariant,the inverse λ 7→ bλ ∈ S(1, g) of every CN , 0 ≤ N ≤ ∞,map λ 7→ aλ ∈ S(1, g) comprised of invertible elementson L2 is again of class CN .

The talk is based on collaborative works with StevanPilipovic

———

51

Abstracts

Aragona Algebras

Walter RodriguesDepartamento de Ciencias Exatas e Naturais, Universi-dade Federal Rural do Semi-arido, Av. Francisco Mota,572, Bairro Costa e Silva, CEP: 59625-900 - Mossoro -RN - BRAZILwalterm@ufersa.edu.br

By introducing Distribution Theory, Schwartz wasamong the first to provide us with a well developedtheory of generalized functions. This is a linear theorywhich does not admit multiplications among the newobjects. Several attempts were made to find a settingwhich permits such multiplications while maintainingthe essence of Schwartz’ theory. This was achieved byColombeau. Contributions to this new theory, made bymany prominent researchers such as J. Aragona, H. Bia-gioni, M. Oberguggenberger, M. Kunzinger, E. Rosingerand Pilipovic among others, permitted that it becamea well established theory. Algebraic and topological as-pects of the theory was developed by many other re-searchers. The rings in which these new generalizedfunctions take values are not fields and so the search foralgebras of generalized functions having point values ina field became an interesting question. Examples in thisdirection were given by Vernaev and Todorov. These al-gebras are more in the direction of nonstandard analysiswhich is not such a surprise since Colombeau’s algebrashave a non-standard flavor. The starting point of thispaper is the part of article from Khelif and Scarpale-zos. There the similarity of some maximal ideals in G(Ω)with maximal ideals in rings of continuous functions isnoted. We develop all the machinery needed to carry outthe details of the observation made in that paragraph.This is done in the setting of the full algebras of gen-eralized functions. Along the way, we show that mostmodels of algebras of generalized functions are quotientsof Colombeau algebras. We name the arising algebras,Aragona algebras, in honor of Jorge Aragona. The re-sults obtained are then used to study idempotents andideals in the full algebra. While preparing the final draftof this paper we received the manuscript of Khelif andScarpalezos with results in the same direction as oursfor the simplified algebra.

This is joint work with Juriaans, O. S., Garcia,A. R. G., Silva, J. C. and is in honor of Professor Al-fredo Jorge Aragona Vallejo.

———

Stochastic Fourier Integral Operators

Martin SchwarzTechnikerstraße 13, Arbeitsbereich fur Technische Math-ematik, 6020 Tirol, AUSTRIAmartin.schwarz@uibk.ac.at

In this talk we will define Fourier integral operators(FIOs) with stochastic phase functions and stochasticamplitude functions. In general, one cannot easily provemeasurability of stochastic FIOs. However, if one makesa small restriction the the phase function, one can showthat the FIO depends continuously on the phase andamplitude function. This means that it is measurable.Finally, we will present some examples of stochasticFourier integral operators coming from stochastic hy-perbolic differential equations.

———

Some results on the zero set of Generalized Holomor-phic functions

Jailson Calado SilvaUniversity of Sao Paula (USP) / Federal Universityof Maranhao (UFMA), Rua do Matao Butanta / RuaJose Leao, 484, Centro, Cidade Balsas, MA, 05508090,BRAZILjailson.calado@ufma.br

A class of generalized functions was introduced byColombeau. These are quotient of differentiable netssatisfying a growth condition induced by the parameterset. Here we are interested in those generalized func-tions which are nets of holomorphic functions. We takethis as a definition but in fact it is a theorem.It is well known that a holomorphic function of one vari-able can not have an accumulation point of zeros or polesin its domain. This is know as the Identity Theorem.This is no longer true in the context of generalized func-tions as the following simple example shows: let e2=eand f(z) = ez. Then zn = (1 − e)αn, α = [ε −→ ε] is asequence of zeros of f converging to 0.Colombeau and Gale proved that if f is a generalizedholomorphic function defined on Ω is zero on a subdo-main of Ω than f is identically zero on Ω. These resultswere generalized by Khelif and Scarpalezos and they alsogive an example of a generalized function whose zero setis a dense Gδ set.In this paper we consider sets one uniqueness E of theunit circle, i.e., the boundary ∂D, of the unit disc. Bythis we mean that if a function f which is continuous onthe closure D of D and holomorphic in D is zero on Ethen f is identically zero. We prove the following

Theorem. Let f be a generalized function defined on Dand holomorphic in D and E ⊂ ∂D a set of uniqueness.If fE = 0 as a generalized function then f ≡ 0.

This is joint work with Rodrigues, W. M., Juriaans, O.S. and Garcia, A.R.G and is in honor of Professor Al-fredo Jorge Aragona Vallejo.

———

Hyperseries of Colombeau generalized numbers

Diksha TiwariUniversity of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, AUSTRIAdiksha.tiwari@univie.ac.at

This talk is the natural succeeding step after the studyof hypernatural numbers and hyperlimits (see talk ofA. Mukhammadiev Supremum, infimum and hyperlim-its of Colombeau generalized numbers).

Since the ring ρR of Robinson-Colombeau is non-Archimedean, a classical series

∑+∞n=0 an of generalized

numbers an ∈ ρR is convergent if and only if an → 0in the sharp topology. Therefore, this property does notpermit us to generalize several classical results, mainlyin the study of analytic generalized functions and inthe study of sigma-additivity in integration of general-ized functions. We first see that the reduction of thenotion of convergent hyperseries to that of the hyper-limit of N ∈ σN 7→

∑Nn=0 an ∈

ρR is possible only

if (an)n ∈(RN×I)

ρ/ ∼ρ=: ρRs, where for (anε)n,ε,

52

Abstracts

(anε)n,ε ∈ RN×I , we have (anε)n,ε ∈(RN×I)

ρif and

only if

∃Q ∈ N ∀0ε ∀n ∈ N : |anε| ≤ ρ−Qε ,

and (anε)n,ε ∼ρ (anε)n,ε if and ony if

∀q ∈ N ∀0ε ∀n ∈ N : |anε − anε| ≤ ρqε.

We therefore need to ask for uniform moderateness andnegligibility properties with respect to n ∈ N. Us-ing these notions, we recover classical examples suchas ρ∑

n∈σ N kn = 1

1−k for all k ∈ ρR, 0 < k < 1;ρ∑

n∈ρNxn

n!= ex for all x ∈ ρR finite, and conver-

gence of ρ∑n∈ρN

1np

if p > 1. Also classical resultscan be proved, such as: algebraic operations on seriesand generalization of classical results for Cauchy prod-uct; an → 0 (hyperlimit) if the series converges; rela-tions with inequalities, absolute convergence, adding orremoving an hyperfinite number of terms to/from anhyperseries and all the classical convergence tests. Wefinally briefly discuss about Cesaro summability and theformalization of classical divergent series using the hy-perfinite sum

∑Nn=0 an, where N ∈ ρN is an infinite

hypernatural number.This is a joint work with P. Giordano (Wolfgang PauliInstitute, Vienna).

———

Axiomatic Approach to Colombeau Theory

Todor D. TodorovMathematics Department, Califonia Polytechnic StateUniversity, 1 Grand Ave., 93407 San Luis Obispo, CAUSAttodorov@calpoly.edu

We present a differential algebra of generalized functionsover a field of generalized scalars by means of severalaxioms in terms of general algebra and topology. Ourdifferential algebra is of Colombeau type in the sensethat it contains a copy of the space of Schwartz dis-tributions, and the set of regular distributions with C∞-kernels forms a differential subalgebra. We discuss theuniqueness of the field of scalars as well as the con-sistency of our axioms. The talk is directed mostlyfor mathematicains who are not necessarily experts inColombeau theory.Key words: Algebraically closed field, real closed field,non-Archimedean ordered field, infinitesimals, saturatedfield, Cantor-complete field, valuation field, Schwartzdistributions, Colombeau algebra.

———

f-Frequently hypercyclic C0-semigroups

Daniel VelinovSs. Cyril and Methodius University, Faculty of CivilEngineering, Partizanski Odredi 24, P.O. box 560, 1000Skopje, MACEDONIAvelinov.daniel@gmail.com

The main subject of the talk is f -frequently hypercyclicand q-frequently hypercyclic (q ≥ 1) C0-semigroupsdefined on complex sectors in the setting of separa-ble Frechet spaces. Results on generalized frequentlyhypercyclic translation semigroups and generalized fre-quently hypercyclic semigroups induced by semiflows on

weighted spaces will be presented. Several illustrativerelated applications will be given.

———

Harmonic representations of generalized functions

Jasson VindasDepartment of Mathematics: Analysis, Logic and Dis-crete Mathematics, Ghent University, Krijgslaan 281, B9000 Gent, BELGIUMjasson.vindas@ugent.be

The purpose of this talk is to discuss an alternative wayto construct linear generalized functions via the bound-ary values of harmonic functions. The procedure basi-cally works for all classical generalized function sheavesbetween distributions and hyperfunctions, namely, forsheaves of ultradistributions and infrahyperfunctions. Inparticular, our ideas lead to a new elementary proofof Hormander’s support theorem for quasianalytic func-tionals. Several interesting new concepts will be intro-duced, such as that of “almost harmonic extensions” offunctions. The talk is based on collaborative works withAndreas Debrouwere and Ricardo Estrada.

———

Some classes of stochastic evolution equations withWick-type nonlinearities

Milica ZigicUniversity of Novi Sad, Faculty of Sciences, Departmentof Mathematics and Informatics, Novi Sad, SERBIAmilica.zigic@dmi.uns.ac.rs

We study nonlinear stochastic evolution equations withWick-power and Wick-polynomial type nonlinearitiesset in the framework of white noise analysis. In par-ticular, these equations are of the form

ut(t, ω) = Au(t, ω) +

n∑k=0

aku♦k(t, ω) + f(t, ω),

u(0, ω) = u0(ω), ω ∈ Ω,

for t ∈ (0, T ], where u(t, ω) is an X−valued generalizedstochastic process; X is a certain Banach algebra andA corresponds to a densely defined infinitesimal gener-ator of a C0−semigroup. These equations include thestochastic Fujita equation, the stochastic Fisher-KPPequation and the stochastic FitzHugh-Nagumo equa-tion among many others. By implementing the the-ory of C0−semigroups and evolution systems into thechaos expansion theory in infinite dimensional spaces,we prove existence and uniqueness of solutions for thisclass of SPDEs. Additionally, we treat the linear nonau-tonomous case and provide several applications featuredas stochastic reaction-diffusion equations that arise inbiology, medicine and physics.The talk is based on collaborations with T. Levajkovic,S. Pilipovic and D. Selesi.

———

53

Abstracts

S.10 Geometric & Regularity Properties ofSolutions to Elliptic and Parabolic PDEs

OrganisersPierre Bousquet, Lorenzo Brasco,Rolando Magnanini

Scope of the session: In the study of partial differen-tial equations, the quest for a precise description of thequalitative and quantitative features of their solutionshave attracted much attention from the early begin-nings. In recent years, once the classical question ofwell-posedness has been quite understood, research ontopological and/or geometric properties of solutions toparabolic and elliptic PDEs have become more intense.The study of these aspects is tightly connected to that ofRegularity Theory. Indeed, in many situations, one hasto deal with solutions obtained by variational or viscos-ity methods - possibly in degenerate or singular regimes- which have to be intended in a suitable weak sense.Thus, a thorough analysis of the regularity of their so-lutions has to be undertaken in order to handle theirgeometric properties.A list of topics currently and actively investigated inthis field includes, to name a few: positivity; a priori es-timates and sharp constants; critical points of solutions;symmetry and non-symmetry of ground states; rigidityresults for overdetermined boundary value problems andentire solutions; stability of symmetric configurations;geometric properties of level sets of solutions; interplaybetween the curvature of the domain and the geometryof the relevant solutions.The session is intended to put together young and expertresearchers in these topics.

—Abstracts—

Minkowski inequality and nonlinear potential theory - 2

Virginia AgostinianiUniversita degli Studi di Verona - Dipartimento di In-formatica, Strada le Grazie 15, 37134 Verona, ITALYvirginia.agostiniani@univr.it

In this talk, we first recall how some monotonicity formu-las can be derived along the level set flow of the capaci-tary potential associated with a given bounded domainΩ. A careful analysis is required in order to preservethe monotonicity across the singular times, leading inturn to a new quantitative version of the Willmore in-equality. Remarkably, such analysis can be carried outwithout any a priori knowledge of the size of the sin-gular set. Hence, the same order of ideas applies to thep-capacitary potential of Ω, whose critical set, for p 6= 2,is not necessarily negligible. In this context, a gener-alized version of the Minkowski inequality is deduced.Joint works with M. Fogagnolo and L. Mazzieri.

———

Asymptotic regularity for a random walk over ellipsoids

Angel ArroyoUniversity of Jyvaskyla, Jyvaskyla, FINLANDangel.a.arroyo@jyu.fi

In 1979 Krylov and Safonov proved the Harnack inequal-ity for solutions of the elliptic PDE in non-divergenceform TrA(x) ·D2u(x) = 0, where A(·) is a symmetricand uniformly elliptic matrix-valued function with mea-surable coefficients in Ω ⊂ Rn. Then the interior Holderregularity of solutions followed as an immediate conse-quence. On the other hand, it turns out that viscositysolutions to this equation can be asymptotically charac-terized by means of a mean value property over a suit-able family of ellipsoids. This establishes a connectionbetween the PDE and a stochastic process describing arandom walk in which the next step is chosen inside an(space-dependent) ellipsoid. Then the value functionsof this process satisfy a dynamic programming principle(DPP): a functional equation which reads as

uε(x) =1

|Eε(x)|

∫Eε(x)

uε(y) dy,

where Eε(x) stands for an ellipsoid centered at x withsize controlled by ε > 0 and shape determined by A(x).We show that, provided with some uniform bound forthe distortion of the ellipsoids, the solutions of the DPPare (locally) asymptotically Holder continuous, that is

|uε(x)− uε(y)| ≤ C(|x− y|α + εα

)for all x, y ∈ Br ⊂ Ω, where C > 0 and 0 < α < 1 donot depend on ε.Moreover, assuming some uniform bound for the maxi-mum ratio between the eigenvalues of A(x) and lettingε→ 0, this method provides an alternative proof for theinterior Holder regularity of certain viscosity solutions toTrA · D2u = 0. (Joint work with Mikko Parviainen,University of Jyvaskyla).

———

Local regularity for viscosity solutions of quasilinearparabolic equations in nondivergence form

Amal AttouchiDepartment of Mathematics and Statistics, P.O. Box35 (MaD), University of Jyvaskyla, 40014 Jyvaskyla,Keski-Suomi, FINLANDamal.a.attouchi@jyu.fi

In this talk we present some recent results on the reg-ularity for a class of degenerate or singular parabolicequations in nondivergence from. We will see how touse the viscosity theory tools to prove Holder estimatesfo the gradient and provide qualitative properties of theviscosity solutions. Our equations are mainly modeledon the parabolic p-Laplace operator but similar equa-tions can also be included.

———

Short-time asymptotics for solutions of the evolutionarygame theoretic p-laplacian and Pucci operators

Diego BertiInstituto de Ciencias Matematicas, CSIC, UAM, 28049Madrid, SPAINdiego.berti@icmat.es

Consider a domain Ω ⊂ RN with boundary Γ 6= ∅ andu the solution of ut − F

(∇u,∇2u

)= 0 in Ω × (0,∞)

such that u = 1 on Γ at all times and u = 0 in Ω at timet = 0. Here, we study the cases in which the differentialoperator F is either the game-theoretic p-laplacian orthe Pucci operator.

54

Abstracts

We present two kinds of results concerning asymptoticformulas for small values of t. First, we associate appro-priate rescalings of the values of u(x, t) to the distanceof x from the boundary Γ, generalizing what obtainedby S. R. S. Varadhan when F is the Laplace operator.Secondly, we connect the asymptotic behavior of q-means on balls touching the boundary to a suitable func-tion of principal curvatures of Γ at the touching point.These results generalize and extend formulas for the heatcontent, obtained by R. Magnanini and S. Sakaguchi inthe linear case.Asymptotic formulas also hold for ε → 0, if oneconsiders the solution uε of the elliptic equationε2F

(∇uε,∇2uε

)− uε = 0 in Ω such that uε = 1 on

Γ.All results are based on joint work with R. Magnanini(Universita di Firenze).

———

On the pairs of domains that solve a two-phase overde-termined problem of Serrin-type

Lorenzo CavallinaTohoku University, 6-3-09 Aoba, Aramaki-aza Aoba-ku,980-8579 Sendai, Miyagi, JAPANcava@ims.is.tohoku.ac.jp

Let (D,Ω) be a pair of sufficiently smooth, bounded do-mains in the N -dimensional Euclidean space (N ≥ 2).For two given positive constants σc 6= σs, we defineσ = σ(x) as the piece-wise constant function that takesthe value σc in D and σs in Ω \ D respectively. Weconsider the following elliptic overdetermined problemof Serrin-type:

−div(σ∇u) = 1 in Ω, u = 0 on ∂Ω, σs∂nu = c on ∂Ω

Here ∂nu denotes the outward normal derivative of u,and c is a real constant. Notice that the overdeterminedproblem (1) does not admit a solution in general, but issolvable only for determinate pairs (D,Ω). We addressthe basic questions of (local) existence and uniquenessof such pairs and study their geometrical properties.

———

Some recent results in the study of fractional meancurvature flow

Eleonora CintiDipartimento di Matematica, Universita di Bologna,Bologna, ITALYeleonora.cinti5@unibo.it

We study a geometric flow driven by the fractional meancurvature. The notion of fractional mean curvaturearises naturally when performing the first variation ofthe fractional perimeter functional. More precisely, weshow the existence of surfaces which develope neckpinchsingularities in any dimension n ≥ 2. Interestingly,in dimension n = 2 our result gives a counterexampleto Greyson Theorem for the classical mean curvatureflow. We also present a very recent result, in the volumepreserving case, establishing convergence to a sphere.The results have been obtained in collaboration with C.Sinestrari and E. Valdinoci.

———

Regularity and rigidity results for nonlocal minimalgraphs

Matteo CozziUniversity of Bath, Department of Mathematical Sci-ences, BA2 7AY Bath, UKm.cozzi@bath.ac.uk

Nonlocal minimal surfaces are hypersurfaces of Eu-clidean space that minimize the fractional perimeter, ageometric functional introduced in 2010 by Caffarelli,Roquejoffre, and Savin in connection with phase transi-tion problems displaying long-range interactions. In thistalk, I will introduce these objects, describe the mostimportant progresses made so far in their analysis, anddiscuss the most challenging open questions. I will thenfocus on the particular case of nonlocal minimal graphsand present some recent results obtained on their regu-larity and classification in collaboration with X. Cabre(ICREA and UPC, Barcelona), A. Farina (Universite dePicardie, Amiens), and L. Lombardini (UWA, Perth).

———

An optimization problem in thermal insulation

Francesco Della PietraDipartimento di Matematica e Applicazioni”R.Caccioppoli” - Universita degli studi di Napoli Fed-erico II, Via Cintia - Monte S. Angelo, 80126 Napoli,ITALYf.dellapietra@unina.it

Given a Lipschitz domain D ⊂ Rn, and Ω = D + δB,δ > 0, let us consider the miminum problem

Iβ,δ(D) = min

∫Ω\D|∇ϕ|2dx+ β

∫∂Ω

ϕ2dHn−1, ϕ ∈ H1(Ω), ϕ ≥ 1 in D

with β > 0. The minimum u of the functional Iβ,δ(D)satisfies the following boundary value problem:

∆u = 0 in Ω \ D,u = 1 su ∂D,∂u∂ν

+ βu = 0 su ∂Ω.

(0.2)

The equation (0.2) is related to a thermal insulatingproblem of a given domain D surrounded by an insu-lator Σ = Ω \D. The coefficient β > 0 depends on thephysical properties of the insulator and the conditionsoutside of Ω; moreover, the insulator is distributed insuch a way that its thickness is constant. The main aimis to determine the extremals of the functional Iβ,δ(D)among the sets D in some suitable classes of domains.For example, in the planar case we consider the maxi-mization problem

supD⊂R2 : P (D)=k

Iβ,δ(D)

where δ, β and k are fixed positive numbers, and P (D)denotes the perimeter of D. This is joint work withCarlo Nitsch (Universita di Napoli Federico II) andCristina Trombetti (Universita di Napoli Federico II).

———

Strong unique continuation for higher order fractionalSchrodinger equations

Marıa Angeles Garcıa-FerreroMax-Planck-Institut fur Mathematik in den Naturwis-senschaften Inselstras. 22, 04103 Leipzig, Germanymaria.garcia@mis.mpg.de

55

Abstracts

In this talk I will show the strong unique continua-tion property for solutions of higher order fractionalSchrodinger operators, including the case of variable co-efficients and the presence of Hardy type gradient po-tentials. The proof relies on a generalised Caffarelli-Silvestre extension for the higher order fractional Lapla-cian and on Carleman estimates. I will also discuss ap-plications of the unique continuation results in the con-text of nonlocal Calderon problems. This is a joint workwith Angkana Ruland.

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On a maximizing problem for the Moser-Trudinger typeinequality with inhomogeneous constraints

Wadade HidemitsuKanazawa University, Kakuma, 920-1192 Kanazawa,Ishikawa, JAPANwadade@se.kanazawa-u.ac.jp

In this talk, we consider the existence or non-existenceof a maximizer for the Moser-Trudinger type inequalityin RN of the form :

DN,α(a, b) :=

supu∈W1,N (RN ),

‖∇u‖aLN (RN )

+‖u‖bLN (RN )

=1

∫RN

ΦN(α|u|

NN−1

)dx,

where N ≥ 2, a, b > 0, α ∈ (0, αN ], ΦN (t) := et −∑N−2j=0

tj

j!, αN := Nω

1/(N−1)N−1 and ωN−1 denotes the sur-

face area of the unit ball in RN . We show the existenceof the threshold α∗ = α∗(a, b,N) ∈ [0, αN ] such thatDN,α(a, b) is not attained if α ∈ (0, α∗), while is at-tained if α ∈ (α∗, αN ). We also provide the conditionson (a, b) such that the inequality α∗ ∈ (0, αN ) holds.In their proofs, we analyze the behavior of a maximiz-ing sequence of DN,α(a, b) to exclude the vanishing phe-nomenon which causes a non-compactness of the func-tional. This result is a joint work with Prof. NorihisaIkoma in Keio University and Prof. Michinori Ishiwata inOsaka University, which is the sequel of the works in Ishi-wata, Math.Ann.(2011), Ruf, J.Funct.Anal.(2005), Li-Ruf, Indiana Univ.Math.J.(2008) and do O-Sani-Tarsi,Commun.Contemp.Math.(2016).

———

Quasilinear parabolic equations: from Sturm attractorsto Ginzburg-Landau patterns

Phillipo LappicyBRAZILlappicy@hotmail.com

The study of the Ginzburg-Landau equations plays akey role in the dynamics of nonlinear fields. It servesas a normal form for PDEs near the Hopf instability,and is capable of displaying a vast array of patterns.We prove the existence of two types of patterns on thesphere: vortex and spirals solutions. On the other hand,the study of global attractors in the space of vortex solu-tions provides information about asymptotic dynamicsof the system. Indeed, we show that such global attrac-tors consist only of the vortex solutions (as equilibria)and their transition waves (as heteroclinics). Moreover,we construct explicitly the global attractors. In particu-lar, we state necessary and sufficient conditions in order

to determine which pair of vortex solutions admits atransition wave with pinned vortices.

———

Extremals for Morrey’s inequality

Erik Lindgren12636 Hagersten, SWEDENerik.kristian.lindgren@gmail.com

A celebrated result in the theory of Sobolev spaces isMorrey’s inequality, which establishes in particular thatfor a bounded domain Ω ⊂ Rn and p > n, there is c > 0such that

c‖u‖pL∞(Ω) ≤∫

Ω

|Du|pdx, u ∈W 1,p0 (Ω).

Interestingly enough the equality case of this inequalityhas not been thoroughly investigated (unless the under-lying domain is Rn or a ball). I will discuss unique-ness properties of extremals of this inequality. Theseextremals are minimizers of the nonlinear Rayleigh quo-tient

inf

∫Ω|Du|pdx‖u‖pL∞(Ω)

: u ∈W 1,p0 (Ω) \ 0

.

In particular, I will present the result that in convex do-mains, extremals are determined up to a multiplicativefactor. I will also explain why convexity is not necessaryand why stareshapedness is not sufficient for this resultto hold. The talk is based on results obtained with RyanHynd.

———

On Blaschke-Santalo diagrams involving the torsionalrigidity and the first eigenvalue of the Dirichlet Lapla-cian

Ilaria LucardesiUniversite de Lorraine, IECL, B.P. 70239, 54506Vandoeuvre-les-Nancy, FRANCEilaria.lucardesi@univ-lorraine.fr

In this talk we will present some recent results on someBlaschke-Santalo diagrams associated to the shape func-tionals λ1 and T , the first Dirichlet eigenvalue and thetorsional rigidity, respectively, under suitable geometricconstraints. This is a joint work with D. Zucco (Politec-nico di Torino, Italy).

———

The sharp quantitative isocapacitary inequality

Michele MariniSISSA - Scuola Internazionale Superiore di Studi Avan-zati, Via Bonomea, 265, 34136 Trieste, Trieste, ITALYmmarini@sissa.it

The well-known isocapacitary inequality states that ballsminimize the capacity among all sets of the same givenvolume.In the talk we prove a sharp quantitative form of thisclassical result. Namely, we show that the difference be-tween the capacity of a set and that of a ball with thesame volume bounds the square of the Fraenkel asym-metry of the set. This provides a positive answer to aconjecture of Hall, Hayman, and Weitsman.

———

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Minkowski inequality and nonlinear potential theory - 1

Lorenzo MazzieriUniversita degli Studi di Trento - Dipartimento diMatematica, via Sommarive 14, 38123 Povo (TN),ITALYlorenzo.mazzieri@unitn.it

In this talk, we first recall how some monotonicity formu-las can be derived along the level set flow of the capaci-tary potential associated with a given bounded domainΩ. A careful analysis is required in order to preservethe monotonicity across the singular times, leading inturn to a new quantitative version of the Willmore in-equality. Remarkably, such analysis can be carried outwithout any a priori knowledge of the size of the sin-gular set. Hence, the same order of ideas applies to thep-capacitary potential of Ω, whose critical set, for p 6= 2,is not necessarily negligible. In this context, a gener-alized version of the Minkowski inequality is deduced.Joint works with M. Fogagnolo and L. Mazzieri.

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Hyperbolic solutions to Bernoulli’s free boundary prob-lem

Michiaki OnoderaDepartment of Mathematics, Tokyo Institute of Tech-nology, 2-12-1 Ookayama, Meguro-ku, 152-8551 Tokyo,JAPANonodera@math.titech.ac.jp

Bernoulli’s free boundary problem is an overdeterminedproblem in which one seeks an annular domain such thatthe capacitary potential satisfies an extra boundary con-dition. This problem arises as the Euler-Lagrange equa-tion for minimizing capacity among all subsets of equalvolume in a prescribed container. There exist two differ-ent types of solutions: elliptic and hyperbolic solutions.Elliptic solutions are “stable” solutions and tractable byvariational methods and maximum principles, while hy-perbolic solutions are “unstable” solutions of which thequalitative behavior is less known. I will present a recentjoint work with Antoine Henrot in which we show theexistence and qualitative behavior of foliated hyperbolicsolutions by a new flow approach.

———

Regularity for the normalized p-Poisson problem

Eero RuosteenojaDepartment of Mathematical Sciences, NTNU, NO-7491Trondheim, NORWAYeero.k.ruosteenoja@ntnu.no

We discuss the local regularity of the normalized p-Poisson problem

−∆Np u = f in Ω.

The normalized (or game-theoretic) p-Laplacian

∆Np u := |Du|2−p∆pu = ∆u+ (p− 2)∆N

∞u

is in non-divergence form and arises for example fromstochastic games. This is a joint work with Amal At-touchi and Mikko Parviainen.

———

Existence and regularity of Faber-Krahn minimizers ina Riemannian manifold

Pieralberto SicbaldiUniversidad de Granada, Granada, SPAIN / Universited’Aix-Marseille, Marseille, FRANCEpieralberto@ugr.es

We show a recent result about the existence and the reg-ularity of sets whose minimize the first eigenvalue of theLaplace-Beltrami operator under volume constraint.Joint work with J. Lamboley.

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On a thin coating problem and its related inverse prob-lem

Yachimura ToshiakiJAPANyachimura@ims.is.tohoku.ac.jp

In this talk, we consider an inverse problem related to aRobin eigenvalue problem and determine the Robin co-efficients by the knowledge of the first eigenvalue and themeasurement on some part of the boundary. This prob-lem is closely related to the problem of thin insulatorcoating for a heat conductor. We prove the uniquenessof the inverse problem and establish the identification byusing a Neumann tracking type functional. This is jointwork with Matteo Santacesaria (University of Genoa).

———

A Quantitative Weinstock Inequality

Leonardo TraniUniversita degli Studi di Napoli Federico II, Diparti-mento di Matematica e Applicazioni ”R. Caccioppoli”,Complesso Universitario di Monte S. Angelo, Via Cintia45, 80126 Napoli, ITALYleonardo.trani@unina.it

The aim of the talk is to present a result, obtained ina joint work with N. Gavitone, D. A. La Manna andG. Paoli, about a quantitative version of the Weinstockinequality in Rn with n ≥ 2 for the first non trivialSteklov-Laplacian eigenvalue in the class of convex sets.The key role is played by a quantitative isoperimetricinequality which involves the boundary momentum, thevolume and the perimeter of a convex set.

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On the logarithmic epiperimetric inequality

Bozhidar VelichkovUniversita degli Studi di Napoli Federico II, Naples,ITALYbozhidar.velichkov@gmail.com

This talk will be dedicated to the logarithmic epiperi-metric inequality, which is a new tool for the regularityof the free boundaries and the analysis of the singular-ities arising in the context of variational free boundaryproblems. In this talk we will consider the case of theclassical obstacle problem and we will prove that thesingular part of the free boundary is contained into theunion of C1,log-regular manifolds.

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S.11 Geometries Defined by DifferentialForms

OrganisersMahir Bilen Can, Sergey Grigorian, Sema Salur

Scope of the session: The subject of geometric struc-tures on Riemannian manifolds is a very important, andintensely studied, subject in Differential Geometry. Inparticular, geometric structures such as contact geome-try, symplectic geometry, calibrated geometry, and spe-cial holonomy geometries, are all defined by differen-tial forms, and turn out to be very closely related toone another. Ever since the foundational work of Har-vey and Lawson in 1982, the relationship between cali-brated geometry and special holonomy has been of fun-damental importance in Differential Geometry, as anyG−invariant p−form on Rn induces a calibration on aconnected Riemannian manifold (Mn, g) with holonomyG ⊂ O(n). Moreover, recent directions in research showthat there are manifolds that lie at the intersection ofcontact geometry and special holonomy, in particularG2 and Spin(7) holonomy manifolds, and that manyideas from both contact geometry and symplectic geom-etry have analogues to manifolds with special holonomy.Many of these objects have complexifications which areamenable to algebraic geometric techniques. In turn,real algebraic analogues provide a new frontier for re-search.The cross-pollination of ideas from different areas of ge-ometry takes roots in papers such as those of Fernandezand Gray from the 1960’s wherein manifolds with specialholonomy are treated as analogues of Kahler manifolds;moreover, because of the introduction in recent years ofsuch powerful tools in symplectic and contact geome-tries such as Floer homology theories and its relatives,it is more important than ever to understand the con-nections between these fields as this will yield deeperresults about manifolds with special holonomy and cal-ibrated geometries than have been possible up to thispoint.Our goal is to bring together a group of both junior andsenior mathematicians who are experts in these differ-ent geometric structures in order to create a venue wherethey can interact and exchange ideas and developments,and to further the understanding of the relationshipsbetween these geometries, as well as the implicationsin gauge theory, (real) algebraic geometry, geometricPDE’s, and mathematical physics.

—Abstracts—

The Asymptotic Nilpotent Variety of G2

Mahir Bilen CanTulane University, 70118 New Orleans, LA, USAmahirbilencan@gmail.com

The asymptotic semigroup of a complex semisimple Liegroup H is the algebraic variety that is obtained from Hby taking its horospherical contraction. This semigroup,which is originally introduced by Vinberg, is closely re-lated to the canonical compactification of the adjointform of H. In this talk, by building on the works ofPutcha, Renner, Rittatore, and Therkelsen, we will in-troduce and study the nilpotent variety associated with

the asymptotic semigroup of H. We will demonstrate ourresults on the exceptional Lie group H:=G2.

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A compact G2-calibrated manifold with first Betti num-ber b1 = 1.

Alexei KovalevDPMMS, University of Cambridge, Centre for Mathe-matical Sciences, Wilberforce Road, CB3 0WB Cam-bridge, UKA.G.Kovalev@dpmms.cam.ac.uk

We construct a compact formal 7-manifold with a closedG2-structure and with first Betti number b1 = 1, whichdoes not admit any torsion-free G2-structure, that is, itdoes not admit any G2-structure such that the holon-omy group of the associated metric is a subgroup ofG2. We also construct associative calibrated (hencevolume-minimizing) 3-tori with respect to this closedG2-structure and, for each of those 3-tori, we show a3-dimensional family of non-trivial associative deforma-tions. We also construct a fibration of our 7-manifoldover (S2×S1) with generic fibre a (non-calibrated) coas-sociative 4-torus and some singular fibers. Joint workwith Marisa Fernandez, Anna Fino and Vicente Munoz.

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Ancient solutions in Lagrangian mean curvature flow

Jason LotayUniversity of Oxford, Woodstock Road, OX26GG Ox-ford, England, UKjason.lotay@maths.ox.ac.uk

Motivated by mirror symmetry, the Thomas-Yau con-jecture states that the convergence of almost calibratedLagrangian mean curvature flow (LMCF) to a specialLagrangian is governed by a “stability condition”, asone has for Hermitian-Yang-Mills flow. Singularities ofalmost calibrated LMCF are necessarily Type II, andso modelled by ancient solutions to the flow. I will de-scribe classification results for almost calibrated ancientsolutions to LMCF, which give us new understandingof singularity formation for the flow. This is joint workwith Ben Lambert and Felix Schulze.

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Generalizing holomorphic bundles to almost complex 6-manifolds

Goncalo OliveiraUniversidade Federal Fluminense, Niteroi, Rio deJaneiro, BRAZILgalato97@gmail.com

(Joint work with Gavin Ball) The notion of a holomor-phic bundle and Hermitian-Yang-Mills connections isone that proved to be very fruitful in complex geometry.There are some natural generalizations of these notionsin almost complex geometry such as those of pseudo-holomorphic and pseudo-Hermitian-Yang-Mills connec-tions. In this talk, I will focus on a system of partial dif-ferential equations, the DT-instanton equations, whosesolutions give a further generalization of the notion of aHermitian-Yang-Mills connection in the setting of real 6dimensional almost Hermitian manifolds.

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Extremal length in higher dimensions and complex sys-tolic inequalities

Tommaso PaciniUniversity of Torino, ITALYtommaso.pacini@unito.it

Extremal length is a classical tool in 1-dimensional com-plex analysis for building conformal invariants. We pro-pose a higher-dimensional generalization for complexmanifolds defined in terms of complex volume forms, andprovide some ideas on how to estimate and calculate it.We also show how to formulate natural geometric in-equalities in this context in terms of a complex analogueof the classical Riemannian notion of systole.

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Generalized geometry in physics and mechanics

Vladimir SalnikovCNRS / La Rochelle University, LaSIE, Avenue MichelCrepeau, 17042 La Rochelle, FRANCEvladimir.salnikov@univ-lr.fr

In this talk I will present various instances of the gen-eralized geometry, appearing naturally in some fields ofcontemporary physics and mechanics. From the math-ematical perspective, I will mostly discuss Dirac struc-tures - those generalize simultaneously Poisson, symplec-tic geometry and more. I will also recall the descriptionof them in terms of graded geometry and differentialgraded manifolds. From the “applied” point of view, onthe one hand, it turns out to be a convenient languagefor theoretical physics [1-3]. It permits to study somesigma models, define the gauging procedure in terms ofequivariant cohomology, and relate the algebra of sym-metries of the respective functionals to multi-symplecticgeometry. On the other hand, for mechanics one canalso spell-out the Dirac structures in the context of dis-sipative and coupled systems [4]. Time permitting, I willalso explain that those can be important for structurepreserving integrators.References: [1] V.Salnikov, T.Strobl, Dirac Sigma Mod-els from Gauging, Journal of High Energy Physics,11/2013 ; 2013(11) [2] A.Kotov, V.Salnikov, T.Strobl,2d Gauge Theories and Generalized Geometry, Journalof High Energy Physics, 2014:21, 2014 [3] V.Salnikov,Graded geometry in gauge theories and beyond, Jour-nal of Geometry and Physics, Volume 87, 2015 [4]V.Salnikov, A.Hamdouni, From modelling of systemswith constraints to generalized geometry and back tonumerics, Z Angew Math Mech. 2019

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Almost formality of closed G2- and Spin(7) manifolds

Hong Van LeInstitute of Mathematics, Zitna 25, 14000 Praha,CZECH REPUBLIChvle@math.cas.cz

We introduce the notion of Poincare DGCAs of Hodgetype, which is a subclass of Poincare DGCAs encom-passing the de Rham algebras of closed orientable mani-folds. We show that a (r−1) connected Poincare DGCAof Hodge type A∗ of dimension n ≤ 5r− 3 is A∞-quasi-isomorphic to an A3-algebra and prove that the only ob-struction to the formality of A∗ is a distinguished Harri-son cohomology class [µ3] ∈ Harr3,−1(H∗(A∗), H∗(A∗)).

Moreover, the cohomology class [µ3] and the DGCAisomorphism class of H∗(A∗) determine the A∞-quasi-isomorphism class of A∗. This can be seen as a Harrisoncohomology version of the Crowley-Nordstrom resultson rational homotopy type of (r − 1)-connected (r > 1)closed manifolds of dimension up to 5r− 3. We also de-rive the almost formality of closed G2-manifolds, whichhave been discovered recently by Chan-Karigiannis-Tsang from our results and the Cheeger-Gromoll split-ting theorem. This is a joint work with DomenicoFiorenza, Kotaro Kawai, and Lorenz Schwachhofer inarXiv:1902.08406.

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On a sub-Riemannian space associated to a Lorentzianmanifold

Dimiter VassilevUniversity of New Mexico, Department of Mathematicsand Statistics MSC01 1115, 87131 Albuquerque, NM,USAvassilev@unm.edu

I will present a certain construction of a sub-Riemannianspace naturally associated to a Lorentzian manifold.Some additional structures and relations between ge-ometric properties of the corresponding spaces will beexplored.

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Geometric flows of closed forms

Luigi VezzoniDipartimento di Matematica ”G. Peano”, via Carlo Al-berto 10, 10124 Torino, ITALYluigi.vezzoni@unito.it

The talk focuses on geometric flows of closed forms onsmooth manifolds. This class of flows includes some in-teresting flows in literature, as the Laplacian flow in G2-geometry. In the talk it will be described a general cri-terium to establish if a flow of closed forms is well-posedand it will be discussed the stability around static solu-tions.The talk is mainly based on the following papers writtenin collaboration with Lucio Bedulli[1] L. Bedulli, L. Vezzoni, A parabolic flow of balancedmetrics, J. Reine Angew. Math. 723 (2017), 79-99.[2] L. Bedulli, L. Vezzoni, Stability of geometric flows ofclosed forms, arXiv:1811.09416.

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Classifying Spaces of Diffeological Groups

Jordan WattsCentral Michigan University, Department of Mathemat-ics, 214 Pearce Hall, 48859 Mount Pleasant, Michigan,USAjordan.watts@cmich.edu

Fix an irrational number A, and consider the action ofthe group of pairs of integers on the real line defined asfollows: the pair (m,n) sends a point x to x+m+ nA.Since the orbits of this action are dense, the quotienttopology on the orbit space is trivial, and continuousreal-valued functions are constant. Can we give thespace any type of useful ”smooth” group structure?The answer is ”yes”: its natural diffeological group

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structure. It turns out this group is not just some patho-logical example, but has many interesting associatedstructures, and is of interest to many areas of mathemat-ics. In particular, it shows up in geometric quantisationand the integration of certain Lie algebroids as the struc-ture group of certain principal bundles, the main topicof this talk.We will perform Milnor’s construction in the realm ofdiffeology to obtain a diffeological classifying space fora diffeological group G, such as the irrational torus. Af-ter mentioning a few hoped-for properties, we then con-struct a connection 1-form on the G-bundle EG→ BG,which will naturally pull back to a connection 1-formon sufficiently nice principal G-bundles. We then lookat what this can tell us about irrational torus bundles,their connection 1-forms, and corresponding curvatureforms. [Joint work with Jean-Pierre Magnot.]

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Contact Structures on G2 Manifolds

Emily Windes500 Joseph C. Wilson, 14627 Rochester, NY, USAewindes@u.rochester.edu

In 2011, Arikan, Cho, and Salur introduced the con-cept of A− and B−compatible contact structures on G2

manifolds. I will explain the relationship between sym-plectic, (almost) contact, and G2 structures on mani-folds with G2 structures from this new perspective. Iwill give some examples of compatible structures andexplain some new results about these relationships.

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S.12 Harmonic Analysis and PartialDifferential Equations

OrganisersVladimir Georgiev, Tohru Ozawa,Michael Ruzhansky, Jens Wirth

Scope of the session: The session will focus on the in-terplay between harmonic analysis and the theory ofpartial differential equations, highlight recent advancesand bring together experts in both fields. Topics cov-ered in the session will be sharp constants in variationalinequalities, relations to solvability/stability of (non-linear) partial differential equations, dispersive proper-ties and scattering of equations mathematical physics,symmetry based concepts, aspects of harmonic analysison Lie groups and related operator quantisations.

—Abstracts—

On Kohn-Nirenberg symbols of operators on theHeisenberg group

Jonas BrinkerInstitut fur Analysis, Dynamik und Modellierung - Uni-versitat Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart,Baden-Wurttemberg, GERMANYjonas.brinker@mathematik.uni-stuttgart.de

We examine Kohn-Nirenberg symbols in relation totheir operators on the Heisenberg group. Our goal is

to gain criterias for an operator, such that its sym-bol is uniformly bounded. Estimates of this form areof importance to Beals-Cordes type characterizationsof certain algebras of pseudodifferential operators, de-fined by Michael Ruzhansky and Vernonique Fischer.The used methods involve a Schwartz-kernel representa-tion of both the Kohn-Nirenberg quantization and theFourier transform on the Heisenberg group with respectto alternative spaces of test functions. These test func-tions are either defined to be usual Schwartz functions,which are orthogonal to all polynomials in the centercoordinates of the Heisenberg group, or defined to beoperator valued Schwartz functions vanishing rapidly inzero.Also we find, analogously to the compact group case, arepresentation of the Kohn-Nirenberg symbols with thehelp of Fourier multipliers given by vector valued, slowlyincreasing functions.

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Focusing energy-critical inhomogeneous NLS

Yonggeun ChoDepartment of Mathematics, Chonbuk National Univer-sity, 567 Baekjedaero, 54886 Duckjingu, Jeolkabukdo,REPUBLIC OF KOREAchangocho@jbnu.ac.kr

In this talk we consider the global well-poseness of fo-cusing energy-critical inhomogeneous NLS. Based on theKenig-Merle method we find a sufficient condition on theinitial data associated with inhomogeneous term. Thisgives us GWP and scattering in H1. For this purpose wedevelope long time perturbation, profile decomposition,and rigidity theorem.

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Critical well-posedness for the modified Korteweg-deVries equation and self-similar dynamics

Simao CorreiaFaculdade de Ciencias da Universidade de Lisboa,Edifıcio C6, Campo Grande, 1749-016 Lisboa, PORTU-GALsfcorreia@fc.ul.pt

We consider the modified Korteweg-de Vries equationover the real line

ut + uxxx = ±(u3)x.

This equation arises, for example, in the theory of waterwaves and vortex filaments in fluid dynamics. A partic-ular class of solutions to (mKdV) are those which do notchange under scaling transformations, the so-called self-similar solutions. Self-similar solutions blow-up whent → 0 and determine the asymptotic behaviour of theevolution problem at t = +∞.The known local well-posedness results for the (mKdV)fail when one considers critical spaces, where the normis scaling-invariant. This also means that self-similarsolutions lie outside of the scope of these results. Con-sequently, the dynamics of (mKdV) around self-similarsolutions are currently unknown.In this talk, we will show existence and uniqueness ofsolutions to the (mKdV) lying on a critical space whichincludes both regular and self-similar solutions. After-wards, we present several results regarding global exis-tence, asymptotic behaviour at t = +∞ and blow-upphenomena at t = 0.

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Old and New Trends in Hausdorff Operator

Radouan DaherLaboratory T.A.G.M.D, Departement of Mathematics,Faculty of Sciences Aın Chock University Hassan II,Casablanca, MOROCCOrjdaher024@gmail.com

Liflyand and Moricz studied the Hausdorff operator onthe real Hardy space and proved the boundedness prop-erty and the commuting relations of this operator andHilbert transformations. The main objective of this pa-per is extending these results to the two contexts ofDunkl theory and the Jacobi hypergroup. (This talk isbased on the joint work with T. Kawazoe and F. Saadi)

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Global almost radial solutions to supercritical dispersiveequations outside the unit ball

Piero D’AnconaDipartimento di Matematica - Sapienza, Universita diRoma - P.Moro 2 - 00185 Roma (Italy)dancona@mat.uniroma1.it

I consider a defocusing semilinear wave equation withpower nonlinearity, on the exterior of the unit ball, withDirichlet boundary conditions, in dimension n ≥ 3. Forany radial initial data, the solution is global and de-cays as t−1. I prove that the radial solutions are stablefor small perturbations of the initial data in a weightedSobolev norm of order O(n) of Christodoulou type, andfor sufficiently high powers p >O(n). This produces afanily of global large solutions to the supercritical waveequation on the exterior of the ball, with arbitrarily highpower nonlinearity. The almost radial solutions thusconstructed are unique among energy class solutionswhich satisfy an energy inequality, by a weak-strongstability result of Struwe type. Similar results hold forthe defocusing supercritical nonlinear Schrodinger equa-tion.

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Remark on blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients

Kazumasa FujiwaraMathematical Institute, Tohoku University, 6-3 Aoba,Aramaki, Aoba-ku, Sendai, Miyagi, 980-8578 JAPANk-fujiwara@asagi.waseda.jp

In this talk, we study the blow-up for solutions to thehalf Ginzburg-Landau-Kuramoto equation with an odeargument The key tools in the proof are appropriatecommutator estimates and the essential self-adjointnessof the symmetric uniformly elliptic operator with roughmetric.

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Uniqueness and nondegeneracy of ground states fornonlinear Schrodinger equations with attractive inverse-power potential

Noriyoshi FukayaTokyo University of Science, 1-3 Kagurazaka, 162-8601Shinjuku-ku, Tokyo, JAPANfukaya@rs.tus.ac.jp

We consider the uniqueness and nondegeneracy ofground states for stationary nonlinear Schrodinger equa-tions with a focusing power-type nonlinearity and an at-tractive inverse-power potential. In this talk, we provethat all ground states are positive up to phase rota-tion, radial, and decreasing. Moreover, by extendingthe results of Shioji and Watanabe (2016), we prove theuniqueness and nondegeneracy of the positive radial so-lutions.

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Uniqueness of standing-waves solutions to the non-linear Schroedinger equations for combined power-typenon-linearities

Daniele GarrisiUniversity of Nottingham-Ningbo, 199 Taikang EastRoad, 315100 Ningbo, Zheijang, PEOPLE’S REPUB-LIC OF CHINAdaniele.garrisi@nottingham.edu.cn

We illustrate recent results of orbital stability for non-linear dispersive equations. One of the problems we willbe focusing on is the stability of standing-wave solutionsto the non-linear Schrodinger equation

i∂tϕ(t, x) + ∆xϕ(t, x)−G(ϕ(t, x)) = 0.

The stability can be deduced from properties of the setof minima of the functional

E(u) =1

2

∫Rn|∇u(x)|2dx+

∫RnG(u(x))dx

on the constraint S = u ∈ H1r,+(Rn) | ‖u‖2L2 = 1.

The functions H1r,+(Rn) are radially symmetric (with

respect to the origin), H1 and positive. Orbital stabilityholds when there are finitely many minima. It is knownfrom (M. K. Kwong, ARMA 1989) and (H. Berestyckiand P. L. Lions, ARMA, 1983), that if G(s) = −a|s|p,there is exactly one minimum. In this talk we present anextension of this uniqueness result to combined power-type non-linearities, G(s) = −a|s|p + b|s|q and spatialdimension n = 1. We also provide a condition on Gwhich guarantees that minima of the functional E on Sare non-degenerate. As a consequence, there are onlyfinitely many of them.

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Scattering solutions for the focusing nonlinearSchrodinger equation with a potential

Masaru HamanoSaitama University, Shimo-Okubo 255, Sakura-ku,Saitama city, Saitama 338-8570, JAPANm.hamano.733@ms.saitama-u.ac.jp

This talk is based on a joint work with Masahiro Ikedafrom RIKEN AIP, Japan. We consider the nonlinearSchrodinger equation with a linear potential term:

(NLSV )

i∂tu+ ∆u− V u = −|u|p−1u, (t, x) ∈ R×R3,

u(0, x) = u0(x) ∈ H1(R3),

where V = V (x) is a real-valued potential. We deter-mine the conditions for the time behaviors of a solu-tion to (NLSV ), scattering, blowing-up, and growing-up.The conditions are expressed in terms of some quanti-ties which consist of mass and energy. Sharp criteria

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are given by these quantities of the ground state of thestandard NLS, without the potential. If we state moreprecisely, we give a sufficient condition of the potentialand initial data for scattering solutions, and this is amain topic in this talk. Also, we introduce sufficientconditions of the potential and initial data such thatblowing-up or growing-up holds at least. Moreover if weadd some conditions, we can prove that the solutionsmust blow-up.

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Probabilistic well-posedness of the mass-critical NLSwith radial data below Rd

Gyeongha HwangDepartment of Mathematics, Yeungnam University, 280Daehak-Ro, 38541 Gyeongsan-si, Gyeongsangbuk-do,KOREAghhwang@yu.ac.kr

In this talk, we consider the Cauchy problem of themass-critical nonlinear Schrodinger equation (NLS) withradial data below L2(Rd). We prove almost sure localwell-posedness along with small data global existenceand scattering. Furthermore, we also derive conditionalalmost sure global well-posedness of the defocusing NLSunder the assumption of a probabilistic a priori energybound. The main ingredient is to establish the proba-bilistic radial Strichartz estimates.

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Test function method for blow-up phenomena of semi-linear wave equations and their weakly coupled systems

Masahiro Ikedayour postal addressmasahiro.ikeda@riken.jp

In this talk we consider the wave equations with powertype nonlinearities including time-derivatives of un-known functions and their weakly coupled systems. Wepropose a framework of test function method and givea simple proof of the derivation of sharp upper boundof lifespan of solutions to nonlinear wave equations andtheir systems. We point out that for respective criti-cal case, we use a family of self-similar solution to thestandard wave equation including Gauss’s hypergeomet-ric functions which are originally introduced by Zhou(1992). However, our framework is much simpler thanthat. As a consequence, we found new (p, q)-curve forthe system ∂2

t u − ∆u = |v|q, ∂2t v − ∆v = |∂tu|p and

lifespan estimate for small solutions for new region.

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The Strichartz estimates for the damped wave equationand its application to a nonlinear problem

Takahisa InuiDepartment of Mathematics, Graduate School of Sci-ence, Osaka University, Machikaneyamacho 1-1, 560-0043 Toyonaka, Osaka, JAPANinui@math.sci.osaka-u.ac.jp

We give the Strichartz estimates for the damped waveequation ∂2

t φ−∆φ+∂tφ = 0. Moreover, we apply themto a nonlinear damped wave equation.

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Blow-up and lifespan estimate to a nonlinear waveequation in Schwarzschild spacetime

Ningan LaiDepartment of Mathematics, Lishui University, 323000Lishui, Zhejiang, PEOPLE’S REPUBLIC OF CHINAhyayue@gmail.com

Luk (2013, Journal Eur. Math. Soc.) proved global ex-istence for semilinear wave equations in Kerr spacetimewith small angular momentum(aM)

gKφ = F (∂φ),

when the quadratic nonlinear term satisfies the null con-dition. In this work, we will show that if the null con-dition does not hold, at most we can have almost globalexistence for semilinear wave equations with quadraticnonlinear term in Schwarzschild spacetime, which is thespecial case of Kerr with a = 0

gSφ = (∂tφ)2,

where gS denotes the D’Alembert operator associ-ated with Schwarzschild metric. What is more, if thepower of the nonlinear term is replaced with p satisfying3/2 ≤ p < 2, we still can show blow-up, no matter howsmall the initial data are. We do not have to assumethat the support of the data should be far away fromthe event horizon.

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On the scattering problem for the nonlinear Schrodingerequation with a potential in 2D

Chunhua LiYanbian University, 133002 Yanji City, Jilin, PEOPLE’SREPUBLIC OF CHINAsxlch@ybu.edu.cn

This is a joint work with Vladimir Georgiev. Weconsider the scattering problem for the nonlinearSchrodinger equation with a potential in 2D. Appropri-ate resolvent estimates are proved and applied to esti-mate the operator A(s) appearing in commutator rela-

tions. The equivalence between the operators (−∆V )s2

and (−∆)s2 in L2 norm sense for 0 ≤ s < 1 is inves-

tigated by using free resolvent estimates and Gaussianestimates for the heat kernel of the Schrodinger operator−∆V . Our main result guarantees the global existenceof solutions and time decay of the solutions assuminginitial data have small weighted Sobolev norms. More-over, the global solutions obtained in the main resultscatter.

———

Asymptotic decay results for critical subelliptic equa-tions

Annunziata LoiudiceDepartment of Mathematics, University of Bari, ViaOrabona, 4, 70125 Bari, ITALYannunziata.loiudice@uniba.it

In this talk we shall present some recent results on theasymptotic behavior of finite energy solutions to criti-cal growth equations in the subelliptic setting. We dealwith the semilinear case with Hardy perturbation and

62

Abstracts

the quasilinear case. The involved operators are sub-Laplacians on Carnot groups and Grushin-type opera-tors. As a remarkable consequence of our analysis, weobtain the exact rate of decay of the extremal functionsfor some relevant functional inequalities on Stratified Liegroups, whose analytic expression is not known.

———

Energy estimate for wave equation with bounded time-dependent coefficient

Tokio MatsuyamaChuo University, 1-13-27, Kasuga, 112-8551 Bunkyo-ku,Tokyo, JAPANtokio@math.chuo-u.ac.jp

In this talk we present a good energy estimate for waveequation with time-dependent coefficient. To this end,we employ asymptotic integrations method, and investi-gate oscillation property of amplitude functions in rep-resentation formula of solutions. This talk is based onthe joint work with Michael Ruzhansky (Ghent Univ.and Queen Mary Univ.).

———

Strong instability for standing wave solutions to thesystem of the quadratic NLKG

Hayato MiyazakiNational Institute of Technology, Tsuyama College, 624-1 Numa, 708-0824 Tsuyama, Okayama, JAPANmiyazaki@tsuyama.kosen-ac.jp

We consider the instability for standing wave solutionsto the system of the quadratic nonlinear Klein-Gordonequations. In the single case, namely the nonlinearKlein-Gordon equation with power type nonlinearity,stability and instability for standing wave solutions havebeen extensively studied. On the other hand, in the caseof our system, there would be no result concerning thestability and instability for the standing wave as far aswe know. In this talk, we give a strong instability resultfor the standing wave to our system. The proof is basedon the techniques in Ohta and Todorova (2007). Newingredient is to need the mass resonance condition intwo or three space dimensions whose cases are the masssub-critical case.

———

Spectral theory for magnetic Schrodinger

Kiyoshi Mochizuki1-13-27, Kasuga, Bunkyo, 112-8551 Tokyo, JAPANmochizuk@math.chuo-u.ac.jp

In this talk we present a new approach to growthestimates of generalized eigenfunctions for magneticSchrodinger operators in exterior domain with explodingor oscillating potentials, apply them to show the princi-ple of limiting absorption.

———

Fractional wave equation with discontinuous coeffi-cients

Ljubica OparnicaDepartment of Mathematics: Analysis, Logic and Dis-crete Mathematics, University of Gent, BELGIUM, and

Faculty of Education in Sombor, University of Novi Sad,SERBIAOparnica.Ljubica@UGent.be; ljubica.oparnica@gmail.com

We consider wave propagation in viscoelastic mediahaving non-constant density ρ = ρ(x), obeying Zenerconstitutive law giving relation between stress and strainand with non-constant Young modulus of elasticityE = E(x). If for ρ and E we allow also discontinu-ities, for example jumps modeling media consisting oftwo or more different materials, it would imply partialintegro-differential equations with distributional coeffi-cients. Existence of very week solution for such equationis going to be proved.

Work is done in collaboration with Michael Ruzhansky.

———

A new approach on the Calderon problem for discon-tinuous complex conductivities

Ivan PomboUniversidade de Aveiro, Portugalivanpombo@ua.pt

In this talk we present the inverse conductivity problemfor complex conductivities with jumps. Such materialshave never been considered in the literature where stillthe case of Lipschitz conductivities are assumed. For thestudy of this problem we model it as an interior trans-mission problem. To treat this problem several new con-cepts are required, such as an adaptation of the notion ofscattering data, and the definition of admissible points,which permit the enlargement of the set of CGO incidentwaves. This will allow us to prove the reconstruction ofthe conductivity.Given that this are early results in this direction, wealso present some of the footwork necessary to proceedfurther in this direction.

———

Boundary value problem for the biharmonic operator ina ball

Petar PopivanovInstitute of Mathematics and Informatics, BulgarianAcademy of Sciences, 1113 Sofia, BULGARIAangela.slavova@gmail.com

This talk deals with several boundary value problems(bvp) for the biharmonic operator in the unit ball. Theyare divided into two different classes: (a) bvp satisfyingShapiro-Lopatinskii condition and (b) bvp violating thelatter requirement. In the case (a) existence results ofFredholm type hold, while in the case (b) non-Fredholmresults could appear. A bvp is of non-Fredholm type ifit possesses infinite dimensional kernel or cokernel. Forseveral examples explicit formulas for the correspondingsolutions are given.

———

Local smoothing of Fourier integral operators and Her-mite functions

Peetta Kandy RatnakumarHarish-Chandra Research Institute, Chhatnaag Road,Jhunsi, 211019 Prayagraj, Uttar Pradesh, INDIAratnapk@hri.res.in

63

Abstracts

In this talk, we discuss a local smoothing estimate forFourier integral operators of the form

Ff(x, t) =

∫R2

ei(x·ξ+t|ξ|) a(x, t, ξ) f(ξ) dξ

for a wide class of symbols a ∈ Sm(R2 × R× R2), m ≤ 0.Our result generalises the well known local smoothing es-timate of Mockenhaupt, Seeger and Sogge appeared in1992, to a global result with respect to the space vari-able. We use only a mild decay assumption on the am-plitude function. The novelty in our approach is theuse of Hermite functions in the study of Fourier integraloperators. This is a joint work with Ramesh Manna.

———

Harmonic and Anharmonic Oscillators on the Heisen-berg Group

David RottensteinerUniversity of Vienna, A-1060 Vienna, AUSTRIAdavid.rottensteiner@univie.ac.at

We propose a canonical version of the harmonic oscilla-tor on the Heisenberg group Hn in terms of the represen-tation theory of the Dynin-Folland group Hn,2, a 3-stepstratified Lie group, whose generic representations act onL2(Hn). The classical relation between the harmonic os-cillator on Rn and the sum of squares in the first stratumof the Heisenberg Lie algebra hn has an analog, whichwe turn into a definition: we define the harmonic oscil-lator on Hn as the image of the sub-Laplacian LHn,2

under the generic unitary irreducible representation πof the Dynin-Folland group which has formal dimen-sion dπ = 1. More generally, this approach gives riseto a large class of so-called anharmonic oscillators byemploying positive Rockland operators on Hn,2. Em-ploying methods developed by Elst and Robinson, weobtain spectral estimates for the harmonic and anhar-monic oscillators on Hn. The second part of the talk fo-cuses on useful Lp-Lq-estimates for spectral multipliersof the sub-Laplacian LHn,2 and, more generally, of gen-eral Rockland operators on general graded groups. As aby-product, we recover Sobolev embeddings on gradedgroups, and obtain explicit hypoelliptic heat semigroupestimates. This is joint work with Prof. Michael Ruzhan-sky (Ghent University, Queen Mary University of Lon-don).

———

Geometric Hardy inequalities in the half-space on strat-ified groups

Bolys SabibtekInstitute of mathematics and mathematical modeling,125 Pushkin street, 050010 Almaty, KAZAKHSTANsabytbek.bolys@gmail.com

In this talk, we present the geometric Hardy inequalitiesin the half-space on stratified groups, as a consequencewe derive the geometric Hardy-Sobolev inequality on theHeisenberg group.

———

On a wave equation with singular dissipation

Mohammed Elamine SebihUniversity of Mascara, 29000 Mascara, ALGERIAsebihmed@yahoo.fr

We consider a wave equation with singular dissipationterm and provide information on the behaviour of veryweak solutions near the singularity.

———

Product formulas and convolutions for solutions ofSturm-Liouville equations

Ruben SousaCMUP, Departamento de Matematica - Faculdade deCiencias da Universidade do Porto, Rua do Campo Ale-gre 687, 4169-007 Porto, PORTUGALrubensousa@fc.up.pt

The Fourier transform, which lies at the heart of theclassical theory of harmonic analysis, is generated bythe eigenfunction expansion of the Sturm-Liouville op-

erator − d2

dx2. This naturally raises a question: is it

possible to generalize the main facts of harmonic analy-sis to integral transforms of Sturm-Liouville type? Inthis talk we introduce a novel unified framework forthe construction of product formulas and convolutionsassociated with a general class of regular and singularSturm-Liouville boundary value problems. This uni-fied approach is based on the application of the Sturm-Liouville spectral theory to the study of the associatedhyperbolic equation. As a by-product, an existence anduniqueness theorem for degenerate hyperbolic Cauchyproblems with initial data at a parabolic line is estab-lished. We will show that each Sturm-Liouville convolu-tion gives rise to a Banach algebra structure in the spaceof finite Borel measures in which various probabilisticconcepts and properties can be developed in analogywith the classical theory. We will discuss whether theconvolution structure satisfies the basic axioms of thetheory of hypergroups. Examples will be given, showingthat many known convolution-type operators — such asthe Hankel, Jacobi and Whittaker convolutions — canbe constructed using this general approach.

———

A trial to construct specific self-similar solutions fornonlinear wave equations

Mitsuru SugimotoGraduate School of Mathematics, Nagoya University,Furo-cho, Chikusa-ku, 464-8602 Nagoya, Aichi, JAPAN

sugimoto@math.nagoya-u.ac.jp

An attempt to construct self-similar solutions to nonlin-ear wave equations u = |u|p and u = |u|p−1u willbe explained. The existence of self-similar solutions hasbeen already established by Pecher (2000), Kato-Ozawa(2003), etc, based on the standard fixed point theorem.In this talk, we will discuss it by a constructive methodbased on a relation to the hypergeometric differentialequation.

———

Local well-posedness for higher order Benjamin-Onotype equations

Tomoyuki TanakaFurocho, Chikusaku, 464-8602 Nagoya, Aichi, JAPANd18003s@math.nagoya-u.ac.jp

64

Abstracts

In this talk, we consider the local well-posedness forhigher order Benjamin-Ono type equations, especiallyfourth order equations. The proof is based on the en-ergy method with correction terms. Our equations haveat most three derivatives in nonlinear terms, so that weneed to cancel out all derivative losses by introducingcorrection terms into the energy. We also employ theBona-Smith approximation technique in order to showthe continuity of the flow and the continuous depen-dence.

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Endpoint Strichartz estimates for Schrodinger equationon exterior domains

Koichi TaniguchiNagoya University, Furocho, Chikusaku, 464-8602Nagoya, JAPANkoichi-t@math.nagoya-u.ac.jp

In this talk we consider the initial-boundary value prob-lem of Schrodinger equation with Dirichlet or Neumannboundary condition on the exterior domain of a geomet-ric non-trapping obstacle. Our goal is to prove endpointStrichartz estimates for solutions to the problem. Forthis purpose, we start by introducing suitable local ex-tension operators, and then combine these extension op-erators with Strichartz and smoothing estimates for thefree Schrodinger equation, the known local smoothingestimates for the exterior boundary value problem, andsome commutator estimates between polynomial weightsand fractional differential operators. This talk is basedon the joint work with Professor Vladimir Georgiev(University of Pisa).

———

Decay and Scattering in energy space for the solutionof generalised Hartree equation

Mirko TarulliFaculty of Applied Mathematics and Informatics, Tech-nical University of Sofia, Kliment Ohridski Blvd. 8, 1000Sofia, and IMI BAS, Acad. Georgi Bonchev Str., Block8, 1113 Sofia, BULGARIA, and Universita Degli Studidi Pisa, Dipartimento di Matematica, Largo Bruno Pon-tecorvo 5 I - 56127 Pisa, ITALY.tarulli@mail.dm.unipi.it

We prove decay with respect some Lebesgue norms fora class of Schrodinger equations with non-local nonlin-earities by showing new Morawetz inequalities and es-timates. As a straightforward product we obtain large-data scattering in the energy space for the solutions tothe defocusing generalized Hartree equations with mass-energy intercritical nonlinearities in any space dimen-sions.

———

Very weak solutions of wave equation for LandauHamiltonian with irregular electromagnetic field

Niyaz TokmagambetovAl-Farabi Kazakh National Universitym, 71 Al-Farabiave., 050040 Almaty, KAZAKHSTANtokmagam@list.ru

In this paper, we study the Cauchy problem forthe Landau Hamiltonian wave equation, with the

time-dependent irregular (distributional) electromag-netic field and similarly irregular velocity. For suchequations, we describe the notion of a ‘very weak so-lution’ adapted to the type of solutions that exist forregular coefficients. The construction is based on con-sidering Friedrichs–type mollifier of the coefficients andcorresponding classical solutions, and their quantitativebehaviour in the regularising parameter. We show thateven for distributional coefficients, the Cauchy problemdoes have a very weak solution, and that this notionleads to classical or distributional type solutions underconditions when such solutions also exist.

———

Blowing-up solutions of the time-fractional dispersivepartial differential equations

Berikbol Torebek9000 Ghent, BELGIUMtorebek@math.kz

This paper is devoted to the study of initial-boundary value problems for time-fractional ana-logues of Korteweg-de Vries-Benjamin-Bona-Mahony-Burgers, Rosenau-Korteweg-de Vries-Benjamin-Bona-Mahony-Burgers, Ostrovsky and time-fractional modi-fied Korteweg-de Vries-Burgers equations on a boundeddomain. Sufficient conditions for the blowing-up of so-lutions in finite time of aforementioned equations arepresented. We also discuss the maximum principle andin uence of gradient non-linearity on the global solv-ability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our studyis the Pohozhaev nonlinear capacity method. We alsoprovide some illustrative examples.

———

Inverse Scattering Problems on Quantum Graphs

Igor Trushin3-1-1 Asahi, Matsumoto City, Nagano Prefecture, 390-8621 Japantrushin@math.shinshu-u.ac.jp

I investigate inverse scattering problems for a Sturm-Liouvile operator on the metric graph consisting of afinite number of half-lines joined with either a loop or afinite number of finite intervals.The scattering matrix, part of the negative eigenvaluesand corresponding normalizing coefficients are taken asa scattering data.The main goal of this research is to reconstruct thecoefficients of Sturm-Liouvile operator on the basis ofthe given scattering data. We have deduced Marchenkoequation which allowed us to prove the uniqueness theo-rems, provided a reconstruction procedure for the coeffi-cients on the half-lines and investigated the conditionalstability of the inverse scattering problem.Later on I have used the asymptotic behaviour of thescattering matrix to investigate the geometry of thegraph.

———

Optimal symmetric range for Hilbert transform and itsnon-commutative extensions

Kanat Tulenov71 Al-Farabi, 050040 Almaty, KAZAKHSTANtulenov@math.kz

65

Abstracts

We identify the optimal range of the Calderon opera-tor and that of the classical Hilbert transform in theclass of symmetric quasi-Banach spaces. Further con-sequences of our approach concern the optimal rangeof the triangular truncation operator, of operator Lip-schitz functions and commutator estimates in ideals ofcompact operators.

———

Local uniqueness of ground states for generalizedChoquard model

George VenkovFaculty of Applied Mathematics and Informatics, Tech-nical University of Sofia, Kliment Ohridski Blvd. 8, 1000Sofia, BULGARIAgvenkov@tu-sofia.bg

We consider the generalized Choquard equation de-scribing trapped electron gas in 3 dimensional case.The study of orbital stability of the energy minimizers(known as ground states) depends essentially in the localuniqueness of these minimizers. In equivalent way onecan optimize the Gagliardo–Nirenberg inequality subjectto the constraint fixing the L2 norm. The uniqueness ofthe minimizers for the case p = 2, i.e. for the case ofHartree–Choquard is well known. The main difficultyfor the case p > 2 is connected with possible lack ofcontrol on the Lp norm of the minimizers.

———

Index transforms with the squares of Kelvin functions

Semyon YakubovichUniversity of Porto, Campo Alegre str. 687, 4169-007Porto, PORTUGALsyakubov@fc.up.pt

New index transforms, involving squares of Kelvin func-tions, are investigated. Mapping properties and inver-sion formulas are established for these transforms inLebesgue spaces. The results are applied to solve aboundary value problem on the wedge for a fourth orderpartial differential equation.

———

Hypoelliptic functional inequalities and applications

Nurgissa YessirkegenovImperial College London and Institute of Mathematicsand Mathematical Modelling, 125 Pushkin str., 050010Almaty, KAZAKHSTANn.yessirkegenov15@imperial.ac.uk

In this talk we will give a review of our recent re-search on hypoelliptic functional inequalities. Wemanaged to link the integral versions of Hardy in-equalities on homogeneous groups to their hypoel-liptic versions through the Riesz and Bessel ker-nels of the Rockland operators (hypoelliptic left-invariant homogeneous differential operators, follow-ing the Helffer-Nourrigat’s resolution of the Rocklandconjecture in the 80s). Consequently, this leads togeneral hypoelliptic versions of Hardy-Sobolev, Hardy-Littlewood-Sobolev, Trudinger-Moser, Caffarelli-Kohn-Nirenberg, Gagliardi-Nirenberg and other inequalities(https://arxiv.org/abs/1805.01064). We will then con-centrate also on discussing their best constants, ground

states for higher order hypoelliptic Schrodinger typeequations, and the solutions to the corresponding varia-tional problems (https://arxiv.org/abs/1704.01490).

———

Global regularity for Einstein-Klein-Gordon system withU(1)× R isometry group

Yi ZhouSchool of Mathematical Sciences, Fudan University, 220Handan Road, 200433 Shanghai, PEOPLE’S REPUB-LIC OF CHINAyizhou@fudan.edu.cn

In this paper we prove the global regularity of the3+1 dimensional Einstein-Klein Gordon system with aU(1)× R isometry group. We reduce the Cauchy prob-lem of the Einstein-Klein Gordon system to a 2+1 di-mensional system. We prove that the energy of thissystem cannot concentrate near the first possible singu-larity, and hence is small. Then, we show that the globalregularity holds for the reduced 2+1 system with initialdata of small energy.

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S.13 Integral Transforms and ReproducingKernels

OrganisersZouhaır Mouayn

Scope of the session: Over the years, motivated by ap-plications in initial and boundary value ordinary andpartial differential equations, many integral transformshave been constructed, leading to rich mathematical the-ories with an outstanding effectiveness in physical andsignal analytic applications. A notorious famous exam-ple is the use of the Fourier transform in heat conductionproblems and signal analysis of band-limited functions.Today, a number of transforms have reached a classi-cal status, namely those associated with the names ofLaplace, Fourier, Mellin, Hankel, Radon and Hilbert.They are all widely used in physics and signal analy-sis. More recently, many fractional integral transformsbecame popular in image reconstruction, pattern recog-nition and acoustic signal processing. A notable classof integral transforms, developed with the goal of un-derstanding non-stationary phenomena in several ge-ometries is provided by the so-called coherent statestransforms CST (including Gabor and wavelet trans-forms, by specifying the underlying geometries to eu-clidean and hyperbolic and the groups structure to theHeisenberg and affine cases). They have been used inSchroedinger equations with specific potential, in theanalysis of higher Landau level eigenspaces, in the analy-sis of nonstationary signals, in analytic and polyanalyticfunction theory, in frame theory and, more recently, indeterminantal point processes (for instance, in Ginibre-type models for higher Landau levels and in the so-calledWeyl-Heisenberg ensemble). On the background of allthese applications is the fact that the range of CSTis a reproducing kernel Hilbert space. Another usefultransform defined via reproducing kernels is the Berezintransform, which shows up naturally in quantization-dequantization procedures. Our aim in this session is

66

Abstracts

to discuss old and new integral transforms: their ori-gin, their mathematical properties (in particular, of theirreproducing kernel structure) and their potential useto solve problems in mathematics, signal analysis andphysics.

—Abstracts—

Models and classifications in fractional calculus

Arran FernandezDepartment of Mathematics, Eastern MediterraneanUniversity, William Shakespeare Street, 99628 Fama-gusta, North Cyprus, via Mersin-10, TURKEYarran.fernandez@emu.edu.tr

Fractional calculus is a branch of analysis in which onecan differentiate or integrate functions a non-integernumber of times. Generalising the basic operations ofcalculus to non-integer orders allows a wider range ofbehaviours to be captured. As it turns out, the range isvery wide indeed, because there is no unique way to de-fine fractional derivatives and integrals: a whole host ofdifferent possible models have been proposed. This hasled to several different points of view within the field.From the mathematical perspective, the key idea is gen-eralisation. The proliferation of definitions of fractionalcalculus gives rise to many unique challenges and prop-erties which are not found in classical calculus. For ex-ample, most fractional derivatives are nonlocal and donot satisfy a semigroup law. There is also a desire fordefinition: what exactly makes a particular operator afractional derivative or not?From the applications perspective, the key idea is mod-elling. Many processes which arise in real life can bemore accurately modelled using fractional operators –of various types – than using classical calculus. For ex-ample, viscoelastic materials may be described by frac-tional differential equations with order between thoseof the equations describing viscosity and elasticity. Anydefinition of fractional calculus which is useful physicallyis worth analysing mathematically.After summarising the different approaches found withinthe fractional calculus community, I will attempt tounify them by introducing some broad classes of frac-tional operators. Especially important is the class de-fined by a convolution integral with an analytic functionof a fractional power: this is broad enough to cover manydifferent types of fractional calculus, yet still clearly con-nected to the classical Riemann–Liouville fractional cal-culus. It can be studied on a general level, as well asdivided into subclasses with specific desired properties.

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Paley-Wiener and Wiener’s Tauberian results for a classof oscillatory integral operators

Rita GuerraCIDMA, Department of Mathematics, University ofAveiro, Aveiro, PORTUGALritaguerra@ua.pt

The main aim of this talk is to obtain Paley–Wiener andWiener’s Tauberian results associated with a class of os-cillatory integral operators, which depends on cosine andsine kernels, as well as to introduce a consequent newconvolution associated with that class of operators. Ad-ditionally, a new Young-type inequality for the obtained

convolution is proven, and a new Wiener-type algebra isalso associated with this convolution. This is based ona joint work with L. P. Castro and N. M. Tuan.

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On the boundedness of composition operators on re-producing kernel Hilbert spaces with analytic positivedefinite functions

Isao IshikawaAdachi-ku, 1200022 Tokyo, JAPANisao.ishikawa@riken.jp

In this talk, I would like to explain our result thatboundedness of composition operators of maps impliesthe maps are affine maps in certain situations. Com-position operators (Koopman operators) are classicallyinvestegated in complex analysis, but, they have beengetting popular in the context of machine leraning anddata analysis these days. Besides, reproducing kernelHilbert spaces with analytic positive definite functionson euclidean spaces are utilized in many fields in enge-neering and statistics. On the other hand, although itis important to prove the relation between the proper-ties of maps and those of composition operators of themaps to guruantee theoretical validity, such relation iscurrently not known very well. In some important sit-uation, we prove that a map become an affine map ifits composition operator is bounded on an RKHS as-sociated with analytic positive definite functions on eu-clidean spaces. This is the joint work with MasahiroIkeda (RIKEN/Keio University) and Yoshihiro Sawano(Tokyo metropolitan University/RKEN).

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A harmonic interpolation sequence on the real unit ball

El Aıdi MohammedUniversidad Nacional de Colombia, Sede Bogota,Avenida Carrera 30 numero 45-03, Bogota, Cundina-marca, COLOMBIAmelaidi@unal.edu.co

We provide sufficient conditions for interpolating a se-quence by a function in a weighted Bergman type spaceof harmonic functions on the real unit ball.

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Nonlinear coherent states associated with a measureon the positive real half line

Zouhaır MouaynSultan Moulay Slimane University, 23000 Beni Mellal,MOROCCOmouayn@gmail.com

We construct a class of generalized nonlinear coherentstates by means of a newly obtained class of 2D com-plex orthogonal polynomials. The associated coherentstates transform is discussed. A polynomials realiza-tion of the basis of the quantum states Hilbert spaceis also given. Here, the entire structure owes its exis-tence to a certain measure on the positive real half line,of finite total mass, together with all its moments. Weillustrate this construction with the example of the mea-sure rβe−rdr, which leads to a new generalization of thetrue-polyanalytic Bargmann transform.

———

67

Abstracts

Distributed-order non-local optimal control

Faical NdairouCIDMA, University of Aveiro, 3810-193 Aveiro, POR-TUGALfaical@ua.pt

Distributed-order fractional non-local operators havebeen introduced and studied by Caputo at the endof the 20th century. They generalize fractional orderderivatives/integrals in the sense that such operatorsare defined by a weighted integral of different ordersof differentiation over a certain range. The subject ofdistributed-order non-local derivatives is currently un-der strong development due to its applications in mod-eling some complex real world phenomena. Fractionaloptimal control theory deals with the optimization ofa performance index functional subject to a fractionalcontrol system. One of the most important results inclassical and fractional optimal control is the Pontrya-gin Maximum Principle, which gives a necessary opti-mality condition that every solution to the optimizationproblem must verify. In our work, we extend the frac-tional optimal control theory by considering dynamicalsystems constraints depending on distributed-order frac-tional derivatives. Precisely, we prove a weak version ofPontryagin’s maximum principle and a sufficient opti-mality condition under appropriate convexity assump-tions.

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Convolution integral equations related to Fourier sineand cosine transforms and Hermite functions

Anabela SilvaCIDMA - Center for Research and Development inMathematics and Applications, University of Aveiro,3810-193 Aveiro, PORTUGALanabela.silva@ua.pt

In this talk, we present new convolutions on the positivehalf-line for Lebesgue integrable functions. Factoriza-tion identities for those convolutions are derived, uponthe use of Fourier sine and cosine transforms and Her-mite polynomials. These results allow us to consider(systems of) convolution integral equations, to analyseconditions for their solvability and, under such condi-tions, to obtain their solutions.

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Noetherian Solvability of an Operator Singular IntegralEquation with Carleman Shift in Fractional Spaces

Nurlan YerkinbayevAlmaty, st. Pushkin 125, 050000 Almaty, KAZA-KHSTANnurlan.erkinbaev@mail.ru

Let Γ be a Lyapunov closed contour of class C1ν ,

2p−1 <

ν ≤ 1, 1 < p < 2. We consider in Besov’s spaceB(Γ) ≡ Brp,1(Γ), r = 1

p, 1 < p < 2,the singular integral

equation

Lϕ ≡ a(t)ϕ(t) + b(t)ϕ[α(t)] +c(t)

πi

∫Γ

ϕ(τ)

τ − tdτ

+d(t)

πi

∫Γ

ϕ(τ)

τ − α(t)dτ +

∫Γ

K(t, τ)ϕ(τ)dτ = g(t)

where a(t), b(t), c(t), d(t)andg(t) belong to spaceB(Γ), α(t) is Carleman’s shift, i.e. homeomorphicallymaps Γ onto itself, preserving or changing its orientationto Γ and α[α(t)] = t. It is supposed, that there existsa derivative α′(t), belonging to space Hµ(Γ) of Holdercontinuous functions with indicator 2

p− 1 < ν ≤ 1.The

kernels Kj(t, τ), j = 1, 2, have such weak singularitiesthat the corresponding integral operators are completelycontinuous in B(Γ). We note, that B(Γ) embedded inthe class of continuous functions C(Γ), but not em-bedded in Hµ(Γ)0 < µ ≤ 1, is a commutative Banachalgebra with a unit with the usual operations of addi-tion and multiplication of functions [1, 2]. Equation(1) is considered in [3] in spaces Hµ(Γ)0 < µ ≤ 1, andLp(Γ), 1 < p < ∞. We have obtained the conditions ofthe noetherian solvability of the equation (1) in B(Γ)and its index formula. The results are used [4].1. Besov OV, Ilyin V.P., Nikolsky S.M. Integral rep-resentations of functions and embedding theorems. M.Science - fml, 1996. 345 p. 2. Bliev N.K. Singular inte-gral operators with Cauchy kernel in fractional spaces.Sib. mat. Journal, 2006. V. 47, No. 1, p. 37 - 45. 3.Litvinchuk G.S. Boundary value problems and singularintegral equations with shift. M. Science - fml. 1977,448 p. 4. Bliev N.K. The system of singular integralequations with Cauchy kernel in Besov spaces. DocladyAN RK, 2007, ?5, p. 5 - 9.Joint talk with Bliev N.K., bliev.nazarbay@mail.ru

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S.14 Partial Differential Equations onCurved Spacetimes

OrganisersAnahit Galstyan, Makoto Nakamura,Karen Yagdjian

Scope of the session: The session will discuss variousproblems for partial differential equations on curvedspacetimes. The topics will include the representationand qualitative properties of the solutions of initial valueand boundary value problems for linear and nonlinearPDEs in the curved spacetimes, global existence andblow up of the solutions, and numerical simulations, etc.

—Abstracts—

Computational analysis of a nonlinear wave equationwith black hole embedded in an expanding universe

Andras BaloghUniversity of Texas Rio Grande Valley, School of Math-ematical and Statistical Sciences, 78539 Edinburg, TX,USAandras.balogh@utrgv.edu

Numerical simulations are presented for a nonlinearwave equation with singularity that models black holeand with time dependent coefficient that models anexpanding universe. The high performance Pycudacomputations use an explicit fourth order Runge-Kuttascheme on the temporal discretization and fourth or-der finite difference discretization in the 3-dimensionalspace. Bubble formation and the properties of solutions

68

Abstracts

with compact support are examined both inside and out-side of the blackhole.

———

Radiation fields for wave equations

Dean BaskinTexas A&M University, Department of Mathematics,Mailstop 3368, 77843 College Station, TX, USAdbaskin@math.tamu.edu

Radiation fields are rescaled limits of solutions of waveequations near ”null infinity” and capture the radiationpattern seen by a distant observer. They are intimatelyconnected with the Fourier and Radon transforms andwith scattering theory. In this talk, I will define and dis-cuss radiation fields in a few contexts, with an emphasison spacetimes that look flat near infinity. The main re-sult is a connection between the asymptotic behaviorof the radiation field and a family of quantum objectson an associated asymptotically hyperbolic space. Thistalk is based on joint work with Jeremy Marzuola, An-dras Vasy, and Jared Wunsch

———

Local existence of solutions to the Euler–Poisson sys-tem, including densities without compact support

Uwe BrauerUniversidad Complutense de Madrid, Plaza de Ciencias2, 28040 Madrid, SPAINoub@mat.ucm.es

Local existence and uniqueness for a class of solutionsfor the Euler Poisson system is shown. These solutionshave a density ρ which either falls off at infinity or hascompact support. Since the Euler equations degeneratein such a setting, a regularisation has to be consideredwhich complicates the proof considerably. The solutionshave finite mass, finite energy functional and includethe static spherical solutions for γ = 6

5. The result is

achieved by using weighted Sobolev spaces of fractionalorder and a new non linear estimate which allows to es-timate the physical density by the regularised non linearmatter variable. With these tools at hand we then provethe existence of solutions to the Euler–Poisson–Makinosystem by using a fixed-point argument. Gamblin alsohas studied this setting but using very different func-tional spaces. However, we believe that the functionalsetting we use is more appropriate to describe a physi-cal isolated body and more suitable to study the New-tonian limit. The talk is based on a collaboration withL. Karp.

———

About critical exponents in semi-linear de Sitter models

Marcelo EbertDepartamento de Computacao e Matematica, Universi-dade de Sao Paulo (USP), Av. dos Bandeirantes, 3900,14040-901 Ribeirao Preto, Sao Paulo, BRAZILebert@ffclrp.usp.br

In this paper we consider the Cauchy problem for semi-linear de Sitter models. The model of interest is

φtt − e−2t∆φ+ nφt +m2φ = Fp(φ),

(φ(0, x), φt(0, x)) = (f(x), g(x)),

where n is the dimension, m2 is a non-negative con-stant and p is a positive parameter. Our main goal isto verify that, in general, one can not observe a criticalexponent pcrit = pcrit(n) in the family of non-linearitiesFp(φ)p>1 = |φ|pp>1. Moreover, we like to proposeparameter-dependent families Fp(φ)p>0 of non-linearright-hand sides where the correct choice of the parame-ter p might provide a threshold between global (in time)existence of small data solutions (stability of the zerosolution) and blow-up behavior even of small data solu-tions.

———

Semilinear Klein-Gordon Equation in the Friedmann-Lamaitre-Robertson-Walker spacetime

Anahit GalstyanSchool of Mathematical an Statistical Sciences, Univer-sity of Texas RGV, 1201 West University Drive, 78539Edinburg, TX USanahit.galstyan@utrgv.edu

We present a condition on the self-interaction term thatguaranties the existence of the global in time solutionof the Cauchy problem for the semilinear Klein-Gordonequation in the Friedmann-Lamaitre-Robertson-Walker(FLRW) model of the contracting universe. For theKlein-Gordon equation with the Higgs potential we givea lower estimate for the lifespan of solution.

———

Heat Flow of Isometric G2-structures

Sergey GrigorianUniversity of Texas Rio Grande Valley, 1201 W Univer-sity Drive, 78539 Edinburg, TX, USAsergey.grigorian@utrgv.edu

Given a 7-dimensional compact Riemannian manifold(M, g) that admits G2-structure, all the G2-structuresthat are compatible with the metric g are parametrizedby unit sections of an octonion bundle over M. We de-fine a natural energy functional on unit octonion sec-tions and consider its associated heat flow. The criticalpoints of this functional and flow precisely correspond toG2-structures with divergence-free torsion. In this talk,we first derive estimates for derivatives of V (t) alongthe flow and prove that the flow exists as long as thetorsion remains bounded. We will also show a mono-tonicity formula and and an epsilon-regularity result forthis flow. Finally, we show that within a metric class ofG2-structures that contains a torsion-free G2-structure,under certain conditions, the flow will converge to atorsion-free G2-structure.

———

Energy estimates for Klein-Gordon type equations withtime dependent mass

Fumihiko HirosawaYamaguchi University, Faculty of Science, 753-8512 Ya-maguchi, JAPANhirosawa@yamaguchi-u.ac.jp

We consider the energy estimate of the solution to theCauchy problem of Klein-Gordon type equation withtime dependent mass:

∂2t u−∆u+M(t)u = 0, (t, x) ∈ (0,∞)× Rn,u(0, x) = u0(x), ∂tu(0, x) = u1(x), x ∈ Rn,

69

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where M(t) is real valued, oscillating and not necessarilypositive. The main purpose of my talk is to give suffi-cient conditions to M(t) for the energy to be asymptot-ically stable.

———

Continuous dependence on the geometrical initial datafor the Einstein vacuum equations

Lavi KarpDepartment of Mathematics, ORT Braude College, P.O.Box 72, 201982 Karmiel, ISRAELkarplavi@gmail.com

The Cauchy problem of Einstein equations deals withthe construction of a Lorenztian manifold from givengeometric quantities on an initial sub manifold. The so-lutions consists of two major steps, namely, solution ofthe constraint and evolution equations. The constraintequations can be written as an elliptic system on the ini-tial manifold and their solutions provide the initial datafor the evolution equations. Under the harmonic gaugethe evolution equations are reduced to quasilinear waveequations. The common methods to solve these centralproblems in asymptotically flat spacetime is to considerthe constraint equations in a weighted Sobolev spaces,while the quasilinear wave equation are dealt in the or-dinary unweighted Sobolev spaces. Therefore it is im-possible to obtain well–posedness of these equations bythis approach. We treat both central type of equationsin the weighted Sobolev spaces, and hence we are ableto derive the well–posedness of the Cauchy problem forEinstein equations in asymptotically flat spacetime, in-cluding continuous depending of the Lorentzian metricon the initial geometrical data.The talk is based on a joint work with U. Brauer.

———

Asymptotic Behavior of nonlinear Schrodinger equa-tions with radial data

Baoping LiuPeking University, Room 78404 Jing Chun Yuan 78, No5 Yiheyuan Road, 100871 Beijing, PEOPLE’S REPUB-LIC OF CHINAbaoping@math.pku.edu.cn

The asympototic behavior of solutions to dispersiveequations is a very important question. The problem isparticularly difficult if the equation admits solitons. Inthis talk, we will report some of our recent progress inthis direction for nonlinear Schrodinger equations withradial data.

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Remarks on the Navier-Stokes equations in homoge-neous and isotropic spacetimes

Makoto NakamuraYamagata University, Faculty of Science, 990-8560 Ya-magata, Yamagata, JAPANnakamura@sci.kj.yamagata-u.ac.jp

The Navier-Stokes equations are considered in homoge-neous and isotropic spacetimes. The energy estimates ofthe equations are constructed. The effects of the spatialexpansion and contraction are characterized by the dis-sipative and anti-dissipative properties of the equations.

The Cauchy problem for the Navier-Stokes equations isconsidered under the constant density.

———

Solutions of the wave equation bounded at the BigBang

Jose NatarioDepartamento de Matematica, Instituto SuperiorTecnico, Universidade de Lisboa, 001 Lisboa, PORTU-GALjnatar@math.ist.utl.pt

By solving a singular initial value problem, we provethe existence of solutions of the wave equation gφ = 0which are bounded at the Big Bang in the Friedmann-Lemaıtre-Robertson-Walker cosmological models. Moreprecisely, we show that given any function A ∈ H3(Σ)(where Σ = Rn, Sn or Hn models the spatial hypersur-faces) there exists a unique solution φ of the wave equa-tion converging to A in H1(Σ) at the Big Bang, andwhose time derivative is suitably controlled in L2(Σ).

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Semilinear de Sitter model of cosmology - global exis-tence of small data solutions

Michael Gerhard ReissigInstitute of Applied Analysis, Technical UniversityBergakademie Freiberg, Pruferstr. 9, 09596 Freiberg,Saxony, GERMANYreissig@math.tu-freiberg.de

In this talk we consider the Cauchy problem for semi-linear de Sitter models with power non-linearity. Themodel of interest is

φtt − e−2t∆φ+ nφt +m2φ = |φ|p,

(φ(0, x), φt(0, x)) = (f(x), g(x)),

where m2 is a non-negative constant. We study theglobal (in time) existence of small data solutions. Inparticular, we show the interplay between the power p,admissible data spaces and admissible spaces of solu-tions (in weak sense, in sense of energy solutions or inclassical sense).These are joint considerations with Marcelo RempelEbert (University of Sao Paulo).

———

Outgoing Fredholm theory and the limiting absorptionprinciple for asymptotically conic spaces

Andras VasyDepartment of Mathematics, Stanford University, 450Serra Mall, Bldg 380, 94305-2125 Stanford, CA, USAandras@math.stanford.edu

In this talk I will discuss geometric generalizations of Eu-clidean resolvent estimates, such as estimates for the re-solvent of the Laplacian of an asymptotically conic met-ric plus a decaying potential, in a Fredholm frameworkthat focuses on capturing the outgoing asymptotics ofthe resolvent applied to a Schwartz function (outgoingwaves); this is different from even the usual treatmentof the Euclidean problem. More precisely, the setting isthat of perturbations P (σ) of the spectral family of theLaplacian ∆g−σ2 on asymptotically conic spaces (X, g)of dimension at least 3 (with the asymptotic behavior

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at the ‘large end’ of the cone), and the main results arethe limiting absorption principle, as well as uniform esti-mates for P (σ)−1 as σ → 0, on function spaces betweenwhich P (σ) is Fredholm even for real σ 6= 0 and whichcorrespond to finite regularity Lagrangian distributionsassociated to the, conic in the base, Lagrangian givenby the outgoing radial set of the Hamilton flow. Suchresults have immediate applications to the behavior ofthe wave equation on black hole spacetimes.

———

Quantum Field Theory on Low regularity Spacetimes

James VickersUniversity of Southampton, School of Mathematical Sci-ences, SO15 5DQ Southampton, Hampshire, UKJ.A.Vickers@soton.ac.uk

In this talk we develop the theory of quantum fields onlow regularity curved spacetimes. We follow the alge-braic approach as outlined in the book by Bar, Ginouxand Pfaffle suitably adapted to the low regularity set-ting. In particular we will show how to obtain a uniqueglobal H2

loc(M) causal solution to the Cauchy problemfor a normally hyperbolic operator on a globally hyper-bolic C1,1 spacetime. We then show how to constructadvanced and retarded Green operators on suitable func-tion spaces and use this to construct the causal propa-gator G. This is then used to define a symplectic form ωon H1

comp(M)/Ker(G). Finally this is used to constructa representation of the canonical commutation relationson the space of quasi-local C∗-algebras which satisfy theHaag-Kastler axioms. Defining the physical states re-quires a Sobolev space variant of the micro-local spec-trum condition of Radzikowski.

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On the Cauchy problem for hyperbolic operators withtriple characteristics whose coefficients depend only onthe time variable

Seiichiro WakabayashiUniversity of Tsukuba, Tsukuba, 305-0042 Ibaraki,JAPANwkbysh@math.tsukuba.ac.jp

We have considered the Cauchy problem for hyperbolicoperators with double characteristics whose principalsymbols have time-dependent coefficients and given suf-ficient conditions and necessary conditions for C∞ well-posednesss, before. Here we deal with the Cauchy prob-lem for hyperbolic operators with triple characteristicswhose coefficients are real analytic functions of the timevariable. We fatorize the operators as products of hy-perbolic operators whose symbols are polynomials of τof degree ≤ 3, where τ is the dual variable of the timevariable. For hyperbolic operators of third order we de-fined the sub-sub-principal symbols and we proved theCauchy is C∞ well-posed imposing conditions on thesubprincipal symbol and the sub-sub-principal symbol.We shall generalize the conditions for operators of thirdorder and give the sufficient conditions that the Cauchyproblem for hyperbolic operators with triple character-istics is C∞ well-posed. For special operators of thirdorder we shall prove our conditions are necessary for C∞

well-posedness.

———

Navier-Stokes equation with very rough data

Baoxiang WangSchool of Mathematical Sciences, Peking University, Bei-jing 100871, PEOPLE’S REPUBLIC OF CHINAwbx@pku.edu.cn

We study the Cauchy problem for the incompressibleNavier-Stokes equation (NS):

ut −∆u+ u · ∇u+∇p = 0, div u = 0, u(0, x) = u0.

We consider a class of very rough initial data in Es2,2 forwhich the norm are defined by

‖u0‖Es = ‖2s|ξ|u0(ξ)‖L2 , s < 0

and show that NS has a unique global solution if theinitial value u0 ∈ Es, s < 0 and their Fourier transformsare supported in RdI := ξ ∈ Rd : ξi ≥ 0, i = 1, ..., d.Our results imply that NS has a unique global solutionif the initial value u0 is in L2 with supp u0 ⊂ RdI . This isa joint work with Professors Feichtinger, Grochenig andDr Li.

———

Properties of solutions of hyperbolic equations in thecurved space-times

Karen YagdjianSchool of Mathematical and Statistical Sciences, Univer-sity of Texas RGV, 1201 W. University Drive, Edinburg,TX 78539, USAkaren.yagdjian@utrgv.edu

In this talk we describe some quantitative and qualita-tive properties of the solutions of hyperbolic equations inthe curved space-times. This properties are obtained byrepresentation formulas written via the integral trans-form approach that was developed by the author. Spe-cial attention is given to the solutions of the linear andsemilinear equations in the FLRW space-times.

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Global solutions of massive Maxwell-Klein-Gordonequations with large Maxwell field

Shiwu YangBICMR, NO.78 Jingchunyuan, Peking University,100871 Beijing, PEOPLE’S REPUBLIC OF CHINAshiwuyang@bicmr.pku.edu.cn

In this talk, I will discuss the global asymptotic dynam-ics for solutions of massive Maxwell-Klein-Gordon equa-tions with a class of unrestricted large data, which havearbitrary size of Maxwell field. The main difficulties arearising from the largeness of the Maxwell field and thedifferent decay properties for linear Maxwell field andlinear massive Klein-Gordon equation, which will causea loss of decay for solutions of the coupled system atthe causal infinity within a forward light cone. The firstissue can be addressed by using the modified vector fieldmethod developed by Klainerman-Wang-Yang while theloss of decay is overcome by uncovering a hidden nullstructure.

———

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Abstracts

S.15 Partial Differential Equations withNonstandard Growth

OrganisersHermenegildo Borges de Oliveira,Sergey Shmarev

Scope of the session: The aim of this Session is to dis-cuss the theoretical and numerical aspects of the anal-ysis of Partial Differential Equations with nonstandardgrowth. This class of equations embraces the equationswhose nonlinear structure may vary in the space/timedomain and is allowed to depend on the independentvariables or even in the solution itself. During the lastdecades, PDEs of this type have been the focus of atten-tion of many researchers. On the one hand, the inter-est in the study of such equations is explained by theirapplications in the mathematical modelling of the realworld processes, in particular, in the models of fluid dy-namics or in processing of digital images. On the otherhand, their theoretical study leads to challenging prob-lems related to the issues of existence, uniqueness andqualitative properties of the solutions.

—Abstracts—

On a multidimensional boundary value problem for amodel degenerate elliptic-parabolic equation

Gaukhar ArepovaInstitute of Mathematics and Mathematical Modeling,Pushkin str. 25, A25Y6Y7 Almaty, KAZAKHSTANarepova@math.kz

For a degenerate elliptic-parabolic equation bound-ary value problems are mainly studied in the two-dimensional case. The main bibliographies on theseproblems are given in the works of Salakhitdinov M.S.,Dzhuraev T.D., Muratbekov M.B. and Kalmenov T.Sh.In the present work, by using the methods of spectraldecomposition, it is established the unique solvability ofthe classical multidimensional problem of a degenerateelliptic-parabolic equation. In this case, the boundaryconditions are defined only in the domain of the elliptic-ity of the equation and also proved smoothness of solu-tions in the elliptic and parabolic domains.

———

Interior Regularity to the Steady Incompressible ShearThinning Fluids with Non-standard Growth

Hyeong Ohk BaeAjou University, Dasan 422, 16499 Suwon, Gyeonggi-do,KOREAhobae@ajou.ac.kr

We consider weak solutions to the equations of station-ary motion of a class of non-Newtonian fluids which in-cludes the power law model. The power depends onthe spatial variable, which is motivated by electrorhe-ological fluids. We prove the existence of second orderderivatives of weak solutions in the shear thinning cases

———

Well-posedness of the Cosserat-Bingham multi-dimensional fluid equations

Nikolai V. ChemetovDepartamento de Matematica-ICE, Universidade Fed-eral de Amazonas, BRAZILnvchemetov@gmail.com

We consider a model for Cosserat-Bingham fluids. Incontrast to the classical Bingham fluid, the Cosserat-Bingham fluids support local micro-rotations and twotypes of plug zones. We show the global-in-time solv-ability of the initial boundary value problem for theCosserat-Bingham model.Joint work with A. de Araujo (UFV, Brasil), M.Santos(UNICAMP, Brasil).

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Boundary value problem involving p−Laplacian andjumping nonlinearities

QHeung ChoiInha University, Department of Mathematics Education,22212 Incheon, KOREAqheung@inha.ac.kr

We get one theorem which shows existence of solutionsfor boundary value problem involving p-Laplacian andjumping nonlinearities. This theorem is that there ex-ists at least one solution when nonlinearities crossing fi-nite number of eigenvalues. We obtain this result by theeigenvalues and the corresponding normalized eigenfunc-tions of the p−Laplacian eigenvalue problem, variationalreduction method and critical point theory.

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Optimal feedback control of second grade fluids

Fernanda CiprianoNew University of Lisbon, Faculty of Science and Tech-nology, 2829-516 Caparica, PORTUGALcipriano@fct.unl.pt

This talk deals with a feedback optimal control problemfor the stochastic second grade fluids. More precisely, weestablish the existence of an optimal feedback control forthe two-dimensional stochastic second grade fluids, withNavier-slip boundary conditions. In addition, using theGalerkin approximations, we show that the optimal costcan be approximated by a sequence of finite dimensionaloptimal costs. This is a joint work with Diogo Pereira.

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On the numerical simulation of a parabolic equationwith p-Laplacian and memory.

Jose Carlos Matos DuqueUniversity of Beira Interior, Center of Mathematics andApplications, Covilha, PORTUGALjduque@ubi.pt

In this talk we consider the following parabolic equa-tion with the p-Laplacian and a memory term in a onedimensional spatial domain:

ut(x, t) = (|ux(x, t)|p−2ux(x, t))x

+

∫ t

0

g(t− s)(|ux(x, s)|p−2ux(x, s))x ds+ f(x, t).

The main numerical issues concern the accurate estima-tion of the p-Laplacian and the computer memory con-sumption. For the spatial discretization, we consider a

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continuous piecewise polynomial finite element approxi-mation. In order to discretize the time variable we pro-pose the Crank-Nicolson method with the trapezoidalrule to handle the memory term. Finally, several sim-ulations addressing some properties of the solutions arepresented.This is a joint work with Rui Almeida and BelchiorMario.

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Picone identity for variable exponents operators andapplications

Jacques GiacomoniLMAP UMR E2S-UPPA CNRS 5142, FED. IPRA, av-enue universite, 64000 Pau, FRANCEjacques.giacomoni@univ-pau.fr

The talk is concerned with new results about Diaz-Saatype inequality and Picone identity for quasilinear ellip-tic operators with variable exponents. Various applica-tions to uniqueness and global behaviour for quasilinearelliptic and parabolic problems are derived.

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About some generalizations of the parabolic p-Laplacian and the Porous Medium Equation

Eurica HenriquesPolo CMAT-UTAD, UTAD, Departamento deMatematica - ECT - UTAD, 5000-611 Vila Real, POR-TUGALeurica@utad.pt

In this talk we will present and discuss some prop-erties of two generalizations of the two well knownparabolic PDEs, the parabolic p-Laplacian and theporous medium equation, namely

ut −∇ ·(uγ(x,t) ∇u

)= 0,

where −1 < γ− ≤ γ(x, t) ≤ γ+ < ∞ and γ ∈L∞

(0, T ;W 1,p (Ω)

), for some p > max 2, N and

∂t(uq)−∇ ·

(|∇u|p−2∇u

)= 0, 0 < q, p < 1.

———

Multiplicity results for p−Laplacian boundary valueproblem

Tacksun JungKunsan National University, Department of Mathemat-ics, 54150 Kunsan, REPUBLIC OF KOREAtsjung@kunsan.ac.kr

We investigate multiplicity of solutions for one dimen-sional p−Laplacian Dirichlet boundary value problemwith jumping nonlinearites. We obtain three theorems:The first one is that there exists exactly one solutionwhen nonlinearities cross no eigenvalue. The second oneis that there exist exactly two solutions, exactly one so-lutions and no solution depending on the source termwhen nonlinearities cross one first eigenvalue. The thirdone is that there exist at least three solutions, exactlyone solutions and no solution depending on the source

term when nonlinearities cross the first and second eigen-values. We obtain the first theorem and the second oneby eigenvalues and the corresponding normalized eigen-functions of the p−Laplacian eigenvalue problem, andthe contraction mapping principle on p−Lebesgue space(when p ≥ 2). We obtain the third result by Leray-Schauder degree theory.

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A free boundary problem for an inhomogeneous opera-tor with nonstandard growth

Claudia LedermanFacultad de Ciencias Exactas, Universidad de BuenosAires and IMAS - CONICET, 1428 Ciudad Autonomade Buenos Aires, ARGENTINAclederma@dm.uba.ar

We present recent results on a free boundary problem foran inhomogeneous operator with nonstandard growth.This operator has been used in the modelling of elec-trorheological fluids and in image processing. We obtainfree boundary regularity results for weak solutions andwe apply these results to minimizers of suitable energyfunctionals. In our work we overcome deep technical dif-ficulties not present in previous literature for this typeof problems. This is joint work with Noemi Wolanski(Universidad de Buenos Aires and IMAS - CONICET).

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About a new non linear boundary condition for the tur-bulent kinetic energy involved in models of turbulence

Roger LewandowskiIRMAR-UFR de mathematiques, Bat. 22, CampusBeaulieu, Universite Rennes 1, 35000 Rennes, FRANCE

Roger.Lewandowski@univ-rennes1.fr

We first derive from a standard modeling process a newnon linear boundary condition for the turbulent kineticenergy k, that is autonomous in k. We then show thatthe corresponding model of turbulence, coupling k tothe mean velocity and the mean pressure of a turbulentflows, has a distributional solution. This analysis is illus-trated by several numerical simulations in a 3D channelthat shows the accuracy of the model.

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On a nonlocal viscosity p-Laplacian evolutive problem.Existence and long-time behavior

Pedro Marın-RubioDpto. Ecuaciones Diferenciales y Analisis Numerico,Universidad de Sevilla, C/ Tarfia s/n, 41012 Sevilla,SPAINpmr@us.es

We analyze the (global) existence of solutions for thescalar evolutive p-Laplacian problem with nonlocal vis-cosity and extra terms (eventually another nonlinearityand a time-dependent) in the right hand side

du

dt− a(l(u))∆pu = f(u) + h(t) in Ω× (τ, T ),

u = 0 on ∂Ω× (τ, T ),u(x, τ) = uτ (x) in Ω,

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Abstracts

where Ω is a bounded open set of RN , p ≥ 2, a ∈C(R; [m,∞)), m > 0, and l(u) =

∫Ωg(x)u(x)dx for a

given g ∈ L2(Ω).The non-standard nonlocal diffusion term given by thefunction a aims to model both quick aggregation or op-positely, the necessity to leave crowded zones, dependingon the choice of the viscosity function, non-increasingor increasing. To better model certain situations withrough viscosity terms only continuity is assumed for theviscosity function.These two nonlinearities (nonlocal term and p-Laplacian) combined with the fact of an evolutive prob-lem involves extra difficulties to solve it, which are cir-cumvented by a nontrivial change of variables, compact-ness and monotonicity techniques. The regularizing ef-fect of the equation is shown by applying an argumentof a posteriori regularity, even although uniqueness ofsolution for the problem is unknown.Besides that, the long-time behavior of solutions is ana-lyzed via multi-valued dynamical systems and the exis-tence of several families of (pullback) attractors ensured,in L2(Ω), Lp(Ω) and actually in any Banach space Xsuch that the following compact/continuous embeddingsW 1,p

0 (Ω) ⊂⊂ X ⊂ L2(Ω) hold.This is a joint work with Tomas Caraballo and MartaHerrera-Cobos (Universidad de Sevilla).

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A survey of recent results on the characterization ofdecay of solutions to some dissipative equations

Cesar J. NicheDepartamento de Matematica Aplicada - UniversidadeFederal do Rio de Janeiro, Centro de Tecnologia - BlocoC, Cidade Universitaria - Av. Athos da Silveira Ramos,149 - Ilha do Fundao, 21941-909 Rio de Janeiro, BRAZIL

cniche@im.ufrj.br

Solutions to many dissipative equations in Fluid Me-chanics obey inequalities which show that their energydecays in time. In this talk we will describe recent re-sults in which a complete characterization of decay ratesis obtained through a single number associated to theinitial datum, for a large family of equations includingNavier-Stokes and dissipative quasi-geostrophic.

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Existence results for the p(u)−Laplacian problem.

Hermenegildo Borges de OliveiraFCT - Universidade do Algarve, Campus de Gambelas,8005-269 Faro, Portugalholivei@ualg.pt

In this talk, we consider the p−Laplacian problem withthe exponent of nonlinearity p depending on the solu-tion u itself. For the associated boundary-value and ini-tial boundary-value problems, we prove the existence ofweak solutions by using a singular perturbation tech-nique. We will also point out some challenging prob-lems in this class of partial differential equations withnonstandard growth. This talk is based in joint worksin collaboration with Michel Chipot from Zrich Univer-sitt.

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Entropy and kinetic solutions of the genuinely-nonlinearKolmogorov-type equation with partial diffusion andnonlinear source term

Sergey SazhenkovLavrentyev Institute of Hydrodynamics, LavrentyevAve., 15, 630090 Novosibirsk, RUSSIAN FEDERATION

sazhenkovs@yandex.ru

The talk is devoted to a study of the Cauchy–Dirichlet problem for quasilinear Kolmogorov-typeultra-parabolic equation incorporating two time-likevariables and a nonlinear source term:

(1a) ∂tu+ ∂sa(u) + divxϕ(u) = ∆xu+ hγ(x, t, s, u),where (x, t, s) ∈ GT,S ,

(1b) u|t=0 = u(1)0 , u|s=0 ≈ u(2)

0 , u|s=S ≈ u(2)S , u|Ω = 0.

Here GT,S := Ω× (0, T )× (0, S); Ω is a bounded domainof spatial variables x ∈ Rd with a smooth boundary ∂Ω;t ∈ [0, T ] and s ∈ [0, S] are two independent time-likevariables, T, S = const > 0; u = u(x, t, s) is a sought

function; initial and final data u(1)0 , u

(2)0 and u

(2)S , non-

linearities a, ϕ1, . . . , ϕd and source term hγ are givenand smooth. Function a satisfies the special genuinenonlinearity condition:

the set λ ∈ R : ξ1 + a′(λ)ξ2 = 0(2)

has the empty interior for each fixed (ξ1, ξ2) ∈ S1.

In general, function a is non-monotone, which meansthat evolution process in time-like direction s may be re-versible. Condition (2) is the cornerstone of the properentropy and kinetic formulations of problem (1a)-(1b).Justification of existence and uniqueness of entropy andkinetic solutions is the first major result of this study.The second major result relates with the case when hγcollapses to the Dirac delta-function as γ → 0. We showthat there is a limiting point u = lim

γ→0uγ of the family

of solutions to problem (1a)-(1b) such that u solves theimpulsive equation of the form (1a), with the suddensource term δ(t=τ)β(x, s, u) on the place of hγ(x, t, s, u).The presented research was fulfilled in collaboration withDr Ivan Kuznetsov (Novosibirsk State University).

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Solvability of Nonlinear Elliptic Type Equation WithTwo Unrelated Non standard Growths

Ugur SertHacettepe University, Faculty of Science, Department ofMathematics, 06800, Beytepe, Ankara, TURKEYusert8483@gmail.com

In this paper, we study the solvability of the nonlin-ear Dirichlet problem with sum of the operators of in-dependent non standard growths in a bounded domainΩ ⊂ Rn. Here, one of the operators in the sum is mono-tone and the other is weakly compact. We obtain suffi-cient conditions and show the existence of weak solutionsof the problem under consideration by using monotonic-ity and compactness methods together.

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Global regularity of solutions of singular parabolic equa-tions with nonstandard growth

Sergey ShmarevUniversidad de Oviedo, Departamento de Matematicas,

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Abstracts

33007 Oviedo, Asturias, SPAINshmarev@uniovi.es

We present new results on the global regularity prop-erties of solutions to parabolic equations which involvep(x) and p(x, t)-Laplace operators. The solutions of suchequations are usually understood in a weak sense. Inparticular, the time derivative is a distribution whichdoes not belong to any Lebesgue space. We find condi-tions on the data that guarantee the existence of strongsolutions. For these solutions, the second derivatives inspace and the first derivative in time belong to Lebesguespaces with variable exponents prompted by the equa-tion. It is shown that under certain conditions on thedata the solutions of the parabolic problem are contin-uous with respect to time in the sense of Holder andLipschitz. This is joint work with Prof. S.Antontsev.

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On a stochastic p(ω, t, x)-Laplace equation

Petra WittboldUniversitat Duisburg-Essen, 45117 Essen, GERMANYpetra.wittbold@uni-due.de

A stochastic forcing of a non-linear singular/degeneratedparabolic problem of p(ω, t, x)-Laplace type is proposedin the framework of Orlicz-Lebesgue and Sobolev spaceswith variable random exponents. We give a result of ex-istence and uniqueness of the solution, for additive andmultiplicative problems.This is joint work with Guy Vallet and Aleksandra Zim-mermann.

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Sufficient conditions for local regularity to the general-ized Newtonian fluid with shear thinning viscosity

Joerg WolfChung-Ang University, 156-756 Seoul, KOREAjwolf2603@gmail.com

We prove the local regularity of a weak solution u to theequations of a generalized Newtonian fluid with powerlaw 1 < q < 2 if u belongs to a suitable Lebesguespace. This result extends the well-known Serrin con-dition for weak solutions of the Navier-Stokes equationsto the shear-thinning fluids.

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S.16 Pseudo Differential Operators

OrganisersShahla Molahajloo, Man Wah Wong

Scope of the session: This is intended to be a specialsession in the mathematics of pseudo-differential opera-tors to be interpreted in a very broad sense. Topics in-clude functional analysis, operator theory and operators,harmonic analysis, partial differential equations, appli-cations and computation related to pseudo-differentialoperators.

—Abstracts—

Two-wavelet curvelet localization operators and two-wavelet curvelet multipliers

Viorel CatanaUniversity Politehnica of Bucharest, ROMENIAcatana viorel@yahoo.co.uk

The main aim of this talk is to introduce and study thetwo-wavelet curvelet localization operators and the two-wavelet curvelet multipliers.To this end we give a resolution of the identity for-mula for high frequency signals (in fact for images) inL2

2/a0,π−√a0

(R2) = f ∈ L2(R2); f(ξ) = 0, if |ξ| < 2/a0

or |arg ξ| > π−√a0 for some a0 ∈ (0, π2) and anotherone for arbitrarily signals f in L2(R2).Let X = (0, a0) × R2 × [−π, π] be the measure spaceequipped with the measure dµ = da

a3dbdθ. Then in the

following two theorems we give the two resolution iden-tity formulae mentioned above.Theorem 1. Let f ∈ L2

2/a0,π−√a0

(R2). Then

(f, g) =∫X

(f, γabθ)(γabθ, g)dµ, for all g ∈ L2(R2), whereγabθ, γabθ : R2 → C are functions defined by means ofradial windows W, W and angular windows V, V , whichhave some suitable properties.Theorem 2. For all f, g ∈ L2(R2), (f, g) =∫R2(f,Φb)(Φb, g) +

∫X

(f, γabθ)(γabθ, g)dµ, where Φ and

Φ are father wavelets, which satisfies some conditionsand for all b ∈ R2 we defined Φb(x) = Φ(x− b), x ∈ R2.In the following we give some boundedness resultsconcerning the two-wavelet curvelet localization oper-ators Lτ : L2

2/a0,π−√a0

(R2) → L2(R2) defined by

(Tτf, g) =∫Xτ(a, b, θ)(f, γabθ)(γabθ, g)dµ for all f ∈

L22/a0,π−

√a0

(R2) and g ∈ L2(R2), which have their sym-

bols τ in L∞(X) ∪ L1(X). We also state and provesome boundedness results referring to the two-waveletcurvelet multiplers Tσ : L2(R2) → L2(R2) defined by(Tσf, g) =

∫R2 σ(b)(f,Φb)(Φb, g)db, for all f and g in

L2(R2) with their symbols σ in L∞(R2)∪L1(R2). In ad-dition some results concerning the trace class propertyand the trace of the two-wavelet curvelet multipliers aregiven.

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Pseudo-differential operators of Gevrey type and theiraction on classes of modulation spaces

Sandro CoriascoDipartimento di Matematica “G. Peano”, Universitadegli Studi di Torino, V. C. Alberto, n. 10, 10123 Torino,ITALYsandro.coriasco@unito.it

We deduce one-parameter group properties for pseudo-differential operators op(a), where a belongs to symbolclasses of Gevrey type, associated with suitable, non-vanishing, weight functions ω0, and denoted by Γ

(ω0)∗ .

This allows to show that there are pseudo-differentialoperators op(a) and op(b) which are inverses to each

others, and satisfy a ∈ Γ(ω0)∗ and b ∈ Γ

(1/ω0)∗ .

We apply these results to deduce lifting properties forclasses of modulation spaces and construct isomorpismsbetween them. In particular, for any couple of admissi-ble weight functions ω, ω0, we prove that the Toeplitzoperator (or localization operator) Tp(ω0) is an iso-morphism from the weighted modulation space Mp,q

(ω)

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Abstracts

to the weighted modulation space Mp,q(ω/ω0), for every

p, q ∈ (0,∞].This is joint work with A. Abdeljawad and J. Toft.

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Pseudo-Differential Operators and Existence of GaborFrames

Gianluca GarelloDipartimento di Matematica Universita di Torino, ViaCarlo Alberto 10, 10123 Torino, ITALYgianluca.garello@unito.it

In this talk we present, from a a pseudo-differentialpoint of view, the behavior of the frame operator asso-ciated with a Gabor system. In particular we show howan application of the classical boundedness theorem ofCalderon -Vaillancourt yields sufficient conditions for aGabor system to form a frame in L2(Rd).

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Continuity of linear operators on mixed-norm Lebesgueand Sobolev spaces

Ivan IvecUniversity of Zagreb, CROATIAivan.ivec@gmail.com

Mixed-norm Lebesgue spaces found their place in thestudy of some questions in the theory of partial differ-ential equations, as it can be seen from recent interestin continuity of certain classes of pseudodifferential op-erators on these spaces. We present a general frame-work for dealing with continuity of linear operators onthese spaces. This allows us to prove the boundednessof a large class of pseudodifferential operators, and alsothe boundedness of integral operators on mixed-normLebesgue spaces. In some cases, the generalisations tomixed-norm Sobolev spaces are obtained as well, to-gether with applications to some interpolation and com-pactness results.

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C∗-algebras, H∗-algebras and trace ideals of pseudo-differential operators on locally compact, Hausdorff andabelian groups

Vishvesh KumarDepartment of Mathematics: Analysis, Logic and Dis-crete Mathematics, Ghent University, Krijgslaan 281,Building S8, B 9000 Ghent, BELGIUMvishveshmishra@gmail.com

In this talk, we define pseudo-differential operators ona locally compact, Hausdorff and abelian group G asnatural extensions of pseudo-differential operators onRn. In particular, for pseudo-differential operators withsymbols in L2(G × G), where G is the dual group ofG, we give explicit formulas for the products and ad-joints, characterize them as Hilbert–Schmidt operatorson L2(G) and prove that they form a C∗-algebra, whichis also a H∗-algebra. We give a characterization of traceclass pseudo-differential operators in terms of symbolslying in a subspace of L1(G× G) ∩ L2(G× G).

This is a joint work with Prof. M. W. Wong (York Uni-versity).

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Pseudos with limited smoothness applied to strictly hy-perbolic Cauchy problems with coefficients low-regularin time and space

Daniel LorenzTU Bergakademie Freiberg, Akademiestr 6, 09599Freiberg, GERMANYdaniel.lorenz@math.tu-freiberg.de

Pseudodifferential operators are a well-known tool tostudy smooth hyperbolic Cauchy problems. Even if thecoefficients are low-regular in time, we are still ableto obtain well-posedness results by using classical orweighted symbol spaces and the related operators.In this talk, however, we consider strictly hyperbolicCauchy problems with coefficients low-regular in t andx. By this we mean that the coefficients have a modulusof continuity below Lipschitz in t and belong to the Zyg-mund space C∗s in x. Our goal is to make s as small aspossible and still have an energy estimate with no lossof derivatives.To prove an energy estimate for this problem, we applypseudodifferential operators with limited smoothness inx. We discuss how results for mapping properties, ad-joints and sharp Garding’s inequality influence our finalresult. The talk is concluded by some examples.

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Paley-Wiener Theorems of real type in ultradifferen-tiable spaces

Alessandro OliaroDepartment of Mathematics, Universita di Torino, ViaCarlo Alberto, 10, 10123 Torino, TO, ITALYalessandro.oliaro@unito.it

In this talk we present some kind of “real Paley-Wienertheorems” for classes of rapidly decreasing ultradiffer-entiable functions. The word “real” expresses that in-formation about the support of f comes from growthrates associated to the function f on Rd rather than onCd as in the classical (complex) Paley-Wiener theorems.This theory was initiated by Bang and Tuan, and herewe follow the approach of Andersen and De Jeu, whoprove characterizations, relating support to growth, ofthe image of the Schwartz spaces of rapidly decreasingfunctions under the Fourier transform. In this talk weconsider as functional setting the space Sω of ultradiffer-entiable functions of Beurling type and investigate therelations between the support of the Fourier transformof an ultradifferentiable function f and the behaviour off , its derivatives, and its Wigner transform. We intro-duce results of this type for the so-called Gabor trans-form and give a full characterization in terms of Fourierand Wigner transforms for several variables of a Paley-Wiener theorem in this general setting.

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Fredholm Property of Non-Smooth PseudodifferentialOperators

Christine PfeufferUniversitat Halle, Halle (Saale), GERMANYChristine.Pfeuffer@mathematik.uni-halle.de

Hence great effort already was spent to get some con-ditions for the Fredholmness of smooth pseudodiffer-ential operators with symbols in the Hormander-class

76

Abstracts

Smρ,δ(Rn×Rn). Finally Schrohe was able to show that theuniformly ellipticity of certain smooth symbols is a nec-essary and sufficient condition for the Fredholmness ofthe associated pseudodifferential operator considered asa map between certain weighted Bessel potential spaces.In applications also non-smooth pseudodifferential op-erators occur. The goal of this talk is to give sufficientcondiditions for the Fredholm property of non-smoothpseudodifferential operators with symbols in the classCτS0

1,0(Rn × Rn;N).The talk is based on a joint work with H. Abels.

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Analytic pseudo-differential calculus via the Bargmanntransform

Joachim ToftDepartment of Mathematics, Vejdes plats 6,7, LinnaeusUniversity, 35195 Vaxjo, Smaland, SWEDENjoachim.toft@lnu.se

The Bargmann transform is a transform which mapsFourier-invariant function spaces and their duals to cer-tain spaces of formal power series expansions, whichsometimes are convenient classes of analytic functions.In the 70th, Berezin used the Bargmann transform totranslate problems in operator theory into a pseudo-differential calculi, where the involved symbols are an-alytic functions, and the corresponding operators mapsuitable classes of entire functions into other classes ofentire functions. Recently, some investigations on cer-tain Fourier invariant subspaces of the Schwartz spaceand their dual (distribution) spaces have been performedby the author. These spaces are called Pilipovic spaces,and are defined by imposing suitable boundaries on theHermite coefficients of the involved functions or dis-tributions. The family of Pilipovic spaces contains allFourier invariant Gelfand-Shilov spaces as well as otherspaces which are strictly smaller than any Fourier in-variant non-trivial Gelfand-Shilov space. In the sameway, the family of Pilipovic distribution spaces containsspaces which are strictly larger than any Fourier invari-ant Gelfand-Shilov distribution space. In the talk weshow that the Bargmann images of Pilipovic spaces andtheir distribution spaces are convenient classes of an-alytic functions or power series expansions which aresuitable when investigating analytic pseudo-differentialoperators. We also deduce continuity properties for suchpseudo-differential operators when the symbols and tar-get functions possess certain (weighted) Lebesgue esti-mates. We also show that the counter image with re-spect to the Bargmann transform of these results gener-alise some continuity results for (real) pseudo-differentialoperators with symbols in modulation spaces, when act-ing on other modulation space. The talk is based oncollaborations with Nenad Teofanov from the Univer-sity of Novi Sad, and the content of the talk is availableat

N. Teofanov, J. Toft Pseudo-differential calculus in aBargmann setting, Ann. Acad. Sci. Fenn. Math. (toappear).

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Time-frequency analysis and pseudo-differential opera-tors on locally compact groups

Ville Turunen

Aalto University, Department of Mathematics and Sys-tems Analysis, FI-00076, FINLANDville.turunen@aalto.fi

Time-frequency analysis can be described as Fourieranalysis simultaneously both in time and in frequency.Its origins are in quantum mechanics, in signal process-ing in Euclidean spaces, and in pseudo-differential op-erators. In this presentation we show how to generalizetime-frequency analysis to those locally compact groupsthat allow a nice-enough Fourier transform: wide fam-ilies of non-commutative groups can be treated. Ourresults on locally commutative groups shed new lightalso on the analysis in Euclidean spaces.

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Defect distributions applied to differential equationswith polynomial coefficients

Ivana VojnovicDepartment of Mathematics and Informatics, Facultyof Sciences, University of Novi Sad, Trg DositejaObradovica 4, 21000 Novi Sad, SERBIAivana.vojnovic@dmi.uns.ac.rs

H-distributions were introduced by Antonic andMitrovic (2011) for weakly convergent sequences in dualpair of Lp − Lq spaces.Defect distributions are extension of H-distributions.They are connected with weakly convergent sequencesin weighted Sobolev spaces Hs,p

Λ (Rd). We prove exis-tence of defect distributions using weighted type classesof symbols and global pseudo-differential calculus. De-fect distributions related with elliptic symbols give usconnection between weakly and strongly convergent se-quences in appropriate Sobolev spaces.We apply results in order to analyze equations with poly-nomial coefficients. We consider sequence of approxi-mate equations ∑

(α,β)∈V (P)

xβDαun(x) = fn(x)

and obtain information about the set of points whereweakly convergent sequence un is (locally) strongly con-vergent.This is joint work with Stevan Pilipovic.

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Strong continuity on Shubin-Sobolev spaces of semi-groups for evolution equations of quadratic operators

Patrik WahlbergLinnaeus University, Department of Mathematics, SE-35195 Vaxjo, SWEDENpatrik.wahlberg@lnu.se

We consider the initial value Cauchy problem for evolu-tion equations of Schrodinger type defined by a Hamil-tonian that is the Weyl quantization of a quadraticform with nonnegative real part. We show that thecorresponding solution semigroup is strongly continuouson Shubin-Sobolev spaces of integer order. This givesuniqueness and continuity of solutions when the initialdatum is a tempered distribution.

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Abstracts

Normality, Self-Adjointness, Spectral Invariance,Groups and Determinants of Pseudo-Differential Oper-ators on Finite Abelian Groups

K. L. WongSchool of Mathematics and Statistics, Carleton Uni-versity, 1125 Colonel By Dr, Ottawa, ON K1S 5B6,CANADAKelseeWong@cmail.carleton.ca

We give the normality, self-adjointness and spectral in-variance of pseudo-differential operators on finite abeliangroups. We also give a formula for the determinant ofevery element in a group of pseudo-differential operatorson a finite abelian group.

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S.17 Quaternionic and Clifford Analysis

OrganisersSwanhild Bernstein, Uwe Kahler,Irene Sabadini, Franciscus Sommen

Scope of the session: Quaternionic and Clifford analy-sis are higher dimensional analogues of complex analysisand represent a function theory in Rn. In the last yearsquaternionic and Clifford analysis spread into several dif-ferent directions such as continuous and discrete theory,hermitian Clifford analysis, higher spin operators, slicemonogenic functions as well as new applications to var-ious fields. This session aims to present recent advancesin the field of continuous and Clifford analysis as well asits applications in numerical analysis of PDE’s, signaland image processing, operator theory, and physics to abroad audience.

—Abstracts—

A general setting for functions of Fueter variables: dif-ferentiability, rational functions, Fock module and re-lated topics

Daniel AlpaySchmid College of Science and Technology, ChapmanUniversity, Orange, California 92866, USAalpay@chapman.edu

We develop some aspects of the theory of hyperholo-morphic functions whose values are taken in a Banachalgebra over a field – assumed to be the real or the com-plex numbers – and which contains the field. Notably,we consider Fueter expansions, Gleason’s problem, thetheory of hyperholomorphic rational functions, modulesof Fueter series, and related problems. Such a frameworkincludes many familiar algebras as particular cases. Thequaternions, the split quaternions, the Clifford algebras,the ternary algebra, and the Grassmann algebra are afew examples of them.This is joint work with Ismael Paiva and DanieleStruppa.

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Sufficient Conditions for Associated Operators to aSpace of Harmonic-Type functions

Eusebio Ariza GarciaYachay Tech University, 100150 Ibarra, Imbabura,

ECUADOREariza@yachaytech.edu.ec

In this talk, we deal with the second-order differen-tial operator ∆ = div(B∇) and the first-order differen-

tial operator F =

n∑i=0

a(i)(x)∂xiu(x) + b(x)u(x) + c(x),

where B is a symmetric and positive definite matrixin Rn+1×n+1, x = (x0, x1, . . . , xn) ∈ Rn+1, a(i), u ∈C2(Rn+1;An), b, c ∈ C1(Rn+1;An) and An is the classicClifford algebra with structure relations

eiej + ejei = −2δij , i, j = 1, 2, . . . , n.

We give sufficient conditions for operators ∆ and F tobe associated. We also explore the necessary conditionsand, provided a bounded domain Ω is given, we exhibitan interior estimate for harmonic functions in the supre-mum norm. This is could be used to study the existenceand uniqueness of solutions of some particular initialvalue problems.

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A Dirac operator in infinite dimensional analysis

Swanhild BernsteinTU Bergakademie Freiberg, Institute of Applied Analy-sis, Pruferstr. 9, 09599 Freiberg, GERMANYswanhild.bernstein@math.tu-freiberg.de

We aim to construct a Dirac operator in the setting ofinfinite dimensional analysis. Let H be a real Hilbertspace and µ = NQ a non-degenerate Gaussian measureon H. We denote by (ξk) a complete orthonormal sys-tem in H and by (λk) a sequence of positive numberssuch that Qξk = λkξk and by E(H) the space of all ex-ponential functions. For any ϕ ∈ E(H) and any k ∈ Nthen

Dkϕ(x) = limε→0

1

ε[ϕ(x+ εξk)− ϕ(x)] , x ∈ H

and the Dirac operator is defined as

D =

∞∑k=0

ekDk,

where (ek) is a sequence of generating vectors of an infi-nite dimensional Clifford algebra such that eiej +ejei =δij .The mapping

D : (E) ⊂ L2(H,µ)→ L2(H,µ;H), ϕ 7→ Dϕ

is closeable. Moreover, D2 = −∆G, the Gross Lapla-cian.

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On Runge Pairs and Topology of Axially Symmetric Do-mains.

Cinzia BisiDipartimento di Matematica ed Informatica, Universitadegli Studi di Ferrara, Via Machiavelli 30, I-44121, Fer-rara, ITALY.bsicnz@unife.it

Subsequently to the very interesting and seminal pa-per of Colombo, Sabadini and Struppa in 2011, we willpresent a new theorem on Runge Pairs of Axially Sym-metric Domains in the quaternionic setting, that can be

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easily generalized to the Clifford environment. A globalapproach (in which we strongly believe) and a completestudy of the homology groups of these domains, insteadof a slicewise analysis, have permitted us to improve theresults in Colombo, Sabadini and Struppa’s original pa-per and a more deep understanding of the subject. Thisis part of a joint project with J. Winkelmann.

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Special classes of monogenic functions and applicationsin linear elastic fracture mechanics

Sebastian BockBauhaus-Universitat Weimar, Professur AngewandteMathematik, Coudraystraße 13B, 99423 Weimar,Thuringia, GERMANYsebastian.bock@uni-weimar.de

Recently, the classical orthogonal function systems of in-ner and outer monogenic Appell functions were used tofind new classes of monogenic functions with (logarith-mic) line singularities, which extend the known functionclasses in a natural way. In the talk we will discusssome essential properties (orthogonality, Appell prop-erty, three-term recurrence relation) of the classical andthe special classes of monogenic functions as well as theirrelations to each other. Furthermore, on the basis ofthe generalized Kolosov-Muskhelishvili formulas, it willbe shown how these special functions can be used toconstruct analytical near-field solutions of spatial crackfronts in linear elastic fracture mechanics.

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Interpolation by polynomials over division rings

Vladimir BolotnikovDepartment of Mathematics, The College of Williamand Mary, 23185-8795 Williamsburg, VA, USAvladi@math.wm.edu

Polynomials over a division ring F can be evaluated“from the left” and “from the right” giving rise to leftand right interpolation problems. If the interpolationproblem involves interpolation conditions of the same(left or right) type, the results are similar to the commu-tative case: a consistent problem has a unique solutionof a low degree (less than the number of interpolationconditions imposed), and the solution set of the homo-geneous problem is an ideal of F[z]. The problem con-taining both “left” and “right” interpolation conditionsis quite different: there may exist infinitely many low-degree solutions and the solution set of the homogeneousproblem is a quasi-ideal in F[z]. In the talk, we willdiscuss the two-sided Lagrange interpolation formula,two-sided polynomial independence, and extensions ofpolynomials within a given conjugacy class.

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GASPT, Radially Holomorphic Functions, the FueterPotential Method in an Inhomogeneous Medium andProblems of Generalized Joukowski Transformations inR3

Dmitry BryukhovMagnit Ltd, Mira avenue 17, 141196 Fryazino, Moscowregion, RUSSIA FEDERATIONbryukhov@mail.ru

The Fueter potential method in the setting of GASPTis provided. An axially symmetric generalization of theCauchy-Riemann system in R3 = (x0, x1, x2), includ-ing the longitudinal variable x0 and the cylindrical ra-dial variable ρ =

√x2

1 + x22, and further the Cauchy-

Riemann system in the meridian plane, spanned by(x0, ρ), play the main roles. New classes of Fuetermappings of the first kind and the second kind are de-fined. Corresponding classes of irrotational meridionalvelocity fields in an axially symmetric inhomogeneousmedium are described as gradient dynamical systemswith variable dissipation. Problems of the sets of crit-ical points of the scalar velocity potential and prob-lems of singular sets of spatial axially symmetric gen-eralizations of conformal mappings of the second kindare studied. Fueter mappings of the second kind, char-acterized by the Hessian matrix J , allow us to developimportant applications of radially holomorphic functionsu = u0 + Iuρ = u(x), where x = x0 + Iρ. HerewithI = ix1+jx2

ρ, Iuρ = iu1 + ju2 and i2 = j2 = I2 = −1.

The Fueter holomorphic primitive U = U0 +IUρ = U(x)of a radially holomorphic function u = u(x), satisfy-ing conditions U ′ = ∂U

∂x0= u, where IUρ = iU1 + jU2,

plays a key role of Fueter potential in continuum me-chanics. In particular, properties of models of general-ized Joukowski transformations of order n in the formU(x) = 1

2(xn + x−n) are described. The Fueter poten-

tial method assumes three principal invariants IJ , IIJ ,IIIJ and three real roots λ1, λ2, λ3 of the correspondingcharacteristic equation. Theorem. Roots of the char-acteristic equation λ3 − IJλ2 + IIJλ − IIIJ = 0 in theframework of the Fueter potential method are describedby formulas:

λ1 =uρρ, λ2,3 = ±

√(∂uρ∂x0

)2

+

(∂uρ∂ρ

)2

.

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Some applications of the Spectral Theory on the S-spectrum

Fabrizio ColomboPolitecnico di Milano, 20133 Milano, ITALYfabcol27@gmail.com

In this talk we give an overview of the spectral the-ory based on the notion of S-spectrum. This theoryturned out very useful to define new classes of frac-tional diffusion and fractional evolution processes. Wewill show new results on the quaternionic version of theH∞-functional calculus and we use it to compute thefractional powers of vector operators. Such fractionalpowers of generalized gradient operators define non-localFourier’s? laws for the propagation of the heat.

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A right inverse operator for curl + λI and applications

Briceyda Berenice DelgadoBolshaya Sadovaya 105/42, Rostov-on-Don, 344006 Ros-tov, RUSSIAN FEDERATIONbriceydadelgado@gmail.com

In this talk will be presented a general solution of theequation

curl−→w + λ−→w = −→g ,

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Abstracts

where λ is an arbitrary non-zero complex number and−→g belongs to the class of Lp-integrable functions whosedivergence is also Lp-integrable. In other words, a rightinverse operator for the operator curl+λI is constructedin this class of integrable functions. The explicit generalsolution is based on the use of classical integral operatorsof quaternionic analysis as well as on the constructionof metaharmonic conjugate functions. Applications ofthe above result are considered to the nonhomogeneoustime-harmonic Maxwell system.

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On the Bargmann-Fock-Fueter and Bergman-Fueter in-tegral transforms (Joint work with Prof. Krausshar andProf. Sabadini)

Kamal DikiMilano, ITALYkamal.diki@polimi.it

The classical complex Fock space and associated Segal-Bargmann transform define some important mathemat-ical models used in quantum mechanics. In this talk, wepresent some special integral transforms of Bargmann-Fock type in the setting of quaternions. Our construc-tions are based on the well-known Fueter mapping theo-rem. In particular, starting with the normalized Hermitefunctions we construct an Appell system of quaternionicregular polynomials. The ranges of such integral trans-forms are quaternionic reproducing kernel Hilbert spacesof regular functions. Some new integral representationsand generating functions in this setting are obtained inboth the Fock and Bergman cases.

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Sufficient Conditions for the Fractional Vekua equationto be Associated with the fractional Cauchy-Riemannoperator in the Riemann-Liouville sense

Antonio Di TeodoroUniversidad San Francisco de Quito, Diego de Robles yVıa Interoceanica, 170407 Quito, Pichincha, ECUADOR

nditeodoro@usfq.edu.ec

Consider the fractional Cauchy-Riemann operatorsDα,β

+ = Dα

x+0+ iDβ

x+1in the Riemann-Liouville sense.

Where Dα,β+ is its conjugate. With this operator, we

can write the fractional Vekua equation in the Riemann-Liouville sense:

∂tω =

ADα,β+ ω +BDα,β

+ ω + CDα,β+ ω + EDα,β

+ ω + Fω +Gω

+H = Ψω.

where A,B,C,E, F,G and H are complex valued func-tions and t is the time. Ψ is a linear first order operatoracting in the complex plane.On the other hand. A pair (L, l) of fractional differen-tial operators is said to be associated if l(u) = 0 impliesl(Lu) = 0.In this work we formulate sufficient conditions on thecoefficients of operator Ψω under which this operator isassociated to the fractional Cauchy-Riemann operatorin the Riemann-Liouville sense, using a matrix repre-sentations (this is: if Dα,β

+ ω = 0 then Dα,β+ (Ψω) = 0).

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Hyperbolic function theory and hyperbolic Brownianmotion

Sirkka-Liisa ErikssonDepartment of Mathematics and Statistics, Universityof Helsinki, P.O. Box 64, FI-00014, Helsinki, FINLAND

sirkka-liisa.eriksson@helsinki.fi

We study hyperbolic function theory of quaternion val-ued functions depending on three coordinates. A func-tions is called α-hyperbolic harmonic if it is harmonicwith respect to the Riemannian metric

ds2k =

dx20 + dx2

1 + dx22

x2α2

in the upper half space R3+ = (x0, x1, x2, ) ∈ R3 : x2 >

0. An interesting result is that a real valued functionf is α-hyperbolic harmonic if and only if the function

g (x) = x1−α2

3 f (x) is the eigenfunction of the hyperbolicLaplace operator 4h = x2

24 − x2∂∂x2

corresponding to

the eigenvalue 14

((α+ 1)2 − 4

)= 0. This means that

in case α = 1, the 1-hyperbolic harmonic functions areharmonic with respect to the hyperbolic metric of thePoincare upper half-space. Leutwiler started to studythis case around 1990 and noticed that the quaternionicpower function xm (m ∈ Z), is a conjugate gradient of a1-hyperbolic harmonic function.The theory was generalized to total quaternion val-ued functions by defining modified Dirac operators andα-hypermonogenic functions. Recently, we verified aCauchy type integral formula with more concrete kernelsfor a-hypermonogenic functions. In this talk, we are pre-senting some connections of hyperbolic function theoryto the hyperbolic Brownian motion. Hyperbolic Brown-ian motion is defined in terms of stochastic differentialequations. The hyperbolic Brownian motion is a canon-ical diffusion on hyperbolic spaces and its generator isthe half of the hyperbolic Laplace-Beltrami operator.

———

The discrete Cauchy-Kovalevskaya extension originat-ing from fractional central difference operators

Nelson FaustinoUniversidade Federal do ABC, CEP 09210-580 SantoAndre, Sao Paulo, BRAZILnelson.faustino@ufabc.edu.br

On this talk we will report on recently ongoing resultsrelated to discrete time-changed fractional equations onthe space-time lattice hZn × (1− α)τZ of the form

e0

(sinh((1− α)τ∂t)

(1− α)τ

)βΨ(x, t)− e0φα,β (t; τ) Φ0(x)

= −θ(2α− β) DhΨ(x, t)− e2n+1β − 2α

2βτβ∆hΨ(x, t)

(e20 = −1, e2

2n+1 = +1 & 0 ≤ α, β < 1), carrying the

fractional central difference operator(

sinh((1−α)τ∂t)(1−α)τ

)βand two given discretizations Dh resp. ∆h of the Diracand Laplace operator on the lattice hZn satisfying thefactorization property (Dh)2 = −∆h.

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Abstracts

The functions θ(2α − β) resp. φα,β(t; τ) correspond tothe Heaviside step function and to the fractional regu-larization of the sinc function on the limit β → α+:

limβ→α+

φα,β(t; τ) = (1− α)τsin(

πt(1−α)τ

)πt

.

At a first glance, we will show that the solution ofthe above equation seamlessly describes the solutionassociated to the discrete counterpart of the Cauchy-Kovalevskaya extension.Secondly, we will employ Fourier analysis techniques onthe space-time toroidal manifold Rn/ 2π

hZn×R/ 2π

τ(1−α)Z

to represent(

sinh((1−α)τ∂t)(1−α)τ

)βas a discrete convolution

operator and to obtain a representation formula for theaforementioned equation. In case where φα,β(t; τ) equalsto

cos(

πt(1−α)τ

)Γ(β − α+ 1)

Γ(β−α+1

2+ t

(1−α)τ+ 1)

Γ(β−α−1

2− t

(1−α)τ+ 1)

one will also show that the underlying solution maybe represented my means of Fourier-Laplace multipliers,carrying the generalized Mittag-Leffler function Eβ,α(z).Special emphasize will be also given to the discrete Diracsemigroup exp(e0tDh)t≥0 (2α−β ≥ 0 & τ → 0+) andto the cases where the above equation resembles to astochastic process of Wiener type in the limit α→ 1−.

———

Real Paley-Wiener Theorem for Clifford-Fourier Trans-from

Minggang FeiSchool of Mathematical Sciences, University of Elec-tronic Science and Technology of China, Chengdu, PEO-PLE’S REPUBLIC OF CHINAfei@uestc.edu.cn

In the framework of Clifford analysis, we consider severalversions of the real Paley-Wiener theorems for a gener-alized Clifford-Fourier transform. This Clifford-Fouriertransform is given by a similar operator exponential asthe classical Fourier transform but containing generatorsof Lie superalgebra. In addition, some related results fora fractional version of Clifford-Fourier transform are alsopresented at the end of this talk.

———

THE commutative diagram of signal processing from aClifford perspective

Brigitte Forster-HeinleinUniversitat Passau, Fakultat fur Informatik und Math-ematik, 94032 Passau, GERMANYbrigitte.forster@uni-passau.de

The commutative diagram of signal processing consistsof the Fourier Series, the Paley-Wiener theorem andthe Sampling Theorem. In the L2-setting, the the-ory is well known, for the classical case as well as forthe case of nonharmonic Fourier series and entire func-tions of exponential type with various growth condi-tions. For Lp(R)-spaces, p 6= 2, the commutative dia-gram was established by Maergoiz in 2006. For entire

functions with polygonal indicator diagram partial re-sults are given by Levin/Ljubarski in 1975 and Semm-ler/F. in 2015. Recently, the Clifford community devel-oped Paley-Wiener theorems for multivariate functionson Clifford-Hilbert-spaces, see e.g. Kou/Tian 2002 andFranklin/Hogan/Larkin 2017. In the presentation, weshow the developments of THE commutative diagramwith respect of these aspects and give first results for theClifford setting. This is research in progress in collab-oration with Jeff Hogan, University of Newcastle, Aus-tralia.

———

A Clifford Construction of Multidimensional ProlateSpheroidal Wave Functions

Hamed Baghal GhaffariUniversity Dr, Callaghan NSW 2308, 2298 Newcastle,NSW, AUSTRALIAhamed.baghalghaffari@uon.edu.au

We review the one dimensional prolate spheroidal wavefunctions (PSWFs) including their construction and im-portance as bandlimited functions. We then, recall thehigher dimension version of PSWFs, the radial parts ofwhich are the eigenfunctions of the operator

Mc(u)(t) = (1− t2)d2u

dt− 2t

du

dt+ (−c2t2 +

14−N2

t2)u.

which has a singularity at the origin, causing instabil-ities. We investigate the construction of multidimen-sional PSWFs using techniques from Clifford analysis.The Clifford-Legendre differential equation is given by

∂2xC

0n,m(Pk)(x)− 2x∂xC

0n,m(Pk)(x)

−C(0, n,m, k)C0n,m(Pk)(x) = 0,

where ∂x is Dirac operator on Rm. We will find the Bon-net formula for the Clifford-Legendre polynomials andthen we define the following operator

Lcf(x) = ∂x((1− |x|2)∂xf(x)) + 4π2c2|x|2f(x),

which is Hermitian. We refer the eigenfunctions ofthe latter operator as ”Clifford multidimensionalPSWFs.” By the use of the Bonnet formula, we willtry to compute the Clifford multidimensional PSWFs.Then we will prove that they are also the eigenfunctionsof a time-frequency limiting operator in L2(B(1),R2).We will define the time-frequency limiting operator Gcby

Gcf(x) = χB(1)(x)

∫B(1)

e2πic〈x,y〉f(y)dy,

and show that Lc and Gc commute, a phenomenonknown as the ”lucky accident.”

———

Slice regular functions in several variables

Riccardo GhiloniVia Sommarive n. 14, 38123 Povo-Trento, Trento (TN),ITALYriccardo.ghiloni@unitn.it

In this talk we present basic concepts and results of anew function theory in several hypercomplex variables,including quaternions, octonions and Clifford algebras.This new function theory have been obtained jointlywith Alessandro Perotti (University of Trento).

———

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Abstracts

Compressed Sensing with Slice Monogenic Signals

Narciso GomesFaculdade de Ciencias e Tecnologia, Campus de Pal-marejo, 0279 Praia, Ilha de Santiago, CAPE VERDE

narciso.gomes@docente.unicv.edu.cv

Here we propose the reconstruction of signals by basis-Pursuit, i.e., by `1-minimization and the sampling modelis random points with high probability by using the sliceregular monogenic functions.

———

Radial integration operators in infinite domains andtheir applications in quaternion analysis

Yuri Grigor’ev, Andrei Iakovlev58 Belinsky str, Yakutsk, Republic of Sakha (Yakutia),Russia, 677027grigyum@yandex.ru

Radial integration operators Iα in bounded star-shapeddomains are the effective tool in the mathematicalphysics and quaternion analysis. The first three-dimensional analogue of the Kolosov-Muskhelishvili for-mulae was obtained by means of such operators whena general solution of the Lame equation for the spatialtheory of elasticity was expressed in terms of two regu-lar quaternionic functions. In this report we introducenew radial integration operators Jα acting in an infinitespace with a star-shaped cavity. It is shown that opera-tors Jα have properties similar to Iα. By means of theseoperators Jα some problems of quaternion analysis aresolved. Three-dimensional quaternionic analogue of theKolosov-Muskhelishvili formulae in infinite space with astar-shaped cavity is constructed. As an application, theboundary value problem of equilibrium of infinite elasticspace with spherical cavity is solved in a closed form.

———

A distributional approach to integration over smoothsurfaces of lower dimension embedded in Rm

Ali Guzman AdanGhent University, Department of Mathematical Analy-sis, Krijgslaan 281, 9000 Gent, Oost-Vlaanderen, BEL-GIUMali.guzmanadan@ugent.be

Methods of integration have been amply developed forparametrically defined (m − k)-surfaces in Rm, i.e. de-fined as the image of a map from a subset of Rm−kto Rm. In this case, the use of differential forms playsan important role. However, some methods of calcula-tion are also necessary in the case where an (m − k)-surface is implicitly defined by means of k equationsP1(x) = · · · = Pk(x) = 0. In this talk, we illustratea distributional method for oriented and non-orientedintegration over implicit surfaces. This method followsfrom the link between differential forms and the Diracdistribution concentrated on manifolds. This approachwill be applied to compute integrals over real Stiefelmanifolds and to obtain a distributional Cauchy theo-rem for the tangential Dirac operator.

———

Quaternionic splines, wavelets and prolates

Jeffrey HoganSchool of Mathematical and Physical Sciences – SR233,University of Newcastle – University Drive, 2308Callaghan, NSW, AUSTRALIAjeff.hogan@newcastle.edu.au

We investigate the use of Clifford analysis in the con-struction of

• quaternionic splines on the line

• quaternionic wavelets on the plane, and

• bandpass prolate spheroidal wave functions in ar-bitrary dimensions.

The first topic generalises constructions of fractionaland complex splines and the work has been donejointly with Peter Massopust (Munich). The secondtopic is joint work with David Franklin and AndrewMorris (Newcastle) and involves the Clifford-Fouriertransform and modern techniques of optimisation, es-pecially the Douglas-Rachford algorithm, to constructquaternion-valued, orthogonal, smooth, compactly sup-ported wavelets. The third topic is joint wok with JoeLakey (Las Cruces) and involves the analysis of Clifford-based generalised translation operators acting on ban-dlimited functions f : Rn 7→ Rn (where Rn is the naturalClifford algebra associated with Rn).

———

Time-Frequency Analysis in the Unit Ball

Mathias Ionescu-TiraGERMANYmathias.tira@mail.de

As an appropriate analogue of the Euclidean short-timeFourier transform we study a windowed version of theHelgason-Fourier transform on a hyperbolic manifold:the complex unit ball. This voice transform is con-structed by means of translations (composition with spe-cial Moebius transforms) and modulations (multiplica-tion with the kernel of the Helgason-Fourier transform)of a window function. In absence of a related grouprepresentation, the algebraic tools available in the tra-ditional setting have to be replaced in this context byregularity and decay conditions imposed on the fixedwindow. Under such assumptions, the voice transformcan be defined for distributions and displays propertiessimilar to those in the group setting. Modulation spaces(or more general, coorbit spaces) are defined in termsof weighted Lebesgue spaces, and, using partitions ofunity, atomic decompositions and Banach frames are ob-tained.

———

On the hypercomplex Airy function in the context ofquantum Yang-Mills theory

Igor KanatchikovNational Quantum Information Center, Ul. Gen Wl.Andersa, 81-831 Sopot, Polandkanattsi@gmail.com

I outline the quantum pure Yang-Mills theory whichis obtained by means of the precanonical quantization

82

Abstracts

based on the De Donder-Weyl (DW) Hamiltonian the-ory. It leads to a Dirac-like equation on the Clifford-algebra-valued wave function with the mass term re-placed by a the Clifford-algerba-valued DW Hamiltonianoperator which controls the mass spectrum of the the-ory. We show that the eigenvalue problem for the DWHamiltonian operator can be exactly solved in terms ofa certain hypercomplex generalization of the Airy func-tion expressed as the contour integral. We discuss someproperties of the solution and the estimation of the gapin the spectrum of the DW Hamiltonian operator.

———

Spectral Decomposition of Clifford algebras

Yakov KrasnovBar Ilan University, 5290002 Ramat Gan, ISRAELkrasnov@math.biu.ac.il

In this talk we find a Peirce decompositions of a complexClifford algebra Cln with respect to the complete set ofan idempotents. We also show that the above spectraldecomposition holds when Cln is nondegenerate.

———

Applications of slice-holomorphic functions to automor-phic forms and Bergman kernels

Soren KraußharUniversitat Erfurt, Fachgebiet Mathematik,Erziehungswissenschaftliche Fakultat, Nordhauser Str.63, 99089 Erfurt, GERMANYsoeren.krausshar@uni-erfurt.de

In this talk we describe the symmetry behavior of slice-holomorphic functions. In particular, we determinethe maximal invariance group taking slice-holomorphicfunctions into themselves. Concretely, we discusswhich kinds of Mobius transformations exactly mapslice-holomorphic functions onto other slice-holomorphicfunctions. We prove a conformal covariance propertyof the Cauchy integral formula. We discuss the prob-lem of isometry between related functions spaces. Fur-thermore, we explain the explicit construction of theBergman kernel for slice-holomorphic functions on theunit ball, the half unit ball and the quater unit ball.We round off our talk by giving explicit constructions ofautomorphic forms in the slice holomorphic setting andexplain their relations to Clifford-holomorphic automor-phic forms and classical holomorphic forms in severalcomplex variables. This is joint work with F. Colombo,K. Diki and I. Sabadini from the Depatment of Mathe-matics of the Politecnico di Milano,

———

Nonstable quaternionic and orthogonal complexes for kDirac operators

Lukas KrumpMathematical Institute of Charles University, Facultyof Mathematics and Physics, Sokolovska 83, 186 75Praha 8, CZECH REPUBLICkrump@karlin.mff.cuni.cz

We study the operator Dk = (∂1, . . . , ∂k), sometimescalled ”k Dirac operators” or ”Dirac in k variables”. Itacts on functions defined on the product Rm×. . .×Rm of

k copies of the Euclidean space with values in the basicspinor module SA of the (complexified) Clifford algebraCm.The question whether there is an analogue of the Dol-beault complex (called BGG-complex) for the operatorDk, has been studied for many years. The answer isknown in several cases and it is well known that therealso second order operators appear in the complex.The BGG-complex for Dk in dimension 4 is an exam-ple of so-called nonstable case and it can exist in twoversions, according to the choice of a symmetry group.This holds only in the dimension 4, where we can chooseeither quaternionic or orthogonal symmetry.We compare both versions of BGG-complexes and showthat the orthogonal complex, obtained by Penrose trans-form, is a refinement of the Baston quaternionic com-plex, obtained by branching rules.

———

Fischer decomposition in non-stable cases

Roman LavickaFaculty of Mathematics and Physics, Charles University,Sokolovska 83, 186 75 Praha, CZECH REPUBLIClavicka@karlin.mff.cuni.cz

For spinor valued polynomials in k vector variables of di-mension m, the Fischer decomposition has been recentlyproved in the stable rangem ≥ 2k, see arXiv:1708.01426.This decomposition was conjectured by F. Colombo,F. Sommen, I. Sabadini and D. Struppa in 2004. In thiscase, the symmetry is given by the orthogonal groupO(m) and uniqueness of the Fischer decomposition isequivalent to irreducibility of generalized Verma mod-ules for the Howe dual partner osp(1|2k) generated byspherical monogenics. Here spherical monogenics arepolynomial solutions of the Dirac equation in k vectorvariables. In this talk, we shall deal with two possible ex-tensions of this result. First we shall discuss non-stablecases and second a generalization for polynomials in kvector variables on superspaces. In both cases, structureof generalized Verma modules for osp(1|2k) plays an im-portant role.The talk is based on results obtained jointly withV. Soucek.

———

Application of the discrete potential theory to problemsof mathematical physics

Anastasiia LegatiukCourdraystr. 13B, R. 204, 99425 Weimar, GERMANYanastasiia.legatiuk@uni-weimar.de

Discrete potential theory is a natural extension of thecontinuous theory to functions defined on lattices. Theidea of the discrete potential theory is to transfer allimportant aspects of the continuous theory directly tothe discrete level. Direct formulation on the discretelevel is advantageous for applications in mathematicalphysics, since important physical quantities, such as forexample conservation laws or asymptotic conditions, aremodelled and satisfied on the discrete level and not onlyapproximated as in conventional approaches. Thus, inthis contribution, we present recent developments in thediscrete potential theory on rectangular lattices and itsapplication to planar boundary value problems of math-ematical physics in bounded domains.

83

Abstracts

———

On application of script geometry to finite element ex-terior calculus

Dmitrii LegatiukChair of Applied Mathematics, Bauhaus UniversityWeimar, Coudraystr. 13B, 99425 Weimar, GERMANYdmitrii.legatiuk@uni-weimar.de

Finite element method is probably the most popular nu-merical method used in different fields of applicationsnowadays. While approximation properties of the clas-sical finite element method, as well as its various mod-ifications, are well understood, stability of the methodis still a crucial problem in practice. Therefore, alterna-tive approaches based not on an approximation of con-tinuous differential equations, but working directly withdiscrete structures associated with these equations, havegained an increasing interest in recent years. Finite el-ement exterior calculus is one of such approaches. Thefinite element exterior calculus utilises tools of algebraictopology, such as de Rham cohomology and Hodge the-ory, to address the stability of the continuous problem.By its construction, the finite element exterior calculusis limited to triangulation based on simplicial complexes.However, practical applications often require triangula-tions containing elements of more general shapes. There-fore, it is necessary to extend the finite element exteriorcalculus to overcome the restriction to simplicial com-plexes. In this paper, the script geometry, a recentlyintroduced new kind of discrete geometry and calculus,is used as a basis for the further extension of the finiteelement exterior calculus.

———

Some Properties and Application of Teodorescu Oper-ators Associated with the Helmholtz Equation and theTime-harmonic Maxwell Equations

Wang LipingNo. 20 East South Second Ring Road, Hebei NormalUniversity, 050024 Shijiazhuang, Hebei Province, PEO-PLE’S REPUBLIC OF CHINAwlpxjj@163.com

In this paper, firstly, we discuss some properties ofTeodorescu operator Tψ,α related to the Helmholtz equa-tion and give the integral representation of the solutionto the Riemann boundary value problem related to theHelmholtz equation. Then, by using of the previousconclusions, we show some properties of the Teodor-escu operator associated with the N matrix operatorand give the integral representation of the solution tothe Riemann boundary value problem associated withthe N matrix operator. Finally, by using the relation-ship between the corresponding Cauchy type integral ofthe N matrix operator and the corresponding Cauchytype integral of the Time-harmonic Maxwell equations,we give the integral representation of the solution to theRiemann boundary value problem for homogeneous par-tial differential equations related to the Time-harmonicMaxwell equations.

———

On the structure of the singularities of bicomplex holo-morphic functions

Maria Elena Luna-Elizarraras

Golomb St 52, 5810201 Holon, ISRAELeluna 10@hotmail.com

The paper Luna-Elizarraras, Perez-Regalado, Shapiro“On the Laurent Series for bicomplex holomorphic func-tions”, Complex Variables and Elliptic Equations, Vol62, Issue 9, 2017, 1266-1286, gives a detailed descriptionof the structure and the main properties of the Laurentseries in bicomplex analysis. The aim of this talk isto apply the ideas and the results of it for developingthe theory of singular points for bicomplex holomorphicfunctions. A classification of such points is given; thebehaviour of bicomplex holomorphic functions near suchpoints is described; the peculiarities of this situation aresingled out. The talk is based on is a joint work withC.O. Perez-Regalado and M. Shapiro

———

Clifford-Valued Cone and Hex Splines

Peter MassopustCentre of Mathematics, Boltzmannstrasse 3, 85748Garching b. Munich, Bavaria, GERMANYmassopust@ma.tum.de

We introduce an extension of cone splines and boxsplines hypercomplex orders. These new families of mul-tivariate splines are defined in the Fourier domain alongcertain s-directional meshes and include as special cases3-directional box splines and hex splines found in theliterature. These cone and hex splines of hypercom-plex order generalize univariate complex and hypercom-plex B-splines. Explicit time domain representations arederived for these splines on 3-directional meshes. Wepresent some properties of these two multivariate splinefamilies, such as recurrence, decay and refinement. Fi-nally, we show that a bivariate hex spline and its integerlattice translates form a Riesz basis of its linear span.

———

Radon-Type transforms for holomorphic functions inthe Lie ball

Teppo MertensUGent, BELGIUMteppo.mertens@ugent.be

In this talk we will discuss certain Radon-type trans-forms and their reproducing kernels which are studiedin the paper by Sabadini and Sommen. Extending theirwork we will see some new results I found in joint workwith F. Sommen.

———

Immersions of Riemann surfaces

Craig NolderFlorida State University, 1017 Academic Way, 32306Tallahassee, Florida, USAcraiganolder@hotmail.com

We discuss the theory of immersions of Riemann sur-faces into R3 as developed by Kamberov, Norman, Peditand Pinkall. Their use of the imaginary quaternions iselegant and adds to the theory. We discuss extendingthe theory to Klein surfaces. This requires includingan anti-holomorphic involution of the surface associatedwith an almost dianalytic structure. We also consider

84

Abstracts

the use of the imaginary split quaternions which spanthe Minkowski space R1,2.

———

A group classification of linear fractional partial differ-ential equation with Laplacian on a plane

Heikki OrelmaTampere University, Mechanics and mathematics, Kale-vantie 4, 33100 Tampere, FINLANDHeikki.Orelma@tuni.fi

In this talk we consider fractional deformations of sec-ond order partial differential operators defined on theplane (such as Laplace and Weinstein-Leutwiler equa-tions). We study basic concepts of local Lie symmetriesfor operators in the Riemann-Liouville case. Then westudy good chosen examples to give classifications foroperators. This lead us a canonical way to find properfractional generalization for PDE’s.This is a joint work with Dr. Aleksej Kasatkin (Ufa)and Dr. Nelson Vieira (Aveiro).

———

An Almansi type decomposition for slice-regular func-tions on Clifford algebras

Alessandro PerottiDept. of Mathematics, Via Sommarive 14, 38123 Trento,ITALYalessandro.perotti@unitn.it

The aim of the talk is to present an Almansi type de-composition for polynomials with Clifford coefficients,and more generally for slice-regular functions on Cliffordalgebras. The classical result by Emilio Almansi, pub-lished in 1899, dealt with polyharmonic functions, theelements of the kernel of an iterated Laplacian. Here weconsider polynomials of the form P (x) =

∑dk=0 x

kak,with Clifford coefficients ak ∈ R0,n, an get an analogousdecomposition related to zonal polyharmonics. We showthe relation between such decomposition and the Dirac(or Cauchy-Riemann) operator and extend the resultsto slice-regular functions.

———

Slice Analysis of Several Variables

Guangbin RenDepartment of Mathematics, University of Science andTechnology of China, Hefei, Anhui 230026, PEOPLE’SREPUBLIC OF CHINArengb@ustc.edu.cn

Slice analysis of several variables involves a canonicalextension of the theory of several complex variables tothe non-commutative and non- associative realm viaslice technique. The classical theory is a canonical liftof the holomorphic theory from one complex variableto one quaternionic variable and it impels the devel-opment of quaternionic Hilbert spaces. In particular,it gives arise the study of spherical-spectrum and hasdemonstrated potential applications in quantum physicssince the spherical-spectrum of a self-adjoint operatorfor quaternions turns out to be real in contrast to thespectrum non-real. Instead of quaternions, the theory

has been successfully extended to octonions, Clifford al-gebras, and even any alternative algebra but only re-stricted to the case of one variable. In this talk we ex-tend the theory of several complex variables through theslice technique to several octonionic variables. And weextend the theory of Riemann domains through the slicetechnique to quaternions and it yields a canonical topol-ogy called slice topology in order to be in consistencywith the slice structure. We also initiate to study thetheory of slice Dirac operator for octonions. The sliceDirac operator is a slice counterpart of the Dirac op-erator over quaternions. It originates from a new slicestructure of octonions with quaternions as a slice and ex-tends the theory of stem functions from abelian settingto nonabelian setting.

———

Bargmann-Radon transform for axially monogenic func-tions

Hu RenClifford Research Group, Department of MathematicalAnalysis, Ghent University, Building S8, Krijgslaan 281,9000 Gent, BELGIUMren.hu@ugent.be

In our work we study a class of monogenic functionscalled axially monogenic functions. First we present theexplicit form for the general Cauchy-Kowalewski exten-sion for axially monogenic functions. Then we determinethe Bargmann-Radon transform for these functions, re-lying on Funk-Hecke theorem in the process.

———

Multivariable quaternionic functions and power series

Irene SabadiniPolitecnico di Milano, Dipartimento di Matematica, Mi-lano, ITALYirene.sabadini@polimi.it

In the talk we consider functions in several quaternionicvariables, based on a non-commutative version of Glea-son’s problem. Our study is based on converging seriesof monomials in several noncommuting variables. Somequestions can be tackled in this framework and we showhow to solve the Gleason problem, and an interpolationproblem. We also discuss Schur multipliers.(Joint work with K. Abu-Ghanem, D. Alpay, F.Colombo).

———

Domains of monogenicity in several quaternionic vari-ables

Tomas SalacCharles University, Sokolovska 83, 18675 Prague,CZECH REPUBLICsalac@karlin.mff.cuni.cz

The n-Cauchy-Fueter operator plays the role of theCauchy-Riemann operator in the analysis of severalquaternionic variables. It is well known that the Har-tog’s phenomenon holds in the quaternionic vector spaceHn of dimension n and so it is natural to introduce socalled domains of monogenicity in Qn, these are the do-mains of holomorphy for the n-Cauchy-Fueter operator.

85

Abstracts

In the talk I will give sufficient and necessary conditionsfor a domain in Hn to be a domain of monogenicity.

———

Analyticity in the sense of Hausdorff and classes of hy-perholomorphic quaternionic functions

Michael ShapiroGolomb St 52, 5810201 Holon, ISRAELshapiro1945@outlook.com

The notion of the Hausdorff analyticity for functionswith values in real algebras will be considered togetherwith the Hausdorff derivative for such functions. We an-alyze what both give for the real quaternions and willcompare with the results for several well-known classesof quaternion-valued functions which have the intrin-sic notions of the (hyper-) analyticity and the (hyper-)derivative.The talk is based on a joint work with M.E. Luna-Elizarraras and V. Shpakivskyi.

———

Two-sided Fourier Transform in Clifford Analysis and itsApplication

Haipan Shi050000 Shijiazhuang, Hebei, PEOPLE’S REPUBLICOF CHINAshihaipan226@163.com

In this paper, we first define a two-sided Clifford Fouriertransform(CFT) and its inverse transformation on L1

space. Then we study the differential of the two-sidedCFT, the k-th power of Fh, Plancherel identity andtime-frequency shift of the two-sided CFT.Finally we discuss the uncertainty principle of the two-sided CFT and give an application of the two-sided CFTto the partial differential equation.

———

Zeros of slice functions over dual quaternions: theoryand applications

Caterina StoppatoDipartimento di Matematica e Informatica “U. Dini”,Universita degli Studi di Firenze, Viale Morgagni 67/A,I-50134 Firenze, Italystoppato@math.unifi.it

Within the theory of slice functions over alternative∗-algebras, this talk will focus on the algebra of dualquaternions DH. The zero sets of slice functions, sliceregular functions and polynomials over this algebra havepeculiar properties. A detailed study of these propertiesis possible thanks to the characterization of zero divisorswithin DH.In addition to its intrinsic interest, this study has use-ful applications to the open problem of factorizing mo-tion polynomials over dual quaternions. The polyno-mials in this class, introduced by Hegedus, Schicho, andSchrocker in 2013, correspond to rational rigid body mo-tions in the Euclidean 3-space. Their factorizations cor-respond to linkages producing the same motions, so theirclassification is relevant to mechanism science.This is a joint work with Graziano Gentili and TomasoTrinci.

———

Dirichlet and Bloch Spaces in different Contexts

Luis Manuel Tovar SanchezEscuela Superior de Fısica y Matematicas, Edificio 9,Unidad Profesional A.L.M. Zacatenco del IPN Ciudadde Mexico, Caldas 562 Depto 101 Col. Valle del Te-peyac, Mexico, MEXICOtovar@esfm.ipn.mx

What are the diferences and similarities when we con-sider Bloch and Dirichlet Spaces in different contexts: a)Analytic functions in the unit open complex ball b) Holo-morphic functions in the unit ball of Cn c) Monogenicfunctions in the three dimensional unit ball d) Mono-genic functions in the four dimensional unit ball e) Bi-complex holomorphic functions in the bidisk In this talkwe present a discussion about that situations

———

Ternary Regular Functions

Mihaela VajiacChapman University, Keck 365, One University Drive,92866 Orange CA, USmbvajiac@chapman.edu

This talk will present a study of ternary regular func-tions, as derived from the general theory of hypercom-plex analysis. We introduce the notion of regularity,ternary differential operators, and ternary laplacian inthis case

———

Analysis of the k-Cauchy-Fueter complexes

Wei WangZhejiang University, Dept. of Math., Zhe Da road 38,310027 Hangzhou, Zhejiang, PEOPLE’S REPUBLICOF CHINAwwang@zju.edu.cn

The k-Cauchy-Fueter complexes are quaternionic coun-terparts of ∂-complex in SCV. We consider the non-homogeneous k-Cauchy-Fueter equations over a domainunder the compatibility condition, which naturally leadsto a Neumann problem. The method of solving the ∂-Neumann problem in SCV is applied to this Neumannproblem. We introduce notions of k-plurisubharmonicfunctions and k-pseudoconvex domains, establish theL2 estimate and solve this Neumann problem over k-pseudoconvex domains in R4. 0-Cauchy-Fueter complexcan be applied to the quaternionic Monge-Ampere op-erator, and allows us to introduce the notion of a closedpositive current in the quaternionic case and to extendseveral results in complex pluripotential theory to thequaternionic pluripotential theory.

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Important Properties of Clifford Mobius Transformationand Its Application

Yonghong XieHebei Normal University, No. 20 South Second RingEast Road, 050024 Shijiazhuang, Hebei, PEOPLE’S RE-PUBLIC OF CHINAxyh1973@126.com

Important properties of Clifford Mobius transformationand its application are discussed in this paper. First,

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some important properties of Clifford Mobius transfor-mation are studied. Then, the application of CliffordMobius transformation is given, that is, the generalizedJørgensen inequality.

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A new Cauchy integral formula in the complex Cliffordanalysis

Heju YangHebei University of Science and Technology, No. 26 Yux-iang Street, 050018 Shijiangzhuang, Hebei, PEOPLE’SREPUBLIC OF CHINAearnestqin7384@163.com

In this talk, we construct an analogue of Bochner-Martinelli Kernel based on theory of functions of severalcomplex variables in complex Clifford analysis, whichhas generalized complex differential forms with Cliffordbasis vectors. Using these complex differential forms, weobtain the Stoke’s formula of complex Clifford functionswhich are defined on a domain Ω ⊂ Cn+1 and take val-ues in a complex Clifford algebra Cl1,n(C). Then, wegive a Stoke’s formula which has a classical form andan analogue of Cauchy-Pompieu formula which is rep-resented by Bochner-Martinelli Kernel, and establish ananalogue of Cauchy integral formula in complex Clif-ford analysis. It is possible to promote these results tocomplex manifold’s corresponding results in the Cliffordanalysis using the representation by generalized complexdifferential forms.

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S.18 Recent Progress in Evolution Equations

OrganisersMarcello D’Abbicco, Marcelo Rempel Ebert

Scope of the session: The goal of the session “Re-cent progress in evolution equations” is to discuss thestate-of-the-art of qualitative properties of solutions oflinear and nonlinear evolution models. We have inmind results for dispersive equations, hyperbolic andp-evolution equations and parabolic equations as well.Among other things well-posedness, asymptotic profile,dissipative estimates, blow-up behavior, critical expo-nents, influence of low regular coefficients are of interest.

—Abstracts—

Modified different nonlinearities for weakly coupled sys-tems of semilinear effectively damped waves with differ-ent time-dependent coefficients in the dissipation terms

Mohammed Djaouti AbdelhamidUniversite hassiba Benbouali Chlef Algeria, ALGERIAdjaouti abdelhamid@yahoo.fr

We prove the global existence of small data solutionin all space dimension for weakly coupled systems ofsemi-linear effectively damped wave, with different time-dependent coefficients in the dissipation terms. More-over, nonlinearity terms f(t, u) and g(t, v) satisfyingsome properties of the parabolic equation. We studythe problem in several classes of regularity.

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Decay of the data and regularity of the solution inSchrodinger equations

Alessia AscanelliDipartimento di Matematica e Informatica, Universitadi Ferrara, via Machiavelli, 30, 44121 Ferrara, ITALYalessia.ascanelli@unife.it

We deal with the global Cauchy problem for aSchrodinger type equation(

Dt −∆ +

n∑j=1

aj(t, x)Dxj + b(t, x)

)u(t, x) = 0,

(t, x) ∈ [0, T ] × Rn, under the decay condition|=aj(t, x)| ≤ C/(1 + |x|2)σ/2, σ ∈ (0, 1); this conditionis known to produce a unique solution with Gevrey reg-ularity of index s ≥ 1 and loss of an infinite number ofderivatives with respect to the data g(x) = u(0, x), forevery s ≤ 1/(1−σ). We consider the case s > 1/(1−σ)and we explain how the space where a unique solutionexists depends on the decay and regularity of an initialdata g ∈ Hm, m ≥ 0. As a byproduct, we show that ifg ∈ Hm has a decay as |x| → ∞, then the solution has(at least locally) the same regularity as g but a differentbehavior at infinity.

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The influence of oscillations on energy estimates fordamped wave models with time-dependent propagationspeed and dissipation

Halit Sevki AslanInstitut fur Angewandte Analysis, Technische Univer-sitat Bergakademie Freiberg, Pruferstrasse 9, 09596Freiberg, Sachsen, GERMANYhalitsevkiaslan@gmail.com

In this talk we discuss the Cauchy problem for dampedwave models with time-dependent propagation speedand dissipation. The model of interest is

utt − λ2(t)ω2(t)∆u+ ρ(t)ω(t)ut = 0,

u(0, x) = u0(x), ut(0, x) = u1(x).

The coefficients λ = λ(t) and ρ = ρ(t) are shape func-tions and ω = ω(t) is a (bounded) oscillating functiontaking account of Cm properties of the coefficients witha stabilization condition. Moreover, the damping termρ(t)ω(t)ut is assumed to be an “effective like”damping.By introducing these properties simultaneously, moreprecise analysis is applicable.Our goal is to derive higher order energy estimates forsolutions to the Cauchy problem. We will explain howthe interplay between the shape functions and very fastoscillating behavior of the coefficients will influence en-ergy estimates. Finally, our interest is to apply the de-rived linear estimates to study global existence of smalldata solution to semi-linear models.The results of this talk are based on collaborations withMichael Reissig (TU Bergakademie Freiberg).

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Regularity of linear partial differential operators withpolynomial coefficients

Chiara BoitiDipartimento di Matematica e Informatica, University

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of Ferrara, Via Machiavelli, 30, 44121 Ferrara, ITALYchiara.boiti@unife.it

Let S be the classical Schwartz space and S ′ its dualspace of tempered distributions. A linear operator A :S ′ → S ′ is said to be globally regular if

Au ∈ S ⇒ u ∈ S, ∀u ∈ S ′.

Here we study regularity of linear partial differential op-erators A(x,D) with polynomial coefficients. A well-known sufficient condition for regularity is the conditionof global hypoellipticity given by Shubin, but it is farfrom being necessary and studying regularity for non-global hypoelliptic operators is not trivial, in general.Here we study regularity of linear p.d.o. with polyno-mial coefficients by using a Wigner type transform andthen Cohen classes of the form Q[w] := σ ∗Wig[w] fora kernel σ ∈ S ′. We find classes of regular (but nothypoelliptic) operators, and these classes are quite largebecause of the freedom in the choice of the kernel σ.

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Semilinear p-evolution equations in weighted Sobolevspace

Marco CappielloDipartimento di Matematica - Universita degli Studi diTorino - Via Carlo Alberto 10, 10123 Torino, ITALYmarco.cappiello@unito.it

We study the Cauchy problem associated to a class ofsemilinear p-evolution equations in a suitable scale ofweighted Sobolev spaces. Applications concern KdV-type equations and semilinear Schrodinger-type equa-tions.

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Existence and asymptotic properties for dissipativesemilinear second order evolution equations with frac-tional Laplacian operators

Ruy Coimbra CharaoFederal University of Santa Catarina, Department ofMathematics, Florianopolis, Santa Catarina, BRAZIL

ruy.charao@ufsc.br

In this work we study the existence and uniqueness ofsolutions for a generalized second order semilinear evolu-tion equation with Laplace operators depending on frac-tional powers. We also study the decay rates of the so-lutions and associated energy in the sense of L2 norm.For the associated linear equation, using an asymptoticexpansion of the solution of the problem in the Fourierspace we show optimality of the decay rates obtainedfor certain powers of the involved Laplacian operators .Moreover, we study the case of super damping using animprovement of the standard case and we prove optimal-ity of the decay rates for the L2 norm. Our results gener-alize some previous works that deal with particular casesof fractional exponents of the Laplacian operators. Keywords: Plate/Boussinesq type equation; Existence anduniqueness; Fractional Laplacian; Decay rates; Asymp-totic expansion; Super-damping; Optimal decay rates.Joint with: Juan Torres Espinoza (UFSC-Brazil) andRyo Ikehata (Hiroshima University.

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Weakly coupled systems of semilinear elastic waveswith different damping mechanisms in 3D

Wenhui ChenInstitut fur Angewandte Analysis, Technische Univer-sitat Bergakademie Freiberg, Pruferstrasse 9, 09596Freiberg, Sachsen, GERMANY809934675@qq.com

We consider the following Cauchy problem for weaklycoupled systems of semilinear damped elastic waves witha power source nonlinearity in three dimensions:

Utt − a2∆U −(b2 − a2)∇div U + (−∆)θUt

= F (U), (t, x) ∈ (0,∞)× R3,

where U =(U (1), U (2), U (3)

)Twith b2 > a2 > 0 and

θ ∈ [0, 1]. Our interests are some qualitative proper-ties of solutions to the corresponding linear model withvanishing right-hand sides and the influence of the valueof θ on the exponents p1, p2, p3 in the nonlinear term

F (U) =(|U (3)|p1 , |U (1)|p2 , |U (2)|p3

)Tto get results for

the global (in time) existence of small data solutions.

The results of this talk are based on collaborations withMichael Reissig (TU Bergakademie Freiberg). [1] ChenW, Reissig M. Weakly coupled systems of semilinearelastic waves with different damping mechanisms. Math.Methods Appl. Sci. (accepted for publication)

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Asymptotics for a semilinear damped plate equationwith time-dependent coefficients

Marcello D’AbbiccoUniversity of Bari, Via Orabona 4, 70124 Bari, BA,ITALYmarcello.dabbicco@uniba.it

In this talk, we will discuss the asymptotic profile ofthe solution to a semilinear damped plate equation withtime-dependent coefficients

utt + ∆2u− λ(t)∆u+ ut = |u|p, t > 0, x ∈ Rn

assuming small initial data. Here λ(t) is a nonincreas-ing function whose decay speed at infinity regulates ifthe asymptotic profile is described by a heat equation,or by a fourth order parabolic equation.

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L1 estimates for oscillating integrals and applicationto parabolic like semi-linear structurally damped σ-evolution models

Tuan Anh DaoInstitut fur Angewandte Analysis, Technische Univer-sitat Bergakademie Freiberg, Pruferstrasse 9, 09596Freiberg, Sachsen, GERMANYanh.daotuan@hust.edu.vn

In the talk we discuss L1 estimates for oscillating inte-grals by applying the theory of modified Bessel functionscombined with Faa di Bruno’s formula and their applica-tions to the Cauchy problems for semi-linear structurallydamped σ-evolution models. The model of interest is

utt + (−∆)σu+ µ(−∆)δut = f(u, ut),

u(0, x) = u0(x), ut(0, x) = u1(x)

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where σ ≥ 1, µ > 0, δ ∈ (0, σ2

) and the function f(u, ut)stands for the power nonlinearities |u|p and |ut|p. Thenovelty is to prove the global (in time) existence of smalldata Sobolev solutions to the above semi-linear modelsfrom suitable function spaces basing on Lq spaces byusing Lq ∩ Lm − Lq estimates not necessarily on theconjugate line and some modern tools from HarmonicAnalysis. Moreover, we show how the flexibility of pa-rameters m and q brings some benefits to relax therestrictions to the admissible exponents p.

The results of this talk are based on collaborations withMichael Reissig (TU Bergakademie Freiberg).

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On regularity of solutions to two-dimensional Zakharov-Kuznetsov equation

Andrey FaminskiyRUDN University, Miklukho-Maklaya str. 6, 117198Moscow, RUSSIAN FEDERATIONafaminskii@sci.pfu.edu.ru

The initial and initial-boundary value problems indifferent domains are considered for two-dimensionalZakharov-Kuznetsov equation ut + bux + uxxx + uxyy +uux = 0. Results both on solubility of these problemsin regular functional spaces and on internal regularity ofweak solutions are established.

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On Traveling Solitary Waves for nonlinear Half-WaveEquations

Vladimir GeorgievUniversity of Pisa, Department of Mathematics, LargoBruno Pontecorvo 5, 56127 Pisa, ITALYgeorgiev@dm.unipi.it

We consider nonlinear half-wave equations with focus-ing power-type nonlinearity of order p. with exponents1 < p < ∞ for d = 1 and 1 < p < (d + 1)/(d − 1) ford ≥ 2. We study traveling solitary waves of the form

u(t, x) = eiωtQv(x− vt)

with frequency ω ∈ R, velocity v ∈ Rd, and some finite-energy profile Qv ∈ H1/2(Rd), Qv 6≡ 0. We prove thattraveling solitary waves for speeds |v| ≥ 1 do not ex-ist. As a second main result, we show that small datascattering fails to hold for the focusing half-wave equa-tion in any space dimension. The proof is based on theexistence and properties of traveling solitary waves forspeeds |v| < 1. Finally, we discuss the energy-criticalcase when p = (d+ 1)/(d− 1) in dimensions d ≥ 2.

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Time-dependent propagation speed vs strong damping

Marina GhisiUniversity of Pisa, Dipartimento di Matematica, LargoPontecorvo 5, 56127 Pisa, ITALYmarina.ghisi@unipi.it

We consider a degenerate abstract wave equation with atime-dependent propagation speed. We investigate theinfluence of a strong dissipation, namely a friction term

that depends on a power of the elastic operator. Wediscover a threshold effect. If the propagation speed isregular enough, then the damping prevails, and thereforethe initial value problem is well-posed in Sobolev spaces.Solutions also exhibit a regularizing effect analogous toparabolic problems. As expected, the stronger is thedamping, the lower is the required regularity. On thecontrary, if the propagation speed is not regular enough,there are examples where the damping is ineffective, andthe dissipative equation behaves as the non-dissipativeone.

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L1 −L1 estimates for the strongly damped plate equa-tion

Giovanni GirardiUniversita degli Studi di Bari “Aldo Moro”, ViaOrabona 4, 70125 Bari, Puglia, ITALYgiovanni.girardi@uniba.it

We will derive L1 estimates for the solution for thestrongly damped plate equation,

utt + ∆2u+ ∆2ut = 0 x ∈ Rn, t ∈ R+,

u(0, x) = u0(x), ut(0, x) = u1(x).

In particular, we will prove that

‖u(t, ·)‖L1 ≤ C (1 + t)n4(‖u0‖L1 + (1 + t)

12 ‖u1‖L1

),

for any t ≥ 0, in space dimension n ≥ 5.

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Infinite dimensional Duffing-like evolution equations

Massimo GobbinoUniversity of Pisa, Dipartimento di Ingegneria Civile eIndustriale, Largo Lucio Lazzarino, 56126 Pisa, ITALYmassimo.gobbino@unipi.it

We consider an abstract evolution equation inspired bysome physical models for damped oscillations of a beamsubject to external loads or magnetic fields, and shakenby a transversal force. This leads to an equation of or-der two with linear damping, a linear term (the elasticrestoring force), a nonlinear term of Duffing type (due tothe external load or the magnetic fields), and a time de-pendent forcing term (the transversal force). The Duff-ing form of the nonlinear term implies that, when thereis no external force, the system has three stationary po-sitions, two stable and one unstable, and all solutionsare asymptotic for large times to one of these stationarysolutions. We show that this pattern extends to the casewhere the external force is bounded and small enough,in the sense that all solutions remain close, as time goesto infinity, to one of the three stationary solutions tothe unforced equation, within a distance depending onthe size of the forcing term. Moreover, two solutionsthat are eventually close to the same stationary pointare actually asymptotic to each other, and therefore inthis regime solutions have only three possible asymp-totic profiles. One interesting feature if that boundaryconditions have an important influence on the functionalanalytic setting. The case of hinged endpoints is easierto deal with because the operator involved in the elasticpart commutes with the operator in the Duffing term.More difficult is the case of clamped endpoints, becausenow the operators do not commute, and new functionaltools are needed in the analysis.

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Some properties of time dependent propagation speedof the wave equation for the estimates of low frequencyenergy

Fumihiko HirosawaYamaguchi University, Faculty of Science, 753-8512 Ya-maguchi, JAPANhirosawa@yamaguchi-u.ac.jp

We consider the energy estimate of the solution to theCauchy problem of the wave equation with time depen-dent propagation speed:

∂2t u− a(t)2∆u = 0, (t, x) ∈ (0, T ]× Rn,u(0, x) = u0(x), ∂tu(0, x) = u1(x), x ∈ Rn,

where a(t) is not Lipschitz continuous at t = T henceit may occur some loss of regularity as t → T − 0. Themain purpose of my talk is to introduce a new propertyof a(t) which is effective in particular for the estimate toof the low frequency energy.

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On a criterion for log-convex decay in non- selfadjointdynamical systems

Jon JohnsenAalborg University, Department of Mathematics,Skjernvej 4A, DK-9220 Aalborg, DENMARKjjohnsen@math.aau.dk

Some recent results on the well-posedness of a large classof variational evolution equations are first recalled as amotivation. Then the short-time and global behaviourare studied in an autonomous case defined by a non-self-adjoint generator inducing a uniformly bounded holo-morphic semigroup in a Hilbert space. A general neces-sary and sufficient condition is introduced under whichthe norm of the solution is shown to be a log-convex andstrictly decreasing function of time, and differentiablealso at the initial time with a derivative controlled bythe lower bound of the generator. The recently obtainedinjectivity of holomorphic semigroups plays a pervasiverole in the arguments.

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Blow-up for Quasi-Linear Wave Equations with timedependent potential

Sandra LucenteDipartimento Interateneo di Fisica, Universita degliStudi di Bari, Via Giovanni Amendola, 173, 70125 Bari,ITALYsandra.lucente@uniba.it

Many papers deal with nonlinear wave equations withlow order terms, such as damping terms. The mainquestion concerns the influence of the low order termson the critical exponents of global existence or blow-upor scattering. With suitable transformations these equa-tions can be studied as nonlinear wave equations witha time dependent potential in the nonlinear terms. Ina joint project with Vladimir Georgiev we are workingon quasi-linear wave equations having particular struc-ture. In this talk we present a blow-up result in three-dimensional case with polynomial potential. An interac-tion between the decay of the potential and the structure

of the quasilinear term appears. Coming back to nonlin-ear wave equation with scale invariant damping, we canadd some missing tiles on the analysis of such problem.

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Sigma-evolution models with low regular time-dependent structural damping: effective and non-effective dissipation

Cleverson da LuzFederal University of Santa Catarina, Department ofMathematics, Florianopolis, Santa Catarina, BRAZIL

cleverson.luz@ufsc.br

In this work, we consider σ−evolution models under ef-fects of a damping term represented by the action of afractional Laplacian operator and a time-dependent co-efficient b(t)(−∆)θut. The objective of this work is toobtain Lp−Lq decay rates, with 1 ≤ p ≤ 2 ≤ q ≤ ∞, forthe solution and its first derivative in time, consideringlow regularity in the coefficient b = b(t). More precisely,considering a suitable t0, we take b(t) “confined” in thecurve g(t) := (1 + t)α lnγ(1 + t) for t ≥ t0. In thiscontext, when compared with previous results that as-sume more regularity in the function b, we obtain thesame decay rates for the solution when γ = 0 and wereach better rates for γ 6= 0. For the first derivativein time of the solution, we derive improved rates evenfor the case γ = 0. Written in a equivalent manner, animportant conjecture was asserting: “For θ = 0, whenthe coefficient of the damping is effective, without fur-ther assumptions on derivatives of the coefficient is stillpossible to achieve the same known decay rates for theproblem”. In the present work we provide an answerto the conjecture, showing, in addition, that there areother situations in which the conjecture remains valid,for example θ 6= 0 or even in the case that the dampingis non-effective.

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Critical exponent for the semilinear damped wave equa-tion with polynomial decaying propagation speed

Jorge MarquesUniversity of Coimbra, Av. Dias da Silva 165, 3004-512Coimbra, PORTUGALjmarques@fe.uc.pt

We consider the Cauchy problem on R+0 × Rn for the

semilinear damped wave equation

utt(t, x)− a2(t)∆u(t, x) + b(t)ut(t, x) = |u|p

with decreasing in time coefficients, the propagationspeed a(t) = (1+t)−`, 0 < ` < 1, the scale-invariant dis-sipation b(t) = β(1 + t)−1, β > 0, and a nonlinear powerterm of order p > 1. The solution u0 of the correspond-ing linear problem will be represented in the explicitform using Fourier multipliers operators with multipliersexpressed in terms of special functions. Our main goalis to prove the global existence of u in time when initialdata belongs to the spaces H1(Rn) and L2(Rn). Weare focused in finding the critical exponent pc(n) suchthat if 1pc(n) the global solution has the same long timebehavior of u0. In order to estimate u we use Duhamel’sprinciple to represent u and then we apply L2 − L2 es-timates of u0.

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Global solutions for the semilinear diffusion equation inthe de Sitter spacetime

Makoto NakamuraYamagata University, Faculty of Science, 990-8560 Ya-magata, Yamagata, JAPANnakamura@sci.kj.yamagata-u.ac.jp

The Cauchy problem for the semilinear diffusion equa-tion is considered in the de Sitter spacetime with the spa-tial zero-curvature. Global solutions and their asymp-totic behaviors for small initial data are obtained forpositive and negative Hubble constants. The effects ofthe spatial expansion and contraction are studied on theproblem.

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Some estimates of solutions of perturbed Helmholtzequations

Hideo NakazawaDepartment of Mathematics, Nippon Medical School, 1-7-1, Kyonan-cho, 180-0023 Musashino, Tokyo, JAPANnakazawa-hideo@nms.ac.jp

In this talk, we consider the stationary problem ofwave equations with dissipation in an exterior domainΩ ⊂ RN (N = 3). For the estimate of solutions of thisproblem, the existing results require an assumption ofthe smallness of the coefficient function of dissipation.To overcome this difficulty, we introduce new inequali-ties needed to estimate the solution.

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Recent progress in semilinear wave equations in thescale-invariant case

Alessandro PalmieriUniversita di Pisa, Dipartimento di Matematica LargoBruno Pontecorvo 5, 56127 Pisa, ITALYalessandro.palmieri.math@gmail.com

In this talk, we provide an overview of semilinear wavemodels with scale-invariant damping and mass with dif-ferent nonlinear terms. In particular, we shall emphasizehow the critical conditions for the exponents in the non-linear terms turn out to be shifts of the critical exponents(respectively, critical curves in the case of weakly cou-pled systems) for classical wave and damped wave mod-els. This phenomenon is strictly related to the thresholdnature of the time-dependent coefficients for the damp-ing and mass terms in the scale-invariant case.This talk is based on joint works with Michael Reis-sig (TU Bergakademie Freiberg), Wenhui Chen (TUBergakademie Freiberg), Wanderley Nunes do Nasci-mento (Federal University of Rio Grande do Sul) andZiheng Tu (Zhejiang University).

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Some recent results on periodic waver waves

Fabio PusateriDepartment of Mathematics University of Toronto,Toronto, ON M5S 2E4, CANADAfabiop@math.toronto.edu

We will consider the full water waves problem, mod-eling the motion of waves like those on the surface of

the deep ocean, and discuss the question of long-timestability for small solutions. While several results havebeen proven in recent years for the problem set on thewhole Euclidean space, the motion of periodic waves isless understood. I will present two recent results in thisdirection: a result on the long-time stability of 1d peri-odic interfaces under the influence of gravitational forceswhich settles a conjecture by Zakharov and Dyachenkoon the almost integrability of the equations (joint workwith M. Berti and R. Feola) and a result on the full 3dperiodic gravity-capillary problem (joint work with A.Ionescu).

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Lp-Lq estimates for the damped wave equation and thecritical exponent for the nonlinear problem with slowlydecaying data

Yuta WakasugiEhime University, Graduate School of Science and En-gineering, 7908577 Matsuyama, Ehime, JAPANwakasugi.yuta.vi@ehime-u.ac.jp

We study the Cauchy problem of the damped wave equa-tion

∂2t u−∆u+ ∂tu = 0

and give sharp Lp-Lq estimates of the solution for 1 ≤q ≤ p < ∞ (p 6= 1) with derivative loss. This is an im-provement of the so-called Matsumura estimates. More-over, as its application, we consider the nonlinear prob-lem with initial data in (Hs ∩Hβ

r ) × (Hs−1 ∩ Lr) withr ∈ (1, 2], s ≥ 0, and β = (n − 1)| 1

2− 1

r|, and prove

the local and global existence of solutions. In particu-lar, we prove the existence of the global solution withsmall initial data for the critical nonlinearity with thepower 1 + 2r

n.

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S.19 Spectral Theory of Partial DifferentialEquations

OrganisersJames Kennedy, Davide Buoso

Scope of the session: This session seeks to bring to-gether researchers working on spectral-theoretic aspectsof partial differential equations, including but not lim-ited to those coming from mathematical physics. Thiscovers both the analysis of eigenvalues of partial differ-ential operators with discrete spectrum, as well as prop-erties of operators with essential spectrum, and non-local and non-self-adjoint problems. The emphasis isanticipated to be on topics such as perturbation the-ory and spectral dependence on parameters, weightsetc., non-convergence results and ”term-coming-from-nowhere” effects, spectral asymptotics, location of thespectrum and isolated eigenvalues, variational methods,and spectral geometry and shape optimization, but sub-missions are welcome from all related areas.

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Some sharp spectral inequalities

Giuseppina di BlasioUniversita degli Studi della Campania ”L. Vanvitelli”,viale Lincoln, 5, 81100 Caserta, CE, ITALYgiuseppina.diblasio@unicampania.it

We prove upper and lower bounds for Dirichlet eigenval-ues of linear and nonlinear operators.

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Semiclassical bounds for spectra of biharmonic opera-tors

Davide BuosoEPFL, Station 8, 1015 Lausanne, SWITZERLANDdavide.buoso@epfl.ch

We consider Riesz means of the eigenvalues of the bi-harmonic operator subject to various boundary condi-tions, and we improve the known sharp semiclassicalbounds in terms of the volume of the domain with asecond term with the expected power. We obtain suchestimates by means of the averaged variational prin-ciple. This method intrinsically also yields two-sidedbounds for individual eigenvalues, which are semiclassi-cally sharp. Based on a joint work with L. Provenzanoand J. Stubbe.

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Spectral theory for beams with intermediate piers andrelated nonlinear evolution problems

Maurizio GarrionePolitecnico di Milano, ITALYmaurizio.garrione@polimi.it

We prove a complete spectral theorem for the lineareigenvalue problem for beams with intermediate piers,explicitly determining the expression of the eigenfunc-tions and discussing in detail their nodal properties.This provides a suitable tool in order to study the quali-tative properties of the solutions of different fourth-ordernonlinear evolution problems (possibly with nonlocal ef-fects) motivated by the dynamics of suspension bridges.In this framework, we discuss the optimal placement ofthe piers in terms of suitable notions of stability. Jointwork with Filippo Gazzola (Dipartimento di Matemat-ica, Politecnico di Milano).

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On the structure of the spectrum of regular boundaryvalue problems for differential equations

Tynysbek Kal’menovInstitute of Mathematics and Mathematical Modeling,Pushkin str. 125, 050010 Almaty, KAZAKHSTANkalmenov.t@mail.ru

So far, an example of a regular boundary value prob-lem for differential equations, whose spectrum (exceptfor the empty set) is a finite set, has not been found.In the present work, using the method of anti-aprioritest function estimates, it was possible to establish thatthe spectrum of differential operators (with constant co-efficients in a certain subdomain) generated by regular

boundary conditions is either empty or infinite. In thecase of hypoellipticity of the considered differential op-erator with variable coefficients everywhere, the absenceof its finite spectrum is also established.

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On homogenization of periodic parabolic systems

Yulia MeshkovaChebyshev Laboratory, St. Petersburg State University,14th Line V.O., 29B, 199178 Saint Petersburg, RUS-SIAN FEDERATIONy.meshkova@spbu.ru

The talk is devoted to homogenization of periodic dif-ferential operators. In L2(Rd;Cn), we consider matrixelliptic second order differential operator Bε. It is as-sumed that Bε is positive definite. Coefficients of theoperator are periodic with respect to some lattice in Rdand depend on x/ε. For the semigroup e−Bεt, t > 0,we obtain approximation in the L2-operator norm withthe error estimates of the order O(ε) and O(ε2) for fixedtime:

‖e−Bεt − e−B0t‖L2→L2 6 C1ε(t+ ε2)−1/2e−C2t,

‖e−Bεt − e−B0t − εK(ε; t)‖L2→L2 6 C3ε

2t−1e−C2t.

Here B0 is the effective operator with constant coef-ficients and K(ε; t) is the corrector. We apply spec-tral approach to homogenization problems developed byM. Birman and T. Suslina. The method of investiga-tion is based on the scaling transformation, the Floquet-Bloch theory, and the analytic perturbation theory.

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Eigenvalues of the Third Boundary Value Problem forthe Heat Equation

Alip MohammedDepartment of Mathematics, College of Arts and Sci-ences, Khalifa University, Abu Dhabi, UAEmaimaiti.alifu@ku.ac.ae

The talk will present some challenges and insights indealing with the eigenvalue problem of the third bound-ary condition for the heat equation on a line segment. Itis shown that for the full third boundary condition, un-like the mixed boundary condition, in general one cannothave unique solution. One could have one or two solu-tions or no solution at all. The most significant chal-lenge occurs in determining the number of solutions ,i.e., how to spot the condition which separates the onesolution case from the two solution case. The problem isstudied utilizing separation of variables method. In allcases, solvability conditions and explicit solutions areprovided.

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On the negative spectrum of generalized Hamiltoniansdefined on metric graphs

El Aıdi MohammedUniversidad Nacional de Colombia, Sede Bogota,Avenida Carrera 30 numero 45-03, Bogota, Cundina-marca, COLOMBIAmelaidi@unal.edu.co

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On a connected compact metric graph, we show that thenumber of negative eigenvalues, of a generalized Hamil-tonian, is bounded (up to a multiplicative constant) byan integral of the potential.

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Coercive estimates for a second-order differential equa-tion with oscillating drift

Kordan OspanovL.N. Gumilyov Eurasian National University, Faculty ofMechanics and Mathematics, Satpaev Str., 2, 010008Astana, KAZAKHSTANospanov kn@enu.kz

We study the following differential equation:

−y′′ + r (x) y′ + q (x) y = f(x), (1)

where x ∈ R = (−∞, +∞) and f ∈ Lp(R), 1 < p <+∞. We assume that r, q are, respectively, continuouslydifferentiable and continuous functions. The purpose ofthis work is to find some conditions for the coefficients rand q such that for any f ∈ Lp(R) there exists a uniquesolution y of the equation (1) and the following estimateholds: ∥∥ y′′ ∥∥

p+∥∥ ry′ ∥∥

p+ ‖ qy ‖p ≤ C ‖Ly ‖p , (2)

where ‖ · ‖p is the norm in Lp(R).The equation (1) and its multidimensional generaliza-tions with unbounded coefficients have used in stochas-tic analysis, biology and financial mathematics. Forthis reason, interest in these equations has considerablygrown in recent years. A number of researches were de-voted to the case that the coefficient r are controlled byq. Without the dominating potential q, the case thatr growth at most as |x|ln(1 + |x|) were considered byLunardi A. and Vespri V. (1997), Metafune G. (2001),Pruss J., Rhandi A. and Schnaubelt R. (2006), HieberM. and others (2009).In the present work, we study the equation (1) in as-sumpsion that the coefficient r can quickly grow andfluctuate, and it does not depend on q.

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Quantitative aspects for homogenisation problems inRn

Marcus WaurickDepartment of Mathematics and Statistics, Universityof Strathclyde, Livingstone Tower, 26 Richmond Street,Glasgow G1 1XH, Scotland, UKmarcus.waurick@strath.ac.uk

In this talk we provide a methodology to obtain opti-mal in order operator norm estimates for homogenisa-tion problems for PDE-systems in Rn. Assuming thatthe operators in question admit a so-called fibre decom-position, we use elementary Hilbert space methods tocompare PDEs with highly oscillatory periodic coeffi-cients with a suitable homogenised PDE. The results areapplicable to elliptic systems or static Maxwell’s equa-tions. With these results, estimates for time-dependentproblems can be derived yielding quantitative homogeni-sation error estimates for both parabolic and hyperbolicproblems.The talk is based on joint work with Shane Cooper(Durham). Most of the results can be found in [Cooper,

S. and Waurick, M. Fibre Homogenisation; Journal ofFunctional Analysis 276(11):3363–3405, 2019].

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Spectral stability for the Maxwell equations in a cavity

Michele ZaccaronUniversita degli Studi di Padova, ITALYmichele.zaccaron@math.unipd.it

Given a simply connected Lipschitz domain Ω in R3, weconsider the following eigenvalue problem arising in thetheory of electromagnetism

curl curl v = λv, on Φ(Ω),div v = 0, on Φ(Ω),v × ν = 0, on ∂Φ(Ω),

in the unknown vector fields v (the eigenvectors) andλ ∈ R (the eigenvalues). Here Φ is taken in a suit-able class of diffeomorphism in R3 and ν denotes theunit outer normal to ∂Φ(Ω). We discuss the stabilityof eigenvalues and eigenvectors upon the perturbationof Φ. In particular, we prove the analytic dependenceof the symmetric functions of the eigenvalues, and weprovide Hadamard-type formulas for the correspondingdifferentials.Based on a joint work with Pier Domenico Lamberti.

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Conditions of maximal regularity for a third-order dif-ferential equation with fast growing intermediate coef-ficients

Yeskabylova ZhuldyzFaculty of Mechanics and Mathematics, L. N. GumiliovEurasian National University, Astana, 010008, KAZA-KHSTANjuli e92@mail.ru

In this report we consider the third-order differentialequation with variable coefficients of the form

Ly := −y′′′

+ r(x)y′′

+ q(x)y′

+ s(x)y = f(x), (1)

where x ∈ R = (−∞,+∞) , f ∈ L2(R), and r , q and sare smooth functions. We discuss the correct solvability(1) and the possibility of performing for its solution y ofthe following estimate

‖ − y′′′‖2 + ‖ry

′′‖2 + ‖qy

′‖2 + ‖sy‖2 ≤ C‖f‖2. (2)

If (2) is fulfilled, then the solution y of the equation (1)is called the maximally L2 -regular, and the inequality(2) is called the of maximal L2 -regularity estimate.Maximal regularity is an important tool in the theory oflinear and nonlinear differential equations. The questionof the maximal regularity for the second order differen-tial equations was investigated by P.C. Kunstmann, W.Arendt, M. Duelli, G. Metafune, D. Pallara, J. Pruss,R. Schnaubelt, A. Rhandi and etc.The problems of the solid state physics, atmosphericalphysics, electrohydrodynamics, as well as the processesflowing in environments, that able to remember lead tothe third-order differential equations (1). The case ofbounded domains and smooth coefficients are well un-derstood and sufficiently well described in the knownliterature. In the singular case, although the solution of

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the odd-order equation (1) is smooth, but it may not be-long to any Sobolev space, or its belongs to the Sobolevspace is not known in advance. These facts causes somedifficulties for study of (1).In the case that r = q = 0 the maximal regularity of thesolution of (1) is previously investigated by M.B. Murat-bekov, M. Otelbaev, A. Birgebaev and etc. Their resultsare applicable to the case (1) that r and q are controlledby s. In the present paper, we investigate the case, thatr and q in (1) do not depend on s.

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S.20 Theory and Applications ofBoundary-domain Integral andPseudodifferential Operators

OrganisersSergey Mikhailov, David Natroshvili

Scope of the session: The goal of this session is to dis-cuss recent progress in the theory of boundary-domainintegral and pseudodifferential operators and their ap-plications in Mathematical Physics, Solid and Fluid Me-chanics, Wave Scattering Problems, Engineering Math-ematics, etc.

—Abstracts—

Two-operator BDIEs for Variable-Coefficient NeumannProblem with General Data

Tsegaye AyeleDepartment of Mathematics at Addis Ababa University,P.O.Box: 1176, Addis Ababa, ETHIOPIAtsegaye.ayele@aau.edu.et

The Neumann problem for the second-order scalar el-liptic differential equation with variable coefficient andwith the PDE right-hand side from H−1(Ω) is consid-ered. Applying the two-operator approach, the prob-lem is reduced to two different systems of two-operatorboundary-domain integral equations (BDIEs). It isproved that both BDIE systems are equivalent to theoriginal BVP. Invertibility of Boundary-Domain Integraloperators is investigated.

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Boundary-Domain Integral Equation systems to theMixed BVP for Compressible Stokes Equations withVariable Viscosity in 2D

Tsegaye AyeleDepartment of Mathematics at Addis Ababa University,P.O.Box: 1176, Addis Ababa, ETHIOPIAtsegaye.ayele@aau.edu.et

The Boundary-Domain Integral Equations (BDIEs) forthe mixed boundary value problem for a compressibleStokes system of partial differential equations with vari-able coefficient in 2D is considered . An appropriateparametrix (Levi function) is used to reduce this bound-ary value problem (BVP) to the BDIE systems. Al-though the theory of BDIEs in 3D is well developed,the BDIEs in 2D need a special consideration due totheir different equivalence properties. As a result, we

need to set conditions on the domain or on the corre-sponding function spaces to ensure the invertibility ofparametrix-based integral layer potentials and hence theunique solvability of BDIE systems. Equivalence of theBDIE systems to the mixed BVP and invertibility of thematrix operators associated with the BDIE systems areproved.(Based on a joint work with Mulugeta A. Dagnaw andSergey E. Mikhailov)

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Uniform Convergence of Double Vilenkin-Fourier Series

Lasha BaramidzeIlia Vekua Institute of Applied Mathematics (VIAM)of Ivane Javakhishvili Tbilisi State University (TSU),GEORGIAlashabara@gmail.com

My talk is about the uniform convergence problem forthe rectangular partial sums of double Fourier serieson bounded Vilenkin group of the functions of partialbounded oscillation.

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Two-operator Boundary-Domain Integral Equations forvariable coefficient Neumann boundary value problemin 2D

Solomon Tesfaye BekeleAddis Ababa University, Natural and computationalSciences college, 1176 Addis Ababa, ETHIOPIAso te2004@yahoo.com

The Neumann boundary value problem (BVP) for thesecond order “stationary heat transfer” elliptic partialdifferential equation with variable coefficient is consid-ered in two-dimensional bounded domain. Using anappropriate parametrix (Levi function) and applyingthe two-operator approach, this problem is reduced tosome systems of boundary-domain integral equations(BDIEs). The two-operator BDIEs in 2D have specialconsideration due to their different equivalence proper-ties as compared to higher dimensional case due to thelogarithmic term in the parametrix for the associatedpartial differential equation. Consequently, we need toset conditions on the domain or function spaces to insurethe invertibility of the corresponding layer potentials,and hence the unique solvability of BDIEs. Equivalenceof the two operator BDIE systems to the original Neu-mann BVP, BDIEs solvability, uniqueness/non unique-ness of the solution, as well as Fredholm property and in-vertibility of the BDIE operator are analysed. Moreover,the two operator boundary domain integral operators forthe Neumann BVP are not invertible, and appropriatefinite-dimensional perturbations are constructed leadingto invertibility of the perturbed operators.

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Dynamical interaction problems of acoustic waves andthermo-electro-magneto elastic structures

George ChkaduaDepartment of Mathematics, Georgian Technical Uni-versity, 77 M.Kostava, 0160 Tbilisi, GEORGIAg.chkadua@gmail.com

We investigate the dynamical transmission problemsarising in the model of fluid-solid acoustic interaction

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when thermo-electro-magneto elastic body is embeddedin an unbounded fluid domain. The physical kinematicand dynamic relations mathematically are described byappropriate boundary and transmission conditions. Weinvestigate the uniqueness of solutions to the dynami-cal transmission problems and analyze the correspond-ing transmission pseudo-oscillation problems, which areobtained from the dynamical problems by the Laplacetransform.The solvability of the transmission pseudo-oscillationproblems are analyzed by the potential method andtheory of pseudodifferential equations in appropriateSobolev-Slobodetskii spaces. Further using the inverseLaplace transform we obtain solvability of the dynami-cal transmission problems. This work was supported byShota Rustaveli National Science Foundation of Georgia(SRNSF) (Grant number FS-18-126).

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Approximate Solution for Thin Plates in an Infinite Do-main

Christian ConstandaDepartment of Mathematics, The University of Tulsa,800 S. Tucker Drive, 74104 Tulsa, Oklahoma, USAchristian-constanda@utulsa.edu

Many times, solutions of boundary value problems for amathematical model cannot be computed explicitly, butcan be approximated to within acceptable tolerances bymeans of expansions in a complete set of functions in aHilbert space such as L2. The usefulness of the methodincreases when the choice of these functions is based onthe structure of the layer potentials generated by theproblem. This type of expansion may encounter difficul-ties in the case of an infinite domain, where the solutionmust fit a certain prescribed far-field pattern. Often,such a pattern requires the a priori knowledge of what isknown as a rigid displacement in elasticity theory, whichis not normally readily available. In this paper, we ad-dress the question of numerical solvability for the Dirich-let problem associated with the bending of a thin elasticplate with transverse shear deformation that occupiesthe complement to the plane of a finite domain boundedby a closed curve. The generalized Fourier series proce-dure constructed for this model is illustrated by meansof a computational example where the domain is theoutside of a circle.Joint work with Dale Doty, dale-doty@utulsa.edu

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Boundary-Domain Integral Equation Systems to theDirichlet BVP for Stokes Equations with variable vis-cosity in 2D

Mulugeta Alemayehu DagnawAddis Ababa University, Arat killo, 1176 Addis Ababa,ETHIOPIAmalemayehu3@gmail.com

The Dirichlet boundary value problem for the steady-state Stokes system of partial differential equations fora compressible viscous fluid with variable viscosity co-efficient is considered in two- dimensional bounded do-main. Using an appropriate parametrix, this problemis reduced to a system of direct segregated Boundary-Domain Integral Equations (BDIEs). The BDIEs in the

two-dimensional case have special properties in compar-ison with the three dimension because of the logarith-mic term in the parametrix for the associated partialdifferential equation. Consequently, we need to set con-ditions on the function spaces or on the domain for theinvertibility of corresponding parametrix-based hydro-daynamic single layer potential and hence the uniquesolvability of BDIEs. Equivalence of the BDIE systemsto the Dirichlet BVP and invertibility of the correspond-ing boundary-domain integral operators are shown.

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Boundary-Domain Integral Equation Systems to theNeumann BVP for a compressible Stokes System withvariable viscosity in 2D

Mulugeta Alemayehu DagnawAddis Ababa University, Arat killo, 1176 Addis Ababa,ETHIOPIAmalemayehu3@gmail.com

The Neumann boundary value problem for the steady-state Stokes system of partial differential equations fora compressible viscous fluid with variable viscosity co-efficient is considered in two- dimensional bounded do-main. Using an appropriate parametrix, this problemis reduced to a system of direct segregated Boundary-Domain Integral Equations (BDIEs). The BDIEs in thetwo-dimensional case have special properties in compar-ison with the three dimension because of the logarith-mic term in the parametrix for the associated partialdifferential equation. Consequently, setting conditionson the function spaces ensures the invertibility of corre-sponding parametrix-based layer potentials and hencethe unique solvability of BDIEs. Equivalence of theBDIE systems to the Neumann BVP and invertibilityof the corresponding boundary-domain integral opera-tors are shown.

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Mixed Boundary Value Problems for an Elliptic Equa-tion in a Lipschitz Domain

Roland DuduchavaThe University of Georgia & Andrea Razmadze Math-ematical Institute, M. Kostava str. 77a, 0171 Tbilisi,GEORGIARolDud@gmail.com

The mixed boundary value problems for an elliptic equa-tion in a planar Lipschitz domain Ω ⊂ R2 is investigated.The BVP is considered in a non-classical setting when asolution is sought in the Bessel potential spaces Hsp(Ω),s > 1/p, 1 < p < ∞. By applying a localization theBVP is reduced to the investigation of the same prob-lems at model angular domains, as many as the angularpoints on the boundary of the domain.The model problems in an angular domain are inves-tigated using the potential method by reducing themto an equivalent boundary integral equation (BIE) inthe Sobolev-Slobodeckii space on a semi-infinite axesWs−1/pp (R+), which is of Mellin convolution type. By

applying the recent results on Mellin convolution equa-tions in the Bessel potential spaces obtained by V. Di-denko & R. Duduchava in 2016, explicit conditions of theunique solvability of this BIE in the Sobolev-SlobodeckiiWrp(R+) and Bessel potential Hrp(R+) spaces for arbi-

trary r are found and used to write explicit conditions

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for the Fredholm property and unique solvability of theinitial model BVPs for the Helmholtz equation in theabove mentioned non-classical setting.

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Boundary Domain Integral Equations for DiffusionEquation in Non-homogeneous Media with DirichletBoundary Conditions based on a New Family of Para-metrices

Carlos Fresneda-PortilloSenior Lecturer at Oxford Brookes University, WheatleyCampus, OX33 1HX Oxford, Oxfordshire, UKc.portillo@brookes.ac.uk

The Dirichlet problem for the diffusion equation in nothomogeneous media is reduced to a system of directsegregated parametrix-based Boundary-Domain IntegralEquations (BDIEs). We use a parametrix different fromthe one studied in (Chkadua et al, 2009) . Mappingproperties of the potential type integral operators ap-pearing in these equations are presented in appropriateSobolev spaces. We prove the equivalence between theoriginal BVP and the corresponding BDIE system. Theinvertibility and Fredholm properties of the boundary-domain integral operators are also analysed. Possibleextension of these results to exterior domains is also dis-cussed.

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Mellin pseudodifferential operators with non-regularsymbols and their applications

Yuri KarlovichUniversidad Autonoma del Estado de Morelos, Centrode Investigacion en Ciencias, Universidad 1001, Col.Chamilpa, 62209 Cuernavaca, Morelos, MEXICOkarlovich@uaem.mx

The talk is devoted to applications of Mellin pseudodif-ferential operators with non-regular symbols to studyingthe Fredholmness in Banach algebras of singular integraloperators with piecewise quasicontinuous coefficients onweighted Lebesgue spaces with Muckenhoupt weights.Applications to Banach algebras of convolution type op-erators are also considered.

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Layer potentials and exterior boundary problems inweighted Sobolev spaces for the Stokes system withL∞ elliptic coefficient tensor

Mirela KohrBabes-Bolyai University, Faculty of Mathematics andComputer Science, 1 M. Kogalniceanu Str., 400084 Cluj-Napoca, ROMENIAmkohr@math.ubbcluj.ro

We obtain well-posedness results in Lp-based weightedSobolev spaces for transmission and exterior boundaryvalue problems for the anisotropic Stokes system withL∞ strongly elliptic coefficient tensor in bounded andcomplementary exterior Lipschitz domains of Rn, n ≥ 3.First, we use a variational approach that reduces two lin-ear transmission problems for the anisotropic Stokes sys-tem to equivalent mixed variational formulations withdata in Lp-based weighted Sobolev and Besov spaces.

We show that such a mixed variational formulation iswell-posed in the space H1

p(Rn)n × Lp(Rn), n ≥ 3, forany p in an open interval containing 2. These resultsare used to define the Newtonian and layer potentialoperators for the considered anisotropic Stokes system.Various mapping properties of these operators are alsoobtained. A similar variational approach is used to showthe well-posedness of the exterior Dirichlet problem andof the exterior mixed problem for the Stokes system withL∞ elliptic coefficient tensor in an exterior Lipschitz do-main Ω− of Rn, n ≥ 3, and in Lp-based weighted Sobolevspaces H1,p(Ω−)n × Lp(Ω−). For p = 2, the solutionsof the exterior Dirichlet and Neumann problems are ob-tained in terms of Newtonian and layer potentials. Thistalk is based on joint work with Sergey E. Mikhailov(Brunel University London) and Wolfgang L. Wendland(Stuttgart University).

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Solution of the Riemann boundary value problem in thecase when the free term belongs to the grand variableexponent Lebesgue space Lp(t),θ(Γ) when min

Γp(t) = 1

Vakhtang KokilashviliA. Razmadze Mathematical Institute of I. JavakhishviliTbilisi State University, 6 Tamarashvili Str., 0177 Tbil-isi, 0177 Tbilisi, GEORGIAvakhtangkokilashvili@yahoo.com

In our talk we plan to present our results concerning thefollowing topics: i) Singular integrals in grand variableexponent Lebesgue spaces; ii) An extension of notion ofLebesgue integral; iii) Generalized Cauchy type integralsand their properties; iv) Solution of Riemann BVP; v)Boundary singular integral equation of Harmonic Anal-ysis in mixed weighted grand function spaces. We intendto give some application to the approximation theory.Acknowledgement. The work was supported byShota Rustaveli National Science Foundation grant (No.DI-18-118).

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Singular integrals in weighted grand variable exponentLebesgue spaces

Alexander MeskhiA. Razmadze Mathematical Institute, I. JavakhishviliTbilisi State University, 6. Tamarashvili Str., 0177 Tbil-isi, Department of Mathematics, Faculty of Informaticsand Control Systems, Georgian Technical University, 77,Kostava St., 0175, Tbilisi, GEORGIAa.meskhi@gtu.ge

The boundedness of maximal and singular integral oper-ators is established in weighted grand variable exponentLebesgue spaces with power-type weights. The sameproblem for commutators of singular integrals is alsostudied. These spaces unify two non-standard classesof function spaces, namely, grand Lebesgue and variableexponent Lebesgue spaces. The spaces and operatorsare defined, generally speaking, on quasi-metric measurespaces with doubling measure. Exponents of spaces sat-isfy log-Holder continuity condition.Acknowledgement. The work was supported by theShota Rustaveli National Science Foundation grant (No.DI-18-118).

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Analysis of Boundary-Domain Integral Equations forthe Variable-Viscosity Compressible Robin-Stokes BVPin Lp-based Spaces

Sergey E. MikhailovBrunel University London, Dept. of Mathematics, UB83PH Uxbridge, UNITED KINGDOMmastssm@brunel.ac.uk

We consider Boundary-Domain Integral Equations(BDIEs) associated with the Robin boundary valueproblem for the stationary compressible Stokes systemin Lp-based Sobolev spaces in a bounded Lipschitz do-main in Rn, n ≥ 3, with the variable viscosity coeffi-cient. First, we introduce a parametrix and constructthe corresponding parametrix-based variable-coefficientStokes Newtonian and layer integral potential operatorswith densities and the viscosity coefficient in Lp-basedSobolev or Besov spaces. Then we generalize variousproperties of these potentials, known for the Stokes sys-tem with constant coefficients, to the case of the Stokessystem with variable coefficients. Next, we show thatthe Robin boundary value problem for the Stokes systemwith variable coefficients is equivalent to a BDIE system.Then we analyse the Fredholm properties of the BDIEsystems in Lp-based Sobolev and Besov spaces and fi-nally prove their invertibility in corresponding spaces.This is a joint work with Mirela Kohr and MassimoLanza de Cristoforis.

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Investigation of nonclassical transmission problems ofthe thermo-electro-magneto elasticity theory for com-posed bodies by the integral equation method

Maia MrevlishviliDepartment of Mathematics, Georgian Technical Uni-versity, 77 M.Kostava, 0160 Tbilisi, GEORGIAm mrevlishvili@yahoo.com

Investigation of nonclassical transmission problems ofthe thermo-electro-magneto elasticity theory for com-posed bodies by the integral equation method Abstract:We investigate multi-field problems for complex elasticanisotropic structures when in different adjacent com-ponents of the composed body different refined modelsof elasticity theory are considered. In particular, weanalyse the case when we have the generalized thermo-electro-magneto elasticity model (GTEME model) inone region of the composed body and the generalizedthermo-elasticity model (GTE model) in another adja-cent region. Both models are associated with Green-Lindsay’s model [1], [2]. This type of mechanical prob-lems mathematically are described by systems of par-tial differential equations with appropriate transmissionand boundary conditions. In the GTEME model partwe have six dimensional unknown physical field (threecomponents of the displacement vector, electric poten-tial function, magnetic potential function, and tempera-ture distribution function), while in the GTE model partwe have four dimensional unknown physical field (threecomponents of the displacement vector and temperaturedistribution function). The diversity in dimensions ofthe interacting physical fields complicates mathematicalformulation and analysis of the corresponding boundary-transmission problems. We apply the potential methodand the theory of pseudodifferential equations and prove

uniqueness and existence theorems of solutions to differ-ent type boundary-transmission problems in appropriateSobolev spaces. This work was supported by Shota Rus-taveli National Science Foundation of Georgia (SRNSF)(Grant number FS-18-126).References [1] A.E. Green and K.A. Lindsay, Thermoe-lasticity, J. Elast., 2 (1972), 1–7.[2] T. Buchukuri, O. Chkadua, and D.Natroshvili,Mathematical Problems of Generalized Thermo-Electro-Magneto-Elasticity Theory, Memoirs on DifferentialEquations and Mathematical Physics, Volume 68 (2016),1-166.

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Localized boundary-domain integral equations ap-proach with piecewise constant cut-off function for theheat transfer equation with a variable coefficient

David NatroshviliGeorgian Technical University, Department of Mathe-matics, 77 M.Kostava st, 0175 Tbilisi, GEORGIAnatrosh@hotmail.com

Localized boundary-domain integro-differential equa-tions (LBDIDE) systems associated with the Dirichletand Robin boundary value problems (BVP) for the sta-tionary heat transfer partial differential equation (PDE)with a variable coefficient are obtained and analyzed.The parametrix is localized by a characteristic functionof a ball of radius ε which is not a smooth cut-off functionin the whole space. The main results of the present pa-per are equivalence theorems of the LBDIDE systems tothe original variable-coefficient BVPs and unique solv-ability of the LBDIDE systems in the correspondingSobolev spaces.This work was supported by Shota Rustaveli NationalScience Foundation of Georgia (SRNSF) (Grant numberFS-18-126).

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Rescaling and Trace Operators in Fractional SobolevSpaces on Bounded Lipschitz Domains with PeriodicStructure

Julia OrlikFraunhoferplatz 1, 67661 Kaiserslautern, GERMANYorlik@itwm.fhg.de

This paper presents rescaling of the trace operator act-ing on functions from fractional Sobolev type spaces andalso some related results on rescaling the Bessel poten-tial, Riesz potential, and Sobolev-Slobodetskii spaces onbounded Lipschitz domains with periodic structure.

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On the convergence of Cesaro means of negative orderof Vilenkin-Fourier series

Gvantsa ShavardenidzeIlia Vekua Institute of Applied Mathematics (VIAM)of Ivane Javakhishvili Tbilisi State University (TSU), 2University str., 0186 Tbilisi, GEORGIAshavardenidzegvantsa@gmail.com

In 1971 Onnewer and Waterman establish a suffi-cient condition which guarantees uniform convergenceof Vilenkin-Fourier series of continuous function. In mytalk it is considered different classes of functions of gen-eralized bounded oscillation and in the terms of these

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classes there are established sufficient conditions for uni-form convergence of Cesaro means of negative order.

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On Toeplitz operators with matrix almost periodic sym-bols

Ilya SpitkovskyNew York University Abu Dhabi (NYUAD), SaadiyatIsland, 129188 Abu Dhabi, UAEims2@nyu.edu

Fredholmness and invertibility of Toeplitz operators canbe characterized fully in terms of the Wiener-Hopf fac-torization (or its appropriate modification) of their sym-bols, thus the importance of factorability criteria. Scalarcontinuous functions are factorable if and only if theyare invertible; the result which carries over to the matrixcase. Scalar almost periodic case is no different from thecontinuous version: factorability is equivalent to invert-ibility. Things change, however, in the matrix almostperiodic setting: factorability still implies invertibility,but not the other way around. Moreover, the factorabil-ity criterion in this setting is still unknown. In our talk,we will describe several cases in which efficient necessaryand sufficient factorability conditions have been estab-lished. Some more specific open problems will also bediscussed, time permitting.

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S.21: Time-frequency Analysis andApplications

OrganisersLuis Daniel Abreu, Peter Balazs

Scope of the session: This session includes aspects oftime-frequency analysis and applications in its broadestsense, including Gabor frames and multipliers, applica-tions in physics and in the analysis of non-stationaryphenomena (e. g. of acoustic signals), random point pro-cesses and spectral estimation, as well as more theoreticaspects of time-frequency analysis like classical Har-monic Analysis, wavelets and analysis in function spaces(Fock/Bergman, polyanalytic, Besov, Feichtinger’s alge-bra, other modulation and related function spaces).

—Abstracts—

Time-frequency analysis of signals periodic in time andin frequency

Luis Daniel AbreuAcoustics Research Institute, Austrian Academy of Sci-ences, 1040 Vienna, AUSTRIAlabreu@kfs.oeaw.ac.at

Following a well-established tradition in physics, whichgained widespread attention after the 2016 Nobel Prizeaward to Kosterlitz-Haldane-Thouless for the discov-ery of topological states of matter, we will impose pe-riodic boundary conditions in time and frequency, lead-ing to a distributional finite-dimensional Hilbert spaceof Dirac deltas for representations in the signal domain,and to a Hilbert space of functions over a 2-Torus in

the time-frequency domain. The two representationsare linked by the good old Short-Time-Fourier Trans-form, whose action on distributional spaces has beenput on firm ground by a celebrated work of Feichtingerand Grochenig.I will present some foundational aspects of this new the-ory, which is undergoing an open discussion, as tradi-tional in the Acoustics Research Institute.

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Recent Developments in Time-frequency Analysis andApplications

Peter BalazsAcoustics Research Institute, Austrian Academy of Sci-ences, Wohllebengasse 12-14, 1040 Wien, AUSTRIApeter.balazs@oeaw.ac.at

We will give an overview on our research on the topic ofthe special session.We start with a short introduction to time-frequencyrepresentations and the concept of frames. We willtalk about the implementation of time-frequency trans-forms, and in particular mention how to design opti-mal windows for reconstruction. We will present waysto get adapted and adaptive time-frequency representa-tions and how to match it to the human auditory system.We will present how to use a time-frequency representa-tion to detect ultrasonic mouse vocalization, as well asits usage in an inpainting algorithm.We will introduce the notion of multipliers, and talkabout their mathematical properties, in particular in-vertibility and unconditional convergence. We willdemonstrate applications to compressed sensing usingrandom multipliers and to perceptual sparsity by usingan adaptive approach.

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Frame Recycling

Brigitte Forster-HeinleinUniversitat Passau, Fakultat fur Informatik und Math-ematik, 94032 Passau, GERMANYbrigitte.forster@uni-passau.de

Grafakos and Sansing have shown how to obtain di-rectionally sensitive time-frequency decompositions inL2(Rn) based on Gabor systems in L2(R); the key toolis the “ridge idea,” which lifts a function of one variableto a function of several variables. We generalize theirresult by showing that similar results hold starting withgeneral frames for L2((R), both in the setting of discreteframes and continuous frames. This allows to apply thetheory for several other classes of frames, e.g., waveletframes and shift-invariant systems. We will consider ap-plications to the Meyer wavelet and complex B-splines.In the special case of wavelet systems we show how todiscretize the representations using ε-nets. We will closewith a short discussion of partial ridges. This is jointwork with Peter Massopust (TU Munchen), Ole Chris-tensen (DTU Lyngby) and Florian Heinrich (Universityof Passau).

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Current problems in Gabor Analysis

Hans G. FeichtingerFaculty of Mathematics, University Vienna, Oskar-

98

Abstracts

Morgenstern-Platz 1, 1090 Vienna, AUSTRIAhans.feichtinger@univie.ac.at

Talking about the connection between discrete and con-tinuous Gabor analysis.

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Coorbit spaces for warped time-frequency representa-tions

Nicki HolighausAcoustics Research Institute, Austrian Academy of Sci-ences, Wohllebengasse 12-14, 1st Floor, 1040 Wien,AUSTRIAnicki.holighaus@oeaw.ac.at

Continuous frames on a Hilbert space H provide a verypowerful, abstract framework for the analysis of invert-ible representations of functions in terms of the innerproducts with respect to a, possibly uncountable, dictio-nary. If H = L2(Rd) or a subspace thereof, the dictio-nary is often chosen to be time- or frequency-invariant,i.e. to consist of all translations, respectively modu-lations, of a collection of generators gωω ⊂ L2(Rd).The actual construction of good continuous frames withadditional structural properties may, however, be highlynontrivial. In practice, this has lead to a small number ofclassical construction schemes being used almost exclu-sively, despite often providing suboptimal performance.In the context of time-frequency analysis, these clas-sical schemes are embodied by short-time Fourier andwavelet transforms and their, various and lesser known,extensions. The main characteristics of a time-frequencysystem are comprised of the local time-frequency trade-off and the existence and form of discrete frames ob-tained by selecting an appropriate discrete subset of thedictionary. Here, we propose the construction of time-frequency systems with almost arbitrary local frequencyresolution by means of warping. The resulting time-invariant continuous tight frames are constructed froma prototype function θ : Rd → C, that serves the purposeof a mother “wavelet”, and some weighted diffeomor-phism Φ: D → Rd, the warping function, that controlsthe local time-frequency trade-off of the resulting warpedtime-frequency system G(θ,Φ). We determine basic con-ditions on the warping function Φ, or more specifcallythat Jacobian DΦ−1 of its inverse, such that a family ofcoorbit spaces can be associated with G(θ,Φ), providedthat θ is smooth and decays rapidly. It is further shownthat G(θ,Φ) can be sampled in an intuitive way to ob-tain discrete frames not only on the Hilbert space H,but simultaneously for the entire family of consideredcoorbit spaces. Finally, we link the constructed functionspaces with certain decomposition spaces. This is jointwork with Felix Voigtlaender.

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What is the Wigner function closest to a given squareintegrable function?

Joao Nuno PrataGrupo de Fısica Matematica, Departamento deMatematica, Faculdade de Ciencias, Universidade deLisboa, Edifıcio C6, 1749-016 Lisboa, PORTUGALjnprata@FC.UL.PT

One can perform ”time-varying filtering” by multiply-ing a given Wigner function Wψ(x, ω) by a weight-ing function Γ(x, ω), such that the resulting functionWΓψ(x, ω) = Γ(x, ω)Wψ(x, ω) has some optimal time-frequency concentration. However, WΓψ(x, ω) is not, ingeneral, a Wigner function.This raises the problem of determining the Wigner func-tion closest to it. I will discuss an iterative procedure toapproximate the optimal solution to this problem in acontrolled way. This problem can be related to the local-ization problem for Wigner functions, which consists ofdetermining the Wigner function which has the largestor smallest integral on a given measurable bounded set.

This is talk is based on joint work with J.S. Ben-Benjamin, L. Cohen, N.C. Dias and P. Loughlin.

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Continuity properties of Multilinear τ -Wigner Trans-form

Ayse SandikciOndokuz Mayis University, Faculty of Arts and Sciences,Department of Mathematics, 55200 Samsun, TURKEY

ayses@omu.edu.tr

In this work, we introduce a τ -dependent multilinearWigner transform, Wτ , τ ∈ [0, 1]. It also includesvarious types of multilinear time-frequency representa-tions, among the others the classical multilinear Wignerand Rihaczek representations and some properties formultilinear τ -Wigner transform. We then prove theboundedness properties of multilinear τ -Wigner trans-form both on products of Lebesgue spaces and on mod-ulation spaces.

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Planar sets of sampling for the short-time Fourier trans-form - From sufficient conditions to sampling bounds onpolyanalytic Fock spaces

Michael SpeckbacherUniversite de Bordeaux, Institut de Mathematiques deBordeaux, 351, Cours de la Liberation, 33405 Talence,FRANCEspeckbacher@kfs.oeaw.ac.at

In this talk we will discuss several aspects of the follow-ing problem: Given a window function g, under whichconditions does the restriction of the short-time Fouriertransform to Ω ⊂ R2 yield a continuous frame for L2(R),and how does the frame bound depend on Ω. For a ’nice’class of window functions, it is known that the frameproperty is equivalent to Ω being relatively dense at anarbitrary scale. We will show that Ω being relativelydense at specific scales is also a sufficient condition for agreater class of window functions. Moreover, we presentquantitative estimates of the lower frame bound in thecase of Hermite windows using the correspondence be-tween the short-time Fourier transform and the polyan-alytic Fock space.

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99

Abstracts

Compactly supported linear combinations of elementsof Battle–Lemarie spline wavelet systems and decom-positions in weighted function spaces

Elena UshakovaComputing Center of FEB RAS, 65 Kim Yu Chena St.Khabarovsk, 680000, RUSSIA, and Steklov Mathemat-ical Institute of RAS, 8 Gubkina St. Moscow, 119991,RUSSIAelenau@inbox.ru

Battle–Lemarie scaling function φn of natural order nis a polynomial spline obtained by orthogonalisationprocess of Bn−spline, where Bn := Bn−1 ∗ B0 andB0 := χ[0,1). Let ψn be a wavelet function related toφn. A system φn, ψn is an orthonormal spline waveletbasis of order n in L2(R). For any n ∈ N elementsφn and ψn have exponential decay and unbounded sup-ports. The main result of this work is an algorithmresulting in functions Φn and Ψn which are particularfinite linear combinations of integer shifts of φn and ψn,respectively. The crucial point is that Φn and Ψn arecompactly supported on R, and for all n ∈ N systemsΦn,Ψn form Riesz bases in L2(R). Moreover, Φn andΨn are finite linear combinations of basic splines Bn.The result is applied to wavelet decomposition of Besovand Triebel–Lizorkin function spaces with Muckenhouptweights.The research was supported by the Russian ScienceFoundation (project 19-11-00087).

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Coorbit spaces associated to integrably admissible dila-tion groups

Jordy Timo Van VelthovenFaculty of Mathematics, University Vienna, 1090 Vi-enna, AUSTRIAjordy-timo.van-velthoven@univie.ac.at

The talk considers wavelet coorbit spaces associated tointegrably admissible dilation groups. The quasi-regularrepresentation of such groups is integrable, but not nec-essarily a discrete series representation. Results thatwill be presented are the existence of smooth, admissi-ble vectors and the realization of the coorbit space as adecomposition space relative to an induced cover. Thetalk is based on joint work with Hartmut Fuehr (RWTHAachen).

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Invertibility of frame operators on Besov-type decom-position spaces

Felix VoigtlaenderKatholische Universitat Eichstatt-Ingolstadt, Osten-straße 26, 85072 Eichstatt, Bavaria, GERMANYfelix.voigtlaender@gmail.com

We derive an extension of the Daubechies-Walnut cri-terion for the invertibility of frame operators. Thecriterion concerns general reproducing systems (Gaborsystems, wavelets, shearlets, etc.) and Besov-type de-composition spaces as introduced by Feichtinger andGrobner. As an application, we conclude that L2 frameexpansions associated with reproducing systems on suf-ficiently fine lattices extend to Besov-type spaces. Thissimplifies and improves on recent results on the existence

of atomic decompositions, which only provide a particu-lar dual reproducing system with suitable properties. Incontrast, we conclude that the L2 canonical dual frameexpansions extend to many other function spaces, and,therefore, operations on the canonical frame expansionsuch as thresholding are bounded on these spaces.This is joint work with Jose Luis Romero and Jordy vanVelthoven.

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Adaptive Fourier Decomposition in Hp Spaces

Wang YanboFaculty of Mathematics and Computer Science, WuHanTextile University, WuHan City, HuBei Province, PEO-PLE’S REPUBLIC OF CHINAwybwtu@163.com

In this paper, we study decomposition of functions inHardy spaces Hp(1 < p <∞). First, we will give a directapplication of Adaptive Fourier Decomposition(AFD) inH2 to functions in Hp. Then, we study adaptive decom-position by the normalized Szego kernel dictionary underp−norm. Under the proposed decomposition procedure,we give a maximal selection principle at each step andprove the convergence.

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S.22 Wavelet theory and its Related Topics

OrganisersKeiko Fujita, Akira Morimoto

Scope of the session: The theory of the mathematics isimportant, but it is also important to apply it to reallife. We will organize the session “Wavelet theory andits related Topics” intended to discuss not only the puremathematics, but also the applied mathematics relatedto the research in engineering, medicine, acoustics, andthe other various fields.

—Abstracts—

Deficiency of holomorphic curves for hypersurfaces andlinear systems

Yoshihiro AiharaFukushima University, 1 Kanayagawa, 960-1296Fukushima, Fukushima, JAPANaihara@educ.fukushima-u.ac.jp

In this talk, we discuss relations between the deficiencyof holomorphic curves for hypersurfaces and the defi-ciency for the base loci of linear systems in algebraicmanifolds. We also discuss methods constructing holo-morphic curves with deficiencies.

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Holder regularity of anisotropic wavelets and waveletframes: matrix approach

Maria CharinaUniversity of Vienna, Fakulty of Mathematics, 1090 Vi-enna, AUSTRIAmaria.charina@univie.ac.at

100

Abstracts

The compactly supported solution (refinable function)of the functional equation

ϕ(x) =∑k∈Zs

ckϕ(Mx−k), x ∈ Rs, ck ∈ R, M ∈ Zs×s,

can generate systems of multivariate wavelets or frames.In the univariate case, M ≥ 2 is an integer, there areseveral efficient methods for determining the regularityof refinable functions. One of them, the so-called matrixapproach, yields the Holder exponent of ϕ ∈ C(R) and,in addition, provides a detailed analysis of the modulusof continuity of ϕ and of its local regularity. The gen-eralization of the matrix approach to the multivariatecase turned out to be a difficult task in the case of gen-eral dilation matrices M . The special case of isotropicdilation M (all eigenvalues of M are equal in the abso-lute value) is currently fully understood. In this talk wediscuss the challenges of the anisotropic case, i.e. the di-lation matrix M has eigenvalues different in the absolutevalue. We show how the Holder exponent of ϕ ∈ C(Rs)reflects the influence of the invariant subspaces of Mcorresponding to its different in modulus eigenvalues.

This is a joint work with V. Protasov, Moscow StateUniversity, Russia.

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The Template-based Procedure of BiorthogonalTernary Wavelets for Curve Multiresolution Processing

Li DengfengSchool of Mathematics and Computer, Wuhan TextileUniversity, 430200 Wuhan, Hubei Province, PEOPLE’SREPUBLIC OF CHINAwavelet2009@126.com

Curve multiresolution processing techniques have beenwidely discussed in the study of subdivision schemes andmany applications, such as surface progressive trans-mission, compression, etc. The ternary subdivisionschemes are the more appealing one because it can pos-sess high symmetry, smaller topological support andcertain smoothness, simultaneously. So, biorthogonalternary wavelets are discussed in this talk, in which re-finable functions are designed for cure and surface mul-tiresolution processing of ternary subdivision schemes.Moreover, by the help of lifting techniques, the template-based procedure is established for constructing ternaryrefinable systems with certain symmetry, and it alsogives a clear geometric templates of corresponding mul-tiresolution algorithms by several iterative steps. Someexamples with certain smoothness are constructed.

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Optimization in the construction of nearly cardinal andnearly symmetric wavelets

Neil Kristofer DizonThe University of Newcastle Australia, 130 UniversityDrive, Callaghan, 2308 Newcastle, New South Wales,AUSTRALIAneilkristofer.dizon@uon.edu.au

In this talk, we present two approaches to wavelet con-struction that produce nearly cardinal and nearly sym-metric wavelets on the line. The first approach models

wavelet construction as an optimization problem withconstraints derived from the compact support, regular-ity, and quadrature mirror filter (QMF) conditions im-posed on the integer samples of the scaling function andits associated wavelet. The objective function is set toallow for minimization of the scaling function’s deviationfrom symmetry or cardinality. The second method is anextension of the feasibility approach by Franklin, Hogan,and Tam to allow for near symmetry or near cardinalityby considering variables generated from uniform samplesof the QMF, and is solved via the Douglas-Rachford al-gorithm. We also give examples of complex symmetricscaling functions which have been successfully obtainedusing this approach. Finally, we discuss the possibility ofextending these constructions to multivariate wavelets..

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Two-dimensional frames for multidirectional decompo-sitions

Kensuke FujinokiTokai University, 4-1-1 Kitakaname, 259-1292 Hiratsuka,Kanagawa, JAPANfujinoki@tokai-u.jp

In this talk we consider two-dimensional frames onL2(R2). Starting with a review on an approximationtheory and a frame theory, we give our recent study ofwavelet frames for multidirectional expansion of multi-variate functions that have anisotropic features. Resultsof its numerical experiments are also shown.

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Some topics on the Gabor wavelet transformation

Keiko FujitaUniversity of Toyama, Department of Mathematics, Fac-ulty of Science, 840-8555 Toyama, JAPANkeiko@sci.u-toyama.ac.jp

We have studied the blind source separation problemby using some wavelet transformations. Since the Ga-bor function is an exponential type, we can consider theGabor wavelet transformation not only for square inte-grable functions but also for analytic functionals. In thistalk, we will review our previous results, and will con-sider some applications by means of the characteristicsof the Gabor wavelet transform of analytic functionalsor square integrable functions.

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Subdivision schemes on a dyadic half-line

Mikhail KarapetyantsMoscow Institute of Physics and Technology, MoscowMIPT, RUSSIAN FEDERATIONkarapetyantsmk@gmail.com

We consider the subdivision operator on a dyadic half-line. Necessary and sufficient convergence conditions,the connection between the subdivision scheme and therefinement equation, existence criterion of a fractal curveand continuous solution of the refinement equation andsome combinatorial properties of a subdivision schemeare studied. The conjecture on convergence of subdi-vision schemes with non-negative masks is also formu-lated.

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101

Abstracts

The time-frequency analysis on the half space

Yun-Zhang LiCollege of Applied Sciences, Beijing University of Tech-nology, 100124 Beijing, PEOPLE’S REPUBLIC OFCHINAyzlee@bjut.edu.cn

This talk addresses the theory of time-frequency on thespace L2(R+) of square integrable functions defined onthe right half real line. We will give an overview of itshistory, and then focus on our recent study of the frametheory based on a class of dilation-and-modulation sys-tems in L2(R+).

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Application of frame theory to shape from defocus

Anastasia ZakharovaUniversity of La Rochelle, 23, Avenue A. Einstein, BP33060, 17031 La Rochelle, FRANCEanastasia.zakharova@univ-lr.fr

In this talk we study the problem of depth reconstruc-tion from a series of defocused images. Expanding thegeometrical approach describing the intensity of a de-focused image as a result of acting of a linear integraloperator, we interpret the defocusing operator as anal-ysis operator of a continuous frame of a subspace of aHilbert space. The reconstruction can then be realizedwith the help of an oblique frame. We show then thatthe initial minimization problem formulated in terms ofdepth and radiance can be reduced to a minimizationproblem involving only the depth term.

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S.23 Poster Session

Organisersnot appliable

Scope of the session: The Poster Session is directed toparticipants which do not see themselves reflected in theabove listed sessions.

—Abstracts—

Spin Structures on p-Gonal Surfaces via Divisors

Yahya AlmalkiKing Khalid University, Abha, SAUDI ARABIAyahya701@gmail.com

We study spin structures on Riemann and Klein surfacesin terms of divisors. In particular, we take a closer lookat spin structures on hyperelliptic and p-gonal surfacesdefined by divisors supported on their branch points.Moreover, we study invariant spin divisors under au-tomorphisms and anti-holomorphic involutions of Rie-mann surfaces and count them. We generalize a formulathat gives 2-spin divisors proved by Mumford to the caseof m-spin divisors, for an even m, supported on branchpoints of a hyperelliptic surface.

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Analytic Improvement of Frequency Characteristic ofGaN-based HEMT

Asmae BabayaEcole Nationale Superieure d’arts et Metiers, Moulay Is-mail University, MOROCCOasmae.babaya@gmail.com

Gan-based high-electron mobility transistors (HEMTs)have demonstrated microwave power performance andhave a high potential to be used in electronics powerhigh frequency. The present work is dedicated to studythe physical properties of InAlN/GaN HEMT and theapproximate resolution of the Poisson and Schrodingerequations to determine the density of the two dimen-sional electron gas. The influences of the InAlN barrierthickness and the layer compositions on the propertiesof the InAlN / GaN heterojunction have been demon-strated. The improvement of drain current contributeto the improvement of the sheet concentration of twodimensional electron gas produced at the interface of In-AlN barrier and GaN channel layers by spontaneous andpiezoelectric polarizations. In other hand, the increasingof drain current can improve its transconductance whichcontribute to the enhancement of the device’s frequencyperformance. The 2DEG density and strain betweenInAlN and GaN can be controlled by modulating theIn and Al molar fraction. A thinner InAlN barrierlayer with an accurate In and Al molar fraction can in-crease the 2DEG density at the interface of InAlN/GaNHEMT. By using a thinner barrier layer, InAlN/GaNHEMT gives a much higher drain current of 1.63A/mmdue to its higher two electron gas density of 6.7e13 cm-2.

Join work with Adil Saadi and Seddik Bri.

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Harmonic Analysis and the topology of Spheres

David Santiago Gomez CobosCarrera 24 C # 54-60 Sur Int 7 Apto 301, 110621 Bo-gota D.C., COLOMBIAds.gomez11@uniades.edu.co

In this poster we study the harmonic analysis in homo-geneous spaces, based on the theory of representations ofcompact Lie groups, and how we can use this theory tofind topological invariants of those spaces. Specifically,it deals with the case of spheres Sn, which can be seen ashomogeneous spaces by means of the quotient of specialorthogonal groups SO(n+1)/ SO(n), and explicit calcu-lations of its Euler character are presented. The latterwith the help of the Hodge decomposition theorem andthe index theorem for the Dirac-de Rham operator onthe sphere.

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On the series with an affine sum range in a Banachspace

George GiorgobianiMuskhelishvili Institute of Computational Mathematicsof the Georgian Technical University, Grigol Peradze str.#4, 0159 Tbilisi, GEORGIAgiorgobiani.g@gtu.ge

This talk is about the duality between the permutationsand arrangements of signs for the series with terms in

102

Abstracts

infinite dimensional Banach spaces. We discuss one suf-ficient condition for the affinity of the sum range of aseries and give some explicit constructions of the se-quences, satisfying this condition in some concrete Ba-nach spaces. We also consider some classes of bounded,linear operators, which emerge in related problems.

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Colombeau generalized stochastic processes and appli-cations

Snezana GordicFaculty of Education in Sombor, University of Novi Sad,Podgoricka 4, 25000 Sombor, Vojvodina, SERBIAgordicsnezana@gmail.com

Colombeau generalized stochastic processes are de-fined. Probabilistic properties of Colombeau general-ized stochastic processes are investigated with emphasison their stationarity. A necessary condition for the exis-tence of a stationary Gaussian solution to some class ofstochastic partial differential equations is given. As anexample, the stationary Klein-Gordon equation drivenby higher order derivatives of white noise is considered.

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Exact solutions of a nonlinear equation from the theoryof nonstationary processes in semiconductors

Anna KholomeevaLomonosov Moscow State University, 119602 Moscow,RUSSIAN FEDERATIONkholomeeva@cs.msu.ru

We consider a nonlinear partial differential equation

∂

∂t(∆u− u) + ∆u+ div(u∇u) = 0

where u is real-valued function depending on the one-dimensional spatial variable x and on the time variablet¿0. We construct families of exact solutions of the givenequation. Solutions are found in explicit analytical formusing elementary functions and special functions. Thequalitative behavior of the constructed solutions is ana-lyzed. Joint work with dr. A.I. Aristov.The research was supported by the Program of the Pres-ident of Russian Federation for Support of Young PhDs(project no. MK-1829.2018.1).

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Bifurcations of invariant manifolds of nonlinear opera-tors in Banach spaces

Liudmila NovikovaSouthern Federal University, 344038 Rostov-on-Don,RUSSIAN FEDERATIONlvnovikova@sfedu.ru

We find conditions, for an invariant manifold K of an op-eratorNδ in a Banach space, to be asymptotically stable.There is studied the case of the loss of stability of the in-variant manifold K in the direction of a one-dimensionalfoliation R1,δx, x ∈ U. There are given additional con-ditions for the invariant manifold K to be unstable forsmall δ > 0, with a branching off asymptotically stableinvariant manifold Kδ homeomorphic to K.

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Some refinements of Ostrowski’s inequality and an ex-tension to a 2-inner product space

Nicusor MinculeteTransilvania University of Brasov, Str. Iuliu Maniu, Nr.50, 500091 Brasov, ROMENIAminculeten@yahoo.com

The purpose of this paper is to prove certain refinements of Ostrowski’s inequality in an inner productspace. In section 3 we study extensions of Ostrowskitype inequalities in a 2-inner product space. Finally,some applications which are related to the Chebyshevfunctional and the Gruss inequality are presented.

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Heat-like and wave-like behaviour of the lifespan esti-mates for wave equations with scale invariant dampingand mass

Nico Michele SchiavoneUniversity of Rome “La Sapienza”, Department ofMathematics, Piazzale Aldo Moro 5, 00185 Rome,ITALYschiavone@mat.uniroma1.it

We study the upper lifespan estimates for wave equa-tions with scale invariant damping and mass, highlight-ing the competition between heat-like and wave-like be-haviour, the case δ := (µ1−1)2−4µ2 = 0 and a particu-lar condition on the initial data which implies a changein the estimates.

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Global solutions of 3D incompressible MHD systemwith mixed partial dissipation and magnetic diffusion

Yi ZhuEast China University of Science and Technology, 130Meilong Road, Xuhui District, 200237 Shanghai, PEO-PLE’S REPUBLIC OF CHINAzhuyim@ecust.edu.cn

In this talk, we focuses on the 3D incompressible mag-netohydrodynamic (MHD) equations with mixed par-tial dissipation and magnetic diffusion. Our main resultassesses the global stability of perturbations near thesteady solution given by a background magnetic field.The stability problem on the MHD equations with par-tial or no dissipation has attracted considerable interestsrecently and there are substantial developments. Thenew stability result presented here is among the veryfew stability conclusions currently available for ideal orpartially dissipated MHD equations. As a special con-sequence of the techniques introduced in this paper, weobtain the small data global well-posedness for the 3Dincompressible Navier-Stokes equations without verticaldissipation.

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103

Abstracts

104

105

Index

Odemis, O. B., 40Zigic , M., 53

Abdelhamid, M. D., 87Abreu, L. D., 98Abylayeva, A., 41Agostiniani, V., 54Aihara, Y., 100Aksoy, U., 24Akyel, T., 30Alip, M., 30Almalki, Y., 102Almeida, A., 41Alpay, D., 78Alqahtani, O., 38Amma, B. B., 38Antonic, N., 47Arepova, G., 72Ariza Garcia, E., 78Arroyo, A., 54Ascanelli, A., 87Aslan, H. S., 87Atanasova, S., 47Attouchi, A., 54Ayele, T., 94Azeroglu, T. A., 30

Babaya, A., 102Bae, H. O., 72Balazs, P., 98Balogh, A., 68Bandeira, A., 15Bar, O., 35Baramidze, L., 94Baran, W., 35Bargetz, C., 47Baskin, D., 69Begehr, H., 3Bekele, S. T., 94Bernstein, S., 78Berti, D., 54Bilen Can, M., 58Bisi, C., 78Blasco, O., 41Bock, S., 79Boiti, C., 88Bolotnikov, V., 79Borges de Oliveira, H., 72Borsuk, M., 35Bousquet, P., 54Bouzar, C., 48Brasco, L., 54Brauer, U., 69Brinker, J., 60Broucke, F., 48Bryla, M., 35Bryukhov, D., 79Brzdek, J., 38Buoso, D., 91, 92

Caetano, A., 41Cappiello, M., 88Casimiro, A., 27Catana, V., 75Cavallina, L., 55Celebi, A. O., 3, 24, 25Cerejeiras, P., 3Chaabani, A., 25Charao, R. C., 88Charina, M., 100Chemetov, N. V., 72Chen, W., 88Chen, X., 45Cheng, Y., 21Chinchaladze. N., 25Chkadua, G., 94Cho, Y., 60Choi, Q., 72Chung, S-Y., 48Cialdea, A., 30Cieplinski, K., 39Cinti, E., 55Cipriano, F., 72Colombo, C., 79Constanda, C., 95Coriasco, S., 75Correia, S., 60Cozzi, M., 55Cunanan, J., 45Cvetkovic, M., 39Czapla, R., 36

D’Abbicco, M., 87, 88D’Ancona, P., 61Dagnaw, M. A., 95Daher, R., 61Dahlke, S., 41Dalla Riva, M., 30Dao, T. A. , 88Debbouche, A., 39Delgado, B.B., 79Della Pietra, F., 55Denega, I., 30Dengfeng, L., 101di Blasio, G., 92Di Teodoro, A., 80Diki, K., 80Dizon, N. K., 101Dominguez, O., 41Drihem, D., 41Drygas, P., 36Duduchava, R., 95

Ebert, M., 69, 87Erhan, I., 39Eriksson, S.-L., 80

Faminskiy, A., 89Faustino, N., 80

Fei, M., 81Feichtinger, H. G., 99Felisetti, C., 27Fernandez, A., 67Ferreira, M., 3Florentino, C., 27Forster-Heinlein, B., 81, 98Franco, E., 28Fresneda-Portillo, C., 96Fujinoki, K., 101Fujita, K., 100, 101Fujiwara, K., 61Fukaya, N., 61Fulga, A., 39Fyodorov, Y., 21

Gomez Cobos, D. S., 102Gaiko, V., 21Galstyan, A., 68, 69Garcıa-Ferrero, M. A., 56Garcia, A. R. G., 48Garello, G., 76Garrione, M., 92Garrisi, D., 61Georgiev, V., 60, 89Ghaffari, H.B., 81Ghiloni, R., 81Ghisi, M., 89Giacomoni, J., 73Gilbert, R., 3Giorgadze, G., 31Giorgobiani, G., 102Girardi, G., 89Gobbino, M., 89Golberg, A., 3, 30, 31, 54Gomes, N., 82Gordic, S., 103Gothen, P., 28Grochenig, K., 16Green, C., 31Grigor’ev,, Y., 82Grigor’yan, A., 15Grigorian, S., 58, 69Gryshchuk, S., 32Guerra, R., 67Gulua, B., 25Guncaga, J., 36Guzman Adan, A., 82

Hasto, P., 42Hamano, M., 61Haroske, D. D., 41Hasler, M. F., 48Hedenmalm, H., 16Henriques, C., 3Henriques, E., 73Henriques, F., 3Heu, V., 28Hidemitsu, K., 56

105

Abstracts

Hirosawa, F., 69, 90Hoffmann, N., 28Hogan, J., 82Holighaus, N., 99Hritonenko, N., 21, 49Hwang, G., 62Hwang, J., 49

Ikeda, M., 62Inui, T., 62Ionescu-Tira, M., 82Ishikawa, I., 67Ivanov, A., 22Ivec, I., 76

Jacobzon, F., 32Johnsen, J., 90Jung, T., 73

Kahler, U., 3, 78Kaddour, T. H., 39Kal’menov, T., 92Kaminski, A., 49Kanatchikov, I., 82Karapetyants, A., 42Karapetyants, M., 101Karapinar, E., 38Karlovich, Y., 96Karlovych, O., 42Karp, L., 70Kato, K., 45Kato, T., 45Kennedy, J., 3, 91Khan, S. H., 39Kholomeeva, A., 103Kishimoto, N., 46Klishchuk, B., 32Kobayashi, M., 46Kohr, G., 33Kohr, M., 96Kokilashvili, V., 42, 96Konjik, S., 49Kovalev, A., 58Krasnov, Y., 83Kraußhar, S., 83Krump, L., 83Kumar, V., 76Kunzinger, M., 47, 49Kurtyka, P., 36

Lai, N., 62Lanza de Cristoforis, M., 3, 30,

33Lanzara, F., 22Lappicy, P., 56Lavicka, R., 83Lederman, C., 73Legatiuk, A., 83Legatiuk, D., 84Lemos-Paiao, A. P., 22Lewandowski, R., 73Li, C., 62Li, Y-Z., 102Lindgren, E., 56Lindstrom, T., 21, 22Liping, W., 84Liu, B., 70

Liu, Y., 25Loiudice, A., 62Lorenz, D., 76Lotay, J., 58Lu, T-T., 46Lucardesi, I., 56Lucente, S., 90Luna-Elizarraras, M.E., 84Luz, C. da, 90Luzzini, P., 33Lysik, G., 25

Maciaszek, K., 33Magnanini, R., 54Manjavidze, N., 25Marın-Rubio, P., 73Marini, M., 56Marques, J., 90Marreiros, R., 26Massopust, P., 84Matos Duque, J. C., 72Matsuyama, T., 63Mazzieri, L., 57Melnikova, I., 50Mertens, T., 84Meshkova, Y., 92Meskhi, A., 42, 96Mezadek, M. K., 26Mihula, Z., 42Mikhailov, S., 94, 97Mincheva-Kaminska, S., 50Minculete, N., 103Mityushev, V., 3, 35, 36Miyazaki, H., 63Mochizuki, K., 63Mohammed, A., 92Mohammed, E. A., 67, 92Molahajloo, S., 75Morimoto, A., 100Mouayn, Z., 66, 67Moura, S., 43Mrevlishvili, M., 97Munoz-Castaneda, A. L., 28Mukhammadiev, A., 50Muratbekov, M., 26Murzabekova, G., 23Musolino, P., 33

Nakamura, M., 68, 70, 91Nakazawa, H., 91Nasser, M. M. S., 33Natario, J., 70Natroshvili, D., 94, 97Nawalaniec, W., 37Ndairou, F., 68Nedeljkov, M., 50Nedic, M., 34Neves, A., 17Neves, J., 43Neyt, L., 50Niche, C.J., 74Nigsch, E., 51Noi, T., 43Nolder, C., 84Novikov, R., 17Novikova, L., 103

Nurtazina, K., 23

Oberguggenberger, M., 3, 47, 51Oinarov, R., 43Ok, J., 43Oliaro, A., 76Oliveira, G., 58Oliveira, H. B. de , 74Onodera, M., 57Oparnica, L., 63Orelma, H., 85Orlik, J., 97Ortega, A., 28Ospanov, K., 93Ozawa, T., 18, 60

Pacini, T., 59Palmieri, A., 91Peixe, T., 23Peon-Nieto, A., 29Perotti, A., 85Pfeuffer, C., 76Pick, L., 43Pilipovic, S., 3, 47, 51Plaksa, S., 30Polizzi, F., 29Pombo, I., 63Popivanov, P., 63Prangoski, B., 51Prata, J. N., 99Pusateri, F., 91

Qian, T., 3, 44

Rost, G., 23Rafeiro, H., 44Rajabov, N., 26Rajchel, K., 37Ratnakumar, P. K., 64Reissig, M. G., 3, 70Ren, G., 85Ren, H., 85Restrepo, J. E., 44Rodino, L., 3Rodrigues, M.M., 44Rodrigues, W., 52Rogosin, S., 24Romero, M., 29Rosiek, R., 37Rottensteiner, D., 64Ruan, S., 24Ruosteenoja, E., 57Ruzhansky, M., 3, 60Rylko, N., 37

Sabadini, I., 3, 78, 85Sabar, T., 40Sabibtek, B., 64Sadybekov, M., 34Sajka, M., 37Salac, T., 85Salnikov, V., 59Salur, S., 58Samko, N., 44Samko, S., 41Sandikci, A., 99Sazhenkov, S., 74

106

Abstracts

Schaffhauser, F., 29Schiavone, N. M., 103Schmitt, A., 27Schwarz, M., 52Sebih, M. E., 64Sert, U., 74Shapiro, M., 86Shavardenidze, G., 97Shi, H., 86Shmarev, S., 72, 75Shpakivskyi, V., 34Sicbaldi, P., 57Sickel, W., 44Siltanen, S., 18Silva, A., 68Silva, C. J. , 24Silva, J. C., 52Simoes, A., 40Sommen, F., 78Sousa, R., 64Speckbacher, M., 99Spitkovsky, I., 98Stoppato, C., 86Sugimoto, M., 3, 45, 64Suragan, D., 19Sutherland, T., 29Svanadze, M., 38

Tanaka, T., 64Taniguchi, K., 46, 65Tarulli, M., 65Tetunashvili, S., 44Tiwari, D., 52

Todorov, T. D., 53Toft, J., 3, 46, 77Tokmagambetov, N., 65Tomita, N., 46Torebek, B., 65Toshiaki, Y., 57Tovar Sanchez, L. M., 35, 86Trani, L., 57Trushin, I., 65Tsutsui, Y., 47Tulenov, K., 66Turcinova, H., 45Turunen, V., 3, 77

Uddin, I., 40Ushakova, E., 100

Vajiac, M., 86Van Le, H., 59Van Velthoven, J. T., 100Vassilev, D., 59Vasy, A., 70Velichkov, B., 57Velinov, D., 53Venkov, G., 66Vezzoni, L., 59Vickers, J., 71Vieira, N., 3Vindas, J., 3, 53Voigtlaender, F., 100Vojnovic, I., 77

Wahlberg, P., 77

Wakabayashi, S., 71Wakasugi, Y., 91Wang, B., 45, 71Wang, W., 86Wang, Xi., 26Wang, Xu., 47Watts, J., 59Waurick, M., 93Windes, E., 60Wirth, J., 3, 60Wittbold, P., 75Wolf, J., 75Wong, K. L., 78Wong, M. W. , 3, 75

Xie, Y., 86

Yagdjian, K., 68, 71Yakubovich, S., 66Yanbo, W., 100Yang, H., 87Yang, S., 71Yerkinbayev, N., 68Yessirkegenov, N., 66Yun, H., 27

Zaccaron, M., 93Zakharova, A., 102Zhao, L., 47Zhou, Y., 66Zhu, Y., 103Zhuldyz, Y., 93

107