a personal mathematical history

A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalized Hyperanalytic Function Theory The Georgians Free University Berlin and the Hahn-Meitner Institute 1974-75 University of Delaware, Unidel Professor 1975 Some Work in Elasticity, etc. I visit China for the First Time, 1984 Research in Underwater Acoustics The 65th Birthday Conference in Graz The Cold War Ends: No More Underwater Acoustics The Birth of ISAAC I am 63 and I begin to work in Homogenization My 70th Birthday Conference in Frejus, France The 75thBirthdayConferenceinOrlando Medical Mathematics
A Personal Mathematical History
Robert Pertsch-Rusterholz Gilbert Department of Mathematics
I wasn’t always a mathematician
Originally I was a student of physics and obtained a BSc and MSc in Physics before switching to mathematics. The cause was the Heisenberg Uncertainty Principle, which led me to write to Albert Einstein about my uncertainty about Uncertainty. After switching to the Mathematics Department I worked on a project with Hirsh Cohen and it resulted in a paper. Hirsh left Carnegie for Rennselaer. I began working with Zeev Nehari, who had recently joined the Department from Washington University. This was professionally a good move, as Zeev Nehari was one of the preeminent complex analysts during the 1950s.
This is the paper with Hirsh Cohen on free-boundary problems, an area I would return to after many years.
For my PhD thesis I used Integral Operator Methods, a le Stefan Bergman, which were used to determine the analytic and singular behavior of various classes of elliptic partial differential equations. The Bergman-Whittaker Operator generates harmonic functions in R3,
H(X) = B3(f ) :=
ζ ,
from functions of two complex variable f (u, ζ), where X = (x1, x2, x3), u = x1 (ζ − 1/ζ) + ix2 (ζ + 1/ζ) + x3 is an auxillary variable and C is a contour in C1. I proved necessary and sufficient conditions for H(X) to be singular in terms of its analytic associate.
Theorem
Let the defining function for the set of singularities of f (u, ζ)ζ−1
be a global function in C2. Then if h(u, ζ) := S(X; ζ) = 0 is such a function we have that H(X) = B3f is regular for all points X which may be reached by continuation along a curve Γ starting at an initial point of definition X0, provided X and hence, the curve Γ, do not lie on the intersection
S := {X : S(X; ζ) = 0} ∩ {X : Sζ(X; ζ) = 0} .
Figure: Zeev Nehari, nee Wili Weissbach
Figure: Graduation, Carnegie-Mellon 1958
Using the methods of projective differential geometry we were able to prove, among other results,
Theorem
Let H(X) be a harmonic function given by H(X) = B3f where the singulaities of f (u, ζ) have a global representation, h(u, ζ) := ζu − φ(ζ) = 0. Then the singularities of H(X) must lie on a develople in P. Further, if φ(ζ) is a polynomial of degree n, n > 2, then the singularities are twisted nth degree (complex) space curves.
The above results were generalized to harmonic functions in R4
using the operator
H(X) = G4f := − 1
η f (τ, ξ, η) X; = (x1, x2, x3, x4)
where f is a holomorphic function of three complex variables and
τ := x1(1 + 1
1
ξ +
1
η ).
Here the curves Ck are homologous to zero. Kreyzig inverted this operator and referred to G4 as Gilbert’s operator. Using Kreyzig’s inversion the entire three-dimensional theory was carried over to either R4 or C4.
Figure: Erwin Kreyzig
Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor
We were supported by Alexander Weinstein’s AFOSR grant and our NSF grant to make a systematic investigation of the analytic properties of solutions to singular, elliptic partial differential equation of the type blow
∂2u
∂x2 +
∂2u
∂y2 +
2µ
y +
∂ u
in two and higher dimensions.
We developed the so-called envelope method which was applied to potential scattering and to quantum field theory by Gilbert with Aks, Howard and Shieh.
Figure: Expansions of analytic functions
Indiana University Bloomington, Professor 1966-75
I am all of 34 and Full Professor at Indiana University. I have an AFOSR grant which is supporting, in addition to me, David Colton, Lucio Tavernini and Kendall Atkinson. I start the journal Applicable Analysis.
Figure: Kendal Atkinson and Lothar Collatz. During the 1960s Collatz was the premier numerical analyst in Germany.
Figure: Sketch of ”Owls” by L. Collatz. Collatz knew I liked owls and gave me the sketch.
Figure: Max Schiffer visits us at IU.
Analytic Methods in Mathematical Physics
In June 1968 Roger Newton and I organized an AFOSR Conference named Analytic Methods in Mathematical Physics
In his plenary address Maria v. Krzywoblocki presented a review of the integral Operator Method as initiated by Bergman and others, me included.
Figure: Paul Garabedian and David Widder
Figure: A. Wheeler and S. Abyankar
Figure: Bers, Garabedian, Erdelyi, Masani, me
My First Monograph
I summed up my work to this time with the Academic Press monograph
Figure: My first monograph, 1969.
Before leaving the operator approach of Bergman I developed the Method of Ascent MoA. The method of ascent maps a harmonic function of two variables onto solutions of a metaharmonic-type equation.
∂2u
This idea has had applications in potential scattering and acoustical scattering. Robert Carroll referred to this as the Bergman-Gilbert Transmutation.
Generalized Hyperanalytic Function Theory
I then turned away from the Bergman approach to function theoretic methods for pde to that of Bers and Vekua. and to the introduction of an entirely new function theory called Generalized Hyperanalytic Function Theory GHA.
The generalized hyperanalytic theory was developed to handle individual blocks of first order elliptic systems, ie those terms belonging to a single complex eigenvalue. The GHA functions satisfiedequations of the type
Du + Au + Bu = F , D := ∂
∂z + q(z)
ekak(z) etc. where e is a
nilpotent such that en = 0. A complete function theory was developed for this system by Gilbert with Hile, Begehr, Buchanan, Wendland etc. Bojarski also treated this problem using a different approach. Gillbert with Buchanan and Wendland used this function theory to investigate elastic plates and shells.
The Georgians
Ilya Vekua invites me to attend the Muskhelishvili Conference in Tblisi. I meet Ladyshenskya, Mazya and Koshelev and Leray. I already knew Fichera, Vekua and Oleinik from Maryland.
Figure: Gaetano Fichera with Galina Koshelev
Figure: Vladimir Mazya and Koshelev
Free University Berlin and the Hahn-Meitner Institute 1974-75
I spent a sabbatical semester Spring 1974 at the Hahn Meitner Institute for Nuclear Research in Berlin. The Mathematics group was headed up by Wolfgang Haack who happens to be Wolfgang Wendland’s thesis advisor. The possibility to take over the Dinghas Chair at the Free University opened up and I remained in Berlin for another year. This time was spent developing the generalized hyperanalytic function theory with Heinrich Begehr and Manfred Schneider. George Hsiao and I carry on a midnight vigil writing papers, he in Darmstadt and me in Berlin. I visit the Soviet Union and the Georgian Republic again.
Figure: A paper of mine and George’s and Pat’s in Rendiconti”.
I visit the Humboldt Universiy and Martin Luther University and begin to lecture in German.
Figure: Christian Vidic, Hahn-Meitner Institute
Figure: Wolfgag Tutschke and Naas at Humboldt University
Figure: My father who is my present age visits me in Berlin in Spring 1974.
University of Delaware, Unidel Professor 1975
I am 43 and Unidel Professor. The AFOSR allows me to transfer the grant to Delaware. The grant supports my work on generalized hyperanalytic functions and elasticity and David Colton’s work on scattering. Alan Jeffrey and I had a DOE grant here also for a while. I become interested in Underwater Acoustics after spending a summer the Naval Underwater Systems Center in New London, CT. I had served on David Wood’s PhD committee at the University of Rhode Island. David had done work on the Function Theoretic Method for his thesis. This leads to support from NOAH and NSF for this research. David Wood and I wrote a paper on the use of transmutations in underwater acoustics and provided some propagator representations. These were representations in a stratified, acoustic wave guide. This work was continued by me with Steve Xu and published jointly in what now is a well cited set of articles.
Some people at and passing through Delaware during my first few years here.
Figure: Gary Roach and Sam Nadler Figure: Dick Weinacht
Figure: Pat Brown, David Colton, Alan Jeffrey
Figure: Tom Angell Figure: George Hsiao
Some Work in Elasticity, etc.
A reactive shock in a polytropic gas, J. Math. Analysis & Applications 89 (1982), 193-202 (with A. Jeffrey).
Function theoretic solutions to problems of orthotropic elasticity Journal of Elasticity 15 (1985) 143-154.(with Lin)
The two dimensional linear, orthotropic plate: The traction problem, Math. Methods Appl. Scis. 8 (1986), 102-116 ( M. Schneider). GSch= 11
The boundary integral method for two dimensional orthotropic materials, Journal of Elasticity (with R. Magnanini) J. Elasticity 18 (1987) 61-82. GSch= 12
Nonisothermal, nonnewtonial Hele-Shaw flows, Part II: Asymptotics and existence of weak solutions, Nonlinear Analysis 27(1996) 539-559. (with P. Shi).
Journal of Elasticity 15 (1985) 143-154. © 1985 Martinus Nijhoff Publishers, Dordrecht. Printed in The Netherlands.
Function theoretic solutions to problems of orthotropic elasticity * R O B E R T P. GILBERT 1,** and L I N W E I 2
J Department of Mathematics, University of Delaware, Newark, DE 19711, USA 2 Zhongshan University, Guangzhou, People's Republic of China
(Received October 18, 1983)
Abstract The plane strain problem for a two dimensional orthotropic elastic body is investigated. In particular analytic representations for the solution of the displacement boundary value problem and the stress boundary value problems are found. To this end, the Navier equations are reducrd by means of composite transformations to normal form. These are the so-called equations for bianalytic function of the type (hk). The generalized Cauchy integral formula for this function theory is used to obtain representation formulae. A simplified method to solve these problems by bianalytic function theory is given for certain situations of plane strain for an orthotropic elastic body. AMS (MOS): 35A20, 35CO5, 35G15, 35J55.
1. Introduction Funct ion theoretic methods for second order elliptic systems have been investigated by H u a Loo Keng [1], [2] and his coworkers. These results are well known in the east; however, for the benefi t of western readers we shall present as prel iminaries a short in t roduct ion to these ideas.
Let us consider the following second order system
32 32 C 3---~-2 ] ( u ] = 0 , (1.1) A ~ x 2 + 2B 3 - ~ y + 3 y 2 j \ v l
where A, B, C are constant 2 × 2 real matrices. We shall refer to this system as elliptic if the characterist ic equat ion IA~ 2 = 2B~ + C I = 0 possesses only complex eigenvalues. W h e n this characterist ic equat ion has two mult iple (complex) eigenvalues the system (1.1) m a y be reduced to the canonical fo rm El:
3 [ (1 0 ) ~ x + ( 0 0~)3_%][ i3_~+(0 1 3 0 0 ) ~ y ] ( U ) = 0, (1.2)
for ~ 4= 0, - 1. This factors into the first-order system
1 0 ) ~ x + ( 0 - 1 ) ~ y ] ( U ) = ( O ) [ ( 0 1 1 0 ' (1.3)
[(01 -~)~x+( ° 0~)~y](0) = ° o
* Applied Mathematics Institute Technical Report No. 140A, July 1983. ** The work of this author was supported in part by grant no. DE-AC01-81ER-10967 from the Department
of Energy.
143
Journal of Elasticity 22: 81-93, 1989. © 1989 Kluwer Academic Publishers. Printed in the Netherlands. 81
A function theory for thin elastic shells
R O B E R T P. G I L B E R T & Y O N G Z H I XU Department of Mathematical Sciences, University of Delaware, Newark, DE 19717, USA
Received 16 November 1987
Abstract. It is well known in the theory of elastic shells that a first order approximation using the shell thickness as an expansion parameter leads to the membrane theory of shells. The membrane equations have as solutions the generalized analytic functions. These functions have been exhaus- tively studied by Ilya N. Vekua [6], [7] and his students. R.P. Gilbert and J. Hile [3] introduced an extension of these systems to include elliptic systems of 2n equations in the plane and named the solutions of these systems generalized hyperanalytic functions.
It is shown in this paper that the next order approximation to the shell, which permits, moreover, the introduction of bending, may be described in terms of the generalized hyperanalytic functions. It is strongly suspected that the higher order approximations may also be described in terms of corresponding hypercomplex systems.
1. Notation and formulation
In this work we follow the notation used by Dikmen [1]; we view a shell as a
three dimensional body, which we try to reduce to two dimensional consider-
ation by introducing a suitable reference surface. A set of curvilinear coordi-
nates 0 i, (i = 1, 2, 3) are chosen so that a reference surface within the shell may be represented by 0 3 = 0. For purposes of defining our notation let a be a
surface embedded in R 3 which we represent in the form r = r(0~). The vectors
a~ .'=r~ = (ar)/(O0 ~) are the base vectors. The first fundamental form of the surface is given by a~p d0~d0 a where the a~a.-=a~ "ap are the covariant
components of the metric tensor. I f the .4 ~p are the cofactors of the a~p we define the contravariant metric tensor as a~a:=.4~a/a where a is the determi-
nant of a~p. The second fundamental form of the surface is defined through
b~ a dO ~ d0a:= a3 • a~,a dO ~ d0 a = - d r " da3, with a3 = (al × a2)/([al )< a2l). The Christoffel symbols are given as usual by . i F~a ~ .= ~(a~,a + ap~,~ - a~a,~ ), and covariant differentiation is defined by
, r'~ Tx. (1.1) T~I := T~ -
The position vector of a generic point on the shell at time t is given by x = x(O ~, ~, t) where (0 l, 0 2) ~ i'~ is a point on the reference surface and
~ [0, h], in particular x(O ~, O, t ) = r(O~, t). The methods of Dikmen [1] and
A function theory for thin elastic shells, Journal of Elasticity 22:81-93, (1989). ( with Xu).
Wavelet solutions for time harmonic acoustic waves in a finite ocean, J. Comp. Acoustics,1(1)(1993), 31-61. (with W. Lin)
Non-isothermal, non-Newtonial Hele-Shaw flows, Part II: Asymptptotics and existence of weak solutions, Nonlinear Analysis 27(1996) 539-559. (with P. Shi).
An inverse problem for harmonic acoustics in stratified oceans, J. Math. Anal. Appl.176(1)(1993),121-137 (with Y. Xu).
I visit China for the First Time, 1984
Nixon has opened up relations with China. I am in the first group to visit China. They have translated my book into Chinese. 7, 500 copies are sold in China alone!
Figure: Acting like Nixon on the Great Wall.
Figure: 7,500 copies!!!
Research in Underwater Acoustics
Yongzhi Xu and I began our underwater research by extending the Colton-Kirsch theory for R3 to a waveguide.
The propagation problem and far-field problem in a stratified finite-depth ocean Math. Meth in Applied Sciences 12 (1990), 199-208 (with Xu, Yongzhi).
Dense sets and the projection theorem for acoustic harmonic waves in homogeneous finite depth ocean, Math. Meth in Applied Sciences 12 (1990), 69-76 (with Xu, Yongzhi).
Starting fields and far fields in ocean acoustics, Wave Motion 11 (1989), 507-524 (with Xu, Yongzhi)
Armand Wirgin and I with Buchanan, Xu continued by developing the Intersecting Canonical Domain Method for the Waveguide.
Identification of a 3D object in a shallow sea from scattered sound Ondes et vibrations/ Waves and Vibrations,C. R. Sci. Paris, 325 (1997) 383-389. (with T. Scotti, A. Wirgin, and Y. Xu)
Identification of object in an acoustic wave guide inversion II: Robin-Dirichlet conditions, Math. Methods. Appl. Scis. bf 14 (2)185-199 (2006). (with Doo-Sung Lee)
Figure: Armand Wirgin.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES Math. Meth. Appl. Sci. 2006; 29:401–414 Published online 8 December 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/mma.687 MOS subject classication: 35R 30; 73D 50
Identication of objects in an acoustic wave guide inversion II: Robin–Dirichlet conditions
Doo-Sung Lee1 and R. P. Gilbert2;∗;†
1Department of Mathematics; College of Education; Konkuk University; 1 Hwayang-Dong; Kwangjin-Gu; Seoul; Korea
2Department of Mathematical Sciences; University of Delaware; Newark; DE 19716; U.S.A.
Communicated by W. Sproig
SUMMARY
In this paper we investigate the unknown body problem in a wave guide where one boundary has a pressure release condition and the other an impedance condition. The method used in the paper for solving the unknown body inverse problem is the intersection canonical body approximation (ICBA). The ICBA is based on the Rayleigh conjecture, which states that every point on an illuminated body radiates sound from that point as if the point lies on its tangent sphere. The ICBA method requires that an analytical solution be known exterior to a canonical body in the wave guide. We use the sphere of arbitrary centre and radius in the wave guide as our canonical body. We are lead then to analytically computing the exterior solution for a sphere between two parallel plates. We use the ICBA to construct solutions at points ranging over the suspected surface of the unknown object to reconstruct the unknown object using a least-squares matching of computed, acoustic eld against the measured, scattered eld. Copyright ? 2005 John Wiley & Sons, Ltd.
KEY WORDS: inverse problems; Helmholtz equation; special functions; wave guide
1. INTRODUCTION
In this paper we investigate the localization and identication of a three-dimensional object in an acoustic wave guide by inversion of the scattered wave eld. This inverse problem is non-linear in terms of position and shape parameters of the object and fundamentally ill posed. Furthermore, because of the multiple scattering of sound between the object and the top and bottom boundaries of the sea, the associated direct problem is much more complicated than for a body in space. This problem has important practical marine applications such as identication of mineral deposits, submarines, submerged navigational obstacles and wreckage,
∗Correspondence to: R. P. Gilbert, Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, U.S.A.
†E-mail: [email protected]
Copyright ? 2005 John Wiley & Sons, Ltd. Received 13 June 2005
Inverse problems in a waveguide are numerically more difficult than in R3 because of the attenuation of certain modes.
2 1
0 1
2 1
0 1
Figure: The reconstructed spindle
1 0.5
0 0.5
1
0.5
0
0.5
1
Figure: The reconstructed torus
The 65th Birthday Conference in Graz
Some of the people at my 65th birthday conference in GRAZ, Austria.
Figure: Klaus Habetha, Rector University Aachen
Figure: Bogdan Bojarski, Dir. Stefan Banach Institute
Figure: Elena Obalishvili at Graz Figure: Okay Celebi
The Cold War Ends: No More Underwater Acoustics
We were happy to have funding from NOA, NSF and ONR for another 15 years for our research on Inverse Problems in an underwater environment. We collected our research in the following ( my fifth monograph):
The Birth of ISAAC
At the Hong Kong Complex Analysis meeting in 1997 I floated the idea of starting an international society devoted to real and complex analysis. The idea was strongly endorsed by the attendees. In order to get the thing moving I agreed to hold the first meeting at the University of Delaware. The turnout was impressive, approximately 400 attendees and ISAAC the International Society for Analysis, Applications and Computation was born. We decided to elect officers, a board and to approve a constitution. Subsequent meetings were held at Fukuoka University Japan, the Free University Berlin, University of Catania, the Mideast Technical University Ankara, York University Toronto and last year at Imperial College London where we had 600 attendees. Pierre Lions gave a plenary talk at this meeting along with Sir John Ball and Gunter Uhlmann. This next summer the meeting will be at the Steklov Institute Moscow. All are welcome to attend.
7th ISAAC Congress Imperial College London July 13-18, 2009
International
Society for Analysis, its Applications and Computation
Sir John Ball (Oxford) Louis Boutet de Monvel (Paris) Brian Davies (London) Simon Donaldson (London) Carlos Kenig (Chicago) Vakhtang Kokilashvili (Tbilisi)
Nicolas Lerner (Paris) Paul Malliavin (Paris) Vladimir Maz’ya (Liverpool) Bert-Wolfgang Schulze (Potsdam) Gunther Uhlmann (Seattle) Masahiro Yamamoto (Tokyo)
Confirmed Plenary Speakers
Sessions I.1. Complex variables and potential theory I.2. Differential equations: complex methods, applications I.3. Complex-analytical methods in applied sciences I.4. Value distributions of functions II.1. Clifford and quaternion analysis II.2. Methods in Clifford- and Cayley- Dickson algebras III.1. Toeplitz operators and applications III.2. Reproducing kernels III.3. Integral transforms III.4. Spaces of differentiable functions III.5. Analytic function spaces
III.6. Spectral theory IV.1. Pseudo-differential operators IV.2. Dispersive equations IV.3. Control and optimisation of evolutionary systems IV.4. Nonlinear PDE IV.5. Asymptotic and multiscale analysis V.1. Inverse problems V.2. Stochastic analysis V.3. Coercivity and functional inequalities V.4. Dynamical systems V.5. Functional differential equations V.6. Mathematical biology VI. Others
Contact and Further Information Michael Ruzhansky, Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK http://www.isaac2009.org [email protected]
Local Organisers Michael Ruzhansky (Chairman) Dan Crisan Brian Davies Jeroen Lamb
Ari Laptev (President of the EMS) Jens Wirth Boguslaw Zegarlinski
International Advisory Board H. Begehr (Berlin) A. Berlinet (Montpellier) B. Bojarski (Warsaw) E. Bruning (Durban) V. Burenkov (Padova) O. Celebi (Istanbul) R. Gilbert (Newark)
A. Kilbas (Minsk) M. Lanza (Padova) M. Reissig (Freiberg) L. Rodino (Torino) M. Ruzhansky (London) J. Ryan (Fayetteville) S. Saitoh (Aveiro)
B.-W. Schulze (Potsdam) J. Toft (Växjö) M.W. Wong (Toronto) Y. Xu (Louisville) M. Yamamoto (Tokyo) S. Zhang (Newark)
European Math. Society
London Math. Society
International Math. Union
Public Lecture on Nonlinear PDEs (Monday 13 July) Pierre-Louis Lions (Paris)
Figure: The announcement of the London meeting.
Figure: Lee Lorch, Victor Berenkov and Heinrich begehr.
I am 63 and I begin to work in Homogenization
I began to work in Homogenization on my sabbatical to Lyon and St. Etienne in 1995. This work was supported by ONR, NATO and NSF. My first paper is with Grisha Panasenko. My second paper is with Andro Mikelic.
Figure: A. Bourgeat, G. Panaseko and the Gilberts
Homogenizing the acoustic properties of the seabed: Part I, Nonlinear Analysis 40, 185-212, (2000), (with A. Mikelic) (45)
Homogenizing the acoustic properties of the seabed, Part II,(with Th. Clopeau, J.L. Ferrn, and A. Mikelic), Mathematical and Computer Model. 33 (2001) 821-841.
Acoustics of a stratified poroelastic composite, Zeit. f. Analysis u. i. Anwendungen 184 (1999), 977-1001. (1999)977-1001. (with A. Panchenko)
Vibration of two bonded composites: effects of the interface and distinct periodic structures,Int. J. Solids and Structures, 40 (2003), 3177-3193, (with Michael Harik and Alexander Panchenko)
Figure: Chez le maison de Andro Milic a St. Etienne.
Figure: My second homogenization paper.
I begin to work with two of my students, Alex Panchenko and Yvonne Ou, on homogenization.
Figure: Yvonne Ou
Figure: Alex Panchenko.
Figure: Gaetano and Mathilda Fichera with friends.
Figure: Heinrich Begehr, Wolfgang Wendland and friends..
My 70th Birthday Conference in Frejus, France
I was lucky that Armand Wirgin got CNRS funding from the French Government for my 70th birthday celebration in FREJUS on the Cote d’Azure. This resulted a collection of very nice papers
Figure: Armand Wirgin.
Figure: Mazya, Jenny and Lenny Schwarzt Figure: Lin Wei and Andro Mikelic
Figure: Klaus Hackl Figure: Lou Fishman
Figure: The Mazyas and me Figure: Heinrich Begehr and Franco Nicolosi
Figure: Richard Delanghe and jerry Hile
Figure: Jim Buchanan and Lenny Schwarzt
Some more recent papers:
An obstacle problem in non-isothermal and non-Newtonian Hele-Shaw flows Comm. Applied Analysis 8 (4) (2004) 459-489 (with Ming Fang).
Effective acoustic equations for a two-phase medium with microstructure, Mathematical and Computer Model., 39, (2004) 1431-1448. (with A.Panchenko )
A two-dimensional nonlinear theory of anisotropic plates, Mathematical and Computer Modeling 32 (2000) 855-857, (with T. S. Vashakmadze)
Compression moulding II: Existance of a solution for a Hele Shaw type model, Jour. Analysis and Applications 23 783–807 (2004).(with M. Fang)
Yvonne Ou and Alex Panchenko decide to do something about my 75th birthday.
I really like parties!
Figure: D. Khavinson, J. Ryan and G. Auchmutty at my 75th
Figure: Armand Wirgin and me at my 75th
Medical Mathematics
Since the cold war was over and there was no more funding for the type of problems we were interested in we decided first to look into another poroelastic material, namely bone. This has led to an active research program and funding from NSF for the International Program, NATO and the NSF Bio-Math Program. We have been interested in determining the rigidity of bone, modeling the disease osteoporosis and medical-mathematics in general.
A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalize

a personal mathematical history

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