mathematics - 6te.netmathskthm.6te.net/all exams.pdf · mathematics, claiming that it is just a...
TRANSCRIPT
MATHEMATICS
Rough Lecture Plan
• What is Mathematics ?
• Information about various Exams.
• Careers in Mathematics.
• Philosophy of Mathematics.• Philosophy of Mathematics.
What is Mathematics?
• <R, +, *>
R denotes the set of real numbers,
+ denotes the operation of addition,
* denotes the operation of multiplication.* denotes the operation of multiplication.
What is Mathematics?
• Mathematics is a study of Sets with
Structures. There are three basic structures.
Three Basic Structures of Mathematics
• Algebraic Structures
Three Basic Structures of Mathematics
• Algebraic Structures → • An algebraic
structure consists of
one or more sets
closed under one or
more operations more operations
satisfying some axioms.
<R, +>
Three Basic Structures of Mathematics
• Algebraic Structures
• Topological Structures
Three Basic Structures of Mathematics
• Algebraic Structures
• Topological Structures →
• Topological structures allow the formal definition of concepts such as convergence, connectedness, and continuity in every and continuity in every mathematical space where it is possible to do so.
The branch of mathematics that studies topological spaces in their own right is called topology.
Three Basic Structures of Mathematics
• Algebraic Structures
• Topological Structures
• Ordered Structures
Three Basic Structures of Mathematics
• Algebraic Structures
• Topological Structures
• Ordered Structures →
• In Mathematical space:
the relations are defined
to relate or compare
different elements.
These relations These relations
together with the set
under consideration
gives rise to ordered
structure in
Mathematics.
Building
Of
Mathematics
A
L
G
E
B
T
O
P
O
L
O
O
R
D
E
RB
R
A
O
G
Y
R
E
D
Building
Of
Mathematics
A
L
G
E
B
T
O
P
O
L
O
O
R
D
E
RB
R
A
O
G
Y
R
E
D
Building
Of
Mathematics
A
L
G
E
B
T
O
P
O
L
O
O
R
D
E
RB
R
A
O
G
Y
R
E
D
Logic
What is Logic?
• Logic is the basis of Mathematical reasoning.
• It makes possible to assign meaning to
statements in Mathematics.
• The rules of logic allows us to distinguish • The rules of logic allows us to distinguish
between valid and invalid statements in
mathematics.
Rules of Logic
How the empty set is subset of
every set?
To prove A is subset of B, We prove
So to prove φ is subset of S, We prove
Consider the statement
It is the Statement of the form
(*)
It is the Statement of the form
where p is always false. Since p implies q is always true when p is false (regardless of truth value of q), the statement (*) is true.
Definition
The definition of definition consists
of certain axioms. It is studied in
the subject Logic. the subject Logic.
Two properties of Definition
1. Definition is always Fundamental.
2. Definition is if and only if type of 2. Definition is if and only if type of statement.
DEFINITIONFundamental
Level
1. Definition is always Fundamental
Consider the following argument
Where is the Mistake?
The proper way to calculate
product Is to use the DEFINITION
of product of complex numbers.
Definition of product of two
complex numbers is
2. Definition is an If and only If
statement.
Triangle is equilateral
All its sides are equal.
Definition of SET
In formal mathematics, you begin with
some "undefined objects", some axioms or
postulates about those objects, possibly some
definitions, and from those, you prove
theorems. In Euclidean geometry, for example,
the terms "point, line, plane" are undefined
objects.
Euclid tried to define a point as "A point is Euclid tried to define a point as "A point is
that which has no part." Of course as soon as he
said that, we can ask, "What is a part?" If he had
given the definition of "part", it would have had
undefined terms, and the process would never
end.
The only thing we officially know about these
things is whatever the postulates tell us. For example,
there is a postulate that tells us that every two points
lie on exactly one line. From this, we can prove the
following very simple theorem: "If the system contains
at least two points, then it contains a line."
Because of this, more modern mathematicians Because of this, more modern mathematicians
realized that it is impossible to define everything--you
will always be forced into an infinite set of definitions--
so the only solution is to begin with a set of undefined
terms, and then to write down a series of properties of
these terms that you call axioms or postulates and
work from those.
Axiomatic Set Theory
The axioms for set theory now most often
studied and used, although put in their final
form by Skolem, are called the Zermelo-Fraenkel
axioms (ZF). Actually, this term usually excludes
the axiom of choice, which was once more
controversial than it is today. When this axiom is
included, the resulting system is called ZFC.included, the resulting system is called ZFC.
An important feature of ZFC is that every object
that it deals with is a set. In particular, every
element of a set is itself a set. Other familiar
mathematical objects, such as numbers, must be
subsequently defined in terms of sets.
The ten axioms of ZFC are listed below. (Strictly speaking, the axioms of ZFC
are just strings of logical symbols. What follows should therefore be viewed
only as an attempt to express the intended meaning of these axioms in
English. Moreover, the axiom of separation, along with the axiom of
replacement, is actually an infinite schema of axioms, one for each formula.)
Each axiom has further information in its own article.
1. Axiom of extensionality: Two sets are the same if and only if they have the
same elements.
2. Axiom of empty set: There is a set with no elements. We will use {} to 2. Axiom of empty set: There is a set with no elements. We will use {} to
denote this empty set.
3. Axiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as
its only elements.
4. Axiom of union: For any set x, there is a set y such that the elements
of y are precisely the elements of the elements of x.
5. Axiom of infinity: There exists a set x such that {} is in x and whenever y is
in x, so is the union y U {y}.
6. Axiom of separation (or subset axiom): Given any set and
any proposition P(x), there is a subset of the original set containing precisely
those elements x for which P(x) holds.
7. Axiom of replacement: Given any set and any mapping, formally defined as
a proposition P(x,y) where P(x,y) and P(x,z) implies y = z, there is a set
containing precisely the images of the original set's elements.
8. Axiom of power set: Every set has a power set. That is, for any set x there
exists a set y, such that the elements of y are precisely the subsets of x.exists a set y, such that the elements of y are precisely the subsets of x.
9. Axiom of regularity (or axiom of foundation): Every non-empty
set x contains some element y such that x and y are disjoint sets.
10. Axiom of choice: (Zermelo's version) Given a set x of mutually disjoint
nonempty sets, there is a set y (a choice set for x) containing exactly one
element from each member of x.
The axioms of choice and regularity are still controversial today among a
minority of mathematicians.
Set theory foundations for
mathematicsIt is often asserted that axiomatic set theory is an adequate
foundation for current mathematical practice, in the sense
that in principle all proofs produced by the mathematical
community could be written formally in set theory terms. It is
also generally believed that no serious advantage would come
from doing that, in almost all cases: the axiomatic foundations from doing that, in almost all cases: the axiomatic foundations
normally used are sufficiently closely aligned to the underlying
set theory, that full axiomatic translation yields only a little extra,
compared to argument in the usual, traditional informal style.
One area where a gap can appear between practice and easy
formalisation is in category theory, where for example a concept
like 'the category of all categories' requires more careful set-
theoretic handling.
Objections to set theorySince its inception, there have been some mathematicians who
have objected to using set theory as a foundation for
mathematics, claiming that it is just a game which includes
elements of fantasy. Notably,
Henri Poincare said
"set theory is a disease from which mathematics will one day
recover", and
Errett Bishop dismissed set theory as God's mathematics, which Errett Bishop dismissed set theory as God's mathematics, which
we should leave for God to do.
The most frequent objection to set theory is based on the
constructivist view that, loosely, mathematics has something to
do with computation. See mathematical constructivism.
On the other hand this is not really an objection to axiomatic set
theory, as a formal theory. It is a comment on the naive set
theory that is being formalised, and its admission of non-
computational elements.
Set is the most fundamental
concept of Mathematics
that is
Every definition in Mathematics is
ultimately a Set.ultimately a Set.
e.g. definition of function,
sequence, group is a set. (How?)
Concept of a function:
A function f from set A to set B is
the relation from set A to set B
such that every element of set A is
related to unique element of B.
In other words it is the subset of In other words it is the subset of
A B in which every element of B
appears exactly once as first
coordinate. It is a subset of A B
i.e. it is a set.
Sequence
It is a function whose domain is a
set of natural numbers. Since a
function is a set, sequence is a set.function is a set, sequence is a set.
Group
Group = { <G, *>/ * is a binary operation
on set G satisfying p1, p2, p3 }
where
p : * is associativep1 : * is associative
p2 : G has identity with respect to *
p3 : every element of G has an inverse
element with respect to * .
Conclusions
1. Set is the most fundamental
concept of Mathematics.
2. Every Definition in Mathematics
is ultimately a Set.
3. In set theory, the word "set" is
formally undefined.
4. All we know about sets are what
the axioms of set theory tell us. the axioms of set theory tell us.
Various Entrance Exams/Interviews at
Graduate and Post-Graduate level
1. IIT JAM Entrance
Examination
2. TIFR Entrance Examination
7. NBHM Ph.D. Scholarship
Examination
8. GATE Entrance
Examination3. IISC Bangalore (I-Math)
Entrance Examination
4. NBHM MA/M.Sc.
Scholarship Examination
5. JEST Entrance Examination
6. INAT Entrance Examination
Examination
9. IISER Ph.D. Interview
10. IMSC Ph.D. Interview
11. ISI Examination
12. SET Exam
13. NET Exam
Basic Books For these Exams
Analysis
i) Real Analysis
Author: N. L. Carothers
Publisher: Cambridge University Press
ii) Real Analysisii) Real Analysis
Author: H.L. Royden
Publisher: Prentice Hall of India
iii) Analysis on Manifolds
Author: J. R. Munkres
Publisher: Addison Wesley Publishing Company.
iv) Functions of One Complex Variable
Author: John B. Conway
Publisher: Narosa Publishing House
Algebra
• i) An Introduction to AlgebraAuthor- Donald LewisPublisher- Harper and Row, New York.
• ii) A First Course in Abstract Algebra• ii) A First Course in Abstract AlgebraAuthor- John B. FraleighPublisher- Narosa publishing house.
• iii) Linear AlgrbraAuthor- Seymour LipschutzPublisher- Schaum Series
Topology
• i) Topology Author- James R. MunkresPublisher- Prentice Hall of India
• ii) An Introduction to Topology & Modern Analysis • ii) An Introduction to Topology & Modern Analysis Author- G.F. SimmonsPublisher- Tata McGraw-Hill.
• iii) Topology Author- Seymour LipschutzPublisher- Schaum Series
1. IIT JAM Entrance Examination
SYLLABUS
• Sequences, Series and Differential Calculus : Sequences of real numbers. Convergent sequences and series, absolute and conditional convergence. Mean value theorem. Taylor 's theorem. Maxima and minima of functions of a single variable. Functions of two and three variables. Partial derivatives, maxima and minima.
• Integral Calculus : Integration, Fundamental theorem of calculus. Double and Triple, integrals, Surface areas and volumes.
• Differential Equations : Ordinary differential equations of the first order of the form y'=f(x,y). Linear differential equations of second order with constant coefficients. Euler-Cauchy equation. Method of variation of parameters.
• Vector Calculus : Gradient, divergence, curl and Laplacian. Green's, Stokes' and Gauss' theorems • Vector Calculus : Gradient, divergence, curl and Laplacian. Green's, Stokes' and Gauss' theorems and their applications.
• Algebra : Groups, subgroups and normal subgroups, Lagrange's Theorem for finite groups, group homomorphisms and basic concepts of quotient groups, rings, ideals, quotient rings and fields.
• Linear Algebra : Systems of linear equations. Matrices, rank, determinant, inverse. Eigenvalues and eigenvectors. Finite Dimensional Vector Spaces over Real and Complex Numbers, Basis, Dimension, Linear Transformations.
• Real Analysis : Open and closed sets, limit points, completeness of R, Uniform Continuity, Uniform convergence, Power series.
How to Prepare?
• Basic Books on Algebra and Analysis
(except : Functions of One Complex Variable)
1) Calculus Volume II1) Calculus Volume II
Author- Tom M. Apostol
Publisher- John Wiley & Sons.
2) Differential Equations
Author- Gupta, Malik, Mittal
Publisher- Pragati Publications, Meerut.
2. TIFR Entrance Examination
(Tata Institute of Fundamental
Research, Mumbai)
• Website:
http://univ.tifr.res.in/gs2010/index.html
Question Paper Pattern
• There are questions on Algebra, Analysis,
Topology and simple questions on
Combinatorics and on other topics.
• Questions are of only True/False type!• Questions are of only True/False type!
• There is negative marking.
How to Prepare?
• Basic Books on Algebra and Analysis and Topology
• Combinatorics
Author- V. K. Balakrishnan
Publisher- Schaum Series.
3. IISC Bangalore (I-Math) Entrance
Examination
(Indian Institute of Science, Bangalore)
• At IISC Bangalore:
1. Ph. D. Programme (After M.Sc.)1. Ph. D. Programme (After M.Sc.)
2. Integrated Ph.D. Programme (After B.Sc.)
How to get into IISC?
• By Qualifying in one of the following Tests
CSIR-UGC NET for JRF; or UGC-NET for JRF; or
DBT JRF or ICMR JRF; or JEST; or NBHM; or
IISc Entrance Test; or GATE.IISc Entrance Test; or GATE.
How to prepare for IISC Entrance?
• Basic Books on Algebra and Analysis and
Topology
• Topics from Applied Mathematics chosen from
the syllabus
4. NBHM MA/M.Sc. Scholarship
Examination
(National Board for Higher
Mathematics, Department of Atomic Mathematics, Department of Atomic
Energy, Mumbai)
• Website:
http://www.nbhm.dae.gov.in/msc.html
What is NBHM MA/M.Sc. Scholarship?
• NBHM (National Board of Higher
Mathematics) gives scholarship to the
selected students doing MA/M.Sc. In
Mathematics. Around Ten students are Mathematics. Around Ten students are
selected from all over India on the basis of
their performance on written test and
interview. At present this scholarship is Rs.
6000 per month which amounts to Rs.
1,44,000 for two years of post graduation.
Who can apply for this Scholarship?
• Student appearing/appeared in T.Y.B.Sc, M.Sc.
I year or in M.Sc. II year.
How To Prepare?
• There are three sections in the question
paper:
1. Algebra
2. Analysis2. Analysis
3. Topology/Geometry (optional)
Student can choose Topology or Geometry
from third section.
Books
• 1. Algebra
• i) Introduction to AlgebraAuthor- Donald LewisPublisher- Harper and Row, New York.(This is a MUST MUST have book).You just need to go through all the statements and results in this You just need to go through all the statements and results in this book.By doing so you will have Complete, Concise, Clear collection of all the results in Group Theory, Ring Theory, Field Theory, Linear Algebra (Vector Spaces). This collection will help you to solve any question on these topics. Great Book!!!
2. Analysis
• i) Real AnalysisAuthor: H.L. RoydenPublisher: Prentice Hall of India
Prepare only first Nine chapters.
ii) Complex Variables and ApplicationsAuthors- Churchil and BrownPublisher- McGraw HillPrepare only the first Six Chapters of this book. This book is written in simple language and is very useful.
3. Geometry (optional)
• i) Book: Solid GeometryAuthor: M.L. KhannaPublisher: Jai Prakash Nath & Co.,
Prepare only first Eight chapters of this book.
ii) Book: Elements of Coordinate GeometryAuthor: S.l. LoneyPublisher: S. Chand
Prepare only first Fourteen Chapters.
3. Topology (optional)
• i) Book: An Introduction to Topology & Modern AnalysisAuthor: G.F. SimmonsPublisher:Tata McGraw-Hill
Prepare only chapters Three to SixPrepare only chapters Three to Six
ii) Book: General TopologyAuthor: S. LipschutzPublisher: Schaum Series
•
5. JEST Entrance Examination
Joint Entrance Screening Test
• Website: http://jest09.veccal.ernet.in/
What is JEST?
• A number of reputed institutes in the country have come together to conduct a Joint Entrance Screening Test (JEST) for enrolment of students in Ph.D. / Integrated M.Sc.-Ph.D. programs in physics (including various inter-disciplinary research areas in physics) and Theoretical Computer Science.
JEST is conducted for selecting candidates to be interviewed for admission to Ph. D. Programmes in Physics/ Theoretical Computer admission to Ph. D. Programmes in Physics/ Theoretical Computer Science in these institutions in the country.
Applications are invited from motivated students, with consistently good academic record, for appearing in JEST leading to enrolment in a Ph. D. Programme in Physics (including various inter-disciplinary research areas in Physics)/ Theoretical Computer Science, at any of the 21 premier participating institutions.
JEST score forms an important component in the selection of candidates for the Ph.D. programmes in these institutions.
Institutes which allow admission on
qualifying JEST1. ARIES, Nainital
2. IIA, Bangalore
3. IISC, Bangalore
4. IISER MOHALI, Mohalii
5. IISER PUNE, Pune
11. SNBNCBS, Kolkata
12. BARC, Mumbai
13. UGC-DAE CSR, Indore
14. HRI, Allahabad
15. IGCAR, Kalpakkam5. IISER PUNE, Pune
6. IUCAA, Pune
7. JNCASR, Bangalore
8. NCRA, Pune
9. PRL, Ahmedabad
10. RRI, Bangalore
15. IGCAR, Kalpakkam
16. IOP, Bhubaneswar
17. IMSC, Chennai.
18. IPR, Gandhinagar
19. RRCAT, Indore
20. SINP, Kolkata
21. VECC, Kolkata
Eligibility
• Students who expect to complete their final examination by August 2009 are also
eligible to apply.
• Minimum Qualifications & Selection Procedures
• Ph.D. Programme (PHYSICS)
• M.Sc. in Physics or M.Sc. / M.E. / M.Tech. in related disciplines. Candidates with
M.Sc. in Mathematics and B.Tech (in Engg Physics/Applied Physics/Post B.Sc. Hons M.Sc. in Mathematics and B.Tech (in Engg Physics/Applied Physics/Post B.Sc. Hons
only) will also be considered at IIA. Graduates with a B.E. or B.Tech. will also be
considered at IISc, IMSc, IUCAA, JNCASR, NCRA, RRI, IISER Mohali, IISER Pune and
SNBNCBS. At IPR candidates should have a Masters degree in Physics, Engineering
Physics, Applied Physics, etc. with at least 55% aggregate. Talented final year B. Sc.
Graduates will also be considered for pre-selection at IUCAA.
• Ph.D. Programme (Theoretical Computer Science)
• M.Sc./ M.E. / M.Tech. / M.C.A. in Computer Science and related disciplines, and
should be interested in the mathematical aspects of computer science. Talented
B.Sc. / B.E. / B.Tech. graduates will also be considered for integrated M.Sc.- Ph.D.
programme.
1. At HRI, candidates with a Bachelor's degree will also be considered for the integrated M.Sc. -
Ph.D. programme in Physics.
2. At IMSc, graduates with B.Sc. / B.E. / B. Tech. degree will also be considered for admission to
M.Sc. (by Research) programme in Physics and Theoretical Computer Science, as part of an
integrated Ph.D. programme.
3. At SNBNCBS, graduates with B.Sc. (Physics) / B.E. / B.Tech. degree will be considered for the
integrated Post-B.Sc.-Ph.D. programme in subject areas mentioned above.integrated Post-B.Sc.-Ph.D. programme in subject areas mentioned above.
4. At IIA, graduates with B.Sc. (Physics/Mathematics) / B.E. / B.Tech. degree will be considered
for the integrated M.Sc.-Ph.D. programme in subject areas mentioned above.
5. Integrated M.Tech - Ph.D. Programme at IIA
6. At IIA, graduates with M.Sc. (Physics / Applied Physics) / B.E. / B.Tech. degree will be
considered for the integrated M.Tech-Ph.D. programme in subject areas mentioned above.
Details of the programme can be found on the IIA webpages.
Mere qualifying in JEST does not entitle one to get a Research Fellowship. Using the JEST
results, each institute will call a limited number of candidates for its further selection
procedure depending on its requirements. All candidates selected after interview will
receive Research Fellowship from the respective institutes.
How to Prepare?
• Basic Books on Analysis and Topology
1. Fundamentals of Physics
Author- Halliday, Rescnick, Walker
Publisher- John Wiley & Sons.Publisher- John Wiley & Sons.
2. Fundamental Laws of Mechanichs
Author- I.E. Irodov
Publisher- MIR Publishers Moscow/CBS Publisher India
3. Fundamental Laws of Electromagnetism
Author- I.E. Irodov
Publisher- MIR Publishers Moscow/CBS Publisher India
4. Problems in General Physics
Author- I.E. Irodov
Publisher- MIR Publishers Moscow/CBS Publisher India
6. INAT Entrance Examination
IUCAA-NCRA Admission Test
What is INAT Test?
• IUCCA-NCRA Admisson Test is being conducted to select candidate for research scholarship to join Ph.D. programme either at
Inter University for Astronomy and Inter University for Astronomy and Astrophysics (IUCAA), Pune
OR
National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research (NCRA-TIFR), Pune.
Eligibility and Preparation
• Same as that of JEST test.
7. NBHM Ph.D. Scholarship
Examination
• Website:
http://www.nbhm.dae.gov.in/phd.html
What is NBHM Ph.D. Scholarship?
• NBHM (National Board of Higher
Mathematics) gives scholarship to the
selected students for doing Ph.D. in
Mathematics. Around Ten students are Mathematics. Around Ten students are
selected from all over India on the basis of
their performance on written test and
interview. At present this scholarship is Rs.
22,000 per month. The scholarship is given
maximum up to 5 years.
Who can apply for this Scholarship?
• Student appearing/appeared in T.Y.B.Sc., M.Sc.
I year M.Sc. II year or those who have
completed M.Sc.
Pattern of the Exam
• There are five sections, containing ten questions each
1. Algebra
2. Analysis
3. Topology
4. Applied Mathematics4. Applied Mathematics
5. Miscellaneous.
• Answer as many questions as possible.
• The assessment of the paper will be based on the best FOUR sections.
• Question Types: Objective and Short answer.
How to prepare?
• Basic Books on Algebra and Analysis and
Topology
• Selected topics from Applied Mathematics
8. GATE Entrance Examination
• Graduate Aptitude Test in Engineering
What is GATE?
• Clearing GATE (Graduate Aptitude Test in Engineering) examination in Mathematics opens door to do Ph.D. in many National Institutes such as IIT, IISC, IISER etc.
• GATE qualified candidates (in Mathematics) can • GATE qualified candidates (in Mathematics) can also apply for M.Tech. in selected subjects in selected institutes including IIT’s.
• All GATE qualified students who have taken admissions for Ph.D./M.Tech get scholarship from MHRD (Ministry of Human Resource and Development).
Eligibility
• M.Sc. II year appeared or passed.
Syllabus
• Linear Algebra: Finite dimensional vector spaces; Linear transformations and their matrix representations, rank; systems of linear equations, eigen values and eigen vectors, minimal polynomial, Cayley-Hamilton Theroem, diagonalisation, Hermitian, Skew-Hermitian and unitary matrices; Finite dimensional inner product spaces, Gram-Schmidt orthonormalization process, self-adjoint operators.
• Complex Analysis: Analytic functions, conformal mappings, bilinear transformations; complex integration: Cauchy’s integral theorem and formula; Liouville’s theorem, maximum modulus principle; Taylor and Laurent’s series; residue theorem and applications for evaluating real integrals.
• Real Analysis: Sequences and series of functions, uniform convergence, power series, Fourier series, functions of several variables, maxima, minima; Riemann integration, multiple integrals, line, series, functions of several variables, maxima, minima; Riemann integration, multiple integrals, line, surface and volume integrals, theorems of Green, Stokes and Gauss; metric spaces, completeness, Weierstrass approximation theorem, compactness; Lebesgue measure, measurable functions; Lebesgue integral, Fatou’s lemma, dominated convergence theorem.
• Ordinary Differential Equations: First order ordinary differential equations, existence and uniqueness theorems, systems of linear first order ordinary differential equations, linear ordinary differential equations of higher order with constant coefficients; linear second order ordinary differential equations with variable coefficients; method of Laplace transforms for solving ordinary differential equations, series solutions; Legendre and Bessel functions and their orthogonality.
Syllabus
• Algebra: Normal subgroups and homomorphism theorems, automorphisms; Group actions, Sylow’stheorems and their applications; Euclidean domains, Principle ideal domains and unique factorization domains. Prime ideals and maximal ideals in commutative rings; Fields, finite fields.
• Functional Analysis: Banach spaces, Hahn-Banach extension theorem, open mapping and closed graph theorems, principle of uniform boundedness; Hilbert spaces, orthonormal bases, Rieszrepresentation theorem, bounded linear operators.
• Numerical Analysis: Numerical solution of algebraic and transcendental equations: bisection, secant method, Newton-Raphson method, fixed point iteration; interpolation: error of polynomial interpolation, Lagrange, Newton interpolations; numerical differentiation; numerical integration: Trapezoidal and Simpson rules, Gauss Legendre quadrature, method of undetermined parameters; Trapezoidal and Simpson rules, Gauss Legendre quadrature, method of undetermined parameters; least square polynomial approximation; numerical solution of systems of linear equations: direct methods (Gauss elimination, LU decomposition); iterative methods (Jacobi and Gauss-Seidel); matrix eigenvalue problems: power method, numerical solution of ordinary differential equations: initial value problems: Taylor series methods, Euler’s method, Runge-Kutta methods.
• Partial Differential Equations: Linear and quasilinear first order partial differential equations, method of characteristics; second order linear equations in two variables and their classification; Cauchy, Dirichlet and Neumann problems; solutions of Laplace, wave and diffusion equations in two variables; Fourier series and Fourier transform and Laplace transform methods of solutions for the above equations.
• Mechanics: Virtual work, Lagrange’s equations for holonomic systems, Hamiltonian equations.
• Topology: Basic concepts of topology, product topology, connectedness, compactness, countabilityand separation axioms, Urysohn’s Lemma.
Syllabus
• Probability and Statistics: Probability space, conditional probability, Bayestheorem, independence, Random variables, joint and conditional distributions, standard probability distributions and their properties, expectation, conditional expectation, moments; Weak and strong law of large numbers, central limit theorem; Sampling distributions, UMVU estimators, maximum likelihood estimators, Testing of hypotheses, standard parametric tests based on normal, X2 , t, F – distributions; Linear regression; Interval estimation.
• Linear programming: Linear programming problem and its formulation, convex sets and their properties, graphical method, basic feasible solution, simplex
• Linear programming: Linear programming problem and its formulation, convex sets and their properties, graphical method, basic feasible solution, simplex method, big-M and two phase methods; infeasible and unbounded LPP’s, alternate optima; Dual problem and duality theorems, dual simplex method and its application in post optimality analysis; Balanced and unbalanced transportation problems, u -u method for solving transportation problems; Hungarian method for solving assignment problems.
• Calculus of Variation and Integral Equations: Variation problems with fixed boundaries; sufficient conditions for extremum, linear integral equations of Fredholm and Volterra type, their iterative solutions.
Question Paper Pattern
Patterns of Question papers Negative Marks for wrong Answer
Q.1 to Q.20 : Will carry one mark
each (sub-total 20 marks).
1/3 mark will be deducted for each wrong
answer.
Q.21 to Q.50 : Will carry two marks
each (sub-total 60 marks)
2/3 mark will be deducted for each wrong
answer.
Question pairs (Q.57, Q.58) and (Q.59, There will be negative marks only for Question pairs (Q.57, Q.58) and (Q.59,
Q.60) will be linked answer questions.
The answer to the second question of the
last two pairs will depend on the answer
to the first question of the pair.
If the first question in the linked pair is
wrongly answered or is un-attempted,
then the answer to the second question
in the pair will not be evaluated. Each
question will carry two marks
There will be negative marks only for
wrong answer to the first question of the
linked answer question pair i.e. for Q.57
and Q.59, 2/3 mark will be deducted for
each wrong answer. There is no negative
marking for Q.58 and Q.60.
How to prepare?
• Basic Books on Algebra and Analysis and Topology
1) Differential Equations
Author- Gupta, Malik, MittalAuthor- Gupta, Malik, Mittal
Publisher- Pragati Publications, Meerut.
2) For functional analysis:
Book: An introduction to Topology and Modern Analysis
Author- G.F. Simmons
Publisher- Tata McGraw Hill
Books
3) For Numerical Analysis:
Book: Numerical Methods for Mathematics, Science and Engineering.
Author- John H. Mathews.Author- John H. Mathews.
Publisher- Prentice Hall of India
4) For Partial Differential Equations:
Book: Partial Differential Equations
Author- W. E. Williams
Publisher- Clarendon Press Oxford
Books
5) For Linear Programming Theory:
I used internet resources for preparing
theory of various methods of LPP.
6) For Linear Programming Problems:6) For Linear Programming Problems:
Book: Operations Research
Author- V.K. Kapoor
Publisher-
Books
7) For Mechanics:
Book: Classical Mechanics
Author- N.C. Rana, P.C. Joag
Publisher- Tata McGraw HillPublisher- Tata McGraw Hill
8) For Calculus of Variation and Integral Equations
Book:
Author-
Publisher-
Books
9) For Probability and Statistics
Book:
Author-
Publisher-Publisher-
9. IISER Ph.D. Interview
(Indian Institute of Science Education
and research)
About IISER
• The Government of India, through the Ministry of Human Resource Development (MHRD), has established five Indian Institutes of Science Education and Research (IISER). These institutes are located in Bhopal, Mohali, Pune, Kolkata and Thiruvanantapuram. The IISERs represent a unique Thiruvanantapuram. The IISERs represent a unique initiative in India where teaching and education are totally integrated with state-of-the-art research nurturing both curiosity and creativity in an intellectually vibrant atmosphere of research. Each IISER is an autonomous institution awarding its own Masters and Doctoral degrees.
•
IISER Ph.D. Interview- Eligibility
• To appear for interview candidate should have
cleared any one of these exams:
1. UGC CSIR NET with JRF
2. GATE (Minimum percentile 95)2. GATE (Minimum percentile 95)
3. NBHM Ph.D. Scholarship Examination
For more information, visit the IISER’s
websites:
1. IISER Pune:
Website: www.iiserpune.ac.in
2. IISER Bhopal:
Website: www.iiserbhopal.ac.in
3. IISER Mohali:3. IISER Mohali:
Website: www.iisermohali.ac.in
4. IISER Kolkata:
Website: www.iiserkol.ac.in
5. IISER Thiruvanantapuram:
Website: www.iisertvm.ac.in
10. IMSC Ph.D. Interview
Institute of Mathematical Sciences,
Chennai
• Website: www.imsc.res.in
IMSCInstitute of Mathematical SciencesChennai
At IMSC Chennai:
1. Ph. D. Programme (After M.Sc.)
2. Integrated Ph.D. Programme (After B.Sc.)
Eligibility
• Applicants for the PhD programme should
have completed a Masters degree in
Mathematics or Statistics by the time they
actually join the programme.actually join the programme.
• Applicants for the integrated PhD programme
should have completed a Bachelors degree in
Mathematics or Statistics by the time they
actually join the programme.
How to get into IMSC?
• Admission to both of these programmes (Ph.Dand integrated Ph.D.) is based on the candidate's performance in an interview. Candidates are selected for the interview by screening applications. This screening is screening applications. This screening is typically based on the applicant's performance in the PhD Scholarship screening test of the NBHM.
• There is NO separate entrance test of IMSC.
IMSC Scholarships
• A monthly stipend of Rs. 22,000 for the first
two years and Rs. 24,000 in subsequent years
(subject to satisfactory performance),
together with an annual contingency grant of together with an annual contingency grant of
Rs. 20,000, housing and medical facilities will
be provided.
11. ISI Examination
INDIAN STATISTICAL INSTITUTE
Kolkata• Website: www.isical.ac.in
About ISI
• The Indian Statistical Institute (I.S.I.), founded by Professor Prasanta Chandra Mahalanobis, grew out of the Statistical Laboratory set up by him in the Presidency College in Kolkata. In 1932 the Institute was registered as a non-profit making learned society for the advancement of statistics in India. Within a few years the Institute's achievements in research that included innovative projects on sample surveys of agricultural crops and socio-economic after-effects of the Bengal famine (1943-44) as well as pathbreaking research publications of Professor R.C. Bose on experimental designs in the Annals of Eugenics (1939), brought recognition in India and abroad. The Institute is now considered as one of the foremost centres in the and abroad. The Institute is now considered as one of the foremost centres in the world for training and research in statistics and related sciences. Under the leadership of Professor P. C. Mahalanobis, the Institute also initiated and promoted the interaction of statistics with natural and social sciences to unfold the role of statistics as a key technology which explicated the twin aspects of statistics - its general applicability and its dependence on other disciplines for its own development. In keeping with this long tradition, the Institute has been engaged in developing statistical theory and methods and their practical applications in various branches of science and technology.
ISI Exam for JRF/Scholarship
• There will be two tests RM-1, and RM-2 of 2 hours duration each in the forenoon and in the afternoon.
• 1) Topics for MI (Forenoon examination) : Real Analysis, Lebesgue Integration, Complex Analysis, Ordinary Differential Equations and General Topology.Differential Equations and General Topology.
• 2) Topics for MII (Afternoon examination) : Algebra, Linear Algebra, Functional Analysis, Elementary Number Theory and Combinatorics.
• Candidates will be judged based on their performance in both the tests.
How to prepare?
• Basic Books on Algebra and Analysis and Topology
1) Differential Equations
Author- Gupta, Malik, MittalAuthor- Gupta, Malik, Mittal
Publisher- Pragati Publications, Meerut.
2) For functional analysis:
Book: An introduction to Topology and Modern Analysis
Author- G.F. Simmons
Publisher- Tata McGraw Hill
Books
3) For Elementary Number Theory:
Book: An Introduction to Number Theory
Author- I. Niven, H. Zuckerman
Publisher- Wiley Eastern LimitedPublisher- Wiley Eastern Limited
4) For Combinatorics:
Book: Combinatorics
Author- V. K. Balakrishnan
Publisher- Schaum Series
12. SET Exam
13. CSIR NET/JRF Exam
12. SET Exam
13. CSIR NET/JRF Exam
Toughest Exam/Interview
Toughest Exam/Interview
• TIFR Interview (after
clearing written test)
Toughest Exam/Interview
• TIFR Interview (after
clearing written test)
Easiest Exam
Toughest Exam/Interview
• TIFR Interview (after
clearing written test)
Easiest Exam
SET Exam
The Joy of Learning Mathematics
A great discovery solves a great problem but
there is a grain of discovery in the solution of
any problem. Your problems may be modest ;
but if it challenges your curiosity and brings into
play your inventive faculties, and if you solve it play your inventive faculties, and if you solve it
by your own means , you may experience the
tension and enjoy the triumph of victory. Such
experiences at a susceptible age may create a
taste for mental work and leave their imprint on
mind and character for a lifetime.
Thus a student whose college curriculum
includes some Mathematics has a singular
opportunity. This opportunity is lost, of course, if
he regards mathematics as a subject in which he
has to earn so and so much marks and which he
should forget after the final examination as
quickly as possible. The opportunity may be lost quickly as possible. The opportunity may be lost
even if the student has some natural talent for
Mathematics because he, as everybody else,
must discover his talents or tastes; he cannot
know that he likes cake if he has never tasted
cake.
You may mange to find out, however, a
Mathematics problem may be as much fun as a
crossword puzzle, or that vigorous mental work
may be an exercise as desirable as a fast game of
football. Having tasted the pleasure in football. Having tasted the pleasure in
Mathematics you will not forget it easily and
then there is a good chance that Mathematics
will become something for you: a hobby, or a
tool of your profession, or your profession or a
great ambition.
Teaching Mathematics
A teacher of Mathematics has great
Opportunity.
If He:
Fills his allotted time with drilling his
students in routine operations-students in routine operations-
Then He:
Kills their interest, hampers their
intellectual development, and misuses the
opportunity.
But –
If He:
Challenges the Curiosity of students by
setting them problems proportionate to their
knowledge, and helps them to solve their
problems with stimulating questions problems with stimulating questions
Then He:
Stimulates the interest in Mathematics
and gives some means of Independent thinking.
Are You interested in Mathematics?
It is Not necessary
that you are interested in Mathematics
But it is Necessary
that you are interested in Something.
You Must “Discover” that “Something.”
But:
Before You Say
that you are Not interested in Mathematics
You, as everybody else, You, as everybody else,
must discover Your Talents or Tastes;
You cannot know that you like cake if you
have never tasted cake.
Stop Saying that
Mathematics
is Difficult.is Difficult.
If Mathematics is Difficult ,
What is Easy ?
Is Playing Cricket Easy?
Is Acting Easy?
Is Singing Easy?
Careers in Mathematics:
MATEMATICS TEACHINGRESEARCH
Teaching
The task of Teaching Mathematics is not
quite easy.
It demands time, practice , devotion and It demands time, practice , devotion and
sound principles.
I hate 'teaching'....I love lecturing,
and have lectured a
great deal to extremely able great deal to extremely able
classes.
-G. H. Hardy
Who is good teacher of Mathematics?
Beauty of Mathematics
“Mathematics, rightly viewed, possesses not
only truth, but supreme beauty — a beauty cold
and austere, like that of sculpture, without
appeal to any part of our weaker nature,
without the gorgeous trappings of painting or
music, yet sublimely pure, and capable of a
stern perfection such as only the greatest art stern perfection such as only the greatest art
can show. The true spirit of delight, the
exaltation, the sense of being more than Man,
which is the touchstone of the highest
excellence, is to be found in mathematics as
surely as poetry.”
- Bertrand Russell
The good teacher is one who can
convey this Beauty to the students.
What is an Effective teaching in
Mathematics?
It results in
1. an increase in mathematics
communication among students;
2. an improvement in student ability to 2. an improvement in student ability to
read mathematics;
3. an increase in the understanding of
mathematics by the students;
.
4. a greater use of mathematics vocabulary by
students;
5. an increased student interest in the class
because of the
variety in ‘teachers’ throughout the course;variety in ‘teachers’ throughout the course;
6. a positive student attitude toward the class
and mathematics; and
7. a more ‘student-centred’ classroom
Research
Who should do the research?Who should do the research?
If
One feels that his or her natural talent
is not suitable for some topic or even for
the subject Mathematics,
Then
He or She should not proceed for the He or She should not proceed for the
research
Because the most probably he or she
would not contribute anything meaningful
to the subject.
Keep other options open.
A Man of Mathematics need not be in the field of
Mathematics.
A student having Mathematical background
performs well in various competitive exams.
The Mathematics people have Analytical Minds
and they do Well in other fields such as software
development.
Go Ahead and look for other options. Explore the
possibilities and select the one where you can perform
well.
Software
Management (M.B.A)
Banking MATEMATICS Software
Development
Banking
Exams
Other competitive exams
M.P.S.C., U.P.S.C.
Philosophy of MathematicsOn the one hand, philosophy of mathematics is
concerned with problems that are closely related to
central problems of metaphysics At first blush,
mathematics appears to study abstract entities. This
makes one wonder what the nature of mathematical makes one wonder what the nature of mathematical
entities consists in and how we can have knowledge of
mathematical entities. If these problems are regarded
as intractable, then one might try to see if
mathematical objects can somehow belong to the
concrete world after all.
On the other hand, it has turned out that to
some extent it is possible to bring mathematical
methods to bear on philosophical questions
concerning mathematics. The setting in which
this has been done is that of mathematical
logic when it is broadly conceived as comprising logic when it is broadly conceived as comprising
proof theory, model theory, set theory, and
computability theory as subfields. Thus the
twentieth century has witnessed the
mathematical investigation of the consequences
of what are at bottom philosophical theories
concerning the nature of mathematics.
When professional mathematicians are
concerned with the foundations of their subject,
they are said to be engaged in foundational
research. When professional philosophers
investigate philosophical questions concerning
mathematics, they are said to contribute to the
philosophy of mathematics. Of course the
distinction between the philosophy of distinction between the philosophy of
mathematics and the foundations of
mathematics is vague, and the more interaction
there is between philosophers and
mathematical logicians working on questions
pertaining to the nature of mathematics, the
better.
The Beauty of Mathematics is the
BEAUTY OF ITS CREATOR who
made it perfect as He is the only
one who is PERFECT. LET’S PRAISE
HIM.
LET’S ADORE HIM.LET’S ADORE HIM.
AND LET’S LISTEN AND DO WHAT
HE TELLS US TO DO.
-Unknown
Locus of Mathematical Reality
“ I believe that Mathematical Reality lies
outside us,that our function is to
discover or observe it,discover or observe it,
and that
the theorems which we prove,
and which we describe grandiloquently as our
'creations‘ are simply our notes of our
observations. ”
-G. H. Hardy