Mathematics

FIRST SEMESTER
Math 611 Analysis VII
Integration: concept of measurability, simple functions, elementary properties of measures, integration of positive functions, Lebesgue’s monotone convergence theorem, integration of complex functions, Lebesgue’s dominated convergence theorem.
Positive Borel Measures: Riesz representation theorem, regularity properties of Borel measures, Lebesgue measures, continulity properties of measurable functions
Lp-spaces: convex functions and inequalities, the Lp-spaces, approximation by continuous functions.
Texts
[1] Swe. K. M. (1995) Lectures on Functional Analysis, Mathematics Association,
University of Mawlamyine
[2] Rudin. W. (1966) Real and Complex Analysis, McGraw- Hill, New York.

References
[1] Royden.H.L. (1968) Real Analysis. 2nd Printing, Macmillan.
[2] Friedmen .A. (1982) Foundations of Modern Analysis , Dover
Publications, Inc, New York.
[3] Taylor A.E, General Theory of Function and Integration.

Math 612 Abstract Algebra IV
Ring: Polynomial rings, polynomials over the rationals, field of quotients of an integral domain
Fields: Examples of fields, a brief excursion into vector spaces, field extensions
Text: Herstein, I. N. (1996), Abstract Algebra, Prentice-Hall.
(Chapter 4 Ring Theory [Sections 4.5, 4.6, 4.7]
Chapter 5 Fields [Sections 5.1, 5.2, 5.3])

Math 613
(a) Qualitative Theory of Ordinary Differential Equations
Lyapunov’s second method, applications of ODE
Text: Brauer. F. and Nohel, J. A. (1969), The Qualitative Theory of Ordinary
Differential Equations: An Introduction, W. A. Benjamin, Inc.
(Chapter 5 Lyapunov’s Second Method
Chapter 6 Some Applications)

(OR)

(b) Dynamical Systems
1. Linear differential systems
Case of constant coefficients. Existence and uniqueness theorem in the general case, resolvent.
2. Nonlinear differential systems
Analytical aspects: existence and uniqueness theorem, maximal solutions, estimations of the time of existence, Gronwell lemma.
Geometric aspects: flow, phase portrait and qualitative study of differential systems, Poincare first return map, invariance submanifolds, Poincare-Bendixson theorem. Perturbations of a differential system.
3. Stability of invariant sets
First integrals and Lyapunov functions. Stability of fixed points. Stability of periodic orbits.
4. Local study in the neighborhood of a fixed point
Stable and unstable manifolds of a hyperbolic fixed point. Hartman-Grobman theorem.

Text:
[1] Hirsch, Smale and R. L. Devancy. (2004), Differential equations, dynamical systems and an introduction to chaos, Elevesier.
[2] Arnold. V. I., (1992) Oridinary differential equations, third edition, Springer Verlag
Network models, a maximal flow algorithm, the max flow, min cut theorem, matching,

Reference:
[1] Verhulst, F. (1980) Nonlinear equations and Dynamical Systems. Springer-Verlag
[2] Teschl. G. Oridinary Differential Equations and Dynamical Systems

Math 614 Discrete Mathematics III
Network models, a maximal flow algorithm, the max flow, min cut theorem, matching,
combinatorial circuits, properties of combinatorial circuits, Boolean algebras, Boolean functions and synthesis of circuits, applications
Text: Johnsonbaugh, R. (1990), Discrete Mathematics, Macmillan.
(Chapter 8 Network Models
Chapter 9 Boolean Algebras and Combinatorial Circuits)

Math 615 Numerical Analysis III
Polynomial approximation, interpolation, quadrature formulas, solution of non linear equations, optimization
Text:
[1] Quarteoni, A. and Sacco, R. and Saleri, F. (2007) Numerical mathematics, Springer-Verlag

References:
[1] Ciarlet, P. G. (1989) Introduction to numerical linear algebra and optimization, Cambridge University Press
[2] Trefethen Lioyd N. (2013) Approximation theory and approximation practice, SIAM.

(OR)
Solution of linear systems of equations
Text: Plato. R. (2003), Concise Numerical Mathematics, AMS.
(Chapter 4 Solution of Linear Systems of Equations)

Math 616 Physical Applied Mathematics I
General theory of stress and strain: Definitions of stress, stress vector and components of stress tensor, state of stress at a point, symmetry of stress tensor, transformation of stress components, principal stresses and principal directions, principal direction of stress tensors.
Nature of strain, transformation of the rates of strain components, relation between stress and rate of strain in two dimensional case, the rate of strain quadratic, translation, rotation and deformation.
Viscous fluid: The Navier-Stokes equations of motion of a viscous fluid, the energy equation conservation of energy.
Text
[1] M.D.RAISINGHANIA(2010)
( Chapter 13, Chapter 14)

Math 617 Stochastics Process III
Foundations of probability of theory, limit theorems, probability distributions, probability measures and spaces

References:
[1] Cramer, H. & Leadbetter, M. R. (1946) Stationary and related stochastic
processes, Princeton University Press
[2] Feller, W. (1970) An Introduction to Probability Theory and Its Applications, 3rd
Edition, Vol I, John Wiley & Sons
[3] Feller, W. (1966) An Introduction to Probability Theory and Its Applications,
Vol II, John Wiley & Sons
[4] Hida, T. (1980) Brownian Motion, Springer-Verlag
[5] Hida, T. & Hitsuda, M. (1991) Gaussian Processes, AMS
SECOND SEMESTER
Math 621 Analysis VIII
Banach Spaces: Banach Spaces. Examples: c1 , c0 , C(X) Continuous Linear Transformation. Functionals. Dual Space N * of a Normed Space N. The Hahn Banach Theorem. Duals of Natural Imbedding of N in N ** . Reflexive Spaces. Weak Topology. Weak* topology. The Open Mapping Theorem. The Closed Graph Theorem. The Uniform Boundedness Theorem. The Conjugate of an Operator.
Hilbert Space: Inner Product Space. Hilbert Space. Examples: , l2 , L2. Schwarz Inequality. Orthogonal Complement, Orthonormal Sets. Bessel’s Inequality. Parscal’s Equation. The Conjugate Space H* of a Hilbert Space H. Representation of Functionals in H*. The Adjoint of an Operator. Self-Adjoint Operator. Normal and Unitary Operators-Projectory.
Texts
[1] Swe, K. M. (1995) Lectures on Functional Analysis. University of Mawlamyine, Mathematics Association.
[2] Simmons, G.F. (1963) Introduction to Topology and Modern Analysis, McGraw-Hill, New York.
Reference
[1] Kreyszig, E.(1978) Introductory Functional Analysis with Applications, John Wiley
& Sons, New York.

Math 622 Linear Algebra IV
Eigenvectors and eigenvalues, polynomials and matrices, triangulations of matrices and linear maps
Text: Lang, S. (2004), Linear Algebra, Third Edition, Springer.
(Chapter 8 Eigenvectors and Eigenvalues
Chapter 9 Polynomials and Matrices
Chapter 10 Triangulations of Matrices and Linear Maps)

Math 623
(a) Partial Differential Equations
Integral curves and surfaces of vector fields, theory and applications of quasi-linear and linear equations of first order, series solutions, linear partial differential equations, equations of mathematical Physics
Text: Zachmanoglou, E. C. and Thoe, D. W. (1976), Introduction to Partial
Differential Equations with Applications, Dover Publications, Inc.
(Chapter 2 Integral Curves and Surfaces of Vector Fields
Chapter 3 Theory and Applications of Quasi-Linear and Linear Equations of
First Order
Chapter 4 Series Solutions
Chapter 5 Linear Partial Differential Equations
Chapter 6 Equations of Mathematical Physics)

(OR)

(b) Differential Geometry
The course of differential geometry is an introduction of methods of differential calculus on submanifolds. We address the following points: Inverse function theorem. Implicit function theorem. Local normal forms for maps of constant rank. Definition of submanifolds. Examples. Tangent bundle. Vector fields. Lie bracket. Lie groups. Local geometry of a hypersurface in the Euclidian soace. First and second fundmental form. Gauss curvature. Egregium theorem.

Text:
[1] M. P. Do Carmo, Differential Geometry of curves and surfaces

Math 624 Graph Theory II
Connectivity, Euler tours and Hamilton cycles
Text: Bondy, J. A., and Murty, U. S. R (1984), Graph Theory with Applications,
Springer- Verlag.
(Chapter 3 Connectivity
Chapter 4 Euler Tours and Hamilton Cycles)

Math 625 Numerical Analysis IV
Nonlinear system of equations, explicit one-step methods for initial value problems in ordinary differential equations
Text: Plato. R. (2003), Concise Numerical Mathematics, AMS
(Chapter 5 Nonlinear System of Equations
Chapter 7 Explict One-Step Methods for Initial Value Problems in Ordinary
Differential Equations)

Math 626 Physical Applied Mathematics II
Viscous fluid: Diffusion of vorticity, equations for vorticity and circulation, dissipation of energy, vorticity equation of a vortex filament.
Laminar flow of viscous incompressible fluids: Plane coquette flow, generalized plane Couette flow, plane Poiseuille flow, the Hegen-Poiseuille flow, laminar steady flow of incompressible viscous fluid in tubes of cross-section other than circular.
Text
[1] M.D.RAISINGHANIA(2010)
( Chapter 15, Chapter 16)

Math 627 Stochastics Process IV
Higher dimensional distributions and infinite dimensional distributions, stochastic processes: Principle classes, Canonical representations of Gaussian process, multiple Markov Gaussian processes

References:
[1] Cramer, H. & Leadbetter, M. R. (1946) Stationary and related stochastic
processes, Princeton University Press
[2] Feller, W. (1970) An Introduction to Probability Theory and Its Applications, 3rd
Edition, Vol I, John Wiley & Sons
[3] Feller, W. (1966) An Introduction to Probability Theory and Its Applications,
Vol II, John Wiley & Sons
[4] Hida, T. (1980) Brownian Motion, Springer-Verlag
[5] Hida, T. & Hitsuda, M. (1991) Gaussian Processes, AMS

Math 628 PDE and Approximations
Introduction to the study of elliptic boundary value problems (modeling, mathematical analysis in the 1D case), of parabolic (heat equation) and hyperbolic (wave equation) problems. Introduction to the finite difference method for these 3 (model) problems and numerical simulations
Text:
[1] Le Dret H,m Lucquin B. (2016) Partial differential equations: modeling, analysis and numerical approximation, Birkhauser

References:
[1] Atkinson, K. E. and Han, W. (2009) Theoretical numerical analysis, Springer
[2] Quarteoni, A. and Sacco, R. and Saleri, F. (2007) Numerical mathematics, Springer-Verlag

Math 629 Applied Probability and Statistics 2
Markov chains, the random counterpart of the recursive sequences, and Martingales which are the mathematical tradition of the notion of equitable dynamics in economics. The aim of the course is to introduce the main concepts of the theory but also to furnish quantitative methods to use these models for concrete applications.
Text:
[1] A. N. Sirjaev. (1984) Probability, Springer

Reference:
[1] Williams, D. (1991) Probability with martingales