Mathematics Department is located in right wing of the Science Building and near University of Yangon library. Head of the department of Mathematics is Professor Dr. Cho Win. His research works are in Qualitative theory of Differential Equations. Our department is very proud for working with Professor Dr. Aung Kyaw. He is specialize in Graph Theory, Combinatorial Mathematics.

Our department has conducted the programs with following number of faculties:

- professors – 3
- associate professors – 4
- lecturers – 35
- assistant lecturers and – 19
- tutors – 10

We offers the following programs.

* Bachelor of Science in Mathematics (BSc)

* Bachelor of Honours in Mathematics (BSc Honours)

* Master of Science in Mathematics (MSc)

* Master of Research in Mathematics (MRes)

* Doctor of Philosophy in Mathematics (PhD)

Our department collaborates with Graduate School of Mathematics, Nagoya University, Japan. Two of our students are now making research in Nagoya University. Professors from Nagoya University came to conduct lectures this year.

2. Algebra

3. Ordinary Differential Equation

4. Partial Differential Equation

5. Graph Theory

6. Combinatorics and Number Theory

**FIRST SEMESTER**

**Math 1101 Algebra and Analytic Geometry**

__Algebra__: logic and sets, polynomial functions, permutations, combinations, binomial theorem, mathematical induction.

__Analytic Geometry__: lines, circles, conic sections, translation of coordinate axes, rotation of coordinate axes.

**Texts:**

[1] Lipschutz, S. (1981), *Set Theory and Related Topics*, SOS, McGraw-Hill.

(Chapter 14 Algebra of Propositions

Chapter 15 Quantifiers)

[2] Vance, E. P. (1983), *Modern Algebra and Trigonometry*, Addison-Wesley.

(Chapter 9 Polynomial Functions

Chapter 11 Permutations, Combinations and the Binomial Theorem

Chapter 12 Mathematical Induction)

[3] Anton, H (1995), *Calculus with Analytic Geometry*, 5^{th} Edition, John Wiley & Sons.

(Chapter 1 Coordinates, Graphs, Lines

Section 1.4 Lines

Section 1.5 Distances; Circles; Equations of the Form

Chapter 12 Topics in Analytic Geometry

Section 12.1 Introduction to the Conic Sections

Section 12.2 The Parabola; Translation of Coordinate Axes Section 12.3 The Ellipse

Section 12.4 The Hyperbola

Section 12.5 Rotation of Axes; Second-Degree Equations)

**Math 1102 Trigonometry and Differential Calculus**

__Trigonometry__: inverse trigonometric functions.

__Calculus__: limits, continuity, exponential, logarithmic, hyperbolic functions and their inverses, differentiation of inverse trigonometric functions, differentiation of exponential, logarithmic, hyperbolic functions and their inverse, L’Hopital’s rule, Taylor series.

**Text:** Thomas, G. B. (2009), *Calculus*, 12^{th} Edition, Addison-Wesley.

(Chapter 2 Limits and Continuity

Section 2.1 Rates of Change and Tangents to Curves

Section 2.2 Limit of a Function and Limit Laws

Section 2.3 The Precise Definition of a Limit

Section 2.4 One-Sided Limits

Section 2.5 Continuity

Section 2.6 Limits Involving Infinity; Asymptotes of Graphs

Chapter 7 Transcendental Functions

Section 7.1 Inverse Functions and Their Derivatives

Section 7.2 Natural Logarithms [Except Integration]

Section 7.3 Exponential Functions [Except Integration]

Section 7.5 Indeterminate Forms and L’Hospital’s Rule

Section 7.6 Inverse Trigonometric Functions [Except Integration]

Section 7.7 Hyperbolic Functions [Except Integration]

Chapter 10 Infinite Sequences and Series

Section 10.8 Taylor and Maclaurin Series)

**Math 1001 Mathematics I (For Science Students)**

permutations, combinations, binomial theorem, mathematical inductions;

lines, circles, conic sections;

limits, continuity, inverse trigonometric functions, exponential, logarithmic, hyperbolic functions and their inverse functions, L’Hopital’s rule, differentiation of inverse trigonometric functions.

**Texts:**

** **[1] Vance, E. P. (1983), *Modern Algebra and Trigonometry*, Addison-Wesley.

(Chapter 11 Permutations, Combinations and the Binomial Theorem

Chapter 12 Mathematical Induction)

** **[2] Anton, H (1995), *Calculus with Analytic Geometry*, 5^{th} Edition, John Wiley & Sons.

(Chapter 1 Coordinates, Graphs, Lines

Section 1.4 Lines

Section 1.5 Distances; Circles; Equations of the Form

Chapter 12 Topics in Analytic Geometry

Section 12.1 Introduction to the Conic Sections

Section 12.2 The Parabola; Translation of Coordinate Axes Section 12.3 The Ellipse

Section 12.4 The Hyperbola

Section 12.5 Rotation of Axes; Second-Degree Equations)

[3] Thomas, G. B. (2009), *Calculus*, 12^{th} Edition, Addison-Wesley.

(Chapter 2 Limits and Continuity

Section 2.1 Rates of Change and Tangents to Curves

Section 2.2 Limit of a Function and Limit Laws

Section 2.3 The Precise Definition of a Limit

Section 2.4 One-Sided Limits

Section 2.5 Continuity

Section 2.6 Limits Involving Infinity; Asymptotes of Graphs

Chapter 7 Transcendental Functions

Section 7.1 Inverse Functions and Their Derivatives

Section 7.2 Natural Logarithms [Except Integration]

Section 7.3 Exponential Functions [Except Integration]

Section 7.5 Indeterminate Forms and L’Hospital’s Rule

Section 7.6 Inverse Trigonometric Functions [Except Integration]

Section 7.7 Hyperbolic Functions [Except Integration]

**Math 1002 Mathematics I (For Arts Students)**

permutations, combinations, binomial theorem, variables and graphs, frequency distributions, mean, median, mode and other measures of central tendency, standard deviation and other measures of dispersion, elementary probability theory.

**Texts:**

** **[1] Vance, E. P. (1983), *Modern Algebra and Trigonometry*, Addison-Wesley.

(Chapter 11 Permutations, Combinations and the Binomial Theorem)

** **[2] Spiegel. M. R. (1961), *Statistics*, SOS, McGraw-Hill.

(Chapter 1 Variables and Graphs Chapter 2 Frequency Distributions

Chapter 3 Mean, Median, Mode and Other Measures of Central Tendency

Chapter 4 The Standard Deviation Chapter 6 Elementary probability theory)

**SECOND SEMESTER**

**Math 1103 Algebra and Analytical Solid Geometry**

__Algebra__: determinants, matrices, complex numbers.

__Polar Coordinates System__: polar coordinates, graphing in polar coordinates, areas and lengths in polar coordinates.

__Analytical Solid Geometry__: three dimensional Cartesian coordinate system, lines, planes, quadric surfaces.

**Texts:**

[1] Kolman, B. (1988), *Introductory Linear Algebra with Applications*, Macmillan.

(Chapter 1 Linear Equations and Matrices

Chapter 2 Determinants)

[2] Brown, J. W. and Churchill, R. V. (2009), *Complex Variables and Applications*, McGraw-Hill.

(Chapter 1 Complex Numbers)

[3] Thomas, G. B. (2009), *Calculus*, 12^{th} Edition, Addison-Wesley.

(Chapter 11 Parametric Equations and Polar Coordinates

Section 11.3 Polar Coordinates

Section 11.4 Graphing in Polar Coordinates

Section 11.5 Areas and Lengths in Polar Coordinates

Chapter 12 Vectors and the Geometry of Space

Section 12.6 Cylinders and Quadric Surfaces)

[4] Narayan, S. (1999), *Analytical Solid Geometry*, S. Chand.

(Chapter 1 Coordinates

Chapter 2 The Plane

Chapter 3 Right Line)

**Math 1104 Differential and Integral Calculus**

extreme values of functions, the mean value theorem, monotonic functions and the first derivatives test, methods of integration, improper integrals, applications of integration, partial differentiation, ordinary differential equation of first order.

**Texts:**

[1] Thomas, G. B. (2009), *Calculus*, 12^{th} Edition, Addison-Wesley.

(Chapter 4 Applications of derivatives

Section 4.1 Extreme Values of Functions

Section 4.2 The Mean Value Theorem

Section 4.3 Monotonic Functions and the First Derivatives Test

Chapter 5 Integration

Chapter 6 Applications of the definite integrals

Section 6.1 Volumes Using Cross-Sections

Section 6.2 Volumes Using Cylindrical Shells

Section 6.3 Arc Length

Section 6.4 Areas of Surfaces of Revolution

Chapter 8 Techniques of Integration

Section 8.1 Integration by Parts

Section 8.2 Trigonometric Integrals

Section 8.3 Trigonometric Substitutions

Section 8.4 Integration of Rational Functions by Partial Fractions

Section 8.7 Improper Integrals

Chapter 9 First-order differential equations

[2] Thomas, G. B. (1977), *Calculus and Analytic Geometry*, 4^{th} Edition, Addison-Wesley.

** **(Chapter 15 Partial Differentiation

Section 15.1, 15.2, 15.3

Chapter 20 Section 20.6

**Math 1003 Mathematics II (For Science Students)**

__Algebra__: determinants, matrices, complex numbers;

__Polar Coordinates System__: polar coordinates, graphing in polar coordinates, areas and lengths in polar coordinates;

methods of integration, improper integrals, partial differentiation, ordinary differential equation of first order.

**Texts:**

[1] Kolman, B. (1988), *Introductory Linear Algebra with Applications*, Macmillan.

(Chapter 1 Linear Equations and Matrices

Chapter 2 Determinants)

[2] Brown, J. W. and Churchill, R. V. (2009), *Complex Variables and Applications*, McGraw-Hill.

(Chapter 1 Complex Numbers)

[3] Thomas, G. B. (2009), *Calculus*, 12^{th} Edition, Addison-Wesley.

(Chapter 5 Integration

Chapter 8 Techniques of Integration

Section 8.1 Integration by Parts

Section 8.2 Trigonometric Integrals

Section 8.4 Integration of Rational Functions by Partial Fractions

Section 8.7 Improper Integrals

Chapter 9 First-order differential equations

Chapter 11 Parametric Equations and Polar Coordinates

Section 11.3 Polar Coordinates

Section 11.4 Graphing in Polar Coordinates

Section 11.5 Areas and Lengths in Polar Coordinates)

[4] Thomas, G. B. (1977), *Calculus and Analytic Geometry*, 4^{th} Edition, Addison-Wesley.

** **(Chapter 15 Partial Differentiation

Section 15.1, 15.2, 15.3

Chapter 20 Section 20.6)

**Math 1004 Mathematics II (For Arts Students)**

inverse trigonometric functions and their derivatives, determinants, matrices, solving system of linear equations, lines, circles, conic sections.

**Texts:**

[1] Kolman, B. (1988), *Introductory Linear Algebra with Applications*, Macmillan.

(Chapter 1 Linear Equations and Matrices

Chapter 2 Determinants)

[2] Thomas, G. B. (2009), *Calculus*, 12^{th} Edition, Addison-Wesley.

(Chapter 7 Transcendental Functions

Section 7.1 Inverse Functions and Their Derivatives

Section 7.6 Inverse Trigonometric Functions [Except Integration]

[3] Anton, H (1995), *Calculus with Analytic Geometry*, 5^{th} Edition, John Wiley & Sons.

(Chapter 1 Coordinates, Graphs, Lines

Section 1.4 Lines

Section 1.5 Distances; Circles; Equations of the Form

Chapter 12 Topics in Analytic Geometry

Section 12.1 Introduction to the Conic Sections

Section 12.2 The Parabola; Translation of Coordinate Axes Section 12.3 The Ellipse

Section 12.4 The Hyperbola

Section 12.5 Rotation of Axes; Second-Degree Equations)

FIRST SEMESTER

Math 2101 Complex Variables I

analytic functions, elementary functions, integrals, residues and poles.

Text: Brown, J. W. and Churchill, R. V. (2009), Complex Variables and Applications,

8th edition, McGraw-Hill.

(Chapter 2 Analytic Functions

Chapter 3 Elementary Functions

Chapter 4 Integrals

` Chapter 6 Residues and Poles)

Math 2102 Calculus of Several Variables

Functions of two or more variables, partial derivatives, directional derivatives, chain rule for partial derivatives, total differential, maxima and minima, exact differentials, derivatives of integrals, double integrals in Cartesian coordinates and polar coordinates, triple integrals in cartesian coordinates, cylindrical coordinates and spherical coordinates, applications of multiple integrals.

Text: Thomas, G. B. (2009), Calculus, 12th Edition, Addison-Wesley.

(Chapter 14 Partial Derivatives

Chapter 15 Multiple Integrals)

Math 2103 Vector Algebra and Statics

Vector Algebra: dot products, cross products, triple scalar products, triple vector products.

Statics: statics of a particle, tension of a string, friction, moment and couples, centre of gravity, statics of a rigid body in a plane, jointed rods, virtual works, stability.

Texts: [1] Chorlton, L.(1967), Textbook of Fluid Dynamics, Van Nostrand.

(Chapter 1 Vector Analysis [Sections 1.1 to 1.7])

[2] Humphery, D. (1964), Statics, Longmans.

(Chapter 1 Statics of a Particle

Chapter 2 Statics of a Rigid Body–Parallel Forces–Monents–Couples

Chapter 3 Forces in a Plane Acting on a Rigid Body

Chapter 5 Friction

Chapter 7 Center of Gravity

Chapter 9 Virtual Work–Stability–Miscellaneous Examples)

Math 2104 Discrete Mathematics I

Counting Methods and the Pigeonhole Principle: basic principle, permutations and combinations, algorithms for generating permutations and combinations, generalized permutations and combinations, binomial coefficients and combinatorial identities, the pigeonhole principle.

Recurrence Relations: introduction, solving recurrence relations, applications to the analysis of algorithms.

Text: Johnsonbaugh, R. (1990), Discrete Mathematics, Macmillan.

(Chapter 4 Counting Methods and the Pigeonhole Principle

Chapter 5 Recurrence Relations)

Math 2105 Theory of Sets I

cardinal numbers, partially and totally ordered sets.

Text: Lipschutz, S. (1981), Set Theory and Related Topics, SOS, McGraw-Hill.

(Chapter 9 Cardinal Numbers

Chapter 10 Partially and Totally Ordered Sets)

Math 2106 Spherical Trigonometry and Its Applications

Spherical Trigonometry: the spherical triangle, length of small circle arc, terrestrial latitude and longitude, the fundamental formula of spherical trigonometry, the sine formula, formula C, the four parts formula, the trigonometrical ratios for small angles.

Celestial Sphere: altitude and azimuth, declination and hour angle, diagram for the southern hemisphere, circumpolar stars, the standard or geocentric celestial sphere, right ascension and declination, the earth’s orbit, celestial latitude and longitude, sidereal time, mean solar time, hour angle of a heavenly body, rising and setting, Twilight.

Text: Smart, W. M & Greene, R. (1986) , Text-Book on Spherical Astronomy, Cambridge University Press.

(Chapter 1 Spherical Trigonometry [Sections 1 to 8, 15]

Chapter 2 Celestial Sphere)

Math 2001 Mathematics I (For Science Students)

coordinates, the plane, partial derivatives, the chain rule for partial derivatives, the total differential, maxima and minima, double integrals, area by double integration, triple integrals Volume, the dot and cross product.

Statistics: a quick review on mean, median, mode and other measures of central tendency, standard deviation and other measures of dispersion.

Probability: introduction to probability, finite sample spaces, conditional probability and independence, one-dimensional random variables.

Texts:

[1] Narayan, S. (1999), Analytical Solid Geometry, S. Chand & Co., LTD.

(Chapter 1 Coordinates

Chapter 2 The Plane)

[2] Spiegel, M.R.(1963), Vector Analysis, SOS, McGraw-Hill.

(Chapter 2 The dot and cross product)

[3] Thomas, G. B. (2009), Calculus, 12th Edition, Addison-Wesley.

(Chapter 14 Partial Derivatives

Section 14.3 Partial Derivatives

Section 14.4 The Chain Rule

Section 14.6 Tangent Planes and Differentials

Section 14.7 Extreme Values and Saddle Points

Chapter 15 Multiple Integrals

Section 15.1 Double and Iterated Integrals over Rectangles

Section 15.3 Area by Double Integration

Section 15.5 Triple Integrals in Rectangular Coordinates

[4] Spiegel, M.R., (1961) Statistics, SOS, McGraw-Hill.

(Chapter 1 Variables and Graphs,

Chapter 2 Frequency Distributions

Chapter 3 Mean, Median, Mode and Other Measures of Central

Tendency

Chapter 4 The Standard deviation)

[5] Francis, A., (1979) Advanced Level Statistics, Stanley Thrones Ltd., London.

(Chapter 4 Probability [Sections: 4.3, 4.4]

Chapter 5 Random variables and Probability Distributions [Section: 5.3])

Math 2002 Mathematics I (For Arts Students)

methods of integration, binomial distribution, normal distribution, Poisson distribution, method of least square, regression.

Texts:

[1] Spiegel. M. R. (1961), Statistics, SOS, McGraw-Hill.

(Chapter 7 The Binomial, Normal and Poisson Distributions

Chapter 13 Curve Fitting and the Method of Least Squares)

[2] Thomas, G. B. (2009), Calculus, 12th Edition, Addison-Wesley.

(Chapter 5 Integration

Chapter 8 Techniques of Integration

Section 8.1 Integration by Parts

Section 8.4 Integration of Rational Functions by Partial Fractions

Math 2003 Mathematics I (For Industrial Chemistry Students)

three-dimensional cartesian coordinate system, lines, planes, quadric surfaces, partial derivatives, chain rule, total differentials, maxima and minima, multiple integrals.

Statistics: a quick review on mean, median, mode and other measures of central tendency, standard deviation and other measures of dispersion.

Probability: introduction to probability, finite sample spaces, conditional probability and independence, one-dimensional random variables.

Texts:

[1] Thomas, G. B. (2009), Calculus, 12th Edition, Addison-Wesley.

(Chapter 14 Partial Derivatives

Section 14.3 Partial Derivatives

Section 14.4 The Chain Rule

Section 14.6 Tangent Planes and Differentials

Section 14.7 Extreme Values and Saddle Points

Chapter 15 Multiple Integrals

Section 15.1 Double and Iterated Integrals over Rectangles

Section 15.3 Area by Double Integration

Section 15.5 Triple Integrals in Rectangular Coordinates

[2] Narayan, S., (1999), Analytical Solid Geometry, S. Chand.

(Chapter 1 Coordinates, Chapter 2 The Plane, Chapter 3 Right Line)

[3] Francis, A., (1979) Advanced Level Statistics ,Stanley Thornes Ltd., London.

(Chapter 4 Probability [Sections: 4.3, 4.4]

Chapter 5 Random variables and Probability Distributions [Section: 5.3])

[4] Spiegel, M.R., (1961) Statistics, SOS, McGraw-Hill.

(Chapter 1 Variables and Graphs,

Chapter 2 Frequency Distributions

Chapter 3 Mean, Median, Mode and Other Measures of Central Tendency

Chapter 4 The Standard deviation)

SECOND SEMESTER

Math 2107 Linear Algebra I

vector spaces, subspaces, bases and dimensions, sums and direct sums, linear mapping, kernel and image of linear map and their dimension, compositions of linear mapping.

Text: Lang, S. (2004), Linear Algebra, Third Edition, Springer.

(Chapter 1 Vector Spaces

Chapter 2 Matrices

Chapter 3 Linear Mappings)

Math 2108 Ordinary Differential Equations

second-order linear differential equations, higher-order linear differential equations, system of differential equations.

Text: Kreyszig, E (2011), Advanced Engineering Mathematics, Tenth Edition, John Wiley.

(Chapter 2 Second-Order Linear Differential Equations

Chapter 3 Higher-Order Linear Differential Equations

Chapter 4 System of Differential Equations. Phase Plane, Stability

[Sections 4.1, 4.2, 4.3, 4.6])

Math 2109 Vector Calculus and Dynamics

Vector Calculus: gradient, divergence, curl, line integrals, green’s theorem, divergence theorem, Stoke’s theorem.

Dynamics: kinematics of a particle, relative velocity, mass, momentum and force, Newton’s law of motion, work, power and energy, simple harmonic motions, kinematics of a particle in two dimensions, kinetic of a particle in two dimensions.

Texts: [1] Chorlton, L.(1967), Textbook of Fluid Dynamics, Van Nostrand.

(Chapter 1 Vector Analysis [Sections 1.8 to 1.19])

[2] Humphery, D. and Topping, J. (1964), Intermediate Mechanics, Vol I.

(Dynamics), Longmans.

(Chapter 1 Speed and velocity [Section: Relative velocity]

Chapter 3 Force, momentum, laws of motion

[Sections: mass, momentum, Newton’s law of motion]

Chapter 4 Work, power and energy

Chapter 8 Simple harmonic motions

Chapter 9 Motion of a particle in two dimensions)

Math 2110 Discrete Mathematics II

Graph Theory: introduction, paths and cycles, hamiltonian cycles and the traveling salesperson problem, a shortest path algorithm, representations of graphs, isomorphism of graphs, planar graphs.

Trees: introduction, terminology and characterizations of trees, Spanning trees, minimal spanning trees, binary trees.

Text: Johnsonbaugh, R. (1990), Discrete Mathematics, Macmillan.

(Chapter 6 Graph Theory, Chapter 7 Trees)

Math 2111 Theory of Sets II

well-ordered sets, ordinal numbers, axiom of choice, Zorn’s lemma, well-ordering theorem

Text: S. Lipschutz, (1981) Set Theory and Related Topics, SOS, McGraw-Hill.

(Chapter 11 Well-ordered sets. Ordinal Numbers

Chapter 12 Axiom of Choice, Zorn’s Lemma, Well-ordering Theorem)

Math 2112 Astronomy

Planetary Motions: Kepler’s three laws, Newton’s law of gravitation, the masses of the planets, the dynamical principles of orbital motion, the equation of the orbit, velocity of a planet in its orbit, components of the linear velocity perpendicular to radius vector and to the major axis, the true and eccentric anomaly expressed as a series in terms of e and the eccentric anomaly, the equation of the centre, the orbit in the space, the orbital and synodic periods of a planet, the earth’s orbit, the sun’s orbit, the moon’s orbit.

Time: sidereal time, the mean sun, the sidereal year and the tropical year, relation between mean and sidereal time, the calendar, the Julian date, the equation of time, the seasons.

Text: Smart, W. M. & Greene, R. (1986), Text-Book on Spherical Astronomy, Cambridge University Press.

(Chapter 5 Planetary Motions)

Math 2004 Mathematics II (For Science Students)

right line, second order linear differential equations, higher order linear differential equations, power series method, vector differentiation, gradient, divergence, curl.

Texts:

[1] Kreyszig, E (2011), Advanced Engineering Mathematics, Tenth Edition, John

Wiley.

(Chapter 2 Second-Order Linear Differential Equations

Chapter 3 Higher-Order Linear Differential Equations

Chapter 5 Series Solutions of ODEs)

[2] Narayhan, S. (1999), Analytical Solid Geometry, S. Chand & Co., Ltd.

(Chapter 3 Right Line)

[3] Spiegel, M.R.(1963), Vector Analysis, SOS, McGraw-Hill.

(Chapter 3 Vector Differentiation

Chapter 4 Gradient, Divergence, Curl)

Math 2005 Mathematics II (For Arts Students)

three-dimensional cartesian coordinate system, lines, planes, quadric surfaces, spherical trigonometry.

Texts: [1] Narayhan Math 2005 Mathematics II (For Arts Students)

three-dimensional cartesian coordinate system, lines, planes, quadric surfaces, spherical trigonometry.

Texts: [1] Narayhan, S. (1999), Analytical Solid Geometry, S. Chand & Co., Ltd.

(Chapter 1 Co-ordinates

Chapter 2 The Plane

Chapter 3 Right Line)

[2] Thomas, G. B. (2009), Calculus, 12th Edition, Addison-Wesley.

(Chapter 12 Vectors and the Geometry of Space

Section 12.6 Cylinders and Quadric Surfaces)

[3] Smart, W. M & Greene, R. (1986) Text-Book on Spherical Astronomy, Cambridge University Press.

(Chapter 1 Spherical Trigonometry [Sections 1 to 8, 15])

Math 2006 Mathematics II (For Industrial Chemistry Students)

dot products, cross products, triple scalar products, triple vector products, complex functions, complex integrals, higher order linear ordinary differential equations.

linear programming: general discussion, mathematical background, the general linear programming problem, the simplex computational procedure.

Texts:

[1] Spiegel, M.R.(1974), Vector Analysis, SOS, McGraw-Hill.

(Chapter 2)

[2] Brown, J. W. and Churchill, R. V. (2009), Complex Variables and Applications,

8th edition, McGraw-Hill.

(Chapter 2 Analytic Functions

Chapter 4 Integrals)

[3] Kreyszig, E (2011), Advanced Engineering Mathematics, Tenth Edition, John Wiley.

(Chapter 2 Second-Order Linear Differential Equations

Chapter 3 Higher-Order Linear Differential Equations)

[4] Gass, S.I., (1964) Linear Programming, Third Edition, McGraw-Hill.

(Chapter 1 General Discussion

Chapter 2 Mathematical Background

Chapter 3 The General Linear-programming Problem

Chapter 4 The Simplex Computational Procedure)

FIRST SEMESTER

Math 3101 Analysis I

elements of set theory, numerical sequences and series.

Text: Rudin, W. (1976), Principle of Mathematical Analysis, McGraw-Hill.

(Chapter 2 Basic Topology

Chapter 3 Numerical Sequences and Series)

Math 3102 Linear Algebra II

linear maps and matrices, determinants.

Text: Lang, S. (2004), Linear Algebra, Third Edition, Springer.

(Chapter 4 Linear Maps and Matrices

Chapter 6 Determinants)

Math 3103 Differential Equations

series solutions of differential equations, special functions, laplace transforms.

Text: Kreyszig, E (2011), Advanced Engineering Mathematics, Tenth Edition, John Wiley.

(Chapter 5 Series Solutions of ODEs. Special Functions

Chapter 6 Laplace Transforms)

Math 3104 Differential Geometry

concept of a curve, curvature and torsion, the theory of curves, concept of a surface.

Text: Lipschutz, M. M. (1969), Differential Geometry, SOS, McGraw-Hill.

(Chapter 3 Concept of a Curve

Chapter 4 Curvature and Torsion

Chapter 5 The Theory of Curves

Chapter 8 Concept of a Surface)

Math 3105 Tensor Analysis

curvilinear coordinates, tensor analysis

Text: Spiegel, M. R. (1974), Vector Analysis, SOS, McGraw-Hill.

(Chapter 7 Curvilinear Coordinates

Chapter 8 Tensor Analysis)

Math 3106 Number Theory I

divisibility theory, congruences, Fermat’s little theorem, Euler’s generalization of FLT, Wilson’s theorem, Eulers-function.

Text: Malik, S. B. (1995), Basic Number Theory. Vikas Publishing House PVT Ltd.

(Chapter 2 Divisibility Theory

Chapter 3 Congruences

Chapter 4 Some Special Congruences and Euler’s φ-Functions)

SECOND SEMESTER

Math 3107 Analysis II

continuity, differentiation

Text: Rudin, W. (1976), Principle of Mathematical Analysis, McGraw-Hill.

(Chapter 4 Continuity, Chapter 5 Differentiation)

Math 3108 Linear Algebra III

scalar products and orthogonality, matrices and bilinear maps.

Text: Lang, S. (2004), Linear Algebra, Third Edition, Springer.

(Chapter 5 Scalar Products and Orthogonality

Chapter 7 Symmetric, Hermitian and Unitary Operators)

Math 3109 Mechanics

Impulsive Forces, central force motion, kinematics of plane rigid bodies, kinetics of plane rigid bodies, impact, dynamics of a particle in three dimensions, dynamics of system of particles, moment of inertia, polar coordinates, orbits.

Texts: [1] Humphery, D. and Topping, J. (1964), Intermediate Mechanics, Vol I.

(Dynamics), Longmans.

(Chapter 5 Impulsive Forces

Chapter 10 Dynamics of a rigid body)

[2] Ramsey, A.S., (2009) Dynamics, Digitally printed version 2009, Cambridge

University. (Chapter 12 Polar Coordinates. Orbits )

Math 3110 Probability and Statistics

introduction to probability theory, random variables, mean, median, mode, standard deviation, correlation, regression.

Texts: [1] Ross. S. M. (2010), Introduction to Probability Models, Tenth Edition,

Elsevier.

(Chapter 1 Introduction to Probability Theory,

Chapter 2 Random Variables)

[2] Spiegel. M. R. (1961), Statistics, SOS, McGraw-Hill.

(Chapter 3 Mean. Median. Mode

Chapter 4 The Standard Deviation, Chapter 14 Correlation Theory)

Math 3111 Complex Variables II

conformal mapping, application of conformal mapping.

Text: Brown, J. W. and Churchill, R. V. (2009), Complex Variables and Applications,

8th edition, McGraw-Hill.

(Chapter 9 Conformal Mapping, Chapter 10 Application of Conformal Mapping)

Math 3112 Number Theory II

primitive roots, quadratic congruence and quadratic reciprocity law, perfect numbers and Fermat’s numbers, representation of integers as sum of squares.

Text: Malik, S. B.(1995), Basic Number Theory. Vikas Publishing House PVT Ltd.

(Chapter 6 Primitive Roots

Chapter 7 Quadratic Congruences and Quadratic Reciprocity Law

Chapter 8 Perfect Numbers and Fermat’s Numbers

Chapter 10 Representation of Integers as Sum of Squares)

FIRST SEMESTER

Math 4101 Analysis III

methods of mathematical research, the Riemann-Stieltjes integral.

Text: [Zaw Win (2011), Some Strategies of Mathematical Research, Math

Department, Yangon University.]

Rudin, W., (1976), Principle of Mathematical Analysis, McGraw-Hill.

(Chapter 6 The Riemann-Stieltjes Integral )

Math 4102 Numerical Analysis I

numerical methods in general, numerical methods in linear algebra.

Text: Kreyszig, E (2011), Advanced Engineering Mathematics, Tenth Edition, John Wiley.

(Chapter 19 Numerics in General

Chapter 20 Numeric Linear Algebra

[Sections 20.1, 20.2, 20.3, 20.4])

Math 4103 Linear Programming

basic properties of linear programs, the simplex method, duality, dual simplex method and primal dual algorithms,

Text: Luenberger, D. G.(1971), Linear and Nonlinear Programming,

Addison-Wesley.

(Chapter 2 Basic Properties of Linear Programs

Chapter 3 The Simplex Method

Chapter 4 Duality)

Math 4104 Partial Differential Equations

fourier series, integrals, and transforms, partial differential equations.

Text: Kreyszig, E (2011), Advanced Engineering Mathematics, Tenth Edition, John Wiley.

(Chapter 11 Fourier Analysis

Chapter 12 Partial Differential Equations)

Math 4105 Stochastic Process I

conditional probability and conditional expectation, Markov chains .

Text: Ross. S. M. (2010), Introduction to Probability Models, Tenth Edition,

Elsevier.

(Chapter 3 Conditional Probability and Conditional Expectation

Chapter 4 Markov Chains [Sections 4.1 to 4.4])

Math 4106 Fundamentals of Algorithms and Computer Programming

the idea of an algorithm, pseudo code descriptions of algorithms, efficiency of algorithms, algorithms for arithmetic and algebra, coding and implementations of algorithms in some programming languages

Text: Grossman J.W, (1990), Discrete Mathematics, Macmillan.

(Chapter 4 Algorithms)

SECOND SEMESTER

Math 4107 Analysis IV

sequences and series of functions

Text: Rudin, W. (1976), Principle of Mathematical Analysis. McGraw-Hill.

(Chapter 7 Sequences and Series of Functions)

Math 4108 General Topology I

topology of the line and plane, topological spaces, bases and subbases.

Text: Lipschutz. S. (1965), Theory and Problems of General Topology, SOS,

McGraw-Hill.

(Chapter 4 Topology of the Line and Plane

Chapter 5 Topological Spaces, Chapter 6 Bases and Subbases)

Math 4109 Abstract Algebra I

semigroups, monoids and groups, homomorphisms and subgroups, cyclic groups, cosets and counting, normality, quotient groups, and homomorphisms.

Text: Herstein, I.N. (1996), Abstract Algebra (3rd Edition). Prentice-Hall.

(Chapter 2 Groups [Sections 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7])

Math 4110 Hydromechanics

density and specific gravity, theorems on fluid-pressure under gravity, pressure of heavy fluids, thrusts on curved surfaces and floating bodies, stability of floating bodies, equation of continuity, equation of motion, some three dimensional flows.

Texts: [1] Chorlton, F. (1967), Textbook of Fluid Dynamics, Van Nostrand.

(Chapter 2 Kinematics of Fluids in Motion

Chapter 3 Equations of Motion of a Fluid

Chapter 4 Three Dimensional Flows Sections 4.1 to 4.4)

[2] Ray, M. and Sharma, H.S. (1961), A Text Book of Hydrostatics, Premier

Publishing.

(Chapter 2 Density and Specific Gravity

Chapter 3 Theorems on Fluid-Pressure under Gravity

Chapter 4 Pressures on Plane Surfaces

Chapter 5 Center of Pressure

Chapter 6 Thrusts on Curved Surfaces

Chapter 7 Floating Bodies

Chapter 9 Stability of Floating Bodies)

Math 4111 Stochastic Process II

Markov chains, the exponential distribution and the Poisson process.

Text: Ross. S. M. (2010), Introduction to Probability Models, Tenth Edition,

Elsevier.

(Chapter 4 Markov Chains [Sections 4.5 to 4.11]

Chapter 5 The Exponential Distribution and the Poisson Process)

Math 4112 Integer Programming

formulations of integer programming, branch and bound algorithms, cutting plane methods.

Texts: [1] Bronson, R. (1983), Theory and Problems of Operations

Research, SOS, McGraw-Hill.

[2] Wolsey, L. (1998), Integer Programming, John Wiley.

(Chapter 1 Formulations)

FIRST SEMESTER

Math 3201 Analysis I

elements of set theory, numerical sequences and series.

Text: Rudin, W. (1976), Principle of Mathematical Analysis, McGraw-Hill.

(Chapter 2 Basic Topology

Chapter 3 Numerical Sequences and Series)

Math 3202 Linear Algebra II

linear maps and matrices, determinants.

Text: Lang, S. (2004), Linear Algebra, Third Edition, Springer.

(Chapter 4 Linear Maps and Matrices

Chapter 6 Determinants)

Math 3203 Differential Equations

series solutions of differential equations, special functions, laplace transforms.

Text: Kreyszig, E (2011), Advanced Engineering Mathematics, Tenth Edition, John Wiley.

(Chapter 5 Series Solutions of ODEs. Special Functions

Chapter 6 Laplace Transforms)

Math 3204 Differential Geometry

concept of a curve, curvature and torsion, the theory of curves, concept of a surface.

Text: Lipschutz, M. M. (1969), Differential Geometry, SOS, McGraw-Hill.

(Chapter 3 Concept of a Curve

Chapter 4 Curvature and Torsion

Chapter 5 The Theory of Curves

Chapter 8 Concept of a Surface)

Math 3205 Tensor Analysis

curvilinear coordinates, tensor analysis

Text: Spiegel, M. R. (1974), Vector Analysis, SOS, McGraw-Hill.

(Chapter 7 Curvilinear Coordinates

Chapter 8 Tensor Analysis)

Math 3206 Number Theory I

divisibility theory, congruences, Fermat’s little theorem, Euler’s generalization of FLT, Wilson’s theorem, Eulers-function.

Text: Malik, S. B. (1995), Basic Number Theory. Vikas Publishing House PVT Ltd.

(Chapter 2 Divisibility Theory

Chapter 3 Congruences

Chapter 4 Some Special Congruences and Euler’s φ-Functions)

SECOND SEMESTER

Math 3207 Analysis II

continuity, differentiation

Text: Rudin, W. (1976), Principle of Mathematical Analysis. McGraw-Hill.

(Chapter 4 Continuity, Chapter 5 Differentiation)

Math 3208 Linear Algebra III

scalar products and orthogonality, matrices and bilinear maps.

Text: Lang, S. (2004), Linear Algebra, Third Edition, Springer.

(Chapter 5 Scalar Products and Orthogonality

Chapter 7 Symmetric, Hermitian and Unitary Operators)

Math 3209 Mechanics

Impulsive Forces, central force motion, kinematics of plane rigid bodies, kinetics of plane rigid bodies, impact, dynamics of a particle in three dimensions, dynamics of system of particles, moment of inertia, polar coordinates, orbits.

Texts: [1] Humphery, D. and Topping, J. (1964), Intermediate Mechanics, Vol I.

(Dynamics), Longmans.

(Chapter 5 Impulsive Forces

Chapter 10 Dynamics of a rigid body)

[2] Ramsey, A.S., (2009) Dynamics, Digitally printed version 2009, Cambridge

University. (Chapter 12 Polar Coordinates. Orbits )

Math 3210 Probability and Statistics

introduction to probability theory, random variables, mean, median, mode, standard deviation, correlation, regression.

Texts: [1] Ross. S. M. (2010), Introduction to Probability Models, Tenth Edition,

Elsevier.

(Chapter 1 Introduction to Probability Theory,

Chapter 2 Random Variables)

[2] Spiegel. M. R. (1961), Statistics, SOS, McGraw-Hill.

(Chapter 3 Mean. Median. Mode

Chapter 4 The Standard Deviation, Chapter 14 Correlation Theory)

Math 3211 Complex Variables II

conformal mapping, application of conformal mapping.

Text: Brown, J. W. and Churchill, R. V. (2009), Complex Variables and Applications,

8th edition, McGraw-Hill.

(Chapter 9 Conformal Mapping, Chapter 10 Application of Conformal Mapping)

Math 3212 Number Theory II

primitive roots, quadratic congruence and quadratic reciprocity law, perfect numbers and Fermat’s numbers, representation of integers as sum of squares.

Text: Malik, S. B.(1995), Basic Number Theory. Vikas Publishing House PVT Ltd.

(Chapter 6 Primitive Roots

Chapter 7 Quadratic Congruences and Quadratic Reciprocity Law

Chapter 8 Perfect Numbers and Fermat’s Numbers

Chapter 10 Representation of Integers as Sum of Squares)

FIRST SEMESTER

Math 4201 Analysis III

methods of mathematical research, the Riemann-Stieltjes integral.

Text: [Zaw Win (2011), Some Strategies of Mathematical Research, Math

Department, Yangon University.]

Rudin, W., (1976), Principle of Mathematical Analysis, McGraw-Hill.

(Chapter 6 The Riemann-Stieltjes Integral )

Math 4202 Numerical Analysis I

numerical methods in general, numerical methods in linear algebra.

Text: Kreyszig, E (2011), Advanced Engineering Mathematics, Tenth Edition, John Wiley.

(Chapter 19 Numerics in General

Chapter 20 Numeric Linear Algebra

[Sections 20.1, 20.2, 20.3, 20.4])

Math 4203 Linear Programming

basic properties of linear programs, the simplex method, duality, dual simplex method and primal dual algorithms,

Text: Luenberger, D. G.(1971), Linear and Nonlinear Programming,

Addison-Wesley.

(Chapter 2 Basic Properties of Linear Programs

Chapter 3 The Simplex Method

Chapter 4 Duality)

Math 4204 Partial Differential Equations

fourier series, integrals, and transforms, partial differential equations.

Text: Kreyszig, E (2011), Advanced Engineering Mathematics, Tenth Edition, John Wiley.

(Chapter 11 Fourier Analysis

Chapter 12 Partial Differential Equations)

Math 4205 Stochastic Process I

conditional probability and conditional expectation, Markov chains .

Text: Ross. S. M. (2010), Introduction to Probability Models, Tenth Edition,

Elsevier.

(Chapter 3 Conditional Probability and Conditional Expectation

Chapter 4 Markov Chains [Sections 4.1 to 4.4])

Math 4206 Fundamentals of Algorithms and Computer Programming

the idea of an algorithm, pseudo code descriptions of algorithms, efficiency of algorithms, algorithms for arithmetic and algebra, coding and implementations of algorithms in some programming languages

Text: Grossman J.W, (1990), Discrete Mathematics, Macmillan.

(Chapter 4 Algorithms)

SECOND SEMESTER

Math 4207 Analysis IV

sequences and series of functions

Text: Rudin, W. (1976), Principle of Mathematical Analysis, McGraw-Hill.

(Chapter 7 Sequences and Series of Functions)

Math 4208 General Topology I

topology of the line and plane, topological spaces, bases and subbases.

Text: Lipschutz. S. (1965), Theory and Problems of General Topology, SOS,

McGraw-Hill

(Chapter 4 Topology of the Line and Plane

Chapter 5 Topological Spaces, Chapter 6 Bases and Subbases)

Math 4209 Abstract Algebra I

semigroups, monoids and groups, homomorphisms and subgroups, cyclic groups, cosets and counting, normality, quotient groups, and homomorphisms.

Text: Herstein, I.N. (1996), Abstract Algebra (3rd Edition). Prentice-Hall.

(Chapter 2 Groups [Sections 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7])

Math 4210 Hydromechanics

density and specific gravity, theorems on fluid-pressure under gravity, pressure of heavy fluids, thrusts on curved surfaces and floating bodies, stability of floating bodies, equation of continuity, equation of motion, some three dimensional flows.

Texts: [1] Chorlton, F. (1967), Textbook of Fluid Dynamics, Van Nostrand.

(Chapter 2 Kinematics of Fluids in Motion

Chapter 3 Equations of Motion of a Fluid

Chapter 4 Three Dimensional Flows Sections 4.1 to 4.4)

[2] Ray, M. and Sharma, H.S. (1961), A Text Book of Hydrostatics, Premier

Publishing.

(Chapter 2 Density and Specific Gravity

Chapter 3 Theorems on Fluid-Pressure under Gravity

Chapter 4 Pressures on Plane Surfaces

Chapter 5 Center of Pressure

Chapter 6 Thrusts on Curved Surfaces

Chapter 7 Floating Bodies

Chapter 9 Stability of Floating Bodies)

Math 4211 Stochastic Process II

Markov chains, the exponential distribution and the Poisson process.

Text: Ross. S. M. (2010), Introduction to Probability Models, Tenth Edition,

Elsevier.

(Chapter 4 Markov Chains [Sections 4.5 to 4.11]

Chapter 5 The Exponential Distribution and the Poisson Process)

Math 4212 Integer Programming

formulations of integer programming, branch and bound algorithms, cutting plane methods.

Texts: [1] Bronson, R. (1983), Theory and Problems of Operations

Research, SOS, McGraw-Hill.

[2] Wolsey, L. (1998), Integer Programming, John Wiley.

(Chapter 1 Formulations)

FIRST SEMESTER

Math 5201 Analysis V

power series, the exponential and logarithmic functions, the trigonometric functions,

fourier series

Text: Rudin, W. (1976), Principle of Mathematical Analysis. McGraw-Hill.

(Chapter Some Special Functions)

Math 5202 General Topology II

continuity and topological equivalence, metric and normed spaces

Text: Lipschutz, S. (1965), Theory and Problems of General Topology, SOS.

(Chapter 7 Continuity and Topological Equivalence

Chapter 8 Metric and Normed Spaces)

Math 5203 Abstract Algebra II

symmetric, alternating, and dihedral groups, categories, products, coproducts, and free objects, direct products and direct sums, free groups, free products, generators and relations.

Text: Herstein, I.N. (1996), Abstract Algebra (3rd Edition). Prentice-Hall.

(Chapter 2 Groups [Sections 2.8, 2.9, 2.10, 2.11]

Chapter 3 The Symmetric Groups [Sections 3.1, 3.2, 3.3])

Math 5204 Hydrodynamics I

axisymmetric flow, Stoke’s stream function, some two-dimensional flows, general motion of cylinder

Text: Chorlton, F. (1967), Textbook of Fluid Dynamics, Van Nostrand.

(Chapter 4 Three Dimensional Flows [Section 4.5]

Chapter 5 Some Two-Dimensional Flows)

Math 5205 Numerical Analysis II

numerical methods in linear algebra, numerical methods for differential equations.

Text: Kreyszig, E (2011), Advanced Engineering Mathematics, Tenth Edition, John Wiley.

(Chapter 20 Numeric in Linear Algebra

[Sections 20.5, 20.6, 20.7, 20.8, 20.9]

Chapter 21 Numerics for ODEs and PDEs)

Math 5206 Qualitative Theory of Ordinary Differential Equations I

systems of differential equations, linear systems with an introduction to phase space analysis.

Text: Brauer.F.and Nohel, J. A.(1969), The Qualitative Theory of Ordinary Differential

Equations: An Introduction, W. A. Benjamin, Inc.

(Chapter 1 System of Differential Equations

Chapter 2 Linear Systems with an Introduction to Phase Space Analysis)

SECOND SEMESTER

Math 5207 Analysis VI

continuous transformations of metric spaces, euclidean spaces, continuous functions of several real variables, partial derivatives, linear transformations and determinants, the inverse function theorem, the implicit function theorem, functional dependence.

Text: Rudin, W. (1974), Principle of Mathematical Analysis, McGraw-Hill.

(Chapter 9 Functions of Several Variables)

Math 5208 General Topology III

separation axioms, compactness, concept of product topology and examples.

Text: Lipschutz. S. (1965), Theory and Problems of General Topology, SOS,

McGraw-Hill.

(Chapter 10 Separation Axioms

Chapter 11 Compactness

Chapter 12 Product Spaces Section: Product topology)

Math 5209 Abstract Algebra III

free abelian groups, finitely generated abelian groups, the Krull-Schmidt theorem, the action of a group on a set, the Sylow theorems.

Text: Herstein, I.N. (1996), Abstract Algebra (3rd Edition). Prentice-Hall.

(Chapter 3 Ring Theory [Sections 4.1, 4.2, 4.3, 4.4])

Math 5210 Hydrodynamics II

two-dimensional vortex motion, water waves

Texts: [1] Acheson, D. J. (1990), Elementary Fluid Dynamics, Oxford University

Press.

[2] O’Neill, M. E., Chorlton, F. (1986), Ideal and Incompressible Fluid Dynamics, Ellis, Harwood Ltd.

(Chapter 8)

Math 5211 Graph Theory

graphs and subgraphs, trees

Text: Bondy, J. A., and Murtyh, U. S. R (1984), Graph Theory with Applications,

Springer- Verlag.

(Chapter 1 Graphs and Subgraphs

Chapter 2 Trees)

Math 5212 Qualitative Theory of Ordinary Differential Equations II

existence theory, stability of linear and almost linear systems

Text: Brauer. F. and Nohel, J. A. (1969), The Qualitative Theory of Ordinary

Differential Equations: An Introduction, W. A. Benjamin, Inc.

(Chapter 3 Existence Theory

Chapter 4 Stability of Linear and Almost Linear Systems)

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