- B.Sc. / B.Sc. (Hons) in Mathematics
- M.Sc. in Mathematics

## Mathematics

#### Programmes Offered

## Curriculum

#### B.Sc.in Mathematics

#### B.Sc. (Honours) in Mathematics

#### M.Sc. in Mathematics

### Course Descriptions

*Math 1101: Algebra and Alalytic Geometry*

Algebra: Logic and sets, Polynomial functions, permutions, combinations, binomial theorem, mathematical induction.
Analytic Geometry: Lines, circle, conic section, translation of coordinate axes, rotation of coordinate axes.

*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.

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

Permutations, combinationsm,binomial theorem, mathematical inductions: lines, circles, conic sections; Limits, continuity, inverse trigonometric functions, exponential, logarithic, hyperbolic functions and their inverse functions, L' Hopital's rule, differentiation of inverse trigonometry.

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

Permutations, combinations,binomial theorem, variable and graph, frequency distributions, mean, mediam, mode and other measures of central tendency, standard deviation and other measures of dispersion, elementary probaility theory.

*Math 1103: Algebra and Analytical Solid Gemmetry*

Algebra: determinants, matrics,complex numbers.
Polar Coordinate System: polar coordinates, graphing in polar coordinate systm, ares and lengths in polar coordinates
Analytical Solid Gemoetry: three dimensional Cartesian coordinate system, lines, planes quadric, surfaces

*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 intergration, partial differentiation, ordinary differential equation of first order.

*Math 1003: Mathematics II ( for Science Students)*

Algebra: determinants, matrices, complex numbers;
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.

*Math 1004: Mathematics II ( for Arts Students)*

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

*Math 2101: Complex Variablea I*

Analytic functions, elementary functions,integrals, 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 differential, 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.

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

*Math 2104: Discrete Mathematics I*

Counting Methods and the Pigeonhole Principle: basic principle, permutations and combinations, algorithmas 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.

*Math 2105: Theory of Sets I*

Cardinal numbers, 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 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.

*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.

*Math 2002: Mathematics I ( for Arts Students)*

Methods of integration, binomial distribution, normal distribution, Possion distribution, method of least square, regression.

*Math 2107: Linear Algebra I*

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

*Math 2108: Ordinary Differential Equations*

General and particular solution: Slope fields and solution curve, Separable; Exact: Linear: Homgeneous; Reducible second order equations; Bernoulli equation.

*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.

*Math 2110: Discrete Mathematics II*

Graph Theory: introduction, paths and cycles, hamiltoniancycles and the traveling salesperson problems, a shortest path algorithm, representations of graphs, isomorphism of graphs, planargraphs.
Trees: introduction, terminology and characterizations of trees, Spanning tress, minimal spanning trees, binary trees.

*Math 2111: Theory of Sets II*

Well- ordered sets, ordinalnumbers, 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 season.

*Math 2004: Mathematics II ( for Science Students)*

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

*Math 2005: Mathematics II ( for Arts Students)*

Three-dimensional Cartesian coordinate system, lines, planes, quadric surfaces, spherical trigonometry.

*Math 3101: Analysis I*

Elements of set theory, numerical sequences and series.

*Math 3102: Linear Algebra II*

Linear maps and matrices, determinants.

*Math 3103: Differential Equations*

Power Series Methods,Laplace Transforms and Inverse Transforams, Elementary of series, Method of Frobenius, Bessel’s equation.

*Math 3104: Differential Geometry*

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

*Math 3105: Tensor Analysis*

Curvilinear coordinates, tensor analysis.

*Math 3106: Number Theory I*

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

*Math 3107: Analysis II*

Continuity, differentiation

*Math 3108: Linear Algebra III*

Scalar products and orthogonality, matrices and bilinear maps.

*Math 3109: Mechanics*

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

*Math 3110: Probability and Statistics*

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

*Math 3111: Complex Variablea II*

Conformal mapping, 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.

*Math 4101: Analysis III*

methods of mathematical research, the Riemann – Stieltjes integral.

*Math 4102: Numerical Analysis I*

Numerical methods in general, numerical methods in linear algebra.

*Math 4103: Linear Programming*

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

*Math 4104: Partial Differential equation*

First order partial di_erential equations[3]
2nd order linear equations in two independent variables
Fourier Series
The method of separation of variables
Canonical form of hyperbolic/ parbolic, elliptic;
Heat equation in one dimension
Wave equation in one dimension

*Math 4105: Stochastics Process I*

Conditional probability and conditional expectation, Markov chains.

*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.

*Math 4107: Analysis IV*

Sequences and series of functions.

*Math 4108: General Topology I*

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

*Math 4109: Abstract Algebra I*

Definitions and examples of groups, some simple remarks, subgroups, Lagrange’s theorem, homomorphisms and normal subgroups, factor groups, the homomorphism theorems.

*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.

*Math 4111: Stochastics Process II*

Markov chains, the exponential distribution and Poisson process.

*Math 4112: Integer Programming*

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

*Math 3201: Analysis I*

Elements of set theory, numerical sequences and series.

*Math 3202: Linear Algebra II*

Linear maps and matrices, determinants

*Math 3203: Differential Equations*

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

*Math 3204: Differential Geometry*

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

*Math 3205: Tensor Analysis*

Curvilinear coordinates, tensor analysis.

*Math 3206: Number Theory I*

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

*Math 3207: Analysis II*

Continuity, differentiation

*Math 3208: Linear Algebra III*

Scalar products and orthogonality, matrices and bilinear maps.

*Math 3209: Mechanics*

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

*Math 3210: Probability and Statistics*

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

*Math 3211: Complex Variablea II*

Conformal mapping, 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 squares.

*Math 4201: Analysis III*

methods of mathematical research, the Riemann – Stieltjes integral.

*Math 4202: Numerical Analysis I*

Numerical methods in general, numerical methods in linear algebra.

*Math 4203: Linear Programming*

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

*Math 4204: Partial Differential equations*

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

*Math 4205: Stochastics Process I*

Conditional probability and conditional expectation, Markov chains.

*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.

*Math 4207: Analysis IV*

Sequences and series of functions.

*Math 4208: General Topology I*

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

*Math 4209: Abstract Algebra I*

Definitions and examples of groups, some simple remarks, subgroups, Lagrange’s theorem, homomorphisms and normal subgroups, factor groups, the homomorphism theorems.

*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.

*Math 4211: Stochastics Process II*

Markov chains, the exponential distribution and Poisson process.

*Math 4212: Integer Programming*

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

*Math 5201: Analysis V*

Power series, the exponential and logarithmic functions, the trigonometric functions, fourier series.

*Math 5202: General Topology II*

Continuity and topological equivalence, metric and normed spaces.

*Math 5203: Abstract Algebra II*

Cauchy’s theorem, Direct products, finite abelian groups, conjugacy and Sylow’s theorem, preliminaries on symmetric groups, cycle decompositions, odd and even permutations.

*Math 5204: Hydrodynamics I*

Axisymmetric flow, Stoke’s stream function, some two – dimensional flows, general motion of cylinder.

*Math 5205: Numerical Analysis II*

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

*Math 5206: Qualitative Theory of Differential Equations I*

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

*Math 5207: Analysis VI*

Continuous transformations of metric space, euclidean spaces, continuous functions of serversl real variables, partial derivatrives, linear transformations and determinants, the inverse functions theorem, the implicit functions theorem, functional dependence.

*Math 5208: General Topology III*

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

*Math 5209: Abstract Algebra III*

Definitions and examples of rings,some simple results on ring, ideals, homomorphisms, and quotient rings,maximal ideals.

*Math 5210: Hydrodynamics II*

Vortex motion: Vorticity, vortex line, vortex tube and vortex filament, rectilinear vortices, two vortex filaments, vortex pair, vortex doublet, motion of any vortex, image of a vortex filament in aplane, Vortex inside an infinite circular cylinder, vortex outside a circular cylinder, image of a vortex outside/ inside a circular cylinder, vortex rows, Karman vortex street, Rankine’s combined vortex.
Waves: Mathematical representation of a wave motion, standing or stationary waves, types of liquid waves, surface waves, the energy of progressive waves, the energy of stationary waves, wave at the interface of two liquids, waves at the interface of two liquids with upper surface free, group velocity.

*Math 5211: Graph Theory*

Graphs and subgraphs, trees

*Math 5212: Qualitative Theory of Ordinary Differential Equations II*

existence theory, stability of linear and almost linear systems

*Math 611: Analysis I*

Integration: concept of measurability, simple functions, elementary properties of measures, intergration of positive functions, lebesgue’s monotone convergence theorem, integration of complex functions, lebesgue’s dominated convergence theorem.
Positive Borel Measure: Riesz representation theorem, regularity properties of Borel measure, Lebesgue measure, continulity properties of messurable functions.
Lp –spaces: convex functions and inequalities, the Lp-spaces, approximation by continuous functions,

*Math 612: Abstract Algebra*

Ring: Polynomial ring, polynomials over the rationals, field of quotients of an integral domain.
Fields: Examples of fields a brief excursion into vector spaces, field extensions.

*Math 613: Differential Equations I*

(a) Qualitative Theory of Ordinary Differential Equations
Lyapunov’s second method, applications of ODE
(b) Dynamical Systems
(1) Linear differential systems
Case of contast coefficients, Existence and uniqueness theorem in the general case, resolvent.
(2) Nonlinear differential systems
Analytical aspect: existence and uniqueness theorem, maximal solutions, estimations of the time of existence, Gronwell lemma.
Geometric aspect: flow, phase portrait and qualitative study of differential systems, Poincare first return map, invariance submanifolds, Poincare–Bendixson theorem. Perturbationd 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 hyperbolic fixed poine. Hartman – Grobman theorem.

*Math 614: Discrete Mathematics*

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.

*Math 615: Numerical Analysis I*

Polynomial approximation, interpolation, quadrature formulas, solution of non linear equations, optimization.

*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 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.

*Math 617: Stochastics Process I*

Foundations of probability of theory, limit theorems, probability distributions, probability measures and spaces.

*Math 621: Analysis II*

Banach Space: Banach Spaces. Example: c1,co, , , , C (X) continuous Linear Trasformation.
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, , , 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.

*Math 622: Linear Algebra*

Eigenvectors and eigenvalues, polynomials and matrices, triangulations of matrics and linear maps

*Math 623: Differential Equations II*

(a) Partial Differential Equations
Integral curves and surface 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.
(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 submainfolds. Examples, Tangent buldle. Vector fields.Lie bracket. Lie group. Local geometry of a hypersurface in the Euclidian soace. First and second fundamental form. Gauss curvature. Egregium theorem.

*Math 624: Graph Theory*

Connectivity, Eular tours and Hamilton cycles.

*Math 625: Numerical Analysis II*

Nonlinear system of equations, explicit 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 fluid: 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.

*Math 627: Stochastics Process II*

Higher dimensional distributions and infinite dimensional distributions, stochastic process: Principle classes, Canonical representations of Gaussian process, multiple Markov Gaussian process.

*Math 628: PDE and Approximations*

Introduction to the study of elliptic boundary value problems (modeling, mathematical analysis in the ID case) of parabolic (heat equation) and hyperbolic (wave equation) problems. Introduction to the finite difference method for these 3 (model) problems and numerical simulations.

*Math 629: Applied Probability and Statisstics*

Markov chain, the random counter part of the recurise 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 furnish quantitative methods to use these models for concete applications.

#### Dr. Ye Ye Mar

Professor(Head)

#### Dr. Than Than Oo

Professor

#### Dr. Myo Myo Hla

Associate Professor

#### Dr. Aye Aye Min

Associate Professor

#### Dr. Thein Swe Myint

Associate Professor

__Lecturers__

- U Hlaing Win
- Daw Khin Myo Wai
- Daw Yi Yi Khaing
- Daw Htay Htay Khaing
- Daw Thuzar
- Daw Yu Yu Htar
- Daw Myat Thit Soe
- Daw May Nwet Zin
- Dr. Khin Mar Aye
- Daw Thae Nandar Khaing
- Dr. Chaw Su Linn
- Daw Thet Thet Mar
- Daw Zin Zin Yu
- Dr. Mya Hnin Si
- Dr. Zinmar Htet

__Assistant Lecturers__

- Daw Zarni Thein
- U Saw Nay Wah

__Tutor__

- Dr. Cho Zin Latt

*Mission*

- Collecting technical information, doing researches, distributing information, applying and reviewing for the society
- Producing intellectually wise, discerning and responsible human resources

*Vision*

Bago University’s main goal is to produce highly qualified, competent, undergraduate human resources for the purpose of building a modern, developed nation.

*Contact*

**Bago University**

Yangon-Mandalay Highway Road, (8/9) ward, Oakthar Myothit, Bago, Bago Region, Myanmar.

**Tel:**(+95) 052 2230288