# Course Listings

**MATH 91 - Pre-calculus:**

Set theory. Real numbers. Inequalities. Straight lines and circles. Trigonometry. Complex Numbers.

**MATH 100 - Elementary Mathematics:**

Set theory. Real numbers. Inequalities. Straight lines and circles. Complex Numbers. Systems of linear equation, Elementary matrices and their applications.

**MATH 101 - Calculus I:**

Limits, Continuous functions. The derivative, formulas of differentiation, differentials. Related rates, Extrema. Rolle’s and mean value theorems. Graph sketching Optimization. Indefinite integrals, definite integrals, fundamental theorem of calculus. Applications of definite integrals to find area and volume.

**MATH 102 - Calculus II:**

Logarithmic and exponential functions. Inverse trigonometric and hyperbolic functions. Techniques of integration. Indeterminate forms and improper integrals. Conic sections. Plane curves and polar coordinates.

**MATH 103 - Biomathematics (Only for Biology students):**

Functions and their graphs, the derivative, applications of the derivative, simple integrations, law of growth and decay.

**MATH 105 - Applied Calculus (Only for students from the College of women):**

Trigonometry, analytical geometry in two dimensions, linear equations, functions and their graphs, derivatives and its geometric interpretation, simple integration and its applications.

**MATH 115 - Finite Mathematics (Non Mathematics Major) :**

Algebra of Sets. Simple coordinate systems and graphs. Geometric approach to linear programming. Basic ideas of simplex method. Probability and applications to Medical Sciences. Statistics.

**MATH 211 - Calculus III:**

Sequences, infinite series, partial differentiation and its applications. Multiple integrals and its applications, Green's and Divergence theorems.

**MATH 226 - Euclidean Plane Geometry:**

Historical background, fundamentals of plane Euclidean geometry, triangles, quadrilaterals, circles, locus problems, geometric constructions, transformations, symmetry, analytic geometry.

**MATH 240 - Ordinary Differential Equations:**

Origin of differential equations in science. Differential equations of first order. Linear differential equations of higher order. Power series and Laplace transform methods. Applications.

**MATH 250 - Introduction to the Foundations of Mathematics :**

Algebra of propositions, mathematical induction, operations on sets, binary relations, equivalence relations and partitions, denumerable sets, Cardinal numbers, partial order. Boolean algebra.

**MATH 261 - Introduction to Abstract Algebra I:**

Groups: definition and elementary properties. Subgroups. Cyclic groups. Factor groups and homomorphisms. Ring and fields: definitions and elementary properties.

**MATH 262 - Introduction to Abstract Algebra II:**

Rings and fields. Integral domains. Fermat's and Eulers theorems. The field of quotients of an integral domain, rings of polynomials, factorization of polynomials over a field. Non commutative examples. Homomorphism and factor rings. Prime and Maximal ideals. Principal ideal domains, definition of unique factorization domains. Principal ideal domains are unique factorization domains. Euclidean domains. Introduction to extension fields.

**MATH 313 - Applied Mathematical Analysis:**

Gamma and Beta functions. Hypergeometric functions. Bessel functions. Orthogonal polynomials. Fractional calculus.

**MATH 315 - Mathematical Electromagnetic Theory:**

Electrostatics. Dielectrics. Magnetostatics. Induced magnetism. Steady currents. Maxwell's equations. Quasi-stationary Fields. Electromagnetic waves. Notion of electric charges.

**MATH 316 - Elements of Mathematical Modelling:**

The basic idea of mathematical modeling of real world problems. Modeling using graphs; proportionality; fitting, and optimization, experimental modeling. Dimensional analysis and similitude, simulation modeling; modeling using the derivative, interactive dynamic systems.

**MATH 325 - Combinatorics:**

Pairing. Stirling's formula. Recurrence relations and generating functions. The inclusion exclusion principle, Boolean functions. Switching circuits. Applications to digital computers.

**MATH 326 - Graph Theory:**

Graphs and digraphs. Connectivity. Eulerian and Hamiltonian problems. Trees. Planarity. Matrix representation. Shortest path problem. Colouring. Applications.

**MATH 327 - Theory of Numbers:**

The fundamental theorem of arithmetic. Congruences. The theorems of Fermat and Wilson. Quadratic residues. The quadratic reciprocity law. Diophantine equations.

**MATH 329 - Introduction to Real Analysis:**

Topology of the real number system. Functions. Limits of functions. Limit theorems for functions. Continuity. Types of discontinuity. Continuity theorems. The derivative. Roll's and the mean value theorems. Riemann integral. Properties of the definite integral. Sequences of real numbers. Theorems on sequences. Infinite series. Continuity of functions of several variables.

**MATH 330 - Functions of Complex Variables:**

The algebra of complex numbers. Analytic functions, integration of complex functions. Cauchy's theorem and the Cauchy integral formula. Power series, Laurent series, Calculus of residues.

**MATH 340 - Theory of Differential Equations :**

Series solutions near regular singular points. Nonlinear differential equations. Systems of linear differential equations. The theory of first-order equations. Stability of equilibria. Boundary value problems.

**MATH 343 - Classical Mechanics:**

Rectilinear motion of a particle with variable acceleration. Simple harmonic motion. Damped and forced oscillations. Motion of a particle in a plane using Cartesian, polar and intrinsic coordinates. Central orbits. Kepler's laws. Moments and products of intertia. Kinetic energy and angular momentum of rigid body. The general equations of motion. Two-dimensional problems. Generalized coordinates and Lagrange equations of motion. Small oscillations.

**MATH 345 - Abstract Algebra:**

Homomorphism. Isomorphism theorems. Class equation. Cauchy theorem for finite groups. Direct sums and Jordan Holder theorem. Structure of groups of orders up to 8. Integral domains. Division rings. Subrings. Ideals. Fields of quotients. Polynomial rings. Eienstein criteria.

**MATH 346 - Group Theory :**

Permutation groups, conjugate classes in Sn, the simplicity of An, Direct products, the fundamental theorem of finite abelian groups. The Sylow theorems, groups of order 2p, pq and orders up to 10. Jordan-Holder theorem, solvable groups.

**MATH 352 - Numerical Analysis:**

Numerical solution of nonlinear equations. Numerical solution of linear algebraic systems: direct and iterative methods. Polynomial interpolation. Least-squares approximation. Numerical integration.

**MATH 354 - Methods in Applied Mathematics:**

Vector calculus, Asymptotic expansions, Calculus of variations, Applications of ODEs to sciences.

**MATH 355 - Operational Mathematics:**

Equation of Laplace, Fourier and Hankel transforms, Boundary value problems, applications.

**MATH 361 - Theoretical Fluid Dynamics:**

Equations of motion. Two-dimensional motion. Streaming motions. Sources and sinks. Theorem of Schwarz and Christoffel. Stoke’s stream function and three-dimensional motion.

**MATH 362 - Computational Linear Algebra:**

Systems of linears equations: direct methods: elimination and its matrix analysis, special structured systems, Thomas and Choleski algorithms, iterative methods: Jacobi and Gauss-Seidel, SOR, optimal SOR. The linear least-squares problem: normal equations, QR-and SVD-decompositions, error analysis. Matrix eigenvalue problem: power method, deflation, Given and Householder orthogonal transformations, the QR-algorithm, reduction to Hessenberg form; eigenvalue problem for symmetric matrices.

**MATH 363 - Advanced Linear Algebra :**

Vector spaces, subspaces, linear independence. Linear transformations and their representations via matrices. Kernel and range, dimension theorem and isomorphism theorem. Direct sums of vector spaces. Duality and the interpretation of the adjoint matrix. Characteristic polynomiaI and trace. An introduction to the theory of linear transformations, polynomials in a linear transformation, eigenspaces, cyclic spaces, nilpotent and semisimple transformations. Applications to canonical forms.

**MATH 370 - Geometry I:**

Axioms for the Euclidean plane, transformations in the plane, distance, isometries, dilatations, similarities, inversions. Projective plane. Desargues’ theorem. Harmonic sequences. Pappus theorem. Homogeneous coordinates.

**MATH 375 - Introduction to Topology:**

Metric spaces: open and closed sets, continuous functions, product of metric spaces, complete metric spaces. Topological spaces: open and closed sets, continuous functions, product of topological spaces, connectedness, compactness, separation and countability axioms.

**MATH 401 - Analysis:**

Sequences and series of functions, the Weierstrass approximation theorem, Picard's existence theorem and equicontinuous families of functions. Lebesgue measure and the Lebesgue integral of bounded and unbounded functions. Convergence theorems. Fourier series.

**MATH 402 - Set Theory:**

Formation of set theory. Propositional calculus. Quantification. Calculus of classes. Set of natural numbers. Relations and maps. Axiom of choice. Zorn’s dilemma. Well-ordering principle. Ordinals and cardinals.

**MATH 403 - Advanced Linesr Algebra:**

Linear transformations, change of bases matrices, dual spaces, irreducible polynomials, vector spaces as F[x]

modules, characteristic polynomial decomposition companion matrices, rational and Jordan canonical forms

**MATH 404 - Analysis of Functions of Several Variables:**

Sets and functions in R^n, continuity and compactness, differentiability, Jacobian matrix and gradient, inverse and implicit function theorems, extrema, Hessian matrix, Taylor’s formula, integration in R^n, vector fields, integral theorems.

**MATH 406 - Algebraic Coding Theory:**

Polynomial rings, field extensions, finite fields, computations in finite fields, irreducible polynomials. Linear machines, shift registers, codes over GF(g), Hamming codes, cyclic codes, BC codes, Reed-Solomon codes, encoding cyclic codes, BCH decoding as an FSR problem, algorithmic BCH decoding (only illustrations).

**MATH 407 - Tensor Analysis:**

Orthogonal Transformations. Cartesian tensors. Generalised N-dimensional spaces. Contravariant and covariant tensors. Tensor algebra. Tensor densities. Covariant differentiation. The Riemann Christoffel curvature tensor. Metrical connection. The Laplacian. Einstein tensor. Geodesics.

**MATH 408 - Topology:**

Topological spaces, product spaces, quotient spaces, convergence, separation axioms, first and second countable spaces, compactness, connectedness, fundamental groups.

**MATH 409 - Geometry II:**

Analytic projective geometry. Affine, hyperbolic and elliptic geometries.

**MATH 410 - Theory of Relativity:**

Review of tensor analysis. The principle of special relativity. Lorentz transformations. minkowski geometry. Relativistic mechanics and electrodynamics. Introduction to the general theory of relativity. The principle of equivalence, metric in a gravitational field, Einstein's law of gravitation. Shwarzchild solution.

**MATH 412 - Theory of Elasticity:**

Analysis of stress and strain. Principal axes. Compatibility equations. Generalised Hooke's law.Beltrami-Michell equations. The fundamental boundary value problems. Applications.

**MATH 415 - Partial Differential Equations:**

First and second order linear partial differential equations. The method of characteristics. Separation of variables and its application to linear partial differential equations. Fourier series. Sturm-Liouville theory, Green’s functions.

**MATH 416 - Mathematical Modelling:**

Selection of mathematical models based on differential equations and linear systems. Analysis of models, development of solution algorithms and sensitivity of models to perturbations in data. Computer simulation for visualization and interpretation of physical phenomenon governed by these models.

**MATH 438 - Applications of Graph Theory:**

Network flows. Timetable. Matching. Travelling salesman problem. Some computer algorithms.

**MATH 440 - Integral Equations:**

Basic concepts. Fredholm and Volterra integral equations. Integral transformations. Approximation methods.

**MATH 450 - Ring Theory:**

Field quotients. Ideals, prime ideals, Direct sums. Modules, submodules, homomorphisms, free modules, vector spaces. Subdirect sums of rings, primitive rings, prime radical, density-Theorem (without proof), Jacobson radical, chain conditions, Wedderburn Artin Theorem.

**MATH 454 - Advanced Methods in Applied Mathematics:**

Fourier transform and inversion techniques, applications in signal processing. Laplace and Mellin transforms. Discrete linear systems and filters, discrete Fourier transforms. Fast Fourier transform. Perturbation methods.

**MATH 468 - Numerical Ordinary Differential Equations:**

Numerical integration.Some basic rules. Gaussian and Romberg integration. Numerical solution of ordinary differential equations. Initial value problems. One step and multistep methods (convergence, absolute stability). Boundary value problems.

**MATH 469 - Numerical Partial Differential Equations:**

Numerical solution of partial equations selected topics to be chosen from: Theory of approximation, variational methods, Eigenvalue problems, further linear algebra, integral equations.

**MATH 470 - Calculus of Variations:**

Fundamental problems. Euler’s equation. Boundary conditions. Second variation. Isoperimetrical problem. Transformations. Variational principles. Applications.

**MATH 480 - Differential Geometry:**

Curves in space. Frenet-Serrt formulas. Curvature and torsion. Surfaces. First and second fundamental forms. Intrinsic geometry.

**MATH 481 - Functional Analysis:**

Metric spaces, Properties of normed spaces, Inner product spaces. Bounded and continuous linear operators, Linear functionals, Hilbert adjoint operators, Fundamental theorems for normed and Banach spaces.

**MATH 485 - Finite Planes**

Finite incidence structures. Incidence matrices. Block design. Projective and affine planes. Coolineations. Coordination and ternary rings. The Bruck-Ryser theorem.

**MATH 495 - Mathematical Logic**

Propositional calculus. First order logic. First order recursive arithmetic. Arithmatization of syntax. The incompleteness theorem and other applications.