## BSc Degree in Mathematics - Course Description

Degree Plan | Requirements | Course Descriptions

 MATH 001 Preparatory Mathematics I (3-1-4)

Concepts and manipulations in algebra. Introduction to concepts of calculus. Preparation for rigorous study of mathematics.

 MATH 002 Preparatory Mathematics II (3-1-4)

Concepts and manipulations in algebra. Trigonometry. Elementary analytic geometry. Introduction to concepts of calculus. Preparation for rigorous study of mathematics.

Prerequisite:  MATH 001 or its equivalent.

 MATH 101 Calculus  I (4-0-4)

Limits and continuity of functions of a single variable. Differentiability. Techniques of differentiation. Implicit differentiation. Local extrema, first and second derivative tests for local extrema. Concavity and inflection points. Curve sketching. Applied extrema problems. The Mean Value Theorem and applications.

Prerequisite: One year preparatory mathematics or its equivalent

 MATH 102 Calculus II (4-0-4)

Definite and indefinite integrals of functions of a single variable. Fundamental Theorem of Calculus. Techniques of integration. Hyperbolic functions. Applications of the definite integral to area, volume, arc length and surface of revolution. Improper integrals. Sequences and series: convergence tests, integral, comparison, ratio and root tests. Alternating series. Absolute and conditional convergence. Power series. Taylor and Maclaurin series.

Prerequisite: MATH 101

 MATH 105 Finite  Mathematics (3-0-3)

Linear equations and inequalities. Systems of linear equations. Basic material on matrices. Elementary introduction to linear programming. Counting techniques. Permutations and combinations. Probability for finite sample space. Basic concepts in statistics. Topics in the mathematics of finance.

Prerequisite:  One year preparatory mathematics or its equivalent

 MATH 106 Applied Calculus (3-0-3)

The derivative. Rules for differentiation. Derivative of logarithmic, exponential, and trigonometric functions. Differentials. Growth and decay models. Definite and indefinite integrals. Techniques of integration. Integrals involving logarithmic, exponential and trigonometric functions. Integration by tables. Area under a curve and between curves. Functions of several variables. Partial derivatives and their applications to optimization.

Prerequisite: One year preparatory mathematics or its equivalent

 MATH 201 Calculus III (3-0-3)

Polar coordinates, polar curves, area in polar coordinates. Vectors, lines, planes and surfaces. Cylindrical and spherical coordinates. Functions of two and three variables, limits and continuity. Partial derivatives, directional derivatives. Extrema of functions of two variables. Double integrals, double integrals in polar coordinates. Triple integrals, triple integrals in cylindrical and spherical coordinates.

Prerequisite: MATH 102

 MATH 202 Elements of Differential Equations (3-0-3)

First order and first degree equations. The homogeneous differential equations with constant coefficients. The methods of undetermined coefficients, reduction of order, and variation of parameters. The Cauchy-Euler equation. Series solutions. Systems of linear differential equations. Applications.

Prerequisite: MATH 102

 MATH 208 Introduction to Differential Equations & Linear Algebra (3-0-3)

Systems of linear equations. Rank of matrices. Eigenvalues and eigenvectors. Vector spaces, subspaces, bases, dimensions. Invertible matrices. Similar matrices. Diagonalizable matrices. Block diagonal and Jordan forms. First order differential equations: separable and exact. The homogeneous differential equations with constant coefficients. Wronskian. Non-homogeneous differential equations. Methods of undetermined coefficients and variation of parameters. Systems of differential equations. Non-homogeneous systems.

Note: Not to be taken for credit with MATH 202 or MATH 225

Prerequisite: MATH 102

 MATH 210 Introduction to Sets and Structures (3-0-3)

Elementary logic. Methods of proof. Set theory. Relations and functions. Finite and infinite sets. Equivalence relations and congruence. Divisibility and the fundamental theorem of arithmetic. Well-ordering and axiom of choice. Groups, subgroups, symmetric groups, cyclic groups and order of an element, isomorphisms, cosets and Lagrange's Theorem.

Note: Not to be taken for credit with ICS 253

Prerequisite: MATH 102

 MATH 225 Introduction to Linear Algebra (3-0-3)

Matrices and systems of linear equations. Vector spaces and subspaces. Linear independence. Basis and dimension. Inner product spaces. The Gram-Schmidt process. Linear transformations. Determinants. Diagonalization. Real quadratic forms.

Prerequisite: MATH 102

 MATH 302 Engineering  Mathematics (3-0-3)

Vector analysis including vector fields, gradient, divergence, curl, line and surface integrals, Gauss' and Stokes' theorems. Introduction to complex variables. Vector spaces and subspaces. Linear independence, basis and dimension. Solution of linear equations. Orthogonality. Eigenvalues and eigenvectors. Applications to systems of differential equations.

Note: Not to be taken for credit with MATH 225 or MATH 333

Prerequisite: MATH 201

 MATH 310 Logic and Set Theory (3-0-3)

The Propositional Logic, First-order predicate calculus. Truth and Models. Soundness and Completeness for Propositional Logic. Deduction. Models of Theories. Interpretations. Soundness and Completeness Theorems for first-order logic. The Compactness Theorem. Nonstandard models. Naive Set Theory. Zermelo-Fraenkel Axioms. Wellorders and Ordinal Numbers. ON as a proper class. Arithmetic of Ordinals. Transfinite  Induction and Recursion. Cardinality. Goodstein Sequences.

Prerequisite: MATH 210

 MATH 315 Development of Mathematics (3-0-3)

History of numeration: Egyptian, Babylonian, Hindu and Arabic contributions. Algebra: including the contributions of Al-Khwarizmi and Ibn Kura. Geometry: areas, approximation of , the work of Al-Toussi on Euclid's axioms. Analysis. The calculus: Newton, Leibniz, Gauss. The concept of limit: Cauchy, Laplace. An introduction to some famous old open problems.

Prerequisite: MATH 102 or MATH 106

 MATH  323 Modern Algebra I (3-0-3)

Review of basic group theory including Lagrange's Theorem. Normal subgroups, factor groups, homomorphisms, fundamental theorem of finite Abelian groups. Examples and basic properties, integral domains and fields, ideal and factor rings, homomorphisms. Polynomials, factorization of polynomials over a field, factor rings of polynomials over a field. Irreducibles and unique factorization, principal ideal domains.

Prerequisite: MATH 210 or (ICS 253, ICS 254)

 MATH 325 Linear Algebra (3-0-3)

Theory of vector spaces and linear transformations. Direct sums. Inner product spaces. The dual space. Bilinear forms. Polynomials and matrices. Triangulation of matrices and linear transformations. Hamilton-Cayley theorem.

Prerequisite: MATH 225

 MATH 333 Methods of Applied Mathematics I (3-0-3)

Special functions. Bessel's functions and Legendre polynomials. Vector analysis including vector fields, divergence, curl, line and surface integrals, Green's, Gauss' and Stokes' theorems. Sturm-Liouville theory. Laplace transforms. Fourier series and transforms. Introduction to partial differential equations and boundary value problems in rectangular, cylindrical and spherical coordinates.

Prerequisite: MATH 201, MATH 202 or MATH 208

 MATH 341 Advanced Calculus I (3-0-3)

The real number system. Continuity and limits. Uniform continuity. Differentiability of functions of one variable. Definition, existence and properties of the Riemann integral. The fundamental theorem of calculus. Sequences and series of real numbers.

Prerequisite: MATH 210 or ICS 253

 MATH  353 Euclidean and Non-Euclidean Geometry (3-0-3)

Classical Euclidean and non-Euclidean geometries. Matrix representations of transformations in R3. Isometries. Transformation and symmetric groups. Similarity and affine transformations.

Prerequisite: MATH 210

 MATH 371 Introduction to Numerical Computing (3-0-3)

Floating-point arithmetic and error analysis. Solution of non-linear equations. Polynomial interpolation. Numerical integration and differentiation. Data fitting. Solution of linear algebraic systems. Initial and boundary value problems of ordinary differential equations.

Note: Not to be taken for credit with CISE 301

Prerequisite: MATH 201, ICS 101 or ICS 102 or ICS 103

 MATH  399 Summer Training (0-0-2)

Students are required to spend one summer working in industry prior to the term in which they expect to graduate. Students are required to submit a report and make a presentation on their summer training experience and the knowledge gained. The student may do his summer training by doing research and other academic activities.

Prerequisite: ENGL 214, Junior Standing, Approval of the Department

 MATH  423 Modern Algebra II (3-0-3)

Finite and finitely generated Abelian groups. Solvable groups. Nilpotent groups. Sylow theorems. Factorization in integral domains. Principal ideal domains. Fields. Field extensions. Finite fields. An introduction to Galois theory.

Prerequisite: MATH 323

 MATH  424 Applied Algebra (3-0-3)

Boolean algebras. Symmetry groups in three dimensions. Polya-Burnside method of enumeration. Monoids and machines. Introduction to automata theory. Error correcting codes.

Prerequisite: MATH 323

 MATH  427 Number Theory (3-0-3)

Divisibility and primes. Congruences. Positive roots. Quadratic reciprocity. Arithmetic functions. Diophantine equations. Applications (e.g. cryptography or rational approximations).

Prerequisite: MATH 210 or Senior Standing

 MATH  432 Applied Matrix Theory (3-0-3)

Review of the theory of linear systems. Eigenvalues and eigenvectors. The Jordan canonical form. Bilinear and quadratic forms. Matrix analysis of differential equations. Variational principles and perturbation theory: the Courant minimax theorem, Weyl's inequalities, Gershgorin's theorem, perturbations of the spectrum, vector norms and related matrix norms, the condition number of a matrix.

Prerequisite: MATH 208 or MATH 225 or MATH 302

 MATH  433 Methods of Applied Mathematics II (3-0-3)

Introduction to linear spaces and Hilbert spaces. Strong and weak convergence. Orthogonal and orthonormal systems. Integral Equations: Fredholm and Volterra equations. Green's Function: Idea of distributions, properties of Green's function and construction. Any one of the following topics: Asymptotic Methods: Laplace method, Steepest descent method, Perturbation Theory: regular and singular perturbations, Integral Transforms: Fourier, Laplace, Mellin and Hankel transforms.

Prerequisite: MATH 333

 MATH  434 Calculus of Variations and Optimal Control (3-0-3)

Introduction to the calculus of variations. Euler-Lagrange, Weierstrass, Legendre and Jacobi necessary conditions. Formulation of optimal control problems. Bolza, Mayer and Lagrange formulations. Variational approach to optimal control. Pontryagin maximum principle.

Prerequisite: MATH 202 or MATH 208

 MATH  435 Ordinary Differential Equations (3-0-3)

First order scalar differential equations. Initial value problems. Existence, uniqueness, continuous dependence on initial data. Linear systems with constant coefficients. The exponential matrix. Asymptotic behavior of linear and almost linear systems. Two dimensional autonomous systems. Critical points and their classifications. Phase plane analysis. Introduction to the theory of Lyapunov stability.

Prerequisite: (MATH 202, MATH 225) or MATH 208

 MATH  437 Partial  Differential  Equations (3-0-3)

First order quasilinear equations. Lagrange method and Characteristics. Classification of linear second order PDEs. Brief review of separation of variables. The one dimensional wave equation: its solution and characteristics. Cauchy problem for the wave equation. Laplace's equation: The maximum principle, uniqueness theorems. Green's function. Neumann's function. The heat equation in one dimension.

Prerequisite: MATH 333

 MATH  441 Advanced Calculus II (3-0-3)

Theory of sequences and series of functions. Real functions of several real variables: limit, continuity, differentiability. Taylor's theorem. Maxima and minima, Lagrange multipliers rule. Elementary notion of integration on RN. Change of variables in multiple integrals, Fubini's theorem. Implicit and inverse function theorems. Convergence and divergence of improper integrals- Differentiation under the integral sign.

Prerequisite: MATH 341

 MATH  443 Advanced Calculus III (3-0-3)

Functions of bounded variation. The Riemann-Stieltjes integral. Implicit and inverse function theorems. Lagrange multipliers. Change of variables in multiple integrals. Vector functions and fields on  Rn. Line and surface integrals. Green's theorem. Divergence theorem. Stokes' theorem.

Prerequisite: MATH 441

 MATH  445 Introduction to Complex Variables (3-0-3)

The theory of complex analytic functions, Cauchy's integral theorem, contour integrals, Laurent expansions, the residue theorem with applications, evaluation of improper real integrals and series,  conformal mappings.

Prerequisite: MATH 201

 MATH  451 Differential Geometry (3-0-3)

Curves in 3-dimensional Euclidean space: the Frenet frame and formulae, curvature and torsion, natural equations. Surfaces in 3-dimensional Euclidean space: tangent plane, first fundamental form and isometries, second fundamental forms, normal and principal curvatures, Gaussian and mean curvatures, geodesics. Geometry of the sphere and the disc (with Poincare metric).

Prerequisite: MATH 208 or MATH 225 or MATH 302

 MATH  453 Introduction to Topology (3-0-3)

Topological Spaces: Basis for a topology, The order topology. The subspace topology. Closed sets and limit points. Continuous functions. The product topology, The metric topology. Connected spaces. Compact spaces. Limit point compactness. The countability axioms. The separation axioms. Complete metric spaces.

Prerequisite: MATH 341

 MATH  463 Combinatorics (3-0-3)

Enumerative techniques, Recurrence relations, Generating functions, Principle of inclusion-exclusion, Introduction to graph theory, selected topics (e.g. Ramsey Theory, Optimization in graphs and networks, Combinatorial designs, Probabilistic methods.)

Prerequisite: MATH 201

 MATH  467 Graph Theory (3-0-3)

Graphs and digraphs. Degree sequences, paths, cycles, cut-vertices, and blocks. Eulerian graphs and digraphs. Trees, incidence matrix, cut-matrix, circuit matrix and adjacency matrix. Orthogonality relation. Decomposition, Euler formula, planar and nonplanar graphs. Menger's theorem. Hamiltonian graphs.

Prerequisite: MATH 208 or MATH 225 or MATH 302

 MATH  471 Numerical Analysis I (3-0-3)

Floating-point, round-off analysis. Solution of linear algebraic systems: Gaussian elimination and LU decomposition, condition of a linear system, error analysis of Gaussian elimination, iterative improvement. Least squares and singular value decomposition. Matrix eigenvalue problems.

Prerequisite: MATH 371 or CISE 301

 MATH  472 Numerical Analysis II (3-0-3)

Approximation of functions: Polynomial interpolation, spline interpolation, least squares theory, adaptive approximation. Differentiation. Integration: basic and composite rules, Gaussian quadrature, Romberg integration, adaptive quadrature. Solution of ODEs: Euler, Taylor series and Runge-Kutta methods for IVPs, multistep methods for IVPs, systems of higher-order ODEs. Shooting, finite difference and collocation methods for BVPs. Stiff equations.

Prerequisite: MATH 371 or CISE 301

 MATH  474 Linear & Nonlinear Programming (3-0-3)

Formulation of linear programs. Basic properties of linear programs. The simplex method. Duality. Necessary and sufficient conditions for unconstrained problems. Minimization of convex functions. A method of solving unconstrained problems. Equality and inequality constrained optimization. The Lagrange multipliers theorem. The Kuhn-Tucker conditions. A method of solving constrained problems.

Prerequisite: MATH 201

 MATH  475 Wavelets and Applications (3-0-3)

Wavelets. Wavelet transforms. Multiresolution analysis. Discrete wavelet transform. Fast wavelet transform. Wavelet decomposition and reconstruction. Applications such as boundary value problems, data compression, etc.

Prerequisite: MATH 225 or MATH 302

 MATH  490 Seminar in Mathematics (1-0-1)

This course provides a forum for the exchange of mathematical ideas between faculty and students under the guidance of the course instructor. Students are expected to do research on a mathematical problem of their choice or the instructor's. The instructor arranges weekly presentations by himself, other faculty members and/or students, of lectures or discussions on topics or problems of general interest. The course culminates in the presentation by each student of at least one written report on a selected topic or problem, reflecting some independent work and evidence of familiarity with the mathematical literature. With the permission of the instructor, students may work with other faculty members in the preparation of written reports.

Prerequisite: Any two of {MATH 323, MATH 333, MATH 341, MATH 371}

 MATH  498 Topics in Mathematics I (1-3, 0, 1-3)

Variable contents. Open for Senior students interested in studying an advanced topic in mathematics.

Note: May be repeated for a maximum of three credit hours total.

Prerequisite: Senior Standing, Permission of the Department Chairman upon recommendation of the instructor.

 MATH  499 Topics in Mathematics II (1-3, 0, 1-3)

Variable contents. Open for Senior students interested in studying an advanced topic in mathematics.

Note: May be repeated for a maximum of three credit hours total.

Prerequisite: Senior Standing, Permission of the Department Chairman upon recommendation of the instructor.