## B. Sc. Degree In Mathematics - Course Descriptions

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 131           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 132      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 201

MATH 232          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.
Prerequisite: MATH 102

MATH 260       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 280
Prerequisite: MATH 102

MATH 280       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.
Corequisite: MATH 201

MATH 301         Methods of Applied Mathematics         (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 202 or MATH 260

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 280 or MATH 301
Prerequisite: MATH 201

MATH 305             evelopment 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 132

MATH 311          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 232

MATH 321          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 322           Quantitative Methods for Actuaries         3-0-3)
Algorithms; simplex and dual method; linear and quadratic programming; Solution of non-linear equations; finite differences; cubic splines; individual risk models; life tables. Floating-point arithmetic and error analysis. Interpolation. Polynomial interpolation. Numerical integration and differentiation. Data fitting. Solution of linear algebraic systems. Initial and boundary value problems of ordinary differential equations. This course section is designed to meet the Actuarial Science course degree requirement.
Note: Not to be taken for credit with Math 321 or CISE 301
Prerequisite: MATH 201, ICS 102 or ICS 103

MATH  330     Euclidean and Non-Euclidean Geometry      3-0-3)
Axiomatic approach to Euclidean geometry. Use of logic in mathematical reasoning. Hilbert’s formulation. Removal of the parallel axiom. The discovery of non-Euclidean geometries. Independence of the parallel postulate. The question of the geometry of physical space. Geometric transformations and invariance under groups of transformations. Hyperbolic geometry.
Prerequisite: MATH 232

MATH  345        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 232

MATH 355        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 280

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

MATH  401          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 301

MATH  411           Advanced Calculus II           (3-0-3)
Theory of sequences and series of functions. Continuity and differentiability of functions of several variables. Partial derivatives. The Chain rule. Taylor’s theorem. Maxima and minima. Integration of functions of several variables. Convergence and divergence of improper integrals. Derivative of functions defined by improper integrals.
Prerequisite: MATH 311

MATH  412           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 411

MATH  421           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. The Urysohn lemma. The Urysohn metrization theorem. Complete metric spaces.
Prerequisite: MATH 311

MATH  425       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 260 or MATH 280 or MATH 302

MATH  430      Introduction to Complex Variables        (3-0-3)
Complex numbers and the complex plane. Arguments and roots, roots of unity. De Moivre’s theorem. Basic topological definitions. Analytic functions. Limits. Continuity. Differentiability. Cauchy-Riemann conditions. Elementary functions. Branch cuts. Convergence of complex series. Complex integration. Cauchy’s theorem. Cauchy’s integral formula. Morera’s and Liouville’s theorems. Taylor’s and Laurent’s series. Residues and poles. Rouche’s theorem. Fundamental theorem of algebra. Evaluation of improper integrals. Meromorphic functions. Basic concepts of conformal mapping.
Prerequisite: MATH 201

MATH  431      Introduction to Measure Theory and Fucntion Analysis          (3-0-3)
Lebesgue integrable functions. Fatou’s lemma. Dominated convergence theorem. Measurable functions. Measurable sets, non-measurable sets. Egoroff ’s theorem. Convergence in measure. Lp-spaces, Riesz-Fischer theorem, geometry of Hilbert spaces. Orthonormal sequences. Fourier series. Bounded linear functionals. Hahn- Banach theorem. Linear functionals on Hilbert and Lp-spaces.
Prerequisite: MATH 311

MATH  440       Differential Geometry         (3-0-3)
Manifolds in Rn and their orientability. Tensor fields. Curves in 3-dimensional Euclidean space: the Frenet frame and formulae, curvature and torsion, natural equations. Surfaces in 3-dimensional Euclidean space: the first and second fundamental forms, the classification of surfaces, the fundamental theorem.
Prerequisite: MATH 260 or MATH 280 or MATH 302

MATH  442            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

MATH  450         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 345

MATH  452       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 345

MATH  455            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 232 or Senior Standing

MATH  460      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 260 or MATH 280 or MATH 302

MATH  465          Ordinary Differential Equations        (3-0-3)
Existence, uniqueness and continuation of solutions to initial value problems: scalar, 1st order systems and linear systems. Linear systems: solution matrix, fundamental solution matrix. Variation of constants method. Phase space analysis. Autonomous systems. Definitions of Stability. Stability for linear and almost linear systems. Basic concepts of Liapunov’s method.
Prerequisite: MATH 202, MATH 280

MATH  470         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 301

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 321 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 321 or CISE 301

MATH  480      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: Junior Standing

MATH  485     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 301 or EE 207 or CISE 315

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. 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 301, MATH 311, MATH 321, MATH 345

MATH  495           Industrial Mathematics     (3-0-3)
Industrial and environmental problems. Theoretical foundations and computational methods involving ordinary and partial differential equations.
Prerequisite: MATH 301 or EE 207, MATH 321 or CISE 301

MATH  499     Topics in Mathematics        (1-3, 0, 1-3)
Variable contents. Open for Senior students interested in studying an advanced topic in mathematics with a departmental faculty member.
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.