## BSc Degree in Mathematics - Course Description

Degree Plan | Requirements | Course Descriptions

 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) 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 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 to linear models of first and second order. ​ ​ 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. Nonhomogeneous differential equations. Methods of undetermined coefficients and variation of parameters. Systems of differential equations. Non-homogeneous systems. Applications to linear models of first and second order.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. Applications as mini Projects. ​ ​ 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 openproblems. 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 336 Mathematical Models in Biology (3-0-3) Growth models, Single species and interacting population dynamics. Dynamics of infectious diseases. Modeling enzyme dynamics. Some fatal diseases models. Programing software for numerical simulations. ​ ​ Prerequisite 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 (2-2-3) 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; Using computer software as a computational platform. Note: Not to be taken for credit with CISE 301 Prerequisite Math 201 ​ ​ ​ 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 405 Learning from Data (3-0-3) Basic vector and matrix operations, Factorizations, Basic Probability Theory, Inference, LeastSquare Estimation, Maximum Likelihood Estimation, Gradient Descent, Linear Regression and Neural Networks. ​ ​ Prerequisite MATH 102 or MATH 106 and STAT 201 or 212, or 319 or ISE 205, and ICS 103 ​ ​ ​ 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 436 Discrete Models (3-0-3) Difference equations and discrete dynamical systems, linear and nonlinear models, linear and nonlinear systems, stability and well-posedness, models and numerical experiments (from different fields of science and engineering). ​ ​ Prerequisite Math 202 or Math 208 ​ ​ ​ MATH 437 Partial Differential Equations (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. Nonhomogeneous differential equations. Methods of undetermined coefficients and variation of parameters. Systems of differential equations. Non-homogeneous systems. ​ ​ 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 ​ 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 inclusionexclusion, 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 ​ 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 477 Foundation of Scientific Computing (3-0-3) Concepts of numerical mathematics, approximation tools, system of equations, least squares, numerical differentiation and integration, quadrature on different geometries, Runge-Kutta and multistep methods for initial value problems, finite difference methods for initial and boundary value problems. Applications to steady-state and time-dependent problems.  ​ Prerequisite Math 102 ​ ​ ​ MATH 481 Computational Inverse Problem (3-0-3) Regression, Least squares, Maximum likelihood estimation, Rank deﬁciency, Ill-conditioning, Generalized and Truncated SVD solutions, regularizations (Tikohonov, spectral filtering), proximal and primal-dual iterative schemes, Nonlinear inverse (gradient-based and global optimization methods), OCCAM method. ​ ​ Prerequisite Math 405 or consent of the instructor ​ 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 repeatefor 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 ​