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

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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

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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

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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

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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

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Math 208

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.

Prerequisite

Math 102

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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

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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

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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 

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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

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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

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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

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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 or MATH 208

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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 208

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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

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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

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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 CIE 301 (old code: CISE 301)

Prerequisite

Math 201

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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

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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

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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

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MATH 427

Number Theory

(3-0-3)

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

Prerequisite

MATH 210 or Senior Standing

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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

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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

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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 208

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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 225 or Math 208

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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 208

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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}

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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 repeat for a maximum of three credit hours total.

Prerequisite

Senior Standing, Permission of the Department Chairman upon recommendation of the instructor

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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