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.

**Pre-Requisites:**
MATH101

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 in cylindrical and spherical coordinates

**Pre-Requisites:**
MATH102

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.

**Pre-Requisites:**
MATH102

Systems of linear equations. Vector spaces đť‘…đť‘›: subspaces, bases, dimensions. Rank of matrices. Eigenvalues and eigenvectors. Similar matrices. Diagonalizable matrices. Matrix exponential. First order differential equations: separable, linear, exact, substitutions methods. Applications to linear models of first order. 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. Series

**Pre-Requisites:**
MATH102

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, isomorphism, cosets and Lagrange's Theorem : Note: Not to be taken for credit with ICS 253

**Pre-Requisites:**
MATH102

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 form

**Pre-Requisites:**
MATH102

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 firms.

**Co-Requisites:**
MATH 201

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

**Pre-Requisites:**
MATH201

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.

**Pre-Requisites:**
MATH210 Or MATH232

History of numeration: Egyptian, Babylonian, Hindu and Arabic contributions. Algebra: including the contributions of Al-Khwarizmi and Ibn Kura. Geometry: areas, approximation of z, 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.

**Pre-Requisites:**
MATH102 Or MATH106

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. Irreducible and unique factorization, principal ideal domains.

**Pre-Requisites:**
MATH210 Or MATH232 Or (ICS253 And ICS254)

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.

**Pre-Requisites:**
MATH225 Or MATH280

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

**Pre-Requisites:**
MATH201 And (MATH202 Or MATH208 Or MATH260)

Growth models, Single species and interacting population dynamics. Dynamics of infectious diseases. Modeling enzyme dynamics. Some fatal diseases models. Programing software for numerical simulations.

**Pre-Requisites:**
MATH202 Or MATH208 Or MATH260

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

**Pre-Requisites:**
MATH210 Or MATH232 Or ICS253

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

**Pre-Requisites:**
MATH210 Or MATH232

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.

**Pre-Requisites:**
MATH201

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.

**Pre-Requisites:**
ENGL214

Review of basic vector and matrix operations, Orthogonality, Projection, Eigendecomposition, Factorizations, Covariance, Multivariate Gaussian, Minimum Problems, Lagrange Multipliers, Linear Programming, Least-Square Estimation, Maximum Likelihood Estimation, Gradient Descent. Applications to Machine Learning using Linear Regression and Neural Networks.

**Pre-Requisites:**
(MATH102 Or MATH106) And (STAT201 Or STAT212 Or STAT319 Or ISE205) And (ICS103 Or ICS104)

Formulation of strategic and cooperative games in energy industry, such as oil & gas and electric power companies, and portfolio analysis. Dominant, optimal strategies and Nash equilibrium. Coalition formation in cooperative games is used to represent OPEC to investigate their formation. Games in characteristic function format. Concepts of solutions for games. Pareto optimal solutions, core, and Shapely value. Other cases for allocation of resources, design, supply chain will be modelled in the context of game theory.

**Pre-Requisites:**
ISE303 Or STAT361

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

**Pre-Requisites:**
MATH323 Or MATH345

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

**Pre-Requisites:**
MATH323 Or MATH345

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

**Pre-Requisites:**
MATH210 Or MATH232

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

**Pre-Requisites:**
MATH208 Or MATH225 Or MATH302 Or MATH260 Or MATH280

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.

**Pre-Requisites:**
MATH333 Or MATH301

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

**Pre-Requisites:**
MATH202 Or MATH208 Or MATH260

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.

**Pre-Requisites:**
MATH208 Or MATH260 Or (MATH202 And (MATH225 Or MATH280) )

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).

**Pre-Requisites:**
MATH202 Or MATH208 Or MATH260

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 theorem. Green's function. Neumann's function. The heat equation in one dimension.

**Pre-Requisites:**
MATH333 Or MATH301

Theory of sequences and series of functions. Real functions of several real variables: limi4 continuity,. differentiability. Taylor's theorem. Maxima and minima, Lagrange multipliers rule. Elementary notion of integration on .R N 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.

**Pre-Requisites:**
MATH341 Or MATH311

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.

**Pre-Requisites:**
MATH441 Or MATH411

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

**Pre-Requisites:**
MATH201

Curves ln 3-dimenslonai 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).

**Pre-Requisites:**
MATH208 Or MATH260 Or MATH225 Or MATH280 Or MATH302

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.

**Pre-Requisites:**
MATH341 Or MATH311

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.)

**Pre-Requisites:**
MATH201

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. Mengerâ€™s theorem. Hamiltonian graphs.

**Pre-Requisites:**
MATH208 Or MATH260 Or MATH225 Or MATH280 Or MATH302

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.

**Pre-Requisites:**
MATH321 Or SE301 Or CISE301 Or MATH371 Or CIE301

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.

**Pre-Requisites:**
MATH321 Or SE301 Or CISE301 Or MATH371 Or CIE301

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

**Pre-Requisites:**
MATH201

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

**Pre-Requisites:**
MATH225 Or MATH280 Or MATH302

Concepts of numerical mathematics, approximation tools, system of equations, least squares, numerical differential and integration, quadrature on different geometries, Runge-Kutta and multistep methods for and boundary value problems. Applications to steady-state and time-dependent problems

**Pre-Requisites:**
MATH102

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. Prerequislte:Any two of MATH323, MATH333, MATH341, MATH371

**Pre-Requisites:**
( (MATH323 Or MATH345) And (MATH333 Or MATH301) ) Or (MATH323 And MATH341) Or (MATH323 And MATH371) Or (MATH333 And MATH341) Or (MATH333 And MATH371) Or (MATH341 And MATH371)