| Systems of linear equations, determinants, vectors, geometry, linear transformations, matrices and graphs, number fields, applications.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| MTH 110 Discrete Mathematics | |
| This course covers the fundamentals of discrete mathematics with a focus on proof methods. Topics include: propositional and predicate logic, notation for modern algebra, naive set theory, relations, functions and proof techniques.
| | Lect: 3 hrs./Lab: 1 hr | | back to top |
| MTH 125 Mathematics for the Health Sciences | |
| Basic Algebra, trigonometric functions, radicals and exponents and a basic introduction calculus.
| | Lect: 4 hrs. | | back to top |
| MTH 128 Introductory Mathematics | |
| Factoring and Fractions. Functions (linear, quadratic, simple trigonometric, exponential and logarithmic). Differential calculus: limits, tangent lines, rates of change, derivatives and applications. Other topics: fundamental trigonometric identities, trigonometric equations. This course is graded on a pass/fail basis.
| | Lect: 4 hrs. | | back to top |
| MTH 131 Modern Mathematics I | |
| Differentiation; applications of differentiation; Newton’s method; differentials; integration; applications of integration; Linear Algebra: systems of linear equations, Gauss elimination, matrices; vectors, dot product, cross product.
| | Lect: 4 hrs./Lab 1 hr. | | back to top |
| Limits, continuity, differentiability, rules of differentiation. Absolute and relative extrema, inflection points, asymptotes, curve sketching. Applied max/min problems, related rates. Definite and indefinite integrals, Fundamental Theorem of Integral Calculus. Areas, volumes. Transcendental functions (trigonometric, logarithmic, hyperbolic and their inverses).
| | Lect: 4 hrs./Lab: 1 hr. | | back to top |
| Review. Systems of linear equations and matrices. Determinants. Vector spaces. Inner product spaces. Eigenvalues and eigenvectors.
| | Lect: 4 hrs. | | back to top |
| Graphical presentation, frequency distribution, descriptive statistics, probability theory, normal distribution, sampling distribution, binomial distribution, Poisson distribution, t distribution, estimation, hypotheses tests.
| | Lect: 3 hrs. | | back to top |
| MTH 207 Calculus & Computational Methods I | |
| Calculus of functions of one variable and related numerical topics. Derivatives of algebraic, trigonometric and exponential functions, techniques of integration, numerical integration.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| MTH 210 Discrete Mathematics II | |
| This course is a continuation of Discrete Mathematics I. Topics include: recursion, induction, regular expressions and finite state automata, efficiency of algorithms, graph theory, introduction to number theory and counting.
|
| Prerequisite: MTH 110. |
| Lect: 3 hrs./Lab: 1 hr. | | back to top |
| MTH 231 Modern Mathematics II | |
| Applications of integration and integration techniques. Introduction to partial derivatives. Introduction to ordinary differential equations. Sequences and series, convergence, Taylor series. Elementary Linear Algebra: lines and planes in 3-space, determinants.
| | Prerequisite: MTH 131. | | Lect: 4 hrs./Lab: 1 hr. | | back to top |
| Integration techniques. L’Hôpital’s Rule. Improper integrals. Partial derivatives. Infinite sequences and series, power series. First-order differential equations, with applications.
|
| Prerequisite: MTH 140. |
| Lect: 4 hrs. | | back to top |
| MTH 25A/B Algebra and Introductory Calculus | |
| Basic algebra, the straight line, functions (quadratic, simple trigonometric, exponential and logarithmic), vectors in the real and complex planes and applications. Trigonometric identities, sine and cosine of the sum and difference of two angles and double angle formulae, graphical and algebraic solution of systems of linear and non-linear equations, analytic geometry, conic sections, differential calculus (limits, tangent lines, rates of change, derivatives and applications). Only available to ATSG students.
| | Lect: 4 hrs. | | back to top |
| MTH 281 Differential Equations | |
| Review of first-order ordinary differential equations and applications; Higher-order linear differential equations; Methods of Undetermined Coefficients and Variation of Parameters; Series solutions; Laplace Transforms; Systems of Differential Equations; Operator Methods; Application of MAPLE to solution of differential equations; Applications of differential equations in automatic control of chemical processes.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| Scalar and vector fields. Gradient, divergence and curl operators. Line, surface and volume integrals. Theorems of Green, Gauss and Stokes. Introduction to Transport equations with applications.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| MTH 304 Probability and Statistics I | |
| Brief Introduction to Statistics. Description of Numerical Data. Elements of Probability Theory. Discrete Probability Distribution. (Hyper-geometric, Binomial, Poisson). Normal Distribution and its applications. Sampling Distributions. The t-distribution and the X² distribution. Confidence Interval and Hypothesis Testing concerning the mean, variance and proportion of a single population. Confidence Interval and Hypothesis Testing concerning the mean and proportion of two populations. The F-distribution. SAS will be used in this course.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| MTH 309 Differential Equations | |
| Ordinary differential equations with applications, Laplace transforms, linear systems of differential equations with applications.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| MTH 310 Calculus & Computational Methods II | |
| Integration techniques, improper integrals, sequences, infinite series, power series, partial derivatives, maxima and minima.
| | Lect: 3 hrs/ Lab: 1 hr. | | back to top |
| MTH 312 Differential Equations & Vector Calculus | |
| Second and higher order differential equations. Applications to electric circuits. Directional derivative. Line, surface and volume integrals. Green’s theorem, Stoke’s theorem and divergence theorem. Vector fields, coordinate systems.
| | Lect: 4 hrs. | | back to top |
| MTH 314 Discrete Mathematics for Engineers | |
| Sets and relations, proposition and predicate logic, functions and sequences, elementary number theory, mathematical reasoning, combinatorics, graphs and trees, finite-state machines, Boolean algebra.
| | Lect: 3 hrs. | | back to top |
| MTH 322 Chaos, Fractals and Dynamics | |
| Fractals; drawing fractals, fractal dimension, Julia sets. Discrete dynamical systems; Logistic equation, period-doubling bifurcations. The Henon map. Nonlinear ordinary differential equations; phase portraits, stability, periodic orbits, averaging methods and bifurcations. Nonlinear oscillations.
| | Prerequisite: MTH 231. | | Lect: 3 hrs. | | back to top |
| MTH 330 Calculus and Geometry | |
| Derivatives and the chain rule. Multiple integrals, curves and surfaces in 3-space. Div, grad and curl operators, line and surface integrals, theorems of Green, Gauss and Stokes. Linear Algebra: linear transformations, matrix representations and change of coordinates.
| | Prerequisite: MTH 231 or (MTH 108 and MTH 310). | | Lect: 4 hrs. | | back to top |
| Additional applications of Integration. Partial differentiation. Unconstrained extrema and the Hessian matrix. Constrained extrema and Lagrange multipliers. Curves and Surfaces. Multiple integration. Line and surface integrals. Theorems of Gauss, Green and Stokes. Fourier series. Laplace transforms and their application to second-order and other differential equations.
| | Lect: 4 hrs./Lab: 1 hr. | | back to top |
| MTH 380 Probability and Statistics I | |
| Brief Introduction to Statistics. Description of Numerical Data. Probability. Discrete Probability Distributions. Normal Distribution and its applications. Sampling Distributions. Large Sample Estimation. Large Sample Tests of Hypotheses. Inference from Small Samples. A statistics computer package will be used in this course.
| | Lect: 3 hrs. | | back to top |
| MTH 401 Differential Equations | |
| First-order differential equations with applications. Linear higher-order differential equations with applications. Laplace transform methods. Simultaneous Differential Equations. Use of Maple to solve differential equations.
| | Lect: 3 hrs. | | back to top |
| MTH 404 Probability and Statistics II | |
| A continuation of the introductory topics covered in MTH 304. Contingency Tables. Goodness of fit tests. Type I and Type II errors. Correlation. Regression. ANOVA One and two-way. A statistics computer package may be used in this course.
| | Lect: 3 hrs. | | back to top |
| This is a first course in automata theory and formal languages. Topics include: regular grammars and finite state automata, context-free grammars and pushdown automata, pumping lemmas, Turing machines and computable languages, Chomsky hierarchy, Church’s thesis, halting problem, NP-completeness.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| Description of numerical data. Elements of probability theory. Discrete probability distributions (hypergeometric, binomial, Poisson). Normal distribution. t-distribution. X² distribution. Confidence interval and hypothesis testing concerning mean, variance and proportion for one and two populations. F-distribution. Correlation. Simple linear regression (if time permits).
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| MTH 430 Dynamic Systems Differential Equations | |
| First-order differential equations with applications; linear higher-order differential equations with applications; simultaneous Eigenvalues and Eigenvectors.
| | Prerequisite: MTH 131. | | Lect: 3 hrs. | | back to top |
| MTH 480 Probability and Statistics II | |
| A continuation of the introductory topics covered in MTH 380. Contingency Tables. Goodness of fit tests. Type I and Type II errors. Correlation. Regression. ANOVA One and two-way. A statistics computer package will be used in this course.
| | Lect: 3 hrs. | | back to top |
| MTH 500 Introduction to Stochastic Processes | |
| Probability of a function of several variables, martingales, conditional expectations, maximum likelihood estimators, random walks, stochastic processes (stationary and ergodic). Applications of statistical processes in science.
| | Prerequisite: MTH 480. | | Lect: 3 hrs. | | back to top |
| MTH 501 Numerical Analysis I | |
| Errors and floating point arithmetic. Solutions of non-linear equations including fixed point integration and Bairstow-Lin’s method. Matrix computations and solution of systems of linear equations. Interpolation. Finite difference methods. Least squares fit. Cubic spline interpolation. Numerical integration. Numerical solution of ordinary differential equations. Taylor series method. Euler method. This course will include laboratory classes using electronic calculators and computer terminals.
| | Lect: 4 hrs. | | back to top |
| MTH 503 Operations Research I | |
| Linear Programming and the Simplex Algorithm. Sensitivity analysis, duality, and the dual simplex algorithm. Transportation and Assignment Problems, Network models. Integer programming.
| | Lect: 3 hrs. | | back to top |
| Multiple Integrals, curves and surfaces in 3-space. Div, grad and curl operators, line and surface integrals, theorems of Green, Gauss, and Stokes, numerical methods, integral transforms.
| | Lect: 3 hrs./Lab: 2 hrs. | | back to top |
| MTH 510 Numerical Analysis | |
| Review of Taylor’s formula, truncation error and roundoff error. Solutions of Non linear Equations in one variable. Linear Equations. LU-decompostion. Eigenvalues and eigenvectors. Jacobi, Gauss-Seidel methods. Interpolation and curve fitting. Numerical integration. Numerical solution of ordinary differential equations. (Initial value problems.)
| | Lect: 3 hrs. | | back to top |
| MTH 514 Probability and Stochastic Processes | |
| Introduction to probability theory and stochastic processes. Topics covered include: elements of probability theory, conditional probability sequential experiments, random variables and random vectors, probability density, function cumulative density functions, functions of random variables, expected values of random variables, transform methods in random variable, reliability of systems, joint and marginal probability, correlation, confidence intervals, stochastic processes, stationary and ergodic processes, power spectral density, sample processes.
| | Lect: 3 hrs. | | back to top |
| Projective plane and 3-space. Cross-ratio, perspectivity, conics and quadrics, poles and polars. Line geometry in projective 3-space. Euclidean, elliptic and hyperbolic interpretation of projective results. Inversive geometry and the complex projective line. Classification of motions in the Euclidean, elliptic, Gaussian and hyperbolic cases.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| MTH 599 Foundations of Mathematical Thought | |
| A one semester course on the nature of mathematical thought. Mathematics is commonly believed to enjoy a degree of certainty which sets it apart from other disciplines. Moreover, this certainty is often confused with veracity, and a science gains respectability as its quantitative component increases. This course will explore the nature and extent of this certainty in mathematics. There are no specific pre-requisites but a previous course in Philosophy or other course requiring logical reasoning is recommended. (UL)
| | Lect: 3 hrs. | | back to top |
| MTH 601 Numerical Analysis II | |
| Numerical solutions for initial value and boundary value problems for ordinary differential equations. Runge-Kutta, Multi-step, Hybrid methods. Convergence criteria. Error analysis aspects. Shooting, finite- difference, Rayleigh-Ritz methods. Matrix eigenvalue problem. Jacobi, Givens, Householder, Power methods. Numerical double interpolation and multiple integration. Non-linear systems of equations. Numerical solutions to partial differential equations. This course will include laboratory classes using electronic calculators and computer terminals.
| | Lect: 4 hrs. | | back to top |
| MTH 603 Operations Research II | |
| Nonlinear programming, decision making, inventory models, Markov chains, queuing theory, dynamic programming, Simulation.
| | Lect: 3 hrs. | | back to top |
| Introduction to graph theory and its applications with an emphasis on algorithmic structure. Topics may include graphs, digraphs and subgraphs, representation of graphs, breadth first and depth first search, connectivity, paths, trees, circuits and cycles, planar graphs flows and networks, matchings, colourings, hypergraphs, intractability and random algorithms.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| Linear congruencies and systems, primitive roots and prime certificates, applications to data encryption for security. Legendre and Jacobi symbols. Euler and Mobius functions, quadratic reciprocity, sums of two, three and four squares, quadratic forms and class groups, partitions, efficient algorithms and their computer implementation.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| DeMoivre’s theorem. Roots and Powers of complex numbers. Functions of a complex variable. Limits and continuity. Cauchy-Riemann equations. Exponential, trigonometric, hyperbolic and logarithmic functions. Conformal transformations. Integration in the complex plane. Residue theorem and some of its applications. Laplace and Fourier transforms.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| An advanced course in Fourier Methods dealing with the application of Fourier series, Fourier transforms, convolution, correlation, discrete and fast Fourier transforms.
| | Lect: 3 hrs. | | back to top |
| MTH 712 Differential Equations II | |
| Series solutions of differential equations. Bessel’s equation and Bessel functions. Legendre’s differential equation. Derivation of some partial differential equations (P.D.E.). Solution of P.D.E.’s using separation of variables.
| | Lect: 3 hrs. | | back to top |
| MTH 714 Logic & Computability | |
| Propositional and predicate calculus, first order theories, models and review of semantics of logic, resolution proof, completeness, consistency, independence, undecideability. Logic programming. Effective computability, evidence for Church’s Thesis. Review of Turing machines, reducibility, halting problem, Rice’s theorem, decideability of various formal language problems.
| | Lect: 3 hrs./Lab 1 hr. | | back to top |
| Students will learn the basics of design theory, with particular emphasis on error correcting and detecting codes. Such codes are widely used in network communications. The student will also be exposed to other applications of design such as scheduling and routing problems. Topics covered are introduction to codes Hamming distance, minimum distance; Error correction and detection. Perfect codes. Dual codes; Finite geometries Linear codes; Designs Latin squares and Transversal Designs. Shannon’s Theorem. Authentication codes. Threshold schemes. One time pad; Block Designs from geometries. Triple systems. Block designs and their codes. Scheduling problems; Codes Assumus Mattson Theorem. Hamming designs/codes. Reed Muller codes. Golay codes. Codes from triple systems.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| MTH 814 Computational Complexity | |
| Order of Growth notation, time and space complexities of DTMs and NDTMs, intractability, basic complexity classes, P=NP?, reducibility and completeness, NP-completeness, Cook’s theorem, hierarchy results, circuit complexity, probabilistic algorithms, models for parallel computation.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| This course will consider the mathematics of modern cryptographic schemes, including commonly used public and private key systems. The main uses; authentication, validation and encryption will be discussed. System vulnerabilities will also be considered. Topics covered include: Introduction to Ciphers. Authentication, validation and encryption. Public vs. private keys. Finite fields. Properties of a good cipher; Simple Ciphers ROT n, Matrix schemes. Probabilistic attacks, brute force; Authentication and Validation MD5, Digital Signatures, integrity checks, hash functions; Private key encryption Block ciphers, 3DES, IDEA, AES (Rijndael); Public key encryption RSA, Rabin-Williams, Integer Factorization problem (IFP). DSA, Diffie, Hellman, Discrete logarithm problem (DFP). ECC, Elliptic Curves, Elliptic curve discrete logarithm problem (ECDLP) (if time permits).
| | Lect: 3hrs./Lab: 1 hr. | | back to top |
| Elementary principles of counting: permutations, combinations, circular arrangements. Partitions; derangements, number of integer solutions of a Diophantine equation with unit coefficients; Bell numbers. Introduction to the generating function method, exponential generating functions. Solutions of recurrence equations. Principle of inclusion and exclusion; Stirling numbers. groups of permutations and applications to counting problems; orbit numbers, Polya’s counting formula. Designs, latin squares, orthogonal Latin squares. Hadamard matrices. Matroids. Other optional topics may include: posets and Zorn’s lemma, Ramsey’s Theorem, finite geometries.
| | Lect: 3 hrs./Lab: 1 hr. | | back to top |
| Continuous and discrete image representation. Sampling and reconstruction. Quantization. Convolution. Transforms: Fourier, Sine, Cosine, wavelet. Time/Frequency domains. Image enhancement/restoration. Edge detection, feature extraction, segmentation, registration.
| | Prerequisite: MTH 710. | | Lect: 3 hrs./Lab: 1 hr. | | back to top |
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