# Mathematics (MATH) Courses

Evening and summer sections of large-enrollment, multi-section service courses are offered on a regular basis. For other courses, evening and summer sections are offered only upon sufficient demand. Students should contact the department well in advance (at least a semester) to request such course offering.

#### Lower Division

**MATH 015 PRE-ALGEBRA (3-0-0)(F,S).** Fundamental algebraic skills needed for MATH 25. Review of arithmetic (fractions, negative numbers, and percents), an introduction to graphing, and an introduction to variables, simplifying algebraic expressions, and solving linear equations.

**MATH 025 ELEMENTARY ALGEBRA (3-0-0)(F,S,SU).** Brief review of arithmetic operations and their properties. Positive integer exponents, variables, algebraic expressions, solution of linear equations, definition of absolute value. Expansion of product of two binomials, factorization of quadratics, solution of quadratic equations by factoring. Two-dimensional Cartesian coordinate systems, slope, equations of lines, solution of 2-by-2 linear systems. Simple “word problems.”

**MATH 108 INTERMEDIATE ALGEBRA (3-0-3)(F,S,SU).** Radicals, negative and rational exponents, completing the square, quadratic formula. Linear and quadratic inequalities (including absolute value); simple systems of equations and inequalities. Multiplication of polynomials; basic factorization techniques. Manipulation of rational expressions, compound fractions, rationalization of denominator (or numerator). Introduction to the concept of function, graphs of functions and equations. Introduction to exponential and logarithmic expressions. Math 108 is NOT a FM course, and cannot be taken for credit after any MATH course numbered MATH 143 or higher. PREREQ: MATH 25 or satisfactory placement score.

**MATH 123 MATH IN MODERN SOCIETY (3-0-3)(F,S,SU)(FM).** Survey of quantitative reasoning topics including deductive and inductive reasoning, benchmarks, and sense of scale. Topics will be applied in a conceptual way to interpretation of graphical information, descriptive and inferential statistics, elementary probability, and exponential growth. PREREQ: MATH 25 or satisfactory placement score.

**MATH 143 COLLEGE ALGEBRA (3-0-3)(F,S,SU)(FM).** Emphasis on the concept of functions as mathematical entities; domain, range, algebraic operations, composition, inverses, graphing. Polynomial functions, division of polynomials, roots, factor theorem, complex numbers, fundamental theorem of algebra. Rational functions and asymptotes. Logarithmic and exponential functions. Multi-level algebraic manipulation of functional expressions – e.g., difference quotients. Conic sections and other topics from analytic geometry as time permits. Mathematical modeling based on Business and Science applications using algebraic functions will be prominent. PREREQ: MATH 108 or satisfactory placement score.

**MATH 144 PRECALCULUS II: TRIGONOMETRIC FUNCTIONS (2-0-2)(F,S,SU). **Right-triangle and circular function approaches to trigonometry. Trigonometric identities. Graphs of trigonometric functions; amplitude, frequency, phase shift. Inverse trigonometric functions and their graphs. Polar coordinates and polar representations of complex numbers. PRE/COREQ: MATH 143 or MATH 149 or satisfactory placement score.

**MATH 149 PRECALCULUS: FUNCTIONS FOR BUSINESS (3-0-3)(F,S,SU)(FM). **Provides algebra and function skills for business. Topics include: polynomial functions, composition of functions, inverse functions, logarithmic functions, exponential functions, linear programming, counting methods and an introduction to probability. This course will instruct using Excel as the primary technology tool. PREREQ: MATH 108 or satisfactory placement score or COREQ: MATH 149P.

**MATH 149P PRECALCULUS: FUNCTIONS FOR BUSINESS LAB (0-2-1)(F,S).** Co-requisite lab component paired with MATH 149. Intensive study of mathematical skills, concepts and strategies to support and supplement MATH 149. PREREQ: satisfactory placement score or MATH 025 with a grade of A- or better. COREQ: MATH 149.

**MATH 153 STATISTICAL REASONING (3-0-3)(F/S/SU)(FM).** Pre-calculus treatment of descriptive statistics, confidence intervals, hypothesis testing, simple linear regression and correlation, and basic data collection concepts. Emphasis on reasoning, interpretation, and communicating ideas in the context of a wide variety of disciplines with computer software for computations. Carries no credit after MATH 254. PREREQ: MATH 25 or satisfactory placement score.

**MATH 157 NUMBER AND OPERATIONS FOR TEACHERS (4-0-4)(F,S,SU).** Laboratory-based course that addresses number meanings, representations, operations, algorithms, and properties. Interpreting mathematical reasoning and developing non-standard algorithms are central themes. PREREQ: MATH 108 or satisfactory placement score.

**MATH 160 SURVEY OF CALCULUS (4-0-4)(F,S,SU)(FM).** A survey of the essentials of calculus, intended mainly for students in business and social sciences; emphasis on applications to such areas. Basic concepts and computational techniques for functions, derivatives, and integrals, with emphasis on polynomial, rational, exponential and logarithmic functions. Very brief introduction to calculus of functions of several variables. MATH 160 cannot be taken for credit after MATH 170 or MATH 171. PREREQ: MATH 143 or MATH 149 or satisfactory placement score.

**MATH 170 CALCULUS I (4-0-4)(F,S,SU)(FM).** Informal limits. Derivatives and antiderivatives, including trigonometric, exponential, and logarithmic functions. The relationship between a function, its derivative, and its antiderivative. Integration and the fundamental theorem of calculus. Applications of calculus to physical models, geometry, approximation, and optimization. Credit cannot be earned for both MATH 170 and MATH 171. PREREQ: MATH 143 or MATH 149, or satisfactory placement score. PRE/COREQ: MATH 144 or satisfactory placement score.

**MATH 171 CALCULUS I: THEORY AND APPLICATIONS (4-0-4)(F,S,SU)(FM).** Limits and continuity. Derivatives and antiderivatives, including trigonometric, exponential, and logarithmic functions. The relationship between a function, its derivative, and its antiderivative. Integration and the fundamental theorem of calculus. Applications of calculus to physical models, geometry, approximation, and optimization. Includes elements of the theory of calculus and the language of mathematics. Credit cannot be earned for both MATH 170 and MATH 171. PREREQ: MATH 143 and MATH 144, or satisfactory placement score.

**MATH 175 CALCULUS II: CONCEPTS AND APPLICATIONS (4-0-4)(F,S,SU).** A continuation of MATH 170. Techniques of integration and calculation of antiderivatives. Applications of integration to physical models, including calculation of volume, moment, mass, and centroid. Informal convergence of sequences and series of real numbers. Taylor series, Taylor polynomials, and applications to approximation. Vectors, parametric curves, and polar coordinates. Credit cannot be earned for both MATH 175 and MATH 176. PREREQ: MATH 143, MATH 144 and MATH 170, or corresponding satisfactory placement score.

**MATH 176 CALCULUS II: THEORY AND APPLICATIONS (4-0-4)(F,S,SU). **A continuation of MATH 171. Techniques of integration and calculation of antiderivatives. Applications of integration to physical models, including volume, moment, mass, and centroid. Convergence of sequences and series of real numbers. Taylor series, Taylor polynomials, and applications to approximation. Vectors, parametric curves, and polar coordinates. Includes elements of the theory of calculus and the language of mathematics. Credit cannot be earned for both MATH 175 and MATH 176. PREREQ: MATH 143, MATH 144, and MATH 171, or corresponding satisfactory placement score.

**MATH 187 DISCRETE AND FOUNDATIONAL MATHEMATICS (3-0-3)(F,S).** An introduction to the language and methods of reasoning used throughout mathematics. Topics include propositional and predicate logic, elementary set theory, proof techniques including mathematical induction, functions and relations, combinatorial enumeration, permutations and symmetry. COREQ: MATH 170 or MATH 171.

**MATH 189 DISCRETE MATHEMATICS (4-0-4)(F,S,SU).** Content drawn from propositional and predicate logic; proof logic, induction and recursion, elementary set theory; functions and relations; combinatorial enumeration; graph theory and basic elementary number theory. Intended for computer science majors. Credit cannot be granted for both MATH 187 and MATH 189. PREREQ: MATH 170.

**MATH 211 GEOMETRIC REASONING IN THE SECONDARY CLASSROOM (3-0-3)(F/S). **Laboratory-based course that involves the study of geometry in relation to teaching secondary mathematics. Topics include: congruence, inductive and deductive reasoning, dynamic geometry technology, transformations, and applications of geometry and measurement. It is recommended that this course be taken prior to MATH 311. PREREQ: MATH 143 or MATH 157.

**MATH 250 ALGEBRAIC AND PROPORTIONAL REASONING FOR TEACHERS (3-0-3)(F/S).** Laboratory-based course that addresses the development of algebraic and proportional reasoning. Interpreting middle grades’ student mathematical reasoning, developing conjectures and sensible arguments are central themes. PREREQ: MATH 143 or MATH 157.

**MATH 254 INTRODUCTION TO STATISTICS (3-0-3)(F,S,SU)(FM).** Pre-calculus treatment of descriptive statistics, confidence intervals, hypothesis testing, simple linear regression, correlation, introduction to probability. Emphasis on reasoning, problem solving, communicating ideas, and applications to a wide variety of disciplines. Use of computer statistics packages and calculators to handle computations. Carries no credit after MATH 360 or MATH 361. PREREQ: MATH 108 or satisfactory placement score.

**MATH 257 GEOMETRY AND MEASUREMENT FOR TEACHERS (4-0-4)(F,S)(FM). **Laboratory-based course that addresses geometric reasoning and models, along with principles of measurement. Interpreting mathematical reasoning, developing conjectures and sensible arguments are central themes. PREREQ: MATH 157.

**MATH 261 STATISTICAL REASONING IN THE SECONDARY CLASSROOM (3-0-3)(F/S). **Laboratory-based course that addresses the statistical processes of formulating questions, collecting and analyzing data, and interpreting results. Through activities and projects, students will use modern statistical methods while learning about social and classroom issues affecting the teaching and learning of secondary level statistics. PREREQ: MATH 143 or MATH 157.

**MATH 275 MULTIVARIABLE AND VECTOR CALCULUS (4-0-4)(F,S,SU). **Vector algebra and geometry, functions of several variables, partial and directional derivatives, gradient, chain rule, optimization, multiple and iterated integrals. Parametric curves and surfaces, vector fields, divergence and curl, line and surface integrals, Green’s, Stokes’ and divergence theorems. Use of software such as Maple or Mathematica for visualization, exploration and solutions of “real-world” problems. PREREQ: MATH 175 or MATH 176.

**MATH 287 COMMUNICATION IN THE MATHEMATICAL SCIENCES (3-0-3)(F,S).** Integrates mathematics content with the opportunity to develop proof writing and communication skills important in the mathematical sciences. Content is drawn from discrete and foundational math and elementary analysis. Introduction to and engagement with written and verbal communication practices characteristic to mathematical sciences. Introduction to and use of technologies that support communication in the mathematical sciences. PREREQ: ENGL 102, MATH 187.

**MATH 291 PUTNAM PRACTICE I (1-0-1)(F). **Solving problems from previous Putnam examinations and related problems. May be repeated once for credit. (Pass/Fail.)

#### Upper Division

**MATH 301 INTRODUCTION TO LINEAR ALGEBRA (3-0-3)(F,S). **Linear algebra from a matrix perspective with applications from the applied sciences. Topics include the algebra of matrices, methods for solving linear systems of equations, eigenvalues and eigenvectors, matrix decompositions, vector spaces, linear transformations, least squares, and numerical techniques. PREREQ: MATH 175 or MATH 176.

**MATH 305 INTRODUCTION TO ABSTRACT ALGEBRA AND NUMBER THEORY (3-0-3)(F,S).** Division algorithm. Greatest common divisor and Euclidean algorithm. Solving linear modular equations, Chinese Remainder Theorem, Primitive roots, solving modular quadratic equations. Introduction to group theory: motivation, definitions and basic properties. Finite cyclic groups, permutation groups, isomorphisms, Lagrange’s Theorem. PREREQ: MATH 187 or MATH 189.

**MATH 307 FOUNDATIONS OF CRYPTOLOGY (3-0-3)(F).** Introduction to security (authentication, confidentiality, message integrity and non-repudiation) and the mathematical mechanisms to achieve them. Topics include concepts in cryptography and cryptanalysis, symmetric key systems, public key systems, key management, public-key infrastructure (PKI), digital signatures, authentication schemes and non-repudiation. PREREQ: MATH 187 or MATH 189.

**MATH 308 INTRODUCTION TO ALGEBRAIC CRYPTOLOGY (3-0-3)(S).** Introduction to groups, fields and polynomial rings. The study of enciphering/deciphering and cryptanalysis of the Elliptic Curve, LUC, and NTRU public key cryptosystems. Group based authentication and digital signature schemes and anonymity protocols. PREREQ: MATH 187 or MATH 189.

**MATH 311 FOUNDATIONS OF GEOMETRY (3-0-3)(S). **Euclidean, non-Euclidean, and projective geometries from an axiomatic point of view. PREREQ: MATH 175 or MATH 176, and MATH 187 or MATH 189.

**MATH 314 FOUNDATIONS OF ANALYSIS (3-0-3)(F,S).** The real number system, completeness and compactness, sequences, continuity, foundations of the calculus. PREREQ: MATH 176 and MATH 287.

**MATH 333 DIFFERENTIAL EQUATIONS WITH MATRIX THEORY (4-0-4)(F,S,SU).** Use of differential equations to model phenomena in sciences and engineering. Solution of differential equations via analytic, qualitative and numerical techniques. Linear and nonlinear systems of differential equations. Introduction to matrix algebra, determinants, eigenvalues, and solutions of linear systems. Laplace transforms. PREREQ: MATH 175 or MATH 176.

**MATH 337 (CS 330)(ECE 337)(ENGR 337)(ME 337) INTRODUCTION TO SECURITY IN CYBER-PHYSICAL SYSTEMS (3-0-3)(F).** Overview of systems security: hardware, software, encryption, and physical security. Includes multiple modules: system security, physical issues in security, hardware and firmware security issues, industrial control, and all things connected to the internet. PREREQ: CS 117 or CS 121, PHYS 211, and MATH 187 or MATH 189 or MATH 360 or MATH 361.

**MATH 360 ENGINEERING STATISTICS (3-0-3)(F,S).** Calculus based survey of statistical techniques used in Engineering. Data collection and organization, basic probability distributions, sampling, confidence intervals, hypothesis testing, process control, simple regression techniques, design of experiments. Emphasis on examples and applications to engineering, including product reliability, robust design and quality control. Credit cannot be earned for both MATH 360 and MATH 361. PREREQ: MATH 175 or MATH 176.

**MATH 361 PROBABILITY AND STATISTICS I (3-0-3)(F,S,SU). **Calculus- based treatment of probability theory, random variables, distributions, conditional probability, central limit theorem, descriptive statistics, estimation, tests of hypotheses, and regression. Differs from MATH 360 by providing more thorough coverage of theoretical foundations and wider variety of applications drawn from natural and social sciences as well as engineering. Credit cannot be earned for both MATH 360 and MATH 361. PREREQ: MATH 175 or MATH 176.

**MATH 365 INTRODUCTION TO COMPUTATIONAL MATHEMATICS (3-0-3)(F,S).** Uses Matlab and Maple software packages from a problem-oriented perspective with examples from the applied sciences. Matrix computations, solving linear systems, interpolation, optimization, least squares, discrete Fourier analysis, dynamical systems, computational efficiency, and accuracy. Emphasis on critical thinking and problem solving using both numerical and symbolic software. PREREQ: MATH 301 or MATH 333.

**MATH 370 FUNCTIONS AND MODELING (3-0-3)(F/S).** Laboratory-based course that involves the study of mathematical modeling in relation to teaching secondary mathematics. Mathematical topics include data collection, rate of change, and applications of polynomial, exponential, logarithmic, and trigonometric functions. Course also includes investigating research on student thinking and the use of technology. PREREQ: MATH 171 and one of the following: MATH 211 or MATH 250 or MATH 261.

**MATH 387 INTRODUCTION TO COMBINATORICS (3-0-3)(S)(Odd years). **Covers basic enumerative techniques and fundamentals of graph theory. Additional content may include further topics in enumerative techniques or graph theory, extremal combinatorics, Ramsey theory, the probabilistic method, or combinatorial algorithms. PREREQ: MATH 187 or MATH 189.

**MATH 401 SENIOR THESIS IN THE MATHEMATICAL SCIENCES (1-0-1)(F,S)(FF).** Independent mathematical work in an active and modern subject area of the mathematical sciences, guided by an official research faculty member in the department of mathematics and culminating in a written thesis presented in an appropriate public forum. PREREQ: Senior standing and PERM/INST.

**MATH 402 LOGIC AND SET THEORY (3-0-3)(S).** Structured as three five-week components: formal logic, set theory, and topics to be determined by the instructor. The logic component includes formalization of language and proofs, the completeness theorem, and the Lowenheim-Skolem theorem. The set theory component includes orderings, ordinals, the transfinite recursion theorem, and the Axiom of Choice and some of its equivalents. PREREQ: MATH 314.

**MATH 403 LINEAR ALGEBRA (3-0-3)(F). **Concepts of linear algebra from a theoretical perspective. Topics include vector spaces and linear maps, dual vector spaces and quotient spaces, eigenvalues and eigenvectors, diagonalization, inner product spaces, adjoint transformations, orthogonal and unitary transformations, Jordan normal form. PREREQ: MATH 187 or MATH 189, and MATH 301.

**MATH 405 ABSTRACT ALGEBRA (3-0-3)(F)(Odd years). **Topics in group theory, ring theory and field theory with emphasis on finite and solvable groups, polynomials and factorization, extensions of fields. PREREQ: MATH 301 and MATH 305.

**MATH 406 NUMBER THEORY (3-0-3)(S). **Quadratic residues, Representing numbers as sums of squares, Continued fractions, Diophantine equations Including Pell’s equation, arithmetic functions and Mobius Inversion, the distribution of prime numbers, primality testing, factoring natural numbers. PREREQ: MATH 305.

**MATH 408 ADVANCED ASYMMETRIC CRYPTOGRAPHY AND CRYPTANALYSIS (3-0-3)(F). **An in-depth study of asymmetric cryptography, post-quantum cryptography, digital signatures, and analysis of cryptographic security. Elliptic curves and elliptic curve isogenies. Lattices and the shortest vector problem. Post-quantum cryptosystems such as NTRU and cryptographic schemas based on Short Integer Solution and Learning With Errors. PREREQ: CS 567 or MATH 305 or MATH 307 or MATH 308.

**MATH 409 SYMMETRIC KEY CRYPTOGRAPHY AND CRYPTANALYSIS (3-0-3)(S). **An in-depth study of modern block and stream ciphers, lightweight cryptography, hash functions, analysis cryptographic security, and current advances in cryptanalysis. Finite fields, vector spaces, enumerative combinatorics. Algebraic structures of symmetric key cryptosystems. Security proofs including algebraic, linear, and differential cryptanalysis. Analysis of side-channel attacks. PREREQ: CS 567 or MATH 305 or MATH 307 or MATH 308.

**MATH 411 INTRODUCTION TO TOPOLOGY (3-0-3)(F)(Even years). **Sets, metric and topological spaces, product and quotient topology, continuous mappings, connectedness and compactness, homeomorphisms, fundamental group, covering spaces. PREREQ: MATH 314.

**MATH 414 REAL ANALYSIS (3-0-3)(S). **Covers Riemann integration, the fundamental theorem of calculus, sequences and series of functions, multivariable calculus. Additional topics may include Fourier series, analysis of metric spaces, the Baire property, and advanced topology of Euclidean space. PREREQ: MATH 275 and MATH 314.

**MATH 426 COMPLEX VARIABLES (3-0-3)(S)(Odd years).** Complex numbers, functions of a complex variable, analytic functions, infinite series, infinite products, integration, proofs and applications of basic results of complex analysis. Topics include the Cauchy integral formulas, the residue theorem, the Riemann mapping theorem and conformal mapping. PREREQ: MATH 275.

**MATH 427 INTRODUCTION TO APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS (3-0-3)(F). **Introduction to applied mathematics in science and engineering: Vector calculus, Fourier series and transforms, series solutions to differential equations, Sturm-Liouville problems, wave equation, heat equation, Poisson equation, analytic functions, and contour integration. PREREQ: MATH 275 and MATH 333.

**MATH 433 ORDINARY DIFFERENTIAL EQUATIONS (3-0-3)(S)(Odd years). **Theory of linear and nonlinear ordinary differential equations and their systems, including Dynamical systems theory. Properties of solutions including existence, uniqueness, asymptotic behavior, stability, singularities and boundedness. PREREQ: MATH 333.

**MATH 436 PARTIAL DIFFERENTIAL EQUATIONS (3-0-3)(S)(Even years).** Theory of partial differential equations and boundary value problems with applications to the physical sciences and engineering. Detailed analysis of the wave equation, the heat equation, and Laplace’s equation using Fourier series and other tools. PREREQ: MATH 275 and MATH 333.

**MATH 456 LINEAR PROGRAMMING (3-0-3)(SU)(On Demand).** Linear optimization problems and systems of linear inequalities. Algorithms include simplex method, two-phase method, duality theory, and interior point methods. Programming assignments. PREREQ: MATH 301.

**MATH 462 PROBABILITY AND STATISTICS II (3-0-3)(F). **Provides a solid foundation in the mathematical theory of statistics. Topics include probability theory, distributions and expectations of random variables, transformations of random variables, moment-generating functions, basic limit concepts and brief introduction to theory of estimation and hypothesis testing: point estimation, interval estimation and decision theory. PREREQ: MATH 275, MATH 301, and MATH 361.

**MATH 465 NUMERICAL METHODS I (3-0-3)(F).** Approximation of functions, solutions of equations in one variable and of linear systems. Polynomial, cubic spline, and trigonometric interpolation. Optimization. Programming assignments. PREREQ: MATH 365 or PERM/INST.

**MATH 471 DATA ANALYSIS (3-0-3)(F). **Applications of statistical data analysis in various disciplines, introduction to statistical software, demonstration of interplay between probability models and statistical inference. Topics include introduction to concepts of random sampling and statistical inference, goodness of fit tests for model adequacy, outlier detection, estimation and testing hypotheses of means and variances, analysis of variance, regression analysis and contingency tables. PREREQ: MATH 361.

**MATH 472 COMPUTATIONAL STATISTICS (3-0-3)(S).** Introduction to the trend in modern statistics of basic methodology supported by state-of-art computational and graphical facilities, with attention to statistical theories and complex real world problems. Includes: data visualization, data partitioning and resampling, data fitting, random number generation, stochastic simulation, Markov chain Monte Carlo, the EM algorithm, simulated annealing, model building and evaluation. A statistical computing environment will be used for students to gain hands-on experience of practical programming techniques. PREREQ: MATH 361.

**MATH 480 SENIOR PROJECT (3-4 credits)(On Demand).** Research on a mathematical problem in the form of a thesis, or work on an applied problem which could be provided by local industry. PREREQ: Senior standing.

**MATH 488 SENIOR OUTCOMES ASSESSMENT (0-0-0)(F,S). **Required to graduate. Senior Mathematics and Applied Mathematics students will take an outcome assessment examination. Senior Mathematics Secondary Education students will submit a portfolio and should take MATH 488 during their student teaching. (Pass/Fail.) PREREQ: Senior standing.

**MATH 491 PUTNAM PRACTICE II (1-0-1)(F). **Solving problems from previous Putnam examinations and related problems. May be repeated once for credit. (Pass/Fail.)

**MATH 498 SEMINAR IN MATHEMATICS (1-0-1)(F/S). **Seminars by mathematicians on a wide range of subjects, including advanced mathematical topics selected from texts, mathematical journals, and current research. Format may include student presentation and discussion. Students will attend seminars, write summaries, and search for relevant literature. May be repeated once for credit. (Pass/Fail.) PREREQ: PERM/INST.