Mathematics Minor

This is the first in a two-course sequence intended for students majoring in mathematics, science, or engineering. It emphasizes the understanding of concepts, and using them to solve physical problems. The course covers functions, limits, continuity, the derivative, rules of differentiation, applications of the derivative, Riemann sums, definite integrals, and indefinite integrals. (Prerequisites: MATH-111 or (NMTH-220 and NMTH-260 or NMTH-272 or NMTH-275) or equivalent courses with a minimum grade of B-, or a score of at least 60% on the RIT Mathematics Placement Exam.) Lecture 4 (Fall, Spring).

This is the first in a two-course sequence intended for students majoring in mathematics, science, or engineering. The course includes the same topics as MATH-181, but the focus of the workshop component is different. Whereas workshops attached to 181 emphasize concept development and real-world applications, the workshops of MATH-181A emphasize skill development and provide just-in-time review of precalculus material as needed. The course covers functions, limits, continuity, the derivative, rules of differentiation, applications of the derivative, Riemann sums, definite integrals, and indefinite integrals. (Prerequisite: B- or better in MATH-111 or B- or better in ((NMTH-260 or NMTH-272 or NMTH-275) and NMTH-220) or a math placement exam greater than or equal to 60.) Lecture 6 (Fall, Spring).

This is the first course in a three-course sequence (COS-MATH-171, -172, -173). This course includes a study of precalculus, polynomial, rational, exponential, logarithmic and trigonometric functions, continuity, and differentiability. Limits of functions are used to study continuity and differentiability. The study of the derivative includes the definition, basic rules, and implicit differentiation. Applications of the derivative include optimization and related-rates problems. (Prerequisites: Completion of the math placement exam or C- or better in MATH-111 or C- or better in ((NMTH-260 or NMTH-272 or NMTH-275) and NMTH-220) or equivalent course.) Lecture 5 (Fall, Spring).

This is the second course in three-course sequence (COS-MATH-171, -172, -173). The course includes Riemann sums, the Fundamental Theorem of Calculus, techniques of integration, and applications of the definite integral. The techniques of integration include substitution and integration by parts. The applications of the definite integral include areas between curves, and the calculation of volume. (Prerequisites: C- or better in MATH-171 or 1016-171T or 1016-281 or 1016-231 or equivalent course.) Lecture 5 (Fall, Spring).

This is the second in a two-course sequence. It emphasizes the understanding of concepts, and using them to solve physical problems. The course covers techniques of integration including integration by parts, partial fractions, improper integrals, applications of integration, representing functions by infinite series, convergence and divergence of series, parametric curves, and polar coordinates. (Prerequisites: C- or better in MATH-181 or MATH-181A or equivalent course.) Lecture 4 (Fall, Spring).

This course introduces students to ideas and techniques from discrete mathematics that are widely used in Computer Science. Students will learn about the fundamentals of propositional and predicate calculus, set theory, relations, recursive structures and counting. This course will help increase students’ mathematical sophistication and their ability to handle abstract problems. (Co-requisites: MATH-182 or MATH-182A or MATH-172 or equivalent courses.) Lecture 3, Recitation 1 (Fall, Spring).

This course prepares students for professions that use mathematics in daily practice, and for mathematics courses beyond the introductory level where it is essential to communicate effectively in the language of mathematics. It covers various methods of mathematical proof, starting with basic techniques in propositional and predicate calculus and set theory, and then moving to applications in advanced mathematics. (Prerequisite: MATH-182 or equivalent course.) Lecture 3, Recitation 4 (Fall, Spring).

This course is principally a study of the calculus of functions of two or more variables, but also includes the study of vectors, vector-valued functions and their derivatives. The course covers limits, partial derivatives, multiple integrals, and includes applications in physics. Credit cannot be granted for both this course and MATH-221. (Prerequisite: C- or better MATH-173 or MATH-182 or MATH-182A or equivalent course.) Lecture 3 (Fall, Spring, Summer).

This course is principally a study of the calculus of functions of two or more variables, but also includes a study of vectors, vector-valued functions and their derivatives. The course covers limits, partial derivatives, multiple integrals, Stokes' Theorem, Green's Theorem, the Divergence Theorem, and applications in physics. Credit cannot be granted for both this course and MATH-219. (Prerequisite: C- or better MATH-173 or MATH-182 or MATH-182A or equivalent course.) Lecture 4 (Fall, Spring, Summer).

This course is an honors version of MATH-221. It includes an introduction to vectors, surfaces, and multivariable functions. It covers limits, partial derivatives and differentiability, multiple integrals, Stokes’ Theorem, Green’s Theorem, the Divergence Theorem, and applications. Unlike MATH-221, students in this course will often be expected to learn elementary skills and concepts from their text so that in-class discussion can focus primarily on extending techniques, interpreting results, and exploring mathematical topics in greater depth; homework exercises and projects given in this class will require greater synthesis of concepts and skills, on average, than those in MATH-221. Students earning credit for this course cannot earn credit for MATH-219 or MATH-221. (Prerequisites: C or better in MATH-182 or MATH-173 or MATH-182A and Honors program status or at least a 3.2 cumulative GPA.) Lecture 4 (Fall).

This course is an introduction to the study of ordinary differential equations and their applications. Topics include solutions to first order equations and linear second order equations, method of undetermined coefficients, variation of parameters, linear independence and the Wronskian, vibrating systems, and Laplace transforms. (Prerequisite: MATH-173 or MATH-182 or MATH-182A or equivalent course.) Lecture 3, Recitation 1 (Fall, Spring, Summer).

This is an introductory course in linear algebra and ordinary differential equations in which a scientific computing package is used to clarify mathematical concepts, visualize problems, and work with large systems. The course covers matrix algebra, the basic notions and techniques of ordinary differential equations with constant coefficients, and the physical situation in which they arise. (Prerequisites: MATH-172 or MATH-182 or MATH-182A and students in CHEM-BS or CHEM-BS/MS or ISEE-BS programs.) Lecture 4 (Spring).

This course is an introduction to the basic concepts of linear algebra, and techniques of matrix manipulation. Topics include linear transformations, Gaussian elimination, matrix arithmetic, determinants, vector spaces, linear independence, basis, null space, row space, and column space of a matrix, eigenvalues, eigenvectors, change of basis, similarity and diagonalization. Various applications are studied throughout the course. (Prerequisites: MATH-190 or MATH-200 or MATH-219 or MATH-220 or MATH-221 or MATH-221H or equivalent course.) Lecture 3 (Fall, Spring).

This honors course introduces the basic concepts and techniques of linear algebra. Concepts are addressed at a higher level than the standard course in linear algebra, and the topic list is somewhat broader. Topics include linear independence and span, linear functions, solving systems of linear equations using Gaussian elimination, the arithmetic and algebra of matrices, basic properties and interpretation of determinants, vector spaces, the fundamental subspaces of a linear function, eigenvalues and eigenvectors, change of basis, similarity and diagonalization. Students will learn to communicate explanations of mathematical facts and techniques by participating in a collaborative workshop format, and will learn to use MATLAB to solve matrix equations. (Prerequisites: MATH-219 or MATH-221 or MATH-221H or equivalent course and Honors program status or at least a 3.2 cumulative GPA.) Lecture 3 (Spring).

This course introduces sample spaces and events, axioms of probability, counting techniques, conditional probability and independence, distributions of discrete and continuous random variables, joint distributions (discrete and continuous), the central limit theorem, descriptive statistics, interval estimation, and applications of probability and statistics to real-world problems. A statistical package such as Minitab or R is used for data analysis and statistical applications. (Prerequisites: MATH-173 or MATH-182 or MATH 182A or equivalent course.) Lecture 3, Recitation 1 (Fall, Spring, Summer).

This course is an introduction to mathematical techniques and algorithms for (pseudo-)random number generation and simulation. Randomness and simulation are major tools used in mathematical modeling, statistical and data analysis, computing, and engineering. This course will provide both a solid mathematical foundation in these topics and computational experience utilizing them in practice. (Prerequisites: (MATH-190 or MATH-200) and MATH-251 and (GCIS-123 or CSCI-141) or equivalent courses.) Lecture 3 (Fall).

This course presents the general linear programming problem. Topics include a review of pertinent matrix theory, convex sets and systems of linear inequalities, the simplex method of solution, artificial bases, duality, parametric programming, and applications. (Prerequisites: MATH-241 or MATH-241H or equivalent course.) Lecture 3 (Spring).

This course provides a study of the theory of optimization of non-linear functions of several variables with or without constraints. Applications of this theory in business, management, engineering and the sciences are considered. Algorithms for practical applications will be analyzed and implemented. The course may require the use of specialized software to analyze problems. Students taking this course will be expected to complete applied projects and/or case studies. (Prerequisites: (MATH-219 or MATH-221 or MATH-221H) and MATH-311 or equivalent course.) Lecture 3 (Spring).

Classical game theory models conflict and cooperation between rational decision-making agents with hidden parameters. Topics include matrix games, Nash equilibria, the minimax theorem, prisoner’s dilemma, and cooperative games. Applications can include adaptive or statistical decision theory, artificial intelligence (online learning, multi-agent systems), biology (evolutionary games, signaling behavior, fighting behavior), economics and business (auctions, bankruptcy, bargaining, pricing, two-sided markets), philosophy (ethics, morality, social norms), and political science (apportionment, elections, military strategy, stability of government, voting). (Prerequisites: MATH-241 or MATH-241H or equivalent course.) Lecture 3 (Fall).

Combinatorial games are two-player games with perfect information and no randomness or element of chance (such as Go, Chess, and Checkers). The course covers basic techniques of game theory, outcome classes, sums of games, the algebra of games, and top-down induction. Analyses will emphasize no-draw games terminating in a finite number of moves such as Nim, Domineering, Hackenbush, Chomp, and Amazons. (Prerequisites: MATH-190 or MATH-200 or equivalent course.) Lecture 3 (Fall).

This course provides an introduction to boundary value problems. Topics include Fourier series, separation of variables, Laplace's equation, the heat equation, and the wave equation in Cartesian and polar coordinate systems. (Prerequisites: (MATH-231 or MATH-233) and (MATH-219 or MATH-221) or equivalent courses.) Lecture 3 (Fall, Spring).

The course revisits the equations of spring-mass system, RLC circuits, and pendulum systems in order to view and interpret the phase space representations of these dynamical systems. The course begins with linear systems followed by a study of the stability analysis of nonlinear systems. Matrix techniques are introduced to study higher order systems. The Lorentz equation will be studied to introduce the concept of chaotic solutions. (Prerequisites: (MATH-231 and (MATH-241 or MATH-241H)) or MATH-233 or equivalent courses.) Lecture 3 (Spring).

This course introduces the mathematical theory of enumeration of discrete structures. Topics include enumeration, combinatorial proofs, recursion, inclusion-exclusion, and generating functions. (Prerequisites: MATH-190 or MATH-200 or equivalent course.) Lecture 3 (Spring).

This course will introduce, explain and employ both the classical and modern basic techniques of cryptography. Topics will include the Vignère cipher, affine ciphers, Hill ciphers, one-time pad encryption, Enigma, public key encryption schemes (RSA, Diffie-Hellman, El-Gamal, elliptic curves), and hash functions. The course will include an introduction to algebraic structures and number theoretic tools used in cryptography. (Prerequisites: MATH-190 or MATH-200 or equivalent course.) Lecture 3 (Spring).

The course introduces students to basic techniques of classical and modern cryptography, and learn about the significant impact of codes and ciphers on historical events. Topics will include the Vignère cipher, affine ciphers, Hill ciphers, one-time pad encryption, Enigma, public key encryption schemes (RSA, Diffie-Hellman, elliptic curves), and cryptographic hash functions. The course will include an introduction to algebraic structures and number theoretic tools used in cryptography. Students in this honors course will also study and explore historical source documents to get first-hand exposure to critical aspects of cryptanalysis from the early to mid-20th century. (Prerequisites: MATH-190 or MATH-200 or equivalent course and Honors program status or at least a 3.3 cumulative GPA.) Lecture 3 (Fall).

This course covers the algebra of complex numbers, analytic functions, Cauchy-Riemann equations, complex integration, Cauchy's integral theorem and integral formulas, Taylor and Laurent series, residues, and the calculation of real-valued integrals by complex-variable methods. (Prerequisites: MATH-219 or MATH-221 or equivalent course.) Lecture 3 (Fall, Spring).

This is a second course in linear algebra that provides an in-depth study of fundamental concepts of the subject. It focuses largely on the effect that a choice of basis has on our understanding of and ability to solve problems with linear operators. Topics include linear transformations, similarity, inner products and orthogonality, QR factorization, singular value decomposition, and the Spectral Theorem. The course includes both computational techniques and the further development of mathematical reasoning skills. (Prerequisites: MATH-241 or MATH-241H or equivalent course.) Lecture 3 (Spring, Summer).

This course covers the theory of graphs and networks for both directed and undirected graphs. Topics include graph isomorphism, Eulerian and Hamiltonian graphs, matching, covers, connectivity, coloring, and planarity. There is an emphasis on applications to real world problems and on graph algorithms such as those for spanning trees, shortest paths, and network flows. (Prerequisites: MATH-190 or MATH-200 or equivalent course.) Lecture 3 (Fall).

This course provides an introduction to the study of the set of integers and their algebraic properties. Topics include prime factorization and divisibility, linear Diophantine equations, congruences, arithmetic functions, primitive roots, and quadratic residues. (Prerequisites: MATH-190 or MATH-200 or equivalent course.) Lecture 3 (Spring).

This course covers numerical techniques for the solution of nonlinear equations, interpolation, differentiation, integration, and the solution of initial value problems. (Prerequisites: (MATH-231 and (MATH-241 or MATH-241H)) or MATH-233 or equivalent courses.) Lecture 3 (Fall).

This course covers numerical techniques for the solution of systems of linear equations, eigenvalue problems, singular values and other decompositions, applications to least squares, boundary value problems, and additional topics at the discretion of the instructor. (Prerequisites: (MATH-220 or MATH-221 or MATH-221H or 1055-359 (Honors Multivariable Calculus)) and (MATH-231 and MATH-341) or equivalent courses.) Lecture 3 (Spring).

This course is an investigation and extension of the theoretical aspects of elementary calculus. Topics include mathematical induction, real numbers, sequences, functions, limits, and continuity. The workshop will focus on helping students develop skill in writing proofs. (Prerequisites: (MATH-190 or MATH-200 or 1055-265) and (MATH-220 or MATH-221 or MATH-221H or 1016-410 or 1016-328) or equivalent courses.) Lec/Lab 4 (Fall, Spring).

This course is a continuation of MATH-431. It concentrates on differentiation, integration (Riemann and Riemann-Stieltjes integrals), power series, and sequences and series of functions. (Prerequisites: MATH-431 or equivalent course) Lecture 3 (Spring).

This course covers basic set theory, number theory, groups, subgroups, cyclic and permutation groups, Lagrange and Sylow theorems, quotient groups, and isomorphism theorems. Group Theory finds applications in other scientific disciplines like physics and chemistry. (Prerequisites: (MATH-190 or MATH-200 or 1055-265) and (MATH-241 or MATH-241H) or equivalent courses.) Lec/Lab 4 (Fall, Spring).

This course covers the basic theory of rings, integral domains, ideals, modules, and abstract vector spaces. It also covers the key constructions including direct sums, direct products, and field extensions. These topics serve as the foundation of mathematics behind advanced topics such as algebraic geometry and various applications like cryptography and coding theory. (Prerequisites: MATH-441 or equivalent course.) Lecture 3 (Spring).

This course defines metric spaces and topological spaces. For metric spaces it examines continuity spaces of continuous functions and completeness in Euclidean spaces. For topological spaces it examines compactness, continuous functions, and separation axioms. (Prerequisites: MATH-432 or equivalent course.) Lecture 3 (Spring).

This course explores Poisson processes and Markov chains with an emphasis on applications. Extensive use is made of conditional probability and conditional expectation. Further topics, such as renewal processes, Brownian motion, queuing models and reliability are discussed as time allows. (Prerequisites: (MATH-241 or MATH-241H) and MATH-251 or equivalent courses.) Lecture 3 (Spring).