# Mathematics Courses

NOTE: This page is maintained separately from the University Catalog. In case of discrepancy, the content of the University Catalog prevails.
** MATH 105: Fundamentals of Mathematics (3 cr).** Integers, rational numbers, exponents, order of operations. Functions in context, and their algebraic and graphical representation. Linear and quadratic equations. Introduction to the graphing calculator. (This course does not count toward the core curriculum requirement in mathematics.)

** MATH 113: Mathematics of Finance (3 cr). **Simple and compound interest, discounting, annuities, amortization and sinking funds, stocks, bonds, insurance. (Prerequisite: MATH 105 or by placement.)

** MATH 115: Topics in Applied Mathematics (3 cr). ** Topics in the application of elementary mathematics to real world problems: management science, voting schemes, theory of games, population growth, other models. (Prerequisite: MATH 105 or by placement.)

** MATH 116: Elementary Statistics (3 cr). ** Description of sample data. Simple probability, theoretical distributions, normal and binomial estimation. Tests of hypotheses, correlation, regression. (Prerequisite: MATH 105 or by placement.)

** MATH 120: Elementary Functions (3 cr). ** Graphs and properties of functions, including polynomial functions, exponential functions, logarithmic functions, inverse functions and composition of functions. Applications to real world situations using algebraic, numerical, and graphical methods. (Prerequisite: MATH 105 or by placement.)

** MATH 125: Mathematical Perspectives (3 cr). ** Exploration of easily accessible, engaging, and thematically connected mathematical ideas as a vehicle to lead students to experiences that are characteristic of the mathematical enterprise. (Prerequisite: MATH 105 or by placement.)

** MATH 147: Calculus from an Historical Perspective (3 cr).** An overview of concepts from differential and integral calculus through excerpted readings in English translation of original texts which emphasizes connections with developments in science and philosophy. (Prerequisite: MATH 120 or by placement.)

** MATH 150: Elements of Calculus I (3 cr). ** Modeling data with polynomial functions, exponential functions, and logistic functions. Rates of change and the derivative. Application of the derivative including optimization and inflection points. Result of cumulative change and the definite integral. (Prerequisite: MATH 120 or by placement.)

** MATH 151: Elements of Calculus II (3 cr).** Modeling with trigonometric functions, functions of several variables, contour maps, partial derivatives, and optimization with and without constraints. (Prerequisite: MATH 150.)

** MATH 154: Milestones in Mathematics (3 cr). ** Charts milestones in various branches of mathematics through the reading of original sources: number theory, infinity, Euclidean and non-Euclidean geometry, and algebra are all possible threads of development. (Prerequisite: MATH 120 or by placement.)

** MATH 156: General Statistics (3 cr). ** Descriptive statistics, basic probability, normal distribution, confidence intervals, regression, correlation, hypothesis tests, and analysis of categorical data. (Prerequisite: MATH 120 or by placement.)

** MATH 158: General Statistics II (3 cr). ** Building upon the introductory material presented in MATH 156 (or equivalent), this is a second course in statistical methods and data analysis. The course objectives are: 1) To study in detail the distinctions between observational studies and controlled experiments, the questions they can address and what types of statistical methods are appropriate for analyzing them 2) To learn some basic statistical tools used to analyze data, such as: 2 sample t-tests, analysis of categorical data and Goodness-of-Fit tests, multiple comparison procedures, multiple regression, analysis of variance (ANOVA), nonparametric methods (such as the randomization test and the rank-sum test), and log transformations 3) To get hands-on experience analyzing data and computing with data (using R) 4) To gain experience in interpreting the results of a statistical analysis and communicating the results to others. (Prerequisite: MATH 156, or MATH 116 with minimum grade of B.)

** MATH 169: Precalculus (3 cr). ** This is a study of linear, polynomial, rational, exponential, logarithmic, and trigonometric functions from symbolic, graphical, and numerical perspectives. Topics include algebraic and analytic properties of functions; sums, differences, products, quotients, and composites of functions; inverse functions; and functions as models. (Prerequisite: By placement.)

** MATH 170: Calculus I (4 cr).** Limits and continuity. Transcendental functions. The derivative, techniques of differentiation, and applications of the derivative. Parametric equations. The definite integral, numerical integration, antiderivatives, and method of substitution. (Prerequisite: MATH 169 or by placement.)

** MATH 171: Calculus II (4 cr).** Numerical integration, applications of the definite integral, techniques of integration, and improper integrals. Taylor polynomials. Sequences and series. Polar coordinates. (Prerequisite: MATH 170.)

** MATH 201: Foundations of Arithmetic - ECED (3 cr). ** Concepts necessary for understanding the structure of arithmetic and its algorithms (with whole numbers, integers, fractions and decimals), number patterns, and introductory probability and statistics. (Restrictions: Must be enrolled in one of these programs: Master of Education, B.S. in Education.)

** MATH 202: Geometry and Measurement - ECED (3 cr). ** Concepts necessary for an understanding of basic geometry: shapes in one, two, and three dimensions, scientific measurement and dimensional analysis, congruence and similarity of figures, compass and straightedge constructions, transformations, and coordinate geometry. Use of computer software to explore geometric concepts. (Restrictions: Must be enrolled in one of these programs: Master of Education, B.S. in Education.)

** MATH 211: Foundations of Arithmetic - MCED (3 cr).** Concepts necessary for understanding the structure of arithmetic, its algorithms and properties (with whole numbers, integers, rational and irrational numbers), basic set theory and introductory number theory. (Restrictions: Must be enrolled in one of these programs: Master of Education, B.S. in Education.)

** MATH 212: Geometry and Measurement - MCED (3 cr).** Concepts necessary for an understanding of basic geometry: shapes in one, two, and three dimensions, scientific measurement and dimensional analysis, congruence and similarity of figures, compass and straightedge constructions, transformations, coordinate geometry, conjecture and proof, perspective drawing and introductory trigonometry. Use of computer software to explore geometric concepts. (Restrictions: Must be enrolled in one of these programs: Master of Education, B.S. in Education.)

** MATH 213: Algebra Concepts - MCED (3 cr).** Development of algebraic problem solving, polynomials, linear, quadratic and exponential equations and functions, pattern representation, sequences and series. Use of technology and manipulative materials in the teaching of algebra. (Restrictions: Must be enrolled in one of these programs: Master of Education, B.S. in Education.)

** MATH 214: Mathematical Problem Solving - MCED (3 cr). ** Problem solving, drawing from a wide range of school mathematics topics, logic, combinatorics, and basic probability theory. (Prerequisites: MATH 211, MATH 212, MATH 213.)

** MATH 220: Calculus III (4 cr).** Vectors, lines and planes. Functions of several variables, partial derivatives and applications, gradient and directional derivative. Multiple integrals, line integrals, Green’s Theorem. (Prerequisite: MATH 171.)

** MATH 222: Applied Linear Algebra (3 cr).** An introduction to elementary linear algebra with an emphasis on application and interpretation. Topics include systems of linear equations and their solutions, matrix algebra, linear transformations, determinants, inverses, eigenvalues and eigenvectors, orthogonality. Selected applications to physical and social sciences. (Prerequisite: MATH 169 or by placement.)

** MATH 225: Foundations of Higher Mathematics (3 cr). **Propositional and predicate logic; methods of proof, including direct approaches, contradiction, contraposition, mathematical induction; sequences, recursion, recurrence relations; set theory; functions and relations. Primary emphasis on proof-writing. (Prerequisite: By placement.)

** MATH 230: Introduction to Ordinary Differential Equations (3 cr).** Modeling with ordinary differential equations. Analytical, qualitative, and numerical techniques for first-order equations, first-order nonlinear systems, and linear systems. (Prerequisite: MATH 171.)

** MATH 240: Linear Algebra (3 cr). ** Systems of linear equations, Gaussian elimination, echelon forms, basic and free variables, algebraic structure of solutions; vector arithmetic, partitioning of matrices, invertibility; linear transformations and their matrices, LU and PLU factorization; vector spaces and subspaces, bases, coordinates, dimension, rank; change of basis and similarity of matrices; determinants, cofactor expansion, elementary matrices, Cramer’s Rule; eigenvectors and eigenvalues, the characteristic equation; diagonalization; inner products, length and orthogonality of vectors, projections, the Gram-Schmidt process. Additional topics may include the dual space, linear functionals, least squares approximation, QR factoriztion, Riesz’ Representation Theorem, the Principal Axis Theorem, the Spectral Theorem. (Prerequisite: MATH 225; MATH 230 recommended.)

** MATH 256: Introduction to Probability and Statistics (3 cr).** Introduction to probability, descriptive statistics, the central limit theorem, estimation, hypothesis testing, regression and correlation, goodness of fit and linear models. (Prerequisite: MATH 171.)

** MATH 257: Data Modeling (3 cr). ** This is a course on applied statistics, which is supported by a statistical free-software R. The course aims to prepare a student to be a successful research manager, who can churn through huge data points and show where patterns emerge. The fundamental statistical methods as applied to practical problems will be taught in greater detail, so that the student will be able to extract meaningful statistics from raw data. The content includes methods of curve fitting, transformations of data, various regression techniques for both linear and nonlinear regressions, the generalized linear model, model selections and diagnostics, analysis of the categorical data, analysis of variance, and distribution free procedures. (Prerequisite: MATH 256.)

** MATH 280: Combinatorics (3 cr)**. An introduction to counting techniques of discrete objects. The enumeration of sets, permutations and combinations, the binomial and multinomial theorem will serve as an appetizer; counting methods including the inclusion-exclusion principle; the pigeonhole principle, generating functions, and recurrence relations will be the main course. Applications of combinatorial techniques and problem solving will be emphasized. [Optional: finite geometries, permutation groups, latin squares, designs, and codes.] (Prerequisite: MATH 225.)

** MATH 300: History of Mathematics (3 cr).** Some of the highlights in the historical development of mathematics with special attention given to the invention of non-Euclidean geometry and its importance for mathematics and Western thought. (Prerequisites: MATH 220, MATH 240.)

** MATH 301: Geometry (3 cr). ** Axiom systems, models and finite geometries, convexity, transformations, Euclidean constructions, and the geometry of triangles and circles. Introduction to projective and non-Euclidean geometries. (Prerequisite: MATH 225.)

** MATH 302: Number Theory (3 cr). ** Divisibility and primes, linear congruences, quadratic residues and reciprocity. Diophantine equations, multiplicative functions, distribution of primes. (Prerequisite: MATH 240.)

** MATH 303: Mathematical Logic (3 cr).** Axiomatic development of propositional calculus, functional complete sets of operators, axiomatic development of the first order function calculus, the existential operator, the algebra of logic. (Prerequisite: MATH 225.)

** MATH 311: Probability Theory (3 cr). ** Sample spaces, basic axioms of probability, conditional probability and independence, Bayes’ theorem, random variables, expectation and variance, discrete and continuous distributions, central limit theorem. (Prerequisite: MATH 171.)

** MATH 312: Statistical Inference (3 cr). ** Parameter estimation, sufficient statistics, maximum likelihood principle, confidence intervals, hypothesis testing, linear regression, nonparametric methods, analysis of variance, simulation. (Prerequisite: MATH 256 or MATH 311.)

** MATH 316: Cryptology (3 cr). ** In this course we will be exploring cryptology - the making and breaking of secret ciphers and codes. We will start with classical ciphers: shift, affine, Vigenere, substitution, Hill, one-time pads, Enigma, etc. Then after a brief introduction to number theory, we will start on modern methods: the RSA algorithm, DES, and AES: Rijndael. If time permits, we will also explore discrete logarithms and elliptic curves. All codes will be placed in historical perspective by exploring the political and military contexts in which they were devised, through readings in The Code Book. (Prerequisite: MATH 225.)

** MATH 321: Numerical Analysis (3 cr).** Accuracy, function evaluation and approximation, systems of linear equations, nonlinear equations, numerical differentiation and integration, and solutions to differential equations. (Prerequisite: CSCI 170 and MATH 171.)

** MATH 325: Mathematical Modeling (3 cr). ** The synthesis, formulation and solution of various problems in applied mathematics and related fields. (Prerequisite: MATH 230.)

** MATH 330: Graph Theory (3 cr). **Graphs, subgraphs, trees, isomorphism, Eulerian and Hamiltonian paths, planarity, digraphs, connectivity, and chromatic number. Other topics may be included. (Prerequisite: MATH 225.)

** MATH 340: Abstract Algebra I (3 cr).** Groups, isomorphism, homomorphism, normal subgroups, rings, ideals, fields. (Prerequisites: MATH 225, MATH 240.)

** MATH 341: Abstract Algebra II (3 cr).** A continuation of MATH 340. Topics may include Boolean algebra, lattice theory, combinational group theory, coding theory, Galois theory, commutative rings. (Prerequisite: MATH 340.)

** MATH 360: Elementary Topology (3 cr).** Metric spaces, topological spaces, continuity, convergence, compactness, connectedness, and separation axioms. (Prerequisite: MATH 240.)

** MATH 370: Real Analysis (3 cr). ** Rigorous development of calculus of functions of a single variable. The real number system, topology of the real line, continuity, uniform continuity, the derivative, the Riemann integral, sequences and series of real numbers, and uniform convergence. (Prerequisites: MATH 220 and MATH 225.)

** MATH 372: Applied Analysis (3 cr). ** Special functions, orthogonal sets of functions. Sturm-Liouville theory. Partial Differential Equations. Fourier series, integrals and transforms. (Prerequisite: MATH 230.)

** MATH 380: Introduction to Complex Variables (3 cr). ** Complex numbers, analytic functions, complex integration, series representation of analytic functions, the calculus of residues. (Prerequisites: MATH 220 and MATH 225.)

** MATH 385: Secondary Mathematics from an Advanced Perspective (3 cr).** The Mathematical Education of Teachers, a 2001 report of the Conference Board of the Mathematical Sciences, recommended that prospective teachers of high school mathematics take a capstone course in which “conceptual difficulties, fundamental ideas, and techniques of high school mathematics are examined from an advanced prospective”. This course is intended to fulfill such a role for Xavier students pursuing secondary licensure. The primary learning mechanisms will be reading, problem solving, and communicating and justifying one’s mathematical thinking to others, both verbally and in written form. Although the mathematical content of this ocurse is rooted in high school mathematics, we will approach it from a sophisticated undergraduate level that emphasizes the many interconnections among high school mathematics topics and includes analyses that reveal important insights and understandings not always considered in other undergraduate mathematics courses. (Prerequisite: MATH 340.)

** MATH 391: Mathematics Seminar 1 (1 cr). ** Juniors (MATH 391) and seniors (MATH 393) meet together in the spring semester. Students will read selections from the mathematical literature, explore how to write mathematics effectively, learn how to use technical word processing tools, practice how to communicate mathematical ideas and give oral presentations. (Restriction: Must be enrolled as a major in mathematics.)

** MATH 392: Mathematics Seminar 2 (1 cr). ** Each senior will meet with a faculty advisor to work on an individual research project. (Restriction: Must be enrolled as a major in mathematics.)

** MATH 393: Mathematics Seminar 3 (1 cr). ** The student will write a paper and give a formal presentation describing the project developed during MATH 392. (Prerequisite: MATH 392; Restriction: Must be enrolled as a major in mathematics.)

** MATH 397: Special Study (0-6 cr). ** Credit by special arrangement. Area to be specified. (Prerequisite: MATH 225.)