# Mathematics

The Department of Mathematics offers a B.S. degree in mathematics, a B.S. degree in Data Science and a double major in mathematics and secondary education that includes teaching certification. The objective of the bachelor’s degree program in mathematics is to prepare students for professional careers in a variety of industries and for graduate programs leading to the M.S. and Ph.D. Students also may opt for advanced degrees in education, business administration, law, or medicine.

Well-versed in a wide range of mathematical concepts, from algebraic topology to statistics and data science, the faculty in Saint Joseph's Department of Mathematics are dedicated to sharing their knowledge and research experience with students. Classroom sizes are small and hands-on, providing students with the opportunity to work closely with mathematics faculty and prepare for successful careers after graduation.

**MAT 101 Mathematical Explorations I (3 credits)**

For humanities majors, the course covers set theory and a number of its applications, topics from logic including propositions, truth tables, number systems, and elementary geometry. Other topics may be covered at instructor's discretion.

*Restrictions:* Enrollment is limited to PLS/HDC level students.

*Attributes:* Undergraduate

**MAT 102 Mathematical Explorations II (3 credits)**

This is a second course for humanities majors. The course covers elementary probability, including independent and dependent events, conditional probability, binomial probability, and certain applications in a wide variety of situations. MAT 101 is not required for MAT 102.

*Restrictions:* Enrollment is limited to PLS/HDC level students.

*Attributes:* Undergraduate

**MAT 103 Quantitative Appl in Business (3 credits)**

Topics in finite mathematics: matrices, solving linear systems, optimization using linear programming, simplex algorithm. Pre-calculus topics: linear, quadratic, exponential, and logarithmic functions and their graphs, mathematical models, and certain applications.

*Restrictions:* Enrollment is limited to PLS/HDC level students.

*Attributes:* Undergraduate

**MAT 107 Contemporary Topics in Math (3 credits)**

This course is designed to enable the student to recognize, understand and apply various mathematical concepts and principles that are the foundation for many things that we take for granted in our everyday lives, such as Voting, Traveling, Finances, Government and the wonders of Nature.

*Attributes:* Undergraduate

**MAT 110 Fundamental Math for Educators (3 credits)**

This course is designed to ensure that pre-service educators have a deep understanding of the essential mathematical core standards and competencies required to enter the teaching profession. Students will develop basic mathematical skills, will be able to employ problem solving strategies, will be able to communicate mathematical concepts, and will be able to construct and evaluate mathematical arguments.

*Attributes:* Undergraduate

**MAT 111 The Mathematics of Patterns (3 credits)**

This course focuses on mathematics as the science of identifying, understanding and describing patterns. Patterns that occur in nature and empirical studies can be identified and modeled using fundamental ideas such as functions (mathematical rules), probability (long term behavior), exploratory data analysis (statistics) and geometry. Through a series of guided investigations students will master the reasoning used to identify the patterns, the mathematical model used to describe the pattern and the computational techniques necessary to further explore and apply the pattern in new situations. This course is designed specifically for students intending to become elementary or middle school teachers.

*Attributes:* Undergraduate

**MAT 118 Introduction to Statistics (3 credits)**

Introduction to statistics and probability: measures of central tendency, variability, correlation, regression, chance and randomness, random variables, probability distributions, law of large numbers, central limit theorem. Students will be required to use a computer software package to solve various statistical problems. Designed for Social Science majors. Students may NOT receive credit for this course and for MAT 128.

*Attributes:* Undergraduate

**MAT 120 The Mathematics of Modeling (3 credits)**

This course focuses on functions, graphs, and algebraic techniques. Topics include an introduction to functions and graphs, linear, exponential, logarithmic, and trigonometric functions. Functions are used for solving multidisciplinary application problems.

*Prerequisites:* Math Placement with a score of MA120

*Attributes:* Undergraduate

**MAT 121 Math Modeling for MS Teachers (3 credits)**

Designed for students who will become middle school teachers, this course will explore mathematical topics in the context of building of building models to solve problems. The emphasis will be on using multiple representations to develop mathematical models that describe some phenomena and learning the mathematical techniques necessary for working with the model in order to effectively answer questions about the situation being modeled. Students will interpret results given the context of the model and develop their communication skills for explaining mathematics.

**MAT 122 Trigonometry (3 credits)**

Topics include Angle Measurements; Triangles; Trigonometric and Inverse Trigonometric Functions and Graphs; Solving Trigonometric Equations; Essential Trigonometric Identities; Laws ofSine, Cosine, and Tangent; Vectors; Parametric Equations; Polar Coordinates. This course will emphasize application and modeling problems related to the topics.

**MAT 123 Differential Calculus (3 credits)**

Review of mathematical models using polynomial, rational, exponential and logarithmic functions with business applications. Introduction to differential calculus including limits, rates of change and the derivative, optimization using the derivative. Students may NOT receive credit for both this course and for any of the following courses: MAT 155 or MAT 161.

*Attributes:* Undergraduate

**MAT 128 Applied Statistics (3 credits)**

Introduction to statistics and probability: design of a study, measures of central tendency, variability, correlation, regression; probability, random variables, probability distributions, central limit theorem; inferential statistics, hypothesis testing, etc. Students will be required to use a computer software package to solve various statistical problems. Data analysis projects will be assigned. Students may NOT receive credit for both this course and for MAT 118.

*Attributes:* Undergraduate

**MAT 130 Whole Truth about Whole Number (3 credits)**

This course involves studying properties of natural numbers and integers. Topics include divisibility, prime numbers, the Euclidean Algorithm and cryptography for putting messages into code.

*Prerequisites:* Math Placement with a score of BEAUT or Math Placement with a score of MA155 or Math Placement with a score of MA161 or Math Placement with a score of MA162

*Attributes:* Math Beauty, Undergraduate

**MAT 131 Linear Methods (3 credits)**

This course studies basic properties and applications of matrices and vectors. Then, matrices and vectors will be used in a variety of applications, including vector geometry, elementary graph theory, solving word problems involving systems of linear equations, least-squares functions, and geometric transformations. The course also covers some topics in basic logic, including logical operators, the conditional, truth tables, quantifiers, and syllogisms. Students in this course will be required to have a graphing calculator that can perform standard matrix operations.

*Prerequisites:* Math Placement with a score of BEAUT or Math Placement with a score of MA155 or Math Placement with a score of MA161 or Math Placement with a score of MA162

*Attributes:* Math Beauty, Undergraduate

**MAT 132 Math of Games & Politics (3 credits)**

This course will focus on both computational and theoretical aspects of probability theory, game theory and social choice theory. Topics include expected value, counting methods and conditional probability, dominant strategies, combinatorial games, Nash equilibria, social dilemmas and, for zero sum games, saddle points and the Minimax theorem. Social choice theory topics include voting methods, weighted voting, fairness criteria and impossibility theorems.

*Prerequisites:* Math Placement with a score of BEAUT or Math Placement with a score of MA155 or Math Placement with a score of MA161 or Math Placement with a score of MA162

*Attributes:* Math Beauty, Undergraduate

**MAT 134 Math of Uncertainty:Rules/Prob (3 credits)**

This course provides students with an in-depth introduction to probability and its many real-life applications. Students will study counting techniques including permutations, combinations, binomial coefficients, occupancy problems and runs within random orderings and will prove combinatorial identities. Students will study topics in probability including sample spaces, DeMorgan's Laws, conditional probability, independent events, Bayes Theorem, random variables and expected value. Students will examine many of the classical problems in probability theory including Prisoner's Dilemma, Gambler's Ruin and the Birthday Problem as well as lotteries, card games and random walks.

*Prerequisites:* Math Placement with a score of BEAUT or Math Placement with a score of MA155 or Math Placement with a score of MA161 or Math Placement with a score of MA162

*Attributes:* Math Beauty, Undergraduate

**MAT 135 Sounding Number: Music & Math (3 credits)**

Music has many connections to mathematics. The ancient Greeks discovered that chords with pleasing sounds are related to simple ratios of integers. Other connections include equations describing the sounds of musical instruments, the mathematics of digital recording, the use of symmetry in composition, and the systematic exploration of patterns by African and Indian drummers. This course introduces basic concepts in trigonometry, set and group theory, and combinatorics and investigates their applications in the analysis, recording, and composition of music. Along the way, we consider the role of creativity in mathematics and the ways in which mathematics has inspired musicians. The course will involve hands-on laboratory work in audio engineering and music composition.

*Prerequisites:* Math Placement with a score of BEAUT or Math Placement with a score of MA155 or Math Placement with a score of MA161 or Math Placement with a score of MA162

*Attributes:* Math Beauty, Undergraduate

**MAT 138 Symmetry (3 credits)**

"Symmetry" is a ubiquitous concept in modern mathematics and science. Certain shapes and images seem more symmetric than others, yet is not immediately obvious how to best measure and understand an object's symmetry. In fact, the quest to more precisely quantify the concept of symmetry has been a driving force in science and mathematics, and will form the central theme of this course.

*Prerequisites:* Math Placement with a score of BEAUT or Math Placement with a score of MA155 or Math Placement with a score of MA161 or Math Placement with a score of MA162

*Attributes:* Math Beauty, Undergraduate

**MAT 150 First Year Seminar (3 credits)**

This course investigates several beautiful topics within mathematics. Depending on the instructor, these topics might include: prime numbers, the different sizes of infinity, the Platonic solids, the fourth dimension, fractals, chaos, probability, and the math of voting.

*Attributes:* First-Year Seminar, Undergraduate

**MAT 155 Fundamentals of Calculus (3 credits)**

This course covers differential calculus and the beginning of integral calculus. Topics include limits, continuity, differentiation, applications of derivatives, indefinite and definite integrals, and the fundamental theorem of calculus.

*Prerequisites:* MAT 120 or Math Placement with a score of MA162 or Math Placement with a score of MA161 or Math Placement with a score of MA155 or MA 107

*Restrictions:* Students cannot enroll who have a major in Actuarial Science, Chemistry, Mathematics or Physics.

*Attributes:* Math Beauty, Undergraduate

**MAT 161 Calculus I (4 credits)**

Limits; slopes, rates of change and the derivative; techniques of differentiation; implicit differentiation; derivatives of transcendental functions; related rates; linear approximation; L'Hosptial's Rule; the Mean Value Theorem; applications of differentiation (including curve sketching and optimization); introduction to integration; the Fundamental Theorem of Calculus. Students may NOT receive credit for both this course and for any of the following courses: MAT 123 or MAT 155.

*Prerequisites:* MAT 120 or Math Placement with a score of MA162 or Math Placement with a score of MA161

*Attributes:* Math Beauty, Undergraduate

**MAT 162 Calculus II (4 credits)**

Areas between curves, volumes, average value of a function, integration by parts, improper integrals, approximate integration, arc length, area of a surface of revolution, differential equations and applications, parametric curves, polar coordinates, and Taylor polynomials.

*Prerequisites:* MAT 161 or MAT 155 or Math Placement with a score of MA162 or MA 110 or MA 122

*Attributes:* Math Beauty, Undergraduate

**MAT 170 Special Topics in Mathematics (3 credits)**

Topics will vary according to the semester in which the class is offered.

*Attributes:* Undergraduate

**MAT 210 Theory of Numbers (3 credits)**

Division Algorithm; Mathematical induction; Euclidean algorithm; fundamental theorem of arithmetic; linear Diophantine equations; modular arithmetic; number theoretic functions; prime numbers; Fermat's last theorem; quadratic residues, primitive roots, Chinese Remainder theorem. This course fulfills the GEP Mathematics requirement but is at a slightly more advanced level than courses in the range of MAT 130 - MAT 139.

*Attributes:* Math Beauty, Undergraduate

**MAT 213 Calculus III (4 credits)**

Study of vectors, curves, functions of several variables, partial derivatives, tangent planes, maximum and minimum values, Lagrange multipliers, double integrals, triple integrals, spherical and cylindrical coordinates, vector fields, line integrals, sequences, series, convergence tests, power series, and Taylor series.

*Prerequisites:* MAT 162 or MA 221

*Attributes:* Math Beauty, Undergraduate

**MAT 226 Introduction to Linear Algebra (3 credits)**

Linear systems, matrices, linear transformations, determinants, vector spaces, dimension, orthogonality, eigenvalues, eigenvectors, and diagonalization.

*Prerequisites:* MAT 155 or MAT 161

*Attributes:* Math Beauty, Undergraduate

**MAT 232 Chaos, Fractals & Dynamic Syst (3 credits)**

Introduction to dynamical systems: one dimensional dynamics; attracting, repelling, periodic and chaotic orbits; bifurcation; dynamics in the complex plane, Julia sets, the Mandelbrot set; two dimensional dynamics. Introduction to fractals: self-similarity, iterated function systems, fractal dimension.

*Prerequisites:* MAT 162

*Attributes:* Math Beauty, Undergraduate

**MAT 233 History of Mathematics (3 credits)**

Development of mathematical ideas over 2500 years, beginning with Greek geometry and including Euclid, Archimedes, Newton, Euler, Gauss, and Poincare.

*Prerequisites:* MAT 161 or MAT 155

*Attributes:* Math Beauty, Undergraduate

**MAT 238 Differential Equations (3 credits)**

Solution of ordinary differential equations using analytic, numerical, and qualitative techniques. Modeling via differential equations, systems of differential equations. Laplace transforms; discrete dynamical systems. Use of a computer software package is required.

*Prerequisites:* MAT 162 or MA 221

*Attributes:* Math Beauty, Undergraduate

**MAT 250 Fundamentals of Mathematics (3 credits)**

The course consists of the following elements. Fundamentals of Logic: propositional logic, predicate logic, rules of inference and proofs; Fundamentals of Algebra: sets, functions, and relations; Fundamentals of Counting and Set Theory: natural numbers, mathematical induction, cardinality of sets, infinity; Fundamentals of Analysis: building number systems, basics of epsilon-delta calculus, continuity, convergence, and uniform convergence.

*Prerequisites:* MAT 161 or MAT 155

*Attributes:* Math Beauty, Undergraduate

**MAT 270 Special Topics in Mathematics (3 credits)**

Topics will vary according to the semester in which the class is offered.

*Attributes:* Undergraduate

**MAT 290 Career Prep Seminar (1 credit)**

What will YOU do with your degree in Mathematics, Computer Science, Information Technology, or Actuarial Science? Learn how to explore the variety of professional options open to you based on your major and on your unique personality and individual traits. This professional development seminar will help you build practical skills through a series of hands on assignments, a detailed self-assessment, goal-planning for a successful future, and multiple points of engagement with alumni. This one-credit course meets once a week to provide instruction and support in topics including internship search and application, resume/cover letter prep, post-grad options, professional communication and networking/interviewing . Is an internship right for you? Do you know the best way to search for one? Why do I need to network - how will that help me? Register now to get the answers to these and many other questions and invest in your future!

*Attributes:* Undergraduate

**MAT 311 Numerical Analysis (3 credits)**

An introduction to numerical methods for solving a variety of problems. Included will be root finding, numerical integration and differentiation, polynomial approximation, ordinary differential equations, discussion of convergence issues, error analysis and machine arithmetic, introduction to Python programming.

*Prerequisites:* MAT 162

*Attributes:* Math Beauty, Undergraduate

**MAT 313 Mathematical Optimization (3 credits)**

The course covers basic ideas in optimization beginning with linear programming, the simplex method and duality and finishes with non-linear optimization and algorithms and conditions leading to a solution of non-linear problems.

*Prerequisites:* MAT 226

*Attributes:* Math Beauty, Undergraduate

**MAT 316 Operations Research (3 credits)**

The course will cover some of the basic models and techniques used in operations research. Topics include: linear programming, the simplex method, duality, network problems, transportation problems, and time permitting, game theory.

*Prerequisites:* MAT 226

*Attributes:* Math Beauty, Undergraduate

**MAT 321 Probability (3 credits)**

The first part of a two-semester sequence, this course includes discrete probability and counting methods, conditional probability and independence, Bayes' Theorem, discrete and continuous random variables, expectation, variance, moment-generating functions,special probability distributions, joint distributions, marginal and conditional distributions, independent random variables, covariance and correlation, conditional expectations, and distributions of functions of random variables.

*Prerequisites:* MAT 213 (may be taken concurrently)

*Attributes:* Math Beauty, Undergraduate

**MAT 322 Mathematical Statistics (3 credits)**

Random samples, sample size, statistics and sampling distributions, the Central Limit Theorem, methods of point estimation including moment matching, percentile matching, maximum likelihood estimation, main properties of point estimators, asymptotic properties of MLE, evaluation of goodness of a point estimator, Rao-Blackwell theorem, UMVUE, interval estimation, hypothesis testing, power of tests, the Neyman-Pearson lemma, regression analysis, analysis of variance, categorical data analysis (Chi-square test). Data analysis projects will be assigned.

*Prerequisites:* MAT 321

*Attributes:* Undergraduate

**MAT 328 Design of Experiments (3 credits)**

In this course we discuss proper design and analysis of experiments, including the role of randomization, selecting sample sizes, and allocating treatments to experimental units. Designs covered include completely randomized designs, designs with factorial treatment structure, random and mixed effects designs, complete and incomplete blocked designs, Latin squares, confounding, split plots, fractional factorials, and response surfaces. Examples and exercises are taken from a broad range of subject areas. Appropriate computer programs are used for analysis of real data sets.

*Prerequisites:* MAT 118 or MAT 128 or MAT 213 or DSS 210

*Attributes:* Undergraduate

**MAT 332 Geometry (3 credits)**

An axiomatic treatment of the foundations of geometry. Axioms of incidence, order, congruence, Bolyai- Lobachevsky parallel axiom, angle of parallelism. A rigorous development of selected topics in non-Euclidean geometry.

*Prerequisites:* MAT 155 or MAT 161

*Attributes:* Math Beauty, Undergraduate

**MAT 334 Combinatorics & Graph Theory (3 credits)**

Introduction to combinatorics and graph theory and to methods by which each theory is applied to the other. Topics include basic counting formulas; generating functions; the principle of inclusion-exclusion; counting labeled trees (Cayley's Theorem, Kirchhoff's Theorem, Prüfer's Theorem); directed Euler circuits; Pólya-deBrujin theory; Möbius inversion.

*Prerequisites:* MAT 162

*Attributes:* Math Beauty, Undergraduate

**MAT 336 Logic & Foundations (3 credits)**

Cantorian set theory and the crisis in foundations (Cantor's paradox, Russell's paradox); the intuitionist challenge and the formalist response; formal logic and meta mathematics (Propositional Calculus, Predicate Calculus, formal number theory); Goedel's incompleteness theorems of 1931.

*Prerequisites:* MAT 162

*Attributes:* Math Beauty, Undergraduate

**MAT 340 Math Methods Phys Sci (3 credits)**

The course condenses the material of several full-semester mathematics courses, such as complex analysis, Fourier analysis, and group theory, with the focus on mathematical methods used in the physical and related sciences.

*Prerequisites:* MAT 213

*Attributes:* Math Beauty, Undergraduate

**MAT 370 Special Topics in Mathematics (3 credits)**

Topics will vary according to the semester in which the class is offered.

*Attributes:* Undergraduate

**MAT 403 Abstract Algebra (3 credits)**

Group theory, including finite groups, subgroups, cyclic groups, permutation groups, group isomorphisms, and cosets; introduction to rings and fields, including integral domains, polynomial rings, unique factorization domains and Euclidean domains.

*Prerequisites:* MAT 250

*Attributes:* Undergraduate

**MAT 404 Abstract Algebra II (3 credits)**

A more in-depth treatment of rings and fields including integral domains, fields, field extensions, homomorphisms, and the insolvability of the quintic by radicals. Galois theory.

*Prerequisites:* MAT 403

*Attributes:* Math Beauty, Undergraduate

**MAT 409 Real Analysis (3 credits)**

Elementary topology of Euclidean spaces, including open, closed and compact sets; convergence of sequences and series; least upper bound axiom and its equivalents; sequences of functions, pointwise and uniform convergence, continuity, differentiation and integration of sequences.

*Prerequisites:* MAT 250

*Attributes:* Math Beauty, Undergraduate

**MAT 410 Complex Analysis (3 credits)**

Analytic functions; complex integration; singularities.

*Prerequisites:* MAT 213

*Attributes:* Math Beauty, Undergraduate

**MAT 415 Differential Geometry (3 credits)**

The local and global theory of curves and surfaces in Euclidean space. Topics include Frenet frames, orientation, geodesics, the second fundamental form, and Gauss curvature.

*Prerequisites:* MAT 213 and MAT 226

*Attributes:* Math Beauty, Undergraduate

**MAT 418 Topology of Point Sets (3 credits)**

Open and closed sets, closure and interior, continuity, metric spaces, connectivity, compactness; the Heine- Borel and Bolzano-Weierstrass Theorems. The Classification of Surfaces may also be covered.

*Prerequisites:* MAT 409

*Attributes:* Undergraduate

**MAT 420 Convex Analysis (3 credits)**

This course covers the algebraic properties of affine sets, convex sets, cones, affine and convex functions, quasi-convex and pseudo-convex functions, topological properties of convex sets and functions, separation theorems, duality correspondences, Caratheodory’s Theorem, extreme points and faces of convex sets, polyhedral convex sets and functions, systems of linear inequalities, and related topics.

*Prerequisites:* MAT 226 and MAT 409

*Attributes:* Math Beauty, Undergraduate

**MAT 423 Applied Statistical Methods (3 credits)**

Statistical models, design and analysis of experiments, regression, Monte Carlo methods, and other advanced topics in statistics.

*Prerequisites:* MAT 162

*Attributes:* Math Beauty, Undergraduate

**MAT 424 Regression and Time Series (3 credits)**

The first part of the course covers Generalized Linear Models (GLMs). Topics include exponential family, important link functions, estimations (maximum likelihood estimation, generalized moment matching), diagnostic tests for model validations (graphical methods, chi-square statistics, t and F tests, AIC and BIC, likelihood ratio test), applications of GLMs on real data, prediction and confidence intervals. It also includes penalized regression (ridge and lasso regression, k-nearest neighbors algorithm). The second part of the course covers Time Series Analysis. Topics include an introduction to discrete stochastic processes, random walks, stationary processes, autocorrelation functions, and partial autocorrelation functions, various time series models (exponential smoothing, autoregressive (AR) model, moving average (MA) model, ARMA model), autoregressive conditional heteroskedastic (ARCH) model, generalized ARCH (GARCH) model, variants of GARCH, predictions and their confidence intervals using time series models.

*Prerequisites:* MAT 322

*Attributes:* Math Beauty, Undergraduate

**MAT 470 Topics in Mathematics (1-3 credits)**

Topics will vary according to the semester in which the class is offered.

*Attributes:* Undergraduate

**MAT 471 Independent Study (3 credits)**

**MAT 472 Independent Study (3 credits)**

**MAT 481 Data Science Capstone (3 credits)**

The capstone course provides students with a comprehensive learning experience that integrates ideas and experiencesgained from the three core disciplines of mathematics, statistics and computer science, and applies them to their chosenapplication domain. Working with a family advisor, students will engage in the process of solving a real-world data scienceproblem.

**MAT 491 Mathematics Internship I (3 credits)**

The course goals are: to gain first-hand experience of the daily activities of professionals in mathematics and related fields, to verify an interest in a particular area of mathematics, to develop and hone skills required for mathematical professions, to establish contacts outside the academic community who will facilitate a career in mathematics. An internship journal and an academic paper are also required.

**MAT 492 Mathematics Internship II (3 credits)**

The course goals are: to gain first-hand experience of the daily activities of professionals in mathematics and related fields, to verify an interest in a particular area of mathematics, to develop and hone skills required for mathematical professions, to establish contacts outside the academic community who will facilitate a career in mathematics. An internship journal and an academic paper are also required.

**MAT 493 Independent Research (3 credits)**

Students need to complete the application form for independent study (available in the Dean's Office) and have the approval of the department chair and Associate Dean in order to register. Honors Research (6 credits) must be elected in junior year to allow adequate research time. Students need to complete the application form for independent study (available in the Dean's Office) and have the approval of the department chair, Associate Dean and the Honors Program Director in order to register. Honors Students must complete this sequence.

*Attributes:* Math Beauty, Undergraduate

**MAT 494 Independent Research (3 credits)**

Students need to complete the application form for independent study (available in the Dean's Office) and have the approval of the department chair and Associate Dean in order to register. Honors Research (6 credits) must be elected in junior year to allow adequate research time. Students need to complete the application form for independent study (available in the Dean's Office) and have the approval of the department chair, Associate Dean and the Honors Program Director in order to register. Honors Students must complete this sequence.

*Attributes:* Math Beauty, Undergraduate

**MAT 704 Statistics for Research (3 credits)**

This class covers statistical inference on two samples, design of experiments, repeated measures, analysis of covariance,multiple regression, categorical data analysis, and factor analysis.