The Department of Mathematics and Computer Science offers a B.S. degree in mathematics, a B.S. degree in actuarial science, and a five-year B.S./M.S. program in mathematics and secondary education. 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. A creative imagination is required for success. Students also may opt for advanced degrees in education, business administration, law, or medicine.

The Department also offers B.S. in computer science and M.S. in computer science.

Advisor: Dr. Klingsberg

With the approval of the department, students may minor in mathematics. Upon acceptance, they will be assigned an advisor within the department who will assist the students in selecting courses appropriate for their area of interest.

Students desiring a minor in mathematics must take MAT 1371 or MAT 1391, and any five other mathematics courses numbered higher than 1371, with approval of the departmental advisor. The grade in each course must be C+ or better, or the average of the six classes must be at least 2.7.

Advisor: Dr. Fillebrown

The combined B.S./M.S. program in mathematics and secondary mathematics education offers students the opportunity to complete both an undergraduate major, a minor in secondary education, Commonwealth of Pennsylvania Secondary Teacher Certification in mathematics, and a M.S. in Secondary Mathematics Education. This curriculum emphasizes the unique character of mathematics learning theory, technology, and techniques for effective teaching in the discipline. The program strengthens the traditional certification programs by incorporating modern teaching/learning strategies and an opportunity to participate in research in the field.

Undergraduate mathematics majors at Saint Joseph’s University who apply for the five-year combined B.S./M.S. program will be evaluated for acceptance into the program after the completion of the first semester of junior year. Applicants must submit a complete undergraduate transcript, three letters of recommendation, and a personal essay describing the student’s goals. A cumulative grade point average of 3.0 or better is required. For more information, contact the Department of Mathematics and Computer Science or the Graduate Arts and Sciences office. Courses in mathematics education are listed in the Graduate Catalog. They appear in a separate section under Education and are designated by MED.

Advisor: Dr. Fillebrown

Students who are pursuing certification in Secondary School Mathematics Teaching minor in education. For this option, consult the department’s web page, the Secondary Education sequence listed in the Education section of this Catalog, or consult with the Departmental advisor, Dr. Fillebrown. The optional five-year B.S./M.S. program is described above.

Advisor: Dr. Cavaliere

Actuarial Science is an inter-college major with combined coursework from the Haub School of Business and the College of Arts and Sciences. The Actuarial Science major resides within the Mathematics and Computer Science Department. Please see the separate major listing Actuarial Science at the beginning of the College of Arts and Sciences section of this catalog. A minor in Actuarial Science is also available.

Requirements for departmental honors are found under Honors Program and in the brochure published annually by the Director of the Honors Program.

Note: Certain courses substantially overlap with other courses in the curriculum and therefore no more than one course in the overlapping sets may be taken for credit.

Students may not take overlapping courses for credit. Please note the following five overlapping course groupings:

• MAT 1161, MAT 1251, MAT 1351
• MAT 1261, MAT 1361
• MAT 1371, MAT 1391
• MAT 1391, MAT 1461
• MAT 1571, MAT 1591, CSC 1671

MATHEMATICS OFFERINGS

MAT 1011 Excursions in Mathematics I, 3 credits
An introduction to modern mathematical concepts, this course strengthens the student’s understanding of fundamental mathematics. Topics covered include logic, number theory, relations and functions, and geometry. Additional topics may be covered at the discretion of the instructor.  MAT 1011 is not required for MAT 1021.

This course will strengthen and expand the student’s understanding of fundamental mathematics through comparative study of the mathematics of many cultures.  We strive to appreciate the contributions of all cultures to the development of mathematics, and explore the connections between mathematics, art, and music.  Topics covered include number systems; methods of arithmetic, including use of the abacus; basic algebra and geometry, including a comparative study of the Pythagorean Theorem; the mathematics or drumming; the infinite in mathematics and art, and probability.  The course will be particularly appropriate for students interested in non-European history or culture, and majors in elementary education or fine arts.

MAT 1021 Excursions in Mathematics II, 3 credits
The second term of the introductory, terminal course covers topics in combinatorics and graph theory, probability, statistics, and the history of mathematics.  Other topics may be covered at the discretion of the instructor.
MAT 1011 is not a prerequisite for MAT 1021.

MAT 1151 Finite Mathematics with Applications in Business, 3 credits

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

MAT 1161 Brief Business Calculus, 3 credits
Introduction to calculus: mathematical models using polynomial, rational, exponential and logarithmic
functions, rates of change and the derivative, optimization using the derivative, integration.
Prerequisite: MAT 1151 or permission of instructor.

MAT 1171 Topics in Contemporary Mathematics, 3 credits
Topics in discrete mathematics: graphs and routing problems, Euler and Hamilton circuits, minimum
cost spanning trees, linear and exponential growth models, voting systems, voting power, introduction to counting and probability theory.
Designed for Social Science majors.

MAT 1181 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.  Prerequisite:  MAT 1011, 1151, or 1171

MAT 1201 Pre-Calculus Mathematics, 4 credits
The sole purpose is to prepare the student for courses in Calculus, i.e. MAT 1251-1261 or 1351-1361. Algebra:
factoring, simultaneous equations, polynomial, rational, and exponential functions, binomial theorem, word
problems. Trigonometry: basic functions, formulae, graphs. Analytic Geometry: straight line, circle, conic sections.

MAT 1251 Applied Calculus I, 3 credits
This course covers the derivatives of polynomial, rational, exponential, and logarithmic functions.  The emphasis is on applications of the derivative.  It is especially directed toward Biology, Business and Social Science Majors in order to provide a valuable and useful device to help them solve problems.
Prerequisite:  MAT 1201 or adequate performance on calculus readiness test.

MAT 1261 Applied Calculus II, 3 credits
This course covers the definite integra, techniques of integration, solving differential equations and the calculus of several variables.  The emphasis is on applications on the Business and Social Sciences.
Prerequisite:  MAT 1251 or MAT 1351.

MAT 1281 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.
 

MAT 1351 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’Hospital’s Rule; the Mean Value Theorem; applications of differentiation (including curve sketching and optimization); introduction to integration; the Fundamental Theorem of Calculus.
Prerequisite:  MAT 1201 or adequate performance on calculus readiness test.

MAT 1361 Calculus II, 4 credits
Techniques of integration; applications of integration; improper integrals; exponential growth, infinite sequences and series; power series and Taylor series.
Prerequisite:  A grade of C or better in MAT 1351 or permission of department.

MAT 1371 Calculus III, 4 credits
Vector geometry in R² and R³; polar coordinates; introduction to the calculus of vector-valued functions (velocity, speed, acceleration, curvature, parametric equations); differentiation of functions of several variables (partial derivatives, the differential, chain rules, directional derivatives); applications of differentiation (linear approximation, optimization, the method of Lagrange multipliers); integrals of functions of several variables; applications of integration.  Also, if time permits, cylindrical and spherical coordinates; some surface integrals; athe Change of Variable theorem.
Prerequisite:  A grade of C or better in MAT 1361 or permission of department.

MAT 1381 Differential Equations, 4 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.
Prerequisite:  MAT 1371, or MAT 1391, or equivalent.

MAT 1391 Calculus III and Linear Algebra, 4 credits
A combined course covering the following topics:  Linear Algebra; row reduction, and reduced row-echelon form; matrix and vector operations; linear transformations from  Rⁿ  to  R^m and their standard matrices; linear independence and bases of subspaces of  Rⁿ ; column space, mull space, rank; determinants; eigenvalues, eigenvectors, and the characteristic equation.  Calculus III: polar coordinates; introduction to the calculus of vector-valued functions (velocity, speed, acceleration, parametric equations); differentiation of functions of several variables (partial derivatives, the differential, chain rules, directional derivatives); applications of differentiation (linear approximation, optimization, the method of Lagrange multipliers); introduction to integrals of functions of several variables.  Also, if time permits, cylindrical and spherical coordinates.
Prerequisite:  A grade of C or better in MAT 1361 or permission of department.

MAT 1421 Advanced Calculus, 4 credits
This course covers three main groups of topics. I) Calculus of functions from Rⁿ to R^k: Differentials, Inverse
Function Theorem, Implicit Function Theorem, Jacobian, change of variables. II) Vector calculus. Vector
fields, line and surface integrals, Green’s Theorem, Stokes’s Theorem, the Divergence Theorem. III) Sequences
and series of functions. Pointwise and uniform convergence; continuity, differentiation and integration of
sequences and series; Fourier series; Taylor series for functions of several variables.
Prerequisite:  MAT 1371, or MAT 1391, or equivalent.

MAT 1431 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.

MAT 1461 Introduction to Linear Algebra, 4 credits
Linear systems, vector spaces, dimension, linear transformations, matrices, inner product, orthogonality, characteristic polynomials, diagonalization, eigenvalues, and eigenvectors.
Prerequisite: MAT 1371 or equivalent.

MAT 1571 Discrete Structures, 3 credits
An introduction to the basic concepts of discrete mathematics essential both to mathematics and many of its
applications. Topics include logic, sets, relations, functions, recurrence equations, combinatorics, graphs.
Techniques of mathematical proof will be developed.
Prerequisite: MAT 1361.

MAT 1591 The Fundamental Ideas of Mathematics, 3 credits

MAT 1611 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.

MAT 1651 Problem Solving, 3 credits
The course is designed to involve students in an active way in the mathematical process by having them participate in the major activity of both pure and applied mathematics: the solving of problems. Problems will be chosen from many areas of mathematics, and an attempt will be made to develop general approaches to and general paradigms for problem solving.
Prerequisite: MAT 1441 or MAT 1571

MAT 2011 Numerical Analysis and Computer Techniques, 3 credits
An introduction to numerical methods for solving a variety of problems. Included will be rootfinding, numerical integration and differentiation, polynomial approximation, and solutions of equations, systems of equations, ordinary differential equations, and discussion of convergence issues, error analysis and machine arithmetic.
Prerequisite: MAT 1371 or MAT 1391.  Enrollment in or completion of MAT 1461 or MAT 1391 would be beneficial.

MAT 2021 Chaos, Fractals and Dynamical Systems, 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.
Prerequisite: MAT 1251-1261 or MAT 1351-1361

MAT 2031 Abstract Algebra  I, 3 credits
Groups, Coset decomposition, homomorphism and counting techniques used to obtain structure theorems for
finite groups.
Prerequisite: MAT 1431 or MAT 1591.

MAT 2041 Abstract Algebra II, 3 credits                                                                                                              
Rings, integral domains, fields, field extensions,. homomorphisms, the insolvability of the quintic by radicals.
Galois theory.
Prerequisite: MAT 2031.

MAT 2091 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.  Topics which overlap MAT 1421 will be covered in more depth in this course.  
Prerequisite: MAT 1371.

MAT 2101 Complex Analysis, 3 credits

MAT 2121 Geometry, 3 credits

MAT 2131 Mathematical Programming, 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.
Prerequisite: MAT 1461 or MAT 1391 or permission of instructor.

MAT 2151 Differential Geometry, 3 credits
Tangent vectors and directional derivatives; mappings and differential forms on R³; Frenet fields; covariant derivatives; frame fields; Cartan structural equations; orientation; Euclidean geometry; surfaces in R³; calculus on surfaces; integration of forms; shape operators; normal and Guassian curvature; geometry of surfaces.
Prerequisite: MAT 2091

MAT 2201 Advanced Numerical Analysis, 3 credits
Eigenvalue and eigenvector problems; partial differential equations; boundary value problems; integral equations; error and stability analysis.
Prerequisite: MAT 2011

MAT 2211 Probability and Statistics I, 3 credits

MAT 2221 Probability and Statistics II, 3 credits

MAT 2231 Applied Statistical Methods, 3 credits
Statistical models, design and analysis of experiments, Monte Carlo methods, and other advanced topics in statistics.
Prerequisite: MAT 1261 or 1361; and MAT 1281 or MAT 2221

MAT 2321 (CSC 2721) Signal Processing, 3 credits
Time-domain response and convolution; frequency-domain response using Fourier series, Fourier transform, Laplace transform; discrete Fourier series and transforms; sampling; z-transform; filter design.
Prerequisite: Math 1371 and 1461, or Math 1391.

MAT 2361 Logic and 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 mathematics (Propositional Calculus, Predicate Calculus, formal number theory); Goedel’s incompleteness theorems of 1931.

MAT 2441 Combinatorics and 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, Prufer's Theorem); directed Euler circuits; Polya-deBrujin theory; Mobius inversion.

MAT 2461 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 ppermitting, game theory.
Prerequisites: MAT 1461 or MAT 1391 or  permission of instructor

MAT 2551 Topology of Point Sets, 3 credits
Set theory, mappings, topological and metric spaces, sequences, continuity, connectivity, and compactness.
Prerequisite: MAT 2091 or equivalent

MAT 2601 Topics in Abstract Algebra, 3 credits

MAT 2621 Topics in Real Variables, 3 credits

MAT 2651 Problem Solving Capstone, 3 credits

MAT 2661 Capstone Seminar Project, 1 credit
A capstone course required for graduation for students entering the program before Fall 2005. Each student, under the guidance of a faculty mentor, will undertake an independent project culminating in a presentation. The topic may be suggested by the student, chosen by the mentor, or undertaken as an extension of material covered in a mathematics course. The venue for the presentation will be chosen jointly by the student and the mentor. Students should register for this course in the spring semester of the senior year. Pass-Fail.

MAT 2911-2921 Independent Study, 6 credits

MAT 2931-2941 Directed Readings, 6 credits

MAT 2951-2961 Research, 6 credits

MAT 2953-2963 Honors Research and Independent Study, 6 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.