Bellarmine University: Mathematics

NOTE: Math. 116, 117, 125, and 205 assume that the student has taken a college preparatory course in high school mathematics, typically including two years of algebra. Students who have had only one year of high school algebra should take Math. 105. A student separated from mathematics for some time, even with two years of algebra, might be advised to take a college algebra course.

This course is an investigation of our numeration system. The NCTM Standards guide the course through an introduction to problem solving, sets, functions, ancient numeration systems, and place value. Next, a thorough examination of addition, subtraction, multiplication, and division reveals why these operations behave the way they do and what interconnections exist between these operations. Finally, the counting numbers are exended to include fractions and decimals, and the arithmetic of fractions and decimals is investigated at a deep level. Every fall.

This course begins with an investigation of ratios, rates, and proportions, leading to percentages, uncertainty, and chance. This is followed by an examination of geometry. A study of the basic shapes of one, two, and three dimensions is followed by an investigation of the basic ways these shapes can be transformed: translation, reflection, and rotation. Length, area, surface area, and volume complete the geometric content of this course. Every spring.

Taught as a preparatory course to remedy deficiencies in algebra. Properties of real numbers, linear equations and inequalities, quadratic equations and inequalities, polynomial functions, exponential and logarithmic functions, algebraic functions, systems of linear equations. Math. 105 does not fulfill the general education requirement in mathematics. Students with credit for Math. 116, 117, or 125 may not enroll in Math. 105. Every fall.

This course will examine mathematical issues at a non-technical level. The course will emphasize conjecture and investigation by the students. The students will be expected to communicate mathematics through reading, writing, and presenting their mathematical ideas. Every spring.

A study of elementary functions, their graphs and applications, including polynomials, rational and algebraic functions, exponential, logarithmic, and trigonometric functions. This course is taught with graphing calculators. (Prerequisites: two years of high school algebra and one year of geometry.) Every fall.

Limits and continuity of functions; the concept of derivative; calculating derivatives; applications of derivatives such as optimization and related rates; integration through the Fundamental Theorem. The course includes computer-based explorations. (Prerequisite: Math. 116 or its equivalent.) Every semester.

Applications of integration such as area, volume, and arc length; techniques of integration and improper integrals; approximation of integrals; infinite sequences and infinite series. The course includes computer-based explorations. (Prerequisite: Math. 117 or its equivalent.) Every spring.

An introduction to topics involving discrete sets of objects. These include number systems, sets and their operations, propositional logic, quantification, algorithms, functions, recursion, relations, and graph theory. The course contains an introduction to proof methodology, including mathematical induction, based on the preceding topics. This course makes extensive use of the computer for exploration and discovery of the concepts. (There is no formal prerequisite for this course. However, students should either have taken a precalculus course or be enrolled in Math. 116 concurrently.) Every fall.

Applications of polynomial, rational, exponential, and logarithmic functions. Limits, derivatives, and integrals with applications to business and economics. (Prerequisite: Math. 105 or its equivalent.) Every semester.

Descriptive statistics; graphical representation and numerical summaries of data. Elementary probability. Basic concepts of sampling and experimental design. Linear correlation and regression. Interval estimates and hypothesis testing, including chi-square and ANOVA. Two years of high school algebra and one year of high school geometry, or their equivalent, are strongly recommended as preparation for this course. Every semester.

This course covers basic ideas of matrix theory and linear algebra, including applications in Mathematics and other disciplines. This course begins with systems of linear equations, then explores matrices and their relation to systems of linear equations. This includes elementary row operations, the arithmetic of matrices, inverting a matrix, special types of matrices, the determinant of a matrix. Other topics covered are vector spaces (mainly Euclidean space) and linear transformations, including eigenvalues and eigenvectors of a linear transformation. (Prerequisites: Math. 117 and sophomore standing, or permission of instructor.) Every fall.

A study of the concepts from Calculus I and II in the multivariable case. This includes partial derivatives, multiple integrals, and vector calculus. The course makes extensive use of computer explorations and cooperative learning. (Prerequisite: Math. 118.) Every fall.

an exploration of fundamental concepts involving natural numbers, integers, rational numbers, real numbers, and complex numbers, and their operations. We will examine field properties, cardinality issues, and ordering properties, with other topics as time allows. The course will emphasize conjecture and proof. Students will develop, write, and present their proofs. (Prerequisites: Math. 118 and 120.) Every spring.

The objective of the course is to introduce applications and solution methods for equations which include derivatives. Maple software will be used extensively. The following topics will be covered: basic definitions and terminology; direction fields, phase portraits; first-order differential equations; modeling with first-order differential equations; higher-order differential equations; modeling with initial-value problems and boundary-value problems; the Laplace transform; the Dirac delta function; systems of first-order differential equations; numerical solutions of ordinary differential equations.. (Prerequisite: Math. 118.) Fall, odd years.

A survey of topics in advanced geometry from three historical perspectives: synthetic, analytic, and transformational. Topics include advanced results in Euclidean geometry, axiomatics of Euclidean geometry, axioms and results in non-Euclidean geometry, an introduction to projective geometry, the use of coordinates, and insights gained from transformations. (Prerequisite: Math. 215 or permission of instructor.) Spring, odd years.

These courses provide a foundation of classical probability theory and mathematical statistics to help prepare students for the Actuarial Exams as well as for further study in probability and statistics. Math. 314 will focus primarily on probability theory. Topics covered include combinatorics, basic axioms and theorems, random variables and probability distributions, expectation, moment, moment generating functions, and functions of random variables. Math. 315 will build on the probability theory from Math. 314 to develop understanding of mathematical statistics. Topics covered include generation and properties of point estimators, confidence intervals, tests of hypothesis, regression, and an introduction to the analysis of variance. (Prerequisite for Math. 314 is Math 118. Prerequisites for Math. 315 are Math. 314 and 216.) Math. 314 fall, odd years; Math. 315. spring, odd years.

Topics in graph theory, including circuits, coloring, trees and searching. Enumeration methods, including permutations and combinations, the inclusion-exclusion principle, generating functions and recurrence relations. (Prerequisite: Math. 118, 120, and 215.) Fall, odd years.

An examination of addition and multiplication, and how their properties resemble other operations in other settings. With a single operation the notion of group is available; adding a second operation extends this to rings and fields. Basic properties of groups, rings, and fields will be examined, including the Fundamental Theorem of Homomorphisms. Applications will be included as time allows. (Prerequisites: Math. 215 and 231.) Fall, even years.

This course is a continuation of Math. 215. It negins with a brief review of topics from the earlier course. The course then develops more deeply the theory of linear transformations on vector spaces and examines its applications. Topics include inner product spaces, orthogonality, eigenvalues and eigenvectors, and diagonizable linear operators. (Prerequisite: Math. 215.) Spring, odd years.

This course includes an introduction to insurance and risk management, an introduction to the actuarial profession, actuarial applications of calculus and probability, and preparation for the Society of Actuaries Exam I. (Prerequisites: Math. 216 and 314.) Spring, even years.

This course covers the basics of applying statistics to real world situations. This includes the design and conducting of experiments, data analysis with SPSS, interpretation of results, and critical evaluation of published research. Topics include experimental design, tests of significance, regression, tests for normality, canonical correlation analysis, discriminate analysis, and non-parametric methods. (Prerequisites: a calculus course, 117 or 125; and a probability course, 205, 314, or 352.) Spring, odd years.

This course is valuable to students who wish to take the Actuarial Exams or to study probability and statistics. Generally speaking, a stochastic process (random process) is a sequence of observations X1, X2, … whose values cannot be predicted precisely beforehand, but for which probabilities of the values can be specified at any particular time. Topics include probability theory, a brief introduction to statistics, Markov chains, queuing theory, Markovian decision processes, game theory, and decision analysis. (Prerequisite: Math. 118.) Fall, even years.

This course develops a practical knowledge of the theory of interest in both finite and continuous time. This knowledge includes how these concepts are used in the various annuity functions and how to apply the concepts of present and accumulated value for various streams of cash flows as a basis for future use in: reserving, valuation, pricing, duration, asset/liability management, investment income, capital budgeting, and contingencies. (Prerequisite: Math. 118.) Spring, odd years.

These courses provide a rigorous critical study of calculus of one real variable. Topics include the real number system and its properties, the theory of sequences, limits of functions, continuity, derivatives, integrals, and infinite series. Mathematical writing and the mathematical proof will be emphasized. (Prerequisites: Math. 403 requires Math. 118 and 231; Math. 404 requires Math. 403.) Fall, odd years; spring, even years.

A numerical method is used to solve a problem approximately when an exact solution cannot be found. The following topics will be covered: properties of a floating point number system and IEEE754; types of errors; stability and conditioning; solution of equations in one variable; interpolation and polynomial approximation; numerical differentiation and integration; numerical solution of ordinary differential equations; and direct methods for solving linear systems. (Prerequisite: Math. 118.) Fall, even years.

In this course we consider familiar concepts encountered in calculus in the new setting of functions of a complex variable. Topics covered include: the arithmetic, algebra, and geometry of the complex number system ad complex plane; elementary functions of a complex variable; the derivative (analyticity and harmonicity); the integral (line and contour integrals); the topological aspects of the complex plane needed to develop the theory of differentiability and integrability (including Cauchy’s Theorem, Cauchy’s Integral Formula, and the Maximum Modulus Theorem); and series representations for functions. (Prerequisite: Math. 216.) Fall, even years.

Selected topics in mathematics, intended to draw together and unify the various subject areas of the mathematics program. Emphasis given to research, written and oral reports. (Prerequisite: permission of instructor.) As required.

Guided reading or research in an area of special interest under the direction of a faculty member. As required.

An introduction to deterministic optimization. Topics include linear programming, sensitivity analysis, duality theory, network analysis, integer programming, and game theory. (Prerequisite: Math. 215.) Fall, even years.

This course will examine a subject not typically included in our curriculum. Students are encouraged to suggest topics of interest for a possible course offering. (Prerequisite: permission of instructor.) As required.

A capstone course for majors in mathematics and actuarial science. Professional readings will be assigned. Satisfactory completion of a major writing project is required. (Prerequisite: Senior standing or permission of instructor.) Every spring.

This course is an investigation at the master’s level of topics from the P-5 mathematics curriculum. It is not a study of how to do mathematics, but why the way we do mathematics works. The NCTM Standards guide the course through a study of problem-solving, sets, functions, ancient numeration systems and numeration in various bases. The four basic operations of addition, subtraction, multiplication, and division are thoroughly examined. The counting numbers are extended to include fractions, decimals and negative numbers. Next, an investigation of ratio, rates, and proportion leads to percentages, uncertainty, and chance. This is followed by an examination of geometry. A study of the basic shapes of one, two, and three dimensions is followed by an investigation of the three basic transformations: translation, reflection, and rotation. Length, area, and volume complete the geometric content of this course. As required.
*Enrollment restricted to graduate Education majors.

Basic statistical principles and use of computer software packages in statistical analysis. Topics include descriptive statistics and graphical techniques of data presentation, sampling, hypothesis testing, regression and correlation, analysis of variance and covariance. (Prerequisite: Graduate standing in Nursing or by permission.) As required.