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# Undergraduate Courses

# Mathematical Sciences Undergraduate Courses

### MA123 Applied Calculus for Business I

Presents basic concepts of functions, graphs and differential calculus. Special emphasis is placed on business applications such as break-even analysis, depreciation, marginal profit/revenue/cost and optimization. Topics include the notion of a function; properties of linear, quadratic, exponential and logarithmic functions; and basic techniques of differential calculus.

### MA123L Applied Calculus for Business I with Lab

Same content as MA 123 with one additional class period per week.

### MA126 Applied Calculus for Business II

Prerequisite(s): MA 123 or MA 123L

Continues MA 123. Presents the basics of math of finance, integral calculus and probability. Specific emphasis is placed on business applications. Math of finance topics include simple/compound interest, present/future value, annuities and amortization. Other topics include evaluating indefinite and definite integrals using substitution, improper integrals and an introduction to probability.

### MA126L Applied Calculus for Business II with Lab

Prerequisite(s): MA 123 or MA 123L

Same content as MA 126 with one additional class period per week.

### MA131 Calculus I

Prerequisite(s): Solid foundation in algebra and trigonometry.

Note: Students who have completed MA 123 may not receive credit for MA 131.

Presents a thorough treatment of differential calculus that assumes a solid foundation in algebra and trigonometry. Topics include limits and continuity; the differentiation of single-variable functions; implicit and logarithmic differentiation; curve sketching; optimization; and applications to business, economics and the social and natural sciences.

### MA139 Calculus II

Prerequisite(s): MA 131

Note: Students who have completed MA 126 may not receive credit for MA 139

Continues MA 131. Presents a thorough treatment of integral calculus. Topics include integrating single-variable functions, including indefinite, definite and improper integrals by substitution, parts and partial fraction expansion; an introduction to ordinary differential equations; and applications to probability, business, economics and the social and natural sciences.

### MA141 Accelerated Calculus with Business Applications

Prerequisite(s): One year of high school calculus; enrollment in the Bentley University Honors Program. Students who take this course may not receive credit for other non-honors freshman courses.

Note: Fulfills the mathematics general education requirement when followed by honors MA 249 (Mathematics II) or any other MA elective.

Reviews techniques of single-variable differential and integral calculus. Class time is devoted primarily to business applications such as depreciation, present/future value, capitalized cost and internal rate of return. Written reports and oral presentations for weekly projects are required.

### MA205 Chaos, Fractals and Dynamics

Prerequisite(s): MA 126 or MA 139 or MA 141

Introduces basic concepts of dynamical systems through lectures, slides, films and computer experimentation. Students predict system behavior based on mathematical calculations and on observation of computer results (no computer programming experience is necessary). Topics include iteration of functions, Julia sets, Mandelbrot sets, chaos and fractals.

### MA207 Matrix Algebra with Applications

Prerequisite(s): 3 credits of math

Includes such topics as matrix algebra operations, simultaneous linear equations, linear programming, Markov chains, game theory, graph theory, linear economic models, least square approximation and cryptography. Business applications are emphasized and computer solutions (using MATLAB and/or Excel) are used for selected problems.

### MA215 Mathematics of Sports

Prerequisite(s): GB 213 and (MA 126, MA 139 or MA 141)

Mathematics and sports will help students understand how analytic ideas can aid in understanding athletic competitions and improving individual and team performances. The mathematical topics will include some with a statistical component ( expectations, probability and risk/reward judgments) and some with a deterministic bent (optimization, ranking and validation.) A variety of software packages will be used to demonstrate the many ways that a mathematical point of view can inform participants and fans alike.

### MA223 Linear Models for Business Decision-Making

Prerequisite(s): 3 credits of math

An introduction to linear optimization models as they apply to problems in business and economics. The potential and limitations of various models are discussed. Emphasis is placed on developing models from written descriptions and interpreting model solutions, typically computer-generated. Specific topics include linear and integer programming models.

### MA225 Probability Models for Business Decision-Making

Prerequisite(s): GB 210 or GB 213

An introduction to probabilistic models as they apply to management, economic and business administration problems. The potential and limitations of various models are discussed. Emphasis is placed on developing models from written descriptions and interpreting model solutions, typically computer-generated. Specific topics include an introduction to basic probability, decision analysis, queuing models and simulation.

### MA227 Mathematical Modeling in Environmental Management

Prerequisite(s): MA 123 or MA 131 or MA 141

An interdisciplinary course that introduces a number of environmental management issues arising frequently in business settings and for which quantitative models are important tools in their resolution. Problem areas include air pollution, surface and groundwater contamination, waste management, risk analysis and public health. Students investigate case studies using library and online research sources. Computer modeling is based on spreadsheet programs and commercial packages. The course may include a number of field trips to business and government facilities where such models are used for technical and regulatory purposes.

### MA233 Calculus III

Prerequisite(s): MA 139 or MA 141

Includes such topics as sequences and series (including geometric and Taylor series); multivariable differential and integral calculus; vector calculus; and applications to business, economics and the social and natural sciences.

### MA235 Differential Equations

Prerequisite(s): MA 139 or MA 141

An introductory course in ordinary differential equations with application to the social and natural sciences. First-order differential equations, second-order linear equations with constant coefficients and first-order linear systems are examined. The emphasis is on formulation of equations (modeling), analytical and graphical solution techniques and interpretation of solutions (prediction). Solution techniques include the methods of integrating factors, undetermined coefficients and variation of parameters. Linear first-order and second-order difference equations with applications are also introduced. Computer experiments are carried out in MATLAB and PHASER.

### MA239 Linear Algebra

Prerequisite(s): MA 139 or MA 141

This course includes topics on matrices, determinants, systems of linear equations and Gaussian elimination, vector spaces, linear independence, inner products, orthonormal bases, Gram-Schmidt process, QR-Factorization, the least-squares method, eigenvalues and eigenvectors. Applications to social and natural sciences as well as the connection with other mathematical disciplines is discussed. MATLAB is used throughout the course.

### MA243 Discrete Probability

Prerequisite(s): 3 credits of math

Relates to problems of a probabilistic nature in business, economics, management science and the social sciences. Includes such topics as set notation, permutations, combinations, mutually exclusive and independent events, conditional probability, Bayes' Theorem, expectation and dispersion, Markov chains and decision-making. Introduces the common discrete distributions: binomial, hypergeometric, geometric, negative binomial and Poisson. Simulation may be used where appropriate.

### MA249 Case Studies in Mathematics

Prerequisite(s): (MA141 or MA 139) and enrollment in the Bentley University Honors Program

This course addresses a variety of real-life problems drawn from multiple disciplines, such as economics, finance, marketing, environmental sciences, criminology and epidemiology. The relevant mathematics in each problem is emphasized, including material drawn from differential equations and series/sequences. Written reports and oral presentations for each problem are required. Field trips and guest speakers are used where appropriate. Research from traditional and electronic sources is expected.

### MA252 Mathematical Statistics

Prerequisite(s): (MA 139 or MA 141) and (GB 210 or GB 213)

Note: Students may not take both MA 252 and EC 361 for credit

The course focuses on the statistical concepts which form the basis for advanced topics in regression analysis, notably the construction of multiple regression models, time-series models and an analysis of the residuals. Students apply these concepts to large, multi-dimensional data sets using advanced software such as SAS or SPSS and gain experience in becoming more informed decision-makers through the interpretation of the software results. Emphasis is also placed on being able to communicate the statistical results to a general audience.

### MA261 Numerical Methods

Prerequisite(s): MA 139 or MA 141

Focuses on the numerical evaluation of functions, derivatives, integrals and the numerical approximation of solutions to algebraic and differential equations. Computer solutions to problems are used where appropriate.

### MA263 Continuous Probability for Risk Management

Prerequisite(s): (GB 210 or GB 213) and (MA 139 or MA 141)

This course focuses on concepts and techniques of continuous probability and their applications to risk management in insurance and finance. Among other topics, the most commonly used single- and multi-variable continuous probability distributions are addressed. Concepts are illustrated with a large number of applied risk management problems. Calculus tools such as single and double integration are used extensively.

### MA267 Discrete Mathematics

Prerequisite(s): MA 126 or MA 139 or MA 141

In contrast to the continuous real number line from calculus, "discrete" mathematical structures are made up of distinct, separate parts. The instructor chooses a few topics to cover from the many available discrete mathematics topics, including mathematical language and syntax, proofs and logic, circuits, cryptography, graphs (i.e., relationships among people, agencies, machines, etc.), number theory, combinations and permutations, etc. The relationship of mathematics to computer science features prominently.

### MA280 Selected Topics in the Mathematical Sciences

Prerequisite(s): Prerequisites will be announced at registration depending on the particular topic being addressed.

Note: With department approval, MA 280 may be taken more than once.

Examines a particular area of mathematics or its applications. May include such topics as the use of mathematical models in environmental science, the history of mathematics, elementary measure theory or financial mathematics. The topic will be announced prior to registration.

### MA305 Mathematical Logic

Prerequisite(s): Completion of any freshman mathematics sequence

Note: This course is also listed as PH 305; it can be used as either a philosophy or mathematical sciences elective depending on which designator the student chooses at registration.

Mathematics analyzes the world in a precise, quantitative way. Mathematical logic applies that same precise analysis to mathematics itself. Analysis of mathematical formulas, how they are constructed and how they relate, lead to the two most famous formal reasoning systems, classical propositional logic and classical predicate logic. Arguments constructed through formal reasoning in these systems are compared to informal reasoning. Examples of logic in algebra and the foundations of calculus lead to consideration of historically important questions such as, "Do we know that the generally accepted rules for reasoning are correct, or reliable?" This leads to the study of historical roots of non-classical logics and their relationship to computer science.

### MA307 The Mathematics of Computer Graphics

Prerequisite(s): MA 123 or MA 131 or MA 141

This course introduces mathematics for analyzing and describing images and scenes. Manipulations of two-and three-dimensional figures and spaces are analyzed using geometry, vectors, matrices and polynomials. A significant aspect of the course involves using these mathematical methods to generate images and animations that are both attractive and informative.

### MA309 Game Theory

Prerequisite(s): 6 credits of math

Game theory is the study of strategic behavior of rational actors who are aware of the interdependence of their actions. Course topics include the extensive form tree representation and the key concepts of strategy space and strategy profile. The normal form game representation is developed and illustrated with classical games such as the Prisoner's dilemma and Hawk-Dove. The discrete probability model is developed and applied to the concepts of player beliefs and mixed strategies. Solution concepts for games such as dominance and iterated dominance, best response curves, Nash equilibrium and security strategies are developed and compared. Additional topics may also be included, such as evolutionary games and fair division strategies.

### MA310 Actuarial Topics in Probability and Risk Management

Prerequisite(s): MA 233 and (MA 243 or MA 263)

An advanced course focused on further developing fundamental tools in discrete and continuous probability necessary for the analysis and solution of risk management problems. Significant time is spent examining complex problems and determining which mathematical technique(s) to apply. Success in mastering the techniques presented requires a substantial commitment to independent study. Students doing well in this course should be prepared to take the Society of Actuaries Exam P (Probability) or Casualty Actuarial Society Exam 1.

### MA335 Financial Calculus and Derivative Pricing

Prerequisite(s): (MA 139 or MA 141) and (GB 210 or GB 213)

This course provides an introduction to the basic mathematical concepts underlying the famous Black-Scholes-Merton option pricing formula and the associated financial market model, including model limitations and alternatives. Selected topics from ordinary differential equations, probability theory and statistics are used to develop and analyze the economic concepts. Hedging strategies and portfolio sensitivity parameters associated with options are also developed and discussed.

### MA343 The Mathematics of Discrete Options Pricing

Prerequisite(s): 6 credits of math

This course is devoted to basic principles and techniques of no-arbitrage discrete derivative pricing. Using elementary probability and linear algebra, the binomial option pricing model is developed. No-arbitrage option pricing and hedging are addressed using binomial trees. Real market data is used to explore the computational aspects of options pricing. This course should be of interest to strong math students who would like to see how fundamental mathematics is applied to a significant area of finance and to strong finance and economics students who would like to better understand the concepts behind the standard options pricing models.

### MA357 Mathematical Theory of Interest

Prerequisite(s): MA 139 or MA 141

Note: We recommend that students preparing for Exam FM/2 also take MA 335

The theory of interest addresses the critical financial question of determining the value of a stream of cash flows. This is a problem-solving intensive course aimed at preparing the highly motivated student for the interest theory portion of the Society of Actuaries Exam FM and the Casualty Actuary Society Exam 2. Emphasis is placed on learning efficient and effective techniques for solving interest theory problems.

### MA370 Models for Financial Economics

Prerequisite(s): MA 233 and MA 335

This course will develop the student’s knowledge of the theoretical basis of certain actuarial models and the application of those models to insurance and other financial risks. The topics covered include rational valuation of derivative securities using the binomial as well as the Black-Scholes option pricing models; risk management techniques (such as delta-hedging); interest rate models as well as elements of Stochastic Calculus. This is an intensive problem-solving course aimed at helping highly motivated students prepare for Exam MFE, the financial economics portion of the third exam offered by the Society of Actuaries (SOA) and the Casualty Actuarial Society (CAS). The ideal candidate will have passed Exam P/1 and/or Exam FM/2 prior to taking this course and be willing to invest the extensive time and effort required to pass Exam MFE.

### MA375 Models Life Contingencies I

Prerequisite(s): MA 310 and MA 357

Note: Formerly MA 402

The goal of this course is to develop student knowledge of the theoretical basis of life contingent acturial models and the application of these models to insurance and other financial risks. Specific topics include the mathematics of survival distributions, life tables, life insurances, life annuities, benefit premiums and premium reserves. Emphasis will be placed on developing familiarity with the theory behind these acturial models. This is an intensive problem-solving course aimed at helping highly motivated students prepare for Exam MLC, the life contingent modeling exam offered by the Society of Actuaries (SOA).

### MA376 Models Life Contingencies II

Prerequisite(s): MA 375

Note: Formerly MA 402

The goal of this course is to develop student knowledge of the theoretical basis of life contingent acturial methods and the application of those models to insurance and other financial risks. Specifics topics include discrete and continuous Markov models, multiple decrement models, multiple life models, universal life models and profit tests. Emphasis will be placed on developing familiarity with the theory behind these acturial models. This is an intensive problem-solving course aimed at helping highly motivated students prepare for Exam MLC, the life contingent modeling exam offered by the Scoiety of Actuaries (SOA).

### MA401 Directed Study in Mathematical Sciences

Prerequisite(s): Junior- or senior-level standing and department chairperson's permission

Permits superior students to study special topics. *(May be repeated for credit.)*

### MA402 Seminar in Mathematical Sciences

Prerequisite(s): Junior- or senior-level standing and department chairperson's permission

Note: Not offered regularly. Check with department chair for availability.

Permits small-group study of selected topics by advanced students. *(May be repeated for credit.)*

### MA421 Internship in Mathematical Sciences

Prerequisite(s): Junior-level standing, 3.0 cumulative average, and permission of mathematical sciences internship coordinator.

An internship provides the student with an opportunity to gain on-the-job experience and apply principles and issues raised in the academic discipline to a work environment. The student is required to attend pre-internship workshops sponsored by the Center for Career Services, meet regularly with a faculty adviser, and develop a final paper or special project.

### ST242 Applied Business Statistics

Prerequisite(s): (MA 126 or MA 139 or MA 141) and (GB 210 or GB 213)

Note: Students may take both MA 252 and ST 242 for credit.

Presents a practical development of several advanced methods of statistical inference that are useful in a wide range of business contexts. Topics include multiple regression and correlation analysis, analysis of variance, contingency tables and the chi square test for independence and statistical decision theory. Considers additional topics such as time series analysis and forecasting, non-parametric statistics, index numbers and survey sampling. The computer is used throughout the course as a means to efficiently solve practical problems.