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# Undergraduate Courses

Undergraduate Courses

#### Mathematics

- MA 090 Pre-Calculus
Partial fractions, polynomials, Remainder Theorem, Fundamental Theorem of Algebra, Descartes' Rule, exponential and log functions, trigonometric functions, trigonometry of triangles, right triangles, laws of sines and cosines, and conic sections.

- MA 115 Calculus I
An introduction to differential and integral calculus for functions of one variable. Begins with limits and continuity, and ends with integration techniques and applications of the definite integral. As of Fall 2012, MA 115 is replaced by the sequence MA 121 and MA 122.

- MA 116 Calculus II
Improper integrals, infinite series. Taylor series, vector operations in 3D, calculus for functions of two and three variables including graphical representations, partial derivatives, the gradient, optimization, iterated integrals in rectangular and polar coordinates and applications of double integrals. As of Spring 2013 MA116 is replaced by the sequence MA123 and MA124.

- MA 117 Calculus for Business and Liberal Arts
Limits, the derivatives of functions of one variable, differentiation rules, and applications of the derivative. Definite integrals for functions of one variable, antiderivatives, the Fundamental Theorem, integration techniques, and applications of the integral.

- MA 119 Multivariable Calculus & Finite Mathematics
The first third of this course introduces students to calculus for functions of several variables and requires that students are familiar with the main results and techniques from one-variable calculus. The applied problems emphasize optimization problems for functions of two and three variables. The second part of the course reviews the use of matrices in representing systems of linear equations and then returns to the theme of optimization with an introduction to Linear Programming. The final third of the course teaches set notation and theory, basic counting principles, and an introduction to discrete probability. Throughout the course, motivating examples are drawn from applications in business, engineering, and the social sciences.

- MA 120 Introduction to Calculus
The first part of the course reviews algebra and precalculus skills. The second part of the course introduces students to topics from differential calculus, including limits, rates of change and differentiation rules. This is a seven week course.

- MA 121 Differential Calculus
Limits, the derivatives of functions of one variable, differentiation rules, applications of the derivative. This is a seven week course.

- MA 122 Integral Calculus
Definite integrals of functions of one variable, antiderivatives, the Fundamental Theorem, integration techniques, improper integrals, applications. This is a seven week course.

- MA 123 Series, Vectors, Functions, and Surfaces
Taylor polynomials and series, functions of two and three variables, linear functions, implicit functions, vectors in two and three dimensions. This is a seven week course.

- MA 124 Calculus of Two Variables
Partial derivatives, the tangent plane and linear approximation, the gradient and directional derivatives, the chain rule, implicit differentiation, extreme values, application to optimization, double integrals in rectangular coordinates. This is a seven week course.

- MA 134 Discrete Mathematics
This course provides the background necessary for advanced study of mathematics or computer science. Topics include propositional calculus, predicates and quantifiers, elementary set theory, countability, functions, relations, proof by induction, elementary combinatorics, elements of graph theory, mends, and elements of complexity theory.

- MA 188 Seminar in Mathematical Sciences
Selected topics in the Mathematical Sciences and applications of mathematics. Topics typically reflect the research interests of the department faculty. The course may be repeated for credit. Pass/Fail.

- MA 221 Differential Equations
Ordinary differential equations of first and second order, homogeneous and non-homogeneous equations; improper integrals, Laplace transforms; review of infinite series, series solutions of ordinary differential equations near an ordinary point; boundary-value problems; orthogonal functions; Fourier series; separation of variables for partial differential equations.

- MA 222 Probability and Statistics
Introduces the essentials of probability theory and elementary statistics. Lectures and assignments greatly stress the manifold applications of probability and statistics to computer science, production management, quality control, and reliability. Contents include: descriptive statistics, pictorial and tabular methods, and measures of location and of variability; sample space and events, probability axioms, and counting techniques; conditional probability and independence, and Bayes' formula; discrete random variables, distribution functions and moments, and binomial and Poisson distributions; continuous random variables, densities and moments, normal, gamma, and exponential and Weibull distributions unions; distribution of the sum and average of random samples; the Central Limit Theorem; confidence intervals for the mean and the variance; hypothesis testing and p-values, and applications for the mean; simple linear regression, and estimation of and inference about the parameters; and correlation and prediction in a regression model.

- MA 227 Multivariable Calculus
Review of matrix operations, Cramer’s rule, row reduction of matrices; inverse of a matrix, eigenvalues and eigenvectors; systems of linear algebraic equations; matrix methods for linear systems of differential equations, normal form, homogeneous constant coefficient systems, complex eigenvalues, nonhomogeneous systems, the matrix exponential; double and triple integrals; polar, cylindrical and spherical coordinates; surface and line integrals; integral theorems of Green, Gauss and Stokes.

- MA 230 Multivariable Calculus and Optimization
This course starts with some fundamental notions in multivariate analysis and geometry as well as basic notions and results of convex analysis: (gradient, Jacobian and Hessian, closed and open sets, convex sets, convex hulls, convex cones, polyhedral sets, convex functions, and convexity criteria). These notions are used to present the theory and methods of nonlinear optimization: necessary and sufficient conditions of optimality for nonlinear optimization problems with and without constraints, and duality theory. Numerical methods for unconstrained and constrained problems with differentiable functions include, gradient methods, Newton method, conjugate gradients, gradient projection, reduced gradient, simplex method, penalty methods, dual methods. Optimization problems from statistics, engineering, and business will serve as examples.

- MA 232 Linear Algebra
This course introduces basic concepts of linear algebra from a geometric point of view. Topics include the method of Gaussian elimination to solve systems of linear equations; linear spaces and dimension; independent and dependent vectors; norms, inner product, and bases in vector spaces; determinants, eigenvalues and eigenvectors of matrices; symmetric, unitary, and normal matrices; matrix representations of linear transformations and orthogonal projections; the fundamental theorems of linear algebra; and the least-squares method and LU-decomposition. Prerequisites: Sophomore or higher class standing.

- MA 234 Complex Variables with Applications
An introduction to functions of a complex variable. The topics covered include complex numbers, analytic and harmonic functions, complex integration, Taylor and Laurent series, residue theory, and improper and trigonometric integrals.

- MA 236 Introduction to Mathematical Reasoning
This course introduces students to first order logic and to fundamental discoveries about the nature and limits of mathematics which have emerged in the last hundred years. The course begins with a concrete treatment of first order logic and culminates with the unsolvability of the halting problem and the Church-Turing Theorem on the undecidability of first order logic.

- MA 281 Honors Mathematical Analysis III
Covers the same material as that dealt with in MA 221, but with more breadth and depth.

- MA 282 Honors Mathematical Analysis IV
Covers the same material as that dealt with in MA 227, but with more breadth and depth. By invitation only.

- MA 293 Supplementary Topics of Differential Equations
This course is designed for the completion of transferring credits for MA 221 Differential Equations. The transfer students, who need to learn some topics of MA 221 not included in the courses taken elsewhere, may enroll in this course only once with permission of an undergraduate adviser in the Math Department, and are required to complete this course under the guidance of the MA 221 course coordinator. The students who pass this course will receive the full transfer credits for MA 221. The students who fail will then be required to enroll in the full course of MA 221 at Stevens. Pass/Fail.

- MA 294 Supplementary Topics of Calculus IV
This course is designed for the completion of transferring credits for MA 227 Multivariable Calculus. The transfer students, who need to learn some topics of MA 227 not included in the courses taken elsewhere, may enroll in this course only once with permission of an undergraduate adviser in the Math Department. The students are required to complete this course under the guidance of the MA 227 course coordinator. The students who pass this course will receive the full transfer credits for MA 227. The students who fail will then be required to enroll in the full course of MA 227 at Stevens. Pass/Fail.

- MA 331 Intermediate Statistics
An introduction to statistical inference and to the use of basic statistical tools. Topics include descriptive and inferential statistics; review of point estimation, method of moments, and maximum likelihood; interval estimation and hypothesis testing; simple and multiple linear regression; analysis of variance and design of experiments; and nonparametric methods. Selected topics, such as quality control and time series analysis, may also be included. Statistical software is used throughout the course for exploratory data analysis and statistical inference based in examples and in real data relevant for applications.

- MA 335 Introduction to Number Theory
This is an introductory course to number theory. Topics include divisibility, prime numbers and modular arithmetic, arithmetic functions, the sum of divisors and the number of divisors, rational approximation, linear Diophantine equations, congruences, the Chinese Remainder Theorem, quadratic residues, and continued fractions.

- MA 336 Modern Algebra
A rigorous introduction to group theory and related areas with applications as time permits. Topics include proof by induction, greatest common divisor, and prime factorization; sets, functions, and relations; definition of groups and examples of other algebraic structures; and permutation groups, Lagrange's Theorem, and Sylow's Theorems. Typical application: error correcting group codes. Sample text: Numbers Groups and Codes, Humphries and Prest, Cambridge U.P.

- MA 346 Numerical Methods
This course begins with a brief introduction to writing programs in a higher level language, such as Matlab. Students are taught fundamental principles regarding machine representation of numbers, types of computational errors, and propagation of errors. The numerical methods include finding zeros of functions, solving systems of linear equations, interpolation and approximation of functions, numerical integration and differentiation, and solving initial value problems of ordinary differential equations.

- MA 360 Intermediate Differential Equations
This course offers more in-depth coverage of differential equations. Topics include ordinary differential equations as finite-dimensional dynamical systems; vector fields and flows in phase space; existence/uniqueness theorems; invariant manifolds; stability of equilibrium points; bifurcation theory; Poincaré-Bendixson Theorem and chaos in both continuous and discrete dynamical systems; and applications to physics, biology, economics, and engineering.

- MA 361 Intermediate Partial Differential Equations
This course offers a rigorous approach to classical partial differential equations. It begins with definitions, properties, and derivations of some basic equations of mathematical physics followed by the topics: solving of first order equations with the method of characteristics; classification of second order equations; the heat equation and wave equation; Fourier series and separation of variables; Green's functions and elliptic theory; examples of the first and second order nonlinear partial differential equations.

- MA 410 Differential Geometry
This course is an introduction to the geometry of curves and surfaces. Topics include tangent vectors, tangent bundles, directional derivatives, differential forms, Euclidean geometry and calculus on surfaces, Gaussian curvatures, Riemannian geometry, and geodesics.

- MA 441 Introduction to Mathematical Analysis
This course introduces students to the fundamentals of mathematical analysis at an adequate level of rigor. Topics include fundamental mathematical logic and set theory, the real number systems, sequences, limits and completeness, elements of topology, continuity, derivatives and related theorems, Taylor expansions, the Riemann integral, and the Fundamental Theorem of Calculus.

- MA 442 Real Variables
This course introduces principles of real analysis and the modern treatment of functions of one and several variables. Topics include metric spaces, the Heine-Borel theorem in R-n, Lebesgue measure, measurable functions, Lebesgue and Stieltjes integrals, Fubini's theorem, abstract integration, L-p classes, metric and Banach space properties, and Hilbert space.

- MA 450 Optimization models in finance
This course introduces the students to mathematical models and computational methods for static and dynamic optimization problems occurring in finance. The models involve knowledge of probability, optimality conditions, duality, and basic numerical methods. Special attention will be paid to portfolio optimization and to risk management problems.

- MA 460 Chaotic Dynamics, with Computations and Applications
This course introduces students to the concepts behind the modern theory of dynamical systems, particularly chaotic systems. Although the course is mathematical in nature, the emphasis is on the underpinning ideas and applications, rather than a systematic exposition of results. Topics include: standard examples and definitions, solutions of ODEs as dynamical systems, flows, and maps; fixed points of linear maps, periodic orbits, limit cycles, and asymptotic stability; rudiments of hyperbolicity; and symbolic dynamics and the Horse Shoe. Further topics may include: fundamentals of topological dynamics, fundamentals of ergodic theory, attractors, and fractals. A good part of the assigned work involves computer experimentation and computations.

- MA 461 Special Problems I
Individual projects in pure and applied mathematics.

- MA 462 Special Problems II
Individual projects in pure and applied mathematics.

- MA 463 Seminar in Mathematics I
Seminar in selected topics, such as: combinatorial topology, differential geometry, finite groups, number theory, or statistical techniques.

- MA 464 Seminar in Mathematics II
Seminar in selected topics such as: combinatorial topology, differential geometry, finite groups, number theory, or statistical techniques.

- MA 498 Senior Research Project I
Students will do a research project under the guidance of a faculty advisor. Senior standing and prior approval are required. Topics may be selected from any area of mathematics with the instructor's approval. Each student will be required to present results in both a written and oral report. The written report may be in the form of a senior thesis.

- MA 499 Senior Research Project II
Students will do a research project under the guidance of a faculty advisor. Senior standing and prior approval are required. Topics may be selected from any area of mathematics with the instructor's approval. Each student will be required to present results in both a written and oral report. The written report may be in the form of a senior thesis.

Contact

Prof. Patrick Miller*Undergraduate Chair*

[email protected]

Ms. Linda Habermann*Administrative Assistant*

[email protected]