Mathematics 2009-2010
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Course Listing
Requirements for the Major in Mathematics:
Twelve courses in the mathematics department including Math 113 (or both 110 and 112), 115, 117, 199, 332, 336, 340, 497 or 498–99; at least one course chosen from 219, 221, 224, 234, and 235; and Physics 120. It is also recommended that two courses with substantial mathematical content be taken outside the department and that majors considering graduate work take one of French, German, or Russian as a foreign language. Advanced placement credit may be used to satisfy at most two of the twelve required math courses.
Mathematics Requirements for Any Interdepartmental Major Having Mathematics as a Component:
Eight mathematics courses, including Math 113 (or both 110 and 112), 115, 199, and either two courses from List 1 or one from List 1 and one from List 2 below. Advanced placement credit may be used to satisfy at most two of the eight required courses.
— List 1: Math 325, 330, 332, 336, 340, 432, 436, 448, 480.
— List 2: Math 127, 219, 221, 224, 234, 235
Requirements for a Minor in Mathematics:
Six courses in the department including (1) calculus through Math 115; (2) Math 199; (3) at least one course having Math 199 as a prerequisite; (4) at least one additional course chosen from Math 117, 119, 127, 128, 130, 138, or any 200, 300, or 400-level course. Advanced placement credit may be used to satisfy at most one of the six required courses.
Course registration guidelines:
Nearly all mathematics courses are petition courses, so if you plan to take a math course, be sure to follow the instructions for petitioning, and do so on time. If you miss the petition period, it may be very difficult to get into a math course, particularly those numbered below 200. In the event that you wish to try to enroll in a petition course after the petition period is over, you should use the form available at www.math.union.edu/courses/requests/ on the math department’s web site.
Requirements for Honors in Mathematics:
Candidates for honors in mathematics must fulfill the college-wide criteria for honors. In addition, they must have a grade point average of at least 3.5 in mathematics courses numbered 199 and above, complete a two-term honors thesis with a grade of A or A-, and take at least two of the following courses: 330, 432, 436, 448, 480.
Requirements for Students Seeking Secondary School Certification as Part of a Four-Year Bachelor’s Program:
PSY 246, EDS 500A, EDS 500B, and EDS 500C and two terms of a foreign language. PSC 281 is strongly recommended. Mathematics majors must take 12 mathematics department courses including Math 113 (or both 110 and 112), 115, 117, 128, 199, 224, 332, 336, 340, and 497 or 498–99. They must also take MBA 506, Computer Science 105, and Physics 120.
The college recommends that any undergraduate seeking New York State secondary teacher certification should consider attending the Master of Arts in Teaching program in their fifth year. Mathematics majors who wish to apply to a master’s program in teaching are advised to incorporate Math 128, 224, 332, and 336, plus Statistics and Computer Science, into their undergraduate program.
General Education Courses
Calculus continues to be the most common way for both science and non-science majors to meet the Quantitative and Mathematical Reasoning requirement at Union. The following courses (51 through 60) represent alternatives that also fulfill that requirement. These courses normally are not open to students who have passed calculus courses. Note that there also are courses in other departments (e.g., Computer Science) that can be used to fulfill the QMR requirement.
51. Cryptology: The Mathematics of Secrecy (Spring). The course will focus on the mathematical aspects of public-key cryptography, the modern science of creating secret ciphers (codes), which is largely based on number theory. Additional topics will be taken from cryptanalysis (the science of breaking secret ciphers) and from contributions that mathematics can make to data security and privacy.
53. Visualizing the Fourth Dimension (Not offered 2009–10). An investigation of the idea of higher dimensions and some of the ways of understanding them. The classic novel, Flatland, is the starting point; discussions, writing, projects and interactive computer graphics are used to extrapolate ideas from two and three dimensions to their analogues in four dimensions and higher.
54. Number Theory: From Clock Arithmetic to Unbreakable Codes (Fall, Spring). An introduction to the beauty and use of numbers. Topics chosen from divisibility tests, prime numbers, perfect numbers, unbreakable codes, Fermat’s theorem, the golden section, calendars, magic squares, quadratic reciprocity, and others.
55. Ancient Greek Mathematics (Fall). Ancient Greek mathematicians invented the notion of abstraction (in mathematics and other fields), absolute precision, and proof. The approach to mathematics that we take today can be traced back to these Greek mathematicians. After examining some pre-Greek mathematical traditions, we study Greek mathematics, beginning with Thales and Pythagoras. Topics include the intellectual crisis caused by the discovery that not all magnitudes are commensurable; Plato and his academy; Euclid and his Elements; the three special construction problems (trisecting an angle, squaring a circle, doubling a cube); and the greatest of the Greek mathematicians, Archimedes.
57. Game Theory and its Applications in the Humanities and Social Sciences (Not offered 2009–10). A self-contained introduction to the mathematical theory of conflict. Examples and applications include parlor games, auctions, games from the Bible and games commenting on the existence of superior beings, game-theoretic analyses in literature, philosophical questions and paradoxes arising from game theory, and game-theoretic models of international conflict. Not open to students who have passed Math 199.
58. Applications of Mathematics to Economics I (Not offered 2009–10). Linear and exponential functions, matrix algebra and linear programming with applications to the social sciences. Some sections include the use of computer spread-sheets for computations and graphical analysis. Not open to students who have passed a college calculus course.
59. Applications of Mathematics to Economics II (Not offered 2009–10). Differential and integral calculus with applications in the social sciences. Students who wish to continue the calculus after Math 59 should enroll in Math 112. Prerequisite: Math 58. Not open to students who have passed a college calculus course.
60. Mathematics and Politics (Winter) (Same as Political Science 123). A mathematical treatment (not involving calculus or statistics) of escalation, political power, social choice, and international conflict. No previous study of political science is necessary, but PSC 111 or 112 would be relevant.
Courses
100, 101, 102. Calculus with Precalculus (100 – Fall; 101 – Winter; 102 – Spring). This sequence covers the same material as Math 110 and Math 112, but it is spread out over three terms. There is an additional emphasis placed on review of fundamental precalculus concepts. Math 100 alone does not fulfill the Quantitative and Mathematical Reasoning requirement.
104. Introduction to Statistics: Analysis of Data (Not offered 2009–10). An introductory course on the concepts and application of probability and the analysis of sampling data. Topics include an introduction to numerical and graphical descriptions of data, probability, random variables, linear regression, sampling theory, and inference. Applicable for Environmental Studies, not open to students who received credit for Math 52 or PSY 200.
110. Calculus I: Differential Calculus (Fall, Winter). Calculus of one real variable. Differentiation of algebraic functions, and applications. Not intended for students who have passed a calculus course or Math 59.
112. Calculus II: Integral Calculus (Winter, Spring). Integral calculus of functions of a single variable, the fundamental theorem, formal integration and applications, calculus of logarithmic, exponential, and inverse trigonometric functions. Prerequisite: Math 110.
113. AP Calculus (Fall, Winter). Self-contained treatment of the main topics in Math 110 and Math 112. Intended for freshmen who have been introduced to (but have not yet mastered) the basics of differential and integral calculus.
115. Calculus III: Differential Vector Calculus and Matrix Theory (Fall, Winter, Spring). Geometry of 3-space, differential calculus of functions of several variables, linear systems, matrices. Prerequisite: Math 102, 112, or 113.
117. Calculus IV: Integral Vector Calculus (Fall, Winter, Spring). Double and triple integrals, line integrals and Green’s theorem, divergence and curl, divergence theorem and Stokes’ theorem. Prerequisite: Math 115.
127. Numerical Methods (Winter). Newton’s method, numerical differentiation and integration, solution of ordinary differential equations, error estimates. Prerequisites: Math 115 and fluency in some mathematical programming language.
128. Probability (Winter). Probability theory and applications. Prerequisite: Math 102, 112, or 113.
130. Ordinary Differential Equations (Winter, Spring). Linear differential equations and power series. Not open to students who have passed Math 234. Prerequisite: Math 115.
138. Methods of Applied Mathematics I (Not offered 2009–10) An introduction to the mathematical techniques and analysis of ordinary differential equations, partial differential equations, and complex variables. The emphasis is on the equations arising from physical, biological, and economic phenomena. Prerequisite: Math 130 or 234.
180. Linear Algebra and Matrix Theory (Not offered 2009-10). Vector spaces, matrices, and solutions of linear systems of equations, linear transformations, eigenvectors, and eigenvalues. Not open to students who have passed Math 340. Prerequisite: Math 115 or equivalent.
197. Discrete Mathematics for Computer Science (Winter). An introduction to fundamental concepts and methods of proof in mathematics and computer science. Topics include elementary logic, functions, relations, sets, and basic combinatorics.
199. Introduction to Logic and Set Theory (Fall, Winter, Spring). Designed to enable the student to develop the ability to understand and communicate mathematical arguments. Logic and set theory form the core. Selected topics are covered at the discretion of the instructor. For those considering any form of mathematics major, the department recommends that Math 199 be taken by fall term of the sophomore year, if possible. Prerequisite: Math 102, 112, or 113.
219. Topics in Discrete Mathematics (Fall). Topics may include graph theory, partially ordered sets, algebraic coding theory, computational complexity, number theory. Prerequisite: Math 199 or permission of the instructor.
221. Mathematical Cryptology (Spring). An in-depth look at the mathematical theory underlying modern methods to accomplish the secret transmission of messages, as well as other tasks related to data security, privacy, and authentication. Math 221 normally is closed to students who have passed Math 235 or Math 51. Prerequisite: Math 199 or permission of the instructor.
224. Geometry (Winter). Topics in Projective, Affine, Euclidean, and/or non-Euclidean geometries. Prerequisite: Math 199 or permission of the instructor.
234. Differential Equations (Spring). Topics include systems of ordinary differential equation, series solutions, asymptotic solutions, integral equations. Not open to students who have passed Math 130. Prerequisite: Math 115 and Math 199, or permission of the instructor.
235. Number Theory (Not offered 2009–10). Properties of natural numbers including divisibility, prime numbers, congruences, special number theoretic functions and quadratic reciprocity. Math 235 normally is closed to students who have passed Math 221. Prerequisite: Math 199 or permission of the instructor.
238. Methods of Applied Mathematics II (Not offered 2009–10). Provides a more rigorous development of the mathematical techniques and analysis of ordinary differential equations, partial differential equations, and calculus of variations. The emphasis is on the equations arising from physical, biological, and economic phenomena. Prerequisite: Math 138.
325. Knot Theory (Not offered 2009–10). An introduction to the mathematical study of knots, including colorability, chirality, genus, and the Jones polynomial. Course will also explore the relationship between mathematical knots and structures in molecular chemistry and biology, and physics. Not open to students who have passed Math 225. Prerequisite: Math 221, 235, 332, or 340, or permission of the instructor.
330. Complex Analysis (Fall). An introduction to analytic functions of a complex variable. Prerequisite: One 200-level course having 199 as a prerequisite or permission of the instructor.
332. Abstract Algebra I (Spring). Algebraic structures including groups, rings and fields. Prerequisite: One 200-level course having 199 as a prerequisite or permission of the instructor.
336. Real Variable Theory (Fall). A study of point sets on the real line and of real functions defined on these sets. Prerequisite: Math 332 or Math 340 or permission of the instructor.
340. Linear Algebra (Winter). Vector spaces, linear transformations, inner product and dual spaces, eigenvalues and eigenvectors, special topics. Prerequisite: Math 115 and one 200-level course having 199 as a prerequisite, or permission of the instructor.
432. Abstract Algebra II (Winter). Continuation of Math 332. Certain topics will be selected for more intensive study. Prerequisite: Math 332.
436. Topology (Not offered 2009–10). Topological spaces, connectedness, compactness, continuous mappings and homeomorphisms. Prerequisite: One 300-level course or permission of the instructor.
448. Differential Geometry (Not offered 2009–10). A study of curves and surfaces in 3-space. Topics include arc length, curvature, torsion, the Frenet trihedron, the first and second fundamental forms, normal curvature, and Gaussian curvature. Prerequisite: Math 117 and Math 340, or permission of the instructor.
480. Foundations of Mathematics (Spring). Propositional and predicate logic, Gödel completeness theorem, introduction to recursion theory. Prerequisite: Math 332 or permission of the instructor.
Independent Studies and Thesis
295H-96H. Two-Term Math Honors Independent Study
490-96. Independent Study in Mathematics (Fall, Winter, Spring). Independent study in a particular area of mathematics under the supervision of a faculty member.
497. One-Term Senior Thesis (Fall, Winter)
498-99. Two-Term Senior Thesis (Fall-Winter)