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Saratoga Springs,
New York, 12866
SKIDMORE PHONE
518-580-5000
Mathematics
Chair of the Department of Mathematics and Computer Science:
Pierre von Kaenel
Professors: *Robert DeSieno, Mark Hofmann, R. Daniel
Hurwitz, Mark E. Huibregtse, Alice M. Dean, Gove W. Effinger, Pierre von Kaenel
Associate Professors: Una Bray, David C. Vella
Assistant Professor: Thomas O'Connell
THE MATHEMATICS MAJOR: Students majoring in mathematics fulfill
the departmental requirements by completing
nine courses in
mathematics or computer science: eight of the courses must be
at the 200 level or above, to include
MA200,
202,
MA215 or
MC215,
MA303,
319,
376,
and two additional courses, at least one of which is at the 300 level.
Students must also complete CS106.
Under exceptional circumstances, and only with the consent of the department,
MA371,
372,
381, or
382
may be counted as the additional 300-level course.
CS318
may not be counted toward the major. Courses counting toward
the major may not be taken satisfactory/unsatisfactory.
HONORS: Students wishing to qualify for departmental honors
in the mathematics major must:
complete all departmental requirements for the mathematics major
and have a GPA of 3.5 or higher for all course work (MA, MC, and CS) taken in the department;
have a GPA of 3.0 for all course work taken at Skidmore;
file with the department, by the end of the official add-drop period
of the spring semester of the senior year, a declaration of intention to qualify for honors; and
submit an honors thesis or project to be read by a review committee,
and give an oral presentation of the thesis or project to the department.
The review committee will evaluate the thesis or project to
determine if it is of the exceptional quality which merits honors;
the committee's recommendation will be submitted to the department
for final adjudication.
INTERDEPARTMENTAL MAJORS: The department offers an economics-mathematics
major in cooperation with the Department of Economics and a
business-mathematics major in cooperation with the Department
of Business. See
Interdepartmental Majors.
THE MATHEMATICS MINOR: Students minoring in mathematics fulfill
the departmental requirements by completing
MA113 and
200 or the equivalent;
MA215 or
MC215;
MA303 or
319;
and two more 3- or 4-credit courses in mathematics at the 200
level or above. MC306 may be substituted as one of the elective
courses.
PI MU EPSILON, New York Alpha Theta Chapter:
Incorporated in 1914, Pi Mu Epsilon is a national honorary society
whose purpose is the promotion of scholarly activity in mathematics.
Undergraduate students are qualified for membership if they
meet one of the following criteria: 1) upperclassmen who have
completed at least two years of college mathematics, including
calculus, with at least a B average and who are in the top third
of their class in general college work; 2) sophomores, majoring
in or intending to major in mathematics, who have completed
at least three semesters of college mathematics, including one
year of calculus, with a straight A record and who are in the
top quarter of their class in general college work.
Note regarding mathematics and science majors:
Students who plan to take
calculus should take the online placement exam prior to the beginning
of classes. Based on this exam, the Department will recommend which
course in the sequence the student should begin with.
Most frequently, prospective mathematics or science majors elect
MA111 in the fall semester and
113 in the spring semester of their first year.
Prospective mathematics or science majors with weak preparation in mathematics may be recommended to take
MA108 in the fall semester followed by
MA109 in the spring.
First-year students with advanced placement who take
MA113 in the fall should register for
MA200 at the same time if they plan to continue into
MA202 in the spring.
Students interested in learning how to use computers to solve
problems in the quantitative disciplines should consider the courses:
CS102,
103,
106, and
MS104.
MA 100. QUANTITATIVE
REASONING 3
Study of practical arithmetic and geometry, data gathering and analysis, introductory probability
and statistics, size and bias in sampling, hypothesis testing,
confidence intervals and their use in statistical analysis,
linear relationships, interpolation and extrapolation, correlation,
linear and exponential growth with practical applications. The
course is primarily intended to fulfill the first part of the
quantitative reasoning requirement (QR1). Prerequisite:
placement by department or permission of instructor. The
Department
Note: Courses numbered MS104, MA215 or MC215, 302, 306, 316, and MA102
through MA382 have as a prerequisite QR1 or permission of the department.
MA 102. MATHEMATICS
IN CONTEXT 3
A set of courses exploring interesting
questions from a variety of disciplines with the aid of mathematics;
primarily intended for students seeking to fulfill the College's
requirement in Quantitative Reasoning (QR2). Courses including
the following are offered periodically depending on faculty
availability. A student may take more than one of these courses
for credit.
A. Statistical Controversies.
It is often said that one can prove anything using statistics. Indeed,
the reading of any newspaper or news magazine presents one with
bold statements about important topics (economic, political,
and health issues, etc.) based on statistical studies, together
with strong opposition to those statements—in a phrase,
statistical controversies. This course aims to study many such
controversies to try to understand how they arise, what statistics
lie behind them, and how at least some of them might be avoided
by proceeding with greater care. Intended for students with
little or no experience with statistics.
B. Modeling Epidemics.
This course several mathematical techniques for modeling epidemics,
including differential equations and statistical methods.
D. Mathematics in Context:
Mathematical Models for Business and Economics—Efficient Planning.
What is the best way to allocate scarce resources?
In the business world, linear programming is often used to answer this question.
Phone companies, oil refineries, airlines, and many other businesses use linear
programming to schedule transportation networks, plan portfolios, control inventories,
and maximize profits. Since its invention in the 1940s, linear programming has become
increasingly important, and according to some estimates, well over $100 million in human
and computer time is spent yearly in the formulation and solution of linear programming
problems. In this course, the student will be introduced to linear programming problems
and their solution, related topics such as sensitivity analysis and integer programming,
and various other mathematical models useful in business and economics, such as Leontief
economic models, basic probability theory, and Markov chains.
MS 104. INTRODUCTION
TO STATISTICS 4
An introduction to fundamental
concepts in statistical reasoning. Students will consider contexts,
both historical and modern, in which statistical approaches
arose and methodologies developed. Topics considered will include
organization and analysis of data, the drawing of inferences
from these data, and the careful presentation of these inferences.
Examples will be drawn from a variety of disciplines. (Fulfills
QR2 requirement.) The Department
MS 104E. INTRODUCTION
TO STATISTICS 4
An introduction to fundamental
concepts in statistical reasoning. Students will consider contexts,
both historical and modern, in which statistical approaches
arose and methodologies developed. Topics considered will include
organization and analysis of data, the drawing of inferences
from these data, and the careful presentation of these inferences.
Examples will be drawn primarily from Environmental Studies. (Fulfills
QR2 requirement.) The Department
MA 107. CONCEPTS
OF MATHEMATICS 3
An introductory course for liberal
arts and education majors or anyone seeking a general, nontechnical
overview of mathematics. Topics covered include set theory,
review of number systems, geometry concepts, basic concerns
of probability and statistics, and introductory number theory.
(Fulfills QR2 requirement.) The Department
MA 108. CALCULUS
WITH ALGEBRA I 3
An introduction to derivatives, integrals, and their
applications. Primarily for students who are not adequately
prepared for MA111, this course (together with MA109) covers
the same material as MA111 but integrates the material
requisite to calculus with the calculus itself. Note that
MA108 alone can not be used as a substitute for MA111.
Successful completion of MA108 and MA109 is equivalent to
completion of MA111. (Fulfills QR2 requirement.) Prerequisite:
Permission of instructor. The Department
MA 109. CALCULUS
WITH ALGEBRA II 3
A continuation of MA108. A study of exponential, logarithmic,
and trigonometric functions and their applications in
differential and integral calculus. Successful completion of
MA108 and MA109 is equivalent to completion of MA111.
(Fulfills QR2 requirement.) Prerequisite:
MA108. The Department
MA 111. CALCULUS
I 4
Derivatives, integrals and their
applications. Techniques of differentiation. Integration and
differentiation of exponential, logarithmic and trigonometric
functions. Prerequisite: high school preparation including
trigonometry or consent of department.
(Fulfills QR2 requirement.) The Department
MA 111W. CALCULUS
I 4
Derivatives, integrals and their
applications. Techniques of differentiation. Integration and
differentiation of exponential, logarithmic and trigonometric
functions. Prerequisite: high school preparation including
trigonometry or consent of department. (Fulfills QR2 requirement;
meets expository writing requirement for students who placed at
EN105 level or who have completed
EN103.) The Department
MA 113. CALCULUS
II 4
Inverse trigonometric functions
and hyperbolic functions. Systematic study of integration. Series
and Taylor series. Polar coordinates. Indeterminate forms, L'Hôpital's
rule and improper integrals. (Fulfills QR2 requirement.) Prerequisite:MA111, or both
MA108 and
109, or consent of department. The Department
MA 113W. CALCULUS
II 4
Inverse trigonometric functions
and hyperbolic functions. Systematic study of integration. Series and Taylor series. Polar coordinates. Indeterminate forms, L'Hôpital's
rule and improper integrals. (Fulfills QR2 requirement; meets expository
writing requirement for students who placed at EN105 level or
who have completed
EN103.) Prerequisite:MA111, or both
MA108 and
109, or consent of department. The Department
MA 125, 126. PROBLEM
SOLVING IN MATHEMATICS 1, 1
Introductory level. Students
will work collaboratively on problems posed in various undergraduate
mathematics journals and other sources. Solutions to journal
problems will be submitted to the journal editors for acknowledgment
and possible publication. Problems are taken from all areas
of specialty within mathematics. During fall semesters, students
will have an opportunity to compete in the annual William Lowell
Putnam Mathematical Competition. Prerequisite: QR1. Must be taken S/U.
May be repeated for credit. The
Department
MA 200. LINEAR ALGEBRA 4
Vector spaces, matrices and linear
transformations, determinants, solution of linear equations.
Prerequisite: high school preparation including trigonometry
or consent of department. Offered fall semester. (Fulfills QR2
requirement.) The Department
GM 201. GLOBAL SECURITY
IN AN AGE OF INVENTION 3
The evolution of two twentieth-century technological achievements,
nuclear weapons and digital computing. This course explores
the roles of scientists and the institutions that foster these
achievements, and examines the cooperation of scientists and
policy makers who convert these technologies into instruments
of international politics and global competition. The course
is intended to help students understand the relationships between
technological developments and selection of policies that shape
international affairs. Prerequisites:
GO103, QR1, and one course in
computer science or laboratory science. R. DeSieno
MA 202. CALCULUS III 4
Multivariable calculus. Prerequisite:MA108 and
109, and
MA113 and
200 or consent of department. Offered spring
semester. The Department
MA 204. PROBABILITY
AND STATISTICS 3
Elementary probability, discrete and continuous random variables, theory of expectation, analysis
of distribution functions. (Fulfills QR2 requirement.) Prerequisite:MA111, or both
MA108 and
109 or consent of department.
Spring 2007 and alternate years. The Department
MA 214. THEORY OF NUMBERS 3
Topics in classical and modern number theory including congruences, Diophantine equations,
quadratic residues. (Fulfills QR2 requirement.) Prerequisite:MA111, or both
MA108 and
109, or
MA200 or consent of department.
Spring 2006 and alternate years. The Department
MC 215. MATHEMATICAL
REASONING AND DISCRETE STRUCTURES 4
An introduction to mathematical reasoning in the context of studying
discrete structures fundamental to both mathematics and computer
science. Topics include elementary logic and sets, methods of proof
including mathematical induction, algorithms and their analysis,
functions and relations, elementary combinatorics, discrete probability,
and graph theory. (Fulfills QR2 requirement.) Prerequisites: QR1, and
CS106 or
MA113, or permission of the
instructor. The Department
MA 225, 226. PROBLEM SOLVING
IN MATHEMATICS 1,1
Intermediate level. Students will work collaboratively on problems posed in various undergraduate
mathematics journals and other sources. Solutions to journal
problems will be submitted to the journal editors for acknowledgment
and possible publication. Problems are taken from all areas
of specialty within mathematics. During fall semesters, students
will have an opportunity to compete in the annual William Lowell
Putnam Mathematical Competition. Prerequisite: QR2. Must be taken
S/U. May be repeated for credit. The Department
MA 270. DIFFERENTIAL
EQUATIONS 4
An introduction to the theory and applications of differential equations. Prerequisite:MA113 and
200.
Offered spring semester. The Department
MA 276. SELECTED
TOPICS IN MATHEMATICS 3
Topics that complement the established
lower level course offerings in mathematics will be selected.
Emphasis will be on the nature of mathematical thought. May
be repeated for credit. Offered on sufficient demand. The
Department
MC 302. GRAPH
THEORY 3
An introduction to the theory and applications of graphs. Topics may include graphs and digraphs,
connectivity, trees, Euler and Hamiltonian cycles, and graph embeddings. Prerequisite:
MC115 or
MC215 or
MA200 or permission of instructor.
Fall 2007 and alternate years. The Department
MA 303. ADVANCED CALCULUS 4
Rigorous treatment of foundational issues in analysis. Topics may include set theory, the real
number system, sequences, series, limits and continuity, theory
of differentiation and integration, and elementary notions of
topology. Prerequisite:MA113 and
MA215 or
MC215 or consent of the department.
Offered fall semester. The Department
MC 306. THEORY OF COMPUTATION 3
A study of the major theoretical models of computation. Topics include automata, nondeterminism,
regular and context-free languages, Turing machines, unsolvability, and
computational complexity. Prerequisite:
MC115 or
MC215 and
CS106, or permission of instructor.
Beginning in fall 2005, MC306 will have
CS210 as a prerequisite. The Department
MA 309. ELEMENTS OF MODERN GEOMETRY 4
Study of various topics in modern geometry, with emphasis on the axiomatic method.
Fall 2006 and alternate years. Prerequisite:MA113 and
MA215 or
MC215
or consent of instructor. The Department
MA 310. HISTORY OF MATHEMATICS 3
Study of the development of mathematical ideas. Prerequisite:MA113 and
MA215 or
MC215 or permission of the instructor.
Offered on sufficient demand. The Department
MA 311. DIFFERENTIAL GEOMETRY 3
An introduction to differential geometry in a classical setting: the study of n-surfaces, embedded
in Euclidean space. Offered on sufficient demand. Prerequisite:MA200,
202,
MA215 or
MC215, or consent of the department.
MA270 recommended. The Department
MA 313. INTRODUCTION TO TOPOLOGY 3
Selected topics in topology such as metric spaces, point set topology of Euclidean spaces, introduction
to algebraic topology. Prerequisite:MA113 and
MA215 or
MC215 or consent of the department.
Spring 2007 and alternate years. The Department
MC 316. NUMERICAL
ALGORITHMS 3
An introduction to using computation to obtain approximate solutions to mathematical problems. A
variety of algorithms are studied, as are the limitations of
using computational methods. Topics include algorithms for solving
equations, systems, and differential equations; approximating
functions and integrals; curve fitting; round-off errors and
convergence of algorithms. Prerequisites:CS106, and
MA111, or both
MA108 and
109, or permission of instructor.
Offered on sufficient demand. The Department
MA 319. ABSTRACT ALGEBRA I 4
Survey of algebraic structures; groups, rings, fields, vector spaces, and linear transformations.
Prerequisite:MA200 and
MA215 or
MC215 or consent of the department.
Offered fall semester. The Department
MA 320. ABSTRACT ALGEBRA II 3
Selected topics in advanced algebra. Prerequisite:MA319 or consent of department.
Spring 2006 and alternate years. The Department
MA 323. REAL ANALYSIS 3
Selected topics in real analysis. Prerequisite:MA303 or consent of department.
Offered alternate years. The Department
MA 324. COMPLEX
ANALYSIS 3
Analytic functions, complex integration,
complex sequences and series, and conformal mapping. Prerequisite:MA303 or consent of department.
Offered alternate years. The Department
MA 325, 326. PROBLEM SOLVING
IN MATHEMATICS 1,1
Advanced level. Students will work collaboratively on problems posed in various undergraduate
mathematics journals and other sources. Solutions to journal
problems will be submitted to the journal editors for acknowledgment
and possible publication. Problems are taken from all areas
of specialty within mathematics. During fall semesters, students
will have an opportunity to compete in the annual William Lowell
Putnam Mathematical Competition. Prerequisite: QR2. Must be taken S/U.
May be repeated for credit. The Department
MA 371, 372. INDEPENDENT
STUDY 3, 3
Special study in mathematics outside the regular department offerings. Prerequisite:
consent of department. The Department
MA 376. SEMINAR 3
Research, discussion, and presentation of selected topics at an advanced level, to provide a capstone
experience for the mathematics major; primarily intended for seniors. Prerequisites:MA303 and 319 or consent of the department.
This course may be repeated for credit with permission
of the department. The Department
MA 381, 382. SENIOR THESIS 3, 3
Optional for mathematics majors. Recommended for those working toward professional careers or
graduate study in mathematics, and required for those seeking
to satisfy the criteria for departmental honors. The Department
MA 399. INTERNSHIP IN MATHEMATICS 3 or 6
Professional experience at an advanced level for juniors and seniors with substantial academic
experience in mathematics. With faculty sponsorship and departmental
approval, students may extend their educational experience in
pure or applied mathematics. This course may not be used to
satisfy the requirements of any major or minor in the department. Prerequisites:MA200, two additional courses in mathematics
at the 200 level or above, and permission of the department. Non-liberal arts.
