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Mathematics
Chair of the Department of Mathematics and Computer Science:
Mark Hofmann
Mathematics and Computer Science Faculty:
Professors: *Robert DeSieno, R. Daniel
Hurwitz, Mark E. Huibregtse, Alice M. Dean, Gove W. Effinger
Associate Professors: Richard Speers,
Pierre von Kaenel, Mark Hofmann, Una Bray, David C. Vella
Assistant Professor: Thomas O'Connell
Lecturer: Elizabeth Heston
THE MATHEMATICS MAJOR: Students majoring in mathematics fulfill
the departmental requirements by completing eight courses in
mathematics or computer science at the 200 level or above, to
include MA200,
202,
215,
303,
319,
376,
and two additional courses, at least one of which is at the
300 level. 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: 1) complete all departmental
requirements for the mathematics major and have a grade-point
average of 3.5 or higher for all course work (MA, MC, and CS)
taken in the department; 2) have a grade-point average of 3.0
for all course work taken at Skidmore; 3) 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 4) 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;
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: 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 choose
to take MA105
in the fall semester followed by 111
in the spring. First-year students with advanced placement who
take MA113
in the fall should register for 200
at the same time if they plan to continue into 202
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, MC115, 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
uses several mathematical techniques for modeling epidemics,
including differential equations and statistical methods.
C.
Serious Games: Conflict, Voting and Power.
Mathematics is often applied to situations where the players
(which can be individuals, teams, corporations, or entire nations)
have conflicting interests. Players attempt to determine the
best course of action without knowing what their opponent(s)
will do. In this course, students study game theory, a field
of mathematics that was developed to analyze conflict and competition.
Game theory can be applied in a wide variety of situations,
such as choosing a location to open a business or understanding
tactical choices made in the Cuban missile crisis and the Yom
Kippur war. In addition to game theory, students explore other
social and political issues susceptible to mathematical analysis,
such as the assessment of the fairness of various voting schemes
or the measurement of political power.
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 105. PRE-CALCULUS
MATHEMATICS 4 Study of the real number system,
elementary functions and their graphs, and coordinate geometry.
Primarily for students who intend to take calculus but who have
not had sufficient preparation. (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 110. SKILLS
FOR CALCULUS 1 Study of the algebra and geometry
of elementary functions used in calculus. Primarily for students
who are concurrently enrolled in MA111 who have not had adequate
preparation. Prerequisites: QR1 and placement by the
department or permission of instructor. Offered S/U only. 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. Prerequisite: MA111 or
consent of department. (Fulfills QR2 requirement.) 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. Prerequisite: MA111 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
MC 115. INTRODUCTION
TO DISCRETE MATHEMATICS 3 An introduction to the study
of discrete (as opposed to continuous) mathematical systems.
These include systems that are essential in computer science
as well as in more advanced mathematics courses. Mathematical
reasoning and algorithms are fundamental themes of the course.
Topics include logic and sets, complexity of algorithms, computer
arithmetic, arrays, mathematical proofs and induction, elementary
combinatorics, and discrete probability, graphs, and trees.
Prerequisite: high school preparation including intermediate
algebra or consent of department. (Fulfills QR2 requirement.) 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. Offered
S/U only. 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:MA111, 113, 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. Prerequisite: MA111 or consent
of department. Spring 2005 and alternate years. (Fulfills QR2
requirement.) The Department
MA 214. THEORY
OF NUMBERS 3 Topics in classical and modern
number theory including congruences, Diophantine equations,
quadratic residues. Prerequisite: MA111 or 200 or consent
of department. Spring 2004 and alternate years. (Fulfills QR2
requirement.) The Department
MA 215. BRIDGE
TO ADVANCED MATHEMATICS 3 An introduction to mathematical
proof and to concepts of abstract mathematics, including elementary
logic, methods of proof, set theory, functions, and relations.
Prerequisite: one mathematics course numbered MA113 or
above, or permission of the instructor. Offered spring semester. 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. Offered
S/U only. 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 MA200 or permission
of instructor. Fall 2004 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 215
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,
computational complexity, and NP-completeness. Prerequisite:MC115 and CS106, or permission of instructor. The
Department
MA 309. ELEMENTS
OF MODERN GEOMETRY 4 Study of various topics in modern
geometry, with emphasis on the axiomatic method. Fall 2004 and
alternate years. Prerequisite: MA113 and 215 or consent
of instructor. The Department
MA 310. HISTORY
OF MATHEMATICS 3 Study of the development of mathematical
ideas. Prerequisite:MA113 and 215 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, 215, 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 215 or
consent of the department. Spring 2005 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 permission of instructor. Offered in 2003 and alternate years. The
Department
MA 319. ABSTRACT ALGEBRA
I 4 Survey of algebraic structures;
groups, rings, fields, vector spaces, and linear transformations.
Prerequisite:MA200 and 215 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
2004 and alternate years. The Department
MA 323. REAL
ANALYSIS 3 Selected topics in real analysis.
Prerequisite: MA303 or consent of department. Spring
2004 and 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. Spring 2005 and 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. Offered
S/U only. 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
numbered 115 or higher, and permission of the department. Non-liberal
arts.
