## Mathematics (Math)

This discipline is in the Division of Science and Mathematics.

*Objectives—*The mathematics and statistics curriculum is
designed to help students develop competence in mathematical and
statistical techniques and methods. The curriculum aims to sharpen
students' mathematical intuition and abstract reasoning as well
as their reasoning from numerical data. It also encourages and
stimulates the type of independent thinking requiring research
beyond the confines of the textbook. The mathematics program aims
to provide students with the basic knowledge and skills to make
mathematical contributions to modern society, whether in the form
of pure mathematics or mathematics applied in other disciplines.
The program seeks to enable students to see and communicate how
the development of mathematics has been part of the development
of several civilizations and is intimately interwoven with the
cultural and scientific development of these societies. The statistics
program aims to provide an effective operational knowledge of
the theory and methods of statistics and the application of the
statistical methods in a liberal arts environment. It seeks to
enhance students' critical thinking in domains involving judgments
based on data. The curriculum prepares students to enter graduate
school, pursue careers in applied mathematics or statistical fields,
or teach mathematics and statistics.

*Requirements for a Major* include the calculus sequence
Math 1201-1202-1203-3202; 3250, 3300, 3310, 3410, 3450, 3610,
3611; one of 3320, 3411; plus a minimum of 8 additional credits
in 3xxx mathematics courses or Math 1760. No required courses
may be taken S-N. Up to 10 credits of coursework with the grade
of D are allowed to meet the major requirements if offset by an
equivalent number of credits of A or B.

Majors should begin with Math 1112 or 1201. Students with questions
about placement are encouraged to discuss them with members of
the mathematics faculty.

Students planning to pursue graduate work in mathematics should
complete Math 3320, 3411, and 3550.

Students planning to work in applied mathematics and statistics
after graduation are advised to complete some electives from Math
1760, 3203, 3205, 3270, 3370, 3620, 3630, 3640, 3690, 3750, and
3790.

Students interested in statistics can design an area of concentration
in consultation with the statistics faculty. It is suggested that
the designed academic program in statistics include Math 3610,
3611, 3620, 3630, 3640, and 3690. Students designing their area
of concentration in statistics are encouraged to enrich their
degree by taking Econ 3400, Psy 3600 or Soc 3230.

*Requirements for a Minor* include the calculus sequence
Math 1201-1202-1203; 1760 or 3250; and a minimum of 16 additional
credits in mathematics courses numbered 3000 through 3800 in at
least two areas, such as algebra, analysis, geometry, and applied
mathematics. Required courses may be taken S-N, but it is not
recommended. Up to 10 credits of coursework with the grade of
D are allowed to meet the minor requirements if offset by an equivalent
number of credits of A or B.

*Requirements for Teacher Preparation—*Students interested
in teaching licensure in mathematics must complete the following
requirements: a mathematics major including Math 3800, 3500 or
3520; a course on computer programming; and required education
courses including methods (MthE 3940) and student teaching in
mathematics.

The teaching licensure minor in mathematics requires a minor in
mathematics including Math 3250, a course on computer programming,
and required professional education courses including methods
(MthE 3940) and student teaching in mathematics.

**Math 1100s. Concepts in Mathematics. **(E10; 5 cr; no cr
for students who are concurrently enrolled in or have received
cr for courses numbered 1201 or above; prereq 2 yrs high school
mathematics)

Sets, concepts of logic, systems of numeration, mathematical systems,
sets of numbers, algebra, geometry, and probability.

**Math 1106f. The Nature of Mathematics. **(E10; 5 cr; prereq
high school higher algebra and geometry)

Nature and processes of abstraction in mathematics. Survey of
the historical development of the content of high school mathematics
followed by a study of the elementary concepts underlying several
aspects of modern mathematics.

**Math 1110f,w. College Algebra. **(5 cr; no cr for students
who are concurrently enrolled in or have received cr for courses
numbered 1140 or above [excluding 1150]; prereq high school higher
algebra)

Sets, absolute values, inequalities, functions and graphs, exponential,
logarithmic, and classical trigonometry, arithmetic of complex
numbers, and elementary systems of equations.

**Math 1112f,s. Pre-Calculus Mathematics. **(5 cr; no cr for
students who have received cr for courses numbered 1140 or above
[excluding 1150]; prereq high school higher algebra and geometry)

Polynomial, rational, exponential, logarithmic and trigonometric
functions; trigonometric identities and equations; polar coordinates
and topics from analytic geometry; systems of equations, determinants
and matrices; arithmetic, geometric, and simple infinite series;
binomial theorem.

**Math 1120Hw. Honors Problem Solving Seminar. **(E10; 4 cr;
prereq # for students not in Honors Program)

Process of solving problems and its typical mental operations.
Illustrations of the methods used by outstanding problem solvers;
practice in applying these successful methods to a variety of
questions drawn from logic, higher arithmetic, algebra, and geometry.

**Math 1140f,s. Survey of Calculus. **(5 cr; no cr for students
who are concurrently enrolled in or have received cr for 1202
or 1203; prereq high school higher algebra or 1112)

Short course for students in social sciences, biological sciences,
and other areas requiring a minimal amount of calculus. Topics
include basic concepts of functions, derivatives, integrals, exponential
and logarithmic functions, maxima and minima, partial derivatives,
applications.

**Math 1150f,w,s. Introduction to Statistics. **(E10; 5 cr;
no elective cr for mathematics majors; no cr for students who
are concurrently enrolled in or have received cr for Math 3605;
prereq high school higher algebra)

Scope, nature, tools, language, and interpretation of elementary
statistics. Descriptive statistics; graphical and numerical representation
of information; measures of location, dispersion, position, and
dependence; exploratory data analysis. Elementary probability
theory, discrete and continuous probability models. Inferential
statistics, point and interval estimation, tests of statistical
hypotheses. Inferences involving one and two populations, ANOVA,
regression analysis, and chi-square tests; use of statistical
computer packages.

**Math 1201f,w-1202w,s-1203f,s. Calculus I-II-III. **(C2; E10,
1201 and 1202 only; 5 cr per qtr; prereq high school higher algebra,
geometry, trigonometry or 1112 for 1201; 1202 or # for 1203)

The concepts and properties of differentiation, antidifferentiation
and definite integration and their interconnection by the Fundamental
Theorem. Partial derivatives. Techniques of differentiation and
integration using rational and transcendental functions. Applications
involving mathematical modeling, solution of simple differential
equations and Taylor's Theorem. Rigorous treatment of limits.
Use and theory of convergence of power series. Vector geometry,
derivative of vectors and path integrals. Computer algebra system
used.

**Math 1201Hf-1202Hw-1203Hs. Honors Calculus I-II-III. **(C2;
E10, 1201H and 1202H only; 5 cr per qtr; prereq high school higher
algebra, geometry, trigonometry or 1112 for 1201H, # for students
not in Honors Program)

Definite integral and properties. Differentiation and properties.
Antiderivatives. Fundamental theorem. Partial derivatives. Differentiation
and integration techniques. Applications including Taylor's theorem.
Differential equations: modeling, solution techniques for first
order, higher order linear and systems. Applications. Rigorous
treatment of limits and associated theorems. Series: convergence,
power series, Fourier series. Vector geometry, derivatives of
vectors, path integrals and applications. Computer algebra system
used and the art of problem solving emphasized.

**Math 1760s. Discrete Mathematics. **(C2, E10; 5 cr; prereq
#)

Logic, sets, the nature of proof, mathematical induction, recurrence
relations, relations, functions, permutations and combinations,
graph theory, trees, network models, partial ordering. Boolean
algebra, algebraic structures.

**Math 3202f,w. Calculus IV. **(4 cr; prereq 1203 or #)

Multivariable calculus. Vectors, three-dimensional analytic geometry,
partial differentiation, multiple integration, vector fields,
and applications.

**Math 3203f,s. Differential Equations. **(4 cr; prereq 1203
or #)

First order and second order differential equations with methods
of solution and applications, systems of equations, series solutions,
the fundamental existence theorem and its application to numerical
solutions of first order equations, the qualitative theory of
differential equations.

**Math 3205s. Vector Analysis. **(4 cr; prereq 3202)

Vector algebra, vector calculus, space curves, gradient, divergence
and curl, line and surface integrals, divergence theorem, Green
and Stokes theorems, applications.

**Math 3250f. Foundations of Mathematics. **W, E10; 4 cr; prereq
#)

Elements of set theory and logic, relations and functions, introductions
to the theory of ordinal and cardinal numbers. Emphasis on the
nature of proof and the axiomatic approach. Recommended for sophomores.

**Math 3270w. Operations Research. **(C2; 4 cr; prereq 1140
or 1202)

Topics include linear and other mathematical programming, sensitivity
analysis, network and transportation models.

**Math 3300w,s. Linear Algebra. **(4 cr; prereq 1203 or #)

Matrix algebra, systems of linear equations, finite dimensional
vector spaces, linear transformations, determinants, characteristic
values and vectors, bilinear and quadratic forms, diagonalization
of matrices, applications.

**Math 3300Hf. Honors Linear Algebra. **(4 cr; prereq 1203
or #, # for students not in Honors Program; intended for high-ability
students; students may not receive cr for both 3300 and 3300H)

Linear equations, matrix algebra, vector spaces, inner product
spaces, linear transformations, orthogonal, unitary and Hermitian
matrices, unitary and similarity transformations, characteristic
values and vectors, including Cayley-Hamilton theorem and minimal
polynomial, diagonalization, and topic chosen from bilinear, quadratic
and Hermitian forms, Jordan canonical form and matrix analysis.

**Math 3310w. Abstract Algebra I. **(4 cr; prereq 3250, 1203
or #)

Modern algebra with emphasis on rigor and axiomatic development.
Three systems of major consideration are groups, including the
fundamental isomorphism theorems, rings, and fields.

**Math 3320s. Abstract Algebra II. **(4 cr; prereq 3310 or
#; not offered 1998-99)

Selected topics from abstract algebra and/or linear algebra.

**Math 3350s. Number Theory. **(E10; 4 cr; prereq #; not offered
1997-98)

Whole numbers, divisibility properties of integers, prime numbers,
congruences, quadratic residues, applications.

**Math 3370s. Combinatorial Mathematics. **(4 cr; prereq 1140
or 1202 or #; not offered 1998-99)

For students in mathematics, computer science, natural sciences,
and related areas in social sciences. Selected topics from permutations
and combinations, generating functions, recurrence relations,
0-1 matrices, partitions, inclusion and exclusion, graphs, trees
and circuits, bipartite graphs, planar graphs, and networks.

**Math 3410w. Mathematical Analysis I.** (4 cr; prereq 1203,
3250)

Axiomatic method, nature of proof, ordering of the real numbers,
sequences, completeness of the real numbers, continuity, differentiability.

**Math 3411s. Mathematical Analysis II. **(4 cr; prereq 3410;
not offered 1997-98)

Riemann integral, improper integrals, infinite series, Riemann
Stieltjes integral, sequences of functions, metric and Euclidean
spaces.

**Math 3450s. Elements of Complex Variables. **(4 cr; prereq
3202 or #)

Complex numbers, derivatives and integrals of analytic functions,
elementary functions and geometry of complex numbers. Cauchy integral
theorem and formula, Laurent expansions, evaluation of contour
integrals by residues, conformal mapping, applications.

**Math 3500s. College Geometry. **(4 cr; prereq #)

Advanced topics of Euclidean geometry including the circle and
triangle, constructions with ruler and compass, incidence relations,
harmonic points and lines, and other topics. Introduction to non-Euclidean
geometries.

**Math 3520. Modern Geometry. **(3-5 cr; prereq 3202 or #;
offered when feasible)

Selected topics from projective geometry, affine geometry, non-Euclidean
geometries, differential geometry.

**Math 3550. Topology. **(3-5 cr; prereq 3250, 3202; offered
when feasible)

Selected topics from point set topology and/or algebraic topology.

**Math 3605f. Statistical Methods. **(C2; 5 cr; prereq soph
standing, 1202 or 1140)

Descriptive statistics, elementary probability theory; laws of
probability, random variables, discrete and continuous probability
models, functions of random variables, mathematical expectation.
Statistical inference; point estimation, classical and Bayesian
methods of estimation, interval estimation, tests of hypotheses.
Other statistical methods; linear regression and correlation,
ANOVA, nonparametric statistics, statistical quality control,
use of statistical computer packages.

**Math 3610w. Introduction to Probability Theory and Stochastic
Processes. **(4 cr; prereq 1202 or #)

Probability theory; set theory, axiomatic foundations, conditional
probability and independence, Bayes' rule, random variables. Transformations
and expectations; expected values, moments and moment generating
functions. Common families of distributions; discrete and continuous
distributions. Multiple random variables; joint and marginal distributions,
conditional distributions and independence, covariance and correlation,
multivariate distributions. Properties of random sample and central
limit theorem. Markov chains, Poisson processes, birth and death
processes, and queuing theory.

**Math 3611s. Introduction to Mathematical Statistics.** (4
cr; prereq 3610)

Principles of data reduction; sufficiency principle, likelihood
principle, invariance principle. Point estimation; methods of
finding and evaluating estimators. Hypothesis testing; methods
of finding and evaluating tests. Interval estimation; methods
of finding and evaluating interval estimators. Linear regression
and ANOVA.

**Math 3620w. Elementary Statistical Data Analysis. **(C2,
E10; 4 cr; prereq 1150 or 3605 or #)

Nature and objectives of statistical data analysis, exploratory
and confirmatory data analysis techniques. Some types of statistical
procedures; formulation of models, examination of the adequacy
of the models. Some special models; simple regression, correlation
analysis, multiple regression analysis, analysis of variance,
use of statistical computer packages.

**Math 3630f. Discrete Statistical Multivariate Analysis. **(C2;
4 cr; prereq 1150 or 3605 or 3610 or #; not offered 1998-99)

Analysis of categorical data. Loglinear models for two- and higher-dimensional
contingency tables; model selection, ordered categories, fixed
margins and logit models, casual analysis involving logit and
loglinear models, fixed and random zeroes, use of statistical
computer packages.

**Math 3640f. Applied Continuous Statistical Multivariate Analysis.
**(C2; 4 cr; prereq 1150 or 3605 or 3610, 3300 or #; not offered
1997-98)

Aspects of multivariate analysis, random vectors, sample geometry
and random sampling, multivariate normal distribution, inferences
about mean vector, MANOVA. Analysis of covariance structures:
principal components, factor analysis. Classification and grouping
techniques: discrimination and classification, clustering, use
of statistical computer packages.

**Math 3690s. Topics in Statistics. **(C2; 4 cr; prereq 1150
or 3605 or 3610 or #)

Topics selected from nonparametric methods, linear and nonlinear
regression analysis, ANOVA, design of experiments, sampling methods,
time series analysis, simulation and statistical computing.

**Math 3710f,w,s. Problem-Solving Seminar.** (1-3 cr per qtr;
repeatable; prereq #)

Development of problem-solving skills through practice and discussion.
Topics and hours arranged with a faculty member.

**Math 3750w. Numerical Methods.** (C2; 4 cr; prereq 3203,
3300, CSci 1300; not offered 1997-98)

Same as CSci 3750. Finite differences; interpolation; numerical
integration; numerical solutions of differential, algebraic, and
transcendental equations.

**Math 3790. Topics in Applied Mathematics. **(1-4 cr; repeatable
with #; prereq #; offered when feasible)

Treatment of advanced applied mathematics not included in the
regular curriculum.

**Math 3800w. History of Mathematics. **(4 cr; prereq #)

Survey of mathematics and mathematicians from ancient to recent
times.

**Math 3901f,w,s. Senior Project. **(1-3 cr; prereq for cr
is sr standing; S-N only)

Individual study of a topic outside the usual course offerings.
Each student prepares a project under the direction of a faculty
member and presents a written and oral report.

**Math 3950f, 3951w, 3952s. Directed Study.** (1-5 cr per qtr;
prereq #)

**Math 3960Hf, 3961Hw, 3962Hs. Senior Honors Project. **(1-5
cr per qtr; prereq participation in Honors Program, #)

A substantial scholarly or creative work (at the undergraduate
level) within the discipline. Successful completion of the senior
honors project is one of the requirements for graduating from
UMM "with honors."