141B. (GQ) CALCULUS
AND BIOLOGY II (4) Derivatives, integrals, applications
from biology; sequences and series; analytic geometry; polar
coordinates. Students may take only one course for credit from
MATH 141 and 141B. Prerequisite: MATH 140B.
200. (GQ) PROBLEM
SOLVING IN MATHEMATICS (3) Mathematical ways of thinking,
number sequences, numeracy, symmetry, regular polygons, plane
curves, methods of counting, probability and data analysis.
For elementary education students only.
220. (GQ) MATRICES
(2-3) Systems of linear equations appear everywhere in mathematics
and its applications. MATH 220 will give students the basic
tools necessary to analyze and understand such systems.
The initial portion
of the course teaches the fundamentals of solving linear systems.
This requires the language and notation of matrices and fundamental
techniques for working with matrices such as row and column
operations, echelon form, and invertibility. The determinant
of a matrix is also introduced; it gives a test for invertibility.
In the second part
of the course the key ideas of eigenvector and eigenvalue
are developed. These allow one to analyze a complicated matrix
problem into simpler components and appear in many disguises
in physical problems. The course also introduces the concept
of a vector space, a crucial element in future linear algebra
courses.
This course is
completed by a wide variety of students across the university,
including students majoring in engineering programs, the sciences,
and mathematics. (In the case of many of those students, MATH
220 is a required course in their degree program.)
The course is offered
at a number of locations throughout the Penn State system during
both fall and spring semesters. (With additional sections offered
during the summer at a few locations). Prerequisite: MATH 110
or 140.
230. CALCULUS
AND VECTOR ANALYSIS (4) Three-dimensional analytic geometry;
vectors in space; partial differentiation; double and triple
integrals; integral vector calculus. Students who have passed
either MATH 231 or 232 may not schedule MATH 230 for credit.
Prerequisite: MATH 141.
231. CALCULUS
OF SEVERAL VARIABLES (2) Analytic geometry in space; partial
differentiation and application. Students who have passed MATH
230 may not schedule this course. Prerequisite: MATH 141.
232. INTEGRAL
VECTOR CALCULUS (2) Multidimensional analytic geometry,
double and triple integrals; potential fields; flux; Green's
divergence and Stokes' theorems. Students who have passed MATH
230 may not schedule this course. Prerequisite: MATH 231.
250. ORDINARY
DIFFERENTIAL EQUATIONS (3) First- and second-order equations;
numerical methods; special functions; Laplace transform solutions;
higher order equations. Students who have passed MATH 251 may
not schedule this course for credit. Prerequisite: MATH 141.
251. ORDINARY
AND PARTIAL DIFFERENTIAL EQUATIONS (4) First- and second-order
equations; special functions; Laplace transform solutions; higher
order equations; Fourier series; partial differential equations.
Prerequisite: MATH 141.
310. ELEMENTARY
COMBINATORICS (3) Fundamental techniques of enumeration
and construction of combinatorial structures, permutations,
recurrences, inclusion-exclusion, permanents, 0, 1-matrices,
Latin squares, combinatorial designs. Prerequisite: MATH 220.
311W. CONCEPTS
OF DISCRETE MATHEMATICS (3) Introduction to mathematical
proofs; elementary number theory and group introduction to theory.
Students who have passed CMPSC 260 may not schedule this course
for credit. Prerequisite: MATH 141.
312. CONCEPTS
OF REAL ANALYSIS (3) An introduction to rigorous analytic
proofs involving properties of real numbers, continuity, differentiation,
integration, and infinite sequences and series. Prerequisite:
MATH 141.
318. (STAT)
ELEMENTARY PROBABILITY (3) Combinatorial analysis, axioms
of probability, conditional probability and independence, discrete
and continuous random variables, expectation, limit theorems,
additional topics. Students who have passed either MATH (STAT)
414 or MATH (STAT) 418 may not schedule this course for credit.
Prerequisite: MATH 141.
319. (STAT)
APPLIED STATISTICS IN SCIENCE (3) Statistical inference:
principles and methods, estimation and testing hypotheses, regression
and correlation analysis, analysis of variance, computer analysis.
Students who have passed MATH (STAT) 415 may not schedule this
course for credit. Prerequisite: MATH (STAT) 318 or knowledge
of basic probability.
401. INTRODUCTION
TO ANALYSIS I (3) Review of calculus, properties of real
numbers, infinite series, uniform convergence, power series.
Students who have passed MATH 403 may not schedule this course.
Prerequisite: MATH 230 or 231.
403. CLASSICAL
ANALYSIS I (3) Topology of R(n), compactness, continuity
of functions, uniform convergence, Arzela-Ascoli theorem in
the plane, Stone-Wierstrass theorem. Prerequisite: MATH 312.
404. CLASSICAL
ANALYSIS II (3) Differentiation of functions from R(n) to
R(m), implicit function theorem, Riemann integration, Fubini's
theorem, Fourier analysis. Prerequisite: MATH 403.
405. ADVANCED
CALCULUS FOR ENGINEERS AND SCIENTISTS I (3) Vector calculus,
linear algebra, ordinary and partial differential equations.
Students who have passed MATH 411 OR 412 may not schedule this
course for credit. Prerequisites: MATH 231; MATH 250 or 251.
406. ADVANCED
CALCULUS FOR ENGINEERS AND SCIENTISTS II (3) Complex analytic
functions, sequences and series, residues, Fourier and Laplace
transforms. Students who have passed MATH 421 may not take this
course for credit. Prerequisite: MATH 405.
411. ORDINARY
DIFFERENTIAL EQUATIONS (3) Linear ordinary differential
equations; existence and uniqueness questions; series solutions;
special functions; eigenvalue problems; Laplace transforms;
additional topics and applications. Prerequisites: MATH 230
or 231; MATH 250 or 251.
412. FOURIER
SERIES AND PARTIAL DIFFERENTIAL EQUATIONS (3) The purpose
of MATH 412 is to introduce students to the origins, theory,
and applications of partial differential equations. Several
basic physical phenomena are considered - including flows, vibrations,
and diffusions - and used to derive the relevant equations.
The fundamentals of the mathematical theory of partial differential
equations are motivated and developed for the students through
the systematic exploration of these classic physical systems
and their corresponding equations: the Laplace, wave , and heat
equations.
In addition to
treating the physical origins of the equations, this course
focuses on solving evolution equations as initial value problems
on unbounded domains (the Cauchy problem), and also on solving
partial differential equations on bounded domains (boundary
value problems). There is not one but many techniques for solving
these equations, and the course presents some aspect of the
expansion in orthogonal functions (including Fourier series),
eigenvalue theory, functional analysis, and the use of separation
of variables, Fourier transforms, and Laplace transforms to
solve PDEs by converting them to ordinary differential equations.
This course currently
serves as cross-section of students at the university with interests
or the need for this advanced subject mathematics, including
students majoring in the engineering programs, meteorology,
physics, and mathematics. This typically includes mathematics
majors with interests in applied mathematics.
The course is offered
at the University Park campus, typically once per year, and
sometimes at a few other locations throughout the Penn State
system. Prerequisites: MATH 230 or 231; MATH 250 or 251.
414. (STAT)
INTRODUCTION TO PROBABILITY THEORY (3) Probability spaces,
discrete and continuous random variables, transformations, expectations,
generating functions, conditional distributions, law of large
numbers, central limit theorems. Students may take only one
course from MATH (STAT) 414 and 418 for credit. Prerequisite:
MATH 231.
415. (STAT)
INTRODUCTION TO MATHEMATICAL STATISTICS (3) A theoretical
treatment of statistical inference, including sufficiency, estimation,
testing, regression, analysis of variance, and chi-square tests.
Prerequisites: MATH (STAT) 414.
416. (STAT)
STOCHASTIC MODELING (3) Review of distribution models, probability
generating functions, transforms, convolutions, Markov chains,
equilibrium distributions, Poisson process, birth and death
processes, estimation. Prerequisites: MATH (STAT) 318, 414,
or MATH 230.
417. QUALITATIVE
THEORY OF DIFFERENTIAL EQUATIONS (3) The main objective
of the course is the qualitative theory of ordinary differential
equations such as existence and uniqueness of solutions, dependence
on initial data and parameters, and basic stability of solutions
for both linear and nonlinear equations. It is designed to introduce
students to modern concepts including the bifurcation theory,
intermittent (transitional) and chaotic behavior of solutions
and dynamical system approach to differential equations. Along
the way, a number of applications are discussed and students
get familiar with some basic examples illustrating main principles
of the theory, such as the Lorenz attractor, predator-prey models,
etc.
The course is completed
by students majoring in engineering programs, the sciences,
and mathematics, and is offered at a few locations in the Penn
State system. At the University Park campus the course is typically
offered once per year in the spring semester. Prerequisites:
MATH 220, 250.
418. (STAT)
PROBABILITY (3) Fundamentals and axioms, combinatorial probability,
conditional probability and independence, probability laws,
random variables, expectation; Chebyshev's inequality. Students
may take only one course from MATH (STAT) 414 and 418 for credit.
Prerequisite: MATH 231.
419. (PHYS)
THEORETICAL MECHANICS (3) Principles of Newtonian, Lagrangian,
and Hamiltonian mechanics of particles with applications to
vibrations, rotations, orbital motion, and collisions. Prerequisites:
MATH 230 or 232; MATH 250 or 251; PHYS 203 or 204.
420. ELEMENTARY
INTRODUCTION TO CHAOTIC DYNAMICS AND FRACTAL GEOMETRY ( 3)
An introduction to the theory of fractals for undergraduates
in mathematics, science, engineering, economics, and computer
science. Prerequisite: MATH 140 , MATH 141 , MATH 220 or MATH
110 , MATH 111 , MATH 220
421. COMPLEX
ANALYSIS (3) Infinite sequences and series; algebra and
geometry of complex numbers; analytic functions; integration;
power series; residue calculus; conformal mapping, applications.
Prerequisites: MATH 230, 232, or 405; MATH 401 or 403.
422. WAVELETS
AND FOURIER ANALYSIS: THEORY AND APPLICATIONS (3) Fundamental
mathematical issues of the theory of wavelets for senior undergraduate
and graduate students in mathematics, engineering, physics, and computer
science. Prerequisites: MATH 312,
MATH 401, MATH 403, MATH 405, or MATH 412.
426. INTRODUCTION
TO MODERN GEOMETRY (3) Plane and space curves; space surfaces;
curvature; intrinsic geometry of surfaces; Gauss-Bonnet theorem;
covariant differentiation; tensor analysis. Prerequisite: MATH
401 or 403.
427. FOUNDATIONS
OF GEOMETRY (3) Euclidean and various non-Euclidean geometries
and their development from postulate systems. Prerequisite:
MATH 230 or 231.
429. INTRODUCTION
TO TOPOLOGY (3) Metric spaces, topological spaces, separation
axioms, product spaces, identification spaces, compactness,
connectedness, fundamental group. Prerequisite: MATH 311W.
435. BASIC ABSTRACT
ALGEBRA (3) Elementary theory of groups, rings, and fields.
Prerequisite: MATH 311W.
436. LINEAR
ALGEBRA (3) Vector spaces and linear transformations, canonical
forms of matrices, elementary divisors, invariant factors; applications.
Prerequisite: MATH 311W.
437. ALGEBRAIC
GEOMETRY (3) The geometric study of algebraic equations
is one of the oldest and deepest parts of mathematics, and it
lies at the heart of modern developments in geometry, algebra,
number theory and physics. Students completing MATH 437 will
understand many new algebraic and geometric ideas by studying
examples of curves defined by equations of degrees 2 and 3 in
the plane.
First come conics
(given by equations of degree 2 in two variables). Rigid motions,
similarities, and affine transformations give different classifications
of them. New ideas then show how to get a conic through any
five points and prove Pascal's theorem about six points on a
conic. Special cases suggest extension of the usual plane to
the projective plane, with "points at infinity," homogeneous
coordinates, and projective transformations.
The main part of
the course turns to equations of degree 3 and their singularities,
flex points, tangents, and degeneracies. Several new ideas,
both algebraic and analytic, are brought in to prove the existence
of complex flex points on nonsingular cubic and then real flex
points on nonsingular real cubic. There is then a classification
of complex projective cubics by a single parameter and finally
a full classification of all real projective cubics.
As time permits,
relations to further topics are sketched: addition of points
on a nonsingular cubic, Mordell's theorem, doubly periodic functions,
and Fermat's last theorem.
The course is typically
taken by mathematics majors and is offered at the University
Park campus about once every two years. Prerequisite:
MATH 230 or 231.
441. MATRIX
ALGEBRA (3) Determinants, matrices, linear equations, characteristic
roots, quadratic forms, vector spaces. Students who have passed
MATH 436 may not schedule this course. Prerequisite: MATH 220.
450. MATHEMATICAL
MODELING ( 3) Constructing mathematical models of physical
phenomena; topics include pendulum motion, polymer fluids, chemical
reactions, waves, flight, and chaos. Prerequisite: MATH 405
or MATH 412
451. (CSE) NUMERICAL
COMPUTATIONS (3) Algorithms for interpolation, approximation,
integration, nonlinear equations, linear systems, fast Fourier
transform, and differential equations emphasizing computational
properties and implementation. Students may take only one course
for credit from MATH 451 and 455. Prerequisites: CMPSC 201C,
201F, or CSE 103; MATH 230 or 231.
455. (CSE) INTRODUCTION
TO NUMERICAL ANALYSIS I (3) Floating point computation,
numerical rootfinding, interpolation, numerical quadrature,
direct methods for linear systems. Students may take only one
course for credit from MATH 451 and 455. Prerequisites: CMPSC
201C, 201F, or CSE 103; MATH 220; MATH 230 or 231.
456. (CSE) INTRODUCTION
TO NUMERICAL ANALYSIS II (3) Polynomial and piecewise polynomial
approximation, matrix least squares problems, numerical solution
of eigenvalue problems, numerical solution of ordinary differential
equations. Prerequisite: MATH 455.
457. INTRODUCTION
TO MATHEMATICAL LOGIC (3) Propositional logic, first-order
predicate logic, axioms and rule of inference, structures, models,
definability, completeness, compactness. Prerequisites: MATH
311W or PHIL 212; 3 additional credits in philosophy.
459. COMPUTABILITY
AND UNSOLVABILITY (3) An introduction to the theory of recursive
functions; solvable and unsolvable decision problems; applications.
Prerequisite: MATH 311W.
461. (PHYS)
THEORETICAL MECHANICS (3) Continuation of MATH (PHYS) 419.
Theoretical treatment of dynamics of a rigid body, theory of
elasticity, aggregates of particles, wave motion, mechanics
of fluids. Prerequisite: MATH (PHYS) 419.
465. NUMBER
THEORY I (3) MATH 465 - Number Theory serves as an upper-level
introduction to the fundamentals of elementary number theory.
A major emphasis in the course is placed on the role that the
prime numbers play in the study of properties of the integers
along with the related topics of divisibility and factorization
of integers. Additional topics covered in the course include
congruences (and the theorems of Euler and Fermat which are
classics in this area), properties of arithmetic functions including
those which are multiplicative, and other topics such as Pythagorean
triples and representations of numbers as sums of squares.
This course is
completed by a wide variety of students across the university,
especially those majoring in mathematics. (In many other options
in the MTHBS degree, MATH 465 can be used to satisfy one of
the major requirements.) The course is also taken quite frequently
by non-mathematics majors who wish to use the course to satisfy
and upper-level requirement for the mathematics minor.
The course is offered
at a few locations in the Penn State system. At University Park,
MATH 465 is typically offered once a year. Prerequisite: MATH
230 or 231.
467. (CSE) FACTORIZATION
AND PRIMALITY TESTING (3) Prime sieves, factoring, computer
numeration systems, congruences, multiplicative functions, primitive
roots, cryptography, quadratic residues. Students who have passed
MATH 465 may not schedule this course. Prerequisite: CSE 260
or MATH 311W.
468. MATHEMATICAL
CODING THEORY (3) Shannon's theorem, block codes, linear
codes, Hamming codes, Hadamard codes, Golay codes, Reed-Muller
codes, bounds on codes, cyclic codes. Prerequisites: MATH 311W,
advanced calculus.
469. MATHEMATICS
OF ALGORITHMS (3) Binomial identities; recurrence relations,
operator methods; asymptotic methods. Prerequisite: advanced
calculus.
470. ALGEBRA
FOR TEACHERS (3) An introduction to algebraic structures
and to the axiomatic approach, including the elements of linear
algebra. Designed for teachers and prospective teachers. Students
who have passed MATH 435 may not schedule this course. Prerequisite:
MATH 311W.
471. GEOMETRY
FOR TEACHERS (4) Problem solving-oriented introduction to
Euclidean and non-Euclidean geometries; construction problems
and geometrical transformations via "Geometer's Sketchpad"
software. Intended primarily for those seeking teacher certification
in secondary mathematics. Students who have passed MATH 427
may not schedule this course. Prerequisite: MATH 311W.
483. APPLIED
MODERN ALGEBRA II (3) Semigroups, groups, permutation groups,
machines, Polya enumeration theory, switching functions, de
Bruijn's theorem, fast adders. Prerequisite; MATH 311W.
484. LINEAR
PROGRAMS AND RELATED PROBLEMS (3) Introduction to theory
and applications of linear programming; the simplex algorithm
and new methods of solution; duality theory. Prerequisites:
MATH 220; MATH 230 or 231.
485. GRAPH THEORY
(3) Introduction to the theory and applications of graphs
and directed graphs. Emphasis on the fundamental theorems and
their proofs. Prerequisite: MATH 311W.
486. MATHEMATICAL
THEORY OF GAMES (3) Basic theorems, concepts, and methods
in the mathematical study of games of strategy; determination
of optimal play when possible. Prerequisite: MATH 220.
493. MATHEMATICS
RECITATION INSTRUCTOR TRAINING (1 per semester, maximum of 3)
Instruction and practice in the role of recitation instructor.
Prerequisites: 18 credits in mathematics.
