In the Department of Mathematics and Statistics
In the College of Sciences
OFFICE: Geology/Mathematics/Computer Science 413 mathematics education; in climate mathematics, computational math-
TELEPHONE: 619-594-6191 ematics, control theory, dynamical systems, financial mathematics,
mathematics of communication, mathematical physics, modeling and
optimization within applied mathematics.
Faculty Opportunities for research in mathematics education are available
Mathematics and Applications through research facilities in the Center for Research in Mathematics
Samuel S. P. Shen, Ph.D., Professor of Mathematics, and Science Education.
Chair of Department The department hires qualified graduate students as teaching
José E. Castillo, Ph.D., Professor of Mathematics associates. These positions serve as an important stepping stone on
(M.S. Computational Science Graduate Adviser) the path to a career in the teaching of mathematics at various levels.
T. Marc Dunster, Ph.D., Professor of Mathematics (Coordinator)
John D. Elwin, Ph.D., Professor of Mathematics, Emeritus Admission to Graduate Study
Tunc Geveci, Ph.D., Professor of Mathematics All students must satisfy the general requirements for admission to
Robert D. Grone, Ph.D., Professor of Mathematics the university with classified graduate standing, as described in Part
(Associate Chair of Department) Two of this bulletin.
Stefen Hui, Ph.D., Professor of Mathematics Students applying for admission should electronically submit the
(M.S. Applied Mathematics Graduate Adviser) university application available at http://www.csumentor.edu along
F. David Lesley, Ph.D., Professor of Mathematics, Emeritus with the $55 application fee.
Joseph M. Mahaffy, Ph.D., Professor of Mathematics All applicants must submit admissions materials to SDSU
Antonio Palacios, Ph.D., Professor of Mathematics Graduate Admissions.
Peter Salamon, Ph.D., Professor of Mathematics Graduate Admissions
(Coordinator and M.S. Applied Mathematics Graduate Adviser) The following materials should be submitted as a complete
Peter Blomgren, Ph.D., Associate Professor of Mathematics package directly to:
Ricardo Carretero, Ph.D., Associate Professor of Mathematics Graduate Admissions
(M.S. Dynamical Systems Graduate Adviser) Enrollment Services
Stephen J. Kirschvink, Ph.D., Associate Professor of Mathematics San Diego State University
Michael O’Sullivan, Ph.D., Associate Professor of Mathematics San Diego, CA 92182-7416
Roxana N. Smarandache, Ph.D., Associate Professor of Mathematics
(1) Official transcripts (in sealed envelopes) from all
J. Carmelo Interlando, Ph.D., Assistant Professor of Mathematics
postsecondary institutions attended;
Vadím Ponomarenko, Ph.D., Assistant Professor of Mathematics
(M.A. Mathematics Graduate Adviser) Note:
• Students who attended SDSU need only submit tran-
Mathematics Education scripts for work completed since last attendance.
Joanne Lobato, Ph.D., Professor of Mathematics • Students with international coursework must submit both
B. Ricardo Nemirovsky, Ph.D., Professor of Mathematics the official transcript and proof of degree. If documents
Janet Sue Bowers, Ph.D., Associate Professor of Mathematics are in a language other than English, they must be
(M.A.T.S. Graduate Adviser) accompanied by a certified English translation.
Andrew G. Izsák, Ph.D., Associate Professor of Mathematics (2) GRE scores (http://www.ets.org, SDSU institution code 4682);
Chris L. Rasmussen, Ph.D., Associate Professor of Mathematics (3) TOEFL score, if medium of instruction was in a language other
Susan D. Nickerson, Ph.D., Assistant Professor of Mathematics than English (http://www.ets.org, SDSU institution code 4682).
Associateships Advancement to Candidacy
Graduate teaching associateships in mathematics are available to All students must satisfy the general requirements for
a limited number of qualified students. Application blanks and advancement to candidacy as described in Part Two of this bulletin. In
additional information may be secured from the chair of the addition, the student must have passed a qualifying examination in
department. some programs.
The Department of Mathematics and Statistics offers graduate
Specific Requirements for the Master of
study leading to the Master of Arts degree in mathematics, the Master Arts Degree in Mathematics
of Arts degree for teaching service with a concentration in mathemat- (Major Code: 17011)
ics, the Master of Science degree in applied mathematics, the Master In addition to meeting the requirements for classified graduate
of Science degree in statistics (see the Statistics section of this bulletin standing and the basic requirements for the master’s degree as
for a description of the statistics program and courses), the Master of described in Part Two of this bulletin, the student must meet the
Science degree in Applied Mathematics with a Concentration in Math- following requirements:
ematical Theory of Communications Systems, and the Master of
Science degree in Applied Mathematics with a Concentration in 1. Complete 30 units of approved 500, 600, and 700 level
Dynamical Systems. courses, of which at least 24 units must be in mathematics. At
Faculty active in research direct theses and research projects in least 21 units must be at the 600 level or above. Mathematics
most general areas of the mathematical sciences: in complex 600, 601, and 602 may not be part of this degree. No more than
analysis, differential equations, number theory, numerical analysis; in six units of Mathematics 797 and 798 will be accepted toward
cognitive science, computer education and problem solving within the degree.
262 SDSU GRADUATE BULLETIN 2009-2010
2. Among the 30 units of coursework, students must include at least recommended electives include Mathematics 542, 623, 637, 668,
two courses in the area of algebra chosen from courses Mathe- 693A, 693B, 797; Computer Science 553; Physics 580. Depending on
matics 623, 627A, 627B, and at least two courses in analysis the student’s interests and background, electives from other
chosen from courses Mathematics 630A, 630B, 631A, 631B. departments may be approved by the adviser.
3. Before entering the program, students should have completed Concentration in Mathematical Theory
the following courses or their equivalents: Mathematics 521B, of Communication Systems
524, 532, 534B. If a student has not had these courses before
entering the program, they must be taken during the first year. This concentration focuses on the area of mathematics relevant to
(A maximum of two of these courses may be applicable toward the transmitting and processing of information by digital or analog
the degree course requirements.) methods. In addition to meeting the requirements for classified
standing in the Master of Science program in applied mathematics,
4. With departmental approval, students may select Plan A and students pursuing this concentration should also have completed
complete Mathematics 799A or Plan B requiring a written com- Mathematics 521A or its equivalent before entering the program.
prehensive examination based on materials to be selected by Students must complete Mathematics 525, 626, 668; two courses
the department from among Mathematics 623, 627A, 627B, selected from Mathematics 528, 625 or 667, and two courses selected
630A, 630B, 631A, 631B. from Mathematics 623, 627A, 627B, 630A-630B, 631A-631B. Two
additional courses in mathematics or in a related area may be
Plan A is encouraged for most students since it provides an selected with the approval of the program adviser. Either Mathematics
introduction to independent reading and is a natural pathway to 797 (Research) or 799A (Thesis) are required of students in this
independent research. degree program.
Specific Requirements for the Master of Communications Systems Certificate
Science Degree in Applied Mathematics The Communication Systems Certificate provides mathematicians
and engineers with the specialized training in the areas of coding,
(Major Code: 17031) cryptography, and signal processing relevant for the understanding of
In addition to meeting the requirements for classified graduate modern communication systems. This certificate is designed for
standing and the basic requirements for the master’s degree individuals who need the knowledge this certificate program provides
described in Part Two of this bulletin, the student must meet the to participate in projects in the area of communication systems and
following requirements: signal processing.
1. Have completed before entering the program, the following This is an advanced academic certificate at the postbaccalaureate
level. The admission requirement is a bachelor’s degree in mathemat-
courses or their equivalents: Mathematics 524, 534A, 534B,
ics, engineering, or a closely related field. Individuals with knowledge
537, 541; Statistics 551A. At most one of these courses can be of the background materials through work or self-study may also be
counted towards the degree course requirements. Program- accepted into this program at the discretion of the program director.
ming proficiency in a computer language is also a prerequisite. Course requirements for the certificate program are the following
Admission to the program as conditionally classified may be courses completed with a grade point average of 3.0 or above:
granted without some of the coursework above, contingent on Mathematics 522, 525, 626, 667, and 668.
the student removing any deficiencies by the end of the first For information on the application process, contact the
year in the program. Department of Mathematics and Statistics or call 619-594-6191.
2. Complete a minimum of 30 units of approved 500-, 600-, and
700-numbered courses. All programs must include at least 21 Courses Acceptable on Master’s Degree
units in mathematical science (with the possible exception of a
student whose main interest is mathematical modeling) and at
Programs in Applied Mathematics,
least 18 units selected from 600- and 700-numbered courses. Mathematics, and Statistics (MATH)
No more than six units in Mathematics 797 and 798 will be Refer to Courses and Curricula and Regulations of the Division of Gradu-
accepted for credit toward the degree. A program of study ate Affairs sections of this bulletin for explanation of the course numbering
must be approved by the graduate adviser. system, unit or credit hour, prerequisites, and related information.
3. The student must select Plan A and complete Mathematics UPPER DIVISION COURSES
799A, Thesis. The student must also have an oral defense of
their thesis or research, open to the public. NOTE: Proof of completion of prerequisites required for all upper
division courses: Copy of transcript.
Concentration in Dynamical Systems
MATH 509. Computers in Teaching Mathematics (3)
This concentration focuses on interdisciplinary applications of Two lectures and three hours of laboratory.
dynamical systems and nonlinear modeling in biology, chemistry, Prerequisite: Mathematics 252.
engineering, and physics. Students with interests in modeling and
Solving mathematical tasks using an appropriate computer
analyzing real life problems through mathematics will benefit from this
interface, and problem-based curricula. Intended for those interested
concentration. To enter the program, students must possess a
in mathematics teaching.
bachelor’s degree with a strong mathematical background. In addition
to completing the specific requirements for the Master of Science MATH 510. Introduction to the Foundations of Geometry (3)
degree in applied mathematics, students pursuing this concentration Prerequisite: Mathematics 122 or 151.
will complete the following 15 units of core courses: Mathematics 531, The foundations of Euclidean and hyperbolic geometries. Highly
537, 538, 636, and 638; 12 units of electives and three units of recommended for all prospective teachers of high school geometry.
Mathematics 799A (Thesis/Project). Possible electives include
Mathematics 696, Special Topics in Dynamical Systems (Nonlinear MATH 511. Projective Geometry (3)
Waves, Pattern Formation, Applied Bifurcation of Dynamical Systems, Prerequisite: Mathematics 254.
Nonlinear Time Series, Numerical Experiments and Methods in Geometry emphasizing relationships between points, lines, and
Dynamical Systems, Fractal Geometry, Mathematical Biology/Neural conics. Euclidean geometry and some non-Euclidean geometries as
Modeling) to be offered depending on demand and resources. Other special cases of projective geometry.
SDSU GRADUATE BULLETIN 2009-2010 263
MATH 521A. Abstract Algebra (3) MATH 538. Discrete Dynamical Systems and Chaos (3)
Prerequisites: Mathematics 245 and 254. Prerequisites: Minimum grade of C in Mathematics 151; Mathe-
Abstract algebra, including elementary number theory, groups, matics 254 or 342A, 342B.
and rings. One- and two-dimensional iterated maps, equilibria and their
MATH 521B. Abstract Algebra (3) stability, sensitive dependence on initial conditions, Lyapunov
Prerequisite: Mathematics 521A. exponents, horseshoe maps, period doubling, chaotic attractors,
Poincare maps, stable/unstable manifolds, bifurcations. Applications
Continuation of Mathematics 521A. Rings, ideals, quotient rings, in biology, chemistry, physics, engineering, and other sciences.
unique factorization, noncommutative rings, fields, quotient fields, and
algebraic extensions. MATH 541. Introduction to Numerical Analysis and Computing (3)
MATH 522. Number Theory (3) Prerequisites: Mathematics 254 or 342A; and Computer Science
Prerequisite: Mathematics 245. 106 or 107 or 205.
Theory of numbers to include congruences, Diophantine Solution of equations of one variable, direct methods in numerical
equations, and a study of prime numbers; cryptography. linear algebra, least squares approximation, interpolation and uniform
MATH 523. Mathematical Logic (3)
Prerequisite: Mathematics 245. MATH 542. Introduction to Numerical Solutions of Differential
Propositional logic and predicate calculus. Rules of proof and Equations (3)
models. Completeness and the undecidability of arithmetic. Not open Prerequisites: Mathematics 337 and 541.
to students with credit in Philosophy 521. Initial and boundary value problems for ordinary differential
MATH 524. Linear Algebra (3) equations. Partial differential equations. Iterative methods, finite
difference methods, and the method of lines.
Prerequisites: Mathematics 245 and 254; or 342A.
Vector spaces, linear transformations, orthogonality, eigenvalues MATH 543. Numerical Matrix Analysis (3)
and eigenvectors, normal forms for complex matrices, positive Prerequisite: Mathematics 541.
definite matrices and congruence. Gaussian elimination, LU factorizations and pivoting strategies.
MATH 525. Algebraic Coding Theory (3) Direct and iterative methods for linear systems. Iterative methods for
Prerequisite: Mathematics 254. diagonalization and eigensystem computation. Tridiagonal,
Linear codes, perfect and related codes, cyclic linear codes, BCH Hessenberg, and Householder matrices. The QR algorithm.
codes, burst error-correcting codes. MATH 544. Computational Finance (3)
MATH 528. Information Theory and Data Compression (3) Prerequisite: Statistics 550 or 551A.
Prerequisites: Mathematics 245 and 254. Risk evaluation. Numerical procedures for evaluating financial
Fundamental of discrete probability and information theory: joint derivatives. Monte Carlo simulation techniques.
and conditional distributions, Bayes' theorem, entropy, channel
capacity. Noiseless coding theorem and data compression MATH 561. Applied Graph Theory (3)
algorithms: Huffman codes, arithmetic coding, Ziv-Lempel codes. Prerequisite: Mathematics 245 or 254.
Information theory in error correction coding and cryptography. Undirected and directed graphs, trees, Hamiltonian circuits,
MATH 531. Partial Differential Equations (3) classical problems of graph theory including applications to linear
Prerequisites: Mathematics 252 and 337.
Boundary value problems for heat and wave equations: eigen- MATH 562. Mathematical Methods of Operations Research (3)
function expansions, Sturm-Liouville theory and Fourier series. Prerequisites: Mathematics 252 and 254.
D'Alembert's solution to wave equation; characteristics. Laplace's Theory and applications concerned with optimization of linear and
equation, maximum principles, Bessel functions. non-linear functions of several variables subject to constraints,
MATH 532. Functions of a Complex Variable (3) including simplex algorithms, duality, applications to game theory, and
Prerequisite: Mathematics 252. descent algorithms.
Analytic functions, Cauchy-Riemann equations, theorem of MATH 579. Combinatorics (3)
Cauchy, Laurent series, calculus of residues, and applications.
Prerequisite: Mathematics 245.
MATH 533. Vector Calculus (3) Permutations, combinations, generating functions, recurrence
Prerequisite: Mathematics 254 or 342A. relations, inclusion-exclusion counting. Polya's theory of counting,
Scalar and vector fields; gradient, divergence, curl, line and other topics and applications.
surface integrals: Green's, Stokes' and divergence theorems. Green's
identities. Applications to potential theory or fluid mechanics or MATH 580. Risk Management: Stocks and Derivative Securities (3)
electromagnetism. Prerequisite: Statistics 550 or 551A.
MATH 534A. Advanced Calculus I (3) Theory of derivative securities with focus on evolution of stock
prices and pricing of options.
Prerequisites: Mathematics 245 and 254; or 342A.
Completeness of the real numbers and its consequences, MATH 581. Risk Management: Portfolio Selection and Other
sequences of real numbers, continuity, differentiability and integra- Features of Finance Markets (3)
bility of functions of one real variable. Prerequisite: Statistics 550 or 551A.
MATH 534B. Advanced Calculus II (3) Derivatives and term structures, method of principal components,
Prerequisite: Mathematics 534A. theory of portfolio optimization, some numerical methods.
Series and sequences of functions and their applications, MATH 596. Advanced Topics in Mathematics (1-4)
functions of several variables and their continuity, differentiability and
integrability properties. Prerequisite: Consent of instructor.
Selected topics in classical and modern mathematical sciences.
MATH 537. Ordinary Differential Equations (3) May be repeated with the approval of the instructor. See Class
Prerequisite: Mathematics 337. Schedule for specific content. Limit of nine units of any combination of
Theory of ordinary differential equations: existence and 296, 496, 596 courses applicable to a bachelor's degree. Maximum
uniqueness, dependence on initial conditions and parameters, linear credit of six units of 596 applicable to a bachelor's degree. Credit for
systems, stability and asymptotic behavior, plane autonomous 596 and 696 applicable to a master's degree with approval of the
systems, series solutions at regular singular points. graduate adviser.
264 SDSU GRADUATE BULLETIN 2009-2010
GRADUATE COURSES MATH 638. Continuous Dynamical Systems and Chaos (3)
Prerequisites: Mathematics 337 or 537 and Mathematics 254 or
MATH 623. Linear Algebra and Matrix Theory (3) 342A, 342B.
Prerequisite: Mathematics 524. Nonlinear systems of differential equations, potential fields,
Characteristic and minimal polynomials, Cayley-Hamilton theorem, periodic solutions, Lyapunov functions. Chaos in differential
canonical forms, hermitian matrices, Sylvester's law, norms, singular equations, Lyapunov exponents, chaotic attractors, Poincare maps.
values, stability, non-negative matrices. Lorenz and Rossler attractors, forced oscillators, Chua's circuit, stable
MATH 625. Algebraic Coding Theory (3) manifolds. Bifurcations. Applications in science and engineering.
Prerequisites: Mathematics 525 and Mathematics 521B or 522. MATH 639. Nonlinear Waves (3)
Algebraic theory of error correction codes and decoding Prerequisite: Mathematics 531 or 537.
algorithms used in modern communications systems. Reed-Solomon Linear waves, dissipation, dispersion. Conservation laws. Water
codes and algebraic decoding algorithms. Code duality, MacWilliam's waves. KdV equation, solitary waves, cnoidal waves. Scattering and
identities and the linear programming bound. Probabilistic decoding inverse scattering. Perturbation theory. Nonlinear Schroedinger
of convolutional codes, low-density parity-check codes and turbo equation, dark and bright solitons, vortex solutions. Variational
codes. techniques, modulational instability, stability.
MATH 626. Cryptography (3) MATH 667. Mathematical Aspects of Systems Theory (3)
Prerequisites: Mathematics 521A and 522. Prerequisites: Mathematics 524 and 537.
Design of secure cryptosystems with applications. Classical and Linear and nonlinear systems, nonlinear differential equations,
public key cryptosystems. Primality testing, factoring, discrete log equilibrium equations. Linearization, state transition matrix, stability
problem, and knapsack problem. theory, feedback control systems.
MATH 627A. Modern Algebra I (3) MATH 668. Applied Fourier Analysis (3)
Prerequisites: Mathematics 524, 534A; 532 or 534B.
Prerequisite: Mathematics 521B.
Discrete and continuous Fourier transform methods with applica-
Group theory, including isomorphism theorems, permutation tions to statistics and communication systems.
groups, and simplicity of An, finite abelian groups, and Sylow
theorems. Rings, ideals, principal ideal domains, and unique MATH 693A. Advanced Numerical Analysis (3)
factorization. Prerequisites: Mathematics 524 and 542 or 543.
Numerical optimization, Newton's methods for nonlinear equations
MATH 627B. Modern Algebra II (3) and unconstrained minimization. Global methods, nonlinear least
Prerequisite: Mathematics 627A. squares, integral equations.
Modules and the Wedderburn-Artin theorem, field extensions,
splitting fields, Galois theory, finite fields, the fundamental theorem of MATH 693B. Advanced Numerical Analysis (3)
algebra. Prerequisites: Mathematics 531, 537, and 693A.
Methods for differential equations. Elliptic and parabolic partial
MATH 630A-630B. Functions of a Real Variable (3-3) differential equations. Stiff ordinary differential equations.
Prerequisites: Mathematics 524 and 534B. Mathematics 630A is
MATH 696. Selected Topics in Mathematical Sciences (3)
prerequisite to Mathematics 630B.
Prerequisite: Graduate standing.
Lebesgue measure and integration, metric spaces, Banach Intensive study in specific areas of mathematical sciences. May be
spaces, Hilbert spaces, spectral theory. repeated with new content. See Class Schedule for specific content.
MATH 631A-631B. Functions of a Complex Variable (3-3) Credit for 596 and 696 applicable to a master's degree with approval
Prerequisites: Mathematics 532 and 534B. Mathematics 631A is of the graduate adviser.
prerequisite to 631B. MATH 720. Seminar (1-3)
Theory of analytic functions. Elementary functions and power Prerequisite: Consent of instructor.
series, Cauchy’s theorem and its consequences. Entire functions, An intensive study in advanced mathematics. May be repeated
conformal mappings, Riemann mapping theorem. Harmonic with new content. See Class Schedule for specific content. Maximum
functions. credit six units applicable to a master's degree.
MATH 635. Pattern Formation (3) MATH 790. Practicum in Teaching of Mathematics (1) Cr/NC
Prerequisites: Mathematics 337 or 531 and Mathematics 254 or Prerequisite: Award of graduate teaching associateship in
342A, 342B. mathematics.
Linear stability, marginal stability curves, classification. One Supervision in teaching mathematics. Lecture writing, style of
dimensional patterns, bifurcations. Two dimensional patterns, square lecture presentation and alternatives, test and syllabus construction,
and hexagonal patterns, spirals, defects. Diffusion driven instability, and grading system. Not applicable to an advanced degree. Required
Turing patterns. Spatio-temporal chaos. Applications in biology, for first semester GTA's.
chemistry, and physics. MATH 797. Research (1-3) Cr/NC/RP
MATH 636. Mathematical Modeling (3) Prerequisite: Six units of graduate level mathematics.
Prerequisites: Mathematics 254 and 337 or Mathematics 342A and Research in one of the fields of mathematics. Maximum credit six
342B or Engineering 280. units applicable to a master's degree.
Advanced models from the physical, natural, and social sciences. MATH 798. Special Study (1-3) Cr/NC/RP
Emphasis on classes of models and corresponding mathematical Prerequisite: Consent of staff; to be arranged with department
structures. chair and instructor.
MATH 637. Theory of Ordinary Differential Equations (3) Individual study. Maximum credit six units applicable to a master's
Prerequisite: Mathematics 537. degree.
Existence, uniqueness, and continuation of solutions from an MATH 799A. Thesis or Project (3) Cr/NC/RP
advanced standpoint. Linear systems and their stability and asymp- Prerequisites: An officially appointed thesis committee and
totic behavior, regular and irregular singularities, and regular advancement to candidacy.
boundary value problems. Preparation of a project or thesis for the master's degree.
SDSU GRADUATE BULLETIN 2009-2010 265
MATH 799B. Thesis or Project Extension (0) Cr/NC MATH 799C. Comprehensive Examination Extension (0) Cr/NC
Prerequisite: Prior registration in Thesis or Project 799A with an Prerequisite: Completion or concurrent enrollment in degree
assigned grade symbol of RP. program courses.
Registration required in any semester or term following assignment Registration required of students whose only requirement is
of RP in Course 799A in which the student expects to use the facilities completion of the comprehensive examination for the master's degree
and resources of the university; also student must be registered in the Registration in 799C limited to two semesters.
course when the completed thesis or project is granted final approval.
For additional courses useful to mathematicians see the sec-
Mathematics and Science Education
266 SDSU GRADUATE BULLETIN 2009-2010
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