Mathematics
Courses
For course descriptions not found in the 2006-2007 General
Catalog, please contact the department for more information.
All Prerequisites listed below may be replaced by an equivalent
or higher-level course. The listings of quarters in which courses
will be offered are only tentative. Please consult the Department
of Mathematics to determine the actual course offerings each year.
Lower-Division
3C. Pre-Calculus (4) Functions and
their graphs. Linear and polynomial functions, zeroes, inverse functions,
exponential and logarithm, trigonometric functions and their inverses.
Emphasis on understanding algebraic, numerical and graphical approaches
making use of graphing calculators. (No credit given if taken after
Math. 4C, 1A/10A, or 2A/20A.) Prerequisite: two or more years
of high school mathematics or equivalent.
4C. Pre-Calculus for Science and Engineering (4) Review
of polynomials. Graphing functions and relations: graphing rational
functions, effects of linear changes of coordinates. Circular functions
and right triangle trigonometry. Reinforcement of function concept:
exponential, logarithmic, and trigonometric functions. Vectors.
Conic sections. Polar coordinates. Three lectures, one recitation.
(No credit given if taken after Math. 1A/10A or 2A/20A. Two units
of credit given if taken after Math. 3C.) Prerequisite: Math
Placement Exam qualifying score, or Math. 3C with a grade of C or
better.
10A. Calculus (4) Differentiation
and integration of algebraic functions. Fundamental theorem of calculus.
Applications. (No credit given if taken after Math. 2A/20A. Formerly
numbered Math. 1A.) Prerequisite: Math Placement Exam qualifying
score, or AP Calculus AB score of 2, or SAT II Math. 2C score of
650 or higher, or Math. 3C with a grade of C or better, or Math.
4C with a grade of C– or better.
10B. Calculus (4) Further applications
of the definite integral. Calculus of trigonometric, logarithmic,
and exponential functions. Complex numbers. (No credit given if
taken after Math. 2B/20B. Formerly numbered Math. 1B.) Prerequisite:
AP Calculus AB score of 3, 4, or 5, or Math. 10A with a grade of
C– or better, or Math. 20A with a grade of C– or better.
10C. Calculus (4) Vector geometry,
velocity, and acceleration vectors. (No credit given if taken after
Math. 2C/20C. Formerly numbered Math. 1C.) Prerequisite: AP Calculus
BC score of 3, 4, or 5, or Math. 10B with a grade of C– or
better, or Math. 20B with a grade of C– or better.
11. Elementary Probability and Statistics (4) Events
and probabilities, combinatorics, conditional probability, Bayes
formula. Discrete random variables: mean, variance; binomial, multinomial,
Poisson distributions. Continuous random variables: densities, mean,
variance; normal, uniform, exponential distributions. Sample statistics,
confidence intervals, regression. Applications. Intended for biology
and social science majors. Prerequisites: Math. 10A-B or Math.
20A-B.
15A. Discrete Mathematics (4) Basic
discrete mathematical structures: sets, relations, functions, sequences,
equivalence relations, partial orders, number systems. Methods of
reasoning and proofs: propositional logic, predicate logic, induction,
recursion, pigeonhole principle. Infinite sets and diagonalization.
Basic counting techniques; permutations and combinations. Applications
will be given to digital logic design, elementary number theory,
design of programs, and proofs of program correctness. Equivalent
to CSE 20. Credit not offered for both Math. 15A and CSE 20. Prerequisites:
CSE 8A or CSE 8B or CSE 11. CSE 8B or CSE 11 may be taken concurrently
with Math. 15A/CSE 20.
15B. Mathematics for Algorithm and Systems (4) This
course introduces mathematical tools for the qualitative and quantitative
analysis of algorithms and computer systems. Topics to be covered
include basic enumeration and counting techniques; recurrence relations;
graph theory; asymptotic notation; elementary applied discrete probability.
Equivalent to CSE 21. Prerequisite: Math. 15A or CSE 20 or 160A;
CSE 12 is strongly recommended for CSE 21.
18. Computer Animated Statistics (4) Students
will acquire the basics of statistical analysis by working with
computer-simulated models rather than abstract mathematical language.
Topics include hypothesis testing, maximum likelihood estimation,
sampling, chi-square tests and construction of confidence intervals.
Prerequisite: Math. 1B or 10B or 20B.
20A. Calculus for Science and Engineering (4) Foundations
of differential and integral calculus of one variable. Functions,
graphs, continuity, limits, derivative, tangent line. Applications
with algebraic, exponential, logarithmic, and trigonometric functions.
Introduction to the integral. (Two credits given if taken after
Math. 1A/10A and no credit given if taken after Math. 1B/10B or
Math. 1C/10C. Formerly numbered Math. 2A.) Prerequisite: Math
Placement Exam qualifying score, or AP Calculus AB score of 2 or
3, or SAT II Math. 2C score of 650 or higher, or Math. 4C with
a
grade of C– or better, or Math. 10A with a grade of C– or
better.
20B. Calculus for Science and Engineering (4) Integral
calculus of one variable and its applications, with exponential,
logarithmic, hyperbolic, and trig-onometric functions. Methods
of integration. Infinite series. Polar coordinates in the plane
and complex exponentials. (Two units of credits given if taken
after Math. 1B/10B or Math. 1C/10C.)Prerequisite:
AP Calculus AB score of 3, 4, or 5, or AP Calculus BC score
of 3, or Math. 20A with a grade of C– or better, or Math.
10B with a grade of C– or better, or Math. 10C with a grade
of C– or better.
20C. Calculus and Analytic Geometry for Science and Engineering
(4) Vector geometry, vector functions
and their derivatives. Partial differentiation. Maxima and minima.
Double integration, Two units of credit given if taken after Math.
10C. Formerly numbered Math. 21C. Prerequisite: AP Calculus BC
score of 3, 4, or 5, or Math. 20B with a grade of C– or better.
20D. Introduction to Differential Equations
(4) Ordinary
differential equations: exact, separable, and linear; constant
coefficients, undetermined coefficients. Variations of parameters.
Series solutions. Systems. Laplace transforms. Techniques for engineering
sciences. Computing symbolic and graphical solutions using Matlab.
Formerly numbered Math. 21D. May be taken as repeat credit for
Math. 21D.Prerequisite: Math. 20C (or Math. 21C)
with a grade of C– or better.
20E. Vector Calculus (4) Change
of variable in multiple integrals, Jacobian Line integrals, Greens
theorem. Vector fields, gradient fields, divergence, curl. Spherical
and cylindrical coordinates. Taylor series in several variables.
Surface integrals, Stokes theorem. Gauss theorem and
its applications. Conservative fields. (Zero units given if Math.
2F previously. Formerly numbered Math. 2F) Prerequisite: Math.
20C (or Math. 21C) with a grade of C– or better. 20F. Linear Algebra (4) Matrix algebra,
solution of systems of linear equations by Gaussian elimination,
determinants. Linear and affine subspaces, bases of Euclidean spaces.
Eigenval-ues and eigenvectors, quadratic forms, orthogonal matrices,
diagonalization of symmetric matrices. Applications. Computing symbolic
and graphical solutions using Matlab. (Zero units given if Math.
2EA previously. Formerly numbered 2EA.) Prerequisite: Math.
20C (or Math. 21C) with a grade of C– or better. 87. Freshman Seminar (1) The Freshman
Seminar Program is designed to provide new students with the opportunity
to explore an intellectual topic with a faculty member in a small
seminar setting. Freshman seminars are offered in all campus departments
and undergraduate colleges, and topics vary from quarter to quarter.
Enrollment is limited to 15 to 20 students, with preference given
to entering freshman. Prerequisite: none.
Upper-Division
100A-B-C. Modern Algebra (4-4-4) An
introduction to the methods and basic structures of higher algebra:
sets and mappings, the integers, rational, real and complex numbers,
groups, rings (especially polynomial rings) and ideals, fields,
real and complex vector spaces, linear transformations, inner product
spaces, matrices, triangular form, diagonalization. Both 100 and
103 cannot be taken for credit. Three lectures, one recitation.
Prerequisites: Math. 20F, and Math. 109 or consent of instructor.
(F,W,S)
102. Applied Linear Algebra (4) A
second course in linear algebra from a computational yet geometric
point of view. Elementary Hermitian matrices, Schurs theorem,
normal matrices, and quadratic forms. Moore-Penrose generalized
inverse and least square problems. Vector and matrix norms. Characteristic
and singular values. Canonical forms. Determinants and multilinear
algebra. Three lectures, one recitation. Prerequisite: Math.
20F. (W)
103A-B. Modern Applied Algebra (4-4) Abstract
algebra with applications to computation. Set algebra and graph
theory. Finite state machines. Boolean algebras and switching theory.
Lattices. Groups, rings and fields: applications to coding theory.
Recurrent sequences. Three lectures, one recitation. Both 100 and
103 cannot be taken for credit. Prerequisites: Math. 20F and
Math. 109 (may be taken concurrently). (F,W)
104A-B-C. Number Theory (4-4-4) Topics
from number theory with applications and computing. Possible topics
are: congruences, reciprocity laws, quadratic forms, prime number
theorem, Riemann zeta function, Fermats conjecture, diophantine
equations, Gaussian sums, algebraic integers, unique factorization
into prime ideals in algebraic number fields, class number, units,
splitting of prime ideals in extensions, quadratic and cyclotomic
fields, partitions. Possible applications are Fast Fourier Transform,
signal processing, coding, cryptography. Three lectures. Prerequisite:
consent of instructor.
107A-B. Computer Algebra (4) An introduction
to algebraic computation. Compu-tational aspects of groups, rings,
fields, etc. Data representation and algorithms for symbolic computation.
Polynomials and their arithmetic. The use of a computer algebra
system as an experimental tool in mathematics. Programming using
algebra systems. Prerequisite: prior or concurrent enrollment
in the Math. 100 or 103 sequence.
109. Mathematical Reasoning (4) This
course uses a variety of topics in mathematics to introduce the
students to rigorous mathematical proof, emphasizing quantifiers,
induction, negation, proof by contradiction, naive set theory, equivalence
relations and epsilon-delta proofs. Required of all departmental
majors. Prerequisite: Math. 20F.
110. Introduction to Partial Differential Equations (4) Fourier
series, orthogonal expansions, and eigenvalue problems. Sturm-Liouville
theory. Separation of variables for partial differential equations
of mathematical physics, including topics on Bessel functions and
Legendre polynomials. Prerequisites: Math. 20D (or 21D) and 20F,
or consent of instructor. (F,S)
111A-B. Mathematical Modeling (4-4) An
introduction to mathematical modeling in the physical and social
sciences, concentrating on one or more topics that vary from year
to year. Students work on independent or group projects. May be
repeated for credit when topics change. Prerequisites: Math.
20D and Math. 20F, or consent of instructor.
120A. Elements of Complex Analysis (4) Complex
numbers and functions. Analytic functions, harmonic functions, elementary
conformal mappings. Complex integration. Power series. Cauchys
theorem. Cauchys formula. Residue theorem. Three lectures,
one recitation. Prerequisite or co-registration: Math. 20E, or
consent of instructor. (F,W)
120B. Applied Complex Analysis (4) Applications
of the Residue theorem. Conformal mapping and applications to potential
theory, flows, and temperature distributions. Fourier transformations.
Laplace transformations, and applications to integral and differential
equations. Selected topics such as Poissons formula. Dirichlet
problem. Neumanns problem, or special functions. Three lectures,
one recitation. Prerequisite: Math. 120A. (W,S)
130A. Ordinary Differential Equations (4) Linear
and nonlinear systems of differential equations. Stability theory,
perturbation theory. Applications and introduction to numerical
solutions. Three lectures. Prerequisites: Math. 20D/21D and 20F.
(F)
130B. Ordinary Differential Equations (4) Existence
and uniqueness of solutions to differential equations. Local and
global theorems of continuity and differentiabillity. Three lectures.
Prerequisites: Math. 20D/21D and 20F, and Math. 130A. (W)
131. Variational Methods in Optimization (4) Maximum-minimum
problems. Normed vector spaces, functionals, Gateaux variations.
Euler-Lagrange multiplier theorem for an extremum with constraints.
Calculus of variations via the multiplier theorem. Applications
may be taken from a variety of areas such as the following: applied
mechanics, elasticity, economics, production planning and resource
allocation, astronautics, rocket control, physics, Fermats
principle and Hamiltons principle, geometry, geodesic curves,
control theory, elementary bang-bang problems. Three lectures, one
recitation. Prerequisites: Math. 20D/21D and 20F or consent of
instructor. (S)
132A. Elements of Partial Differential Equations and Integral
Equations (4) Basic concepts and classification
of partial differential equations. First order equations, characteristics.
Hamilton-Jacobi theory, Laplaces equation, wave equation,
heat equation. Separation of variables, eigenfunction expansions,
existence and uniqueness of solutions. Three lectures. Prerequisite:
Math. 110 or consent of instructor. (W)
132B. Elements of Partial Differential Equations and Integral
Equations (4) Relation between differential
and integral equations, some classical integral equations, Volterra
integral equations, integral equations of the second kind, degenerate
kernels, Fredholm alternative, Neumann-Liouville series, the resolvent
kernel. Three lectures. Prerequisite: Math. 132A. (S)
140A-B-C. Foundations of Analysis (4-4-4) Axioms,
the real number system, topology of the real line, metric spaces,
continuous functions, sequences of functions, differentiation, Riemann-Stieltjes
integration, partial differentiation, multiple integration, Jacobians.
Additional topics at the discretion of the instructor: power series,
Fourier series, successive approximations of other infinite processes.
Three lectures, one recitation. Prerequisites: Math. 20F and
Math. 109 or consent of instructor. Credit cannot be obtained for
both Math. 140A-B and 142A-B. (F,W,S)
142A-B. Advanced Calculus (4-4) The
number system. Functions, sequences, and limits. Continuity and
differentiability. The Riemann integral. Transcendental functions.
Limits and continuity. Infinite series. Sequences and series of
functions. Uniform convergence. Taylor series. Improper integrals.
Gamma and Beta functions. Fourier series. Three lectures. Prerequisites:
Math. 20F and Math. 109 (concurrent enrollment in Math. 109 allowed.)
Credit cannot be obtained for both Math. 140A-B and 142A-B.
150A. Differential Geometry (4) Differential
geometry of curves and surfaces. Gauss and mean curvatures, geodesics,
parallel displacement, Gauss-Bonnet theorem. Three lectures. Prerequisites:
Math. 20E with a grade of C- or better and Math. 20F with a grade
of C- or better, or consent of instructor. (F)
150B. Calculus on Manifolds (4) Calculus
of functions of several variables, inverse function theorem. Further
topics, selected by instructor, such as exterior differential forms,
Stokes theorem, manifolds, Sards theorem, elements of
differential topology, singularities of maps, catastrophes, further
topics in differential geometry, topics in geometry of physics.
Three lectures. Prerequisite: Math. 150A. (W)
151. Topics in Geometry (4) A topic,
selected by the instructor, from Euclidean geometry, non-Euclidean
geometry, projective geometry, algebraic geometry, or other geometries.
May be repeated for credit with a different topic. Three lectures.
Prerequisite: consent of instructor. (S)
152. Applicable Mathematics and Computing (4) This
course will give students experience in applying theory to real
world applications such as Internet and wireless communication problems.
The course will incorporate talks by experts from industry and students
will be helped to carry out independent projects. Topics include
graph visualization, labelling, and embeddings, random graphs and
randomized algorithms. May be taken 3 times for credit. Prerequisites:
Math. 20D or 21D, and 20F or consent of instructor.
153. Geometry for Secondary Teachers (4) Two-
and three-dimensional Euclidean geometry is developed from one set
of axioms. Pedagogical issues will emerge from the mathematics and
be addressed using current research in teaching and learning geometry.
This course is designed for prospective secondary school mathematics
teachers. Prerequisite: Math. 109.
154. Discrete Mathematics and Graph Theory (4) Basic
concepts in graph theory. Combinatorial tools, structures in graphs
(Hamiltonian
cycles, perfect matching). Properties
of graphics and applications in basic algorithmic problems (planarity,
k-colorability, traveling salesman problem). Prerequisites:
Math. 20F and Math. 109, or consent of instructor.
155A. Computer Graphics (4) Bezier
curves and control lines, de Casteljau construction for subdivision,
elevation of degree, control points of Hermite curves, barycentric
coordinates, rational curves. Three lectures, one recitation, and
approximately eight laboratory hours per week. Prerequisites:
Math. 20F and programming experience. [Warning: There are duplicate
credit restrictions on this course. See section on Duplication of
Credit.] (F)
155B. Topics in Computer Graphics (4) Spline
curves, spline interpolation, affine and affine cross ratios, polar
forms (blossoming), the Oslo algorithm for knot insertion, NURBS
and geometric continuity. Three lectures, one recitation, and approximately
eight laboratory hours per week. Prerequisite: Math. 155A or
consent of instructor. (W)
160A-B. Elementary Mathematical Logic (4-4) An
introduction to recursion theory, set theory, proof theory, and
model theory. Turing machines. Undecidability of arithmetic and
predicate logic. Proof by induction and definition by recursion.
Cardinal and ordinal numbers. Completeness and compactness theorems
for propositional and predicate calculi. Three lectures. Prerequisite:
Math. 100A, 103A, 140A, or consent of instructor.
161. Mathematical Computing (2 or 4) Programming
in higher level mathematical language such as Mathematica: Lists,
Functions, Expressions, Recursion, Iteration, graphics, packages.
Application to diverse areas of mathematics such as differential
equations, dynamical systems, fractals, chaos, probability, financial
models. Prerequisite: Math. 20A-B, 21C-D, 20E-F or equivalent.
>163. History of Mathematics (4) Topics
will vary from year to year in areas of mathematics and their development.
Topics may include the evolution of mathematics from the Babylonian
period to the eighteenth century using original sources, a history
of the foundations of mathematics and the development of modern
mathematics. Prerequisite: Math. 20B or consent of instructor.
(S)
165. Introduction to Set Theory (4) Sets,
relations, and functions. Partial, linear, and well-orders. The
axiom of choice, proof by induction and definition by recursion.
Cardinal and ordinal numbers and their arithmetic. Prerequisite:
Math. 100A or 140A or 103, or consent of instructor.
166. Intro to the Theory of Computation (4) Introduction
to formal languages; regular languages; regular expressions, finite
automata, minimization, closure properties, decision algorithms,
and non-regular languages; context-free languages, context-free
grammars, push-down automata, parsing theory, closure properties,
and noncontext-free languages; computable languages; turing machines,
recursive functions, Churchs thesis, undecidability and the
halting problem. Equivalent to CSE 105. Prerequisites: CSE 8B
or 9B or 10 or 65 or 62B AND CSE 20 or 160A or Math. 15A or 109
or 100A or 103A.
168A. Topics in Applied Mathematics-Computer Science (4) Topics
to be chosen in areas of applied mathematics and mathematical aspects
of computer science. May be repeated once for credit with different
topics. Three lectures, one recitation. Prerequisite: consent
of instructor. (W,S)
170A. Numerical Linear Algebra (4) Analysis
of numerical methods for linear algebraic systems and least squares
problems. Orthogonalization methods. Ill conditioned problems. Eigenvalue
and singular value computations. Three lectures, one recitation.
Prerequisites: Math. 20F and knowledge of programming. (F,S)
170B. Numerical Analysis (4) Rounding
and discretization errors. Calculation of roots of polynomials and
nonlinear equations. Interpolation. Approximation of functions.
Three lectures, one recitation. Prerequisites: Math. 20F and
knowledge of programming. (W)
170C. Numerical Ordinary Differential Equations (4) Numerical
differentiation and integration. Ordinary differential equations
and their numerical solution. Basic existence and stability theory.
Difference equations. Boundary value problems. Three lectures, one
recitation. Prerequisite: Math. 170B or consent of instructor.
(S)
171A-B. Mathematical ProgrammingNumerical Optimization
(4-4) Mathematical optimization and applications.
Linear programming, the simplex method, duality. Nonlinear programming,
Kuhn-Tucker theorem. Selected topics from integer programming, network
flows, transportation problems, inventory problems, and other applications.
Three lectures, one recitation. Prerequisites: Math. 20F and
knowledge of programming.
172. Numerical Partial Differential Equations (4) Finite
difference methods for the numerical solution of hyperbolic and
parabolic partial differential equations; finite difference and
finite element methods for elliptic partial differential equations.
Three lectures. Prerequisites: Math. 170A or Math. 110 and programming
experience. (S)
173. Mathematical SoftwareScientific Programming (4) Development
of high quality mathematical software for the computer solution
of mathematical problems. Three lectures, one recitation. Prerequisites:
Math. 170A or Math. 174 and knowledge of FORTRAN. (W)
174. Numerical Methods in Science and Engineering (4) Floating
point arithmetic, linear equations, interpolation, integration,
differential equations, nonlinear equations, optimization, least
squares. Students may not receive credit for both Math. 174 and
Physics 105 or MAE 153 or 154. Students may not receive credit for
Math. 174 if Math. 170 A,B, or C has already been taken. Prerequisites:
Math. 20D (21D) and Math. 20F.
176. Advanced Data Structures (4) Descriptive
and analytical presentation of data structures and algorithms. Lists,
tables, priority queues, disjoint subsets, and dictionaries data
types. Data structuring techniques include linked lists, arrays,
hashing, and trees. Performance evaluation involving worst case,
average and expected case, and amortized analysis. Crecit not offered
for both Math. 176 and CSE 100. Equivalent to CSE 100. Prerequisites:
CSE 12, CSE 21, or Math. 15B, and CSE 30, or consent of instructor.
180A. Introduction to Probability (4) Probability
spaces, random variables, independence, conditional probability,
distribution, expectation, joint distributions, central-limit theorem.
Three lectures. Prerequisites: Math. 20D/21D. [Warning: There
are duplicate credit restrictions on this course. See section on
Duplication of Credit.] (F)
180B. Introduction to Probability (4) Random
vectors, multivariate densities, covariance matrix, multivariate
normal distribution. Random walk, Poisson process. Other topics
if time permits. Three lectures. Prerequisites: Math. 180A and
Math. 20E. (W)
180C. Introduction to Probability (4) Markov
chains in discrete and continuous time, random walk, recurrent events.
If time permits, topics chosen from stationary normal processes,
branching processes, queuing theory. Three lectures. Prerequisite:
Math. 180B. (S)
181A. Introduction to Mathematical Statistics (4) Random
samples, linear regression, least squares, testing hypotheses, and
estimation. Neyman-Pearson lemma, likelihood ratios. Three lectures,
one recitation. Prerequisites: Math. 180A and 20F. [Warning:
There are duplicate credit restrictions on this course. See section
on Duplication of Credit.] (W)
181B. Introduction to Mathematical Statistics (4) Goodness
of fit, special small sample distribution and use, nonparametric
methods. Kolmogorov-Smirnov statistics, sequential analysis. Three
lectures. Prerequisite: Math. 181A. (S)
181C. Mathematical Statistics (4) Nonparametric
Statistics. Topics covered may include the following: Classical
rank test, rank correlations, permutation tests, distribution free
testing, efficiency, confidence intervals, nonparametric regression
and density estimation, resampling techniques (bootstrap, jackknife,
etc.) and cross validations. Prerequisites: Math. 181A, 181B
previously or concurrently.
181E. Mathematical Statistics (4) Time
Series. Analysis of trends and seasonal effects, autoregressive
and moving averages models, forecasting, informal introduction
to
spectral analysis. Prerequisites: Math. 181A, 181B previously
or concurrently.
183. Statistical Methods (4) Introduction
to probability. Discrete and continuous random variablesbinomial,
Poisson and Gaussian distributions. Central limit theorem. Data
analysis and inferential statistics: graphical techniques, confidence
intervals, hypothesis tests, curve fitting. (Credit not offered
for Math. 183 if Econ. 120A, Math. 180A, or Math. 181A previously
or concurrently.) Prerequisite: Math. 20C (21C). (F,S)
184A. Mathematical Foundations of Computer Science (4) Enumeration
of combinatorial structures. Ranking and unranking. Graph theory
with applications and algorithms. Recursive algorithms. Circuit
design. Inclusion-exclusion. Generating functions. Polya theory.
Three lectures, one recitation. Prerequisite: Math. 15B or CSE
21 or Math. 109 or consent of instructor. (W,S)
186. Probability Statistics for Bioinformatics (4) This
course will cover an introduction to probability and statistics,
the use of discrete and random variables, different types of distributions,
data analysis and inferential statistics, likelihood estimators
and scoring matrices with applications to biological problems. Introduction
to probability, Binomial, Poisson, and Gaussian distributions, central
limit theorem, applications to sequence and functional analysis
of genomes and genetic epidemiology. Prerequisite: Math. 20A,
Math. 20B, Math. 20C (21C).
187. Introduction to Cryptography (4) An
introduction to the basic concepts and techniques of modern cryptography.
Classical cryptanalysis. Probabilistic models of plaintext. Monalphabetic
and polyalphabetic substitution. The one-time system. Caesar-Vigenere-Playfair-Hill
substitutions. The Enigma. Modern-day developments. The Data Encryption
Standard. Public key systems. Security aspects of computer networks.
Data protection. Electronic mail. Three lectures, one recitation.
Prerequisite: programming experience. (S)
188. Design and Analysis of Algorithms (4) Introduction
to the design and analysis of efficient algorithms. Basic techniques
for analyzing the time requirements of algorithms. Algorithms for
sorting, searching, and pattern matching, algorithms for graphs
and networks. NP-complete problems. Equivalent to CSE 101. Prerequisites:
CSE 100 or Math. 176A for Math. 188; CSE 12, CSE 21, and CSE 100
for CSE 101.
190. Introduction to Topology (4) Topological
spaces, subspaces, products, sums and quotient spaces. Compactness,
connectedness, separation axioms. Selected further topics such as
fundamental group, classification of surfaces, Morse theory, topological
groups. May be repeated for credit once when topics vary, with consent
of instructor. Three lectures. Prerequisite: Math. 109 or consent
of instructor. (W)
191. Topics in Topology (4) Topics
to be chosen by the instructor from the fields of differential algebraic,
geometric, and general topology. Three lectures. Prerequisite:
Math. 190 or consent of instructor. (S)
193A. Actuarial Mathematics (4) Probabilistic
Foundations of Insurance. Short-term risk models. Survival distributions
and life tables. Introduc-tion to life insurance. Prerequisite:
Math. 180A or 183, or consent of instructor.
193B. Actuarial Mathematics (4) Life
Insurance and Annuities. Analysis of premiums and premium reserves.
Introduction to multiple life functions and decrement models as
time permits. Prerequisite: Math. 193A.
194. The Mathematics of Finance (4) Introduction
to the mathematics of financial models. Basic probabilistic models
and associated mathematical machinery will be discussed, with emphasis
on discrete time models. Concepts covered will include conditional
expectation, martingales, optimal stopping, arbitrage pricing,
hedging,
European and American options. Prerequisites: Math. 20D (21D),
Math. 20F, and Math. 180A or 183.
195. Introduction to Teaching in Mathematics (4) Students
will be responsible for and teach a class section of a lower-division
mathematics course. They will also attend a weekly meeting on teaching
methods. (Does not count towards a minor or major.) Five lectures,
one recitation. Prerequisite: consent of instructor. (F,W,S)
196. Student Colloquium (1) A
variety of topics and current research results in mathematics
will be presented
by guest lecturers and students under faculty direction. Prerequisite:
upper-division status.
197. Mathematics Internship (2 or 4) An
enrichment program which provides work experience with public/private
sector employers. Subject to the availablility of positions, students
will work in a local company under the supervision of a faculty
member and site supervisor. Units may not be applied towards major
graduation requirements. Prerequisites: completion of 90 units,
2 upper-division mathematics courses, an overall 2.5 UCSD G.P.A.,
consent of mathematics faculty coordinator, and submission of written
contract. Department stamp required.
199. Independent Study for Undergraduates (2 or 4) Independent
reading in advanced mathematics by individual students. Three periods.
(P/NP grades only.) Prerequisite: permission of department. (F,W,S)
199H. Honors Thesis Research for Undergraduates (2-4) Honors
thesis research for seniors participating in the Honors Program.
Research is conducted under the supervision of a mathematics faculty
member. Prerequisites: admission to the Honors Program in mathematics,
department stamp.
Graduate
200A-B-C. Algebra (4-4-4) Group actions,
factor groups, polynomial rings, linear algebra, rational and Jordan
canonical forms, unitary and Hermitian matrices, Sylow theorems,
finitely generated abelian groups, unique factorization, Galois
theory, solvability by radicals, Hilbert Basis Theorem, Hilbert
Nullstellensatz, Jacobson radical, semisimple Artinian rings. Prerequisite:
consent of instructor.
201A. Basic Topics in Algebra (4) Recommended
for all students specializing in algebra. Basic topics include categorical
algebra, commutative algebra, group representations, homological
algebra, nonassociative algebra, ring theory. Prerequisites:
Math. 200A-B-C or consent of instructor. (F,W,S)
202B-C. Applied Algebra (4-4) Algebra
from a computational perspective using Maple, Mathematica and Matlab.
Groups, rings, linear algebra, rational and Jordan forms, unitary
and Hermitian matrices, matrix decompositions, perturbation of eigenvalues,
group representations, symmetric functions, fast Fourier transform,
commutative algebra, Grobner basis, finite fields. Prerequisite:
consent of instructor.
203A-B-C. Algebraic Geometry (4-4-4) Places,
Hilbert Nullstellensatz, varieties, product of varieties: correspondences,
normal varieties. Divisors and linear systems; Riemann-Roch theorem;
resolution of singularities of curves. Grothendieck schemes; cohomology,
Hilbert schemes; Picard schemes. Prerequisites: Math. 200A-B-C.
(F,W,S)
204. Topics in Number Theory (4) Topics
in analytic number theory, such as zeta functions and L-functions
and the distribution of prime numbers, zeros of zeta functions and
Siegels theorem, transcendence theory, modular forms, finite
and infinite symmetric spaces. Prerequisite: consent of instructor.
205. Topics in Algebraic Number Theory (4) Topics
in algebraic number theory, such as cyclotomic and Kummer extensions,
class number, units, splitting of primes in extensions, zeta functions
of number fields and the Brauer-Siegel Theorem, class field theory,
elliptic curves and curves of higher genus, complex multiplication.
Prerequisite: consent of instructor.
207A-B-C. Topics in Algebra (4-4-4) In
recent years, topics have included number theory, commutative algebra,
noncommutative rings, homological algebra, and Lie groups. May be
repeated for credit with consent of adviser. Prerequisite: consent
of instructor.
208. Seminar in Algebra (1-4) Prerequisite:
consent of instructor. (S/U grades permitted.)
209. Seminar in Number Theory (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
210A. Mathematical Methods in Physics and Engineering (4) Complex
variables with applications. Analytic functions, Cauchys theorem,
Taylor and Laurent series, residue theorem and contour integration
techniques, analytic continuation, argument principle, conformal
mapping, potential theory, asymptotic expansions, method of steepest
descent. Prerequisites: Math. 20DEF, 140A/142A or consent of
instructor.
210B. Mathematical Methods in Physics and Engineering (4) Linear
algebra and functional analysis. Vector spaces, orthonormal bases,
linear operators and matrices, eigenvalues and diagonalization,
least squares approximation, infinite-dimensional spaces, completeness,
integral equations, spectral theory, Greens functions, distributions,
Fourier transform. Prerequisite: Math. 210A or consent of instructor.
(W)
210C. Mathematical Methods in Physics and Engineering (4) Calculus
of variations: Euler-Lagrange equations, Noethers theorem.
Fourier analysis of functions and distributions in several variables.
Partial differential equations: Laplace, wave, and heat equations;
fundamental solutions (Greens functions); well-posed problems.
Prerequisite: Math. 210B or consent of instructor. (S)
211. Fourier Analysis on Finite Groups (4) Applied
group representations. Emphasis on the integers, mod n, finite matrix
groups. Applications may include: the fast Fourier tranform, digital
signal processing, finite difference equations, spectral graph theory,
error-correcting codes, vibrating systems, finite wavelet tranforms.
Prerequisite: none.
212A. Introduction to the Mathematics of Systems and Control
(4) Linear and nonlinear systems, and
their input-output behavior, linear continuous time and discrete-time
systems, reachability and controllability for linear systems, feedback
and stabilization, eigenvalue placement, nonlinear controllability,
feedback linearization, disturbance rejection, nonlinear stabilization,
Lyapunov and control-Lyapunov functions, linearization principle
for stability. Prerequisites: Math. 102 or equivalent, Math.
120A or equivalent, Math. 142A or equivalent.
212B. Introduction to the Mathematics of Systems and Control
(4) Observability notions, linearization
principle for observability. Realization theory for linear systems,
observers and dynamic feedback, detectability, external stability
for linear systems, frequency-domain considerations, dynamic programming,
quadratic cost, state estimation and Kalman filtering, nonlinear
stabilizing optimal controls, calculus of variations, and the Maximum
Principle. Prerequisite: Math. 212A.
212C. Introduction to the Mathematics of Systems and Control
(4) Topics of current interest on systems
theory, control, and estimation to be chosen by instructor. Prerequisite:
Math. 212B.
216. Topics in Pure Mathematics (4) This
course brings together graduate students, postdocs, and faculty
to examine a current research topic of broad interest. Previously
covered topics include: noncommutative geometry, Loop groups, geometric
quantization. Prerequisite: consent of instructor.
217A. Topics in Applied Mathematics (4) In
recent years, topics have included applied complex analysis, special
functions, and asymptotic methods. May be repeated for credit with
consent of adviser. Prerequisite: consent of instructor.
220A-B-C. Complex Analysis (4-4-4) Complex
numbers and functions. Cauchy theorem and its applications, calculus
of residues, expansions of analytic functions, analytic continuation,
conformal mapping and Riemann mapping theorem, harmonic functions.
Dirichlet principle, Riemann surfaces. Prerequisites: Math. 140A-B
or consent of instructor. (F,W,S)
221A-B. Topics in Several Complex Variables (4-4) Formal
and convergent power series, Weierstrass preparation theorem; Cartan-Ruckert
theorem, analytic sets; mapping theorems; domains of holomorphy;
proper holomorphic mappings; complex manifolds; modifications. Prerequisites:
Math. 200A and 220A-B-C or consent of instructor.
229. Computing Technology for Mathematics (2) Preparation
for making effective use of computer technology in research and
teaching of mathematics. UNIX basics, document preparation using
TEX, Internet resources, HTML, computer technology in teaching. Prerequisite: graduate status in mathematics.
231A-B-C. Partial Differential Equations (4-4-4) Existence
and uniqueness theorems. Cauchy-Kowalewski theorem, first order
systems. Hamilton-Jacobi theory, initial value problems for hyperbolic
and parabolic systems, boundary value problems for elliptic systems.
Greens function, eigenvalue problems, perturbation theory.
Prerequisites: Math. 210A-B or 240A-B-C or consent of instructor.
233. Singular Perturbation Theory for Differential Equations
(4) Multivariable techniques, matching
techniques and averaging techniques, including various approaches
to proofs of asymptotic correctness, for singular perturbation problems
including initial value problems with nonuniformities at infinity,
initial value problems with initial nonuniformities, two point boundary
value problems, and problems for partial differential equations.
Applications taken from celestial mechanics, oscillation problems,
fluid dynamics, elasticity, and applied mechanics. Prerequisites:
Math. 130A-B or 132A-B or consent of instructor. (S/U grades
permitted.) (S)
237A-B-C. Topics in Differential Equations (4-4-4) May
be repeated for credit with consent of adviser. Prerequisite:
consent of instructor.
240A-B-C. Real Analysis (4-4-4) Lebesgue
integral and Lebesgue measure, Fubini theorems, functions of bounded
variations, Stieltjes integral, derivatives and indefinite integrals,
the spaces L and C, equi-continuous families, continuous linear
functionals general measures and integrations. Prerequisites:
Math. 140A-B-C. (F,W,S)
241A-B. Functional Analysis (4-4) Metric
spaces and contraction mapping theorem; closed graph theorem; uniform
boundedness principle; Hahn-Banach theorem; representation of continuous
linear functionals; conjugate space, weak topologies; extreme points;
Krein-Milman theorem; fixed-point theorems; Riesz convexity theorem;
Banach algebras. Prerequisites: Math.240A-B-C or consent of instructor.
247A-B-C. Topics in Real Analysis (4-4) In
recent years, topics have included Fourier analysis, distribution
theory, martingale theory, operator theory. May be repeated for
credit with consent of adviser. Prerequisite: consent of instructor.
248. Seminar in Real Analysis (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
250A-B-C. Differential Geometry (4-4-4) Differential
manifolds, Sard theorem, tensor bundles, Lie derivatives, DeRham
theorem, connections, geodesics, Riemannian metrics, curvature tensor
and sectional curvature, completeness, characteristic classes. Differential
manifolds immersed in Euclidean space. Prerequisite: consent
of instructor. (F,W,S)
251A-B-C. Lie Groups (4-4-4) Lie
groups, Lie algebras, exponential map, subgroup subalgebra correspondence,
adjoint group, universal enveloping algebra. Structure theory of
semi-simple Lie groups, global decompositions, Weyl group. Geometry
and analysis on symmetric spaces. Prerequisites: Math. 200 and
250 or consent of instructor. (F,W,S)
256. Seminar in Lie Groups and Lie Algebras (2 to 4) Various
topics in Lie groups and Lie algebras, including structure theory,
representation theory, and applications. Prerequisite: consent
of instructor. (F,W,S)
257A. Topics in Differential Geometry (4) In
recent years, topics have included Morse theory and general relativity.
May be repeated for credit with consent of adviser. Prerequisite:
consent of instructor.
257B. Topics in Differential
Geometry (4) In recent years, topics
have included Morse theory and general relativity. May be repeated
for credit
with consent of adviser. Math. 257A must be taken before Math.
257B. Prerequisite: consent of instructor.
257C. Topics in Differential
Geometry (4) In recent years, topics
have included Morse theory and general relativity. May be repeated
for credit
with consent of adviser. Math. 257B must be taken before Math.
257C. Prerequisite: consent of instructor.
258. Seminar in Differential Geometry (1-4) Various
topics in differential geometry. Prerequisite: consent of instructor.
259A-B-C. Geometrical Physics (4-4-4) Manifolds,
differential forms, homology, deRhams theorem. Riemannian
geometry, harmonic forms. Lie groups and algebras, connections in
bundles, homotopy sequence of a bundle, Chern classes. Applications
selected from Hamiltonian and continuum mechanics, electromagnetism,
thermodynamics, special and general relativity, Yang-Mills fields.
Prerequisite: graduate standing in mathematics, physics, or engineering,
or consent of instructor.
261A-B. Combinatorial Algorithms (4-4) Lexicographic
order, backtracking, ranking algorithms, isomorph rejection, sorting,
orderly algorithms, network flows and related topics, constructive
Polya theory, inclusion-exclusion and seiving methods, Mobius inversion,
generating functions, algorithmic graph theory, trees, recursion,
depth firstsearch and applications, matroids. Prerequisites:
CSE 160A-B or Math.184A-B or consent of instructor. (F,W,S)
262A-B. Topics in Combinatorial Mathematics (4-4) Development
of a topic in combinatorial mathematics starting from basic principles.
Problems of enumeration, existence, construction, and optimization
with regard to finite sets. Some familiarity with computer programming
desirable but not required. Prerequisites: Math. 100A-B-C.
263. History of Mathematics (4) Mathematics
in the nineteenth century from the original sources. Foundations
of analysis and commutative algebra. For algebra the authors studied
will be Lagrange, Ruffini, Gauss, Abel, Galois, Dirichlet, Kummer,
Kronecker, Dedekind, Weber, M. Noether, Hilbert, Steinitz, Artin,
E. Noether. For analysis they will be Cauchy, Fourier, Bolzano,
Dirichlet, Riemann, Weierstrass, Heine, Cantor, Peano, Hilbert.
Prerequisites: Math. 100A-B, Math. 140A-B.(S)
264A-B-C. Combinatorics (4-4-4) Topics
from partially ordered sets, Mobius functions, simplicial complexes
and shell ability. Enumeration, formal power series and formal languages,
generating functions, partitions. Lagrange inversion, exponential
structures, combinatorial species. Finite operator methods, q-analogues,
Polya theory, Ramsey theory. Representation theory of the symmetric
group, symmetric functions and operations with Schur functions.
(F,W,S)
267A-B-C. Topics in Mathematical Logic (4-4-4) Topics
chosen from recursion theory, model theory, and set theory. May
be repeated with consent of adviser. Prerequisite: consent of
instructor. (S/U grades permitted.)
268. Seminar in Logic (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
269. Seminar in Combinatorics (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
270A-B-C. Numerical Mathematics (4-4-4) Error
analysis of the numerical solution of linear equations and least
squares problems for the full rank and rank deficient cases. Error
analysis of numerical methods for eigenvalue problems and singular
value problems. Error analysis of numerical quadrature and of the
numerical solution of ordinary differential equations. Prerequisites:
Math. 20F and knowledge of programming.
271A-B-C. Numerical Optimization (4-4-4) Formulation
and analysis of algorithms for constrained optimization. Optimality
conditions; linear and quadratic programming; interior methods;
penalty and barrier function methods; sequential quadratic programming
methods. Prerequisite: consent of instructor. (F,W,S)
272A-B-C. Numerical Partial Differential Equations (4-4-4) The
numerical solution of elliptic, parabolic, and hyperbolic partial
differential equations; discretization and solution techniques.
Prerequisite: consent of instructor. (F,W,S)
273A-B-C. Scientific Computation (4-4-4) Continuum
mechanics models of physical and biological systems, finite element
methods and approximation theory, complexity of iterative methods
for linear and nonlinear equations, continuation methods, adaptive
methods, parallel computing, and scientific visualization. Project-oriented;
theoretical and software development projects designed around problems
of current interest in science and engineering. Prerequisite:
experience with Matlab and C, some background in numerical analysis,
or consent of instructor. (F,W,S)
277A. Topics in Numerical Mathematics (4) Topics
vary from year to year. May be repeated for credit with consent
of adviser. Prerequisite: consent of instructor.
278. Seminar in Numerical Mathematics (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
280A-B-C. Probability Theory (4-4-4) Probability
measures; Borel fields; conditional probabilities, sums of independent
random variables; limit theorems; zero-one laws; stochastic processes.
Prerequisites: advanced calculus and consent of instructor. (F,W,S)
281A. Mathematical Statistics (4) Statistical
models, sufficiency, efficiency, optimal estimation, least squares
and maximum likelihood, large sample theory. Prerequisites: advanced
calculus and basic probablilty theory or consent of instructor.
281B. Mathematical Statistics (4) Hypothesis
testing and confidence intervals, one- sample and two-sample problems.
Bayes theory, statistical decision theory, linear models and regression.
Prerequisites: advanced calculus and basic probablilty theory
or consent of instructor.
281C. Mathematical Statistics (4) Nonparametrics:
tests, regression, density estimation, bootstrap and jackknife.
Introduction to statistical computing using S plus. Prerequisites:
advanced calculus and basic probablilty theory or consent of instructor.
282A-B. Applied Statistics (4-4) Sequence
in applied statistics. First quarter: general theory of linear models
with applications to regression analysis. Second quarter: analysis
of variance and covariance and experimental design. Third quarter:
further topics to be selected by instructor. Emphasis throughout
is on the analysis of actual data. Prerequisite: Math. 181B or
equivalent or consent of instructor. (S/U grades permitted.)
283. Statistical Methods in Bioinformatics (4) This
course will cover material related to the analysis of modern genomic
data; sequence analysis, gene expression/functional genomics analysis,
and gene mapping/applied population genetics. The course will focus
on statistical modeling and inference issues and not on database
mining techniques. Prerequisites: one year of calculus, one statistics
course or consent of instructor.
285A-B. Stochastic Processes (4-4) Elements
of stochastic processes, Markov chains, hidden Markov models, Poisson
point processes, renewal processes martingales, Brownian motion,
Gaussian processes, Kalman filter. Other topics to be selected by
instructor depending on interest of class. Prerequisites: Math.
180A ( or equivalent basic probablilty course) or consent of instructor.
286. Stochastic Differential Equations (4) Review
of continuous martingale theory. Stochastic integration for continuous
semimartingales. Existence and uniqueness theory for stochastic
differential equations. Strong Markov property. Selected applications.
Prerequisite: Math. 280A-B or consent of instructor.
287A. Time Series Analysis (4) Discussion
of finite parameter schemes in the Gaussian and non-Gaussian context.
Estimation for finite parameter schemes. Stationary processes and
their spectral representation. Spectral estimation. Prerequisite:
Math. 181B or equivalent or consent of instructor.
287B. Multivariate Analysis (4)
288. Seminar in Probability and Statistics (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
289A-B. Topics in Probability and Statistics (4-4) In
recent years, topics have included Markov processes, martingale
theory, stochastic processes, stationary and Gaussian processes,
ergodic theory. May be repeated for credit with consent of adviser.
290A-B-C. Topology (4-4-4) Point
set topology, including separation axioms, compactness, connectedness.
Algebraic topology, including the fundamental group, covering spaces,
homology and cohomology. Homotopy or applications to manifolds as
time permits. Prerequisites: Math. 100A-B-C and Math. 140A-B-C.
(F,W,S)
291A-B-C. Topics in Topology (4) In
recent years, topics have included generalized cohomology theory,
spectral sequences, K-theory, homotophy theory. Prerequisites:
consent of instructor.
292. Seminar in Topology (1-4) Various
topics in topology. Prerequisites: consent of instructor.
294. The Mathematics of Finance (4) Introduction
to the mathematics of financial models. Hedging, pricing by arbitrage.
Discrete and continuous stochastic models. Martingales. Brownian
motion, stochastic calculus. Black-Scholes model, adaptations to
dividend paying equities, currencies and coupon-paying bonds, interest
rate market, foreign exchange models. Prerequisite: Math. 180A
(or equivalent probability course) or consent of instructor.
295. Special Topics in Mathematics (1 to 4) A
variety of topics and current research results in mathematics will
be presented by staff members and students under faculty direction.
296. Student Colloquium (1 to 2) A
variety of topics and current research in mathematics will be presented
by guest lecturers and students under faculty direction. Prerequisites:
for one unitupper-division status or consent of instructor
(may only be taken P/NP), or graduate status (may only be taken
S/U); for two unitsconsent of instructor, standard grading
option allowed.
297. Mathematics Graduate Research Internship (2-4) An
enrichment program which provides work experience with public/private
sector employers and researchers. Under supervision of a faculty
adviser, students provide mathematical consultation services. Prerequisites:
consent of instructor.
299. Reading and Research (1 to 12) Independent
study and research for the doctoral dissertation. One to three credits
will be given for independent study (reading) and one to nine for
research. Prerequisite: consent of instructor. (S/U grades permitted.)
Teaching of Mathematics
500. Apprentice Teaching (1 to 4) Supervised
teaching as part of the mathematics instructional program on campus
(or, in special cases such as the CTF program, off campus). Prerequisite:
consent of adviser. (S/U grades only.)
Mathematics Courses
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