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Courses - MATH Upper Division

303. Reflections in Space and Time (3)

Prerequisites: GE Foundation requirements, at least one GE Exploration course, upper-division standing.
An experimentally-driven investigation of the mathematical nature of symmetry and patterns. Considers the pervasive appearance and deep significance of symmetry and patterns in art and science.
(Lecture 3 hrs.)

309. Complexity and Emergence (3)

Prerequisites: GE Foundation requirements, at least one GE Exploration course, upper-division standing.
Introduction to complexity science. Qualitative and computational exploration of emergent properties in dynamical systems, fractals, algorithms, networks, self-organizing behavior and selected topics.
Letter grade only (A-F)

310. History of Early Mathematics (3)

Prerequisite/Corequisite: At least one of MATH 224 or MATH 233 or MATH 247.
History of mathematics through seventeenth century, including arithmetic, geometry, algebra, and beginnings of calculus. Interconnections with other branches of mathematics. Writing component; strongly recommended students enrolling have completed the G.E. A.1 requirement.
(Lecture 3 hrs.)

323. Introduction to Numerical Analysis (4)

Prerequisites: MATH 224, and a course in computer programming.
Numerical solution of nonlinear equations, systems of linear equations, and ordinary differential equations. Interpolating polynomials, numerical differentiation, and numerical integration. Computer implementation of these methods.
(Lecture-discussion 3 hrs., problem session 2 hrs.)

341. Number Theory (3)

Prerequisite: A grade of "C" or better in MATH 233
Divisibility, congruences, number theoretic functions, Diophantine equations, primitive roots, continued fractions. Writing proofs.
(Lecture 3 hrs.)

347. Linear Algebra (3)

Prerequisites: MATH 233 and MATH 247.
In-depth study of linear transformations, vector spaces, inner product spaces, quadratic forms, similarity and the rational and Jordan canonical forms. Writing proofs.
(Lecture 3 hrs.)

355. College Geometry (3)

Prerequisite: MATH 247.
Transformations, motions, similarities, geometric objects, congruent figures, axioms of geometry and additional topics in Euclidean and non-Euclidean geometry. Writing proofs.
(Lecture 3 hrs.)

361A. Introduction to Mathematical Analysis I (3)

Prerequisites: MATH 224, and MATH 233 or MATH 247.
Rigorous study of calculus and its foundations. Structure of the real number system. Sequences and series of numbers. Limits, continuity and differentiability of functions of one real variable. Writing proofs.
(Lecture 3 hrs.)

361B. Introduction to Mathematical Analysis II (3)

Prerequisite: A grade of "C" or better in MATH 361A.
Riemann integration. Topological properties of the real number line. Sequences of functions. Metric spaces. Introduction to calculus of several variables. Writing proofs.
(Lecture 3 hrs.)

364A. Ordinary Differential Equations I (3)

Prerequisites: MATH 224.
Prerequisite/Corequisite: MATH 247.
First order differential equations; undetermined coefficients and variation of parameters for second and higher order differential equations, series solution of second order linear differential equations; systems of linear differential equations; applications to science and engineering.
(Lecture 3 hrs.)

364B. Ordinary Differential Equations II (3)

Prerequisite: MATH 364A or MATH 370A.
Existence-uniqueness theorems; Laplace transforms; difference equations; nonlinear differential equations; stability, Sturm-Liouville theory; applications to science and engineering.
(Lecture 3 hrs.)

370A. Applied Mathematics I (3)

Prerequisite: A grade of "C" or better in MATH 123. Excludes freshmen.
First order ordinary differential equations, linear second order ordinary differential equations, numerical solution of initial value problems, Laplace transforms, matrix algebra, eigenvalues, eigenvectors, systems of differential equations, applications.
Not open for credit to mathematics majors. (Lecture 3 hrs.)

370B. Applied Mathematics II (3)

Prerequisite: MATH 364A or MATH 370A
Arithmetic of complex numbers, functions of a complex variable, contour integration, residues, conformal mapping; Fourier series; separation of variables for partial differential equations. Applications. Not open for credit to mathematics majors. .
Not open for credit to mathematics majors. (Lecture 3 hrs.)

380. Probability and Statistics (3)

Prerequisite: MATH 224.
Frequency interpretation of probability. Axioms of probability theory. Discrete probability and combinatorics. Random variables. Distribution and density functions. Moment generating functions and moments. Sampling theory and limit theorems.
Letter grade only (A-F). (Lecture 3 hrs.) Not open for credit to student with credit in STAT 380.

409. Strategies for Graduate Admission (1)

Prerequisite: Senior or graduate standing.
The nature and expectations of doctoral programs in Mathematics and related fields. Intensive preparation for GRE mathematics subject exams.
Credit/No Credit grading only. Does not satisfy Mathematics major requirements. (Lecture-discussion 1 hr.)

410. History of Modern Mathematics (3)

Prerequisites: MATH 247, MATH 310 and at least three of the following: MATH 233, MATH 341, MATH 355, MATH 361A, MATH 380.
History of mathematics from seventeenth century onward. Development of calculus, analysis, and geometry during this time period. Other topics discussed may include history of probability and statistics, algebra and number theory, logic, and foundations.
(Lecture 3 hrs.)

423. Intermediate Numerical Analysis (3)

Prerequisites: MATH 247 and MATH 323.
Numerical solutions of systems of equations, calculation of eigenvalues and eigenvectors, approximation of functions, solution of partial differential equations. Computer implementation of these methods.
(Lecture 3 hrs.)

444. Introduction to Abstract Algebra (3)

Prerequisites: MATH 233 and MATH 247 and a grade of "C" or better in at least one of MATH 341 or MATH 347.

Groups, subgroups, cyclic groups, symmetric groups, Lagrange's theorem, quotient groups. Homomorphisms and isomorphisms of groups. Rings, integral domains, ideals, quotient rings, homomorphisms of rings. Fields. Writing proofs.
(Lecture 3 hrs.)

451. Differential Geometry (3)

Prerequisite: MATH 364A or MATH 370A.
Structure of curves and surfaces in space, including Frenet formulas of space curves; frame fields and connection forms; geometry of surfaces in Euclidean three space; Geodesics and connections with general theory of relativity.
(Lecture 3 hrs.)

456. Dynamics and Geometry of Chaos (3)

Prerequisites: MATH 247, MATH 361A, or consent of instructor.
An introduction to discrete dynamical systems in one and two dimensions. Theory of iteration: attracting and repelling periodic points, symbolic dynamics, chaos, and bifurcation. May include a computer lab component.
(Lecture 3 hrs)

461. Introduction to Complex Analysis (3)

Prerequisite: MATH 361A.
Theory and applications of complex variables. Analytic functions, integrals, power series and applications.
(Lecture 3 hrs.)

463. Multivariable Calculus (3)

Prerequisites: MATH 224, MATH 247, and MATH 361B.
Topology of Euclidean spaces. Partial derivatives. Derivatives as linear transformations. Inverse and implicit function theorems. Jacobians, vector calculus, Green's and Stokes' theorems. Variational problems.
(Lecture 3 hrs.)

470. Introduction to Partial Differential Equations (3)

Prerequisite: MATH 364A or MATH 370A.
First and second order equations, characteristics, Cauchy problems, elliptic, hyperbolic, and parabolic equations. Introduction to boundary and initial value problems and their applications.
(Lecture 3 hrs.)

472. Fourier Analysis (3)

Prerequisite: MATH 364A or MATH 370A.
Theory of Fourier series and Fourier transforms. Physics and engineering applications. Parseval's and Plancherel's identities. Convolution. Multi-dimensional transforms and partial differential equations. Introduction to distributions. Discrete and fast Fourier transforms.
(Lecture 3 hrs.)

473. Scientific Computing (3)

Prerequisites: MATH 323 and MATH 364A or MATH 370A.
Introduction to programming languages. Analysis and implementation of numerical algorithms for linear systems, linear and nonlinear regression, differentiation, integration, optimization and fast convolution using FFT. Basic algorithms for differential equations. Letter grade only (A-F).

474. Mathematics of Financial Derivatives (3)

Prerequisites: MATH 364A or MATH 370A, MATH 380, or consent of instructor.
Options, futures, and other financial derivatives; arbitrage; risk-neutral valuation; binomial trees; the log-normal hypothesis; the Black-Scholes-Merton formula and applications; the Black-Scholes-Merton partial differential equation; American options; exotic options; bond models and interest rate derivatives; credit risk and credit derivatives.

479. Mathematical Modeling (3)

Prerequisites:MATH 247, MATH 323; MATH 364A or MATH 370A; and consent of instructor.
Introduction to mathematical modeling in the applied sciences, including validation and practical use of various modeling methodologies. Mathematical models in physics, chemistry, biology, and other natural sciences. Applications of computational mathematics in computer science, engineering, finance, and related disciplines. Letter grade only (A-F).

485. Mathematical Optimization (3)

Prerequisites: MATH 247 and at least one of MATH 323, MATH 347 or MATH 380.
Linear and nonlinear programming: simplex methods, duality theory, theory of graphs, Kuhn-Tucker theory, gradient methods and dynamic programming.
(Lecture 3 hrs.)

491. Honors Seminar in Problem Solving (1)

Prerequisite: Consent of instructor.
Challenging problems from many fields of mathematics, taken largely from national and worldwide collegiate and secondary school competitions. Students required to participate in at least one national competition.
May be repeated to a maximum of 3 units. (Lecture-discussion 1 hr.)

495. Topics in Modern Mathematics (3)

Prerequisite: Consent of instructor.
Topics of current interest from mathematics literature.

496. Special Problems (1-3)

Prerequisite: Consent of instructor.
Student investigations in mathematics, applied mathematics, mathematics education, or statistics. May include reports and reviews from the current literature, as well as original investigations.
May be repeated to a maximum of 3 units. Letter grade only (A-F).

497. Directed Studies (1-6)

Prerequisites: Junior or senior standing and consent of instructor.
Readings in areas of mutual interest to student and instructor which are not a part of any regular course. A written report or project may be required.
May be repeated to a maximum of 6 units.

498H. Senior Thesis - Honors (3)

Prerequisites: Admission to Honors in the Major in Mathematics or to the University Honors Program, and consent of instructor.
Planning, preparation, completion, and oral presentation of a written thesis in mathematics, applied mathematics, mathematics education, or statistics.
Not available to graduate students. Letter grade only (A-F).

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