### Topological Methods in Group Theory: 243 (Graduate Texts in Mathematics)

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Calculus of functions of several variables, vector-valued functions of one variable, scalar and vector fields, integration along paths, double and triple integrals, integration over surfaces, properties, and applications of integrals, and classical integration theorems of vector calculus. Honors version of Mathematics 6A. MATH 6B. Integral theorems of vector calculus continuation , infinite series, Fourier series, integrals and transforms, partial differential equations.

MATH 7H. Repeat Comments: May be repeated for credit to a maximum of 6 units. Emphasizing fundamental concepts and applications. Intended for highly motivated and well prepared students. MATH 8. Introduction to the elements of propositional logic, techniques of mathematical proof, and fundamental mathematical structures, including sets, functions, relations, and other topics as time permits.

Mastery of this material is essential for students planning to major in mathematics. MATH 34A.

### Introduction

Introduction to differential and integral calculus with applications to modeling in the biological sciences. MATH 34B. Continued study of differential and integral calculus with differential andintegral calculus with applications. Introduction to mathematical modeling with differential equations. Calculus of several variables including an introduction to partial derivatives. MATH Lectures and discussions on special topics. Designed for transfer students only. Upper Division. MATH A. Enrollment Comments: Course cannot be used to satisfy any mathematics major or minor requirements.

This class teaches ways to think about and explain elementary school mathematics. Topics include: cultural and base-n number systems, algorithms, elementary number theory, probability, and graphing. MATH B. Completes the explanation of elementary school mathematics by discussing geometry and algebra. Discusses the pedagogy with the California mathematics framework, the NCTM standards, and "replacement units". Enrollment Comments: Not open for credit to students who have completed Mathematics A.

A conceptual rather than an axiomatic development starting with the natural numbers and progressing through the integral, rational, real, and complex number systems. The historical implications of these developments in number systems. Especially suitable for prospective middle and high school teachers. The theory of operations within rings and fields and the foundations of the real number system. Ideals, quotient rings, and factorization theorems. The history and the historical implications of these developments in mathematical systems.

Topics in plane and solid geometry. The axioms of pure, euclidean, projective, and noneuclidean geometry. Transformational geometry isometries, dilitations, involutions, perspectivities, and projectivities. The history and the historical implications of these developments in geometry. Permutation groups, cyclic groups, theory of finite groups, group homomorphisms and isomorphisms, and Abelian groups. Applications to number theory and geometry. Numerical methods for the solution of nonlinear equations Newton method , for integration quadrature formulas and composite integration , and for the initial value problem for ordinary differential equations Euler and Kutta methods.

Numerical methods for the solution of systems of linear equations direct and iteractive methods , and the finite difference methods for boundary value problems for ordinary and partial differential equations. MATH C. Topics in approximation theory; numerical methods for finding eigenvalues of a matrix; and advanced topics in numerical methods for ordinary and partial differential equations.

Abstract vector spaces subspaces. Span and linear independence. Basis and dimension. Linear maps. Eigenvalues and eigenvectors. Diagonalization, inner product spaces, projections, least- squares approximations, invariant factors and elementary divisors, canonical forms, topics from advanced matrix theory, applied linear algebra, and group representation theory. An introduction to mathematical logic with applications in computer science and mathematics. Topics include propositional and predicate calculi; models; proof systems, decidability and undecidability, automated theorem-proving, unification, logic programming, and program verification.

An introduction to algebraic structures with an emphasis on groups. Rings, fields, Galois theory. An introduction to hyperbolic geometry with some discussion of other non- euclidean systems. Recommended Preparation: At least one quarter of programming experience. Enrollment Comments: Designed for majors.

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Introduction to programming with specific examples in scientific computing. The algorithms used will be analyzed in detail in Math A-B-C. The emphasis will be the translation of numerical algorithms into actual working code. Recommended Preparation: Students are encouraged to take both A and B in the same academic year as topics may very from year to year. Divisibility, congruences, primitive roots an indices, quadratic residues and the quadratic reciprocity law, number-theoretic functions, Diophantine equations, the distribution of primes, number-theorhetic methods in cryptography, quadratic forms, continued fractions, and the approximation of real numbers, algebraic number theory, partitions.

Recommended Preparation: Students are encouraged to take both A and B in the same academic year as topics may vary from year to year. Divisibility, congruences, primitive roots and indices, quadratic residues and the quadratic reciprocity law, number-theoretic functions, Diophantine equations, the distribution of primes, number-theoretic methods in cryptography, quadratic forms, continued fractions, and the approximation of real numbers, algebraic number theory, partitions.

Recommended Preparation: Completion of additional courses may prove useful, depending on the topics to be considered. Consult the department or instructor for details. Selected topics in number theory at the direction of the instructor.

Elementary counting principles, binomial coefficients, generating functions, recurrence relations, the principle of inclusion and exclusion, distributions and partitions, systems of distinct representatives, applications to computation. Introduction to methods of proof in analysis. This course is intended to follow Mathematics 8 and to introduce students to the level of sophistication of upper-division mathematics.

The real number system, elements of set theory, continuity, differentiability, Riemann integral, implicit function theorems, convergence processes, and special topics. Existence, uniqueness, and stability; the geometry of phase space; linear systems and hyperbolicity; maps and diffeomorphisms. Hyperbolic structure and chaos; center manifolds; bifurcation theory; and the Feigenbaum and Ruelle-Takens cascades to strange attractors. Complex numbers, functions, differentiability, series extensions of elementary functions, complex integration, calculus of residues, conformal maps, mapping functions, applications.

Wave, heat, and potential equations. Fourier series; generalized functions; and numerical methods.

## Graduate Texts in Mathematics_百度百科

Computer Science 8 or 16, or Engineering 3 with a minimum grade of C. Linear programming, the simplex method, duality, applications to the transportation and assignment problems, sensitivity analysis, problem formulation.

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Network analysis: shortest route, minimal spanning tree and maximal flow problems; PERT including the critical path method; dynamic programming; game theory; integer programming, nonlinear programming. Elements of graph and network theory including paths, circuits, trees, coloring, planarity, matching theory, Hall's theorem, applications to scheduling theory, flows in networks, Menger's theorem, and other topics as time permits. Either Math A or Math Metric spaces, continuity, compactness, classification of surfaces, Euler characteristics, and fundamental groups. Additional topics at the discretion of the instructor.

Curves and surfaces in three-dimensional Euclidean space, first and second fundamental forms, Gaussian and mean curvature, geodesics, Gauss-Bonnet theorem, and non-Euclidean geometry. Curves and surfaces in three-dimensional Euclidean space, first and second fundamental forms, Gaussian and mean curvature, geodesics, Gauss-Bonnet theorem, and non- Euclidean geometry.

### Graduate Texts in Mathematics | Awards | LibraryThing

Describes mathematical methods for estimating and evaluating asset pricing models, equilibrium and derivative pricing, options, bonds, and the term-structure of interest rates. Also introduces finance optimization models for risk management and financial engineering. Focuses on the representations, strategies, and language learners use to conceptualize and develop fundamental ideas of mathematics. Includes advanced mathematical problem solving and its implications for teaching and learning at the secondary level.

Continuation of Math A or ED An examination of the major achievements in mathematical thinking throughout history. Topics may include the history of numerical systems in early civilizations, the development of formal proof, mathematical contributions from diverse populations and the impact of technological innovations on mathematics. Concurrent enrollment in Math Enrollment Comments: May be repeated for credit to a maximum of 8 units. Information about the special topics to be presented may be obtained from the office of the Department of Mathematics. Enrollment Comments: May be repeated for credit up to a maximum of four units.

Credit will not be given toward upper-division Mathematics major requirements. Faculty sponsored academic internship in industrial or research firms. Enrollment Comments: Enrollment normally limited to 12 or fewer students. Participants will select a math-related book or papers, read the section before the next meeting and discuss reading at the meeting. Readings may include biographies of mathematicians, histories or popularizations of mathematics, textbooks, and readings in mathematical physics or biology.

Enrollment Comments: Up to four units may apply to the major. Students must have aminimum overall grade-point average of 3. Independent research under the supervision of a faculty member which will result in a senior thesis. Student will concentrate on reading and gathering material for a thesis. Open to senior majors only; consent of department and instructor. Enrollment Comments: Students must have a minimum overall grade-point average of 3. Up to four units may apply towards the major.

Student will concentrate on writing a thesis. Independent Studies in Mathematics STAFF Prerequisite: Upper-division standing; completion of 2 upper-division courses in math; consent of instructor and department.

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Granas, J. Jaworowski, Reminiscences of Karol Borsuk, Topol. Grove, P. Petersen, Bounding homotopy types by geometry, The Annals of Mathematics , Hilton, Duality in homotopy theory: a retrospective essay,]. Pure Appl. Algebra 19 , Hilton, On some contributions of Karol Borsuk to homotopy theory, Topol. James ed. Kahn, G. Kuperberg, A smooth counterexample to the Seifert conjecture, Ann. Kuperberg, W. Kuperberg, P. Mine, C. Segal, Shape theory, North-Holland Publ.

## Course Listings and Descriptions

III 17 , Sanjurjo, Stability, attraction and shape: A topological study of flows, vol. Segal, Borsuk's shape theory, Topol. Shchepin, Finite-dimensional bicompact absolute neighborhood retracts are metrizable, Dokl. Nauk SSSR , Simon, S. Thurston, On proof and progress in mathematics, Bull.

Viro, O. Ivanov, N. Netsvetaev, V. West, Borsuk's influence on infinite-dimensional topology, Topol.

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