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Various topics in the mathematics of biological systems.

### M.S. in Mathematics Department

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. Introduction to varied topics in several complex variables. In recent years, topics have included formal and convergent power series, Weierstrass preparation theorem, Cartan-Ruckert theorem, analytic sets, mapping theorems, domains of holomorphy, proper holomorphic mappings, complex manifolds and modifications.

Continued development of a topic in several complex variables.

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Topics include formal and convergent power series, Weierstrass preparation theorem, Cartan-Ruckert theorem, analytic sets, mapping theorems, domains of holomorphy, proper holomorphic mappings, complex manifolds and modifications. 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. Introduction to varied topics in differential equations. In recent years, topics have included Riemannian geometry, Ricci flow, and geometric evolution. Continued development of a topic in differential equations. Topics include Riemannian geometry, Ricci flow, and geometric evolution. 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.

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 A-B-C or consent of instructor. In recent years, topics have included Fourier analysis in Euclidean spaces, groups, and symmetric spaces. Various topics in functional analysis. Convex sets and functions, convex and affine hulls, relative interior, closure, and continuity, recession and existence of optimal solutions, saddle point and min-max theory, subgradients and subdifferentials.

Recommended preparation: course work in linear algebra and real analysis. Optimality conditions, strong duality and the primal function, conjugate functions, Fenchel duality theorems, dual derivatives and subgradients, subgradient methods, cutting plane methods. Convex optimization problems, linear matrix inequalities, second-order cone programming, semidefinite programming, sum of squares of polynomials, positive polynomials, distance geometry. Introduction to varied topics in real analysis. In recent years, topics have included Fourier analysis, distribution theory, martingale theory, operator theory.

May be taken for credit six times with consent of adviser. Continued development of a topic in real analysis. Topics include Fourier analysis, distribution theory, martingale theory, operator theory. Various topics in real analysis. 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. Lie groups, Lie algebras, exponential map, subgroup subalgebra correspondence, adjoint group, universal enveloping algebra. Structure theory of semisimple Lie groups, global decompositions, Weyl group. Geometry and analysis on symmetric spaces. Prerequisites: MATH and or consent of instructor. Various topics in Lie groups and Lie algebras, including structure theory, representation theory, and applications.

Introduction to varied topics in differential geometry. In recent years, topics have included Morse theory and general relativity. Continued development of a topic in differential geometry. Topics include Morse theory and general relativity. May be taken for credit three times with consent of adviser.

Various topics in differential geometry. 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. Prerequisites: graduate standing in mathematics, physics, or engineering, or consent of instructor. Propositional calculus and first-order logic. Feasible computability and complexity. Peano arithmetic and the incompleteness theorems, nonstandard models.

Introduction to the probabilistic method. Combinatorial applications of the linearity of expectation, second moment method, Markov, Chebyschev, and Azuma inequalities, and the local limit lemma. Introduction to the theory of random graphs. Introduction to probabilistic algorithms. Game theoretic techniques. Applications of the probabilistic method to algorithm analysis.

Markov Chains and Random walks. Applications to approximation algorithms, distributed algorithms, online and parallel algorithms. Advanced topics in the probabilistic combinatorics and probabilistic algorithms. Random graphs. Spectral Methods. Network algorithms and optimization. Statistical learning. Introduction to varied topics in combinatorial mathematics.

In recent years topics have included problems of enumeration, existence, construction, and optimization with regard to finite sets. Recommended preparation: some familiarity with computer programming desirable but not required. Continued development of a topic in combinatorial mathematics. Topics include problems of enumeration, existence, construction, and optimization with regard to finite sets. Topics from partially ordered sets, Mobius functions, simplicial complexes and shell ability. Enumeration, formal power series and formal languages, generating functions, partitions.

## Books by Carl Ludwig Siegel

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. Introduction to varied topics in mathematical logic. Topics chosen from recursion theory, model theory, and set theory. Continued development of a topic in mathematical logic.

Various topics in logic.

## Several complex variables - Wikipedia

Various topics in combinatorics. 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. Iterative methods for large sparse systems of linear equations. Unconstrained and constrained optimization. The Weierstrass theorem, best uniform approximation, least-squares approximation, orthogonal polynomials.

www.emanuellive.com/wp-content/apple-konum-takip.php Polynomial interpolation, piecewise polynomial interpolation, piecewise uniform approximation. Numerical differentiation: divided differences, degree of precision. Numerical quadrature: interpolature quadrature, Richardson extrapolation, Romberg Integration, Gaussian quadrature, singular integrals, adaptive quadrature. Linear methods for IVP: one and multistep methods, local truncation error, stability, convergence, global error accumulation. Finite difference, finite volume, collocation, spectral, and finite element methods for BVP; a priori and a posteriori error analysis, stability, convergence, adaptivity.

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.

Survey of discretization techniques for elliptic partial differential equations, including finite difference, finite element and finite volume methods. A priori error estimates. Mixed methods. Convection-diffusion equations. Systems of elliptic PDEs.

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Survey of solution techniques for partial differential equations. Basic iterative methods. Preconditioned conjugate gradients. Multigrid methods. Hierarchical basis methods.