The lattice of subgroups. Normal subgroups and simple groups. The isomoprhisms theorems. Lagrange theorem. Group actions. Permutations representations, Cayley's theorem and action of a group on a set of cosets, order of orbits, index of stabilizer, class equation. Sylow theorems. Frobenius' proof, Wieland's proof, simplicity of the alternating group. New groups from old. The isomorphism types of group of order less than The direct product, internal and external.

The direct sum, internal and external. The semidirect product. Classes of groups: nilpotent groups, solvable groups, free groups, generators and relations. A primary focus is on the use of Taylor's theorem to analyze the methods. The analysis will be emphasized here instead of computation. Carefully chosen model or prototype problems will be examined in order to furnish theorems and insight into the behavior of the approximation methods. Wave equation: vibrating strings, and membranes. Sturm-Liouville eigenvalue problems. Non-homogenous problems. Text: Introduction to Probability Models , 11e free.

Detailed study of discrete and continuous time Markov chains and Poisson processes, with introduction to one or more of the following: martingales, Brownian motion, random walks, renewal theory. Then the analogous continuous time theory is developed. Concepts and techniques from probability and stochastic processes are introduced, including Brownian motion, martingales and stochastic calculus, in order to derive the martingale risk-neutral approach to solving the Black-Scholes p.

This course will be useful for students preparing for the Financial Economics segment of Actuarial Exam M.

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Topics include: Elementary stochastic differential equations SDE ; Monte-Carlo simulations; Ito chain rule; Log-Normal market model; derivation of the Black-Scholes partial differential equation PDE - pricing and hedging in complete markets; statistics of SDEs; statistical and implied volatility; local volatility pricing models and numerical PDEs; American options and free boundary problems; optimal portfolio theory; introduction to pricing and hedging in incomplete markets. Vector spaces, basis and dimension, the fundamental subspaces of a matrix.

Linear transformations, matrix representations, change of bases. Orthogonality, Gram-Schmidt method, QR factorization, projections, least squares. Determinants, properties and applications. Eigenvalues and eigenvectors, diagonalization of a matrix, similarity transformations, symmetric matrices, applications to difference equations and differential equations. The Jordan form.

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Projects will encompass calculus, linear algebra, and differential equations. Specific topics include: correlation coefficient, statistical inference of parameters, checking model assumptions, variable selection, transformations of variables and diagnostics. Mathematical and interpretational aspects of the models will be covered along with statistical estimation, confidence intervals and multiple hypothesis testing. SAS statistical software will be used. It will then discuss generalized linear models, including logistic and Poisson regression.

Finally various topics in survival analysis will be covered: namely Kaplan-Meier curves and log-rank statistics, Weibull regression, and Cox proportional hazard regression. Examples from medicine and engineering will be given. SAS and S-plus statistical software will be used. Specific topics include: Foundation of Bayesian Approach, Prior and Posterior distributions; Choice of Priors: subjective and non-subjective or default approaches; Inference using posterior distribution for standard models; and Hierarchical models, and their applications.

WinBUGS will be introduced. Topics include, but are not limited to, the one- and two-sample location problems including the Wilcoxon signed-rank and rank-sum test, Spearman correlation coefficient, one- and two-way Analysis-of-Variance tests, and Kolmogorov-Smirnov test for testing different distributions. In addition, the multiple comparisons issue will be discussed, specifically by comparing several treatments with and without a control treatment.

Null distributions of test statistics will be discussed in the small sample and asymptotic cases, with and without ties. Topics covered include: importing external files, subsetting and merging data files, performing statistical procedures, graphics, matrix calculations, and macros and functions. It covers the theory of ordinary differential equations, with an emphasis on applications. Basic concepts, special types of differential equations of the first order,and problems that lead to them.

Linear differential equations of order greater than one and problems that lead to them. Linear vector spaces. Systems of differential equations, linearization of first order systems, problems giving rise to systems. Existence and uniqueness theorem for first order differential equations. Existence and uniqueness theorem for a system of first order differential equations and for linear and nonlinear differential equations of order greater than one.

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Other supplementary topics: state variable description of systems, fundamental matrix, state transition matrix, matrix exponential, stability of linear systems. Time permitting: Operators and Laplace transforms, series methods. Measures and measurable functions, Lusin and Egoroff theorems, Lebesgue integral, Fatou's lemma, monotone and dominated convergence. Convergences: uniform, a. Product measures, Fubini and Tonelli theorems.

Radon-Nikodym theorem. Absolute continuity, bounded variation, and the fundamental theorem of calculus on the real line. Fundamental theory: existence and uniqueness, continuity and differentiability of solutions in initial conditions, extending solutions, global solutions. Nonlinear systems: nonlinear sinks and sources, hyperbolicity, stability, limit sets, gradient and Hamiltonian systems, other topics at instructor's discretion.

Laplace equation: mean-value property, smoothness, maximum principle, uniqueness of solutions, Harnack inequality, Liouville theorem. Poisson Equation: Fundamental solution, Greens functions, energy methods. Heat Equation: Fundamental solution, maximum principle, uniqueness of solutions on a bounded domain, Duhamel's principle, energy methods. Wave equation: Fundamental solutions in 1, 2, and 3 dimensions, energy methods, finite propagation speed.

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## Read Mathematical Modeling of Biosensors: An Introduction for Chemists and Mathematicians (Springer

Nonlinear first-order equations: Characteristic ODEs, local existence of smooth solutions, conservation law equations, shocks, rarefaction, integral solutions. Additional topics traveling waves, Fourier transform, etc. SAS will be extensively used to apply these concepts with real data. Laws of large numbers, weak convergence. Characteristic functions, central limit theorem. Conditional probability, conditional expectation. Students will be expected to have a strong background in theoretical mathematics or statistics.

Linear maps between Banach spaces.

## Biological Sciences, Chemistry and Mathematical Sciences - The University of Nottingham

Introduction to Banach Algebras. The spectrum and spectral radius of an element. Abelian algebras, Gelfand theory of representations. Time permitting, spectral measures adn functional calculus, or else compact and Fredholm operators. Correspondence between ideals and varieties, Zariski topology, Hilbert's nullstellensatz. Hilbert's basis theorem, Polynomial and rational functions.

Projective varieties. Projective space and varieties, maps between projective varieties, adjunction of roots, finite fields. Tangent spaces, smoothness and dimension, localization and the tangent space at a point, smooth and singular points, dimension of a variety. Optional topics: Elliptic Curves. Plane curves. Classification of smooth cubics. Group structure of an elliptic curve.

Theory of Curves. Divisors on curves, Bezout's theorem, Linear systems on curves Computational algebraic geometry, Groebner basis algorithm, existence and uniqueness of Groebner bases, implementation of the algorithm. Computer approximations to the solutions of the PDE problems that arise in these applications are usually required. This course will focus on the finite element method FEM and will use energy Hilbert space techniques. The first part of the course will cover error analysis for ordinary differential equations from the Atkinson text Chapter 6 and also iterative methods for matrices Sections 8.