Matrix Logic. Theory and Applications

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Functions of several variables, continuous transformations, Jacobians, chain rule, implicit function theorem, inverse function theorem, extreme, optimization and Lagrange multiplier technique. Applications in Operations Research. Applications to engineering and physical science. Course covers basic material essential for signal processing, filtering, transmission, waveguides, and other related problems. Applications include spectral analysis of electronic signals, e. Dyadic representation in Cartesian and general components. Calculus of tensor fields in curvilinear coordinates.

Derivation and application of the basic equations of heat conduction, rigid body mechanics, elasticity, fluid mechanics, electromagnetism, Newtonian and Einsteinian orbital mechanics. MA Numerical Methods for Partial Differential Equations Winter Course designed to familiarize the student with analytical techniques as well as classical finite difference techniques in the numerical solution of partial differential equations. In addition to learning applicable algorithms, the student will be required to do programming. Second, to develop working proficiency by designing, implementing, and evaluating the performance of several parallel algorithms.

These include, but are not limited to, numerical quadrature, matrix computations, sorting, network analysis, and dynamic programming.

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Prerequisites: MA or MA and ability to program in a high-level language. The course content varies and the credit varies. This course is intended to reflect study for the beginning graduate student in an area for which no formal course is taught. Credit for this course may be granted more than one time to an individual student. Prerequisites: Consent of instructor.

Course Catalog Descriptions

MA Applied Modern Algebra and Number Theory As Required This course is devoted to aspects of modern algebra and number theory that directly support applications, principally in communication. The algebraic emphasis is on ring and field theory, with special emphasis on the theory of finite fields, as well as those aspects of group theory that are important in the development of coding theory. Elements of number theory include congruences and factorization. These include error correcting codes and cryptography. MA Introduction to Real Analysis Summer The objective of this course is for students to achieve a solid understanding of the basic concepts, theorems, and proofs in introductory real analysis, including: limits, sequences, series, continuity, uniform convergence and uniform continuity, differentiation, and Riemann integration.

This is a mathematics course in the pure sense. Proofs will be emphasized, and the student will learn how to reproduce, understand, create and enjoy mathematical proofs. Singularities of analytic functions; contour integration and residues; applications of residues to real integrals and Laplace transforms, zeros of analytic functions, infinite product representation for analytic functions; maximum modulus theorems for analytic and harmonic functions; conformal mapping.

Applications include interference effects in optics and problems from heat flow and fluid flow. MA Theory of Numerical Computation As Required Analysis of computational methods used for the solution of problems from the areas of algebraic equations, polynomial approximation, numerical differentiation and integration, and numerical solutions of ordinary differential equations. Topics include generating functions, recurrence relations, elements of Ramsey theory, theorems of Burnside and Polya, and balanced incomplete block designs.

Topics include graph coloring, Eulerian and Hamiltonian graphs, perfect graphs, matching and covering, tournaments, and networks. MA Thesis Topics Seminar As Required Explores in depth discrete dynamical systems and the thesis topics of students enrolled in the Applied Mathematics degree program. Fulfills the ESR to provide students with the experience of organizing and presenting applied mathematical ideas to students and faculty, including a classroom environment.

MA Advanced Topics in Numerical Analysis V-0 The subject matter will vary according to the abilities and interest of those enrolled.

MA Numerical Solution of Ordinary Differential Equations As Required Adams formulas, Runge-Kutta formulas, extrapolation methods, implicit formulas for stiff equations; convergence and stability, error estimation and control, order and stepsize selection, applications. MA Numerical Solution of Partial Differential Equations As Required Finite difference methods for parabolic, elliptic, and hyperbolic equations, multi-grid methods; convergence and stability, error estimation and control, numerical solution of finite difference equations, applications.

Prerequisites: MA, MA suggested. MA Mathematical Foundations of Galerkin Methods As Required Variational formulation of boundary value problems, finite element and boundary element approximations, types of elements, stability, eigenvalue problems. Prerequisites: MA, MA or equivalent. Rounding errors and introduction to stability analysis.

Stable algorithms for solving systems of linear equations, linear least squares problems and eigen problems. Iterative methods for linear systems.

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Structured problems from applications in various disciplines. MA Distributed Scientific Computing As Required General principles of parallel computing, parallel techniques and algorithms, solution of systems of linear equations, eigenvalues and singular value decomposition, domain decomposition and application e. MA Calculus of Variations As Required Euler equation, Weierstrass condition, Legendre condition, numerical procedures for determining solutions, gradient method, Newton method, Transversability condition, Rayleigh Ritz method, conjugate points.

Concepts are related to geometric principles whenever possible. Prerequisites: MA programming experience desirable. MA Stability, Bifurcation and Chaos As Required Differential equations and dynamical systems, equilibrium of autonomous systems, stability, Liapunov's method, examples of chaos, local bifurcations of vector fields and maps, chaotic dynamical systems.

MA Principles and Techniques of Applied Mathematics I Fall Selected topics from applied mathematics to include: Dimensional Analysis, Scaling, Stability and Bifurcation, Perturbation Methods— regular and singular with boundary layer analysis, as well as, asymptotic expansions of integral, integrals equations, Green's functions of boundary value problems, and distribution theory. Selected topics include: calculus of variations, Hamiltonian Mechanics, distribution theory and Green's Functions in two and three dimensions, and discrete models.

It includes the following topics: classification of second order equations; initial value and boundary value problems for hyperbolic, parabolic, and elliptic partial differential equations; existence and uniqueness of linear elliptic and parabolic PDEs; nonlinearparabolic and elliptic PDEs; Hamilton-Jacobi equations; systems of conservation laws and nonlinear wave equations; transform methods and Green's functions. Prerequisites: MA, and MA strongly recommended. MA Linear and Nonlinear Waves As Required Analysis of the two main classes of wave motion, hyperbolic waves and linear dispersive waves.

Topics covered include: kinematic waves, shock waves, shock structure and shock fitting, Burger's equation, the wave equation, linear dispersive waves, wave patterns and water waves. The effects of a third point mass and a distributed mass. Expansion of the disturbing potential in series of Legendre functions. Variation of parameter equations for osculating orbital elements. Perturbation and numerical solution techniques.

Statistical orbit determination. Codes used by the military to maintain the catalog of artificial satellites and space debris. Prerequisites: SS or equivalent. Applications to problems in engineering and physics. MA Asymptotic and Perturbation Methods I As Required Advanced course in the application of approximate methods to the study of integrals and differential equations arising in physical problems.

Topics covered include: asymptotic sequences and expansions, integrals of a real variable, contour integrals, limit process expansions applied to ordinary differential equations, multiple variable expansion procedures and applications to partial differential equations. MA Analytical Methods for Fluid Dynamics As Required The basic fluid dynamic equations will be derived, and a variety of analytical methods will be applied to problems in viscous flow, potential flow, boundary layers, and turbulence.

Applications in aeronautics will be discussed. Prerequisites: MA or MA MA Numerical Methods for Fluid Dynamics As Required Numerical methods exclusively will be applied to fluid dynamics problems in viscous flow, potential flow, boundary layers, and turbulence. Credit may be granted for taking this course more than once. This course focuses on the emerging field of network science.

The students will learn about the ongoing research in the field by listening to conference presentations prerecorded or in person if possible to attend conferences , and applying the learned knowledge to a research project. If the research project is used towards a funded research, then 1 students' visit to sponsors will be included during the quarter if feasible or virtual conversations with sponsors are facilitated, and 2 a final presentation to the sponsor is anticipated upon the completion of the project. There is no set content for the course, as students will be exposed to current topics in the network science field at the time.

Prerequisites: consent of instructor Prof.

Ralucca Gera. MA Cooperation and Competition Spring The course will develop game theoretic concepts in evaluations of the importance of players in bargaining situations and of elements in networks. Topics covered include cooperative and noncooperative games, bargaining, the Shapley Value, and coalitions. The course will study applications to military problems and applications to economics, political science, and biology.

There will be extensive reading from the literature.

MA3042 Linear Algebra (4-0) As Required

Prerequisites: MA, OA, and an introductory course in probability. MA Structure and Analysis of Complex Networks Winter The course focuses on the emerging science of complex networks and their applications, through an introduction to techniques and models for understanding and predicting their behavior.

Matrix Logic and Mind: A Probe Into a Unified Theory of Mind and Matter

The topics discussed will be building mainly on graph theory concepts, and they will address the mathematics of networks, their applications to the computer networks and social networks, and their use in research. The students will learn the fundamentals of dynamically evolving complex networks, study current research in the field, and apply their knowledge in the analysis of real network systems through a final project.

DoD applications include security of critical communication infrastructure. MA Combinatorial and Cryptographic Properties of Boolean Functions As Required The course will discuss the Fourier analysis of Boolean functions and the relevant combinatorics with an eye toward cryptography and coding theory. Particular topics will include avalanche features of Boolean functions, correlation immunity and resiliency, bentness, trade-offs among cryptographic criteria and real-life applications in the designs of stream and block ciphers.

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