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Mathematics |
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MATH 5000 - Instructional Issues for the Professional Mathematician 3 hours
Focus on various instructional issues from the perspective of the professional mathematician. Some major topics include course planning, the content of a course syllabus, lecture styles, the preparation and mechanics of lectures, the conduct of problem solving sessions, classroom management, the student-instructor relationship, examination formats, the preparation, administration and grading of examinations and the management of teaching assistants and graders.
Prerequisite(s): Consent of department.
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MATH 5010 - Mathematical Logic and Set Theory 3 hours
Rigorous development of first-order logic, basic model theory, completeness and incompleteness theorems, decidable and undecidable theories, axioms of set theory, ordinal and cardinal numbers, the axiom of choice, the continuum hypothesis, constructible sets, and basic descriptive set theory.
Prerequisite(s): None.
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MATH 5020 - Mathematical Logic and Set Theory 3 hours
Rigorous development of first-order logic, basic model theory, completeness and incompleteness theorems, decidable and undecidable theories, axioms of set theory, ordinal and cardinal numbers, the axiom of choice, the continuum hypothesis, constructible sets, and basic descriptive set theory.
Prerequisite(s): None.
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MATH 5050 - Linear Programming 3 hours
Convex polyhedra, simplex method, duality theory, network flows, integer programming, ellipsoidal method, applications to modeling and game theory.
Prerequisite(s): Consent of department.
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MATH 5110 - Introduction to Analysis 3 hours
Rigorous development for the real case of the theories of continuous functions, differentiation, Riemann integration, infinite sequences and series, uniform convergence and related topics; an introduction to the complex case.
Prerequisite(s): None.
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MATH 5120 - Introduction to Analysis 3 hours
Rigorous development for the real case of the theories of continuous functions, differentiation, Riemann integration, infinite sequences and series, uniform convergence and related topics; an introduction to the complex case.
Prerequisite(s): None.
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MATH 5200 - Topics in Dynamical Systems 3 hours
Dynamical systems in one and higher dimensions. Linearization of hyperbolic fixed points. Hamiltonian systems and twist maps. The concept of topological conjugacy and structural stability. Anosov diffeomorphisms, geodesic flow and attractors. Chaotic long-term behavior of these hyperbolic systems. Measures of complexity.
Prerequisite(s): Consent of department.
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MATH 5210 - Numerical Analysis 3 hours
Rigorous mathematical analysis of numerical methods: norms, error analysis, linear systems, eigenvalues and eigenvectors, iterative methods of solving non-linear systems, polynomial and spline approximation, numerical differentiation and integration, numerical solution or ordinary and partial differential equations.
Prerequisite(s): FORTRAN programming or consent of department.
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MATH 5220 - Numerical Analysis 3 hours
Rigorous mathematical analysis of numerical methods: norms, error analysis, linear systems, eigenvalues and eigenvectors, iterative methods of solving non-linear systems, polynomial and spline approximation, numerical differentiation and integration, numerical solution or ordinary and partial differential equations.
Prerequisite(s): FORTRAN programming or consent of department.
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MATH 5270 - Mathematical Theory of Computation 3 hours
Mathematical models of computation and algorithms, Church-Turing thesis, recursive functions, complexity measures, algorithm analysis, complexity classes and hierarchies, the P versus NP problem, NP-completeness.
Prerequisite(s): Consent of department.
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MATH 5290 - Numerical Methods 3 hours
Non-theoretical development of various numerical methods for use with a computer to solve equations, solve linear and non-linear systems of equations, find eigenvalues and eigenvectors, approximate functions, approximate derivatives and definite integrals, solve differential equations and solve other such problems of a mathematical nature. Errors due to instability of method and those due to the finite-precision computer will be studied.
Prerequisite(s): A programming language and consent of department.
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MATH 5310 - Functions of a Real Variable 3 hours
Sets and operations; descriptive set properties; cardinal numbers; order types and ordinals; metric spaces; the theory of Lebesque measure; metric properties of sets.
Prerequisite(s): None.
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MATH 5320 - Functions of a Real Variable 3 hours
Set functions and abstract measure; measurable functions; types of continuity; classification of functions; the Lebesque integral; Dini derivatives and the fundamental theorem of calculus.
Prerequisite(s): None.
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MATH 5350 - Markov Processes 3 hours
The ergodic theorem; regular and ergodic Markov chains; absorbing chains and random walks; mean first passage time; applications to electric circuits, entropy, genetics, games, decision theory and probability.
Prerequisite(s): None.
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MATH 5400 - Introduction to Functions of a Complex Variable 3 hours
Algebra of complex numbers and geometric representation; analytical functions; elementary functions and mapping; real-line integrals; complex integration; power series; residues, poles, conformal mapping and applications.
Prerequisite(s): None.
Only one course, MATH 5400, MATH 5500 or MATH 5600 , may be used towards satisfying the course work requirements for a graduate degree in mathematics.
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MATH 5410 - Functions of a Complex Variable 3 hours
Theory of analytic functions from the Cauchy-Riemann and Weierstrass points of view.
Prerequisite(s): None.
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MATH 5420 - Functions of a Complex Variable 3 hours
Theory of analytic functions from the Cauchy-Riemann and Weierstrass points of view.
Prerequisite(s): None.
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MATH 5450 - Calculus on Manifolds 3 hours
Introduction to differential geometry and topology. Topics include implicit and inverse function theorems, differentiable manifolds, tangent bundles, Riemannian manifolds, tensors, curvature, differential forms, integration on manifolds and Stokes’ theorem.
Prerequisite(s): Consent of department.
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MATH 5460 - Differential Equations 3 hours
Calculation of solutions to systems of ordinary differential equations, study of algebraic and qualitative properties of solutions, study of partial differential equations of mathematical physics, iterative methods for numerical solutions of ordinary and partial differential equations and introduction to the finite element method.
Prerequisite(s): MATH 5110 -MATH 5120 and linear algebra.
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MATH 5470 - Differential Equations 3 hours
Calculation of solutions to systems of ordinary differential equations, study of algebraic and qualitative properties of solutions, study of partial differential equations of mathematical physics, iterative methods for numerical solutions of ordinary and partial differential equations and introduction to the finite element method.
Prerequisite(s): MATH 5110 -MATH 5120 and linear algebra.
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MATH 5500 - Introduction to the Theory of Matrices 3 hours
Congruence (Hermitian); similarity; orthogonality, matrices with polynomial elements and minimal polynomials; Cayley-Hamilton theorem; bilinear and quadratic forms; eigenvalues.
Prerequisite(s): None.
Only one course, MATH 5400 , MATH 5500 or MATH 5600 , may be used towards satisfying the course work requirements for a graduate degree in mathematics.
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MATH 5520 - Modern Algebra 3 hours
Groups and their generalizations; homomorphism and isomorphism theories; direct sums and products; orderings; abelian groups and their invariants.
Prerequisite(s): MATH 3510 or equivalent.
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MATH 5530 - Selected Topics in Modern Algebra 3 hours
Ring and field extensions, Galois groups, ideals and valuation theory.
Prerequisite(s): None.
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MATH 5600 - Introduction to Topology 3 hours
Point set topology; connectedness, compactness, continuous functions and metric spaces.
Prerequisite(s): None.
Only one course, MATH 5400 , MATH 5500 or MATH 5600, may be used towards satisfying the course work requirement for a graduate degree in mathematics.
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MATH 5610 - Topology 3 hours
Rigorous development of abstract topological spaces, mappings, metric spaces, continua, product and quotient spaces; introduction to algebraic methods.
Prerequisite(s): None.
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MATH 5620 - Topology 3 hours
Rigorous development of abstract topological spaces, mappings, metric spaces, continua, product and quotient spaces; introduction to algebraic methods.
Prerequisite(s): None.
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MATH 5700 - Selected Topics in Contemporary Mathematics 3 hours
Topics of current interest that vary from year to year.
Prerequisite(s): Consent of department.
May be repeated for credit as topics vary.
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MATH 5810 - Probability and Statistics 3 hours
Important densities and stochastic processes; measure and integration; laws of large numbers; limit theorems.
Prerequisite(s): None.
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MATH 5820 - Probability and Statistics 3 hours
Markov processes and random walks; renewal theory and Laplace transforms; characteristic functions; infinitely divisible distribution; harmonic analysis.
Prerequisite(s): None.
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MATH 5900 - Special Problems 1–3 hours
Prerequisite(s): None.
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MATH 5910 - Special Problems 1–3 hours
Prerequisite(s): None.
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MATH 5940 - Seminar in Mathematical Literature 1–3 hours
Prerequisite(s): None.
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MATH 5950 - Master’s Thesis 3 or 6 hours
To be scheduled only with consent of department. 6 hours credit required. No credit assigned until thesis has been completed and filed with the graduate school.
Prerequisite(s): None.
Continuous enrollment required once work on thesis has begun. May be repeated for credit.
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MATH 6000 - Millican Colloquium 1 hour
Departmental colloquium. New research developments are presented by nationally and internationally recognized mathematicians from the U.S. and abroad. Topics vary weekly and can cover any of the subdisciplines of mathematics.
Prerequisite(s): None.
May be repeated for credit. Pass/no pass only.
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MATH 6010 - Topics in Logic and Foundations 3 hours
Mathematical logic, metamathematics and foundations of mathematics.
Prerequisite(s): None.
May be repeated for credit.
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MATH 6020 - Logic Seminar 1 hour
Weekly seminar series covering contemporary topics in logic and set theory. Talks are given by UNT faculty and graduate students as well as by prominent visitors from other institutions. Topics vary weekly.
Prerequisite(s): None.
May be repeated for credit. Pass/no pass only.
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MATH 6110 - Topics in Analysis 3 hours
Measure and integration theory, summability, complex variables and functional analysis.
Prerequisite(s): None.
May be repeated for credit.
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MATH 6130 - Infinite Processes 3 hours
Topics selected from infinite series, infinite matrices, continued fractions, summation processes and integration theory.
Prerequisite(s): None.
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MATH 6150 - Functional Analysis 3 hours
Normed linear spaces; completeness, convexity and duality. Topics selected from linear operators, spectral analysis, vector lattices and Banach algebras.
Prerequisite(s): None.
May be repeated for credit.
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MATH 6170 - Differential Equations 3 hours
Existence, uniqueness and approximation of solutions to linear and non-linear ordinary, partial and functional differential equations. Relationships with functional analysis. Emphasis is on computer-related methods.
Prerequisite(s): None.
May be repeated for credit.
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MATH 6200 - Topics in Ergodic Theory 3 hours
Basic ergodic theorems. Mixing properties and entropy. Oseledec’s multiplicative ergodic theorem and Lyapunov exponents. Applications to dynamical systems. Rational functions and Julia sets. Wandering across Mandelbrot set. Sullivan’s conformal measure. Thermodynamical formalism and conformal measures applied to compute Hausdorff measures and packing measures of attractors, repellors and Julia sets. Dimension invariants (Hausdorff, box and packing dimension) of these sets.
Prerequisite(s): Consent of department.
May be repeated for credit.
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MATH 6210 - Dynamical Systems Seminar 1 hour
Weekly seminar series covering contemporary topics in dynamical systems and ergodic theory. Talks are given by UNT faculty and graduate students as well as by prominent visitors from other institutions. Topics vary weekly.
Prerequisite(s): None.
May be repeated for credit. Pass/no pass only.
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MATH 6220 - Logic and Dynamics Seminar 1 hour
Weekly seminar series covering contemporary topics in logic and dynamical systems. Talks are given by UNT faculty and graduate students as well as by prominent visitors from other institutions. Topics vary weekly.
Prerequisite(s): None.
May be repeated for credit. Pass/no pass only.
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MATH 6310 - Topics in Combinatorics 3 hours
Selected topics of current interest in combinatorics such as enumeration, combinatorial optimization, Ramsey theory, topological graph theory, random methods in combinatorics (random graphs, random matrices, randomized algorithms, etc.), combinatorial designs, matroids, formal languages and combinatorics on words, combinatorial number theory, combinatorial and symbolic methods in dynamical systems.
Prerequisite(s): None.
May be repeated for credit.
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MATH 6510 - Topics in Algebra 3 hours
Groups, rings, modules, fields and other algebraic structures; homological and categorical algebra. Multiplicative and additive number theory, diophantine equations and algebraic number theory.
Prerequisite(s): None.
May be repeated for credit.
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MATH 6520 - Algebra Seminar 1 hour
Weekly seminar series covering topics in algebra. Talks are given by UNT faculty and graduate students as well as by prominent visitors from other institutions. Topics vary weekly.
Prerequisite(s): None.
May be repeated for credit. Pass/no pass only.
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MATH 6610 - Topics in Topology and Geometry 3 hours
Point set and general topology, differential geometry and global geometry.
Prerequisite(s): None.
May be repeated for credit.
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MATH 6620 - Algebraic Topology 3 hours
Topics from algebraic topology such as fundamental group, singular homology, fixed point theorems, cohomology, cup products, Steenrod powers, vector bundles, classifying spaces, characteristic classes and spectral sequences.
Prerequisite(s): MATH 5530 , MATH 5620 .
May be repeated for credit.
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MATH 6700 - Selected Topics in Advanced Mathematics 3 hours
Topics of current interest that vary from year to year.
Prerequisite(s): Consent of department.
May be repeated for credit as topics vary.
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MATH 6710 - Topics in Applied Mathematics 3 hours
Optimization and control theory, perturbation methods, eigenvalue problems, generalized functions, transform methods and spectral theory.
Prerequisite(s): None.
May be repeated for credit.
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MATH 6810 - Probability 3 hours
Probability measures and integration, random variables and distributions, convergence theorems, conditional probability and expectation, martingales, stochastic processes.
Prerequisite(s): None.
May be repeated for credit.
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MATH 6900 - Special Problems 1–3 hours
Prerequisite(s): None.
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MATH 6910 - Special Problems 1–3 hours
Prerequisite(s): None.
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MATH 6940 - Individual Research Variable credit
To be scheduled by the doctoral candidate engaged in research.
Prerequisite(s): None.
May be repeated for credit.
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MATH 6950 - Doctoral Dissertation 3, 6 or 9 hours
To be scheduled only with consent of department. 9 hours credit required. No credit assigned until dissertation has been completed and filed with the graduate school.
Prerequisite(s): None.
Doctoral students must maintain continuous enrollment in this course subsequent to passing qualifying examination for admission to candidacy. May be repeated for credit.
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