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Research
Resources
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All Graduate Math Courses
This is not necessarily the official description for the courses. For the official descriptions, consult the 2012 - 2013 graduate catalog.
Description: Taught on Mondays and Wednesdays. The Monday classes cover the theory and practice of teaching mathematics at the college level: basic skills, grading methods, cooperative learning, active learning, use of technology, classroom problems, history of learning theory, reflective practice. The Wednesday classes cover the IT resources required for someone to become an effective member of our department.
Prerequisites: Open to graduate students in mathematics, others with consent of instructor. May not be used to satisfy degree requirements in mathematics.
Offered: Fall
Credits: 1
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Description: Advanced topics in analysis. With a change of content, this course is repeatable to a maximum of twelve credits.
Credits: 3
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Description: Advanced topics in analysis. With a change of content, this course is repeatable to a maximum of twelve credits.
Credits: 3
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Description: Advanced topics in probability theory, theory of random processes, mathematical statistics, and related fields. With a change of content, this course is repeatable to a maximum of twelve credits.
Credits: 3
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Description: Advanced topics chosen from group theory, ring theory, number theory, Lie theory, combinatorics, commutative algebra, algebraic geometry, homological algebra, and representation theory. With change of content, this course may be repeated to a maximum of twelve credits.
Prerequisites: MATH 5211
Credits: 3
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Description: Topics include, but are not restricted to, recursion theory (degree structures, hyperarithmetic hierarchy, applications to computable algebra, reverse mathematics), model theory (quantifier elimination, o-minimality, types, categoricity, indiscernible), set theory (ordinals, cardinals, Martin's axiom, constructible sets, forcing), and proof theory (deductive systems, cut elimination and applications, ordinal analysis). With a change of content, this course is repeatable to a maximum of twelve credits.
Prerequisites: MATH 5260
Credits: 3
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Description: Advanced topics in geometry and topology. With a change of content this course is repeatable to a maximum of twelve credits.
Credits: 3
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Description: Advanced topics in geometry and topology. With a change of content this course is repeatable to a maximum of twelve credits.
Credits: 3
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Description: Advanced topics from the theory of ordinary or partial differential equations. Other possible topics: integral equations, optimization theory, the calculus of variations, advanced approximation theory. With a change of content, this course is repeatable to a maximum of twelve credits.
Credits: 3
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Description: Advanced topics from the theory of ordinary or partial differential equations. Other possible topics: integral equations, optimization theory, the calculus of variations, advanced approximation theory. With a change of content, this course is repeatable to a maximum of twelve credits.
Prerequisites: Instructor consent required.
Credits: 3
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Description: Complex plane, Riemann sphere, polar coordinates and Euler's formula, complex differentiable functions and Cauchy-Riemann equations, harmonic conjugates, conformal maps. Elementary functions: exp, sin, cos, log, Log, powers. Integration along simple curves, simply connected domains, Cauchy-Goursat theorem, Cauchy integral formula. Power series and the disk of convergence, Taylor and Laurent series, classification of singularities. Cauchy's residue theorem and its use in evaluating real-valued integrals. Not open to students who have passed MATH 3146. Open for master's credit but not doctoral credit toward degree in Mathematics.
Credits: 3
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Description: Construction of real numbers, completeness, infima, suprema. Sequences of real numbers, liminf and limsup, limits, big-O and little-o notations. Continuity of functions of one real variable, Intermediate Value theorem, continuous functions on closed bounded sets. Sequences and series of functions of one real variable, uniform convergence. Differentiation of functions of one real variable, Rolle's and Mean Value theorems. Taylor series. Riemann integration of functions of one real variable and the Fundamental Theorem of Calculus. Not open to students who have passed MATH 3150. Open for master's credit but not doctoral credit toward degree in mathematics.
Offered: Spring
Credits: 3
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Description: Real and Complex Number Systems. Basic topology of metric spaces, Bolzano-Weierstrass and Heine-Borel theorems. Sequences and series of functions on compact metric spaces, uniform convergence, Arzela-Ascoli and Stone-Weierstrass theorems. Differentiation and integration of vector valued functions of several real or complex variables. Inverse and implicit function theorems. Contraction mapping and Picard-Lindelof theorems. Differential forms.
Offered: Spring
Credits: 3
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Description: Abstract integration: Lebesgue integration theory, outer measures and Caratheodory's theorem, Fatou's lemma, monotone and dominated convergence theorems. Measure theory: positive Borel measures, Riesz representation theorem for positive linear functionals on C(K), complex measures, Hahn-Jordan and Lebesgue decompositions, Radon-Nykodim theorem and differentiation of measures, Riemann-Stieltjes integral, the Lebesgue measure on Rd. Lp spaces: Cauchy-Bunyakovsky-Schwarz, Hoelder, Minkowski and Jensen inequalities, L2 and Lp spaces as Hilbert and Banach spaces, Riesz representation theorem for bounded functionals on Lp. Integration on product spaces, Fubini's and Tonelli's theorems, Fourier transform and Plancherel theorem. For prelim preparation, see the prelim study guide.
Prerequisites: MATH 5110
Offered: Spring
Credits: 3
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Description: Complex plane, Riemann sphere, Euler's formula, complex differentiable functions and Cauchy-Riemann equations, conformal maps, linear fractional transformations. Integration along simple rectifiable curves, Cauchy-Goursat and Morera's theorems, Cauchy integral formula, Cauchy estimates and Schwarz lemma. Power series and the disk of convergence, Taylor and Laurent series, classification of singularities. The argument principle, winding numbers and Rouche's theorem. Cauchy's residue theorem and its use in evaluating real-valued integrals. Maximum modulus, Liouville and Picard theorems, the Fundamental Theorem of Algebra, Schwarz reflection principle. Harmonic functions and harmonic conjugates. Normal families and Montel's theorem. The Riemann mapping theorem. A practical purpose of the class is to prepare students to take the qualifying exams.
For prelim preparation, see the prelim study guide.
Prerequisites: MATH 5110
Credits: 3
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Description: Advanced topics of contemporary interest. These include Riemann surfaces, Kleinian groups, entire functions, conformal mapping, several complex variables, and automorphic functions, among others. With a change of content this course may be repeatable to a maximum of twelve credits.
Prerequisites: MATH 5120
Credits: 3
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Description: Theory of Banach spaces: duality, reflexivity, weak and weak* topologies. Hahn-Banach, Banach-Steinhaus, Banach-Alaoglu theorems. Krein-Milman theorem. Linear operators: compact, integral, trace class, Fredholm, Hilbert-Schmidt, Toepliz, Volterra. Commutative Banach and C*-algebras, Gelfand transform and the spectral theorem for bounded normal operators. Compact self-adjoint operators with applications to the classical Sturm-Liouville theory. Other topics in functional analysis at the choice of the instructor (e.g. unbounded self-adjoint operators, distributions, Banach algebra L1, Kaplansky density theorem, Gelfand-Naimark-Segal construction, introduction to von Neumann algebras and non-commutative integration, introduction to unbounded self-adjoint operators and the role of the Fourier transform).
Prerequisites: MATH 5111
Credits: 3
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Description: Spectral theory of unbounded self-adjoint and normal operators on Hilbert spaces. Quadratic forms. Examples and counterexamples of self-adjoint operators. Spectral theory of differential operators with constant coefficients. Unitary and positivity preserving operator semigroups, resolvents, Trotter product formula, Hille-Yosida theorem. Other topics in functional analysis at the choice of the instructor (e.g. applications to probability and quantum mechanics, introduction to von Neumann algebras and non-commutative integration).
Prerequisites: MATH 5111
Credits: 3
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Description: Basic properties of Fourier series, convergence of Fourier series, applications of Fourier series. Fourier transform and distributions. Fourier transform in Lp-spaces. Hardy-Littlewood maximal inequality. Marcinkiewicz and Riesz-Thorin interpolation theorem. Hilbert and Riesz transforms, singular integrals, Calderon-Zygmund operators. Other topics in harmonic and Fourier analysis at the choice of the instructor (e.g. Littlewood-Paley theory, Marcinkiewicz multiplier theorem, fast Fourier transform, wavelets).
Prerequisites: MATH 5111
Credits: 3
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Description: Harmonic analysis on various spaces such as Euclidean spaces, and abelian and non-abelian locally compact groups. Pontryagin duality, the Peter-Weyl theorem, various Fourier transforms and connections to unitary representation theory.With a change of content, this course is repeatable to a maximum of six credits.
Prerequisites: MATH 5111
Credits: 3
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Description: Foundation of probability theory, monotone classes and pi-lambda theorem, Kolmogorov extension theorem and infinite product spaces, Kolmogorov zero-one law, a.s. convergence, convergence in probability and in Lp of random variables, Borell-Cantelli lemma. Convergence of series of independent random variables: the theorems of Kolmogorov and Levy. Weak convergence of probability measures: characteristic functions, Levy-Cramer continuity theorem, tightness and Prohorov's theorem. The Central Limit Theorem: the Lindeberg-Feller theorem, the Levy-Khintchine formula, stable laws. Conditional expectation. Discrete time (sub- and super) martingales: Doob's maximal inequality, Optional Stopping Theorem, uniform integrability, and the a.s. convergence theorem for L1 bounded martingales, convergence in Lp. Definition, existence and basic properties of the Brownian Motion. Other topics in probability theory at the choice of the instructor (e.g. Markov chains, Birkhoff-Khinchine and Kigman ergodic theorems, Levy's arcsine law, Law of Iterated Logarithm, convergence to stable laws).
For prelim preparation, see the prelim study guide.
Prerequisites: MATH 5111
Credits: 3
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Description: The course material changes with each occurrence of the course and may be taken for credit repeatedly with the instructor's permission. Continuous time random processes, Kolmogorov's continuity theorem. Brownian Motion: the Donsker invariance principle, Holder continuity, quadratic variation. Continuous-time martingales and square integrable martingales. Markov processes and the strong Markov property. Properties of Brownian Motion: strong Markov property, Blumenthal zero-one law, Law of Iterated Logarithm. Stochastic integration with respect to continuous local martingales, Ito's formula, Levy's Characterization of Brownian motion, Girsanov transformation, Stochastic Differential Equations with Lipschitz Coefficients. Other topics in probability theory and stochastic processes at the choice of the instructor (e.g. connection to PDEs, local time, Skorokhod's embedding theorem, zeros of the BM, empirical processes, concentration inequalities and applications in non-parametric statistics, infinite dimensional analysis, processes with stationary independent increments and infinitely divisible processes, jump measures and Levy measures, random orthogonal measures, symmetric Markov processes, stochastic analysis for jump process). With a change of content, this course is repeatable to a maximum of six credits.
Prerequisites: MATH 5160
Credits: 3
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Description: Group theory, ring theory and modules, and universal mapping properties. For prelim preparation, see the prelim study guide.
Offered: Fall
Credits: 3
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Description: Linear and multilinear algebra, Galois theory, category theory, and commutative algebra.
Prerequisites: MATH 5210
Offered: Spring
Credits: 3
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Description: This course is an introduction to quivers and their representations with a focus on applications to the representation theory of finite dimensional algebras over a field. A quiver is a set of points and a set of arrows between the points, and a representations of a quiver consist of one vector space for each point and one linear map for each arrow. These representations can be thought of modules over a corresponding finite dimensional algebra defined in terms of paths in the quiver.
Topics covered in the course include: representations of quivers, direct sums, morphisms, kernels, exact sequences, projective and injective representations, Auslander-Reiten quiver, algebras, modules, idempotents, path algebras, Auslander-Reiten theory. We will use the language of category theory throughout the course.
Prerequisites: MATH 5210
Offered: Fall
Credits: 0
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Description: Algebraic integers, ideal class group, ramification, Frobenius elements in Galois groups,
Dirichlet's unit theorem, localization, and completion. Further topics (zeta-functions, function fields, non-maximal orders) as time permits.
Prerequisites: MATH 5211
Credits: 3
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Description: The LU, QR, symmetric, polar, and singular value matrix decompositions. Schur and Jordan normal forms. Symmetric, positive-definite, normal and unitary matrices. Perron-Frobenius theory and graph criteria in the theory of non-negative matrices.
Offered: Fall
Credits: 3
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Description: Predicate calculus, completeness theorem, compactness and applications, Lowenheim-Skolem theorems, formal theories with applications to algebra, Goedel's incompleteness theorems. Further topics chosen from: axiomatic set theory (ordinals, cardinals, infinite combinatorics, independence), model theory (quantifier elimination, types), recursion theory (reducibilities, degree structures, arithmetic hierarchy, Post's problem) or proof theory (deductive systems, cut elimination).
Prerequisites: MATH 5210
Credits: 3
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Description: Topological spaces, maps, induced topologies, separation axioms, compactness, connectedness, classification of surfaces, the fundamental group and its applications, covering spaces. For prelim preparation, see the prelim study guide.
Prerequisites: MATH 5110, which may be taken concurrently.
Offered: Fall
Credits: 3
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Description: This course is offered either as an introduction to algebraic topology or differential topology. Differential topology: review of topology; smooth manifolds; vector fields, tangent and cotangent bundles; submersions, immersions and embeddings, submanifolds; Lie groups; differential forms, orientations, integration on manifolds; De Rham cohomolgy, singular cohomology and de Rham theorem; other topics in geometry at the choice of the instructor (e.g. intrinsic Riemannian geometry of surfaces). Algebraic topology: simplical homology, singular homology; Eilenberg-Steenrod axioms; Mayer-Vietoris sequences; CW-homology; cohomology; cup products; Hom and Tensor products; Ext and Tor; the Universal Coefficient Theorems; Cech cohomology and Steenrod homology; Poincare, Alexander, Lefschetz, Alexander-Pontryagin dualities. With a change of content, this course is repeatable to a maximum of six credits.
Prerequisites: MATH 5310
Credits: 3
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Description: This course is an introduction to algebraic varieties: affine and projective varieties, dimension of varieties and subvarieties, algebraic curves, singular points, divisors and line bundles, differentials, intersections.
Prerequisites: MATH 5211 and MATH 5310, which may be taken concurrently.
Credits: 3
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Description: This course introduces further concepts and methods of modern algebraic geometry, including schemes and cohomology.
Prerequisites: MATH 5320
Credits: 3
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Description: This course is an introduction to the study of differentiable manifolds on which various differential and integral calculi are developed. The topics include covariant derivatives and connections, geodesics and exponential map, Riemannian metrics, curvature tensor, Ricci and scalar curvature.
Prerequisites: MATH 5310
Credits: 3
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Description: Banach spaces, linear operator theory and application to differential equations, nonlinear operators, compact sets on Banach spaces, the adjoint operator on Hilbert space, linear compact operators, Fredholm alternative, fixed point theorems and application to differential equations, spectral theory, distributions.
Credits: 3
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Description: Banach spaces, linear operator theory and application to differential equations, nonlinear operators, compact sets on Banach spaces, the adjoint operator on Hilbert space, linear compact operators, Fredholm alternative, fixed point theorems and application to differential equations, spectral theory, distributions.
Credits: 3
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Description: Existence and uniqueness of solutions, stability and asymptotic behavior. If time permits: eigenvalue problems, dynamical systems, existence and stability of periodic solutions.
Prerequisites: MATH 5111
Credits: 3
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Description: Convergence of Fourier series, Legendre and Hermite polynomials, existence and uniqueness theorems, two-point boundary value problems and Green's functions.
Prerequisites: MATH 5111 and 5140 are helpful but not required.
Credits: 3
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Description: Solution of first and second-order partial differential equations with applications to engineering and science.
Prerequisites: Not open to students who have passed MATH 3435. Not open for graduate credit toward degrees in mathematics.
Credits: 3
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Description: Cauchy-Kovalevskaya Theorem, classification of second-order equations, systems of hyperbolic equations, the wave equation, the potential equation, the heat equation in Rn.
Prerequisites: MATH 5120
Credits: 3
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Description: The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.
Prerequisites: MATH 5110, which may be taken concurrently.
Offered: Fall
Credits: 3
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Description: The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.
Prerequisites: MATH 5510
Offered: Spring
Credits: 3
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Description: Numerical solution of elliptic, parabolic and hyperbolic partial differential equations by finite element solution methods. Applications.
Credits: 3
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Description: Numerical solution of elliptic, parabolic and hyperbolic partial differential equations by finite element solution methods. Applications.
Prerequisites: MATH 5520
Credits: 3
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Description: Development of mathematical models emphasizing linear algebra, differential equations, graph theory and probability. In-depth study of the model to derive information about phenomena in applied work.
Credits: 3
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Description: Development and computer-assisted analysis of mathematical models in chemistry, physics, and engineering. Topics include chemical equilibrium, reaction rates, particle scattering, vibrating systems, least squares analysis, quantum chemistry and physics.
Credits: 4
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Description: Theory of linear programming: convexity, bases, simplex method, dual and integer programming, assignment, transportation, and flow problems. Theory of nonlinear programming: unconstrained local optimization, Lagrange multipliers, Kuhn-Tucker conditions, computational algorithms.
Credits: 3
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Description: The mathematics of measurement of interest, accumulation and discount, present value, annuities, loans, bonds, and other securities. Not open to students who have passed MATH 2620.
Credits: 3
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Description: The continuation of Math 5620, focusing on the mathematics of finance: measurement of financial risk and the opportunity cost of capital, the mathematics of capital budgeting and securities valuation, mathematical analysis of financial decisions and capital structure, and option pricing theory. Provides VEE credit in the Corporate Finance subject area for Society of Actuaries and Casualty Actuarial Society requirements. Not open to students who have passed MATH 3650.
Credits: 4
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Description: Survival distributions, claim frequency and severity distributions, life tables, life insurance, life annuities, net premiums, net premium reserves, multiple life functions, and multiple decrement models. Not open to students who have passed MATH 3630.
Prerequisites: MATH 2620 or MATH 5620, which may be taken concurrently.
Offered: Fall
Credits: 4
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Description: Lecture. Survival distributions, claim frequency and severity distributions, life tables, life insurance, life annuities, net premiums, net premium reserves, multiple life functions, and multiple decrement models. Not open to students who have passed MATH 3631.
Prerequisites: MATH 5630.
Credits: 4
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Description: Analysis, estimation, and validation of lifetime tables
Prerequisites: MATH 5630 or STAT 3445.
Credits: 3
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Description: Introduction to the use of mathematical and statistical techniques to solve a wide variety of organizational problems. Topics include linear programming, project scheduling, queuing theory, decision analysis, dynamic and integer programming and computer simulation. Not open to students who have passed MATH 4535, STAT 4535, or STAT 5535.
Credits: 3
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Description: Individual and collective risk theory, distribution theory, ruin theory, stoploss, reinsurance and Monte Carlo methods. Emphasis is on problems in insurance. For prelim preparation, see the prelim study guide.
Offered: Fall
Credits: 3
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Description: Survival models, mathematical graduation, or demography.
Credits: 3
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Description: Credibility theory or advanced theory of interest.
Credits: 3
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Description: An introduction to the standard models of modern financial mathematics including martingales, the binomial asset pricing model, Brownian motion, stochastic integrals, stochastic differential equations, continuous time financial models,
completeness of the financial market, the Black-Scholes formula, the fundamental theorem of finance, American options, and term structure models.
Offered: Spring
Credits: 3
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Description: An introduction to tensor algebra and tensor calculus with applications chosen from the fields of the physical sciences and mathematics.
Credits: 3
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Description: An introduction to tensor algebra and tensor calculus with applications chosen from the fields of the physical sciences and mathematics.
Prerequisites: MATH 5710
Credits: 3
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Description: Vector algebra and vector calculus with particular emphasis on invariance. Classification of vector fields. Solution of the partial differential equations of field theory.
Credits: 3
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Description: Vector algebra and vector calculus with particular emphasis on invariance. Classification of vector fields. Solution of the partial differential equations of field theory.
Prerequisites: MATH 5720
Credits: 3
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Description: Students who have well defined mathematical problems worthy of investigation and advanced reading should submit to the department a semester work plan.
Prerequisites: Instructor consent required.
Credits: 1-6
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Description: Participation in internship and paper describing experiences.
Credits: 1 to 3
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Description: Seminar. Participation and presentation of mathematical papers in joint student faculty seminars. Variable topics
Credits: 1
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Description: Seminar.
Credits: 1
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Description: Seminar.
Prerequisites: MATH 5211
Credits: 1
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Description: Seminar.
Prerequisites: MATH 5260
Credits: 1
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Description: Seminar.
Credits: 1
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Description: Seminar.
Prerequisites: MATH 5310
Credits: 1
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Prerequisites: MATH 5360
Credits: 1
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Description: (Doctoral Level).
Credits: 3
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University of Connecticut | 
