Mathematics (MATH)

The terms indicated are expected but are not guaranteed. For the courses offered during any given term, consult the Schedule of Classes.

040x Basic Mathematical Skills (4, FaSp) Intensive review of arithmetic and algebra. Not available for degree credit. Graded CR/NC.

100x Mathematics: The Third Culture (4) Current applications of mathematical concepts; selections from algebra, geometry, combinatorics, logic, set theory, probability, statistics, calculus, and mathematical programming. Not available for major credit to science majors. Prerequisite: MATH 040x or placement exam.

108 Introductory College Mathematics (4, FaSpSm) Equations and inequalities; systems of linear equations; functions; graphs; exponential, logarithmic, and trigonometric functions; polynomial and rational functions; analytic geometry. Prerequisite: MATH 040x or placement exam.

116 Mathematics for the Social Sciences I (4, FaSp) Finite mathematics with application to the social sciences; elementary set theory and logic; counting techniques; probability; statistics; matrices and systems of linear equations. Selected topics. Prerequisite: MATH 040x or placement exam.

117 Introduction to Mathematics for Business and Economics (4, FaSpSm) Functions, graphs, polynomial and rational functions, exponential and logarithmic functions, matrices, systems of linear equations, elementary probability. Prerequisite: MATH 040x or skill level exam in Math.

118x Fundamental Principles of the Calculus (4, FaSpSm) Derivatives; extrema. Definite integral; fundamental theorem of the calculus. Functions of several variables; partial derivatives; multiple integrals; Lagrange multipliers. Not available for credit toward a degree in mathematics. Prerequisite: MATH 117 or placement exam.

125 Calculus I (4, FaSpSm) Limits; continuity, derivatives and applications; antiderivatives; the fundamental theorem of calculus; exponential and logarithmic functions. Prerequisite: MATH 108 or placement exam.

126 Calculus II (4, FaSpSm) A continuation of MATH 125: trigonometric functions; applications of integration; techniques of integration; indeterminate forms; infinite series; Taylor series; polar coordinates. Prerequisite: MATH 125.

190 Accelerated Math Tutorial (2, FaSp) Supervised individual studies in advanced topics from real analysis, modern algebra, and multi-variable calculus. Intended for students in the Accelerated Math Program only.

200 Elementary Mathematics from an Advanced Standpoint (4, FaSp) An explication of arithmetic and geometry, including the algebraic operations, number bases, plane and solid figures; and coordinate geometry. Prerequisite: MATH 040x or placement exam.

208x Elementary Probability and Statistics (4, Fa) Descriptive statistics, probability concepts, discrete and continuous random variables, mathematical expectation and variance, probability sampling, Central Limit Theorem, estimation and hypothesis testing, correlation and regression. Not available for major credit to mathematics majors. Prerequisite: MATH 118x or MATH 125.

216 Mathematics for the Social Sciences II (4) Continuation of MATH 116. Further topics in probability, elementary functions, graphing, and analytic geometry; elementary calculus; derivatives, integrals of simple functions.

218 Probability for Business (4, FaSpSm) Combinatorial probability, discrete and continuous distributions, expectation and variance, joint distributions, independence. Models for random phenomena, foundations of simulation and stochastic modeling. Prerequisite: MATH 118x.

225 Linear Algebra and Linear Differential Equations (4, FaSpSm) Matrices, systems of linear equations, vector spaces, linear transformations, eigenvalues, systems of linear differential equations. Prerequisite: MATH 126.

226 Calculus III (4, FaSpSm) A continuation of MATH 126; vectors, vector valued functions; differential and integral calculus of functions of several variables; Green's theorem. Prerequisite: MATH 126.

245 Mathematics of Physics and Engineering I (4, FaSpSm) First-order differential equations; second-order linear differential equations; determinants and matrices; systems of linear differential equations; Laplace transforms. Prerequisite: MATH 226.

270 Proof Techniques and Mathematical Structures (4, Fa) Proof types and techniques, statement calculus and axioms, number systems, basic set theory, order and equivalence relations, functions, cardinality, algebraic structures.

390 Special Problems (1-4) Supervised, individual studies. No more than one registration permitted. Enrollment by petition only.

406 Probability and Statistics for Secondary Teachers (4) Descriptive statistics, analysis of sample data, statistical inference, probability, distributions and applications in the physical, biological, and social sciences. Credit applicable only to the M.A. (teacher's option).

407 Probability Theory (4, FaSp) Probability spaces, discrete and continuous distributions, moments, characteristic functions, sequences of random variables, laws of large numbers, central limit theorem, special probability laws. Prerequisite: MATH 226.

408 Mathematical Statistics (4, Sp) Principles for testing hypotheses and estimation, small sample distributions, correlation and regression, nonparametric methods, elements of statistical decision theory. Prerequisite: MATH 407.

410 Fundamental Concepts of Modern Algebra (4, FaSp) Sets; relations; groups; homomorphisms; symmetric groups; Abelian groups; Sylow's theorems; introduction to rings and fields. Prerequisite: MATH 225.

425ab Fundamental Concepts of Analysis (a: 4, FaSpSm; b: 4, Sp) a: The real number system, limits, continuity, derivatives and integrals, infinite series. b: Implicit function theorems, Jacobians, transformations, multiple integrals, line integrals. Prerequisite: MATH 226; MATH 425a before MATH 425b.

430 Theory of Numbers (4, Fa) Introduction to the theory of numbers, including prime factorization, congruences, primitive roots, N-th power residues, number theoretic functions, and certain diophantine equations. Prerequisite: MATH 126.

432 Applied Combinatorics (4, Sp) Mathematical induction, counting principles, arrangements, selections, binomial coefficients, generating functions, recurrence relations, inclusion-exclusion, symmetric groups, graphs, Euler and Hamiltonian circuits, trees, graph algorithms; applications. Prerequisite: MATH 225 or MATH 226 or departmental approval.

434 Geometry and Transformations (4, Fa) Incidence and separation properties of planes and spaces. Geometric inequalities, models of Riemannian and hyperbolic geometry. Isometrics, Jordan measure, constructions, and affine geometry.

435 Vector Analysis and Introduction to Differential Geometry (4, Sp) Vectors, elements of vector analysis, applications to curves and surfaces, standard material of differential geometry. Prerequisite: MATH 226.

440 Topology (4, Fa) Cardinals, topologies, separation axioms. Compactness, metrizability, function spaces; completeness; Jordan curve theorem. Recommended preparation: upper division MATH course.

445 Mathematics of Physics and Engineering II (4, FaSpSm) Vector field theory; theorems of Gauss, Green, and Stokes; Fourier series and integrals; complex variables; linear partial differential equations; series solutions of ordinary differential equations. Prerequisite: MATH 245.

447 Variational Methods (4) Euler-Lagrange equations, Hamilton's principle, eigenvalue problems in linear algebra and differential equations, Rayleigh-Ritz and finite-element methods; examples from geometry, physics, and numerical analysis. Prerequisite: MATH 225 or MATH 245 or departmental approval.

450 History of Mathematics (4, Fa) Evolution of mathematical ideas and techniques as seen through a study of the contributions of eminent mathematicians to the formulation and solution of celebrated problems. Prerequisite: MATH 225 or MATH 245; recommended preparation: upper division MATH course.

458 Numerical Methods (4, Sp) Rounding errors in digital computation; solution of linear algebraic systems; Newton's method for nonlinear systems; matrix eigenvalues; polynomial approximation; numerical integration; numerical solution of ordinary differential equations. Prerequisite: MATH 225 or MATH 245.

460 Introduction to Mathematical Logic (4) Development of the first order predicate calculus. Fundamental metamathematical notions.

461 Linear Inequalities and Linear Programming (4) Linear equations and inequalities; the simplex method and variations; linear and integer programming problems; flows in networks. Prerequisite: MATH 225 or MATH 245.

462 Axiomatic Set Theory (4) Detailed development of general set theory via the Zermelo-Fraenkel axiom system. Special attention to the role of the axiom of choice. Prerequisite: MATH 460.

465 Ordinary Differential Equations (4, Sp) Linear systems, phase plane analysis, existence and uniqueness, sta-bility of linear and almost linear systems, Lyapunov's method, nonlinear oscillations, flows, invariant surfaces, and bifurcation. Prerequisite: MATH 225 or MATH 245.

466 Dynamic Modeling (4, Fa) Formulation and study of models arising in population dynamics, growth of plankton, pollution in rivers, highway traffic, morphogenesis and tidal dynamics: stability, oscillations, bifurcations, chaos. The lab will consist of computer simulation of models using commercially available software. Prerequisite: MATH 225 or MATH 245 or departmental approval.

471 Topics in Linear Algebra (4, Sp) Polynomial rings, vector spaces, linear transformations, canonical forms, inner product spaces. Prerequisite: MATH 225; recommended preparation: MATH 410.

475 Introduction to Theory of Complex Variables (4, Sp) Limits and infinite series; line integrals; conformal mapping; single-valued functions of a complex variable; applications. Primarily for advanced students in engineering. Prerequisite: MATH 226.

490x Directed Research (2-8, max 8, FaSpSm) Individual research and readings. Not available for graduate credit. Prerequisite: departmental approval.

499 Special Topics (2-4, max 8) Lectures on advanced material not covered in regularly scheduled courses. No more than two registrations allowed. Prerequisite: departmental approval.

501 Numerical Analysis and Computation (3, FaSp) Linear equations and matrices, Gauss elimination, error estimates, iteration techniques; contrac-tive mappings, Newton's method; matrix eigenvalue problems; least-squares approximation, Newton-Cotes and Gaussian quadratures; finite difference methods. Prerequisite: linear algebra and calculus.

502ab Numerical Analysis (a: 3, Fa; b: 3, Sp) Computational linear algebra; solution of general nonlinear systems of equations; approximation theory using functional analysis; numerical solution of ordinary and partial differential equations. Prerequisite: MATH 425a and MATH 471.

504ab Numerical Solution of Ordinary and Partial Differential Equations (a: 3, Sp; b: 3, Fa) a: Initial value problems; multistep methods, stability, convergence and error estimation, automatic stepsize control, higher order methods, systems of equations, stiff problems; boundary value problems; eigenproblems. Prerequisite: MATH 501 or MATH 502a or departmental approval. b: Computationally efficient schemes for solving PDE numerically; stability and convergence of difference schemes, method of lines; fast direct and iterative methods for elliptic equations. Prerequisite: MATH 501 or MATH 502a or departmental approval.

505ab Applied Probability (a: 3, Fa; b: 3, Sp) a: Populations, permutations, combinations, random variables, distribution and density functions conditional probability and expectation, binomial, Poisson, and normal distributions; laws of large numbers, central limit theorem. Prerequisite: departmental approval. b: Markov processes in discrete or continuous time; renewal processes; martingales; Brownian motion and diffusion theory; random walks, inventory models, population growth, queueing models, shot noise. Prerequisite: departmental approval.

506 Stochastic Processes (3) Basic concepts of stochastic processes with examples illustrating applications; Markov chains and processes; birth and death processes; detailed treatment of 1-dimensional Brownian motion. Prerequisite: MATH 407.

507ab Theory of Probability (a: 3, Fa; b: 3, Sp) a: Probability spaces; distributions and characteristic functions; laws of large numbers, central limit problems; stable and infinitely divisible laws; conditional distributions. Prerequisite: MATH 525a or MATH 570. b: Dependence, martingales, ergodic theorems, second-order random functions, harmonic analysis, Markov processes.

508 Filtering Theory (3) Theory of random differential equations and stochastic stability; optimum linear and nonlinear filtering, with discussion of asymptotic behavior of filter. Prerequisite: MATH 507a.

509 Stochastic Differential (3) Brownian motion, stochastic integrals, the Ito formula, stochastic differential equations, analysis of diffusion processes, Girsanov transformation, Feynmann-Kac formula, applications. Prerequisite: MATH 505ab or MATH 507ab.

510ab Algebra (a: 3, Fa; b: 3, Sp) a: Group Theory: Isomorphism theorems, group actions, Sylow's theorems, simple and solvable groups; Field Theory: Galois correspondence, radical extensions, algebraic and transcendental extensions, finite fields. b: Commutative Algebra: Integrality, Hilbert Basis theorem, Hilbert Nullstellensatz; Modules: modules over PIDs, chain conditions, tensor products; Noncommutative Rings: Jacobson radical, Artin-Wedderburn theorem, Maschke's theorem. Prerequisite: MATH 410, MATH 471.

511L Data Analysis (4) (Enroll in PM 511L)

520 Complex Analysis (3, Sp) Theory of analytic functions - power series and integral representations, calculus of residues, harmonic functions, normal families, approximation theorems, conformal mapping, analytical continuation. Prerequisite: MATH 425ab.

525ab Real Analysis (a: 3, Fa; b: 3, Sp) a: Measure and integration over abstract measure spaces, Radon-Nikodym theorem, Fubini's theorem, convergence theorems, differentiation. Prerequisite: MATH 425ab. b: Metric spaces, contraction principle, category, Banach spaces, Riesz representation theorem, properties of Lp Hilbert spaces, orthogonal expansions, Fourier series and transforms, convolutions. Prerequisite: MATH 525a.

530 Analytic Number Theory (3, Fa) Additive and multiplicative number theory, exponential and character sums, cyclotomy, the Riemann zeta function, primes in an arithmetic progression, the prime number theorem, sieve methods. Prerequisite: MATH 520.

532 Combinatorial Analysis (3, Fa) Inversion formulas, generating functions and recursions, partitions, Stirling numbers, distinct representatives, Ramsey's theorem, graph theory, block designs, difference sets, finite geometrics, Latin squares, Hadamard matrices.

533 Combinatorial Analysis and Algebra (3, Sp) Advanced group theory; algebraic automata theory; graph theory; topics in combinatorial analysis.

535ab Differential Geometry (a: 3, Fa; b: 3, Sp) Elementary theory of manifolds, Lie groups, homogeneous spaces, fiber bundles and connections. Riemannian manifolds, curvature and conjugate points, second fundamental form, other topics. Prerequisite: MATH 440.

540 Topology (3, Sp) Initial and final topologies, function spaces, algebras in C(Y), homotopy, fundamental group, fiber spaces and bundles, smashes, loop spaces, groups of homotopy classes, cw-complexes. Prerequisite: MATH 440.

541ab Introduction to Mathematical Statistics (a: 3, Sp; b: 3, Fa) a: Exponential families, sufficiency. Estimation: methods of estimation, maximum likelihood, least squares, comparison of estimators, unbiased estimation, optimality, theory, information inequality, asymptotic efficiency, confidence intervals. Prerequisite: MATH 505a or MATH 407 or MATH 408. b: Testing: Neyman-Pearson lemma, consistency, power, linear models, regression, analysis of variance, discrete data, nonparametric methods. Prerequisite: MATH 541a.

542L Analysis of Variance and Design (3, Sp) Least squares estimation in the linear model, analysis of variance and covariance, F-test, multiple comparisons, multiple regression, selection of variables; introduction to experimental design. Includes laboratory. Prerequisite: MATH 225, MATH 226, and MATH 208x.

543L Nonparametric Statistics (3) Distribution-free methods for comparisons of two or more samples, tests of randomness, independence, goodness of fit; classification, regression. Comparison with parametric techniques. Includes laboratory. Prerequisite: MATH 226, MATH 208x.

544L Multivariate Analysis (3) (Enroll in PM 544L)

545L Introduction to Time Series (3, Fa) Transfer function models; stationary, nonstationary processes; moving average, autoregressive models; spectral analysis; estimation of mean, autocorrelation, spectrum; seasonal time series. Includes laboratory. Prerequisite: MATH 225, MATH 226, and MATH 208x.

546 Statistical Computing (3) (Enroll in PM 546)

547 Methods of Statistical Inference (3, Fa) Statistical decision theory: game theory, loss and risk functions; Bayes, minimax, admissible rules; sufficiency, invariance, tests of hypotheses, optimality properties. Inference for stochastic processes. Prerequisite: MATH 407 or MATH 408.

548 Sequential Analysis (3) Sequential decision procedures: sequential probability-ratio tests, operating characteristic, expected sample size, two-stage procedures, optimal stopping, martingales, Markov processes; applications to gambling, industrial inspection. Prerequisite: MATH 407 or MATH 408.

549 Bayesian Statistics (3) Bayes risk, Bayes decision functions, prior, posterior distributions, factorization, conjugate families, likelihood function, improper prior distributions, convergence of posterior distributions; Bayes methods of estimation, testing. Prerequisite: MATH 225, MATH 226 and MATH 208x.

550 Sample Surveys (3, Sp) Theory of sampling and design of sample surveys; bias and precision; finite populations; stratification; cluster sampling; multistage, systematic sampling; non-sampling errors. Prerequisite: MATH 208x.

551L Analysis of Discrete Observations (3, Sp) Standard discrete distributions, probability generating functions, branching processes, birth, death processes; goodness of fit, contingency tables, chi-square, likelihood ratio tests; regression, probit, logit models. Laboratory. Prerequisite: MATH 225, MATH 226 and MATH 208x.

555ab Partial Differential Equations (a: 3, Fa; b: 3, Sp) Second-order partial differential equations of elliptic, parabolic, and hyperbolic type; in particular, potential and wave equations. Prerequisite: MATH 425ab.

565ab Ordinary Differential Equations (a: 3, Fa; b: 3, Sp) Existence, uniqueness and continuation of solutions, differential inequalities, linear systems, Sturm-Liouville theory, boundary value problems, Poincare-Bendixson theory, periodic solutions, perturbations, stability, fixed point techniques. Prerequisite: MATH 425ab.

570ab Methods of Applied Mathematics (a: 3, FaSp; b: 3, Sp) a: Metric spaces, fundamental topological and algebraic concepts, Banach and Hilbert space theory. Prerequisite: MATH 425a or departmental approval. b: Hilbert spaces, normal, self-adjoint and compact operators, geometric and spectral analysis of linear operators, elementary partial differential equations. Prerequisite: MATH 570a.

572 Applied Algebraic Structures (3, Fa) Elementary predicate logic, model theory, axiomatic set theory; relations, functions, equivalences; algebraic and relational structures; graph theory; applications of lattices, Boolean algebras; groups, rings, field. Prerequisite: departmental approval.

574 Applied Matrix Analysis (3, Fa) Equivalence of matrices; Jordon canonical form; functions of matrices; diagonalization; singular value decomposition; applications to linear differential equations, stability theory, and Markov processes. Prerequisite: departmental approval.

576 Applied Complex Analysis and Integral Transforms (3, Fa) Review of basic complex analysis; integral transforms of Laplace, Fourier, Mellin, and Hankel; applications to solutions of ordinary and partial differential equations; Wiener-Hopf technique. Prerequisite: MATH 475 or MATH 520.

578 DNA and Protein Sequence Analysis (3, Sp) Genetic and physical mapping of genomes, restriction mapping of DNA, reconstruction of evolutionary trees from sequence data, algorithms and statistics for sequence comparisons, secondary structure. Prerequisite: MATH 425a and MATH 407, or departmental approval.

580 Introduction to Functional Analysis (3, Fa) Basic functional analysis in Banach and Hilbert spaces. Weak topologies, linear operators, spectral theory, calculus of vector-valued functions. Banach algebras. Prerequisite: MATH 525ab.

585ab Mathematical Theory of Optimal Control (a: 3, Fa; b: 3, Sp) a: Deterministic control: calculus of variations; optimal control; Pontryagin principle; multiplier rules and abstract nonlinear programming; existence and continuity of controls; problem of Mayer; dynamic programming. Prerequisite: MATH 570 and MATH 525a. b: Stochastic control: Markov diffusion processes; backward-forward equations; linear systems; Kalman-Bucy filtering; optimal stochastic control; stochastic dynamic programming; linear regulator problem; separation principle. Prerequisite: MATH 585a and either MATH 505ab or MATH 507ab.

587ab Mathematical Models of Neurons and Neural Networks (a: 3, Fa; b: 3, Sp) a: Dynamics of discrete and analog neural networks; qualitative and numerical analysis; computer simulation; learning algorithms and convergence; Kolmagorov theory of feed-forward networks. Prerequisite: MATH 465 and either MATH 501 or MATH 502a. b: Nernst-Planck and Goldman-Hodgkin-Katz equations; Hodgkin-Huxley theory; cable theory; compartment models of dendritic structures; McCulloch-Pitts networks; perceptron theory. Prerequisite: MATH 587a.

590 Directed Research (1-12, FaSpSm) Research leading to the master's degree. Maximum units which may be applied to the degree to be determined by the department. Graded CR/NC.

594abz Master's Thesis (2-2-0, FaSpSm) Credit on acceptance of thesis. Graded IP/CR/NC.

597 Mathematical Packages and Scientific Computing (3) Use of mathematical software packages in scientific computing (linear algebra, differential equations, etc.): reliability; portability; influence of different computer architectures; techniques for large, sparse systems. Prerequisite: MATH 501 or MATH 502a or departmental approval.

599 Special Topics (2-4, max 8, FaSpSm) Course content will be selected each semester to reflect current trends and developments in the field of mathematics.

600 Topics in Numerical Analysis (3, max 12)

601 Optimization Theory and Techniques (3, SpSm) Necessary and sufficient conditions for existence of extrema with equality constraints; gradient methods; Ritz methods; eigenvalue problems; optimum control problems; inequality constraints; mathematical programming. Prerequisite: MATH 502ab.

602 Galerkin Approximation Methods in Partial Differential Equations (3) Galerkin methods of approximating solutions of elliptic boundary value problems in one and several dimensions; includes the use of spline functions and triangularizations. Prerequisite: departmental approval.

605 Topics in Probability (3, max 12)

610 Topics in Algebra (3, max 12)

611 Homological Algebra (3) Functors, modules, gradings, resolutions, homology, derived functors, spectral sequences, Kunneth theorem, homology and cohomology of groups, bar resolutions, homological dimension, relative homological algebra. Prerequisite: MATH 510ab.

612 Topics in Commutative Ring Theory (3, max 12) Localization, structure of Noetherian rings, integral extensions, valuation theory, graded rings, characteristic functions, local algebra, dimension theory. Prerequisite: MATH 510ab.

613 Topics in Noncommutative Ring Theory (3, max 12) Jacobson radical, nil radical, nil rings and nil-potence, chain conditions, polynomial identity and group rings. Goldie theorems, current research. Prerequisite: MATH 510ab.

620 Topics in Complex Analysis (3, max 12)

621 Riemann Surfaces (3) Topology of Riemann surfaces, Dirichlet problem, existence of analytic functions, function field of a Riemann surface, algebraic curves, uniformization theory, the Riemann-Roch theorem. Prerequisite: MATH 520, MATH 440.

625 Topics in Real Analysis (3, max 12)

626 Linear Integral Equations (3) Equation of Volterra and Fredholm type, adjoint theory and Fredholm alternative, spectral theory and Sturm-Liouville problems, singular equations and Wiener-Hopf theory. Prerequisite: MATH 525ab.

627 Nonlinear Integral Equations (3) Volterra and Fredholm equations of Hammerstein type. Fixed point and implicit function theorems, equations of Urysohn type, dependence on parameters and bifurcation theory. Prerequisite: MATH 525ab.

630 Topics in Number Theory (3, max 12)

635 Topics in Differential Geometry (3, max 12) Topics to be chosen from the following: geometry of complex manifolds, relations between topology and curvature, homogeneous spaces, symmetric spaces, geometry of submanifolds. Prerequisite: MATH 535ab.

640 Algebraic Topology (3) Eilenberg-MacLane spaces, spectra, extraordinary homology and cohomology, homology in fiber spaces, manifolds, duality, characteristic classes, sheaves. Prerequisite: MATH 540.

641 Topics in Topology (3, max 12)

650 Seminar in Statistical Consulting (3)

665 Topics in Ordinary Differential Equations (3, max 12)

677 Mathematical Biosciences (3, max 12) A survey of the types of mathematical problems that arise in biology and medicine with emphasis upon the use of digital and analog computers. Prerequisite: MATH 465.

680 Nonlinear Functional Analysis (3) Calculus in Banach spaces, degree theory, fixed point theorems. Study of compact, monotone, accretive and nonexpansive operators. Prerequisite: MATH 580.

681 Selected Topics in Functional Analysis (3, max 12) Course content will vary with professor and academic year offered. It will include topics of current interest in both linear and nonlinear functional analysis and their applications.

685 Topics in Mathematical Control Theory (3, max 12)

689 Topics in Mathematical Physics (3, max 12)

700 Seminar in Numerical Analysis (3)

705 Seminar in Probability (3)

710 Seminar in Algebra (3)

725 Seminar in Analysis (3)

730 Seminar in Number Theory (3)

735 Seminar in Differential Geometry (3)

740 Seminar in Topology (3)

761 Seminar in Programming and Computability (3)

765 Seminar in Ordinary Differential Equations (3)

780 Seminar in Functional Analysis (3)

790 Research (1-12, FaSpSm) Research leading to the doctorate. Maximum units which may be applied to the degree to be determined by the department. Graded CR/NC.

794abcdz Doctoral Dissertation (2-2-2-2-0, FaSpSm) Credit on acceptance of dissertation. Graded IP/CR/NC.

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