For a list of all courses offered by the Department of Mathematics, click here.
Graduate course options are listed in numerical order, below. The Math 400 courses in this list are undergraduate level courses that may be taken for graduate credit, with an advisor's permission. 500 level courses are offered for the graduate student, though they can be can be taken by an undergraduate with an advisor's permission. "Topics in" courses (e.g., 541) are modeled depending on the interests of students and faculty each year; listed below are only a few examples of the contents of these special courses.
Letters after the name of a course mean:
F = offered each fall
FS = offered every semester
FO = offered each fall in odd-numbered years
FE = offered each fall in even-numbered years
S= offered each spring
SO = offered each spring in odd-numbered years
SE = offered each spring in even-numbered years
Any course not marked in this way is offered only irregularly.
An asterisk (*) before the name of a course indicates that it is statistics-related.
List of Courses:
- Mathematics 407 An Introduction to Differential Geometry SO
A study of properties of curves and surfaces in 3-dimensional Euclidean space. The course is essentially a modern recounting of a seminal paper of Gauss. Prerequisites: Math 233 and Math 309.
- * Mathematics 408 Nonparametric Statistics SO
Statistical methods that make few or no assumptions about the data distribution. Permutation tests of different types; nonparametric confidence intervals and correlation coefficients; jackknife and bootstrap resampling; nonparametric regressions. If there is time, topics chosen from density estimation and kernel regression. Short computer programs will be written in a language like R or C. Prerequisite: Math 3200 and Math 493, or permission of instructor.
- Mathematics 410 Introduction to Fourier Series and Integrals
The basic theory of Fourier series and Fourier integrals including different types of convergence. Applications to certain differential equations. Prerequisites: Math 233 and 309.
- Mathematics 4111 Introduction to Analysis F
The real number system and the least upper-bound property; metric spaces (completeness, compactness and connectedness); continuous functions (in R^n; on compact spaces; on connected spaces); C(X) (pointwise and uniform convergence; Weierstrass approximation theorem); differentiation (mean value theorem; Taylor’s theorem); the contraction mapping theorem; the inverse and implicit function theorems. Prerequisite: Math 310 or permission or instructor.
- Mathematics 4121 Introduction to Lebesgue Integration S
Riemann integration; measurable functions; measures; Lebesgue measure; the Lebesgue integral; integrable functions; L^p spaces; modes of convergence; decomposition of measures; product measures. Prerequisite: Math 4111 or permission of the instructor.
- Mathematics 415 Partial Differential Equations FE
An introduction to the theory of P.D.E.'s and their solution. Elliptic, parabolic, and hyperbolic equations. Heat and wave equations. Separation of variables, Fourier series, Bessel functions, orthogonal polynomials. Numerical methods: relaxation, method of characteristics, finite elements, finite differences. Prerequisite: Math 217 and 309 or permission of instructor.
- Mathematics 416 Complex Variables FO
Analytic functions, line integrals, the Cauchy integral formula, power series, residues, poles, conformal mapping, and applications. Prerequisite: ESE 317 or ESE 318 (formerly SSM 317), Math 318, or Math 411.
- Mathematics 4171 Topology I (formerly Math 417) F
An introduction to set theory, metric spaces, and general topology. Connections to tools useful in analysis are made as appropriate. Prerequisite: Math 4111 or permission of instructor.
- Mathematics 4181 Topology II (formerly Math 418) SE
A continuation of Math 4171, covering selected topics in topology. Prereq: Math 4171.
- * Mathematics 420 Experimental Design SE
Learn how to design an experimental study, carry out an appropriate statistical analysis of the data, and properly interpret and communicate the analyses.
- Mathematics 429 Linear Algebra F
Introduction to the linear algebra of finite-dimensional vector spaces. Includes systems of equations, matrices, determinants, inner product spaces, spectral theory. Prerequisite: Math 310 or permission of instructor. Math 309 is not an explicit prerequisite but students should already be familiar with such basic topics from matrix theory as matrix operations, linear systems, row reduction and Gaussian elimination. Material on these topics in early chapters of the text will be covered very quickly.
- Mathematics 430 Modern Algebra S
An introduction to groups, rings, and fields. Includes permutation groups, group and ring homomorphisms, field extensions, connections with linear algebra. Prereq: Math 429.
- Math 4301 (former 425C) (POLI SCI 584C) Multilevel Models in Quantitative Research F
This course covers statistical model development with explicitly defined hierarchies. Such multilevel specifications allow researchers to account for different structures in the data and provide for the modeling of variation between defined groups. The course begins with simple nested linear models and proceeds on to non-nested models, multilevel models with dichotomous outcomes, and multilevel generalized linear models. In each case, a Bayesian perspective on inference and computation is featured. The focus on the course will be practical steps for specifying, fitting, and checking multilevel models with much time spent on the details of computation in the R and Bugs environments. PREREQ: Math 2200, Math 3200, Poli Sci 582, or equivalent.
- * Mathematics 434 Survival Analysis FO
Life table analysis and testing, mortality and failure rates, Kaplan-Meier or product-limit estimators, hypothesis testing and estimation in the presence of random arrivals and departures, and the Cox proportional hazards model. Techniques of survival analysis are used in medical research, industrial planning and the insurance industry. Prerequisites: Math 309 and 3200, or permission of the instructor.
- Math 4351 Number Theory and Cryptography
The course will cover many of the basics of elementary number theory, providing a base from which to approach modern algebra, algebraic number theory and analytic number theory. It will also introduce one of the most important real-world applications of mathematics, namely the use of number theory and algebraic geometry in public key cryptography. Topics from number theory involve divisibility (Euclidean algorithm, primes, Fundamental Theorem of Arithmetic), congruences (modular arithmetic, Chinese Remainder Theorem, primality testing and factorization). Topics from cryptography will include RSA encryption, Diffie-Hellman key exchange and elliptic curve cryptography. Topics about algebraic numbers may be include if time permits. Prerequisites: Math 233, 309 and 310 (or permission of instructor).
- Mathematics 436 Algebraic Geometry
Introduction to affine and projective algebraic varieties; the Zariski topology; regular and rational mappings; simple and singular points; divisors and differential forms; genus; the Riemann-Roch theorem. Prerequisites: Math 318, 429 and 430, or permission of the instructor.
- Mathematics 437 An Introduction to Algebraic Topology
- * Mathematics 439 Linear Statistical Models FE
Review of matrix theory relevant for the course; multivariate data; multivariate normal distributions; (univariate) multiple linear regressions; multivariate two-sample and multiple-sample tests (Hotelling's T2 test and MANOVAs); principal component analysis; linear discriminant models; multivariate linear regressions; canonical correlations; other topics as time permits. The computer package SAS will be used for examples and homework problems. Prerequisite: Math3200 or equivalent.
- * Math 4392 Advanced Linear Statistical Models SO
Review of basic linear models relevant for the course; generalized linear models including logistic and Poisson regression (heterogeneous variance structure, quasilikelihood); linear mixed-effects models (estimation of variance components, maximum likelihood estimation, restricted maximum likelihood, generalized estimating equations), generalized linear mixed-effects models for discrete data, models for longitudinal data, optional multivariate models as time permits. The computer software R will be used for examples and homework problems. Implementation in SAS will be mentioned for several specialized models. Prerequisites: Math 439 and a course in linear algebra (such as Math 309 or 429), or consent of instructor.
- * Math 459 Bayesian Statistics S
Introduces the Bayesian approach to statistical inference for data analysis in a variety of applications. The topics include: comparison of Bayesian and frequentist methods, Bayesian model specification, choice of priors, computational methods such as rejection sampling, and stochastic simulation (Markov chain Monte Carlo), empirical Bayes method, hands-on Bayesian data analysis using appropriate software. Prerequisite: Math 493 and either Math 3200 or 494; or permission of the instructor. Some programming experience may also be helpful (consult with the instructor).
- * Mathematics 475 Statistical Computation F
Applied statistics using SAS. An introduction to SAS and SAS programming; contingency tables and Mantel-Haenszel tests; general linear models and matrix operations; simple, multilinear, and stepwise regressions; ANOVAs with nested and crossed interactions; ANOVAs and regressions with vector-valued data (MANOVAs). Topics chosen from discriminant analysis, principal components analysis, logistic regression, survival analysis, and generalized linear models. Prior acquaintance with SAS at the level introduced in Math 3200 is assumed. Prerequisites: Math 3200 and 493 (or 493 concurrently), or permission of instructor.
- Mathematics 481 Group Representations
Ideas and techniques in representation theory of finite groups and Lie groups.
- Mathematics 493 Probability F
Mathematical theory and application of probability at the advanced undergraduate level; a calculus based introduction to probability theory. Topics include the computational basics of probability theory, combinatorial methods, conditional probability including Bayes´ theorem, random variables and distributions, expectations and moments, the classical distributions, and the central limit theorem. Prereq: Math 318 or 308. Note: Math 3200 is frequently also taken before Math 493, and Math 3200 is a prerequisite to continue in the spring into Math 494.
- * Mathematics 494 Mathematical Statistics S
Theory of estimation, minimum variance and unbiased estimators, maximum likelihood theory, Bayesian estimation, prior and posterior distributions, confidence intervals for general estimators, standard estimators and distributions such as the Student-t and F-distribution from a more advanced viewpoint, hypothesis testing, the Neymann-Pearson Lemma (about best possible tests), linear models, and other topics as time permits. Prereq: Math 3200 and Math 493.
- Mathematics 495 Stochastic Processes SO
Content varies with each offering of the course. Past offerings have included such topics as random walks, Markov chains, Gaussian processes, empirical processes, Markov jump processes, and a short introduction to martingales, Brownian motion and stochastic integrals. Prerequisites: Math 318 and 493, or permission of instructor.
- * Mathematics 496 Topics in Statistics
Topic varies with each offering.
- Mathematics 496A Topics in Algebra
The topic varies with each offering. Past titles include: Topics in Algebra: Matrix Groups.
- Mathematics 499 Study for Honors FS
Prerequisite: Senior standing, a distinguished performance in upper level mathematics courses, and permission of the Chair of the Undergraduate Committee.
- Mathematics 500 Independent Work FS
Register for section corresponding to supervising instructor. Credit variable, max 6 units. Prerequisites: Senior standing and permission of the instructor.
- Math 501C Methods of Theoretical Physics I (=L31 Physics 501) F
The first part of a two-semester course reviewing the mathematical methods essential for the study of physics. Theory of functions of a complex variable, residue theory; review of ordinary differential equations; introduction to partial differential equations; integral transforms. Prerequisite: undergraduate differential equations (Math 217), or permission of instructor.
- Math 502C Methods of Theoretical Physics II (=L31 Physics 502) S
Continuation of Phys 501. Introduction to function spaces; self-adjoint and unitary operators; eigenvalue problems, partial differential equations, special functions; integral equations; introduction to group theory. Prerequisite: Phys 501, or permission of instructor.
- Mathematics 5021 Complex Analysis I F
An intensive course in complex analysis at the introductory graduate level. Math 5021-5022 form the basis for the Ph.D. qualifying exam in complex variables. Prerequisite: Math 4111, 4171 and 418, or permission of the instructor.
- Mathematics 5022 Complex Analysis II S
Continuation of math 5021. Normal families. Riemann mapping theorem. Poisson integral. Jensen's theorem. Analytic continuation. Univalent functions. Integral and meromorphic functions. Elliptic functions. Prerequisite: Math 5021, or permission of the Department.
- Mathematics 5031 Algebra I F
An introductory graduate level course on the basic structures and methods of algebra. Detailed survey of group theory including the Sylow theorems and the structure of finitely generated Abelian groups, followed by a study of basic ring theory and the Galois theory of fields. Math 5031-5032 form the basis for the Ph.D. qualifying exam in algebra. Prerequisite: Math 430 or the equivalent, or permission of the instructor
- Mathematics 5032 Algebra II S
Continuation of Math 5031. Multilinear Algebra: Matrices, bilinear forms, tensor product, symmetric algebra, exterior algebra, semi-simple rings, Wedderburn's theorem, Representation of finite groups, Clifford Algebra. Homological Algebra and Category theory: Categories, Functors, Adjoint functors, Injective modules, Complexes, Derived functors, Koszul complex. Prerequisite: Math 5031, or permission of the Department.
- Mathematics 5041 Geometry I F
An introductory graduate level course including differential calculus in n-space; differentiable manifolds; vector fields and flows; differential forms and calculus on manifolds; elements of Lie groups and Lie algebras; Frobenius theorem; elements of Riemannian geometry. Math 5041and 5042 or 5043 form the basis for the Ph.D. qualifying exam in geometry / topology. Prerequisites: Math 4121, 429, and 418, or permission of the instructor.
- Mathematics 5042 Geometry II SE
Continuation of Math 5041. Tensor fields and their behavior under mappings. Tensor product and exterior product. The calculus of differential forms. Stokes' Theorem for manifolds with boundary. Covariant differentiation on Riemannian manifolds, parallel transport, geodesics. Examples from classical differential geometry and Lie groups. Riemannian curvature and symmetric spaces. Further topics from the geometry of Riemannian manifolds and Lie Groups. Prerequisite: Math 5041. Math 5042 and Math 5043 are offered in alternate spring semesters as a sequel to Math 5041.
- Mathematics 5043 Algebraic Topology SO
Algebraic topology, including homology, cohomology, the fundamental group, and covering spaces. Math 5043 and Math 5042 are offered in alternate spring semesters as a sequel to Math 5041. Prerequisites: Math 5041 and a course in abstract algebra. Math 5042 and Math 5043 are offered in alternate spring semesters as a sequel to Math 5041.
- Mathematics 5051 Measure Theory and Functional Analysis I F
An introductory graduate level course including the theory of integration in abstract and Euclidean spaces, and an introduction to the basic ideas of functional analysis. Math 5051-5052 form the basis for the Ph.D. qualifying exam in analysis. Math 4111, 4171, and 418, or permission of the instructor.
- Mathematics 5052 Measure Theory and Functional Analysis II S
Continuation of Math 5051. Topological groups and Haar measure. Topological spaces. Hahn-Banach theorem. Weak topologies. The closed graph theorem. The Banach--Steinhaus theorem. Locally convex spaces and the Krein--Milman theorem. Measures on locally compact spaces. Prerequisite: Math 5051.
- * Mathematics 5061: Theory of Statistics I F
An introductory graduate level course. Probability spaces; derivation and transformation of probability distributions; generating functions and characteristic functions; law of large numbers, central limit theorem; exponential family; sufficiency, uniformly minimum variance unbiased estimators, Rao-Blackwell theorem, information inequality; maximum likelihood estimation; estimating equation; Bayesian estimation; minimax estimation; basics of decision theory. Prerequisite: Math 493 or equivalent.
- * Mathematics 5062: Theory of Statistics II S
Continuation of Math 5061. Bayes estimates, minimaxity, admissibility; maximum likelihood estimation, consistency, asymptotic efficiency; confidence regions; Neyman-Pearson theory of hypothesis testing, uniformly most powerful tests; likelihood ratio tests and large-sample approximation; decision theory. Prereq: Math 5061 or permission of instructor.
- Mathematics 513-514 Theory of Differential Equations I, II
- Mathematics 515-516 Partial Differential Equations
A mathematically rigorous study of partial differental equations. Existence, uniqueness, and regularity for solutions of partial differential equations. Elliptic, parabolic, and hyperbolic equations. Boundary value problems. Distributions. Pseudodifferential operators and Fourier integral operators. Functional analysis techniques. Fixed point theorems. Double and single layer potentials. Prerequisites: Math 5022 and 5052, or permission of instructor.
- Mathematics 517-518 Hilbert Spaces I, II
Operators on Hilberts spaces. Hardy and Bergman spaces. Spectral theorem. Toeplitz operators. C*-algebras and von Neumann algebras. Ext and K-theory
- Mathematics 519-520 Topics in Harmonic Analysis
The topics covered here vary from year to year. We treat the Fourier transform, convergence techniques, Hardy spaces, singular integrals, pseudodifferential operators, interpolation theory, domains with Lipschitz boundary, Ap weights, Lusin integrals,and other modern topics. Discrete Fourier transforms: "fast" factored implementations, real-valued versions such as Hartley, sine, and cosine transforms, and implementation issues such as accuracy and memory requirements. Local trigonometric bases: existence and construction or orthonormal bases for L2(R) consisting of compactly-supported smooth functions. Discrete local trigonometric transforms and their implementations. Discrete wavelet transforms: multiresolution analysis, conjugate quadrature filters, Mallat's algorithm, Daubechies' filters. Phase shifts, frequency localization, and periodization. The discrete wavelet transform and its inverse. Implementation issues such as memory requirements and boundary artifacts. Discrete wavelet packets: definitions, frequency localization, libraries of orthonormal bases, information cost functions, and the best-basis transform. Separable mulitdimensional wavelet packets, basis labeling, anisotropy, and algorithmic complexity. Time-frequency analysis: the idealized time-frequency plane, graph tilings and arbitrary tilings. Phase shifts and deviation from linear phase, and implementations with wavelet packets and local trigonometric functions. Applications: Transform coding image compression, fast approximate factor analysis, speech analysis and recognition, statistical "de-noising", and functional calculi. Prerequisites: We shall refer to theorems from advanced calculus at the level of Math 411 and Math 412, and linear algebra at the level of Math 309. Familiarity with a computer programming language is recommended.
- Mathematics 521-522 Topics in Complex Variables
The topic varies with each offering of the course. Past titles include: Dirichlet Series and Conformal Mapping in the Plane. Prerequisites: check with instructor.
- Mathematics 523 Topics in Analysis
The topic varies with each offering of the course. Past topics include: Dirichlet Space, Bounded Analytic Functions, C* Algebras and Quantum Field Theory, Wavelets, and Introduction to Ergodic Theory. Prerequisites: check with instructor.
- * Mathematics 523C Information Theory and Statistics (ESE 523) FO
Discrete source and channel model, definition of information rate and channel capacity, coding theorems for sources and channels, encoding and decoding of data for transmission over noisy channels. Corequisite: ESE 520.
- Mathematics 524-525 Measure Theory and Related Topics I, II
- Mathematics 527-528 Topics in Functional Analysis
The focus varies with each offering of the course. Past titles include: C* Algebras and von Neumann Algebras, Lipschitz Algebras. Also included: Pick Interpolation: Interpolation by bounded analytic functions on the disk and bidisk; Operator theory aspects of Nevanlinna-Pick interpolation.
- Mathematics 531-532 Theory of Algebraic Numbers I, II
- * Mathematics 533 Mathematical Statistics I
Semiparametric Statistics: Semiparametric estimation efficiency; examples of semiparametric models such as Cox model and Proportional odds model. Prerequisites: Math 5051 (measure theory) and Math 493 (or other equivalent probability course), or permission of the instructor.
- * Mathematics 534 Mathematical Statistics II
- Mathematics 535-536 Topics in Combinatorics
The topic varies with each offering of the course. Past titles include: The Charney-Davis Conjecture, Geometric Combinatorics, Spectral Graph Theory, Algebraic Combinatorics (as L24 571 in Fall 2009), Algebraic and Topological Combinatorics, and Theory of Groups I, II.
- Mathematics 537-538 Topics in Algebra
The focus varies with each offering of the course. Past topics include: Algebraic Groups, Representation Theory, Complex Abelian Varieties and Theta Functions, Polynomial Automorphisms, and Commutative Algebra.
- Mathematics 539 Topics in Algebraic Geometry I
Topics vary with each offering of the course. Currently: Hodge Theory. An introduction to transcendental methods in algebraic geometry. Topics will include: Kaehler geometry and the Hodge theorem; Hodge structure of projective hypersurfaces; variations of Hodge structure, the period mapping, and mirror symmetry; algebraic cycles, normal functions, and the Hodge Conjecture. Prerequisites: Math 5022, Math 5032, and Math 5042, or permission of the instructor. Past tiles for this class include: Vector Bundles and Vanishing Theorems and Higher Dimensional Geometry.
- Mathematics 5392 Topics in Algebraic Geometry II
This course is may be offered as a sequel to Math 539, the second course in a two-semester sequence in algebraic geometry. The topics which will be covered vary. Prerequisite: Math 539 or permission of the instructor.
- Mathematics 540C Advanced Algorithms
Provides a broad coverage of fundamental algorithm design techniques with the focus on developing efficient algorithms for solving combinatorial and optimization problems. The topics covered include: greedy algorithms, dynamic programming, linear programming, NP-completeness, approximation algorithms, lower bound techniques, and on-line algorithms. Throughout this course there is an emphasis on correctness proofs and the ability to apply the techniques taught to design efficient algorithms for problems from a wide variety of application areas. Prerequisites: CSE 240/CS 201 and CSE 241.
- Mathematics 541 Topics in Applied Mathematics
Topic and prerequisites vary with each offering of the course. Past titles for this class include: Wavelet Algorithms.
- Mathematics 542C Algorithms for Nonlinear Optimization
The course provides an in-depth coverage of modern algorithms for the numerical solution of multidimensional optimization problems. Unconstrained optimization techniques including Gradient methods, Newton´s methods, Quasi-Newton methods, and conjugate methods will be introduced. The emphasis is on constrained optimization techniques: Lagrange theory, Lagrangian methods, penalty methods, sequential quadratic programming, primal-dual methods, duality theory, nondifferentiable dual methods, and decomposition methods. The course will also discuss applications in engineering systems and use of state-of-the-art computer codes. Special topics may include large-scale systems, parallel optimization, and convex optimization.
- Mathematics 543-544 Geometry and Manifold Theory F, Topics in Differential Geometry and Manifold Theory S
This course covers sheaf theory, complex vector bundles, Chern classes, elliptic operator theory, Hodge theory on compact complex manifolds, Hodge-Riemann bilenar relations on Kaehler manifolds, Kodaira-Nakano vanishing theorem, and Kodaira embedding theorem. The focus varies with each offering of the course. Past titles include: The Geometry of Physics, Geometry and Manifold Theory: Hyperbolic Geometry, Geometry and Manifold Theory: Symplectic Geometry,Geometry and Manifold Theory: Complex Manifolds, Geometry and Manifold Theory: Method of Moving Frames,Geometry and Manifold Theory: Classical Mechanics on Manifolds.
- Mathematics 545-546 Topics in Riemannian Geometry
A first course on Riemannian geometry, nonpositive curvature in geometry and group theory. The focus varies with each offering of the course. Past titles include: Topics in Riemannian Geometry: Geometry of Physics, Topics in Riemannian Geometry: deRham-Hodge, Discrete Subgroups of Lie Groups, Geometry and Probability: An introduction to stochastic calculus on manifolds.
- Math 547 Topics in Geometry S
- Mathematics 551 Advanced Probability I
Probability theory based on measure theory. Strong law of large numbers, central limit theorem, martingales, applications of martingales, Markov processes, Brownian motion. Prerequisites: Math 5051 (Measure Theory) or permission of the instructor.
- Mathematics 552 Advanced Probability II
A possible sequel to Mathematics 551 that is usually suplanted by Mathematics 553 or 554.
- *Mathematics 553-554 Topics in Advanced Probability
Topics vary with each offering of the course. Past titles include: Brownian Motion on Riemannian Manifolds andStochastic Calculus.
- Mathematics 560 Topics in Topology
Topics vary with each offering of the course. Current title: 4-Manifolds.
- Mathematics 561-562 Topics in General Topology
Topics vary with each offering of the course. Past titles include: Bass-Serre Theory.
- Mathematics 563-564 Topics in Lie Groups and Lie Algebras
Topics vary with each offering of the course. Past titles include: Introduction to Lie Algebras and Lie Groups.
- Mathematics 565-566 Algebraic Topology I, II
- Mathematics 567-568 Several Complex Variables I, II
This is an introduction to the function theory of several complex variables. Comparisons and contrasts are drawn with the one-variable theory. The idea of domain of holomorphy, pseudoconvexity, and related ideas is developed. The Levi problem is studied. The Cauchy-Riemann equations, sheaves, and other tools for solving the Levi problem are treated. Zeros of holomorphic functions, holomorphic mappings, invariant metrics, and other more advanced topics are treated as time permits. Prerequisites: Math 5021-5022, or permission of instructor.
- Mathematics 569-570 Topics in Homological Algebra
- Mathematics 572 Topics in Set Theory and Logic
Topics vary with each offering of the course. Past titles include: Forcing and Independence.
- Mathematics 581 Introduction to Computational Analysis
This course will develop methods of numerical analysis and signal processing with origins in classical harmonic analysis. Though applied to quantized data, these methods take advantage of properties of the underlying continuous functions.
- Mathematics 582 Advanced Methods of Computational Analysis
- Mathematics 590 Research FS
Contact department directly for details on faculty/sections and enrollment. Credit variable, max 3 units.
- Mathematics 595-596 Seminar
Register for section corresponding to supervising instructor. Credit variable, max 3 units.
- Mathematics 597 Teaching Seminar
Techniques of mathematical instruction for prospective teaching assistants. Prerequisites: Permission of Instructor