**MAT 106 Intermediate Algebra. (3)**

*fall, spring, summer*

Topics from basic algebra such as linear equations, polynomials, factoring, exponents, roots, and radicals. Prerequisite: 1 year of high school algebra.

**MAT 114 College Mathematics. (3)**

*fall, spring, summer*

Applications of basic college-level mathematics to real-life problems. Appropriate for students whose major does not require MAT 117 or 170. Prerequisite: MAT 106 or 2 years of high school algebra.

*General Studies: MA*

**MAT 117 College Algebra. (3)**

*fall, spring, summer*

Linear and quadratic functions, systems of linear equations, logarithmic and exponential functions, sequences, series, and combinatorics. Prerequisite: MAT 106 or 2 years of high school algebra.

*General Studies: MA*

**MAT 119 Finite Mathematics. (3)**

*fall, spring, summer*

Topics from linear algebra, linear programming, combinatorics, probability, and mathematics of finance. Prerequisite: MAT 117 (or its equivalent).

*General Studies: MA*

**MAT 170 Precalculus. (3)**

*fall, spring, summer*

Intensive preparation for calculus (MAT 260, 270, and 290). Topics include functions (including trigonometric), matrices, polar coordinates, vectors, complex numbers, and mathematical induction. Prerequisite with a grade of “B” or higher: MAT 106. Prerequisite with a grade of “C” or higher: MAT 117 or two years of high school algebra.

*General Studies: MA*

**MAT 210 Brief Calculus. (3)**

*fall, spring, summer*

Differential and integral calculus of elementary functions with applications. Not open to students with credit for MAT 260, 270, or 290. Prerequisite: MAT 117 (or its equivalent).

*General Studies: MA*

**MAT 242 Elementary Linear Algebra. (2)**

*fall, spring, summer*

Introduces matrices, systems of linear equations, determinants, vector spaces, linear transformations, and eigenvalues. Emphasizes development of computational skills. Prerequisite: 1 semester of calculus or instructor approval.

**MAT 243 Discrete Mathematical Structures. (3)**

*fall, spring, summer*

Logic, sets, functions, elementary number theory and combinatorics, recursive algorithms, and mathematical reasoning, including induction. Emphasizes connections to computer science. Prerequisite: 1 semester of calculus or computer programming.

**MAT 251 Calculus for Life Sciences. (3)**

*fall and spring*

Differential and integral calculus of elementary functions. Introduces differential and difference equations. Emphasizes applications to the life sciences. Not open to students with credit for MAT 210, 260, or 270. Prerequisite: MAT 170 (or its equivalent).

*General Studies: MA*

**MAT 260 Technical Calculus I. (3)**

*selected semesters*

Analytic geometry, differential, and integral calculus of elementary functions, emphasizing physical interpretation and problem solving. Not open to students with credit for MAT 210, 270, or 290. Prerequisite: MAT 170 (or its equivalent).

*General Studies: MA*

**MAT 261 Technical Calculus II. (3)**

*selected semesters*

Continuation of MAT 260. Prerequisite: MAT 260 or instructor approval.

*General Studies: MA*

**MAT 262 Technical Calculus III. (3)**

*selected semesters*

Infinite series, an introduction to differential equations and elementary linear algebra. Prerequisite: MAT 261 (or its equivalent).

*General Studies: MA*

**MAT 270 Calculus with Analytic Geometry I. (4)**

*fall, spring, summer*

Real numbers, limits and continuity, and differential and integral calculus of functions of 1 variable. Not open to students with credit for MAT 290. The sequence MAT 270 and 271 may be substituted for MAT 290 to satisfy requirements of any curriculum. Fee. Prerequisite with a grade of “C” or higher: MAT 170 or satisfactory score on placement examination.

*General Studies: MA*

**MAT 271 Calculus with Analytic Geometry II. (4)**

*fall, spring, summer*

Methods of integration, applications of calculus, elements of analytic geometry, improper integrals, sequences, and series. Not open to students with credit for MAT 291. The sequence MAT 270, 271, 272 may be substituted to satisfy requirements for MAT 290 and 291. Fee. Prerequisite with a grade of “C” or higher: MAT 270 (or its equivalent).

*General Studies: MA*

**MAT 272 Calculus with Analytic Geometry III. (4)**

*fall, spring, summer*

Vector-valued functions of several variables, multiple integration, and introduction to vector analysis. The sequence MAT 270, 271, 272 may be substituted to satisfy requirements for MAT 290 and 291. Fee. Prerequisite with a grade of “C” or higher: MAT 271 (or its equivalent).

*General Studies: MA*

**MAT 274 Elementary Differential Equations. (3)**

*fall, spring, summer*

Introduces ordinary differential equations, adapted to the needs of students in engineering and the sciences. Prerequisites: MAT 271 (or its equivalent); MAT 272 (or its equivalent) recommended.

*General Studies: MA*

**MAT 275 Modern Differential Equations. (3)**

*fall and spring*

Introduces differential equations, theoretical and practical solution techniques. Applications. Problem solving using Matlab. Lecture, computing lab. Fee. Pre- or corequisite: MAT 271 (or its equivalent).

*General Studies: MA*

**MAT 290 Calculus I. (5)**

*selected semesters*

Differential and integral calculus of elementary functions; topics from analytic geometry essential to the study of calculus. Prerequisite: MAT 170 (or its equivalent).

*General Studies: MA*

**MAT 291 Calculus II. (5)**

*selected semesters*

Further applications of calculus, partial differentiation, multiple integrals, and infinite series. Prerequisite: MAT 290 (or its equivalent).

**MAT 294 Special Topics. (1 – 4)**

*selected semesters*

**MAT 300 Mathematical Structures. (3)**

*fall and spring*

Logic and set theory, induction, functions, order and equivalence relations, cardinality. Emphasizes writing proofs. Prerequisite: 1 semester of calculus or instructor approval.

*General Studies: L*

**MAT 310 Introduction to Geometry. (3)**

*spring*

Congruence, area, parallelism, similarity and volume, and Euclidean and non-Euclidean geometry. Prerequisite: MAT 272 (or its equivalent).

**MAT 340 Theory of Interest. (3)**

*fall and spring*

Compound interest, discount rates, annuities, present values, depreciation, and bond valuations. Prerequisites: MAT 243 (or 300 or instructor approval); 1 semester of calculus.

**MAT 342 Linear Algebra. (3)**

*fall, spring, summer*

Linear equations, matrices, determinants, vector spaces, bases, linear transformations and similarity, inner product spaces, eigenvectors, orthonormal bases, diagonalization, and principal axes. Pre- or corequisite: MAT 272 (or its equivalent).

**MAT 343 Applied Linear Algebra. (3)**

*fall and spring*

Solving linear systems, matrices, determinants, vector spaces, bases, linear transformations, eigenvectors, norms, inner products, decompositions, applications. Problem solving using Matlab. Lecture, computing lab. Fee. Pre- or corequisite: MAT 271 (or its equivalent).

**MAT 351 Mathematical Methods for Genetic Analysis. (3)**

*fall and spring*

Discrete mathematics, probability, statistics, and associated computer packages. Applications to genomics, bioinformatics, forensics, and DNA/protein sequence patterns. Fee. Prerequisite: MAT 251 or 270 or instructor approval.

*General Studies: CS*

**MAT 362 Advanced Mathematics for Engineers and Scientists. (3)**

*fall, spring, summer*

Vector analysis, Fourier analysis, and partial differential equations. Prerequisites: MAT 272 and 274 (or their equivalents).

**MAT 370 Intermediate Calculus. (3)**

*fall and spring*

Theory behind basic 1-variable calculus: continuity, derivative, Riemann integral, sequences, and series. Not open to students who have received a “C” or higher in MAT 371. Students may not count both MAT 370 and 371 toward a mathematics degree. Prerequisites: MAT 272, 300.

**MAT 371 Advanced Calculus I. (3)**

*fall and spring*

Real numbers, completeness, sequences/series, continuity, uniform theorems, derivative, Riemann integral, pointwise/uniform convergence, Taylor’s theorem. Students may not count both MAT 370 and 371 toward a mathematics degree. Prerequisite: MAT 272 or 300 or instructor approval.

**MAT 372 Advanced Calculus II. (3)**

*spring*

Open, closed, compact sets in R^{n} continuity, differentiation, partial differentiation, integration in R^{n}. Inverse/implicit function theorems. Not open to students with credit for MAT 460. Prerequisite: MAT 371. Pre- or corequisite: MAT 342.

**MAT 410 Introduction to General Topology. (3)**

*once a year*

Topological spaces, metric spaces, compactness, connectedness, and product spaces. Prerequisite: MAT 300 or 371 or instructor approval.

**MAT 415 Introduction to Combinatorics. (3)**

*fall*

Topics include proof techniques, permutations, combinations; counting techniques including recurrence relaxations, generating functions, inclusion-exclusion; Ramsey theory and combinatorial designs. Prerequisites: both MAT 300 (or 243) and 342 (or 242) or only instructor approval.

**MAT 416 Introduction to Graph Theory. (3)**

*spring*

Topics include trees, cycles, matchings, planarity, connectivity, hamiltonicity, colorings, graph algorithms, and other advanced topics. Prerequisites: both MAT 300 (or 243) and 342 (or 242) or only instructor approval.

**MAT 419 Introduction to Linear Programming. (3)**

*spring*

Simplex method, duality, and network flows. Applications to game theory, geometry, combinatorics, graph theory, and posets. Prerequisites: a combination of CSE 100 (or 200 or 210) and MAT 300 (or 243) and 342 (or 242) or only instructor approval.

*General Studies: CS*

**MAT 420 Scientific Computing. (3)**

*fall*

Surveys and applies programming languages, libraries, and scientific visualization tools. Programming assignments emphasize software development skills. Lecture, lab. Fee. Prerequisites: a combination of CSE 200 and MAT 274 and 342 (or their equivalents) or only instructor approval.

**MAT 421 Applied Computational Methods. (3)**

*fall and spring*

Numerical methods for quadrature, differential equations, roots of nonlinear equations, interpolation, approximation, linear equations, floating-point arithmetic, and roundoff error. Prerequisites: both MAT 271 (or its equivalent) and fluency in computer programming (preferably FORTRAN) or only instructor approval.

*General Studies: CS*

**MAT 423 Numerical Analysis I. (3)**

*fall*

Analysis and algorithms for numerical solutions linear/nonlinear equations, direct solvers, iterative procedures, optimization. Determination of eigenvalues. Elementary computer arithmetic. Prerequisites: both MAT 342 and fluency in computer programming or only instructor approval.

*General Studies: CS*

**MAT 425 Numerical Analysis II. (3)**

*spring*

Analysis of and algorithms for numerical interpolation, integration, and differentiation. Numerical solution of ordinary differential equations, and method of lines. Those seeking a methods survey course should take MAT 421. Prerequisites: both MAT 274 and fluency in computer programming or only instructor approval. MAT 371 recommended.

*General Studies: CS*

**MAT 427 Computer Arithmetic. (3)**

*selected semesters*

Number systems, hardware/software arithmetic, overflow, significance, rounding, multiple precision, and automatic error control; impact on languages, architectures, robust programming, and software development. Prerequisite: only CSE 100 (or 200) or both MAT 421 and 423 (or 425) or only instructor approval.

*General Studies: CS*

**MAT 442 Advanced Linear Algebra. (3)**

*fall*

Fundamentals of linear algebra, dual spaces, invariant subspaces, canonical forms, bilinear and quadratic forms, and multilinear algebra. Prerequisites: both MAT 300 and 342 or only instructor approval.

**MAT 443 Introduction to Abstract Algebra. (3)**

*fall*

Introduces concepts of abstract algebra. Not open to students with credit for MAT 444. Prerequisites: both MAT 300 and 342 or only instructor approval.

**MAT 444 Intermediate Abstract Algebra. (3)**

*spring*

Basic theory of groups, rings, and fields, including an introduction to Galois theory. Appropriate as preparation for MAT 543. Prerequisite: MAT 443 or graduate standing or instructor approval.

**MAT 445 Theory of Numbers. (3)**

*spring*

Prime numbers, unique factorization theorem, congruences, Diophantine equations, primitive roots, and quadratic reciprocity theorem. Prerequisites: both MAT 300 and 342 or only instructor approval.

**MAT 447 Cryptography. (3)**

*fall and spring*

Block ciphers, stream ciphers, congruence arithmetic, information theory, public key cryptosystems, key exchange, electronic signatures. Prerequisites: MAT 242 (or 342), 300.

**MAT 451 Mathematical Modeling. (3)**

*spring*

Detailed study of 1 or more mathematical models that occur in the physical or biological sciences. May be repeated for credit with instructor approval. Prerequisites: both MAT 242 (or 342) and 274 or only instructor approval.

*General Studies: CS*

**MAT 452 Introduction to Chaos and Nonlinear Dynamics. (3)**

*fall*

Properties of nonlinear dynamical systems; dependence on initial conditions; strange attractors; period doubling; bifurcations; symbolic dynamics; Smale-Birkhoff theorem; and applications. Prerequisites: MAT 274, 342 (or 242); MAT 371 recommended.

**MAT 455 Introduction to Fractals and Applications. (3)**

*spring*

Fractals; self-similar structures, fractals with iterated function systems of maps, computing fractals, fractal dimensions, chaotic dynamics on fractals, applications. Prerequisites: MAT 274, 342 (or 242); MAT 371 recommended.

**MAT 460 Vector Calculus. (3)**

*spring*

Vectors, curvilinear coordinates, Jacobians, implicit function theorem, line and surface integrals, Green’s, Stokes’, and divergence theorems. Not open to students with credit for MAT 372. Prerequisites: MAT 242 (or 342), 272, 274.

**MAT 461 Applied Complex Analysis. (3)**

*fall and summer*

Analytic functions, complex integration, Taylor and Laurent series, residue theorem, conformal mapping, and harmonic functions. Prerequisite: MAT 272 (or its equivalent).

**MAT 462 Applied Partial Differential Equations. (3)**

*spring*

Second-order partial differential equations, emphasizing Laplace, wave, and diffusion equations. Solutions by the methods of characteristics, separation of variables, and integral transforms. Prerequisites: MAT 242 (or 342), 274.

**MAT 472 Intermediate Real Analysis I. (3)**

*fall*

Introduces analysis in metric spaces with emphasis on the real line. Appropriate as preparation for MAT 570. Prerequisites: MAT 300, 342.

**MAT 473 Intermediate Real Analysis II. (3)**

*spring*

Analysis in R^{n}: implicit function theorem, introduction to manifolds, Lebesque integration, change of variables formula, convergence theorems for integrals. Prerequisite: MAT 472 or instructor approval.

**MAT 475 Differential Equations. (3)**

*fall*

Asymptotic behavior of solutions of linear and nonlinear ordinary differential equations, stability, Sturm-Liouville problems, boundary value problems, and singular point behavior of autonomous systems. Prerequisites: MAT 242 (or 342), 274.

**MAT 476 Partial Differential Equations. (3)**

*spring*

First-order quasilinear, second-order linear (wave, Laplace, heat). Characteristics, harmonic functions, maximum principles, Fourier series, separation of variables. Prerequisites: MAT 274 (or 475), 372 (or 472).

**MAT 484 Internship. (1 – 12)**

*selected semesters*

**MAT 485 History of Mathematics. (3)**

*selected semesters*

Topics from the history of the origin and development of mathematical ideas. Prerequisite: MAT 272 (or its equivalent).

**MAT 502 Neural Modeling. (3)**

*fall and spring*

Mathematical modeling electrochemical processes in nerve. Cable theory, neuronal branching, spines, bifurcation analysis of excitable membrane models. Prerequisite: MAT 274.

**MAT 503 Mathematical Cell Physiology. (3)**

*fall and spring*

Mathematical modeling of dynamical aspects of cell physiology. Diffusion, membrane transport, intracellular calcium channel kinetics, calcium oscillations and waves. Lecture, computing lab.

**MAT 504 Mathematical Aspects of Biotechnology. (3)**

*fall and spring*

Bacterial growth, bacterial genetics, gene expression, stochiometry of metabolic pathways, random walks, diffusion processes, biofilms. Prerequisite: instructor approval.

**MAT 505 Perturbation Methods. (3)**

*selected semesters*

Nonlinear oscillations, strained coordinates, renormalization, multiple scales, boundary layers, matched asymptotic expansions, turning point problems, and WKBJ method. Cross-listed as MAE 505. Credit is allowed for only MAE 505 or MAT 505.

**MAT 514 Enumerative Combinatorics I. (3)**

*fall*

First semester of a systematic development of enumerative combinatorics, including elementary counting techniques, sieve methods, and partially ordered sets. Prerequisite: graduate standing or instructor approval.

**MAT 515 Enumerative Combinatorics II. (3)**

*spring*

Second semester of a systematic development of enumerative combinatorics, including lattices, exponential structures, symmetric functions, and selected special topics. Prerequisite: MAT 514 or instructor approval.

**MAT 516 Graph Theory I. (3)**

*fall*

First semester of a systematic development of graph theory, including matchings, connectivity, arboricity, planarity, coloring, network flows. Prerequisite: graduate standing or instructor approval.

**MAT 517 Graph Theory II. (3)**

*spring*

Second semester of a systematic development of graph theory, including dense and sparse graphs, Ramsey theory, hamiltonicity, random graphs, minors. Prerequisite: MAT 516 or instructor approval.

**MAT 518 Combinatorial Optimization I. (3)**

*fall*

First semester of a systematic development of combinatorial optimization, including linear programming, duality, primal-dual algorithms, network flow algorithms, weighted matchings. Prerequisite: graduate standing or instructor approval.

**MAT 519 Combinatorial Optimization II. (3)**

*spring*

Second semester of a systematic development of combinatorial optimization, including matroid algorithms, theory of NP-completeness, polynomial time approximation, dynamic programming. Prerequisite: MAT 518 or instructor approval.

**MAT 520 Numerical Linear Algebra. (3)**

*fall*

Direct solution of linear systems, iterative methods, eigenvalues and eigenvectors, singular value decomposition, the QR algorithm, error propagation, arithmetic, and stability. Prerequisites: both MAT 342 and 423 (or 421) or only instructor approval.

**MAT 521 Iterative Methods. (3)**

*spring*

Numerical methods for solving linear/nonlinear systems of equations (symmetric, nonsymmetric). Iterative methods for linear systems, conjugate gradients, multigrid methods, preconditioning, Krylov methods. Prerequisites: both MAT 371 and 423 (or 421) or only instructor approval.

**MAT 523 Numerical Optimization. (3)**

*selected semesters*

Linear programming, unconstrained nonlinear minimization, line search algorithms, conjugate gradients, quasi-Newton methods, constrained nonlinear optimization, gradient projection, and penalty methods. Prerequisite: MAT 342 or 371 or 460 or 520 (or its equivalent) or instructor approval.

**MAT 524 Parallel Numerical Algorithms. (3)**

*selected semesters*

Algorithms for massively parallel, hypercube architectures; “parallel” FORTRAN; solution of linear, nonlinear systems; partial differential equations;
iterative methods; multigrid; domain decomposition. Prerequisites: both MAT 371 and 423 (or 421) or only instructor approval.

**MAT 530 Numerical Solution of Ordinary Differential Equations. (3)**

*fall*

One-step, linear multistep methods; consistency, order, stability, convergence; discretization, roundoff errors, error estimation, adaptive strategy; implementation, software for nonstiff equations. Prerequisites: both MAT 371 and 423 (or 421) or only instructor approval.

**MAT 531 Numerical Solution of Stiff Differential Systems. (3)**

*spring*

Runge-Kutta methods, order conditions, construction of highly stable methods, order stars, error estimation, stepsize selection, contractivity properties, linear multistep methods. Prerequisites: both MAT 371 and 423 (or 421) or only instructor approval.

**MAT 533 Computational Elliptic and Parabolic Partial Differential Equations. (3)**

*fall*

Parabolic and elliptic equations, finite difference, finite element methods, stability, consistency, convergence, practical aspects, applications, software. Prerequisites: both MAT 371 and 423 (or 421) or only instructor approval.

**MAT 534 Computational Hyperbolic Partial Differential Equations. (3)**

*spring*

Numerical solutions of hyperbolic PDEs, finite difference methods, well-posedness, stability, consistency, convergence, adaptive grids; Maxwell’s equations, elastic wave propagation; Navier-Stokes. Prerequisites: both MAT 371 and 423 (or 421) or only instructor approval.

**MAT 535 Spectral Methods for Partial Differential Equations. (3)**

*selected semesters*

Spectral, pseudospectral theory; Galerkin, collocation methods; Tau-methods, global approximation properties, stability; convergence; solutions for linear, nonlinear systems. Prerequisites: both MAT 371 and 423 (or 421) or only instructor approval.

**MAT 543 Abstract Algebra. (3)**

*fall*

Groups, modules, rings and fields, Galois theory, homological algebra, and the representation theory. Prerequisite: MAT 444 or instructor approval.

**MAT 544 Abstract Algebra. (3)**

*spring*

Continuation of MAT 543. Prerequisite: MAT 543 or instructor approval.

**MAT 551 Linear Operators and Integral Equations. (3)**

*spring*

Bounded linear and compact operators on Hilbert spaces. Linear integral equations, Fredholm and Hilbert-Schmidt theory, and approximate methods. Distributions. Prerequisites: MAT 242 and 462 (or their equivalents).

**MAT 555 Fractal Geometry. (3)**

*selected semesters*

Geometry and analysis of fractal sets; definitions of dimensions; calculating dimensions; projections, products of fractals; random fractals; multifractal measures; and applications. Prerequisites: MAT 371, 455; MAT 472 recommended.

**MAT 560 Dynamical Systems Methods in Fluid Dynamics. (3)**

*fall*

Applies modern dynamical systems methods to fluid mechanics: bifurcations, normal forms, nonlinear dynamics, pattern formation, mixing, and Lagrangian chaos. Prerequisite: graduate standing or instructor approval.

**MAT 562 Nonlinear Analysis of PDEs in Fluids. (3)**

*spring*

Sobolev spaces; incompressible Euler and Navier-Stokes equations; weak and strong solutions; attractors and the connection with turbulence; geophysical applications. Prerequisite: graduate standing or instructor approval.

**MAT 570 Real Analysis. (3)**

*spring*

Lebesgue integration, selected function spaces, differentiation, abstract measure theory, and elements of functional analysis. Prerequisite: MAT 372 or instructor approval.

**MAT 571 Real Analysis. (3)**

*fall*

Continuation of MAT 570. Prerequisite: MAT 570 or instructor approval.

**MAT 572 Complex Analysis. (3)**

*fall*

Analytic functions, series and product representations, entire and meromorphic functions, normal families, Riemann mapping theorem, harmonic functions, and Riemann surfaces. Prerequisite: MAT 371 or instructor approval.

**MAT 573 Complex Analysis. (3)**

*spring*

Continuation of MAT 572. Prerequisite: MAT 572 or instructor approval.

**MAT 574 Theory of Ordinary Differential Equations. (3)**

*selected semesters*

Systems, existence proofs, singularities, asymptotic behavior of solutions, boundedness of solutions, eigenvalues and eigenfunctions, and perturbation theory. Prerequisite: MAT 372 or instructor approval.

**MAT 575 Theory of Ordinary Differential Equations and Dynamical Systems. (3)**

*selected semesters*

Geometric approach to ODEs and dynamical systems; (un)stable, center manifolds; structural stability; normal forms; averaging; chaos; persistence. May be repeated for credit with instructor approval. Prerequisites: both MAT 452 and 475 or only MAT 574 or only instructor approval.

**MAT 576 Theory of Partial Differential Equations. (3)**

*selected semesters*

Existence and uniqueness theorems, boundary value and initial value problems, characteristics, Green’s functions, maximum principle, distributions, and weak solutions. Prerequisite: knowledge of Lebesgue integration or instructor approval.

**MAT 577 Theory of Partial Differential Equations. (3)**

*selected semesters*

Continuation of MAT 576. Prerequisite: MAT 576 or instructor approval.

**MAT 578 Functional Analysis. (3)**

*selected semesters*

Locally convex, normed, and Hilbert spaces. Linear operators, spectral theory, and application to classical analysis. Prerequisite: MAT 472 or 571 or instructor approval.

**MAT 579 Functional Analysis. (3)**

*selected semesters*

Continuation of MAT 578. Prerequisite: MAT 578 or instructor approval.

**MAT 591 Seminar. (1 – 12)**

*selected semesters*

Topics may include the following:

• | Algebra. (1 – 3) |

• | Analysis. (1 – 3) |

• | Applied Mathematics. (1 – 3) |

• | Combinatorial Mathematics. (1 – 3) |

• | Mathematical Logic. (1 – 3) |

• | Numerical Analysis. (1 – 3) |

• | Topology. (1 – 3) |

**Omnibus Courses.** For an explanation of additional courses offered but not specifically listed, see “Omnibus Courses.”

Page Last Updated: | Visits to this page: | Web Site Feedback | Copyright © Arizona Board of Regents |