### Courses of Instruction

###### Mathematics (MATH)

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

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

**MATH 108 Precalculus (4, FaSp)** Equations and inequalities; functions; graphs; polynomial and rational functions; exponential, logarithmic, and trigonometric function; analytic geometry. *Prerequisite:* MATH 040*x* or passing of placement exam.

**MATH 116 Mathematics for the Social Sciences (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.

**MATH 117 Introduction to Mathematics for Business and Economics (4, FaSp)** Functions, graphs, polynomial and rational functions, exponential and logarithmic functions, matrices, systems of linear equations. *Prerequisite:* MATH 040*x* or math placement exam.

**MATH 118x Fundamental Principles of the Calculus (4, FaSpSm)** Derivatives; extrema. Definite integral; fundamental theorem of calculus. Extrema and definite integrals for functions of several variables. Not available for credit toward a degree in mathematics. *Prerequisite:* MATH 117 or math placement exam.

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

**MATH 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.

**MATH 127 Enhanced Calculus I (4, Fa)** Applications of integration, review of techniques of integration, infinite sequences and series, some beginning linear algebra, ordinary differential equations. Designed for students who earn a score of 4 or 5 on the Advanced Placement Calculus AB Examination, or a score of 3 or 4 on the BC Examination. Admission to course by departmental approval. (Duplicates credit in MATH 126.)

**MATH 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 040*x* or math placement exam.

**MATH 208x Elementary Probability and Statistics (4, FaSp)** 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 118*x* or MATH 125.

**MATH 218 Probability for Business (4, FaSpSm)** Basic probability, discrete and continuous distributions, expectation and variance, independence. Sampling, estimation, confidence intervals, hypothesis testing. *Prerequisite:* MATH 118*x* or MATH 125.

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

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

**MATH 227 Enhanced Calculus II (4, Sp)** A continuation of MATH 127; vectors and vector spaces, functions of several variables, partial differential equations, optimization theory, multiple integration; Green's Stokes', divergence theorems. *Prerequisite:* MATH 127 or MATH 225.

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

**MATH 370 Applied Algebra (4, Sp)** Induction, Euclidean algorithm, factorization, congruence classes, Rings, RSA algorithm, Chinese remainder theorem, codes, polynomials, fundamental theorem of algebra, polynomial multiplication, Fourier transform, and other topics. *Prerequisite:* MATH 226; MATH 225 or MATH 245.

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

**MATH 395 Seminar in Problem Solving (2, max 8)** Systematic approach to solving non-standard and competition level math problems on inequalities, infinite sums and products, combinatorics, number theory, and games. *Recommended Preparation:* MATH 126.

**MATH 400 Foundations of Discrete Mathematics (4, Fa)** Methods of proof, predicate calculus, set theory, order and equivalence relations, partitions, lattices, functions, cardinality, elementary number theory and combinatorics. *Prerequisite:* MATH 225 or MATH 226.

**MATH 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.

**MATH 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.

**MATH 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.

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

**MATH 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.

**MATH 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.

**MATH 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.

**MATH 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.

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

**MATH 445 Mathematics of Physics and Engineering II (4, FaSp)** 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.

**MATH 450 History of Mathematics (4, Sp)** 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.

**MATH 458 Numerical Methods (4, Fa)** 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.

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

**MATH 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.

**MATH 467 Theory and Computational Methods for Optimization (4)** Methods for static, dynamic, unconstrained, constrained optimization. Gradient, conjugate gradient, penalty methods. Lagrange multipliers, least squares, linear, nonlinear dynamic programming. Application to control and estimation. *Prerequisite:* MATH 226; MATH 225 or MATH 245.

**MATH 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.

**MATH 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.

**MATH 490x Directed Research (2-8, max 8, FaSpSm)** Individual research and readings. Not available for graduate credit.

**MATH 499 Special Topics (2-8, max 8)** Lectures on advanced material not covered in regularly scheduled
courses. No more than two registrations allowed.

**MATH 500 Graduate Colloquium (2)** Lectures directed to mathematics graduate students by faculty of the department and by outside speakers. Problem solving workshops. Graded CR/NC.

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

**MATH 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 425*a* and MATH 471.

**MATH 503 Stochastic Calculus for Finance (3, Sp)** Stochastic differential equations. Bellman equation. Applications to option pricing. Kolmogorov equations and derivative securities. State prices, equivalent martingale measure. Optimal stopping, American options. Exotic options. *Prerequisite:* MATH 506 or MATH 507*a*.

**MATH 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 502*a* 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 502*a*.

**MATH 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, queuing models, shot noise.

**MATH 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.

**MATH 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 525*a* or MATH 570. *b:* Dependence, martingales, ergodic theorems, second-order random functions, harmonic analysis, Markov processes.

**MATH 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 507*a*.

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

**MATH 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.

**MATH 511abL Data Analysis (4-4)** (Enroll in PM 511*abL*)

**MATH 512 Financial Informatics and Simulation (Computer Labs and Practitioner Seminar) (3, FaSp)** Experimental laboratory trading for financial markets using double auctions: handling statistical packages for data analysis. Practical training in virtual market environments, using financial trading system software.

**MATH 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 425*ab*.

**MATH 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 425*ab*. *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 525*a*.

**MATH 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.

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

**MATH 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.

**MATH 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.

**MATH 541ab Introduction to Mathematical Statistics (a: 3, Sp; b: 3, Fa)** *a:* Parametric families of distributions, sufficiency. Estimation: methods of moments, maximum likelihood, unbiased estimation. Comparison of estimators, optimality, information inequality, asymptotic efficiency. EM algorithm, jackknife and bootstrap. *Prerequisite:* MATH 505*a* or MATH 407 or MATH 408. *b:* Hypothesis testing, Neyman-Pearson lemma, generalized likelihood ratio procedures, confidence intervals, consistency, power, jackknife and bootstrap. Monte Carlo Markov chain methods, hidden Markov models. *Prerequisite:* MATH 541*a*.

**MATH 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 208*x*.

**MATH 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 208*x*.

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

**MATH 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 208*x*.

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

**MATH 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.

**MATH 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.

**MATH 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 208*x*.

**MATH 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 425*ab*.

**MATH 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 425*ab*.

**MATH 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 425*a* 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 570*a*.

**MATH 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.

**MATH 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.

**MATH 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.

**MATH 577ab Computational Molecular Biology Laboratory (a: 2, Sp; b: 2, Fa)** (Enroll in BISC 577*ab*)

**MATH 578ab Computational Molecular Biology (3-3, FaSp)** Applications of the mathematical, statistical and computational sciences to data from molecular biology. *a:* Algorithms for genomic sequence data: sequence and map assembly and alignment, RNA secondary structure, protein structure, gene-finding, and tree construction. *Prerequisite:* CSCI 570; *Recommended Preparation:* familiarity with the concepts of basic molecular biology as covered in BISC 320. b: Statistics for genomic sequence data: DNA sequence assembly, significance of alignment scores, hidden Markov models, genetic mapping, models of sequence evolution, and microarray analysis. *Prerequisite:* MATH 505*a*, MATH 541*a*.

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

**MATH 585 Mathematical Theory of Optimal Control (3, Fa)** 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 525*a*.

**MATH 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.

**MATH 592 Computational Molecular Biology Internship (3)** Industrial or genome-centered internship for students in the Computational Molecular Biology master's program. Real-world experience in applications. Open to M.S., Computational Molecular Biology students only.

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

**MATH 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.

**MATH 600 Topics in Numerical Analysis (3, max 12)**

**MATH 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 502*ab*.

**MATH 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.

**MATH 605 Topics in Probability (3, max 12)**

**MATH 610 Topics in Algebra (3, max 12)**

**MATH 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 510*ab*.

**MATH 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 510*ab*.

**MATH 620 Topics in Complex Analysis (3, max 12)**

**MATH 625 Topics in Real Analysis (3, max 12)**

**MATH 630 Topics in Number Theory (3, max 12)**

**MATH 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 535*ab*.

**MATH 641 Topics in Topology (3, max 12)**

**MATH 650 Seminar in Statistical Consulting (3)**

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

**MATH 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.

**MATH 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.

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

**MATH 689 Topics in Mathematical Physics (3, max 12)**

**MATH 700 Seminar in Numerical Analysis (3)**

**MATH 705 Seminar in Probability (3)**

**MATH 710 Seminar in Algebra (3)**

**MATH 725 Seminar in Analysis (3)**

**MATH 730 Seminar in Number Theory (3)**

**MATH 735 Seminar in Differential Geometry (3)**

**MATH 740 Seminar in Topology (3)**

**MATH 761 Seminar in Programming and Computability (3)**

**MATH 765 Seminar in Ordinary Differential Equations (3)**

**MATH 780 Seminar in Functional Analysis (3)**

**MATH 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.

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