Graduate Course Descriptions

The following directory lists the graduate courses which the University expects to offer, although the University in no way guarantees that all such courses will be offered in any given academic year, and reserves the right to alter the list if conditions warrant. Click on the links below for a list of courses in that subject area. You may then click “View Classes” to see scheduled classes for individual courses.

5000. Mathematical Pedagogy

1.00 credits

Prerequisites: None.

Grading Basis: Graded

The theory and practice of teaching mathematics at the college level. Basic skills, grading methods, cooperative learning, active learning, use of technology, classroom problems, history of learning theory, reflective practice. Open to graduate students in Mathematics, others with consent of instructor. May not be used to satisfy degree requirements in mathematics.

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5010. Topics in Analysis I

3.00 credits | May be repeated for credit.

Prerequisites: None.

Grading Basis: Graded

Advanced topics in analysis.

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5011. Topics in Analysis II

3.00 credits | May be repeated for credit.

Prerequisites: Prerequisite: MATH 5010 (RG385).

Grading Basis: Graded

Advanced topics in analysis.

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5016. Topics in Probability

3.00 credits | May be repeated for credit.

Prerequisites: None.

Grading Basis: Graded

Advanced topics in probability theory, theory of random processes, mathematical statistics, and related fields.

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5020. Topics in Algebra

3.00 credits | May be repeated for credit.

Prerequisites: Prerequisite: MATH 5211 (RG 381)

Grading Basis: Graded

Advanced topics chosen from group theory, ring theory, number theory, Lie theory, combinatorics, commutative algebra, algebraic geometry, homological algebra, and representation theory.

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5026. Topics in Mathematical Logic

3.00 credits | May be repeated for credit.

Prerequisites: Prerequisite: MATH 5260 (RG 386)

Grading Basis: Graded

Topics include, but are not restricted to, Computability Theory, Model Theory, and Set Theory.

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5030. Topics in Geometry and Topology I

3.00 credits | May be repeated for credit.

Prerequisites: None.

Grading Basis: Graded

Topics in Geometry and Topology I. Advanced topics in geometry and topology. This course may be repeated with each change of topic.

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5031. Topics in Geometry and Topology II

3.00 credits | May be repeated for credit.

Prerequisites: Prerequisite: MATH 5030 (RG 387)

Grading Basis: Graded

Advanced topics in Geometry and Topology.

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5040. Topics in Applied Analysis I

3.00 credits | May be repeated for credit.

Prerequisites: None.

Grading Basis: Graded

Advanced topics from the theory of ordinary or partial differential equations. Other possible topics: integral equations, optimization theory, the calculus of variations, advanced approximation theory.

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5041. Topics in Applied Analysis II

3.00 credits | May be repeated for credit.

Prerequisites: None.

Grading Basis: Graded

Advanced topics from the theory of ordinary or partial differential equations. Other possible topics: integral equations, optimization theory, the calculus of variations, advanced approximation theory.

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5046. Introduction to Complex Variables

3.00 credits

Prerequisites: Not open to students who have passed MATH 3146

Grading Basis: Graded

Functions of a complex variable, integration in the complex plane, conformal mapping. Open for master's credit but not doctoral credit toward degree in Mathematics.

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5050. Analysis

3.00 credits

Prerequisites: Antipre MATH 3150

Grading Basis: Graded

Introduction to the theory of functions of a real variable. Open for master's credit but not doctoral credit toward degree in Mathematics.

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5070. Topics in Scientific Computation

3.00 credits | May be repeated for credit.

Prerequisites: None.

Grading Basis: Graded

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5110. Introduction to Modern Analysis

3.00 credits

Prerequisites: None.

Grading Basis: Graded

Metric spaces, sequences and series, continuity, differentiation, the Riemann-Stielties integral, functions of several variables.

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5111. Measure and Integration

3.00 credits

Prerequisites: Prerequisite: MATH 5110 (RG 375)

Grading Basis: Graded

General theory of measure and Lebesgue integration, L^p-spaces.

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5120. Complex Function Theory I

3.00 credits

Prerequisites: Prerequisite: MATH 5110 (RG 375)

Grading Basis: Graded

An introduction to the theory of analytic functions, with emphasis on modern points of view.

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5121. Topics in Complex Function Theory

3.00 credits | May be repeated for a total of 12 credits.

Prerequisites: Prerequisite: MATH 5120 (RG 388)

Grading Basis: Graded

Advanced topics of contemporary interest. These include Riemann surfaces, Kleinian groups, entire functions, conformal mapping, several complex variables, and automorphic functions, among others.

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5130. Functional Analysis I

3.00 credits

Prerequisites: Prerequisite: MATH 5111 (RG4837)

Grading Basis: Graded

Normed linear spaces and algebras, the theory of linear operators, spectral analysis.

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5131. Functional Analysis II

3.00 credits | May be repeated for a total of 6 credits.

Prerequisites: Prerequisite: MATH 5111 (RG4837)

Grading Basis: Graded

Normed linear spaces and algebras, the theory of linear operators, spectral analysis.

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5140. Fourier Analysis

3.00 credits

Prerequisites: Prerequisite: MATH 5111 (RG4837)

Grading Basis: Graded

Foundations of harmonic analysis developed through the study of Fourier series and Fourier transforms.

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5141. Fourier Analysis on Groups

3.00 credits

Prerequisites: Prerequisite: MATH 5111 (RG4837)

Grading Basis: Graded

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5160. Probability Theory and Stochastic Processes I

3.00 credits

Prerequisites: None.

Grading Basis: Graded

Convergence of random variables and their probability laws, maximal inequalities, series of independent random variables and laws of large numbers, central limit theorems, martingales, Brownian motion.

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5161. Probability Theory and Stochastic Processes II

3.00 credits | May be repeated for a total of 12 credits.

Prerequisites: Prerequisite: MATH 5160 (RG383)

Grading Basis: Graded

Contemporary theory of stochastic processes, including stopping times, stochastic integration, stochastic differential equations and Markov processes, Gaussian processes, and empirical and related processes with applications in asymptotic statistics.

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5210. Abstract Algebra I

3.00 credits

Prerequisites: None.

Grading Basis: Graded

Group theory, ring theory and modules, and universal mapping properties.

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5211. Abstract Algebra II

3.00 credits

Prerequisites: Prerequisite: MATH 5210 (RG380).

Grading Basis: Graded

Linear and multilinear algebra, Galois theory, category theory, and commutative algebra.

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5220. Introduction to Representation Theory

3.00 credits

Prerequisites: Prerequisite: MATH 5210 (RG380).

Grading Basis: Graded

Semi-simple rings, Jacobson radical, density theory, Wedderburn's Theorem, representations and characters of groups, orthogonality relations, Burnside's theorem.

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5230. Algebraic Number Theory

3.00 credits

Prerequisites: Prerequisite: MATH 5211 (RG 381)

Grading Basis: Graded

Algebraic integers, ideal class group, ramification, Frobenius elements in Galois groups, Dirichlet's unit theorem, localization, and completion. Further topics (zeta-functions, function fields, non-maximal orders) as time permits.

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5250. Modern Matrix Theory and Linear Algebra

3.00 credits

Prerequisites: None.

Grading Basis: Graded

The LU, QR, symmetric, polar, and singular value matrix decompositions. Schur and Jordan normal forms. Symmetric, positive-definite, normal and unitary matrices. Perron-Frobenius theory and graph criteria in the theory of non-negative matrices.

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5260. Mathematical Logic I

3.00 credits

Prerequisites: Prerequisite: MATH 5210 (RG380).

Grading Basis: Graded

Predicate calculus, completeness, compactness, Lowenheim-Skolem theorems, formal theories with applications to algebra, Godel's incompleteness theorem. Further topics chosen from: axiomatic set theory, model theory, recursion theory, computational complexity, automata theory and formal languages.

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5310. Introduction to Geometry and Topology I

3.00 credits

Prerequisites: Prerequisite: MATH 5110, which may be taken concurrently. (RG 376)

Grading Basis: Graded

Topological spaces, maps, induced topologies, separation axioms, compactness, connectedness, classification of surfaces, the fundamental group and its applications, covering spaces.

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5311. Introduction to Geometry and Topology II

3.00 credits | May be repeated for a total of 12 credits.

Prerequisites: Prerequisite: MATH 5310 (RG 377)

Grading Basis: Graded

Smooth manifolds, vector fields, differential forms, de Rham cohomology, homology theory, singular (co)homology, Poincare duality.

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5320. Algebraic Geometry I

3.00 credits

Prerequisites: Prerequisite: MATH 5211 and MATH 5310, which may be taken concurrently (RG393).

Grading Basis: Graded

This course is an introduction to algebraic varieties: affine and projective varieties, dimension of varieties and subvarieties, algebraic curves, singular points, divisors and line bundles, differentials, intersections.

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5321. Algebraic Geometry II

3.00 credits

Prerequisites: Prerequisite: MATH 5320 (RG394).

Grading Basis: Graded

This course introduces further concepts and methods of modern algebraic geometry, including schemes and cohomology.

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5360. Differential Geometry

3.00 credits

Prerequisites: None.

Grading Basis: Graded

This course is an introduction to the study of differentiable manifolds on which various differential and integral calculi are developed. The topics include covariant derivatives and connections, geodesics and exponential map, Riemannian metrics, curvature tensor, Ricci and scalar curvature.

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5410. Introduction to Applied Mathematics I

3.00 credits

Prerequisites: None.

Grading Basis: Graded

Banach spaces, linear operator theory and application to differential equations, nonlinear operators, compact sets on Banach spaces, the adjoint operator on Hilbert space, linear compact operators, Fredholm alternative, fixed point theorems and application to differential equations, spectral theory, distributions.

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5411. Introduction to Applied Mathematics II

3.00 credits

Prerequisites: None.

Grading Basis: Graded

Banach spaces, linear operator theory and application to differential equations, nonlinear operators, compact sets on Banach spaces, the adjoint operator on Hilbert space, linear compact operators, Fredholm alternative, fixed point theorems and application to differential equations, spectral theory, distributions.

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5420. Ordinary Differential Equations

3.00 credits

Prerequisites: Prerequisite: MATH 5111 (RG4837)

Grading Basis: Graded

Existence and uniqueness of solutions, stability and asymptotic behavior. If time permits: eigenvalue problems, dynamical systems, existence and stability of periodic solutions.

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5430. Applied Analysis

3.00 credits

Prerequisites: Not open to students who have passed MATH 3430. May not be used for credit for Mathematics graduate degrees (RG608).

Grading Basis: Graded

Convergence of Fourier Series, Legendre and Hermite polynomials, existence and uniqueness theorems, two-point boundary value problems and Green's functions.

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5435. Introduction to Partial Differential Equations

3.00 credits

Prerequisites: Not open for credit to students who have passed MATH 3435

Grading Basis: Graded

Solution of first and second order partial differential equations with applications to engineering and science.

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5440. Partial Differential Equations

3.00 credits

Prerequisites: Prerequisite: MATH 5120 (RG384).

Grading Basis: Graded

Cauchy Kowalewsky Theorem, classification of second-order equations, systems of hyperbolic equations, the wave equation, the potential equation, the heat equation in Rn.

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5510. Numerical Analysis and Approximation Theory I

3.00 credits

Prerequisites: Prerequisite: MATH 5110, which may be taken concurrently. (RG 376)

Grading Basis: Graded

The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.

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5511. Numerical Analysis and Approximation Theory II

3.00 credits

Prerequisites: Prerequisite: MATH 5510 (RG379).

Grading Basis: Graded

The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.

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5520. Finite Element Solution Methods I

3.00 credits

Prerequisites: None.

Grading Basis: Graded

Numerical solution of elliptic, parabolic and hyperbolic partial differential equations by finite element solution methods. Applications.

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5521. Finite Element Solution Methods II

3.00 credits

Prerequisites: Prerequisite: MATH 5520 (RG389).

Grading Basis: Graded

Numerical solution of elliptic, parabolic and hyperbolic partial differential equations by finite element solution methods. Applications.

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5580. Optimization

3.00 credits

Prerequisites: None.

Grading Basis: Graded

Theory of linear programming: convexity, bases, simplex method, dual and integer programming, assignment, transportation, and flow problems. Theory of nonlinear programming: unconstrained local optimization, Lagrange multipliers, Kuhn-Tucker conditions, computational algorithms. Concrete applications.

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5600. Fundamentals of Financial Mathematics

3.00 credits

Prerequisites: None.

Grading Basis: Graded

The risk-neutral model for pricing and hedging derivative financial instruments within the context of binomial and trinomial models of the stock price process.

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5620. Financial Mathematics I

3.00 credits

Prerequisites: Not open to students who have passed MATH 2620 (RG606).

Grading Basis: Graded

The mathematics of measurement of interest, accumulation and discount, present value, annuities, loans, bonds, and other securities.

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5630. Long-Term Actuarial Mathematics I

4.00 credits

Prerequisites: Prerequisite: MATH 2620 or MATH 5620, which may be taken concurrently. Not open to students who have passed MATH 3630 (RG397).

Grading Basis: Graded

Long-term insurance products, survival and longevity models, life tables, life insurance, life annuities, premium calculations, reserves.

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5631. Long-Term Actuarial Mathematics II

4.00 credits

Prerequisites: Prerequisite: MATH 5630. Not open to students who have passed MATH 3631 (RG398).

Grading Basis: Graded

A continuation of Long-Term Actuarial Mathematics I. Topics include multiple state models, multiple decrements, multiple lives, profit and loss analysis, pension plans and funding, retirement benefits, long-term health and disability.

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5637. Statistics for Actuarial Modeling

4.00 credits

Prerequisites: Not open to students who have passed MATH 3636 or 3637.

Grading Basis: Graded

Data analysis for actuaries, linear models including generalized linear models, time series, principal component analysis, decision trees, cluster analysis, statistical computing with R, actuarial applications.

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5638. Predictive Analytics for Actuaries

3.00 credits

Prerequisites: MATH 5637

Grading Basis: Graded

Models for predictive analytics, model building, selection, estimation, validation and diagnostics, and limitations; actuarial applications, and communication of results.

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5639. Actuarial Loss Models

3.00 credits

Prerequisites: Not open to students who have passed MATH 3639.

Grading Basis: Graded

Loss distribution models for claim frequency and severity, aggregate risk models, coverage modifications, risk measures, construction and selection of parametric models, introduction to simulation.

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5640. Short-Term Insurance Ratemaking

3.00 credits

Prerequisites: MATH 5639. Not open to students who have passed MATH 3640.

Grading Basis: Graded

Credibility theory, pricing for short-term insurance coverages, reinsurance, experience rating, risk classification, introduction to Bayesian statistics.

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5641. Short-Term Insurance Reserving

3.00 credits

Prerequisites: MATH 5639. Not open to students who have passed MATH 3641.

Grading Basis: Graded

Techniques for estimating unpaid claims, use of claims triangles, underlying statistical theory behind the techniques, basic adjustments to data and estimation techniques to account for internal and external environments, estimating recoveries, model adequacy and reasonableness.

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5650. Financial Mathematics II

4.00 credits

Prerequisites: Not open to students who have passed MATH 3650 (RG2754).

Grading Basis: Graded

The continuation of Math 5620, focusing on the mathematics of finance: measurement of financial risk and the opportunity cost of capital, the mathematics of capital budgeting and securities valuation, mathematical analysis of financial decisions and capital structure, and option pricing theory. Provides VEE credit in the Corporate Finance subject area for Society of Actuaries and Casualty Actuarial Society requirements.

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5660. Advanced Financial Mathematics

3.00 credits

Prerequisites: None.

Grading Basis: Graded

An introduction to the standard models of modern financial mathematics including martingales, the binomial asset pricing model, Brownian motion, stochastic integrals, stochastic differential equations, continuous time financial models, completeness of the financial market, the Black-Scholes formula, the fundamental theorem of finance, American options, and term structure models.

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5661. Yield Curve Models

3.00 credits

Prerequisites: None.

Grading Basis: Graded

The theory and practice of stochastic models to analyze and value interest rate derivatives, and practical issues in the markets where they are traded.

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5670. Financial Programming and Modeling

3.00 credits

Prerequisites: None.

Grading Basis: Graded

Optimization; linear and non-linear programming; data mining and machine learning in a financial context.

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5671. Financial Data Mining and Big Data Analytics

3.00 credits

Prerequisites: None.

Grading Basis: Graded

Data structures and algorithms; regression; classification; clustering; recommender systems; anomaly detection; Big Data tools; databases.

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5800. Investigation of Special Topics

1.00 - 6.00 credits | May be repeated for a total of 36 credits.

Prerequisites: None.

Grading Basis: Graded

Students who have well defined mathematical problems worthy of investigation and advanced reading should submit to the department a semester work plan.

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5850. Graduate Field Study Internship

1.00 - 3.00 credits

Prerequisites: None.

Grading Basis: Graded

Participation in internship and paper describing experiences.

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6000. Seminar in Current Mathematical Literature

1.00 - 6.00 credits | May be repeated for a total of 12 credits.

Prerequisites: None.

Grading Basis: Graded

Participation and presentation of mathematical papers in joint student faculty seminars. Variable topics.

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6010. Seminar in Analysis

1.00 - 6.00 credits

Prerequisites: None.

Grading Basis: Satisfactory/Unsatisfactory

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6020. Seminar in Algebra

1.00 - 6.00 credits

Prerequisites: Prerequisite: MATH 5211 (RG 381)

Grading Basis: Satisfactory/Unsatisfactory

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6026. Seminar in Mathematical Logic

1.00 - 6.00 credits | May be repeated for a total of 12 credits.

Prerequisites: MATH 5260 (RG3453).

Grading Basis: Satisfactory/Unsatisfactory

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6027. Seminar in Set Theory

1.00 - 6.00 credits

Prerequisites: Prerequisite: MATH 5310 (RG402).

Grading Basis: Satisfactory/Unsatisfactory

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6030. Seminar in Topology

1.00 - 6.00 credits

Prerequisites: Prerequisite: MATH 5321 (RG401).

Grading Basis: Satisfactory/Unsatisfactory

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6036. Seminar in Geometry

1.00 - 6.00 credits

Prerequisites: Prerequisite: MATH 5360 (RG400).

Grading Basis: Satisfactory/Unsatisfactory

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6040. Seminar in Applied Mathematics

1.00 - 6.00 credits

Prerequisites: None.

Grading Basis: Satisfactory/Unsatisfactory

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