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Course Criteria
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3.00 Credits
Factorization of integers, Fundamental Theorem of Arithmetic, congruences, Wilson’s theorem. Fermat’s theorem, arithmetic functions, perfect numbers, Law of Quadratic Reciprocity. Diophantine equations, Pythagorean triples, sums of squares.
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3.00 Credits
Fundamental properties of integers and polynomials. Elementary properties of groups, rings, integral domains, fields, and lattices. Prerequisite: linear algebra.
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3.00 Credits
Numerical solution of linear and nonlinear equations, interpolation and polynomial approximation, non-numerical differentiation and integration, least-squares curve fitting and approximation theory, numerical solution of differential equations, errors and floating point arithmetic. Application of the theory to problems in science, engineering, and economics. Student use of the computer is emphasized. Prerequisite: computer programming and linear algebra, differential equations.
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3.00 Credits
Vector analysis including directional derivatives, transformation of coordinates, divergence and curl. Line integrals, surface integrals, and divergence theorem. Stokes’ theorem. Functions of a complex variable, including limits, derivatives, and Cauchy-Riemann equations. Exponential, trigonometric, hyperbolic, and logarithmic functions. Complex integrals, Cauchy’s integral theorem and formula. Taylor and Laurent series. Calculus of residues. Prerequisite: ordinary differential equations. No credit for students who have completed 259.
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3.00 Credits
Initial- and boundary-value problems for partial differential equations using separation of variables in conjunction with Fourier series and integrals. Explicit solutions of problems involving the heat equation, the wave equation, and Laplace’s equation. Prerequisite: ordinary differential equations, linear algebra.
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3.00 Credits
Transformations of the plane, groups of transformations, reflections, glide reflections, classification of the isometries of the plane, frieze groups, analysis of frieze patterns, wall paper groups, and analysis of wall paper patterns. Especially recommended for prospective teachers of mathematics. Prerequisite: linear algebra.
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3.00 Credits
Open sets, closed sets, continuity, compactness, and connectivity. Subspaces, product spaces, and quotient spaces. Knot theory, topology of surfaces, and applications.
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3.00 Credits
Manifolds in n-dimensional Euclidean space, smooth maps; inverse and implicit function theorems. Regular value theorem, immersions and submersions, Sard’s theorem, and transversality. Degree of a map; winding numbers and the Fundamental Theorem of Algebra; intersection theory modulo 2. Prerequisite: multivariable calculus, linear algebra.
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3.00 Credits
Applications of calculus and probability to actuarial science. The foundations of financial mathematics, including the theory of interest. Prerequisite: multivariable calculus. Pre- or corequisite: 216, 218, or 247.
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3.00 Credits
Probabilistic analysis of insurance. Single- life models, including time-value of benefits, life annuities, premiums, and benefit reserves: Multiple-decrement models; Multiple-life models. Probabilistic topics: Markov chains and Poisson processes. Prerequisite: 216, 218, or 247; and 246a.
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