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Course Criteria
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3.00 Credits
Discrete mathematics, inductive reasoning, counting problems, binomial coefficients and Pascal's triangle, Fibonacci numbers, combinatorial probability, divisibility and primes, partitions, and generating functions. Prerequisite: 240 or instructor' s consent.
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3.00 Credits
Introduction to graph theory: graphs, trees, matchings, planar graphs, colorings. Additional topics as time permits. Prerequisites: 291-1, 300, 306, or equivalent.
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1.00 Credits
Discrete probability spaces, random variables, expected value, combinatorial problems. Special distributions, independence, and conditional probability. Weak law and central limit theorem. 2. Convolution, central limit theorem, Markov processes in discrete time, recurrence, and transience. 3. Markov processes in continuous time, stationary process, second-order process, stochastic differential equations. Students may not receive credit for both 310-1 and 383 or 385. Prerequisites: 234, 240.
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3.00 Credits
Rigorous analysis in Euclidean space, beginning with one and proceeding to several variables. Properties of the real numbers, limits and continuity, differentiation and integration, sequences and series, the inverse and implicit function theorems. Applications to Fourier series. Primarily for undergraduates; open to graduate students only with departmental consent. Students may not receive credit for both 320-1 and 321-1 or both 320-2 and 321-2. Prerequisite: 234, 240, 300, or 291-1,2,3 or instructor's consent.
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3.00 Credits
Rigorous analysis in Euclidean space and on metric spaces. Metric space topology, properties of Euclidean spaces, limits and continuity, differentiation and integration, sequences and series, the inverse and implicit function theorems. Lebesgue integration with applications. Faster paced and more abstract than 320-1,2, this sequence covers more topics in more depth and aims at intensive development of students' ability to analyze and create mathematical proofs. Students may not receive credit for both 320-1 and 321-1 or both 320-2 and 321-2. Prerequisite: grade of A- or above in 291 or 334, A in 300, B or above in 331, or consent of department.
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3.00 Credits
Complex numbers, functions of a complex variable, theory of analytic functions, series development, analytic continuation, contour integration, conformal mapping. Students may not receive credit for both 325 and 360-3 or ESAM 311-3. Prerequisite: 250.
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3.00 Credits
Groups and their structure; elementary ring theory; polynomial rings. 2. Continuation of ring theory. 3. Field theory and Galois theory. Prerequisite: 291-1,2,3, 300, or instructor's consent. Students may not receive credit for corresponding quarters of both 330 and 331.
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1.00 Credits
Groups and their structure, including the Sylow theorems. 2. Ring theory; polynomial rings. Module theory, including applications to canonical form theorems of linear algebra. 3. Field theory; Galois theory. 331 differs from 330 in that it covers more topics in more depth and aims at intensive development of students' ability to analyze and create mathematical proofs. Prerequisite: consent of department. Students may not receive credit for corresponding quarters of both 330 and 331.
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3.00 Credits
Abstract theory of vector spaces and linear transformations. Complex vector spaces, unitary and Hermitian matrices. Jordan canonical form. Selected applications as time permits. Students who took 330-1 (formerly 337-1) prior to 2004 C05 may not also take 334 for credit toward the major without departmental consent.
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1.00 Credits
Divisibility and primes, congruences, quadratic reciprocity, Diophantine problems. 2. Additional topics in analytic and algebraic number theory. Prerequisite: 230.
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