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Course Criteria
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3.00 Credits
PQ: MATH 25000 or 25400, or CMSC 27100, or consent of instructor. Experience with mathematical proofs. Methods of enumeration, construction, and proof of existence of discrete structures are discussed in conjunction with the basic concepts of probability theory over a finite sample space. Enumeration techniques are applied to the calculation of probabilities, and, conversely, probabilistic arguments are used in the analysis of combinatorial structures. Other topics include basic counting, linear recurrences, generating functions, Latin squares, finite projective planes, graph theory, Ramsey theory, coloring graphs and set systems, random variables, independence, expected value, standard deviation, and Chebyshev's and Chernoff' s inequalities . L. Babai. Spring.
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3.00 Credits
PQ: CMSC 15300. This course covers the basics of the theory of finite graphs. Topics include shortest paths, spanning trees, counting techniques, matchings, Hamiltonian cycles, chromatic number, extremal graph theory, Turan's theorem, planarity, Menger' s theorem, the max-flow/min-cut theorem, Ramsey theory, directed graphs, strongly connected components, directed acyclic graphs, and tournaments. Techniques studied include the probabilistic method . Spring.
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3.00 Credits
PQ: Familiarity with basic discrete mathematics/statistics/algorithms and biology is helpful but not required. This course serves as a general introduction to the basic algorithms used to understand current problems in biology. Topics may include sequence alignment algorithms to study DNA and protein sequences, algorithms and experiments for protein structure prediction, dynamics, and folding, clustering and machine learning methods for gene expression analysis, computational models of RNA structure, and DNA computing and self-assembly. N. Hinrichs. Spring. Not offered 2009 C10; will be offered 201 0 -11.
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3.00 Credits
PQ: MATH 25400 or 25700. This course introduces mathematical logic. Topics include propositional and predicate logic and the syntactic notion of proof versus the semantic notion of truth (e.g., soundness, completeness). We also discuss the G del completeness theorem, the compactness theorem, and applications of compactness to algebraic problems. Autumn.
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3.00 Credits
PQ: MATH 27700 or equivalent. Topics include number theory, Peano arithmetic, Turing compatibility, unsolvable problems, G del's incompleteness theorem, undecidable theories (e.g., the theory of groups), quantifier elimination, and decidable theories (e.g., the theory of algebraically closed fields). Winter.
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3.00 Credits
PQ: One year of calculus, two quarters of physics at any level, and PHYS 25000 or prior programming experience. For course description, see Physics. Winter. L.
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3.00 Credits
PQ: CMSC 15300, or MATH 25000 or 25500. This course is a basic introduction to computability theory and formal languages. Topics include automata theory, regular languages, context-free languages, and Turing machines. Autumn.
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3.00 Credits
PQ: CMSC 27100, or MATH 25000 or 25500; and experience with mathematical proofs. Computability topics are discussed (e.g., the s-m-n theorem and the recursion theorem, resource-bounded computation). This course introduces complexity theory. Relationships between space and time, determinism and non-determinism, NP-completeness, and the P versus NP question are investigated. Spring.
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3.00 Credits
PQ: Consent of instructor. This course covers current topics in scientific computing. Autumn, Winter, Spring.
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3.00 Credits
PQ: A year of calculus (MATH 15300 or higher), a quarter of linear algebra (MATH 19620 or higher), and CMSC 10600 or higher; or consent of instructor. Basic processes of numerical computation are examined from both an experimental and theoretical point of view. This course deals with numerical linear algebra, approximation of functions, approximate integration and differentiation, Fourier transformation, solution of nonlinear equations, and the approximate solution of initial value problems for ordinary differential equations. We concentrate on a few widely used methods in each area covered. T. Dupont. Autumn. Not offered 2009 C10; will be offered 201 0 -11.
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