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Course Criteria
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3.00 Credits
PQ: Consent of instructor. This course emphasizes mathematical discovery and rigorous proof, illustrated on a variety of accessible and useful topics, including basic number theory, asymptotic growth of sequences, combinatorics and graph theory, discrete probability, and finite Markov chains. The course includes aan introduction to linear algebra. L. Babai. Autumn.
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3.00 Credits
PQ: Linear algebra, basic combinatorics, or consent of instructor. Methods of enumeration, construction, and proof of existence of discrete structures are discussed. The course emphasizes applications of linear algebra, number theory, and the probabilistic method to combinatorics. Applications to the theory of computing are indicated, and open problems are discussed. This course is offered in alternate years. L. Babai. Spring.
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3.00 Credits
PQ: Advanced knowledge of mathematics and consent of instructor. This course covers constructive combinatorial techniques in areas such as enumerative combinatorics, invariant theory, and representation theory of symmetric groups. Constructive techniques refer to techniques that have algorithmic flavor, such as those that are against purely existential techniques based on counting. K. Mulmuley. Spring. Not offered 2009 C10; will be offered 201 0 -11.
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3.00 Credits
N. Hinrichs. Spring. Not offered 2009 C10; will be offered 201 0 -11.
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3.00 Credits
PQ: Consent of instructor. Current topics in bioinformatics are covered in this course. Autumn, Winter, Spring.
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3.00 Credits
PQ: Consent of instructor. This course introduces concepts of systems biology. We also discuss computational methods for analysis, reconstruction, visualization, modeling, and simulation of complex cellular networks (e.g., biochemical pathways for metabolism, regulation, and signaling). Students explore systems of their own choosing and participate in developing algorithms and tools for comparative genomic analysis, metabolic pathway construction, stoichiometeric analysis, flux analysis, metabolic modeling, and cell simulation. We also focus on understanding the computer science challenges in the engineering of prokaryotic organisms. R. Stevens. Autumn. Not offered 2009 C10; will be offered 201 0 -11.
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3.00 Credits
PQ: STAT 34300 or consent of instructor. This course covers topics in numerical methods and computation that are useful in statistical research (e.g., simulation, random number generation, Monte Carlo methods, quadrature, optimization, matrix methods). Autumn.
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3.00 Credits
PQ: MATH 25500 or consent of instructor. CMSC 38000 is concerned with recursive (computable) functions and sets generated by an algorithm (recursively enumerable sets). Topics include various mathematical models for computations (e.g., Turing machines and Kleene schemata, enumeration and s-m-n theorems, the recursion theorem, classification of unsolvable problems, priority methods for the construction of recursively enumerable sets and degrees). CMSC 38100 treats classification of sets by the degree of information they encode, algebraic structure and degrees of recursively enumerable sets, advanced priority methods, and generalized recursion theory. This course is taught in alternate years. R. Soare. Winter, Spring.
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3.00 Credits
PQ: Consent of instructor. This course covers the basic mathematical theory behind numerical solution of partial differential equations. We investigate the convergence properties of finite element, finite difference and other discretization methods for solving partial differential equations, introducing Sobolev spaces and polynomial approximation theory. We emphasize error estimators, adaptivity, and optimal-order solvers for linear systems arising from PDEs. Special topics include PDEs of fluid mechanics, max-norm error estimates, and Bananch-space operator-interpolation techniques. T. Dupont. Spring.
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3.00 Credits
PQ: Consent of instructor. Part one consists of models for defining computable functions: primitive recursive functions, (general) recursive functions, and Turing machines; the Church-Turing Thesis; unsolvable problems; diagonalization; and properties of computably enumerable sets. Part two deals with Kolmogorov (resource bounded) complexity: the quantity of information in individual objects. Part three covers functions computable with time and space bounds of the Turing machine: polynomial time computability, the classes P and NP, NP-complete problems, polynomial time hierarchy, and P-space complete problems. A. Razborov. Winter.
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