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Math 223: Introduction to Object Oriented Programming and C++ Prerequisites: CS221 or consent of the instructor Frequency: Annually, Fall/Spring Term Credit: (3-0)3 Content: Programming paradigms. Comprehensive introduction to C++ language, giving a detailed grounding in C++ programming and fundamentals. Program structure; functions, flow of control, array and pointers, abstraction. Object-orientation; classes, operator overloading, inheritance. Mathematical application examples. Goals: The goal of this course is to eliminate the obstacles in designing mathematical applications, using the most suitable programming technique: Object oriented programming. In order to engage students for step by step transition through procedural abstractions, data abstraction, and object-orientation, the most popular scientific programming language C++ is given. Course Outline: Suggested textbook: D.M. Capper; Introducing C++ for Scientist, Engineers, and Mathematicians.
Math 258: Differential Equations II Prerequisites: Math 255 Frequency: Annually, Spring Term Credit: (3-0)3 Content: Existence and uniqueness theorems for IVP; first order equations, systems and higher order equations. Structure of linear problems. Boundary value problems and eigenvalue problems. Oscillation and comparison theorems. Goals: Suggested textbook: Albert Erkip; Introduction to Theoretical Aspects of Ordinary Differential Equations, Matematik Vakfi Yayini, ODTÜ
Math 301 Introduction to Probability Theory (3-0)3 Events and probability. Combinatorial problems. Independence and conditional probability. Measure theoretical approach to probability. Random variables and distribution functions. Marginal distributions and conditional distributions. Moments and characteristic functions. Convergence of random variables. Law of large numbers. Prerequisit: Advanced Calculus Textbook: B. Harris, Theory of Probability, Addison- Wesley, 1966. References for reading:
Allan,Gut. An intermediate course in probability /Springer-Verlag, 1995 Y. G. Sinai. Probability theory: an introductory course /Springer-Verlag 1992. Harold J. Larson. Introduction to Probability / Addison – Wesley 1994. Courses Objectives: Probability theory is one of the most important branches of Mathematics providing powerful tools for applied sciences and having strong implications in the development of Mathematics. The majority of mathematical models in Physics, Chemistry, Biology, Engineering Sciences, Computer Sciences, Economic and Social Sciences are probabilistic models. On the other hand, Probability Theory is a source of new problems in Number Theory, Numerical Analysis, Functional Analysis, Potential Theory, etc. Therefore, this course is higly justified as an initiation to mathematical modeling. Course Outline:
Justification of the course proposal:
Math 303: History of Mathematical Concepts I Prerequisites: Consent of the instructor. Frequency: Annually, Fall Term Credit: (3-0)3 Content: Mathematics in Egypt and Mesopotamia, Ionia and pythagoreans, paradoxes of Zeno and the heroic age. Mathematical works of Plato, Aristotle, Euclid of Alexandria, Archimedes, Appolonius and Diophantus. Mathematics in China and India. Goals: It is intendent to provide an adequate explanation of how mathematics came to occupy its position as a primary cultured force in civilization. Suggested textbooks: D.M. Burton; The History of Mathematics (An introduction)WCB Wbm.C.Brown Publishers, 1988 C.B.Boyer, U.C.Merzbach; A History of Mathematics, John Wiley & Sons., 1989
Math 304: History of Mathematical Concepts II Prerequisites: Consent of the instructor. Frequency: Annually, Spring Term Credit: (3-0)3 Content: Mathematics of the renaissance, islamic contributions. Solution of the cubic equation and consequences. Invention of logarithms. Time of Fermat and Descartes. Development of the limit concept. Newton and Leibniz. The age of Euler. Contributions of Gauss and Cauchy. Non-Euclidean geometries. The arithmetization of analysis. The rise of abstract algebra. Aspects of the twentieth century. Goals: This course is a continuation of Math 303 Suggested textbooks: D.M. Burton; The History of Mathematics (An introduction)WCB Wbm.C.Brown Publishers, 1988 C.B.Boyer, U.C.Merzbach; A History of Mathematics, John Wiley & Sons., 1989
Math 319: Lebesgue Integral Prerequisites: Math 252 Frequency: Annually, Fall Term Credit: (3-0)3 Content: Review of Riemann integration. Sets of (Lebesgue) measure zero in Rn and characterization of Riemann integrable functions. Lebesgue integrable functions and the Lebesgue integral in Rn. Convergence theorems, theorems of Lusin and Egorov. Fubini’s theorem, convolutions. Differentiablity properties of functions and integrals. Selected applications of the Lebesgue theory. Goals: Course Outline: Week 1: Preliminaries: Topological Concepts in R, Continuous functions on Metric Spaces. Week 2: Riemann Integral, Sets of Measure Zero, Existence and Deficiencies of Riemann Integral. Week 3: Step Functions and their integrals, Two Fundamental Lemma, The Class L+ Week 4: The Lebesgue Integral. Monotone Convergence and Dominated Convergence Theorems. Week 5: The space L. Measurable Functions. Lebesgue Measure. Week 6: s -Algebra and Borel Sets. Nonmeasurable sets. Structure of Measurable sets. More about measurable functions. Week 7: Egoroff’s theorem. Steinhaus’ theory. The Cauchy Functional Equations. Week 8: Lebesgue Outer and Inner Measures. The Integral on Measurable Sets. The integral on Infinite intervals. Week 9: Lebesgue measure on R. Finite Additive Measure, the Double Lebesgue Integral. Week 10: The Fubini theorem. The Complex Integral. Week 11: Nonhere differentiable Functions. The Dini derivatives. The Rising sum Lemma and Differentiability of Monotone Functions. Week 12: Functions of Bounded Variation. Absolute Countinuity. Week 13: The Fundamental Theorem of Calculus The Lp space. Week 14: The Lp space (continued) Approximations by Continuous Functions. Suggested textbook: Soo Bong Chae: Lebesgue Integration (2nd Edition) Springer Verlag 1995.
Math 320: Set Theory Prerequisites: Consent of the instructor. Frequency: Upon request, Spring Term Credit: (3-0)3 Content: Language and axioms of set theory. Ordered pairs, relations and functions. Order relation and well ordered sets. Ordinal numbers, transfinite induction, arithmetic of ordinal numbers. Cardinality and arithmatic of cardinal numbers. Axiom of choice, generalized continuum hypothesis. Goals: Suggested textbooks: Levy; Basic Set Theory, J.Malitz; Introduction to Mathematical Logic.
Math 321: Automata and Languages Prerequisites: Consent of the instructor Frequency: Upon request, Spring Term Credit: (3-0)3 Content: Automata, finite state automata. Minimal and Reduced automatas, Transformation monoid. Languages, Phrase structure grammers, Regular and Rational languages, Context free languages. Variteis, F-varieties, Star free languages and aperiodic monoids. Goals: The student will have first contact to the theory that enables him to understand recent papers and that is a foundation for more advanced topics as solvability and other decision problems. Suggested textbook: J.M.Howie; Automata and languages, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1991.
Math 341: Graph Theory Prerequisites: Consent of the Instructor or Math 112 Frequency: Upon request, Fall Term Credit: (3-0)3 Content: Graphs, vareties of graphs, connectedness, extremal graphs, blocks, trees, partitions, line graphs, planarity, Kuratowsky’s theorem, colorability, Chromatice numbers, Five Color theorem, Four Color conjecture. Goals: The student should know and apply class’cal results of the theory. Also the ability of the student to model a problem into mathematical language will be improved. Suggested textbooks: F. Harary; Graph Theory, Addison Wesley Publishing Company, Reading Mass.-Menlo Park, Calif.-London 1969
Math 344: Introduction to Universal Algebra Prerequisites: Consent of the Instructor Frequency: Upon request, Spring Term Credit: (3-0)3 Content: Lattices: distributive and modular lattices, complete and algebraic lattices, Boolean lattices. Semigroups: Green’s equivalence, semilattice decomposition, completely simple semigroups. Universal Algebra: algebraic lattices and subinverse, congurence and quotients algebras, free algebras. Goals: The student should understand the most general and fundamental notions of universal algebra-these include results that apply to all types of algebras-and he sholud learn how sets of algebras are studied. Suggested textbooks: J.W. Howie; An introduction to semigroup theory. L.M.S. Monographs, No.7. Academic Press (Harcourt Brace Jovanovich, Publishers), London-New York, 1976. S.Burris, H.P. Sankappanavar; A course in universal algebra. Graduate Texts in Mathematics, 78. Springer - Verlag, New York-Berlin, 1981.
Prerequisites: Consent of the instructor Frequency: Annually, Fall Term Credit: (3-0)3 Content: Fourier series. The Fourier transform, inverse Fourier transform. The Laplace transform. The inversion integral for the Laplace transform (complex contour integration). Applications of Laplace transform to linear ordinary, partial differential and integral equations. The Z-transform. The inversion integral for the Z-transform. Applications of Z-transform to difference equations and linear networks. Goals: Suggested textbooks:
Math 365: Elementary Number Theory I Prerequisites: Consent of the Instructor Frequency: Annually, Fall Term Credit: (3-0)3 Content: Divisibility, Congruences, Euler, Chinese Remainder and Wilson’s Theorems. Arithmetical functions. Primitive roots. Quadratic residues and quadratic reciprocity. Diophantine equations. Goals: Suggested textbook: D.M.Burton; Elementary Number Theory, Wn.C.Brown Publishel, 1989
Math 366: Elementary Number Theory II Prerequisites: Math 365 or consent of the Instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: Arithmetic in Quadratic fields. Factorization theory, Continued fractions, Periodicity. Transcendental numbers. Goals: Suggested textbook:
Prerequisites: Math 367 Frequency: Annually, Spring Term Credit: (3-0)3 Content: Field Extensions, Splitting field of a polynomial, Multiple roots, Galois group, Criteria for solvability by radicals, Galois group as permutation groups of the roots of polynomials of degree n, Constructible n-gons, Transcendence of e, Finite fields. Goals: Suggested textbook:
Math 373: Geometry I Prerequisites: Consent of the instructor Frequency: Upon request, Fall Term Credit: (3-0)3 Content: Foundations: The parallel axiom, models, Hilbert’s theorem. Triangles: Theorems of Menelaus and Ceva, classical remarkable points. Circles: Power of a point with respect to a circle, coaxal systems of circles, inversive geometry. Conic sections: Focus and directrix, reflection property, theorems of Poncelet. Goals: Outlining foundational issues very briefly, this course aims at providing the student with a robust knowledge of Euclidean geometry in its classical setting. Suggested textbooks: D.Pedoe; A Course of Geometry for Colleges and Universities. H.S.M.Coxeter, S.L.Greitzer; Geometry Revisited L.S.Shively; An Introduction to Modern Geometry
Math 374: Geometry II Prerequisites: Math 262 and Math 373 Frequency: Annually, Spring Term Credit: (3-0)3 Content: Projective spaces over division rings. Theorems of Desargues and Pappus. Harmonic ranges and pencils, collineations, correlations, involutions, polarities. Affine geometry via “the line at infinity”. Euclidean geometry with “circular points at infinity”. Conic sections and quadratic surfaces. Goals: The subject of this course is projective spaces in their unifying capacities presented mainly by means of homogenous coordinates with occasional incursions into axiomatics. Suggested textbooks: H.S.M.Coxeter; Projective Geometry W.T.Fishback; Projective and Euclidean Geometry B.C.Patterson; Projective Geometry
Math 375: Periodic Distributions and Fourier Series Prerequisites: Math 349 Frequency: Annually, Fall Term Credit: (3-0)3 Content: Properties of periodic functions, convolution, approximation, Weierstrass approximation theorem. Periodic distributions, operations on periodic distributions. Hilbert spaces, L2, orthogonal expansions, Fourier series. Applications of Fourier series. Goals: Suggested textbook:
Math 381: Numerical Analysis I Prerequisites: Math 262 and Ceng 221 Frequency: Annually, Fall Term Credit: (3-0)3 Content: Basic concepts: Convergence, stability, error analysis and conditioning. Solving systems of linear equations: Matrix algebra, the LU and Cholosky factorization, pivoting, norms and error analysis in Gaussian elimination. Matrix eigenvalue problem, power method, orthogonal factorizations and least squares problems. Solutions of nonlinear equations. Biscestion, Newton’s, secant and fixed point iteration methods. Goals: This course forms the basis for all the other undergraduate and graduate numerical analysis courses. The concept of error, convergence and stability analysis is provided. The basic numerical methods for solving linear and nonlinear equations, eigenvalue problems are given. Suggested textbook: D. Kincaid, W. Cheney; Numerical Analysis, Thomson Information/Publishing Group
Math 382: Numerical Analysis II Prerequisites: Math 381 Frequency: Annually, Spring Term Credit: (3-0)3 Content: Approximating functions: polynomial interpolation, divided differences, Hermite interpolation, Spline interpolation, The B-Splines, Taylor Series, least square approximation. Numerical differentiation and integration: Numerical differentiation and Richardson extrapolation, numerical differentiation based on interpolation, Gaussian quadrature, Romberg integration, adaptive quadrature, Bernoulli polynomials and Euler-Maclaurin formula. Goals: This course gives interpolation, differentiation and integration techniques numerically together with error, convergence and stability analysis. Suggested textbook: D. Kincaid, W. Cheney; Numerical Analysis, Thomson Information/Publishing Group
Math 385: Special Functions of Applied Mathematics I Prerequisites: Math 255 or Math 253 or Math 254 or Consent of the instructor Frequency: Annually, Fall Term Credit: (3-0)3 Content: Gamma and Beta functions; definitions and basic properties, Pochhammer’s symbol. Hypergeometric series; elementary series manipulations, linear transformations. Hypergeometric differential equation; Ordinary and Confluent hypergeometric functions. Generalized hypergeometric functions; the contiguous function relations. Bessel function; the functional relationships, Bessel’s differential equation. Orthogonality of Bessel functions. Goals: This course is intended primarily for the student of mathematics, physics or engineering who wishes to study the “special” functions in connection with the use of “hypergeometric functions”. Suggested textbook:
Math 386: Special Functions of Applied Mathematics II Prerequisites: Math 385 or Consent of the instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: Legendre functions: Generating Function of Legendre polynomials, Special values, Recurrence relations, Murphy and Rodrigues formulas. Integrals involving Legendre polynomials, series of Legendre polynomials. Legendre differential equation. Associated Legendre functions. Hermite functions: Generating function of Hermite polynomials, recurence relations. Hermite’s differential equations. Laguerre functions, Laguerre differential equations, Associated Laguerre polynomials. Goals: This course is intended primarily for the student of mathematics, physics or engineering who wishes to use the “orthogonal” polynomials associated with the names of Legendre, Hermite and Languerre. It aims at providing in a compact form most of the properties of these polynomials in the simplest possible way. Suggested textbook:
Math 387: Advanced Object Oriented Programming Prerequisites: Math 223 or consent of the instructor Frequency: Annually, Fall Term Credit: (3-0)3 Content: Further features of the C++ Language and principles of Object Oriented Programming, including dynamic data-structures, stream input and output, encapsulation, inheritance, polymorphism, and further C ++ language constructs. If time permits, some object-oriented design and analysis models and methodology will be given. Covers, case studies of OOP in Mathematics. Goals: Give concept and technipues of Object Oriented Programming and C++ on High Level. Suggested textbook:
Math 388: Data Structures Prerequisites: Math 387, Math 223 or consent of the Instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: Review of stacks, queues, linked lists. Algorithms for searching and sorting. More complicated data structures such as multi-linked lists, trees, graphs. Throughout the course there will be an emphasis on the efficiency of the various algorithms. Goals: Suggested textbooks: W.Ford; Data Structures C ++, Prentice-Hall, 1996 L.Yedidyah; Data Structures using C and C ++, Prentice-Hall, E.Horowitz; Fundamentals of Data Structures in C ++, Computer Science Press, New York, 1995
Prerequisites: Frequency: Annually, Spring semester. Credit: (3-0)3 Content: Introductory information about reduce. Structure of programs, built in prefix operators. Procedures. A computer Algebra system. How to use a Computer Algebra systems. Representations of polynomials, rational functions, algebraic functions, matrices and series. Advanced algorithms. g.c.d. in several variables. Other applications of modular methods. P-adic Methods. Formal integration and differential equations. Goals: Computer algebra have already been applied to a large number of different areas in science and engineering. The most extensive use has occured in the fields where the algebraic calculations necessary are extremely tedious and time consuming, such as general relativity, celestrial mechanics. Before the advent of computer algebra such hand calculations took many months to complete and were error prone. Personal workstations can now perform much larger calculations without error in a matter of minutes. Given the potential of computer algebra in scientific problem solving, it is clear that this course is needed to narrow to gap between the department’s numerical analysis courses and computer literacy. Course Outline: (1 Week ) Introductory information about reduce (2 Weeks ) Structure of programs. Built-in prefix oper. Procedures. (1 Week ) A Computer Algebra system. How to use a Comp. Alg. Systems. (2 Weeks) Representations of poly., rational func., algeb. func., matrices and series. (1 Week) Simplifications of poly. equ. and real poly. systems. (2 Weeks) Advanced algorithms, g.c.d. in several variables. (1 Week) Other applications of modular methods. (2 Weeks) P-adic methods. (2 Weeks) Formal integration and differential equations. Suggested textbooks: J.H. Davenport, Y.Siret, E. Tournier; Computer Algebra, Academic Press, 1993.
Math 395: Symbolic Programing Languages (Prolog) Prerequisites: Math 223, Ceng 221 or consent of the instructor Frequency: Annually, Fall Term Credit: (3-0)3 Content: An overview of language. Syntax and meaning of programs. Lists, operators and arithmetic structures. Controlling, backtracking. Unification. Input and output. More built in procedures. Programming style and techniques. Recursion. Operators on data structures. Advanced tree representations. Goals: Suggested textbooks: N.C. Rowe; Artificial Intelligence through Prolog, Prentice-Hall, 1988 L.Sterling; The art of Prolog: Advanced programing, MIT Press, 1987 I. Bratko; Prolog Programming for artificial, Addison-Wesley, 1990 W.F. Clocksin ; Programming in Prolog, Springer, 1987 U.Nilsson; Logic, programming, and prolog, John-Wiley, 1995
Math 396: Artificial Intelligence and Applications Prerequisites: Math 387, Math 223, Ceng 221 or consent of the instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: Basic problem-solving strategies. A Heuristic search principle. Problem reduction and AND/OR graphs. Expert systems and knowledge representation. An expert system shell. Planning. Language processing with grammer rules. Machine learning. Game playing. Logic and uncertainty. Meta pragramming. Goals: Suggested textbook: N.C. Rowe; Artificial Intelligence through Prolog, Prentice-Hall, 1988
Math 400: Basic Distribution Theory Prerequisites: Math 319 Frequency: Annually, Spring Term Credit: (3-0)3 Content: Test functions. Spaces E,D,S and their duals. Differentiation, localization, convolution, Fourier transform. Supports and singular supports. Kernels and tensor products. Homogeneous distributions. Fundamental solutions of differential operators. Outline of distribution theory on manifolds. Goals: Suggested textbook:
Prerequisites: Real Analysis Frequency: Annually,Fall Term Credit: (3-0)3 Content: Events and probability. Combinatorial problems and equally likely events. Probability spaces. Independence and finite product spaces. Random variables and distribution functions. Integration of random variables. Lp - spaces. Convergence of random variables. Conditional expectation. Canonical space of a stochastic process. Markov chains. Martingales. Goals: Probability Theory is one of the most important branches of mathematics providing powerful tools for applied sciences and having strong implications in the development of mathematics. On the other hand, Probability Theory is a source of new problems in Number Theory, Numerical Analysis, Functional analysis, Potential Theory, Differential Geometry... Therefore, this course is highly justified as a fundamental prerequisit for mathematical modeling and for advanced level courses of algebra and analysis. Course Outline: (1 Week) Probability on a Boolean algebra of events, combinatorial problems and equally likely outcomes. (2 Weeks) Probability measure on a s -algebra. Monotone class theorem. Conditional probability. Bayes formula. Extension of a probability. Independence and finite product spaces. (1 Week) Random variables, distribution functions, induced probability (2 Weeks) Integration of random variables. Lp - spaces. (2 Weeks) Notions of convergence. (1 Week) Conditional expectation and conditional distribution. (1 Week) Infinite product spaces. Canonical space of a process. (2 Weeks) Countable state space Markov chains (2 Weeks) Discrete time martingales. Suggested textbook: J.C. Taylor; An Introduction to Measure and Probability. Springer, 1997. Harris; Theory of Probability, Addison-Wesley, 1966
Prerequisites: Consent of the instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: The importance of optimization, examples of optimization problems, basic definition and facts on convex analysis. Theory of linear programming and convex programming, simplex method and its applications, introduction to nonlinear programming, search methods, basic ideas of classical variational calculus, optimal control theory, Pontraygin’s maximum principle and dynamic programming linear theory of optimal control. Goals: Suggested textbook:
Math 403: Foundations of Mathematics Prerequisites: Consent of the instructor Frequency: Biannually, Fall/Spring Term Credit: (3-0)3 Content: Apriosits programme, mathematical empiricism, scientific and mathematical change, formalists, intuitionists, constructivist and logicist views. Goals: Suggested textbook: Math 404: Introduction to Vector Lattices and Applications (3-0)3 Catalog Description: Basic elementary properties of Riesz spaces.
Sublattices, ideals and Bands. Order convergence and properties. Order
units. Normed Riesz spaces. Elementary duality results; Disjoint sequences,
order continuity of the norm. Operators on Riesz spaces.
Springer, 1997.
Most important elements of the analysis courses that we deal with in undergraduate courses are natural examples of the vector lattices. To give few examples: Rn, matrices, space of measurable functions, continuous functions, integrable functions and all sequence spaces are examples of vector lattices. This makes vector lattices as one of the most important and common space in mathematical analysis. Some nice tools of this theory can be used to explain highly complicated economic issues. Course Outline:
Justification of the course proposal: This is a course that will be helpfull to mathematics students who would like to pursue in an active field of research as well as those who are study industrial engineering and Economics. It is also designed to show mathematics students that elementary mathematics can be used to solve real life problems. Overlapping and Complementing : none Effective Date: Spring 1998-1999 Frequency: Every other year Opinion of the Departmental Academic Board: 23-8 in support Math 405: Combinatorics Prerequisites: Consent of the Instructor Frequency: Credit: (3-0)3 Content: Basic counting: Permutations, r-permutations, combinations, multinomial coefficients, occupancy problems, good algorithms. Generating functions: Power series, operating on generating functions, applications to counting, binomial theorem, exponential generating functions, probability generating functions. Recurrence relations: Simple recurrences, linear recurrence relations, characteristic equations, solving recurrences using generating functions, simultaneous equations, recurrences involving convolutions. Divide and conquer algorithms. Experimental design: Blockdesign, balanced incomplete blockdesign. Applications: coding theory, Hadamard designs. Goals: The aim of the course is to introduce various counting technics to the students so that they are able to apply them in standard problems to modify and calibrate them to special cases. Moreover the student should be able to recognize parts of a problem that is combinatorial in nature. Course Outline:
Suggested textbook: Fred. S. Roberts: Applied Combinatorics, Allyn&Bacon, 1984 (ISBN: 0130393134) Reference Books: R. L. Graham, M. Grötschel, L. Lovasz, Handbook
of Combinatorics, Vol. I, II, Elsevier, 1995 (ISBN: 0262071703)
Prerequisites: Frequency: Annually, Spring Semester. Credit: (3-0)3 Content: Model theory is a new branch of mathematical logic which deals with the connection between a formal language and its interpretation. The main goal is to present model theory of first order predicate logic which is the simplest language that has applications to the main body of mathematics. Course Outline: Week 1: First order language Week 2: Logical axioms and derivation rules Week 3: Deduction theorem Week 4: Structures and satisfactation Week 5: Completeness and consistency Week 6: Elementary equivalence and imbedding Week 7: Compactness theorem Week 8: Löwenheim-Skolem theorem Week 9: Interpolotion and definability Week 10: Extensions of first order logic Suggested textbooks: H.D. Ebblinghaus, J. Flum, W. Thomas; Mathematical Logic MATH 410 Modeling Mathematical Methods and Scientific Computing (2-2)3 Catalogue Description: Introduction to numerical and symbolical computational tools. Balance equations, continuous system models and partial differential equations. Introduction to numerical methods for ordinary and partial differential equations. Case studies from mechanics, fluid dynamics, heat and mass transfer, electrical engineering. Introduction to stochastic process and differential equations. Models from mathematical finance. Prerequisite courses: Consent of the Department Textbook(s): - Modeling Mathematical Methods and Scientific Computing, N.
Bellomo,
L. Preziosi, CR Press,
Course objectives: This course will give an introduction to the problems that arise in mathematical modeling by combining modem aspects of computing. Students will learn about the nature of mathematical modeling, starting with a physical model, representing it mathematically, simplifying and solving the resulting model and interpreting the results. They should also learn how to work in teams, communicate and present their results. Course outline: 1. Week: Introduction to numerical and symbolical computational tools
Faculty member(s) proposing the course: Prof. Dr. Bülent
Karasözen
Justification for course proposal:
Overlapping and Complimenting Courses Compliments the courses: None Effective date: Spring 2002 Frequency: Once in a year Opinion of the Departmental Board: Affirmative
Math 420: Elementary Point Set Topology Prerequisites: Math 251 Frequency: Annually,Fall Term Credit: (3-0)3 Content: Topological Spaces; Basis, subbasis subspaces. Closed sets, limit points. Hausdorff Spaces. Continuous functions, homeomorphisms. Product topology (with emphasis on finite number of factors). Connected spaces, components, path connectedness, path components. Compactness, sequential compactness, compactness in metric spaces. Definition of regular and normal spaces. Urysohn’s Lemma, Tietsze Extension Theorem. Goals: The purpose of this course is to expose the students to the basic concepts of topology and a few related ideas which are among the core material of mathematics and which form the language of higher level courses in most areas of mathematics. Course Outline: (3 Weeks) Introductory concepts (5 Weeks) Topological spaces and continuity (3 Weeks) Connectesness and compactness (3 Weeks) Countability and seperation axioms Suggested textbooks: J.Munkres; “Topology, a first course” G. Simmons; “An Introduction to Topology and Analysis.
Math 422: Elementary Geometric Topology Prerequisites: Math 252 or consent of the instructor Frequency: Annually,Spring Term Credit: (3-0)3 Content: Topology of subsets of Euclidean space. Topological surfaces. Surfaces in Rn. Surfaces via glueing, connected sum and the classification of compact connected surfaces. Simplicial complexes and simplicial surfaces (simplicial complexes with underlying spaces that are topological surfaces). Euler characteristic. Goals: The aim of this course is to introduce the student to some algebraic and differential topological ideas at an early stage emphasizing unity with geometry and more generally introduce the student to the relation of the modern axiomatic approach in mathematics to geometric intuition. Course Outline: (2 Weeks) Topology of subsets of Euclidean space (4 Weeks) Topological surfaces: Surfaces in IRn, surfaces via gluenig, connected sums and classification of compact connected surfaces (3 Weeks) Simplices and simplicial complexes (3 Weeks) Simplicial surfaces, Euler characteristic proof of classification of compact surfaces. Suggested textbook: Ethan D. Bloch, A first course in Geometric Topology and Differential Geometry
Math 441: Mechanics I Prerequisites: Math 255 Frequency: Upon request, Spring Term Credit: (3-0)3 Content: Statics of rigid bodies, equations of equilibrium, statics of suspended strings and cables. Kinematics of a particle. Translation, rotation of rigid body about an axis and about a fixed point, relative motion and Coriolis acceleration. Dynamics of a particle, harmonic oscillators, motion of a simple pendulum, flight of a projectile, motion under the action of central forces. The solar system, dynamics of a system of particles, motion of a body with varying mass. Goals: Suggested textbook: B.Nuriyev; Lectures on Mechanics I, METU, Department of Mathematics, 1995
Math 442: Mechanics II Prerequisites: Math 358 and Math 441 Frequency: Upon request, Spring Term Credit: (3-0)3 Content: Analytical statics; principle of virtual work; equilibrium of conservative forces; stability of equilibrium. Lagrange’s equation of first and second-kind; Hamilton’s canonical equations; variational principles of mechanics; Poisson brackets, canonical transformations and generating functions; the Hamilton-Jacobi equation; action and angle variables; Completely integrable systems; canonical perturbation theory; Kolmogorov-Moser-Arnolds theorem. Fundamentals of continuum mechanics. Goals: Suggested textbook: B.Nuriyev; Lectures on Mechanics II, METU, Department of Mathematics, 1995
Prerequisites: Consent of the instructor Frequency: Annually,Spring Term Credit: (3-0)3 Content: Primal Algebras: Maltsev Conditions. Characterisation
of finite primal algebras. Term Conditions and Commutator. Characterisation
of abelian algebras. Generalized term conditions. Finite Algebras (Tame
congruence theory): Minimal algebras. Induced algebras. ?-minimal sets.
Trace algebras. Tame congruences. Types of minimal algebras.
Course Outline: (1 Week) Review of Math 344 (1 Weeks) Definition of primal algebras, examples and Post’s Theorem (1 Week) Maltsev conditions and congruence permutable varieties (1 Week) Characterization of finite primal algebras (1 Week) Term conditions, abelian algebras, term conditions with respect to a pair of congruences (1 Week) Commutator and polynomial equivalence. Nilpotent algebras (1 Week) Theorem of Gumm (1 Week) Minimal algebras, induced algebras and idempotent mappings (1 Week) minimal sets, a-minimal sets (1 Week) L-endomorphisms, extensive ant strong extensive mappings (1 Week) Tame-congruences and polynomial isomorphisms (1 Week) algebras of unary polynomials, Theorem of Post and Palfy (1 Week) Types of minimal algebras (1 Week) Theorem of Palfy Suggested textbooks:.Burris, H.P: Sankappanavar: A course in
universal algebra. Springer, New York, 1981.
Math 450: Potential Theory in The Complex Plane Prerequisites: Math 353 Frequency: Upon request, Spring Term Credit: (3-0)3 Content: Harmonic functions, Subharmonic functions, Potential Theory, The Dirichlet Problem, Capacity, Some Applications. Goals: Potential Theory in the plane contais all the essential ingredients of the subject and yet it is relatively easy and quick to treat. It worths mastering. There is also another goal, (hinted at by the use of the word “Complex” in the title): It is to emphasize the very close connection between potential theory and complex analysis; conformal mapping can be used to speed up and simplify proofs of some results in potential theory. Going the other way, theorems in potential theory have a multitude of applications an complex analysis. Suggested textbook: T. Ransford; Potential Theory in the Complex Plane, London Mathematical Society, Student Texts 28, 1995 Math 452:Introduction to Functional Analysis Prerequisites:Consent of the instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: Normed linear spaces, Banach spaces. Hahn-Banach Theorem and consequences, Baire category Theorem. Uniform boundedness principle. Open Mapping and Closed graph theorems. Selected Topics and applications. Goals: The aim of this course is to assist the student in learning ideas about vector spaces and linear operators and to lead them to an appreciation of the unifiying power of the abstract vector space point of view in surveying the problems of algebra, analysis and the theory of integration, differential and integral equations. Course Outline: (1 Week) Vector Spaces, Hamel bases (4 Weeks) Topological Spaces and Metric Spaces(Convergence of sequences; open, closed compact subsets, completion of a metric space) (2 Weeks) Norm spaces, Banach spaces, bounded operators between normed spaces (2 Weeks) Hahn-Banach theorem and consequences (2 Weeks) Baire’s theorem, uniform boundedness principle, open mapping and closed graph theorems. (2 Weeks) Selected topics and some applications Suggested textbook: T. Terzio?lu; Fonksiyonel Analizin Yöntemleri, TUB?TAK yay?n?,
Math 453: Introduction to Complex Analysis Prerequisites:Math 353 Frequency: Annually, Fall Term Credit: (3-0)3 Content: Riemann mapping theorem and Schwarz-Christofel transformations zn, z1/n. Elementary Riemann surfaces. Applications of conformal mapping: (flows, heat conduction, electrostatistics,...) Analytic continuation. Argument principle, Rouche’s theorem. Mapping properties of analytic functions (inverse function theorem, open mapping theorem, maximum modulus theorem). Goals: Course Outline: Suggested textbook:
Prerequisites: Math 353 Frequency: Annually, Spring Term Credit: (3-0)3 Content: Cauchy theorems; maximum principle; Schwarz lemma; argument principle; normal families; Riemann mapping theorem; isolated singularities. Riemannian metrics; isometries; Poincare metric; Schwarz lemma. Curvature; Liouville theorem; spherical metric; Montel theorem; Picard theorems. Caratheodory metric; Kobayashi metric; their completeness; automorphisms of domains; hyperbolicity. Functions of several complex variables; ball and polydisc; their inequivalence. Goals: This course intends to show how geometry and analysis interact by giving a unifying picture of complex analysis and differential geometry. Geometric methods not only provide simpler proofs of analytic results, but also cause a deeper understanding of more advanced topics. These ideas will bring about new insight for future studies in mathematics. Course Outline: (2,5 Weeks) Principal ideas of complex analysis (3 Weeks) Basic notions of differential geometry (3 Weeks) Curvature and applications (3,5 Weeks) Invariant metrics (2 Weeks) A brief view of several complex variables. Suggested textbook: S. G. Krantz; Complex Analysis: The Geometric Viewpoint, The Mathematical Association of America, 1990
Math 456: Fourier Analysis and Wavelets Prerequisite.Math 349 or consent of the department. Catalog Description. Orthogonality
and modes of
convergence.
Textbook. A First Course in Wavelets with Fourier Analysis, A. Boggess & F. J. Narcowich, Prentice Hall, Upper Saddle River, 2001. References. 1. Wavelets Made Easy,
Y.
Nievergelt,
Birkhauser, Boston, 1999.
Course objectives 1. To present
applications of
analysis to concrete areas such as waves,
signals, etc.
3 weeks Fourier series and their convergence (1.1-1.3) 2 weeks Fourier transform and its applications (2.1-2.4) (Avoid rigorous proofs, but do applications, and omit uncertainty principle.) 1 week Discrete Fourier transform (3.1-3.1.2, 3.2.1) (Omit Z-transform, and touch upon fast Fourier transform only if time allows.) 3 weeks Haar wavelets (4.1-4:4) 3 weeks Multiresolution analysis and Daubechies wavelets (5.1-5.3, 6.1) (Do only the simplest Daubechies wavelets to have a second example.) Math 457: Calculus on Manifolds Prerequisites:Math 262 and Math 252 Frequency: Annually, Fall Term Credit: (3-0)3 Content: Review of differentiation, inverse and implicit function theorems, integration on subsets of Euclidean space, tensor, differential forms, integration on chains, integration on manifolds. Stokes’ theorem. Goals: Suggested textbook: M.Spivak; Calculus on Manifolds, W.A. Benjamin, Inc. 1965
Math 461: Rings and Modules Prerequisites:Math 367 Frequency: Annually, Fall Term Credit: (3-0)3 Content: Classical theory of rings, ideal theory, isomorphism theorems, selected topics in commutative rings. The group ring. Localization. Submodules, intersections and sums of modules, internal direct sums, factor modules (and factor rings). Homomorphisms of modules (and rings). Classical isomorphism theorems. The endomorphism ring of a module. Connection between the internal and external direct sums. Free modules, free and divisible abelian groups. Tensor product of modules. Finitely generated modules over principal ideal domains. Goals: Course Outline: Suggested textbook: T.W.Hungerford; Algebra, Reinhart & Winstor,
1974
Math 463: Introduction to Group Theory Prerequisites:Math 367 Frequency: Annually, Fall Term Credit: (3-0)3 Content: Group, subgroup, normal subgroup, cyclic subgroup, coset, quotient group. Commutator subgroup, center, homomorphism and isomorphism Theorems (invariant subgroup, wreath products), Abelian groups. Free abelian group, rank of an abelian group Divisible abelian group, Periodic Abelian group, Sylow Theorems and their applications, Soluble groups, Nilpotent groups. Goals: Suggested textbook:
Math 464: Introduction to Representation Theory Prerequisites:Math 463 or Consent of the Instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: Group representations, FG-Modules, Machke Theorem, Irreducible modules and Group algebras, Characters, inner products of characters, the number of irreducible characters, character table, induced modules and characters, Algebraic integers and real representations. Goals: Suggested textbook:
Math 466 — Groups and Geometry (3-0)3 Catalogue Description: Symmetry. Isometries of Rn. the Euclidean group, symmetry groups of regular polygons and polyhedra, classification of finite subgroups of the three dimensional rotation group. Frieze groups, crystals,wallpaper groups, groups acting on trees. Reflection groups, root systems, classification of finite reflection groups, crystallographic root systems and Weyl groups. Prerequisite Courses: Math 367 Textbooks:
Reference Books:
Course Objectives:
Course Outline:
Math 470: File Structures Prerequisites: Math 387 or consent of the instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: Update, merge and sort algorithms for sequential files. Direct access files with hashing. Indexed sequential files. Multikey organization. Operations involving data from several files.If time permits some database concepts may be included. Goals: Suggested textbooks: M.J. Folk; File Structures, Addison-Wesley, 1992 P.L. Livadas; File Structures: Theory and Practice, Prentice-Hall, 1990 B.J. Salzberg; File Structures: An Analytical Approach, Prentice-Hall, 1988
Math 471: Hyperbolic Geometry Prerequisites:Math 252. Frequency: Annually, Fall Term Credit: (3-0)3 Content:Parallel postulate and the need for non-Euclidean geometry, models of the hyperbolic plane, Möbius group, classification of Möbius transformations, classical geometric notions such as length, distance, isometry, parallelism, convexity, area, trigonometry in the hyperbolic plane, groups acting on the hyperbolic plane, fundamental domains. Textbook. Hyperbolic Geometry, 3. W. Anderson, Springer, London, 1999. References. I. Hyperbolic Geometry, J. W. Cannon, W. J. Floyd,
R.
Kenyon, & W. JR.
Parry, in Flavors of Geometry, S. Levy, ed., Cambridge Univ,, Cambridge,
1997.
Goals: To give a unifying picture of analysis and geometry and explore some interconnections. Course Outline: 2 weeks Parallel postulate and the need for
non-Euclidean geometry
Math 473 Ideals, Varieties and
Algorithms
Frequency: Annually, Fall Term Credit: (3-0)3 Content: Ideals in Polynomials rings, Affine Varieties, Parametrizati$ Division algorithm in k[x1 ,...,xn], Monomial Ideals, Hilbert Basis Theorem and Groebner Basis, Buchberger’s algorithm, Applications, Elimination Theory, Geometry of Elimination, Resultants, Extension Theorems.Hilbert’s Nullstellensatz, Ideal-Variety correspondence, Irreducible varieties, Decomposition of varieties in to irreducibles. Goals: The aim of this course is the study of systems of polynomial equations in many variables, and asking questions such as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is called an ideal in the polynomial ring k[x1 ,...,xn]. There is a close relationship between these ideals and the varieties which reveal the intimate link between algebra and geometry. Until recently, these topics involved a lot of abstract mathematics and were only taught in graduate school. With today’s technology, and recent algorithms for manipulating systems of polynomial equations, it is possible to do some substantial mathematics, including Hilbert Basis Theorem, Elimination Theory, and The Nullstellensatz, and introduce these computational techniques to the undergradu$ The ability to compute efficiently with polynomial equations has made it possible to investigate complicated examples that would be impossible to do by hand. The course assumes that the students will have access to a computer algebra system, such as Mathematica 3, Maple, Reduce. The course does not assume any prior experience with a computer, however, many of the algorithms will be described in pseudocode, which may be unfamiliar to students with no background in programming. Course Outline: (1 ½ Weeks) Geometry-Algebra and Algorithms: Polynomials and Affine Space, Affine Varieties, Parametrization of Affine Varieties, Ideals. (4 ½ Weeks) Groebner Bases: Ordering on Monomials in k[x1 ,...,xn], A division Algorithm in k[x1 ,...,xn$ Dickson’s Lemma, Hilbert Basis Theorem, Groebner Bases. Properties of Groebner Bases, Buchberger’s Algorithm, Applications of Groebner bases: The ideal membership problem, the problem of solving polynomial equations the implicitiza$ problem. (4 ½ Weeks) Elimination Theory: The elimination and Extension Theorems, Geometry of Elimination, Implicitization, Singular points and Envelops, Irreducible polynomials and Unique Factorization and Resultants, (2 ½ Weeks) The Algebra-Geometry Dictionary: Hilbert’s Nullstellensat$ Radical ideals, Radical Membership problem, Ideal-Variety correspondence, Irreducible varieties, Irreducible decomposition of a variety, Zariski Closure. Suggested textbook: D. C. Little, D.O’Shea; Ideals, Varieties, and Algorithms. Undergraduate Text in Mathematics. Springer-Verlag, 1992, QA 564, C688.
Math 474 Introduction to Computational Algebraic Geometry Prerequisites: Math 473 or Consent of the Instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: Polynomial and rational functions on a variety and algorithm$ computations in k[x1 ,...,xn]/I. Invariant theory of finite groups, generators for the ring of invariants, and the geometry of orbits. Projective varieties, the projective elimination theory, the geometry of quadric hypersurfaces, and Bezout’s theorem. Goals: This is basically a continuation of the course Math 473. The aim of this course is to study systems of homogeneous polynomial equations in more than one variables; and introduce students to the computational aspects of algebraic geometry. At the same time, the course will introduce students to The Projective Algebra-Geometry dictionary, The passage from Affine to Projective varieties, Projective Elemination Theory, The dimension of a Variety. Course Outline: (4 Weeks) Polynomial and Rational Functions on a Variety: Polynomial mappings, quotients of polynomial rings, algorithmic computations in k[x1,...,xn]/I, the coordinate ring of an affine variety, rational functions on a variety. (4 Weeks) Projective Algebraic Geometry: Projective plane, projective space and projective varieties, the Projective Algebra-geometry dictionary, The projective closure of an Affine variety, Projective elimination theory, The geometry of Quadric hypersurfaces. (6 Weeks) The Dimension of a Variety: The variety of a Monomial ideal, the complement of a Monomial ideal, The Hilbert Function and the Dimension of a variety, Dimension and Algebraic independence, Dimension, and Nonsingulari$ the Tangent cone. Suggested textbooks: D. Cox, Little, D.O’Shea; Ideals, Varieties and Algorithms Undergraduate Text in Mathematics Springer-Verlag, 1992.
Prerequisites: Math 367 and Math 371 Frequency: Annually, Spring Term Credit: (3-0)3 Content: Affine and Projective plane curves, Local properties of plane curves, Multiple points, Intersection numbers, Bezout’s theorem, Noether’s fundamental theorem, Applications to some enumerative geometry problems, Riemann-Roch theorem. Goals: Suggested textbook: W.Fulton; Algebraic Curves, W.A.Benjamin Inc. 1976
Math 478: Mathematical Aspects of Crytography Prerequisites: Math 365 or Consent of the Instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: Time estimates for doing arithmetic, some simple cryptosystems, the idea of public key cryptosystems, RSA, Discrete log, Knapsack, primality and factoring, the rho method, Fermat factorization, the continued fraction method. Goals: Suggested textbook: N.Koblitz; A Course in Number Theory and Crytography, Graduate Text in Mathematics, Springer Verlag, 1987
Math 480: Numerical Methods for Differential Equations Prerequisites: Consent of instructors Frequency: Fall Term Credit: (3-0)3, ECTS 6 Content: Initial value problems for ordinary differential equations, Convergnece, stability, Stiffnes, Predictor-corrector methods, Boundary value problems, Hyperbolic and Elliptic differentialy equations, Iterative methods Goals: To introduce and give an understanding of numerical methods for the solution of ordinary and partial differential equations, their derivation, analysis and applicability. Course Outline: 1. week: Initial value problems for ordinary differential equations
Suggested textbook: A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Pres (1996). ISBN: 0521556554 L. W. Johnson, R. D., Riess, Numerical Analysis, 2nd edition, (1982). ISBN: 0201103923 Reference Books: K. W. Morton, D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Pres (1994). ISBN: 0521418550 Math 484: Complexity of Algoritms Prerequisites: Consent of the instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: Recursive algorithms: Quicksort, recursive graph algorithms, fast matrix multiplication, the discrete Fourier transform. NP-complete problems: Turing machines, network flow problems. Information based complexity: Complexity of numerical algorithms (Newton’s method, interpolation, integration, solution of linear equations.) Goals: Suggested textbooks: L.Kronsjö; Algorithms: their complexity and efficiency, Wiley, 1987 H.S. Wilf; Algorithms and Complexity, Prentice-Hall, 1986 C.H. Papadimitriou; Computational Complexity, Addison-Wesley, 1994
Math 486: Fundamentals of Database Systems Prerequisites: Consent of the instructor Frequency: Anually, Spring Term Credit: (3-0)3 Content: A study of concepts and components of programming languages such as data types, control sequence, subprograms, data management. The various ways these can be implemented will be illustrated from different language types. Students will be expected to construct and run programs in several of these languages. Goals: Suggested textbook: L.B.Wilson, R.G.Clark; Comparative programming languages, Addison-Wesley, 1988
Prerequisites: Math 258, Math 357 Frequency: Annually, Fall Term Credit: (3-0)3 Content: Mathematical modelling: Derivation of Wave, heat and Laplace equations Green’s Function: Introduction: Solutions of boundary value problems or eigenvalue problems Dirac d -function Green’s function Asymptotic expansions: Definitions and properties. Integration by parts. Laplace’s method. Method of steepest descents. Method of stationary phase. Regular perturbation Theory: The implicit function theorem. Perturbation of eigenvalues. Nonlinear eigenvalue problems. Oscillation and periodic solutions. Hopf bifurcations(?) Singular perturbation theory: Initial value problems. Boundary value problems Goals: To give the basic ideas on modelling, Green’s function, perturbation and asymptotic analysis. Course Outline: (3 Weeks) Mathematical modelling: Derivation of Wave, heat and Laplace equations (3 Weeks) Green’s Function: Introduction: Solutions of boundary value problems or eigenvalue problems Dirac d -function Green’s function (3 Weeks) Asymptotic expansions. Definitions and properties. Integration by parts. Laplace’s method. Method of steepest descents. Method of stationary phase. (4 Weeks) Regular perturbation Theory. The implicit function theorem. Perturbation of eigenvalues. Nonlinear eigenvalue problems. Oscillation and periodic solutions. Hopf bifurcations(?) (1 Week) Singular perturbation theory. Initial value problems. Boundary value problems Suggested textbooks: R.L.Street; Analysis and solutions of Partial Differential Equations (For Chapter 1 and 2) R. Dennemeyer; An Introduction to Partial Differential Equations and Boundary Value Problems. (For Chapter 2) G. F. Carrier and Carl E. Pearson: Partial Differential Equations, Theory and Technique (For Chapter 2) J.P. Keener: Principles of Applied Mathematics (For Chapter 3,4 and
5)
Math 488: Applied Mathematics II Prerequisites: Math 258 or Math 358 Frequency: Annually, Spring Term Credit: (3-0)3 Content: Introduction to integral equations. Volterra and Fredholm equations. Solutions by Neumann series. Connection with eigenvalue problems. Essentials of calculus of variations, Euler-Lagrange equations, canonical form of the Euler equation, applications to mechanics and mathematical physics. Goals: The main goal of this course is to give an adequate background to treat the applied mathematics problems. In particular Calculus of Variations and Integral Equations are investigated Suggested textbooks: F.B. Hildebrand; Methods of Applied Mathematics, 1965 LE. Elsgolts; Differential equations and the Calculus of Variations, M?R, 1969 Ll.G. Chambers; Integral Equatons, Int. Textbook Comp. 1976
Math 489: Dynamical Systems Prerequisites: Math 258 and Math 261 Frequency: Annually, Fall Term Credit: (3-0)3 Content: Second order differential equations in phase plane. Linear systems and exponential operators, canonical forms. Stability of equilibria. Lyapunov functions. The existence of periodic solutions. Applications to various fields. Goals: Suggested textbook:
MATH 490: Difference Equations Prerequisites: Consent of the instructor Frequency: Credit: (3-0)3 Content: The Difference calculus. Linear difference equations: First order equations, high order equations. Systems of difference equations. Basic theory. Linear periodic systems. Stability theory. Linear approximation. Lyapunov's second method. The Z transform. Asymptotic behaviour of difference equations. Sturmian theory. Oscillation. Goals: Suggested textbooks:
Math 492: Numerical Optimization Prerequisites: Math 261 or consent of the instructor Frequency: Annually, Spring Term Credit: (3-0)3 Content: Unconstrained optimization: Vector and matrix norms, theory of unconstrained optimization, treatment of non-linear systems (simple iteration, Newton’s iteration, Gauss-Newton methods for overdetermined systems), line search based on descent methods, conjugate gradient directions, Newton-like methods, quasi-Newton methods. Constrained optimization: Linear programming, simplex method, feasible directions, Lagrange multipliers, convex programming problems, duality, methods of quadratic problems, active set methods for inequality constrained problems. Goals: Suggested textbooks: R.Fletcher; Practical Methods of Optimization, Wiley, 1987 J.E.Dennis, R.B.Schnabel; Numerical methods for unconstrained optimization and nonlinear equations, Prentice-Hall, 1983. E.M.L.Beal; Introduction to Optimization, Wiley, 1988 P.E. Gill; Numerical-Linear Algebra and Optimization, Addison-Wesley, 1991
Math 493: Philosophy of Mathematics Prerequisites: Consent of the instructor Frequency: Upon to request, Fall/Spring Term Credit: (3-0)3 Content: Philosophical problems about mathematics, Euclidean and non-Euclidean Geometries. The existence of mathematical objects, mathematical truth, Wittgenstein and Lakatos on mathematics. Goals: Suggested textbook:
Prerequisites: Math 387 or consent of the instructor. Frequency: Annually, Spring Term Credit: (3-0)3 Content: The principals of sotfware engineering with special emphasis on mathematical software. By way of class examples, laboratories and/or term projects students will see two or three small software packages evolve through the stages of specification, design, implementation and testing. Goals: Suggested textbook: D.Kahaner, C.Moler,S.Nash; Numerical methods
and software Prentice Hall, 1989.
Math 496: Supervised Independent Study and Research Prerequisites: Consent of the department and the instructor Frequency: Annually, Spring Term Credit: (2-0)2 Content: Individualized reading, and Study/research in mathematics for students of high intellectual promise. Goals:
Math 497: Hilbert Space Techniques Prerequisites: Math 349 Frequency: Annually, Fall Term Credit: (3-0)3 Content: Inner product spaces. Examples of inner product spaces; Hilbert spaces (definition and examples); convergence in Hilbert spaces; orthogonal complements and the projection theorem; linear functionals and the Riesz representation theorem; applications to various branches of mathematics. Goals: Course Outline: (2 Weeks) Preliminaries: Vector spaces and Hamel bases; Metric Spaces, Short Review of the Topology via Sequences: Open, Closed, Compact Subsets; Completion of a Metric Space. (2 Weeks) Metric and Normed Linear Spaces: The Classical Examples; f
, (2 Weeks) Hilbert Spaces: Inner Product Spaces, Orthogonality, Total Sequences, Convex Sets and the Minimizing Vector, Orthogonal Projections. (2 Weeks) Linear operators between normed Linear Spaces: Norm of a Linear Continuous Operator, the Continuous Dual of a Normed Linear Space, Dual of a Hilbert Space and the Riesz-Frechet Theorem; Adjoint Operators, Unitary, Self-Adjoint, Isometric, Normal Operators. (3 Weeks) Spectral Theory: Spectrum, Resolvent, Eigenvalues of a Linear Continuous Operator between Banach Spaces; Basic Properties of Compact Operators, Eigenvalues and Eigenspace of a Compact Operator; Spectral Decomposition Theorem for a Self-Adjoint Compact Operator between Hilbert Spaces; Positive Opertors and the Square Root of a Compact Operator. (2 Weeks) Applications. Suggested textbooks: S.K. Berberian ; Introduction to Hilbert Space, Oxford Univ. Press, 1961 R.E. Moore, J.Wiley ; Computational Functional Analysis A.E. Taylor ; Introduction to Functional Analysis
|
,c,c
,
,
,C[
a,b] ,etc.; Products, Direct Sums, Subspaces,
Quotients, Finite Dimensional Normed Linear Spaces, Riesz’s Lemma; Schauder
Bases and Separability