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Symbolic logic. Set theory. Cartesian product. Relations. Functions. Injective, surjective and bijective functions. Composition of functions. Equipotent sets. Countability of sets. More about relations: equivalence relations, equivalence classes and partitions. Quotient sets. Order relations: Partial order, Total order, Well ordering. Mathematical induction and recursive definitions of functions.

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Basic counting: The sum and product rules, the pigeonhole principle, generalized permutations and combinations. The binomial theorem. Discrete probability. Inclusion-exclusion. Recurrence relations. Introduction to graphs and trees.

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Functions. Limits and Continuity. Tangent lines and derivatives. Chain rule. Implicit differentiation. Inverse functions. Related rates. Linear approximations. Extreme values. Mean Value Theorem and its applications. Sketching graphs. indeterminate forms and L'Hospital's rules. Definite integral, Antiderivatives and the Indefinite integral. Fundamental Theorem of Calculus.

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Substitution. Areas between curves. Integration Techniques, Improper integrals. Arc length. Volumes and surface areas of solids of revolution. Parametric plane curves. Polar coordinates. Arc length in polar coordinates. Sequences and infinite series. Power series. Taylor series. Functions of several variables: limits, continuity, partial derivatives. Chain rule. Directional derivatives. Tangent planes and linear approximations. Extreme values. Lagrange multipliers.

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Fundamental principles of Analytic Geometry. Cartesian coordinates in plane and space. Lines in the plane. Review of trigonometry and polar coordinates. Rotation and translation in the plane. Vectors in plane and space. Lines and planes in 3-space. Basics about conics. Basic surfaces in space, cylinders, surface of revolutions, quadric surfaces. Cylindrical and spherical coordinates.

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Binary operations. Groups. The symmetric group. Subgroups. The order of an element. Cyclic groups. Rings. Integral domains. Subrings. Ideals. Fields: Q, R, C, Zp. The concept of an isomorphism. The ring of integers and the ring of polynomials over a field: Division and Euclidean algorithms. GCD and LCM. Prime factorization. Quotient structures.

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Functions and their graphs. Limits and continuity. Tangent lines and derivative. Chain rule. Implicit differentiation. Inverse functions. Related rates. Linear approximation. Extreme values. Mean Value Theorem and its applications. Sketching graphs. Indeterminate forms and L Hospital s rules. Exponential growth and decay. Definite integral. Fundamental Theorem Calculus. Substitution and areas between curves. Formal definition of natural logarithm function.

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Indefinite Integral. Techniques of integration. Arc length. Volumes and surface areas of solids of revolution. Improper integrals. Sequences and infinite series. Power series. Taylor series. Vectors and analytic geometry in 3-space. Functions of several variables:Limits, continuity, partial derivatives, chain rule, directional derivatives, tangent plane and linear approximations. Extreme values. Lagrange multipliers. Double integrals.

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Sequences and infinite series. Power series. Taylor series. Vectors and analytic geometry in 3-space. Functions of several variables: limits, continuity, partial derivatives. Chain rule. Directional derivatives. Tangent planes and linear approximations. Extreme values. Lagrange multipliers. Double integrals. Double integrals in polar coordinates. General change of variables in double integrals. Surface parametrization and surface area in double integrals. Triple integrals in Cartesian, cylindrical and spherical coordinates. Parametrization of space curves. Line integrals. Path independence. Green s theorem in the plane.

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Well ordering of integers, mathematical and strong induction, Divisibility, Division algorithm, Greatest common divisor, Euclidean algorithm, Linear Diophantine equations, Prime numbers, Fundamental theorem of arithmetic, General information about Goldbach conjecture and gaps between primes and Drichlet`s theorem, Congruence modulo n, Modular arithmetic, Linear congruences, Chinese remainder theorem, Fermat`s little theorem, Wilson`s theorem, Number theoretic functions, Tau and sigma functions, Greatest integer function, Moebius inversion, Euler`s phi function, Euler`s theorem and its applications to cryptography.

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Fundamental Principle of Analytic Geometry, Coordinate Systems in R^2 and R^3 vectors in R^2 and R^3. Dot product, cross product, lines, planes, distance, vector spaces, systems of linear equations, matrices, diagonalization of 2x2 and 3x3 matrices, conic sections and quadric surfaces, classification of quadric surfaces and curves (as an application of diagonalization).

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Logic. Relations and Functions. Matrices and determinants. Inverse of a matrix, matrix polynomials, Cayley-Hamilton theorem. Systems of linear equations, parametric solutions. Counting: principle of inclusion exclusion, pigeonhole principle. Mathematical induction, recursive relations. Permutations, combinations. Discrete probability. Graphs.

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Analytic Geometry in R2 , R3. Functions of one and several variables: Limit, continuity and differentiation. Chain rule, implicit differentiation. Differential calculus, optimization, Lagrange multipliers. The definite integral. The indefinite integral. Logarithmic and exponential functions. Techniques of integration: Integration by substitution, integration by parts, integration by partial fractions.

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Logic,sets,relations and functions; Boolean algebras; Elementary number theory ( Euclidean algorithm, Euler,s and Wilson,s theorem and their applications) ; discrete probability, elementary graph theory and its applications.

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Functions, limit and derivative of a function of a single variable. A thorough discussion of the basic theorems of differential calculus: Intermediate value, extreme value, and the Mean Value Theorem. Applications: Graph sketching and problems of extrema.

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The Riemann Integral. Mean Value Theorem for integrals. Fundamental Theorem of Calculus. Techniques to evaluate anti derivative families various geometric and physical applications. Sequences improper integrals infinite series of constants power series and Taylors series with applications.

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(Only for students of EME 413)Introduces the axiomatic structures in geometry; finite geometries, Euclidean and non-Euclidean geometries. Provides study in geometry and trigonometry including polygons, similar figures, geometric solids, properties of circles, constructions, right triangles, angle measurement in radians and degrees, trigonometric functions and their applications to right triangles, Phytagorean theorem, laws of sine and cosine, graphing of trigonometric functions, trigonometric identities, vectors and coordinate conversions.

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First order equations and various applications. Higher order linear differential equations. Power series solutions: The Laplace transform: solution of initial value problems. Systems of linear differential equations: Introduction Partial Differential Equations.

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Review of Multidimensional Calculus. Derivatives of multivariable functions, continuity of multivariable functions. Fundamental Lemma for differentiability. Chain rule and Taylor`s Theorem for multivariable functions. Jacobian. Inverse and Implicit Function Theorems. Topology of R2 and R3. Riemann-Stieltjes Integral, integrability. Integrability of continuous functions, sequences of integrable functions. Bounded convergence and Riesz Representation Theorems. Theorems of Integral Calculus: Integration in Cartesian spaces. Improper and infinite integrals. Series of functions.

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Topology of R, R2 and R 3 . Functions of several variables; limits and continuity. Partial derivatives, directional derivatives, gradients. Differentials and the tangent plane: the Fundamental Lemma, approximations. The Mean Value, implicit and Inverse function theorems. Extreme values. Introduction to vector differential calculus: the gradient, divergence and curl. Curvilinear coordinates.

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Double Integrals, polar coordinates. Improper double integrals. Change of variables in double integrals. Triple Integrals: Cylindrical and spherical coordinates. Applications. Line integrals: Parametrisation of curves, Greens Theorem. Independence of path, exact differentials. Parametrisation and orientation of surfaces. Surface Integrals. Divergence and Stokes Theorems, applications.

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Existence and uniqueness theorems. First order equations. Trajectories. Higher order linear equations; undetermined coefficients, variation of parameters and operator methods. Power series solutions. Laplace transform solutions of IVPs. Theory of linear systems. Solutions by operator, Laplace and linear algebra methods. Partial differential equations, separation of variables and Fourier series.

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Matrices, determinants and systems of linear equations. Vector spaces, the Euclidian space, inner product spaces, linear transformations. Eigenvalues, diagonalization.

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Matrices and systems of linear equations. Vector spaces; subspaces, sums and direct sums of subspaces. Linear dependence, bases, dimension, quotient spaces. Linear transformations, kernel, range, isomorphism. Spaces of linear transformations, Hom (V,W),V*, V** transpose. Representations of linear transformations by matrices, similarity. Determinants.

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Characteristic and minimal polynomials of an operator, eigenvalues, diagonalizability, canonical forms, Smith normal form, Jordan and rational forms of matrices. Inner product spaces, norm and orthogonality, projections. Linear operators on inner product spaces, adjoint of an operator, normal, self adjoint, unitary and positive operators. Bilinear and quadratic forms.

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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.

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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.

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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.

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Review of Riemann integration. Sets of (Lebesgue) measure zero in Rn and charac-terization of Riemann integrable functions. Lebesgue integrable functions and the Lebesgue integral in Rn. Convergence theorems, theorems of Lusin and Egorov. Fubini`s theorem. Selected applications.

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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 arithmetic of cardinal numbers. Axiom of choice, generalized continuum hypothesis.

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Graphs, varieties of graphs, connectedness, extremal graphs, blocks, trees, partitions, line graphs, planarity, Kuratowsky`s theorem, colorability, chromatice numbers, five color theorem, four color conjecture.

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LUB Property of real numbers. Compactness, connectedness, limits and continuity in metric spaces. Sequences and series of scalars, complete metric spaces, limsup. Sequences and series of functions, uniform convergence, applications.

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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.

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Algebra of complex numbers. Polar representation. Analyticity. Cauchy-Riemann equations. Power series. Elementary functions. Mapping by elementary functions. Linear fractional transformations. Line integral. Cauchy-Theorem. Cauchy integral formula. Taylor`s Series. Laurent series. Residues. Residue theorem. Improper integrals.

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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.

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First order equations; linear, quasilinear and nonlinear equations. Classification of second order linear partial differential equations, canonical forms. The Cauchy problem for the wave equation. Dirichlet and Neumann problems for the Laplace equation, maximum principle. Heat equation on the strip.

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Primitive roots of an integer, Integers n for which Z_n^* is cyclic, Theory of indices, Quadratic residues, Legendre symbol, Quadratic Reciprocity Law, Solving quadratic congruences. Perfect numbers, Mersenne primes, Fermat numbers, Fibonacci numbers, Linear Diophantine equations, Pythagorean triples, Quadratic Diophantine equations, Fermat?s Infinite descent (x^4+-y^4=z^2 equations), Representing integers as sums of squares, Pell`s equation, Finite and infinite continued fractions, Solving Pell`s equation using continued fractions.

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Divisibility, congruences, Euler, Chinese Remainder and Wilson`s Theorems. Arithmetical functions. Primitive roots. Quadratic residues and quadratic reciprocity. Diophantine equations.

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Arithmetic in quadratic fields. Factorization theory. Continued fractions, periodicity. Transcendental numbers.

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Groups. Isomorphism theorems, direct pro-ducts. Groups acting on sets. Class equation. Statements of Sylow theorems and the F.T. on finite abelian groups. Rings, isomorphism theorems. Prime and maximal ideals. Integral domains, field of fractions. Euclidean domains, PIDs, UFDs. Polynomials, polynomials in several variables. Field extensions. Impossibility of certain geometric constructions. Finite fields.

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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.

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Curves in R3: Frenet formulas and Fundamental Theorem. Regular surfaces. Inverse image of regular values. Differentiable functions on surfaces. Tangent plane; the differential of a map, vector fields, the first fundamental form. Gauss map, second fundamental form, normal, principal curvatures, principal and asymptotic directions. Gauss map in local coordinates. Covariant derivative, geodesics.

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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.

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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 quadric surfaces.

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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.

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Convergence, stability, error analysis and conditioning. Solving systems of linear equations: The LU and Cholosky factorization, pivoting, error analysis in Gaussian elimination. Matrix eigenvalue problem, power method, orthogonal factorizations and least squares problems. Solutions of nonlinear equations. Bisection, Newton`s, secant and fixed point iteration methods.

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Approximating functions: polynomial interpolation, divided differences, Hermite interpolation, spline interpolation, the B-splines, Taylor Series, least square app-roximation. Numerical differentiation and integration based on interpolation. Richardson extrapolation, Gaussian quadrature, Romberg integration, adaptive quadrature, Bernoulli polynomials and Euler-Maclaurin formula.

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Gamma and Beta functions. Pochhammer`s symbol. Hypergeometric series. Hypergeomet-ric differential equation; ordinary and con-fluent hypergeometric functions. Generalized hypergeometric functions; the contiguous function relations. Bessel function; the functional relationships, Bessel`s differential equation. Orthogonality of Bessel functions.

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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.

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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 grammar rules. Machine learning. Game playing. Logic and uncertainty. Meta programming.

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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.

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The importance of optimization, basic definition and facts on convex analysis. Theory of linear programming and convex prog-ramming, simplex method and its applications, 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.

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Riesz spaces (vector lattices). Riesz subspaces, ideals and bands. Normed Riesz spaces. Order convergence, relatively uniform convergence and norm convergence. Operators on Riesz spaces.

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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.

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First order language, structures and satisfaction. Completeness and compactness theorems. Isomorphism, elementary equivalence and elementary imbedding. Löwenheim-Skolem theorem. Interpolation and definability. Atomic, universal and saturated models and their characterisation. Extensions of first order logic.

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Strategic games, Nash equilibrium, Bayesian Games, Mixed, Correlated, Evolutionary equilibrium, extensive games with perfect information, bargaining games, reğeated games, extensive games with imperfect information, sequential equilibrium, coalition games, core, stable sets, bargaining aets, shapley value, market games.

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Introduction to numerical and symbolical computational tools. Balance equations, continuos 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.

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Topological Spaces; basis, subbasis, subspaces. Closed sets, limit points. Hausdorff Spaces. Continuous functions, homeomorphisms. Product topology. Connected spaces, compo-nents, path connectedness, path components. Compactness, sequential compactness, compactness in metric spaces. Definition of regular and normal spaces. Urysohn`s Lemma, Tietsze Extension Theorem.

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Convex sets, subdivision problems, isoperimetric inequality, Minkowski sum; polytopes, Dehn-Sommerville equations, scissors equivalence; Erdös distance set problem, line arrangements, counting lattice points; packing, covering and tiling problems

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Topology of subsets of Euclidean space. Topological surfaces. Surfaces in Rn. Surfaces via gluing, 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.

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Basic topology, Surfaces and their triangulations, Complexes, Homology, Persistence homology, Morse functions, Discrete Morse Functions, Applications.

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Basics of topological spaces, fundamental groups covering spaces, topological Galois Theory, functions on surfaces, differential equations, regular singular points and differential equations of Fuchsian type

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Mappings. Time series. Orbits. Periodic orbits. Fixed points and periodic points. Finding periodu points. Eventually periodic points. Families of mappings and bifurcations. Transitivity. Sensitive dependence. Dense periodic points. One-dimensional chaos. Devaney ingredients. The logistic map. Period-three points and chaos. Period-doubling bifurcation, Two-dimensional chaos. The Henon map. The Horseshoe map. Dimensions. Systems of differential equations. Homoclinic chaos. Dynamics on labels, Fractals. Abstract similarity map, Abstract similarity sets, Labeling. Chaos in stochastic processes

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Well-formed formulas of Peano arithmetic, Gödel numbering, primitive recursive functions, Gödel’s Incompleteness Theorems, partial recursive functions, Turing machines, Church-Turing Thesis, decidabilitiy, recursion theorem, s-m-n theorem, padding lemma, recursively enumerable sets, computable approximations, halting problem, creative sets, simple sets, Turing reducibility, Turing degrees, properties of Turing degrees, recursively enumerable degrees, joins, Turing jump.
Selected topics may include: Arithmetical hierarchy, limit lemma, finite extension method, co-infinite extension method, minimal degrees, splitting threes, jump classes, jump inversion, low and hign degrees, finite injury priority method, r.e. permitting, computable dimination, hyperimmune-free degrees.

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Statics of rigid bodies, 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. Dynamics of a particle, harmonic oscillators, motion of a simple pendulum, flight of a projectile, motion under the action of central forces. Dynamics of a system of particles, motion of a body with varying mass.

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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.

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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).

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Orthogonality and modes of convergence. Fourier series, convergence of Fourier series, Fourier transform, Fourier inversion, discrete Fourier transform. Haar and Daubechies wavelets, decomposition and reconstruction, multiresolution analysis. Applications.

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Review of differentiation, inverse and implicit function theorems, integration on subsets of Euclidean space, tensors, differential forms, integration on chains, integration on manifolds. Stokes` theorem.

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Classical theory of rings, ideal theory, isomorphism theorems. The group ring. Localization. Submodules, direct products direct sums, factor modules and factor rings. Homomorphisms. Classical isomorphism the-orems. The endomorphism ring of a module. Free modules, free and divisible abelian groups. Tensor product of modules. Finitely generated modules over principal ideal domains.

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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.

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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.

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General linear groups. Bilinear forms. Projective geometry and projective linear groups. Symplectic and orthogonal geometries. Symplectic groups. Orthogonal groups. Hermitian forms and unitary groups.

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Symmetry. Isometrics of R?, 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 of acting on trees. Reflection groups, root systems, classification of finite reflection groups, crystallographic root systems and Weyl groups.

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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.

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Affine varieties. Groebner bases, monomial ideals and Dickson`s Lemma, Hilbert Basis Theorem. Buchberger`s algorithm. Ideal membership problem. The problem of Elimination Theory. Unique factorization and resultants. Resultant and extension Theorem.

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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. Prerequisite: 2360 367 and 2360 353.

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Hilbert `s axioms. Geometry over fields. Segment arithmetic. Area. Construction problems and field extensions.

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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.

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Initial value problems for ordinary differential equations, Convergence, Stability, Stiffness, Predictor-corrector methods, Boundary value problems, Hyperbolic and Elliptic differential equations, Iterative methods.

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Database concepts. Database Management Systems (DBMS). Relational data model and relational DBMS. Use of ER-diagrams in database design. Normalizing relations. Relational algebra and query languages. Structured Query Language (SQL). Oracle and/or Access will be introduced in a laboratory environment.

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Mathematical modelling of boundary value problems of partial differential equations. Formulation of Dirichlet and Neumann problems. Green`s function. Asymptotic analysis of solutions. Perturbation techniques.

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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.

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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.

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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.

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Philosophical problems about mathematics, Euclidean and non-Euclidean Geometries. The existence of mathematical objects, mathematical truth, Wittgenstein and Lakatos on mathematics.

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Individualized reading, and study/research in mathematics for students of high intellectual promise.

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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.

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Program of research leading to M.S. degree arranged between student and a faculty member. Students register to this course in all semesters starting from the beginning of their second semester while the research program or write-up of thesis is in progress.

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General measure and integration theory. General convergence theorems. Decomposition of measures. Radon-Nikodym Theorems. Outer measure. Caratheodory extension theorem. Product measures. Fubini's theorem. Riesz representation theorem.

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Compact operators, compact operators in Hilbert Spaces, Banach Algebras, The spectral theorem for normal operators, unbounded operators between Hilbert spaces, the spectral theorem for unbounded self-adjoint operators, self-adjoint operators, self-adjoint extensions.

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Groups quotient groups, isomorphism theorems, alternating and dihedral groups, direct products, free groups, generators and relations, free abelian groups, actions. Sylow theorems, nilpotent and solvable groups, normal and subnormal series. Rings, ring homomorphisms, ideals, factorization in commutative rings, rings of quotients, localization, principle ideal domains, Eucladian domains, unique factorization domains, polynomials and formal power series, factorization in polynomial rings.

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Modules; homomorphisms, exact sequences., projective and injective modules, free modules, vector spaces, tensor products, modules over a PID. Fields, field extensions, the fundamental theorem of Galois theory, splitting fields, algebraic closure and normality, the Galois group of a polynomial, finite fields.

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Differentiable manifolds, smooth mappings, tangent, cotangent bundles, differential of a map, submanifolds, immersions, imbeddings, vector fields, tensor fields, differential forms, orientation on manifolds, integration on manifolds, Stoke's theorem.

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The aim of this course is to test the knowledge of the student in the basic areas of mathematics. For this purpose, a written exam is given in the following topics and subtopics: Algebra (A. Groups and Rings B. Modules and Fields), Analysis (A. Real Analysis B. Complex Analysis), Differential Equations (A. Ordinary DE B. Partial DE), Geometry-Topology (A. Geometry B. Topology), Numerical Analysis (A. Numerical Analysis I B. Numerical Analysis II). Each student is required to take the exam in 4 subtopics chosen from 3 distinct topics.

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General research guidelines and general tools: Resources in mathematical research, use of databases, libraries etc. Publications and their types. The nature of graduate studies, aims of an M.S. thesis and a Ph.D. thesis. Writing theses and paper. Journal types. Citation indices and impact factors. Academic integrity and ethics, ethical issues in research and publications, plagiarism and its various forms, code of conduct in graduate studies.

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Abelian groups; torsion, divisible, torsion-free groups, pure subgroups, finitely generated abelian groups. Solvable and nilpotent groups, Hall ?-subgroups. Permutation groups. Representations. Fixed-point free automorphisms. Locally nilpotent groups, locally solvable groups. Finiteness properties. Infinite solvable groups.

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Locally finite groups. Maximal and minimal condition on subgroups, Cernikov groups and automorphisms of Cernikov groups, direct limit inverse limit of groups, linear groups, locally finite simple groups, Hall universal group, centralizers of elements in simple locally finite groups.

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Ring theoretic preliminaries. Group representations and their characters. Characters, integrality and application to the structure theory of finite groups. Product of characters. Induced characters. Reduction and extension of characters. Brauer's theorem on characterization of characters.

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Rings and ideals. Modules. Rings and modules of fractions. Primary decomposition. Integral dependence.

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Introduction to finite fields. Traces, norms and bases, factoring polynomials over finite fields, construction of irreducible polynomials, normal bases, optimal normal bases.

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Basic concepts and examples, linear codes (Hamming, Golay, reed-Muller codes) bounds on codes, cyclic codes (BCH, RS; Quadratic Residue Codes), Goppa codes.

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Ring of integers of an algebraic number field. Integral bases. Norms and traces. The discriminant. Factorization into irreducibles. Euclidean domains. Dedekind domains. Prime factorization of ideals. Minkowski’s theorem. Class-group and class number.

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Valuations. Divisors, repartitions, differentials. Riemann-Roch Theorem. Rational function fields, elliptic and hyperelliptic function fields. Congruence zeta function, the functional equation for the L-functions.

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Dirichlet series, Dirichlet L-functions, Chebychevs y and q functions, prime number theorem, distribution of primes, functional equations.

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Elliptic functions, modular functions, Dedekind eta function, congruencies for the coefficients of the modular function j, Rademacher s series for the partition function, modular forms with multiplicative coefficients, Kronecker s theorem, general Dirichlet series and Bohr s equivalence theorem.

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Topological spaces. Neighborhoods. Basis. Subspace topology, product and quotient topologies. Compactness. Tychonoff's Theorem. Heine-Borel theorem. Separation properties. Urysohn's Lemma and Tietze Extension theorem. Stone-Cech compactification. Alexandroff one point compactification. Convergence of sequences and nets. Connectedness. Metrizability. Complete metric spaces. Baire's theorem.

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Fundamental group, Van Kampen’s Theorem, covering spaces. Singular homology: Homotopy invariance, homology long exact sequence, Mayer-Vietoris sequence, excision. Cellular homology. Homology with coefficients. Simplicial homology and the equivalence of simplicial and singular homology. Axioms of homology. Homology and fundamental groups. Simplicial approximation. Applications of homology.

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Comohology groups, Universal Coefficient Theorem, comohology of spaces. Products in comohology, Kunneth formula. Poincare duality. Universal coefficient theorem for homology. Homotopy groups.

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Manifolds and differentiable structures. Tangent Space. Vector bundles. Immersions, submersions, embeddings. Transversality. Sard's theorem. Whitney Embedding Theorem. The exponential map and tubular neighborhoods. Manifolds with boundary. Thom's tranversality Theorem.

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4-manifolds, surfaces in 4-manifolds, complex surfaces, complex curves and their desingularizations. Elliptic surfaces; classification of complex surfaces and logarithmic transform. Handle decomposition, Heegard splitting and Kirby diagrams. Linking numbers and framings. Kirby calculus, handle moves and Dehn surgery. Spin structures, plumbings and related constructions. Embedded surfaces and branched covers.

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Review of differentiable manifolds and tensor fields. Riemannian metrics, the Levi-civita connections. Geodesics and exponential map. Curvature tensor, sectional curvature. Ricci tensor, scalar curvature. Riemannian submanifolds. Gauss and Codazzi equations.

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Lie groups; principle fibre bundles; almost complex and complex manifolds; Hermitian and Kaehlerian geometry; symmetric spaces.

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Theory of algebraic varieties: Affine and projective varieties, dimension, singular points, divisors, differentials, Bezout's theorem.

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Review of sheaves; quasi-coherent and coherent sheaves, direct and inverse images. Proper morphisms, quasi-coherent sheaves on projective schemes, projective morphisms. Cohomology of sheaves, cohomology of a projective scheme, higher direct images. Geometric applications.

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Analytic functions. Singular points and zeros. The argument principle. Conformal mappings. Riemann Mapping Theorem. Mittag-Lefler Theorem. Infinite products. Canonical products. Analytical continuation. Elementary Riemann surfaces.

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Vector lattices. Basic inequalities, Basic properties, Positive operators. Extension of positive operators. Order projectives. Order continuous operators. Lattice Homomorphisms. Orthomorphism. Banach Lattices with order continuous norms. Weak compactness in Banach Lattices. Embedding Banach spaces. Banach lattices of operators. Compact operators. Weakly compact operators.

Course Content

Review of metric spaces, Normed Linear Spaces, Dual Spaces and Hahn-Banach Theorem, Bidual and Reflexivity, Baires Theorem, Dual Maps, Projections, Hubert Spaces, The spaces Lp(X,m), C(X), Locally Convex Vector Spaces, Duality Theory of lcs, Projective and Inductive topologies.

Course Content

Introduction to topological vector spaces, locally convex topological vector spaces. Inductive and projective limits. Frechet spaces. Montel, Schwartz, nuclear spaces. Bases in Frechet spaces and the quasi-equivalence property. Köthe sequence spaces. Linear topological invariants.

Course Content

Error analysis. Solutions of linear systems: LU factorization and Gaussian elimination, QR factorization, condition numbers and numerical stability, computational cost. Least squares problems: the singular value decomposition (SVD), QR algorithm, numerical stability. Eigenvalue problems: Jordan canonical form and conditioning, Schur factorization, the power method, QR algorithm for eigenvalues. Iterative Methods: construction of Krylov subspace, the conjugate gradient and GMRES methods for linear systems, the Arnoldi and Lanczos method for eigenvalue problems.

Course Content

Interpolation and approximation: Lagrange and Newton interpolation, Hermite interpolation, trigonometric interpolation and Fourier series. Spline interpolation B-splines and recursive algorithms. Numerical differentation and quadrature: Newton-Cotes formulas, Gaussian integration rules. Extrapolation and Romberg integration, adaptive quadrature. Hierarchal and recursive quadrature formulas: Archimedes integration formula. Root finding methods.

Course Content

Cauchy-Kowalevski Theorem. Linear and quasilinear first order equations. Existence and uniqueness theorems for second order elliptic, parabolic and hyperbolic equations. Correctly posed problems, Green's functions.

Course Content

Sobolev spaces: Weak Derivatives, Approximation by Smooth functions, Extentions, Traces, Sobolev Inequalities, The Space H^-1. Second-order Elliptic Equations: Weak Solutions, Lax-Milgram Teorem, Energy estimates, Fredholm Alternative, Regularity, Maximum Principles, Eigenvalues and Eigenfunctions. Linear Evolution Equations: Second-order Parabolic equations, (Weak solutions, Regularity, Maximum principle), Second-order Hyperbolic Equations (Weak Solutions, Regularity, Propagation of disturbances), Hyperbolic Systems of First-order Equations, Semigroup theory.

Course Content

Initial Value Problem: Existence and Uniqueness of Solutions; Continuation of Solutions; Continuous and Differential Dependence of Solutions. Linear Systems: Linear Homogeneous And Nonhomogeneous Systems with Constant and Variable Coefficients; Structure of Solutions of Systems with Constant and Periodic Coefficients; Higher Order Linear Differential Equations; Sturmian Theory, Stability: Lyapunov Stability and Instability. Lyapunov Functions; Lyapunov's Second Method; Quasilinear Systems; Linearization; Stability of an Equilibrium and Stable Manifold for Nonautonomous Differential Equations.

Course Content

Nonlinear Periodic Systems: Limit Sets; Poincare-Bendixon Theorem. Linearization Near Periodic Orbits; Method of Small Parameters in Noncritical Case; Orbital stability. Bifurcation: Bifurcation of Fixed Points; The Saddle-Node Bifurcation; The Transcritical Bifurcation; The Pitchfork Bifurcation; Hopf Bifurcation; Branching of Periodic Solutions for Nonautonomous Systems. Boundary Value Problems: Linear Differential Operators; Boundary Conditions; Existence of Solutions of BVPs; Adjoint Problems; Eigenvalues and Eigenfunctions for Linear Differential Operators; Green's Function of a Linear Differential Operator.

Course Content

General description of impulsive differential equations: Systems with fixed moments of impulses; systems with variable moments of impulses; discontinuous dynamical systems. Linear systems: General properties of solutions; periodic solutions; Floquet theory; adjoint systems. Stability: Stability criterion based on linearization of systems; direct Lyapunov method; B-equivalence; stability of systems with variable time of impulses. Quasilinear systems: Bounded solutions; periodic solutions; quasiperiodic and almost periodic solutions; integral manifolds. Discontinuous dynamical systems and applications.

Course Content

Presentation involving current research given by graduate students and invited speakers.

Course Content

Presentation involving current research given by graduate students and invited speakers.

Course Content

Finite difference method, stability, convergence and error analysis. Initial and boundary conditions, irregular boundaries. Parabolic equations; explicit and implicit methods, stability analysis, error reduction, variable coefficients, derivative boundary conditions, solution of tridiagonal systems. Elliptic equations, iterative methods, rate of convergence. Hyperbolic equations. The Lax-Wendroff method, variable coefficients, systems of conservation laws, stability. Finite volume method.

Course Content

Appell's symbol and hypergeometric series. The gamma function. The beta function. Dirichlet averages. Jacobi polynomials. Elliptic integrals.

Course Content

Weighted residual methods, the boundary element method for Laplace and Poisson equations. The dual reciprocity method, computer implementation.

Course Content

Introduction to fluid behavior. Derivation of continuity, momentum and energy equations. Navies-Stokes equations. Stream function, vorticity. Solutions of creeping, potential, laminar, boundary layer, turbulent flows. Solution of Navier-Stokes equations using finite difference methods in velocity-pressure , stream function-vorticity and stream function forms. Example solutions. Stability, convergence and error analysis.

Course Content

Project carried out under the supervision of a faculty member in a specific area of mathematics. A written report is expected from students about their work.

Course Content

Program of research leading to Ph.D. degree arranged between student and a faculty member. Students register to this course in all semesters starting from the beginning of their second semester while the research program or write-up of thesis is in progress.

Course Content

Generalities on algebras over commutative rings. Group algebras. Morita duality and quasi-Frobenius algebras, Frobenius algebras. Polynomial identity algebras, Artin-Procesi Theorem.

Course Content

Basic Concepts, semisimple Lie Algebras, root systems, isomorphism and conjugacy theorems, existence theorem.

Course Content

Introduction to Numerical Methods, Linear multistep methods. Runge-Kutta methods, stiffness and theory of stability. Numerical methods for Hamiltonian systems, iteration and differential equations.

Course Content

Calculus of variations. Weighted residual methods. Theory and derivation of interpolation functions. Higher order elements. Assembly procedure, insertion of boundary conditions. Finite element formulation of ordinary differential equations, some applications. Error and convergence analysis. Finite element formulation of non-linear differential equations. Time dependent problems. Convergence and error analysis. Solution of the resulting algebraic system of equations. Applications on steady and time dependent problems in applied continuum mechanics.

Course Content

Presentation involving current reserach given by Ph.D. students of invited speakers.

Course Content

Presentation involving current research given by Ph.D. students of invited speakers.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Course Content

ALGEBRAIC SURFACES

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

CODING THEORY

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Course Content

Darboux-Moser-Weinstein theorems, J-holomorphic curves. Symplectic camel and capacities. Symplectization and contactization. Arnold Liouville theorem and Hamiltonian equations. moment map and symplectic reduction. Kaehler manifolds and toric manifolds.

Course Content

Sturm-Liouville operators on a finite interval; the Dirichlet problem; the inverse Dirichlet problem; Uniqueness theorems; the Gelfand-Levitan method; isospectral sets.

Course Content

Brief review of complex manifolds, holomorphic vector bundles, connections, Chern classes. Harmonic forms, elliptic differential operators, main existence and uniqueness results. Harmonic forms applied to Kahler manifolds: Hodge decomposition, Lefschetz decomposition, Hodge Index Theorem. Applications to the topology of algebraic varieties.

Course Content

Advance theory of boundary element method and finite elemnt methods. Fundamental solutions of partial differential equations governing fluid dynamics problems. Use of both of the methods in coupled form for solving the same problem especially in irregular regions. Computer implementations.

Course Content

Weyl groups, Simple Lie Algebras, Chavalley Groups, Unipotent Subgroups, Diagonal and Monomial Subgroups, Relations Involving Generators of a Chevalley Group, Bruhat Decomposition, Automorphisms in Chevalley Groups.

Course Content

Twisted simple groups, Associated Geometrical Structures, Sporadic simple groups.

Course Content

Ordered Banach Spaces, Compressing of constrictors for positive semigroup, Asymptotic domination for semigroups, Positive semigroups in Banach Lattices, Geometry of Banach Lattices and asymptotic properties of semigroups.

Course Content

Mapping and Orbits. Periodic orbits, Linear mappings. Differentiable mappings. Hyberbolicity. Family of mappings Bifurcations. The quadratic map, Symbolic dynamics. Ingredients of chaos: Sensitivity, Dense orbits and transitivity. Schwarzian derivatives, Wiggly iterates. Cantor set and chaos. Higher dimensional dynamics: linear maps. The horseshoe map. Various routes to chaos.

Course Content

Changing The Field, The Schur Index, Projective Representations, Character Degrees, Character Correspondence, Linear Groups, Changing The Characteristics.

Course Content

Linear reflection groups; affine reflection groups. Coxeter groups; Coxeter complexes. Spherical buildings; groups with a BN-pair. Classical groups.

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Content

Algebraic Geometric and Function Field view of Curves over a finite field; Application of this to Coding Theory and other areas.

Course Content

Flows and cascades. Limit sets and compact motions. Recurrence. Minimal sets. Attractors. Poisson stability. Quasi-minimal sets and chaos. Central motions. Almost periodic motions. Lyapunov stability. Birkhoff`s ergodic theorem. Studying chaos with densities. Asymptotic properties of densities. Semi-flows.

Course Content

A short introduction to GAP, Group actions, Orbits and stabilizers, Stabilizer chains and their computation, The Schreier-Sims algorithm, Backtrack, Natural actions and decompositions, Primitive groups, Computing a composition series, Factor groups, Conjugacy classes, Complements, Subgroups, Maximal subgroups.

Course Content

Categories, functors, derived functors, extensions, resolutions, homology and cohomology of complexes. Some applications depending on the consent of the instructor such as modular representation theory or cohomology of groups or Lie algebras, algenraic topology.

Course Content

Basics of graph and group theory, orbitals and rank, graphs admitting a given group, primitivity and double transitivity, eigenvalues of graphs, automorphisms of graphs, vertex transitive and edge transitive graphs, graph homomorphisms, retracts, Cayley graphs, quotient graphs and primitivity, strongly regular graphs.

Course Content

Cayley graph of a group, Generating graph of a group, Intersection graph of a group, Commuting graph of a group, Graphs associated to a set of integers arising from the structure of a given group (character degree graphs, conjugacy class graphs, Grünberg-Kegel graphs), Characterization of Grünberg-Kegel graphs of solvable groups.

Course Content

Polynomial methods in error-correcting codes, Bezouts theorem, incidence geometry, ruled surfaces, polynomial method in differential geometry, Kakeya problem, Harmonic analysis, polynomial method in number theory.

Course Content

Semi-stability, stability, Harder-Narasimhan filtration. Families of sheaves, moduli spaces. Construction methods. The derived category of the abelian category of coherent sheaves. Quiver representations and the Bondal correspondance with the derived category of the abelian category of coherent sheaves.

Course Content

Understanding the behavior of families of algebraic curves is of central interest in many branches of mathematics, including algebraic geometry, mathematical physics, low dimensional topology, as well as certain branches of theoretical physics such as string theory and various gauge theories. Moduli spaces of algebraic curves are among the standard tools for researchers working in these areas. The course aims to cover the essentials of the subject and bring the students to a position so that they can access the modern literature.

Course Content

Review of ordinals, cardinals, transfinite induction and recursion. Basics of infinitary combinatorics, Suslin s hypothesis and trees, the diamond principle,Martin s axiom and their consequences. Models of set theory, relative consistency, absoluteness and reflection. Gödel s constructible universe and the axiom of constructibility. Forcing and its general theory, the forcing theorems. The relative consistency of CH, CH and other applications of forcing.

Course Content

Fundamental number-theoretic algorithms. The Euclidean algorithm and the greatest common divisor. Computations modulo n. Computations in finite fields. Algorithms on polynomials. A survey of algorithms for linear algebra. Algorithms for algebraic number theory. Factoring algorithms.Primality tests.

Course Content

This course is constructed from seminars that will be organised by Graduate School of Natural and Applied Sciences. The seminars will cover technical, cultural, social and educational issues to prepare the graduate students following the PhD programs.