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Item Open Access Integrability and Poisson structures of three dimensional dynamical systems and equations of hydrodynamic type(Bilkent University, 1992) Gümral, HasanShow more We show that the Poisson structure of completely integrable 3 dimensional dynamical systems can be defined in terms of an integrable 1-form. We shall take advantage of this fact and use the theory of foliations in discussing the geometrical structure underlying complete and partial integrability. Techniques for finding Poisson structures are presented and applied to various examples such as the Halphen system which has been studied as the two monopole problem by Atiyah and Hitchin. We shall show that the Halphen system can be formulated in terms of a flat SL{2, /i)-valued connection and belongs to a non-trivial GodbillonVey class. On the other hand, for the Euler top and a special case of 3- species Lotka-Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable biHamiltonian structures. The globally integrable bi-Hamiltonian case is a linear and the sb structure is a quadratic unfolding of an integrable 1-form in 3 -f 1 dimensions. We shall show that the existence of a vector field compatible with the flow is a powerful tool in the investigation of Poisson structure and present some new techniques for incorporating arbitrary constants into the Poisson 1- form. This leads to some extensions, analoguous to q-extensions, of Poisson structure. We shall find that the Kermack-McKendrick model and some of its generalizations describing the spread of epidemics as well as the integrable cases of the Lorenz, Lotka-Volterra, May-Leonard and Maxwell-Bloch systems admit globally integrable bi-Hamiltonian structure. In the second part, we complete the discussion of the Hamiltonian structure of 2-component equations of hydrodynamic type by presenting the Hamiltonian operators for Euler’s equation governing the motion of plane sound waves of finite amplitude and another quasi-linear second order wave equation. There exists a doubly infinite family of conserved Hamiltonians for the equations of gas dynamics which degenerate into one, namely the Benney sequence, for shallow water waves. We present further infinite sequences of conserved quantities for these equations. In the case of multi-component equations of hydrodynamic type, we show that Kodama’s generalization of the shallow water equations admits bi-Hamiltonian structure. We present a simple way of constructing the second Hamiltonian operators for N-component equations admitting some scaling properties. Using dimensional analysis we are led to an Ansatz for both the Hamiltonian operator as well as the conserved quantities in terms of ratios of polynomials. The coefficients of these polynomials are determined from the Jacobi identities. The resulting bi-Hamiltonian structure of Kodama equations consists of generalization of the Cavalcante-McKean’s work for the shallow water waves. The Kodama reduction of the dispersionless-Boussinesq equations and the Lax reduction of the Benney moment equations are shown to be equivalent by a symmetry transformation. They can be cast into the form of a triplet of conservation laws which enable us to recognize a non-trivial scaling symmetry. The choice of the Hamiltonian density lor the second Hamiltonian structure is a crucial step and the analysis of recursion relations becomes necessary. The resulting bi-Hamiltonian structure generates three infinite sequences of conserved densities.Show more Item Open Access Character sums, algebraic function fields, curves with many rational points and geometric Goppa codes(Bilkent University, 1997) Özbudak, FerruhShow more In this thesis we have found and studied fibre products of hyperelliptic and superelliptic curves with many rational points over finite fields. We have applied Goppa construction to these curves to get “good” linear codes. We have also found a nontrivial connection between configurations of affine lines in the affine plane over finite fields and fibre products of Rummer extensions giving “good” codes over F,2. Moreover we have calculated an important parameter of a class of towers of algebraic function fields over finite fields, which are studied recently.Show more Item Open Access Some exact solutions of colliding waves, black holes and their thermodynamical properties in Dilaton gravity(Bilkent University, 1998) Sermutlu, EmreShow more We consider all possible theories in spherically symmetric Riemannian geometry in D-dimensions. We find solutions to such theories, in particular black hole solutions of the low energy limit of the string theory in Ddimensions and study their uniqueness. We find for the first time in literature, exact solutions of colliding Dilaton-Einstein-Maxwell Plane Waves. We investigate thermodynamical properties of four and five dimensional black hole solutions of toroidally compactified string theory. We find the explicit expression of the first law of black hole thermodynamics. We calculate the temperature T, angula,r velocity ÎÎ and the electromagnetic potentials on the horizon using two different methods. Collision of plane waves in dilaton gravity theories and low energy limit of string theory is considered. The formulation of the problem and some exact solutions are presentedShow more Item Open Access The solvability of PVI equation and second-order second-degree Painleve type equations(Bilkent University, 1998) Sakka, AymanShow more A rigorous method was introduced by Fokas and Zhou for studying the Riernaiin-Hilhert problem associated with the Painleve II and IV. The same methodology has been applied to Painleve I, III and V. In this thesis, we applied the same methodology to the Painleve VI equation. VVe showed that The Cauchy problem for the Painleve VI equation admits in general global meromorphic solution in /. Furthermore, a. special solution which can lie written in terms of hypergeometric function is obtained via sob’ing the special case of the Riemann-Hilbert problem. Moreover, an algorithmic method introduced by Fokas and Ablowitz to investigate the transformation properties of Painleve equations and a generalization of it are used to obtain one-to-one correspondence between the Painleve equations and the second-ord(U‘ seconddegree equations of Painleve type.Show more Item Open Access KO-rings and J-groups of lens spaces(Bilkent University, 1998) Kırdar, MehmetShow more In this thesis, we make the explicit computation of the real A'-theory of lens spaces and making use of these results and Adams conjecture, we describe their .7-groups in terms of generators and relations. These computations give nice by-products on some geometrical problems related to lens spaces. We show that J-groups of lens spaces approximate localized J-groups of complex projective spaces. We also make connections of the J-cornputations with the classical cross-section problem and the .James numbers conjecture. Many difficult geometric problems remain open. The results are related to some arithmetic on representations of cyclic groups o\er fields and the Atiyah-Segal isomormhisrn. Eventually, we are interested in representations over rings, in connection with Algebraic K-theory. This turns out to lie a very non-trivial arithmetic problem related to number theory.Show more Item Open Access Symmetries and boundary conditions of integrable nonlinear partial differential equations(Bilkent University, 1999) Gürel, T BurakShow more The solutions of initial-boundary value problems for integrable nonlinear partial differential equations have been one the most important problems in integrable systems on one hand, and on the other hand these kind problems have proved to be very hard especially when considered on half or bounded lines. The proper generalization of the Inverse Spectral Transform or any other possible method in a way that they apply on half or bounded lines, is a complicated problem itself. But one of the other obstacles is the choice of suitable boundary conditions. In this direction, there is a pioneering work of Sklyanin which has motivated us in considering the problem of establishing boundary conditions for integrable partial differential equations. To this end, we try to develop a way to find boundary conditions which would, in turn, be suitable for certain solution techniques. This could have been done in many different ways depending upon what is understood from integrability. Throughout this work, we use the phrase integrability \i\ the sense of generalized symmetries which has proved to be one of the most efficient approaches. We first give a proper definition of compatibility of boundary condition with a symmetry. Then we interpret the well known tools of symmetry approach in a different manner. These tools include the recursion operators and symmetries themselves. After some technical theorems, we pass to examples and consider many integrable equations. Furthermore, we give, a generalization of the method which makes use of the non-homogeneous symmetries. Finally we finish by some discrete equations, including the 2D Toda lattice. It is crucial to note that all the boundary conditions that have been already known to be compatible with the integrability property of the original equation, pass our criterion of compatibility.Show more Item Open Access Monomial curves and the Cohen-Macaulayness of their tangent cones(Bilkent University, 1999) Arslan, Sefa FezaShow more In this thesis, we show that in affine /-space with / > 4, there are monomial curves with arbitrarily large minimal number of generators of the tangent cone and still having Cohen-Macaulay tangent cone. In order to prove this result, we give complete descriptions of the defining ideals of infinitely many families of monomial curves. We determine the tangent cones of these families of curves and check the Cohen-Macaulayness of their tangent cones by using Grobner theory. Also, we compute the Hilbert functions of these families of monomial curves. Finally, we make some genus computations by using the Hilbert polynomials for complete intersections in projective case and by using Riemann-Hurwitz formula for complete intersection curves of superelliptic type.Show more Item Open Access The quasi-equivalence problem and isomorphic classification of Whitney spaces(Bilkent University, 1999) Arslan, BoraShow more We proved that the quasi-equivalence property holds in a subclass of the class of stable nuclear Frechet, Kothe spaces. Also we considered the isomorphic classification of spaces of Whitney functions on some special compact sets in R. As a tool we use linear topological invariants and obtain some conditions.Show more Item Open Access Gibbs measures and phase transitions in one-dimensional models(Bilkent University, 2000) Mallak, SaedShow more In this thesis we study the problem of limit Gibbs measures in one-dimensional models. VVe investigate uniqueness conditions for the limit Gibbs measures for one-dimensional models. VVe construct a one-dimensional model disproving a uniqueness conjecture formulated before for one-dimensional models. It turns out that this conjecture is correct under some natural regularity conditions. VVe also apply the uniqueness theorem to some one-dimensional models.Show more Item Open Access Painleve test and the Painleve equations hierarchies(Bilkent University, 2001) Jrad, FahdShow more Recently there has been a considerable interest in obtaining higher order ordinary differential equations having the Painleve property. In this thesis, starting from the first, the second and the third Painleve transcendents polynomial and non-polynomial type higher order ordinary differential equations having the Painleve property have been obtained by using the singular point analysis.Show more Item Open Access Recursion operator and dispersionless Lax representation(Bilkent University, 2002) Zheltukhin, KostyantynShow more We give a general method for constructing recursion operators for some equations of hydrodynamic type, admitting a dispersionless Lax representation. We consider a polynomial and rational Lax function. We give several examples containing the equations of shallow water waves, polytropic gas dynamics and a degenerate bi-Hamiltonian system with a recursion operator. We also discuss a reduction of N + 1 systems to N systems of some new integrable equations of hydrodynamic type.Show more Item Open Access On sections and tails of power series(Bilkent University, 2002) Zheltukhina, NatalyaShow more The thesis is devoted to the study of connections between properties of a power series and properties of its sections and tails. Power series having sections or tails with multiply positive coefficients are considered and their growth estimates are obtained. Our results strengthen and supplement previous results in this direction, in particular, the well-known P´olya theorem on power series with sections having only negative zeros. The asymptotic zero distribution of linear combinations of sections and tails of the Mittag-Leffler function E1/ρ of order ρ > 1 is studied. Our results generalize and supplement previous results in this direction, in particular, the well-known Szeg¨o result on the linear combinations of sections and tails of the exponential function and the A Edrei, E.B. Saff and R.S. Varga results on sections of E1/ρ.Show more Item Open Access Codes on fibre products of Artin-Schreier and Kummer coverings of the projective line(Bilkent University, 2002-08) Shalalfeh, MahmoudShow more In this thesis, we study smooth projective absolutely irreducible curves defined over finite fields by fibre products of Artin-Schreier and Kummer coverings of the projective line. We construct some curves with many rational points defined by the fibre products of Artin-Schreier and Kummer coverings. Then, we apply Goppa construction to the curves that we have found, and obtain long linear codes with good relative parameters.Show more Item Open Access Code construction on modular curves(Bilkent University, 2003) Kara, OrhunShow more In this thesis, we have introduced two approaches on code construction on modular curves and stated the problems step by step. Moreover, we have given solutions of some problems in road map of code construction. One of the approaches uses mostly geometric and algebraic tools. This approach studies local invariants of the plane model Z0(`) of the modular curve Y0(`) given by the modular equation Φ` in affine coordinates. The approach is based on describing the hyperplane of regular differentials of Z0(`) vanishing at a given Fp 2 rational point. As constructing a basis for the regular differentials of Z0(`), we need to investigate its singularities. We have described the singularities of Z0(`) for prime ` in both characteristic 0 and positive characteristic. We have shown that all singularities of of the affine part, Z0(`), are self intersections. These self intersections are all simple nodes in characteristic 0 whereas the order of contact of any two smooth branches passing though a singular point may be arbitrarily large in characteristic p > 3 where p 6= `. Moreover the self intersections in characteristic zero are double. Indeed, structure of singularities of the affine curve Z0(`) essentially depends on two types of elliptic curves: The singularities corresponding to ordinary elliptic curves and the singularities corresponding to supersingular elliptic curves. The singularities corresponding to ordinary elliptic curves are all double points even though they are not necessarily simple nodes as in the case of characteristic 0. The singularities corresponding to supersingular elliptic curves are the most complicated ones and it may happen that there are more then two smooth branches passing though such kind of a singular point. We have computed the order of contact of any two smooth branches passing though a singular point both for ordinary case and for supersingular case.We have also proved that two points of Z0(`) at ∞ are cusps for odd prime ` which are analytically equivalent to the cusp of 0, given by the equation x ` = y `−1 . These two cusps are permuted by Atkin-Lehner involution. The multiplicity of singularity of each cusp is (`−1)(`−2) 2 . This result is valid in any characteristic p 6= 2, 3. The second approach is based on describing the Goppa codes on modular curve Y (`) as P SL2(F`) module. The main problem in this approach is investigating the structure of a group code as P SL2(F`) module. We propose a way of computing the characters of representations of a group code by using the localization formula. Moreover, we give an example of computing the characters of the code which associated to a canonical divisor on Y (`).Show more Item Open Access Representations of functions harmonic in the upper half-plane and their applications(Bilkent University, 2003) Gergün, SeçilShow more In this thesis, new conditions for the validity of a generalized Poisson representation for a function harmonic in the upper half-plane have been found. These conditions differ from known ones by weaker growth restrictions inside the halfplane and stronger restrictions on the behavior on the real axis. We applied our results in order to obtain some new factorization theorems in Hardy and Nevanlinna classes. As another application we obtained a criterion of belonging to the Hardy class up to an exponential factor. Finally, our results allowed us to extend the Titchmarsh convolution theorem to linearly independent measures with unbounded support.Show more Item Open Access Extension operators for spaces of infinitely differentiable functions(Bilkent University, 2005) Altun, MuhammedShow more We start with a review of known linear continuous extension operators for the spaces of Whitney functions. The most general approach belongs to PawÃlucki and Ple´sniak. Their operator is continuous provided that the compact set, where the functions are defined, has Markov property. In this work, we examine some model compact sets having no Markov property, but where a linear continuous extension operator exists for the space of Whitney functions given on these sets. Using local interpolation of Whitney functions we can generalize the PawÃlucki-Ple´sniak extension operator. We also give an upper bound for the Green function of domains complementary to generalized Cantor-type sets, where the Green function does not have the H¨older continuity property. And, for spaces of Whitney functions given on multidimensional Cantor-type sets, we give the conditions for the existence and non-existence of a linear continuous extension operator.Show more Item Open Access Distance between a maximum point and the zero set of an entire function(Bilkent University, 2006) Üreyen, Adem ErsinShow more We obtain asymptotical bounds from below for the distance between a maximum modulus point and the zero set of an entire function. Known bounds (Macintyre, 1938) are more precise, but they are valid only for some maximum modulus points. Our bounds are valid for all maximum modulus points and moreover, up to a constant factor, they are unimprovable. We consider entire functions of regular growth and obtain better bounds for these functions. We separately study the functions which have very slow growth. We show that the growth of these functions can not be very regular and obtain precise bounds for their growth irregularity. Our bounds are expressed in terms of some smooth majorants of the growth function. These majorants are defined by using orders, types, (strong) proximate orders of entire functions.Show more Item Open Access Modular vector invariants(Bilkent University, 2006) Madran, UğurShow more Vector invariants of finite groups (see the introduction for definitions) provides, in general, counterexamples for many properties of the invariant theory when the characteristic of the ground field divides the group order. Noether number is such property. In this thesis, we improve a lower bound for Noether number given by Richman in 1996: namely, we give a lower bound depending on the Jordan canonical form of an element of order equal to characteristic of the field. This method yields an effective bound by means of simple arithmetic arguments. The results are valid for any faithful representation of the group, including reducible and irreducible ones. Also they are extended to any algebraic field extensions provided the characteristic of the field divides the group order.Show more Item Open Access Solution surfaces and surfaces from a variotional principle(Bilkent University, 2007) Tek, SüleymanShow more In this thesis, we construct 2-surfaces in R 3 and in three dimensional Minkowski space (M3). First, we study the surfaces arising from modified Korteweg-de Vries (mKdV), Sine-Gordon (SG), and nonlinear Schr¨odinger (NLS) equations in R 3 . Second, we examine the surfaces arising from Korteweg-de Vries (KdV) and Harry Dym (HD) equations in M3. In both cases, there are some mKdV, NLS, KdV, and HD classes contain Willmore-like and algebraic Weingarten surfaces. We further show that some mKdV, NLS, KdV, and HD surfaces can be produced from a variational principle. We propose a method for determining the parametrization (position vectors) of the mKdV, KdV, and HD surfaces.Show more Item Open Access The Pauli principe, representation theory, and geometry of flag varieties(Bilkent University, 2008) Altunbulak, MuratShow more According to the Pauli exclusion principle, discovered in 1925, no two identical electrons may occupy the same quantum state. In terms of electron density matrix this amounts to an upper bound for its eigenvalues by 1. In 1926, it has been replaced by skew-symmetry of a multi-electron wave function. In this thesis we give two different solutions to a problem about the impact of this replacement on the electron density matrix, which goes far beyond the original Pauli principle.Show more

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