Theses - Department of Mathematics
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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, HasanWe 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.Item Open Access Homogeneous, anisotropic solutions of topologically massive gravity including a cosmological constant(Bilkent University, 1993) Saygılı, KamuranI'lxact solutions l.o tin' (ic'ld (M|iial.ioiis o( 'Го|)о1о;г;1са,11у ma.ssiv(i gravity wit h a cosmological coiisl.anl. ai<' pix'scntisl. Tlics(' ai<‘ lK)mogint te> the e.xiste'ite:e e)f a. e ritical value' lor the te)|)ole)gical mass whie h is ele'te'rmine'd by the ce)sme)le)gie al e e)nstaid .Item Open Access Unified approach to transformations of Painleve equations(Bilkent University, 1993) Chahardehi, Ali Reza ModaressiIn this thesis, we iind the explicit form of some transformations associated with the second, third, fourth and fifth Painleve equations. These transformations are obtained by using the Schlesinger transformations associated with the linear system of equations of Painleve eciuations.The application of such transformations enables us to generate the new solutions of the given Painleve equation with different values of parameters, from the known solutions.Item Open Access An integrable family of Monge-Ampere equations and their multi-Hamiltonian structure(Bilkent University, 1993) Sarıoğlu, Bahtiyar ÖzgürWe have identified a completely integrable family of Monge-Ampère equations through an examination of their Hamiltonian structure. Starting with a variational formulation of the Monge-Ampère equations we have constructed the first Hamiltonian operivtor through an application of Dirac’s theory of constraints. The completely integrable class of Monge-Ampère equations are then obtained by solving the .Jacobi identities for a sufficiently general form of the second Hamiltonian operator that is compatible with the first. Furthermore, Chern, Levine and Nirenberg have long ago pointed out the distinguished role that the complex homogeneous Monge-Ampère equation plays in the theory of functions of several complex variables. In particular Semmes has called attention to the symplectic structure of the geodesic flow defined by this equation. A new approach to this problem in the framework of dynamical .systems ( with infinitely many degrees of freedom ) shows that it is a completely integrable system. This example exhibits several new features in the theory of integrable systems as well. Namely it is an integrable system in arbitrary dimension and furthermore admits infinitely many symplectic structures. The latter is the key to a proof of integrability through Magri’s theorem which requires only bi-Hamiltonian structure.Item Open Access On Arf rings(Bilkent University, 1994) Arslan, Sefa FezaIn this thesis, we worked with curves which have cusp type singularities. We described the Arf theory, which solves the problem of understanding and finding the multiplicity sequence of a curve branch algebraically. We proposed an algorithm for finding the Arf characters of a given curve branch. We also faced the problem of Frobenius, and proposed an algorithm for the solution of problem of Frobenius in the most general case.Item Open Access Higher dimensional spherically symmetric gravitational theories(Bilkent University, 1994) Sermutlu, EmreWe 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.Item Open Access Review of an approach to obstruction theory(Bilkent University, 1994) Kırdar, MehmetIn this work, we summarized an approach to obstruction tlieory developed by E. Thomas. We gave some illustrative examples to demonstrate the method. These are : 2-plane fields on manifolds revi(iwing works of Thotnas, 3 and 4 fields on (4k-f3)-dimensional manifolds.Item Open Access Regular basis and functor Ext(Bilkent University, 1994) Ertuğrul, ZehraThis work is a study of the relation between the vanishing of Ext functor and the existence of regular bases in the cartesian product and tensor product of some special Kothe spaces. We give some new results concerning Sg Spaces in Chapter 3 and the study in the last chapter is about the existence of pseudo-regular bases in the cartesian product and tensor product of two regular Schwartz Kothe spaces E and F , one of which having property (DN), when Ext(E x F,E x F) vanishes.Item Open Access Aspects of Fibonacci numbers(Bilkent University, 1994) Yücel, GülnihalThis thesis consists of two parts. The first part, which is Chapter 2, is a survey on some aspects of Fibonacci numbers. In this part, we tried to gather some interesting properties of these numbers and some topics related to the Fibonacci sequence from various references, so that the reader may get an overview of the subject. After giving the basic concepts about the Fibonacci numbers, their arithmetical properties are studied. These include divisibility and periodicity properties, the Zeckendorf Theorem, Fibonacci trees and their relations to the representations of integers, polynomials used for deriving new identities for Fibonacci numbers and Fibonacci groups. Also in Chapter 2, natural phenomena related to the golden section, such as certain plants having Fibonacci numbers for the number of petals, or the relations of generations of bees with the Fibonacci numbers are recounted. In the second part of the thesis. Chapter 3, we focused on a Fibonacci based random number sequence. We analyzed and criticized the generator Sfc = k(j>—[k(j)] by applying some standart tests for randomness on it. Chapter 5, the Appendix consists of Fortran programs used for executing the tests of Chapter 3.Item Open Access Isomorphic classification problem and linear topological invariants(Bilkent University, 1995) Arslan, BoraWe consider all possible isomorphisms of cartesian products of Dragilev spaces, and thanks to relations between the Dragilev functions of each factor try to show that if there exists such an isomorphism, then any factor on the left of the isomorphism is nearly isomorphic to the corresponding factor on the right. We also try to get a liecessary condition for the isomorphisms of the tensor products of infinite type Dragilev spaces by the dual of an infinite type Montel power series space.Item Open Access Transformation properties of Painleve VI equation(Bilkent University, 1995) Sakka, AymanIn this thesis, we studied the Schlesinger transformations of Painleve VI equation. We showed that Painleve VI equation admits Schlesinger transformations which relate a given solution of Paileve VI to solution of Painleve VI but with different values of the parameters. Using these transformations we obtained the corresponding Bäcklund transformations for Painleve VI. Also, we showed that the Schlesinger transformations and the corresponding Bäcklund transformations break down if and only if Painleve VI has certain one-parameter family of solutions.Item Open Access Analytic and asymptotic properties of non-symmetric Linnik's probability densities(Bilkent University, 1995) Erdoğan, M. BurakWe prove that the function 1 , a 6 (0 ,2 ), ^ e R, 1 + is a characteristic function of a probability distribution if and only if ( a , 0 e P D = {{a,e) : a € (0,2), \d\ < m in (f^ , x - ^ ) (mod 27t)}. This distribution is absolutely continuous, its density is denoted by p^(x). For 0 = 0 (mod 2tt), it is symmetric and was introduced by Linnik (1953). Under another restrictions on 0 it was introduced by Laha (1960), Pillai (1990), Pakes (1992). In the work, it is proved that p^{±x) is completely monotonic on (0, oo) and is unimodal on R for any (a,0) € PD. Monotonicity properties of p^(x) with respect to 9 are studied. Expansions of p^(x) both into asymptotic series as X —»· ±oo and into conditionally convergent series in terms of log |x|, \x\^ (^ = 0 ,1 ,2 ,...) are obtained. The last series are absolutely convergent for almost all but not for all values of (a, 0) € PD. The corresponding subsets of P D are described in terms of Liouville numbers.Item Open Access Boundary conditions compatible with the generalized symmetries(Bilkent University, 1995) Gürel, T. BurakIn this work evolution type integrable equations and systems are considered. An efficient method is given to construct their boundary conditions and hence boundary value problems which are compatible with the generalized symmetries. This method is applied to some well-known nonlinear partial differential equations.Item Open Access On lower bounds of character sums(Bilkent University, 1995) Özbudak, Ferruh111 ilii.s work wo oxlomlod llio ic.sult.s of S.A. Slc|)aiiov [3], [i] about lower bouiid.s for incomplete clia.ra.cter .suiiks over a prime finite Held to the ea..se of arbitrar}^ linite field Moreover we atso a.pplied Cioppats con.struction to .siiperelliptic curves witli a lot of rational |)oints to construct rather good geometric Goppa codes.Item Open Access On possible deterioration of smoothness under the operation of convolution(Bilkent University, 1996) Uludağ, A. MuhammedWe show that the convolution of two probability densities which are restrictions to R of entire functions can possess infinite essential supremuin on each interval. We also present several sufficient conditions of deterioration of smoothness under the operation of convolution.Item Open Access Partial differential equations possessing the Painleve property(Bilkent University, 1996) Jrad, Fahd111 this tli(\sis, a|)|)lying llie l^viiilovc tost (l(ivolo|)('.(l by VV(hss, 'labor ainl t biriK'vale (VV1X9) investigatc'd the Pa.inleve property of Ibirgers’ ty|)e of ('(piarioiis, KdV type of equations and the KP extc'iisions of th(' KdV i-yi)(' of ('i|na,tions. VVe showed that there a.rc^ iiiiinitely many e(|nations of t,h('S(' t-ypc's poss('ssing tlu^ Painleve propcn'ty and tims we elassiíi(MÍ tlnmi witJi res])ect to Pa.illlevé property.Item Open Access Non-stationary Markov chains(Bilkent University, 1996) Mallak, SaedIn thi.s work, we studierl the Ergodicilv of Non-Stationary .Markov chains. We gave several e.xainples with different cases. We proved that given a sec[uence of Markov chains such that the limit of this sec|uence is an Ergodic Markov chain, then the limit of the combination of the elements of this sequence is again Ergodic (under some condition if the state space is infinite). We also proved that the limit of the combination of an arbitrary sequence of Markov chains on a finite state space is Weak Ergodic if it satisfies some condition. Under the same condition, the limit of the combination of a doubly stochastic sequence of Markov chains is Ergodic.Item Open Access The linear mean value of the remainder term in the problem of asymptotic behaviour of eigenfunctions of the automorphic Laplacian(Bilkent University, 1996) Emirleroğlu, ZernişanThe purpose of this thesis is to obtain the estimate for the average mean value of the remainder term of the asymptotic formula for the quadratic mean value of the Fourier coefficients of the eigenfunctions over the discrete spectrum of the automorphic Laplacian.Item Open Access Asymptotic theory of characters of the symmetric groups(Bilkent University, 1996) Kurtaran, ElifIn this work, we studied the connection between ramified coverings of Riemann surfaces tt : X V oi degree n and characters of symmetric group Sn- We considered asymptotics of characters of as n —> oo and normalized characters of Sn under some restrictions.Item Open Access Character sums, algebraic function fields, curves with many rational points and geometric Goppa codes(Bilkent University, 1997) Özbudak, FerruhIn 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.