Academic literature on the topic 'Toda system'

In this thesis, we mainly consider two problems. First, we study the SU(3) Toda system. Let (M,g) be a compact Riemann surface with volume 1, h₁ and h₂ be a C¹ positive function on M and p1; p2 ∈ ℝ⁺. The SU(3) Toda system is the following one on the compact surface M [Formula and equation omitted] where ∆ is the Beltrami-Laplace operator, αq ≥ 0 for every q ∈ S₁, S₁ ⊂ M, Bq ≥ 0 for every q ∈ S₂,S₂ ⊂ M and q is the Dirac measure at q ∈ M. We initiate the program for computing the Leray-Schauder topological degree of SU(3) Toda system and succeed in obtaining the degree formula for p1 ∈ (0,4π)(4π,8π), p2 ∉ 4πℕ when S₁ = S₂ = 0. Second, we consider the following nonlinear elliptic Neumann problem {∆u-μu +uq =0 in Ω,u > 0 in Ω,au/av=0 on aΩ. where q=n+2/n-2, μ > 0 and Ω is a smooth and bounded domain in ℝn. Lin and Ni (1986) conjectured that for μ small, all solutions are constants. In the second part of this thesis, we will show that this conjecture is false for a general domain in n = 4, 6 by constructing a nonconstant solution.<br>Science, Faculty of<br>Mathematics, Department of<br>Graduate

An efficient housing finance system has significant importance both in meeting the housing needs of individuals and in reinforcing the development of the construction, finance and other related sectors of an economy. Today, developed countries have advanced housing finance systems in which funds flow from savers to home-buyers by the mortgage markets. On the other hand, despite its recognized economic and social importance, housing finance often remains under-developed in developing countries mainly due to the lack of macroeconomic stability. Turkey, being a developing country, has made an important step towards the development of a mortgage system with the passage of the new Mortgage Law by the Parliament. Accordingly, the purpose of this thesis is to examine the applicability of mortgage system in Turkey. For this purpose, housing finance systems of some developed and developing countries are reviewed, and the housing finance system in Turkey is explained. Further, causality between the total amount of housing loans issued, inflation and nominal interest rates in Turkey is analyzed with the Toda-Yamamoto VAR approach. VAR analysis shows the negative impact of nominal interest rates on the total amount of housing loans issued in Turkey. To sum up, considering its economic and social environment, Turkey has adapted best international experiences, and it is possible for a mortgage system to develop in the country by the new mortgage legislation combined with the lower interest rates as inflation declines.

This thesis demonstrates the use of a two color lidar (light detection and ranging) instrument for the purpose of studying atmospheric aerosols. The instrument and the analysis techniques are explained and discussed to provide the necessary back-ground. The calibration is discussed and demonstrated followed by an example of the data analysis. The lidar's combination with a digital camera used to image cloud formations is then discussed and preliminary results are displayed.

Dans cette thèse, nous nous intéressons à plusieurs aspects des systèmes de racines des algèbres de Lie simples. Dans un premier temps, nous étudions les coordonnées des vecteurs propres des matrices de Cartan. Nous commençons par généraliser les travaux de physiciens qui ont montré que les masses des particules dans la théorie des champs de Toda affine sont égales aux coordonnées du vecteur propre de Perron -- Frobenius de la matrice de Cartan. Puis nous adoptons une approche différente, puisque nous utilisons des résultats de la théorie des singularités pour calculer les coordonnées des vecteurs propres de certains systèmes de racines. Dans un deuxième temps, en s'inspirant des idées de Givental, nous introduisons les matrices de Cartan q-déformées et étudions leur spectre et leurs vecteurs propres. Puis, nous proposons une q-déformation des équations de Toda et construisons des 1-solitons solutions en adaptant la méthode de Hirota, d'après les travaux de Hollowood. Enfin, notre intérêt se porte sur un ensemble de transformations agissant sur l'ensemble des bases ordonnées de racines comme le groupe de tresses. En particulier, nous étudions les bases distinguées, qui forment l'une des orbites de cette action, et des matrices que nous leur associons<br>In this thesis, our goal is to study various aspects of root systems of simple Lie algebras. In the first part, we study the coordinates of the eigenvectors of the Cartan matrices. We start by generalizing the work of physicists who showed that the particle masses of the affine Toda field theory are equal to the coordinates of the Perron -- Frobenius eigenvector of the Cartan matrix. Then, we adopt another approach. Namely, using the ideas coming from the singularity theory, we compute the coordinates of the eigenvectors of some root systems. In the second part, inspired by Givental's ideas, we introduce q-deformations of Cartan matrices and we study their spectrum and their eigenvectors. Then, we propose a q-deformation of Toda's equations et compute 1-solitons solutions, using the Hirota's method and Hollowood's work. Finally, our interest is focused on a set of transformations which induce an action of the braid group on the set of ordered root basis. In particular, we study an orbit for this action, the set of distinguished basis and some associated matrices

In this thesis we investigate two examples of infinite dimensional integrable Hamiltonian systems in $1$-space dimension: the Toda chain with periodic boundary conditions and large number of particles, and the Korteweg-de Vries (KdV) equation on $\R$. In the first part of the thesis we study the Birkhoff coordinates (Cartesian action angle coordinates) of the Toda lattice with periodic boundary condition in the limit where the number $N$ of the particles tends to infinity. We prove that the transformation introducing such coordinates maps analytically a complex ball of radius $R/N^\alpha$ (in discrete Sobolev-analytic norms) into a ball of radius $R'/N^\alpha$ (with $R,R'>0$ independent of $N$) if and only if $\alpha\geq2$. Then we consider the problem of equipartition of energy in the spirit of Fermi-Pasta-Ulam. We deduce that corresponding to initial data of size $R/N^2$, $0<R\ll 1$, and with only the first Fourier mode excited, the energy remains forever in a packet of Fourier modes exponentially decreasing with the wave number. Finally we consider the original FPU model and prove that energy remains localized in a similar packet of Fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet. The proof of the theorem on Birkhoff coordinates is based on a new quantitative version of a Vey type theorem by Kuksin and Perelman which could be interesting in itself. In the second part of the thesis we study the scattering map of the KdV on $\R$. We prove that in appropriate weighted Sobolev spaces of the form $H^{N} \cap L^2_M$, with integers $N \geq 2M \geq 8$ and in the case of no bound states, the scattering map is a perturbation of the Fourier transform by a regularizing operator. As an application of this result, we show that the difference of the KdV flow and the corresponding Airy flow is 1-smoothing.