On some Density Theorems in Number Theory and Group Theory

Gowers, dans son article sur les matrices quasi-aléatoires, étudie la question, posée par Babai et Sos, de l'existence d'une constante $c>0$ telle que tout groupe fini possède un sous-ensemble sans produit de taille supérieure ou égale a $c|G|$. En prouvant que, pour tout nombre premier $p$ assez grand, le groupe $PSL_2(\mathbb{F}_p)$ (d'ordre noté $n$) ne posséde aucun sous-ensemble sans produit de taille $c n^{8/9}$, il y répond par la négative. Nous allons considérer le probléme dans le cas des groupes compacts finis, et plus particuliérement des groupes profinis $SL_k(\mathbb{Z}_p)$ et $Sp_{2k}(\mathbb{Z}_p)$. La premiére partie de cette thése est dédiée à l'obtention de bornes inférieures et supérieures exponentielles pour la mesure suprémale des ensembles sans produit. La preuve nécessite d'établir préalablement une borne inférieure sur la dimension des représentations non-triviales des groupes finis $SL_k(\mathbb{Z}/(p^n\mathbb{Z}))$ et $Sp_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Notre théoréme prolonge le travail de Landazuri et Seitz, qui considérent le degré minimal des représentations pour les groupes de Chevalley sur les corps finis, tout en offrant une preuve plus simple que la leur. La seconde partie de la thése à trait à la théorie algébrique des nombres. Un polynome monogéne $f$ est un polynome unitaire irréductible à coefficients entiers qui endengre un corps de nombres monogéne. Pour un nombre premier $q$ donné, nous allons montrer, en utilisant le théoréme de densité de Tchebotariov, que la densité des nombres premiers $p$ tels que $t^q -p$ soit monogéne est supérieure ou égale à $(q-1)/q$. Nous allons également démontrer que, quand $q=3$, la densité des nombres premiers $p$ tels que $\mathbb{Q}(\sqrt[3]{p})$ soit non monogéne est supérieure ou égale à $1/9$.

On Szegö’s Type Theorems

This study is to look the effect of change in the ordering of the Fourier system on Szegö’s classical observations of asymptotic distribution of eigenvalues of finite Toeplitz forms.This is done by checking proofs and Szegö’s properties in the new set up.The Fourier system is unconditional [19], any arbitrary ordering of the Fourier system forms a basis for the Hilbert space L2 [-Π, Π].Here study about the classical Szegö’s theorem.Szegö’s type theorem for operators in L2(R+) and check its validity for certain multiplication operators.Since the trigonometric basis is not available in L2(R+) or in L2(R) .This study discussed about the classes of orderings of Haar System in L2 (R+) and in L2(R) in which Szegö’s Type TheoreT Am is valid for certain multiplication operators.It is divided into two sections. In the first section there is an ordering to Haar system in L2(R+) and prove that with respect to this ordering, Szegö’s Type theorem holds for general class of multiplication operators Tƒ with multiplier ƒ ε L2(R+), subject to some conditions on ƒ.Finally in second section more general classes of ordering of Haar system in L2(R+) and in L2(R) are identified in such a way that for certain classes of multiplication operators the asymptotic distribution of eigenvalues exists.

Lateral boundary contributions to wave-activity invariants and nonlinear stability theorems for balanced dynamics

The slow advective-timescale dynamics of the atmosphere and oceans is referred to as balanced dynamics. An extensive body of theory for disturbances to basic flows exists for the quasi-geostrophic (QG) model of balanced dynamics, based on wave-activity invariants and nonlinear stability theorems associated with exact symmetry-based conservation laws. In attempting to extend this theory to the semi-geostrophic (SG) model of balanced dynamics, Kushner & Shepherd discovered lateral boundary contributions to the SG wave-activity invariants which are not present in the QG theory, and which affect the stability theorems. However, because of technical difficulties associated with the SG model, the analysis of Kushner & Shepherd was not fully nonlinear. This paper examines the issue of lateral boundary contributions to wave-activity invariants for balanced dynamics in the context of Salmon's nearly geostrophic model of rotating shallow-water flow. Salmon's model has certain similarities with the SG model, but also has important differences that allow the present analysis to be carried to finite amplitude. In the process, the way in which constraints produce boundary contributions to wave-activity invariants, and additional conditions in the associated stability theorems, is clarified. It is shown that Salmon's model possesses two kinds of stability theorems: an analogue of Ripa's small-amplitude stability theorem for shallow-water flow, and a finite-amplitude analogue of Kushner & Shepherd's SG stability theorem in which the ‘subsonic’ condition of Ripa's theorem is replaced by a condition that the flow be cyclonic along lateral boundaries. As with the SG theorem, this last condition has a simple physical interpretation involving the coastal Kelvin waves that exist in both models. Salmon's model has recently emerged as an important prototype for constrained Hamiltonian balanced models. The extent to which the present analysis applies to this general class of models is discussed.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 2. pseudoenergy-based theory

This paper represents the second part of a study of semi-geostrophic (SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was based on the pseudomomentum; Part 2 is based on the pseudoenergy. A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear stability theorem analogous to Arnol'd's ‘first theorem’; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number. The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir & Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states. The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 1. pseudomomentum-based theory

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics. In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms. The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent. This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii). In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems

Disturbances of arbitrary amplitude are superposed on a basic flow which is assumed to be steady and either (a) two-dimensional, homogeneous, and incompressible (rotating or non-rotating) or (b) stably stratified and quasi-geostrophic. Flow over shallow topography is allowed in either case. The basic flow, as well as the disturbance, is assumed to be subject neither to external forcing nor to dissipative processes like viscosity. An exact, local ‘wave-activity conservation theorem’ is derived in which the density A and flux F are second-order ‘wave properties’ or ‘disturbance properties’, meaning that they are O(a2) in magnitude as disturbance amplitude a [rightward arrow] 0, and that they are evaluable correct to O(a2) from linear theory, to O(a3) from second-order theory, and so on to higher orders in a. For a disturbance in the form of a single, slowly varying, non-stationary Rossby wavetrain, $\overline{F}/\overline{A}$ reduces approximately to the Rossby-wave group velocity, where (${}^{-}$) is an appropriate averaging operator. F and A have the formal appearance of Eulerian quantities, but generally involve a multivalued function the correct branch of which requires a certain amount of Lagrangian information for its determination. It is shown that, in a certain sense, the construction of conservable, quasi-Eulerian wave properties like A is unique and that the multivaluedness is inescapable in general. The connection with the concepts of pseudoenergy (quasi-energy), pseudomomentum (quasi-momentum), and ‘Eliassen-Palm wave activity’ is noted. The relationship of this and similar conservation theorems to dynamical fundamentals and to Arnol'd's nonlinear stability theorems is discussed in the light of recent advances in Hamiltonian dynamics. These show where such conservation theorems come from and how to construct them in other cases. An elementary proof of the Hamiltonian structure of two-dimensional Eulerian vortex dynamics is put on record, with explicit attention to the boundary conditions. The connection between Arnol'd's second stability theorem and the suppression of shear and self-tuning resonant instabilities by boundary constraints is discussed, and a finite-amplitude counterpart to Rayleigh's inflection-point theorem noted

ON NONSYMMETRIC THEOREMS FOR (H, G)-COINCIDENCES

Let X be a compact Hausdorff space, phi: X -> S(n) a continuous map into the n-sphere S(n) that induces a nonzero homomorphism phi*: H(n)(S(n); Z(p)) -> H(n)(X; Z(p)), Y a k-dimensional CW-complex and f: X -> a continuous map. Let G a finite group which acts freely on S`. Suppose that H subset of G is a normal cyclic subgroup of a prime order. In this paper, we define and we estimate the cohomological dimension of the set A(phi)(f, H, G) of (H, G)-coincidence points of f relative to phi.

Coincidence theorems and its applications to equilibrium problems

Given two maps h : X x K -> R and g : X -> K such that, for all x is an element of X, h(x, g(x)) = 0, we consider the equilibrium problem of finding (x) over tilde is an element of X such that h((x) over tilde, g(x)) >= 0 for every x is an element of X. This question is related to a coincidence problem.

LaSalle`s theorems in impulsive semidynamical systems

We consider a semidynamical system subject to variable impulses and we obtain the LaSalle invariance principle and the asymptotic stability theorem for this semidynamical system. (C) 2009 Elsevier Ltd. All rights reserved.

Limit theorems for sequences of random trees

We consider a random tree and introduce a metric in the space of trees to define the ""mean tree"" as the tree minimizing the average distance to the random tree. When the resulting metric space is compact we have laws of large numbers and central limit theorems for sequence of independent identically distributed random trees. As application we propose tests to check if two samples of random trees have the same law.

Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in De Sitter space

In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space S(1)(n+1)(c), n >= 3, with constant normalized scalar curvature R satisfying n-2/nc <= R <= c totally umbilical? (C) 2008 Elsevier B.V. All rights reserved.