IFS attractors and Cantor sets

We build a metric space which is homeomorphic to a Cantor set but cannot be realized as the attractor of an iterated function system. We give also an example of a Cantor set K in R³ such that every homeomorphism f of R³ which preserves K coincides with the identity on K.     

Derivatives of regularly varying functions in Rd and domains of attraction of stable distributions

In R² the integral of a regularly varying (RV) function f is regularly varying only if f is monotone. Generalization to R² of the one-dimensional result on regular variation of the derivative of an RV-function however is straightforward. Applications are given to limit theory for partial sums of i.i.d. positive random vectors in R²₊.     

Wavelets frame associated to accretive function

We construct a frame associated to accretive function, and prove that it is complete in L²(Rⁿ). As an application, T(b) theorem for some kind of accretive function can be deduced naturally.     

Conditional limiting distribution of Type III elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation RAU where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Berman's result to Type III elliptical random vectors in R^d. Further, we derive an asymptotic approximation for the conditional distribution of such random vectors.     

A system of elliptic equations arising in Chern-Simons field theory

We prove the existence of topological vortices in a relativistic self-dual Abelian Chern-Simons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R². We present two approaches to prove existence of solutions on bounded domains: via minimization of an indefinite functional and via a fixed point argument. We then show that we may pass to the full R² limit from the bounded-domain solutions to obtain a topological solution in R².     

Rings whose multiplicative endomorphisms are power functions

Let R be a commutative ring. For any positive integer m, the power function f:R→R defined by f(x):=x ^m is easily seen to be an endomorphism of the multiplicative semigroup (R,?). In this note, we characterize the commutative rings R with identity for which every multiplicative endomorphism of (R,?) is equal to a power function. Specifically, we show that every endomorphism of (R,?) is a power function if and only if R is a finite field.     

Optimal lower bound estimates for the blow-up rate for the Zakharov system in a nonhomogeneous medium

In this paper we obtain lower bound estimates for the blow-up rate of finite time blow-up solutions to the Cauchy problem for the Zakharov system in a nonhomogeneous medium in two space dimensions. By introducing suitable scale transformations of space and time, and the use of compactness arguments, we derive an optimal lower bound estimate in the energy space H²(R²)×L²(R²)×H¹(R²) for the blow-up rate for t near the finite blow-up time T. Also we give an application to the virial identity for the Zakharov system under study.     

Approximation of measures by Markov processes and homogeneous affine iterated function systems

It is shown that under certain conditions, attractive invariant measures for iterated function systems (indeed for Markov processes on locally compact spaces) depend continuously on parameters of the system. We discuss a special class of iterated function systems, the homogeneous affine ones, for which an inverse problem is easily solved in principle by Fourier transform methods. We show that a probability measureμ onR ⁿ can be approximated by invariant measures for finite iterated function systems of this class if \(\hat \mu (t)/\hat \mu (a^T t)\) is positive definite for some nonzero contractive linear mapa:R ⁿ →R ⁿ . Moments and cumulants are also discussed.     

Ombres Convexité, régularité et sous-harmonicité

Letf:V→R be a function defined on a subsetV ofR ⁿ ×R ^d let?:x→inf{f(x t);t such that(x t)∈V} denote theshadow off and letΦ={(x t)∈V; f(x t)=?(x)} This paper deals with the characterization of some properties of ? in terms of the infinitesimal behavior off near points ζ∈Φ proving in particular a conjecture of J M Trépreau concerning the cased=1 Characterizations of this type are provided for the convexity the subharmonicity or theC ^{1 1} regularity of ? in the interior ofI={x∈ R ⁿ;εR ^d (x t)∈V} and in theC ^{1 1} case an expression forD ²? is given To some extent an answer is given to the following question: which convex function ?:I→R I interval ?R (resp which function √:I→R of classC ^{1 1}) is the shadow of aC ² functionf:I×R→R?     

Stability in the Rotating Bénard Problem and Its Optimal Lyapunov Functions

Stability analysis of the rotating Bénard problem gives a spectral instability threshold of the purely conducting solution that can be expressed as a critical Rayleigh number R ² depending on the Taylor number T ². The definition of a functional which can be used to prove Lyapunov stability up to the threshold of spectral instability (optimal Lyapunov function) is an important step forward both, for a proof of nonlinear stability and for the investigation of the basin of attraction of the equilibrium. In previous works a Lyapunov function was found, but its optimality could be proven only for small T ². In this work we describe the reason why this happens, and provide a weaker definition of Lyapunov function which allows to prove that, for the linearized system, the spectral instability threshold is also the Lyapunov stability threshold for every value of T ².     

Nonlinear stability analysis of a regular vortex pentagon outside a circle

A nonlinear stability analysis of the stationary rotation of a system of five identical point vortices lying uniformly on a circle of radius R ₀ outside a circular domain of radius R is performed. The problem is reduced to the problem of stability of an equilibrium position of a Hamiltonian system with a cyclic variable. The stability of stationary motion is interpreted as Routh stability. Conditions for stability, formal stability and instability are obtained depending on the values of the parameter q = R ²/R ₀ ² .     

Spectral expansion on the entire axis of the green function for a three-layer medium in fundamental functions of a nonself-adjoint Sturm-Liouville operator

We give a new representation of the Green function in the space R ² for the Helmholtz equation with coefficient that is a complex-valued piecewise constant function depending on a single variable and taking three values. This representation has the form of an expansion in fundamental functions, i.e., bounded (on the entire line R ¹) solutions of the Sturm-Liouville equation with a complex coefficient. The spectrum consists of two rays parallel to the real axis in the complex plane of the spectral parameter. The origin of the rays is determined by constants characterizing the coefficient in the equation.     

Canonical decompositions of symmetric submodular systems

Let E be a finite set, R be the set of real numbers and f: 2^E→R be a symmetric submodular function. The pair (E,f) is called a symmetric submodular system. We examine the structures of symmetric submodular systems and provide a decomposition theory of symmetric submodular systems. The theory is a generalization of the decomposition theory of 2-connected graphs developed by Tutte and can be applied to any (symmetric) submodular systems.     

Properties of oblique dual frames in shift-invariant systems

Gramian analysis is used to study properties of a shift-invariant system , where B is an invertible n×n matrix and Φ a finite or countable subset of L²(Rn) under the assumption that the system forms a frame for the closed subspace M of L²(Rn). In particular, the relationship between various features of such system, such as being a frame for the whole space L²(Rn), being a Riesz sequence and having a unique shift-generated dual of type I or II is discussed in details. Several interesting examples are presented.     

Immersion extension-lift over a Morse function

Let V be a compact connected oriented surface with boundary and f:∂V×[0,1)→R a non-singular function such that f|∂V×{0} is a Morse function. Let ι:∂V×[0,1)→V be a collaring of ∂V and π:R²→R an orthogonal projection. In this paper, we study existence of an orientation preserving immersion F:V→R² such that π°F°ι=f. We also study image homotopy classes of F when we fix f and study relation between two image homotopy classes when f is deformed under a Morse homotopy.     

Invariant manifolds of dynamical systems close to a rotation: Transverse to the rotation axis

We consider one parameter families of vector fields depending on a parameter ? such that for ?=0 the system becomes a rotation of R²×Rⁿ around {0}×Rⁿ and such that for ?>0 the origin is a hyperbolic singular point of saddle type with, say, attraction in the rotation plane and expansion in the complementary space. We look for a local subcenter invariant manifold extending the stable manifolds to ?=0. Afterwards the analogous case for maps is considered. In contrast with the previous case the arithmetic properties of the angle of rotation play an important role.     

The method of boundary integral equations for a mixed scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R² as cross section. To this end, we solve a scattering problem for the Helmholtz equation in R² where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we have Dirichlet-impedance type boundary condition on Γ and Dirichlet boundary condition on ∂D (∂D∈C²). Applying potential theory, the problem can be reformulated as a boundary integral system. We establish the existence and uniqueness of a solution to the system by using the Fredholm theory.     

Extending a function on a graph

Let G be a graph with vertex set V, and let h be a function mapping a subset U of V into the real numbers R. If ? is a function from V to R, we define δ (?) to be the sum of ∥?(b)? ?(a)∥ over all edges {a, b} of G. A best extension of h is such a function ? with ?(x) = h(x) for X ∈ U and minimum δ (?). We show that such a best extension exists and derive an algorithm for obtaining such an extension. We also show that if instead we minimise the sum of (?(b)??(a))², there is generally a unique best extension, obtainable by solving a system of linear equations.     

Classification of generalized multiresolution analyses

We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions m and matrix-valued filter functions H. Given a natural number valued function m and a system of functions encoded in a matrix H satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function m and filter system H. An equivalence relation on GMRAs is defined and described in terms of their associated pairs (m,H). This classification system is applied to MRAs and other classical examples in L²(Rd) as well as to previously studied abstract examples.     

The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system

The blow-up solutions of the Cauchy problem for the Davey-Stewartson system, which is a model equation in the theory of shallow water waves, are investigated. Firstly, the existence of the ground state for the system derives the best constant of a Gagliardo-Nirenberg type inequality and the variational character of the ground state. Secondly, the blow-up threshold of the Davey-Stewartson system is developed in R³. Thirdly, the mass concentration is established for all the blow-up solutions of the system in R². Finally, the existence of the minimal blow-up solutions in R² is constructed by using the pseudo-conformal invariance. The profile of the minimal blow-up solutions as t→T (blow-up time) is in detail investigated in terms of the ground state.