Inequalities for the Extremal Coefficients of Multivariate Extreme Value Distributions

The extremal coefficients are the natural dependence measures for multivariate extreme value distributions. For an m-variate distribution 2^m distinct extremal coefficients of different orders exist; they are closely linked and therefore a complete set of 2^m coefficients cannot take any arbitrary values. We give a full characterization of all the sets of extremal coefficients. To this end, we introduce a simple class of extreme value distributions that allows for a 1-1 mapping to the complete sets of extremal coefficients. We construct bounds that higher order extremal coefficients need to satisfy to be consistent with lower order extremal coefficients. These bounds are useful as lower order extremal coefficients are the most easily inferred from data.     

Extreme Points and Strongly Extreme Points of Musielak-Orlicz Sequences Spaces

Criteria for extreme points and strongly extreme points in Musielak-Orlicz sequence spaces, equipped with both the Luxemburg norm and the Orlicz norm, are given.     

On Extension and Continuity of p–Additive Functions

In this paper we prove some properties of p–additive functions as well as p–additive set–valued functions.     

Ridged Domains,Embedding Theorems and Poincaré Inequalities

We study the embeddings E : W(X(Ω), Y(Ω)) ↪ Z(Ω), where X(Ω), Y(Ω) and Z(Ω) are rearrangement–invariant Banach function spaces (BFS) defined on a generalized ridged domain Ω, and W denotes a first–order Sobolev–type space. We obtain two–sided estimates for the measure of non–compactness of E when Z(Ω) = X(Ω) and, in turn, necessary and sufficient conditions for a Poincaré–type inequality to be valid and also for E to be compact. The results are used to analyse the example of a trumpet–shaped domain Ω in Lorentz spaces. We consider the problem of determining the range of possible target spaces Z(Ω), in which case we prove that the problem is equivalent to an analogue on the generalized ridge Γ of Ω. The range of target spaces Z(Ω) is determined amongst a scale of (weighted) Lebesgue spaces for “rooms and passages” and trumpet–shaped domains.     

Majorants and Extreme Points of Unit Balls in Bernstein Spaces

The Bernstein space B ^p () (1 $$ " align="middle" border="0"> 0) is the set of functions from L ^p( ) having Fourier transforms (in the sense of generalized functions) with supports in the compact segment [- , ]. Every function f has an analytic continuation onto the complex plane, which is an entire function of exponential type . The spaces B ^p ()\, are conjugate Banach spaces. Therefore, the closed unit ball in B ^p () has a rich set of extreme (boundary) points: coincides with the weakly ^* closed convex hull of its extreme points. Since, for 1< p< , B ^p () is a uniformly convex space, only the balls and have nontrivially arranged sets of extreme points. In this paper, in terms of zeros of entire functions, we obtain necessary and sufficient conditions of extremeness for functions from .     

On Probability and Moment Inequalities for Supermartingales and Martingales

The probability inequality for sum S _n = _j=1 ⁿ X _j is proved under the assumption that the sequence S _k , k= , forms a supermartingale. This inequality is stated in terms of the tail probabilities P(X _j >y) and conditional variances of the random variables X _j , j= . The well-known Burkholder moment inequality is deduced as a simple consequence.     

Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation

In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained.     

On the Points on the Unit Circle with Finite b–Adic Expansions

Let be an arbitrary integer base and let be the number of different prime factors of with , . Further let be the set of points on the unit circle with finite –adic expansions of their coordinates and let be the set of angles of the points . Then is an additive group which is the direct sum of infinite cyclic groups and of the finite cyclic group . If in case of the points of are arranged according to the number of digits of their coordinates, then the arising sequence is uniformly distributed on the unit circle. On the other hand, in case of the only points in are the exceptional points (1, 0), (0, 1), (–1, 0), (0, –1). The proofs are based on a canonical form for all integer solutions of .     

Extreme points of lattice intervals in the Minkowski-Rådström-Hörmander lattice

In this paper we characterize extreme points of any symmetric interval in the Minkowski-Rådström-Hörmander lattice over any Hausdorff topological vector space (Theorem 1). Then we prove that the unit ball in the Minkowski-Rådström-Hörmander lattice over any normed space , dim has exactly two extreme points (Theorem 2).

     

Local exact controllability of the Navier–Stokes system

In this paper we deal with the local exact controllability of the Navier–Stokes system with distributed controls supported in small sets. In a first step, we present a new Carleman inequality for the linearized Navier–Stokes system, which leads to null controllability at any time T>0. Then, we deduce a local result concerning the exact controllability to the trajectories of the Navier–Stokes system.     

Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics

We find necessary density conditions for Marcinkiewicz–Zygmund inequalities and interpolation for spaces of spherical harmonics in with respect to the L^p norm. Moreover, we prove that there are no complete interpolation families for p≠2.     

Lattice Points Inside Rational Simplices and the Casson Invariant of Brieskorn Spheres

We express the number of lattice points inside certain simplices with vertices in Q³ or Q⁴ in terms of Dedekind–Rademacher sums. This leads to an elementary proof of a formula relating the Euler characteristic of the Seiberg–Witten-Floer homology of a Brieskorn Z-homology sphere to the Casson invariant.     

An h–p version of the Chebyshev spectral collocation method for nonlinear delay differential equations

We develop and analyze a spectral collocation method based on the Chebyshev–Gauss–Lobatto points for nonlinear delay differential equations with vanishing delays. We derive an a priori error estimate in the H¹‐norm that is completely explicit with respect to the local time steps and the local polynomial degrees. Several numerical examples are provided to illustrate the theoretical results.     

Examples of fitting structured phase–type distributions

A sub–class of phase–type distributions is defined in terms of a Markov process with sequential transitions between transient states and transitions from these states to absorption. Such distributions form a very rich class; they can be fitted to data, and any structure revealed by the parameter estimates used to develop more parsimonious re–parametrizations. Several example data sets are used as illustrations.     

New exact solutions for seventh‐order Sawada–Kotera–Ito,Lax and Kaup–Kupershmidt equations using Exp‐function method

In this work, Exp‐function method is used to solve three different seventh‐order nonlinear partial differential KdV equations. Sawada–Kotera–Ito, Lax and Kaup–Kupershmidt equations are well known and considered for solve. Exp‐function method can be used as an alternative to obtain analytic and approximate solutions of different types of differential equations applied in engineering mathematics. Ultimately this method is implemented to solve these equations and convenient and effective solutions are obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

Jessen–Wintner type random variables and fractal properties of their distributions

Formulas are given for the Lebesgue measure and the Hausdorff–Besicovitch dimension of the minimal closed set S_ξ supporting the distribution of the random variable ξ = 2^–k τ_k, where τ_k are independent random variables taking the values 0, 1, 2 with probabilities p _0k , p _1k , p _2k , respectively. A classification of the distributions of the r.v. ξ via the metric‐topological properties of S_ξ is given. Necessary and sufficient conditions for superfractality and anomalous fractality of S_ξ are found. It is also proven that for any real number a ₀ ∈ [0, 1] there exists a distribution of the r.v. ξ such that the Hausdorff–Besicovitch dimension of S_ξ is equal to a ₀. The results are applied to the study of the metric‐topological properties of the convolutions of random variables with independent binary digits, i.e., random variables ξⁱ = , where η_k are independent random variables taking the values 0 and 1. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp Integral Inequalities of the Hermite–Hadamard Type

We consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite and Hadamard, Iyengar, Milovanovi and Pecari , and others are obtained as special cases. Our results can also be interpreted as analogues to a theorem of Ostrowski on the deviation of a function from its averages. Furthermore, we establish a generalization of a result of Fink concerning L^p estimates for the remainder of the trapezoidal rule and present the best constants in the error bounds.     

A random evolution related to a Fisher–Wright–Moran model with mutation,recombination and drift

The paper deals with a model of the genetic process of recombination, one of the basis mechanisms of generating genetic variability. Mathematically, the model can be represented by the so‐called random evolution of Griego and Hersch, in which a random switching process selects from among several possible modes of operation of a dynamical system. The model, introduced by Polanska and Kimmel, involves mutations in the form of a time‐continuous Markov chain and genetic drift. We demonstrate asymptotic properties of the model under different demographic scenarios for the population in which the process evolves. Copyright © 2003 John Wiley & Sons, Ltd.     

Singular integral operator,Hardy–Morrey space estimates for multilinear operators and Navier–Stokes equations

After establishing the molecule characterization of the Hardy–Morrey space, we prove the boundedness of the singular integral operator and the Riesz potential. We also obtain the Hardy–Morrey space estimates for multilinear operators satisfying certain vanishing moments. As an application, we study the existence and the uniqueness of the solutions to the Navier–Stokes equations for the initial data in the Hardy–Morrey space ????(p?n) for q as small as possible. Here, the Hardy–Morrey space estimates for multilinear operators are important tools. Copyright © 2010 John Wiley & Sons, Ltd.     

Caffarelli–Kohn–Nirenberg inequalities on Lie groups of polynomial growth

In the setting of a Lie group of polynomial volume growth, we derive inequalities of Caffarelli–Kohn–Nirenberg type, where the weights involved are powers of the Carnot–Caratheodory distance associated with a fixed system of vector fields which satisfy the Hörmander condition. The use of weak spaces is crucial in our proofs and we formulate these inequalities within the framework of Lorentz spaces (a scale of (quasi)‐Banach spaces which extend the more classical Lebesgue spaces) thereby obtaining a refinement of, for instance, Sobolev and Hardy–Sobolev inequalities.