Mathematical Analysis and Applications

Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems established by Rodin in 1990. We prove, that if the growth of a function Φ(t) : [0, ∞) → [0, ∞) is bigger than the exponent, then the strong Φ-summability of a Walsh-Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the author of this paper.

We give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the one-dimensional continuum Anderson model via the fractional moments method. This follows via exponential decay of fractional moments of the Green function, which is shown to hold at arbitrary energy and for any single-site distribution with bounded, compactly supported density.

This paper is concerned with the existence, uniqueness and/or multiplicity, and stability of positive solutions of an indefinite weight elliptic problem with concave or convex nonlinearity. We use mainly bifurcation methods to obtain our results.

We apply the phenomenology of homogeneous, isotropic turbulence to the family of approximate deconvolution models proposed by Stolz and Adams. In particular, we establish that the models themselves have an energy cascade with two asymptotically di¤erent inertial ranges. Delineation of these gives insight into the resolution requirements of using approximate deconvolution models. The approximate deconvolution model's energy balance contains both an enhanced energy dissipation and a modi…cation to the model's kinetic energy. The modi…cation of the model's kinetic energy induces a secondary energy cascade which accelerates scale truncation. The enhanced energy dissipation completes the scale truncation by reducing the model's micro-scale from the Kolmogorov micro-scale.

Investment systems are studied using a framework that emphasize their profiles (the cumulative probability distribution on all the possible percentage gains of trades) and their log return functions (the expected average return per trade in logarithmic scale as a function of the investment size in terms of the percentage of the available capital). The efficiency index for an investment system, defined as the maximum of the log return function, is proposed as a measure to compare investment systems for their intrinsic merit. This efficiency index can be viewed as a generalization of Shannon’s information rate for a communication channel. Applications are illustrated. © 2006 Elsevier Inc. All rights reserved.

In this paper, we develop the sensitivity analysis for generalized set-valued variational inclusions and generalized resolvent equations. We establish the equivalence between the parametric generalized set-valued variational inclusions and parametric generalized resolvent equations, by using the resolvent operator technique without assuming the differentiability of the given data.

The relationship between the reduction of order through point symmetries and integration is explored with particular emphasis on the loss and gain of point Ž. contact for third order symmetries to and from nonlocal symmetries. It is seen that reduction of order can even lead to the loss of all point symmetries at the third order level and their replacement at the second order level from nonlocal symmetries. It is evident that nonlocal symmetries should be given more attention in applications.

The carriage of soil from one plane region to another, under some physical and economical constraints, generates a functional transportation problem. We solve the problem using a discretization scheme. A convergence theorem is proved and we describe a practical application.

In this paper, we propose and study two iteration schemes (modified Halpern's type and HS-iteration schemes). Furthermore, it is proved that if two finite families of quasi-nonexpansive mappings satisfy jointly demiclosedness principle, then under appropriate conditions on the iteration parameters, the schemes so introduced strongly converge to the common fixed points of the mappings. Our main results improve and generalize the results in the literature and many other existing results currently in literature.

Les traumatismes fermés récents des IPP des doigts Acute closed injuries of the digital proximal interphalangeal joints ABSTRACT Acute closed injuries of the digital proximal interphalangeal (PIP) joints are frequent and can leave sequelae because of imprecise diagnosis of the lesions, poorly adapted treatment, or insufficient follow-up. The therapeutic options proposed in this presentation are advocated by all of the participants and are based on their personal experience and evidence reported in the literature. After a brief anatomic review necessary for the understanding of PIP joint pathophysiology we have presented the long-term outcomes of PIP joint injuries, which are central to the therapeutic decision-making process in terms of risk benefit ratio. To facilitate the presentation, we have separated lesions "with" and "without" fracture. A specific chapter is devoted to surgical approaches essential for successful management of these injuries and another to particular problems related to the seldom reviewed topic of traumatic injury of the PIP joint during bone growth.

The study analyzed the algebraic properties of the Euclidean algorithm in details. The analysis included a detailed step by step approach in understanding the algorithm, the extended form of the algorithm, computation of the Greatest Common Divisor (GCD) and its algebraic properties and their applications in algebra an d cryptography. We also showed how the Euclidean algorithm could be applied to trading for the maximization of returns. In our approach, we assumed that gcd[a(x); b(x)] is the monic polynomial of minimal degree within the set G = {s(x)a(x)+t(x)b(x): s(x), t(x) ∈ F[x]} and thus, examining all equations of the form p(x) = s(x)a(x)+t(x)b(x).

Introduced several new axiomatic systems, that are not less general than group theory, and discovered discontinuous analysis.

In this work I introduce and study in details the concepts of funcoids
which generalize proximity spaces and reloids which generalize uniform
spaces, and generalizations thereof. The concept of funcoid is generalized
concept of proximity, the concept of reloid is cleared from superfluous
details (generalized) concept of uniformity.

Also funcoids and reloids are generalizations of binary relations
whose domains and ranges are filters (instead of sets). Also funcoids
and reloids can be considered as a generalization of (oriented) graphs,
this provides us with a common generalization of calculus and discrete
mathematics.

I consider (generalized) limit of arbitrary (discontinuous) function, defined in terms of funcoids.
Definition of generalized limit makes it obvious to define such things as derivative of an arbitrary function, integral of an arbitrary function, etc. It is given a definition of non-dif\-fe\-re\-nti\-able solution of a (partial) differential equation. It's raised the question how do such solutions ``look like'' starting a possible big future research program.

The generalized solution of one simple example differential equation is also considered.

The generalized derivatives and integrals are linear operators. For example $\int_a^b f(x)dx - \int_a^b f(x)dx = 0$ is defined and true for \emph{every} function.

The concept of continuity is defined by an algebraic formula (instead
of old messy epsilon-delta notation) for arbitrary morphisms (including
funcoids and reloids) of a partially ordered category. In one formula
continuity, proximity continuity, and uniform continuity are generalized.

Also I define connectedness for funcoids and reloids.

Then I consider generalizations of funcoids: pointfree funcoids and
generalization of pointfree funcoids: staroids and multifuncoids.
Also I define several kinds of products of funcoids and other morphisms.

I define \emph{space} as an element of an ordered semigroup action, that is a semigroup action conforming to a partial order. Topological spaces, uniform spaces, proximity spaces, (directed) graphs, metric spaces, etc.\ all are spaces. It can be further generalized to ordered precategory actions (that I call interspaces). I build basic general topology (continuity, limit, openness, closedness, hausdorffness, compactness, etc.)\ in an arbitrary space. Now general topology is an algebraic theory.

Before going to topology, this book studies properties of co-brouwerian
lattices and filters.

We study positive shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even on normed vector spaces. We show that for linear operators there is only one chain recurrent set, and this set is a closed invariant subspace. We prove that every chain transitive linear dynamical system with positive shadowing property is frequently hypercyclic and, as a corollary, we obtain that every positive shadowing hypercyclic linear dynamical system is frequently hypercyclic.

In this paper, essentially developing the Stein method, we prove the Kol-Ž. mogorov᎐Stein inequality for any Orlicz norm with the same constants. ᮊ 1996 Academic Press, Inc. p Kolmogorov᎐Stein inequality and its variants are a problem of interest for

In this paper, using proximal-point mapping technique of P-η-accretive mapping and the property of the fixed-point set of set-valued contractive mappings, we study the behavior and sensitivity analysis of the solution set of a parametric generalized implicit quasi-variational-like inclusion involving P-η-accretive mapping in real uniformly smooth Banach space. Further, under suitable conditions, we discuss the Lipschitz continuity of the solution

Given a complex m n matrix A; we index its singular values as 1 (A) 2 (A) ::: and call the value E (A) = 1 (A) + 2 (A) + ::: the energy of A; thereby extending the concept of graph energy, introduced by Gutman. Let 2 m n; A be an m n nonnegative matrix with maximum entry , and kAk 1 n. Extending previous results of Koolen and Moulton for graphs, we prove that

A new generalization of the modified Bessel function of the second kind Kz(x) is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess. The motivation behind introducing this generalization is to have a function which gives a new pair of functions reciprocal in the Koshliakov kernel cos (πz) M2z(4 √ x) − sin (πz) J2z(4 √ x) and which subsumes the self-reciprocal pair involving Kz(x). Its application towards finding modular-type transformations of the form F (z, w, α) = F (z, iw, β), where αβ = 1, is given. As an example, we obtain a beautiful generalization of a famous formula of Ramanujan and Guinand equivalent to the functional equation of a non-holomorphic Eisenstein series on SL2(Z). This generalization can be considered as a higher level analogue of the general theta transformation formula. We then use it to evaluate an integral involving the Riemann Ξ-function and consisting of a sum of products of two confluent hypergeometric functions.

In the last decades, the finite element method (FEM) in fluid mechanics applications has gained substantial momentum. FE analysis was initially introduced to solid mechanics. However, the progress in fluid mechanics problems was slower due to the non-linearities of the equations and inherent difficulties of the classical FEM to deal with instabilities in the solution of these problems. The main goal of this review is to analyze FEM and provide the theoretical basis of the approach mainly focusing on parabolic type of problems applied in fluid mechanics. Initially, we analyze the basics of FEM for the Stokes problem and we provide theorems for uniqueness and error estimates of the solution. We further discuss FE approaches for the solution of the advection–diffusion equation such as the stabilized FEM, the variational multiscale method, and the discontinuous Galerkin method. Finally, we extend the analysis on the non-linear Navier–Stokes equations and introduce recent FEM advancements.

This paper is concerned with a numerical scheme to solve a singularly perturbed convection–diffusion problem. The solution of this problem exhibits the boundary layer on the right-hand side of the domain due to the presence of singular perturbation parameter ε. The scheme ...

A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of products of three Bessel functions.

Sufhcient conditions are established for the pointwise convergence of square partial sums of multiple Fourier series, in any number of dimensions, for functions with discontinuities. The result is used to extend the theory of lattice rules for multiple integration.

This paper is concerned with sequential Monte Carlo methods for optimizing a system under constraints. We wish to minimizef(x), where q"(x) Q 0 (i = l,..., m) must hold. We can calculate the q"(x), but f(x) can only be observed in the presence of noise. A general approach, based on an adaptation of a version of stochastic approximation to the penalty function method, is discussed and a convergence theorem is proved.

We consider a multidimensional Burgers equation on the torus T d and the whole space R d. We show that, in case of the torus, there exists a unique global solution in Lebesgue spaces. For a torus we also provide estimates on the large time behaviour of solutions. In the case of R d we establish the existence of a unique global solution if a Beale-Kato-Majda type condition is satisfied. To prove these results we use the probabilistic arguments which seem to be new.

The classical theory of the Weierstrass transform is extended to a generalized function space which is the dual of a testing function space consisting of purely entire functions with certain growth conditions developed by Kenneth B. Howell. An inversion formula and characterizations for this transform are obtained. A comparative study with the existing literature is also undertaken.

In this paper we consider conical square functions in the Bessel, Laguerre and Schr\"odinger settings where the functions take values in UMD Banach spaces. Following a recent paper of Hyt\"onen, van Neerven and Portal, in order to define our conical square functions, we use $\gamma$-radonifying operators. We obtain new equivalent norms in the Lebesgue-Bochner spaces $L^p((0,\infty ),\mathbb{B})$ and $L^p(\mathbb{R}^n,\mathbb{B})$, $1<p<\infty$, in terms of our square functions, provided that $\mathbb{B}$ is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions.

In this work we study the asymptotic distribution of eigenvalues in one-dimensional open sets. The method of proof is rather elementary, based on the Dirichlet lattice points problem, which enable us to consider sets with infinite measure. Also, we derive some estimates for the the spectral counting function of the Laplace operator on unbounded two-dimensional domains.

The main result of this paper is the generalisation and proof of a conjecture by Gould and Quaintance on the asymptotic behaviour of certain sequences related to the Bell numbers. Thereafter we show some applications of the main theorem to statistics of partitions of a finite set S, i.e., collections B 1 , B 2 , • • • , B k of non-empty disjoint subsets of S such that k i=1 B i = S, as well as to certain classes of partitions of [n].

We consider an infinite locally finite tree T equipped with nearest neighbor transition coefficients, giving rise to a space of harmonic functions. We show that, except for trivial cases, the generic harmonic function on T has dense range in C. By looking at forward-only transition coefficients, we show that the generic harmonic function induces a boundary martingale that approximates in probability all measurable functions on the boundary of T. We also study algebraic genericity, spaceability and frequent universality of these phenomena.

In this work, an attempt is made to understand the dynamics of a modified Leslie-Gower model with nonlinear harvesting and Holling type-IV functional response. We study the model system using qualitative analysis, bifurcation theory and singular optimal control. We show that the interior equilibrium point is locally asymptotically stable and the system under goes a Hopf bifurcation with respect to the ratio of intrinsic growth of the predator and prey population as bifurcation parameter. The existence of bionomic equilibria is analyzed and the singular optimal control strategy is characterized using Pontryagin's maximum principle. The existence of limit cycles appearing through local Hopf bifurcation and its stability is also examined and validated numerically by computing the first Lyapunov number. Optimal singular equilibrium points are obtained numerically for various discount rates.

We find conditions on data guaranteeing global nonexistence of solutions to an inverse source problem for a class of nonlinear parabolic equations. We also establish a stability result on a bounded domain for a problem with the opposite sign on the power type nonlinearity.

This paper deals with the problem of solving a broad class of Fredholm integral equations of the first kind with an arbitrary nondegenerate kernel that is twice differentiable in the variable of integration. A theorem has been proved on a new variational discretized-kernel double series solution for this class of integral equations which may possibly be discontinuous in the other variable. The accuracy of these solution double series, which may also serve as approximants to known functions, is investigated and an inaccuracy measure criterion has been identified for them. An illustration of the manner in which the present double series may converge faster than the corresponding Taylor series or Pade approximants is given. It is also demonstrated that they form a basis for a new real axis approximate method for the inversion of the Laplace transform.

A conjecture of Ulam states that the standard probability measure π on the Hilbert cube I ω is invariant under the induced metric d a when the sequence a = {a i } of positive numbers satisfies the condition ∞ i=1 a 2 i < ∞. This conjecture was proved in [3] when E 1 is a non-degenerate subset of M a. In this paper, we prove the conjecture of Ulam completely by classifying cylinders as non-degenerate and degenerate cylinders and by treating the degenerate case that was overlooked in the previous paper.

In this paper, we consider non-commutative Orlicz spaces as modular spaces and show that they are complete with respect to their modular. We prove some convergence theorems for τ-measurable operators and deal with uniform convexity of non-commutative Orlicz spaces.

Let X1,. .. , Xq be the basis of the space of horizontal vector fields on a homogeneous Carnot group G = (R n , •) (q < n). We consider the following divergence degenerate elliptic system N β=1 q i,j=1 Xi a ij αβ (x)Xju β = q i=1 Xif i α , α = 1, 2, ..., N where the coefficients a ij αβ are real valued bounded measurable functions defined in Ω ⊂ G, satisfying the strong Legendre condition and belonging to the space V M O loc (Ω) (defined by the Carnot-Carathéodory distance induced by the Xi's). We prove interior HW 1,p estimates (2 ≤ p < ∞) for weak solutions to the system.