Showing papers in "Bulletin of The Belgian Mathematical Society-simon Stevin in 2007"
TL;DR: In this article, the Fourier Laplace transform was used to characterize the weight functions for which there exists a weight sequence such that the classes of ultra-differentiable functions defined by imposing conditions on the derivatives of the function in terms of this sequence coincides with a class of ultradifferentiable functions.
Abstract: We characterize the weight sequences $(M_p)_p$ such that the class of ultra-differentiable functions ${\mathcal E}_{(M_p)}$ defined by imposing conditions on the derivatives of the function in terms of this sequence coincides with a class of ultradifferentiable functions ${\mathcal E}_{(\omega)}$ defined by imposing conditions on the Fourier Laplace transform. As a corollary, we characterize the weight functions $\omega$ for which there exists a weight sequence $(M_p)_p$ such that the classes ${\mathcal E}_{(\omega)}$ and ${\mathcal E}_{(M_p)}$ coincide. These characterizations also hold in the Roumieu case. Our main results are illustrated by several examples.
129 citations
TL;DR: In this paper, the authors consider locally symmetric almost Kenmotsu manifold and show that the manifold is locally isometric to the Riemannian product of an n+1-dimensional manifold of constant curvature.
Abstract: We consider locally symmetric almost Kenmotsu manifolds showing that such a manifold is a Kenmotsu manifold if and only if the Lie derivative of the structure, with respect to the Reeb vector field $\xi$, vanishes. Furthermore, assuming that for a $(2n+1)$-dimensional locally symmetric almost Kenmotsu manifold such Lie derivative does not vanish and the curvature satisfies $R_{XY}\xi =0$ for any $X, Y$ orthogonal to $\xi$, we prove that the manifold is locally isometric to the Riemannian product of an $(n+1)$-dimensional manifold of constant curvature $-4$ and a flat $n$-dimensional manifold. We give an example of such a manifold.
122 citations
TL;DR: In this paper, it was shown that the set of complex-valued everywhere-surjective functions on C is a Jgebrable set, which contains an infinitely generated algebra every non-zero element of which is everywhere surjective.
Abstract: We show that the set L of complex-valued everywhere surjective functions on C is aJgebrable. Specifically, L contains an infinitely generated algebra every non-zero element of which is everywhere surjective. We also give a technique to construct, for every n is an element of N, n algebraically independent everywhere surjective functions, f(1), f(2),..., f(n), so that for every non-constant polynomial P is an element of C[z(1), z(2),...,z(n)], P(f(1), f(2),...f(n)) is also everywhere surjective.
85 citations
TL;DR: In recent years, there has been an enormous eort put in the denition and analysis of fractional or fractal operators as discussed by the authors, which are traditionally used in optics, mechanical engineering and signal processing.
Abstract: In recent years, there has been an enormous eort put in the denition and analysis of fractional or fractal operators. Fractional calculus is for example a ourishing eld of active research. In this paper we restrict ourselves to the fractional Fourier operator and friends that are traditionally used in optics, mechanical engineering and signal processing. The book by H.M. Ozaktas, Z. Zalevsky, and M.A. Kutay, The fractional Fourier transform, John Wiley, 2001 gives a state of the art of 2001. Because this eld is still in full expansion, we want to summarize in this survey paper some of the recent developments that appeared in the literature since then, revealing some unexplored
67 citations
TL;DR: In this paper, the boundedness and compactness of integral operators on the Hardy space of the unit ball are discussed. But the authors focus on the integral operators in the unit sphere.
Abstract: Let $g:B\to \mathbb C^1$ be a holomorphic map of the unit ball $B$. We study the integral operators $$ T_gf(z)=\int_0^1f(tz)\Re g(tz)\frac{dt}{t}; \ \ L_gf(z)= \int_0^1 \Re f(tz) g(tz)\frac{dt}{t},\qquad z\in B. $$ The boundedness and compactness of the operators $T_g$ and $L_g$ on the Hardy space $H^2$ in the unit ball are discussed in this paper.
65 citations
TL;DR: The Poisson cluster process structure of a Hawkes process is used to derive non asymptotic estimates of the tail of the extinction time, of the coupling time or of the number of points per interval to derive exponential inequalities for Hawkes processes which can precise the ergodic theorem.
Abstract: We use the Poisson cluster process structure of a Hawkes process to derive non asymptotic estimates of the tail of the extinction time, of the coupling time or of the number of points per interval. This allows us to define a family of independent Hawkes processes ; each of them approximating the initial process on a particular interval. Then we can easily derive exponential inequalities for Hawkes processes which can precise the ergodic theorem.
38 citations
37 citations
TL;DR: In this article, it was shown that for a fixed positive integer n ≥ 2, if a mapping Q : X → Y satisfies the following functional equation, then X and Y are linear spaces.
Abstract: Let X and Y be linear spaces. It is shown that for a fixed positive integer n ≥ 2, if a mapping Q : X → Y satisfies the following functional equation
34 citations
TL;DR: In this article, the authors introduce a new type of inverse w.r.t. monotone bijections that is determined by the direction of the base vectors of the real Euclidean plane.
Abstract: A straightforward generalization of the classical inverse of a real function based on reflections leads to several insuperable difficulties. We introduce a new type of inverse w.r.t. monotone bijections $\phi$ that is determined by the direction of the base vectors of the real Euclidean plane. Inverting a monotone function in the real plane does not necessarily result in a function. Given an increasing real function $f$, Schweizer and Sklar geometrically construct a set of inverse functions. We will largely extend their construction to our new concept of $\phi$-inverses, also incorporating decreasing functions $f$. Furthermore, the geometrical and algebraical aspects of our approach are elaborated comprehensively. Special attention goes to the symmetry of a monotone function $f$ w.r.t. some monotone bijection $\phi$.
30 citations
TL;DR: In this article, it was shown that every real Banach space can be equivalently renormed so that the set of non-norm-attaining functionals is non-dense.
Abstract: We study the set of non–norm–attaining functionals on a Banach space,giving a sufficient condition for thedensity of thisset. We also finda large classof Banach spaces for which the set of norm–attaining functionals is (dense–)lineable. In addition, among other results, we provide a new proof of the factthat every real Banach space can be equivalently renormed so that the set ofnon–norm–attaining functionals is non–dense. 1 Introduction and background Our primary focus in this article will be on the structure of the set of functionalson a real Banach space that do not attain their norm. One reason for this interestarises from geometrical considerations and the connection with the Banach–Mazurconjecture (see [11].) Namely, it was shown in [3] that every transitive and separableBanach space in which the set of non–norm–attaining functionals is not dense is ro-tund. Another motivation comes from an open problem concerning the lineability ofthe set NA(X) of norm–attaining functionals on a Banach space X. Specifically, it isunknown if NA(X) always contains an infinite dimensional, or even a 2−dimensional
28 citations
TL;DR: For 3-dimensional Lagrangian submanifolds of CP 3 (4), the improved Chen inequality was proved in this article, which coincides with a previous version proved in [5].
Abstract: In [7] Oprea gave an improved version of Chen’s inequality for Lagrangian submanifolds of CP n (4). For minimal submanifolds this inequality coincides with a previous version proved in [5]. We consider here those non minimal 3-dimensional Lagrangian submanifolds in CP 3 (4) attaining at all points equality in the improved Chen inequality. We show how all such submanifolds may be obtained starting from a minimal Lagrangian surface in CP 2 (4).
TL;DR: In this article, a classification of parallel surfaces in the groups of rigid motions of Minkowski plane and Euclidean plane equipped with a general left-invariant metric was given.
Abstract: We give a classification of parallel surfaces in the groups of rigid motions of Minkowski plane and Euclidean plane, equipped with a general left-invariant metric Our result completes the classification of parallel surfaces in the eight three-dimensional model geometries of Thurston and in three-dimensional unimodular Lie groups with maximal isometry group
TL;DR: In this paper, it was shown that the C^k-maps considered by Schikhof and De Smedt coincide with those of Bertram, Glockner and Neeb.
Abstract: Various definitions of C^k-maps on open subsets of finite-dimensional vector spaces over a complete valued field have been proposed in the literature. We show that the C^k-maps considered by Schikhof and De Smedt coincide with those of Bertram, Glockner and Neeb. By contrast, Ludkovsky's C^k-maps need not be C^k in the former sense, at least in positive characteristic. We also compare various types of Holder differentiable maps on finite-dimensional and metrizable spaces.
TL;DR: In this paper, it was shown that if a multiplicative (not necessarily linear)-mapping is used, then every $*$-$(varphi, \psi)$-derivation is automatically continuous.
Abstract: Suppose that $\frak A$ is a $C^*$-algebra acting on a Hilbert space $\frak K$, and $\varphi, \psi$ are mappings from $\frak A$ into $B(\frak K)$ which are not assumed to be necessarily linear or continuous A $(\varphi, \psi)$-derivation is a linear mapping $d: \frak A \to B(\frak K)$ such that $$d(ab)=\varphi(a)d(b)+d(a)\psi(b)\quad (a,b\in \frak A)$$ We prove that if $\varphi$ is a multiplicative (not necessarily linear)\ $*$-mapping, then every $*$-$(\varphi,\varphi)$-derivation is automatically continuous Using this fact, we show that every $*$-$(\varphi,\psi)$-derivation $d$ from $\frak A$ into $B(\frak K)$ is continuous if and only if the $*$-mappings $\varphi$ and $\psi$ are left and right $d$-continuous, respectively
TL;DR: In this article, it was shown that the index is unchanged if we pass to a certain b-metric gb( ) and analyzed the limiting behavior as & 0 of the two terms in the formula.
Abstract: Let X be a compact manifold with boundary @X, and suppose that @X is the total space of a fibration Z ! @X ! Y . Let D be a generalized Dirac operator associated to a -metric g on X. Under the assumption that D is fully elliptic we prove an index formula for D . The proof is in two steps: first, using results of Melrose and Rochon, we show that the index is unchanged if we pass to a certain b-metric gb( ). Next we write the b (i.e. the APS) index formula for gb( ); the -index formula follows by analyzing the limiting behaviour as & 0 of the two terms in the formula. The interior term is studied directly whereas the adiabatic limit formula for the eta invariant follows from work of Bismut and Cheeger.
TL;DR: In this paper, it was shown that every separable, topologically mixing analytic uniformly continuous semigroup of operators admits a semigroup on the Banach space, and that the semigroup is strongly analytic.
Abstract: We prove in this note that every separable infinite dimensional complexFr´echet space different from ω, the countably infinite product of lines, admitsa topologically mixing analytic uniformly continuous semigroup of operators.The study of the existence of transitive semigroups on ω, and on its predualϕ is also considered. 1 Notation and preliminaries Let X be a separable infinite-dimensional locally convex space (l.c.s) and let L(X)be the set of linear and continuous operators from X to X. Let ∆ be either N 0 ora concrete sector in the complex plane. For α ∈ [0,π/2]∪ {π} we define the sector∆(α) := {re iθ : r ≥ 0, θ ∈ [−α,α]}.A one-parameter family {T(t)} t∈∆ of bounded linear operators in L(X) is asemigroup if T(0)x = x, and T(t)T(s) = T(t + s) for all t,s ∈ ∆. For the nondiscrete case, we also add the condition lim t→s T(t) = T(s) pointwisely on X for alls ∈ ∆(α). In this case we say that it is a strongly continuous semigroup, or simplya semigroup. If lim t→s T(t) = T(s) holds uniformly on the bounded sets of X wesay that the semigroup is uniformly continuous. If L(X) is endowed with the strongoperator topology, α 6= 0, and the mapping t → T(t) is analytic in the interior of∆(α), then we say that the semigroup is strongly analytic, or simply write analytic.For a full treatment of semigroups defined on Banach spaces we refer the reader to
TL;DR: In this paper, the intermediate value theorem for power series on a Levi-Civita field is presented, and a large class of functions that are given locally by power series and contain all the continuations of real power series is presented.
Abstract: The proof of the intermediate value theorem for power series on a Levi-Civita field will be presented. After reviewing convergence criteria for power series [19], we review their analytical properties [18]. Then we state and prove the intermediate value theorem for a large class of functions that are given locally by power series and contain all the continuations of real power series: using iteration, we construct a sequence that converges strongly to a point at which the intermediate value will be assumed.
TL;DR: In this paper, the authors introduce a new class of functions called the $\mathcal{N}_p$-spaces and study the boundedness and compactness of composition operators on these functions.
Abstract: We introduce a new class of functions, called the $\mathcal{N}_p$-spaces and study the boundedness and compactness of composition operators on $\mathcal{N}_p$-spaces as well as between $\mathcal{N}_p$-spaces and Bergman-type spaces. The paper is intended to give a self-contained introduction the the $\mathcal{N}_p$-spaces.
TL;DR: Euler's Vollstandige Anleitung zur Algebra as mentioned in this paper was published in two volumes by the Academy of Sciences in St- Peterburg in 1770 and was translated into Russian (1768-9), Dutch (1773), French (1774), Latin (1790), English (1797, 1822), and Greek (1800).
Abstract: Christoff Rudolff's Coss as a source Leonard Euler's Vollstandige Anleitung zur Algebra was published in two volumes by the Academy of Sciences in St- Peterburg in 1770 (2). With the exception of Euclid's Elements it is the most printed book on mathematics ((11), xxxiii). It was translated into Russian (1768-9), Dutch (1773), French (1774), Latin (1790), English (1797, 1822) and Greek (1800). One popular German edition from Reclam Verlag sold no less than 108,000 copies between 1883 and 1943 (5). Euler wrote his Algebra originally in German. Based on internal evidence, Fellmann dates the manuscript at 1765/1766 ((3), 108), when he returned from Berlin to St-Petersburg, some years before he went completely blind. In his selection of problems in the Algebra, Euler shows himself familiar with the typical recreational and practical problems of Renaissance and sixteenth-century algebra books. An extensive historical database with algebraic problems (4) imme- diately reveals Euler's use of the Stifel's edition of Rudolff's Coss for his repository of problems. This work, published 1525 in Strassburg (6), was the first German book entirely devoted to algebra. Stifel used many problems from Rudolff in his Arithmetica Integra of 1544 and found the work too important not to publish his own annotated edition (9). The first volume of Euler's Algebra on determinate equations contains 59 num- bered problems. Two thirds of these can be directly matched with the problems from Rudolff. Some are literal reproductions (see table), others were given new values or were slightly reformulated. The second part on indeterminate equations also has 59 problems and although the correlation here is manifestly lower, many problems still originate from Rudolff.
TL;DR: In this article, the curvature properties of generalizations of contact metric manifolds are studied, with special attention to the left (kappa,\mu\right) nullity conditions in the framework of S$-manifolds.
Abstract: Curvature properties of some generalizations of contact metric manifolds are studied, with special attention to $\left(\kappa,\mu\right)$-nullity conditions in the framework of $\cal S$-manifolds.
TL;DR: In this article, the authors discuss problems of extension of vector-valued functions defined on subsets of a domain, which have weak extensions belonging to a space of smooth functions, and look for conditions which ensure that there exists an extension in the corresponding space.
Abstract: We discuss problems of extension of vector-valued functions defined on subsets of a domain $\Omega\subset\mathbb{R}^N$ which have weak extensions belonging to a space $\mathscr{H}(\Omega)$ of smooth functions We look for conditions which ensure that there exists an extension in the corresponding space $\mathscr{H}(\Omega,E)$ of vector-valued functions
TL;DR: In this paper, the inverse estimation of approximation theory by trigonometric polynomials in the weighted Lebesgue spaces was obtained and its application in weighted Smirnov spaces was considered.
Abstract: The improvement of the inverse estimation of approximation theory by trigonometric polynomials in the weighted Lebesgue spaces was obtained and its application in the weighted Smirnov spaces was considered.
TL;DR: In this paper, a numerical solution of one-dimensional Korteweg-de Vries equation with variant boundary conditions by the Fourier pseudospectral method was presented.
Abstract: In this paper, we present a numerical solution of one-dimensional Korteweg-de Vries equation with variant boundary conditions by the Fourier pseudospectral method. Four test problem with known exact solutions were studied to demonstrate the accuracy of the present method. An artificial viscosity was proposed to improve the accuracy of the numerical scheme. The obtained results were compared with the exact solution of each problem and found to be in good agreement with each other.
TL;DR: In this paper, it was shown that a uniform space X is trans-separable if and only if every pointwise bounded uniformly equicontinuous subset of the space of continuous real-valued functions Cc (X) equipped with the compact-open topology is metrizable.
Abstract: The paper shows that a uniform space X is trans-separable if and only if every pointwise bounded uniformly equicontinuous subset of the space of continuous real-valued functions Cc (X) equipped with the compact-open topology is metrizable. This extends earlier results of Pfister and Robertson and also applies to show that if Cc (X) is angelic then X is trans-separable. The precise relation among DCCC spaces and trans-separable spaces has been also determined.
TL;DR: In this article, the existence of forced periodic solutions to symmetric ODEs with and without damping with periodic and symmetric forcings was shown, in both resonance and nonresonance cases.
Abstract: We show the existence of forced periodic solutions to certain symmetric ordinary differential equations. First and second order systems of ordinary differential equations are investigated with and without damping with periodic and symmetric forcings. We study both resonance and nonresonance cases.
TL;DR: In this article, the authors considered a more general situation where they considered a unique Markov additive process ((Jn, Zn)) in place of a compound Markov renewal process (e SNt).
Abstract: In [17] the author considered a compound Markov renewal process ( e SNt) where ((Jn, Sn)) and (( e Jn, e Sn)) are suitable independent Markov additive processes such that (Sn − Sn−1) are positive random variables, and Nt = P n≥1 1Sn≤t In this paper we present the analogous results for a more general situation where we consider a unique Markov additive process ((Jn, Zn)) in place of ((Jn, Sn)) and (( e Jn, e Sn)), and Zn = ( e
TL;DR: In this article, the authors identify those closed subspaces of a non-Archimedean, complete valued field with a normal complement, including projections, adjoint and self-adjoint operators.
Abstract: Let $\mathbb{K}$ be a non-Archimedean, complete valued field. It is known that the supremum norm $\left\Vert \cdot\right\Vert _{\infty}$ on $c_{0}$ is induced by an inner product if and only if the residual class field of $\mathbb{K}$ is formally real. One of the main problems of this inner product is that $c_{0}$ is not orthomodular, as is any classical Hilbert space. Our goal in this work is to identify those closed subspaces of $c_{0}$ which have a normal complement. In this study we also involve projections, adjoint and self-adjoint operators.
TL;DR: In this paper, a new third order finite difference scheme for the solution of initial value problems for hyperbolic conservation laws is presented, which can be used for large time steps which save more time.
Abstract: A new third order finite difference scheme for the solution of initial value problems for hyperbolic conservation laws is presented. The advantages of the scheme are its simplicity, third order accuracy and that it can be used for large time steps which saves more time. The scheme is proved stable for initial and initial boundary value problems for linear case. The technique of making the third order scheme oscillations free (TVD) is carried out. In this paper we extend TVD scheme to two dimension problems. The extension of the TVD scheme to nonlinear system of equations is illustrated by solving shallow water equations. Numerical results are presented and compared with exact solutions and other methods.
TL;DR: In this paper, a wider range of payoffs is considered and the authors partially extend results by Bruss and Ferguson to include a wide range of payoff choices, such as the first-type payoff gives a positive rate of return if the investment is made on the best or the second best option.
Abstract: A decision-maker observes sequentially a given permutation of $n$ uniquely rankable options He has to invest capital into these opportunities at the moment when they appear At each step only relative ranks are known At the end the true rank of the option, at which the investment has been made, is known [Bruss and Ferguson] have considered such problems under the assumption that an investment on the very best opportunity yields a lucrative, possibly time-dependent, rate of return Uninvested capital keeps its risk-free value Wrong investments lose their value In this paper we partially extend results by [Bruss and Ferguson] We confine our study to linear utility but a wider range of payoffs is taken into account Two cases are considered The first-type payoff gives a positive rate of return if the investment is made on the best or the second best option The second-type payoff pays when the investment is at the second best option We motivate these payoff choices A few examples are explicitly solved