Necessary and sufficient conditions for a generalized solution to the initial-boundary value problem for the wave equation to belong to W p 1 with p ≥ 1

We establish necessary and sufficient conditions on the boundary function under which a generalized solution to the initial-boundary value problem for the wave equation with boundary conditions of the first kind belongs to W _p ¹ .     

How to estimate the sum of a series?

<正>Now we’ll learn how to estimate the sum of a series?We have been able to use the integral test to show that a series ∞∑(n=1)a_n is convergent or divergent.But we also don’t know the best way to find the sum of the series.It is difficult to compute the sum of any series.Here is the best way to find an approximation to the sum of the series.You know     

Symbolic dynamics: from the N-centre to the (N + 1)-body problem,a preliminary study

We consider a rotating N-centre problem, with N ≥ 3 and homogeneous potentials of degree ?α < 0, α ∈ [1,2). We prove the existence of infinitely many collision-free periodic solutions with negative and small Jacobi constant and small values of the angular velocity, for any initial configuration of the centres. We will introduce a Maupertuis’ type variational principle in order to apply the broken geodesics technique developed in Soave and Terracini (Discrete Contin Dyn Syst 32:3245–3301, 2012). Major difficulties arise from the fact that, contrary to the classical Jacobi length, the related functional does not come from a Riemaniann structure but from a Finslerian one. Our existence result allows us to characterize the associated dynamical system with a symbolic dynamics, where the symbols are given partitions of the centres in two non-empty sets.     

The Stablility of a Loop Formed the Separatrix of a Saddle Point and the Condition to Produce a Limit Cycle

In this paper,we discuss a loop by the separatrix of a saddle point of two dimensional autonmoussystem.It contains two sections.In section 1,a short proof of the criterion theorem of stability ofthe loop of saparatrix is given wi~h ~he help o~th~ ~h~orjr 0~ i'otatin~ vector field, and some in~ormatioafor the study of stability of that loop in critical case is supplied.In section 2,we give a sufficientcondition under which a limit cycle is produced from the loop of separatrix.     

Does the Balanced Scorecard make a difference to the strategy development process?

A great number of strategy tools are being taught in strategic management modules. These tools are available to managers for use in facilitating strategic decision making and enhancing the strategy development process in their organisations. A number of studies have been published examining which are the most popular tools; however there is little empirical evidence on how their utilisation influences the strategy process. This paper is based on a large scale international survey on the strategy development process, and seeks to examine the impact of a particular strategy tool, the Balanced Scorecard (BSC), upon the strategy process. Recently, it has been suggested that as a strategy tool, the BSC can influence all elements of the strategy process. The results of this study indicate that although there are significant differences in some elements of the strategy process between the organisations that have implemented the BSC and those that have not, the impact is not comprehensive.     

Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given. To cite this article: V. Maz'ya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Correction to the paper “on the curvature of a generalization of a contact metric manifolds”

We considered in Example 3.1 of the paper [1] an S-structure on R^2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.     

On the Suitable Weak Solutions to the Boussinesq Equations in a Bounded Domain

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Large Deviation of a Capacity Related to the Ornstein-Uhlenbeck Operator

We discuss the large deviation estimates of a capacity related to the Ornstein-Uhlenbeck operator and present a refinement of Strassen's form of the law of the iterated logarithm for Brownian metion from the viewpoint of the Dirichlet space theory.     

On the global attraction to solitary waves for the Klein–Gordon equation coupled to a nonlinear oscillator

The long-time asymptotics are analyzed for all finite energy solutions to a model $\mathbf{U}(1)$-invariant nonlinear Klein–Gordon equation in one dimension, with the nonlinearity concentrated at a point. Our main result is that each finite energy solution converges as $t\to \pm \infty$ to the set of ‘nonlinear eigenfunctions’ $\psi (x){\mathrm{e}}^{?\mathrm{i}\omega t}$. To cite this article: A.I. Komech, A.A. Komech, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Ways to square the parabola – a commented picture gallery

The mathematical literature knows several quite different ways to determine the area of a parabolic segment. In this article they are brought together, compared and varied. Finally two new proofs are added to the collection. Each proof is displayed by an expressive figure. A colorful compilation of those figures accompanies the article as an appendix.     

Invariant Measure for the Markov Process Corresponding to a PDE System

In this paper, we consider the Markov process （X^∈（t）, Z^∈（t）） corresponding to a weakly coupled elliptic PDE system with a small parameter ∈ 〉 0. We first prove that （X^∈（t）, Z^∈（t）） has the Feller continuity by the coupling method, and then prove that （X^∈（t）, Z^∈（t）） has an invariant measure μ^∈（·） by the Foster-Lyapunov inequality. Finally, we establish a large deviations principle for μ^∈（·） as the small parameter e tends to zero.     

Functoriality of the Bimodule Associated to a Hilsum–Skandalis Map

We extend the functoriality of the Connes convolution algebra to the category of Hilsum–Skandalis maps between separated smooth étale groupoids. Thereby we establish that Morita equivalent separated smooth étale groupoids have Morita equivalent convolution algebras, analogously to the results of Muhly, Renault and Williams, and others, on the C^*algebras of Morita equivalent groupoids.     

Abelianness of the “missing part” from a sheaf category to a module category

This paper investigates the structure of the"missing part"from the category of coherent sheaves over a weighted projective line of weight type(2,2,n)to the category of finitely generated right modules on the associated canonical algebra.By constructing a t-structure in the stable category of the vector bundle category,we show that the"missing part"is equivalent to the heart of the t-structure,hence it is abelian.Moreover,it is equivalent to the category of finitely generated modules on the path algebra of type An-1.     

Grasping the phantom – a new approach to nonlinear spectral theory

A new definition of a spectrum for nonlinear operators is suggested, which is called phantom. The phantom naturally divides into two subsets which in the linear case correspond to the point spectrum, and to the union of the continuous and residual spectrum. There is also a natural nonlinear analogue to the approximate point spectrum. The phantom may be considered as a variant of the spectrum of M. Furi, M. Martelli, and A. Vignoli (Ann. Mat. Pura Appl. 118 (1978), 229–294), which reflects the behavior of the operator on bounded sets and which is based on a new definition of an eigenvalue that was suggested earlier by the authors (Nonlinear Anal. 40 (2000), 565–576).?To define the phantom, the class of stably 0-epi maps is introduced and studied. This is a subclass of 0-epi maps satisfying a Rouché type theorem. Received: November 15, 1999?Published online: October 2, 2001     

On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation

We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) to use the positive Lax–Oleinik semi-group to propagate singularities.