Initial boundary value problem for generalized logarithmic improved Boussinesq equation

In this paper, we consider the initial boundary value problem for generalized logarithmic improved Boussinesq equation. By using the Galerkin method, logarithmic Sobolev inequality, logarithmic Gronwall inequality, and compactness theorem, we show the existence of global weak solution to the problem. By potential well theory, we show the norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions. Furthermore, for the generalized logarithmic improved Boussinesq equation with damped term, we obtain the decay estimate of the energy. Copyright © 2016 John Wiley & Sons, Ltd.     

A note on the sample average approximation method for stochastic mathematical programs with complementarity constraints

Meng and Xu (2006) [3] proposed a sample average approximation (SAA) method for solving a class of stochastic mathematical programs with complementarity constraints (SMPCCs). After showing that under some moderate conditions, a sequence of weak stationary points of SAA problems converge to a weak stationary point of the original SMPCC with probability approaching one at exponential rate as the sample size tends to infinity, the authors proposed an open question, that is, whether similar results can be obtained under some relatively weaker conditions. In this paper, we try to answer the open question. Based on the reformulation of stationary condition of MPCCs and new stability results on generalized equations, we present a similar convergence theory without any information of second order derivative and strict complementarity conditions. Moreover, we carry out convergence analysis of the regularized SAA method proposed by Meng and Xu (2006) [3] where the convergence results have not been considered.     

Conservative numerical schemes for the Ostrovsky equation

The Ostrovsky equation describes gravity waves under the influence of Coriolis force. It is known that solutions of this equation conserve the L² norm and an energy function that is determined non-locally. In this paper we propose four conservative numerical schemes for this equation: a finite difference scheme and a pseudospectral scheme that conserve the norm, and the same types of schemes that conserve the energy. A numerical comparison of these schemes is also provided, which indicates that the energy conservative schemes perform better than the norm conservative schemes.     

Divergent Solution to the Nonlinear Schr\"{o}dinger Equation with the Combined Power-Type Nonlinearities

In this paper, we consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities, which is mass-critical/supercr-itical, and energy-subcritical. Combing Du, Wu and Zhang'' argument with the variational method, we prove that if the energy of the initial data is negative (or under some more general condition), then the $H^1$-norm of the solution to the Cauchy problem will go to infinity in some finite time or infinite time.     

Trivariate near-best blending spline quasi-interpolation operators

A method to define trivariate spline quasi-interpolation operators (QIOs) is developed by blending univariate and bivariate operators whose linear functionals allow oversampling. In this paper, we construct new operators based on univariate B-splines and bivariate box splines, exact on appropriate spaces of polynomials and having small infinity norms. An upper bound of the infinity norm for a general blending trivariate spline QIO is derived from the Bernstein-Bézier coefficients of the fundamental functions associated with the operators involved in the construction. The minimization of the resulting upper bound is then proposed and the existence of a solution is proved. The quadratic and quartic cases are completely worked out and their exact solutions are explicitly calculated.     

Robust invariant stabilization of nonlinear continuous and discrete systems

Two classes of continuous systems described by differential equations in which coefficients are functions of the state vector are considered. The systems are subject to two scalar controls and a constantly acting scalar perturbation.An analytical synthesis of a control is performed under which the system is invariant in the sense that the scalar output of the system approaches zero as time tends to infinity and does not depend on the perturbation; moreover, the limit norm of the state vector is bounded above by the least upper bound for the norm of perturbation. The case where the coefficients of the system are subject to an uncontrolled additive perturbation is considered. In this case, the limit of the output norm is bounded above by a known function of the perturbation value. The method of synthesis is based on constructing the Lyapunov function as a positive definite quadratic form with Jacobian matrix.     

Saint-venant principle for paraboloidal elastic bodies

We study the asymptotic behavior of solutions to the linear problem of elasticity in a domain Ω with paraboloidal exit at infinity. Properties of solutions and the condition of the existence of solutions depend on a parameter γ∈[0,1] characterizing the velocity of extending the paraboloid (a cylinder and a cone correspond to the cases γ=0¤γ=1 respectively). Asymptotic formulas are deduced for displacement fields generating forces and moments “applied at infinity”. The Saint-Venant principle is verified for “oblong” bodies such as paraboloids, cylinders, and narrow cones. The following question turns out to be a key one: What rigid displacements belong to the energy space obtained by completion of $C_0^\infty (\bar \Omega )^3 $ by the energy norm? The dimension d_γ of the lineal R_γ of rigid energy displacements is computed (in this case, d₀=6, d₁=0, and the function γ?d_γ has jumps at the points γ=1/4, 1/2, 3/4). We also clarify the reasons why it is necessary to distinguish the notions “energy solution” and “solution with finite energy”. We also discuss the phenomenon of a boundary layer that appears near the endpoints of spindle-like rods and is described by energy solutions in paraboloids. As is shown, in order to have the well-posed formulation of the boundary conditions in one-dimensional models of such rods, it is necessary to use the weakened Saint-Venant principle, i.e., replace R₀ with R_γ: for γ>1/4. If we apply the strong principle, we arrive at an overdetermined limit one-dimensional problem. Bibliography: 71 titles.     

Error and extrapolation of a compact LOD method for parabolic differential equations

This paper is concerned with a compact locally one-dimensional (LOD) finite difference method for solving two-dimensional nonhomogeneous parabolic differential equations. An explicit error estimate for the finite difference solution is given in the discrete infinity norm. It is shown that the method has the accuracy of the second-order in time and the fourth-order in space with respect to the discrete infinity norm. A Richardson extrapolation algorithm is developed to make the final computed solution fourth-order accurate in both time and space when the time step equals the spatial mesh size. Numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm.     

Superconvergence results for linear second-kind Volterra integral equations

In this paper, Galerkin method is applied to approximate the solution of Volterra integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We prove that the approximate solutions of the Galerkin method converge to the exact solution with the order \({\mathcal {O}}(h^{r}),\) whereas the iterated Galerkin solutions converge with the order \({\mathcal {O}}(h^{2r})\) in infinity norm, where h is the norm of the partition and r is the smoothness of the kernel. We also consider the multi-Galerkin method and its iterated version, and we prove that the iterated multi-Galerkin solution converges with the order \({\mathcal {O}}(h^{3r})\) in infinity norm. Numerical examples are given to illustrate the theoretical results.     

Improper actions and higher connectivity at infinity

Given an improper action (= cell stabilizers are infinite) of a group G on a CW-complex , we present criteria, based on connectivity at infinity properties of the cell stabilizers under the action of G that imply connectivity at infinity properties for G. A refinement of this idea yields information on the topology at infinity of Artin groups, and it gives significant progress on the question of which Artin groups are duality groups. Received: October 30, 1998     

The blow-up of the conformal mean curvature flow

In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space R~n. This kind of flow is a special case of a general modified mean curvature flow which is of various origination. As the main result, we prove a blow-up theorem concluding that, under the conformal mean curvature flow in R~n, the maximum of the square norm of the second fundamental form of any compact submanifold tends to infinity in finite time. Furthermore, we also prove that the external conformal forced mean curvature flow of a compact submanifold in R~n with the same pinched condition as Andrews-Baker's will be convergent to a round point in finite time.     

The probability of generating certain profinite groups by two elements

In this paper we consider the question of finite generation of profinite groups. We study the class of profinite groups which are inverse limits of wreath products of alternating groups of degree ≥5. We prove that the probability of generating such inverse limits by two elements is strictly positive and tends to 1 as the degree of the first factor tends to infinity. Our method of analysis requires a survey of the maximal subgroups of iterated wreath products of alternating groups. Although we have been unable to classify these precisely we do obtain upper bounds for the number of conjugacy classes of maximal subgroups which we believe to be of independent interest. The author is grateful for financial support received under the FCO-award scheme.     

Spectral methods for weakly singular Volterra integral equations with smooth solutions

We propose and analyze a spectral Jacobi-collocation approximation for the linear Volterra integral equations (VIEs) of the second kind with weakly singular kernels. In this work, we consider the case when the underlying solutions of the VIEs are sufficiently smooth. In this case, we provide a rigorous error analysis for the proposed method, which shows that the numerical errors decay exponentially in the infinity norm and weighted Sobolev space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.     

Asymptotic estimates for the coefficients of bounded univalent functions

In the class of univalent bounded normalized holomorphic functions in the unit disk, we give an asymptotic estimate for the coefficients when the uniform norm of the modulus of the function tends to infinity.     

A numerical study of the null boundary controllability of a convection diffusion equation

In this paper we study numerically the cost of the null controllability of a linear control parabolic 1-D equation as the diffusion coefficient tends to 0. For this linear control parabolic 1-D equation, we know from a prior work by J.-M. Coron and S. Guerrero (2005), that, when the diffusion coefficient tends to 0, for a small controllability time, the norm of the optimal control tends to infinity and that, if the controllability time is large enough, this norm tends to 0. For controllability times which are not covered by this work, we estimate numerically the norm of the optimal control as the diffusion coefficient tends to 0. To cite this article: A. Salem, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Maximal eigenvalue and norm of a product of Toeplitz matrices. Study of a particular case

In this paper we describe the asymptotic behaviour of the spectral norm of the product of two finite Toeplitz matrices as the matrix dimension goes to infinity. These Toeplitz matrices are generated by functions with Fisher–Hartwig singularities of negative order. If these functions are positives the product of the two matrices has positive eigenvalues and it is known that the spectral norm is also the largest eigenvalue of this product.     

Locally minimizing harmonic maps from noncompact manifolds

By introducing the “relative energy”, we develop a new method for finding harmonic maps from noncompact complete Riemannian manifolds with prescribed asympototic behaviour at infinity. This method is an extension of the well known direct method of energy-minimization for compact domains. As an application of our method, we show that the Dirichlet problem at infinity with Hölder continuous boundary data for harmonic maps from a Cartan-Hadarmard manifold with bounded negative curvature into a compact manifold, has a locally minimizing solution which is smooth near infinity.     

One-parameter Family of Positive Solutions for a Nonlinear Integral Equation Arising in Physical Kinetics

The paper is devoted to the question of solvability of a Urysohn type nonlinear integral equation. This equation has an application in the kinetic theory of gases and can be derived from Boltzmann model equation. We prove an existence theorem of one-parameter family of positive solutions in the space of functions possessing linear growth at infinity. Moreover, for each member of this family we find an exact asymptotic formula at infinity. We obtain two-sided estimates for solution, as well as describe an iterative method for construction of solution.We conclude the paper by giving examples of functions that describe nonlinearity and satisfy the conditions of the main theorem.     

Using dual techniques to derive componentwise and mixed condition numbers for a linear function of a linear least squares solution

We prove duality results for adjoint operators and product norms in the framework of Euclidean spaces. We show how these results can be used to derive condition numbers especially when perturbations on data are measured componentwise relatively to the original data. We apply this technique to obtain formulas for componentwise and mixed condition numbers for a linear function of a linear least squares solution. These expressions are closed when perturbations of the solution are measured using a componentwise norm or the infinity norm and we get an upper bound for the Euclidean norm.      

On the derivative of the Perron vector whose infinity norm is fixed

In the recent paper the authors studied the derivaties of the Perron vector at an n × n essentially nonnegative and irreducible matrix A when the Perron vector is subjected to the normalization that one of its components is held a fixed constant in a neighbourhood of A or that the pth norm of the of the eigenvector is held a fixed constant in such a neighborhood. The Perron vector subject to the normalization that its infinity norm is held a fixed constant in a neighborhood of A does not necessarily imply that it is differentiable at A. In this paper we give formulas for the first derivative of this Perron vector where it is differentiable. Our formulas also accommodate left and right derivatives of the eigenvector.