We study boundaryvalueproblems on noncompact cycle-free graphs (i.e., trees) for second-order ordinary, differential equations with a nonlinear dependence on the spectral parameter. We, establish properties of the s...
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We study boundaryvalueproblems on noncompact cycle-free graphs (i.e., trees) for second-order ordinary, differential equations with a nonlinear dependence on the spectral parameter. We, establish properties of the spectrum and analyze the inverse problem of reconstructing the coefficients of a differential equation on the basis of the so-called Weyl functions. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mapping.
In this paper we give a positive answer to the conjecture proposed in [A. El Soufi, M. Jazar, R. Monneau, A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary...
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In this paper we give a positive answer to the conjecture proposed in [A. El Soufi, M. Jazar, R. Monneau, A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincare Anal. Non Lineaire 24 (1) (2007) 17-39] by El Soufi et al. concerning the finite time blow-up for solutions of the problem (1), (2) below. More precisely, we give a direct proof of [A. El Soufi, M. Jazar, R. Monneau, A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincare Anal. Non Lineaire 24 (1) (2007) 17-39, Theorem 1. 1] and the conjecture given for the case p > 2. (C) 2007 Elsevier Masson SAS. All rights reserved.
The construction of a modified Green's function for the internal gravitational wave (IGW) equation in a layer of a stratified medium when there are constant mean shear flows is considered and the basic properties ...
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The construction of a modified Green's function for the internal gravitational wave (IGW) equation in a layer of a stratified medium when there are constant mean shear flows is considered and the basic properties of the corresponding eigenvalueproblems and the modified eigenfunctions and eigenvalues are investigated. It is shown that each mode of the modified Green's function consists of a sum of three terms describing (1) the IGWs that propagate from the source, (2) the effects of a time varying source, localized in a certain neighbourhood of it, and (3) the effects of the displacement of the fluid (an internal discontinuity) caused by the source. The resulting expressions are analysed out for a constant and oscillating source of the generation of IGWs in which each of the terms of Green's function is represented in the form of simple quadratures. (C) 2008 Elsevier Ltd. All rights reserved.
For a linear nonstationary system with skew-symmetric continuously differentiable coefficient matrix of arbitrary even dimension, we construct its quasi-integrals and obtain effective coefficient estimates for their d...
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For a linear nonstationary system with skew-symmetric continuously differentiable coefficient matrix of arbitrary even dimension, we construct its quasi-integrals and obtain effective coefficient estimates for their deviations from the corresponding integrals in the stationary case. For each trajectory of motion described by Such a system bind lying on the sphere of the corresponding radius, these estimates permit precisely indicating a domain on the sphere containing the trajectory on a nonsmall time interval. The estimates can also be used for expanding the multidimensional motion of a mechanical object into multicomponent elements of lower dimension.
The computation of guided modes in an optical fiber is an eigenvalue problem posed in the whole of R-2. To compute the eigenvalues and the associated eigenfunctions, we truncate the domain and we impose a Robin condit...
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The computation of guided modes in an optical fiber is an eigenvalue problem posed in the whole of R-2. To compute the eigenvalues and the associated eigenfunctions, we truncate the domain and we impose a Robin condition on the boundary of the truncated domain. In this Note, we give an error estimate between the solutions of the physical problem and the truncated one.
We consider a weighted difference scheme approximating the heat equation with the nonlocal boundary conditions u(0,t) = 0, partial derivative u/partial derivative x(0,t) + partial derivative u/partial derivative x(1,t...
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We consider a weighted difference scheme approximating the heat equation with the nonlocal boundary conditions u(0,t) = 0, partial derivative u/partial derivative x(0,t) + partial derivative u/partial derivative x(1,t) = 0. We show that in this case the system of eigenfunctions of the main difference operator is not a basis but can be supplemented with associated functions to form a basis. Using the method of expansions in the basis of eigenfunctions and associated functions, we find a necessary and sufficient condition for stability with respect to the initial data in some energy norm. We show that this stability condition cannot be weakened by choosing a different norm. The above-mentioned energy norm is shown to be equivalent to the grid L-2-norm.
We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrodinger operators on graphs. Under a certain genericity condition, we show that the n...
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We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrodinger operators on graphs. Under a certain genericity condition, we show that the number of nodal domains of the n(th) eigenfunction is bounded below by n - l, where l is the number of links that distinguish the graph from a tree. Our results apply to operators on both discrete (combinatorial) and metric (quantum) graphs. They complement already known analogues of a result by Courant who proved the upper bound n for the number of nodal domains. To illustrate that the genericity condition is essential we show that if it is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction. In the Appendix we review the proof of the case l = 0 on metric trees which has been obtained by other authors.
In a recent Letter [R. Friedberg, J.T. Manassah, Phys. Lett. A 372 (2008) 2514] Friedberg and Manassah argue that effect of virtual photons has serious consequences for large atomic samples. Here we show that such eff...
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In a recent Letter [R. Friedberg, J.T. Manassah, Phys. Lett. A 372 (2008) 2514] Friedberg and Manassah argue that effect of virtual photons has serious consequences for large atomic samples. Here we show that such effect is negligible for evolution of a uniformly excited atomic state prepared by absorption of a single photon.(C) 2008 Elsevier B.V. All rights reserved.
We prove a version of the Krasnosel'skii-Krein theorem for differential inclusions with multimappings satisfying certain one-sided constraints. As a corollary. we obtain an analog of the first Bogolyubov theorem f...
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We prove a version of the Krasnosel'skii-Krein theorem for differential inclusions with multimappings satisfying certain one-sided constraints. As a corollary. we obtain an analog of the first Bogolyubov theorem for the inclusion 0 is an element of x' + epsilon F(t,x).
In the present paper, we consider the stability, problem for delay functional-differential equations with finite delay. We suggest a development of the Lyapunov function method involving the use of scalar comparison e...
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In the present paper, we consider the stability, problem for delay functional-differential equations with finite delay. We suggest a development of the Lyapunov function method involving the use of scalar comparison equations and limit, functions and equations. We prove a, localization theorem for the positive limit set of a bounded solution and a theory rein on the asymptotic stability of the zero solution. We present examples of sufficient conditions for the asymptotic stability of solutions of systems of the first, second, and arbitrary orders.
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