In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition of the second kind. We analyze the completeness, the basis proper...
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In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition of the second kind. We analyze the completeness, the basis property, and the minimality of the eigenfunctions depending on the parameter of the problem.
We study uniquely ergodic dynamical systems over locally compact, sigma-compact Abelian groups. We characterize uniform convergence in Wiener/Wintner type ergodic theorems in terms of continuity of the limit. Our resu...
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We study uniquely ergodic dynamical systems over locally compact, sigma-compact Abelian groups. We characterize uniform convergence in Wiener/Wintner type ergodic theorems in terms of continuity of the limit. Our results generalize and unify earlier results of Robinson and Assani respectively. We then turn to diffraction of quasicrystals and show how the Bragg peaks can be calculated via a Wiener/Wintner type result. Combining these results we prove a version of what is sometimes known as the Bombieri/Taylor conjecture. Finally, we discuss various examples including deformed model sets, percolation models, random displacement models, and linearly repetitive systems.
We study the analog of the quantum-mechanical harmonic oscillator on infinite blowups of the Sierpinski Gasket, using the standard Kigami Laplacian. Our main task is to find a class of potentials analogous to 1/2 (x -...
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We study the analog of the quantum-mechanical harmonic oscillator on infinite blowups of the Sierpinski Gasket, using the standard Kigami Laplacian. Our main task is to find a class of potentials analogous to 1/2 (x - x(0))(2) on the line. We describe a class of potentials u with the properties Delta u = 1, u attains a minimum value zero, and u. 8 at infinity. We show how to construct such potentials attaining the minimum value at any prescribed point, and we show how to parameterize the class of potentials by a certain surface in R-3. We obtain estimates for the growth rate of the eigenvalue counting function for -Delta + u. We obtain numerical approximations to the eigenfunctions, and in particular observe that the ground-state eigenfunction resembles a Gaussian function.
He's homotopy-perturbation method (HPM) is presented for solving higher-order boundaryvalueproblems. numerical examples are included to demonstrate the validity and applicability of the technique and a compariso...
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He's homotopy-perturbation method (HPM) is presented for solving higher-order boundaryvalueproblems. numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with the existing results. The method is easy to implement and yields very accurate results. (C) 2008 Elsevier Ltd. All rights reserved.
We study the following third-order p-Laplacian m-point boundaryvalueproblems on time scales (phi(p)(u(Delta del)))(del) + a(t)f(t, u(t)) = 0, t is an element of [0, T](T)(kappa), u(0) = Sigma(m-2)(i=1)b(i)u(xi(i)), ...
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We study the following third-order p-Laplacian m-point boundaryvalueproblems on time scales (phi(p)(u(Delta del)))(del) + a(t)f(t, u(t)) = 0, t is an element of [0, T](T)(kappa), u(0) = Sigma(m-2)(i=1)b(i)u(xi(i)), u(Delta)(T) = 0, phi(p)(u(Delta del)(0)) = Sigma(m-2)(i=1)c(i)phi(p)(u(Delta del)(xi(i))), where phi(p)(s) is p-Laplacian operator, that is, phi(p)(s) = vertical bar s vertical bar(p-2) s, p > 1, phi(-1)(p) = phi(q), 1/p + 1/q = 1, 0 < xi(1) < ... < xi(m-2) < rho(T). We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear term f (t, u) is allowed to change sign. The conclusions in this paper essentially extend and improve the known results. Copyright (C) 2009 F. Xu and Z. Meng.
By constructing an available integral operator and combining Krasnosel'skii-Zabreiko fixed point theorem with properties of Green's function, this paper shows the existence of multiple positive solutions for a...
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By constructing an available integral operator and combining Krasnosel'skii-Zabreiko fixed point theorem with properties of Green's function, this paper shows the existence of multiple positive solutions for a class of m-point second-order Sturm-Liouville-like boundaryvalueproblems on time scales with polynomial nonlinearity. The results significantly extend and improve many known results for both the continuous case and more general time scales. We illustrate our results by one example, which cannot be handled using the existing results. Copyright (C) 2009 Meiqiang Feng et al.
By using the Leray-Schauder continuation theorem, we establish the existence of solutions for m-point boundaryvalueproblems on a half-line x ''(t) + f(t, x(t), x'(t)) = 0, 0 +infinity) x'(t) = 0, whe...
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By using the Leray-Schauder continuation theorem, we establish the existence of solutions for m-point boundaryvalueproblems on a half-line x ''(t) + f(t, x(t), x'(t)) = 0, 0 < t < +infinity, x(0) = Sigma(m-2)(i=1) alpha(i)x(eta(i)), lim(t ->+infinity) x'(t) = 0, where alpha(i) is an element of R, Sigma(m-2)(i=1) alpha(i) not equal 1 and 0 < eta(1) < eta(2) < ... < eta(m-2) < +infinity are given. Copyright (C) 2009 Changlong Yu et al.
This paper establishes exponential inequalities for the probability of deviation between the approximate and the exact solution of an operator equation with an exact second member. These inequalities yield the almost ...
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This paper establishes exponential inequalities for the probability of deviation between the approximate and the exact solution of an operator equation with an exact second member. These inequalities yield the almost complete convergence and the convergence rate of approximate solutions. (c) 2008 Elsevier B.V. All rights reserved.
We introduce a method for treating general boundary conditions in the finite element method generalizing an approach, due to Nitsche (1971), for approximating Dirichlet boundary conditions. We use Poisson's equati...
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We introduce a method for treating general boundary conditions in the finite element method generalizing an approach, due to Nitsche (1971), for approximating Dirichlet boundary conditions. We use Poisson's equations as a model problem and prove a priori and a posteriori error estimates. The method is also compared with the traditional Galerkin method. The theoretical results are verified numerically.
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