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Linear Volterra-type integral equations with kernels having a series expansion in the first variable have series solutions with coefficients given iteratively. Their resolvents may be expanded likewise. The associated homogeneous ...
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Linear Volterra-type integral equations with kernels having a series expansion in the first variable have series solutions with coefficients given iteratively. Their resolvents may be expanded likewise. The associated homogeneous equation Kf = f generally has Frobenius series solutions when the kernel is singular, whereas Kf =0 generally has such solutions regardless of singularity: the proviso in each case is that associated ‘‘indicial equation’’ has solutions.
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This paper presents the technological aspect of application of the ion exchange method in producing gradient refractive index in glass. The possibility of predictable and repeatable producing of the changes in glass refraction wit...
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This paper presents the technological aspect of application of the ion exchange method in producing gradient refractive index in glass. The possibility of predictable and repeatable producing of the changes in glass refraction with the use of this method has been presented, as well as the method of in situ control of the process of diffusion doping of glass based on the measurement of the temperature. This method is based on simultaneous (to the carried process) solving the nonlinear diffusion equation modeling the spatio- temporal changes in normalized concentration of the admixture ions in glass. For this purpose the knowledge of temperature characteristics of diffusion coefficients of exchanged ions is used. The result of such control of diffusion processes is information on the current (temporary) refractive index profile of the resulting waveguide. The presented method of control has been confirmed by experimental results, which concern modeling and measurements of planar waveguide structures of slab type. The proposed methodology can also be used to control the diffusion processes of producing another type of two- and three-dimensional gradient structures. According to the author's knowledge the method mentioned above has not been described in literature before.
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We construct modal outgoing Green's kernels for the simplified Galbrun's equation under spherical symmetry, in the context of helioseismology. The coefficients of the equation are C-2 functions representing the solar interior mode...
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We construct modal outgoing Green's kernels for the simplified Galbrun's equation under spherical symmetry, in the context of helioseismology. The coefficients of the equation are C-2 functions representing the solar interior model S, complemented with an isothermal atmospheric model. We solve the equation in vectorial spherical harmonics basis to obtain modal equations for the different components of the unknown wave motions. These equations are then decoupled and written in Schrodinger form, whose coefficients are shown to be C-2 apart from at most two regular singular points, and to decay like a Coulomb potential at infinity. These properties allow us to construct an outgoing Green's kernel for each spherical mode. We also compute asymptotic expansions of coefficients up to order r(-3) as r tends to infinity, and show numerically that their accuracy is improved by including the contribution from the gravity although this term is of order r(-3). (C) 2021 The Authors. Published by Elsevier Inc.
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The object of our interest is a certain tridiagonal matrix that appears in a variety of problems in statistical mechanics and quantum physics, such as the Brownian motion, random walk on a hypercube, the Ehrenfest urn model, and t...
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The object of our interest is a certain tridiagonal matrix that appears in a variety of problems in statistical mechanics and quantum physics, such as the Brownian motion, random walk on a hypercube, the Ehrenfest urn model, and the Stark effect of the hydrogen atom. The spectral decomposition of this matrix has been studied by a number of authors, among others Sylvester, Cayley, Mazza, Muir, Schrodinger, and Kac. In particular, explicit expressions are known for the eigenvalues and the eigenvectors of the matrix. So the question arises: Does there exist an explicit formula for the singular values? In this paper we find an explicit formula for a subset of the singular values when the order of the matrix is odd. In the process we utilize the method of generating functions, and derive a second-order differential equation. The polynomial solutions of this differential equation provide the elements of the singular vectors. (c) 2006 Elsevier Inc. All rights reserved.
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We consider two linear second-order ordinary differential equations. r = 0 is a regular singular point of these equations. Applying the classical Method of Frobenius, we do not obtain any indicial equation and therefore no solutio...
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We consider two linear second-order ordinary differential equations. r = 0 is a regular singular point of these equations. Applying the classical Method of Frobenius, we do not obtain any indicial equation and therefore no solution, because the differential equations are coupled. In this paper, we present an extended Method of Frobenius on a coupled system of two ordinary differential equations. These equations come from the micropolar theory, which is one of the three kinds of the new 3M physics.
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We derive asymptotics of moments and identify limiting distributions, under the random permutation model on m-ary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functional...
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We derive asymptotics of moments and identify limiting distributions, under the random permutation model on m-ary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the so-called shape functional fall under this framework. The approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. The transfer theorems are used in conjunction with the method of moments to establish limit laws. It is shown that: (i) for small toll sequences (t(n)) [roughly, t(n) = O(n(1/2))] we have asymptotic normality if m = 27; (ii) for moderate toll sequences [roughly, t(n) = omega(n(1/2)) but t(n) = o(n)] we have convergence to nonnormal distributions if m = 26) and typically periodic behavior if m >= m(0) + 1; and (iii) for large toll sequences [roughly, t(n) = omega(n)] we have convergence to normormal distributions for all values of m. (c) 2004 Wiley Periodicals, Inc.
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