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1972

Volume 3, Issue 4, pp. 561-667


Evaluation of Distributions Useful in Kontorovich-Lebedev Transform Theory

G. Z. Forristall and J. D. Ingram

SIAM J. Math. Anal. 3, pp. 561-566 (6 pages)

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The transform methods of Kontorovich and Lebedev seem to offer a direct approach to the solution of many physical problems involving the geometry of angular sectors. This promise has often been thwarted in practice by the coupling of boundary conditions which makes a direct inverse transform impossible. In the present work, we offer a method to meet this difficulty, based on the evaluation of the distributions listed below in terms of tabulated functions: \[ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow 0} \int_0^\infty {K_{i\nu } (\lambda r)K_{i\xi } (r)\frac{{dr}} {{r^{1 - \varepsilon } }}} , \hfill \\ \mathop {\lim }\limits_{\varepsilon \searrow 0} \int_0^\infty {K_{i\nu } (\lambda r)r^{2m - 1 - i\xi + \varepsilon } dr} , \hfill \\ \mathop {\lim }\limits_{\varepsilon \searrow 0} \int_0^\infty {K_{i\xi } (r)I_{i\nu } (\lambda r)\frac{{dr}} {{r^{1 - \varepsilon } }}} . \hfill \\ \end{gathered} \]

Quantitative Estimates for a Nonlinear System of Integrodifferential Equations Arising in Reactor Dynamics

T. A. Bronikowski, J. E. Hall, and J. A. Nohel

SIAM J. Math. Anal. 3, pp. 567-588 (22 pages) | Cited 4 times

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In this paper we obtain precise quantitative estimates for the asymptotic behavior of solutions of a class of nonlinear integrodifferential equations arising in nuclear reactor dynamics. The method uses a Galerkin approximation and certain energy estimates (Lyapunov functions) to obtain suitable bounds for solutions of related ordinary differential equations.

Boundedness Properties of Stationary Linear Operators

James D. Baker

SIAM J. Math. Anal. 3, pp. 589-594 (6 pages)

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A characterization of boundedness given by R. E. Lane provides a foundation for studying stationary, linear transformations which map the left-continuous quasi-continuous functions on the line into functions on the line. Properties of these operators are identified in terms of the transforms of $J(t) = \chi _{(0,\infty )} $ It is shown that the norm of an operator $T$ is the variation of $TJ$, that there is a Stieltjes integral representation of $T$ with this function as the integrator, and that convergence of a sequence of these operators is equivalent to convergence in variation of the sequence of transforms of $J$. An operator which is continuous under pointwise convergence is shown to be bounded on a closed interval, and is characterized by the left-continuity of $TJ$.

Continuous Linear Functionals on Certain $K\{ {M_p } \}$ Spaces

Charles Swartz

SIAM J. Math. Anal. 3, pp. 595-598 (4 pages) | Cited 1 time

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A distribution $T$ is known to be tempered if and only if there is a positive integer $k$ such that ${{T * \phi (x)} / {(1 + | x |^2 )^k }}$ is bounded for any $\phi \in \mathcal{D}$, if and only if there is a positive integer $k$ such that $\{ {1 / {(1 + | x |^2 )^k }}\tau _{ - x} T:x \in R^n \} $ is bounded in $\mathcal{D}'$. The analogues of these characterizations are established for certain of the test spaces $K\{ {M_p } \}$ of I. M. Gelfand and G. E. Shilov.

On Oscillatory Solutions of Certain Fourth Order Linear Differential Equations

M. S. Keener

SIAM J. Math. Anal. 3, pp. 599-605 (7 pages)

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We consider the fourth order linear homogeneous differential equation $y^{(4)} + p_3 (x)y''' + p_2 (x)y'' + p_1 (x)y' + p_0 (x)y = 0$ with continuous coefficients. Our studies center on the oscillatory behavior of solutions of the above equation under the separate disconjugacy conditions $r_{22} = r_{31} = \infty $ and $r_{22} = r_{13} = \infty $.

On the Symmetrized Kronecker Power of a Matrix and Extensions of Mehler’s Formula for Hermite Polynomials

David Slepian

SIAM J. Math. Anal. 3, pp. 606-616 (11 pages) | Cited 4 times

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Mehler’s formula expresses the exponential of a quadratic form in two variables as a series of products of Hermite polynomials. We give several useful generalizations of this formula to the case of $n$ variables, being guided in this work by interpretations in terms of Gaussian variates. Along the way we encounter the symmetrized Kronecker power of a matrix and we present a new generating function and recipe for calculating this quantity.

Lower Bounds for Eigenvalues with Displacement of Essential Spectra

David W. Fox

SIAM J. Math. Anal. 3, pp. 617-624 (8 pages) | Cited 14 times

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New constructions of comparison operators for rigorous lower bounds to eigenvalues of a class of self-adjoint operators are presented. The formulation uses noncompact finite perturbations to displace eigenvalues and essential spectra, and leads to workable numerical procedures. These methods make possible for the first time lower bound calculations for the lower eigenvalues of the Schrödinger operators for atoms and ions having three or more electrons.

Exponential Stability of Solutions of Differential Equations of Sobolev Type

John Lagnese

SIAM J. Math. Anal. 3, pp. 625-636 (12 pages) | Cited 3 times

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Let $\mathcal{M}(x,D)$ and $\mathcal{L}(x,D)$ be linear partial differential operators of order $2m$ with complex-valued coefficients defined on a bounded region $\Omega $ in $R^n$ and suppose $\mathcal{M}$ is elliptic in $\Omega $. Necessary and sufficient conditions are given in order that solutions of $\mathcal{M}(x,D){{\partial u} / {\partial t}} - \mathcal{L}(x,D)u = 0$ in the cylinder $\Omega \times [0,\infty )$ which satisfy general boundary conditions on the wall of the cylinder satisfy inequalities of the form $\| {u(t)} \|_{2m} \leqq Ce^{ - at} \| {u(0)} \|_{2m} $ and $| {u(t)} |_{2m + \rho } \leqq Ce^{ - at} | {u(0)} |_{2m + \rho } $, $t > 0$, with positive constants $a$ and $C$ independent of $u$. $\| \cdot \|_{2m} $ and $| \cdot |_{2m + \rho } $ denote the customary norms in the spaces $H^{2m,2} (\Omega )$ and $C^{2m + \rho } (\bar \Omega )$, $0 < \rho < 1$, respectively.

Characterizations of $\sigma $-Type Zero Polynomial Sets

Arun Verma

SIAM J. Math. Anal. 3, pp. 637-641 (5 pages)

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In this note two characterizations of $\sigma $-type zero polynomial sets are obtained. These characterizations are generalizations of two known results for the Appell polynomial sets.

A Concave Property of the Hypergeometric Function with Respect to a Parameter

Joseph B. Kadane

SIAM J. Math. Anal. 3, pp. 642-643 (2 pages)

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The hypergeometric function is shown to be logarithmically concave in integer values of one of its parameters. The methods used are probabilistic.

A Negative Definite Equilibrium and Its Induced Cone of Global Existence for the Riccati Equation

R. S. Bucy and J. Rodriguez-Canabal

SIAM J. Math. Anal. 3, pp. 644-646 (3 pages) | Cited 1 time

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The domain of global existence for the Riccati equation is shown to include a cone with vertex at the negative definite equilibria and containing the positive definite equilibria, under regularity conditions.

Perturbations in Nonlinear Systems

Noal C. Harbertson

SIAM J. Math. Anal. 3, pp. 647-653 (7 pages)

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If a certain system of nonlinear differential equations has a bounded solution $x(t)$, then for the same system subject to a small perturbation the existence of a solution $y(t)$ which lies “close” to $x(t)$ is established.

The Behavior of Oscillatory Solutions of $x''(t) + p(t)g(x(t)) = 0$

Stephen R. Bernfeld and James A. Yorke

SIAM J. Math. Anal. 3, pp. 654-667 (14 pages) | Cited 1 time

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Various quantitative properties of oscillatory solutions of the scalar second order nonlinear differential equation $x'' + p(t)g(x) = 0$ are obtained under appropriate hypotheses on $p$ and $g$. In particular, letting $\{ {t_i } \}_{i = 1}^\infty $, $0 < t_i < t_{i + 1} $, $t_i \to \infty $, as $i \to \infty $ be the zeros of any solution $x(t)$, we obtain inequalities on $J_i^{{\operatorname{def}}} \equiv \int _{t_i }^{t_{i + 1} } g(x(t))dt$ which yield asymptotic behavior on $x(t)$. For example, it is shown that $\lim_{t \to \infty } \int _0^t g(x(s))ds$ exists and is finite; moreover, assuming an added growth condition on ${{g(x)} / x}$, we have then that $\lim _{t \to \infty } \int _0^t x(s)ds$ exists and is finite.
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