Exact nonnull distribution of Wilks’ statistic: The ratio and product of independent components

The study of the noncentral matrix variate beta type distributions has been sidelined because the final expressions for the densities depend on an integral that has not been resolved in an explicit way. We derive an exact expression for the nonnull distribution of Wilks’ statistic and precise expressions for the densities of the ratio and product of two independent components of matrix variates where one matrix variate has the noncentral matrix variate beta type I distribution and the other has the matrix variate beta type I distribution. We provide the expressions for the densities of the determinant of the ratio and the product of these two components. These distributions play a fundamental role in various areas of statistics, for example in the criteria proposed by Wilks.     

一类反应扩散方程的精确解

运用首次积分法并借助计算机符号计算系统Mathematica得到了一类非线性反应扩散方程u_t-u_(xx) au βu~2 γu~3=0的精确解,其中既包括已知的类孤立波解,又有新的精确解.     

A Multidimensional Nonautonomous Equation Containing a Product of Powers of Partial Derivatives

A class of multidimensional differential equations containing products of powers of partial derivatives of any order is considered. Solutions with additive, multiplicative, and combined separation of variables are obtained. A family of pseudopolynomial solutions expressed in terms of polynomials in independent variables with arbitrary coefficients and functions being solutions of certain ordinary differential equations is also obtained. Solutions of the type of traveling wave and self-similar solutions, as well as families of solutions having the form of a sum or a product of solutions of the type of a traveling wave and self-similar solutions, are found. Finally, solutions that can be represented as functions of more complicated arguments expressed in terms of linear combinations and products of the initial independent variables are found. For all of the obtained solutions, conditions on the righthand side of the equation and its parameters under which these solutions exist are determined.     

On constructing periodic solutions of quasi-linear non-self-contained systems with one degree of freedom, when amplitude equations possess multiple roots

Construction of periodic solutions of quasilinear non-self-contained systems with one degree of freedom, was investigated in [1 and 2]. In [1] the case of simple roots of amplitude equations was considered together with the case of a double root when the solution could be expanded into a series in integral powers of μ. In [2] the case of a double root is investigated in more detail Including expansions of solutions into series in μ^1/2. In the present paper, the case of arbitrary multiple roots for non-self-contained systems is reduced to the corresponding case for self-contained systems, which simplifies computations.     

On the solution of elliptic partial differential equations on regions with corners II: Detailed analysis

In this paper we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations. Previously, we observed that when the boundary value problems are formulated as boundary integral equations of classical potential theory, the solutions are representable by series of elementary functions, to arbitrary order, for all but finitely many values of the angles. Here, we extend this observation to all values of the angles. We show that the solutions near corners are representable, to arbitrary order, by linear combinations of certain non-integer powers and non-integer powers multiplied by logarithms.     

Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem

In this paper, two results concerning the global attractivity and global asymptotic attractivity of the solutions for a nonlinear functional integral equation are proved via a variant of the Krasnoselskii fixed point theorem due to Dhage [B.C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J. 44 (2004) 145–155]. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. A couple of examples are indicated for demonstrating the natural realizations of the abstract results presented in the paper. Our results generalize the attractivity results of Banas and Rzepka [J. Banas, B. Rzepka, An application of measures of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16 (2003) 1–6] and Banas and Dhage [J. Banas, B.C. Dhage, Global asymptotic stability of solutions of a functional integral equations, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.07.038], under weaker conditions with a different method.     

On Complicated Expansions of Solutions to ODES

Polynomial ordinary differential equations are studied by asymptotic methods. The truncated equation associated with a vertex or a nonhorizontal edge of their polygon of the initial equation is assumed to have a solution containing the logarithm of the independent variable. It is shown that, under very weak constraints, this nonpower asymptotic form of solutions to the original equation can be extended to an asymptotic expansion of these solutions. This is an expansion in powers of the independent variable with coefficients being Laurent series in decreasing powers of the logarithm. Such expansions are sometimes called psi-series. Algorithms for such computations are described. Six examples are given. Four of them are concern with Painlevé equations. An unexpected property of these expansions is revealed.     

Hyperbolic integro-differential equations of convolution type

Making use of the theory of Wiener-Hopf operators in the scale of abstract Krein spaces, we prove existence and uniqueness of unbounded solutions for the linear hyperbolic integrodifferential equation (P_o). We extend herewith results obtained in [8] for hyperbolic evolution equations, where the convolution integral was absent. The method utilizes Dunford's functional calculus and permits thus a constructive existence proof for solutions exhibiting an exponential growth rate when time increases. Our approach bases upon the fundamental hypothesis that the spectrum of the time-independent mapping -A shows a parabolic condensation along the negative real axis. This condition completely determines the admissible geometry of the spectral set of the convolution integral operator, and a fortiori the magnitude of the exponential growth rate. The theory works in arbitrary reflexive Banach spaces.     

On the Lyapunov theory of equilibrium figures of celestial bodies

The work of A. M. Lyapunov on the theory of equilibrium figures of celestial bodies is analyzed. The main results are mentioned, such as sufficient conditions for the existence and uniqueness of solutions to the complicated integral and integro-differential equations of the problem; the solution of the stability problem for the MacLaurin and Jacobi ellipsoids; the solution of the existence and stability problem for figures branching from the ellipsoids; the solution of the problem for slowly rotating inhomogeneous bodies in terms of series (called now the Lyapunov series) in powers of a small parameter, which is equal in the first approximation to the centrifugal-to-gravitational force ratio; and the estimation of the convergence radius of the Lyapunov series. Further development of Lyapunov’s ideas and unsolved problems is discussed.     

On some generalizations of the Vandermonde matrix and their relations with the Euler beta-function

A multiple Vandermonde matrix which, besides the powers of variable, also contains their derivatives is introduced and an explicit expression of its determinant is obtained. for the case of arbitrary real powers, when the variables are positive, it is proved that such generalized multiple Vandermonde matrix is positive definite for appropriate enumerations of rows and columns. As an application of these results, some relations are obtained which in the one-dimensional case give the well-known formula for the Euler betafunction.     

实轴上的特征奇异积分方程

该文首先得到了实轴上的特征奇异积分方程的可解性理论.由此,得到了实轴上解具一阶奇异性的特征奇异积分方程的可解性理论,纠正了文献[8]中的错误.     

Perturbed Laguerre unitary ensembles,Hankel determinants,and information theory

This article investigates a key information‐theoretic performance metric in multiple‐antenna wireless communications, the so‐called outage probability. The article is partly a review, with the methodology based mainly on [10], while also presenting some new results. The outage probability may be expressed in terms of a moment generating function, which involves a Hankel determinant generated from a perturbed Laguerre weight. For this Hankel determinant, we present two separate integral representations, both involving solutions to certain non‐linear differential equations. In the second case, this is identified with a particular σ‐form of Painlevé V. As an alternative to the Painlevé V, we show that this second integral representation may also be expressed in terms of a non‐linear second order difference equation. Copyright © 2014 John Wiley & Sons, Ltd.     

A Complete Characterization of Multivariate Normal Stable Tweedie Models through a Monge-Ampère Property

Extending normal gamma and normal inverse Gaussian models, multivariate normal stable Tweedie (NST) models are composed by a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are real independent Gaussian variables with the same variance equal to the fixed component. Within the framework of multivariate exponential families, the NST models are recently classified by their covariance matrices V(m) depending on the mean vector m. In this paper, we prove the characterization of all the NST models through their determinants of V(m), also called generalized variance functions, which are power of only one component of m. This result is established under the NST assumptions of Monge-Ampère property and steepness. It completes the two special cases of NST, namely normal Poisson and normal gamma models. As a matter of fact, it provides explicit solutions of particular Monge-Ampère equations in differential geometry.     

Non-classical laws with conserved quantities from the arbitrary functions included in general solution to elasticity and application

The general solution to static and/or dynamic linear elasticity is a transformation between the displacements and new arbitrary functions, whose conservativeness depends on some independent partial differential equations (PDEs) satisfied by the new arbitrary functions. Zhang's general solutions are mathematically appropriate since the displacements are expressed in terms of two new arbitrary functions, and the sum of the highest order derivative added together from the independent PDEs satisfied by the two new arbitrary functions is the same as that of Navier–Cauchy equations. Therefore, the following points should be emphasized: (i) the independent PDEs come from the Laplace and D'Alembert operators acting on the two new arbitrary functions in static and dynamic general solutions, respectively, and it is found that the two new arbitrary functions are related to the rotations, first strain invariant and distortion; (ii) especially, conservation laws constructed from the equations satisfied by the spatial integrals of functions hold true, although some arbitrary functions of the spatial integrals have been canceled. Based on these facts, since Noether's identity not only can be applied to a Lagrangian but also can be used to construct a functional for widespread PDEs, the functionals relating to the rotations, first strain invariant and distortion are constructed with arbitrary integer order spatial derivative or integral, and the conservation laws follow. This kind of non-classical conservation laws does not come from the Lagrangian density of an elastic body and belongs to the deep-level natures of symmetries of elastic field derived by standard techniques. Availability is shown by two examples, from which the field intensity of a vertical load applied to the surface of an elastic half-space and the path-independent integrals in a coordinate system moving with Galilean transformation are presented for comparison.     

ON THE EXISTENCE AND UNIQUENESS THEOREMS OF SOLUTIONS FOR A CLASS OF THE SYSTEMS OF MIXED MONOTONE OPERATOR EQUATIONS WITH APPLICATION

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Null space distributions—A new approach to finite convolution equations with a Hankel kernel

In some earlier publications it has been shown that the solutions of the boundary integral equations for some mixed boundary value problems for the Helmholtz equation permit integral representations in terms of solutions of associated complicated singular algebraic ordinary differential equations. The solutions of these differential equations, however, are required to be known on some infinite interval on the real line, which is unsatisfactory from a practical point of view. In this paper, for the example of one specific boundary integral equation, the relevant solutions of the associated differential equation are expressed by integrals which contain only one unknown generalized function, the support of this generalized function is no longer unbounded but a compact subset of the real line. This generalized function is a distributional solution of the homogeneous boundary integral equation. By this null space distribution the boundary integral equation can be solved for arbitrary right-hand sides, this solution method can be considered of being analogous to the method of variation of parameters in the theory of ordinary differential equations. The nature of the singularities of the null space distribution is worked out and it is shown that the null space distribution itself can be expressed by solutions of the associated ordinary differential equation.     

The Impedance Boundary Value Problem for the Helmholtz Equation in a Half-Plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L_∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5]. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Differential Algebra and Liouvillian first integrals of foliations

Intuitively, a complex Liouvillian function is one that is obtained from complex rational functions by a finite process of integrations, exponentiations and algebraic operations. In the framework of ordinary differential equations the study of equations admitting Liouvillian solutions is related to the study of ordinary differential equations that can be integrated by the use of elementary functions, that is, functions appearing in the Differential Calculus. A more precise and geometrical approach to this problem naturally leads us to consider the theory of foliations. This paper is devoted to the study of foliations that admit a Liouvillian first integral. We study holomorphic foliations (of dimension or codimension one) that admit a Liouvillian first integral. We extend results of Singer (1992) [20] related to Camacho and Scárdua (2001) [4], to foliations on compact manifolds, Stein manifolds, codimension-one projective foliations and germs of foliations as well.     

Asymptotic existence theorems for formal power series whose coefficients satisfy certain partial differential recursions

We study power series whose coefficients are holomorphic functions of another complex variable and a nonnegative real parameter s, and are given by a differential recursion equation. For positive integer s, series of this form naturally occur as formal solutions of some partial differential equations with constant coefficients, while for s=0 they satisfy certain perturbed linear ordinary differential equations. For arbitrary s?0, these series solve a differential-integral equation. Such power series, in general, are not multisummable. However, we shall prove existence of solutions of the same differential-integral equation that in sectors of, in general, maximal opening have the formal series as their asymptotic expansion. Furthermore, we shall indicate that the solutions so obtained can be related to one another in a fairly explicit manner, thus exhibiting a Stokes phenomenon.     

带一阶导数项多滞量二阶中立型方程解的振动的充要条件

考虑中立型方程其中p_i,r_j(i∈I,j∈J)为任意实常数,q_k,τ_i(k∈K,i∈J)为正常数,而ρ_j,σ_k(j∈J,k∈K)是非负常数。则方程(*)所有解振动的充要条件是它的特征方程无实根。