On the first and second order derivatives of the Perron vector

In a previous work [5] the authors developed formulas for the second order partial derivatives of the Perron root as a function of the matrix entries at an essentially nonnegative and irreducible matrix. These formulas, which involve the group generalized inverse of an associated M-matrix, were used to investigate the concavity and convexity of the Perron root as a function of the entries. The authors now combine the above results together with an approach taken in an earlier joint paper [6] of the second author with L. Elsner and C. Johnson, and they develop formulas for the second order derivatives of an appropriately normalized Perron vector with respect to the matrix entries, which again are given in terms the group generalized inverse of an associated M-matrix. Convexity properties of the Perron vector as a function of the entries of the matrix are then examined. In addition, formulas for the first derivative of the Perron vector resulting from different normalizations of this eigenvector are also given. A by-product of one of these formulas yields that the group generalized inverse of a singular and irreducible M-matrix can be diagonally scaled to a matrix which is entrywise column diagonally dominant.     

On Legendre functions of imaginary degree and associated integral transforms

New integral representations, asymptotic formulas, and series expansions in powers of tanh(t/2) are obtained for the imaginary and real parts of the Legendre function P_iξ(cosht). Coefficients of these series expansions are orthogonal polynomials in the real variable ξ. A number of relations for these orthogonal polynomials are obtained on the basis of the generating function. Several inversion theorems are proven for the integral transforms involving the Legendre function of imaginary degree. In many cases it is preferable to employ these transforms, than Mehler-Fok transforms, since conditions placed on functions are less restrictive.     

Integral representations of the generalized Legendre functions of the second kind Qk m,n(z)

New integral representations for the generalized Legendre functions of the second kind are established and certain integrals, involving these functions, are evaluated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 985–990, July, 1990.     

Reduction of multidimensional first order equations with multi-homogeneous function of derivatives

We present an analysis of solutions to multidimensional first order equation order with several independent variables under assumption that the nonlinear part of the equation is a multi-homogeneous function of derivatives. The reduction of the original equation is performed for the class of solutions depending on linear combinations of prescribed groups of initial variables. We obtain solutions to the reduced equation. We consider also the cases of additional, multiplicational and combined separation of variables.     

On second kind polynomials associated with rational transformations of linear functionals

In this paper the following construction process of orthogonal polynomials on the unit circle is considered: Let u be a regular and hermitian linear functional and let {Φn} be the corresponding orthogonal polynomials sequence. We define a new linear functional L by means the following relation with u:

     

非对称阻尼系统特征对一阶导数与二阶导数的计算

提出了一种计算非对称阻尼系统特征对一阶、二阶导数的方法.该方法利用阻尼系统的特征向量计算特征对的导数,避免了状态空间中特征向量的使用,节省了计算量,且不要求系统所有特征值的互异性.最后以两个非对称阻尼系统进行数值试验,数值结果表明提出的方法是有效的.     

On first and second order stationarity of random coefficient models

We give conditions for first and second order stationarity of mixture autoregressive processes. We obtain a simple condition for positive definiteness of the solution of a generalisation of the Stein’s equation with semidefinite right-hand side and apply it to second order stationarity. The said condition may be of independent interest.     

Exact boundary observability for a kind of second order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global C² solution, we establish the local exact boundary observability for a kind of second order quasilinear hyperbolic systems. As an application, we obtain the one-sided local exact boundary observability for a kind of first order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled.     

On integral representation of Bessel function of the first kind

A first kind Fredholm integral equation with nondegenerate kernel is given, which particular solution is the Bessel function of the first kind. This equation is solved by means of Mellin transform pair.     

The upper and lower second order directional derivatives of a sup-type function

The purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup-type functionS(x) = sup{f(x, t); t T} in terms of the first and second derivatives off(x, t), whereT is a compact set in a metric space and we assume thatf, f/x and ² f/x ² are continuous on ⁿ × T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition forS(x) to be directionally twice differentiable.     

On a representation of fully nonlinear elliptic operators in terms of pure second order derivatives and its applications

We give a method of representing fully nonlinear elliptic operators given by boundedly inhomogeneous functions in terms of operators acting only on pure second order derivatives. We also discuss possible applications of such representations to using .finite di.erence approximations for solving the corresponding equations.     

On the similarity of linear operators in L p to integral operators of the first and second kind

We construct an example of a compact operator of the third kind in L _p (p ≠ 2) not similar to any integral operator of the first or second kind. This example shows that not every linear equation of the third kind in L _p (p ≠ 2) can be reduced by an invertible continuous linear change to an equivalent integral equation of the first or second kind. The example also proves the impossibility of a characterization of integral and Carleman integral operators in L _p (p ≠ 2) in terms of the spectrum and its components.