A Note on the Stability of the Integral-Differential Equation of the Hyperbolic Type in a Hilbert Space

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme approximately solving this problem is presented. The stability estimates for the solution of this difference scheme are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented. The stability estimates for the solutions of these difference schemes are obtained.     

On the choice of the best members of the Kim family and the improvement of its convergence

The best members of the Kim family, in terms of stability, are obtained by using complex dynamics. From this elements, parametric iterative methods with memory are designed. A dynamical analysis of the methods with memory is presented in order to obtain information about the stability of them. Numerical experiments are shown for confirming the theoretical results.     

The application of the principle of minimum cross-entropy to the characterization of the exponential-type probability distributions

Summary Systematic and simple characterizations are presented for several familiar distributions in exponential family by means of the principle of minimum cross-entropy (minimum discrimination information). The suitable prior distributions and the appropriate constraints on expected values are given for the underlying distributions.     

On the Z-eigenvalues of the adjacency tensors for uniform hypergraphs

The adjacency matrices for graphs are generalized to the adjacency tensors for uniform hypergraphs, and some fundamental properties for the adjacency tensor and its Z-eigenvalues of a uniform hypergraph are obtained. In particular, some bounds on the smallest and the largest Z-eigenvalues of the adjacency tensors for uniform hypergraphs are presented.     

Integrability of the discretization of the controlled Euler Top

The Euler equations with two controls are discretized by using the bilinear transformation method. The discrete equations are explicit and exhibit a sufficient number of conserved quantities for integrability. The explicit solution to the discrete controlled Euler top are deduced in terms of elliptic functions.     

Bounds for the expectation of the kth root of a random variable

Exact bounds for the mean value of a fractional moment, such as the sample standard deviation, are considered. These bounds are compared with those found by summation techniques applied to computer extended series. An expansion involving an arbitrary parameter is introduced, and now best bounds are found by optimization. There are applications to bounds for a certain class of quadrature problem.     

On the preconditioning of the block tridiagonal linear system of equations

Two algorithms for computing the inverse factors of general tridiagonal and pentadiagonal matrices are obtained. Then, these algorithms are used for computing a block ILU preconditioner for the block tridiagonal linear system of equations. Some numerical results are given to show the robustness and efficiency of the preconditioner. The performance of the proposed preconditioner is compared with a recently proposed method.     

The Influence of Singular Points of the Laplace Image on the Generalized Spectrum of the Original

Integral representations of the Fourier–Jacobi coefficients are constructed in the form of Riemann–Mellin integrals, and on this basis asymptotic formulas for calculating coefficients for large-order numbers are derived.     

On the integrability of Mizohata structures on the sphereS 2

This paper deals with the integrability problem for structures on the sphere S² which are elliptic on the upper and lower hemispheres, and which are given by a simple fold on the equator. Criteria for standardness are given. Existence and factorization of first integrals are studied.     

On the regularity of the critical point infinity of definitizable operators

In this note necessary and sufficient conditions for the regularity of the critical point infinity of a definitizable operator A are given. Using these criteria it is proved that the regularity of the critical point infinity is preserved under some additive perturbations as well as for some operators which are related to A. Applications to indefinite Sturm-Liouville problems are indicated.     

On the Possibility of Using the Method of Sign-Perturbed Sums for the Processing of Dynamic Test Data

At the present time, the methods for the measurement and prediction of the dynamic strength of materials are complicated and unstandardized. An experimental data processing method based on the incubation time criterion is considered. Only a finite number of measurements containing random errors and limited statistical information are usually available in practice, since dynamic tests are laborious, and every individual test requires a lot of time. This strongly restricts the number of applicable data processing methods unless we are satisfied with approximate and heuristic solutions. The method of sign-perturbed sums (SPS) is used for the estimation of finite-sample confidence regions with a specified confidence probability under the assumption of noise symmetries. It is shown that several experimental points are sufficient to determine the strength parameter with an accuracy acceptable for engineering calculations. The applicability of the proposed method is demonstrated in the processing of a number of experiments on the dynamic fracture of rocks.     

Chaplygin气体Euler方程组的轴对称解

构造了两维Chaplygin气体Euler方程组的三参数、自相似的弱解.在自相似和轴对称的假设下,两维Chaplygin气体Euler方程组可以化为无穷远边值的常微分方程组,由此得到了解的存在性和解的结构.与多方气体不同的是Chaplygin气体的Euler方程组是完全线性退化的.即使在轴向速度大于零的时候解也会出现间断现象.这些解展示了宇宙演化过程中的一些现象,例如黑洞的形成与演化以及宇宙的暴涨和膨胀.     

Coupled problems of the optimization of the elasto-plastic deformation of metals

The mathematical formulation of a coupled optimization problem, which includes several optimal problems which are interrelated via the optimization parameters and functions of state, is analysed. The conditions for the existence of a solution are formulated and a technique for reducing the problem, to a sequence of optimal problems with unknown functions in the optimality and bounding criteria is described. A special minimizing sequence is constructed and its convergence conditions are derived. An example is given to demonstrate the method.     

Efficiencies for window estimates of the parameters of the Cauchy distribution

Higgins and Tichenor [Appl. Math. and Comp. 3 (1977), 113-126] considered “window estimates” of location and reciprocal scale parameters for a general class of distributions and showed them to be asymptotically efficient for the Cauchy distribution. In this study, efficiencies of these estimates for the Cauchy distribution are investigated for small and moderate sample sizes by Monte Carlo methods. For n?40, window estimates of location are nearly optimal, and for n?20, they compare favorably with other easy-to-compute estimates. Window estimates of reciprocal scale are very good even for small samples and are nearly optimal for n?10. Thus, window estimates appear to have high efficiency for moderate as well as large sample sizes. Approximate normality is also investigated. The estimate of location converges rapidly to normality, whereas the estimate of reciprocal scale does not.     

A functional approach to the theory of the structure of the surface of metals

A self-consistent field equation obtained by the method of functional integration is applied to investigate the structure of the lattice near the surface of a simple metal. Formulas are obtained for the displacements of ions relative to the nodes of an ideal lattice near the surface of the metal. Formulas are presented for the surface potential and the work of exit for one-dimensional and three-dimensional models.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 3–11, 1982.     

On the theory of the gyropendulum

Precession equations of motion of the gyropendulum relative to the accompanying Darboux trihedron /1/ and, also, precession equations of the gyropendulum motion relative to the geographic trihedron, considered in /2, 3/, are given a kinematic interpretation. Linear differential equations that define the gyropendulum behavior at finite deflection angles of the rotor axis from the vertical are established for arbitrary motions of its suspension point over the surface of the Earth. These equations have the form of kinematic equations of a solid body spherical motion in terms of Rodrigues-Hamilton parameters, and in the case of stationary base they are in agreement with equations established in /4/. The Liapunov stability ot the gyropendulum equations in both the finite Euler—Krylov angles and in the Rodrigues — Hamilton parameters is proved. Particular cases of integrability in quadratures of the gyropendulum precession equations at finite angles are indicated.     

On the completeness of eigenfunctions of the Frankl problem with the oddness condition

Modified Frankl problems with the oddness condition are considered for the Lavrent’ev-Bitsadze equation. The eigenvalues and eigenfunctions of these problems are found. The completeness of these eigenfunctions in the elliptic part of the domain is proved.     

On the zeros of the Scorer functions

Asymptotic approximations are developed for zeros of the solutions Gi(z) and Hi(z) of the inhomogeneous Airy differential equation . The solutions are also called Scorer functions. Tables are given with numerical values of the zeros.     

几种基于匈牙利算法求解二次分配问题的方法及其分析比较

二次分配问题(Quadratic assignment problem,QAP)属于NP-hard组合优化难题。二次分配问题的线性化模型和下界计算方法,是求解二次分配问题的重要途径。本文以二次分配问题的线性化模型为基础,根据现有QAP对偶上升下界计算方法中的具体操作,提出几种可行的QAP对偶上升计算新方法。最后,通过求解QA-PLIB中的部分实例,深入分析其运行结果,详细讨论了基于匈牙利算法求解二次分配问题的对偶方法中哪些操作可较大程度地提高目标函数最优解的下界增长速度,这为基于匈牙利算法求解二次分配问题的方法的改进奠定了基础。     

On the stability of the one-step exact collocation methods for the numerical solution of the second kind Volterra integral equation

The purpose of this paper is to analyze the stability properties of one-step collocation methods for the second kind Volterra integral equation through application to the basic test and the convolution test equation.Stability regions are determined when the collocation parameters are symmetric and when they are zeros of ultraspherical polynomials.