Some remarks on the Jensen-Hadamard inequalities and applications

We provide some inequalities and integral inequalities connected to the Jensen-Hadamard inequalities for convex functions. In particular, we give some refinements to these inequalities. Some natural applications and further extensions are given.

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Sunto Forniamo alcune diseguaglianze e diseguaglianze integrali connesse alle dise-gueglianze di Jensen-Hadamard per funzioni convesse. In particolare, diamo qualche miglioramento di queste diseguaglianze. Alcune applicazioni naturali ed ulteriori estensioni sono date.

     

关于可达矩阵的求法探讨

在《离散数学》、《图论》课程中 ,用矩阵表示图时 ,涉及到一类重要的矩阵——可达矩阵 ,它是判别图中任意两点是否有通路的重要手段 ,也是求强分图的重要方法 ,但是可达矩阵的求法比较复杂 .本文针对这一问题 ,对可达矩阵的求法进行了改进 ,提出了一种简单可行的算法 .     

q-positive definiteness and related operators

We are in progress of extending the family of ‘q-deformed operators’ considered in the previous papers by joining to them q-subnormal as well as q-formally subnormal ones. It turns out that q-positive definiteness, a notion generalizing Halmos' standard positive definiteness of bounded subnormal operators, is likewise central for our new scheme.     

On acoustic and electric waves diffraction in piezoelectric medium by a permeable half-plane crack

The problem of electric and acoustic waves diffraction by a half-plane crack in a transversal isotropic piezoelectric medium is investigated. The crack is assumed to be electric permeable and free of tractions. The so-called “quasi-hyperbolic approximation” [15] is adopted. Applying Laplace transformations and Wiener–Hopf technique a closed form solution is obtained. By the means of Cagniard–de Hoop method a detailed dynamic full electroacoustic wavefield’s investigation is conducted. Mode conversion between electric and acoustic waves, effect of electroacoustic head wave, Bleustein–Gulyaev surface wave and the wavefield structure depending on the type of the incident wave (acoustic or electric) and its angle of incidence are analyzed in details. The dynamic field intensity factors at the crack tip depending on the angle of incidence and on time are derived explicitly. Numerical analysis is presented.     

Further generalization of the Dubovicki-Milutin theorem

Combining the results of Walczak (Ref. 1) and Lasiecka (Refs. 2 and 3), we extend the well-known Dubovicki-Milutin theorem in two directions: (i) We allow for treatment of several equality constraints in optimization problems; and (ii) We assume less regularity of optimization problems by requiring the existence of external cones instead of tangent cones. An example of an optimization problem is also considered.     

Optimal reconstruction of the solution of the wave equation from inaccurate initial data

In the present paper, we consider the problem of the optimal reconstruction of the solution of the wave equation from the approximate values of the Fourier coefficients of the function specifying the initial form of the string. For an operator defined on the weight space of vectors from l ₂, we present the solution of the more general problem of reconstruction from the approximate values of the coordinates of these vectors.     

On the convergence rate and optimization of a numerical method with splitting of boundary conditions for the stokes system in a spherical layer in the axisymmetric case: Modification for thick layers

The convergence rate of a fast-converging second-order accurate iterative method with splitting of boundary conditions constructed by the authors for solving an axisymmetric Dirichlet boundary value problem for the Stokes system in a spherical gap is studied numerically. For R/r exceeding about 30, where r and R are the radii of the inner and outer boundary spheres, it is established that the convergence rate of the method is lower (and considerably lower for large R/r) than the convergence rate of its differential version. For this reason, a really simpler, more slowly converging modification of the original method is constructed on the differential level and a finite-element implementation of this modification is built. Numerical experiments have revealed that this modification has the same convergence rate as its differential counterpart for R/r of up to 5 × 10³. When the multigrid method is used to solve the split and auxiliary boundary value problems arising at iterations, the modification is more efficient than the original method starting from R/r ～ 30 and is considerably more efficient for large values of R/r. It is also established that the convergence rates of both methods depend little on the stretching coefficient η of circularly rectangular mesh cells in a range of η that is well sufficient for effective use of the multigrid method for arbitrary values of R/r smaller than ～ 5 × 10³.     

凸域内弦的平均长度

本文研究了凸域内弦的平均长度.通过广义支持函数与凸域的弦幂积分,建立了凸域内弦的平均长度的一般公式,并用此公式得出了圆域和矩形域内弦的平均长度.     

Numerical solution of the linear inverse problem for the Euler-Darboux equation

An inverse problem of the reconstruction of the right-hand side of the Euler-Darboux equation is studied. This problem is equivalent to the Volterra integral equation of the third kind with the operator of multiplication by a smooth nonincreasing function. Numerical solution of this problem is constructed using an integral representation of the solution of the inverse problem, the regularization method, and the method of quadratures. The convergence and stability of the numerical method is proved.     

具功能反应的三种群捕食者——食饵系统数学模型稳定性的研究

[1]中提出了具有功能反应的三种群捕食者——食饵系统的两种模型(Ⅰ)、(Ⅱ),并给出了局部渐近稳定的条件。本文研究了另外两种模型(Ⅲ)和(Ⅳ)的局部渐近稳定的条件。