Joint Similarities and Parameterizations for Dilations of Dual g -frame Pairs in Hilbert Spaces

In this paper, we give an operator parameterization for the set of dilations of a given pair of dual g-frames and the set of dilations of pairs of dual g-frames of a given g-frame. In particular, for the dilations of a given pair of dual g-frames, we introduce the concept of joint complementary g-frames and prove that the joint complementary g-frames of a pair of dual g-frames are unique in the sense of joint similarity, which then helps to obtain a sufficient condition such that the complementary g-frames of a g-frame are unique in the sense of similarity and show that the set of dilations of a given dual g-frame pair are parameterized by a set of invertible diagonal operators. For the dilations of pairs of dual g-frames, we prove that the set of dilations of pairs of dual g-frames are parameterized by a set of invertible upper triangular operators.     

Optimization of axisymmetric membrane shells

Problems of the joint optimization of the shape and distribution along the meridian of the thickness of membrane shells of revolution under the action of axisymmetric loads are considered, taking account of the constraints concerning the strength of the shell and the volume of its cavity. General formulations of problems of the optimal design of shells of revolution are given and the optimal shape of a shell and the corresponding thickness distribution are investigated. Results of the exact solution of problems of the optimal design of closed shells of revolution when there is an internal pressure are presented. The simultaneous introduction of two control functions, describing the shape of the shell and the distribution of its thickness, not only ensures a substantial reduction in the mass of a shell but also leads to significant mathematical simplifications, which enable the solution of the optimization problem being considered to be obtained in an analytical form.     

区间数的绝对值与区间值函数的极限

研究了区间数的绝对值和区间值函数的极限问题.首先,讨论了区间数的H-差的性质,得到了H-差的两个运算法则;然后,给出了区间数的绝对值概念,并讨论了区间数绝对值的性质;最后,借助区间数的H-差和绝对值的概念,建立了区间值函数极限概念的一种新的表达方式,给出了极限存在的充分必要条件,证明了极限值的唯一性及对加法运算和数乘运算的封闭性.     

Estimate of the Order of Growth of a Class of Entire Functions on the Real Axis

For a class of entire functions we study the problem of estimation of the order of growth of functions on the real axis. This problem is important for the justification of the integral representation of bounded solutions to certain partial differential equations considered in other papers of the authors. In order to obtain an estimate of the order of growth of a function on the real axis, we use the method of differential equations. The method is based, on one hand, on the construction of a system of first-order ordinary differential equations whose solution is a vector function of traces of function and its derivatives on the real axis. On the other hand, under the respective change of variables in the system of equations, we obtain an estimate of the solution to the system of equations for a large positive values of the argument. The obtained estimate is non-trivial and shows the way a complex parameter of a power series affects the order of growth of a function.     

Configurations of ovals

We survey the known hyperovals in . We then survey the relationship of the study of configurations of ovals in called augmented fans to that of ovoids of , flocks of the quadratic cone of , flocks of a translation oval cone of and spreads of certain generalised quadrangles of order. We also consider the configurations of ovals in called herds and their relationship with flocks of the quadratic cone of.     

Geometry of a five link mechanism with two degrees of freedom

The paper is devoted to the solution of straight and inverse geometrical tasks of five link mechanism with two degrees of freedom. The solution of the mentioned problem is very important in order to determine kinematic parameters of actuators. The problem can be divided into two parts. The first part is considered when we are given the coordinates of the output link of the mechanism and the necessity arises of determining the angles of rotation of actuators. On the other hand, it is very important to determine the position of the output link when the angles of rotation of the actuators are known. Here we consider that the mechanism is composed only of five classes of rotating kinematic pairs and the actuators are situated at the junctions of frames and links of the examined mechanism. The solution of the said problem is based on utilization of homogenous coordinates. On the basis of the obtained equations of motion, one can calculate the trajectories of motion of the output link as well angles of rotation of the actuators by taking into consideration preliminary given kinematic parameters of the mechanism. Here we also obtain equations for calculating values of speed and acceleration of the links of the mechanism. The calculations differ from known methods in simplicity and high performance, which would be useful for programming actuators mounted in the joints of the linkage.     

Complexity Estimation for an Algorithm of Searching for Zero of a Piecewise Linear Convex Function

It is known that the problem of the orthogonal projection of a point to the standard simplex can be reduced to solution of a scalar equation. In this article, the complexity is analyzed of an algorithm of searching for zero of a piecewise linear convex function which is proposed in [30]. The analysis is carried out of the best and worst cases of the input data for the algorithm. To this end, the largest and smallest numbers of iterations of the algorithm are studied as functions of the size of the input data. It is shown that, in the case of equality of elements of the input set, the algorithm performs the smallest number of iterations. In the case of different elements of the input set, the number of iterations is maximal and depends rather weakly on the particular values of the elements of the set. The results of numerical experiments with random input data of large dimension are presented.     

On sums of independent random variables whose distributions belong to the max domain of attraction of max stable laws

In this article we study the max domains of attraction of distributions of sums of independent random variables belonging to the max domains of attraction of max stable laws under linear normalization. These results lead to the study of the max domains of attraction of distributions of products of independent random variables belonging to the max domain of attraction of max stable laws under power normalization.     

Fixed point solutions of generalized equilibrium problems for nonexpansive mappings

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a generalized equilibrium problem in a real Hilbert space. Then, strong convergence of the scheme to a common element of the two sets is proved. As an application, problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem is solved. Moreover, solution is given to the problem of finding a common element of fixed points set of nonexpansive mappings and the set of solutions of a variational inequality problem.     

Viscosity approximation methods for generalized equilibrium problems and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a generalized equilibrium problem (for short, GEP) and the set of fixed points of a nonexpansive mapping in the setting of Hilbert spaces. By using well-known Fan-KKM lemma, we derive the existence and uniqueness of a solution of the auxiliary problem for GEP. On account of this result and Nadler’s theorem, we propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping. Furthermore, it is proven that the sequences generated by this iterative scheme converge strongly to a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping.     

Generalisierung von Kommutationsfunktionen; eine Anwendung im Computermodell einer Lebensversicherungsgesellschaft

Subject of the investigation described in this paper has been: To find the general form (as a function of time elapsed since date of entry) of the variable sum assured to be paid on death or of the variable intensity of payment of a continuous life annuity such, that the calculation of the actual value of the payments can be based on the rise of a finite number of commutation functions. It is shown that the general form of the functions sought for, contains so many parameters which can be freely chosen, that the sums assured represented by functions of time of another type can be approximated very well by appropriate choice of values of the parameters. The last statement is clarified by a demonstration of an approximation of a special type of the variable sum assured payable in case of death playing a role in a forecasting model of a life office in the ordinary branch.     

A criterion for the strong ellipticity of the equilibrium equations of an isotropic non-linearly elastic material

A new criterion for satisfying the condition of strong ellipticity of the equilibrium equations of a homogeneous isotropic elastic medium is obtained, consisting of twelve unconditional and the same number of conditional inequalities. The criterion differs from previously known criteria in the fact that it is more effective for the analytical verification of the condition of strong ellipticity. A number of sufficient criteria of the strong ellipticity of the equations of the non-linear statics of an elastic body is presented. It is shown that the method used in this paper is also effective in the case of an incompressible material. A new constitutive inequality in the non-linear theory of elasticity is proposed that, in particular, ensures the strong ellipticity of the equilibrium equations and the stability of an elastic body in a state of hydrostatic compression. A mechanical interpretation of some implications of the above mentioned inequality is given. Using the criterion obtained, the condition of strong ellipticity is verified for a number of specific models of compressible elastic materials and examples of the violation of this condition are also considered.     

模糊度定义的不一致性及其理论分析

指出模糊度定义的不一致性，并通过实例进行验证；从理论上分析模糊度定义的不一致性产生的原因，指出当模糊度理论在图像处理中应用时模糊度定义的不一致性可能会产生的影响；给出一个比较合理的模糊度定义。文中提出的问题是在工程应用过程中发现的新问题，所做的工作也只是初步的。     

Sharp estimates of integrals in terms of the second modulus of continuity

For a certain class of kernels, the exact constant in the estimate of the integral of the product of two functions in terms of the second modulus of continuity of one of them is obtained. Estimates of best approximations by entire functions of exponential type and by splines in terms of the second modulus of continuity of the second derivative of the approximated function are derived from the results obtained. The constants in these estimates are smaller than the previously known ones.     

Spectral aspects of regularization of the Cauchy problem for a degenerate equation

We study the Cauchy problem for an equation whose generating operator is degenerate on some subset of the coordinate space. To approximate a solution of the degenerate problem by solutions of well-posed problems, we define a class of regularizations of the degenerate operator in terms of conditions on the spectral properties of approximating operators. We show that the behavior (convergence, compactness, and the set of partial limits in some topology) of the sequence of solutions of regularized problems is determined by the deficiency indices of the degenerate operator. We define an approximative solution of the degenerate problem as the limit of the sequence of solutions of regularized problems and analyze the dependence of the approximative solution on the choice of an admissible regularization.     

Optimal control of the rotation of a system of two bodies connected by an elastic rod

Optimal controls of the rotation of a mechanical system consisting of two rigid bodies, joined by an elastic rod, through a specified angle about an axis passing through the centre of mass of one of the bodies are constructed. The problem of the optimal control of the rotation of the system through a given angle with complete suppression of the oscillations of the elastic rod at the minimum of the energy functional of the control moment and the problem of time-optimality for a specified constraint on the energy functional of the control moment are solved.     

Equivalences for blocks of the Weyl groups

We describe the source algebras of the blocks of the Weyl groups of type B and type D in terms of the source algebras of the blocks of the symmetric groups. As a consequence, we show that Puig's conjecture on the finiteness of the number of isomorphism classes of source algebras for blocks of finite groups with a fixed defect group holds for these classes of groups. We also show how certain isomorphisms between subalgebras of block algebras of the symmetric groups can be lifted to block algebras of the Weyl groups of type B.

     

An Analysis of the Decay of Saint-Venant End Effects in a Transversely Isotropic Matrix with an Isotropic Covering

A calculation model is proposed for a numerical analysis of the decay of Saint-Venant end effects in a laminate material of irregular structure to which corresponds a transversely isotropic matrix with an isotropic covering. The elastic properties of the matrix correspond to those of a unidirectional glass-fiber-reinforced plastic. The problem is investigated within the framework of the concept of a representative element of the material. The decay of the end effect in the direction perpendicular to the isotropy plane of a transversely isotropic matrix for the case of symmetric deformation of the material is considered. The source of the end effect is simulated by a piecewise-constant periodic surface load. This load is local for the calculated region and changes within a part of the boundary comparable with the typical size of the structural heterogeneity of the material. The equations of the linear elasticity theory of orthotropic bodies, a model of piecewise-homogeneous media, and quantitative criteria of decay of the end effect are used. Starting from the base system of equations, a discrete problem is constructed and solved. The results of a computing experiment for the choice of a representative element and the results of determining the maximum extent of the end-effect zone for this element are presented.