Extremum principles for a general class of saddle functionals

The usual notion of a saddle functional in the calculus of variations assumes a vex/concave structure over the product space of two inner product spaces. Here the ideas extended to include some convexity in both spaces whilst still retaining an overall saddle property. Dual extremum principles are established for these functionals. Examples include periodic solutions of Duffing's equation, an iterative scheme and a pair of simultaneous partial differential equations which arise in magnetohydrodynamics.     

Generalized concavity and duality with a square root term

Duality results for a class of nondifferentiable mathematical programming problems are given. These results allow for the weakening of the usual convexity conditions required for duality to hold. A pair of symmetric and self dual nondifferentiable programs under weaker convexity conditions are also given. A subgradient symmetric duality is proposed and its limitations discussed. Finally, a pair of nondifferentiable mathematical programs containing arbitrary norms is presented.     

半无限带状各向异性材料的应力效应衰减估计

对带端面负载的半无限带状各向异性弹性材料建立了Saint-Venant原理.应用能量方法确立了应力效应衰减率的下界,得出了用各向异性材料的弹性常量表示的估计显式.最后,为了说明所得结果,对带端面负载、纤维偏离主轴的石墨-环氧树脂的衰减率给出数值实例.     

Catalytical kinetics of butterfly martensite

The stress fields around the first butterfly martensite in an Fe-Ni based alloy were calculated. The calculation results inducate that there are tensile stress fields near the convexity and the two ends of a butterfly martensite. There is a compressive stress field near the concavity. New butterfly martensite is likely induced not only near the convexity but also probably near the concavity. The butterfly martensites induced in tensile stress fields are bigger than those in compressive stress fields. The second butterfly martensite is likely induced near the convexity within 2πm region. All these results agree with the experimental observation.     

周期裂纹削弱的无限长板条的应力分析

作出了周期裂纹削弱的无限长板条的应力分析．假设这些裂纹均在水平位置,又板条承受 y方向的拉伸力p．此时边值问题归结为一个复杂混合边值问题．发现,对此问题言,特征展开变分原理方法 ( eigenfunction expansion variational method,简称为EEVM)是非常有效的．研究了裂纹端的应力强度因子和T-应力．从拉伸力作用下的弹性变形考虑,开裂板条可等价于一不开裂的正交异性板条．还分析了等价正交异性板条的弹性性质．最后给出了算例和数值结果．     

条状功能梯度材料中偏心裂纹对反平面简谐波的散射问题

利用Schmidt方法研究了条状功能梯度材料中偏心裂纹对反平面简谐波的散射问题,裂纹垂直于条状功能梯度材料的边界．通过Fourier变换,问题可以转换为对一对未知变量是裂纹表面位移差的对偶积分方程求解．为了求解对偶积分方程,把裂纹表面的位移差展开为Jacobi多项式级数形式,进而得到了功能梯度参数、裂纹位置以及入射波频率对应力强度因子影响的规律．     

Existence of solutions in variational relation problems without convexity

Two main existence conditions for solutions of variational relation problems are established without convexity. The first one is based on a finite solvability property and the second one on generalized KKM mappings. These conditions unify and strengthen several existing results in the literature on the topic. A model of satisficing process by rejection is considered which gives an economic interpretation of the introduced concepts.     

Ball-Polyhedra

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra.     

Convexities related to path properties on graphs

A feasible family of paths in a connected graph G is a family that contains at least one path between any pair of vertices in G. Any feasible path family defines a convexity on G. Well-known instances are: the geodesics, the induced paths, and all paths. We propose a more general approach for such ‘path properties’. We survey a number of results from this perspective, and present a number of new results. We focus on the behaviour of such convexities on the Cartesian product of graphs and on the classical convexity invariants, such as the Carathéodory, Helly and Radon numbers in relation with graph invariants, such as the clique number and other graph properties.     

Anti-plane analysis of an orthotropic strip weakened by several moving cracks

In this paper several finite cracks with constant length (Yoffe-type crack) propagating in an orthotropic strip were studied. The distributed dislocation technique is used to carry out stress analysis in an orthotropic strip containing moving cracks under anti-plane loading. The solution of a moving screw dislocation is obtained in an orthotropic strip by means of Fourier transform method. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a strip weakened by moving cracks. Finally several examples are solved and the numerical results for the stress intensity factor are obtained. The influences of the geometric parameters, the thickness of the orthotropic strip, the crack size and speed have significant effects on the stress intensity factors of crack tips which are displayed graphically.     

Necessary and Sufficient Convexity Conditions for the Ranges of Vector Measures

The absence of atoms in Lyapunov's Convexity Theorem is a sufficient, but not a necessary condition for the convexity of the range of an n - dimensional vector measure. In this paper algebraic and topological convexity conditions generalizing Lyapunov's Theorem are developed which are sufficient and necessary as well. From these results the converse of Lyapunov's Theorem is derived in the form of a nonconvexity statement which gives insight into the geometric structure of the ranges of vector measures with atoms. Further, a characterization of the one-dimensional faces of a zonoid Z_μ, is given with respect to the generating spherical Borel measure μ. As an application, it is shown that the absence of μ - atoms is a necessary and sufficient convexity condition for the range of the indefinite integral ∫ x dμ where x denotes the identical function on S^n-1.     

Some geometric properties of integral modules

Day's characterization of those spaces 1^p(X_i) which are uniformly convex, in terms of the moduli of convexity of the X_i, is generalized for arbitrary integral modules on measure spaces (K,m) and simplified when m is finite. For this latter purpose a lemma on the moduli of convexity and of smoothness is proved which incidentally gives a further necessary condition for the existence of integral modules in given direct integrals. Further the notions of strict convexity and smoothness of an integral module are related to those of its components.     

Moment problem formulation of the Schrödinger equation

We show that it is possible to project out in an exact manner the lowest eigenstate of Schrödinger equations. Taking into account the nodeless property of the lowest eigenstate one can replace the full Schrödinger equation by a moment problem whose measure is the eigenstate itself. The infinite set of positivity inequalities linked to this moment problem provides a framework which allows to compute sequences of upper and lower bounds to the unknown eigenvalue and eigenfunction.The effective computation is based on deep convexity properties embedded in the set of hierarchical inequalities associated to this moment problem. The convexity allows to get the bounds through linear programming. We illustrate the method with simple one dimensional problems.Laboratoire de la Direction des Sciences de la Matière du Commissariat à l'Energie Atomique.     

A note on the convexity of ruin probabilities

Conditions for the convexity of compound geometric tails and compound geometric convolution tails are established. The results are then applied to analyze the convexity of the ruin probability and the Laplace transform of the time to ruin in the classical compound Poisson risk model with and without diffusion. An application to an optimization problem is given.     

A generalized section theorem and a minimax inequality for a vector-valued mapping 1

A generalized Fan's section theorem has proposed by replacing convexity assumptions with merely topological properties. A generalized reformulation of Browder's fixed point theorem has derived. The Minimax Inequalities for vector-valued mapping in an ordered Banach space have established without the convexity and with convexity, respectively.     

Generalization of some duality theorems in nonlinear programming

Pseudoconvexity of a function on one set with respect to some other set is defined and duality theorems are proved for nonlinear programming problems by assuming a certain kind of convexity property for a particular linear combination of functions involved in the problem rather than assuming the convexity property for the individual functions as is usually done. This approach generalizes some of the well-known duality theorems and gives some additional strict converse duality theorems which are not comparable with the earlier duality results of this type. Further it is shown that the duality theory for nonlinear fractional programming problems follows as a particular case of the results established here.     

A lack of Ricci bounds for the entropic measure on Wasserstein space over the interval

We examine the entropic measure, recently constructed by von Renesse and Sturm, a measure over the metric space of probability measures on the unit interval equipped with the 2-Wasserstein distance. We show that equipped with this measure, Wasserstein space over the interval does not admit generalized Ricci lower bounds in the entropic displacement convexity sense of Lott–Villani–Sturm. We discuss why this is contrary to what one might expect from heuristic considerations.     

外卷边槽钢辊弯成型过程数值模拟

拓展了结构分析有限条法,基于固体力学有限形变理论的更新拉格朗日法(U.L.法),建立了弹塑性大变形样条有限条法,并将其用于模拟外卷边槽钢辊弯成型过程,分析了带材的变形特点,获得了变形带材的三维位移场、应变场和应力场。算例表明,变形带材的角部出现横向压应变峰值,外卷边轧辊前部位出现纵向拉应变峰值。另外,带材的横向变形是主要的,纵向变形较小。     

A variational method for the solution of biharmonic problems for a rectangular domain

We develop a variational method for the solution of biharmonic problems for a rectangular domain where, at one pair of its opposite sides, the unknown function and its normal derivative take zero values, and, at the other pair, certain inhomogeneous conditions are valid. The cases of semiinfinite and finite domain are considered. The method is based on the minimization of a quadratic functional determining the deviation of the solution from the given inhomogeneous conditions in the norm of L ₂. To solve this variational problem, we apply the expansion of the solution in the systems of complex biharmonic functions (the so-called Papkovich homogeneous solutions), which satisfy identically the given homogeneous conditions at the pair of opposite sides of the rectangle. This representation of the solution is somewhat different from that proposed earlier [V. F. Chekurin, “A variational method for the solution of direct and inverse problems of the theory of elasticity for a semiinfinite strip,” Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela, No. 2, 58–70 (1999)]. We consider several variants of inhomogeneous boundary conditions arising in the problems of the two-dimensional theory of elasticity. Finally, we give an example of applying the proposed method for the determination of stress distributions in a rectangular area one of whose sides is rigidly fastened and the opposite one is subjected to the action of normal forces. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 88–98, January–March, 2008.     

Antiplane elastodynamic analysis of a strip weakened by multiple defects

The method of images is utilized to derive the solution of a screw dislocation under time-harmonic conditions for an elastic strip from the solution of infinite planes. The displacement and stress components are obtained for a strip under concentrated antiplane, time-harmonic traction. The dislocation solution is employed to formulate integral equation for a strip weakened by cracks and cavities. The effects of load frequency and crack orientation on the stress intensity factors are studied.