MinMaxDM distributions for an analysis of the tensile strength of a unidirectional composite

An analysis of the tensile strength of some fiber or fiber bundle specimens is presented. The specimens are modeled as chains of links consisting of longitudinal elements (LEs) with different cumulative distribution functions of strength, corresponding to the presence and absence of defects. Each link is considered as a system of parallel LEs a part of which can have defects. In the simplest case, the strength of defective elements is assumed equal to zero. The strength of a link is determined by the maximum average stress the link can sustain under a growing load. To calculate the stress, the randomized Daniels model or the theory of Markov chains is used. The strength of specimens is determined by the minimum strength of links. The concept of MinMaxDM family of distribution functions is introduced. A numerical example of processing experimental results for a monolayer of carbon bundles is presented.     

三维四向编织复合材料刚度和强度的理论预测

以单胞模型为基础,将三维四向编织复合材料中相同走向的纤维束视为单向复合材料,利用桥联模型确定了单向复合材料的柔度矩阵,再将具有不同材料主向的单向复合材料的刚度矩阵通过体积平均,得到了三维四向编织复合材料的总体刚度矩阵,从而得到其工程弹性常数．然后,以单向复合材料为基础,基于等应变假设和桥联模型,确定出材料内各组分（纤维束和基体）的细观应力分布,且对纤维束采用Hoffman失效准则,对基体采用Mises失效准则,预报了三维四向编织复合材料的拉伸强度．     

Minimizing the mass of cylindrical shells formed from a composite material with an elastic filler and designed for strength and stability under a combined loading

The mass of a multilayer cylindrical shell, formed from a composite material with an elastic filler and designed for strength and stability under the combined action of axial compression and external pressure, is minimized. The problem is formulated as one of nonlinear programming and is solved by Rossen's method of projection gradients. The strength of the material is established from analysis of the strength of the layers making up the entire bundle. Failure of an individual layer is determined from Malmeister's criterion. The structure of a shell with different external loads and the dependence of minimal mass on the stiffness of the filler and on the volume coefficient of reinforcement are investigated in numerous examples.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. K. Preikshas Shyaulyaisk Pedagogical Institute. Translated from Mekhanika Polimerov, No. 2, pp. 289–297, March–April, 1976.     

A bundle view of boundary-value problems: generalizing the Gardner-Jones bundle

Holomorphic families of linear ordinary differential equations on a finite interval with prescribed parameter-dependent boundary conditions are considered from a geometrical viewpoint. The Gardner-Jones bundle, which was introduced for linearized reaction-diffusion equations, is generalized and applied to this abstract class of λ-dependent boundary-value problems, where λ is a complex eigenvalue parameter. The fundamental analytical object of such boundary-value problems (BVPs) is the characteristic determinant, and it is proved that any characteristic determinant on a Jordan curve can be characterized geometrically as the determinant of a transition function associated with the Gardner-Jones bundle. The topology of the bundle, represented by the Chern number, then yields precise information about the number of eigenvalues in a prescribed subset of the complex λ-plane. This result shows that the Gardner-Jones bundle is an intrinsic geometric property of such λ-dependent BVPs. The bundle framework is applied to examples from hydrodynamic stability theory and the linearized complex Ginzburg-Landau equation.     

Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations

We introduce a proximal bundle method for the numerical minimization of a nonsmooth difference-of-convex (DC) function. Exploiting some classic ideas coming from cutting-plane approaches for the convex case, we iteratively build two separate piecewise-affine approximations of the component functions, grouping the corresponding information in two separate bundles. In the bundle of the first component, only information related to points close to the current iterate are maintained, while the second bundle only refers to a global model of the corresponding component function. We combine the two convex piecewise-affine approximations, and generate a DC piecewise-affine model, which can also be seen as the pointwise maximum of several concave piecewise-affine functions. Such a nonconvex model is locally approximated by means of an auxiliary quadratic program, whose solution is used to certify approximate criticality or to generate a descent search-direction, along with a predicted reduction, that is next explored in a line-search setting. To improve the approximation properties at points that are far from the current iterate a supplementary quadratic program is also introduced to generate an alternative more promising search-direction. We discuss the main convergence issues of the line-search based proximal bundle method, and provide computational results on a set of academic benchmark test problems.     

Zero-pole interpolation for meromophic matrix functions on an algebraic curve and transfer functions of 2D systems

We formulate and solve the problem of constructing a meromorphic bundle map over a compact Riemann surface X having a prescribed zero-pole structure (including directional information). The output bundle together with the zero-pole data is prespecified while the input bundle and the bundle map are to be determined. The Riemann surface X is assumed to be (birationally) embedded as an irreducible algebraic curve in ² and both input and output bundles are assumed to be equal to the kernels of determinantal representations for X. In this setting the solution can be found as the joint transfer function of a Livsic-Kravitsky two-operator commutative vessel (2D input-output dynamical system). Also developed is the basic theory of two-operator commutative vessels and the correct analogue of the transfer function for such a system (a meromorphic bundle map between input and output bundles defined over an algebraic curve associated with the vessel) together with a state space realization, a Mittag-Leffler type interpolation theorem and the state space similarity theorem for such bundle mappings. A more abstract version of the zero-pole interpolation problem is also presented.     

Natural quotients on split supercotangent bundles and their canonical supersymplectic structures

The Batchelor model of the supercotangent bundle of a given base supermanifold is studied. Under the assumption that the supercotangent bundle splits, two different fibrations over the given base can be globally defined. The total spaces of these fibrations are in turn quotient supermanifolds of the supercotangent bundle, and each of them is equipped with a supersymplectic structure. Their corresponding supersymplectic $\text{2}$-forms are actually exact, and $\text{Z}{}_2$-homogeneous of different degrees. The $\text{Z}{}_2$-homogeneous $\text{1}$-forms from which they come from are natural with respect to Batchelor trivializations. Each of these $\text{1}$-forms can be pulled back to the supercotangent bundle via the quotient maps, and can be added together in the supercotangent bundle to produce a nonhomogeneous $\text{1}$-form there. Such a $\text{1}$-form in the supercotangent bundle is canonical; it is characterized by the fact that the pullback of it under any $\text{1}$-form on the base supermanifold yields the same $\text{1}$-form on the base. The exterior derivative of this canonical $\text{1}$-form is degenerate. Its radical produces an example of an involutive subsheaf, which is not integrable. This phenomenon is explained at the light of Frobenius Theorem for supermanifolds. The radicals of its homogeneous components, on the other hand, taken separately, do produce two globally defined foliations on the supercotangent bundle, and the corresponding spaces of leaves are precisely the two quotients of the supercotangent bundle we started with     

Calibrated Subbundles in Noncompact Manifolds of Special Holonomy

This paper is a continuation of Math. Res. Lett. 12 (2005), 493–512. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of Sⁿ by looking at the conormal bundle of appropriate submanifolds of Sⁿ. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey–Lawson for submanifolds in Rⁿ in their pioneering paper, Acta Math. 148 (1982), 47–157. We also construct calibrated submanifolds in complete metrics with special holonomy G₂ and Spin(7) discovered by Bryant and Salamon (Duke Math. J. 58 (1989), 829–850) on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy. Mathematics Subject Classification (2000): 53-XX, 58-XX.     

A GRASP + ILP-based metaheuristic for the capacitated location-routing problem

In this paper we present a three-phase heuristic for the Capacitated Location-Routing Problem. In the first stage, we apply a GRASP followed by local search procedures to construct a bundle of solutions. In the second stage, an integer-linear program (ILP) is solved taking as input the different routes belonging to the solutions of the bundle, with the objective of constructing a new solution as a combination of these routes. In the third and final stage, the same ILP is iteratively solved by column generation to improve the solutions found during the first two stages. The last two stages are based on a new model, the location-reallocation model, which generalizes the capacitated facility location problem and the reallocation model by simultaneously locating facilities and reallocating customers to routes assigned to these facilities. Extensive computational experiments show that our method is competitive with the other heuristics found in the literature, yielding the tightest average gaps on several sets of instances and being able to improve the best known feasible solutions for some of them.     

Exotic characteristic classes of quaternionic bundles

For aC ^∞ quaternionic vector bundle, the odd-dimensional real Chern classes vanish, and this allows for a construction of secondary (exotic) characteristic classes associated with a pair of quaternionic structures of a given complex vector bundle. This construction is then applied to obtain exotic characteristic classes associated with an automorphismβ of the holomorphic tangent bundle of a Kähler manifold. These results are the complex analoga of those given for the higher order Maslov classes in [V2].     

How the Curvature Generates the Holonomy of a Connection in an Arbitrary Fibre Bundle

For an arbitrary fibre bundle with a connection, the holonomy group of which is a Lie transformation group, it is shown how the parallel displacement along a null-homotopic loop can be obtained from the curvature by integration. The result also sheds some new light on the situation for vector bundles and principal fibre bundles. The Theorem of Ambrose–Singer is derived as a corollary in our general setting. The curvature of the connection is interpreted as a differential 2-form with values in the holonomy algebra bundle, the elements of which are special vector fields on the fibres of the given bundle. Received: May 16, 2006; Revised: July 30, 2006; Accepted: August 2, 2006     

Direct image of parabolic line bundles

Given a vector bundle E, on an irreducible projective variety X, we give a necessary and sufficient criterion for E to be a direct image of a line bundle under a surjective étale morphism. The criterion in question is the existence of a Cartan subalgebra bundle of the endomorphism bundle $\text{End}(E)$. As a corollary, a criterion is obtained for E to be the direct image of the structure sheaf under an étale morphism. The direct image of a parabolic line bundle under any ramified covering map has a natural parabolic structure. Given a parabolic vector bundle, we give a similar criterion for it to be the direct image of a parabolic line bundle under a ramified covering map.     

Geometric methods for construction of quantum gates

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing a universal set of gates for quantum computations: the well-known result that the set of all one-bit gates together with almost any one two-bit gate is universal is considered from the control theory viewpoint. Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold, and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action, and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over the Grassmannian and its relation with the Berry phase is considered and investigated for the trajectories of Hamiltonian dynamical systems. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 44, Quantum Computing, 2007.     

Quasilinear resolutions of non-linear equations

By analogy with the linear vector bundle case, a non-linear partial differential equation on a manifold can be defined as a fibred submanifold R_k of a k-jet bundle. By observing that under natural conditions the first prolongation gives rise to a vector bundle over R_k, (that is, a quasilinear equation), techniques of the linear case are adapted to establish conditions for the formal integrability of the equation.     

Flag bundles,Chern classes and natural lifts

Summary The (total) Chern class of any incomplete flag bundle is determined, thus generalising those found for the complete flag bundle in [3] and for the projective bundle in [1, I] and [6]. We also generalise a formula of D. B. Scott in [4] to any incomplete flag bundle. Entrata in Redazione il 7 settembre 1976.     

Expansion in eigenfunctions of the Sturm-Liouville problem on a bundle graph

In this paper we consider the applicability of the Fourier method for partial differential equations on spatial grids (we choose a bundle graph as a model). This leads to an important problem, namely, to the expansion of a given function in eigenfunctions of the corresponding Sturm-Liouville problem on a grid. We study a model problem which describes a symmetric case, when one considers physically identical one-dimensional continuums on the bundle graph. Such problems arise, for example, in the modeling of oscillating processes of an elastic mast with supporting elastic ties.     

Trigonal gorenstein curves and special linear systems

LetY be a Gorenstein trigonal curve withg:=pa(Y)≥0. Here we study the theory of special linear systems onY, extending the classical case of a smoothY given by Maroni in 1946. As in the classical case, to study it we use the minimal degree surface scroll containing the canonical model ofY. The answer is different if the degree 3 pencil onY is associated to a line bundle or not. We also give the easier case of special linear series on hyperelliptic curves. The unique hyperelliptic curve of genusg which is not Gorenstein has no special spanned line bundle.