Antiplane shear of a strip containing a staggered array of rigid line inclusions

Motivated by the increased use of fibre-reinforced materials, we illustrate how the effective elastic modulus of an Isotropic and homogeneous material can be increased by the insertion of rigid inclusions. Specifically, we consider the two-dimensional antiplane shear problem for a strip of material. The strip is reinforced by introducing two sets of ribbon-like, rigid inclusions perpendicular to the faces of the strip. The strip is then subjected to a prescribed uniform displacement difference between its faces, see Figure 1. It should be noted that the problem posed is equivalent to that of the uniform antiplane shear problem for an infinite two-dimensional material containing a staggered array of rigid inclusions (see [1] for a review of antiplane problems in the literature). The problem is reduced in standard fashion [2–6] to a mixed boundary value problem in a rectangular domain, whose closed form solution given in terms of integrals of Weierstrassian Elliptic functions, is obtained via triple sine series techniques. The effective shear modulus of the reinforced strip can now be calculated and compared with the shear modulus of a strip without inclusions. Also obtained are the stress singularity factors at the end tips of the inclusions. Numerical results are presented for several different reinforcement geometries.     

Largest 4-connected components of 3-connected planar triangulations

Let T_n be a 3-connected n-vertex planar triangulation chosen uniformly at random. Then the number of vertices in the largest 4-connected component of T_n is asymptotic to n/2 with probability tending to 1 as n → ∞. It follows that almost all 3-connected triangulations with n vertices have a cycle of length at least n/2 + o(n).     

On fixing elements in matroid minors

LetF be a collection of 3-connected matroids which is (3, 1)-rounded, that is, whenever a 3-connected matroidM has a minor in F ande is an element ofM, thenM has a minor in F whose ground set contains.e. The aim of this note is to prove that, for all sufficiently largen, the collection ofn-element 3-connected matroids having some minor inF is also (3, 1)-rounded.This research was partially supported by the National Science Foundation under Grant No. DMS-8500494.     

On the boundedness of first derivatives for solutions to the Neumann–Laplace problem in a convex domain

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of the regularity of solutions to the Neumann problem on convex polyhedra are given. Bibliography: 27 titles. Dedicated to Nina Uraltseva with affection on the occasion of her birthday Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 105–112.     

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ballB⊂C ⁿ with its relative logarithmic capacity inC ⁿ with respect to the same ballB. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace ofC ⁿ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity onC ⁿ as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂(C ⁿ . Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic functions. This work was partially supported by the programmes PARS MI 07 and AI.MA 180.     

Upper Bounds to the Number of Vertices in a k-Critically n-Connected Graph

 Let k≤n be positive integers. A finite, simple, undirected graph is called k-critically n-connected, or, briefly, an (n,k)-graph, if it is noncomplete and n-connected and the removal of any set X of at most k vertices results in a graph which is not (n−|X|+1)-connected. We present some new results on the number of vertices of an (n,k)-graph, depending on new estimations of the transversal number of a uniform hypergraph with a large independent edge set. Received: April 14, 2000 Final version received: May 8, 2001     

Uniformly Conditioned Bases of Spectral Subspaces

A condition number of an ordered basis of a finite-dimensional normed space is defined in an intrinsic manner. This concept is extended to a sequence of bases of finite-dimensional normed spaces, and is used to determine uniform conditioning of such a sequence. We address the problem of finding a sequence of uniformly conditioned bases of spectral subspaces of operators of the form T _n  = S _n  + U _n , where S _n is a finite-rank operator on a Banach space and U _n is an operator which satisfies an invariance condition with respect to S _n . This problem is reduced to constructing a sequence of uniformly conditioned bases of spectral subspaces of operators on ? ^n×1. The applicability of these considerations in practical as well as theoretical aspects of spectral approximation is pointed out.     

Thermally Stressed State of Bodies with Two Ellipsoidal Inclusions

Solutions are obtained for the interaction of two ellipsoidal inclusions in an elastic isotropic matrix with polynomial external athermal and temperature fields. Perfect mechanical and temperature contact is assumed at the phase interface. A solution to the problem is constructed. When the perturbations in the temperature field and stresses in the matrix owing to one inclusion are re-expanded in a Taylor series about the center of the second inclusion, and vice versa, and a finite number of expansion terms is retained, one obtains a finite system of linear algebraic equations in the unknown constants. The effect of a force free boundary of the half space on the stressed state of a material with a triaxial ellipsoidal inhomogeneity (inclusion) is investigated for uniform heating. Here it was assumed that the elastic properties of the inclusions and matrix are the same, but the coefficients of thermal expansion of the phases differ. Studies are made of the way the stress perturbations in the matrix increase and the of the deviation from a uniform stressed state inside an inclusion as it approaches the force free boundary.     

The implicit function theorem in a non-Archimedean setting

In this paper, the inverse function theorem and the implicit function theorem in a non-Archimedean setting will be discussed. We denote by N any non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order; and we study the properties of locally uniformly differentiable functions from Nⁿ to N^m. Then we use that concept of local uniform differentiability to formulate and prove the inverse function theorem for functions from Nⁿ to Nⁿ and the implicit function theorem for functions from Nⁿ to N^m with m<n.     

Gradient Estimates for the Perfect and Insulated Conductivity Problems with Multiple Inclusions

In this paper, we study the perfect and the insulated conductivity problems with multiple inclusions imbedded in a bounded domain in ? ⁿ , n ≥ 2. For these two extreme cases of the conductivity problems, the gradients of their solutions may blow up as two inclusions approach each other. We establish the gradient estimates for the perfect conductivity problems and an upper bound of the gradients for the insulated conductivity problems in terms of the distances between any two closely spaced inclusions.     

The Bernstein problem for affine maximal hypersurfaces

In this paper, we prove the validity of the Chern conjecture in affine geometry [18], namely that an affine maximal graph of a smooth, locally uniformly convex function on two dimensional Euclidean space, R ², must be a paraboloid. More generally, we shall consider the n-dimensional case, R ⁿ , showing that the corresponding result holds in higher dimensions provided that a uniform, “strict convexity” condition holds. We also extend the notion of “affine maximal” to non-smooth convex graphs and produce a counterexample showing that the Bernstein result does not hold in this generality for dimension n≥10. Oblatum 16-IV-1999 & 4-XI-1999?Published online: 21 February 2000     

A strongly degenerate system involving an equation of parabolic type and an equation of elliptic type

Abstract

We show the existence of weak solutions in an elliptic region in the self-similar plane to the two-dimensional Riemann problem for the pressure-gradient system of the compressible Euler system. The two-dimensional Riemann problem we study is the interaction of two forward rarefaction waves, which are adjacent to a common vacuum that occupies a sectorial domain of 90 degrees. We assume the origin is on the boundary of the domain. In addition, the domain is open, bounded, and simply connected with a piecewise C ^2,α boundary. We resolve the difficulty that arises from the fact that the origin is on the boundary of the domain.     

A note on the connectivity of 2-regular digraphs

Let D ₂(n) be the probability space of 2-regular digraphs on n vertices with the uniform measure. As n tends to infinity, the probability that the digraphs are 2-connected tends to 1. © 1993 John Wiley & Sons, Inc.     

E-zusammenhängende Räume

The properties of the class ofE-connected spaces are examined. The concept ofE-connectedness was introduced by the author in an earlier article in order to describe the connexion between separation and connectedness. Furthermore the concept is extended to the categoryU of uniform spaces and uniformly continuous maps. Thus a generalization of the concept of uniform connectedness due to MROWKA and PERVIN is obtained. It results that many theorems known from the theory of connected spaces due to HAUSDORFF are also valid forE-connected (topological or uniform) spaces.     

Counting acyclic hypergraphs

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. The number of distinct acyclic uniform hypergraphs withn labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicitformula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.     

The stressed state in an infinite medium having a system of rectilinear thin elastic inclusions with cyclic symmetry

Using the machinery of complex variable theory we study the stressed state in an unbouned medium of cyclically located thin elastic inclusions of finite length. The problem is reduced to solving a system of two singular integro-differential equations. A numerical analysis is carried out for the stress intensity factors in the vicinity of the ends of the inclusions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 87–90.     

Uniform-cost inverse absolute and vertex center location problems with edge length variations on trees

This article considers the inverse absolute and the inverse vertex 1-center location problems with uniform cost coefficients on a tree network T with n+1 vertices. The aim is to change (increase or reduce) the edge lengths at minimum total cost with respect to given modification bounds such that a prespecified vertex s becomes an absolute (or a vertex) 1-center under the new edge lengths. First an O(nlogn) time method for solving the height balancing problem with uniform costs is described. In this problem the height of two given rooted trees is equalized by decreasing the height of one tree and increasing the height of the second rooted tree at minimum cost. Using this result a combinatorial O(nlogn) time algorithm is designed for the uniform-cost inverse absolute 1-center location problem on tree T. Finally, the uniform-cost inverse vertex 1-center location problem on T is investigated. It is shown that the problem can be solved in O(nlogn) time if all modified edge lengths remain positive. Dropping this condition, the general model can be solved in O(r_vnlogn) time where the parameter r_v is bounded by ⌈n/2⌉. This corrects an earlier result of Yang and Zhang.     

Connected,locally 2-connected,K1,3-free graphs are panconnected

A graph G is locally n-connected, n ≥ 1, if the subgraph induced by the neighborhood of each vertex is n-connected. We prove that every connected, locally 2-connected graph containing no induced subgraph isomorphic to K_1,3 is panconnected.     

Inverse sorting problem by minimizing the total weighted number of changes and partial inverse sorting problems

In this paper, we consider two types of inverse sorting problems. The first type is an inverse sorting problem by minimizing the total weighted number of changes with bound constraints. We present an O(n ²) time algorithm to solve the problem. The second type is a partial inverse sorting problem and a variant of the partial inverse sorting problem. We show that both the partial inverse sorting problem and the variant can be solved by a combination of a sorting problem and an inverse sorting problem. Supported by the Hong Kong Universities Grant Council (CERG CITYU 103105) and the National Key Research and Development Program of China (2002CB312004) and the National Natural Science Foundation of China (700221001, 70425004).