Numerical aspects in the SGBEM solution of softening cohesive interface problems

Some numerical aspects of a new symmetric Galerkin boundary integral formulation presented in [A. Salvadori, A symmetric boundary integral formulation for cohesive interface problems, Comput. Mech. 32 (2003) 381–391] connected with a like arc-length technique for softening cohesive interface problems introduced in [F. Freddi, Cohesive interface analysis via boundary integral equations, Ph.D. thesis, University of Bologna, Italy, 2004] are here investigated. Further, if the problem is invariant with respect to a finite group of congruences of the Euclidean space , we propose to reduce the computational cost of the symmetric Galerkin boundary element method (SGBEM) matrix evaluation and linear system solution using suitable restriction matrices strictly related to a system of irreducible matrix representation of . Some numerical simulations are presented and discussed.     

Embedding Into Almost Left Factorizable Restriction Semigroups

The notion of almost left factorizability and the results on almost left factorizable weakly ample semigroups, due to Gomes and the author, are adapted for restriction semigroups. The main result of the paper is that each restriction semigroup is embeddable into an almost left factorizable restriction semigroup. This generalizes a fundamental result of the structure theory of inverse semigroups.     

Basic Homogenization Results for a Biconnected ε-Periodic Structure

We present, in a bounded domain, a model of an l -periodic structure composed of two phases, both being connected but only one reaching the boundary of the domain, avoiding in this way the local type convergences of the homogenization process. In this framework we revise some basic tools of the homogenization theory in porous media: the extension and the restriction operators, the Ne ) as inequality. Moreover, we obtain some compacity properties which reduce the proof of the pressure type convergences from the homogenization of fluid flows through porous media to the expected procedure of a priori estimations and two-scale convergences. As all the properties can be proved without much technical difficulties, avoiding annoying hypotheses and the use of Kolmogorov's criterion of compacity, the present structure seems one of the most convenient realistic models of porous media that can be studied with the methods of homogenization.     

Restricting Riesz-logarithmic-Besov potentials

This paper studies the restriction properties of the Riesz-logarithmic-Besov potential. Moreover, applications to the regularity of the solutions to the fractional Laplace equation with measure data are given.     

Varieties of restriction semigroups and varieties of categories

The variety of restriction semigroups may be most simply described as that generated from inverse semigroups (S, ·, ^?1) by forgetting the inverse operation and retaining the two operations x⁺ = xx^?1 and x* = x^?1x. The subvariety B of strict restriction semigroups is that generated by the Brandt semigroups. At the top of its lattice of subvarieties are the two intervals [B₂, B₂M = B] and [B₀, B₀M]. Here, B₂ and B₀ are, respectively, generated by the five-element Brandt semigroup and that obtained by removing one of its nonidempotents. The other two varieties are their joins with the variety of all monoids. It is shown here that the interval [B₂, B] is isomorphic to the lattice of varieties of categories, as introduced by Tilson in a seminal paper on this topic. Important concepts, such as the local and global varieties associated with monoids, are readily identified under this isomorphism. Two of Tilson's major theorems have natural interpretations and application to the interval [B₂, B] and, with modification, to the interval [B₀, B₀M] that lies below it. Further exploration may lead to applications in the reverse direction.     

Sharp spectral multipliers for operators satisfying generalized Gaussian estimates

Let L   be a non-negative self-adjoint operator acting on L²(X)$L^2(X)$ where X is a space of homogeneous type. Assume that L   generates a holomorphic semigroup e^−tL$e^{-tL}$ which satisfies generalized m-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein–Tomas type estimates. These results are applicable to spectral multipliers for a large class of operators including m-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals.     

Generalized Jacobian for Functions with Infinite Dimensional Range and Domain

In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon–Nikodym property, Clarke’s generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the β-compactness, the β-upper semicontinuity, and a mean-value theorem. A connection with known notions is provided and chain rules are proved using key results developed. This included the vectorization and restriction theorem, and the extension theorem. Therefore, the generalized Jacobian introduced in this paper is proved to enjoy all the properties required of a derivative like-set. Research of the first author is supported by the Hungarian Scientific Research Fund (OKTA) under grant K62316. Research of the second author is supported by the National Science Foundation under grant DMS-0306260.