Epsilon‐stable quasi‐static brittle fracture evolution

We introduce a new definition of stability, ε‐stability, that implies local minimality and is robust enough for passing from discrete‐time to continuous‐time quasi‐static evolutions, even with very irregular energies. We use this to give the first existence result for quasi‐static crack evolutions that both predicts crack paths and produces states that are local minimizers at every time, but not necessarily global minimizers. The key ingredient in our model is the physically reasonable property, absent in global minimization models, that whenever there is a jump in time from one state to another, there must be a continuous path from the earlier state to the later along which the energy is almost decreasing. It follows that these evolutions are much closer to satisfying Griffith's criterion for crack growth than are solutions based on global minimization, and initiation is more physical than in global minimization models. © 2009 Wiley Periodicals, Inc.     

A linearized heat‐conducting fluid and its quasi‐static approximation

A linearized flow of a compressible inviscid heat‐conducting fluid is considered and a comparison is made with its coupled/quasi‐static approximation. Copyright © 2005 John Wiley & Sons, Ltd.     

Existence of solutions for systems of generalized quasi‐equilibrium problems

In this paper, first of all, we consider a generalized game in choice form with 2 constraints and its corresponding equilibrium in choice. We assert new conditions under which the equilibrium in choice exists. As a consequence, we establish the existence of the equilibrium for generalized abstract economies. Then, we apply the obtained theorems to prove the existence of solutions for systems of quasi‐equilibrium problems. We do this by considering new hypotheses for the properties of the involved correspondences. This approach leads us to results which differ a lot from the ones existing in literature.     

Existence and asymptotic behaviour of solutions for a class of quasi‐linear evolution equations with non‐linear damping and source terms

We consider a class of quasi‐linear evolution equations with non‐linear damping and source terms arising from the models of non‐linear viscoelasticity. By a Galerkin approximation scheme combined with the potential well method we prove that when m<p, where m(?0) and p are, respectively, the growth orders of the non‐linear strain terms and the source term, under appropriate conditions, the initial boundary value problem of the above‐mentioned equations admits global weak solutions and the solutions decay to zero as t→∞. Copyright © 2002 John Wiley & Sons, Ltd.     

On the evolution of sharp fronts for the quasi‐geostrophic equation

We consider the problem of the evolution of sharp fronts for the surface quasi‐geostrophic (QG) equation. This problem is the analogue to the vortex patch problem for the two‐dimensional Euler equation. The special interest of the quasi‐geostrophic equation lies in its strong similarities with the three‐dimensional Euler equation, while being a two‐dimen‐sional model. In particular, an analogue of the problem considered here, the evolution of sharp fronts for QG, is the evolution of a vortex line for the three‐dimensional Euler equation. The rigorous derivation of an equation for the evolution of a vortex line is still an open problem. The influence of the singularity appearing in the velocity when using the Biot‐Savart law still needs to be understood. We present two derivations for the evolution of a periodic sharp front. The first one, heuristic, shows the presence of a logarithmic singularity in the velocity, while the second, making use of weak solutions, obtains a rigorous equation for the evolution explaining the influence of that term in the evolution of the curve. Finally, using a Nash‐Moser argument as the main tool, we obtain local existence and uniqueness of a solution for the derived equation in the C^∞ case. © 2004 Wiley Periodicals, Inc.     

A quasi‐static boundary value problem in multi‐surface elastoplasticity: Part 1—Analysis

The quasi‐static evolution of an elastoplastic body with a multi‐surface constitutive law of linear kinematic hardening type allows the modelling of curved stress–strain relations. It generalizes classical small‐strain elastoplasticity from one to various plastic phases. This paper presents the mathematical models and proves existence and uniqueness of the solution of the corresponding initial‐boundary value problem. The analysis involves an explicit estimate for the effective ellipticity constant. Copyright © 2004 John Wiley & Sons, Ltd.     

Non‐isolated quasi‐degrees

We show that non‐isolated from below 2‐c.e. Q ‐degrees are dense in the structure of c.e. Q ‐degrees. We construct a 2‐c.e. Q ‐degree, which can't be isolated from below not only by c.e. Q ‐degrees, but by any Q ‐degree. We also prove that below any c.e. Q ‐degree there is a 2‐c.e. Q ‐degree, which is non‐isolated from below and from above (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of strong minimizers for the Griffith static fracture model in dimension two

We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization to the vectorial situation of the decay estimate by De Giorgi, Carriero, and Leaci. This is based on replacing the coarea formula by a method to approximate $SB{D}^p$ functions with small jump set by Sobolev functions, and is restricted to two dimensions. The other two ingredients will appear in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy $GSB{D}^p$ functions by $SB{V}^p$ ones.     

Weak‐quasi‐Stone algebras

In this paper we shall introduce the variety WQS of weak‐quasi‐Stone algebras as a generalization of the variety QS of quasi‐Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬‐lattices to give a duality for WQS. We prove that a weak‐quasi‐Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the simple and subdirectly irreducible algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A quasi‐static boundary value problem in multi‐surface elastoplasticity: part 2—numerical solution

Multi‐yield elastoplasticity models a material with more than one plastic state and hence allows for refined approximation of irreversible deformations. Aspects of the mathematical modelling and a proof of unique existence of weak solutions can be found in part I of this paper (Math. Models Methods Appl. Sci. 2004). In this part II we establish a canonical time–space discretization of the evolution problem and present various algorithms for the solving really discrete problems. Based on a global Newton–Raphson solver, we carefully study and solve elementwise inner iterations. Numerical examples illustrate the model and its flexibility to allow for refined hysteresis curves. Copyright © 2005 John Wiley & Sons, Ltd.     

Hadwiger's conjecture for quasi‐line graphs

A graph G is a quasi‐line graph if for every vertex v ∈ V(G), the set of neighbors of v in G can be expressed as the union of two cliques. The class of quasi‐line graphs is a proper superset of the class of line graphs. Hadwiger's conjecture states that if a graph G is not t‐colorable then it contains K_{t + 1} as a minor. This conjecture has been proved for line graphs by Reed and Seymour. We extend their result to all quasi‐line graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 17–33, 2008     

Weak quasi‐randomness for uniform hypergraphs

We study quasi‐random properties of k‐uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung‐Graham‐Wilson theorem for quasi‐random graphs. Moreover, let K_k be the complete graph on k vertices and M(k) the line graph of the graph of the k‐dimensional hypercube. We will show that the pair of graphs (K_k,M(k)) has the property that if the number of copies of both K_k and M(k) in another graph G are as expected in the random graph of density d, then G is quasi‐random (in the sense of the Chung‐Graham‐Wilson theorem) with density close to d. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Disjoint quasi‐kernels in digraphs

A quasi‐kernel in a digraph is an independent set of vertices such that any vertex in the digraph can reach some vertex in the set via a directed path of length at most two. Chvátal and Lovász proved that every digraph has a quasi‐kernel. Recently, Gutin et al. raised the question of which digraphs have a pair of disjoint quasi‐kernels. Clearly, a digraph has a pair of disjoint quasi‐kernels cannot contain sinks, that is, vertices of outdegree zero, as each such vertex is necessarily included in a quasi‐kernel. However, there exist digraphs which contain neither sinks nor a pair of disjoint quasi‐kernels. Thus, containing no sinks is not sufficient in general for a digraph to have a pair of disjoint quasi‐kernels. In contrast, we prove that, for several classes of digraphs, the condition of containing no sinks guarantees the existence of a pair of disjoint quasi‐kernels. The classes contain semicomplete multipartite, quasi‐transitive, and locally semicomplete digraphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:251‐260, 2008     

On quasi‐orthogonal cocycles

We introduce the notion of quasi‐orthogonal cocycle. This is motivated in part by the maximal determinant problem for square ‐matrices of size congruent to 2 modulo 4. Quasi‐orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi‐Hadamard groups, relative quasi‐difference sets, and certain partially balanced incomplete block designs, are proved.     

Coloring quasi‐line graphs

A graph G is a quasi‐line graph if for every vertex v, the set of neighbors of v can be expressed as the union of two cliques. The class of quasi‐line graphs is a proper superset of the class of line graphs. A theorem of Shannon's implies that if G is a line graph, then it can be properly colored using no more than 3/2 ω(G) colors, where ω(G) is the size of the largest clique in G. In this article, we extend this result to all quasi‐line graphs. We also show that this bound is tight. © 2006 Wiley Periodicals, Inc. J Graph Theory     

A technique for constructing non‐embeddable quasi‐residual designs

We propose a technique for constructing two infinite families of non‐embeddable quasi‐residual designs as soon as one such design satisfying certain conditions exists. The main tools are generalized Hadamard matrices and balanced generalized weighing matrices. Starting with a specific non‐embeddable quasi‐residual 2‐(27,9,4) design, we construct for every positive integer m a non‐embeddable 2‐(3^m,3^m?1,(3^m?1?1)/2)‐design, and, if r_m=(3^m?1)/2 is a prime power, we construct for every positive integer n a non‐embeddable design. For each design in these families, a symmetric design with the corresponding parameters is known to exist. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 160–172, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.900     

Small non‐embeddable quasi‐derived designs

We establish the existence of non‐embeddable quasi‐derived 2‐designs with the parameters (13, 4, 3), (15, 6, 5), and (16, 6, 5). © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 364–372, 2008     

On the (in)stability of quasi‐static paths of smooth systems: definitions and sufficient conditions

A concept of stability of quasi‐static paths is discussed in this paper that takes into account the existence of fast (dynamic) and slow (quasi‐static) time scales in the evolution of many mechanical systems. The proposed concept is essentially a continuity property with respect to the smallness of the initial perturbations (as in Lyapunov stability) and the smallness of the quasi‐static loading rate (that plays the role of the small parameter in singular perturbation problems). A related concept of attractiveness is also proposed. Several examples illustrate the relevance of the definitions. Sufficient conditions for attractiveness or for instability of quasi‐static paths of smooth systems are proved. Copyright © 2005 John Wiley & Sons, Ltd.     

Dynamic steady‐state crack propagation in quasi‐crystals

The steady propagation of planar cracks in quasi‐crystalline bodies with velocity lower than the one of bulk elastic macroscopic waves is under scrutiny. Closed‐form solutions to the balance laws are provided. Unusual Mach number limits are determined. Numerical experiments describing peculiar aspects of the crack propagation in quasi‐crystals are performed by varying parametrically the coupling coefficient between macroscopic deformation and substructural events. In this way, classes of quasi‐crystals are then compared. Copyright © 2010 John Wiley & Sons, Ltd.     

Separation properties in neighbourhood and quasi‐apartness spaces

We investigate separation properties for neighbourhood spaces in some details within a framework of constructive mathematics, and define corresponding separation properties for quasi‐apartness spaces. We also deal with separation properties for spaces with inequality. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)