This paper introduces a wide generalization of a family of integral lattices defined by Coxeter, which share with the Coxeter lattices the following properties: they are perfect, often with an odd minimum, and have no non-trivial perfect sections with the same minimum.

Let Ω ⊂ R2 denote a bounded Lipschitz domain and consider some portion Γ0 of ∂Ω representing the austenite–twinned-martensite interface which is not assumed to be a straight segment. We prove thatfor an elastic energy density ϖ: R2 → [0 ∞) such that ϖ(0, ±1) = 0. Here, W(Ω) consists of all functions u from the Sobolev class W1, ∞(Ω) such that |uy| = 1 almost everywhere on Ω together with u = 0 on Γ0. We will first show that, for Γ0 having a vertical tangent, one cannot always expect a finite surface energy, i.e. in the above problem, the condition in general cannot be included. This generalizes a result of [12] where Γ0is a vertical straight line. Property (*) is established by constructing some minimizing sequences vanishing on the whole boundary ∂Ω, that is, one can even take Γ0 = ∂Ω. We also show that the existence or non-existence of minimizers depends on the shape of the austenite–twinned-martensite interface Γ0.

For the purpose of this paper, we call a set of positive reals ν a well-spaced set if there is a c > 0 such that

There is an error in the proof of Theorem 2 of my paper [1]. It appears on; age 91, lines 10 and 11: the application of the affine transformation T changes he measure on the Grassmannian G(d, d-i) which is not taken into account. As a result, in the statement of Theorem 2 the coefficient is not correct.

This paper studies the existence and multiplicity of positive solutions of the following problem:

where Ω⊂RN(N≥3) is a smooth bounded domain, , 1 < p < N, and 0 < α < 1, p - 1 < β < p* - 1 (p* = Np/(N - p)) and 0 < γ < N + ((β + 1)(p - N)/p) are three constants. Also δ(x) = dist(x, ∂Ω), a ∈ Lp and λ < 0 is a real parameter. By using the direct method of the calculus of variations, Ekeland's Variational Principle and an idea of G. Tarantello, it is proved that problem (*) has at least two positive weak solutions if λ is small enough.

This article studies the non-homogeneous quadratic Bessel zeta function ζRB(s, v, a), defined as the sum of the squares of the positive zeros of the Bessel function Jv(z) plus a positive constant. In particular, explicit formulas for the main associated zeta invariants, namely, poles and residua ζRB(0, v, a) and ζRB(0, v, a), are given.

Given a function υ ∈ BV (Ω; Rm), we introduce the notion of a minimal lifting of Dυ. We prove that every υ ∈ BV (Ω; Rm) has a unique minimal lifting, and we show that if υk → υ strictly in BV, then the minimal liftings of υk converge weakly as measures to the minimal lifting of υ. As an application, we deduce a result about weak continuity of the distributional determinant Det D2u with respect to strict convergence.

We consider the generalized Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equations with dissipative term. We establish the condition under which the solutions uα,β,γ converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed by Hwang and Tzavaras. First, we obtain the approximate transport equation for the given BBM–Burgers equations. Then, using the averaging lemma, we obtain the convergence.

This paper studies additive properties of the generalized Drazin inverse (g-Drazin inverse) in a Banach algebra and finds an explicit expression for the g-Drazin inverse of the sum a + b in terms of a and b and their g-Drazin inverses under fairly mild conditions on a and b.

A subsemigroup S of a semigroup Q is a left order in Q, and Q is a semigroup of left quotients of S, if every element of Q can be written as a−1b for some a, b∈S with a belonging to a group -class of Q. Necessary and sufficient conditions on a semigroup S are obtained in order that S be a left order in a completely 0-simple semigroup Q. The class of all completely 0-simple semigroups of left quotients of S is related to the set of certain left congruences on S. Axioms are provided for semigroups which occur in the discussion of left orders in completely 0-simple semigroups.

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.

Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.

In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

It is shown that, for integrals of the typewith Ω RN open, bounded and f: Ω × Rm × Rd → [0, + ∞) Carathéodory satisfying a growth condition 0 ≤ f(x, u, υ) ≤ C(1 + |υ|p), for some p ∈ (1, + ∞), a sufficient condition for lower semi-continuity along sequences un → u in measure, υn → υ in Lp, Aυn → 0 in W−1, p is the Ax-quasi-convexity of f(x, u, ·). Here, A is a variable coefficients operator of the formwith A(i) ∈ C∞ (Ω; Ml × d) ∩ W1, ∞, i = 1, …, N, satisfying the conditionand Ax denotes the constant coefficients operator one obtains by freezing x. Under additional regularity conditions on f, it is proved that the condition above is also necessary. A characterization of the Young measures generated by bounded sequences {υn} in Lp satisfying the condition Aυn → 0 in W−1,p, is obtained.

The problem of finding necessary and sufficient condi-tions for the existence of trapped modes in waveguides has been known since 1943. [10]. The problem is the following: consider an infinite strip M in ℝ2(or an infinite cylinder with the smooth boundary in ℝn). The spectrum of the(positive) Laplacian, with either Dirichlet or Neumann boundary conditions, acting on this strip is easily computable via the separation of variables; the spectrum is absolutely continuous and equals [v0,+∞). Here, v0 is the first threshold, i.e., eigenvalue of the cross-section of the cylinder (so v0 = 0 in the case of Neumann conditions). Let us now consider the domain (the waveguide) which is a smooth compact perturbation of M (for example, weinsert an obstacle inside M). The essential spectrum of the Laplacian acting on still equals [v0, +ℝ), but there may be additional eigenvalues, which are often called trapped modes; the number of these trapped modes can be quite large, see examples in [11] and [8].

Voronoĭ conjectured that every parallelotope is affinely equivalent to a Voronoĭ polytope. For some m, a parallelotope is defined by a set of m facet vectors pi, and defines a set of m lattice vectors ti, for 1≤i≤m. It is shown that Voronoĭ's conjecture is true for an n-dimensional parallelotope P if and only if there exist scalars γi, and a positive definite n × n matrix Q such that γipi = Qti for each i. In this case, the quadratic form f(x) = xTQx is the metric form of P.

The set ℳ* of numbers which occur as Mahler measures of integer polynomials and the subset ℳ of Mahler measures of algebraic numbers (that is, of irreducible integer polynomials) are investigated. It is proved that every number α of degree d in ℳ* is the Mahler measure of a separable integer polynomial of degree at most with all its roots lying in the Galois closure F of ℚ(α), and every unit in ℳ is the Mahler measure of a unit in F of degree at most over ℚ This is used to show that some numbers considered earlier by Boyd are not Mahler measures. The set of numbers which occur as Mahler measures of both reciprocal and nonreciprocal algebraic numbers is also investigated. In particular, all cubic units in this set are described and it is shown that the smallest Pisot number is not the measure of a reciprocal number.

A random polytope is the convex hull of n random points in the interior of a convex body K. The expectation of the ith intrinsic volume of a random polytope as n → ∞ is investigated. It is proved that, for convex bodies of differentiability class Kk+1, precise asymptotic expansions for these expectations exist. The proof makes essential use of a refinement of Crofton's boundary theorem.

Let q be a natural number. When the multiplicative iroup (ℤ/qℤ)* is a cyclic group, its generators are called primitive roots. Note that the generators are also elements with the maximum order if (ℤ/qℤ)* is cyclic. Thus, when (ℤ–qℤ)* is not a cyclic goup, we then call an element with: he maximal possible order a primitive root, which was initially introduced by R. Carmichael [1].

The present paper extends the idea of characterizing topological properties of a space X by means of continuous selections for its closed subsets (X) endowed with a “natural” hyperspace topology. In this particular case, it is proved that the property of X to be topologically well-orderable is equivalent to the existence of a selection for (X) which is continuous with respect to the Fell topology.

The aim of this paper is to study the asymptotic behaviour of the solutions of the linearized elasticity system, posed on thin reticulated structures involving several small parameters. We show that this behaviour depends on the relative size of the parameters. In each case, we obtain a limit system where the microstructure and macrostructure appear simultaneously. From it, we get a suitable approximation in L2 of the displacements and the linearized strain tensor.