We characterize and classify the “regular classes of heaps” introduced by the author using ideas of Fan and of Stembridge. The irreducible objects fall into five infinite families with one exceptional case.

Let $m, n\in\Bbb N$, $V$ be a $2m$-dimensional complex vector space. The irreducible representations of the Brauer's centralizer algebra $B_n(-2m)$ appearing in $V^{\otimes{n}}$ are in 1–1 correspondence to the set of pairs $(\,f,\lamda)$, where $f\in\Z$ with $0\leq f\leq [n/2]$, $and$ $\lam\vdash n-2f$ satisfying $\lam_1\leq m$. In this paper, we first show that each of these representations has a basis consists of eigenvectors for the subalgebra of $B_n(-2m)$ generated by all the Jucys-Murphy operators, and we determine the corresponding eigenvalues. Then we identify these representations with the irreducible representations constructed from a cellular basis of $B_n(-2m)$. Finally, an explicit description of the action of each generator of $B_n(-2m)$ on such a basis is also given, which generalizes earlier work of [15] for Brauer's centralizer algebra $B_n(m)$.

In this paper, we prove some Hardy-type inequalities for the degenerate operators, $L_{p,\alpha}u\,{=}\,{\rm div}_L(|\nabla_Lu|^{p-2}\nabla_Lu)$, where $\nabla_Lu\,{=}\,(\frac{\partial u}{\partial z_1},\ldots,\frac{\partial u}{\partial z_n},|z|^\alpha \frac{\partial u}{\partial t_1},\ldots,|z|^\alpha\frac{\partial u}{\partial t_m})$. These inequalities are established for the whole space, the pseudo-ball and the external domain of the pseudo-ball. We also give a generalization of a result in [8]. Finally, a sharp inequality for $L_{\alpha}\,{=}\,L_{2,\alpha}$ is obtained.

Several properties of some quadratic elements of a unitial Banach algebra are studied. Deddens subspaces are also introduced and discussed.

Face 2-colourable triangulations of complete tripartite graphs $K_{n,n,n}$ correspond to biembeddings of Latin squares. Up to isomorphism, we give all such embeddings for $n=3,4,5$ and 6, and we summarize the corresponding results for $n=7$. Closely related to these are Hamiltonian decompositions of complete bipartite directed graphs $K^*_{n,n}$, and we also give computational results for these in the cases $n=3,4,5$ and 6.

In this paper we apply fixed point results in ordered spaces to derive existence and comparison results for discontinuous functional integral equations of Volterra type in ordered Banach spaces. The results obtained are then applied to first order impulsive differential equations.

An integral representation of a relaxed functional arising in the derivation of a nonlinear membrane model accounting for the density of bending moments is obtained in terms of a special class of parametrized probability measures (bending Young measures). A complete characterization of this class is given.

In this paper we study the existence of radially symmetric solitary waves for nonlinear Klein–Gordon equations and nonlinear Schrödinger equations coupled with Maxwell equations. The method relies on a variational approach and the solutions are obtained as mountain-pass critical points for the associated energy functional.

We study the Cauchy problem for the one-dimensional equations of a viscous heat-conducting gas in the Lagrangian mass coordinates with the initial data in the Lebesgue spaces. We prove the existence, the uniqueness and the Lipschitz continuous dependence on the initial data of global weak solutions.

We consider scalar hyperbolic conservation laws with non-convex flux and vanishing, nonlinear and possibly singular, diffusion and dispersion terms. The diffusion has the form R(u, ux)x and we cover, for instance, the singular diffusion (|ux|pux)x, where p ≥ 0 is arbitrary. We investigate the existence, uniqueness and various properties of classical and non-classical travelling waves and of the kinetic function. The latter serves to characterize non-classical shock waves, via an additional algebraic constrain called a kinetic relation. We discover that p = ⅓ is a somewhat unexpected critical value. For p ≤ ⅓, we obtain properties that are qualitatively similar to those we established earlier for regular and linear diffusion. However, for p > ⅓, the behaviour of the kinetic function is very different, as, for instance, non-classical shocks can have arbitrary small strength. The behaviour of the kinetic function near the origin is carefully investigated and depends on whether p < ½, p = ½ or p > ½. In particular, in the special case of the cubic flux-function and for the regularization (|ux|pux)x with p = 0, ½ or 1, the kinetic function can be computed explicitly. When p = ½, the kinetic function is simply a linear function of its argument.

Given a smooth function 𝔏 of a real or complex variable and taking its values in the class of Fredholm operators of index zero in a Banach space, there are some available definitions in the literature of algebraic multiplicity of the family 𝔏 at a point x0 of the parameter at which the operator 𝔏(x0) becomes non-invertible. The purpose of this paper is to suggest an axiomatic for the multiplicity and to prove that the algebraic multiplicity is uniquely determined by a few of its properties, independently of its construction.

Consider the second-order discrete systemwhere f ∈ C (R × Rm, Rm), f(t + M, Z) = f(t, Z) for any (t, Z) ∈ R × Rm and M is a positive integer. By making use of critical-point theory, the existence of M-periodic solutions of (*) is obtained.

We discuss some existence theorems for partial differential inclusions, subject to Dirichlet boundary conditions, of the formwhere Φ is a quasi-affine function and so, in particular, for Φ(Du) = det Du.

We then apply it to minimization problems of the form

We study the linear stability of a vortex sheet in a limit case that corresponds to a transition between a weakly stable regime and a violently unstable regime. We prove an energy estimate that reflects the high degeneracy of the uniform Kreiss–Lopatinskii condition.

In this paper, a host-vector model is considered for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times. Spatial spread in a region is modelled in the partial integro-differential equation by a diffusion term. For the general model, we first study the stability of the steady states using the contracting-convex-sets technique. When the spatial variable is one dimensional and the delay kernel assumes some special form, we establish the existence of travelling wave solutions by using the linear chain trick and the geometric singular perturbation method.

We introduce a new non-classical Riemann solver for scalar conservation laws with concave–convex flux-function. This solver is based on both a kinetic relation, which determines the propagation speed of (under-compressive) non-classical shock waves, and a nucleation criterion, which makes a choice between a classical Riemann solution and a non-classical one. We establish the existence of (non-classical entropy) solutions of the Cauchy problem and discuss several examples of wave interactions. We also show the existence of a class of solutions, called splitting–merging solutions, which are made of two large shocks and small bounded-variation perturbations. The nucleation solvers, as we call them, are applied to (and actually motivated by) the theory of thin-film flows; they help explain numerical results observed for such flows.

We give an alternative self-contained proof of the homogenization theorem for periodic multi-parameter integrals that was established by the authors. The proof in that paper relies on the so-called compactness method for Γ-convergence, while the one presented here is by direct verification: the candidate to be the limit homogenized functional is first exhibited and the definition of Γ-convergence is then verified. This is done by an extension of bounded gradient sequences using the Acerbi et al. extension theorem from connected sets, and by the adaptation of some localization and blow-up techniques developed by Fonseca and Müller, together with De Giorgi's slicing method.

The structure of positive boundary blow-up solutions to semilinear problems of the form −Δu = λf(u) in Ω, u = ∞ on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied for a class of nonlinearities f ∈ C1 ([0, ∞)\{z2}) satisfying f (0) = f(z1) = f (z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1)∪(z2, ∞), f > 0 in (z1, z2). Two positive boundary-layer solutions and infinitely many positive spike-layer solutions are obtained for λ sufficiently large.

We analyse a class of randomized Least Recently Used (LRU) cache replacement algorithms under the independent reference model with generalized Zipf's law request probabilities. The randomization was recently proposed for Web caching as a mechanism that discriminates between different document sizes. In particular, the cache maintains an ordered list of documents in the following way. When a document of size $s$ is requested and found in the cache, then with probability $p_s$ it is moved to the front of the cache; otherwise the cache stays unchanged. Similarly, if the requested document of size $s$ is not found in the cache, the algorithm places it with probability $p_s$ to the front of the cache or leaves the cache unchanged with the complementary probability $(1-p_s)$. The successive randomized decisions are independent and the corresponding success probabilities $p_s$ are completely determined by the size of the currently requested document. In the case of a replacement, the necessary number of documents that are least recently moved to the front of the cache are removed in order to accommodate the newly placed document.

In this framework, we provide explicit asymptotic characterization of the cache fault probability. Using the derived result we prove that the asymptotic performance of this class of algorithms is optimized when the randomization probabilities are chosen to be inversely proportional to document sizes. In addition, for this optimized and easy-to-implement policy, we show that its performance is within a constant factor from the optimal static algorithm.

Let $F(\b{z})=\sum_\b{r} a_\b{r}\b{z^r}$ be a multivariate generating function that is meromorphic in some neighbourhood of the origin of $\mathbb{C}^d$, and let $\sing$ be its set of singularities. Effective asymptotic expansions for the coefficients can be obtained by complex contour integration near points of $\sing$.

In the first article in this series, we treated the case of smooth points of $\sing$. In this article we deal with multiple points of $\sing$. Our results show that the central limit (Ornstein–Zernike) behaviour typical of the smooth case does not hold in the multiple point case. For example, when $\sing$ has a multiple point singularity at $(1, \ldots, 1)$, rather than $a_\b{r}$ decaying as $|\b{r}|^{-1/2}$ as $|\b{r}| \to \infty$, $a_\b{r}$ is very nearly polynomial in a cone of directions.