For and α, we consider sets of numbers x such that for infinitely many n, x is 2−αn-close to some ∑ ni=1ωiλi, where ωi∈{0,1}. These sets are in Falconer’s intersection classes for Hausdorff dimension s for some s such that −(1/α)(log λ /log   2 )≤s≤1/α. We show that for almost all , the upper bound of s is optimal, but for a countable infinity of values of λ the lower bound is the best possible result.

It follows from methods of B. Steinberg, extended to inverse categories, that finite inverse category algebras are isomorphic to their associated groupoid algebras; in particular, they are symmetric algebras with canonical symmetrizing forms.We deduce the existence of transfer maps in cohomology and Hochschild cohomology from certain inverse subcategories. This is in part motivated by the observation that, for certain categories , being a Mackey functor on is equivalent to being extendible to a suitable inverse category containing . We further show that extensions of inverse categories by abelian groups are again inverse categories.

We study bounds on nilpotence in H*(BG), the mod p cohomology of the classifying space of a compact Lie group G. Part of this is a report of our previous work on this problem, updated to reflect the consequences of Peter Symonds's recent verification of Dave Benson's Regularity Conjecture. New results are given for finite p-groups, leading to good bounds on nilpotence in H*(BP) determined by the subgroup structure of the p-group P.

Let be a finite category and let k be a field. We consider the category algebra and show that -mod is closed symmetric monoidal. Through comparing with a co-commutative bialgebra, we exhibit the similarities and differences between them in terms of homological properties. In particular, we give a module-theoretic approach to the multiplicative structure of the cohomology rings of small categories. As an application, we prove that the Hochschild cohomology rings of a certain type of finite category algebras are finitely generated.

After summarizing from previous papers the definitions of the concepts associated with nets, i.e. triples of 6-transpositions in the Monster up to braiding, we give some results.

We study the existence of positive solutions for the nonlinear Schrödinger equation with the fractional Laplacian

Furthermore, we analyse the regularity, decay and symmetry properties of these solutions.

The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his theory which are more directly related to viscosity solutions. Here, on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on weak KAM theory is the still unpublished book Weak KAM theorem in Lagrangian dynamics by Albert Fathi. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired by the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on ℝd and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications, however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centred on the Cauchy problem for the Hamilton–Jacobi equation and the Lax–Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the weak KAM solutions. The evolution operators can also be used for their regularizing properties; this opens an alternative route to dynamical applications.

Almost-everywhere convergence of wavelet transforms of Lp-functions under minimal conditions on wavelets was proved by Rao et al. in 1994. However, results on convergence almost everywhere do not provide any information regarding the exceptional set (of Lebesgue measure zero), where convergence does not hold. We prove that if a wavelet ψ satisfies a single additional condition xψ(x) ∈ L1 (R), then, instead of almost-everywhere convergence, we have a more sophisticated result, i.e. convergence of wavelet transforms everywhere on the entire Lebesgue set of Lp-functions. For example, wavelets with compact support, used frequently in applications, obviously satisfy this extra condition. Moreover, we prove that our conditions on wavelets ensure the Riemann localization principle in L1 for the wavelet transforms.

For an endomorphism φ ∈ EndR(M) of a left R-module RM, we investigate the structure and the polynomial identities of the zero-level centralizer Cen0(φ) and the factor Cen(φ)/Cen0(φ). A double zero-centralizer theorem for Cen0(Cen0(φ)) is also formulated.

We introduce the full expression of the curvature tensor of a real hypersurface M in complex two-plane Grassmannians G2(ℂm+2) from the Gauss equation. We then derive a new formula for the Ricci tensor of M in G2(ℂm+2). Finally, we prove that there does not exist any Hopf real hypersurface in complex two-plane Grassmannians G2(ℂm+2) with parallel Ricci tensor.

The best constant in a mean-value trace inequality for functions of bounded variation on admissible domains Ω ⊂ ℝn is shown to agree with an isoperimetric constant associated with Ω. The existence and form of extremals is also discussed. This result is exploited to compute the best constant in the relevant trace inequality when Ω is a ball. The existence and the form of extremals in this special case turn out to depend on the dimension n. In particular, the best constant is not achieved when Ω is a disc in ℝ2.

We consider the T-periodic problem

where g: [0,T]×]0,+∞[→ℝ exhibits a singularity of a repulsive type at the origin, and an asymptotically linear behaviour at infinity. In particular, for large x, g(t, x) is controlled from both sides by two consecutive asymptotes of the T-periodic Fučik spectrum, with possible equality on one side. Using a suitable Landesman–Lazer-type condition, we prove the existence of a solution.

This paper studies the long-time behaviour of solutions to a one-dimensional strongly nonlinear partial differential equation system arising from phase transitions with microscopic movements. Our system features a strongly nonlinear internal energy balance equation. Uniform bounds of the global solutions and the compactness of the orbit are obtained for the first time using a lemma established recently by Jiang. The existence of global attractors and convergence of global solutions to a single steady state as time goes to infinity are also proved.

We consider a nonlinear, strongly coupled, parabolic system arising in the modelling of burglary in residential areas. This model appeared in Pitcher (Eur. J. Appl. Math., 2010, Vol. 21, pp. 401–419), as a more realistic version of the Short et al. (Math. Models Methods Appl. Sci., 2008, Vol. 18, pp. 1249–1267) model. The system under consideration is of chemotaxis-type and involves a logarithmic sensitivity function and specific interaction and relaxation terms. Under suitable assumptions on the data of the problem, we give a rigorous proof of the existence of a global and bounded, classical solution, thereby solving a problem left open in previous work on this model. Our proofs are based on the construction of approximate entropies and on the use of various functional inequalities. We also provide explicit numerical conditions for global existence when the domain is a square, including concrete cases involving values of the parameters which are expected to be physically relevant.