Volume 5, Issue 4

We introduce two notions of tightness for a set of measurable functions — the finite-tightness and the Jordan finite-tightness with the aim to extend certain compactness results (as biting lemma or Saadoune-Valadier’s theorem of stable compactness) to the unbounded case. These compactness conditions highlight their utility when we look for some alternatives to Rellich-Kondrachov theorem or relaxed lower semicontinuity of multiple integrals. Finite-tightness locates the great growths of a set of measurable mappings on a finite family of sets of small measure. In the Euclidean case, the Jordan finite-tight sets form a subclass of finite-tight sets for which the finite family of sets of small measure is composed by d-dimensional intervals. The main result affirms that each tight set H ⊆ W 1,1 for which the set of the gradients ∇H is a Jordan finite-tight set is relatively compact in measure. This result offers very good conditions to use fiber product lemma for obtaining a relaxed lower semicontinuity condition.

Given a unital C*-algebra $$\mathcal{A}$$ and a right C*-module $$\mathcal{X}$$ over $$\mathcal{A}$$ , we consider the problem of finding short smooth curves in the sphere $$\mathcal{S}_\mathcal{X} $$ = {x ∈ $$\mathcal{X}$$ : 〈x, x〉 = 1}. Curves in $$\mathcal{S}_\mathcal{X} $$ are measured considering the Finsler metric which consists of the norm of $$\mathcal{X}$$ at each tangent space of $$\mathcal{S}_\mathcal{X} $$ . The initial value problem is solved, for the case when $$\mathcal{A}$$ is a von Neumann algebra and $$\mathcal{X}$$ is selfdual: for any element x 0 ∈ $$\mathcal{S}_\mathcal{X} $$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ $$\mathcal{L}_\mathcal{A} (\mathcal{X})$$ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and $$\dot \gamma $$ (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ $$\mathcal{S}_\mathcal{X} $$ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 $$\mathcal{L}_\mathcal{A} (\mathcal{X})$$ f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.

For the dual pair considered, the Cauchy Harish-Chandra Integral, as a distribution on the Lie algebra, is the limit of the holomorphic extension of the reciprocal of the determinant. We compute that limit explicitly in terms of the Harish-Chandra orbital integrals.

Trivial extensions of a certain subclass of minimal 2-fundamental algebras are examined. For such algebras the characterization of components of the Auslander-Reiten quiver which contain indecomposable projective modules is given.

Let G be a hypercyclic group. The most substantial results of this paper are the following. a) If G/G′ is 2-divisible, then G is 2-divisible. b) If G/G′ is a 2′-group, then G is a 2′-group. c) If G/G′ is divisible by finite-of-odd-order, then G/V is divisible by finite-of-odd-order, where V is the intersection of the lower central series (continued transfinitely) of O 2′ (G).

The Banach-Lie algebras ℌκ of all holomorphic infinitesimal isometries of the classical symmetric complex Banach manifolds of compact type (κ = 1) and non compact type (κ = −1) associated with a complex JB*-triple Z are considered and the Lie ideal structure of ℌκ is studied.

In this paper we study two classes of lightlike submanifolds of codimension two of semi-Riemannian manifolds, according as their radical subspaces are 1-dimensional or 2-dimensional. For a large variety of both these classes, we prove the existence of integrable canonical screen distributions subject to some reasonable geometric conditions and support the results through examples.

This note is concerned with the linear Volterra equation of hyperbolic type $$\partial _{tt} u(t) - \alpha \Delta u(t) + \int_0^t {\mu (s)\Delta u(t - s)} ds = 0$$ on the whole space ℝN. New results concerning the decay of the associated energy as time goes to infinity were established.

It is known that the fundamental solution to an elliptic differential equation with analytic coefficients exists, is determined up to the kernel of the differential operator, and has singularities on characteristics of the equation in ℂ2. In this paper we construct a representation of fundamental solution as a sum of functions, each of those has singularity on a single characteristic.

In the present paper we study the unique solvability of two non-local boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations. The uniqueness of the solutions of the considered problems are proven by the “abc” method. Existence theorems for the solutions of these problems are proven by the method of integral equations. The obtained results can be used for studying local and non-local boundary-value problems for mixed-hyperbolic type equations with two and three lines of changing type.

In this paper we consider the problem of detecting pollution in some non linear parabolic systems using the sentinel method. For this purpose we develop and analyze a new approach to the discretization which pays careful attention to the stability of the solution. To illustrate convergence properties we give some numerical results that present good properties and show new ways for building discrete sentinels.