A Kleinian group Γ is a discrete subgroup of PSL2(C). When non-elementary, such a group possesses a unique non-empty minimal closed invariant subset ΛΓ of the Riemann sphere, called the limit set. A Kleinian group acts properly discontinuously on the complement ∆Γ of ΛΓ and so this set is called the domain of discontinuity. Such a group is said to be convex co-compact if it acts co-compactly on...

in this paper, we study the notion of solvable $l$-subgroup of an $l$-group and provide its level subset characterization and this justifies the suitability of this extension. throughout this work, we have used normality of an $l$-subgroup of an $l$-group in the sense of wu rather than liu.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...

In 1933 S. Mazur [4] proved the following Theorem 1. Let (X, ·) be a separable real Banach space. Let f be a real-valued convex continuous function defined on an open convex subset Ω ⊂ X. Then there is a subset A ⊂ Ω of the first category such that f is Gateaux differentiable on Ω \ A. The result of Mazur was a starting point for the theory of differentiability of convex functions (cf. the book...

As well known, the Moreau-Rockafellar-Robinson internal point qualification condition is sufficient to ensure that the infimal convolution of the conjugates of two extended-real-valued convex lower semi-continuous functions defined on a locally convex space is exact, and that the sub-differential of the sum of these functions is the sum of their sub-differentials. This note is devoted to provin...

We consider polygons with the following \pairing property": for each edge of the polygon there is precisely one other edge parallel to it. We study the problem of when such a polygon K tiles the plane multiply when translated at the locations , where is a multiset in the plane. The pairing property of K makes this question particularly amenable to Fourier Analysis. After establishing a necessar...

Any non-empty open convex subset of Rn is the convex hull of a complete submanifold M , of any codimension, but there are obstructions if the geometry of M is, a priori, suitably controlled at infinity. In this paper we develop tools to explore the geometry of ∂[Conv(M)] when the Grassmanian-valued Gauss map of M is uniformly continuous, a condition that, in the C2 case, is weaker than bounding...

We will be concerned with linear positive maps φ : Mm(C) → Mn(C). To fix notation we begin with setting up the notation and the relevant terminology (cf. [7]). We say that φ is positive if φ(A) is a positive element in Mn(C) for every positive matrix from Mm(C). If k ∈ N, then φ is said to be k-positive (respectively k-copositive) whenever [φ(Aij)] k i,j=1 (respectively [φ(Aji)] k i,j=1) is pos...

Fixed a bounded open set Ω of R , we completely characterize the weak* lower semicontinuity of functionals of the form F (u,A) = ess sup x∈A f(x, u(x), Du(x)) defined for every u ∈ W 1,∞(Ω) and for every open subset A ⊂ Ω. Without a continuity assumption on f(·, u, ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it can be represented through a level conv...

In a recent preprint by Amaral & Letchford (2006) convex hulls of sets of matrices corresponding to permutations and certain metrics are studied. For each n-tupel of points x1, . . . , xn ∈ R with |xk − xl| ≥ 1 for k 6= l, we define a metric in form of a symmetric n× n-matrix whose entries are the pairwise distances |xk − xl| between the points. The convex hull of these metrics is denoted by Qn...