The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of pseudoline arrangements with contacts supported by a given network. In this paper, we construct the “brick polytope” of a network, o...

The main purpose of the paper is to find some expansion properties of locally finite metric spaces which do not embed coarsely into a Hilbert space. The obtained result is used to show that infinite locally finite graphs excluding a minor embed coarsely into a Hilbert space. In an appendix a direct proof of the latter result is given. 2000 Mathematics Subject Classification: Primary: 46B20; Sec...

We show that a locally Ahlfors 2-regular and locally linearly locally contractible metric surface is locally quasisymmetrically equivalent to the disk. We also discuss an application of this result to the problem of characterizing surfaces embedded in some Euclidean space that are locally bi-Lipschitz equivalent to a ball in the plane. In memoriam: Juha Heinonen (1960 2007)

If a connected metric space S is locally separable, then S is separable. If a connected, locally connected, metric space S is locally peripherally separable, then S is separable. Furthermore if a connected, locally connected, complete metric space S satisfies certain "flatness" conditions, it is known to be separable. These "flatness" conditions are rather strong and involve both im kleinen and...

We present a constructive proof of Tychonoff’s fixed point theorem in a locally convex space for uniformly continuous and sequentially locally non-constant functions. Keywords—sequentially locally non-constant functions, Tychonoff’s fixed point theorem, constructive mathematics.

In [1,2,3], A. C. Baker and J.W. Baker studied the subspace Ma(S) of the convolution measure algebra M, (S) of a locally compact semigroup. H. Dzinotyiweyi in [5,7] considers an analogous measure space on a large class of C-distinguished topological semigroups containing all completely regular topological semigroups. In this paper, we extend the definitions to study the weighted semigroup ...

Given a separated and locally finitely-presented Deligne-Mumford stack X over an algebraic space S, and a locally finitely-presented OX -module F , we prove that the Quot functor Quot(F/X/S) is represented by a separated and locally finitely-presented algebraic space over S. Under additional hypotheses, we prove that the connected components of Quot(F/X/S) are quasi-projective over S.

A locally truncated geometry with diagram of type affine E7 is studied. One considers a parapolar space e Γ, locally of type A7,4, which is subject to an extra axiom. A covering Γ of this space is constructed. It is proved that Γ is a rank 6, residually connected, locally truncated diagram geometry which is a homomorphic image of a truncated building of affine type E7. Consequently, e Γ is also...

• In Section 4, we define locally homogeneous singularities. A space with locally homogeneous singularities (SLHS) is an analytic space X such that for all x ∈ X , the x-completion of a local ring OxX is isomorphic to an x-completion of associated graded ring (OxX)gr. We show that a complex variety is SLHS if and only if the underlying real analytic variety is SLHS. This allows us to define inv...

Recently, Mohiuddine and Alghamdi introduced the notion of lacunary statistical convergence in a locally solid Riesz space and established some results related to this concept. In this paper, some inclusion relations between the sets of statistically convergent and lacunary statistically convergent sequences are established and extensions of a decomposition theorem, a Tauberian theorem to the l...