The properties of the metric topology on infinite and finite sets are analyzed. We answer whether finite metric spaces hold interest in algebraic topology, and how this result is generalized to pseudometric spaces through the Kolmogorov quotient. Embedding into Lebesgue spaces is analyzed, with special attention for Hilbert spaces, `p, and EN .

Upon their internalization by macrophages, Leishmania promastigotes inhibit phagolysosome biogenesis. The main factor responsible for this inhibition is the promastigote surface glycolipid lipophosphoglycan (LPG). This glycolipid has a profound impact on the phagosome, causing periphagosomal accumulation of F-actin and disruption of phagosomal lipid microdomains. Functionally, this LPG-mediated...

We are interested in making sense of an integral over a singular space, for example the quotient of the torus T 2 by the lines of irrational slope α / ∈ Q. More generally, we would like to study the quotient space of the global unstable manifolds for a hyperbolic dynamical system. In this chapter we present an “easy” version of Connes’ noncommutative integration theory. We start by studying int...

We show that if a Lie group acts properly on a co-oriented contact manifold preserving the contact structure then the contact quotient is topologically a stratified space (in the sense that a neighborhood of a point in the quotient is a product of a disk with a cone on a stratified space). As a corollary we obtain that symplectic quotients for proper Hamiltonian actions are topologically strati...

Recently, Lima and Randrianarivony pointed out the role of the property (β) of Rolewicz in nonlinear quotient problems, and answered a ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. In the present paper, we prove that the modulus of asymptotic uniform smoothness of the range space of a uniform quotient map can be compared with the modulus of (β) of the domain spa...

We present a general framework for Matrix theory compactified on a quotient space R/Γ, with Γ a discrete group of Euclidean motions in R. The general solution to the quotient conditions gives a gauge theory on a noncommutative space. We characterize the resulting noncommutative gauge theory in terms of the twisted group algebra of Γ associated with a projective regular representation. Present A...

Z. Li characterized spaces with a weak-development consisting of point-countable sfp-covers by pseudo-sequence-covering, quotient, and π-s-image of metric spaces. In this paper, we omit ”pseudo-sequence-covering” in the above result, and prove that a space has a weak-development consisting of point-countable sfp-covers iff it is a quotient, and π-s-image of a metric space.

abstract. in this paper we study some relations between the power andquotient power graph of a finite group. these interesting relations motivateus to find some graph theoretical properties of the quotient power graphand the proper quotient power graph of a finite group g. in addition, weclassify those groups whose quotient (proper quotient) power graphs areisomorphic to trees or paths.

In part of the life cycle within their sand fly vector, Leishmania major parasites first attach to the fly's midgut through their main surface adhesin lipophosphoglycan (LPG) and later resynthesize a structurally distinct LPG that results in detachment and eventual transmission. One of these structural modifications requires the addition of alpha1,2-D-arabinopyranose caps to beta1,3-galactose s...

Complex analyses involving multiple, dependent random quantities often lead to graphical models—a set of nodes denoting variables interest, and corresponding edges statistical interactions between nodes. To develop for data, especially towards generative modeling, one needs mathematical representations metrics matching comparing graphs, subsequent tools, such as geodesics, means, covariances. T...