A metric graph is a 1-dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as street maps, neurons, networks of rivers and galaxies. We consider the statistical problem of reconstructing the topology of a metric graph from a random sam...

The presence of a fractional-exponent growth law relating knot energy and knot topology is known to be an essential characteristic for the existence of ‘ideal’ knots. In this paper, we show that the energy infimum EN stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration S/S (nR1) in general dimensions obeys the sharp fractional-exponent growth la...

A net (xγ)γ∈Γ in a locally solid Riesz space (X,τ) is said to be unbounded τ-convergent x if |xγ−x|∧u⟶τ0 for all u∈X+. We recall that there linear topology uτ on X such τ-convergence coincides with uτ-convergence. It turns out characterised as the weakest which τ order bounded subsets. this motivation we introduce, uniform lattice (L,u), uniformity u⁎ L u subsets of L. shown induced by (X,τ), t...

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as lower limit (i.e., inferior) classical derivative where it exists. The lead to significantly shorter proofs basic properties subgradient and including chain rule. establish that sequence maps converges given in L-topology—that is, weakest refinement sup norm ...

If J is a set and Xj are topological spaces for each j ∈ J , let X = ∏ j∈J Xj and let πj : X → Xj be the projection maps. A basis for the product topology on X are those sets of the form ⋂ j∈J0 π −1 j (Uj), where J0 is a finite subset of J and Uj is an open subset of Xj , j ∈ J0. Equivalently, the product topology is the initial topology for the projection maps πj : X → Xj , j ∈ J , i.e. the co...

We study FTOP (L), a fuzzy category related to topology with fuzzy functions in the role of morphisms.

Let L be a linear space of real bounded random variables on the probability space (Ω,A, P0). It is shown that L− L∞ ∩ L∞ = {0}, with the closure in the norm-topology of L∞, if and only if there is a finitely additive probability P on A such that P ∼ P0 and EP (X) = 0 for each X ∈ L. The case where L includes unbounded random variables is investigated as well.

This paper proves a conjecture of Fomin and Shapiro that their combinatorial model for any Bruhat interval for any finite Weyl group is a regular CW complex which is homeomorphic to a ball. The model consists of a stratified space which may be regarded as the link of an open cell in a larger closed cell, all within the totally nonnegative part of the unipotent radical of an algebraic group. A k...

let $fm(x)$ be the  space of  all finite regular borel measures on $x$. a general measure algebra is a subspace  of$fm(x)$,which is an $l$-space and has a multiplication preserving the probability measures. let $clsubseteqfm(x)$ be a general measure algebra on a locallycompact space $x$. in this paper, we investigate the relation between arensregularity of $cl$ and the topology of $x$. we  find...