Related papers: An incomplete real tree with complete segments
Orbifolds of two-dimensional quantum field theories have a natural formulation in terms of defects or domain walls. This perspective allows for a rich generalisation of the orbifolding procedure, which we study in detail for the case of…
We prove that a set of finite perimeter is indecomposable if and only if it is, up to a choice of suitable representative, connected in the 1-fine topology. This gives a topological characterization of indecomposability which is new even in…
We consider all spanning trees of a complete simple graph $\Gamma$ on $n$ vertices that contain a given $m-$forest $F$. We show that the number of such spanning trees, $\tau(F)$, doesn't depend on the structure of $F$ and is completely…
A metric space M is homogeneous if every isometry between finite subsets extends to a surjective isometry defined on the whole space. We show that if M is an ultrametric space, it suffices that isometries defined on singletons extend, i.e…
In this paper we define a field K of characteristic zero with valuation whose value group is the real numbers, and we show that this field of generalised Puiseux series is algebraically closed and complete with respect to the norm induced…
We analyze the interplay between labeled trees and the ultrametric spaces they present. We provide characterizations of labeled trees that generate separable ultrametric spaces and those that generate locally finite ultrametric spaces. In…
Premetrics and premetrisable spaces have been long studied and their topological interrelationships are well-understood. Consider the category ${\bf Pre}$ of premetric spaces and $\epsilon$-$\delta$ continuous functions as morphisms. The…
In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about constant sectional curvature is replaced by the assumption that the…
Inspired by group cohomology, we define several coarse topological invariants of metric spaces. We define the coarse cohomological dimension of a metric space, and demonstrate that if G is a countable group, then the coarse cohomological…
The aim of the present paper is to prove that the family of all closed nonempty subsets of a complete probabilistic metric space $L$ is complete with respect to the probabilistic Pompeiu-Hausdorff metric $H$. The same is true for the…
We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant…
A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in $Q$-polynomial distance-regular…
A Q-system is a unitary version of a separable Frobenius algebra object in a C*-tensor category. In a recent joint work with P. Das, S. Ghosh and C. Jones, the author has categorified Bratteli diagrams and unitary connections by building a…
The text is devoted to explanation of the concept of Topological Quantum Field Theory (TQFT), its application to homological algebra and to the relation with the theory of good section from K.Saito's theory of Primitive forms. TQFT is…
In this work, we analyze the structure of the category of partial representations of a finite group $G$ as a multifusion category, providing an alternative way to describe simple objects and their tensor products. We describe the…
We develop the theory of halving spaces to obtain lower bounds in real enumerative geometry. Halving spaces are topological spaces with an action of a Lie group $\Gamma$ with additional cohomological properties. For $\Gamma=\mathbb{Z}_2$ we…
In this paper we shall give a short proof of the result originally obtained by Ashutosh Kumar that for each $A\subset \mathbb{R}$ there exists $B\subset A$ full in $A$ such that no distance between two distinct points from $B$ is rational.…
The focus of this article is on metric completions of triangulated categories arising in the representation theory of hereditary finite dimensional algebras and commutative rings. We explicitly describe all completions of bounded derived…
Torsion and curvature could play a fundamental role in explaining cosmological dynamics. f(R)-gravity with torsion is an approach aimed to encompass in a comprehensive scheme all the Dark Side of the Universe (Dark Energy and Dark Matter).…
The free abelian group R(Q) on the set of indecomposable representations of a quiver Q, over a field K, has a ring structure where the multiplication is given by the tensor product. We show that if Q is a rooted tree (an oriented tree with…