Related papers: Some notes on trees and paths

This note contains old instead of new results about random walks on groups, which may serve as a small supplement to the author's monograph ``Random Walks on Infinite Graphs and Groups'' (Cambridge Univ. Press 2000/2009). First, we exhibit…

We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a…

The occurrence and the distribution of patterns of trees associated to natural numbers are investigated. Bounds from above and below are proven for certain natural quantities.

These lecture notes are a personal introduction to signed graphs, concentrating on the aspects that have been most persistently interesting to me. They are just a few corners of signed graph theory; I am leaving out a great deal. The…

These notes expound the recent use of the signature transform and rough path theory in data science and machine learning. We develop the core theory of the signature from first principles and then survey some recent popular applications of…

These notes are a self-contained short proof of the stability of persistence diagrams.

Generalized are the investigated in other works of the author transports along paths in fibre bundles to transports along arbitrary maps in them. Their structure and some properties are studied. Special attention is paid to the linear case…

This is a survey of the theory of real trees and their applications.

These notes cover one of the topics programmed for the St Petersburg School in Probability and Statistical Physics of June 2012. The aim is to review recent mathematical developments in the field of random walks in random environment. Our…

The main purpose of this note is to pose a couple of problems which are easily formulated thought some seem to be not yet solved. These problems are of general interest for discrete mathematics including a new twig of a bough of theory of…

The aim of this paper is to revise the literature on different metric locations in the families of paths, cycles, trees and unicyclic graphs, as well as, provide several new results on that matter.

In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this…

These notes concern linear transformations on R^n and C^n, exponentials of linear transformations, and some related geometric questions.

This is a long introduction to the theory of "branch groups": groups acting on rooted trees which exhibit some self-similarity features in their lattice of subgroups.

This note is about variations on a theorem of Bers about short pants decompositions of surfaces. It contains a version for surfaces with boundary but also a slight improvement on the best known bound for closed surfaces.

A few notes about infinite trees in a descriptive set-theoretic setting.

In this short note we discuss recent results on hook length formulas of trees unifying some earlier results, and explain hook length formulas naturally associated to families of increasingly labelled trees.

This paper is devoted to a discussion of specific properties of invariants in the theory of forms.

In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend…

These informal notes concern some basic themes of harmonic analysis related to representations of groups.