Braided Thompson’s groups are finitely presented groups introduced by Brin and Dehornoy which contain the ordinary braid groups Bn, the finitary braid group B∞ and Thompson’s group F as subgroups. We describe some of the metric properties of braided Thompson’s groups and give upper and lower bounds for word length in terms of the number of strands and the number of crossings in the diagrams use...

During inflation vacuum fluctuations on scales smaller than the Hubble radius are magnified into classical perturbations of the scalar field on super Hubble scales. The primordial perturbations arise solely from the zero-point fluctuations of the quantized fields. Although the region which ultimately expanded to become the observed universe may have contained excitations above the vacuum, these...

We study properties of rank metric and codes in rank metric over finite fields. We show that perfect codes do not exist. We derive an equivalent of the Varshamov-Gilbert bound in Hamming metric. We study the asymptotic behavior of the minimum rank distance of codes that are on GV. We show that the packing density of maximum rank distance codes is lower bounded by a function depending on the err...

The metric traveling salesman problem is one of the most prominent APX-complete optimization problems. An important particularity of this problem is that there is a large gap between the known upper bound and lower bound on the approximability (assuming P 6= NP ). In fact, despite more than 30 years of research, no one could find a better approximation algorithm than the 1.5-approximation provi...

Abstract. This paper introduces a new construction for q-ary equidistant code C with rank metric where q is a power of 2. Investigations on structural properties of the proposed code are carried out. The highlight of the paper is that the kernel of the code C happens to be an equidistant constant-weight code of same size as C and is shown to be C+ C. The bounds on number of steps that are requi...

1 Definitions and Examples 1.1 Metric and Normed Spaces Definition 1.1. A metric space is a pair (X, d), where X is a set and d is a function from X ×X to R such that the following conditions hold for every x, y, z ∈ X. 1. Non-negativity: d(x, y) ≥ 0. 2. Symmetry: d(x, y) = d(y, x). 3. Triangle inequality: d(x, y) + d(y, z) ≥ d(x, y) . 4. d(x, y) = 0 if and only if x = y. Elements of X are call...

In this paper, we study properties of rank metric codes in general and maximum rank distance (MRD) codes in particular. For codes with the rank metric, we first establish Gilbert and sphere-packing bounds, and then obtain the asymptotic forms of these two bounds and the Singleton bound. Based on the asymptotic bounds, we observe that asymptotically Gilbert-Varsharmov bound is exceeded by MRD co...

We compute estimates for the word metric of Baumslag–Solitar groups in terms of the Britton’s lemma normal form. As a corollary, we find lower bounds for the growth rate for the groups BS(p, q), with 1 < p ≤ q.

In 2007, Haung and Zhang introduced the notion of cone metric spaces. In this paper, we define an ordered space E, and we discuss some properties and examples. Also, normed ordered space is introduced. We recall properties of R, and we discuss their extension to E. We introduce the notion of E-metric spaces and characterize cone metric space. Afterwards, we get generalizations of notions of con...

Let T (S) be the Teichmüller space of an oriented surface S of finite type. We discuss the action of subgroups of the mapping class group of S on the CAT(0)-boundary of the completion of T (S) with respect to the Weil-Petersson metric. We show that the set of invariant Borel probability measures for the Weil-Petersson flow on moduli space which are supported on a closed orbit is dense in the sp...