Related papers: Enumerating limit groups: A Corrigendum
Limit can be defined by two axioms: 1. Strict inequality between limits implies, ultimately, strict inequality between functions. 2. For constant functions limit is trivial. How can basic results on convergence be derived from these axioms?…
It is known that signature of a Weyl group element is defined in terms of the number of its simple Weyl reflections. Actual calculations hence are not always possible especially for Weyl groups with higher order like $E_8$ Weyl group. By…
We review the number counts to second order concentrating on the terms which dominate on sub horizon scales. We re-derive the result for these terms and compare it with the different versions found in the literature. We generalize our…
We describe the framework for the notion of a restricted inverse limit of categories, with the main motivating example being the category of polynomial representations of the group $GL_{\infty}$. This category is also known as the category…
We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form…
Given a group $G$ we write $h(G)$ to denote the maximum number of times that a field extension of $\mathbb{Q}$ appears as the field of values of a conjugacy class of a group. In this work, we prove that $|G|$ is bounded in terms of $h(G)$.…
Given a group-word w and a group G, the verbal subgroup w(G) is the one generated by all w-values in G. The word w is called concise if w(G) is finite whenever the set of w-values in G is finite. It is an open question whether every word is…
It is shown how one can define vector topological charges for topological exitations of non-linear sigma-models on compact homogeneous spaces T_G and G/T_G (where G is a simple compact Lie group and T_G is its maximal commutative subgroup).…
In the space of marked group, we determine the structure of groups which are limit points of the set of all generalized quaternion groups.
We study the girth of Cayley graphs of finite classical groups G on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word w takes the value 1 when evaluated in G in…
For a finite Coxeter group $W$ and $w$ an element of $W$ the `excess' of $w$ is defined to be $e(w) = \min\{\ell(x) + \ell(y) - \ell(w) \; | \; w=xy, \; x^2 = y^2 = 1\}$ where $\ell$ is the length function on $W$. Here we investigate the…
The purpose of this paper is to revisit the proof of the Gearhardt-Pr\"uss-Hwang-Greiner theorem for a semigroup $S(t)$, following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on $\Vert…
We show that, if $w_1, \ldots , w_6$ are words which are not an identity of any (non-abelian) finite simple group, then $w_1(G)w_2(G) \cdots w_6(G) = G$ for all (non-abelian) finite simple groups $G$. In particular, for every word $w$,…
We indicate a natural generalization of the concept of subgroup commutativity degree of a finite group and a list of open problems on these new concepts.
In this review we consider the concept of limit cycles in the renormalization group flows. The examples of this phenomena in the quantum mechanics and field theory will be presented.
In a previous paper [Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001) 243--270, arXiv:math.GT/0010247], a group H_g of homology cylinders over the oriented surface of genus g is defined. A…
This review summarizes papers which analyze impact of self-citation on research evaluation. We introduce a generalized definition of self-citation and its variants: author, institutional, country, journal, discipline, publisher…
Let $w = w(x_1,..., x_n)$ be a word, i.e. an element of the free group $F =<x_1,...,x_n>$ on $n$ generators $x_1,..., x_n$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{w (g_1,...,g_n)^{\pm 1} | g_i \in…
We determine a reasonable upper bound for the complexity of collection from the left to multiply two elements of a finite soluble, or polycyclic, group by restricting attention to certain polycyclic presentations of the group.
It is well-known that real-world changes constituting distribution shift adversely affect model performance. How to characterize those changes in an interpretable manner is poorly understood. Existing techniques to address this problem take…