Related papers: The extended mapping class group is generated by t…

The simple permutations in two permutation classes --- the 321-avoiding permutations and the skew-merged permutations --- are enumerated using a uniform method. In both cases, these enumerations were known implicitly, by working backwards…

It is shown, that the mapping class group of a surface of the genus g > 1 admits a faithful representation into the matrix group GL (6g-6, Z). The proof is based on a categorical correspondence between the Riemann surfaces and the so-called…

We consider the algebra of invariants of $d$-tuples of $n\times n$ matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic $p$ different from two. It is well-known that this…

Let $S_{g,n}$ be an oriented surface of genus $g$ with $n$ punctures, where $2g-2+n>0$ and $n>0$. Any ideal triangulation of $S_{g,n}$ induces a global parametrization of the Teichm\"uller space $\mathcal{T}_{g,n}$ called the shearing…

An involution is usually defined as a mapping that is its own inverse. In this paper, we study quaternion involutions that have the additional properties of distribution over addition and multiplication. We review formal axioms for such…

The C_{\lambda}-extended oscillator algebra is generated by {1,a,a^{\dagger},N,T}, where T is the generator of the cyclic group C_{\lambda} of order \lambda. It can be realized as a generalized deformed oscillator algebra (GDOA). Its…

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M…

S. Bigelow proved that the braid groups are linear. That is, there is a faithful representation of the braid group into the general linear group of some field. Using this, we deduce from previously known results that the mapping class group…

In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group.…

In this paper, we present an algorithm that enumerates a certain class of signed permutations, referred to as grid signed permutation classes. In the case of permutations, the corresponding grid classes are of interest because they are…

Let $\mathcal{M}(n)$ be the subgroup of $GL(n,\mathbb{Z})$ generated by the particular involutions that are identical to the identity, except for a single line where $-1$ and $+1$ alternate. We study the properties of $\mathcal{M}(n)$, and…

We survey recent developments on mapping class groups of surfaces of infinite topological type.

The plane Cremona group over the finite field $\mathbb{F}_2$ is generated by three infinite families and finitely many birational maps with small base orbits. One family preserves the pencil of lines through a point, the other two preserve…

Let $G$ be a group with involution * and $\sigma\colon G\to\{\pm1\}$ a group homomorphism. The map $\sharp$ that sends $\alpha=\sum\alpha_gg$ in a group ring $RG$ to $\alpha^{\sharp}=\sum\sigma(g)\alpha_gg^*$ is an involution of $RG$ called…

We present four constructions of inversion sequences, and use them to compute the enumeration sequences of 24 classes of pattern-avoiding inversion sequences. This completes the enumeration of inversion sequences avoiding one or two…

We review recent developments of machine learning algorithms pertinent to the inverse renormalization group, which was originally established as a generative numerical method by Ron-Swendsen-Brandt via the implementation of compatible Monte…

We determine, by hierarchy, dependencies between higher order linear symmetries which occur when generating them using recursion operators. Thus, we deduce a formula which gives the number of independent generalized symmetries (basis) of…

Let $C_{k_1}, \ldots, C_{k_n}$ be cycles with $k_i\geq 2$ vertices ($1\le i\le n$). By attaching these $n$ cycles together in a linear order, we obtain a graph called a polygon chain. By attaching these $n$ cycles together in a cyclic…

In this paper, the problem formulated in [8] is solved. We prove, that the group of automorphisms of the category of free associative algebras is generated by semi-inner and mirror automorphisms

In this article we continue to develop the theory of generating symmetries for integrable equations. A technique for computation of generating symmetries using Maple is presented. The technique is based on the standard symmetry method. By…