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We study upper bounds on the order of automorphisms of non-singular curves $X$ satisfying at least one of the following hypothesis: 1) $X$ is an $m$-sheeted covering of exactly one non-singular curve of genus $\gamma$, where $m$ is prime;…
The topological symmetry group $\mathrm{TSG}(\Gamma)$ of an embedding $\Gamma$ of a graph in $S^3$ is the subgroup of the automorphism group of the graph which is induced by homeomorphisms of $(S^3,\Gamma)$. If we restrict to orientation…
In this paper we study the widely considered endomorphisms and weak endomorphisms of a finite undirected path from monoid generators perspective. Our main aim is to determine the ranks of the monoids $wEnd P_n$ and $End P_n$ of all weak…
A group code structure of a linear code is a description of the code as one-sided or two-sided ideal of a group algebra of a finite group. In these realizations, the group algebra is identified with the ambient space, and the group elements…
We revisit the geometry of involutions in groups of finite Morley rank. Our approach unifies and generalises numerous results, both old and recent, that have exploited this geometry; though in fact, we prove much more. We also conjecture…
There are various results in the literature which are part of the general philosophy that a finite group for which a certain parameter (for example, the number of conjugacy classes or the maximum number of elements inverted, squared or…
Maximum distance separable (MDS) codes have significant combinatorial and cryptographic applications due to their certain optimality. Generalized Reed-Solomon (GRS) codes are the most prominent MDS codes. Twisted generalized Reed-Solomon…
We show that the morphisms from the braid group with n strands in the mapping class group of a surface with a possible non empty boundary, assuming that its genus is smaller or equal to n/2 are either cyclic morphisms (their images are…
Gabidulin codes over fields of characteristic zero were recently constructed by Augot et al., whenever the Galois group of the underlying field extension is cyclic. In parallel, the interest in sparse generator matrices of Reed-Solomon and…
Scattered polynomials of a given index over finite fields are intriguing rare objects with many connections within mathematics. Of particular interest are the exceptional ones, as defined in 2018 by the first author and Zhou, for which…
We study partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. These permutations are the linear…
A rigid automorphism of a linking system is an automorphism which restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that…
Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work [2]. Here we consider, more generally, affine Grassmann codes of a given level.…
There has been considerable effort to better understand the generalization capabilities of deep neural networks both as a means to unlock a theoretical understanding of their success as well as providing directions for further improvements.…
We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part…
Rank-metric codes, defined as sets of matrices over a finite field with the rank distance, have gained significant attention due to their applications in network coding and connections to diverse mathematical areas. Initially studied by…
We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In [9], automorphism groups of quandles (up…
We study manifolds endowed with an (almost) even Clifford (hermitian) structure and admitting a large automorphism group. We classify them when they are simply connected and the dimension of the automorphism group is maximal, and also prove…
We show that the fundamental groups of any two closed irreducible non-geometric graph-manifolds are quasi-isometric. This answers a question of Kapovich and Leeb. We also classify the quasi-isometry types of fundamental groups of…
In this paper we describe graded automorphisms and antiautomorphisms of finite order on matrix algebras endowed with a group gradings by a finite abelian group over an arbitrary algebraically closed field of charcteristic different from 2.