Related papers: A Note on Slice Rank and Matchings in Groups

We prove that any multiplicative subgroup G of the prime field f_p with |G| < p^{1/2} satisfies |3G| \gg |G|^2 / \log |G|. Also, we obtain a bound for the multiplicative energy of any nonzero shift of G, namely E^* (G+x) \ll |G|^2 log |G|,…

In \cite[Theorem 2.5]{Bac16} Bachiller proved that if $(G, \cdot, \circ)$ is a brace of order the power of a prime $p$ and the rank of $(G,\cdot)$ is smaller than $p-1$, then the order of any element is the same in the additive and…

A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of…

We consider symmetric tensors of format: $3 \times 3$ over $\mathbb{F}_p$ for $p = 2, 3, 5$; $3 \times 3 \times 3$ over $\mathbb{F}_p$ for $p = 2, 3$; and $3 \times 3 \times 3 \times 3$ over $\mathbb{F}_p$ for $p = 2, 3$. In each case we…

In a finite abelian group $G$, define an additive matching to be a collection of triples $(x_i, y_i, z_i)$ such that $x_i + y_j + z_k = 0$ if and only if $i = j = k$. In the case that $G = \mathbb{F}_2^n$, Kleinberg, building on work of…

In this paper, we establish a bijection between the set of mutation classes of mutation-cyclic skew-symmetric integral 3x3-matrices and the set of triples of integers (a,b,c) which are all greater than 1 and where the product of the two…

One of prospective ways to find new fast algorithms of matrix multiplication is to study algorithms admitting nontrivial symmetries. In the work possible algorithms for multiplication of $3\times3$ matrices, admitting a certain group $G$…

We show that for a subset $A$ of the cyclic group of prime order $p>3$, if the sumset $A+A-2A$ is not the whole group, then $|A|\le \frac27\,p$. Besides combinatorial arguments, we utilize a general technique involving linear programming.

For positive integers $n\geq k\geq t$, a collection $ \mathcal{B} $ of $k$-subsets of an $n$-set $ X $ is called a $t$-packing if every $t$-subset of $ X $ appears in at most one set in $\mathcal{B}$. In this paper, we give some upper and…

We analyze rank decompositions of the $3\times 3$ matrix multiplication tensor over $\mathbb{Z}/2\mathbb{Z}$. We restrict our attention to decompositions of rank $\le 21$, as only those decompositions will yield an asymptotically faster…

Let $G$ be a non-abelian $p$-group of order $p^n$ and $M(G)$ be its Schur multiplier. It is well known result by Green that $|M(G)| \leq p^{\frac{1}{2}n(n-1)}$. So $|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$ for some $t(G) \geq 0$. The groups has…

A matching is a set of edges without common endpoint. It was recently shown that every 1-planar graph (i.e., a graph that can be drawn in the plane with at most one crossing per edge) that has minimum degree 3 has a matching of size at…

We say that a set is a multiplicative 3-Sidon set if the equation $s_1s_2s_3=t_1t_2t_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of…

We prove, in particular, that if A,G are two arbitrary multiplicative subgroups of the prime field f_p, |G| < p^{3/4} such that the difference A-A is contained in G then |A| \ll |\G|^{1/3+o(1)}. Also, we obtain that for any eps>0 and a…

This is the second in a series of papers on rank decompositions of the matrix multiplication tensor. We present new rank $23$ decompositions for the $3\times 3$ matrix multiplication tensor $M_{\langle 3\rangle}$. All our decompositions…

It is proved that there is no an algorithm for multiplication of $3\times3$ matrices of multiplicative length $\leq23$ that is invariant under a certain group isomorphic to $S_4\times S_3$. The proof makes use of description of the orbits…

We study multigraphs whose edge-sets are the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given such a graph $G$ and any $a_1,a_2,a_3\in \mathbb{N}$ with $a_1+a_2+a_3\leq n-2$, we show there exists a matching $M$ of $G$ with…

Suppose that a binary operation $\circ$ on a finite set $X$ is injective in each variable separately and also associative. It is easy to prove that $(X,\circ)$ must be a group. In this paper we examine what happens if one knows only that a…

Suppose $G$ is a undirected simple graph. A $k-$subset of edges in $G$ without common vertices is called a $k-$matching and the number of such subsets is denoted by $p(G,k)$. The aim of this paper is to present exact formulas for $p(G,3)$,…

We determine the structure of the finite non-solvable groups of order divisible by $3$ all whose maximal subgroups of order divisible by $3$ are supersolvable. Precisely, we demonstrate that if $G$ is a finite non-solvable group satisfying…