For a sequence S of terms from an abelian group G length |S|, let \(\Sigma _n(S)\) denote the set all elements that can be represented as sum in some n-term subsequence S. When subsum is very small, \(|\Sigma _n(S)|\le |S|-n+1\), it known partitioned into n nonempty sets \(A_1,\ldots ,A_n\subseteq G\) such _n(S)=A_1+\ldots +A_n\). Moreover, if upper bound strict, then \(|A_i\setminus Z|\le 1\) ...

Let $R$ be a commutative ring with identity‎. ‎A proper ideal $P$ of $R$ is a $(n-1,n)$-$Phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin R$‎, ‎$a_1cdots a_nin Pbackslash P^m$ ($a_1cdots a_nin Pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin P$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎In this paper several results concerning $(n-1,n)$-$Phi_m$-prime and $(n-1,n)$-...

We study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank-fourteen, 2-elementary lattices. Three such lattices exist, namely $H \oplus E_8(-1) A_1(-1)^{\oplus 4}$, D_4(-1)$, D_8(-1) D_4(-1)$. As part of our study, we provide birational models for these as quartic projective hypersurfaces describe the associated coarse moduli spaces in terms suitable modular in...

Abstract In this work we study what call Siegel–dissipative vector of commuting operators $$(A_1,\ldots , A_{d+1})$$ ( A 1 , … d + ) on a Hilbert...

Summary The main purpose of the article is to construct a sophisticated polynomial proposed by Matiyasevich and Robinson [5] that often used reduce number unknowns in diophantine representations, using Mizar [1], [2] formalism. J k ( a <...

In this paper we describe the equations defining multi-Rees algebra $k[x_1,\dots,x_n][I_1^{a_1}t_1,\dots,I_r^{a_r}t_r]$, where ideals are generated by subsets of $x_1,\dots,x_n$. We also show that a family binomials whose leading terms squrefree, form Gr\"{o}bner basis for with lexicographic order. if remove include $x$'s, then remaining toric ideal associated to multi-fiber ring. However binom...

Abstract We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set real numbers, and the differences consecutive elements are all distinct. It known that $|A+B|\geq c|A||B|^{1/2}$ for any finite numbers $B.$ The bound tight up to constant multiplier. give new proof this result using bounds on crossing geometric graphs. construct examples show...

GF(2)-grammars are a recently introduced grammar family that has some unusual algebraic properties and is closely connected to the of unambiguous grammars. By using method formal power series, we establish strong conditions necessary for subsets $$a_1^* a_2^* \cdots a_k^*$$ be described by GF(2)-grammar. further applying established results, settle long-standing open question proving inherent a...

Let K be a field and let $$S=K[x_1,\ldots ,x_n]$$ standard polynomial ring over K. We characterize the extremal Betti numbers, values as well positions, of t-spread strongly stable ideal S. Our approach is constructive. Indeed, given some positive integers $$a_1,\dots ,a_r$$ pairs $$(k_1,\ell _1),\ldots ,(k_r,\ell _r)$$ , we are able to determine under which conditions there exists I S with $$\...

Kontsevich's graph flows are -- universally for all finite-dimensional affine Poisson manifolds infinitesimal symmetries of the spaces brackets. We show that previously known tetrahedral flow and recently obtained pentagon-wheel preserve class Nambu-determinant bi-vectors $P=[\![ \varrho(\boldsymbol{x})\,\partial_x\wedge\partial_y\wedge\partial_z,a]\!]$ on $\mathbb{R}^3\ni\boldsymbol{x}=(x,y,z)...