let $r$ be an infinite ring. here we prove that if $0_r$ belongs to ${x_1x_2cdots x_n ;|; x_1,x_2,dots,x_nin x}$ for every infinite subset $x$ of $r$, then $r$ satisfies the polynomial identity $x^n=0$. also we prove that if $0_r$ belongs to ${x_1x_2cdots x_n-x_{n+1} ;|; x_1,x_2,dots,x_n,x_{n+1}in x}$ for every infinite subset $x$ of $r$, then $x^n=x$ for all $xin r$.

The present article deals with differential equations with spectral parameter from the point of view of formal power series. The treatment does not make use of the notion of eigenvalue, but introduces a new idea: the spectral residue. The article focuses on second-order, self-adjoint problems. In such a setting every potential function determines a sequence of spectral residues. This correspond...

In graph theory, lattices are used when some structural part of the repeats itself finitely or infinitely many times. They have applications in complex analysis and geometry mathematics, also natural chemical theory. As a lattice can be taken as graph, it is possible to use them study large networks. Recently, topological indices pentagonal chains studied here, we double similarly that work. We...

The number cn of weighted partitions of an integer n, with parameters (weights) bk, k ≥ 1, is given by the generating function relationship ∑∞ n=0 cnz n = ∏∞ k=1(1− zk)−bk . Meinardus(1954) established his famous asymptotic formula for cn, as n → ∞, under three conditions on power and Dirichlet generating functions for the sequence bk. We give a probabilistic proof of Meinardus’ theorem with we...

We verify the skew weighing matrix conjecture for orders 2t.7, t ~ 3 a positive integer, by showing that orthogonal (1, k) exist for all t k = 0, 1, .... , 2.7 1 in order 2t.7 We discuss the construction of orthogonal designs using circulant matrices. In particular we construct designs in orders 20 and 28. The weighing matrix conjecture is verified for order 60. Disciplines Physical Sciences an...

[1] Victor V. Batyrev and Benjamin Nill. Multiples of lattice polytopes without interior lattice points. Moscow Mathematical Journal 7:195–207, 2007. [2] Victor V. Batyrev, Benjamin Nill. Combinatorial aspects of mirror symmetry. Contemporary Mathematics, 452:35–66, 2008. [3] Barbara Baumeister, Christian Haase, Benjamin Nill and Andreas Paffenholz. On permutation polytopes. Advances in Mathema...

This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose nodes and links graph present evidence for their interpretation as curvature. Notably, we find relation number well-established curvatures (Ollivier, Forman, combinatorial curvature) show convergence continuous in case Euclidean random graphs. Being both efficient calculate hig...

this study historically analyzes the scientific output of mathematics to describe its structure, notions and scientific origin using the historiographical method. the research data consisted of scientific outputs of mathematics in isi database, science citation index (sci) and social science citation index (ssci) during 8 years (1990 to 2007). the data collection tool was the search engine and ...