Classroom note: On postulates of a group

A semigroup G is a group if it has a left identity and every element has a left inverse. The purpose of this note is to weaken this condition further in two different ways.     

A Zero Divisor Graph Determined by Equivalence Classes of Zero Divisors

We study the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring R. We demonstrate how to recover information about R from this structure. In particular, we determine how to identify associated primes from the graph.     

DECIMAL EXTENSIONS OF THE ADDITIVE GROUP OF INTEGERS

We introduce the concept of decimal extension of the additive group of integers. We characterize these extensions as the linearly ordered abelian groups containing at least one Archimedean element. Finally, we study the additive group of rationals and its subgroups as particular cases of decimal extension.     

On a new type of distance Fibonacci numbers

In this paper, we define a new kind of Fibonacci numbers generalized in the distance sense. This generalization is related to distance Fibonacci numbers and distance Lucas numbers, introduced quite recently. We also study distinct properties of these numbers for negative integers. Their representations and interpretations in graphs are also studied.     

New Versions of Some Classical Stochastic Inequalities

Motivated by an investigation of finding necessary and sufficient conditions for the Kolmogorov strong law of large numbers for the general Gini's mean difference, new versions of the classical Lévy, Ottaviani, and Hoffmann-J\ooTrgensen inequalities are obtained for a sequence of Banach space valued random variables. No geometric conditions are imposed on the Banach space. An application to the general Gini's mean difference in a Banach space setting is presented.     

Anomalous birefringence and dilatation in the Rb(H,D)3(Se0)3))2 near normalincommensurate phase transition: Fluctuations and defects

An attempt is made to extend the method of experimental analysis of fluctuation behaviour at normal (N)—incommensurate (INC) phase transitions in the case of narrow INC-phases. Values of fluctuation parameters (the “effective Ginzburg Numbers”) for RbH₃(SeO₃)₂ and RbD₃(SeO₃)₂ crystals are suggested and compared with those of some other crystals from the A₂BX₄ family (G = 0.01 ÷ 0.036). In addition, in the crossover region the role of defects is clearly demonstrated by comparing the behaviour of incompletely deuterated crystal Rb(H, D)₃(SeO₃)₂ with that of RbH₃(SeO₃)₂.     

Singular numbers of contractions in spaces with an indefinite metric and Yamamoto's theorem

A theorem of Yamamoto on singular numbers of matrices is extended to G-bounded matrices.     

关于完全平方数的一个性质

运用不定方程的理论讨论了完全平方数的一个基本性质,得到了关于完全平方数的几个重要定理.