Abstract Let G G be a group and R R -graded commutative ring with nonzero unity 1. In this article, we introduce the concept of graded weakly 1-absorbing primary ideals which is generalization ideal. A proper ideal P P said to if whenever nonunit elements x , y<...

Let a > b > 1 be integers. We proved that if bn−1 divides an−1 infinitely often, then a is a power of b. In a joint work with Bugeaud, we also proved that the inequality gcd(an−1, bn−1) ¿ exp(2n) holds for large n, whenever a, b are multiplicatively independent. This fact has been generalised to an analogue inequality where an, bn are replaced by arbitrary S-units in a number field; we also dis...

Let F ∈ W 1 , n loc (Ω ; R ) be a mapping with nonnegative Jacobian JF (x) = detDF (x) ≥ 0 for a.e. x in a domain Ω ⊂ R n . The dilatation of F is defined (almost everywhere in Ω) by the formula K(x) = |DF (x)| JF (x) · Iwaniec and Šverák [IS] have conjectured that if p ≥ n − 1 and K ∈ Lploc(Ω) then F must be continuous, discrete and open. Moreover, they have confirmed this conjecture in the tw...

Let {Yn}n∈Z+ be a sequence of random variables in R and let A ⊂ R. Then P{Yn ∈ A for some n} is the hitting probability of the set A by the sequence {Yn}. We consider the asymptotic behavior, as m → ∞, of P{Yn ∈ mA, some n} = P{hitting mA} whenever (1) the probability law of Yn/n satisfies the large deviation principle and (2) the central tendency of Yn/n is directed away from the given set A. ...

Consider lattice paths in Z taking unit steps north (N) and east (E). Fix positive integers r, s and put an equivalence relation on points of Z by letting v,w be equivalent if v − w = l(r, s) for some l ∈ Z. Call a lattice path valid if whenever it enters a point v with an E-step, then any further points of the path in the class of v are also entered with an E-step. Loehr and Warrington conject...

Let Mn(R) be the algebra of all n×n matrices over a unital commutative ring R with 6 invertible. We say that A ∈ Mn(R) is a Jordan product determined point if for every R-module X and every symmetric R-bilinear map {·, ·} : Mn(R)×Mn(R) → X the following two conditions are equivalent: (i) there exists a fixed element w ∈ X such that {x, y} = w whenever x ◦ y = A, x, y ∈ Mn(R); (ii) there exists ...

All rings R considered are commutative and have an identity element. Contessa called R a VNL-ring if a or 1 a has a Von Neumann inverse whenever a 2 R. Sample results: Every prime ideal of a VNL-ring is contained in a unique maximal ideal. Local and Von Neumann regular rings are VNL and if the product of two rings is VNL, then both are Von Neumann regular, or one is Von Neumann regular and the ...