Let $R$ be an associative ring with unity. An element $x \in R$ is called $\mathbb{Z}G$-clean if $x=e+r$, where $e$ is an idempotent and $r$ is a $\mathbb{Z}G$-regular element in $R$. A ring $R$ is called $\mathbb{Z}G$-clean if every element of $R$ is $\mathbb{Z}G$-clean. In this paper, we show that in an abelian $\mathbb{Z}G$-regular ring $R$, the $Nil(R)$ is a two-sided ideal of $R$ and $\fra...

Recently, V.Ginzburg introduced the notion of a principal nilpotent pair (= pn-pair) in a semisimple Lie algebra g [Gi99]. It is a double counterpart of the notion of a regular nilpotent element in g. A pair e = (e1, e2) ∈ g × g is called nilpotent, if [e1, e2] = 0 and there exists a pair h = (h1, h2) of semisimple elements such that [h1, h2] = 0, [hi, ej ] = δijej (i, j ∈ {1, 2}). A pn-pair e ...

We reconstruct glass equations of state for polydisperse hard spheres with the help computer simulations. To perform reconstructions, we assume that hard-sphere have form Zg(?, ?J) = Zg(?J/?), where Zg, ?, and ?J are reduced pressure (PV/NkBT), sphere volume fraction (packing density), jamming density current basin attraction, respectively. Specifically, use X ?iciYi, (?J/?) ? 1 Y 1/(Zg 1). Our...

Let K(α), 0≤α< 1, denote the class of functions g(z)= z+∑∞n=2anzn which are regular and univalently convex of order α in the unit disc U . Pursuing the problem initiated by Robinson in the present paper, among other things, we prove that if f is regular in U , f(0) = 0, and f(z)+zf ′(z) < g(z)+zg′(z) in U , then (i) f(z) < g(z) at least in |z| < r0, r0 = √ 5/3 = 0.745 . . . if f ∈ K; and (ii) f...

where J is the all ones matrix and I is the identity. These matrix conditions are equivalent to the combinatorial conditions that the graph is both inand out-regular, and that the number of directed 2-paths from a vertex x to a vertex y is t if x = y, λ if x → y, and μ otherwise. Recently, these graphs were studied by Klin et al. [2], including some new constructions and a list of feasible para...

Let g be a semisimple Lie algebra over the field of real numbers. G group with g. The Weyl respect to Cartan subalgebra h is defined as W(G,h)=NG(h)/ZG(h). We describe an explicit construction W(G,h) for groups that arise set points connected algebraic groups. show this also gives when adjoint This algorithm important classification regular subalgebras, carrier algebras, and nilpotent orbits as...

Let ZG be the integral group ring of the finite nonabelian group G over the ring of integers Z, and let ∗ be an involution of ZG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (uk,m(x), uk,m(x ∗)), or (uk,m(x), uk,m(y)), formed respectively by Bass cyclic and ∗-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ZG. 1....

Using the Luthar–Passi method, we investigate the Zassenhaus and Kimmerle conjectures for normalized unit groups of integral group rings of the Held and O’Nan sporadic simple groups. We confirm the Kimmerle conjecture for the Held simple group and also derive for both groups some extra information relevant to the classical Zassenhaus conjecture. Let U(ZG) be the unit group of the integral group...

Common somatic mutations in CACNAID and ATP1A1 may define a subgroup of smaller, zona glomerulosa (ZG)-like aldosterone-producing adenomas. We have therefore sought signature ZG genes, which may provide insight into the frequency and pathogenesis of ZG-like aldosterone-producing adenomas. Twenty-one pairs of zona fasciculata and ZG and 14 paired aldosterone-producing adenomas from 14 patients w...