let r be a commutative ring and (r) be its zerodivisor graph. in this article, we studywiener index and energy of γ(zn ) where n = pq or n = p2q and p, q are primes. a matlabcode for our calculations is also presented.

We prove that the sonic surface of axi-symmetric meridional stationary flows is always attached to the accretor, however small, if the adiabatic index of the gas is γ = 5/3. Using local expansions near a point-like accretor, we extend Bondi's classification of spherically symmetric flows to axisymmetric flows, introducing the possibility of angular sectors reached by no flow lines, and singular...

Following the lines of [8], we study the Morse index of the iterated of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λγ on the unit circle with the property that the Morse index of the iterate...

In 1970 Alexander Grothendieck [6] posed the following problem: let Γ1 and Γ2 be finitely presented, residually finite groups, and let u : Γ1 → Γ2 be a homomorphism such that the induced map of profinite completions û : Γ̂1 → Γ̂2 is an isomorphism; does it follow that u is an isomorphism? In this paper we settle this problem by exhibiting pairs of groups u : P ↪→ Γ such that Γ is a direct product...

This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves u+ and u− in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of u+ at covers of γ agrees with the total multiplicity of the positive ends of u − at covers of γ. However, unlike in the usual g...

We derive a new and user friendly simple analytic approximation for determining the effect of intergalactic absorption on the γ-ray spectra of TeV blazars the energy range 0.2 TeV < Eγ < 2 TeV and the redshift range 0.05 < z < 0.4. In these ranges, the form of the absorption coefficient τ(Eγ) is approximately logarithmic. The effect of this energy dependence is to steepen intrinsic source spect...

The unidentified, low-latitude, γ-ray source 3EG J1828+0142 presents high levels of variability and a steep spectral index Γ ∼ 2.7. Here we propose a model for this source where the high-energy emission is produced by a galactic Kerr-Newman black hole. The model takes into account electron-positron annihilation losses in the calculation of the expected spectral energy distribution and can repro...

In this paper we study a new model for patterns in two dimensions, inspired by diblock copolymer melts with a dominant phase. The model is simple enough to be amenable not only to numerics but also to analysis, yet sophisticated enough to reproduce hexagonally packed structures that resemble the cylinder patterns observed in block copolymer experiments. Starting from a sharp-interface continuum...

We consider pairs of finitely presented, residually finite groups P ↪→ Γ for which the induced map of profinite completions P̂ → Γ̂ is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not P is isomorphic to Γ. We construct pairs for which the conjugacy problem in Γ can be solved in quadratic time but the conjugacy problem in P is uns...

Denote Sn(p)=kn−1∑i=1knlog(Xn+1−i,n∕Xn−kn,n)p, where p>0, kn≤n is a sequence of integers such that kn→∞ and kn∕n→0, X1,n≤⋯≤Xn,n are the order statistics iid random variables X1,…,Xn with regularly varying upper tail index 1∕γ. The estimator γ̂(n)=(Sn(p)∕Γ(p+1))1∕p an extension Hill estimator. We investigate asymptotic properties Sn(p) γ̂(n) both for fixed p>0 p=pn→∞. prove consistency limit theor...