Probability that a given element of a group is a commutator of any two randomly chosen group elements

We study the probability of a given element, in the commutator subgroup of a group, to be equal to a commutator of two randomly chosen group elements, and compute explicit formulas for calculating this probability for some interesting classes of groups having only two different conjugacy class sizes. We re-prove the fact that if $G$ is a finite group such that the set of its conjugacy class sizes is $\{1, p\}$, where $p$ is a prime integer, then $G$ is isoclinic (in the sense of P. Hall) to an extraspecial $p$-group.Comment: 14 page

Meshless RBF based pseudospectral solution of acoustic wave equation

Chebyshev pseudospectral (PS) methods are reported to provide highly accurate solution using polynomial approximation. Use of polynomial basis functions in PS algorithms limits the formulation to univariate systems constraining it to tensor product grids for multi-dimensions. Recent studies have shown that replacing the polynomial by radial basis functions in pseudospectral method (RBF-PS) has the advantage of using irregular grids for multivariate systems. A RBF-PS algorithm has been presented here for the numerical solution of inhomogeneous Helmholtz's equation using Gaussian RBF for derivative approximation. Efficacy of RBF approximated derivatives has been checked through error analysis comparison with PS method. Comparative study of PS, RBF-PS and finite difference approach for the solution of a linear boundary value problem has been performed. Finally, a typical frequency domain acoustic wave propagation problem has been solved using Dirichlet boundary condition and a point source. The algorithm presented here can be extended further for seismic modeling with complexities associated with absorbing boundary conditions.Comment: 6 pages, 4 figure

Some bounds on commutativity degree

The relative commutativity degree of a subgroup $H$ of a finite group $G$, denoted by $\Pr(H, G)$, is the probability that an element of $G$ commutes with an element of $H$. In this article we obtain some lower and upper bounds for $\Pr(H, G)$ and their consequences. We also study an invariance property of $\Pr(H, G)$ and its generalizations, under isoclinism of pairs of groups.Comment: 10 page

Emergence of nonlinearity and plausible turbulence in accretion disks via hydromagnetic transient growth faster than magnetorotational instability

We investigate the evolution of hydromagnetic perturbations in a small section of accretion disks. It is known that molecular viscosity is negligible in accretion disks. Hence, it has been argued that Magnetorotational Instability (MRI) is responsible for transporting matter in the presence of weak magnetic field. However, there are some shortcomings, which question effectiveness of MRI. Now the question arises, whether other hydromagnetic effects, e.g. transient growth (TG), can play an important role to bring nonlinearity in the system, even at weak magnetic fields. Otherwise, whether MRI or TG, which is primarily responsible to reveal nonlinearity to make the flow turbulent? Our results prove explicitly that the flows with high Reynolds number (Re), which is the case of realistic astrophysical accretion disks, exhibit nonlinearity by best TG of perturbation modes faster than that by best modes producing MRI. For a fixed wavevector, MRI dominates over transient effects, only at low Re, lower than its value expected to be in astrophysical accretion disks, and low magnetic fields. This seriously questions (overall) persuasiveness of MRI in astrophysical accretion disks.Comment: 6 pages including 1 figure; to appear in the Proceedings of 14th Marcel Grossman Meeting (MG14), Rome, Italy, 12-18 July 2015; based on the talk given in the meeting in the session AC

A sum form functional equation on a closed domain and its role in information theory

This paper is devoted to finding the general solutions of the functional equation \sumin \sumjm h(p_iq_j)=\sumin h(p_i)+\sumjm k_j(q_j)+\lambda\sumin h(p_i)\sumjm k_j(q_j) valid for all complete probability distributions $(p_1,\ldots,p_n)$, $(q_1,\ldots,q_m)$, $0\le p_i\le 1$, $0\le q_j\le 1$, $i=1,\ldots,n$; $j=1,\ldots,m$, \sumin p_i=1, \sumjm q_j=1; $n\ge 3$, $m\ge 3$ fixed integers; \lambda\in\RR, $\lambda\neq 0$ and the mappings h:I\to\RR, k_j:I\to\RR, $j=1,\ldots,m$; $I=[0,1]$, \RR denoting the set of all real numbers. A special case of the above functional equation was treated earlier by L. Losonczi and Gy. Maksa.Comment: 18 pages. Already submitted for revie

An Automated Ac Susceptibility Set up Fabricated Using a Closed-Cycle Helium Refrigerator

We have described here the design and operation of an automated ac susceptibility set up using a closed cycle helium refrigerator. This set up is useful for measuring linear and nonlinear magnetic susceptibilities of various magnetic materials. The working temperature range is 2 K to 300 K. The overall sensitivity of the set up is found to be 10-3 emu

Detection techniques and investigation of different neutrino experiments

Neutrino physics is an experimentally driven field. So, we investigate the different detection techniques available in the literature and study the various neutrino oscillation experiments in a chronological manner. Our primary focus is on the construction and detection mechanisms of each experiment. Today, we know a lot about this mysterious ghostly particle by performing different experiments at different times with different neutrino sources viz. solar, atmospheric, reactor, accelerators and high energy astrophysical; and they have contributed in the determination of neutrino parameters. Yet the problems are far from over. We need to determine more precise values of the already known parameters and unravel the completely unknown parameters. Some of the unknowns are absolute masses of neutrino, types of neutrino, mass hierarchy, octant degeneracy and existence of leptonic CP Phase(s). We analyse the neutrino experiments into the past, present and the future (or proposed). We include SNO, Kamiokande, K2K, MINOS, MINOS+, Chooz, NEMO and ICARUS in the past; while Borexino, Double Chooz, Super-K, T2K, IceCube, KamLAND, NO$\nu$A, RENO and Daya Bay in the present; and SNO+, Hyper-K, JUNO, RENO-50, INO, DUNE, SuperNEMO, KM3NeT, P2O, LBNO and PINGU in the proposed experiments. We also discuss the necessities of upgrading the present ones to those of the proposed ones thereby summarizing the potentials of the future experiments. We conclude this paper with the current status of the neutrinos.Comment: 17 Figures, 4 Table

Critical behavior and magnetic relaxation dynamics of Nd0.4Sr0.6MnO3 nanoparticles

Detailed dc and ac magnetic properties of chemically synthesized Nd0.4Sr0.6MnO3 with different particle size (down to 27 nm) have been studied in details. We have found ferromagnetic state in the nanoparticles, whereas, the bulk Nd0.4Sr0.6MnO3 is known to be an A-type antiferromagnet. A Griffiths-like phase has also been identified in the nanoparticles. Further, critical behavior of the nanoparticles has been studied around the second order ferromagnetic-paramagnetic transition region (|(T-TC)/TC|{\pounds} 0.04) in terms of modified Arrott plot, Kouvel-Fisher plot and critical isotherm analysis. The estimated critical exponents (b,g,d) are quite different from those predicted according to three-dimensional mean-field, Heisenberg and Ising models. This signifies a quite unusual nature of the size-induced ferromagnetic state in Nd0.4Sr0.6MnO3. The nanoparticles are found to be interacting and do not behave like ideal superparamagnet. Interestingly, we find spin glass like slow relaxation of magnetization, aging and memory effect in the nanometric samples. These phenomena have been attributed to very broad distribution of relaxation time as well as to inter-particle interaction. Experimentally, we have found out that the dynamics of the nanoparticle systems can be best described by hierarchical model of spin glasses.Comment: Philosophical Magazine (2013

A note on nn-centralizer finite rings

Let $R$ be a finite ring and let \Cent(R) denote the set of all distinct centralizers of $R$. $R$ is called an $n$-centralizer ring if $|Cent(R)| = n$. In this paper, we characterize $n$-centralizer finite rings for $n \leq 7$.Comment: 12 page

Electron mobility in InxGa1-xN channel HEMTs

In this letter, we report on the theoretical investigations of electron mobility in practically viable designs of InxGa1-xN channel high electron mobility transistors (HEMT). Carriers in such devices are expected to exhibit a higher velocity and hence higher cut-off frequencies (fT) for highly scaled architectures. We estimate that the mobility of two dimensional electron gas (2DEG) is limited by alloy scattering rather than phonon scattering unlike in conventional GaN-channel HEMTs. For indium composition of 0.30, the mobility and sheet resistance are found to be 500 cm2/Vs and 700 ohm per sq. respectively, which can severely affect the parasitic voltage drop in access regions. The results presented here are believed to significantly guide the practical exploration of InxGa1-xN channel HEMTs towards next-generation electronics by enabling careful design of device layouts in highly scaled transistors to minimize parasitic access region voltage drop which results due to significant degradation of 2DEG mobility.Comment: 4 figures, 15 page