Characterization of some simple groups by the multiplicity pattern

Let $G$ be a finite group and let ${\mathrm{Irr}}(G)$ denote the set of all complex irreducible characters of $G.$ Let ${\mathrm{cd}}(G)$ be the set of all character degrees of $G.$ For each positive integer $d,$ the multiplicity of $d$ in $G$ is defined to be the number of irreducible characters of $G$ having the same degree $d.$ The multiplicity pattern ${\mathrm{mp}}(G)$ is the vector whose first coordinate is $|G:G^{\prime }|$ and for $i\ge 1,$ the $(i+1)$ th-coordinate of ${\mathrm{mp}}(G)$ is the multiplicity of the $i$ th-smallest nontrivial character degree of $G.$ In this paper, we show that every nonabelian simple group with at most $7$ distinct character degrees is uniquely determined by the multiplicity pattern.     

Posicast control of a class of fractional-order processes

The number of studies on the control of fractional-order processes—processes having dynamics described by differential equations of arbitrary order—has been increasing in the past two decades and it is now ubiquitous. Various methods have emerged and have been proven to effectively control such processes—usually resulting in fractional-order controllers similar to their conventional integer-order counterparts, which include, but are not limited to fractional PID and fractional lead-lag controllers. However, such methods require a lot of computational effort and fractional-order controllers could be challenging when it comes to their synthesis and implementation. In this paper, we propose a simple yet effective delay-based controller with the use of the Posicast control methodology in controlling the overshoot of a fractional-order process of the class $\mathcal{P}:\left\{ {P\left( s \right) = {1 \mathord{\left/ {\vphantom {1 {\left( {as^\alpha + b} \right)}}} \right. \kern-0em} {\left( {as^\alpha + b} \right)}}} \right\}$ having orders 1 < α < 2. Such controllers have proven to be easy to implement because they only require delays and summers. In this paper, the Posicast control methodology introduced in the past few years is modified to minimize the overshoot of the processes step response to a level that is acceptable in control engineering and automation practices. Furthermore, proof of the existence of overshoot for such class of processes, as well as the determination of the peak-time of the open-loop response of a fractional-order process of the class P is presented. Validation through numerical simulations for a class of fractional-order processes are presented in this paper.     

Electromigration Studies of Carrier-Free 239Np(V) in Aqueos Solutions

Ion mobilities of carrier-free ²³⁹Np(V) have been measured in aqueous solutions, T = 298.1 (1) K. Ion mobilities of ²³⁹NpO₂ ⁺ and its dependencies on pH of acidic inert electrolytes have been measured. In alkaline solutions the stoichimetric hydrolysis constants of NpO₂(OH) as well as NpO₂(OH)₂ ^? have been obtained. Complex formations of ²³⁹Np(V) with oxalate, tartrate, sulphate, acetate and citrate ligands have been studied in neutral solutions.     

Freezing and melting of binary mixtures confined in a nanopore

This paper reports on a Grand Canonical Monte Carlo study of the freezing and melting of Lennard–Jones Ar/Kr mixtures confined in a slit pore composed of two strongly attractive structureless walls. For all molar compositions and temperatures, the pore, which has a width of 1.44?nm, accommodates two contact layers and one inner layer. Different wall/fluid interactions are considered, corresponding to pore walls that have a larger affinity for either Ar or Kr. The solid/liquid phase diagram of the confined mixture is determined and results compared with data for the bulk mixture. The structure of the confined mixture is studied using 2D order parameters and both positional g(r) and bond orientational G₆(r) pair correlation functions. It is found that in the confined solid phase, both the contact and inner layers have a hexagonal crystal structure. It is shown that the freezing temperature of the Ar/Kr confined mixture is higher than the bulk freezing point for all molar compositions. Also, it is found that the freezing temperature becomes larger as the ratio α of the wall/fluid to the fluid/fluid interactions increases, in agreement with previous simulation studies on pure substances confined in nanopores. In the case of pore walls having a stronger affinity for Kr atoms (ε _Ar/W<ε _Kr/W), it is observed that both the contact and inner layers of the confined mixture undergo, at the same temperature, a transition from the liquid phase to the crystal phase. The freezing of Ar/Kr mixtures confined between the walls having a stronger affinity for Ar (ε _Ar/W?>?ε _Kr/W) is more complex: for Kr molar concentration lower than 0.35, we observe the presence of an intermediate state between all layers being 2D hexagonal crystals and all the layers being liquid. This intermediate state consists of a crystalline contact layer and a liquid-like inner layer. It is also shown that the qualitative variations of the increase of freezing temperature with the molar composition depend on the affinity of the pore wall for the different components. These results confirm that, in addition to the parameter α the ratio of the wall/fluid interactions for the two species, η=?_Ar/W/?_Kr/W, is a key variable in determining the freezing and melting behaviour of the confined mixture.     

Pressure-dependent EXAFS mean-square relative displacements of germanium and silicon crystals

In this paper, the statistical moment method (SMM) has been developed to study the pressure dependence of thermodynamic quantities of germanium and silicon crystals. We have derived the analytical expressions of the pressure-dependent parallel mean-square relative displacement (MSRD) or extended X-ray absorption fine structure (EXAFS) Debye–Waller factor, mean-square displacement (MSD) as well as lattice constant and volume change of diamond-type crystals. Numerical calculations performed for these semiconductors up to 11 GPa are found to be in good and reasonable agreement with available experimental data as well as with previous theoretical studies. Our results indicate that the SMM can be efficiently used for determining the relative change of the pressure-dependent MSRDs of germanium and silicon semiconductors. The research also shows the advantage of SMM on studying other thermodynamic properties of materials under high pressures.     

On Huppert’s Conjecture for 3 D 4(q), q ≥ 3

Let G be a finite group and cd(G) be the set of all complex irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G???H × A, where A is an abelian group. In this paper, we verify the conjecture for the family of simple exceptional groups of Lie type ³ D ₄(q), when q?≥?3.     

Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations

We consider the Hardy–Hénon parabolic equation ${u_t-\Delta u =|x|^a |u|^{p-1}u}$ with p > 1 and ${a\in \mathbb{R}}$ . We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial-boundary value problem.