Metric properties of Cayley graphs of alternating groups

A well known diameter search problem for finite groups with respect to its systems of generators is considered. The problem can be formulated as follows: find the diameter of a group over its system of generators. The diameter of a group over a specific system of generators is the diameter of the corresponding Cayley graph. It is considered alternating groups with classic irreducible system of generators consisting of cycles with length three of the form $(1,2,k)$. The main part of the paper concentrates on analysis how even permutations decompose with respect to this system of generators. The rules for moving generators from permutation's decomposition from left to right and from right to left are introduced. These rules give rise for transformations of decompositions, that do not increase their lengths. They are applied for removing fixed points of a permutation, that were included in its decomposition. Based on this rule the stability of system of generators is proved. The strict growing property of the system of generators is also proved, as the corollary of transformation rules and the stability property. It is considered homogeneous theory, that was introduced in the previous author's paper. For the series of alternating groups with systems of generators mentioned above it is shown that this series is uniform and homogeneous. It makes possible to apply the homogeneous down search algorithm to compute the diameter. This algorithm is applied and exact values of diameters for alternating groups of degree up to 43 are computed.

Influence of Alloy Atoms on Substitution Properties of Hydrogen by Helium in ZrCoH3

Intermetallic alloy ZrCo is a good material for storing tritium (T). However, ZrCo is prone to disproportionation reactions during the process of charging and discharging T. Alloying atoms are often added in ZrCo, occupying the Zr or Co site, in order to restrain disproportionation reactions. Meanwhile, T often decays into helium (He), and the purity of T seriously decreases once He escapes from ZrCo. Therefore, it is necessary to understand the influence of alloying atoms on the basic stability property of He. In this work, we perform systematical ab initio calculations to study the stability property of He in ZrCoH3 (ZrCo adsorbs the H isotope, forming ZrCoH3). The results suggest that the He atom will undergo displacements of 0.31 and 0.12 Å when it substitutes for Co and Zr, respectively. In contrast, the displacements are very large, at 0.67–1.09 Å, for He replacing H. Then, we introduce more than 20 alloying atoms in ZrCo to replace Co and Zr in order to examine the influence of alloying atoms on the stability of He at H sites. It is found that Ti, V, Cr, Mn, Fe, Zn, Nb, Mo, Tc, Ru, Ta, W, Re, and Os replacing Co can increase the substitution energy of H by the He closest to the alloying atom, whereas only Cr, Mn, Fe, Mo, Tc, Ru, Ta, W, Re, and Os replacing Co can increase the substitution energy of H by the He next closest to the alloying atom. The influence of the alloying atom substituting Zr site on the substitution energies is inconspicuous, and only Nb, Mo, Ru, Ta, and W increase the substitution energies of H by the He closest to the alloying atom. The increase in the substitution energy may suggest that these alloy atoms are conducive to fix the He atom in ZrCo and avoid the reduction in tritium purity.

Long-time dynamics of a class of nonlocal extensible beams with time delay

This work is devoted to the following nonlocal extensible beam equation with time delay: [Formula: see text] on a bounded smooth domain [Formula: see text]. The main purpose of this paper is to consider the long-time dynamics of the system. Under suitable assumptions, the quasi-stability property of the system is established, based on which the existence and regularity of a finite-dimensional compact global attractor are obtained. Moreover, the existence of exponential attractors is proved.

Identification of van der Pol oscillator parameters

An invariant relations method for parameters estimation is used for the wellknown van der Pol oscillator. The approach is based on dynamical extension of original system and synthesis of appropriate invariant relations, from which the unknowns can be expressed as a functions of the known quantities on the trajectories of extended system. The stability property is formally checked taking into account the oscillatory behavior of the system. The simulation results confirm efficiency of the proposed scheme of nonlinear identificator design.

A Singularly P-Stable Multi-Derivative Predictor Method for the Numerical Solution of Second-Order Ordinary Differential Equations

In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order.