Simplified time-dependent self-consistent field approximation for intramolecular dynamics

The time-dependent Hartree approximation is applied to intramolecular dynamics of polyatomics with smooth, locally quadratic potential surfaces. It is shown that the full quantum solution is obtained from a certain single self-consistent trajectory. An extremely simple model results, pertinent to intramolecular energy transfer, vibrational lineshapes and unimolecular decay.     

Overlap Cycles for Steiner Quadruple Systems

Steiner quadruple systems are set systems in which every triple is contained in a unique quadruple. It is well known that Steiner quadruple systems of order v, or SQS(v), exist if and only if . Universal cycles, introduced by Chung, Diaconis, and Graham in 1992, are a type of cyclic Gray code. Overlap cycles are generalizations of universal cycles that were introduced in 2010 by Godbole, et al. Using Hanani's SQS constructions, we show that for every with there exists an SQS(v) that admits a 1‐overlap cycle.     

Asymptotic behavior of solutions of nonlinear systems with multiple imaginary eigenvalues

The paper is devoted to the analysis of the decay rate of solutions of a nonlinear system with two pairs of purely imaginary eigenvalues. The main result is the power estimate for the norm of solutions. It is proven that the order of such estimate varies for cases of a diagonalizable matrix of linear approximation, and for a matrix containing a Jordan block. Another result of this work provides sufficient asymptotic stability conditions regardless of forms higher than the third order. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semigroups of inverse quotients

We examine, in a general setting, a notion of inverse semigroup of left quotients, which we call left I-quotients. This concept has appeared, and has been used, as far back as Clifford??s seminal work describing bisimple inverse monoids in terms of their right unit subsemigroups. As a consequence of our approach, we find a straightforward way of extending Clifford??s work to bisimple inverse semigroups (a step that has previously proved to be awkward). We also put some earlier work of Gantos into a wider and clearer context, and pave the way for further progress.     

Axiomatisability problems for <Emphasis Type="Italic">S</Emphasis>-posets

Let \(\mathcal{C}\) be a class of ordered algebras of a given fixed type τ. Associated with the type is a first order language L _τ , which must also contain a binary predicate to be interpreted by the ordering in members of \(\mathcal{C}\). One can then ask the question, when is the class \(\mathcal{C}\) axiomatisable by sentences of L _τ ? In this paper we will be considering axiomatisability problems for classes of left S-posets over a pomonoid S (that is, a monoid S equipped with a partial order compatible with the binary operation). We aim to determine the pomonoids S such that certain categorically defined classes are axiomatisable. The classes we consider are the free S-posets, the projective S-posets and classes arising from flatness properties. Some of these cases have been studied in a recent article by Pervukhin and Stepanova. We present some general strategies to determine axiomatisability, from which their results for the classes of weakly po-flat and po-flat S-posets will follow. We also consider a number of classes not previously examined.     

Convergence of discrete and penalized least squares spherical splines

We study the convergence of discrete and penalized least squares spherical splines in spaces with stable local bases. We derive a bound for error in the approximation of a sufficiently smooth function by the discrete and penalized least squares splines. The error bound for the discrete least squares splines is explicitly dependent on the mesh size of the underlying triangulation. The error bound for the penalized least squares splines additionally depends on the penalty parameter.     

Mathematical analysis of the dynamics of COVID-19 in Africa under the influence of asymptomatic cases and re-infection

Coronavirus pandemic (COVID-19) hit the world in December 2019, and only less than 5% of the 15 million cases were recorded in Africa. A major call for concern was the significant rise from 2% in May 2020 to 4.67% by the end of July 15, 2020. This drastic increase calls for quick intervention in the transmission and control strategy of COVID-19 in Africa. A mathematical model to theoretically investigate the consequence of ignoring asymptomatic cases on COVID-19 spread in Africa is proposed in this study. A qualitative analysis of the model is carried out with and without re-infection, and the reproduction number is obtained under re-infection. The results indicate that increasing case detection to detect asymptomatically infected individuals will be very effective in containing and reducing the burden of COVID-19 in Africa. In addition, the fact that it has not been confirmed whether a recovered individual can be re-infected or not, then enforcing a living condition where recovered individuals are not allowed to mix with the susceptible or exposed individuals will help in containing the spread of COVID-19.     

Membership of A ∨ G for classes of finite weakly abundant semigroups

We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.