Group inverse and core inverse in Banach and C*-algebras

Abstract

In this paper, we have focused our study on the acute perturbation of the group inverse for the elements of Banach algebra with respect to the spectral radius. We also give perturbation analysis for the core inverse in C*- algebra. The perturbation bounds for the core inverse under some conditions are presented. Additionally, this paper extends the results obtained in [11, 14].     

Electronic Perturbation of Copper Single-Atom CO2 Reduction Catalysts in a Molecular Way

Fine-tuning electronic structures of single-atom catalysts (SACs) plays a crucial role in harnessing their catalytic activities, yet challenges remain at a molecular scale in a controlled fashion. By tailoring the structure of graphdiyne (GDY) with electron-withdrawing/-donating groups, we show herein the electronic perturbation of Cu single-atom CO₂ reduction catalysts in a molecular way. The elaborately introduced functional groups (−F, −H and −OMe) can regulate the valance state of Cu^δ+, which is found to be directly scaled with the selectivity of the electrochemical CO₂-to-CH₄ conversion. An optimum CH₄ Faradaic efficiency of 72.3 % was achieved over the Cu SAC on the F-substituted GDY. In situ spectroscopic studies and theoretical calculations revealed that the positive Cu^δ+ centers adjusted by the electron-withdrawing group decrease the pK_a of adsorbed H₂O, promoting the hydrogenation of intermediates toward the CH₄ production. Our strategy paves the way for precise electronic perturbation of SACs toward efficient electrocatalysis.     

Magnetohydrodynamic mixed convection in a vertical channel

The problem of combined free and forced convective magnetohydrodynamic flow in a vertical channel is analysed by taking into account the effect of viscous and ohmic dissipations. The channel walls are maintained at equal or at different constant temperatures. The velocity field and the temperature field are obtained analytically by perturbation series method and numerically by finite difference technique. The results are presented for various values of the Brinkman number and the ratio of Grashof number to the Reynolds number for both equal and different wall temperatures. Nusselt number at the walls is determined. It is found that the viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. It is also found that the analytical and numerical solutions agree very well for small values of ε.     

Perturbation methods for nonlinear autonomous discrete-time dynamical systems

Two perturbation methods for nonlinear autonomous discrete-time dynamical systems are presented. They generalize the classical Lindstedt-Poincaré and multiple scale perturbation methods that are valid for continuous-time systems. The Lindstedt-Poincaré method allows determination of the periodic or almost-periodic orbits of the nonlinear system (limit cycles), while the multiple scale method also permits analysis of the transient state and the stability of the limit cycles. An application to the discrete Van der Pol equation is also presented, for which the asymptotic solution is shown to be in excellent agreement with the exact (numerical) solution. It is demonstrated that, when the sampling step tends to zero the asymptotic transient and steady-state discrete-time solutions correctly tend to the asymptotic continuous-time solutions.     

Non-linear vibrations of suspension bridges with external excitation

Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equations. Free and forced vibrations with damping are investigated in detail. Amplitude and phase modulation equations are obtained. The dependence of non-linear frequency on amplitude is described. Steady-state solutions are analyzed. Frequency-response equation is derived and the jump phenomenon in the frequency-response curves resulting from non-linearity is considered. Effects of initial amplitude and phase values, amplitude of excitation, and damping coefficient on modal amplitudes, are determined.     

Robust investment–reinsurance optimization with multiscale stochastic volatility

This paper investigates the investment and reinsurance problem in the presence of stochastic volatility for an ambiguity-averse insurer (AAI) with a general concave utility function. The AAI concerns about model uncertainty and seeks for an optimal robust decision. We consider a Brownian motion with drift for the surplus of the AAI who invests in a risky asset following a multiscale stochastic volatility (SV) model. We formulate the robust optimal investment and reinsurance problem for a general class of utility functions under a general SV model. Applying perturbation techniques to the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation associated with our problem, we derive an investment–reinsurance strategy that well approximates the optimal strategy of the robust optimization problem under a multiscale SV model. We also provide a practical strategy that requires no tracking of volatility factors. Numerical study is conducted to demonstrate the practical use of theoretical results and to draw economic interpretations from the robust decision rules.     

Trace formulas for additive and non-additive perturbations

Trace formulas for pairs of self-adjoint, maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a pair {H^′,H}$\{H^{\prime },H\}$ of maximal accumulative operators has been proved. We investigate also the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H   and H^′=H+V$H^{\prime }=H+V$ are maximal accumulative and V is trace class, we prove the existence of a summable   complex-valued spectral shift function. We also obtain trace formulas for pairs {H,H^?}$\{H,H^{?}\}$assuming only that H and  H^?$H^{?}$are resolvent comparable. In this case the determinant of the characteristic function of H is involved in trace formulas.     

On the symmetry of three identical interacting particles in a one-dimensional box

We study a quantum-mechanical system of three particles in a one-dimensional box with two-particle harmonic interactions. The symmetry of the system is described by the point group D_3d$D_{3d}$. Group theory greatly facilitates the application of perturbation theory and the Rayleigh–Ritz variational method. A great advantage is that every irreducible representation can be treated separately. Group theory enables us to predict the connection between the states for the small box length and large box length regimes of the system. We discuss the crossings and avoided crossings of the energy levels as well as other interesting features of the spectrum of the system.     

Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation

This paper concerns the quadratically-damped Mathieu equation: