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].     

An optimal perturbation bound

Cai and Zhang establish separate perturbation bounds for distances with spectral and Frobenius norms (Cai T, Zhang A. Rate‐optimal perturbation bounds for singular subspaces with applications to high‐dimensional statistics. The Annals of Statistics. 2018; Vol. 46, No. 1: 60?89). We extend their theorem to each unitarily invariant norm. It turns out that our estimation is optimal as well.     

The martingale problem for a class of nonlocal operators of diagonal type

We consider systems of stochastic differential equations of the form $$\begin{array}{c}\hfill d{X}_t^i=\sum _{j=1}^d{A}_{ij}\left(X_{t?}\right)d{Z}_t^j\end{array}$$ for $i=1,?,d$ with continuous, bounded and non‐degenerate coefficients. Here $Z_t^1,?,Z_t^d$ are independent one‐dimensional stable processes with ${\alpha }_1,?,{\alpha }_d\in (0,2)$. In this article we research on uniqueness of weak solutions to such systems by studying the corresponding martingale problem. We prove the uniqueness of weak solutions in the case of diagonal coefficient matrices.     

Anharmonic force fields of cis- and trans-S1 C2H2

We calculate second-order vibrational perturbation theory (VPT2) anharmonic force fields for the cis and trans conformers of S₁ C₂H₂, and compare the results to experiment. The vibrational assignments of recently observed levels belonging to the cis well are of particular interest. A refined estimate of the cis origin position (44,870?±?10?cm^?1) is proposed, and preliminary low-energy fits to the global J?=?K?=?0 trans level structure are also described. The performance of perturbation theory in this isomerizing system is examined, and both surprising successes and failures are encountered. We examine these and their causes, and offer practical suggestions for avoiding the pitfalls of applying perturbation theory to systems with large amplitude motions.     

On one-point functions for sinh-Gordon model at finite temperature

Using fermionic basis we conjecture the exact formulae for the expectation values of local fields in sinh-Gordon model. The conjecture is checked against previously known results.     

Rovibrational and energetic analysis of the hydroxyethynyl anion (CCOH−)

For the first time, high-level structural and rovibrational data are provided for the hyroxyethynyl anion, CCOH^?. CCOH^? is a promising molecule for interstellar detection even though no new anions have been observed in the interstellar medium for the past half-decade. The large dipole moment of the corresponding neutral radical may be key for its creation as has been hypothesised and supported for other anions known to exist in various astronomical environments. Highly accurate quartic force fields are employed where previous benchmarks have produced spectroscopic constants and anharmonic vibrational frequencies within 20 MHz and 1 cm^?1, respectively, of experiment. This same approach is applied here for CCOH^? and its deuterated isotopologue with the goal of assisting laboratory experiments and/or astronomical observers in the potential detection of this anion.     

D-admissible hybrid control of a class of singular systems

This paper mainly studies the problem of designing a hybrid state feedback D-admissible controller for a class of linear and nonlinear singular systems. Based on the relationship between singular discrete systems and singular delta operator systems, several necessary and sufficient conditions for a linear singular delta operator system to be D-admissible (i.e. regular, causal and all finite poles lie in a prescribed circular region) with different representations are derived. Then, the existence conditions and explicit expressions of a desirable D-admissible controller are given by means of matrix inequalities and strict linear matrix inequalities, respectively. We further extend the obtained results to singular delta operator systems with Lipschitz nonlinear perturbations, and the design methods of hybrid controller are presented for the nonlinear case as well. Finally, numerical examples as well as simulations are provided to illustrate the effectiveness of the theoretical outcomes obtained in the paper.