Absence of robust rigidity for circle maps with breaks

We give examples of analytic circle maps with singularities of break type with the same rotation number and the same size of the break for which no conjugacy is Lipschitz continuous. In the second part of the paper, we discuss a class of rotation numbers for which a conjugacy is C¹$C^1$-smooth, although the numbers can be strongly non-Diophantine (Liouville). For the rotation numbers in this class, we construct examples of analytic circle maps with breaks, for which the conjugacy is not C^1+α$C^{1+\alpha }$ smooth, for any α>0$\alpha >0$.     

On bipartite graphs with small number of laplacian eigenvalues greater than two and three

All connected bipartite graphs with exactly two Laplacian eigenvalues greater than two are determined. Besides, all connected bipartite graphs with exactly one Laplacian eigenvalue greater than three are determined.     

Range projections and the Moore–Penrose inverse in rings with involution

In this article we study the existence of range projections in rings with involution, relating it to the existence of the Moore–Penrose inverse. The results are applied to the solution of the equation xbx?=?x in rings with involution, extending the results of Greville for matrices. Simpler new proofs are given of the Moore–Penrose invertibility of regular elements in rings with involution, and of the Ljance's formula.     

On the spectrum of linear combinations of two projections in C*-algebras

In this note, we study the spectrum and give estimations for the spectral radius of linear combinations of two projections in C*-algebras. We also study the commutator of two projections.     

Bases of spaces of matrices satisfying rank conditions

Motivated by the problem concerning the existence of non-singular bilinear maps, vector spaces of matrices consisting of matrices with rank bounded below are investigated. It is shown that bases for such spaces of maximum dimension can be chosen in such a way to consist of matrices of the minimal rank. An estimate of the ranks of matrices in particular types of bases for maximal such spaces is also given. This extends previously known results which were valid only in the case of spaces consisting of matrices of rank not equal to one.     

Exponential Stability in Mean Square for a General Class of Discrete-Time Linear Stochastic Systems

Abstract

The problem of the mean square exponential stability for a class of discrete-time linear stochastic systems subject to independent random perturbations and Markovian switching is investigated. The case of the linear systems whose coefficients depend both to present state and the previous state of the Markov chain is considered. Three different definitions of the concept of exponential stability in mean square are introduced and it is shown that they are not always equivalent. One definition of the concept of mean square exponential stability is done in terms of the exponential stability of the evolution defined by a sequence of linear positive operators on an ordered Hilbert space. The other two definitions are given in terms of different types of exponential behavior of the trajectories of the considered system. In our approach the Markov chain is not prefixed. The only available information about the Markov chain is the sequence of probability transition matrices and the set of its states. In this way one obtains that if the system is affected by Markovian jumping the property of exponential stability is independent of the initial distribution of the Markov chain.

The definition expressed in terms of exponential stability of the evolution generated by a sequence of linear positive operators, allows us to characterize the mean square exponential stability based on the existence of some quadratic Lyapunov functions.

The results developed in this article may be used to derive some procedures for designing stabilizing controllers for the considered class of discrete-time linear stochastic systems in the presence of a delay in the transmission of the data.     

Sonochemically assisted synthesis of zinc-doped maghemite

Nanoparticles of zinc-doped maghemite were prepared using ultrasonic radiation. As a precursor, a suspension of maghemite in an alkaline aqueous solution of zinc nitrate at pH 9 was sonicated. The zinc-doped maghemite nanoparticles were investigated by X-ray diffraction, Mössbauer spectroscopy, high-resolution electron microscopy (HREM) and SQUID magnetometry. The Mössbauer measurements, which cover the temperature range 4.2 K to room temperature, were acquired in zero field and an applied field of 5 T. The results show that by using ultrasound radiation, zinc Zn²⁺ can substitute for Fe³⁺ up to a composition close to zinc ferrite (ZnFe₂O₄), which has a random distribution of Fe³⁺ ions over both A and B sublattices in the spinel structure with an inversity parameter of δ = 0.322. This leads to a maximum saturation magnetization (M_s) of 64.1 emu/g at 300 K and 73.5 emu/g at 2 K.     

Ultrasonic de-agglomeration of barium titanate powder

BaTiO3 (BT) powder, with average particle size of 1.4 microm, was synthesized by solid-state reaction. A high-intensity ultrasound irradiation (ultrasonication) was used to de-agglomerate micro-sized powder to nano-sized one. The crystal structure, crystallite size, morphology, particle size, particle size distribution, and specific surface area of the BT powder de-agglomerated for different ultrasonication times (0, 10, 60, and 180 min) were determined. It was found that the particles size of the BT powder was influenced by ultrasonic treatment, while its tetragonal structure was maintained. Therefore, ultrasonic irradiation can be proposed as an environmental-friendly, economical, and effective tool for the de-agglomeration of barium titanate powders.