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.     

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.     

Analyzing the transition rates of the ionization of atoms by strong fields of a CO<Subscript>2</Subscript> laser including nonzero initial momenta

Here, the method of including nonzero initial momenta for ejected electrons in strong infrared laser fields is further developed [8]. It has been shown that, apart from being natural, including the nonzero initial momenta enables one to go into a deeper analysis of the process of tunnel ionization of atoms in strong laser fields (intensity up to 10¹⁶ W/cm²). This is due to looking closely at Fig. 2, which indicates that all electrons that could be ejected, under the circumstances, are ejected at a field intensity ～10¹³ W/cm², and that the effect of ionization after that is strongly diminished, which can be seen from the slope of the plates on Figs. 2 and 4. This also explains the saturation effect for fields up to 10¹⁶ W/cm² [1, 4, 5, 7], and probably this saturation goes on until the fields raising relativistic effects ～10¹⁸ W/cm² [7]. Opposite to what was believed earlier [7], the atomic field intensities could be increased to values over 10¹⁷ W/cm² only when more than 10 electrons are ejected from the atom, it is shown that the properly calculated ionization of 9 electrons increases the atomic field intensity to ～10¹⁸ W/cm².