Sharp Weak Type Inequality for Fractional Integral Operators Associated with d-Dimensional Walsh–Fourier Series

Suppose that d ≥ 1 is an integer, ${\alpha \in (0,d)}$ is a fixed parameter and let I _α be the fractional integral operator associated with d-dimensional Walsh–Fourier series on [0, 1) ^d . The paper contains the proof of the sharp weak-type estimate $$||I_\alpha(f)||_{L^{d/(d-\alpha),\infty}([0,1)^d)}\leq\frac{2^d-1}{(2^{d-\alpha}-1)(2^\alpha-1)}||f||_{L^1([0,1)^d)}.$$ The proof rests on Bellman-function-type method: the above estimate is deduced from the existence of a certain family of special functions.     

Note on some ideals of associative rings

It is well known that being an ideal (left ideal, right ideal) of a ring is not a transitive relation. Nevertheless in some cases the transitive property does hold. Systematic studies of this subject were started in [11,13]. In this paper we continue these studies.     

Fixed points of a mapping and Hyers–Ulam stability

We show that many general results on Hyers–Ulam stability of some functional equations in a single variable follow immediately from a simple fixed point theorem. The theorem is formulated for self-maps of some subsets of the space of functions from a nonempty set into the set of reals. We also give some applications of that theorem, e.g., in investigations of solutions of some difference equations and functional inequalities.     

Photoelectrochemistry of silicon in HF solution

Photoelectrochemical, photoelectrocatalytic, and electrochemical processes of silicon anodic oxidation and hydrogen evolution in aqueous HF solution are discussed in terms of thermodynamic stability of Si, oxides SiO, SiO₂, and Si surface hydrides. It is shown that photoelectrochemical oxidation of n-type low-resistivity silicon to SiO₂ is catalyzed by Si $^{+}$ photo-hole formation, whereas in the case of p-type Si, the feasibility of this reaction is predetermined by p-type conductivity. It is suggested that anodic oxidation of Si goes through the stage of SiO oxide formation and its subsequent oxidation to SiO₂. Such mechanism accounts for chemical inertness of Si phase in HF solutions as well as for selective, anisotropic, and isotropic etching of Si within E ranges from $-0.5$ to 0.35 V, $0.35-0.8~V,$ and $E > 0.8$ V, respectively. Hydrogen evolution reaction on Si surface proceeds at very large overpotential ( $\geq 0.5$ V) through the stage of surface Si hydride formation: $\mathrm {Si + H_{2}O + e^{-} \rightarrow (SiH)_{surf} + OH^{-}}$ (the rate determining step) and $\mathrm {(SiH)_{surf} + H_{2}O + e^{-} \rightarrow Si + H_{2} + OH^{-}}$ . Illumination-related effects of surface reactions relevant to selective and anisotropic etching and nano/micro-structuring of Si surface are discussed.     

A Way To Discover Maxwell's Equations Theoretically

The Coulomb force, established in the rest frame of a source-charge Q, when transformed to a new frame moving with a velocity V has a form F = q E + q v × B, where E = E′_∥ + γE′_⊥ and B = (1/c ²)v × E and E′ is the electric field in the rest frame of the source. The quantities E and B are then manifestly interdependent. We prove that they are determined by Maxwell's equations, so they represent the electric and magnetic fields in the new frame and the force F is the well known from experiments Lorentz force. In this way Maxwell's equations may be discovered theoretically for this particular situation of uniformly moving sources. The general solutions of the discovered Maxwell's equations lead us to fields produced by accelerating sources.     

Structural and magnetic phase transitions in CoxNi1−xMnGe system under pressure

The interaction of magnetic (T_c) an first-order distortion-type structural (T_D) transitions in the Co_xNi_1?xMnGe system was investigated using an ac susceptibility method under hydrostatic pressure up to 1.3 GPa. For all compositions T_C increases and T_D decreases with pressure. At the first triple point (0.3–0.6 GPa depending on x) these transitions go through a simultaneous magnetic-structural transition, as observed by Anzai and Ozawa in NiMnGe. However, for the composition x = 5, at the second triple point (at 0.8 GPa) the magnetic transition separates from the structural one, with T_C greater than T_D. This appears to be the first example of physically different, first and second-order phase transitions merging at a particular value of an external parameter, i.e. pressure P_TR1 and decoupling again at P_TR2 > P_T1. The (P, T) phase diagrams and the nature of the triple points are discussed using a simple Landau-type theory     

Investigations of crystal and magnetic properties of the Mn5−xFexGe3 intermetallic compounds

The Mn_5?xFe_xGe₃ intermetallic compounds are investigated with X-ray, neutron diffraction, magnetometric and Mössbauer effect methods. It is found that crystal structure of x = 1 compound is of D8₈ type while the structure of x = 3, 4 and 5 compounds is of B8₂ type. All are ferromagnets with collinearly ordered atomic spins. The lattice constants are derived from X-ray diffraction patterns, while magnetometric measurements yield the Curie temperatures and Weiss constants as well as the values of magnetic moments per molecule in ferromagnetic and paramagnetics states. The distributions of Fe and Mn atoms among two non-equivalent crystal sites are determined with the neutron diffraction method and are confirmed by the Mössbauer effect measurements. The parameters of hyperfine interactions are derived from Mössbauer absorption spectra and are attributed to iron atoms in two non-equivalent crystal sites.