The Bohr--P\'al Theorem and the Sobolev Space W21/2W_2^{1/2}

The well-known Bohr--P\'al theorem asserts that for every continuous real-valued function $f$ on the circle $\mathbb T$ there exists a change of variable, i.e., a homeomorphism $h$ of $\mathbb T$ onto itself, such that the Fourier series of the superposition $f\circ h$ converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings $f$ into the Sobolev space $W_2^{1/2}(\mathbb T)$. This refined version of the Bohr--P\'al theorem does not extend to complex-valued functions. We show that if $\alpha<1/2$, then there exists a complex-valued $f$ that satisfies the Lipschitz condition of order $\alpha$ and at the same time has the property that $f\circ h\notin W_2^{1/2}(\mathbb T)$ for every homeomorphism $h$ of $\mathbb T$

Flavor Mixing, Neutrino Masses and Neutrino Oscillations

We study a model for the mass matrices of the leptons. We are ablte to relate the mass eigenvalues of the charged leptons and of the neutrinos to the mxiing angles and can predict the masses of the neutrinos. We find a normal hierarchy -the masses are 0.004 eV, 0.01 eV and 0.05 eV. The atmospheric mixing angle is given by the mass ratios of the charged leptons and of the neutrinos. We find 38 degrees, consistent with the experiments. The mixing element, connecting the first neutrino with the electron, is found to be 0.05.Comment: 4 page

Investigation of the Formation Process of Hazardous and Harmful Production Factors When Cutting a Stone for Construction Works

Stone cutting for construction work is carried out by disk diamond wheels the rotation speed of which, and, consequently, the cutting speed is 35-50 m/s. In view of the high intensity of the cutting process and intensive microchip formation, the process of stone cutting is accompanied by considerable dust formation, which can be both harmful and dangerous in the work.The greatest danger is represented by dust particles, which dimensions are 5 μm or less. These particles have the greatest pathogenic effect on the respiratory system of the human body. In addition, the settling time of these particles is measured in hours. Thus, even after the termination of work, the risk of dust exposure to the human body remains. Insignificant time of inhalation of these particles can lead a person to disability and death. Insurance payments in case of disability and compensation in case of death of an employee for these reasons all over the world amount to multibillion sums, which, even for this reason, makes the issue of combating this phenomenon very urgent.In this paper, the process of generation of hazardous and harmful production factors arising as a result of dust formation during the stone cutting, the regularity of dust distribution by fractions, quantitative indicators of the process for improving equipment, as well as individual and collective means of personnel protection are determined. The shape and dimensions of the cutting grains of the cutting discs are studied in the paper, depending on the grain material and the grain size of the cutting disc. Thecurvature radii of the cutting grains depend on the grain material and the grain size of the cutting disc. The actual number of grains participating in cutting and, consequently, in dust formation is shown.The dimensions of the dust particles depend on the graininess of the cutting discs and the processing regimes. Using the law of normal particle size distribution, the percentage of dust particles is determined depending on processing conditions. The rate of particle settling depends on their size and mass. The formation of a dust-air mixture and its probable concentration and chemical composition of dust depend on the chemical composition of the cutting materials.Regularities are obtained, which can give an opportunity to improve the individual and collective protection of workers from this harmful production factor