A. S. Belov On the estimate the classical cosine-sum

For sums S_n(x) = 1 + \sum_k=1ⁿ (1/k) cos(kx) the estimate S_n(x) > -ln sin(x/2) for x \in (0, \pi) and integer n \ge (\pi)/(2x) - 1/2 is proved. From this estimate, the classical Young's inequality and some others estimates more general than the Young's inequality follow. The sums S_n(x) are partial sums of cosine-series 1 + \sum_n=1^\infty (1/n) cos(nx) that, in the interval (0, 2\pi), converges to the function 1 - \ln |2sin(x/2)| and is the Fourier series of this function. Therefore, the proved estimate for sums S_n(x) is natural.

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L. N. Kuskovskii On the derivatives of potentials

In a real (complex) linear space, we obtain recurrent formulas that allow us to find partial derivatives of any order from the fundamental solution of the Laplace equation.

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E. D. Loginova, D. I. Moldavanskii On free subgroups and on the center of an HNN-extension of groups

Conditions for the absence of noncyclic free subgroups in HNN-extensions of groups and for non-triviality of center of HNN-extensions are obtained.

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B. Ya. Solon m-reducibility of partial functions

The article discusses the standard concept of m-reducibility of sets applied to arithmetic functions. This gives rise to a notion of m-reducibility of functions, which is close to the reducibility of partially recursive functions introduced by A. N. Degtev in 1975.

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