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S. Banach in [1] proved that for any function fL 2(0, 1), f ≁ 0, there exists an ONS (orthonormal system) such that the Fourier series of this function is not summable a.e. by the method (C, α), α > 0.

D. Menshov found the conditions which should be satisfied by the Fourier coefficients of the function for the summability a.e. of its Fourier series by the method (C, α), α > 0.

In this paper the necessary and sufficient conditions are found which should be satisfied by the ONS functions (φ n(x)) so that the Fourier coefficients (by this system) of functions from class Lip 1 or A (absolutely continuous) satisfy the conditions of D. Menshov.

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Motivated by the well known Kadec-Pełczynski disjointification theorem, we undertake an analysis of the supports of non-zero functions in strongly embedded subspaces of Banach functions spaces. The main aim is to isolate those properties that bring additional information on strongly embedded subspaces. This is the case of the support localization property, which is a necessary condition fulfilled by all strongly embedded subspaces. Several examples that involve Rademacher functions, the Volterra operator, Lorentz spaces or Orlicz spaces are provided.

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An extension of von Neumann’s characterization of essentially selfadjoint operators is given among not necessarily densely defined symmetric operators which are assumed to be closable. Our arguments are of algebraic nature and follow the idea of [1].

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We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

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The present note will present a different and direct way to generalize the convexity while keep the classical results for L 1-convergence still alive.

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The Separation Problem, originally posed by K. Bezdek in [1], asks for the minimum number s(O, K) of hyperplanes needed to strictly separate an interior point O in a convex body K from all faces of K. It is conjectured that s(O, K) ≦ 2d in d-dimensional Euclidean space. We prove this conjecture for the class of all totally-sewn neighbourly 4-dimensional polytopes.

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A remarkable class of quadratic irrational elements having both explicit Engel series and continued fraction expansions in the field of Laurent series, mimicking the case of real numbers discovered by Sierpiński and later extended by Tamura, is constructed. Linear integer-valued polynomials which can be applied to construct such class are determined. Corresponding results in the case of real numbers are mentioned.

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R. P. Stanley proved the Upper Bound Conjecture in 1975. We imitate his proof for the Ehrhart rings.

We give some upper bounds for the volume of integrally closed lattice polytopes. We derive some inequalities for the δ-vector of integrally closed lattice polytopes. Finally we apply our results for reflexive integrally closed and order polytopes.

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The modified method of estimation of the resistance of block ciphers to truncated byte differential attack is proposed. The previously known method estimate the truncated byte differential probability for Rijndael-like ciphers. In this paper we spread the sphere of application of that method on wider class of ciphers. The proposed method based on searching the most probable truncated byte differential characteristics and verification of sufficient conditions of effective byte differentials absence.

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Multiplicative complexity is the minimum number of AND-gates required to implement a given Boolean function in (AND, XOR) algebra. It is a good measure of a hardware complexity of an S-box, but an S-box cannot have too low multiplicative complexity due to security constraints. In this article we focus on generic constructions that can be used to find good n×n S-boxes with low multiplicative complexity. We tested these constructions in the specific case when n = 8. We were able to find 8 × 8 S-boxes with multiplicative complexity at most 16 (which is half of the known bound on multiplicative complexity of the AES S-box), while providing a reasonable resistance against linear and differential cryptanalysis.

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