Extreme Values of Euler-Kronecker Constants

In a family of Sn -fields (n ≤ 5), we show that except for a density zero set, the lower and upper bounds of the Euler-Kronecker constants are −(n − 1) log log dK + O(log log log dK ) and loglog dK + O(log log log dK ), resp., where dK is the absolute value of the discriminant of a number field K.

I–Convergence of Arithmetical Functions

Let n > 1 be an integer with its canonical representation, n = p 1 α 1 p 2 α 2 ⋯ p k α k . Put H n = max α 1 … α k , h n = min α 1 … α k , ω n = k , Ω n = α 1 + ⋯ + α k , f n = ∏ d ∣ n d and f ∗ n = f n n . Many authors deal with the statistical convergence of these arithmetical functions. For instance, the notion of normal order is defined by means of statistical convergence. The statistical convergence is equivalent with I d –convergence, where I d is the ideal of all subsets of positive integers having the asymptotic density zero. In this part, we will study I –convergence of the well-known arithmetical functions, where I = I c q = A ⊂ N : ∑ a ∈ A a − q < + ∞ is an admissible ideal on N such that for q ∈ 0 1 we have I c q ⊊ I d , thus I c q –convergence is stronger than the statistical convergence ( I d –convergence).

Estimation of the dimensional variability in the properties of soils in the basement of the atomic power station (APS) territory

A method of the estimating the variability of the physical and mechanical characteristics of soils has been considered. The applying of this method in practice can be used to find out the weakened zones and tectonic structures at the territory of the atomic power stations, and also - traditionally - to estimate the influence on the behavior of the elementary natural technical system «construction - soils». The tasks of the revealing the character of the occurrence of the various possible structural non-homogeneities, which may influence on the atomic power station territory estimation, as well as a general plan allocation, have been estimated in geological parameters fields’ assessment at the one of the constructed atomic power stations territory and environs. The characteristics of the physical properties (humidity, density, zero-air dry unit weight, index fluidity) have been compared on the base of the analyses of the laboratory tests. The analysis has been done for a layer-marker which corresponds to the chalk deposits of the Turonian stage. A selective analysis of the dimensional variability of the soils’ properties has been done for the engineering-geological researches results on the territory of the atomic power-station and, in particular, reactor divisions. It has been concluded, that to make a right forecast, the additional engineering and geological investigations of the atomic power station are needed. Such researches will allow the sustainable definition, or confirmation, of the existing deformation and strength properties of the soils which are in the basement of the buildfngs for the depth, which is deeper than compressible thickness depth.

Universality and Almost Decidability

We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a decidable and generic (i.e. a set of natural density one) set whose intersection with S is decidable. Every decidable set is almost decidable, but the converse implication is false. We prove the existence of infinitely many universal functions whose halting sets are generic (negligible, i.e. have density zero) and (not) almost decidable. One result—namely, the existence of infinitely many universal functions whose halting sets are generic (negligible) and not almost decidable—solves an open problem in [9]. We conclude with some open problems.

Extreme residues of Dedekind zeta functions

In a family ofSd+1-fields (d= 2, 3, 4), we obtain the conjectured upper and lower bounds of the residues of Dedekind zeta functions except for a density zero set. ForS5-fields, we need to assume the strong Artin conjecture. We also show that there exists an infinite family of number fields with the upper and lower bounds, resp.