—Ramification in complete discrete valuation fields is studied. For the case of a perfect residue field, there is a well-developed theory of ramification groups. Hyodo introduced the concept of ramification depth associated with the different of an extension and obtained an inequality that combines the concept of ramification depth in a degree p2 cyclotomic extension with the concept of ramification depth in a degree p subextension. The paper gives a detailed consideration of the structure of degree p2 extensions that can be obtained by a composite of two degree p extensions. In this case, it is not required that the residue field be perfect. Using the concepts of wild and ferocious extensions and the defect of the main unit, degree p2 extensions are classified and more accurate estimates for the ramification depth are obtained. In a number of cases, exact formulas for ramification depth are presented.
The article proves a theorem on the strong law of large numbers for linear functions of concomitants (induced order statistics) for sequences of independent identically distributed two-dimensional random vectors. The result complements previous work by Yang (1981), Gribkova and Zitikis (2017, 2019). The proof is based on the conditional independence property of concomitants Bhattacharya (1974), the strong law of large numbers for functions of order statistics by van Zwet (1980) is used, classical inequalities apply, including Rosenthal’s (1970).
This is the first in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg school of probability and statistics in the period from 1947 to 2017. It is devoted to limit theorems for sums of independent random variables—a traditional subject for St. Petersburg. It refers to the classical limit theorems: the law of large numbers, the central limit theorem, and the law of the iterated logarithm, as well as important relevant problems formulated in the second half of the twentieth century. The latter include the approximation of the distributions of sums of independent variables by infinitely divisible distributions, estimation of the accuracy of strong Gaussian approximation of such sums, and the limit theorems on the weak almost sure convergence of empirical measures generated by sequences of sums of independent random variables and vectors.
This is the fourth article in a series of surveys devoted to the scientific achievements of the Leningrad—St. Petersburg School of Probability and Statistics from 1947 to 2017. It is devoted to studies on the characterization of distributions, limit theorems for kernel density estimators, and asymptotic efficiency of statistical tests. The characterization results are related to the independence and equidistribution of linear forms of sample values, as well as to regression relations, admissibility, and optimality of statistical estimators. When calculating the Bahadur asymptotic efficiency, particular attention is paid to the logarithmic asymptotics of large deviation probabilities of test statistics under the null hypothesis. Constructing new goodness-of-fit and symmetry tests based on characterizations is considered, and their asymptotic behavior is analyzed. Conditions of local asymptotic optimality of various nonparametric statistical tests are studied.
In this work we consider the question of stability loss of shallow orthotropic shell on elastic base. Equations of stability and expression of load parameter are obtained. We compare the critical load for Kirchgoff-Love and Timoshenko models. The results obtained are illustrated by case of shell made of glass-fiber material.
In this work we consider the task of stability of shallow orthotropic shell on elastic base. The dependence of critical load parameter on constants of shell’s elasticity and rigidity of base is considered. As example we analyse the case of spherical shell made of unidirectional glass-fiber material.
The task of stability of filled thin-walled transversally isotropic spherical shell of Timoshenko model with under external pressure and homogeneous heating is considered. The interaction between shell and filler is described by Winkler's model with a constant coefficient. Heating of the filler is not considered. The assumptions of the theory of local stability are accepted. The dependence of the critical load parameter on the parameters of shear, heating and rigidity of the filler is obtained. Various cases of this dependence are investigated. The results are presented in analytical and graphical form. It is found, that during uniform compression of spherical shell the increase of temperature and shift decreases the value of critical load. With low rigidity of the filler the shift exerts main influence, with big rigidity - heating of the shell. Wherein the increase of rigidity of the filler may increase or decrease the value of critical load. The latter depends on interrelations of the parameters of shift and temperature.
The processing of probabilistically uncertain knowledge patterns in intellectual decision support systems falls into three kinds of probabilistic-logic inference, such as reconciliation, a priori and a posteriori inference. The paper presents formulae that allow for putting the process down in terms of matrix-vector language.
This paper contains an explanation of Ramanujan-type formulas with cubic radicals of cubic irrationalities in the situation when these radicals are contained in a pure cubic extension. We give a complete description of formulas of such type, answering the Zippel’s question. It turns out that Ramanujan-type formulas are in some sense unique in this situation. In particular, there must be no more than three summands in the right-hand side and the norm of the irrationality in question must be a cube. In this situation we associate with cubic irrationalities a cyclic cubic polinomial, which is reducible if and only if one can simplify the corresponding cubic radical. This correspondence is inverse to the so-called Ramanujan correspondence defined in the preceding papers, where one associates a pure cubic extension to some cyclic polinomial.