We complete the classification of Ricci solitons within all classes of homogeneous Siklos metrics.

Let be a sequence of identically distributed, weakly independent and weakly Gaussian cylindrical random variables in a separable Banach space . We consider the cylindrical difference equation, , in and determine a cylindrical process which solves the equation. The cylindrical distribution of is shown to be weakly Gaussian and independent of . It is also shown to be strongly Gaussian if the cylindrical distribution of is strongly Gaussian. We determine the characteristic functional of and give conditions under which is unique.

Let be an integral and non-degenerate variety. Recall (A. Bialynicki-Birula, A. Schinzel, J. Jelisiejew and others) that for any the open rank is the minimal positive integer such that for each closed set there is a set with and , where denotes the linear span. For an arbitrary we give an upper bound for in terms of the upper bound for when is a point in the maximal proper secant variety of and a similar result using only points with submaximal border rank. We study when is a Segre variety (points with -rank and ) and when is a Veronese variety (points with -rank or with border rank ).

The aim of this note is to compare work of Formanek [7] on a certain construction of central polynomials with that of Collins [3] on integration on unitary groups. These two quite disjoint topics share the construction of the same function on the symmetric group, which the second author calls Weingarten function. By joining these two approaches we succeed in giving a simplified and very natural presentation of both Formanek and Collins's Theory.

In this paper, we investigate the influence of the certain subgroups of fixed prime power order on the -supersolubility of finite groups. Many recent results are extended.

In this paper, we show that the projection of every compact Riemannian manifold of positive curvature onto a rank one symmetric space is harmonic. As a corollary, an infinite family of distinct harmonic morphisms with minimal circle fibers from the 7-dimensional homogeneous Aloff-Wallach spaces of positive curvature onto the 6-dimensional flag manifolds is given.

In Hopf-Galois theory, every -Hopf-Galois structure on a field extension gives rise to an injective map from the set of -sub-Hopf algebras of into the intermediate fields of . Recent papers on the failure of the surjectivity of reveal that there exist many Hopf-Galois structures for which there are many more subfields than sub-Hopf algebras. In this paper we survey and illustrate group-theoretical methods to determine -Hopf-Galois structures on finite separable extensions in the extreme situation when has only two sub-Hopf algebras. This corresponds to the case when the lack of surjectivity is at its extreme.

Let be a connected graph. The distance between an edge and a vertex is defined as A nonempty set is an edge metric generator for if for any two distinct edges , there exists a vertex such that . An edge metric generating set with the smallest number of elements is called an edge metric basis of , and the number of elements in an edge metric basis is called the edge metric dimension of and it is denoted by . In this paper, we study the edge metric dimension of a blow up of a graph , and also we study the edge metric dimension of the zero divisor graph of the ring of integers modulo . Moreover, the Wiener index and the hyper-Wiener index of the blow up of certain graphs are computed.

Our goal of this paper is to develop an analogue of the theory of group extensions for multiplicative Lie rings. We first define a factor system of pair of multiplicative Lie rings , which use to construct an extension of by . Then we state Schreier's theorem for multiplicative Lie rings. We also use this notation to introduce second cohomology group of multiplicative Lie rings. Finally, we show that the equivalence classes of multiplicative Lie ring extensions can be identified with second cohomology group , where acts on by and .

In this paper, we obtain a measure of inaccuracy between rth concomitant of generalized order statistic and the parent random variable in Morgenstern family. Applications of this result are given for concomitants of order statistics and record values. We also study some results of cumulative past inaccuracy (CPI) between the distribution function of rth concomitant of order statistic (record value) and the distribution function of parent random variable. Finally, we discuss on a problem of estimating the CPI by means of the empirical CPI in concomitants of generalized order statistics.