Regularity of Geometric quadratic stochastic operator generated by 2-partition of infinite points

In this research, we construct a class of quadratic stochastic operator called Geometric quadratic stochastic operator generated by arbitrary 2-partition of infinite points on a countable state space , where . We also study the limiting behavior of such operator by proving the existence of the limit of the sequence through the convergence of the trajectory to a unique fixed point. It is established that such operator is a regular transformation.

Regular Mechanical Transformation of Rotations Into Translations: Part 1. Kinematic Analysis and Definition of the Basic Characteristics

The science that study the processes of motions transformation upon a preliminary defined law between non-coplanar axes (in general case) axes of rotations or axis of rotation and direction of rectilinear translation by three-link mechanisms, equipped with high kinematic joints, can be treated as an independent branch of Applied Mechanics. It deals with mechanical behaviour of these multibody systems in relation to the kinematic and geometric characteristics of the elements of the high kinematic joints, which form them. The object of study here is the process of regular transformation of rotation into translation. The developed mathematical model is subjected to the defined task for studying the sliding velocity vector function at the contact point from the surfaces elements of arbitrary high kinematic joints. The main kinematic characteristics of the studied type motions transformation (kinematic cylinders on level, kinematic relative helices (helical conoids) and kinematic pitch configurations) are defined on the bases of the realized analysis. These features expand the theoretical knowledge, which is the objective of the gearing theory. They also complement the system of kinematic and geometric primitives, that form the mathematical model for synthesis of spatial rack mechanisms.

On the conjugacy and isomorphism problems for stabilizers of Lie group actions

The spaces of subgroups and Lie subalgebras with the group actions by conjugations are considered for real Lie groups. Our approach allows one to apply the properties of algebraically regular transformation groups to finding the conditions when those actions turn out to be type I. It follows, in particular, that in this case the stability groups for all the ergodic actions of such groups are conjugate (for example when the stability groups are compact). The isomorphism of the stability groups for ergodic actions is also established under some assumptions. A number of examples of non-conjugate and non-isomorphic stability groups are presented.

An Inclusion Theorem for Bohr-Hardy Summability Factors

1. Let A denote a sequence to sequence transformation given by the normal matrix A = (ank)(n, k = 0, 1, 2, …), i.e., a lower triangular matrix with ann ≠ 0 for all n. For B = (bnk) we write B ⇒ A if every B limitable sequence is A limitable to the same limit, and say that B is equivalent to A if B ⇒ A and A ⇒ B. If B is normal, then it is well known that the inverse of B exists (we denote it by B-l) and that B ⇒ A if and only if F = AB-1 is a regular transformation, i.e., transforms every convergent sequence into a sequence converging to the same limit. We say that a series ∑ an† is summable A if its sequence of partial sums is A-limitable.