Vol 54, No 6 (2019)
- Year: 2019
- Articles: 7
- URL: https://journals.rcsi.science/1068-3623/issue/view/14109
Algebra
n-torsion Groups
Abstract
A group is called an n-torsion group if it has a system of defining relations of the form rn = 1 for some elements r, and for any of its finite order element a the defining relation an = 1 holds. It is assumed that the group can contain elements of infinite order. In this paper, we show that for every odd n ≥ 665 for each n-torsion group can be constructed a theory similar to that of constructed in S. I. Adian’s well-known monograph on the free Burnside groups. This allows us to explore the n-torsion groups by methods developed in that work. We prove that every n-torsion group can be specified by some independent system of defining relations; the center of any non-cyclic n-torsion group is trivial; the n-periodic product of an arbitrary family of n-torsion groups is an n-torsion group; in any recursively presented n-torsion group the word and conjugacy problems are solvable.
Functional Analysis
Gabor Duals for Operator-valued Gabor Frames on Locally Compact Abelian Groups
Abstract
Motivated by the ordinary Gabor frames in L2(ℝd) and operator-valued frames on abstract Hilbert spaces, we investigate operator-valued Gabor frames associated with locally compact Abelian groups. Necessary and sufficient conditions for an operator-valued Gabor frame to admit a Parseval/tight Gabor dual are given. In particular, we consider a special case, which includes the case of ordinary Gabor frames.
Real and Complex Analysis
A New Family of Starlike Functions in a Circular Domain Involving a q-differential Operator
Abstract
The main purpose of the present paper is to investigate a number of useful properties such as sufficiency criteria, distortion bounds, coefficient estimates, radius of starlikness and radius of convexity for a new subclass of analytic functions, which are defined here by means of a newly defined q-linear differential operator.
On Convergence of Partial Sums of Franklin Series to +∞
Abstract
In this paper, we prove that if {nk} is an arbitrary increasing sequence of natural numbers such that the ratio nk+1/nk is bounded, then the nk-th partial sum of a series by Franklin system cannot converge to +∞ on a set of positive measure. Also, we prove that if the ratio nk+1/nk is unbounded, then there exists a series by Franklin system, the nk-th partial sum of which converges to +∞ almost everywhere on [0, 1].
Double Universal Fourier Series
Abstract
In this paper we construct an integrable function of two variables for which the double Fourier-Walsh series converges both by rectangles and by spheres. Besides, we show that the coefficients of the series on the spectrum are positive and are arranged in decreasing order in all directions. Also, it is proved that after a suitable choice of signs for the Fourier coefficients of the series the spherical partial sums of the obtained series are dense in Lp[0, 1]2, p ∈ (0, 1).
Probability Theory and Mathematical Statistics
Asymptotic Behavior of the Variance of the Best Linear Unbiased Estimator for the Mean of a Discrete-time Singular Stationary Process
Abstract
It is known that for a wide class of discrete-time stationary processes possessing spectral densities f, the variance σn2(f) of the best linear unbiased estimator for the mean depends asymptotically only on the behavior of the spectral density f near the origin, and behaves hyperbolically as n → ∞. In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease of σn2(f) as n → ∞. In particular, we show that a necessary condition for σn2(f) to decrease to zero exponentially is that the spectral density f vanishes on a set of positive measure in any vicinity of zero, and if f vanishes only at the origin, then it is impossible to obtain exponential decay of σn2(f), no mater how high the order of the zero of f at the origin.