In this article we introduce and study the notions of ∆(r)-lacunary strongly summable, ∆(r)-Cesàro strongly summable, ∆(r)-statistically convergent and ∆(r)-lacunary statistically convergent sequences of fuzzy numbers. Consequently we construct the spaces of these four types of difference sequences and investigate the relationship among these spaces. Further we show that the spaces of ∆(r)-lacu...

This note introduces an interesting class of Gauss-Markov Random Fields designated as walk-summable. Several equivalent characterizations of this class of GMRFs are established. Also, several important subclasses of GMRFs are identified as being walk-summable. These include (i) diagonally dominant, (ii) pairwise normalizable, (iii) regular singly-connected, and (iv) regular bipartite with only ...

For n ≥ 2, let Bn denote the unit ball in R, and for p ≥ 1 let L denote the Banach space of p-summable functions on Bn. Let L p h(Bn) denote the subspace of harmonic functions on Bn that are p-summable. When n = 2, we often write D instead of B2, and we let A denote the Bergman space of analytic functions in L. Let ω be a function in L. We are interested in finding the best approximation to ω i...

It is shown that the discrete time convolution of two absolutely summable nowhere zero sequences may be identically equal to zero. DEVELOPMENT Consider two absolutely summable sequences a = {a,,: n E Z } and b = { b,,: n E Z } of real numbers. Further, assume that the sequences a and b are nowhere zero. Does it follow that the discrete time convolution a * b is nowhere zero? Does it follow that...

In the recent papers [1, 2], the author obtained necessary and sufficient conditions for a series ∑ an which is absolutely summable of order k by a weighted mean method, 1 < k ≤ s <∞, to be such that anλn is absolutely summable of order s by a triangular matrix method. In this paper, we obtain sufficient and (different) necessary conditions for a series ∑ an which is absolutely summable |A|k to...

Mazhar (1971) gave the characterization for the series ∑ an n to be summable |N,pn|whenever ∑ an is summable |C,α|k, α≥ 0, k≥ 1. Here we prove two theorems, the first concerns the sufficient conditions and the second the necessary conditions satisfied by { n } in order to have ∑ an n summable |N̄,pn|k whenever ∑ an is summable |C,α|k, k≥ 1. 2000 Mathematics Subject Classification. Primary 40G. L...

Spectral problem for the Dirac operator with regular but not strongly boundary conditions and complex-valued potential summable over a finite interval is considered. The purpose of this paper to find under which root function system forms usual Riesz basis rather than parentheses.

This paper aims to address distributed optimization problems over directed and time-varying networks, where the global objective function consists of a sum locally accessible convex functions subject feasible set constraint coupled inequality constraints whose information is only partially each agent. For this problem, proximal-based algorithm, called proximal primal-dual (DPPD) proposed based ...

We present algorithms which split a rational expression in terms of indefinite nested sums and products into a summable part which can be summed by telescoping and into a non-summable part which is degree-optimal with respect to one of the most nested sums or products. If possible, our algorithms find a non-summable part where all these most nested sums and products are eliminated.