scholarly journals Ideals, Nonnegative Summability Matrices and Corresponding Convergence Notions: A Short Survey of Recent Advancements

Axioms ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 1
Author(s):  
Pratulananda Das
Keyword(s):  
Ideal Convergence ◽  
Survey Article ◽  
Summability Methods ◽  
Short Survey ◽  
Summability Matrices ◽  
Regular Summability

In this survey article, we look into some recent results concerning summability matrices, both regular as well as those which are not regular (called semi-regular) and generated matrix ideals as the overall view of the inter relationship between the notions of ideal convergence and summability methods by regular summability matrices.

On totally regular summability methods

1966 ◽  
Vol 91 (4) ◽  
pp. 348-354 ◽  
Author(s):  
B. Kuttner
Keyword(s):  
Summability Methods ◽  
Regular Summability

Inclusion of sets of regular summability matrices

1964 ◽  
Vol 60 (4) ◽  
pp. 705-712 ◽  
Author(s):  
J. W. Baker ◽  
G. M. Petersen
Keyword(s):  
Finite Subset ◽  
Null Sequence ◽  
Fixed Sequence ◽  
Summable Sequence ◽  
The Matrix ◽  
Summability Matrices ◽  
Image Position ◽  
Regular Summability ◽  
Bounded Sequences

1. In this paper we wish to discuss some problems which arise from a paper by Lorentz and Zeller; see (5). If {μn} is a fixed sequence monotonically increasing to infinity, and every sequence {sn} summed by both of the regular matrices A = (amn) and B = (bmn) and satisfying sn = O{μn) is summed to the same value by both matrices, the matrices are called (μn)-consistent. The two matrices are called consistent if they are (μn)-consistent for all {μn}, μn↗∞; they are b-consistent if the bounded sequences summed by both are summed to the same value by both. The matrix A is said to be (μn)-stronger than the matrix B, if every sequence {μn} that is B summable and satisfying sn = O(μn) is also A summable. The matrix A is stronger than B if every B summable sequence is A summable; A is b-stronger if every bounded B summable sequence is A summable. The symbol A -lim x denotes the value to which the sequence x = {xn} is summed by A; Am(x) is the transformationand A(x) is the sequence {Am(x)}. Let {A(i)}i ∈ I be any family, infinite or finite, of regular summability matrices. This family is called simultaneously consistent if, given any finite subset of I, say F, and any set of sequences {x(i)i ∈ F such that A(i) sums x(i) for each i in F, and such that is the null sequence, then .

Inclusion of sets of regular summability matrices. II

1965 ◽  
Vol 61 (2) ◽  
pp. 381-394 ◽  
Author(s):  
J. W. Baker ◽  
G. M. Petersen
Keyword(s):  
The Matrix ◽  
Summability Matrices ◽  
Image Position ◽  
Regular Summability

1. In a previous paper, ((1)), we have discussed problems which arise in finding a matrix which is in some sense stronger than each of a set of regular summability matrices. We intend in this paper to clear up other problems in this subject. We shall retain the notation and definitions of (1) throughout, but shall later modify one set of definitions in the light of some of our results. In this paper, the term matrix will be reserved for regular summability matrices. Also, u will be used to denote the unit sequence, u = {un} = {1} and h(A) will be used to denote the norm of the matrix A = (amn), i.e.

The Limited Resource View of Self-regulation: Implications for Vocal Behavior Change

2016 ◽  
Vol 1 (3) ◽  
pp. 105-111
Author(s):  
Lisa A. Vinney
Keyword(s):  
Behavior Change ◽  
Self Regulation ◽  
Limited Resource ◽  
Voice Therapy ◽  
Vocal Behavior ◽  
Future Research ◽  
Survey Article ◽  
Short Survey

Successful self-regulation is likely an important construct underlying the learning of new vocal behaviors and lasting vocal behavior change. In this short survey article, review of established and emerging understanding of self-regulatory phenomena, in general and in relationship to vocal behavior, will be discussed. Potential future research avenues, integrating self-regulation experimental paradigms, and the ways in which they may inform and improve voice therapy practice and outcomes will also be highlighted.

Inclusion sets of regular summability matrices. III

1966 ◽  
Vol 62 (3) ◽  
pp. 389-394 ◽  
Author(s):  
J. W. Baker ◽  
G. M. Petersen
Keyword(s):  
Bounded Sequence ◽  
Finite Set ◽  
Summability Matrices ◽  
Image Position ◽  
Summability Matrix ◽  
Regular Summability ◽  
Bounded Sequences

Let A = (am, n) be a (regular summability) matrix. Then will denote the set of bounded sequences which are summed by A. If {Ai} (i = 1, 2, …, N) is a finite set of such matrices, and if consists of every bounded sequence then we shall say that the matrices span the bounded sequences. Ifx = {xn} belongs to then we denote the value to which A sums x by A-lim x. If y = {yn} is any sequence, then the A-transform of y (if it exists) is the sequence {Aμ(y)}, where

scholarly journals On the nonomnipotence of regular summability methods

1978 ◽  
Vol 28 (3) ◽  
pp. 231-232
Author(s):  
S Haber ◽  
O Shisha
Keyword(s):  
Summability Methods ◽  
Regular Summability

On the consistency and relative strength of regular summability methods

1966 ◽  
Vol 62 (3) ◽  
pp. 421-428 ◽  
Author(s):  
J. Copping
Keyword(s):  
Relative Strength ◽  
The Other ◽  
Matrix Summability ◽  
Summability Methods ◽  
Regular Summability

Statement of results. Let A and B be two matrix summability methods, and S a given set of sequences. We shall say that B is S-stronger than A if B sums every sequence which belongs to S and is summed by A. If each of A, B is S-stronger than the other, A and B will be called S-equivalent. If B is S-stronger than A, but A is not S-stronger than B, we say that B is strictly S-stronger than A.

scholarly journals Extreme points for regular summability matrices

1966 ◽  
Vol 18 (3) ◽  
pp. 255-258 ◽  
Author(s):  
Gordon Marshall Petersen
Keyword(s):  
Extreme Points ◽  
Summability Matrices ◽  
Regular Summability
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