scholarly journals Generalized Nörlund and Nörlund-type Means of Sequences of Fuzzy Numbers

2020 ◽  
Vol 13 (5) ◽  
pp. 1088-1096
Author(s):  
Pradosh Kumar Pattanaik ◽  
Susanta Kumar Paikray ◽  
Bidu Bhusan Jena
Keyword(s):  
Fuzzy Numbers ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Slow Oscillation ◽  
Real Numbers ◽  
Fuzzy Real Numbers ◽  
Sequences Of Fuzzy Numbers ◽  
Necessary And Sufficient ◽  
Convergent Sequences

In this article we study some properties of generalized Nörlund and Nörlund-typemeans of sequences of fuzzy real numbers. We establish necessary and sufficient conditions for our purposed methods to transform convergent sequences of fuzzy real numbers into convergent sequences of fuzzy real numbers which also preserve the limit. Finally, we establish some results showing the connection between the generalized N ̈orlund and N ̈orlund-type limits and the usual limits under slow oscillation of sequences of fuzzy real numbers.

New Classes of Statistically Pre-Cauchy Triple Sequences of Fuzzy Numbers Defined by Orlicz Function

2018 ◽  
Vol 85 (3-4) ◽  
pp. 411
Author(s):  
Sangita Saha ◽  
Santanu Roy
Keyword(s):  
Cauchy Sequence ◽  
Fuzzy Numbers ◽  
Orlicz Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
Fuzzy Real Numbers ◽  
Sequences Of Fuzzy Numbers ◽  
Necessary And Sufficient

In this article, the concept of statistically pre-Cauchy sequence of fuzzy real numbers having multiplicity greater than two defined by Orlicz function is introduced. A characterization of the class of bounded statistically pre-Cauchy triple sequences of fuzzy numbers with the help of Orlicz function is presented. Then a necessary and suffcient condition for a bounded triple sequence of fuzzy real numbers to be statistically pre-Cauchy is proved. Also a necessary and sufficient condition for a bounded triple sequence of fuzzy real numbers to be statistically convergent is derived. Further, a characterization of the class of bounded statistically convergent triple sequences of fuzzy numbers is presented and linked with Cesaro summability.

scholarly journals On the Riesz Almost Convergent Sequences Space

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Mehmet Şengönül ◽  
Kuddusi Kayaduman
Keyword(s):  
Sequence Space ◽  
Sufficient Conditions ◽  
Riesz Transforms ◽  
Necessary And Sufficient Conditions ◽  
Infinite Matrices ◽  
Necessary And Sufficient ◽  
Convergent Sequences

The purpose of this paper is to introduce new spaces and that consist of all sequences whose Riesz transforms of order one are in the spaces and , respectively. We also show that and are linearly isomorphic to the spaces and , respectively. The and duals of the spaces and are computed. Furthermore, the classes and of infinite matrices are characterized for any given sequence space and determine the necessary and sufficient conditions on a matrix to satisfy , for all .

scholarly journals GROUPOID -ALGEBRAS WITH HAUSDORFF SPECTRUM

2013 ◽  
Vol 88 (2) ◽  
pp. 232-242 ◽  
Author(s):  
GEOFF GOEHLE
Keyword(s):  
Haar System ◽  
Sufficient Conditions ◽  
Orbit Space ◽  
Necessary And Sufficient Conditions ◽  
Locally Compact ◽  
Fell Topology ◽  
Necessary And Sufficient ◽  
Convergent Sequences

AbstractSuppose that $G$ is a second countable, locally compact Hausdorff groupoid with abelian stabiliser subgroups and a Haar system. We provide necessary and sufficient conditions for the groupoid ${C}^{\ast } $-algebra to have Hausdorff spectrum. In particular, we show that the spectrum of ${C}^{\ast } (G)$ is Hausdorff if and only if the stabilisers vary continuously with respect to the Fell topology, the orbit space ${G}^{(0)} / G$ is Hausdorff, and, given convergent sequences ${\chi }_{i} \rightarrow \chi $ and ${\gamma }_{i} \cdot {\chi }_{i} \rightarrow \omega $ in the dual stabiliser groupoid $\widehat{S}$ where the ${\gamma }_{i} \in G$ act via conjugation, if $\chi $ and $\omega $ are elements of the same fibre then $\chi = \omega $.

scholarly journals On the inclusion of a bounded convergence field in the space of almost convergent sequences

1972 ◽  
Vol 13 (1) ◽  
pp. 82-90 ◽  
Author(s):  
Robert E. Atalla
Keyword(s):  
Sufficient Conditions ◽  
Research Problem ◽  
Necessary And Sufficient Conditions ◽  
General Context ◽  
Regular Matrix ◽  
Reverse Inclusion ◽  
Necessary And Sufficient ◽  
Convergent Sequences

Let T = (tmn) be a regular matrix, and CTbe its bounded convergence field. Necessary and sufficient conditions for CT to contain the space of almost convergent sequences are well known. (See, e.g., [7, p.62]). G. M. Petersen has suggested as a problem for research the discovery of necessary and sufficient conditions for the reverse inclusion: When is CT contained in the space of almost convergent sequences? [7, p. 137, research problem 9]. In this paper we deal with this question in a more general context. First we need some notation.

scholarly journals Cyclic Permutations of Sequences and Uniform Partitions

10.37236/389 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Po-Yi Huang ◽  
Jun Ma ◽  
Yeong-Nan Yeh
Keyword(s):  
Analogous Result ◽  
Cyclic Permutation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fluctuation Theory ◽  
Partial Sums ◽  
Real Numbers ◽  
Necessary And Sufficient ◽  
First Time ◽  
Positive Half

Let $\vec{r}=(r_i)_{i=1}^n$ be a sequence of real numbers of length $n$ with sum $s$. Let $s_0=0$ and $s_i=r_1+\ldots +r_i$ for every $i\in\{1,2,\ldots,n\}$. Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums $s_i$. Define $p(\vec{r})$ to be the number of positive sum $s_i$ among $s_1,\ldots,s_n$ and $m(\vec{r})$ to be the smallest index $i$ with $s_i=\max\limits_{0\leq k\leq n}s_k$. An important problem in fluctuation theory is that of showing that in a random path the number of steps on the positive half-line has the same distribution as the index where the maximum is attained for the first time. In this paper, let $\vec{r}_i=(r_i,\ldots,r_n,r_1,\ldots,r_{i-1})$ be the $i$-th cyclic permutation of $\vec{r}$. For $s>0$, we give the necessary and sufficient conditions for $\{ m(\vec{r}_i)\mid 1\leq i\leq n\}=\{1,2,\ldots,n\}$ and $\{ p(\vec{r}_i)\mid 1\leq i\leq n\}=\{1,2,\ldots,n\}$; for $s\leq 0$, we give the necessary and sufficient conditions for $\{ m(\vec{r}_i)\mid 1\leq i\leq n\}=\{0,1,\ldots,n-1\}$ and $\{ p(\vec{r}_i)\mid 1\leq i\leq n\}=\{0,1,\ldots,n-1\}$. We also give an analogous result for the class of all permutations of $\vec{r}$.

scholarly journals The Hausdorff Moment Problem

1978 ◽  
Vol 21 (3) ◽  
pp. 257-265
Author(s):  
David Borwein
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Real Numbers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Hausdorff Moment Problem

Suppose throughout thatand that {μn}(n≥ 0) is a sequence of real numbers. The (generalized) Hausdorff moment problem is to determine necessary and sufficient conditions for there to be a function x in some specified class satisfying.

Convergence of Hardy Cross’s Balancing Process

1940 ◽  
Vol 7 (4) ◽  
pp. A166-A170
Author(s):  
Rufus Oldenburger
Keyword(s):  
Structural Analysis ◽  
Linear Transformation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Continuous Beam ◽  
Real Numbers ◽  
The Cross ◽  
Real Linear ◽  
Infinite Power ◽  
Necessary And Sufficient

Abstract It can be shown easily that the Cross method of structural analysis may be applied to a given structure in such a manner that the process does not converge. In this paper the author gives necessary and sufficient conditions for the convergence of the Cross method, and exhibits a convergent process of balancing any given structure. In particular he shows that a balancing process can be described by real linear transformation, that is, by a matrix of real numbers, and that the process converges in the sense of this paper if and only if the infinite power of this matrix exists and is zero. The study is restricted to the case of a continuous beam.

scholarly journals Matrix transformations from absolutely convergent series to convergent sequences as general weighted mean summability methods

2000 ◽  
Vol 24 (8) ◽  
pp. 533-538
Author(s):  
Jinlu Li
Keyword(s):  
Classical Result ◽  
Sufficient Conditions ◽  
Convergent Series ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Weighted Mean ◽  
Summability Methods ◽  
Necessary And Sufficient ◽  
Convergent Sequences

We prove the necessary and sufficient conditions for an infinity matrix to be a mapping, from absolutely convergent series to convergent sequences, which is treated as general weighted mean summability methods. The results include a classical result by Hardy and another by Moricz and Rhoades as particular cases.

scholarly journals Measure convergent sequences in Lebesgue spaces and Fatou's lemma

1996 ◽  
Vol 54 (2) ◽  
pp. 197-202 ◽  
Author(s):  
Heinz-Albrecht Klei
Keyword(s):  
Integrable Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Uniform Integrability ◽  
Lebesgue Spaces ◽  
Integrable Functions ◽  
Fatou’S Lemma ◽  
Necessary And Sufficient ◽  
Fatou's Lemma ◽  
Convergent Sequences

Let (fn) be a sequence of positive P-integrable functions such that (∫ fndP)n converges. We prove that (fn) converges in measure to if and only if equality holds in the generalised Fatou's lemma. Let f∞ be an integrable function such that (∥fn − f∞∥1)n converges. We present in terms of the modulus of uniform integrability of (fn) necessary and sufficient conditions for (fn) to converge in measure to f∞.

Generalization of the Hausdorff Moment Problem

1981 ◽  
Vol 33 (4) ◽  
pp. 946-960 ◽  
Author(s):  
David Borwein ◽  
Amnon Jakimovski
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Natural Generalization ◽  
Necessary And Sufficient Conditions ◽  
Real Numbers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Hausdorff Moment Problem ◽  
Positive Integers ◽  
Classical Moment Problem

Suppose throughout that {kn} is a sequence of positive integers, thatthat k0 = 1 if l0 = 1, and that {un(r)}; (r = 0, 1, …, kn – 1, n = 0, 1, …) is a sequence of real numbers. We shall be concerned with the problem of establishing necessary and sufficient conditions for there to be a function a satisfying(1)and certain additional conditions. The case l0 = 0, kn = 1 for n = 0, 1, … of the problem is the version of the classical moment problem considered originally by Hausdorff [5], [6], [7]; the above formulation will emerge as a natural generalization thereof.

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