On ( h(n)) summability methods

1985 ◽  
Vol 97 (2) ◽  
pp. 189-193
Author(s):  
B. Kuttner ◽  
I. L. Sukla
Keyword(s):  
Prime Number ◽  
Sufficient Conditions ◽  
Prime Number Theorem ◽  
Necessary And Sufficient Conditions ◽  
Summability Methods ◽  
Dirichlet Convolution ◽  
Necessary And Sufficient ◽  
Convolution Method

AbstractIn 1967 Segal introduced the Dirichlet convolution (, h(n)), generalizing a method of Ingham developed in studies on the Prime Number Theorem. In this paper we establish necessary and sufficient conditions on the sequence h(n) in order that the convolution method (, h(n)) be conservative. Further conditions are established for the method to be absolutely conservative.

scholarly journals Infinite Paths of Minimal Length on Suborbital Graphs for Some Fuchsian Groups

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Khuanchanok Chaichana ◽  
Pradthana Jaipong
Keyword(s):  
Prime Number ◽  
Fuchsian Group ◽  
Continued Fraction ◽  
Sufficient Conditions ◽  
Minimal Length ◽  
Necessary And Sufficient Conditions ◽  
Fuchsian Groups ◽  
Suborbital Graphs ◽  
Necessary And Sufficient ◽  
Recursive Representation

In this study, we work on the Fuchsian group Hm where m is a prime number acting on mℚ^ transitively. We give necessary and sufficient conditions for two vertices to be adjacent in suborbital graphs induced by these groups. Moreover, we investigate infinite paths of minimal length in graphs and give the recursive representation of continued fraction of such vertex.

Summability methods which include the Riesz typical means. I

1971 ◽  
Vol 69 (1) ◽  
pp. 99-106 ◽  
Author(s):  
Dennis C. Russell
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Alternative Form ◽  
Cesàro Summability ◽  
Abel Summability ◽  
Summability Methods ◽  
Cesaro Summability ◽  
Riesz Summability ◽  
Necessary And Sufficient

A number of special results exist for summability methods B which, include Riesz summability (R,λ,k)—for example, when B is generalized Abel summability (A,λ,ρ) [Kuttner(5)], or Riemann summability (,λ,μ) [Russell(14)], or Riemann-Cesàro summability (,λ,p,α) [Rangachari(12)], or generalized Cesàro summability (C,λ,k) [Meir (9); Borwein and Russell (l)]. The question of necessary and sufficient conditions to be satisfied by an arbitrary method B in order that B ⊇ (R,λ,k) has received an answer only for limited values of λ and k—for example, by Lorentz [(6), Theorem 10] for k = 1; the restrictions on λ in this case were removed by Maddox [(8), Theorem 1]. Thus (apart from the well-known case k = 0) the case k = 1 is the only one for which a complete solution exists, though application of a theorem of Russell [(13), Theorem 1A] yields one form of a result for 0 < k ≤ 1. Maddox's results, however, suggest an alternative form capable of generalization to all k ≥ 0, and in this paper we obtain a complete solution for 0 < k ≤ 1 in that form, without restriction on λ. We first recall the following definitions.

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.

Generalized Markov Projections and Matrix Summability

1979 ◽  
Vol 22 (3) ◽  
pp. 311-316 ◽  
Author(s):  
Robert E. Atalla
Keyword(s):  
Sufficient Conditions ◽  
Markov Operator ◽  
Necessary And Sufficient Conditions ◽  
Averaging Operators ◽  
Matrix Summability ◽  
Markov Operators ◽  
Main Application ◽  
Summability Methods ◽  
Inclusion Relations ◽  
Necessary And Sufficient

In [A1] is defined a class of Markov operators on C(X) (X compact T2), called Generalized Averaging Operators (g.a.o.) which yield an easy solution to the following problem: given a fixed Markov operator T, find necessary and sufficient conditions on any other Markov operator R for the relation ker T ⊂ker R to hold. The main application of this is to inclusion relations between matrix summability methods.

scholarly journals The generators of $3$-class group of some fields of degree $6$ over $\mathbb{Q}$

2021 ◽  
Vol 39 (3) ◽  
pp. 37-52
Author(s):  
Siham Aouissi ◽  
Moulay Chrif Ismaili ◽  
Mohamed Talbi ◽  
Abdelmalek Azizi
Keyword(s):  
Prime Number ◽  
Class Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Let be k=Q(\sqrt[3]{p},\zeta_3), where p is a prime number such that p \equiv 1 (mod  9), and let C_{k,3} the 3-component of the class group of k. In his work [7], Frank Gerth III proves a conjecture made by Calegari and Emerton which gives a necessary and sufficient conditions for C_{k,3} to be of rank two. The present work display a consideration steps towards determination of generators of C_{k,3}, when C_{k,3} is isomorphic to Z/9Z \times Z/3Z.

Characterization of summability methods with high indices

2013 ◽  
Vol 63 (5) ◽  
Author(s):  
Mehmet Sarigol
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Summability Methods ◽  
Necessary And Sufficient ◽  
Crucial Assumption

AbstractIn this paper we establish a set of necessary and sufficient conditions in order that |C, 0|k ⇒ |R, p n|s and |R, p n|k ⇒ |C, 0|s for the case 1 < k ≤ s < ∞. As a corollary, we obtain that a crucial assumption of [BOR, H.: A new result on the high indices theorem, Analysis 29 (2009), 403–405] is omitted and that the other one is not only sufficient but also necessary for his consequence to hold.

scholarly journals On factor relations between weighted and Nörlund means

2018 ◽  
Vol 50 (1) ◽  
pp. 61-69 ◽  
Author(s):  
G. Canan Hazar Güleç ◽  
Mehmet Ali Sarıgöl
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Summability Methods ◽  
Nörlund Means ◽  
Necessary And Sufficient ◽  
Special Case

By $\left( X,Y\right) ,$ we denote the set of all sequences $\epsilon =\left( \epsilon _{n}\right) $ such that $\Sigma \epsilon _{n}a_{n}$ is summable $Y$ whenever $\Sigma a_{n}$ is summable $X,$ where $X$ and $Y$ are two summability methods. In this study, we get necessary and sufficient conditions for $\epsilon \in \left( \left\vert N,q_{n},u_{n}\right\vert _{k},\left\vert \bar{N},p_{n}\right\vert \right) $ and $\epsilon \in \left( \left\vert \bar{N},p_{n}\right\vert ,\left\vert N,q_{n},u_{n}\right\vert _{k}\right) $, $k\geq 1,$ using functional analytic tecniques, where $% \left\vert \bar{N},p_{n}\right\vert $ and $\left\vert N,q_{n},u_{n}\right\vert _{k}$ are absolute weighted and N\"{o}rlund summability methods, respectively, \cite{1}, \cite{5}. Thus, in the special case, some well known results are also deduced.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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