Approximation in Bounded Summability Fields

1968 ◽  
Vol 20 ◽  
pp. 410-415 ◽  
Author(s):  
J. D. Hill ◽  
W. T. Sledd
Keyword(s):  
Strong Summability ◽  
Summability Field ◽  
Image Position ◽  
Real Matrices

This paper deals with several related properties of bounded summability fields of regular, real matrices. For a matrix A = (ank) and a sequence x = {xn}, we write formallyWe denote by m the space of bounded real sequences, and by A* the bounded summability fieldof A. The strong summability field of A is the set

A Further Extension of Cayley's Parameterization

1959 ◽  
Vol 11 ◽  
pp. 48-50 ◽  
Author(s):  
Martin Pearl
Keyword(s):  
Symmetric Matrix ◽  
Arbitrary Field ◽  
Complex Field ◽  
Result Theorem ◽  
Characteristic 2 ◽  
Image Position ◽  
Row Space ◽  
Real Matrices ◽  
Skew Symmetric Matrix

In a recent paper (3)* the following theorem was proved for real matrices.Theorem 1. If A is a symmetric matrix and Q is a skew-symmetric matrix such that A + Q is non-singular, then1is a cogredient automorph (c.a.) of A whose determinant is + 1 and having theproperty that A and I + P span the same row space.Conversely, if P is a c.a. of A whose determinant is + 1 and if P has theproperty that I + P and A span the same row space, then there exists a skew symmetricmatrix Q such that P is given by equation (1).Theorem 1 reduces to the well-known Cayley parameterization in the case where A is non-singular. A similar and somewhat simpler result (Theorem 4) was given for the case when the underlying field is the complex field. It was also shown that the second part of the theorem (in either form) is false when the characteristic of the underlying field is 2. The purpose of this paper is to simplify the proof of Theorem 1 and at the same time, to extend these results to matrices over an arbitrary field of characteristic ≠ 2.

scholarly journals A bounded consistency theorem for strong summabilities

1989 ◽  
Vol 12 (1) ◽  
pp. 39-46 ◽  
Author(s):  
C. S. Chun ◽  
A. R. Freedman
Keyword(s):  
Strong Summability ◽  
Regular Matrix ◽  
Matrix Methods ◽  
Summability Methods ◽  
Summability Field

The study of R-type summability methods is continued in this paper by showing that two such methods are identical on the bounded portion of the strong summability field associated with the methods. It is shown that this “bounded consistency” applies for many non-matrix methods as well as for regular matrix methods.

Strong summability of infinite series on a scale of Abel type summability methods

1967 ◽  
Vol 63 (1) ◽  
pp. 119-127 ◽  
Author(s):  
Babban Prasad Mishra
Keyword(s):  
Infinite Series ◽  
Open Interval ◽  
Strong Summability ◽  
New Method ◽  
Finite Limit ◽  
Summability Methods ◽  
Image Position

Introduction. In a recent paper, Borwein(1) constructed a new method of summability which would read: Letand let {sn} be any sequence of numbers. If, for λ > − 1,is convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we say that the sequence {sn} is Aλ convergent to s and write sn → s(Aλ). The A0 method is the ordinary Abel method.

On the strong summability of Fourier series

1930 ◽  
Vol 26 (4) ◽  
pp. 429-437 ◽  
Author(s):  
R. E. A. C. Paley
Keyword(s):  
Fourier Series ◽  
Strong Summability ◽  
Summability Of Fourier Series ◽  
Image Position

Let f(t) be a function periodic in 2π, and absolutely integrable in the interval (0, 2π). We write

scholarly journals An Analogue for Strong Summability of Abel's Summability Method

1953 ◽  
Vol 9 (1) ◽  
pp. 28-34
Author(s):  
C. F. Harington ◽  
J. M. Hyslop
Keyword(s):  
Sufficient Conditions ◽  
Binomial Coefficient ◽  
Summability Method ◽  
Strong Summability ◽  
Necessary And Sufficient Conditions ◽  
The Difference ◽  
Image Position ◽  
Necessary And Sufficient

Given a series Σan, we define , by the relationwhere is the binomial coefficient . Let . If , the series Σan is said to be summable (C; k) to the sum s. If k > 0, p ≥ 1 and if, as n → ∞,we say that the series Σan is summable [C; k, p] to the sum s, or that the series is strongly summable (C; k) with index p to the sum s. If denotes the difference , it is known that necessary and sufficient conditions for summability [C; k, p], k > 0, p ≥ 1, to the sum s, are that Σan be summable (C; k) to the sum s and that

Improving certain simple eigenvalue bounds

1986 ◽  
Vol 99 (3) ◽  
pp. 507-518
Author(s):  
Seppo Hyydö ◽  
Jorma Kaarlo Merikoski ◽  
Ari Virtanen
Keyword(s):  
Rayleigh Quotient ◽  
Simple Eigenvalue ◽  
Eigenvalue Bounds ◽  
Simple Property ◽  
Image Position ◽  
Real Matrices

Throughout this paper we let A = (aij) be a non-zero n × n matrix-we study real matrices only–with row sums R1,…Rn and eigenvalues λ1,…,λn, ordered λ1≥…≥λn if they are real. We denote E = (1,…,1)T and su A = ΣiΣjaij = ETAE. If A is symmetric, a simple property of the Rayleigh quotient is thatsatisfiesand

Generalized Nörlund means and consistency theorems

1980 ◽  
Vol 87 (2) ◽  
pp. 243-248
Author(s):  
I. J. Maddox
Keyword(s):  
Infinite Matrix ◽  
Nörlund Means ◽  
Summability Field ◽  
General Complex ◽  
Image Position ◽  
Convergent Sequences

It was recently shown in (1), corollary 1, that if (I) = (N, q) then I and (N, q) were consistent. By I we denote the unit infinite matrix, and (N, q) denotes a general complex Nörlund mean. The summability field (A) of any infinite matrix A = (ank) is defined aswhere c is the space of convergent sequences. The summability field of (N, q) is, for simplicity, written as (N, q) instead of ((N, q)).

scholarly journals Note on the Strong Summability of Series

1952 ◽  
Vol 1 (1) ◽  
pp. 16-20 ◽  
Author(s):  
J. M. Hyslop
Keyword(s):  
Binomial Coefficient ◽  
Strong Summability ◽  
Image Position

Given the series ,the n-th Casáro sum of order k is defined by the relationwhere is the binomial coefficient . Let Then Σan is said to be summable (C; K) to the sum s if, as n → ∞, The series is said to be absolutely summable (C; k), or summable | is convergent. The series is said to be strongly summable (C; k) with index p, or summable [Ck, p], to the sum s ifIt is assumed that k and p are positive.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

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