scholarly journals On nonsingularity and group invertibility of combinations of two group invertible matrices

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3675-3687
Author(s):  
Yu Li ◽  
Kezheng Zuo
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Necessary And Sufficient ◽  
Invertible Matrices

Let A and B be two group invertible matrices, we study the rank, the nonsingularity and the group invertibility of A-B, AA#-BB#, c1A + c2B, c1A + c2B + c3AA#B where c1,c2 are nonzero complex numbers. Under some special conditions, the necessary and sufficient conditions of c1A + c2B + c3and c1A + c2B + c3+ c4BA to be nonsingular and group invertible are presented, which generalized some related results of Ben?tez, Liu, Koliha and Zuo [4, 17, 19, 25].

On the Construction of Nonnegative 5×5 Matrices from Spectrum Data

2013 ◽  
Vol 444-445 ◽  
pp. 621-624
Author(s):  
Zhi Bing Liu ◽  
Zhen Tu ◽  
Cheng Feng Xu
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Nonnegative Matrices ◽  
Necessary And Sufficient ◽  
Spectrum Data

This paper studies the construction problems of five order nonnegative matrices from spectrum data. Let be a list of complex numbers with . Necessary and sufficient conditions for the existence of an entry-wise nonnegative 5×5 matrix with spectrum are presented.

scholarly journals A Tauberian theorem for the statistical generalized Nörlund-Euler summability method

2019 ◽  
Vol 11 (2) ◽  
pp. 251-263
Author(s):  
Naim L. Braha
Keyword(s):  
Tauberian Theorem ◽  
Sufficient Conditions ◽  
Summability Method ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Necessary And Sufficient

Abstract Let (pn) and (qn) be any two non-negative real sequences with {{\rm{R}}_{\rm{n}}}: = \sum\limits_{{\rm{k}} = 0}^{\rm{n}} {{{\rm{p}}_{\rm{k}}}{{\rm{q}}_{{\rm{n}} - {\rm{k}}}}} \ne 0\,\,\,\,\left( {{\rm{n}} \in {\rm\mathbb{N}}} \right) With {\rm{E}}_{\rm{n}}^1 − we will denote the Euler summability method. Let (xn) be a sequence of real or complex numbers and set {\rm{N}}_{{\rm{p}},{\rm{q}}}^{\rm{n}}{\rm{E}}_{\rm{n}}^1: = {1 \over {{{\rm{R}}_{\rm{n}}}}}\sum\limits_{{\rm{k}} = 0}^{\rm{n}} {{{\rm{p}}_{\rm{k}}}{{\rm{q}}_{{\rm{n - k}}}}{1 \over {{2^{\rm{k}}}}}\sum\limits_{{\rm{v}} = 0}^{\rm{k}} {\left( {_{\rm{v}}^{\rm{k}}} \right){{\rm{x}}_{\rm{v}}}} } for n ∈ ℕ. In this paper, we present necessary and sufficient conditions under which the existence of the st− limit of (xn) follows from that of {\rm{st - N}}_{{\rm{p}},q}^{\rm{n}}{\rm{E}}_{\rm{n}}^1 − limit of (xn). These conditions are one-sided or two-sided if (xn) is a sequence of real or complex numbers, respectively.

Orthonormal expansions and the principle of uniform boundedness

1991 ◽  
Vol 43 (2) ◽  
pp. 341-347
Author(s):  
S.A. Husain ◽  
V.M. Sehgal
Keyword(s):  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Orthonormal Sequence ◽  
Necessary And Sufficient ◽  
Complex Valued ◽  
Orthonormal Expansions

Let {φν: ν ∈ N (non-negative integers)} ⊆ C[0, 1] be a complete orthonormal sequence of complex-valued functions in L2[0, 1], {λν: ν ∈ N} and {λνμ: ν, μ ∈ N} be sequences of complex numbers. In this paper, the necessary and sufficient conditions are developed for the series to converge and also to exist, in C[0, 1] for each f ∈ L1[0, 1] where .

On Exchange QB∞-Rings

2007 ◽  
Vol 14 (04) ◽  
pp. 613-623 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exchange Ring ◽  
New Class ◽  
Necessary And Sufficient ◽  
Invertible Matrices ◽  
Diagonal Reduction ◽  
Square Matrix

In this paper, we introduce a new class of rings, the QB∞-rings. We investigate necessary and sufficient conditions under which an exchange ring is a QB∞-ring. The modules over an exchange QB∞-ring are studied. Also, we prove that every regular square matrix over an exchange QB∞-ring admits a diagonal reduction by pseudo-invertible matrices.

scholarly journals On the Irreducibility of Perron Representations of Degrees 4 and 5

2018 ◽  
Vol 11 (1) ◽  
pp. 215
Author(s):  
Malak M. Dally ◽  
Mohammad N. Abdulrahim
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Artin Group ◽  
Necessary And Sufficient

We consider the graph $E_{n+1,1}$ with (n+1) generators $\sigma_1,..., \sigma_{n}$, and $\delta$, where $\sigma_{i}$ has an edge with $\sigma_{i+1}$ for $i=1,...,n+1$, and $ \sigma_{1}$ has an edge with $\delta$. We then define the Artin group of the graph $E_{n+1,1}$ for $n=3$ and $n=4$ and consider its reduced Perron's representation of degrees four and five respectively. After we specialize the indeterminates used in defining the representation to non-zero complex numbers, we obtain necessary and sufficient conditions that guarantee the irreducibility of the representations for $n=3$ and $4$ .

Moment Problems and Quasi-Hausdorff Transformations

1968 ◽  
Vol 11 (2) ◽  
pp. 225-236 ◽  
Author(s):  
Dany Leviatan
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Moment Problems ◽  
Right Hand ◽  
The Right ◽  
Necessary And Sufficient

The sequence to sequence quasi - Hausdorff transformations were defined by Hardy [1] 1 1. 19 p. 277 as follows. For a given sequence {μn} (n ≥ 0) of real or complex numbers, define the operator Δ by for k > l. {tm} (m ≥ 0) is called the sequence to sequence quasi-Hausdorff transform by means of {μn} (or, in short, the [QH, μn] transform) of {sn} (n ≥ 0) if if , provided that the sums on the right-hand side converge for all m ≥ 0. Ramanujan in [11] and [12] has defined the series to series quasi-Hausdorff transformation s and has proved necessary and sufficient conditions for the regularity of the two kinds of transformations.

scholarly journals Tauberian conditions under which statistical convergence follows from statistical summability $(EC)_{n}^1$

2018 ◽  
Vol 37 (4) ◽  
pp. 9
Author(s):  
Naim L. Braha ◽  
Ismet Temaj
Keyword(s):  
Finite Number ◽  
Statistical Convergence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Real Numbers ◽  
Tauberian Conditions ◽  
Necessary And Sufficient

Let $(x_k)$, for $k\in \mathbb{N}\cup \{0\}$  be a sequence of real or complex numbers and set $(EC)_{n}^{1}=\frac{1}{2^n}\sum_{j=0}^{n}{\binom{n}{j}\frac{1}{j+1}\sum_{v=0}^{j}{x_v}},$ $n\in \mathbb{N}\cup \{0\}.$  We present necessary and sufficient conditions, under which $st-\lim_{}{x_k}= L$ follows from $st-\lim_{}{(EC)_{n}^{1}} = L,$ where L is a finite number. If $(x_k)$ is a sequence of real numbers, then these are one-sided Tauberian conditions. If $(x_k)$ is a sequence of complex numbers, then these are two-sided Tauberian conditions.

scholarly journals On idempotency of linear combinations of a quadratic or a cubic matrix and an arbitrary matrix

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3161-3185
Author(s):  
Murat Sarduvan ◽  
Nurgül Kalaycı
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Linear Combinations ◽  
Arbitrary Matrix ◽  
Necessary And Sufficient

Let A be a quadratic or a cubic n x n nonzero matrix and B be an arbitrary n x n nonzero matrix. In this study, we have established necessary and sufficient conditions for the idempotency of the linear combinations of the form aA + bB, under the some certain conditions imposed on A and B, where a, b are nonzero complex numbers.

scholarly journals A Tauberian theorem for the generalized Nörlund summability method

2020 ◽  
Vol 27 (1) ◽  
pp. 31-36
Author(s):  
İbrahim Çanak ◽  
Naim L. Braha ◽  
Ümit Totur
Keyword(s):  
Tauberian Theorem ◽  
Sufficient Conditions ◽  
Summability Method ◽  
Necessary And Sufficient Conditions ◽  
Partial Sums ◽  
Complex Numbers ◽  
Necessary And Sufficient

AbstractLet {(p_{n})} and {(q_{n})} be any two non-negative real sequences, with {R_{n}:=\sum_{k=0}^{n}{p_{k}q_{n-k}}\neq 0} ({n\in\mathbb{N}}). Let {\sum_{k=0}^{\infty}a_{k}} be a series of real or complex numbers with partial sums {(s_{n})}, and set {t_{n}^{p,q}:=\frac{1}{R_{n}}\sum_{k=0}^{n}{p_{k}q_{n-k}s_{k}}} for {n\in\mathbb{N}}. In this paper, we present the necessary and sufficient conditions under which the existence of the limit {\lim_{n\to\infty}{s_{n}}=L} follows from that of {\lim_{n\to\infty}t_{n}^{p,q}=L}. These conditions are one-sided or two-sided if {(s_{n})} is a sequence of real or complex numbers, respectively.

scholarly journals ON h-HOLOMORPHY AND h-ANALYTICITY OF FUNCTIONS OF AN h-COMPLEX VARIABLE

2020 ◽  
Vol 133 (4) ◽  
pp. 19-27
Author(s):  
V. A. Pavlovsky ◽  
◽  
I. L. Vasiliev ◽  
Keyword(s):  
Uniqueness Theorem ◽  
Complex Variable ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Necessary And Sufficient

Interest in the study of the properties of functions defined on the set of \textit{h}-complex numbers arose again in connection with existing applications in geometry and mechanics. In this paper, we present necessary and sufficient conditions for \textit{h}-differentiability and \textit{h}-holomorphy of functions of an \textit{h}-complex variable, the theorem on finite increments is proved, sufficient conditions for \textit{h}-analyticity are found, a uniqueness theorem for \textit{h}-analytic functions is proved.

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