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 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].

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 .

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.

scholarly journals Necessary and sufficient conditions under which convergence follows from summability by weighted means

2001 ◽  
Vol 27 (7) ◽  
pp. 399-406 ◽  
Author(s):  
Ferenc Móricz ◽  
Ulrich Stadtmüller
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Real Numbers ◽  
Weighted Mean ◽  
Tauberian Conditions ◽  
Weighted Means ◽  
Necessary And Sufficient

We prove necessary and sufficient Tauberian conditions for sequences summable by weighted mean methods. The main results of this paper apply to all weighted mean methods and unify the results known in the literature for particular methods. Among others, the conditions in our theorems are easy consequences of the slowly decreasing condition for real numbers, or slowly oscillating condition for complex numbers. Therefore, practically all classical (one-sided as well as two-sided) Tauberian conditions for weighted mean methods are corollaries of our two main theorems.

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