scholarly journals Norm of nonnegative and positive matrices

2017 ◽  
Vol 6 (3) ◽  
pp. 98
Author(s):  
Alaa Abu Alroz
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Maximum Modulus ◽  
Upper Bounds ◽  
Nonnegative Matrices ◽  
Lower And Upper Bounds ◽  
Positive Matrix ◽  
Eigen Value ◽  
Eigen Values ◽  
Positive Matrices

The spectral radius r(A) of matrix A is the maximum modulus of the Eigen values. In this paper, the studies about the lower and upper bounds for the spectral radius and the lower bounds for the minimum eigen value of appositive and nonnegative matrices are investigate.The matrix norm, the spectral radius norm,and the column (row) sums of nonnegative and positive matrices are widely used to establish some inequalities for matrices. Then several existing results are improved for these inequalities for nonnegative and positive matrix. Furthermore, the lower and upper bounds of the Perron roots for nonnegative matrices are examined, and some upper bounds are computed.

scholarly journals Lower bounds for maximal cp-ranks of completely positive matrices and tensors

2020 ◽  
Vol 36 (36) ◽  
pp. 519-541
Author(s):  
Werner Schachinger
Keyword(s):  
Lower Bounds ◽  
Generating Function ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Completely Positive ◽  
Completely Positive Matrices ◽  
Positive Matrices ◽  
The Difference

Let $p_n$ denote the maximal cp-rank attained by completely positive $n\times n$ matrices. Only lower and upper bounds for $p_n$ are known, when $n\ge6$, but it is known that $p_n=\frac{n^2}2\big(1+o(1)\big)$, and the difference of the current best upper and lower bounds for $p_n$ is of order $\mathcal{O}\big(n^{3/2}\big)$. In this paper, that gap is reduced to $\mathcal{O}\big(n\log\log n\big)$. To achieve this result, a sequence of generalized ranks of a given matrix A has to be introduced. Properties of that sequence and its generating function are investigated. For suitable A, the $d$th term of that sequence is the cp-rank of some completely positive tensor of order $d$. This allows the derivation of asymptotically matching lower and upper bounds for the maximal cp-rank of completely positive tensors of order $d>2$ as well.

scholarly journals A lower bound for general t-stack sortable permutations via pattern avoidance

2020 ◽  
Vol 22 ◽  
Author(s):  
Pranav Chinmay
Keyword(s):  
Lower Bound ◽  
Recurrence Relation ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Pattern Avoidance ◽  
Lower And Upper Bounds

There is no formula for general t-stack sortable permutations. Thus, we attempt to study them by establishing lower and upper bounds. Permutations that avoid certain pattern sets provide natural lower bounds. This paper presents a recurrence relation that counts the number of permutations that avoid the set (23451,24351,32451,34251,42351,43251). This establishes a lower bound on 3-stack sortable permutations. Additionally, the proof generalizes to provide lower bounds for all t-stack sortable permutations.

scholarly journals Improve Some of Bounds for the Randic Estrada Index of Graphs

2020 ◽  
Author(s):  
Akbar Jahanbani
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Graph Invariants ◽  
Randic Index ◽  
Estrada Index ◽  
Eigenvalues Of Graphs

Let G be a graph with n vertices and let 1; 2; : : : ; n be the eigenvalues of Randic matrix. The Randic Estrada index of G is REE(G) = Ón i=1 ei . In this paper, we establish lower and upper bounds for Randic index in terms of graph invariants such as the number of vertices and eigenvalues of graphs and improve some previously published lower bounds.

On spectra and spectral radius of Signless Laplacian of power graphs of some finite groups

2020 ◽  
pp. 2150090
Author(s):  
Subarsha Banerjee ◽  
Avishek Adhikari
Keyword(s):  
Spectral Radius ◽  
Finite Groups ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Laplacian Spectral Radius ◽  
Power Graph ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
Extremal Values ◽  
A Finite Group

Let [Formula: see text] denote the power graph of a finite group [Formula: see text]. Let [Formula: see text] denote the Signless Laplacian spectral radius of [Formula: see text]. In this paper, we give lower and upper bounds on [Formula: see text] for any [Formula: see text] and find those graphs for which the extremal values are attained. We give a comparison between the bounds obtained and exact value of [Formula: see text] for any [Formula: see text]. We then find the eigenvalues of [Formula: see text] and give lower and upper bounds on the spectral radius of [Formula: see text]. When [Formula: see text] and [Formula: see text] where [Formula: see text] are primes and [Formula: see text] is a positive integer, we obtain sharper bounds on [Formula: see text]. Finally, we make a conjecture on [Formula: see text] for any [Formula: see text].

Lower Bounds for the Probability of Intersection of Several Unions of Events

1998 ◽  
Vol 7 (4) ◽  
pp. 353-364 ◽  
Author(s):  
TUHAO CHEN ◽  
E. SENETA
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Numerical Example ◽  
Bonferroni Inequalities ◽  
Indicator Functions ◽  
Single Set ◽  
Better Than

To bound the probability of a union of n events from a single set of events, Bonferroni inequalities are sometimes used. There are sharper bounds which are called Sobel–Uppuluri–Galambos inequalities. When two (or more) sets of events are involved, bounds are considered on the probability of intersection of several such unions, one union from each set. We present a method for unified treatment of bivariate lower and upper bounds in this note. The lower bounds obtained are new and at least as good as lower bounds appearing in the literature so far. The upper bounds coincide with existing bivariate Sobel–Uppuluri–Galambos type upper bounds derived by the method of indicator functions. A numerical example is given to illustrate that the new lower bounds can be strictly better than existing ones.

scholarly journals Round weighting problem and gathering in radio networks with symmetrical interference

2016 ◽  
Vol 08 (02) ◽  
pp. 1650035
Author(s):  
Jean-Claude Bermond ◽  
Cristiana Gomes Huiban ◽  
Patricio Reyes
Keyword(s):  
Lower Bounds ◽  
Access Network ◽  
Upper Bounds ◽  
Radio Networks ◽  
Interference Model ◽  
Lower And Upper Bounds ◽  
Single Node ◽  
Model Based ◽  
Weighting Problem ◽  
General Graphs

In this paper, we consider the problem of gathering information in a gateway in a radio mesh access network. Due to interferences, calls (transmissions) cannot be performed simultaneously. This leads us to define a round as a set of non-interfering calls. Following the work of Klasing, Morales and Pérennes, we model the problem as a Round Weighting Problem (RWP) in which the objective is to minimize the overall period of non-interfering calls activations (total number of rounds) providing enough capacity to satisfy the throughput demand of the nodes. We develop tools to obtain lower and upper bounds for general graphs. Then, more precise results are obtained considering a symmetric interference model based on distance of graphs, called the distance-[Formula: see text] interference model (the particular case [Formula: see text] corresponds to the primary node model). We apply the presented tools to get lower bounds for grids with the gateway either in the middle or in the corner. We obtain upper bounds which in most of the cases match the lower bounds, using strategies that either route the demand of a single node or route simultaneously flow from several source nodes. Therefore, we obtain exact and constructive results for grids, in particular for the case of uniform demands answering a problem asked by Klasing, Morales and Pérennes.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (4) ◽  
pp. 929-932 ◽  
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

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