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## Sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/07/2015

Relevância na Pesquisa

36.09%

In this paper, we obtain the sharp upper and lower bounds for the spectral
radius of a nonnegative irreducible matrix. We also apply these bounds to
various matrices associated with a graph or a digraph, obtain some new results
or known results about various spectral radii, including the adjacency spectral
radius, the signless Laplacian spectral radius, the distance spectral radius,
the distance signless Laplacian spectral radius of a graph or a digraph.; Comment: 19 pages

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## Application of topological radicals to calculation of joint spectral radii

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/05/2008

Relevância na Pesquisa

46.03%

It is shown that the joint spectral radius $\rho(M)$ of a precompact family
$M$ of operators on a Banach space $X$ is equal to the maximum of two numbers:
the joint spectral radius $\rho_{e}(M)$ of the image of $M$ in the Calkin
algebra and the Berger-Wang radius $r(M)$ defined by the formula \[
r(M)=\underset{n\to\infty}{\limsup}(\sup\left\{\rho(a):a\in M^{n}\right\}
^{1/n}) . \] Some more general Banach-algebraic results of this kind are also
proved. The proofs are based on the study of special radicals on the class of
Banach algebras.

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## Lower bounds of the skew spectral radii and skew energy of oriented graphs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.27%

Let $G$ be a graph with maximum degree $\Delta$, and let $G^{\sigma}$ be an
oriented graph of $G$ with skew adjacency matrix $S(G^{\sigma})$. The skew
spectral radius $\rho_s(G^{\sigma})$ of $G^\sigma$ is defined as the spectral
radius of $S(G^\sigma)$. The skew spectral radius has been studied, but only
few results about its lower bound are known. This paper determines some lower
bounds of the skew spectral radius, and then studies the oriented graphs whose
skew spectral radii attain the lower bound $\sqrt{\Delta}$. Moreover, we apply
the skew spectral radius to the skew energy of oriented graphs, which is
defined as the sum of the norms of all the eigenvalues of $S(G^\sigma)$, and
denoted by $\mathcal{E}_s(G^\sigma)$. As results, we obtain some lower bounds
of the skew energy, which improve the known lower bound obtained by Adiga et
al.; Comment: 16 pages, 2 figures

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## Laplacian and signless Laplacian spectral radii of graphs with fixed domination number

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/10/2013

Relevância na Pesquisa

46.1%

In this paper, we determine the maximal Laplacian and signless Laplacian
spectral radii for graphs with fixed number of vertices and domination number,
and characterize the extremal graphs respectively.

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## Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

36.3%

Recently Blondel, Nesterov and Protasov proved that the finiteness conjecture
holds for the generalized and the lower spectral radii of the sets of
non-negative matrices with independent row/column uncertainty. We show that
this result can be obtained as a simple consequence of the so-called hourglass
alternative earlier used by the author and his companions to analyze the
minimax relations between the spectral radii of matrix products. Axiomatization
of the statements that constitute the hourglass alternative makes it possible
to define a new class of sets of positive matrices having the finiteness
property, which includes the sets of non-negative matrices with independent row
uncertainty. This class of matrices, supplemented by the zero and identity
matrices, forms a semiring with the Minkowski operations of addition and
multiplication of matrix sets, which gives means to construct new sets of
non-negative matrices possessing the finiteness property for the generalized
and the lower spectral radii.; Comment: 17 pages, 38 bibliography references, minor corrections, accepted for
publication in Linear Algebra and its Applications

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## The extremal spectral radii of $k$-uniform supertrees

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/05/2014

Relevância na Pesquisa

56.54%

In this paper, we study some extremal problems of three kinds of spectral
radii of $k$-uniform hypergraphs (the adjacency spectral radius, the signless
Laplacian spectral radius and the incidence $Q$-spectral radius).
We call a connected and acyclic $k$-uniform hypergraph a supertree. We
introduce the operation of "moving edges" for hypergraphs, together with the
two special cases of this operation: the edge-releasing operation and the total
grafting operation. By studying the perturbation of these kinds of spectral
radii of hypergraphs under these operations, we prove that for all these three
kinds of spectral radii, the hyperstar $\mathcal{S}_{n,k}$ attains uniquely the
maximum spectral radius among all $k$-uniform supertrees on $n$ vertices. We
also determine the unique $k$-uniform supertree on $n$ vertices with the second
largest spectral radius (for these three kinds of spectral radii). We also
prove that for all these three kinds of spectral radii, the loose path
$\mathcal{P}_{n,k}$ attains uniquely the minimum spectral radius among all
$k$-th power hypertrees of $n$ vertices. Some bounds on the incidence
$Q$-spectral radius are given. The relation between the incidence $Q$-spectral
radius and the spectral radius of the matrix product of the incidence matrix
and its transpose is discussed.

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## Lower bounds for the spectral radii of adjacency operators on Baumslag-Solitar groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.1%

#Mathematics - Operator Algebras#Mathematics - Group Theory#Mathematics - Probability#46L54 (Primary) 60B15, 60G50 (Secondary)

We will use free probability techniques to find lower bounds for the spectral
radii of the adjacency operators on the Caley graphs of some non-amenable
Baumslag-Solitar groups with the standard generators.; Comment: The preprint is now withdrawn, because of an error in the technique.
(Certain words were counted twice.) However, we are working on producing a
correct argument and we are able to obtain a nontrivial lower bound

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## Counting Spectral Radii of Matrices with Positive Entries

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/05/2013

Relevância na Pesquisa

46.1%

The sum-product conjecture of Erd\H os and Szemer\'edi states that, given a
finite set $A$ of positive numbers, one can find asymptotic lower bounds for
$\max\{|A+A|,|A\cdot A|\}$ of the order of $|A|^{1+\delta}$ for every $\delta
<1$. In this paper we consider the set of all spectral radii of $n\times n$
matrices with entries in $A$, and find lower bounds for the cardinality of this
set. In the case $n=2$, this cardinality is necessarily larger than
$\max\{|A+A|,|A\cdot A|\}$.

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## Spectral Radii of Bounded Operators on Topological Vector Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/04/2000

Relevância na Pesquisa

56.25%

In this paper we develop a version of spectral theory for bounded linear
operators on topological vector spaces. We show that the Gelfand formula for
spectral radius and Neumann series can still be naturally interpreted for
operators on topological vector spaces. Of course, the resulting theory has
many similarities to the conventional spectral theory of bounded operators on
Banach spaces, though there are several important differences. The main
difference is that an operator on a topological vector space has several
spectra and several spectral radii, which fit a well-organized pattern.; Comment: 36 pages

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## Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

36.1%

#Mathematical Physics#Condensed Matter - Mesoscale and Nanoscale Physics#Mathematics - Analysis of PDEs#Mathematics - Spectral Theory#81Q10, 47B80, 81Q15

Bound states of the Hamiltonian describing a quantum particle living on three
dimensional straight strip of width $d$ are investigated. We impose the Neumann
boundary condition on the two concentric windows of the radii $a$ and $ b$
located on the opposite walls and the Dirichlet boundary condition on the
remaining part of the boundary of the strip. We prove that such a system
exhibits discrete eigenvalues below the essential spectrum for any $a,b>0$.
When $a$ and $b$ tend to the infinity, the asymptotic of the eigenvalue is
derived. A comparative analysis with the one-window case reveals that due to
the additional possibility of the regulating energy spectrum the anticrossing
structure builds up as a function of the inner radius with its sharpness
increasing for the larger outer radius. Mathematical and physical
interpretation of the obtained results is presented; namely, it is derived that
the anticrossings are accompanied by the drastic changes of the wave function
localization. Parallels are drawn to the other structures exhibiting similar
phenomena; in particular, it is proved that, contrary to the two-dimensional
geometry, at the critical Neumann radii true bound states exist.; Comment: 25 pages, 7 figures

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## An algorithm computing non solvable spectral radii of $p$-adic differential equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/01/2013

Relevância na Pesquisa

46.1%

We obtain an algorithm computing explicitly the values of the non solvable
spectral radii of convergence of the solutions of a differential module over a
point of type 2, 3 or 4 of the Berkovich affine line.; Comment: 5 pages

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## Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/06/2014

Relevância na Pesquisa

46.1%

It is well known that an ordered normed vector space $X$ with normal cone
$X_+$ has an order-preserving norm that is equivalent to the original norm.
Such an equivalent order-preserving norm is given by \begin{equation} \sharp x
\sharp = \max \{ d(x, X_+), d(x, - X_+)\}, \qquad x \in X. \end{equation} This
paper explores the properties of this norm and of the half-norm $\psi(x) =
d(x,-X_+)$ independently of whether or not the cone is normal. We use $\psi$ to
derive comparison principles for the solutions of abstract integral equations,
derive conditions for point-dissipativity of nonlinear positive maps, compare
Collatz-Wielandt numbers, bounds, and order spectral radii for bounded
homogeneous maps and give conditions for a local upper Collatz-Wielandt radius
to have a lower positive eigenvector.

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## On spectral radius of strongly connected digraphs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/05/2013

Relevância na Pesquisa

36.25%

We determine the digraphs which achieve the second, the third and the fourth
minimum spectral radii respectively among strongly connected digraphs of order
$n\ge 4$, and thus we answer affirmatively the problem whether the unique
digraph which achieves the minimum spectral radius among all strongly connected
bicyclic digraphs of order $n$ achieves the second minimum spectral radius
among all strongly connected digraphs of order $n$ for $n\ge 4$ proposed in
[H. Lin, J. Shu, A note on the spectral characterization of strongly
connected bicyclic digraphs, Linear Algebra Appl. 436 (2012) 2524--2530]. We
also discuss the strongly connected bicyclic digraphs with small and large
spectral radii respectively.

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## Some Spectral Properties of Odd-Bipartite $Z$-Tensors and Their Absolute Tensors

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/03/2015

Relevância na Pesquisa

45.99%

Stimulated by odd-bipartite and even-bipartite hypergraphs, we define
odd-bipartite (weakly odd-bipartie) and even-bipartite (weakly even-bipartite)
tensors. It is verified that all even order odd-bipartite tensors are
irreducible tensors, while all even-bipartite tensors are reducible no matter
the parity of the order. Based on properties of odd-bipartite tensors, we study
the relationship between the largest H-eigenvalue of a $Z$-tensor with
nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor
of that $Z$-tensor. When the order is even and the $Z$-tensor is weakly
irreducible, we prove that the largest H-eigenvalue of the $Z$-tensor and the
largest H-eigenvalue of the absolute tensor of that $Z$-tensor are equal, if
and only if the $Z$-tensor is weakly odd-bipartite. Examples show the
authenticity of the conclusions. Then, we prove that a symmetric $Z$-tensor
with nonnegative diagonal entries and the absolute tensor of the $Z$-tensor are
diagonal similar, if and only if the $Z$-tensor has even order and it is weakly
odd-bipartite. After that, it is proved that, when an even order symmetric
$Z$-tensor with nonnegative diagonal entries is weakly irreducible, the
equality of the spectrum of the $Z$-tensor and the spectrum of absolute tensor
of that $Z$-tensor...

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## The Berger-Wang formula for the Markovian joint spectral radius

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

36.3%

The Berger-Wang formula establishes equality between the joint and
generalized spectral radii of a set of matrices. For matrix products whose
multipliers are applied not arbitrarily but in accordance with some Markovian
law, there are also known analogs of the joint and generalized spectral radii.
However, the known proofs of the Berger-Wang formula hardly can be directly
applied in the case of Markovian products of matrices since they essentially
rely on the arbitrariness of appearance of different matrices in the related
matrix products. Nevertheless, as has been shown by X. Dai the Berger-Wang
formula is valid for the case of Markovian analogs of the joint and the
generalized spectral radii too, although the proof in this case heavily
exploits the more involved techniques of multiplicative ergodic theory. In the
paper we propose a matrix theory construction allowing to deduce the Markovian
analog of the Berger-Wang formula from the classical Berger-Wang formula.; Comment: 13 pages, 29 bibliography references; minor corrections; accepted for
publication in Linear Algebra and its Applications

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## On the spectral radii of bicyclic graphs with fixed independence number

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/02/2014

Relevância na Pesquisa

46.1%

Bicyclic graph is a connected graph in which the number of edges equals the
number of vertices plus one. In this paper, we determine the graph which alone
maximizes the spectral radii among all the bicyclic graphs on $n$ vertices with
fixed independence number.; Comment: 13pages

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## A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.05%

#Mathematics - Functional Analysis#Mathematics - Optimization and Control#Mathematics - Spectral Theory#47H07, 47H08, 91A25

Several notions of spectral radius arise in the study of nonlinear
order-preserving positively homogeneous self-maps of cones in Banach spaces. We
give conditions that guarantee that all these notions lead to the same value.
In particular, we give a Collatz-Wielandt type formula, which characterizes the
growth rate of the orbits in terms of eigenvectors in the closed cone or
super-eigenvectors in the interior of the cone. This characterization holds
when the cone is normal and when a quasi-compactness condition, involving an
essential spectral radius defined in terms of $k$-set-contractions, is
satisfied. Some fixed point theorems for non-linear maps on cones are derived
as intermediate results. We finally apply these results to show that non-linear
spectral radii commute with respect to suprema and infima of families of order
preserving maps satisfying selection properties.; Comment: 24 pages, v2: minor improvements and updated references

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## Mather sets for sequences of matrices and applications to the study of joint spectral radii

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/09/2011

Relevância na Pesquisa

46.16%

The joint spectral radius of a compact set of d-times-d matrices is defined
?to be the maximum possible exponential growth rate of products of matrices
drawn from that set. In this article we investigate the ergodic-theoretic
structure of those sequences of matrices drawn from a given set whose products
grow at the maximum possible rate. This leads to a notion of Mather set for
matrix sequences which is analogous to the Mather set in Lagrangian dynamics.
We prove a structure theorem establishing the general properties of these
Mather sets and describing the extent to which they characterise matrix
sequences of maximum growth. We give applications of this theorem to the study
of joint spectral radii and to the stability theory of discrete linear
inclusions.
These results rest on some general theorems on the structure of orbits of
maximum growth for subadditive observations of dynamical systems, including an
extension of the semi-uniform subadditive ergodic theorem of Schreiber, Sturman
and Stark, and an extension of a noted lemma of Y. Peres. These theorems are
presented in the appendix.

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## Maximizing spectral radii of uniform hypergraphs with few edges

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.16%

In this paper we investigate the hypergraphs whose spectral radii attain the
maximum among all uniform hypergraphs with given number of edges. In particular
we characterize the hypergraph(s) with maximum spectral radius over all
unicyclic hypergraphs, linear or power unicyclic hypergraphs with given girth,
linear or power bicyclic hypergraphs, respectively.

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## Ordering uniform supertrees by their spectral radii

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/02/2015

Relevância na Pesquisa

46.35%

A connected and acyclic hypergraph is called a supertree. In this paper we
mainly focus on the spectral radii of uniform supertrees. Li, Shao and Qi
determined the first two $k$-uniform supertrees with large spectral radii among
all the $k$-uniform supertrees on $n$ vertices [H. Li, J. Shao, L. Qi, The
extremal spectral radii of $k$-uniform supertrees, arXiv:1405.7257v1, May
2014]. By applying the operation of moving edges on hypergraphs and using the
weighted incidence matrix method we extend the above order to the fourth
$k$-uniform supertree.; Comment: arXiv admin note: text overlap with arXiv:1405.7257 by other authors

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