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2. Bounds of modulus of eigenvalues based on Stein equation

Creator:
Hu, Guang-Da and Zhu, Qiao
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
eigenvalues, lower and upper bounds, and Stein equation
Language:
English
Description:
This paper is concerned with bounds of eigenvalues of a complex matrix. Both lower and upper bounds of modulus of eigenvalues are given by the Stein equation. Furthermore, two sequences are presented which converge to the minimal and the maximal modulus of eigenvalues, respectively. We have to point out that the two sequences are not recommendable for practical use for finding the minimal and the maximal modulus of eigenvalues.
Rights:
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3. Bounds on the subdominant eigenvalue involving group inverses with applications to graphs

Creator:
Kirkland, Stephen J., Neumann, Michael, and Shader, Bryan L.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
math, eigenvalues, and graphs
Language:
English
Description:
Let $A$ be an $n\times n$ symmetric, irreducible, and nonnegative matrix whose eigenvalues are $\lambda _1 > \lambda _2 \ge \ldots \ge \lambda _n$. In this paper we derive several lower and upper bounds, in particular on $\lambda _2$ and $\lambda _n$, but also, indirectly, on $\mu = \max _{2\le i \le n} |\lambda _i|$. The bounds are in terms of the diagonal entries of the group generalized inverse, $Q^{\#}$, of the singular and irreducible M-matrix $Q=\lambda _1 I-A$. Our starting point is a spectral resolution for $Q^{\#}$. We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected graphs, where now $Q$ becomes $L$, the Laplacian of the graph. In case the graph is a tree we find a graph-theoretic interpretation for the entries of $L^{\#}$ and we also sharpen an upper bound on the algebraic connectivity of a tree, which is due to Fiedler and which involves only the diagonal entries of $L$, by exploiting the diagonal entries of $L^{\#}$.
Rights:
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4. Controllability of linear impulsive systems - an eigenvalue approach

Creator:
Muni, Vijayakumar S. and George, Raju K.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
controllability, eigenvalues, and impulses
Language:
English
Description:
This article considers a class of finite-dimensional linear impulsive time-varying systems for which various sufficient and necessary algebraic criteria for complete controllability, including matrix rank conditions are established. The obtained controllability results are further synthesised for the time-invariant case, and under some special conditions on the system parameters, we obtain a Popov-Belevitch-Hautus (PBH)-type rank condition which employs eigenvalues of the system matrix for the investigation of their controllability. Numerical examples are provided that demonstrate--for the linear impulsive systems, null controllability need not imply their complete controllability, unlike for the non-impulsive linear systems.
Rights:
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5. Dynamical model and eigenvalues of the turbocharger

Creator:
Zeman, Vladimír and Hlaváč, Zdeněk
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
turbocharger vibrations, eigenvalues, Campbell diagram, and critical speeds
Language:
English
Description:
The paper deals with derivation of the dynamical model of the turbochargers with rotor supported on the two floating ring bearings. The model respects the bearing forces acting upon the journals and floating bearing rings by means of inner and outer oil-films. The gyroscopic effects, external and internal damping of the flexible rotor shaft and the rigid turbine and compressor wheels are respected. The modal analysis and the Campbell diagram is used in the turbocharger linearized model to find the critical speeds. and Obsahuje seznam literatury
Rights:
http://creativecommons.org/licenses/by-nc-sa/4.0/ and policy:public

6. Global structure of positive solutions for superlinear $2m$th-boundary value problems

Creator:
Ma, Ruyun and An, Yulian
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
multiplicity results, Lidstone boundary value problem, eigenvalues, bifurcation methods, and positive solutions
Language:
English
Description:
We consider boundary value problems for nonlinear $2m$th-order eigenvalue problem $$ \begin{aligned} (-1)^mu^{(2m)}(t)&=\lambda a(t)f(u(t)),\ \ \ \ \ 0<t<1, \\ u^{(2i)}(0)&=u^{(2i)}(1)=0,\ \ \ \ i=0,1,2,\cdots ,m-1 . \end{aligned} $$ where $a\in C([0,1], [0,\infty ))$ and $a(t_0)>0$ for some $t_0\in [0,1]$, $f\in C([0,\infty ),[0,\infty ))$ and $f(s)>0$ for $s>0$, and $f_0=\infty $, where $f_0=\lim _{s\rightarrow 0^+}f(s)/s$. We investigate the global structure of positive solutions by using Rabinowitz's global bifurcation theorem.
Rights:
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7. Intertwining of birth-and-death processes

Creator:
Swart, Jan M.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
intertwining of Markov processes, birth and death process, averaged Markov process, first passage time, coupling, and eigenvalues
Language:
English
Description:
It has been known for a long time that for birth-and-death processes started in zero the first passage time of a given level is distributed as a sum of independent exponentially distributed random variables, the parameters of which are the negatives of the eigenvalues of the stopped process. Recently, Diaconis and Miclo have given a probabilistic proof of this fact by constructing a coupling between a general birth-and-death process and a process whose birth rates are the negatives of the eigenvalues, ordered from high to low, and whose death rates are zero, in such a way that the latter process is always ahead of the former, and both arrive at the same time at the given level. In this note, we extend their methods by constructing a third process, whose birth rates are the negatives of the eigenvalues ordered from low to high and whose death rates are zero, which always lags behind the original process and also arrives at the same time.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

8. Nodal solutions for a second-order $m$-point boundary value problem

Creator:
Ma, Ruyun
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
multiplicity results, eigenvalues, bifurcation methods, nodal zeros, and multi-point boundary value problems
Language:
English
Description:
We study the existence of nodal solutions of the $m$-point boundary value problem \[ u^{\prime \prime }+ f(u)=0, \quad 0<t<1, u^{\prime }(0)=0, \quad u(1)=\sum ^{m-2}_{i=1} \alpha _i u(\eta _i) \] where $\eta _i\in \mathbb{Q}$ $(i=1, 2, \cdots , m-2)$ with $0<\eta _1<\eta _2<\cdots <\eta _{m-2}<1$, and $\alpha _i\in \mathbb{R}$ $(i=1, 2, \cdots , m-2)$ with $\alpha _i>0$ and $0<\sum \nolimits ^{m-2}_{i=1} \alpha _i < 1$. We give conditions on the ratio $f(s)/s$ at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.
Rights:
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9. On a bound on algebraic connectivity: the case of equality

Creator:
Kirkland, Stephen J. and Neumann, Michael
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
math, bounds, eigenvalues, and graphs
Language:
English
Description:
In a recent paper the authors proposed a lower bound on $1 - \lambda _i$, where $\lambda _i$, $ \lambda _i \ne 1$, is an eigenvalue of a transition matrix $T$ of an ergodic Markov chain. The bound, which involved the group inverse of $I - T$, was derived from a more general bound, due to Bauer, Deutsch, and Stoer, on the eigenvalues of a stochastic matrix other than its constant row sum. Here we adapt the bound to give a lower bound on the algebraic connectivity of an undirected graph, but principally consider the case of equality in the bound when the graph is a weighted tree. It is shown that the bound is sharp only for certain Type I trees. Our proof involves characterizing the case of equality in an upper estimate for certain inner products due to A. Paz.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public