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Mysteries around the graph Laplacian eigenvalue 4
Get PDFWe describe our current understanding on the phase transition phenomenon of the graph Laplacian eigenvectors constructed on a certain type of unweighted trees, which we previously observed through our numerical experiments. The eigenvalue distribution for such a tree is a smooth bell-shaped curve starting from the eigenvalue 0 up to 4. Then, at the eigenvalue 4, there is a sudden jump. Interestingly, the eigenvectors corresponding to the eigenvalues below 4 are semi-global oscillations (like Fourier modes) over the entire tree or one of the branches; on the other hand, those corresponding to the eigenvalues
above 4 are much more localized and concentrated (like wavelets) around junctions/branching vertices.
For a special class of trees called starlike trees, we obtain a complete understanding of such phase transi-
tion phenomenon. For a general graph, we prove the number of the eigenvalues larger than 4 is bounded from above by the number of vertices whose degrees is strictly larger than 2. Moreover, we also prove that if a graph contains a branching path, then the magnitudes of the components of any eigenvector corresponding to the eigenvalue greater than 4 decay exponentially from the branching vertex toward the leaf of that branch. We have also identi�¯�¬��ed a unique class of trees that can have an eigenvalue exactly
equal to 4
NLEVP: A Collection of Nonlinear Eigenvalue Problems
Get PDFWe present a collection of 52 nonlinear eigenvalue problems
in the form of a MATLAB toolbox.
The collection contains problems from models of real-life applications
as well as ones constructed specifically to have particular
properties.
A classification is given of polynomial eigenvalue problems
according to their structural properties.
Identifiers based on these and other properties can be used to extract
particular types of problems from the collection.
A brief description of each problem is given.
NLEVP serves both to illustrate the tremendous variety of applications
of nonlinear eigenvalue
problems and to provide representative problems for testing,
tuning, and benchmarking of algorithms and codes
Gerschgorin's theorem for generalized eigenvalue problems in the Euclidean metric
Get PDFWe present Gerschgorin-type eigenvalue inclusion sets applicable to generalized eigenvalue problems.
Our sets are defined by circles in the complex plane in the standard Euclidean metric, and are easier to compute than known similar results. As one application we use our results to provide a forward error analysis for a computed eigenvalue of a diagonalizable pencil
Backward stability of iterations for computing the polar decomposition
Get PDFAmong the many iterations available for computing the polar decomposition
the most practically useful are the scaled Newton iteration and
the recently proposed dynamically weighted Halley iteration.
Effective ways to scale these and other iterations are known,
but their numerical stability is much less well understood.
In this work we show that a general iteration
for computing the unitary polar factor
is backward stable under two conditions.
The first condition requires that the iteration is implemented in a mixed
backward--forward stable manner
and the second requires that the mapping
does not significantly decrease the size of any singular value relative to the
largest singular value.
Using this result we show that the
dynamically weighted Halley iteration
is backward stable when it is
implemented using Householder QR factorization
with column pivoting and either row pivoting or row sorting.
We also prove the backward stability of the scaled Newton iteration under
the assumption that matrix inverses are computed in a mixed
backward-forward stable fashion;
our proof is much shorter than a previous one of
Kielbasinski and Zietak.
We also use our analysis to explain the instability of
the inverse Newton iteration and to show that the Newton-Schulz iteration
is only conditionally stable.
This work shows that by carefully blending perturbation analysis with
rounding error analysis it is possible to produce a general result that can
prove the backward stability or
predict or explain the instability (as the case may be)
of a wide range of practically interesting iterations for the polar decomposition
On th Roots of Stochastic Matrices
Get PDFIn Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic th root of a stochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of matrix th roots, and in particular on the existence of stochastic th roots of stochastic matrices. Our contributions include characterization of when a real matrix has a real th root, a classification of th roots of a possibly singular matrix, a sufficient condition for a th root of a stochastic matrix to have unit row sums, and the identification of two classes of stochastic matrices that have stochastic th roots for all . We also delineate a wide variety of possible configurations as regards existence, nature (primary or nonprimary), and number of stochastic roots, and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix
Commuting Involution Graphs for 3-Dimensional Unitary Groups
Get PDFFor a group G and X a subset of G the commuting graph of G on X, denoted by C(G,X), is the graph whose vertex set is X with x, y in X joined by an edge if x =/= y and x and y commute. If the elements in X are involutions, then C(G,X) is called a commuting involution graph. This paper studies C(G,X) when G is a 3-dimensional projective special unitary group and X a G-conjugacy class of involutions, determining the diameters and structure of the discs of these graphs
Interactions among oscillatory pathways in NF-κB signalling
Get PDFBackground:
Sustained stimulation with
tumour necrosis factor α (TNFα) induces substantial
oscillations—observed at both the single cell and
population levels—in the nuclear factor κB (NF-κB)
system. Although the mechanism has not yet been elucidated fully, a
core system has been identified consisting of a negative feedback
loop involving NF-κB (RelA:p50 hetero-dimer) and its
inhibitor IκBα. Many authors have suggested that
this core oscillator should couple to other oscillatory pathways.
Results:
First we analyse single-cell data from
experiments in which the NF-κB system is forced by short trains
of strong pulses of TNFα. Power spectra of the ratio of
nuclear to cytoplasmic concentration of NF-κB suggest that the
cells' responses are entrained by the pulsing
frequency. Using a recent model of the NF-κB system due to Caroline Horton,
we carried out extensive numerical simulations to analyze the
response frequencies induced by trains of pulses of TNFα stimulation
having a wide range of frequencies and amplitudes.
These studies suggest that for sufficiently
weak stimulation, various nonlinear resonances should be
observable. To explore further the possibility of probing alternative
feedback mechanisms, we also coupled the model to sinusoidal signals
signals with a wide range of strengths and frequencies. Our results
show that, at least in simulation, frequencies
other than those of the forcing and the main NF-κB oscillator
can be excited via sub- and superharmonic resonance, producing
quasiperiodic and even chaotic dynamics.
Conclusions:
Our numerical results suggest that
the entrainment phenomena observed in pulse-stimulated
experiments is a consequence of the high intensity of the
stimulation. Computational studies based on current models suggest
that resonant interactions between periodoc pulsatile forcing
and the system's natural frequencies may become evident for
sufficiently weak stimulation.
Further simulations suggest that the nonlinearities of the
NF-κB feedback oscillator mean that even sinusoidally modulated forcing
can induce a rich variety of nonlinear interactions
Multi-Objective Hybrid Intelligent Optimization of Operational Indices for Industrial Processes and Application
No full textTo pursuit the plant-wide optimization of multiple units industrial process, a hybrid
intelligent optimization approach under dynamic environment is proposed. The objective of
optimization is that the production indices de�ned as the performance related to the �nal
product quality, yield, energy and material consumption fall into their target ranges; whilst the
decision variables are operational indices of each unit, which is related to units� intermediate
product quality, e�ciency and consumption. In this context, the domain knowledge of process
engineers are mimicked and combined with the framework in terms of feedback, prediction and
feed-forward schemes so as to realize the required optimization. The e�ectiveness of the proposed
approach has been demonstrated by the practical application results