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Iterative Solution of a Nonsymmetric Algebraic Riccati Equation
Get PDFWe study the nonsymmetric algebraic Riccati equation whose four
coefficient matrices are the blocks of
a nonsingular -matrix or an irreducible singular
-matrix . The solution of practical interest is the minimal nonnegative
solution. We show that Newton's method with zero initial guess can be used to
find this solution without any further assumptions. We also present a
qualitative perturbation analysis for the minimal solution, which is
instructive in designing algorithms for finding more accurate approximations.
For the most practically important case, in
which is an irreducible singular -matrix with zero row sums,
the minimal solution is either stochastic or substochastic and
the Riccati equation can be transformed into a unilateral matrix equation by
a procedure of Ramaswami. The minimal solution of the Riccati equation can
then be found by computing the minimal nonnegative solution of
the unilateral equation using
the Latouche--Ramaswami algorithm.
When the minimal solution of the Riccati equation is stochastic,
we show that the Latouche--Ramaswami algorithm, combined with a shift
technique suggested by He, Mini, and Rhee,
is breakdown-free and is able to find the minimal solution
more efficiently and more accurately than the algorithm without a shift.
When the minimal solution of the Riccati equation is substochastic,
we show how the substochastic minimal solution can be found by computing
the stochastic minimal solution of a related Riccati equation of the same
type
Correlations for pairs of closed geodesics
No full textIn this article we consider natural counting problems for closed geodesics on negatively curved surfaces. We present asymptotic estimates for pairs of closed geodesics, the differences of whose lengths lie in a prescribed family of shrinking intervals. Related pair correlation problems have been studied in both Quantum Chaos and number theory
Classifying projective modules over some semilocal rings
Get PDFWe classify all (finitely generated or not) projective modules
over a class of semilocal ring constructed using nealy simple
uniserial domains. They in turn are connected with noncommutative valuations constructed using embeddings of
right ordered groups into skew fields
Microarray Data Analysis Using Probabilistic Methods
Get PDFAffymetrix microarrays are currently the most widely used microarray technology. Due to the complexity of microarray experiments, the experimental data is very noisy. Many summarization methods have been developed to provide gene expression levels from Affymetrix probe-level data. Most of the currently popular methods do not provide a measure of uncertainty for the estimated expression level of each gene. The use of probabilistic models can overcome this limitation. This thesis extends a previously developed probabilistic model, mgMOS, to obtain an improved model, multi-mgMOS. This new model provides improved accuracy and is more computationally efficient than other alternatives. It also provides a level of uncertainty associated with the measured gene expression level. This probe-level measurement error provides useful information to help in the downstream analysis of gene expression data.
In order to show the advantage of the probe-level probabilistic model, the obtained uncertainty is propagated in two downstream analyses of gene expression data. One is detecting differential gene expression, another is clustering. A Bayesian hierarchical model is proposed to include probe-level measurement error into the detection of differential gene expression from replicated experiments and a standard model-based clustering method is augmented to incorporate probe-level measurement error. Due to the inclusion of the probe-level measurement error, the downstream probabilistic models become more complicated or intractable. In order to perform inference with these augmented models efficiently, various inference approximation approaches are compared in this thesis, including Maximum a Posteriori, Laplace approximation, a variational method and Markov chain Monte Carlo. Results from both benchmark data sets and a real-world data set demonstrate that the incorporation of the probe-level measurement error improves the performance of the downstream probabilistic analysis
The Conditioning of Linearizations of Matrix Polynomials
Get PDFThe standard way of solving the polynomial eigenvalue problem of degree
in matrices
is to ``linearize'' to a pencil in matrices
and solve the generalized eigenvalue problem.
For a given polynomial, , infinitely many linearizations exist
and they can have widely varying eigenvalue condition numbers.
We investigate the conditioning of
linearizations from a vector space of pencils
recently identified and studied by
Mackey, Mackey, Mehl, and Mehrmann.
We look for the best conditioned linearization and
compare the conditioning with that of the original polynomial.
Two particular pencils are shown always to be
almost optimal over linearizations in for eigenvalues of
modulus greater than or less
than 1, respectively,
provided that the problem is not too badly scaled
and that the pencils are linearizations.
Moreover, under this scaling assumption,
these pencils are shown to be
about as well conditioned as the original polynomial.
For quadratic eigenvalue problems that are not too heavily damped,
a simple scaling is shown to convert the problem to one that is well scaled.
We also analyze the eigenvalue conditioning
of the widely used first and second companion linearizations.
The conditioning of the first companion linearization relative to that of
is shown to depend on the coefficient matrix norms,
the eigenvalue, and the left \ev s of the linearization and of .
The companion form is found to be potentially much more ill conditioned than
,
but if the 2-norms of the coefficient matrices are all approximately 1
then the companion form and are guaranteed to have similar
condition numbers.
Analogous results hold for the second companion form.
Our results are phrased in terms of both the standard relative condition number
and the condition number of Dedieu and Tisseur for the problem in
homogeneous form,
this latter condition number having the advantage of applying to zero and
infinite eigenvalues
Structured Eigenvalue Condition Numbers
Get PDFThis paper investigates the effect of structure-preserving perturbations
on the eigenvalues of linearly and nonlinearly structured eigenvalue
problems. Particular attention is paid to structures that form Jordan
algebras, Lie algebras, and automorphism groups of a scalar product.
Bounds and computable
expressions for structured eigenvalue condition numbers are derived for
these classes of matrices, which include
complex symmetric, pseudo symmetric,
persymmetric, skew-symmetric, Hamiltonian, symplectic, and orthogonal
matrices.
In particular we show that under reasonable assumptions on the scalar
product, the structured and unstructured eigenvalue condition numbers
are equal for structures in Jordan algebras.
For Lie algebras, the effect on the condition number of incorporating
structure varies greatly with the structure. We identify Lie algebras
for which structure does not affect the eigenvalue condition number
Structured Eigenvalue Condition Numbers
Get PDFThis paper investigates the effect of structure-preserving perturbations
on the eigenvalues of linearly and nonlinearly structured eigenvalue
problems. Particular attention is paid to structures that form Jordan
algebras, Lie algebras, and automorphism groups of a scalar product.
Bounds and computable
expressions for structured eigenvalue condition numbers are derived for
these classes of matrices, which include
complex symmetric, pseudo symmetric,
persymmetric, skew-symmetric, Hamiltonian, symplectic, and orthogonal
matrices.
In particular we show that under reasonable assumptions on the scalar
product, the structured and unstructured eigenvalue condition numbers
are equal for structures in Jordan algebras.
For Lie algebras, the effect on the condition number of incorporating
structure varies greatly with the structure. We identify Lie algebras
for which structure does not affect the eigenvalue condition number
Transverse flows in rapidly oscillating elastic cylindrical shells
No full textWe analyse the flows in fluid-conveying tubes whose elastic walls perform small-amplitude high-frequency oscillations. We show that the velocity perturbations induced by the wall motion are dominated by their transverse components and use numerical simulations to analyse the two-dimensional flows that develop in the tube's cross-sections. Asymptotic methods are then employed to derive explicit predictions for the flow fields and for the total viscous dissipation, whose magnitude plays an important role in the development of self-excited oscillations.
We show that in cases with fluid–structure interaction, the coupled oscillations are controlled by the ratio of the fluid and wall densities, and by a material parameter that is equivalent to the Womersley number, and indicates the importance of fluid inertia and wall elasticity relative to the fluid's viscosity. We present numerical simulations of the coupled oscillations and use asymptotic techniques to derive explicit predictions for their period and decay rate. Finally, we discuss the implications of our results for the development of self-excited oscillations in three-dimensional collapsible tubes
Stochastic inequality probabilities for adaptively randomized clinical trials
No full textWe examine stochastic inequality probabilities of the form P (X > Y ) and P (X > max (Y, Z )) where X, Y, and Z are random variables with beta, gamma, or inverse gamma distributions. We discuss the applications of such inequality probabilities to adaptively randomized clinical trials as well as methods for calculating their values
Universal homotopy associative, homotopy commutative H-spaces and the EHP spectral sequences
No full textAssume that all spaces and maps are localised at a fixed prime . We study the possibility of generating a universal space from a space which is universal in the category of homotopy associative, homotopy commutative -spaces in the sense that any map to a homotopy associative, homotopy commutative -space extends to a uniquely determined -map . Developing a method for recognising certain universal spaces, we show the existence of the universal space of a certain three-cell complex . Using this specific example, we derive some consequences for the calculation of the unstable homotopy groups of spheres, namely, we obtain a formula for the -differential of the EHP-spectral sequence valid in a certain range