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Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems
Get PDFHyperbolic quadratic matrix polynomials are an important class of
Hermitian matrix polynomials
with real eigenvalues, among which the overdamped quadratics are those
with nonpositive eigenvalues.
Neither the definition of overdamped nor any of the standard
characterizations provides an efficient way to test if a given
has this property.
We show that a quadratically convergent matrix iteration based on
cyclic reduction, previously studied by Guo and Lancaster,
provides necessary and sufficient conditions for to be overdamped.
For weakly overdamped the iteration is shown to be generically linearly
convergent with constant at worst 1/2,
which implies that
the convergence of the iteration is reasonably fast in almost all
cases of practical interest.
We show that the matrix iteration can be implemented in such a way
that when overdamping is detected a scalar is
provided that lies in the gap between the largest and
smallest eigenvalues of the
quadratic eigenvalue problem (QEP) .
Once such a is known, the QEP can be solved by linearizing to a
definite pencil that can be reduced using already available
Cholesky factorizations to a standard Hermitian eigenproblem.
By incorporating an initial preprocessing stage that shifts a
hyperbolic so that it is overdamped,
we obtain an efficient algorithm that identifies and solves a hyperbolic or
overdamped QEP maintaining symmetry throughout and guaranteeing real
computed eigenvalues
Unique Mutational Patterns in the Envelope α2 Amphipathic Helix and Acquisition of Length in gp120 Hypervariable Domains Are Associated with Resistance to Autologous Neutralization of Subtype C Human Immunodeficiency Virus Type 1
No full textAutologous neutralizing antibodies (NAb) against human immunodeficiency virus type 1 generate viral escape variants; however, the mechanisms of escape are not clearly defined. In a previous study, we determined the susceptibilities of 48 donor and 25 recipient envelope (Env) glycoproteins from five subtype C heterosexual transmission pairs to NAb in donor plasma by using a virus pseudotyping assay, thereby providing an ideal setting to probe the determinants of susceptibility to neutralization. In the present study, acquisition of length in the Env gp120 hypervariable domains was shown to correlate with resistance to NAb in donor plasma (P = 0.01; Kendall's tau test) but not in heterologous plasma. Sequence divergence in the gp120 V1-to-V4 region also correlated with resistance to donor (P = 0.0002) and heterologous (P = 0.001) NAb. A mutual information analysis suggested possible associations of nine amino acid positions in V1 to V4 with NAb resistance to the donor's antibodies, and five of these were located within an 18-residue amphipathic helix (α2) located on the gp120 outer domain. High nonsynonymous-to-synonymous substitution (dN/dS) ratios, indicative of positive selection, were also found at these five positions in subtype C sequences in the database. Nevertheless, exchange of the entire α2 helix between resistant donor Envs and sensitive recipient Envs did not alter the NAb phenotype. The combined mutual information and dN/dS analyses suggest that unique mutational patterns in α2 and insertions in the V1-to-V4 region are associated with NAb resistance during subtype C infection but that the selected positions within the α2 helix must be linked to still other changes in Env to confer antibody escape. These findings suggest that subtype C viruses utilize mutations in the α2 helix for efficient viral replication and immune avoidance
On the commutative factorization of n x n matrix Weiner-Hopf kernels with distinct eigenvalues
No full textIn this article, we present a method for factorizing n x n matrix Wiener–Hopf kernels where n > 2 and the factors commute. We are motivated by a method posed by Jones (Jones 1984a Proc. R. Soc. A 393, 185–192) to tackle a narrower class of matrix kernels; however, no matrix of Jones’ form has yet been found to arise in physical Wiener–Hopf
models. In contrast, the technique proposed herein should find broad application. To illustrate the approach, we consider a 3 x 3 matrix kernel arising in a problem from
elastostatics. While this kernel is not of Jones’ form, we shall show how it can be factorized commutatively. We discuss the essential difference between our method and
that of Jones and explain why our method is a generalization. The majority of Wiener–Hopf kernels that occur in canonical diffraction problems are, however, strictly non-commutative. For 2x2 matrices, Abrahams has shown that one can overcome this difficulty using Padé approximants to rearrange a non-commutative
kernel into a partial-commutative form; an approximate factorization can then be derived. By considering the dynamic analogue of Antipov’s model, we show for the first
time that Abrahams’ Padé approximant method can also be employed within a 3x3 commutative matrix form
Modelling conditional covariance in the linear mixed model
No full textWe provide a data-driven method for modelling the conditional, within-subject covariance matrix arising in linear mixed models (Laird and Ware, 1982). Given an agreed structure for the between-subject covariance matrix we use a regression equation approach to model the within-subject covariance matrix. Using an EM algorithm we estimate all of the parameters in the model simultaneously and obtain analytical expressions for the standard errors. By re-analyzing Kenward's (1987) cattle data, we compare our new model with classical menu-selection–based modelling techniques, demonstrating its superiority using the Bayesian Information Criterion. We also conduct a simulation study, which confirms our observational findings. The paper extends our previous covariance modeling work (Pan and MacKenzie, 2003, 2006) to the conditional covariance space of the linear mixed model (LMM)
Pairs of compatible associative algebras, classical Yang-Baxter and quiver representations
Get PDFGiven an associative multiplication in matrix algebra compatible with the usual one or, in other words, a linear deformation of the matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such deformations and construct numerous examples. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures. We also describe an important class of M-structures related to the affine Dynkin diagrams of A, D, E-type. These M-structures and their representations are described in terms of quiver representations
Calculating the -Norm of Large Sparse Systems via Chandrasekhar Iterations and Extrapolations
Get PDFWe describe an algorithm for estimating the -norm of a large linear time invariant dynamical system described by a discrete time state-space model. The algorithm uses Chandrasekhar iterations to obtain an estimate of the -norm and then uses extrapolation to improve these estimates
Magnetic fields in barred galaxies V. Modelling NGC 1365
Get PDFAims. We present a model of the global magnetic field in the barred galaxy NGC 1365 based jointly on the large-scale velocity field
of interstellar gas fitted to Hi and CO observations of this galaxy and on mean-field dynamo theory. The aim of the paper is to present
a detailed quantitative comparison of a galactic dynamo model with independent radio observations.
Methods. We consider several gas dynamical models, based on two rotation curves. We test a range of nonlinear dynamo models
that include plausible variations of those parameters that are poorly known from observations. Models for the cosmic ray distribution
in the galaxy are introduced in order to produce synthetic radio polarization maps that allow direct comparison with those observed
at λλ3.5 and 6.2 cm.
Results. We show that the dynamo model is robust in that the most important magnetic features are controlled by the relatively well
established properties of the density distribution and gas velocity field. The optimal agreement between the synthetic polarization
maps and observations is obtained when a uniform cosmic ray distribution is adopted. These maps are sensitive to the number density
of thermal ionized gas because of Faraday depolarization effects. Our results are compatible with the observed polarized radio intensity
and Faraday rotation measure if the degree of ionization is between 0.01 and 0.2 (with respect to the total gas density, rather than to
the diffuse gas alone).We find some indirect evidence for enhanced turbulence in the regions of strong velocity shear (spiral arms and
large-scale shocks in the bar) and within 1–2 kpc of the galactic centre. We confirm that magnetic stresses can drive an inflow of gas
into the inner 1 kpc of the galaxy at a rate of a few M yr−1.
Conclusions. The dynamo models are successful to some extent in modelling the large scale regular magnetic field in this galaxy.
Our results demonstrate that dynamo models and synthetic polarization maps can provide information about both the gas dynamical
models and conditions in the interstellar medium. In particular, it seems that large-scale deviations from energy equipartition (or
pressure balance) between large-scale magnetic fields and cosmic rays are unavoidable. We demonstrate that the dynamical effects of
magnetic fields cannot be everywhere ignored in galaxy modelling
From Runtime Verification to Evolvable Systems
Get PDFWe consider evolvable computational systems built as hierarchies of evolvable components, where an evolvable component is an encapsulation of a supervisory component and its supervisee. Here, we extend our prior work on a revision-based logical modelling framework for such systems to incorporate programs within each component. We describe mechanisms for combining programs, possibly in different languages, from separate components and outline an operational semantics for programmed evolvable systems. We show how supervisory components extend run-time verifiers/monitors with capabilities for diagnosis and change. We illustrate the logical modelling using an example of an automated bank teller machine
The generating hypothesis in the derived category of a ring
Get PDFWe show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds in the derived category of a ring R if and only if R is von Neumann regular. This extends results of the second author (J. Pure Appl. Algebra 208(2), 2007). We also characterize rings for which the original form (the faithful version) of the generating hypothesis holds in the derived category of R. These must be close to von Neumann regular in a precise sense, and, given any of a number of finiteness hypotheses, must be von Neumann regular. However, we construct an example of such a ring that is not von Neumann regular and therefore does not satisfy the strong form of the generating hypothesis