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Scattering of sound wave by an infinite grating composed of rigid plates
Get PDFA plane sound wave is incident at an angle θ upon an infinite array of rigid plates, equally spaced and lying along the y-axis, where (x, y) are two-dimensional Cartesian coordinates. The boundary value problem is formulated into a matrix Wiener–Hopf equation whose kernel is, when the plates and interstices are of equal length, decomposable into two factors which commute and have algebraic behaviour at infinity. A closed form analytical solution is then obtained following the usual Wiener–Hopf procedure and numerical results are given for various angles of incidence, as well as different spacings
The Generalized Singular Value Decomposition and the Method of Particular Solutions
Get PDFA powerful method for solving planar eigenvalue problems is the
Method of Particular Solutions (MPS), which is also well known under
the name ``point matching method''. The implementation of this
method usually depends on the solution of one of three types of
linear algebra problems: singular value decomposition, generalized
eigenvalue decomposition, or generalized singular value
decomposition. We compare and give geometric interpretations of
these different variants of the MPS. It turns out that the most
stable and accurate of them is based on the
Generalized Singular Value Decomposition. We present results to this
effect and demonstrate the behavior of the generalized singular
value decomposition in the presence of a highly ill-conditioned
basis of particular solutions
Definite Matrix Polynomials and their Linearization by Definite Pencils
Get PDFHyperbolic matrix polynomials
are an important class of Hermitian matrix polynomials
that contain overdamped quadratics as a special case.
They share with definite pencils the spectral property that their eigenvalues
are real and semisimple.
We extend the definition of hyperbolic matrix polynomial
in a way that relaxes the requirement of definiteness of
the leading coefficient matrix,
yielding what we call definite polynomials.
We show that this class of polynomials has an elegant characterization in terms
of definiteness intervals on the extended real line,
and that it includes definite pencils as a special case.
A fundamental question is whether a definite
matrix polynomial can be linearized
in a structure-preserving way.
We show that the answer to this question is affirmative:
is definite if and only if it has a
definite linearization in ,
a certain vector space of Hermitian pencils;
and for definite we give a complete characterization of all the
linearizations in that are definite. For the important
special case of quadratics, we show how a definite quadratic
polynomial can be transformed into a definite linearization with a
positive definite leading coefficient matrix---a form that is
particularly attractive numerically
Something from nothing: bridging the gap between constraint-based and kinetic modelling
Get PDFTwo divergent modelling methodologies have been adopted to increase our understanding of metabolism and its regulation. Constraint-based modelling highlights the optimal path through a stoichiometric network within certain physicochemical constraints. Such an approach requires minimal biological data to make quantitative inferences about network behaviour; however, constraint-based modelling is unable to give an insight into cellular substrate concentrations. In contrast, kinetic modelling aims to characterize fully the mechanics of each enzymatic reaction. This approach suffers because parameterizing mechanistic models is both costly and time-consuming. In this paper, we outline a method for developing a kinetic model for a metabolic network, based solely on the knowledge of reaction stoichiometries. Fluxes through the system, estimated by flux balance analysis, are allowed to vary dynamically according to linlog kinetics. Elasticities are estimated from stoichiometric considerations. When compared to a popular branched model of yeast glycolysis, we observe an excellent agreement between the real and approximate models, despite the absence of (and indeed the requirement for) experimental data for kinetic constants. Moreover, using this particular methodology affords us analytical forms for steady state determination, stability analyses and studies of dynamical behaviour
On the Definition of Two Natural Classes of Scalar Product
Get PDFWe identify two natural classes of scalar product, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations, iterations and mappings.
A variety of different characterizations of these scalar product classes is given.
All the classical examples of scalar products,
each giving rise to important classes of structured matrices, are shown to be both orthosymmetric and unitary
Invariants de classes: examples de non-annulation en dimension supérieure
No full textThe so-called class-invariant homomorphism ψ measures the Galois module structure of torsors—under a finite flat group scheme G—which lie in the image of a coboundary map associated to an isogeny between (Néron models of) abelian varieties with kernel G. When the varieties are elliptic curves with semi-stable reduction and the order of G is coprime to 6, it is known that the homomorphism ψ vanishes on torsion points. In this paper, using Weil restrictions of elliptic curves, we give the construction, for any prime number p > 2, of an abelian variety A of dimension p endowed with an isogeny (with kernel μ p ) whose coboundary map is surjective. In the case when A has rank zero and the p-part of the Picard group of the base is non-trivial, we obtain examples where ψ does not vanish on torsion points
The influence of viscosity on the frozen wave stability: theory and experiment
Get PDFWe present the results of an experimental and linear stability study of the influence of viscosity on the frozen wave (FW) instability, which arises when a vessel containing stably stratified layers of immiscible liquids is oscillated horizontally. Our linear stability model consists of two superposed fluid layers of arbitrary viscosities and infinite lateral extent, subject to horizontal oscillation. The effect of the endwalls of the experimental vessel is simulated by enforcing the conservation of horizontal volume flux, so that the base flow consists of counterflowing layers.
We perform experiments with four pairs of fluids, keeping the viscosity of the lower layer (ν1) constant, and increasing the viscosity of the upper layer (ν2), so that 1.02 × 102 ≤ N1 = ν2/ν1 ≤ 1.21 × 104. We find excellent quantitative agreement between theory and experiment despite the simple model geometry, for both the critical onset parameter and wavenumber of the FW. We show that the model of lyubimov:1987 (Fluid Dyn. vol. 86, 1987, p. 849), which is valid in the limit of inviscid fluids, consistently underestimates the instability threshold for fluids of equal viscosity, but generally overestimates the threshold for fluids of unequal viscosity. We extend the experimental parameter range numerically to viscosity contrasts 1 ≤ N1 ≤ 6 × 104 and identify four regions of N1 where qualitatively different dynamics occur, which are reflected in the non-monotonic dependence of the most unstable wavenumber and the critical amplitude on N1. In particular, we find that increasing the viscosity contrast between the layers leads to destabilization over a wide range of N1, 10 ≤ N1 ≤ 8 × 103. The intricate dependence of the instability on viscosity contrast is due to considerable changes in the time-averaged perturbation vorticity distribution near the interface
Ownership and Control: A Small-World Analysis
Get PDFIn this paper we investigate the ownership and control of British firms using recent techniques from computational graph theory. Specifically, we analyze the `small-world' of ownership and control.
A small-world is a network whose actors are linked by a short chain of acquaintances (short path-lengths), but at the same time have a strongly overlapping circle of friends (high clustering). We simulate a set of corporate worlds using an ensemble of random-graphs introduced by Chung and Lu (2002a,b). We find network structure is more clustered (`clubby') than would be predicted by the random-graph model. Path-lengths, though, are generally not shorter than expected. In addition, we investigate the role of financial institutions: potentially important conduits creating connectivity in corporate networks. We find such institutions give rise to systematically different network topologies
Numerical methods for palindromic eigenvalue problems: Computing the anti-triangular Schur form
Get PDFWe present structure-preserving numerical methods
for the eigenvalue problem of complex palindromic pencils.
Such problems arise in control theory,
as well as from palindromic linearizations
of higher degree palindromic matrix polynomials.
A key ingredient of these methods
is the development of an appropriate condensed form ---
the anti-triangular Schur form.
Ill-conditioned problems with eigenvalues near the unit circle,
in particular near , are discussed.
We show how a combination of unstructured methods
followed by a structured refinement
can be used to solve such problems accurately