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Convective and absolute instability in the incompressible boundary layer on a rotating disk in the prescence of a uniform magnetic field
No full textThe stability of a conducting fluid flow over a rotating disk with a uniform magnetic field applied normal to the disk, is investigated. It is assumed that the magnetic field is unaffected by the motion of the fluid. The mean flow and linear stability equations are solved for a range of magnetic field-strength parameters and the absolute/convective nature of the stability is investigated. It is found that increasing the magnetic field parameter is in general stabilizing
Presentations for semigroups and semigroupoids
No full textWe consider the relationship between the combinatorial properties of semigroupoids in general and semigroups in particular. We show that a semigroupoid is finitely generated [finitely presentable] exactly if the corresponding categorical-at-zero semigroup is finitely generated [respectively, finitely presentable]. This allows us to extend some of the main results of [17], to show that finite generation and presentability are preserved under finite extension of semigroupoids and the taking of cofinite subsemigroupoids. We apply this result to extend the results of [6], giving characterizations of finite generation and finite presentability in Rees matrix semigroups over semigroupoids
Comparison of 3D Image Reconstruction Techniques using Real Electrical Impedance Measurement Data
Get PDFThe present status of electrical impedance tomography (EIT) is such that many successful laboratory
based studies and pilot plant demonstrators are generating interest in developing the technology.
However, while the EIT problem is inherently 3D, many studies make 2D assumptions in order to
simplify the problem. Proportionately less work has been carried out on the true 3D EIT problem,
outside of the mathematical development of the problem. This paper aims to provide a comparison of
the performance of 3D EIT image reconstruction algorithms applied to real data, and assess whether
the extra effort and complexity involved in 3D EIT is worthwhile. Measurement data are taken from a
pilot-scale mixing vessel. The performance of five common EIT reconstruction algorithms is compared
and the best performing algorithm for this arrangement is identified. This work is part of the ongoing
efforts at Manchester to pursue the true 3D EIT problem under the EIDORS-3D framework
Functions Preserving Matrix Groups and Iterations for the Matrix Square Root
No full textFor any matrix automorphism group \G associated with a bilinear
or sesquilinear form, Mackey, Mackey, and Tisseur have recently
shown that the matrix sign decomposition factors of A\in\G also
lie in \G; moreover, the polar factors of lie in \G if the
matrix of the underlying form is unitary. Groups satisfying the
latter condition include the complex orthogonal, real and complex
symplectic, and pseudo-orthogonal groups. This work is concerned
with exploiting the structure of \G when computing the polar and
matrix sign decompositions of matrices in \G. We give sufficient
conditions for a matrix iteration to preserve the group structure
and show that a family of globally convergent rational
Pad\'e-based iterations of Kenney and Laub satisfy these
conditions. The well-known scaled Newton iteration for computing
the unitary polar factor does not preserve group structure, but we
show that the approach of the iterates to the group is precisely
tethered to the approach to unitarity, and that this forces a
different and exploitable structure in the iterates. A similar
relation holds for the Newton iteration for the matrix sign
function. We also prove that the number of iterations needed for
convergence of the structure-preserving methods can be precisely
predicted by running an associated scalar iteration. Numerical
experiments are given to compare the cubically and quintically
converging iterations with Newton's method and to test stopping
criteria. The overall conclusion is that the structure-preserving
iterations and the scaled Newton iteration are all of practical
interest, and which iteration is to be preferred is
problem-dependent
Vector Spaces of Linearizations for Matrix Polynomials
No full textThe classical approach to investigating polynomial eigenvalue problems is linearization,
where the polynomial is converted into a larger matrix pencil with the same eigenvalues. For
any polynomial there are innitely many linearizations with widely varying properties, but in practice
the companion forms are typically used. However, these companion forms are not always entirely
satisfactory, and linearizations with special properties may sometimes be required.
Given a matrix polynomial P, we develop a systematic approach to generating large classes of
linearizations for P. We show how to simply construct two vector spaces of pencils that generalize the
companion forms of P, and prove that almost all of these pencils are linearizations for P. Eigenvectors
of these pencils are shown to be closely related to those of P. A distinguished subspace is then
isolated, and the special properties of these pencils are investigated. These spaces of pencils provide
a convenient arena in which to look for structured linearizations of structured polynomials, as well
as to try to optimize the conditioning of linearizations [7], [8], [12]
A theory for particle size segregation in shallow granular free-surface flows
Get PDFGranular materials composed of a mixture of grain sizes are notoriously prone to segregation during shaking or transport. In this paper, a binary mixture theory is used to formulate a model for kinetic sieving of large and small particles in thin, rapidly flowing avalanches, which occur in many industrial and geophysical free-surface flows. The model is based on a simple percolation idea, in which the small particles preferentially fall into underlying void space and lever large particles upwards. Exact steady-state solutions have been constructed for general steady uniform velocity fields, as well as time-dependent solutions for plug-flow, that exploit the decoupling of material columns in the avalanche. All the solutions indicate the development of concentration shocks, which are frequently observed in experiments. A shock-capturing numerical algorithm is formulated to solve general problems and is used to investigate segregation in flows with weak shear
Standard tableaux and Klyachko's Theorem on Lie representations
Get PDFWe show that for all but two partitions of there exists a standard tableau of shape with major index coprime to . In conjunction with a deep result of Kra{\'s}kiewicz and Weyman this provides a new purely combinatorial proof of Klyachko's famous theorem on Lie representations of the general linear group
Higher derived brackets and homotopy algebras
No full textWe give a construction of homotopy algebras based on “higher derived brackets”. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element Δ. Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of Δ2. This allows to control higher Jacobi identities in terms of the “order” of Δ2. Examples include Stasheff's strongly homotopy Lie algebras and variants of homotopy Batalin–Vilkovisky algebras. There is a generalization with Δ replaced by an arbitrary odd derivation. We discuss applications and links with other constructions
An inverse boundary value problem for harmonic differential forms
Get PDFWe show that the full symbol of the Dirichlet to Neumann map of the k-form Laplace's equation on a Riemannian manifold (of dimension greater than 2) with boundary determines the full Taylor series, at the boundary, of the metric. This extends the result of Lee and Uhlmann for the case k = 0. The proof avoids the computation of the full symbol by using the calculus of pseudo-differential operators parametrized by a boundary normal coordinate and recursively calculating the principal symbol of the difference of boundary operators
Vertices for irreducible characters of a class of blocks
No full textWe observe that Navarro's definition of a vertex for an
irreducible character of a -solvable group may be extended to
irreducible characters in -blocks with defect groups contained
in a normal -solvable subgroup, and show that this definition
is independent of the choice of . We show that the fundamental
properties of Navarro's vertices generalize, and as a corollary
show that the vertices of the irreducible Brauer characters in
blocks of the above form are radical and are intersections of
pairs of Sylow -subgroups