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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
Rings of definable scalars of Verma modules
Get PDFLet M be a Verma module over the Lie algebra, sl2(k), of trace zero 2×2 matrices over the algebraically closed field k. We show that the ring, RM, of definable scalars of M is a von Neumann regular ring and that the canonical map from U(sl2(k)) to RM is an epimorphism of rings. We also describe the Ziegler closure of M. The proofs make use of ideas from the model theory of modules
Bordism classes represented by multiple point manifolds of immersed manifolds
Get PDFWe present a geometrical version of Herbert's Theorem determining the homology classes represented by the multiple point manifolds of a self-tranverse immersion. Herbert's Theorem and generalizations can readily be read off from this result. The simple geometrical proof is based on ideas in Herbert's paper. We also describe the relationship between this theorem and the homotopy theory of Thom spaces
Discontinuous solutions of neutral delay differential equations
No full textIt is well known that the solutions of delay differential and implicit and explicit neutral delay differential equations (NDDEs) may have discontinuous derivatives, but it has not been appreciated (sufficiently) that the solutions of NDDEs—and, therefore, solutions of delay differential algebraic equations—need not be continuous. Numerical codes for solving differential equations, with or without retarded arguments, are generally based on the assumption that a solution is continuous. We illustrate and explain how the discontinuities arise, and present some methods to deal with these problems computationally. The investigation of a simple example is followed by a discussion of more general NDDEs and further mathematical detail
Model-theoretic imaginaries and coherent sheaves
Get PDFCategories of imaginaries (imaginary sorts are the objects and definable functions are the maps) defined using positive existential formulas are shown to be equivalent to categories of finitely presented / coherent functors on the category of models. Localised/relativised versions are also proved. This is linked with interpretation functors between categories of structures.
These results generalise what is already known in the additive case and include an alternative approach to an old result of Makkai and Reyes
Bordism groups of immersions and classes represented by self-intersections
Get PDFA well-known formula of R.J. Herbert's relates the various homology classes represented by the self-intersection immersions of a self-transverse immersion. We prove a geometrical version of Herbert's formula by considering the self-intersection immersions of a self-transverse immersion up to bordism. This clarifies the geometry lying behind Herbert's formula and leads to a homotopy commutative diagram of Thom complexes. It enables us to generalise
the formula to other homology theories. The proof is based on Herbert's but uses the relationship between self-intersections and stable Hopf invariants and the fact that bordism of immersions gives a functor on the category of smooth manifolds and proper immersions
Optimal Scaling for Random walk Metropolis on spherically constrained target densities
Get PDFWe consider the problem of optimal scaling of the proposal variance
for multidimensional Random walk Metropolis (RWM) algorithms. It is
well known, for a wide range of continuous target densities, that
the optimal scaling of the proposal variance leads to an average
acceptance rate of 0.234. Therefore a natural question is, do
similar results for target densities which have discontinuities? In
the current work, we answer in the affirmative for a class of
spherically constrained target densities. Even though the acceptance
probability is more complicated than for continuous target
densities, the optimal scaling of the proposal variance again leads
to an average acceptance rate of 0.234
Scattering of barotropic Rossby waves by the Antarctic Circumpolar Current
No full textThis study examines the interactions between barotropic Rossby waves and a zonal current, with particular reference to the Antarctic Circumpolar Current (ACC). In the high latitude of the Southern Ocean, the effect, which provides the restoring mechanism for Rossby waves, is relatively weak, resulting in barotropic waves that are extremely long, with periods of the order of 1 week or longer. We model the interactions between the current and the waves using the linearized potential vorticity (PV) equation with the inclusion of background PV terms corresponding to a barotropic zonal flow of finite width. An analytical solution is found for the simplest, piecewise-linear flow and numerical solutions obtained for more realistic, smoothly varying flows. The results show that, in general, the long waves are not appreciably modified or reflected by the shear flow, except in the case where the wave is incident at an oblique angle. Wave reflection is also more pronounced for shorter waves, such as Rossby waves having eastward group velocity or the case where the wave frequency is just below the cutoff frequency
Symmetric Linearizations for Matrix Polynomials
Get PDFA standard way of treating the polynomial eigenvalue problem
P(\l)x = 0 is to convert it
into an equivalent matrix pencil---a process known as linearization.
Two vector spaces of pencils
\Ell_1(P) and \Ell_2(P), and their intersection \DL(P),
have recently been defined and studied by
Mackey, Mackey, Mehl, and Mehrmann.
The aim of our work is to gain new insight into these spaces
and the extent to which their constituent pencils inherit
structure from \@.
For arbitrary polynomials we show that every pencil in \DL(P)
is block symmetric
and we obtain a convenient basis for \DL(P) built from block Hankel matrices.
This basis is then exploited to
prove that
the first pencils in a sequence constructed by
Lancaster in the 1960s generate \DL(P).
When is symmetric, we show that
the symmetric pencils in \Ell_1(P) comprise
\DL(P),
while for Hermitian the Hermitian pencils in \Ell_1(P)
form a proper subset of \DL(P) that we explicitly characterize.
Almost all pencils in
each of these subsets are shown to be linearizations.
In addition to obtaining new results, this work provides a self-contained
treatment of some of the key properties of \DL(P)
together with some new, more concise proofs