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Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology
Get PDFIn a recent paper, Belishev and Sharafutdinov consider a compact Riemannian manifold with boundary . They define a generalized Dirichlet to Neumann (DN) operator on all forms on the boundary and they prove that the real additive de Rham cohomology structure of the manifold in question is completely determined by . This shows that the DN map inscribes into the list of objects of algebraic topology. In this
paper, we suppose is a torus acting by isometries on . Given in the Lie algebra of and the corresponding vector field on , one defines Witten's inhomogeneous coboundary operator on invariant forms on . The main
purpose is to adapt Belishev and Sharafutdinov's boundary data to invariant forms in terms of the operator and its adjoint . In other words, we define an operator on invariant forms on the boundary which we call the -DN map and using this we recover the long exact -cohomology sequence of the topological pair from an isomorphism with the long exact sequence formed from our boundary data. We then show that completely determines the free part of the relative and absolute equivariant cohomology groups of when the set of zeros of the corresponding vector field is equal to the fixed point set for the -action. In addition, we partially determine the mixed cup product (the ring structure) of -cohomology groups from . These results explain to what extent the equivariant topology of the manifold in question is determined by the -DN map . Finally, we illustrate the connection between Belishev and Sharafutdinov's boundary data on and ours on
A new bound for the smallest x with \pi(x) > li(x)
Get PDFThe difference \pi(x) - li(x) has been the subject of lively interest since Littlewood's theorem (1914) that \pi(x) - li(x) changes sign infinitely many times. The issue is to find an upper bound for the first crossover.
Two papers on this issue were published in July 2010: Chao-Plymen, Int. J. Number Theory 6 (2010) 681-690, and Saouter-Demichel, Math. Comp. 79 (2010) 2395 - 2405. This dissertation is a very detailed analysis of these two articles. A new theorem (assuming the Riemann Hypothesis) is discovered: see Chapter 7. This theorem (assuming the Riemann Hypothesis) is the best known result
Geometric structure in the tempered dual of the p-adic group SL(4)
Get PDFWe confirm the Aubert-Baum-Plymen conjecture for part of the tempered dual of the p-adic group SL(4). This requires some very detailed representation theory. Of special interest is the case of SL(4,Q_2). Here, there is a tetrahedron of reducibility, and the extended quotient performs a deconstruction: it creates the ordinary quotient and six unit intervals. The six intervals are then assembled into the six edges of a tetrahedron, and create a perfect model of reducibility. The L-packets in this article all conform to the L-packet conjecture in http://eprints.ma.man.ac.uk/1504
Rational Krylov Methods for Operator Functions
Get PDFWe present a unified and self-contained treatment of rational Krylov methods for approximating the product of a function of a linear operator with a vector. With the help of general rational Krylov decompositions we reveal the connections between seemingly different approximation methods, such as the Rayleigh�Ritz or shift-and-invert method, and derive new methods, for example a restarted rational Krylov method and a related method based on rational interpolation in prescribed nodes. Various theorems known for polynomial Krylov spaces are generalized to the rational Krylov case. Computational issues, such as the computation of so-called matrix Rayleigh quotients or parallel variants of rational Arnoldi algorithms, are discussed. We also present novel estimates for the error arising from inexact linear system solves and the approximation error of the Rayleigh�Ritz method. Rational Krylov methods involve several parameters and we discuss their optimal choice by considering the underlying rational approximation problems. In particular, we present different classes of optimal parameters and collect formulas for the associated convergence rates. Often the parameters leading to best convergence rates are not optimal in terms of computation time required by the resulting rational Krylov method. We explain this observation and present new approaches for computing parameters that are preferable for computations. We give a heuristic explanation of superlinear convergence effects observed with the Rayleigh�Ritz method, utilizing a new theory of the convergence of rational Ritz values. All theoretical results are tested and illustrated by numerical examples. Numerous links to the historical and recent literature are included
Micro-chaotic dynamics due to digital sampling in hybrid systems of Filippov type
Get PDFThis paper describes a novel mechanism leading to the sudden onset of chaos that may occur in Filippov type systems when the switching decision is made at discrete times. Such systems can be thought of as models of switched digital control systems. Our findings are explained in detail using a representative planar example. Results are then extended to more general planar switched systems with digital sampling. It is conjectured that this mechanism is also typical of n-dimensional switched systems with digital sampling, and numerical investigations of a third order relay feedback system are presented to support this; the onset of chaos is triggered in the way which is equivalent to that revealed for the planar case
Momentum maps and ...
Get PDFThis Winter School talk aims to give a brief overview of what is a momentum map
Sloshing and slamming oscillations in collapsible channel flow
Get PDFWe consider laminar high-Reynolds-number flow through a finite-length planar channel,
where a portion of one wall is replaced by a thin massless elastic membrane that is held
under longitudinal tension T and subject to an external pressure distribution. The flow
is driven by a fixed pressure drop along the full length of the channel. We investigate
the global stability of two-dimensional Poiseuille flow using a method of matched local
eigenfunction expansions, which is compared to direct numerical simulations. We trace
the neutral stability curve of the primary oscillatory instability of the system, illustrating
a transition from high-frequency ‘sloshing’ oscillations at high T to vigorous ‘slamming’
motion at low T . Small-amplitude sloshing at high T can be captured using a low-order
eigenmode truncation involving four surface-based modes in the compliant segment of the
channel coupled to Womersley flow in the rigid segments. At lower tensions, we show that
hydrodynamic modes contribute increasingly to the global instability and we demonstrate
a change in the mechanism of energy transfer from the mean flow, with viscous effects
being destabilising. Simulations of finite-amplitude oscillations at low T reveal a generic
slamming motion, in which the the flexible membrane is drawn close to the opposite rigid
wall before rapidly recovering. A simple model is used to demonstrate how fluid inertia
in the downstream rigid channel segment, coupled to membrane curvature downstream
of the moving constriction, together control slamming dynamics
Smith Forms of Palindromic Matrix Polynomials
Get PDFMany applications give rise to matrix polynomials whose coefficients have a kind of reversal symmetry,
a structure we call palindromic.
Several properties of scalar palindromic polynomials
are derived,
and together with properties of compound matrices,
used to establish the Smith form of regular and singular -palindromic matrix polynomials
over arbitrary fields.
The invariant polynomials are shown to
inherit palindromicity,
and their structure is described in detail.
Jordan structures of palindromic matrix polynomials
are characterized,
and necessary conditions for the existence
of structured linearizations established.
In the odd degree case, a constructive procedure
for building palindromic linearizations
shows that the necessary conditions
are sufficient as well.
The Smith form for -palindromic polynomials
is also analyzed.
Finally, results for palindromic matrix polynomials
over fields of characteristic two are presented
The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds
Get PDFWe present an error analysis for the pathwise approximation of a general semilinear
stochastic evolution equation in d dimensions. We discretise in space by a
Galerkin method and in time by a stochastic exponential integrator. We show that
for spatially regular (smooth) noise the number of nodes needed for the noise can
be reduced and that the rate of convergence degrades as the regularity of the noise
reduces (and the noise is rougher