332 research outputs found
Model Reduction of Non-densely Defined Piecewise-Smooth Systems in Banach Spaces
In this paper a model reduction technique is introduced for piecewise-smooth
(PWS) vector fields, whose trajectories fall into a Banach space, but the
domain of definition of the vector fields is a non-dense subset of the Banach
space. The vector fields depend on a parameter that can assume different
discrete values in two parts of the phase space and a continuous family of
values on the boundary that separates the two parts of the phase space. In
essence the parameter parametrizes the possible vector fields on the boundary.
The problem is to find one or more values of the parameter so that the solution
of the PWS system on the boundary satisfies certain requirements. In this paper
we require continuous solutions. Motivated by the properties of applications,
we assume that when the parameter is forced to switch between the two discrete
values, trajectories become discontinuous. Discontinuous trajectories exist in
systems whose domain of definition is non-dense. It is shown that under our
assumptions the trajectories of such PWS systems have unique forward-time
continuation when the parameter of the system switches. A finite-dimensional
reduced order model is constructed, which accounts for the discontinuous
trajectories. It is shown that this model retains uniqueness of solutions and
other properties of the original PWS system. The model reduction technique is
illustrated on a nonlinear bowed string model.Comment: 11 figures, 55 pages. Accepted for publication in Journal of
Nonlinear Scienc
On Uniqueness of the Jump Process in Quantum Measurement Theory
We prove that, contrary to the standard quantum theory of continuous
observation, in the formalism of Event Enhanced Quantum Theory the stochastic
process generating individual sample histories of pairs (observed quantum
system, observing classical apparatus) is unique. This result gives a rigorous
basis to the previous heuristic argument of Blanchard and Jadczyk. Possible
implications of this result are discussed.Comment: 31 pages, LaTeX, article; e-mail contact [email protected]
Hopf bifurcation from fronts in the Cahn-Hilliard equation
We study Hopf bifurcation from traveling-front solutions in the Cahn-Hilliard
equation. The primary front is induced by a moving source term. Models of this
form have been used to study a variety of physical phenomena, including pattern
formation in chemical deposition and precipitation processes. Technically, we
study bifurcation in the presence of essential spectrum. We contribute a simple
and direct functional analytic method and determine bifurcation coefficients
explicitly. Our approach uses exponential weights to recover Fredholm
properties and spectral flow ideas to compute Fredholm indices. Simple mass
conservation helps compensate for negative indices. We also construct an
explicit, prototypical example, prove the existence of a bifurcating front, and
determine the direction of bifurcation
Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework
We analyze temporal approximation schemes based on overlapping domain
decompositions. As such schemes enable computations on parallel and distributed
hardware, they are commonly used when integrating large-scale parabolic
systems. Our analysis is conducted by first casting the domain decomposition
procedure into a variational framework based on weighted Sobolev spaces. The
time integration of a parabolic system can then be interpreted as an operator
splitting scheme applied to an abstract evolution equation governed by a
maximal dissipative vector field. By utilizing this abstract setting, we derive
an optimal temporal error analysis for the two most common choices of domain
decomposition based integrators. Namely, alternating direction implicit schemes
and additive splitting schemes of first and second order. For the standard
first-order additive splitting scheme we also extend the error analysis to
semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which
also contains numerical experiments. Version 3 and 4: Only comments added.
Version 2, page 2: Clarified statement on stability issues for ADI schemes
with more than two operator
Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems
We show global well-posedness and exponential stability of equilibria for a
general class of nonlinear dissipative bulk-interface systems. They correspond
to thermodynamically consistent gradient structure models of bulk-interface
interaction. The setting includes nonlinear slow and fast diffusion in the bulk
and nonlinear coupled diffusion on the interface. Additional driving mechanisms
can be included and non-smooth geometries and coefficients are admissible, to
some extent. An important application are volume-surface reaction-diffusion
systems with nonlinear coupled diffusion.Comment: 21 page
Exponential dichotomy and smooth invariant center manifolds for semilinear hyperbolic systems
Es wird gezeigt, dass ein Satz über die Abbildung spektraler Lücken, welcher exponentielle Dichotomie charakterisiert, für eine allgemeine Klasse (SH) von semilinearen hyperbolischen Systemen von partiellen Differentialgleichungen in einem Banach-Raum X von stetigen Funktionen gilt. Dies beantwortet ein Schlüsselproblem für die Existenz und Glattheit invarianter Mannigfaltigkeiten semilinearer hyperbolischer Systeme. Unter natürlichen Annahmen an die Nichtlinearitäten wird gezeigt, dass schwache Lösungen von (SH) einen glatten Halbfluß im Raum X bilden. Für Linearisierungen werden hochfrequente Abschätzungen für Spektren sowie Resolventen unter Verwendung von reduzierten (block)diagonal Systemen hergestellt. Darauf aufbauend wird der Abbildungssatz für spektrale Lücken im kleinen Raum X bewiesen: Eine offene spektrale Lücke des Generators wird exponentiell auf eine offene spektrale Lücke der Halbruppe abgebildet und umgekehrt. Es folgt, dass ein Phänomen wie im Gegenbeispiel von Renardy nicht auftreten kann. Unter Verwendung der allgemeinen Theorie implizieren die Ergebnisse die Existenz von glatten Zentrumsmannigfaltigkeiten für (SH). Die Ergebnisse werden auf traveling wave Modelle für die Dynamik von Halbleiter Lasern angewandt. Für diese werden Moden Approximationen (Systeme von gewöhnlichen Differentialgleichungen, welche die Dynamik auf gewissen Zentrumsmannigfaltigkeiten approximativ beschreiben) hergeleitet und gerechtfertigt, die generische Bifurkation von modulierten Wellen aus rotierenden Wellen wird gezeigt. Globale Existenz und glatte Abhängigkeit von nichtautonomen traveling wave Modellen werden betrachtet, außerdem werden Moden Approximationen für solche nichtautonomen Modelle rigoros hergeleitet. Insbesondere arbeitet die Theorie für die Stabilitäts- und Bifurkationsanalyse von Turing Modellen mit korellierter Zufallsbewegung. Ferner beinhaltet die Klasse (SH) neutrale und retardierte funktionale Differentialgleichungen.A spectral gap mapping theorem, which characterizes exponential dichotomy, is proven for a general class of semilinear hyperbolic systems of PDEs in a Banach space X of continuous functions. This resolves a key problem on existence and smoothness of invariant manifolds for semilinear hyperbolic systems. It is shown that weak solutions to (SH) form a smooth semiflow in X under natural conditions on the nonlinearities. For linearizations high frequency estimates of spectra and resolvents in terms of reduced diagonal and blockdiagonal systems are given. Using these estimates a spectral gap mapping theorem in the small Banach space X is proven: An open spectral gap of the generator is mapped exponentially to an open spectral gap of the semigroup and vice versa. Hence, a phenomenon like in Renardy''s counterexample cannot appear for linearizations of (SH). By the general theory the results imply existence of smooth center manifolds for (SH). Moreoever, the results are applied to traveling wave models of semiconductor laser dynamics. For such models mode approximations (ODE systems which approximately describe the dynamics on center manifolds) are derived and justified, and generic bifurcations of modulated waves from rotating waves are shown. Global existence and smooth dependence of nonautonomous traveling wave models with more general solutions, which possess jumps, are considered, and mode approximations are derived for such nonautonomous models. In particular the theory applies to stability and bifurcation analysis for Turing models with correlated random walk. Moreover, the class (SH) includes neutral and retarded functional differential equations
Introduction to Modern Canonical Quantum General Relativity
This is an introduction to the by now fifteen years old research field of
canonical quantum general relativity, sometimes called "loop quantum gravity".
The term "modern" in the title refers to the fact that the quantum theory is
based on formulating classical general relativity as a theory of connections
rather than metrics as compared to in original version due to Arnowitt, Deser
and Misner. Canonical quantum general relativity is an attempt to define a
mathematically rigorous, non-perturbative, background independent theory of
Lorentzian quantum gravity in four spacetime dimensions in the continuum. The
approach is minimal in that one simply analyzes the logical consequences of
combining the principles of general relativity with the principles of quantum
mechanics. The requirement to preserve background independence has lead to new,
fascinating mathematical structures which one does not see in perturbative
approaches, e.g. a fundamental discreteness of spacetime seems to be a
prediction of the theory providing a first substantial evidence for a theory in
which the gravitational field acts as a natural UV cut-off. An effort has been
made to provide a self-contained exposition of a restricted amount of material
at the appropriate level of rigour which at the same time is accessible to
graduate students with only basic knowledge of general relativity and quantum
field theory on Minkowski space.Comment: 301 pages, Latex; based in part on the author's Habilitation Thesis
"Mathematische Formulierung der Quanten-Einstein-Gleichungen", University of
Potsdam, Potsdam, Germany, January 2000; submitted to the on-line journal
Living Reviews; subject to being updated on at least a bi-annual basi
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