Final Lift Actions Associated with Topological Functors

For a topological functor U:EB, the fiber U ^–1(b), bB, is a cocomplete poset and the left action, induced by final lift, of the endomorphism monoid B(b,b) on U ^–1(b) is cocontinuous. It is shown that every cocontinuous left action of B(b,b) on any cocomplete poset can be realized as the final lift action associated to a canonically defined topological functor over B. If B is a Grothendieck topos and b=, the subobject classifier, then B(,) inherits both a monoidal and a cocomplete poset structure. In the case B= Sets, all cocontinuous left actions of B(,) on itself are explicitly described and each is shown to arise as the final lift action associated to a specific subcategory of a certain fixed category, referred to as the category of LR-spaces. Relationships between these LR-spaces and several other well known topological categories are also considered.     

Dimerization of the keto tautomer of acetohydroxamic acid-infrared matrix isolation and theoretical study

Dimerization of the keto tautomer of acetohydroxamic acid has been studied using FTIR matrix isolation spectroscopy and DFT(B3LYP)/6-31+G(d,p) calculations. Analysis of CH3CONHOH/Ar matrix spectra indicates formation of two dimers in which two intramolecular CO...HON bonds within two interacting acetohydroxamic acid molecules are retained. A chain dimer I is stabilized by the intermolecular CO...HN hydrogen bond, whereas the cyclic dimer II is stabilized by two intermolecular NH...O(H)N bonds. Twelve vibrations were identified for dimer I and six vibrations for dimer II; the observed frequency shifts show a good agreement with the calculated ones for the structures I and II. Both dimers have comparable binding energies (DeltaE(ZPE)(CP)I, II=-7.02, -6.34 kcal mol-1) being less stable than calculated structures III and IV (DeltaE(ZPE)(CP)III, IV=-9.50, -8.87 kcal mol-1) in which one or two intramolecular hydrogen bonds are disrupted. In the most stable 10-membered cyclic dimer III, two intermolecular CO...HON hydrogen bonds are formed at expense of intramolecular hydrogen bonds of the same type. The formation of the less stable (AHA)2 dimers in the studied matrixes indicates that the formation of (AHA)2 is kinetically and not thermodynamically controlled.     

Chiral supergravity and anomalies

Similarly as in the Ashtekar approach, the translational Chern-Simons term is, as a generating function, instrumental for a chiral reformulation of simple supergravity. After applying the algebraic Cartan relation between spin and torsion, the resulting canonical transformation induces not only a decomposition of the gravitational fields into selfdual and anti-selfdual modes, but also a splitting of the Rarita-Schwinger fields into their chiral parts in a natural way. In some detail, we also analyze the consequences for axial and chiral anomalies.     

Comparison of Inertial Manifolds and Application to Modulated Systems

We consider two dissipative systems having inertial manifolds and give estimates which allow us to compare the flows on the two inertial manifolds. As an example of a modulated system we treat the Swift–Hohenberg equation , ∈ ℝ, with periodic boundary conditions on the interval . Recent results in the theory of modulation equation show that the solutions of this equation can be described over long time scales by those of the associated Ginzburg–Landau equation ∈ ℂ, with suitably generalized periodic boundary conditions on . We prove that both systems have an inertial manifold of the same dimension and that the flows on these finite dimensional manifolds converge against each other for .     

Reduction of PDEs on domains with several unbounded directions: A first step towards modulation equations

Bifurcation problems in unbounded domains often lead to spontaneous pattern formation. A basic periodic pattern, derivable from the linearized system, is modulated by nonlinear effects on a slow time and space scale. The modulation of the amplitude is usually described by equations of Ginzburg-Landau type.To study such problems we develop a generalized Lyapunov-Schmidt reduction procedure which allows to treat the case of continuous spectra. Thus, we are able to reduce parabolic systems in plate-like domains to a lower dimensional problem, which is, to first order, a partial differential equation in the unbounded variables only, but contains also non-local terms in the nonlinearity. Using a modulation ansatz with slow time and space variables, the expansion in terms of the bifurcation parameter transfers the non-local terms into local ones involving derivatives of the amplitude function. Thus, we recover the classical modulation equations of Ginzburg-Landau type.

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Zusammenfassung Bifurkation in unbeschränkten Gebieten führt häufig auf spontane Musterbildung. Dabei wird ein periodisches Grundmuster, das aus dem linearen System ableitbar ist, durch nichtlineare Effekte auf großen Raum- und Zeitskalen moduliert. Die Modulation der Amplitude wird meist durch die Ginzburg-Landau-Gleichung beschrieben.Für die Untersuchung solcher System entwickeln wir ein Verallgemeinerung des Lyapunov-Schmidtschen Reduktionsverfahren, das die Behandlung eines kontinuierlichen Spektrums erlaubt. Damit können wir parabolische Systeme in Plattengebieten auf ein niedrigdimensionaleres Problem zurückführen, das im Linearteil eine partielle Differentialgleichung bezüglich der unbeschränkten Variablen ist, das aber im Nichtlinearteil auch nichtlokale Terme enthält. Unter Verwendung des Modulationsansatzes mit großen Zeit- und Raumskalen führt die Entwicklung nach dem Bifurkations-parameter zu lokalen Termen, die Ableitungen der Amplitudenfunktion enthalten. Wir erhalten genau die klassischen Modulationsgleichungen vom Ginzburg-Landau-Typ.

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Dedicated to my teacher Klaus Kirchgässner on the occasion of his sixtieth birthday

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Research partially supported by Deutsche Forschungsgemeinschaft (DFG) under Ki 131/5-1 and by the European Community under SC 1-CT91-0670.     

Diffusive Mixing of Stable States in the Ginzburg–Landau Equation

The Ginzburg–Landau equation on the real line has spatially periodic steady states of the form , with and . For , , we construct solutions which converge for all t>0 to the limiting pattern as . These solutions are stable with respect to sufficiently small perturbations, and behave asymptotically in time like , where is uniquely determined by the boundary conditions . This extends a previous result of [BrK92] by removing the assumption that should be close to zero. The existence of the limiting profile is obtained as an application of the theory of monotone operators, and the long-time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term. Received: 22 January 1998 / Accepted: 19 April 1998