Generic Morse-Smale property for the parabolic equation on the circle

In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. In \cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showing that any connecting orbit between two hyperbolic equilibria with distinct Morse indices or between a hyperbolic equilibrium and a hyperbolic periodic orbit is automatically transverse. We also show that, generically with respect to $f$, there does not exist any connection between equilibria with the same Morse index. The above properties, together with the existence of a compact global attractor and the Poincar\'e-Bendixson property, allow us to deduce that, generically with respect to $f$, the non-wandering set consists in a finite number of hyperbolic equilibria and periodic orbits . The main tools in the proofs include the lap number property, exponential dichotomies and the Sard-Smale theorem. The proofs also require a careful analysis of the asymptotic behavior of solutions of the linearized equations along the connecting orbits

Die Entwicklung des russischen Demokratieverständnisses 2008-2012: Gab es einen demokratischen Aufbruch?

Aufgrund der überraschend ausgebrochenen größten Demonstrationen der Wladimir Putin-Ära, welche einen Protest gegen die weitgehend als Betrug wahrgenommenen Duma-Wahlen vom Dezember 2011 darstellten, spekulierten viele Beobachter, dass es in Russland zu einem demokratischen Aufbruch kommen könnte. Ein Vergleich der unmittelbar nach den Parlaments- und Präsidentschaftswahlen in den Jahren 2008 und 2012 in Russland durchgeführten Meinungsumfragen offenbart jedoch nur wenige Anzeichen eines solchen Aufbruchs und zeigt, dass die Unterstützung für die Demokratie gleich geblieben ist. Die Meinungsumfragen zeigen auch, dass die Idee des "Aufbruchs" unpassend sein könnte, weil die "Demokratie", die von vielen Russen unterstützt wird, gleichzeitig mit einem "starken Anführer" vereinbar ist, der ohne "Checks and Balances" herrscht. Sie unterstützen das, was Guillermo O'Donnell bekanntermaßen eine "delegative Demokratie" nannte, in der die Menschen frei und regelmäßig Staatsoberhäupter wählen, von welchen dann erwartet wird, dass sie ihre breite Macht ohne Einschränkungen ausüben, um Probleme zu lösen und Weiterentwicklung zu fördern

Averaging, Conley index continuation and recurrent dynamics in almost-periodic parabolic equations

We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the Conley index of an isolated invariant set. As the frequency of the oscillations tends to infinity, we prove that every isolated invariant set of the averaged autonomous equation can be continued to an isolated invariant set of the skew-product semiflow associated to the non-autonomous equation. Finally, we illustrate some examples in which the Conley index can be explicitely computed and can be exploited to detect the existence of recurrent dynamics in the equation.Comment: 19 page

Rotation periods of exoplanet host stars

The stellar rotation periods of ten exoplanet host stars have been determined using newly analysed Ca II H & K flux records from Mount Wilson Observatory and Stromgren b, y photometric measurements from Tennessee State University's automatic photometric telescopes (APTs) at Fairborn Observatory. Five of the rotation periods have not previously been reported, with that of HD 130322 very strongly detected at Prot = 26.1 \pm 3.5 d. The rotation periods of five other stars have been updated using new data. We use the rotation periods to derive the line-of-sight inclinations of the stellar rotation axes, which may be used to probe theories of planet formation and evolution when combined with the planetary orbital inclination found from other methods. Finally, we estimate the masses of fourteen exoplanets under the assumption that the stellar rotation axis is aligned with the orbital axis. We calculate the mass of HD 92788 b (28 MJ) to be within the low-mass brown dwarf regime and suggest that this object warrants further investigation to confirm its true nature.Comment: Accepted for publication in MNRAS. 15 pages, 11 figure

Female first-line supervisors : perceptions, problems, and performance

Thesis. 1979. M.S.--Massachusetts Institute of Technology. Alfred P. Sloan School of Management.MICROFICHE COPY AVAILABLE IN ARCHIVES AND DEWEY.Bibliography: leaves 264-269.by Henry Clifford Hale and Homi Kersasp Patel.M.S

On the reduction principle for differential equations with piecewise constant argument of generalized type

In this paper we introduce a new type of differential equations with piecewise constant argument (EPCAG), more general than EPCA. The Reduction Principle is proved for EPCAG. The structure of the set of solutions is specified. We establish also the existence of global integral manifolds of quasilinear EPCAG in the so called critical case and investigate the stability of the zero solution

The numerical solution of forward–backward differential equations: Decomposition and related issues

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 234,(2010), doi: 10.1016/j.cam.2010.01.039This journal article discusses the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions

A condition on delay for differential equations with discrete state-dependent delay

Parabolic differential equations with discrete state-dependent delay are studied. The approach, based on an additional condition on the delay function introduced in [A.V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Analysis: Theory, Methods and Applications, 70 (11) (2009), 3978-3986] is developed. We propose and study a state-dependent analogue of the condition which is sufficient for the well-posedness of the corresponding initial value problem on the whole space of continuous functions $C$. The dynamical system is constructed in $C$ and the existence of a compact global attractor is proved