Functional Inequalities: New Perspectives and New Applications

This book is not meant to be another compendium of select inequalities, nor does it claim to contain the latest or the slickest ways of proving them. This project is rather an attempt at describing how most functional inequalities are not merely the byproduct of ingenious guess work by a few wizards among us, but are often manifestations of certain natural mathematical structures and physical phenomena. Our main goal here is to show how this point of view leads to "systematic" approaches for not just proving the most basic functional inequalities, but also for understanding and improving them, and for devising new ones - sometimes at will, and often on demand.Comment: 17 pages; contact Nassif Ghoussoub (nassif @ math.ubc.ca) for a pre-publication pdf cop

Interval Oscillation Criteria for Second-Order Forced Functional Dynamic Equations on Time Scales

This paper is concerned with oscillation of second-order forced functional dynamic equations of the form (r(t)(xΔ(t))γ)Δ+∑i=0n‍qi(t)|x(δi(t))|αisgn  x(δi(t))=e(t) on time scales. By using a generalized Riccati technique and integral averaging techniques, we establish new oscillation criteria which handle some cases not covered by known criteria

Classical Supersymmetric Mechanics

We analyse a supersymmetric mechanical model derived from (1+1)-dimensional field theory with Yukawa interaction, assuming that all physical variables take their values in a Grassmann algebra B. Utilizing the symmetries of the model we demonstrate how for a certain class of potentials the equations of motion can be solved completely for any B. In a second approach we suppose that the Grassmann algebra is finitely generated, decompose the dynamical variables into real components and devise a layer-by-layer strategy to solve the equations of motion for arbitrary potential. We examine the possible types of motion for both bosonic and fermionic quantities and show how symmetries relate the former to the latter in a geometrical way. In particular, we investigate oscillatory motion, applying results of Floquet theory, in order to elucidate the role that energy variations of the lower order quantities play in determining the quantities of higher order in B.Comment: 29 pages, 2 figures, submitted to Annals of Physic

Oscillation of forced impulsive differential equations with pp-Laplacian and nonlinearities given by Riemann-Stieltjes integrals

In this article, we study the oscillation of second order forced impulsive differential equation with $p$-Laplacian and nonlinearities given by Riemann-Stieltjes integrals of the form \begin{equation*} \left( p(t)\phi _{\gamma }\left( x^{\prime }(t)\right) \right) ^{\prime}+q_{0}\left( t\right) \phi _{\gamma }\left( x(t)\right)+\int_{0}^{b}q\left( t,s\right) \phi _{\alpha \left( s\right) }\left(x(t)\right) d\zeta \left(s\right) =e(t), t\neq \tau _{k}, \end{equation*} with impulsive conditions \begin{equation*} x\left( \tau _{k}^{+}\right) =\lambda _{k}~x\left( t_{k}\right), x^{\prime }\left( \tau _{k}^{+}\right) =\eta _{k}~x^{\prime }\left( \tau_{k}\right), \end{equation*} where \phi _{\gamma }\left( u\right) :=\left\vert u\right\vert ^{\gamma } \mbox{{\rm sgn}\,}u, $\gamma, b\in \left( 0,\infty \right),$ $\alpha \in C\left[ 0,b\right)$ is strictly increasing such that $0\leq \alpha \left( 0\right) <\gamma <\alpha \left( b-\right)$, and $\left\{ \tau_{k}\right\}_{k\in {\mathbb{N}}}$ is the the impulsive moments sequence. Using the Riccati transformation technique, we obtain sufficient conditions for this equation to be oscillatory

High-frequency oscillations in low-dimensional conductors and semiconductor superlattices induced by current in stack direction

A narrow energy band of the electronic spectrum in some direction in low-dimensional crystals may lead to a negative differential conductance and N-shaped I-V curve that results in an instability of the uniform stationary state. A well-known stable solution for such a system is a state with electric field domain. We have found a uniform stable solution in the region of negative differential conductance. This solution describes uniform high-frequency voltage oscillations. Frequency of the oscillation is determined by antenna properties of the system. The results are applicable also to semiconductor superlattices.Comment: 8 pages, 3 figure

A note on the dependence of solutions on functional parameters for nonlinear sturm-liouville problems

We deal with the existence and the continuous dependence of solutions on functional parameters for boundary valued problems containing the Sturm-Liouville equation. We apply these result to prove the existence of at least one solution for a certain class of optimal control problems

Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging

We introduce a new class of integrators for stiff ODEs as well as SDEs. These integrators are (i) {\it Multiscale}: they are based on flow averaging and so do not fully resolve the fast variables and have a computational cost determined by slow variables (ii) {\it Versatile}: the method is based on averaging the flows of the given dynamical system (which may have hidden slow and fast processes) instead of averaging the instantaneous drift of assumed separated slow and fast processes. This bypasses the need for identifying explicitly (or numerically) the slow or fast variables (iii) {\it Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and easily turned into one of the integrators in this paper by turning the large coefficients on over a microscopic timescale and off during a mesoscopic timescale (iv) {\it Convergent over two scales}: strongly over slow processes and in the sense of measures over fast ones. We introduce the related notion of two-scale flow convergence and analyze the convergence of these integrators under the induced topology (v) {\it Structure preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be made to be symplectic, time-reversible, and symmetry preserving (symmetries are group actions that leave the system invariant) in all variables. They are explicit and applicable to arbitrary stiff potentials (that need not be quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy and stability over four orders of magnitude of time scales. For stiff Langevin equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs reversible, quasi-symplectic on all variables and conformally symplectic with isotropic friction.Comment: 69 pages, 21 figure

Dynamics of biologically informed neural mass models of the brain

This book contributes to the development and analysis of computational models that help brain function to be understood. The mean activity of a brain area is mathematically modeled in such a way as to strike a balance between tractability and biological plausibility. Neural mass models (NMM) are used to describe switching between qualitatively different regimes (such as those due to pharmacological interventions, epilepsy, sleep, or context-induced state changes), and to explain resonance phenomena in a photic driving experiment. The description of varying states in an ordered sequence gives a principle scheme for the modeling of complex phenomena on multiple time scales. The NMM is matched to the photic driving experiment routinely applied in the diagnosis of such diseases as epilepsy, migraine, schizophrenia and depression. The model reproduces the clinically relevant entrainment effect and predictions are made for improving the experimental setting.Die vorliegende Arbeit stellt einen Beitrag zur Entwicklung und Analyse von Computermodellen zum Verständnis von Hirnfunktionen dar. Es wird die mittlere Aktivität eines Hirnareals analytisch einfach und dabei biologisch plausibel modelliert. Auf Grundlage eines Neuronalen Massenmodells (NMM) werden die Wechsel zwischen Oszillationsregimen (z.B. durch pharmakologisch, epilepsie-, schlaf- oder kontextbedingte Zustandsänderungen) als geordnete Folge beschrieben und Resonanzphänomene in einem Photic-Driving-Experiment erklärt. Dieses NMM kann sehr komplexe Dynamiken (z.B. Chaos) innerhalb biologisch plausibler Parameterbereiche hervorbringen. Um das Verhalten abzuschätzen, wird das NMM als Funktion konstanter Eingangsgrößen und charakteristischer Zeitenkonstanten vollständig auf Bifurkationen untersucht und klassifiziert. Dies ermöglicht die Beschreibung wechselnder Regime als geordnete Folge durch spezifische Eingangstrajektorien. Es wird ein Prinzip vorgestellt, um komplexe Phänomene durch Prozesse verschiedener Zeitskalen darzustellen. Da aufgrund rhythmischer Stimuli und der intrinsischen Rhythmen von Neuronenverbänden die Eingangsgrößen häufig periodisch sind, wird das Verhalten des NMM als Funktion der Intensität und Frequenz einer periodischen Stimulation mittels der zugehörigen Lyapunov-Spektren und der Zeitreihen charakterisiert. Auf der Basis der größten Lyapunov-Exponenten wird das NMM mit dem Photic-Driving-Experiment überein gebracht. Dieses Experiment findet routinemäßige Anwendung in der Diagnostik verschiedener Erkrankungen wie Epilepsie, Migräne, Schizophrenie und Depression. Durch die Anwendung des vorgestellten NMM wird der für die Diagnostik entscheidende Mitnahmeeffekt reproduziert und es werden Vorhersagen für eine Verbesserung der Indikation getroffen