Fractional Supersymmetry and Quantum Mechanics

We present a set of quantum-mechanical Hamiltonians which can be written as the $F^{\,\rm th}$ power of a conserved charge: $H=Q^F$ with $[H,Q]=0$ and $F=2,3,...\, .$ This new construction, which we call {\it fractional}\/ supersymmetric quantum mechanics, is realized in terms of \pg\ variables satisfying \t^F=0. Furthermore, in a pseudo-classical context, we describe {\it fractional}\/ supersymmetry transformations as the $F^{\,\rm th}$ roots of time translations, and provide an action invariant under such transformations.Comment: 12 pages, plain TEX, McGill/92-54, to appear in Phys. Lett. B (minor corrections and references updated

Fractional Superspace Formulation of Generalized Super-Virasoro Algebras

We present a fractional superspace formulation of the centerless parasuper-Viraso-ro and fractional super-Virasoro algebras. These are two different generalizations of the ordinary super-Virasoro algebra generated by the infinitesimal diffeomorphisms of the superline. We work on the fractional superline parametrized by $t$ and $\theta$, with $t$ a real coordinate and $\theta$ a paragrassmann variable of order $M$ and canonical dimension $1/F$. We further describe a more general structure labelled by $M$ and $F$ with $M\geq F$. The case $F=2$ corresponds to the parasuper-Virasoro algebra of order $M$, while the case $F=M$ leads to the fractional super-Virasoro algebra of order $F$. The ordinary super-Virasoro algebra is recovered at $F=M=2$. The connection with $q$-oscillator algebras is discussed.Comment: 9 pages, McGill/92-30 (small corrections and elimination of the parameter "alpha"

Describability via ubiquity and eutaxy in Diophantine approximation

We present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical ubiquity techniques, as well as the mass and the large intersection transference principles, thereby leading to a thorough description of the properties in terms of Hausdorff measures and large intersection classes associated with general gauge functions. The sets issued from eutaxic sequences of points and optimal regular systems may naturally be described within this framework. The discussed applications include the classical homogeneous and inhomogeneous approximation, the approximation by algebraic numbers, the approximation by fractional parts, the study of uniform and Poisson random coverings, and the multifractal analysis of L{\'e}vy processes.Comment: 94 pages. Notes based on lectures given during the 2012 Program on Stochastics, Dimension and Dynamics at Morningside Center of Mathematics, the 2013 Arithmetic Geometry Year at Poncelet Laboratory, and the 2014 Spring School in Analysis held at Universite Blaise Pasca

Random wavelet series based on a tree-indexed Markov chain

We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Holder exponent form a set with large intersection.Comment: 25 page