57,553 research outputs found
Fractional Supersymmetry and Quantum Mechanics
We present a set of quantum-mechanical Hamiltonians which can be written as
the power of a conserved charge: with and
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 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 operators and special functions. II. Legendre functions
Most of the special functions of mathematical physics are connected with the
representation of Lie groups. The action of elements of the associated Lie
algebras as linear differential operators gives relations among the functions
in a class, for example, their differential recurrence relations. In this
paper, we apply the fractional generalizations of these operators
developed in an earlier paper in the context of Lie theory to the group SO(2,1)
and its conformal extension. The fractional relations give a variety of
interesting relations for the associated Legendre functions. We show that the
two-variable fractional operator relations lead directly to integral relations
among the Legendre functions and to one- and two-variable integral
representations for those functions. Some of the relations reduce to known
fractional integrals for the Legendre functions when reduced to one variable.
The results enlarge the understanding of many properties of the associated
Legendre functions on the basis of the underlying group structure.Comment: 26 pages, Latex2e, reference correcte
Fermi and Bose pressures in statistical mechanics
I show how the Fermi and Bose pressures in quantum systems, identified in
standard discussions through the use of thermodynamic analogies, can be derived
directly in terms of the flow of momentum across a surface by using the quantum
mechanical stress tensor. In this approach, analogous to classical kinetic
theory, pressure is naturally defined locally, a point which is obvious in
terms of the stress-tensor but is hidden in the usual thermodynamic approach.
The two approaches are connected by an interesting application of boundary
perturbation theory for quantum systems. The treatment leads to a simple
interpretation of the pressure in Fermi and Bose systems in terms of the
momentum flow encoded in the wave functions. I apply the methods to several
problems, investigating the properties of quasi continuous systems, relations
for Fermi and Bose pressures, shape-dependent effects and anisotropies, and the
treatment of particles in external fields, and note several interesting
problems for graduate courses in statistical mechanics that arise naturally in
the context of these examples.Comment: RevTeX4, 18 pages. Submitted to American Journal of Physic
Adaptive p-value weighting with power optimality
Weighting the p-values is a well-established strategy that improves the power
of multiple testing procedures while dealing with heterogeneous data. However,
how to achieve this task in an optimal way is rarely considered in the
literature. This paper contributes to fill the gap in the case of
group-structured null hypotheses, by introducing a new class of procedures
named ADDOW (for Adaptive Data Driven Optimal Weighting) that adapts both to
the alternative distribution and to the proportion of true null hypotheses. We
prove the asymptotical FDR control and power optimality among all weighted
procedures of ADDOW, which shows that it dominates all existing procedures in
that framework. Some numerical experiments show that the proposed method
preserves its optimal properties in the finite sample setting when the number
of tests is moderately large
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