Self-Similar Random Processes and Infinite-Dimensional Configuration Spaces

We discuss various infinite-dimensional configuration spaces that carry measures quasiinvariant under compactly-supported diffeomorphisms of a manifold M corresponding to a physical space. Such measures allow the construction of unitary representations of the diffeomorphism group, which are important to nonrelativistic quantum statistical physics and to the quantum theory of extended objects in d-dimensional Euclidean space. Special attention is given to measurable structure and topology underlying measures on generalized configuration spaces obtained from self-similar random processes (both for d = 1 and d > 1), which describe infinite point configurations having accumulation points

On the Fock space for nonrelativistic anyon fields and braided tensor products

We realize the physical N-anyon Hilbert spaces, introduced previously via unitary representations of the group of diffeomorphisms of the plane, as N-fold braided-symmetric tensor products of the 1-particle Hilbert space. This perspective provides a convenient Fock space construction for nonrelativistic anyon quantum fields along the more usual lines of boson and fermion fields, but in a braided category. We see how essential physical information is thus encoded. In particular we show how the algebraic structure of our anyonic Fock space leads to a natural anyonic exclusion principle related to intermediate occupation number statistics, and obtain the partition function for an idealised gas of fixed anyonic vortices.Comment: Added some references, more explicit formulae for the discrete case and remark on partition function. 25 pages latex, no figure

Conformal symmetry transformations and nonlinear Maxwell equations

We make use of the conformal compactification of Minkowski spacetime $M^{\#}$ to explore a way of describing general, nonlinear Maxwell fields with conformal symmetry. We distinguish the inverse Minkowski spacetime $[M^{\#}]^{-1}$ obtained via conformal inversion, so as to discuss a doubled compactified spacetime on which Maxwell fields may be defined. Identifying $M^{\#}$ with the projective light cone in $(4+2)$-dimensional spacetime, we write two independent conformal-invariant functionals of the $6$-dimensional Maxwellian field strength tensors -- one bilinear, the other trilinear in the field strengths -- which are to enter general nonlinear constitutive equations. We also make some remarks regarding the dimensional reduction procedure as we consider its generalization from linear to general nonlinear theories.Comment: 12 pages, Based on a talk by the first author at the International Conference in Mathematics in honor of Prof. M. Norbert Hounkonnou (October 29-30, 2016, Cotonou, Benin). To be published in the Proceedings, Springer 201

Some Variations on Maxwell's Equations

In the first sections of this article, we discuss two variations on Maxwell's equations that have been introduced in earlier work--a class of nonlinear Maxwell theories with well-defined Galilean limits (and correspondingly generalized Yang-Mills equations), and a linear modification motivated by the coupling of the electromagnetic potential with a certain nonlinear Schroedinger equation. In the final section, revisiting an old idea of Lorentz, we write Maxwell's equations for a theory in which the electrostatic force of repulsion between like charges differs fundamentally in magnitude from the electrostatic force of attraction between unlike charges. We elaborate on Lorentz' description by means of electric and magnetic field strengths, whose governing equations separate into two fully relativistic Maxwell systems--one describing ordinary electromagnetism, and the other describing a universally attractive or repulsive long-range force. If such a force cannot be ruled out {\it a priori} by known physical principles, its magnitude should be determined or bounded experimentally. Were it to exist, interesting possibilities go beyond Lorentz' early conjecture of a relation to (Newtonian) gravity.Comment: 26 pages, submitted to a volume in preparation to honor Gerard Emch v. 2: discussion revised, factors of 4\pi corrected in some equation

A FOCUS ON CONTENT: THE USE OF RUBRICS IN PEER REVIEW TO GUIDE STUDENTS AND INSTRUCTORS

Students who are solving open-ended problems would benefit from formative assessment, i.e., from receiving helpful feedback and from having an instructor who is informed about their level of performance. Open-ended problems challenge existing assessment techniques. For example, such problems may have reasonable alternative solutions, or conflicting objectives. Analyses of open-ended problems are often presented as free-form text since they require arguments and justifications for one solution over others, and students may differ in how they frame the problems according to their knowledge, beliefs and attitudes.This dissertation investigates how peer review may be used for formative assessment. Computer-Supported Peer Review in Education, a technology whose use is growing, has been shown to provide accurate summative assessment of student work, and peer feedback can indeed be helpful to students. A peer review process depends on the rubric that students use to assess and give feedback to each other. However, it is unclear how a rubric should be structured to produce feedback that is helpful to the student and at the same time to yield information that could be summarized for the instructor.The dissertation reports a study in which students wrote individual analyses of an open-ended legal problem, and then exchanged feedback using Comrade, a web application for peer review. The study compared two conditions: some students used a rubric that was relevant to legal argument in general (the domain-relevant rubric), while others used a rubric that addressed the conceptual issues embedded in the open-ended problem (the problem-specific rubric).While both rubric types yield peer ratings of student work that approximate the instructor's scores, feedback elicited by the domain-relevant rubric was redundant across its dimensions. On the contrary, peer ratings elicited by the problem-specific rubric distinguished among its dimensions. Hierarchical Bayesian models showed that ratings from both rubrics can be fit by pooling information across students, but only problem-specific ratings are fit better given information about distinct rubric dimensions

Fractional supersymmetric Quantum Mechanics as a set of replicas of ordinary supersymmetric Quantum Mechanics

A connection between fractional supersymmetric quantum mechanics and ordinary supersymmetric quantum mechanics is established in this Letter.Comment: Paper accepted for publication in Physics Letters

On Galilean invariance and nonlinearity in electrodynamics and quantum mechanics

Recent experimental results on slow light heighten interest in nonlinear Maxwell theories. We obtain Galilei covariant equations for electromagnetism by allowing special nonlinearities in the constitutive equations only, keeping Maxwell's equations unchanged. Combining these with linear or nonlinear Schroedinger equations, e.g. as proposed by Doebner and Goldin, yields a Galilean quantum electrodynamics.Comment: 12 pages, added e-mail addresses of the authors, and corrected a misprint in formula (2.10