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 $D$ 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 $D^\mu$ 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

the role of shock waves in modulation of galactic cosmic rays

The understanding of modulation of the galactic cosmic rays has considerably progressed by the exploration by space probes of major heliospheric structures, such as the Corotating Interaction Regions, the neutral sheet, and the compression regions of intense heliospheric magnetic fields. Also relevant in this context were the detections in the outer heliosphere of long lasting Forbush type decreases of cosmic ray intensity. The results of recent theoretical studies on the changes in intensity and energy, at different location from the Sun, induced by the passage of shocks across the heliosphere are presented. In this version of the research, the simplest cases of modulation of uGV and 2GV particles by single or several shocks during periods of positive and negative solar field polarity are reviewed. The results of the theoretical aspects of the search is reported. The comparison of the theoretical predictions with space probe data allows conclusions to be drawn on the role of shocks on the modulation on both the 11 and 22 year galactic cosmic ray cycles in the outer heliosphere and on the plausibility of the models and parameters used

On the Representation Theory of Orthofermions and Orthosupersymmetric Realization of Parasupersymmetry and Fractional Supersymmetry

We construct a canonical irreducible representation for the orthofermion algebra of arbitrary order, and show that every representation decomposes into irreducible representations that are isomorphic to either the canonical representation or the trivial representation. We use these results to show that every orthosupersymmetric system of order $p$ has a parasupersymmetry of order $p$ and a fractional supersymmetry of order $p+1$.Comment: 13 pages, to appear in J. Phys. A: Math. Ge

Localizing the Latent Structure Canonical Uncertainty: Entropy Profiles for Hidden Markov Models

This report addresses state inference for hidden Markov models. These models rely on unobserved states, which often have a meaningful interpretation. This makes it necessary to develop diagnostic tools for quantification of state uncertainty. The entropy of the state sequence that explains an observed sequence for a given hidden Markov chain model can be considered as the canonical measure of state sequence uncertainty. This canonical measure of state sequence uncertainty is not reflected by the classic multivariate state profiles computed by the smoothing algorithm, which summarizes the possible state sequences. Here, we introduce a new type of profiles which have the following properties: (i) these profiles of conditional entropies are a decomposition of the canonical measure of state sequence uncertainty along the sequence and makes it possible to localize this uncertainty, (ii) these profiles are univariate and thus remain easily interpretable on tree structures. We show how to extend the smoothing algorithms for hidden Markov chain and tree models to compute these entropy profiles efficiently.Comment: Submitted to Journal of Machine Learning Research; No RR-7896 (2012

A double continuum hydrological model for glacier applications

Peer reviewedPublisher PD

Unusual Closed Traumatic Avulsion of Both Flexor Tendons in Zones 1 and 3 of the Little Finger.

Closed tendon avulsion of both flexor tendons in the same finger is an extremely rare condition. We encountered the case of a patient who presented a rupture of the flexor digitorum profundus in zone 1 and flexor digitorum superficialis in zone 3 in the little finger. This occurrence has not been reported previously. We hereby present our case, make a review of the literature of avulsion of both flexor tendons of the same finger, and propose a treatment according to the site of the ruptures

Deep Bilateral Learning for Real-Time Image Enhancement

Performance is a critical challenge in mobile image processing. Given a reference imaging pipeline, or even human-adjusted pairs of images, we seek to reproduce the enhancements and enable real-time evaluation. For this, we introduce a new neural network architecture inspired by bilateral grid processing and local affine color transforms. Using pairs of input/output images, we train a convolutional neural network to predict the coefficients of a locally-affine model in bilateral space. Our architecture learns to make local, global, and content-dependent decisions to approximate the desired image transformation. At runtime, the neural network consumes a low-resolution version of the input image, produces a set of affine transformations in bilateral space, upsamples those transformations in an edge-preserving fashion using a new slicing node, and then applies those upsampled transformations to the full-resolution image. Our algorithm processes high-resolution images on a smartphone in milliseconds, provides a real-time viewfinder at 1080p resolution, and matches the quality of state-of-the-art approximation techniques on a large class of image operators. Unlike previous work, our model is trained off-line from data and therefore does not require access to the original operator at runtime. This allows our model to learn complex, scene-dependent transformations for which no reference implementation is available, such as the photographic edits of a human retoucher.Comment: 12 pages, 14 figures, Siggraph 201

The Scattering Approach to the Casimir Force

We present the scattering approach which is nowadays the best tool for describing the Casimir force in realistic experimental configurations. After reminders on the simple geometries of 1d space and specular scatterers in 3d space, we discuss the case of stationary arbitrarily shaped mirrors in electromagnetic vacuum. We then review specific calculations based on the scattering approach, dealing for example with the forces or torques between nanostructured surfaces and with the force between a plane and a sphere. In these various cases, we account for the material dependence of the forces, and show that the geometry dependence goes beyond the trivial {\it Proximity Force Approximation} often used for discussing experiments.Comment: Proceedings of the QFEXT'09 conference (Oklahoma, 2009

Remote Sensing of River Discharge: A Review and a Framing for the Discipline

Remote sensing of river discharge (RSQ) is a burgeoning field rife with innovation. This innovation has resulted in a highly non-cohesive subfield of hydrology advancing at a rapid pace, and as a result misconceptions, mis-citations, and confusion are apparent among authors, readers, editors, and reviewers. While the intellectually diverse subfield of RSQ practitioners can parse this confusion, the broader hydrology community views RSQ as a monolith and such confusion can be damaging. RSQ has not been comprehensively summarized over the past decade, and we believe that a summary of the recent literature has a potential to provide clarity to practitioners and general hydrologists alike. Therefore, we here summarize a broad swath of the literature, and find after our reading that the most appropriate way to summarize this literature is first by application area (into methods appropriate for gauged, semi-gauged, regionally gauged, politically ungauged, and totally ungauged basins) and next by methodology. We do not find categorizing by sensor useful, and everything from un-crewed aerial vehicles (UAVs) to satellites are considered here. Perhaps the most cogent theme to emerge from our reading is the need for context. All RSQ is employed in the service of furthering hydrologic understanding, and we argue that nearly all RSQ is useful in this pursuit provided it is properly contextualized. We argue that if authors place each new work into the correct application context, much confusion can be avoided, and we suggest a framework for such context here. Specifically, we define which RSQ techniques are and are not appropriate for ungauged basins, and further define what it means to be ‘ungauged’ in the context of RSQ. We also include political and economic realities of RSQ, as the objective of the field is sometimes to provide data purposefully cloistered by specific political decisions. This framing can enable RSQ to respond to hydrology at large with confidence and cohesion even in the face of methodological and application diversity evident within the literature. Finally, we embrace the intellectual diversity of RSQ and suggest the field is best served by a continuation of methodological proliferation rather than by a move toward orthodoxy and standardization