The Screen representation of spin networks: 2D recurrence, eigenvalue equation for 6j symbols, geometric interpretation and Hamiltonian dynamics

This paper treats 6j symbols or their orthonormal forms as a function of two variables spanning a square manifold which we call the "screen". We show that this approach gives important and interesting insight. This two dimensional perspective provides the most natural extension to exhibit the role of these discrete functions as matrix elements that appear at the very foundation of the modern theory of classical discrete orthogonal polynomials. Here we present 2D and 1D recursion relations that are useful for the direct computation of the orthonormal 6j, which we name U. We present a convention for the order of the arguments of the 6j that is based on their classical and Regge symmetries, and a detailed investigation of new geometrical aspects of the 6j symbols. Specifically we compare the geometric recursion analysis of Schulten and Gordon with the methods of this paper. The 1D recursion relation, written as a matrix diagonalization problem, permits an interpretation as a discrete Schr\"odinger-like equations and an asymptotic analysis illustrates semiclassical and classical limits in terms of Hamiltonian evolution.Comment: 14 pages,9 figures, presented at ICCSA 2013 13th International Conference on Computational Science and Applicatio

Boundary Conditions on Internal Three-Body Wave Functions

For a three-body system, a quantum wave function $\Psi^\ell_m$ with definite $\ell$ and $m$ quantum numbers may be expressed in terms of an internal wave function $\chi^\ell_k$ which is a function of three internal coordinates. This article provides necessary and sufficient constraints on $\chi^\ell_k$ to ensure that the external wave function $\Psi^\ell_m$ is analytic. These constraints effectively amount to boundary conditions on $\chi^\ell_k$ and its derivatives at the boundary of the internal space. Such conditions find similarities in the (planar) two-body problem where the wave function (to lowest order) has the form $r^{|m|}$ at the origin. We expect the boundary conditions to prove useful for constructing singularity free three-body basis sets for the case of nonvanishing angular momentum.Comment: 41 pages, submitted to Phys. Rev.

Uniform Approximation from Symbol Calculus on a Spherical Phase Space

We use symbol correspondence and quantum normal form theory to develop a more general method for finding uniform asymptotic approximations. We then apply this method to derive a result we announced in an earlier paper, namely, the uniform approximation of the $6j$-symbol in terms of the rotation matrices. The derivation is based on the Stratonovich-Weyl symbol correspondence between matrix operators and functions on a spherical phase space. The resulting approximation depends on a canonical, or area preserving, map between two pairs of intersecting level sets on the spherical phase space.Comment: 18 pages, 5 figure

Prussian blue analogues for potassium-ion batteries: insights into the electrochemical mechanisms

A comprehensive description of the electrochemical mechanisms of the Prussian Blue Analogue (PBA) K1.67Mn0.65Fe0.35[Fe(CN)6]0.92\ub70.45H2O is obtained by combining several complementary ex situ and operando physico-chemical characterisation techniques. This particular PBA, which shows very good electrochemical performance as a cathode material in potassium-ion batteries (PIBs), undergoes three successive redox reactions during the (de-)potassiation that are hereby identified by ex situ57Fe M\uf6ssbauer spectroscopy and operando Mn and Fe K-edge X-ray absorption spectroscopy. These reactions come along with notable modifications of the crystal structure, which are followed in real time by operando X-ray diffraction. The correlation of these results, interpreted with the support of chemometric methods, also reveals the limitations of this PBA, probably related to the deactivation of the Mn undergoing extensive reversible Jahn-Teller distortion during cycling as well as possible dissolution in the electrolyte. These results underline that optimisation of the chemical composition of PBAs is a crucial step towards the preparation of reliable and stable PBA-based cathodes for PIBs

Asymptotes in SU(2) Recoupling Theory: Wigner Matrices, 3j3j Symbols, and Character Localization

In this paper we employ a novel technique combining the Euler Maclaurin formula with the saddle point approximation method to obtain the asymptotic behavior (in the limit of large representation index $J$) of generic Wigner matrix elements $D^{J}_{MM'}(g)$. We use this result to derive asymptotic formulae for the character $\chi^J(g)$ of an SU(2) group element and for Wigner's $3j$ symbol. Surprisingly, given that we perform five successive layers of approximations, the asymptotic formula we obtain for $\chi^J(g)$ is in fact exact. This result provides a non trivial example of a Duistermaat-Heckman like localization property for discrete sums.Comment: 36 pages, 3 figure

Generating Functions for Coherent Intertwiners

We study generating functions for the scalar products of SU(2) coherent intertwiners, which can be interpreted as coherent spin network evaluations on a 2-vertex graph. We show that these generating functions are exactly summable for different choices of combinatorial weights. Moreover, we identify one choice of weight distinguished thanks to its geometric interpretation. As an example of dynamics, we consider the simple case of SU(2) flatness and describe the corresponding Hamiltonian constraint whose quantization on coherent intertwiners leads to partial differential equations that we solve. Furthermore, we generalize explicitly these Wheeler-DeWitt equations for SU(2) flatness on coherent spin networks for arbitrary graphs.Comment: 31 page

Quantum three-body system in D dimensions

The independent eigenstates of the total orbital angular momentum operators for a three-body system in an arbitrary D-dimensional space are presented by the method of group theory. The Schr\"{o}dinger equation is reduced to the generalized radial equations satisfied by the generalized radial functions with a given total orbital angular momentum denoted by a Young diagram $[\mu,\nu,0,...,0]$ for the SO(D) group. Only three internal variables are involved in the functions and equations. The number of both the functions and the equations for the given angular momentum is finite and equal to $(\mu-\nu+1)$.Comment: 16 pages, no figure, RevTex, Accepted by J. Math. Phy

Portuguese cacholeira blood sausage: A first taste of its microbiota and volatile organic compounds

Among typical Portuguese sausages, the cacholeira blood sausage undoubtedly represents one of the most popular preparations. To the authors’ knowledge, a lack of information on both the microbiota and the volatile organic compounds (VOCs) of this blood-containing sausage emerges from the available scientific literature. This study represents the first characterization of physico-chemical, microbiological and volatile traits of Portuguese cacholeira blood sausage. To this end, ready-to-eat cacholeira blood sausages were collected from two production batches manufactured in summer (batch 1) and autumn (batch 2). Viable counts showed active microbial communities mainly composed by lactic acid bacteria, coagulase negative cocci, enterococci and eumycetes. The metataxonomic approach showed a simple bacterial composition, which was dominated by Lactobacillus sakei in both the analyzed batches (1 and 2) considered. Carnobacterium, Enterococcus, Kluyvera, Lactococcus and Serratia were found as minor genera. The mycobiota varied according to the production season. Batch 1 was dominated by Starmerella apicola, Debaryomyces hansenii and Candida tropicalis, whereas batch 2 was dominated by D. hansenii. Moreover, Aspergillus spp., Kurtzmaniella zeylanoides, Saccharomyces cerevisiae, Kurtzmaniella santamariae, Brettanomyces bruxellensis and Pichia kluyveri were detected in both the batches as minority species. Seventy-two volatile compounds were identified, including esters, phenols, terpenoids, acids, alcohols, ketones, aldehydes, lactones, furans, sulphur and nitrogen compounds. Significant differences were seen in the amount of some compounds, as a feasible consequence of differences in the raw materials, artisan production and seasonality

Dietary supplements for obesity

Obesity and associated complications including diabetes, cardiometabolic dysfunction, disability, malignancy and premature mortality are considered epidemic. Research on obesity is therefore of worldwide importance. The development of obesity is a multifactorial phenomenon with contributions from biological, behavioral, genetic and environmental factors. Obesity and its associated issues require various lifestyle modifications and treatment options such medication, exercise, diet, surgery, pharmacological therapy and dietary supplements. Dietary supplements are considered an attractive alternative to traditional therapy due to their low toxicity profile and their accessibility to the general population. Dietary supplements may include one or more dietary ingredients. In this narrative review, we analyze the effects on obesity and obesity-related issues of various natural components. For example, there are a myriad of supplements that have been used as dietary supplements for weight loss such as minerals, vitamins, amino acids, metabolites, herbs, and plant extracts. This narrative review aims to present the benefits and side-effects of several ingredients of dietary supplements for weight loss and treatment of obesity. In particular, the mechanism of action, results of clinical trials, and possible side effects will be presented for the following ingredients: β-Glucans, bitter orange, calcium, vitamin D, chitosan, chromium, cocoa, coleus forskohlii, conjugate linoleic acid, ephedra sinica, fucoxanthin, garcinia cambogia, glucomannan, green coffee, green tea, guar gum, raspberry, hoodia gordonii, irvingia gabonensis, phenylpropylamine, pyruvate, white kidney bean