Advancing Shannon entropy for measuring diversity in systems

From economic inequality and species diversity to power laws and the analysis of multiple trends and trajectories, diversity within systems is a major issue for science. Part of the challenge is measuring it. Shannon entropy H has been used to re-think diversity within probability distributions, based on the notion of information. However, there are two major limitations to Shannon's approach. First, it cannot be used to compare diversity distributions that have different levels of scale. Second, it cannot be used to compare parts of diversity distributions to the whole. To address these limitations, we introduce a re-normalization of probability distributions based on the notion of case-based entropy Cc as a function of the cumulative probability c. Given a probability density p(x), Cc measures the diversity of the distribution up to a cumulative probability of c, by computing the length or support of an equivalent uniform distribution that has the same Shannon information as the conditional distribution of ^pc(x) up to cumulative probability c. We illustrate the utility of our approach by re-normalizing and comparing three well-known energy distributions in physics, namely, the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distributions for energy of sub-atomic particles. The comparison shows that Cc is a vast improvement over H as it provides a scale-free comparison of these diversity distributions and also allows for a comparison between parts of these diversity distributions

R-Matrix Formulation of the Quantum Inhomogeneous Groups Iso_qr(N) and Isp_qr(N)

The quantum commutations $RTT=TTR$ and the orthogonal (symplectic) conditions for the inhomogeneous multiparametric $q$-groups of the $B_n,C_n,D_n$ type are found in terms of the $R$-matrix of $B_{n+1},C_{n+1},D_{n+1}$. A consistent Hopf structure on these inhomogeneous $q$-groups is constructed by means of a projection from $B_{n+1},C_{n+1},D_{n+1}$. Real forms are discussed: in particular we obtain the $q$-groups $ISO_{q,r}(n+1,n-1)$, including the quantum Poincar\'e group.Comment: 14 pages, latex, no figure

Screening of Nuclear Reactions in the Sun and Solar Neutrinos

We quantitatively determine the effect and the uncertainty on solar neutrino production arising from the screening process. We present predictions for the solar neutrino fluxes and signals obtained with different screening models available in the literature and by using our stellar evolution code. We explain these numerical results in terms of simple laws relating the screening factors with the neutrino fluxes. Futhermore we explore a wider range of models for screening, obtained from the Mitler model by introducing and varying two phenomenological parameters, taking into account effects not included in the Mitler prescription. Screening implies, with respect to a no-screening case, a central temperat reduction of 0.5%, a 2% (8%) increase of Beryllium (Boron)-neutrino flux and a 2% (12%) increase of the Gallium (Chlorine) signal. We also find that uncertainties due to the screening effect ar at the level of 1% for the predicted Beryllium-neutrino flux and Gallium signal, not exceeding 3% for the Boron-neutrino flux and the Chlorine signal.Comment: postscript file 11 pages + 4 figures compressed and uuencoded we have replaced the previous paper with a uuencoded file (the text is the same) for any problem please write to [email protected]

Peculiarities of the Canonical Analysis of the First Order Form of the Einstein-Hilbert Action in Two Dimensions in Terms of the Metric Tensor or the Metric Density

The peculiarities of doing a canonical analysis of the first order formulation of the Einstein-Hilbert action in terms of either the metric tensor $g^{\alpha \beta}$ or the metric density $h^{\alpha \beta}= \sqrt{-g}g^{\alpha \beta}$ along with the affine connection are discussed. It is shown that the difference between using $g^{\alpha \beta}$ as opposed to $h^{\alpha \beta}$ appears only in two spacetime dimensions. Despite there being a different number of constraints in these two approaches, both formulations result in there being a local Poisson brackets algebra of constraints with field independent structure constants, closed off shell generators of gauge transformations and off shell invariance of the action. The formulation in terms of the metric tensor is analyzed in detail and compared with earlier results obtained using the metric density. The gauge transformations, obtained from the full set of first class constraints, are different from a diffeomorphism transformation in both cases.Comment: 13 page

(η5‐Cyclopentadienyl)Tricarbonylmanganese(I) Complexes

Since the initial preparation of (η5‐C5H5)Mn(CO)3 by Fischer and Jira in 1954, many methods have been developed for the synthesis of this compound (often referred to as “cymantrene”) and its C5 substituted derivatives. Interestingly, each of these preparations has at least one significant drawback from a preparative perspective (thallium reagents, high pressures, poor yields, or reactants that require significant preparative investment). Surprisingly missing from this collection of preparative methods is the direct reaction between commercially available Mn(CO)5Br and NaC5H5. In 1988, however, Smart and coworkers found MnBr(CO)3(pyridine)2 to be an effective reagent in making a substituted indacenylmanganese tricarbonyl complex. As this pyridine complex also serves as an effective manganese starting material, this chapter discusses the high‐yield syntheses of cymantrene and its derivatives such as (η5‐C5Me5)Mn(CO)3

Mathematical diversity of parts for a continuous distribution

The current paper is part of a series exploring how to link diversity measures (e.g., Gini-Simpson index, Shannon entropy, Hill numbers) to a distribution’s original shape and to compare parts of a distribution, in terms of diversity, with the whole. This linkage is crucial to understanding the exact relationship between the density of an original probability distribution, denoted by p(x), and the diversity D in non-uniform distributions, both within parts of a distribution and the whole. Empirically, our results are an important advance since we can compare various parts of a distribution, noting that systems found in contemporary data often have unequal distributions that possess multiple diversity types and have unknown and changing frequencies at different scales (e.g. income, economic complexity ratings, rankings, etc.). To date, we have proven our results for discrete distributions. Our focus here is continuous distributions. In both instances, we do so by linking case-based entropy, a diversity approach we developed, to a probability distribution’s shape for continuous distributions. This allows us to demonstrate that the original probability distribution g 1, the case-based entropy curve g 2, and the slope of diversity g 3 (c (a, x) versus the c(a, x)*lnA(a, x) curve) are one-to-one (or injective). Put simply, a change in the probability distribution, g 1, leads to variations in the curves for g 2 and g 3. Consequently, any alteration in the permutation of the initial probability distribution, which results in a different form, will distinctly define the graphs g 2 and g3 . By demonstrating the injective property of our method for continuous distributions, we introduce a unique technique to gauge the level of uniformity as indicated by D/c. Furthermore, we present a distinct method to calculate D/c for different forms of the original continuous distribution, enabling comparison of various distributions and their components

Networks from gene expression time series: characterization of correlation patterns

This paper describes characteristic features of networks reconstructed from gene expression time series data. Several null models are considered in order to discriminate between informations embedded in the network that are related to real data, and features that are due to the method used for network reconstruction (time correlation).Comment: 10 pages, 3 BMP figures, 1 Table. To appear in Int. J. Bif. Chaos, July 2007, Volume 17, Issue