Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better

Joel Cohen offers a historical and prospective analysis of the relationship between mathematics and biolog

Spreading lengths of Hermite polynomials

The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (weighted L^q-norms) of Hermite polynomials and subsequently for the Renyi and Tsallis entropies, as well as for the Renyi spreading lengths. Sharp bounds for the Shannon length of these polynomials are also given by means of an information-theoretic-based optimization procedure. Moreover, it is computationally proved the existence of a linear correlation between the Shannon length (as well as the second-order Renyi length) and the standard deviation. Finally, the application to the most popular quantum-mechanical prototype system, the harmonic oscillator, is discussed and some relevant asymptotical open issues related to the entropic moments mentioned previously are posed.Comment: 16 pages, 4 figures. Journal of Computational and Applied Mathematics (2009), doi:10.1016/j.cam.2009.09.04

Theoretical size distribution of fossil taxa: analysis of a null model

BACKGROUND: This article deals with the theoretical size distribution (of number of sub-taxa) of a fossil taxon arising from a simple null model of macroevolution. MODEL: New species arise through speciations occurring independently and at random at a fixed probability rate, while extinctions either occur independently and at random (background extinctions) or cataclysmically. In addition new genera are assumed to arise through speciations of a very radical nature, again assumed to occur independently and at random at a fixed probability rate. CONCLUSION: The size distributions of the pioneering genus (following a cataclysm) and of derived genera are determined. Also the distribution of the number of genera is considered along with a comparison of the probability of a monospecific genus with that of a monogeneric family

Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution

A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $\lambda_n$ and a particle dies with instantaneous rate $\mu_n$. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics

Mathematizing Darwin

Ernst Mayr called the first part of the evolutionary synthesis the ‘Fisherian synthesis’ on account of the dominant role played by R.A. Fisher in forging a mathematical theory of natural selection together with J.B.S. Haldane and Sewall Wright in the decade 1922–1932. It is here argued that Fisher’s contribution relied on a close reading of Darwin’s work to a much greater extent than did the contributions of Haldane and Wright, that it was synthetic in contrast to their analytic approach and that it was greatly influenced by his friendship with the Darwin family, particularly with Charles’s son Leonard

Hypersensitivity to D. pteronyssinus in librarians

Autori su na 4 bolesnika s kroničnom opstruktivnom plućnom bolesti, zaposlena u knjižnici, proučavali preosjetljivost tipa I i tipa III (Arthusova reakcija, alergijski pneumonitis). Budući da su uz preosjetljivost tipa I našli i preosjetljivost tipa III, preporučuju da se za otkrivanje tipa preosjetljivosti upotrijebe i »in vitro« testovi (RIST, RAST, dvostruka imunodifuzija i imunoelektroforeza).Hypersensitivity to D. pteronyssinus was studied in four librarians with chronic obstructive lung disease. Medical histories, clinical data and the results of the diagnostic in vitro and in vivo tests are presented. The subjects had elevated total IgE and specific IgE antibodies. Three subjects had type I hypersensitivity and one subject type III. Type III hypersensitivity (Arthus\u27 reaction) was confirmed by double diffusion-in-gel determination of serum precipitins to D. pteronyssinus. The author suggests that to detect type III hypersensitivity to D. pteronyssinus both the in vitro and in vivo allergological tests (RIST, RAST, double immunodiffusion test) be used in order to complement a clinical examination and lung function tests