On the Generalized Borel Transform and its Application to the Statistical Mechanics of Macromolecules

We present a new integral transform called the Generalized Borel Transform (GBT) and show how to use it to compute some distribution functions used to describe the statistico-mechanical behavior of macromolecules. For this purpose, we choose the Random Flight Model (RFM) of macromolecules and show that the application of the GBT to this model leads to the exact expression of the polymer propagator (two-point correlation function) from which all the statistical properties of the model can be obtained. We also discuss the mathematical simplicity of the GBT and its applicability to polymers with other topologies.Comment: 21 pages, 2 figure

A Closed General Solution of the Probability Distribution Function for Three‐Dimensional Random Walk Processes

A closed general solution of the probability distribution function for three‐dimensional random walk processes is derived. In addition: (1) For the particular case of equal‐length displacements, the exact solution is compared with the Gaussian approximation for n=3, 5, and 10 steps. (2) The general solution is utilized in calculating the probability distribution of gamma‐ray energies resulting in the Cl35 (n, γ) Cl36 process. (3) For five unequal steps of fractional length: 0.582, 0.135, 0.131, 0.092, and 0.060 (which is somewhat characteristic of the fractional energies of gamma rays resulting from neutron capture), the exact solution is compared with a Gaussian, a modified Gaussian, and a five equal‐step approximation. (4) There are presented the specific solutions for all possible unequal‐length random displacements involving n=2, 3, and 4 steps.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70032/2/JCPSA6-34-2-535-1.pd

Toward a Background Independent Quantum Theory of Gravity

Any canonical quantum theory can be understood to arise from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics. This geometric perspective offers a novel, background independent non-perturbative formulation of quantum gravity. We invoke a quantum version of the equivalence principle, which requires both the statistical and symplectic geometries of canonical quantum theory to be fully dynamical quantities. Our approach sheds new light on such basic issues of quantum gravity as the nature of observables, the problem of time, and the physics of the vacuum. In particular, the observed numerical smallness of the cosmological constant can be rationalized in this approach.Comment: Awarded Honorable Mention, 2004 Gravity Research Foundation Essay Competition; 8 pages, LaTe

Shape restricted regression with random Bernstein polynomials

Shape restricted regressions, including isotonic regression and concave regression as special cases, are studied using priors on Bernstein polynomials and Markov chain Monte Carlo methods. These priors have large supports, select only smooth functions, can easily incorporate geometric information into the prior, and can be generated without computational difficulty. Algorithms generating priors and posteriors are proposed, and simulation studies are conducted to illustrate the performance of this approach. Comparisons with the density-regression method of Dette et al. (2006) are included.Comment: Published at http://dx.doi.org/10.1214/074921707000000157 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Interpretation of Hot‐Atom Data

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71353/2/JCPSA6-39-10-2770-1.pd

Failure to Bond Rupture and Nuclear Recoil Following (n, γ) Activation

Following the absorption of a thermal neutron by 127I or 79Br, the neutron‐binding energy is frequently released in the form of a gamma‐ray cascade. As a result of partial cancellation of gamma‐ray momenta, a small fraction of the activated halogens will not receive sufficient recoil momentum to rupture from their parent compound. The gas‐phase failures to bond rupture following 127I(n, γ) 128I, and 79Br(n, γ) 80Br activation were found experimentally to be: CH3I—1.09, CD3I—0.68, CF3I—0.12, CH2I2—0.068, C2H5I—0.082, n‐C3H7I—0.66, i‐C3H7I—0.30, CH3Br—0.25, CD3Br—0.20, CH2Br2—0.12, CF3Br—0.11, CF2Br2—0.093, CHClBr2—0.087, CCl3Br—0.066, CHBr3—0.05, CBr4—0.03, C2H5Br—0.33, and 1,1‐C2H4Br2—0.17%. These data are correlated with the calculated recoil energies required for bond rupture (preceding article). Using as a basis the distribution of net gamma‐ray energies calculated by the random‐walk method for the 35Cl(n, γ) 36Cl process, the kinetic‐energy distributions of the dissociated 128I or 80Br are approximated. These data suggest that the extent of hot‐atom reaction of 128I or 80Br with CH4 should not depend upon the parent molecule from which the activated halogen dissociates.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69788/2/JCPSA6-36-4-954-1.pd

Profiling time course expression of virus genes---an illustration of Bayesian inference under shape restrictions

There have been several studies of the genome-wide temporal transcriptional program of viruses, based on microarray experiments, which are generally useful in the construction of gene regulation network. It seems that biological interpretations in these studies are directly based on the normalized data and some crude statistics, which provide rough estimates of limited features of the profile and may incur biases. This paper introduces a hierarchical Bayesian shape restricted regression method for making inference on the time course expression of virus genes. Estimates of many salient features of the expression profile like onset time, inflection point, maximum value, time to maximum value, area under curve, etc. can be obtained immediately by this method. Applying this method to a baculovirus microarray time course expression data set, we indicate that many biological questions can be formulated quantitatively and we are able to offer insights into the baculovirus biology.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS258 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Thermodynamic curvature measures interactions

Thermodynamic fluctuation theory originated with Einstein who inverted the relation $S=k_B\ln\Omega$ to express the number of states in terms of entropy: $\Omega= \exp(S/k_B)$. The theory's Gaussian approximation is discussed in most statistical mechanics texts. I review work showing how to go beyond the Gaussian approximation by adding covariance, conservation, and consistency. This generalization leads to a fundamentally new object: the thermodynamic Riemannian curvature scalar $R$, a thermodynamic invariant. I argue that $|R|$ is related to the correlation length and suggest that the sign of $R$ corresponds to whether the interparticle interactions are effectively attractive or repulsive.Comment: 29 pages, 7 figures (added reference 27

Energy‐Distribution Function for Hot Atoms Produced by Nuclear Transformations

The energy‐distribution function for hot atoms produced by nuclear transformations is examined in terms of the fraction of the initial hot‐atom energy. Energy dependent and independent asymmetric scattering is specifically considered. For gaseous tritium generated by the 3He(n, p) process, the asymptotic solution probably can serve as a reasonable approximation of the distribution function in the energy range 2–20 eV. For hot atoms produced with an initial distribution of energies, such as 79Br(n, γ)‐produced 80Br, the asymptotic solution will be less valid than for tritium, but may still be a valid approximation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70454/2/JCPSA6-41-6-1595-1.pd

Blue reflectance in tarantulas is evolutionarily conserved despite nanostructural diversity

Slight shifts in arrangement within biological photonic nanostructures can produce large color differences, and sexual selection often leads to high color diversity in clades with structural colors. We use phylogenetic reconstruction, electron microscopy, spectrophotometry, and opticalmodeling to showan opposing pattern of nanostructural diversification accompanied by unusual conservation of blue color in tarantulas (Araneae: Theraphosidae). In contrast to other clades, blue coloration in phylogenetically distant tarantulas peaks within a narrow 20-nm region around 450 nm. Both quasi-ordered and multilayer nanostructures found in different tarantulas produce this blue color. Thus, even within monophyletic lineages, tarantulas have evolved strikingly similar blue coloration through divergent mechanisms. The poor color perception and lack of conspicuous display during courtship of tarantulas argue that these colors are not sexually selected. Therefore, our data contrast with sexual selection that typically produces a diverse array of colors with a single structuralmechanismby showing that natural selection on structural color in tarantulas resulted in convergence on similar color through diverse structural mechanisms