Ln Gamma Research Articles

Sharp Smith\u2019s bounds for the gamma function

Among various approximation formulas for the gamma function, Smith showed that \t\t\tΓ(x+12)∼S(x)=2π(xe)x(2xtanh12x)x/2,x→∞,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\Gamma \\biggl( x+\\frac{1}{2} \\biggr) \\thicksim S ( x ) =\\sqrt{2 \\pi } \\biggl( \\frac{x}{e} \\biggr) ^{x} \\biggl( 2x\\tanh \\frac{1}{2x} \\biggr) ^{x/2}, \\quad x\\rightarrow \\infty , $$\\end{document} which is a little-known but accurate and simple one. In this note, we prove that the function xmapsto ln Gamma ( x+1/2 ) - ln S ( x ) is strictly increasing and concave on ( 0,infty ) , which shows that Smith’s approximation is just an upper one.

Prediction of infinite dilution activity coefficients of chlorinated organic compounds in aqueous solution from quantum-chemical descriptors.

A quantitative structure-property relationship (QSPR) model is developed to correlate the natural logarithm of infinite dilution activity coefficients, ln (gamma(infinity)), of 45 chlorinated organic compounds in aqueous solution from quantum-chemical descriptors. The best correlation equation contains five theoretical molecular descriptors. All descriptors were obtained from the chemical structure of the compounds and have definite physical meaning corresponding to different intermolecular interactions. The model predicts ln (gamma(infinity)) with a correlation coefficients of 0.949 and a standard error of 0.442 ln units. The obtained QSPR equation may be applied to the prediction of gamma(infinity) of other chlorinated organic compounds not present in the data set used for the development of the present model. Copyright 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1851-1856, 2001

Thermoelastic properties of MgSiO3perovskite using the Debye approach

MgSiO 3 perovskite is shown to be a Debye-like mineral by the determination of specific heat, C v , entropy, S, and thermal pressure, Delta P Th , using the Debye theory up to 1800 K. Sound velocities and bulk moduli at ambient conditions published by Yeganeh-Haeri were used to find the ambient acoustic Debye temperature, Theta ac D . The variation of Theta ac D with T was assumed to be a curve parallel to the Theta ac D vs. T curves previously found for Al 2 O 3 , MgO, and MgSiO 3 , enabling Theta ac D (T) to be given up to 1800 K. To determine C p , the thermal expansivity, alpha , and the isothermal bulk modulus, K T , are needed. After considering several sets of alpha (T), the alpha (T) data of Funamori and his colleagues were chosen. Using the ambient K T and the values of (theta K T /theta T) P vs. T reported by Jackson and Rigden, K T (T) up to 1800 K was found. Then C P (T) up to 1800 K was found assuming quasiharmonicity in C v . The data behind the C P (T) calculation are also sufficient to find the Gruneisen parameter, gamma (T), and the Anderson-Gruneisen parameters, delta T and delta S , up to 1800 K. The value of q = (theta ln gamma /theta ln V) T was found, and with gamma and rho , Delta P Th vs. V and T was determined. The three sound velocities, v s , v p , and v b = K (sub s/rho ) , were then determined to 1800 K. From v s and v p , Poisson's ratio and the isotropic shear modulus, G, were found to 1800 K. MgSiO 3 perovskite is one of a small, select group of Debye-like minerals for which thermoelastic properties and the equation of state (EOS) are calculable from acoustic data.

Regulation and function of extracellular matrix intestinal epithelial restitution in vitro.

Repair of epithelial injury in the gastrointestinal tract is initially accomplished by migration of epithelial cells from the wound edge ("restitution"). To assess expression and function of the extracellular matrix (ECM) in the restitution phase after epithelial injury, in vitro studies using wounded monolayers or a rat intestinal epithelium-derived cell line (IEC-6) were undertaken. IEC-6 cells expressed fibronectin (FN) mRNA and protein in large amounts and lesser quantities of laminin-beta 1 (LN beta 1) and LN gamma 1. Collagen IV (Col IV) was weakly expressed, and LN alpha 1 was not detected. After wounding a significant decrease in FN, LN beta 1, LN gamma 1, and Col IV alpha 1 mRNA steady-state levels was observed; mean content 24 h after wounding was reduced by 75-90%. FN, LN, and Col IV proteins were also reduced. The downregulation of these ECM transcripts and proteins could be substantially prevented by transforming growth factor-beta 1, a restitution-promoting growth factor. In addition to changes of expression, the distribution of FN and LN was also altered in migrating cells after wounding, as assessed by immunofluorescence. Arg-Gly-Asp peptides that recognize the major cell attachment site on FN and antibodies recognizing the main noncollagenous domain of Col IV inhibited cell migration, but immunoneutralizing anti-LN antisera did not affect restitution. In conclusion, although paradoxically downregulated after wounding, ECM proteins, in particular FN and Col IV molecules, are able to enhance intestinal epithelial restitution.

LEARNING TEMPORAL SEQUENCES FROM EXAMPLES IN A LOCAL FEEDBACK NEURAL NETWORK

Statistical-mechanical techniques are used to study temporal sequence association in a local feedback neural network consisting of an input layer of context units and a single output unit. Each context unit has a feedback connection onto itself such that it accumulates an exponentially decaying moving average or trace of previous inputs to that unit. The formation of these traces allows the network to extract temporal information over an interval delta, where delta = 1/magnitude of ln gamma and gamma is the decay-rate of the moving average. The particular problem of learning a rule from examples is considered where the rule is generated by a teacher local feedback network. The replica method and other mean-field theory techniques are used to determine how the resulting generalization error of the network varies as a function of sequence length M and decay-rate gamma.

Hot-carrier effects in non-radiative multiphonon capture by deep traps in semiconductors

For any lattice temperature T<or approximately=103K and different charge states Z of the centre the authors describe the dependences of non-radiative multiphonon carrier capture coefficients CZ(T; E) on the carrier energy E by products of the familiar Sommerfeld factors sZ(E) with corresponding energy-loss factors C0(T; E). As the carrier energy E is increased these 'neutral parts' C0(T; E) of capture coefficients decrease or increase (exponentially) in the alternative regimes of small and large lattice relaxation. Their maximum enhancements Cmax0/C0(0;0), which can be produced in the latter regime, at low lattice temperatures, are shown to be approximately exp(S(1+ gamma ln gamma - gamma )) (approaching eS in the gamma to 0 limit) where S represents the Huang-Rhys factor and gamma the ratio between thermal depth and lattice relaxation energy. For values of S of the order of 10 this 'enhanced hot-carrier capture due to large lattice relaxation' effect can thus amount to several orders of magnitude and should hence be of considerable importance for interpretations of corresponding hot-carrier capture data.

Activated tunnelling decay of metastable states

The authors consider a particle in a potential well of depth U0 and level splitting omega <<U0 and study the temperature dependence of the lifetime of a metastable state in the limit of weak friction gamma . The number of levels participating in the relaxation is shown to depend on gamma , whereas the tunnelling transition goes through the highest of these levels. The case of a decay via two or three levels is also considered. For a large number of participating levels the temperature dependence of the lifetime is of the conventional activation type with the activation energy T* approximately=2 omega ln( gamma tau 0)/ln(U0/ omega ) ( tau 0 is the lifetime of the ground state). The crossover from the activated to the pure tunnelling decay takes place at T approximately omega /ln(U0/ omega ).

Magnetostriction and deformation potentials in graphite

Magnetostriction measurements provide a powerful tool for determining the deformation potentials and band structure in the extreme quantum limit in diamagnetic materials. The longitudinal magnetostriction of a highly oriented pyrolitic graphic sample has been measured along the c axis. The observed data were used for computing the deformation potential: D3(2)=( delta / delta e3) ln gamma 2. The numerical results are in good agreement with theoretical estimates and with other data obtained from pressure-dependent magnetoreflection experiments. It is also shown, for the first time, that the magnetic-energy levels derived from the McClure Hamiltonian are valid up to 14 T.