A contact process with a single inhomogeneous site

The one-dimensional basic contact process is a Markov process for which particles give birth on vacant nearest neighbor sites at rate >0 and particles die at rate one. We introduce a one-dimensional contact process with a single inhomogeneous site: the evolution is as above except that a particle located at the origin does not die. Let _c be the critical value of the basic contact process. We show that for _c the upper invariant measures of the inhomogeneous contact process and the basic contact process coincide except at a finite number of sites. The behavior at = _c is much more intersting: the upper invariant measure of the inhomogeneous contact process concentrates on configurations with infinitely many particles, while it is known that the critical basic contact process dies out. So a single inhomogeneity may provoke a perturbation unbounded in space. As a byproduct of our analysis we prove that the connectivity probabilities of the critical basic contact process are not summable. We also give a biological interpretation of this model.     

On the phase transition to sheet percolation in random Cantor sets

Thed-dimensional random Cantor set is a generalization of the classical middle-thirds Cantor set. Starting with the unit cube [0, 1] ^d , at every stage of the construction we divide each cube remaining intoM ^d equal subcubes, and select each of these at random with probabilityp. The resulting limit set is a random fractal, which may be crossed by paths or (d–1)-dimensional sheets. We examine the critical probabilityp _s(M, d) marking the existence of these sheet crossings, and show that p_s(M,d)1–p_c(M ^d) asM, where p_c(M ^d) is the critical probability of site percolation on the lattice (M ^d) obtained by adding the diagonal edges to the hypercubic lattice ^d. This result is then used to show that, at least for sufficiently large values ofM, the phases corresponding to the existence of path and sheet crossings are distinct.     

Critical points of real entire functions and a conjecture of Pólya

Let be a nonconstant real entire function of genus and assume that all the zeros of are distributed in some infinite strip , . It is shown that (1) if has nonreal zeros in the region , and has nonreal zeros in the same region, and if the points and are located outside the Jensen disks of , then has exactly critical zeros in the closed interval , (2) if is at most of order , , and minimal type, then for each positive constant there is a positive integer such that for all has only real zeros in the region , and (3) if is of order less than , then has just as many critical points as couples of nonreal zeros.

     

Thermodynamic properties of water-β-cyclodextrin-dodecylsurfactant ternary systems

Densities, heat capacities and conductivities of water-surfactant--cyclodextrin (-CD) ternary systems were determined at 25°C. The surfactants studied were sodium dodecylsulfate (NaDS) and dodecyltrimethylammonium bromide (DTAB). From conductivity data, apparent critical micelle concentrations (cmc^*) and degree of ionization of micelles were obtained at a fixed -CD concentration (m_CD). From the cmc^* value and that in water (cmc) the stoichiometry of the surfactant--CD complex was calculated. At a given m_CD, the apparent molar volume V_,CD and heat capacity C_,CD of -CD in the two surfactants were calculated as functions of surfactant concentration m_S. For both NaDS and DTAB, V_,CD increases with m_S up to about the cmc beyond which it decreases to a constant value at high m_S, the opposite is observed for C_,CD. With NaDS, a jump in the C_,CD vs, m_S trend was detected and ascribed to a structural NaDS micellar transition. The apparent molar volume V_S and heat capacity C_S of NaDS and DTAB in the water--CD mixture 0.017 m were also obtained. From these properties and those in pure water, the volume V_S and heat capacity C_S of transfer of the surfactant from water to water+-CD mixture as functions of m_S were calculated. For both surfactants, the V_S vs. m_S trends increase to the cmc and then decrease in a monotonic manner, whereas C_S increases regularly with m_S in the pre-micellar region and is essentially constant in the post-micellar region. The V_S vs. m_S trends were qualitatively explained in terms of dispersed, complexed and micellized surfactant contributions.     

Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem inR 2

Let Ω be a bounded smooth domain inR ². Letf:R→R be a smooth non-linearity behaving like exp{s ²} ass→∞. LetF denote the primitive off. Consider the functionalJ:H ₀ ¹ (Ω)→R given by

The resolution of the Gibbs phenomenon for spherical harmonics

Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to functions with discontinuities, causing undesirable oscillations over the entire sphere.

Recently, methods for removing the Gibbs phenomenon for one-dimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first expansion coefficients (either Fourier or Gegenbauer) of a piecewise analytic function is enough to recover an exponentially convergent approximation to the point values of in any subinterval in which the function is analytic.

Here we take a similar approach, proving that knowledge of the first spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function in any subinterval , where the function is analytic. Thus we entirely overcome the Gibbs phenomenon.

     

The oscillatory behavior of the high-temperature expansion of Dyson's hierarchical model: A renormalization group analysis

We calculate 800 coefficients of the high-temperature expansion of the magnetic susceptibility of Dyson's hierarchical model with a Landau-Ginzburg measure. Log-periodic corrections to the scaling laws appear as in the case of an Ising measure. The period of oscillation appears to be a universal quantity given in good approximation by the logarithm of the largest eigenvalue of the linearized RG transformation, in agreement with a possibility suggested by Wilson and developed by Niemeijer and van Leeuwen. We estimate to be 1.300 (with a systematic error of the order of 0.002), in good agreement with the results obtained with other methods, such as the -expansion. We briefly discuss the relationship between the oscillations and the zeros of the partition function near the critical point in the complex temperature plane.     

Conformal invariance and surface defects in the two-dimensional Ising model. Exact results

The surface critical behavior of the two-dimensional Ising model with homogeneous perturbations in the surface interactions is studied on the one-dimensional quantum version. A transfer-matrix method leads to an eigenvalue equation for the excitation energies. The spectrum at the bulk critical point is obtained using anL ^–1 expansion, whereL is the length of the Ising chain. It exhibits the towerlike structure which is characteristic of conformal models in the case of irrelevant surface perturbations (h _s /J _s 0) as well as for the relevant perturbationh _s =0 for which the surface is ordered at the bulk critical point leading to an extraordinary surface transition. The exponents are deduced from the gap amplitudes and confirmed by exact finite-size scaling calculations. Both cases are finally related through a duality transformation.     

Rapidly Estimating Natural Gas Compressibility Factor

Natural gases containing sour components exhibit different gas compressibility factor(Z) behavior than do sweet gases.Therefore,a new accurate method should be developed to account for these differences.Several methods are available today for calculating the Z-factor from an cquation of state. However,these equations are more complex than the foregoing correlations,involving a large number of parameters,which require more complicated and longer computations.The aim of this study is to develop a simplified calculation method for a rapid estimating Z-factor for sour natural gases containing as much as 90% total acid gas.In this article,two new correlations are first presented for calculating the pseudo- critical pressure and temperature of the gas mixture as a function of the gas specific gravity.Then,a simple correlation on the basis of the standard gas compressibility factor chart is introduced for a quick estimation of sweet gases' compressibility factor as a function of reduced pressure and temperature.Finally,a new corrective term related to the mole fractions of carbon dioxide and hydrogen sulfide is developed.     

Thermodynamics ofN,N,N-octylpentyldimethyl-ammonium chloride in water-urea mixtures

Specific conductivities, densities, heat capacities, and enthalpies of dilution at 25‡C were measured forN,N,N-octylpentyldimethylammonium chloride (OPAC) in water-urea mixtures at various urea concentrations m_u as functions of the surfactant concentration m_s. From conductivity data, the cmc and the degree of the counterion dissociation Β of the OPAC micelles were calculated. The cmc increases linearly with increasingm _u while Βvs. m_u is a smooth concave curve. From the experimental thermodynamic data, the apparentY _Φ and partialY ₂ molar properties (volumes, heat capacities, and relative enthalpies) are derived as functions of m_u andm _s . The effect of urea on the dependences of the different properties on m_s are discussed. From data in the premicellar region the standard partial molar volumesV ₂ ⁰ and heat capacitiesC _p2 ⁰ were evaluated. It was observed thatV ₂ ⁰ increases linearly withm _u whileC _p2 ⁰ decreases. The properties of OPAC in the dispersed and micellized forms at the cmc were obtained and, therefore, the thermodynamic functions of micellization were calculated on the basis of the pseudo-phase transition model.