Counting Hamilton cycles in sparse random directed graphs

Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if $p\ge(\log n+\omega(1))/n$ then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically $n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically $n!(\log n/n(1+o(1)))^{n}$ directed Hamilton cycles

A leap of faith: *Scale, critical realism and *emergence in the geography of religion

This dissertation explores the role of scale in human geography through a study involving a critical realist investigation of the geography of religious adherence. Using the contributions of a critical realist framework of stratification, emergence, and pluralistic methodologies, religious adherence is studied at the scales of the individual adherent, the church, and within local associations of churches. Analysis was performed through a study of two denominational congregations and an independent congregation in Harrison County, West Virginia and used a combination of surveys and in-depth interviews with religious adherents, pastors and local denominational leaders. The conceptual framework of this dissertation stands in contrast to traditional studies of the geography of religious adherence which rely on the quantification of denominationally collected attendance statistics aggregated to the scale of county boundaries and displayed as choropleth maps. Importantly, the traditional approach lacks the capacity to jump scale and is only valuable for making general assumptions at regional or national scales. Furthermore, these studies are embedded with the scaled problems associated with ecological fallacy and the Modifiable Areal Unit Problem.;This study demonstrates that the geography of religious adherence in Harrison County is emergent and irreducible. Emergent congregational and denominational powers and properties are facilitated through scaled structures and hierarchies, with mechanisms rooted in, but not reducible to, the scale of the adherent. Because questions pertaining to adherents, churches and church hierarchies are unique to the powers and mechanisms functioning at each stratum, methodological pluralism is required to understand a robust geography of religion. In contrast to traditional GOR studies, a critical realist approach has the capacity to reveal the scaled linkages and complex processes that operate between adherents, congregations and denominations. By incorporating ecclesiastical emergence into GOR, religionists gain a valuable tool to examine the substantial ways in which religion impacts social, economic and environmental life. This study also makes contributions to the broader debate about scale in human geography by suggesting that a framework of emergence provides a valuable contribution and addition to acknowledging and understanding the complex dimensions of scale

On pressure and temperature waves within a cavitation bubble

The presented work is about the detailed pressure, temperature and velocity distribution within a plane, cylindrical and spherical cavitation bubble. The review of Plesset & Prosperetti (1977) and more recently the review of Feng & Leal (1997) describe the time behavior of the gas within a spherical bubble due to forced harmonic oscillations of the bubble wall. We reconsider and extend those previous works by developing from the conversation laws and the ideal gas law a boundary value problem for the distribution of temperature and velocity amplitude within the bubble. This is done for a plane, cylindrical, or spherical bubble. The consequences due to shape differences are discussed. The results show that an oscillating temperature boundary layer is formed in which the heat conduction takes places. With increasing dimensionless frequency, i.e. Péclet number, the boundary-layer thickness decreases and compression modulus approaches its adiabatic value. This adiabatic behaviour is reached at lower frequencies for the plane geometry in comparison with cylindrical and spherical geometry. This is due to the difference in the volume specific surface, which is 1, 2, 3 times the inverse bubble height/radius for the plane, cylindrical and spherical bubble respectively. For the plane bubble the analysis ends up in an eigenvalue problem with four eigenvalues and modes. The analytical result is not distinguishable from the numerical result for the plane case gained by a finite element solution. Interestingly if the diffusion time for the temperature distribution is of the order of the traveling time of a pressure wave no adiabatic behavior is observed. A parameter map for the different regimes is given. Since only the behavior of the gas within the bubble is considered the analysis is independent of the surface tension coefficient and the inertia of the surrounding liquid. For the plane bubble since there is no curvature there is no pressure change over the free surface. Despite of this a plane bubble is manly academic, since due to inertia the pressure within the fluid would have to be infinity if the liquid volume around the bubble is unbounded.http://deepblue.lib.umich.edu/bitstream/2027.42/84253/1/CAV2009-final57.pd

Analysis of spin density wave conductivity spectra of iron pnictides in the framework of density functional theory

The optical conductivity of LaFeAsO, BaFe$_2$As$_2$, SrFe$_2$As$_2$, and EuFe$_2$As$_2$ in the spin-density wave (SDW) state is investigated within density functional theory (DFT) in the framework of spin-polarized generalized gradient approximation (GGA) and GGA+U. We find a strong dependence of the optical features on the Fe magnetic moments. In order to recover the small Fe magnetic moments observed experimentally, GGA+$U_{\rm eff}$ with a suitable choice of negative on-site interaction $U_{\rm eff}=U-J$ was considered. Such an approach may be justified in terms of an overscreening which induces a relatively small U compared to the Hund's rule coupling J, as well as a strong Holstein-like electron-phonon interaction. Moreover, reminiscent of the fact that GGA+$U_{\rm eff}$ with a positive $U_{\rm eff}$ is a simple approximation for reproducing a gap with correct amplitude in correlated insulators, a negative $U_{\rm eff}$ can also be understood as a way to suppress magnetism and mimic the effects of quantum fluctuations ignored in DFT calculations. With these considerations, the resulting optical spectra reproduce the SDW gap and a number of experimentally observed features related to the antiferromagnetic order. We find electronic contributions to excitations that so far have been attributed to purely phononic modes. Also, an orbital resolved analysis of the optical conductivity reveals significant contributions from all Fe 3d orbitals. Finally, we observe that there is an important renormalization of kinetic energy in these SDW metals, implying that the effects of correlations cannot be neglected.Comment: 8 pages, 4 figures; recalculated spectra for U_eff=-1.9 eV for better comparison to experimental results, added discussion of the role of U and J in LDA+

Where two fractals meet: the scaling of a self-avoiding walk on a percolation cluster

The scaling properties of self-avoiding walks on a d-dimensional diluted lattice at the percolation threshold are analyzed by a field-theoretical renormalization group approach. To this end we reconsider the model of Y. Meir and A. B. Harris (Phys. Rev. Lett. 63:2819 (1989)) and argue that via renormalization its multifractal properties are directly accessible. While the former first order perturbation did not agree with the results of other methods, we find that the asymptotic behavior of a self-avoiding walk on the percolation cluster is governed by the exponent nu_p=1/2 + epsilon/42 + 110epsilon^2/21^3, epsilon=6-d. This analytic result gives an accurate numeric description of the available MC and exact enumeration data in a wide range of dimensions 2<=d<=6.Comment: 4 pages, 2 figure

Dynamical Scaling Behavior of Percolation Clusters in Scale-free Networks

In this work we investigate the spectra of Laplacian matrices that determine many dynamic properties of scale-free networks below and at the percolation threshold. We use a replica formalism to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum for a general type of tree-like networks. Close to the percolation threshold we find characteristic scaling functions for the density of states rho(lambda) of scale-free networks. rho(lambda) shows characteristic power laws rho(lambda) ~ lambda^alpha_1 or rho(lambda) ~ lambda^alpha_2 for small lambda, where alpha_1 holds below and alpha_2 at the percolation threshold. In the range where the spectra are accessible from a numerical diagonalization procedure the two methods lead to very similar results.Comment: 9 pages, 6 figure

Entropy-induced separation of star polymers in porous media

We present a quantitative picture of the separation of star polymers in a solution where part of the volume is influenced by a porous medium. To this end, we study the impact of long-range-correlated quenched disorder on the entropy and scaling properties of $f$-arm star polymers in a good solvent. We assume that the disorder is correlated on the polymer length scale with a power-law decay of the pair correlation function $g(r) \sim r^{-a}$. Applying the field-theoretical renormalization group approach we show in a double expansion in $\epsilon=4-d$ and $\delta=4-a$ that there is a range of correlation strengths $\delta$ for which the disorder changes the scaling behavior of star polymers. In a second approach we calculate for fixed space dimension $d=3$ and different values of the correlation parameter $a$ the corresponding scaling exponents $\gamma_f$ that govern entropic effects. We find that $\gamma_f-1$, the deviation of $\gamma_f$ from its mean field value is amplified by the disorder once we increase $\delta$ beyond a threshold. The consequences for a solution of diluted chain and star polymers of equal molecular weight inside a porous medium are: star polymers exert a higher osmotic pressure than chain polymers and in general higher branched star polymers are expelled more strongly from the correlated porous medium. Surprisingly, polymer chains will prefer a stronger correlated medium to a less or uncorrelated medium of the same density while the opposite is the case for star polymers.Comment: 14 pages, 7 figure

Star copolymers in porous environments: scaling and its manifestations

We consider star polymers, consisting of two different polymer species, in a solvent subject to quenched correlated structural obstacles. We assume that the disorder is correlated with a power-law decay of the pair correlation function g(x)\sim x^{-a}. Applying the field-theoretical renormalization group approach in d dimensions, we analyze different scenarios of scaling behavior working to first order of a double \epsilon=4-d, \delta=4-a expansion. We discuss the influence of the correlated disorder on the resulting scaling laws and possible manifestations such as diffusion controlled reactions in the vicinity of absorbing traps placed on polymers as well as the effective short-distance interaction between star copolymers.Comment: 13 pages, 3 figure

Multifractality of Brownian motion near absorbing polymers

We characterize the multifractal behavior of Brownian motion in the vicinity of an absorbing star polymer. We map the problem to an O(M)-symmetric phi^4-field theory relating higher moments of the Laplacian field of Brownian motion to corresponding composite operators. The resulting spectra of scaling dimensions of these operators display the convexity properties which are necessarily found for multifractal scaling but unusual for power of field operators in field theory. Using a field-theoretic renormalization group approach we obtain the multifractal spectrum for absorbtion at the core of a polymer star as an asymptotic series. We evaluate these series using resummation techniques.Comment: 18 pages, revtex, 6 ps-figure