Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdos-Renyi random graphs. In this updated version we incorporate an erratum to be published in a forthcoming issue of Probab. Theory Relat. Fields. This results in a modification of Theorem 1.2 as well as Proposition 3.1.Comment: 16 pages. v4 incorporates an erratum to be published in a forthcoming issue of Probab. Theory Relat. Field

Bioactive compounds in the stem bark of Albizia coriaria (Welw. ex Oliver)

Albizia coriaria was investigated for the bioactive compounds present in its stem bark. The plant was selected on the basis of its widespread use in traditional herbal medicine. Extraction of the plant material was done with ethyl acetate, methanol and water and the bioactivity of each extract was tested against  Pseudomonas aeruginosa and Escherichia coli. Separation and purification of the compounds in the most active (ethyl acetate) extract was done using a  combination of chromatographic techniques. The compounds were identified by 1D and 2D -1H and 13C NMR techniques as well as Mass spectrometry (MS). Six compounds, namely: Lupeol (1), Lupenone (2), Betulinic acid (3), Acacic acid lactone (4), (+) – Catechin (5) and Benzyl alcohol (6) were identified and  characterized from the ethyl acetate extract. The results of the bioactivity tests carried out in this study indicated that A. coriaria has potential antimicrobial activity. Four of the characterized compounds (1, 2, 3 & 5) have a wide range of biological activity reported in literature. This justifies the use of this plant in traditional medicine and indicates a promising potential for the development of medicinal agents from A. coriaria stem bark.Keywords: Biological activity, Lupeol, Lupenone, Betulinic acid, Acacic acid lactone, (+) – Catechin

A new isoflavone from stem bark of Millettia dura

A new isoflavone (7,3’-dimethoxy-4’,5’-methylenedioxyisoflavone) and three known isoflavones [isoerythrinin A 4’-(3-methylbut-2-enyl) ether, isojamaicin and nordurlettone] were isolated from the stem bark of Millettia dura (Leguminosae). The structures were determined by spectroscopic methods. KEY WORDS: Millettia dura, Leguminosae, Isoflavone, 7,3’-Dimethoxy-4’,5’-methylenedioxyisoflavone, Isoerythrinin A 4’-(3-methylbut-2-enyl) ether, Isojamaicin,Nordurlettone_Bull. Chem. Soc. Ethiop. 2003, 17(1), 113-115

The Alexander-Orbach conjecture holds in high dimensions

We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes a conjecture of Alexander and Orbach. En route we calculate the one-arm exponent with respect to the intrinsic distance.Comment: 25 pages, 2 figures. To appear in Inventiones Mathematica

Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(\alpha\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(\alpha\wedge2)$ for percolation, where $d$ denotes the dimension and $\alpha$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)Comment: 43 pages, many figures. Version v2 with various (minor) changes (in particular in Sections 1.4 and A.1), and Sect. 4 is shortened. Journal of Statistical Physics (to appear

Risk measures for direct real estate investments with non-normal or unknown return distributions

The volatility of returns is probably the most widely used risk measure for real estate. This is rather surprising since a number of studies have cast doubts on the view that volatility can capture the manifold risks attached to properties and corresponds to the risk attitude of investors. A central issue in this discussion is the statistical properties of real estate returns—in contrast to neoclassical capital market theory they are mostly non-normal and often unknown, which render many statistical measures useless. Based on a literature review and an analysis of data from Germany we provide evidence that volatility alone is inappropriate for measuring the risk of direct real estate. We use a unique data sample by IPD, which includes the total returns of 939 properties across different usage types (56% office, 20% retail, 8% others and 16% residential properties) from 1996 to 2009, the German IPD Index, and the German Property Index. The analysis of the distributional characteristics shows that German real estate returns in this period were not normally distributed and that a logistic distribution would have been a better fit. This is in line with most of the current literature on this subject and leads to the question which indicators are more appropriate to measure real estate risks. We suggest that a combination of quantitative and qualitative risk measures more adequately captures real estate risks and conforms better with investor attitudes to risk. Furthermore, we present criteria for the purpose of risk classification

Sleep deprivation increases oleoylethanolamide in human cerebrospinal fluid

This study investigated the role of two fatty acid ethanolamides, the endogenous cannabinoid anandamide and its structural analog oleoylethanolamide in sleep deprivation of human volunteers. Serum and cerebrospinal fluid (CSF) samples were obtained from 20 healthy volunteers before and after a night of sleep deprivation with an interval of about 12 months. We found increased levels of oleoylethanolamide in CSF (P = 0.011) but not in serum (P = 0.068) after 24 h of sleep deprivation. Oleoylethanolamide is an endogenous lipid messenger that is released after neural injury and activates peroxisome proliferator-activated receptor-α (PPAR-α) with nanomolar potency. Exogenous PPAR-α agonists, such as hypolipidemic fibrates and oleoylethanolamide, exert both neuroprotective and neurotrophic effects. Thus, our results suggest that oleoylethanolamide release may represent an endogenous neuroprotective signal during sleep deprivation

High-Dimensional Incipient Infinite Clusters Revisited

The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second-named author and Járai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also get estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls. Keywords: Percolation; Incipient infinite cluster; Lace expansion; Critical behavio