Random subgraphs of the 2D Hamming graph: the supercritical phase

We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on n vertices. Let p be the edge probability, and write ${p={(1+\varepsilon)}/{(2(n-1))}}$ for some ${\varepsilon \in \mathbb{R}}$ . In Borgs et al. (Random Struct Alg 27:137–184, 2005; Ann Probab 33:1886–1944, 2005), the size of the largest connected component was estimated precisely for a large class of graphs including H(2,n) for ${\varepsilon \leq \Lambda V^{-1/3}}$ , where Λ >  0 is a constant and V = n ² denotes the number of vertices in H(2,n). Until now, no matching lower bound on the size in the supercritical regime has been obtained. In this paper we prove that, when ${\varepsilon \gg (\log{V})^{1/3} V^{-1/3}}$ , then the largest connected component has size close to ${2 \varepsilon V}$ with high probability. We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of p are supercritical. Barring the factor ${(\log{V})^{1/3}}$ , this identifies the size of the largest connected component all the way down to the critical p window.     

A new approach to the giant component problem

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np = 1 + ω(n)n^?1/3, where ω(n) → ∞ arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie–Weiss Model. For β < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1 ? β)]^?1 n log n. For β = 1, we prove that the mixing time is of order n ^3/2. For β > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).     

The Supermarket Model with Bounded Queue Lengths in Equilibrium

Journal of Statistical Physics - In the supermarket model, there are n queues, each with a single server. Customers arrive in a Poisson process with arrival rate $$\lambda n$$ , where $$\lambda =...     

Chemical Composition and Antibacterial Activity of Liquid and Volatile Phase of Essential Oils against Planktonic and Biofilm-Forming Cells of Pseudomonas aeruginosa

Pseudomonas aeruginosa is an opportunistic pathogen causing life-threatening, hard-to-heal infections associated with the presence of a biofilm. Essential oils (EOs) are promising agents to combat pseudomonal infections because of the alleged antimicrobial activity of their volatile fractions and liquid forms. Therefore, the purpose of this paper was to evaluate the antibacterial efficacy of both volatile and liquid phases of seven EOs (thyme, tea tree, basil, rosemary, eucalyptus, menthol mint, lavender) against P. aeruginosa biofilm and planktonic cells with the use of a broad spectrum of analytical in vitro methods. According to the study results, the antibacterial activity of EOs in their liquid forms varied from that of the volatile fractions. Overall, liquid and volatile forms of rosemary EO and tea tree EO displayed significant antibiofilm effectiveness. The outcomes indicate that these particular EOs possess the potential to be used in the therapy of P. aeruginosa infections.     

Differential Glycosite Profiling—A Versatile Method to Compare Membrane Glycoproteomes

Glycosylation is the most prevalent and varied form of post-translational protein modifications. Protein glycosylation regulates multiple cellular functions, including protein folding, cell adhesion, molecular trafficking and clearance, receptor activation, signal transduction, and endocytosis. In particular, membrane proteins are frequently highly glycosylated, which is both linked to physiological processes and of high relevance in various disease mechanisms. The cellular glycome is increasingly considered to be a therapeutic target. Here we describe a new strategy to compare membrane glycoproteomes, thereby identifying proteins with altered glycan structures and the respective glycosites. The workflow started with an optimized procedure for the digestion of membrane proteins followed by the lectin-based isolation of glycopeptides. Since alterations in the glycan part of a glycopeptide cause mass alterations, analytical size exclusion chromatography was applied to detect these mass shifts. N-glycosidase treatment combined with nanoUPLC-coupled mass spectrometry identified the altered glycoproteins and respective glycosites. The methodology was established using the colon cancer cell line CX1, which was treated with 2-deoxy-glucose—a modulator of N-glycosylation. The described methodology is not restricted to cell culture, as it can also be adapted to tissue samples or body fluids. Altogether, it is a useful module in various experimental settings that target glycan functions.     

A Comparison of Conventional and Ultrasound-Assisted BCR Sequential Extraction Methods for the Fractionation of Heavy Metals in Sewage Sludge of Different Characteristics

The purpose of this study was to determine the heavy metal (HM: Cd, Cr, Cu, Ni, Pb, Zn, and Hg) content in particular chemical fractions (forms) of sewage sludge with different characteristics (primary and dewatered sludge) using conventional (CSE) and ultrasound-assisted (USE) BCR sequential extraction methods (Community Bureau of Reference, now the Standards, Measurements and Testing Programme). The concentrations of HMs were determined using inductively coupled plasma optical spectrometry (ICP-OES). Only mercury was assayed with cold vapor atomic absorption spectrometry (CVAAS). Ultrasound treatment was conducted in the ultrasonic bath (Sonic 5, Polsonic). The optimal sonication time (30 min) was determined using ERM-CC144 (Joint Research Center; JCR) certified reference material. The conducted experiment revealed that the use of ultrasound waves shortened the extraction time to 4 h and 30 min (Stages I to III). The recoveries (R_M) of heavy metals ranged from 62.8% to 130.2% (CSE) and from 79.8% to 135.7% (USE) for primary sludge, and from 87.2% to 113.2% (CSE) and from 87.8% to 112.0% (USE) for dewatered sludge. The only exception was Hg in dewatered sludge. The conducted research revealed minor differences in the concentrations and fractionation patterns for Cd, Ni, and Zn extracted from sludge samples by the tested methods. However, it was confirmed that the above findings do not significantly affect the results of a potential ecological risk assessment (with minor exceptions for Cd and Zn in the primary sludge), which is extremely essential for the natural use of sludge, and especially dewatered sludge (the final sludge). The shorter extraction time and lower energy consumption prove that ultrasound-assisted extraction is a fast and simple method for HM fractionation, and that it provides an alternative to the conventional procedure. Therefore, it can be considered a “green method” for the assessment of the bioavailability and mobility of heavy metals in solid samples.     

Hetero-Diels-Alder Reactions of In Situ-Generated Azoalkenes with Thioketones; Experimental and Theoretical Studies

The hetero-Diels-Alder reactions of in situ-generated azoalkenes with thioketones were shown to offer a straightforward method for an efficient and regioselective synthesis of scarcely known N-substituted 1,3,4-thiadiazine derivatives. The scope of the method was fairly broad, allowing the use of a series of aryl-, ferrocenyl-, and alkyl-substituted thioketones. However, in the case of N-tosyl-substituted cycloadducts derived from 1-thioxo-2,2,4,4-tetramethylcyclobutan-3-one and the structurally analogous 1,3-dithione, a more complicated pathway was observed. By elimination of toluene sulfinic acid, the initially formed cycloadducts afforded 2H-1,3,4-thiadiazines as final products. Advanced DFT calculations revealed that the observed high regioselectivity was due to kinetic reaction control and that the (4 + 2)-cycloadditions of sterically less unhindered thiones occurred via highly unsymmetric transition states with shorter C..S-distances (2.27–2.58 Å) and longer N..C-distances (3.02–3.57 Å). In the extreme case of the sterically very hindered 2,2,4,4-tetramethylcyclobutan-1,3-dione-derived thioketones, a zwitterionic intermediate with a fully formed C‒S bond was detected, which underwent ring closure to the 1,3,4-thiadiazine derivative in a second step. For the hypothetical formation of the regioisomeric 1,2,3-thiadiazine derivatives, the DFT calculations proposed more symmetric transition states with considerably higher energies.     

On-line routing of random calls in networks

 We consider a random sequence of calls between nodes in a complete network with link capacities. Each call first tries the direct link. If that link is saturated, then the `first-fit dynamic alternative routing algorithm' sequentially selects up to d random two-link alternative routes, and assigns the call to the first route with spare capacity on each link, if there is such a route. The `balanced dynamic alternative routing algorithm' simultaneously selects d random two-link alternative routes; and the call is accepted on a route minimising the maximum of the loads on its two links, provided neither of these two links is saturated. We determine the capacities needed for these algorithms to route calls successfully, and find that the second `balanced' algorithm requires a much smaller capacity. Our results strengthen and extend the discrete-time results presented in [10]. Received: 10 January 2002 / Revised version: 7 July 2002 / Published online: 28 March 2003