The isoperimetric constant of the random graph process

The isoperimetric constant of a graph $G$ on $n$ vertices, $i(G)$, is the minimum of $\frac{|\partial S|}{|S|}$, taken over all nonempty subsets $S\subset V(G)$ of size at most $n/2$, where $\partial S$ denotes the set of edges with precisely one end in $S$. A random graph process on $n$ vertices, $\widetilde{G}(t)$, is a sequence of $\binom{n}{2}$ graphs, where $\widetilde{G}(0)$ is the edgeless graph on $n$ vertices, and $\widetilde{G}(t)$ is the result of adding an edge to $\widetilde{G}(t-1)$, uniformly distributed over all the missing edges. We show that in almost every graph process $i(\widetilde{G}(t))$ equals the minimal degree of $\widetilde{G}(t)$ as long as the minimal degree is $o(\log n)$. Furthermore, we show that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically $\Theta(\log n)$, the ratio between the isoperimetric constant and the minimum degree falls from 1 to 1/2, its final value

Ensemble dependence of the critical behavior of a system with long-range interaction and quenched randomness

We propose a hybrid model governed by the Blume-Emery-Griffiths (BEG) Hamiltonian with a mean-field-like interaction, where the spins are randomly quenched such that some of them are "pure" Ising and the others admit the BEG set of states. It is found, by varying the concentration of the Ising spins, that the model displays different phase portraits in concentration-temperature parameter space, within the canonical and the microcanonical ensembles. Phenomenological indications that these portraits are rich and rather unusual are provided.Comment: 6 pages, 4 figure

Constructing cost-effective infrastructure networks

The need for reliable and low-cost infrastructure is crucial in today's world. However, achieving both at the same time is often challenging. Traditionally, infrastructure networks are designed with a radial topology lacking redundancy, which makes them vulnerable to disruptions. As a result, network topologies have evolved towards a ring topology with only one redundant edge and, from there, to more complex mesh networks. However, we prove that large rings are unreliable. Our research shows that a sparse mesh network with a small number of redundant edges that follow some design rules can significantly improve reliability while remaining cost-effective. Moreover, we have identified key areas where adding redundant edges can impact network reliability the most by using the SAIDI index, which measures the expected number of consumers disconnected from the source node. These findings offer network planners a valuable tool for quickly identifying and addressing reliability issues without the need for complex simulations. Properly planned sparse mesh networks can thus provide a reliable and a cost-effective solution to modern infrastructure challenges

Minocycline Synergizes with N-Acetylcysteine and Improves Cognition and Memory Following Traumatic Brain Injury in Rats

Background: There are no drugs presently available to treat traumatic brain injury (TBI). A variety of single drugs have failed clinical trials suggesting a role for drug combinations. Drug combinations acting synergistically often provide the greatest combination of potency and safety. The drugs examined (minocycline (MINO), N-acetylcysteine (NAC), simvastatin, cyclosporine A, and progesterone) had FDA-approval for uses other than TBI and limited brain injury in experimental TBI models. Methodology/Principal Findings: Drugs were dosed one hour after injury using the controlled cortical impact (CCI) TBI model in adult rats. One week later, drugs were tested for efficacy and drug combinations tested for synergy on a hierarchy of behavioral tests that included active place avoidance testing. As monotherapy, only MINO improved acquisition of the massed version of active place avoidance that required memory lasting less than two hours. MINO-treated animals, however, were impaired during the spaced version of the same avoidance task that required 24-hour memory retention. Coadministration of NAC with MINO synergistically improved spaced learning. Examination of brain histology 2 weeks after injury suggested that MINO plus NAC preserved white, but not grey matter, since lesion volume was unaffected, yet myelin loss was attenuated. When dosed 3 hours before injury, MINO plus NAC as single drugs had no effect on interleukin-1 formation; together they synergistically lowered interleukin-1 levels. This effect on interleukin-1 was not observed when th

Measurement of the polar-angle distribution of leptons from W boson decay as a function of the W transverse momentum in proton-antiproton collisions at sqrt{s}=1.8 TeV

We present a measurement of the coefficient alpha_2 of the leptonic polar-angle distribution from W boson decays, as a function of the W transverse momentum. The measurement uses an 80+/-4 pb^{-1} sample of proton-antiproton collisions at sqrt{s}=1.8 TeV collected by the CDF detector and includes data from both the W->e+nu and W->mu+nu decay channels. We fit the W boson transverse mass distribution to a set of templates from a Monte Carlo event generator and detector simulation in several ranges of the W transverse momentum. The measurement agrees with the Standard Model expectation, whereby the ratio of longitudinally to transversely polarized W bosons, in the Collins-Soper W rest frame, increases with the W transverse momentum at a rate of approximately 15% per 10 GeV/c.Comment: 47 pages, 16 figures, submitted to Physical Review