On the independence complex of square grids

The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendley et al., that for some rectangular grids, with toric boundary conditions, the alternating number of independent sets is extremely simple. More precisely, under a coprimality condition on the sides of the rectangle, the number of independent sets of even and odd cardinality always differ by 1. In physics terms, this means looking at the hard-particle model on these grids at activity −1. This conjecture was recently proved by Jonsson. Here we produce other families of grid graphs, with open or cylindric boundary conditions, for which similar properties hold without any size restriction: the number of independent sets of even and odd cardinality always differ by 0, ±1, or, in the cylindric case, by some power of 2. We show that these results reflect a stronger property of the independence complexes of our graphs. We determine the homotopy type of these complexes using Forman’s discrete Morse theory. We find that these complexes are either contractible, or homotopic to a sphere, or, in the cylindric case, to a wedge of spheres. Finally, we use our enumerative results to determine the spectra of certain transfer matrices describing the hard-particle model on our graphs at activity −1. These results parallel certain conjectures of Fendley et al., proved by Jonsson in the toric case.     

Small cliques in random graphs

For d ≥ 1, a d-clique in a graph G is a complete d-vertex subgraph not contained in any larger complete subgraph of G. We investigate the limit distribution of the number of d-cliques in the binomial random graph G(n, p), p = p(n), n→∞.     

Aldehyde

Ohne Zusammenfassung     

Asymptotic bias of some election methods

Consider an election where N seats are distributed among parties with proportions p ₁,…,p _m of the votes. We study, for the common divisor and quota methods, the asymptotic distribution, and in particular the mean, of the seat excess of a party, i.e. the difference between the number of seats given to the party and the (real) number Np _i that yields exact proportionality. Our approach is to keep p ₁,…,p _m fixed and let N→∞, with N random in a suitable way. In particular, we give formulas showing the bias favouring large or small parties for the different election methods.     

Dynamic adjustment: an electoral method for relaxed double proportionality

We describe an electoral system for distributing seats in a parliament. It gives proportionality for the political parties and close to proportionality for constituencies. The system suggested here is a version of the system used in Sweden and other Nordic countries with permanent seats in each constituency and adjustment seats to give proportionality on the national level. In the national election of 2010 the current Swedish system failed to give proportionality between parties. We examine here one possible cure for this unwanted behavior. The main difference compared to the current Swedish system is that the number of adjustment seats is not fixed, but rather dynamically determined to be as low as possible and still insure proportionality between parties.     

Spread‐out percolation in ℝd

Fix d ≥ 2, and let X be either ℤ^d or the points of a Poisson process in ℝ^d of intensity 1. Given parameters r and p, join each pair of points of X within distance r independently with probability p. This is the simplest case of a “spread‐out” percolation model studied by Penrose [Ann Appl Probab 3 (1993) 253–276], who showed that, as r → ∞, the average degree of the corresponding random graph at the percolation threshold tends to 1, i.e., the percolation threshold and the threshold for criticality of the naturally associated branching process approach one another. Here we show that this result follows immediately from of a general result of [3] on inhomogeneous random graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Pionic Fusion Study of the halo analogue state6Li

The isobaric analogue state of the ground state of the halo nucleus⁶He,⁶Li* has been studied in a pionic fusion experiment at the CELSIUS storage ring facility in Uppsala, Sweden. This was the first of two proposed experiments with the aim to study the high momentum part of the wave functions of the halo nuclei⁶He and⁶Li (0⁺, T=1). The Li nuclei were produced in inverse kinematics reactions with alpha particles incident on a deuterium cluster jet target. The⁶Li ions were detected in a zero-degree spectrometer situated in the fourth quadrant of the CELSIUS ring. The measurement was done for three different beam energies corresponding to 5.4, 2.2 and 1.5 MeV above threshold in the c.m. The cross-sections slowly increase with beam energy from 95 nb at the lowest energy to about 250 nb at 5.4 MeV above threshold.     

Enzymes immobilized in mesoporous silica: A physical–chemical perspective

Mesoporous materials as support for immobilized enzymes have been explored extensively during the last two decades, primarily not only for biocatalysis applications, but also for biosensing, biofuels and enzyme-controlled drug delivery. The activity of the immobilized enzymes inside the pores is often different compared to that of the free enzymes, and an important challenge is to understand how the immobilization affects the enzymes in order to design immobilization conditions that lead to optimal enzyme activity. This review summarizes methods that can be used to understand how material properties can be linked to changes in enzyme activity. Real-time monitoring of the immobilization process and techniques that demonstrate that the enzymes are located inside the pores is discussed by contrasting them to the common practice of indirectly measuring the depletion of the protein concentration or enzyme activity in the surrounding bulk phase. We propose that pore filling (pore volume fraction occupied by proteins) is the best standard for comparing the amount of immobilized enzymes at the molecular level, and present equations to calculate pore filling from the more commonly reported immobilized mass. Methods to detect changes in enzyme structure upon immobilization and to study the microenvironment inside the pores are discussed in detail. Combining the knowledge generated from these methodologies should aid in rationally designing biocatalyst based on enzymes immobilized in mesoporous materials.     

Ionic-charge dependence of the intermolecular coulombic decay time scale for aqueous ions probed by the core-hole clock

Auger electron spectroscopy combined with theoretical calculations has been applied to investigate the decay of the Ca 2p core hole of aqueous Ca(2+). Beyond the localized two-hole final states on the calcium ion, originating from a normal Auger process, we have further identified the final states delocalized between the calcium ion and its water surroundings and produced by core level intermolecular Coulombic decay (ICD) processes. By applying the core-hole clock method, the time scale of the core level ICD was determined to be 33 ± 1 fs for the 2p core hole of the aqueous Ca(2+). The comparison of this time constant to those associated with the aqueous K(+), Na(+), Mg(2+), and Al(3+) ions reveals differences of 1 and up to 2 orders of magnitude. Such large variations in the characteristic time scales of the core level ICD processes is qualitatively explained by different internal decay mechanisms in different ions as well as by different ion-solvent distances and interactions.