Partition Lattice q-Analogs Related to q-Stirling Numbers

We construct a family of partially ordered sets (posets) that are q-analogs of the set partition lattice. They are different from the q-analogs proposed by Dowling [5]. One of the important features of these posets is that their Whitney numbers of the first and second kind are just the q-Stirling numbers of the first and second kind, respectively. One member of this family [4] can be constructed using an interpretation of Milne [9] for S[n, k] as sequences of lines in a vector space over the Galois field F _q. Another member is constructed so as to mirror the partial order in the subspace lattice.     

Heat transfer in the plastic phases I and II of cyclopentane

Thermal conductivity of solid cyclopentane C₅H₁₀ has been measured at isochoric conditions in the plastic phases I and II for samples of different densities. Isochoric thermal conductivity is nearly constant in phase II and increases with temperature in phase I. Such behaviour is attributed to weakening of the translational orientational coupling which, in turn, leads to a decrease of phonon scattering on rotational excitations. The experimental data are described in terms of a modified Debye model of thermal conductivity with allowance for heat transfer by both low-frequency phonons and diffusive modes.     

The pentacene films on the glass/ITO surface: structure and optical properties

Abstract

The structure, morphology and optical transmittance spectra of pentacene films on the (glass/ITO) surface were studied. The films were grown by two methods - the thermal vacuum deposition (TVD) and pulsed laser deposition (PLD). The electron diffraction pattern from thermally deposited pentacene films confirms their polycrystalline structure while the diffraction pattern of PLD-coated layers has a diffusion character. The results obtained showed that layers deposited by the TVD method has an optical spectrum that is characteristic for the pentacene film in contrary to the layers deposited by the PLD method. It is found a sensitivity of the optical transmittance of pentacene films to the ammonia action, which may be used for development the optical gas sensor.     

Bijective proofs of two broken circuit theorems

We prove, by means of explicit bijections, theorems of Whitney and Stanley that express the coefficients of the chromatic polynomial of a graph G and the number of acyclic orientations of G in terms of numbers of sets of edges that contain no broken circuits of G.     

Enumeration of trees by inversions

Mallows and Riordan “The Inversion Enumerator for Labeled Trees,” Bulletin of the American Mathematics Society, vol. 74 [1968] pp. 92-94) first defined the inversion polynomial, J_n(q) for trees with n vertices and found its generating function. In the present work, we define inversion polynomials for ordered, plane, and cyclic trees, and find their values at q = 0, ± 1. Our techniques involve the use of generating functions (including Lagrange inversion), hypergeometric series, and binomial coefficient identities, induction, and bijections. We also derive asymptotic formulae for those results for which we do not have a closed form. © 1995 John Wiley & Sons, Inc.     

Congruences via abelian groups

Given a group G acting on a set S, Möbius inversion over the lattice of subgroups can be used to obtain congruences relating the number of elements of S stabilized by each subgroup. By taking S to be a set of subsets, partitions, or permutations congruences for binomial and multinomial coefficients, Stirling numbers of both kinds, and various other combinatorial sequences are derived. Congruences for different moduli are obtained by varying the order of G.     

On the 1/3–2/3 Conjecture

Let (P, ≤) be a finite poset (partially ordered set), where P has cardinality n. Consider linear extensions of P as permutations x₁x₂?x_n in one-line notation. For distinct elements x, y ∈ P, we define ?(x ? y) to be the proportion of linear extensions of P in which x comes before y. For \(0\leq \alpha \leq \frac {1}{2}\), we say (x, y) is an α-balanced pair if α ≤ ?(x ? y) ≤?1 ? α. The 1/3–2/3 Conjecture states that every finite partially ordered set which is not a chain has a 1/3-balanced pair. We make progress on this conjecture by showing that it holds for certain families of posets. These include lattices such as the Boolean, set partition, and subspace lattices; partial orders that arise from a Young diagram; and some partial orders of dimension 2. We also consider various posets which satisfy the stronger condition of having a 1/2-balanced pair. For example, this happens when the poset has an automorphism with a cycle of length 2. Various questions for future research are posed.     

Peroxisome proliferator-activated receptor alpha antagonism inhibits hepatitis C virus replication

Hepatitis C virus (HCV) is a global health problem and a leading cause of liver disease. Here, we demonstrate that the replication of HCV replicon RNA in Huh-7 cells is inhibited by a peroxisome proliferator-activated receptor (PPAR) antagonist, 2-chloro-5-nitro-N-(pyridyl)benzamide (BA). Downregulation of PPARgamma with RNA interference approaches had no effect on HCV replication in Huh-7 cells, whereas PPARalpha downregulation inhibited HCV replication. Fluorescence and coherent anti-Stokes Raman scattering (CARS) microscopy demonstrate a clear buildup of lipids upon treatment with BA. These observations are consistent with the misregulation of lipid metabolism, phospholipid secretion, cholesterol catabolism, and triglyceride clearance events associated with the inhibition of PPARalpha. The inhibition of HCV replication by BA may result from disrupting lipidation of host proteins associated with the HCV replication complex or, more generally, by disrupting the membranous web where HCV replicates.