Euler's partition theorem and the combinatorics of ℓ-sequences

Euler's partition theorem states that the number of partitions of an integer N into odd parts is equal to the number of partitions of N in which the ratio of successive parts is greater than 1. It was shown by Bousquet-Mélou and Eriksson in [M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions II, Ramanujan J. 1 (2) (1997) 165–185] that a similar result holds when “odd parts” is replaced by “parts that are sums of successive terms of an ℓ-sequence” and the ratio “1” is replaced by a root of the characteristic polynomial of the ℓ-sequence. This generalization of Euler's theorem is intrinsically different from the many others that have appeared, as it involves a family of partitions constrained by the ratio of successive parts.In this paper, we provide a surprisingly simple bijection for this result, a question suggested by Richard Stanley. In fact, we give a parametrized family of bijections, that include, as special cases, Sylvester's bijection and a bijection for the lecture hall theorem. We introduce Sylvester diagrams as a way to visualize these bijections and deduce their properties.In proving the bijections, we uncover the intrinsic role played by the combinatorics of ℓ-sequences and use this structure to give a combinatorial characterization of the partitions defined by the ratio constraint. Several open questions suggested by this work are described.     

Influence of poly(ethylene glycol) on the partition of a charged spin probe

The influence of poly(ethylene glycol) on the partition of a charged long chain spin probe between membranes and an external phase is studied. The partition coefficient is derived from the ESR spectra. Membranes of different properties are used (egg lecithin liposomes, erythrocytes) to differentiate between the influence of the external phase and the specific properties of the membrane.The partition coefficient is decreased in an exponential manner on increasing the PEG concentration, which indicates a lowering of the thermodynamic stability of the membranes. The determination of the change in the difference of the chemical potential is dependent on the PEG concentration.The membrane destabilization induced by PEG is caused in an indirect manner by a change of the chemical potential difference as result of the changed water structure and the osmotic pressure, surface tension and hydration of the membrane. This destabilization could be connected with the high fusogenic activity of PEG.     

Colorful flowers

For a set A let k[A] denote the family of all k-element subsets of A. A function f:k[A]→C is a local coloring if it maps disjoint sets of A into different elements of C. A family F⊆k[A] is called a flower if there exists E∈[A]^k−1 so that |F∩F^′|=E for all F,F^′∈F, F≠F^′. A flower is said to be colorful if f(F)≠f(F^′) for any two F,F^′∈F. In the paper we find the smallest cardinal γ such that there exists a local coloring of k[A] containing no colorful flower of size γ. As a consequence we answer a question raised by Pelant, Holický and Kalenda. We also discuss a few results and conjectures concerning a generalization of this problem.     

On a cycle partition problem

Let G be any graph and let c(G) denote the circumference of G. We conjecture that for every pair c₁,c₂ of positive integers satisfying c₁+c₂=c(G), the vertex set of G admits a partition into two sets V₁ and V₂, such that V_i induces a graph of circumference at most c_i, i=1,2. We establish various results in support of the conjecture; e.g. it is observed that planar graphs, claw-free graphs, certain important classes of perfect graphs, and graphs without too many intersecting long cycles, satisfy the conjecture.This work is inspired by a well-known, long-standing, analogous conjecture involving paths.     

The intrinsic XFEM for two‐fluid flows

In two‐fluid flows, jumps and/or kinks along the interfaces are present in the resulting velocity and pressure fields. Standard methods require mesh manipulations with the aim that either element edges align with the interfaces or that the mesh is sufficiently refined near the interfaces. In contrast, enriched methods, such as the extended finite element method (XFEM), enable the representation of arbitrary jumps and kinks inside elements. Thereby, optimal convergence can be achieved for two‐fluid flows with meshes that remain fixed throughout the simulation. In the intrinsic XFEM, in contrast to other enriched methods, no more unknowns are present in the approximation than in a standard finite element approximation. In this work, the intrinsic XFEM is employed for the simulation of incompressible two‐fluid flows. Numerical results are shown for a number of test cases and prove the success of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Shadows and intersections: Stability and new proofs

We give a short new proof of a version of the Kruskal-Katona theorem due to Lovász. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation of Lovász's theorem that answers a question of Frankl and Tokushige.     

Physicochemical investigation of sulfonamide-derived compounds interacting with conventional cationic surfactants in aqueous media

The sulfonamide core in compounds is effective synthons, which offer exciting perspectives in chemotherapeutic and pharmacological research. In this study, we investigated the molecular interaction of potent sulfonamide derivatives (SDs), 2-benzenesulfonamido-3-methylbutyric acid (BSB) and tetraaquabis{3-methyl-2-[(phenylsulfonyl)amino]butanoate}copper(II) (Cu-BSB), with cationic surfactants, cetyltrimethyl ammonium bromide (CTAB) and ethylhexadecyl dimethyl ammonium bromide (EHDAB), using UV-visible spectroscopy and steady-state fluorescence measurements. Various mathematical models were applied to quantify the effect of cationic surfactants on physicochemical characteristics of BSB and Cu-BSB. These interactions were confirmed by estimating the partition coefficient (K_x), binding capacities (K_b), Stern-Volmer quenching constant (K_sv), and related Gibbs free energies (ΔG_b). The in-depth mechanism revealed that the binding mode in SD–surfactants combinational system is spontaneous and quenching exists in static mode initiated by ground-state complex formation. We believe that the true knowledge of host–guest interaction mechanisms concerning model membrane with entrapped moiety can help in better understanding of molecular recognition in related phospholipid membrane models.     

Preparative separation of major xanthones from mangosteen pericarp using high‐performance centrifugal partition chromatography

Mangosteen fruit pericarp (MFP) is a rich source of xanthones, which has shown remarkable pharmacological activities. To isolate xanthones, previous methods included labor intensive and time‐consuming solid‐phase extractions (Sephadex LH20, silica gel) and sequential solvent extraction. In this study, major xanthones (α‐ and γ‐mangostins) in MFP were isolated at high purity in one step utilizing high‐performance centrifugal partition chromatography with solvent system composed of petroleum ether, ethyl acetate, methanol and water (10:5:5:1). In one run, 200 mg crude extract of MFP was injected and 55.4 mg α‐mangostin and 12.4 mg γ‐mangostin were obtained with the purity of 93.6 and 98.4%, respectively. The yields of them were 86.3 and 76.3%, respectively. As α‐ and γ‐mangostins are reported to show potent antioxidant, anti‐inflammatory and anticancer activities, this method can be used for the large‐scale production of them for future in vitro and in vivo biological studies.