Greene-Kleitman's theorem for infinite posets

It is proved that if (P) is a poset with no infinite chain and k is a positive integer, then there exist a partition of P into disjoint chains C _i and disjoint antichains A ₁, A ₂, ..., A _k, such that each chain C _i meets min (k, |C _i|) antichains A _j. We make a dual conjecture, for which the case k=1 is: if (P) is a poset with no infinite antichain, then there exist a partition of P into antichains A _i and a chain C meeting all A _i. This conjecture is proved when the maximal size of an antichain in P is 2.     

Complexes containing the unit Rh2 P PM (M = Cr or Mn, P P = bis(diphenylphosphino) alkane); the independence of the metal centres

Treatment of (η⁵-C₅H₅)₂Rh₂(CO)η¹-Ph₂P(CH₂)_n PPh₂(μ-η¹:η¹-CF₃C₂CF₃) (I) with (η⁵-CH₃C₅H₄)Mn(CO)₂(thf) or Cr(CO)₅(thf) gives the hetero-trinuclear products (η-C₅H₅)₂Rh₂(CO)(μ-CF₃C₂CF₃)μ:η¹:η¹-Ph₂P(CH₂)_nPPh₂(η-CH₃C₅H₃C₄)Mn(CO₂) (II, n = 1–4) and (η⁵-C₅H₅)₂Rh₂(CO)(μ-CF₃C₂CF₃)μ:η¹:η¹-Ph₂P(CH₂)nPPh₂Cr(CO)₅ (IV, n = 1–4) in good yields. In these products, the configuration of the CO and bisphosphine units on the Rh-Rh bond is trans. Related reactions between (η⁵:η⁵-C₅H₄CH₂C₅H₄)Rh₂(CO)η¹-Ph₂P(CH₂)nPPh₂(μ-η¹:η¹- (V) and the same solvated manganese and chromium complexes give (η⁵:η⁵-C₅H₄CH₂C₅H₄)Rh₂(CO)(μ-CF₃C₂CF₃)μ:η¹:η¹-Ph₂P(CH₂) (VI, n = 1, 2 or 4) and (η⁵:η⁵-C₅H₄CH₂C₅H₄)Rh₂(CO)(μ-CF₃C₂CF₃)μ:η¹:η¹-Ph₂P(CH₂) (VIII). The complexes (VI) and (VIII) have a mutually cis arrangement of CO and the bisphosphine on the Rh-Rh bond. Attempts to induce the complexes (IV), (V), (VI), and (VIII) to form clusters by loss of CO and Rh-M bond formation were not successful. Treatment with trimethylamine oxide or sunlight irradiation generally resulted in loss of the hetero-metal and formation of the dirhodium phosphine oxide complexes (III, n = 2 or 4) and (VII, n = 2, 3).     

New coordination modes of iminodiacetamide type ligands in palladium(II) complexes: crystallographic and DFT studies

The reactions of N-alkyliminodiacetamide derivatives, namely N-ethyliminodiacetamide (CH₃CH₂N(CH₂CONH₂)₂; Etimda) and N-isopropyliminodiacetamide (CH₃)₂CHN(CH₂CONH₂)₂; i-Primda), with sodium tetrachloropalladate(II) in aqueous solutions were investigated. Three new palladium(II) complexes, [Pd(Etimda_?H)₂]?2H₂O (1), [Pd(i-Primda_?H)₂]?2H₂O (2) and [PdCl₂(i-Primda)] (3), were obtained and characterized by X-ray structural analysis, infrared spectroscopy and thermal analysis (TGA). The square planar coordination environments around the palladium(II) ions in complexes 1 and 2 consist of two N,N′-bidentate N-alkyliminodiacetamidato ligands, with imino N atoms in trans-position. The complex 3 also exhibits a square planar coordination environment around Pd(II), but with two chloride ions and one neutral N-isopropyliminodiacetamide ligand bound in an N,O′-bidentate coordination mode. The described coordination modes, as well as the presence of deprotonated amide groups in ligands in 1 and 2, are found for the first time in palladium(II) complexes with iminodiacetamide type ligands. The molecular geometries and infrared spectra of these three complexes were also modelled using DFT calculations, at the BP86-D3/def2-TZVPP/PCM level of theory. The RMSD values suggest a good agreement of the calculated and experimental geometries. A QTAIM analysis suggests a qualitative correlation between bond lengths and energy densities, also supported by an NBO analysis. The dimer interaction energy between complex units was estimated at about ?15 kcal/mol for all complexes.     

On a theorem of lovász on covers inr-partite hypergraphs

A theorem of Lovász asserts that (H)/*(H)r/2 for everyr-partite hypergraphH (where and * denote the covering number and fractional covering number respectively). Here it is shown that the same upper bound is valid for a more general class of hypergraphs: those which admit a partition (V ₁, ...,V _k ) of the vertex set and a partitionp ₁+...+p _k ofr such that |eV _i |p _i r/2 for every edgee and every 1ik. Moreover, strict inequality holds whenr>2, and in this form the bound is tight. The investigation of the ratio /* is extended to some other classes of hypergraphs, defined by conditions of similar flavour. Upper bounds on this ratio are obtained fork-colourable, stronglyk-colourable and (what we call)k-partitionable hypergraphs.Supported by grant HL28438 at MIPG, University of Pennsylvania, and by the fund for the promotion of research at the Technion.This author's research was supported by the fund for the promotion of research at the Technion.     

Strictly positive definite functions on spheres in Euclidean spaces

In this paper we study strictly positive definite functions on the unit sphere of the -dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are actually strictly positive definite. The study of this problem was initiated recently by Xu and Cheney (Proc. Amer. Math. Soc. 116 (1992), 977--981), where certain sufficient conditions were derived. A new approach, which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres, is presented here. The relevant interpolation problem is subsequently analyzed by three different complementary methods. The first is based on the de Boor-Ron general ``least solution for the multivariate polynomial interpolation problem'. The second, which is suitable only for , is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials, and reduces the problem to the structure of the integer zeros of bounded univariate exponentials. Finally, the last method invokes the realization of harmonic polynomials as the polynomial kernel of the Laplacian, thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels.

     

Patterns and Structures in Disk Packings

Using a computational procedure that imitates tightening of an assembly of billiard balls, we have generated a number of packings of n equal and non-equal disks in regions of various shapes. Our experiments are of three major types. In the first type, the values of n are in thousands, the initial disk configuration is random and a priori one expects the generated packings to be random. In fact, the packings turn out to display non-random geometric patterns and regular features, including polycrystalline textures with "rattlers" typically trapped along the grain boundaries. An experiment of the second type begins with a known or conjectured optimal disk packing configuration, which is then "frustrated" by a small perturbation such as variation of the boundary shape or a relative increase of the size of a selected disk with respect to the sizes of the other disks. We present such frustrated packings for both large n (~ 10, 000) and small n (~ 50 to 200). Motivated by applications in material science and physics, the first and second type of experiments are performed for boundary shapes rarely discussed in the literature on dense packings: torus, a strip cut from a cylinder, a regular hexagon with periodic boundaries. Experiments of the third type involve the shapes popular among mathematicians: circles, squares, and equilateral triangles the boundaries of which are hard reflecting walls. The values of n in these experiments vary from several tens to few hundreds. Here the obtained configurations could be considered as candidates for the densest packings, rather than random ones. Some of these conjecturally optimal packings look regular and the regularity often extends across different values of n. Specifically, as n takes on an increasing sequence of values, n = n(1), n(2), ...n(k), ..., the packings follow a well-defined pattern. This phenomenon is especially striking for packings in equilateral triangles, where (as far as we can tell from our finite computational experiments), not only are there an infinite number of different patterns, each with its own different sequence n(1), n(2), ...n(k), ..., but many of these sequences seem to continue indefinitely. For other shapes, notably squares and circles, the patterns either cease to be optimal or even cease to exist (as packings of non-overlapping disks) above some threshold value n(k0) (depending on the pattern). In these cases, we try to identify the values of n(k0).     

Experimental investigations in combining primal dual interior point method and simplex based LP solvers

The use of a primal dual interior point method (PD) based optimizer as a robust linear programming (LP) solver is now well established. Instead of replacing the sparse simplex algorithm (SSX), the PD is increasingly seen as complementing it. The progress of PD iterations is not hindered by the degeneracy or stalling problem of SSX, indeed it reaches the near optimum solution very quickly. The SSX algorithm, in contrast, is not affected by the numeral instabilities which slow down the convergence of the PD near the optimal face. If the solution to the LP problem is non-unique, the PD algorithm converges to an interior point of the solution set while the SSX algorithm finds an extreme point solution. To take advantage of the attractive properties of both the PD and the SSX, we have designed a hybrid framework whereby crossover from PD to SSX can take place at any stage of the PD optimization run. The crossover to SSX involves the partition of the PD solution set to active and dormant variables. In this paper we examine the practical difficulties in partitioning the solution set, we discuss the reliability of predicting the solution set partition before optimality is reached and report the results of combining exact and inexact prediction with SSX basis recovery.     

Simultaneous determination of carbohydrates and simmondsins in jojoba seed meal (Simmondsia chinensis) by gas chromatography

Separate methods for the analyses of soluble carbohydrates in different plants and simmondsins in jojoba seed meal are described. A reliable gas chromatographic procedure for the simultaneous quantification of D-pinitol, myo-inositoL sucrose, 5-alpha-D-galactopyranosyl-D-pinitol. 2-alpha-D-galactopyranosyl-D-pinitol, simmondsin, 4-demethylsimmondsin, 5-demethylsimmondsin and 4,5-didemethylsimmondsin as trimethylsilyl derivatives in jojoba seed meal has been developed. The study of different extraction mixtures allowed for the quantitative recovery of the 9 analytes by a mixture of methanol-water (80:20, v/v) in the concentration range between 0.1 and 4%. Comparison of the separation parameters on three different capillary stationary phases with MS detection allowed for the choice of the optimal gas chromatographic conditions for baseline separation of the analytes.     

Mass spectrometry assisted assignment of NMR resonances in 15N labeled proteins

Application of nuclear magnetic resonance (NMR) methods for the structural characterization to larger and more complex protein systems can be facilitated through the development of new methods for resonance assignment. Here, a novel approach that relies on integration of NMR and mass spectrometry (MS) methods is explored. The approach relies on the fact that both NMR and MS are able to monitor rates of exchange of amide protons for water deuterons. Correlating the rates can connect cross-peak positions from NMR data with fragment masses from MS data to support sequential assignment. The example provided is for a small model protein, ubiquitin, but the potential for application to large, more difficult to express proteins is clear.     

Simultaneous measurement of metabolic stability and metabolite identification of 7-methoxymethylthiazolo[3,2-a]pyrimidin-5-one derivatives in human liver microsomes using liquid chromatography/ion-trap mass spectrometry

A liquid chromatography/mass spectrometry (LC/MS) method using an atmospheric pressure chemical ionisation source was used to measure the metabolic stability and metabolite identification of 7-methoxymethylthiazolo[3,2-a]pyrimidin-5-one derivative (1) in human liver microsomes. After 15 min incubation with human liver microsomes, compound 1 exhibited metabolic turnover of 44%. Data-dependent tandem mass spectrometry (MS/MS) scanning was used to generate product ion spectra from the protonated ions of the compound and its metabolites. An unusual metabolite at m/z 407 corresponding to the [M-24+H]+ ion was identified for compound 1. Interestingly, the formation of the [M-24+H]+ ion was not observed in the analogues wherein the fused thieno double bond was substituted (2) and the thieno group replaced by a fused benzo derivative (3). Compounds 2 and 3 exhibited metabolic turnovers of 24 and 30%, yielding oxidative metabolites corresponding to [M+16] and [M+32]+, respectively. Based on these facts the mechanism for [M-24]+ formation in compound 1 through an initial epoxide formation on the double bond of the fused thieno ring followed by hydrolytic ring opening and deacylation is envisaged.