Level-spacing distributions and the Airy kernel

Scaling level-spacing distribution functions in the bulk of the spectrum in random matrix models ofN×N hermitian matrices and then going to the limitN leads to the Fredholm determinant of thesine kernel sin(x–y)/(x–y). Similarly a scaling limit at the edge of the spectrum leads to theAiry kernel [Ai(x)Ai(y)–Ai(x)Ai(y)]/(x–y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, Môri, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlevé transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.     

On orthogonal and symplectic matrix ensembles

The focus of this paper is on the probability,E (O;J), that a setJ consisting of a finite union of intervals contains no eigenvalues for the finiteN Gaussian Orthogonal (=1) and Gaussian Symplectic (=4) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary (=2) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painlevé II function.     

Enhancement of the cubic molecular hyperpolarizabilities of nickel benzendithiol by donor and acceptor groups

In this paper the enhancement of the values of the cubic molecular hyperpolarizability of the nickel benzendithiol system, resulting from the selective addition of acceptor and donor groups to the ligand structure, is presented. The polarization of the molecular electronic structure is calculated by the finite field method. Significant increases in the values of are obtained from bridging two p-nitroaniline molecules with the square planar nickel structure. These results show that structures with significant values of x ⁽²⁾ that are derived from the inclusion of donor and acceptor groups might also be included into the ligands of the nickel dithiolene complex to give rise to materials with enhanced values of x ⁽³⁾.     

Newton's method and periodic solutions of nonlinear wave equations

We prove the existence of periodic solutions of the nonlinear wave equation satisfying either Dirichlet or periodic boundary conditions on the interval [O, π]. The coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation. Using a version of Newton's method, we show that this equation has solutions provided the nonlinearity g(x, u) satisfies certain generic conditions of nonresonance and genuine nonlinearity. © 1993 John Wiley & Sons, Inc.     

Development of a custom high-throughput preparative liquid chromatography/mass spectrometer platform for the preparative purification and analytical analysis of compound libraries

Solution-phase parallel synthesis has had a profound impact on the speed of compound synthesis delivering relatively pure compounds (>80%) in short order. However, to develop structure activity relationships (SAR) for a compound series, each library member should preferably be >95% pure. Historically, achieving and quantifying such high-purity criteria for each library member proved to be the slow step for most lead discovery groups. To address this issue, significant modifications have been made to a commercial Agilent preparative LC/MS system to allow for the general mass-guided purification of diverse compound libraries. The custom modifications include (1) the "DMSO slug" approach for the purification of samples with poor solubility; (2) an active splitter to reduce system back-pressure, reduce the delay volume, and allow for a variable split ratio; (3) a sample loading pump for the quick purification of large, dilute samples; (4) a preparative column-selection valve to quickly change column selectivity or sample loading; and (5) an analytical injector with a separate flow path for crude reaction or fraction analyses.     

Combustion Analyses of Air-Sensitive Complexes

The combustion method described in this paper makes possible the determination of carbon, hydrogen, and nitrogen in highly air-sensitive compounds. A rigorous procedure was developed to ensure isolation from air when these compounds are analyzed. (π-C₅H₅)₂Zr(CH₃)₂, which is known to hydrolyze in air, was used to assess the method′s capability of isolating the compounds from air prior to combustion. The procedural changes described resulted in improved precisions for the determination of carbon, hydrogen, and nitrogen for (π-C₅H₅)₂Zr(CH₃)₂ by at least a factor of 4.     

Backbone cleavages of [M - H](-) anions of peptides. Cyclisation of citropin 1 peptides involving reactions between the C-terminal [CONH](-) residue and backbone amide carbonyl groups. A new type of beta cleavage: a joint experimental and theoretical study

This paper reports the study of backbone cleavages in the collision-induced negative-ion mass spectra of the [M - H](-) anions of some synthetic modifications of the bioactive amphibian peptide citropin 1 (GLFDVIKKVASVIGGL-NH(2)). The peptides chosen for study contain no amino acid residues which could effect facile side-chain cleavage, i.e. Ser (-CH(2)O, side-chain cleavage) and Asp (-H(2)O) are replaced by Ala or Lys. We expected that such peptides should exhibit standard and pronounced peaks due to alpha cleavage ions (and to a lesser extent beta cleavage ions) in their collision-induced negative-ion spectra. This expectation was realised, but the spectra also contained peaks formed by a new series of cleavage anions. These are produced following cyclisation of the C-terminal CONH(-) moiety at carbonyl functions of amide groups along the peptide backbone; effectively transferring the NH of the C-terminal CONH(-) group to other amino acid residues. We have called the product anions of these processes beta' ions, in order to distinguish them from standard beta ions. Some beta' ions also fragment directly to some other beta' ions of smaller mass. The reaction coordinates of alpha,beta and beta' backbone processes have been calculated at the HF/6-31G*//AM1 level theory for simple model systems. The initial cyclisation step of the beta' sequence is barrierless and exothermic. Subsequent steps have a maximum barrier of +40 kcal mol(-1), with the overall reaction being endothermic by some 30 kcal mol(-1) at the level of theory used. These calculations take no account of the complexity of the conformationally flexible peptide system, and it is surprising that each of the two reacting centres can 'find' each other in such a large system.     

(Fe(III)(OH(2))(2))(3)(A-alpha-PW(9)O(34))(2)](9-) on cationic silica nanoparticles,a new type of material and efficient heterogeneous catalyst for aerobic oxidations

Polyoxometalates (POMs) electrostatically bind to silica nanoparticles coated with cationic aluminum oxide "(Si/AlO2)n+" to form a new type of material (the anionic POMs replace Cl- counterions associated with the cationic surface sites). Association of a new approximately D3h POM of formula [(FeIII(OH2)2)3(A-alpha-PW9O34)2]9- (1) with the cationic nanoparticles (to form "K81/(Si/AlO2)") was studied in detail. Elemental analysis, particle sizes from both laser light scattering and TEM before and after association of 1, the size of 1 from X-ray crystallography, and other methods provide mutually consistent data that indicate about 58 K8[(FeIII(OH2)2)3(A-alpha-PW9O34)2]- monoanions associate with the average nanoparticle (diameter of the K81/(Si/AlO2) product = approximately 17 nm). While heterogeneity of the cationic sites and roughness of the (Si/AlO2)n+ surfaces make the associated POMs structurally nonuniform, the equivalent of approximately 1 monolayer of 1 is present in K81/(Si/AlO2). Remarkably, while 1, the precursor (Si/AlO2)n+, and the components of 1, each alone, are inactive as catalysts for O2/air-based oxidation of sulfides or aldehydes in solution, K81/(Si/AlO2) is an active catalyst for both reactions (facile reaction with air at low temperature).     

Novel one-dimensional structures and solution behaviour of copper(II) bromide and chloride complexes of a new pentapyridyldiamine ligand

Copper(II) bromide and chloride complexes of the new heptadentate ligand 2,6-bis(bis(2-pyridylmethyl)amino)methylpyridine (L) have been prepared. For the bromide complexes, chains of novel, approximately C2-symmetric, chiral [Cu2(L)Br2]2+ 'wedge-shaped' tectons are found. The links between the dicopper tectons and the overall chirality and packing of the chains are dictated by the bromide ion content, not the counter anion. In contrast, the chloride complexes exhibit linked asymmetric [Cu2(L)Cl3]+ tectons with distinct N3CuCl2 and N4CuCl2 centres in the solid. The overall structures of the dicopper bromide and chloride units persist in solution irrespective of the halide. The redox chemistry of the various species is also described.     

Structural studies of some body-centered cubic phases of mixed oxides involving Bi2O3: The structures of Bi25FeO40 and Bi38ZnO60

The structures of the compounds initially reported to be 7·Bi₂O₃·ZnO and 96·Bi₂O₃·4Fe₂O₃, have been determined by X ray methods. Three dimensional, absorption corrected diffractometer data were used and atomic parameters were refined by least-squares procedures. The structures are isomorphous, cubic witha = 10.194(3)and10.179(3)A?, respectively, and space group I23. Each Bi³⁺ ion is surrounded by five oxygen atoms that form an incomplete octahedral arrangement with BiO distances ranging from 2.07–2.60A?. The6s²inert electron pair completes the octahedron. The Bi³⁺ ions are vibrating anisotropically. Tetrahedral sites in the structures contain 61 and 46 electrons, respectively. These values are consistent with a statistical distribution of Zn²⁺ and Bi⁵⁺ ions or Fe³⁺ and Bi⁵⁺ ions on these sites. Molar ratios are derived that agree with the observed distributions of electron density and give rise to perfectly stoichiometric systems, devoid of cationic or anionic vacancies. The compositions studied correspond to Bi³⁺₂₄Bi⁵⁺Fe³⁺O₄₀ and Bi³⁺₃₆Bi⁵⁺₂ZnO₆₀ and they are optical enantiomorphs.It is proposed that a reduction in the percentage composition of Bi₂O₅ leads to metastable phases, in which all atomic positions remain fully occupied but some tetrahedral sites contain Bi³⁺ ions. The end product of the series is γ-Bi₂O₃ in which 50% of these sites contain Bi³⁺ and the remainder Bi⁵⁺ ions. We believe that γ-Bi₂O₃ is Bi³⁺₂₅Bi⁵⁺O₄₀.