Universality at the edge of the spectrum for unitary,orthogonal, and symplectic ensembles of random matrices

We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1), and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e^?V(x) where V is a polynomial, V(x) = κ_2mx^2m + · · ·, κ_2m > 0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2 and 1.4 below. For the same class of weights, a proof of universality in the bulk of the spectrum is given in [12] for the unitary ensembles and in [9] for the orthogonal and symplectic ensembles. Our starting point in the unitary case is [12], and for the orthogonal and symplectic cases we rely on our recent work [9], which in turn depends on the earlier work of Widom [46] and Tracy and Widom [42]. As in [9], the uniform Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials found in [12] plays a central role. The formulae in [46] express the correlation kernels for β = 1, 4 as a sum of a Christoffel‐Darboux (CD) term, as in the case β = 2, together with a correction term. In the bulk scaling limit [9], the correction term is of lower order and does not contribute to the limiting form of the correlation kernel. By contrast, in the edge scaling limit considered here, the CD term and the correction term contribute to the same order: this leads to additional technical difficulties over and above [49]. © 2006 Wiley Periodicals, Inc.     

Towards novel difluorinated sugar mimetics; syntheses and conformational analyses of highly-functionalised difluorinated cyclooctenones

Highly-functionalised difluorinated cyclooctenones were synthesised from trifluoroethanol using either metallated difluoroenol acetal or carbamate chemistry, followed by a [2,3]-Wittig rearrangement or aldol reaction. Efficient RCM reactions afforded the title compounds which showed rather restricted fluxional behaviour by VT (19)F NMR. Topological characterisation by molecular modelling and NOESY/ROESY experiments offered a number of challenges, but allowed the identification of two favoured boat-chair conformers which interconverted by pseudorotation with relatively large activation barriers.     

Forced lattice vibrations: Part I

This is the First part of a two-part series on forced lattice vibrations in which a semi-infinite lattice of one-dimensional particles {x_n}_n≧1 is driven from one end by a particle x₀. This particle undergoes a given, periodically perturbed, uniform motion, _x0(_t) = _at + _h(_yt), where a and γ are constants and h(·) has period 2π. For a wide variety of restoring forces F (i.e., F′ > 0), numerical calculations indicate the existence of a sequence of thresholds γ₁ = γ₁(a, h, F) > γ₂ = γ₂(a,h,F) > … > γ_k = γ_k(a,h,F) > …, γ_k → 0, as k → ∞. If γ_k > γ > γ_k+1, a k-phase wave that is well described by the wave form, emerges and travels through the lattice. The goal of this series is to describe the emergence and calculate some properties of these wave forms. In Part I the authors first consider the case where F(x) = e^x (i.e., Toda forces) but h is arbitrary, and show how to compute a basic diagnostic (see J(λ), formula (1.26)) for the system in terms of the solution of an associated scalar Riemann-Hilbert problem, once a certain finite set of numbers is known. In another direction, the authors consider the case where F(x) is restoring but arbitrary, and h is small. Here the authors prove a general result, asserting that if there exists a sufficiently ample family of traveling-wave solutions of the doubly infinite lattice, then it is possible to construct time-periodic k-phase wave solutions with asymptotics in n of type (iii) for the driven system (i). In Part II, the authors prove that sufficiently ample families of traveling-wave solutions of the system (iv) exist in the cases γ > γ₁ and γ₁ > γ > γ₂ for general restoring forces F. In the case with Toda forces, F(x) = e^x, the authors prove that sufficiently ample families of traveling-wave solutions.     

The infrared spectra (600-140 cm−1 of the imidazole,pyrazine and pyrimidine adducts of cobalt(II), nickel(II) and zinc(II) acetylacetonates

Summary The complexes M(acac)₂(imidazole)₂ (M = Co or NO and [M(acac)₂B]_n (M = Co, Ni or Zn; B = pyrazine or pyrimidine) have been prepared and their i.r. spectra determined over the 600–140 cm^–1. range. The metal-oxygen and metal-nitrogen stretching frequencies, (M-O) and v(M-N), are assigned on the basis of the band shifts induced by deuteriation of the adducted base and by substitution of the metal ion. Three or fourv(M-O) bands are observed within the 600-200 cm^–1 range. The twov(M-O) bands of higher frequency are considered to the coupled with internal ligand modes. TwovM-N) bands are observed within the 280–170 cm^–1. range. The metal-ligand stretching frequencies are in good agreement with the values previously established for these vibrations in the [M(imidazole)₆]²⁺ and Ni(acac)₂(pyridine)₂ complexes.     

Study of grasshopper meiosis-associated proteins using two-dimensional polyacrylamide gel electrophoresis

Premeiotic and meiotic whole testes from grasshoppers were compared for the presence of meiosis associated proteins using one- and two-dimensional sodium dodecyl sulphate-polyacrylamide gel electrophoresis. One-dimensional sodium dodecyl sulphate-polyacrylamide gels detected differences between premeiotic and meiotic samples but two-dimensional gels gave more precise results. Isoelectric focusing revealed only one meiosis-associated protein, while nonequilibrium pH gradient electrophoresis detected five more. It is not known whether these proteins relate to the nuclear aspects of meiosis, or associated cellular changes. These proteins have been electrophoretically purified and monoclonal antibodies are being prepared.     

Foliations on $$\mathbb {P}^2$$ P 2 with only one singular point

Geometriae Dedicata - We prove a criterion for Benjamini-Schramm convergence of periodic orbits of Lie groups. This general observation is then applied to homogeneous spaces and the space of...     

MVF Sensor Enables Analysis of Nucleic Acids with Stable Secondary Structures

One major challenge in nucleic acids analysis by hybridization probes is a compromise between the probe's tight binding and sequence‐selective recognition of nucleic acid targets folded into stable secondary structures. We have been developing a four‐way junction (4WJ)‐based sensor that consists of a universal stem‐loop (USL) probe immobilized on an electrode surface and two adaptor strands (M and F). The sensor was shown to be highly selective towards single base mismatches at room temperature, able to detect multiple targets using the same USL probe, and have improved ability to detect folded nucleic acids. However, some nucleic acid targets, including natural RNA, are folded into very stable secondary and tertiary structures, which may represent a challenge even for the 4WJ sensors. This work describes a new sensor, named MVF since it uses three probe stands M, V and F, which further improves the performance of 4WJ sensors with folded targets. The MVF sensor interrogating a 16S rRNA NASBA amplicon with calculated folding energy of ?32.82 kcal/mol has demonstrated 2.5‐fold improvement in a signal‐to‐background ratio in comparison with a 4WJ sensor lacking strand V. The proposed design can be used as a general strategy in the analysis of folded nucleic acids including natural RNA.     

Towards stable analogues of inositol phosphates: stereoselective syntheses of (alpha,alpha-difluoromethyl)phosphonic acid (DFMPA)-containing cyclohexanes

Diels-Alder and conjugate addition reactions were used to prepare precursors to a range of fully functionalized and deoxy inositol phosphate analogues.     

On the distribution of the length of the longest increasing subsequence of random permutations

The authors consider the length, , of the longest increasing subsequence of a random permutation of numbers. The main result in this paper is a proof that the distribution function for , suitably centered and scaled, converges to the Tracy-Widom distribution of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest descent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel for the Poissonization of the distribution function of .