On homotopy rigidity of the functor ΣΩ on co-H-spaces

In this paper we study the homotopy rigidity property of the functors ΣΩ and Ω. Our main result is that both functors are homotopy rigid on simply-connected p-local finite co-H-spaces. The result is obtain by a subtle interplay of homotopy decomposition techniques, modular representation theory and the counting principle.     

On string topology of classifying spaces

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.     

Scanning tunneling microscopy/spectroscopy on Au nanoparticles assembled using lauryl amine (LAM) and octadecane thiol (ODT)

In this report, we demonstrate scanning tunneling microscopy and spectroscopy on thin films of lauryl amine (LAM) and octadecane thiol (ODT) protected gold nanoparticles. We show that the zero current in the I-V curves (measure of Coulomb blockade (CB) of the nanoparticles) depends on the properties of the spacer molecule. In both the cases the gap voltage and the tunneling current at which the images are obtained are quite different which is further confirmed from the fitting performed based on the orthodox theory. The values for the capacitance and charging energy obtained from the fitting for ODT capped particles are comparable to the values obtained using spherical capacitor model. In contrast, values of these parameters were found to differ for LAM capped nanoparticles. While imaging, ODT capped nanoparticles were observed to drag along the scan direction leading to ordering of particles. Images of LAM capped gold nanoparticles show local ordering in self-assembly of particles although no evidence of large scale ordering in spatial Fourier transform was seen. These observations suggest that nanoparticles with larger CB would be imaged nonevasively in contrast to small CB systems for which tip induced effects will be dominant. In both the systems the current was found to rise faster than theoretical curves based on the orthodox theory suggesting that mechanism of charge transfer in this case may involve field emission rather than tunneling through a rectangular barrier. An attempt has been made to explain charge transfer based on Fowler-Nordheim (F-N) plots of the I-V curves.     

Twisted Poincaré Duality for some Quadratic Poisson Algebras

We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier. This research of the first author was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme. The second author was supported by EPSRC Grant EP/D034167/1.     

The Sixth Painlevé Equation as Similarity Reduction of $${\widehat{\mathfrak{gl}}_3}$$ Generalized Drinfel’d–Sokolov Hierarchy

Scaling symmetry of -type Drinfel’d–Sokolov hierarchy is investigated. Applying similarity reduction to the hierarchy, one can obtain the Schlesinger equation with (n + 1) regular singularities. Especially in the case of n = 3, the hierarchy contains the three-wave resonant system and the similarity reduction gives the generic case of the Painlevé VI equation. We also discuss Weyl group symmetry of the hierarchy.      

Essential and continuous spectrum of symmetric difference equations

Essential and continuous spectrum of symmetric difference equations have been investigated. It has been shown that the deficiency indices and the existence of these components of the spectrum are determined by the growth conditions of the coefficients. In particular, the deficiency indices are superimposition of those clusters determined by the coefficient growth. Finally, we have proved the neccessary and sufficient conditions for the existence of essential spectrum of selfadjoint subspace extensions using subspace theory and asymtotic summation.     

A Note on the regularity of autonomous quasilinear elliptic and parabolic systems

An analogy of E.Giusti lemma at infinity was introduced in [5]. Its slight modification enables us to prove the estimates of Holder norms for BMO weak solutions of autonomous elliptic and parabolic systems provided all such solutions are continuous at the same point. The sharpness of this condition is illustrated by a counterexample     

Multiplicity of the second Schrödinger eigenvalue on closed surfaces

We improve – roughly by a factor 2 – the known bound on the multiplicity of the second eigenvalue of Schr?dinger operators (i.e. Laplace plus potential) on closed surfaces. This gives four new topological types of surfaces for which Colin de Verdière's conjecture relating the maximal multiplicity to the chromatic number of the surface is verified. The proof goes by defining a space of "nodal splittings” of the surface, equipped with a double covering to which a Borsuk-Ulam type theorem is applied. Received: 19 June 2001; revised version: 18 March 2002 /Published online: 17 June 2002     

Small compact perturbation of strongly irreducible operators

An operatorT onH is called strongly irreducible ifT is not similar to any reducible operators. In this paper, we shall say yes to answer the following question raised by D. A. Herrero.Given an operatorT with connected spectrum (T) and a positive number , can we find a compact operatorK with K < such thatT+K is strongly irreducible?Supported by National Natural Science Foundation of China(19901011), Mathematical Center of State Education Commission of China and 973 Project of China