Distribution of energy levels of quantum free particle on the Liouville surface and trace formulae

We consider the Weyl asymptotic formula

Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations

We study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system. In that application x lies in R, but here we study primarily the periodic case × R S¹. We establish existence, uniqueness, and regularity of solutions, locally in time, for positive initial data in H¹(S¹), and prove the existence of several families of Lyapunov functions for the evolution. From the latter we establish a sharp connection between existence globally in time and positivity preservation: if [0], T*) is a maximal half open interval of existence for a positive solution of the equation, with T* < ∞, then lim_tT* w(t,·) exists in C¹(S¹) but vanishes at some point. We show further that if T* > (1 + √3)/16π² √3 then T* = ∞ and lim_t∞ w(t,.) exists and is constant. We discuss also some explicit solutions and propose a generalization to higher dimensions. © 1994 John Wiley & Sons, Inc.     

Energy-level statistics of model quantum systems: Universality and scaling in a lattice-point problem

We investigate the statistics of the numberN(R, S) of lattice pointsnZ ², in an annular domain (R, w)=(R+w)A\RA, whereR, w>0. HereA is a fixed convex set with smooth boundary andw is chosen so that the area of (R, w) isS. The statistics comes fromR being taken as random (with a smooth density) in some interval [c ₁ T,c ₂,T],c ₂>c ₁>0. We find that in the limitT the variance and distribution of N=N(R; S)–S depend strongly on howS grows withT. There is a saturation regimeS/T, asT, in which the fluctuations in N coming from the two boundaries of are independent. Then there is a scaling regime,S/Tz, 0<z<, in which the distribution depends onz in an almost periodic way going to a Gaussian asz0. The variance in this limit approachesz for genericA, but can be larger for degenerate cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area.     

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring $R(\mathrm{\Lambda },V)$ and we describe $R(\mathrm{\Lambda },V)$ explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.     

Development of a new parallelized,optical biosensor platform for label-free detection of autoimmunity-related antibodies

Autoimmune diseases are characterized by the presence of autoantibodies in serum of affected patients. The heterogeneity of autoimmune relevant antigens creates a variety of different antibodies, which requires a simultaneous detection mode. For this reason, we developed a tool for parallelized, label-free, optical detection that accomplishes the characterization of multiple antigen–antibody interactions within a single measurement on a timescale of minutes. Using 11-aminoundecyltrimethoxysilane, we were able to immobilize proteinogenic antigens as well as an amino-functionalized cardiolipin on a glass surface. Assay conditions were optimized for serum measurements with a single spot antigen chip on a single spot 1-λ detection system. Minimized background signal allows a differentiation between patients and healthy controls with a good sensitivity and specificity. Applying polarized imaging reflectometric interference spectroscopy, we evaluated samples from three APS patients and three control subjects for this proof-of-principle and already obtained good results for β2-glycoprotein I and cardiolipin.     

Size does matter! Label-free detection of small molecule–protein interaction

This review is focused on methods for detecting small molecules and, in particular, the characterisation of their interaction with natural proteins (e.g. receptors, ion channels). Because there are intrinsic advantages to using label-free methods over labelled methods (e.g. fluorescence, radioactivity), this review only covers label-free techniques. We briefly discuss available techniques and their advantages and disadvantages, especially as related to investigating the interaction between small molecules and proteins. The reviewed techniques include well-known and widely used standard analytical methods (e.g. HPLC-MS, NMR, calorimetry, and X-ray diffraction), newer and more specialised analytical methods (e.g. biosensors), biological systems (e.g. cell lines and animal models), and in-silico approaches.