The Isotropic Semicircle Law and Deformation of Wigner Matrices

We analyze the spectrum of additive finite‐rank deformations of N × N Wigner matrices H. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue d_i of the deformation crosses a critical value ± 1. This transition happens on the scale . We allow the eigenvalues d_i of the deformation to depend on N under the condition . We make no assumptions on the eigenvectors of the deformation. In the limit N → ∞, we identify the law of the outliers and prove that the nonoutliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a Gaussian matrix ensemble. A key ingredient in our proof is the isotropic local semicircle law, which establishes optimal high‐probability bounds on where m(z) is the Stieltjes transform of Wigner's semicircle law and v , w are arbitrary deterministic vectors.© 2013 Wiley Periodicals, Inc.     

Analytical model of dislocation nucleation on a near-surface void under tensile surface stress

We have analyzed in detail the mechanism leading to tip growth on a surface which operates via nucleation of dislocations on a near-surface void under tensile surface stress. We derived a simplified analytical model describing the relevant physical factors related to the observed linearity between the void radius and the maximum depth of the void for the growth to occur. The model is based on the direct numerical calculation of atomic level stresses in the simulated system. Based on the present model we can estimate this maximum depth for a void of a certain size under a given stress in the size range which is beyond the feasibility of the molecular dynamics simulation method.     

Optimal control strategies for coupled quantum dots

Semiconductor quantum dots are ideal candidates for quantum information applications in solid-state technology. However, advanced theoretical and experimental tools are required to coherently control, for example, the electronic charge in these systems. Here we demonstrate how quantum optimal control theory provides a powerful way to manipulate the electronic structure of coupled quantum dots with an extremely high fidelity. As alternative control fields we apply both laser pulses as well as electric gates, respectively. We focus on double and triple quantum dots containing a single electron or two electrons interacting via Coulomb repulsion. In the two-electron situation we also briefly demonstrate the challenges of timedependent density-functional theory within the adiabatic local-density approximation to produce comparable results with the numerically exact approach.     

Critical Mandelbrot Cascades

We study Mandelbrot’s multiplicative cascade measures at the critical temperature. As has been recently shown by Barral et al. (C R Acad Sci Paris Ser I 350:535–538, 2012), an appropriately normalized sequence of cascade measures converges weakly in probability to a nontrivial limit measure. We prove that these limit measures have no atoms and give bounds for the modulus of continuity of the cumulative distribution function of the measure. Using the earlier work of Barral and Seuret (Adv Math 214:437–468, 2007), we compute the multifractal spectrum of the measures. We also extend the result of Benjamini and Schramm (Commun Math Phys 289:653–662, 2009), in which the KPZ formula from quantum gravity is validated for the high temperature cascade measures, to the critical and low temperature cases.     

Delocalization and Diffusion Profile for Random Band Matrices

We consider Hermitian and symmetric random band matrices H = (h _xy ) in ${d\,\geqslant\,1}$ d ? 1 dimensions. The matrix entries h _xy , indexed by ${x,y \in (\mathbb{Z}/L\mathbb{Z})^d}$ x , y ∈ ( Z / L Z ) d , are independent, centred random variables with variances ${s_{xy} = \mathbb{E} |h_{xy}|^2}$ s x y = E | h x y | 2 . We assume that s _xy is negligible if |x ? y| exceeds the band width W. In one dimension we prove that the eigenvectors of H are delocalized if ${W\gg L^{4/5}}$ W ? L 4 / 5 . We also show that the magnitude of the matrix entries ${|{G_{xy}}|^2}$ | G x y | 2 of the resolvent ${G=G(z)=(H-z)^{-1}}$ G = G ( z ) = ( H - z ) - 1 is self-averaging and we compute ${\mathbb{E} |{G_{xy}}|^2}$ E | G x y | 2 . We show that, as ${L\to\infty}$ L → ∞ and ${W\gg L^{4/5}}$ W ? L 4 / 5 , the behaviour of ${\mathbb{E} |G_{xy}|^2}$ E | G x y | 2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.     

The Polyanalytic Ginibre Ensembles

For integers n,q=1,2,3,…?, let Pol _n,q denote the ${\mathbb{C}}$ -linear space of polynomials in z and $\bar{z}$ , of degree ≤n?1 in z and of degree ≤q?1 in $\bar{z}$ . We supply Pol _n,q with the inner product structure of $$\begin{aligned} L^2 \bigl({\mathbb{C}},\mathrm{e}^{-m|z|^2} {\mathrm{d}}A \bigr),\quad \mbox {where } {\mathrm{d}}A(z)=\pi^{-1}{\mathrm{d}}x {\mathrm{d}}y,\ z= x+ {\mathrm{i}}y; \end{aligned}$$ the resulting Hilbert space is denoted by Pol _m,n,q . Here, it is assumed that m is a positive real. We let K _m,n,q denote the reproducing kernel of Pol _m,n,q , and study the associated determinantal process, in the limit as m,n→+∞ while n=m+O(1); the number q, the degree of polyanalyticity, is kept fixed. We call these processes polyanalytic Ginibre ensembles, because they generalize the Ginibre ensemble—the eigenvalue process of random (normal) matrices with Gaussian weight. There is a physical interpretation in terms of a system of free fermions in a uniform magnetic field so that a fixed number of the first Landau levels have been filled. We consider local blow-ups of the polyanalytic Ginibre ensembles around points in the spectral droplet, which is here the closed unit disk $\bar{\mathbb{D}}:=\{z\in{\mathbb{C}}:|z|\le1\}$ . We obtain asymptotics for the blow-up process, using a blow-up to characteristic distance m ^?1/2; the typical distance is the same both for interior and for boundary points of $\bar{\mathbb{D}}$ . This amounts to obtaining the asymptotical behavior of the generating kernel K _m,n,q . Following (Ameur et al. in Commun. Pure Appl. Math. 63(12):1533–1584, 2010), the asymptotics of the K _m,n,q are rather conveniently expressed in terms of the Berezin measure (and density) For interior points |z|<1, we obtain that ${\mathrm{d}}B^{\langle z\rangle}_{m,n,q}(w)\to{\mathrm{d}}\delta_{z} $ in the weak-star sense, where δ _z denotes the unit point mass at z. Moreover, if we blow up to the scale of m ^?1/2 around z, we get convergence to a measure which is Gaussian for q=1, but exhibits more complicated Fresnel zone behavior for q>1. In contrast, for exterior points |z|>1, we have instead that ${\mathrm{d}}B^{\langle z\rangle}_{m,n,q}(w) \to{\mathrm{d}}\omega(w,z, {\mathbb{D}}^{e}) $ , where ${\mathrm{d}}\omega(w,z,{\mathbb{D}}^{e})$ is the harmonic measure at z with respect to the exterior disk ${\mathbb{D}}^{e}:= \{w\in{\mathbb{C}}:\, |w|>1\}$ . For boundary points, |z|=1, the Berezin measure ${\mathrm{d}}B^{\langle z\rangle}_{m,n,q}$ converges to the unit point mass at z, as with interior points, but the blow-up to the scale m ^?1/2 exhibits quite different behavior at boundary points compared with interior points. We also obtain the asymptotic boundary behavior of the 1-point function at the coarser local scale q ^1/2 m ^?1/2.     

Preference Programming with incomplete ordinal information

This paper extends possibilities for analyzing incomplete ordinal information about the parameters of an additive value function. Such information is modeled through preference statements which associate sets of alternatives or attributes with corresponding sets of rankings. These preference statements can be particularly helpful in developing a joint preference representation for a group of decision-makers who may find difficulties in agreeing on numerical parameter values. Because these statements can lead to a non-convex set of feasible parameters, a mixed integer linear formulation is developed to establish a linear model for the computation of decision recommendations. This makes it possible to complete incomplete ordinal information with other forms of incomplete information.     

A structure-activity relationship study of catechol-O-methyltransferase inhibitors combining molecular docking and 3D QSAR methods

A panel of 92 catechol-O-methyltransferase (COMT) inhibitors was used to examine the molecular interactions affecting their biological activity. COMT inhibitors are used as therapeutic agents in the treatment of Parkinson's disease, but there are limitations in the currently marketed compounds due to adverse side effects. This study combined molecular docking methods with three-dimensional structure-activity relationships (3D QSAR) to analyse possible interactions between COMT and its inhibitors, and to incite the design of new inhibitors. Comparative molecular field analysis (CoMFA) and GRID/GOLPE models were made by using bioactive conformations from docking experiments, which yielded q2 values of 0.594 and 0.636, respectively. The docking results, the COMT X-ray structure, and the 3D QSAR models are in agreement with each other. The models suggest that an interaction between the inhibitor's catechol oxygens and the Mg2+ ion in the COMT active site is important. Both hydrogen bonding with Lys144, Asn170 and Glu199, and hydrophobic contacts with Trp38, Pro174 and Leu198 influence inhibitor binding. Docking suggests that a large R1 substituent of the catechol ring can form hydrophobic contacts with side chains of Val173, Leu198, Met201 and Val203 on the COMT surface. Our models propose that increasing steric volume of e.g. the diethylamine tail of entacapone is favourable for COMT inhibitory activity.     

Interactions between europium(III) ion and methyl glycofuranosides in aqueous solution

The interactions between Eu(III) ion and some methyl glycofuranosides have been studied luminometrically in aqueous solution. The measurements were based on the delayed fluorescence of the Eu(III) ion known to be environmentally sensitive. The reciprocal lifetimes, i.e., the decay constants of this fluorescence, depend on the number of OH bonds in the primary hydration sphere of the ion. These were determined in aqueous glycofuranoside solutions of various concentrations. These data enable us to discuss the effect of ligand configuration on the binding sites in the formed complexes. The formation constants for these complexes have been evaluated with the aid of decay rate equations.     

Plasticity of histamine H<Subscript>3</Subscript>receptor expression and binding in the vestibular nuclei after labyrinthectomy in rat

Background  

In rat, deafferentation of one labyrinth (unilateral labyrinthectomy) results in a characteristic syndrome of ocular and motor postural disorders (e.g., barrel rotation, circling behavior, and spontaneous nystagmus). Behavioral recovery (e.g., diminished symptoms), encompassing 1 week after unilateral labyrinthectomy, has been termed vestibular compensation. Evidence suggesting that the histamine H₃ receptor plays a key role in vestibular compensation comes from studies indicating that betahistine, a histamine-like drug that acts as both a partial histamine H₁ receptor agonist and an H₃ receptor antagonist, can accelerate the process of vestibular compensation.