Commensurated Subgroups of Arithmetic Groups, Totally Disconnected Groups and Adelic Rigidity

The paper establishes a substantial number of cases of a conjecture regarding commensurated subgroups of S-arithmetic groups made by Margulis and Zimmer in the late 1970s. New results in the structure theory of totally disconnected groups are established along the way and are of independent interest. Other ideas in the argument motivate a sweeping conjecture, presented in the last section of the paper, which naturally unifies in an adelic setting deep results and fundamental conjectures in the rigidity theory of arithmetic groups.     

Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon

For a general subcritical second-order elliptic operator P   in a domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$ (or noncompact manifold), we construct Hardy-weight W which is optimal   in the following sense. The operator P−λW$P-\lambda W$ is subcritical in Ω   for all λ<1$\lambda <1$, null-critical in Ω   for λ=1$\lambda =1$, and supercritical near any neighborhood of infinity in Ω   for any λ>1$\lambda >1$. Moreover, if P   is symmetric and W>0$W>0$, then the spectrum and the essential spectrum of W⁻¹P$W^{-1}P$ are equal to [1,∞)$[1,\infty )$, and the corresponding Agmon metric is complete. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation Pu=0$Pu=0$, the existence of which depends on the subcriticality of P in Ω.     

Optical and Magnetic Properties of Conjugate Structures of PbSe Quantum Dots and γ‐Fe2O3 Nanoparticles

An investigation of the optical and magnetic properties of a unique hydrogen‐linked conjugate nanostructure, comprised of superparamagnetic γ‐Fe₂O₃ nanoparticles (NPs) and near‐infrared PbSe nanocrystal quantum dot (NQD) chromophores, is reported. The results show retention of the NQDs’ emission quantum efficiency and radiative lifetime, and only a small red shift of its band energy, upon conjugation to the dielectric surroundings of γ‐Fe₂O₃ NPs. The study also shows the sustainability of the superparamagnetism of the NPs after conjugation, with only a slight decrease of the ferromagnetic–superparamagnetic transition temperature with respect to that of the individual NPs. Thus, the conjugate nanostructure can be considered as a useful medical platform when PbSe NQDs act as fluorescent tags, while the γ‐Fe₂O₃ NPs are used as a vehicle driven by an external magnetic field for targeted delivery of tags or drugs.     

Constrained graph layout by stress majorization and gradient projection

The adoption of the stress-majorization method from multi-dimensional scaling into graph layout has provided an improved mathematical basis and better convergence properties for so-called “force-directed placement” techniques. In this paper we explore algorithms for augmenting such stress-majorization techniques with simple linear constraints using gradient-projection optimization techniques. Our main focus is a particularly simple class of constraints called “orthogonal-ordering constraints” but we also discuss how gradient-projection methods may be extended to solve more general linear “separation constraints”. In addition, we demonstrate several graph-drawing applications where these types of constraints can be very useful.     

The electronic origin of the dual fluorescence in donor-acceptor substituted benzene derivatives

The origin of the dual fluorescence of DMABN (dimethylaminobenzonitrile) and other benzene derivatives is explained by a charge transfer model based on the properties of the benzene anion radical. It is shown that, in general, three low-lying electronically excited states are expected for these molecules, two of which are of charge transfer (CT) character, whereas the third is a locally excited (LE) state. Dual fluorescence may arise from any two of these states, as each has a different geometry at which it attains a minimum. The Jahn-Teller induced distortion of the benzene anion radical ground state helps to classify the CT states as having quinoid (Q) and antiquinoid (AQ) forms. The intramolecular charge transfer (ICT) state is formed by the transfer of an electron from a covalently linked donor group to an anti-bonding orbital of the pi-electron system of benzene. The change in charge distribution of the molecule in the CT states leads to the most significant geometry change undergone by the molecule which is the distortion of the benzene ring to a Q or AQ structure. As the dipole moment is larger in the perpendicular geometry than in the planar one, this geometry is preferred in polar solvents, supporting the twisted intramolecular charge transfer (TICT) model. However, in many cases the planar conformation of CT excited states is lower in energy than that of the LE state, and dual fluorescence can be observed also from planar structures.     

Negative curvature from a cohomological viewpoint and cocycle superrigidity

In the framework of general negatively curved spaces, we present new superrigidity results and introduce new techniques based on bounded cohomology. This applies to irreducible lattices, and more generally to cocycles, of products of arbitrary locally compact groups. Together with a new vanishing result for higher rank groups, this also generalizes and unifies all previously known results in that direction. The non-vanishing results provide a large class of examples for our results on orbit equivalence rigidity (Monod and Shalom, Ann. of Math., in press). We prove the ‘toy-case’ of actions on trees. To cite this article: N. Monod, Y. Shalom, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Infinite sequential games with perfect but incomplete information

Infinite sequential games, in which Nature chooses a Borel winning set and reveals it to one of the players, do not necessarily have a value if Nature has 3 or more choices. The value does exist if Nature has 2 choices. The value also does not necessarily exist if Nature chooses from 2 Borel payoff functions. Similarly, if Player 1 chooses the Borel winning set and does not reveal his selection to Player 2, then the game does not necessarily have a value if there are 3 or more choices; it does have a value if there are only 2 choices. If Player 1 chooses from 2 Borel payoff functions and does not reveal his choice, the game need not have a value either.     

On Nonexistence of any λ0-invariand positive harmonic function,a counter example to stroock' conjecture

Consider a sub-Markovian semigroup such that λ₀, the border number between recrrence and transience, equals zero. In 1982, D. W. Stroock conjectured that under general hypotheses on the semi-group the corresponding process always admits an invariant measure.

In this paper we present an example of a second order elliptic operator P with a generalized principal eigenvalue λ₀ which equals zero such that the parabolic equation does not admit any positive invariant P—harmonic function and also any invariant measure. This gives a counter example to Stroock's conjecture for diffusion processes. We also present an example of a complete Riemannian manifold M which does not admit any positive invariant harmonic function while λ₀, the bottom of the spectrum of M, is zero. This gives a partial answer to a question of Stroock and Sullivan.     

Sharp Two-Sided Heat Kernel Estimates of Twisted Tubes and Applications

We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian ${-\Delta^D_\Omega}$ in locally twisted three-dimensional tubes Ω. In particular, we show that for any fixed x the heat kernel decays for large times as ${{\rm e}^{-E_1t} t^{-3/2}}$ , where E ₁ is the fundamental eigenvalue of the Dirichlet Laplacian on the cross section of the tube. This shows that any, suitably regular, local twisting speeds up the decay of the heat kernel with respect to the case of straight (untwisted) tubes. Moreover, the above large time decay is valid for a wide class of subcritical operators defined on a straight tube. We also discuss some applications of this result, such as Sobolev inequalities and spectral estimates for Schrödinger operators ${-\Delta^D_\Omega-V}$ .