Fast thermo-optical excitability in a two-dimensional photonic crystal

We experimentally demonstrate excitability in a semiconductor two-dimensional photonic crystal. Excitability is a nonlinear dynamical mechanism underlying pulselike responses to small perturbations in systems possessing one stable state. We show that a band-edge photonic crystal resonator exhibits class II excitability, resulting from the nonlinear coupling between the high-Q optical mode, the charge-carrier density, and the fast (sub-micros) thermal dynamics. In this context, the critical slowing down of the electro-optical dynamics close to the excitable threshold can delay the optical response by an amount comparable to the duration of the output pulse (5 ns). The latter results from a short thermal dynamical excursion along a high local intensity manifold of the phase space.     

Quantitative real-space analysis of self-assembled structures of magnetic dipolar colloids

We present the first real-space analysis on a single-particle level of the dipolar chains and branched clusters self-assembling in magnetic fluids in zero field. Spatial correlations and chain-length distributions directly obtained from tracked particle positions in vitrified films of synthetic magnetic (Fe3O4) dispersions provide a quantitative test for simulations and theory of dipolar fluids. A pertinent example is the cluster-size distribution that can be analyzed with a one-dimensional aggregation model to yield a dipolar attraction energy that agrees well with the dipole moment found from independent magnetization measurements.     

Global rough solutions to the critical generalized KdV equation

We prove that the initial value problem (IVP) for the critical generalized KdV equation ut+uxxx+x(u⁵)=0 on the real line is globally well-posed in Hs(R) if s>3/5 with the appropriate smallness assumption on the initial data.     

Existence and multiplicity results for a fourth order mean field equation

In this article we consider the following fourth order mean field equation on smooth domain Ω?R⁴:

     

Existence of positive solutions for some nonlinear parabolic systems in exteriors domains

We prove some existence results of positive continuous solutions to the semilinear parabolic system , in an unbounded domain D with compact boundary subject to some Dirichlet conditions, where λ and μ are nonnegative parameters. The functions f, g are nonnegative continuous monotone on (0,∞) and the potentials p, q are nonnegative and satisfy some hypotheses related to the parabolic Kato class J^∞(D).     

Representations of quantum groups defined over commutative rings II

In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules.     

Fuzzy complex Grassmannians and quantization of line bundles

We construct by purely representation-theoretic methods fuzzy versions of an arbitrary complex Grassmannian M=Gr _n (ℂ ^n+m ), i.e., a sequence of matrix algebras tending SU(n+m)-equivariantly to the algebra of smooth functions on M. We also show that this approximation can be interpreted in terms of the Berezin-Toeplitz quantization of M. Furthermore, we use branching rules to prove that the quantization of every complex line bundle over M is given by a SU(n+m)-equivariant truncation of the space of its L ²-sections.     

On principal components regression with Hilbertian predictors

We demonstrate that, in a regression setting with a Hilbertian predictor, a response variable is more likely to be more highly correlated with the leading principal components of the predictor than with trailing ones. This is despite the extraction procedure being unsupervised. Our results are established under the conditional independence model, which includes linear regression and single-index models as special cases, with some assumptions on the regression vector. These results are a generalisation of earlier work which showed that this phenomenon holds for predictors which are real random vectors. A simulation study is used to quantify the phenomenon.

     

Vector subspaces and operators with the Stone condition

We introduce the notion of an operating function on a subset of a F{\Phi} -algebra E. Then we use this notion to generalize results from Huijsmans and de Pagter (Proc. Lond. Math. Soc. 48:161–174, 1984) about the connection between vector subspaces and subalgebras of E. In the second part we investigate the analogous problem for operators.     

The Jacobi-Dunkl transform on ℝ and the convolution product on new spaces of distributions

In this work, we consider the Jacobi-Dunkl operator Λ _α,β , a 3 b 3 \frac-12\alpha\geq\beta\geq\frac{-1}{2} , a 1 \frac-12\alpha\neq\frac{-1}{2} , on ℝ. The eigenfunction Y_l^a,b\Psi_{\lambda}^{\alpha,\beta} of this operator permits to define the Jacobi-Dunkl transform. The main idea in this paper is to introduce and study the Jacobi-Dunkl transform and the Jacobi-Dunkl convolution product on new spaces of distributions