Infinite dimensional integrals beyond Monte Carlo methods: yet another approach to normalized infinite dimensional integrals

An approach to (normalized) infinite dimensional integrals, including normalized oscillatory integrals, through a sequence of evaluations in the spirit of the Monte Carlo method for probability measures is proposed. in this approach the normalization through the partition function is included in the definition. For suitable sequences of evaluations, the ("classical") expectation values of cylinder functions are recoveredComment: Submitted as a communication in the ICMSQUARE conference, september 201

Filling minimality of Finslerian 2-discs

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this result for Finsler metrics with area defined as the two-dimensional Holmes-Thompson volume. This implies a generalization of Pu's isosystolic inequality to Finsler metrics, both for Holmes-Thompson and Busemann definitions of Finsler area.Comment: 16 pages, v2: improved introduction and formattin

Local monotonicity of Riemannian and Finsler volume with respect to boundary distances

We show that the volume of a simple Riemannian metric on $D^n$ is locally monotone with respect to its boundary distance function. Namely if $g$ is a simple metric on $D^n$ and $g'$ is sufficiently close to $g$ and induces boundary distances greater or equal to those of $g$, then $vol(D^n,g')\ge vol(D^n,g)$. Furthermore, the same holds for Finsler metrics and the Holmes--Thompson definition of volume. As an application, we give a new proof of the injectivity of the geodesic ray transform for a simple Finsler metric.Comment: 13 pages, v3: minor corrections and clarifications, to appear in Geometriae Dedicat

Noise auto-correlation spectroscopy with coherent Raman scattering

Ultrafast lasers have become one of the most powerful tools in coherent nonlinear optical spectroscopy. Short pulses enable direct observation of fast molecular dynamics, whereas broad spectral bandwidth offers ways of controlling nonlinear optical processes by means of quantum interferences. Special care is usually taken to preserve the coherence of laser pulses as it determines the accuracy of a spectroscopic measurement. Here we present a new approach to coherent Raman spectroscopy based on deliberately introduced noise, which increases the spectral resolution, robustness and efficiency. We probe laser induced molecular vibrations using a broadband laser pulse with intentionally randomized amplitude and phase. The vibrational resonances result in and are identified through the appearance of intensity correlations in the noisy spectrum of coherently scattered photons. Spectral resolution is neither limited by the pulse bandwidth, nor sensitive to the quality of the temporal and spectral profile of the pulses. This is particularly attractive for the applications in microscopy, biological imaging and remote sensing, where dispersion and scattering properties of the medium often undermine the applicability of ultrafast lasers. The proposed method combines the efficiency and resolution of a coherent process with the robustness of incoherent light. As we demonstrate here, it can be implemented by simply destroying the coherence of a laser pulse, and without any elaborate temporal scanning or spectral shaping commonly required by the frequency-resolved spectroscopic methods with ultrashort pulses.Comment: To appear in Nature Physic

On Alternative Supermatrix Reduction

We consider a nonstandard odd reduction of supermatrices (as compared with the standard even one) which arises in connection with possible extension of manifold structure group reductions. The study was initiated by consideration of the generalized noninvertible superconformal-like transformations. The features of even- and odd-reduced supermatrices are investigated on a par. They can be unified into some kind of "sandwich" semigroups. Also we define a special module over even- and odd-reduced supermatrix sets, and the generalized Cayley-Hamilton theorem is proved for them. It is shown that the odd-reduced supermatrices represent semigroup bands and Rees matrix semigroups over a unit group.Comment: 22 pages, Standard LaTeX with AmS font

Theory and computation of directional nematic phase ordering

A computational study of morphological instabilities of a two-dimensional nematic front under directional growth was performed using a Landau-de Gennes type quadrupolar tensor order parameter model for the first-order isotropic/nematic transition of 5CB (pentyl-cyanobiphenyl). A previously derived energy balance, taking anisotropy into account, was utilized to account for latent heat and an imposed morphological gradient in the time-dependent model. Simulations were performed using an initially homeotropic isotropic/nematic interface. Thermal instabilities in both the linear and non-linear regimes were observed and compared to past experimental and theoretical observations. A sharp-interface model for the study of linear morphological instabilities, taking into account additional complexity resulting from liquid crystalline order, was derived. Results from the sharp-interface model were compared to those from full two-dimensional simulation identifying the specific limitations of simplified sharp-interface models for this liquid crystal system. In the nonlinear regime, secondary instabilities were observed to result in the formation of defects, interfacial heterogeneities, and bulk texture dynamics.Comment: first revisio

Non-isothermal model for the direct isotropic/smectic-A liquid crystalline transition

An extension to a high-order model for the direct isotropic/smectic-A liquid crystalline phase transition was derived to take into account thermal effects including anisotropic thermal diffusion and latent heat of phase-ordering. Multi-scale multi-transport simulations of the non-isothermal model were compared to isothermal simulation, showing that the presented model extension corrects the standard Landau-de Gennes prediction from constant growth to diffusion-limited growth, under shallow quench/undercooling conditions. Non-isothermal simulations, where meta-stable nematic pre-ordering precedes smectic-A growth, were also conducted and novel non-monotonic phase-transformation kinetics observed.Comment: First revision: 20 pages, 7 figure