SPDE in Hilbert Space with Locally Monotone Coefficients

In this paper we prove the existence and uniqueness of strong solutions for SPDE in Hilbert space with locally monotone coefficients, which is a generalization of the classical result of Krylov and Rozovskii for monotone coefficients. Our main result can be applied to different types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equation, stochastic 2-D Navier-Stokes equation, stochastic $p$-Laplace equation and stochastic porous media equation with some non-monotone perturbations.Comment: 20 pages, add Remark 3.1 for stochastic Burgers equatio

Superradiance Lattice

We show that the timed Dicke states of a collection of three-level atoms can form a tight-binding lattice in momentum space. This lattice, coined the superradiance lattice (SL), can be constructed based on electromagnetically induced transparency (EIT). For a one-dimensional SL, we need the coupling field of the EIT system to be a standing wave. The detuning between the two components of the standing wave introduces an effective uniform force in momentum space. The quantum lattice dynamics, such as Bloch oscillations, Wannier-Stark ladders, Bloch band collapsing and dynamic localization can be observed in the SL. The two-dimensional SL provides a flexible platform for Dirac physics in graphene. The SL can be extended to three and higher dimensions where no analogous real space lattices exist with new physics waiting to be explored.Comment: 6pages, 4 figure

Harnack Inequality and Strong Feller Property for Stochastic Fast-Diffusion Equations

This paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results.Comment: to appear in Journal of Mathematical Analysis and Application

Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise

The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework.Comment: 31 pages, published in Appl. Math. Opti

Separation of Normal and Premalignant Cervical Epithelial Cells Using Confocal Light Absorption and Scattering Spectroscopic Microscopy Ex Vivo

Confocal light absorption and scattering spectroscopic (CLASS) microscopy can detect changes in biochemicals and the morphology of cells. It is therefore used to detect high-grade cervical squamous intraepithelial lesion (HSIL) cells in the diagnosis of premalignant cervical lesions. Forty cervical samples from women with abnormal Pap smear test results were collected, and twenty cases were diagnosed as HSIL; the rest were normal or low-grade cervical squamous intraepithelial lesion (LSIL). The enlarged and condensed nuclei of HSIL cells as viewed under CLASS microscopy were much brighter and bigger than those of non-HSIL cells. Cytological elastic scattered light data was then collected at wavelengths between 400 and 1000 nm. Between 600 nm to 800 nm, the relative elastic scattered light intensity of HSIL cells was higher than that of the non-HSIL. Relative intensity peaks occurred at 700 nm and 800 nm. CLASS sensitivity and specificity results for HSIL and non-HSIL compared to cytology diagnoses were 80% and 90%, respectively. This study demonstrated that CLASS microscopy could effectively detect cervical precancerous lesions. Further study will verify this conclusion before the method is used in clinic for early detection of cervical cancer