On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains

We construct a bounded C¹ domain Ω in for which the regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in such that the solution of in Ω and either on or on is contained in but not in for any . An analogous result holds for Sobolev spaces with .     

The regularity of the multiple higher-order poles solitons of the NLS equation

Based on the inverse scattering method, the formulae of one higher-order pole solitons and multiple higher-order poles solitons of the nonlinear Schrödinger equation (NLS) equation are obtained. Their denominators are expressed as , where is a matrix frequently constructed for solving the Riemann-Hilbert problem, and the asterisk denotes complex conjugate. We take two methods for proving is invertible. The first one shows matrix is equivalent to a self-adjoint Hankel matrix , proving . The second one considers the block-matrix form of , proving . In addition, we prove that the dimension of is equivalent to the sum of the orders of pole points of the transmission coefficient and its diagonal entries compose a set of basis.     

A Robust Near-Infrared Calibration Model for the Determination of Chlorophyll Concentration in Tree Leaves with a Calibration Transfer Method

A method based on piecewise direct standardization was developed to directly predict leaf chlorophyll concentrations by correction of near-infrared spectra to construct a robust calibration model. Chinar, camphor, and gingko leaves collected from two growth intervals were evaluated. Spectral pretreatment methods and wavelength selection were investigated. The first derivative combined with stability competitive adaptive reweighted sampling before piecewise direct standardization provided the best performance. Under the optimized parameters, the root mean square error of prediction was significantly reduced by using piecewise direct standardization. This study demonstrates that the calibration model may be used to rapidly characterize chlorophyll concentrations across species and growth intervals.     

Global persistence of geometrical structures for the Boussinesq equation with no diffusion

Our main aim is to investigate the temperature patch problem for the two-dimensional incompressible Boussinesq system with partial viscosity: the initial temperature is the characteristic function of some simply connected domain with 𝒞^1,𝜀 Hölder regularity. Although recent results ensure that the 𝒞¹ regularity of the patch persists for all time, whether higher order regularity is preserved has remained an open question. In the present paper, we give a positive answer to that issue. We also study the higher dimensional case, after prescribing an additional smallness condition involving critical Lebesgue or weak-Lebesgue norms of the data, so as to get a global-in-time statement. All our results stem from general properties of persistence of geometrical structures (of independent interest), that we establish in the first part of the paper.     

Une extension d'une inégalité de burkholder,davis, gundy pour les processus à α-variation bornée et applications

An increasing process associated in a natural sense with each process of bounded x-variation leads us to an extension of an inequality of Burkholder, Davis, Gundy and allows us to determine the regularity of such processes