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Register a new userGeneration of arbitrary full Poincar{e} beams on the hybrid-order Poincar{e} sphere
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We propose that the full Poincar{e} beam with any polarization geometries can be pictorially described by the hybrid-order Poincar{e} sphere whose eigenstates are defined as a fundamental-mode Gaussian beam and a Laguerre-Gauss beam. A robust and efficient Sagnac interferometer is established to generate any desired full Poincar{e} beam on the hybrid-order Poincar{e} sphere, via modulating the incident state of polarization. Our research may provide an alternative way for describing the full Poincar{e} beam and an effective method to manipulate the polarization of light.
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The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincar{e} constant. First we revisit E. Milmans result [40] on the link between weak (Poincar{e} or concentration) inequalities and Cheegers inequality in the logconcave cases, in particular extending localization ideas and a result of Latala, as well as providing a simpler proof of the nice Poincar{e} (dimensional) bound in the inconditional case. Then we prove alternative transference principle by concentration or using various distances (total variation, Wasserstein). A mollification procedure is also introduced enabling, in the logconcave case, to reduce to the case of the Poincar{e} inequality for the mollified measure. We finally complete the transference section by the comparison of various probability metrics (Fortet-Mourier, bounded-Lipschitz,...).
We propose and experimentally demonstrate a novel interferometric approach to generate arbitrary cylindrical vector beams on the higher order Poincare sphere. Our scheme is implemented by collinear superposition of two orthogonal circular polarizations with opposite topological charges. By modifying the amplitude and phase factors of the two beams, respectively, any desired vector beams on the higher order Poincare sphere with high tunability can be acquired. Our research provides a convenient way to evolve the polarization states in any path on the high order Poincare sphere.
Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincare inequality, have been handy for estimating the approximation errors in different scales. The purpose of this paper is to study a generalized Poincar e inequality, where the measurement function is of subsampled type, with a small but non-zero lengthscale that will be made precise. Our analysis identifies this inequality as a basic tool for function recovery problems. We discuss and demonstrate the optimality of the inequality concerning the subsampled lengthscale, connecting it to existing results in the literature. In application to function approximation problems, the approximation accuracy using different basis functions and under different regularity assumptions is established by using the subsampled Poincare inequality. We observe that the error bound blows up as the subsampled lengthscale approaches zero, due to the fact that the underlying function is not regular enough to have well-defined pointwise values. A weighted version of the Poincar e inequality is proposed to address this problem; its optimality is also discussed.
We define Poincar{e} profiles of Dirichlet type for graphs of bounded degree, in analogy with the Poincar{e} profiles (of Neumann type) defined in [HMT19]. The obvious first definition yields nothing of interest, but an alternative definition yields a spectrum of profiles which are quasi-isometry invariants and monotone with respect to subgroup inclusion. Moreover, in the extremal cases $p=1$ and $p=infty$, they detect the Fo lner function and the growth function respectively.
In this paper, we consider a problem in calculus of variations motivated by a quantitative isoperimetric inequality in the plane. More precisely, the aim of this article is the computation of the minimum of the variational problem $$inf_{uinmathcal{W}}frac{displaystyleint_{-pi}^{pi}[(u)^2-u^2]dtheta}{left[int_{-pi}^{pi}|u| dthetaright]^2}$$ where $uin mathcal{W}$ is a $H^1(-pi,pi)$ periodic function, with zero average on $(-pi,pi)$ and orthogonal to sine and cosine.
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