We prove some new $L^p$ estimates for maximal Bochner-Riesz operator in the plane.
Phase III randomized clinical trials play a monumentally critical role in the evaluation of new medical products. Because of the intrinsic nature of uncertainty embedded in our capability in assessing the efficacy of a medical product, interpretation
of trial results relies on statistical principles to control the error of false positives below desirable level. The well-established statistical hypothesis testing procedure suffers from two major limitations, namely, the lack of flexibility in the thresholds to claim success and the lack of capability of controlling the total number of false positives that could be yielded by the large volume of trials. We propose two general theoretical frameworks based on the conventional frequentist paradigm and Bayesian perspectives, which offer realistic, flexible and effective solutions to these limitations. Our methods are based on the distribution of the effect sizes of the population of trials of interest. The estimation of this distribution is practically feasible as clinicaltrials.gov provides a centralized data repository with unbiased coverage of clinical trials. We provide a detailed development of the two frameworks with numerical results obtained for industry sponsored Phase III randomized clinical trials.
Non-Line-of-Sight (NLOS) imaging allows to observe objects partially or fully occluded from direct view, by analyzing indirect diffuse reflections off a secondary, relay surface. Despite its many potential applications, existing methods lack practica
l usability due to several shared limitations, including the assumption of single scattering only, lack of occlusions, and Lambertian reflectance. We lift these limitations by transforming the NLOS problem into a virtual Line-Of-Sight (LOS) one. Since imaging information cannot be recovered from the irradiance arriving at the relay surface, we introduce the concept of the phasor field, a mathematical construct representing a fast variation in irradiance. We show that NLOS light transport can be modeled as the propagation of a phasor field wave, which can be solved accurately by the Rayleigh-Sommerfeld diffraction integral. We demonstrate for the first time NLOS reconstruction of complex scenes with strong multiply scattered and ambient light, arbitrary materials, large depth range, and occlusions. Our method handles these challenging cases without explicitly developing a light transport model. By leveraging existing fast algorithms, we outperform existing methods in terms of execution speed, computational complexity, and memory use. We believe that our approach will help unlock the potential of NLOS imaging, and the development of novel applications not restricted to lab conditions. For example, we demonstrate both refocusing and transient NLOS videos of real-world, complex scenes with large depth.
Using Guths polynomial partitioning method, we obtain $L^p$ estimates for the maximal function associated to the solution of Schrodinger equation in $mathbb R^2$. The $L^p$ estimates can be used to recover the previous best known result that $lim_{t
to 0} e^{itDelta}f(x)=f(x)$ almost everywhere for all $f in H^s (mathbb{R}^2)$ provided that $s>3/8$.
In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation begin{equation*} (-Delta)^{frac{alpha}{2}} u=lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{alpha}-2}u, quadtext{in},,Omega, u=0,text{on},,partialOmega, end
{equation*} where $Omegasubset mathbb{R}^{N}(Ngeq 2)$ is a bounded domain with smooth boundary, $0<alpha<2$, $(-Delta)^{frac{alpha}{2}}$ stands for the fractional Laplacian operator, $fin C(Omegatimesmathbb{R},mathbb{R})$ may be sign changing and $lambda$ is a positive parameter. We will prove that there exists $lambda_{*}>0$ such that the problem has at least two positive solutions for each $lambdain (0,,,lambda_{*})$. In addition, the concentration behavior of the solutions are investigated.
We establish the existence and multiplicity of positive solutions to the problems involving the fractional Laplacian: begin{equation*} left{begin{array}{lll} &(-Delta)^{s}u=lambda u^{p}+f(u),,,u>0 quad &mbox{in},,Omega, &u=0quad &mbox{in},,mathbb{R}^
{N}setminusOmega, end{array}right. end{equation*} where $Omegasubset mathbb{R}^{N}$ $(Ngeq 2)$ is a bounded smooth domain, $sin (0,1)$, $p>0$, $lambdain mathbb{R}$ and $(-Delta)^{s}$ stands for the fractional Laplacian. When $f$ oscillates near the origin or at infinity, via the variational argument we prove that the problem has arbitrarily many positive solutions and the number of solutions to problem is strongly influenced by $u^{p}$ and $lambda$. Moreover, various properties of the solutions are also described in $L^{infty}$- and $X^{s}_{0}(Omega)$-norms.
This paper is devoted to prove the existence and nonexistence of positive solutions for a class of fractional Schrodinger equation in RN of the We apply a new methods to obtain the existence of positive solutions when f(u) is asymptotically linear with respect to u at infinity.
We study the bilinear Hilbert transform and bilinear maximal functions associated to polynomial curves and obtain uniform $L^r$ estimates for $r>frac{d-1}{d}$ and this index is sharp up to the end point.
In this paper, we consider a discrete restriction associated with KdV equations. Some new Strichartz estimates are obtained. We also establish the local well-posedness for the periodic generalized Korteweg-de Vries equation with nonlinear term $ F(u)
p_x u$ provided $Fin C^5$ and the initial data $phiin H^s$ with $s>1/2$.
In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgains level set result on Strichartz estimates associated with Schrodinger equations on torus. Some sharp estimates on $L^{frac{2(d+2)}{d}}$ norm
of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on $L^2(mathbb Z)$.
