13 papers
Theoretical Properties of Covariate-Adaptive Randomization with a Diverging Number of Covariates
Yuhang Tao, Li-Xin Zhang
Covariate-adaptive randomization procedures are widely used in clinical trials to improve covariate balance. In modern applications, experimenters often have access to many covaria…
Sharp Lifespan Dichotomies and Threshold Phenomena for Semilinear Heat Equations Driven by the Logarithmic Laplacian
Huyuan Chen, Rui Chen, Daniel Hauer +1
We study nonnegative mild solutions of \[ \partial_tu+(-Î)^{\ln}u=f(u) \qquad \text{in }(0,T)\times\mathbb R^N, \] with initial datum \(u(0,\cdot)=μu_0\), \(μ>0\). Unlike the cl…
The Fractional-Logarithmic Laplacian: Potentials, Regularity, and Critical Compact Embeddings
Rui Chen
We develop potential-theoretic and \(L^p\)-regularity results for the fractional--logarithmic Laplacian \((-Î)^{s+\ln}\) and its inhomogeneous counterpart \((λI-Î)^{s+\ln}\), \(…
Nonlocal Characterizations of Stochastic Completeness on Complete Riemannian Manifolds
Rui Chen, Bobo Hua
In this paper, we first prove that the following generalized conservation principle holds on complete Riemannian manifolds: for every \(0<s<1\) and \(t>0\), \[ T_t^{(s)}\mathbf 1+\…
The Fractional-Logarithmic Laplacian:Fundamental Properties and Eigenvalues
Huyuan Chen, Rui Chen, Daniel Hauer
In this paper, we introduce, for the first time, the fractional--logarithmic Laplacian \( (-Î)^{s+\log} \), defined as the derivative of the fractional Laplacian \( (-Î)^t \) at…
The Conformal Fractional--Logarithmic Laplacian on the Sphere: Yamabe Problems and Sharp Inequalities
Huyuan Chen, Rui Chen, Daniel Hauer
In this paper, we introduce the conformal fractional--logarithmic Laplacian on the unit sphere, defined as the derivative of the conformal fractional Laplacian with respect to the…