arXiv subjects
Publications and source records attributed to Shiliang Zhao.
We prove the existence of a bounded Sobolev extension operator $E:W^{p,1}\left( B,P \right) \rightarrow W^{p,1}\left( \ell^{2} ,P \right)$ using a completely new method, where $B\subset \ell^{2}$ is the unit ball and $P$ is any non-trivial centered Gaussian measure on $\ell^{2}$. This solves an open problem posed in the literatures.
The classical ``$H=W$" theorem establishes the identity between two function spaces on an arbitrary nonempty open set in the Euclidean spaces: the space $W$ defined via weak derivatives, and the space $H$ defined as the closure of smooth functions within $W$ space. Extending this result to infinite-dimensional spaces is challenging due to the lack of a nontrivial translation-invariant measure and the proliferation of infinite sums inherent to infinite dimensions. In this paper, by adapting several techniques developed in our previous works, we prove that smooth functions are dense in the Sobolev space of functions on arbitrary non-empty open set in $\ell^2$, thereby establishing an infinite-dimensional counterpart of ``$H=W$". Such density results reduce the problem of deriving a priori $L^2$ estimates for differential operators -- originating from the classical Fredholm alternative and Carleman estimates -- to the simpler case of smooth functions. If approximation by smooth cylindrical functions is possible, the problem can be reduced to calculus. Unfortunately, this does not hold for every open set in $\ell^2$. However, we prove that such an approximation does hold on open sets that satisfy the segment condition.
Let $(M, g)$ be a smooth n-dimensional Riemannian manifold for $n\ge 2$. Consider the conformal perturbation $\tilde{g}=h g$ where $h$ is a smooth bounded positive function on $M$. Denote by $\tilde{p}_t(x,y)$ the heat kernel of manifolds $(M, \tilde{g})$. In this paper, we derive the upper bounds and gradient estimates of $\tilde{p}_t(x,y)$.
In this paper, first we consider the uniform complex time heat kernel estimates of $e^{-z(-Δ)^{\fracα{2}}}$ for $α>0, z\in \mathbb{C}^+$. When $\fracα{2}$ is not an integer, generally the heat kernel doest not have the Gaussian upper bounds for real time. Thus the Phragmén-Lindelöf methods fail to give the uniform complex time estimates. Instead, our first result gives the asymptotic estimates for $P(z, x)$ as $z$ tending to the imaginary axis. Then we prove the uniform complex time heat kernel estimates. Finally we also show the uniform estimates of analytic semigroup generated by $H=(-Δ)^{\fracα{2}}+V$ where $V$ belongs to higher order Kato class.
In this paper, the well-posedness of Cauchy's problem of fractional Schrödinger equations with a power type nonlinearity on $n$-dimensional manifolds with nonnegative Ricci curvature is studied. Under suitable volume conditions, the local solution with initial data in $H^{[\frac{n}{2}]+1}$ will blow up in finite time no matter how small the initial data is, which follows from a new weight function and ODE inequalities. Moreover, the upper-bound of the lifespan can be estimated.
A big challenge of person re-identification (Re-ID) using a multi-branch network architecture is to learn diverse features from the ID-labeled dataset. The 2-branch Batch DropBlock (BDB) network was recently proposed for achieving diversity between the global branch and the feature-dropping branch. In this paper, we propose to move the dropping operation from the intermediate feature layer towards the input (image dropping). Since it may drop a large portion of input images, this makes the training hard to converge. Hence, we propose a novel double-batch-split co-training approach for remedying this problem. In particular, we show that the feature diversity can be well achieved with the use of multiple dropping branches by setting individual dropping ratio for each branch. Empirical evidence demonstrates that the proposed method performs superior to BDB on popular person Re-ID datasets, including Market-1501, DukeMTMC-reID and CUHK03 and the use of more dropping branches can further boost the performance.
Learning diverse features is key to the success of person re-identification. Various part-based methods have been extensively proposed for learning local representations, which, however, are still inferior to the best-performing methods for person re-identification. This paper proposes to construct a strong lightweight network architecture, termed PLR-OSNet, based on the idea of Part-Level feature Resolution over the Omni-Scale Network (OSNet) for achieving feature diversity. The proposed PLR-OSNet has two branches, one branch for global feature representation and the other branch for local feature representation. The local branch employs a uniform partition strategy for part-level feature resolution but produces only a single identity-prediction loss, which is in sharp contrast to the existing part-based methods. Empirical evidence demonstrates that the proposed PLR-OSNet achieves state-of-the-art performance on popular person Re-ID datasets, including Market1501, DukeMTMC-reID and CUHK03, despite its small model size.
In this paper, we consider the Cauchy's problem of global existence and scattering behavior of small, smooth, and localized solutions of cubic fractional Schrödinger equations in one dimension, \begin{equation*} \mathrm{i} \partial_t u- (-Δ)^{\fracα{2}} u=c_*|u|^2u, \end{equation*} where $α\in (\frac{1}{3},1), c_* \in \mathbb{R}$. Our work is a generalization of the result due to Ionescu and Pusateri \cite{IP}, where the case $α=\frac{1}{2}$ was considered. The highlight in this paper is to give a modified dispersive estimate in weighted Sobolev spaces for cubic fractional Schrödinger equations, which could be used for $ α\in (\frac{1}{3},1)$. Based on this modified dispersive estimate, we prove the global existence and modified scattering behavior of solutions combining space-time resonance and bootstrap arguments.
In this paper, we consider the Cauchy problem for the fractional Schrödinger equation $i D_t^αu + (-Δ)^{\fracβ{2}} u =0$ with $0<α<1$, $β>0$. We establish the dispersive estimates for the solutions. In particular, we prove that the decay rates are sharp.
The semilinear space-time fractional Schrödinger equation is considered. First, we give the explicit form for the fundamental solutions by using the Fox $H$-functions in order to to establish some $L^s$ decay estimates. After that, we give some space-time estimates for the mild solutions from which the local well-posedness is derived on some proper Banach space.