arXiv subjects
Publications and source records attributed to Xuanyu Li.
We consider a non-interacting Bose gas governed by a general one-particle Hamiltonian in the canonical ensemble. Our main results are large-deviation estimates for the number of particles in the Bose--Einstein condensate. We show that the left- and right-tail probabilities decay at different exponential rates as the number of particles tends to infinity.
We show the existence of non-trivial self-expanding harmonic map flows starting from non-energy-minimizing 0-homogeneous maps to a regular ball or a closed hemisphere. In particular, given a non-minimizing but stationary 0-homogeneous harmonic map $u_0$ to a closed hemisphere, we construct infinitely many different weak solutions to harmonic map flow starting from $u_0$, all of which satisfy the parabolic monotonicity formula. We also use a concatenation argument to solve a non-trivial smooth harmonic map flow starting from a given non-minimizing regular 0-homogeneous harmonic map, thereby yielding the non-uniqueness of harmonic map flow with monotonicity formula. This answers a question of Struwe.
In this paper, we show that the codimension of the singular set of a stable stationary harmonic map to a round $k$-sphere is at least $k+1$ when $k$ is between 3 and 6, and is at least 7 when $k$ is at least 7. The result is sharp in the sense that there exist energy minimizing 0-homogeneous maps in the aforementioned critical dimensions. We also establish non-trivial index bounds for the non-constant harmonic maps from $n$-spheres to $k$-spheres, provided that $n$ is less than $k$.
We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $Σ$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The genus of the surface $Σ$ is bounded from above by the Heegaard genus $h$ of $\mathcal{M}$. Starting from a family of sweep-outs of $\mathcal{M}$ by surfaces of genus $h$, we apply a min-max construction for a family $\{E_{H,σ}\}_σ$ of perturbations of the energy involving the second fundamental form of the immersions to produce almost-critical points $u_k$ of $E_{H,σ}$. We then show, following ideas introduced by Rivière and developed by Pigati and Rivière, that the maps $u_k$ converge to a "CMC-parametrized varifold". This limiting object is then shown to be a smooth, branched immersion with the prescribed mean curvature $H$.
Many weak instrumental variables (IVs) are routinely used in the health and social sciences to improve identification and inference of the treatment effect of interest, along with a broad collection of data on potential confounding factors in the hope that the IV assumptions hold within each data stratum. We propose a new debiased continuous-updating generalized method of moments estimator with multi-task learning of the IV propensity scores to simultaneously address the biases from a diverging number of weak IVs as well as first-step regularized estimation of nuisance regression functions in high-dimensional potential confounding factors. We develop a new multi-task learning theory for generalized linear models under a general sub-Gaussian design to establish valid inference in the many weak IVs asymptotic regime under appropriate sparsity conditions. We evaluate the proposed method via extensive Monte Carlo studies and an empirical application to investigate the returns to education.
Given two Riemannian manifolds $M$ and $N\subset\mathbb{R}^L$, we consider the energy concentration phenomena of the penalized energy functional $$E_ε(u)=\int_M\frac{\vert\nabla u\vert^2}{2}+\frac{F(u)}{ε^2},u\in W^{1,2}(M,\mathbb{R}^L),$$ where $F(x)$=dist$(x,N)$ in a small tubular neighborhood of $N$ and is constant away from $N$. It was shown by Chen-Struwe that as $ε\rightarrow0$, the critical points $u_ε$ of $E_ε$ with energy bound $E_ε(u_ε)\leqslantΛ$ subsequentially converge weakly in $W^{1,2}$ to a weak harmonic map $u:M\rightarrow N$ . In addition, we have the convergence of the energy density $$\left(\frac{\vert\nabla u_ε\vert^2}{2}+\frac{F(u_ε)}{ε^2}\right)dx\rightarrow\frac{\vert\nabla u_ε\vert^2}{2}dx+ν,$$ and the defect measure $ν$ above is $(dimM-2)$-rectifiable. Lin-Wang showed that if $N$ is a sphere or dim$M$=2, then the density of $ν$ can be expressed by the sum of energies of harmonic spheres. In this paper, we prove this result for an arbitrary $M$ using the idea introduced by Naber-Valtorta.
Diffusion auction design for combinatorial settings is a long-standing challenge. One difficulty is that we cannot directly extend the solutions for simpler settings to combinatorial settings (like extending the Vickrey auction to VCG in the traditional settings). In this paper, we propose a different approach to leverage the diffusion auctions for single-item settings. We design a combinatorial diffusion auction framework which can use any desirable single-item diffusion auction to produce a combinatorial auction to satisfy incentive compatibility (IC), individual rationality (IR), and weak budget balance (WBB).
Let $Γ$ be a smooth, closed, oriented, $(n-1)$-dimensional submanifold of $\mathbb{R}^{n+1}$. It was shown by Chodosh-Mantoulidis-Schulze that one can perturb $Γ$ to a nearby $Γ'$ such that all minimizing currents with boundary $Γ'$ are smooth away from a set with Hausdorff dimension less than $n-9$. We prove that the perturbation can be made such that the singular set of the minimizing current with boundary $Γ'$ has Minkowski dimension less than $n-9$.
In this paper, we propose a pre-configured error pattern ordered statistics decoding (PEPOSD) algorithm and discuss its application to short cyclic redundancy check (CRC)-polar codes. Unlike the traditional OSD that changes the most reliable independent symbols, we regard the decoding process as testing the error patterns, like guessing random additive noise decoding (GRAND). Also, the pre-configurator referred from ordered reliability bits (ORB) GRAND can better control the range and testing order of EPs. Offline-online structure can accelerate the decoding process. Additionally, we also introduce two orders to optimize the search order for testing EPs. Compared with CRC-aided OSD and list decoding, PEPOSD can achieve a better trade-off between accuracy and complexity.