arXiv subjects
Publications and source records attributed to Ling Xiao.
In this paper, we study the $σ_k$ curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We prove that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show that after rescaling, the flow converges to a self-expander.
This paper presents a self-adaptive training (SAT) model for fashion compatibility prediction. It focuses on the learning of some hard items, such as those that share similar color, texture, and pattern features but are considered incompatible due to the aesthetics or temporal shifts. Specifically, we first design a method to define hard outfits and a difficulty score (DS) is defined and assigned to each outfit based on the difficulty in recommending an item for it. Then, we propose a self-adaptive triplet loss (SATL), where the DS of the outfit is considered. Finally, we propose a very simple conditional similarity network combining the proposed SATL to achieve the learning of hard items in the fashion compatibility prediction. Experiments on the publicly available Polyvore Outfits and Polyvore Outfits-D datasets demonstrate our SAT's effectiveness in fashion compatibility prediction. Besides, our SATL can be easily extended to other conditional similarity networks to improve their performance.
In this paper, we study fully nonlinear curvature flows of noncompact spacelike hypersurfaces in Minkowski space. We prove that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show that after rescaling the flow converges to the future timelike hyperboloid, which is a self-expander.
In [19], we prove that if an entire, spacelike, convex hypersurface $\mathcal{M}_{u_0}$ has bounded principal curvatures, then the $σ_k^{1/α}$ (power of $σ_k$) curvature flow starting from $\mathcal{M}_{u_0}$ admits a smooth convex solution $u$ for $t>0.$ Moreover, after rescaling, the flow converges to a convex self-expander $\tilde{\mathcal{M}}=\{(x, \tilde{u}(x))\mid x\in\mathbb{R}^n\}$ that satisfies $σ_k(κ[\tilde{\mathcal{M}}])=(-\left )^α.$ Unfortunately, the existence of self-expander for power of $σ_k$ curvature flow in Minkowski space has not been studied before. In this paper, we fill the gap.
In this paper, we investigate the existence and uniqueness of convex, entire, spacelike hypersurfaces of constant $σ_k$ curvature with prescribed set of lightlike directions $\mathcal{F}\subset\mathbb{S}^{n-1}$ and perturbation $q$ on $\mathcal{F}$. We prove that given a closed set $\mathcal{F}$ in the ideal boundary at infinity of hyperbolic space and a perturbation $q$ that satisfies some mild conditions, there exists a complete entire spacelike constant $σ_k$ curvature hypersurface $\mathcal{M}_u$ with prescribed set of lightlike directions $\mathcal{F}$ satisfying when $\frac{x}{|x|}\in\mathcal{F},$ as $|x|\rightarrow\infty,$ $u(x)-|x|\rightarrow q\left(\frac{x}{|x|}\right).$ This result is new even for the case of constant Gauss curvature. We also prove that when the Gauss map image is a half disc $\bar{B}_1^+$ and the perturbation $q\equiv 0,$ if a CMC hypersurface $\mathcal{M}_u$ satisfies $|u(x)-V_{\bar{\mathcal{B}}_+}(x)|$ is bounded, then $u(x)$ is unique.
In this paper, we study the flow of closed, starshaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^ασ_2^{1/2},$ where $σ_2^{1/2}$ is the normalized square root of the scalar curvature, $α\geq 2,$ and $r$ is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the starshapedness is preserved. Moreover, after normalization, we show that the flow converges exponentially fast to a sphere centered at origin. When $α<2,$ a counterexample is given for the above convergence.
We prove three results in this paper. First, we prove for a wide class of functions $φ\in C^2(\mathbb{S}^{n-1})$ and $ψ(X, ν)\in C^2(\mathbb{R}^{n+1}\times\mathbb{H}^n),$ there exists a unique, entire, strictly convex, spacelike hypersurface $M_u$ satisfying $σ_k(κ[M_u])=ψ(X, ν)$ and $u(x)\rightarrow |x|+φ\left(\frac{x}{|x|}\right)$ as $|x|\rightarrow\infty.$ Second, when $k=n-1, n-2,$ we show the existence and uniqueness of entire, $k$-convex, spacelike hypersurface $M_u$ satisfying $σ_k(κ[M_u])=ψ(x, u(x))$ and $u(x)\rightarrow |x|+φ\left(\frac{x}{|x|}\right)$ as $|x|\rightarrow\infty.$ Last, we obtain the existence and uniqueness of entire, strictly convex, downward translating solitons $M_u$ with prescribed asymptotic behavior at infinity for $σ_k$ curvature flow equations. Moreover, we prove that the downward translating solitons $M_u$ have bounded principal curvatures.
In this paper, we prove the existence of smooth, entire, strictly convex, spacelike, constant $σ_k$ curvature hypersurfaces with prescribed lightlike directions in Minkowski space. This is equivalent to prove the existence of smooth, entire, strictly convex, spacelike, constant $σ_k$ curvature hypersurfaces with prescribed Gauss map image. We also show that there doesn't exist any entire, convex, strictly spacelike, constant $σ_k$ curvature hypersurfaces. Moreover, we generalize the result in \cite{RWX} and construct strictly convex, spacelike, constant $σ_k$ curvature hypersurface with bounded principal curvature, whose image of the Gauss map is the unit ball.
We have designed and implemented a straightforward method to deterministically measure the temperature of the selected segment of a cold atom ensemble, and we have also developed an upgrade in the form of nondestructive thermometry. The essence is to monitor the thermal expansion of the targeted cold atoms after labeling them through manipulating the internal states, and the nondestructive property relies upon the nearly lossless detection via driving a cycling transition. For cold atoms subject to isotropic laser cooling, this method has the unique capability of addressing only the atoms on the optical detection axis within the enclosure, which is exactly the part we care about in major applications such as atomic clock or quantum sensing. Furthermore, our results confirm the sub-Doppler cooling features in isotropic laser cooling, and we have investigated the relevant cooling properties. Meanwhile, we have applied the recently developed optical configuration with the cooling laser injection in the form of hollow beams, which helps to enhance the cooling performance and accumulate more cold atoms in the central regions.
We prove that, in Minkowski space, if a spacelike, $(n-1)$-convex hypersurface $M$ with constant $σ_{n-1}$ curvature has bounded principal curvatures, then $M$ is convex. Moreover, if $M$ is not strictly convex, after an $\mathbb{R}^{n,1}$ rigid motion, $M$ splits as a product $M^{n-1}\times\mathbb{R}.$ We also construct nontrivial examples of strictly convex, spacelike hypersurface $M$ with constant $σ_{n-1}$ curvature and bounded principal curvatures.
We prove that any complete immersed two-sided mean convex translating soliton $Σ\subset \mathbb{R}^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in $\mathbb{R}^3$ is the axisymmetric "bowl soliton". We also show that if the mean curvature of $Σ$ tends to zero at infinity, then $Σ$ can be represented as an entire graph and so is the "bowl soliton". Finally we classify all locally strictly convex graphical translating solitons defined over strip regions.
Inspired by [6, 7], we study the boundary regularity of constant curvature hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$, which have prescribed asymptotic boundary at infinity. Through constructing the boundary expansions of the solutions, we derive the optimal regularity of the solutions. Moreover, we obtain an equivalent condition that guarantees the smoothness of the solutions.
In this paper, we study the motion of level sets by general curvature. The difficulty of this setting is that a general curvature function is only well defined in an admissible cone. In order to extend the existence of a weak solution of a general curvature flow to outside the cone we introduce a new approximation function $\hat{f}^n$ (see (3.1)). Moreover, using the idea in [4], we give an elliptic approach for the Ben-Andrews' non-collapsing result in fully nonlinear curvature flows; we hope this approach can be generalized to a wider class of elliptic equations.
In this paper, we prove long time existence and convergence results for a class of general curvature flows with Neumann boundary condition. This is the first result for the Neumann boundary problem of non Monge-Ampere type curvature equations. Our method also works for the corresponding elliptic setting.
In [5], Sáez and Schnürer studied the graphical mean curvature flow of complete hypersurfaces defined on subsets of Euclidean space. They obtained long time existence. Moreover, they provided a new interpretation of weak mean curvature flow. In this paper, we generalize their results to a general curvature setting. Our key ingredient is the existence result of general curvature flow with boundary conditions, which is proved in Section 4.
The medium-long term frequency stability of the integrating sphere cold atom clock was improved.During the clock operation, Rb atoms were cooled and manipulated using cooling light diffusely reflected by the inner surface of a microwave cavity in the clock. This light heated the cavity and caused a frequency drift from the resonant frequency of the cavity. Power fluctuations of the cooling light led to atomic density variations in the cavity's central area, which increased the clock frequency instability through a cavity pulling effect. We overcame these limitations with appropriate solutions. A frequency stability of 3.5E-15 was achieved when the integrating time ? increased to 2E4 s.
We present a modified scheme for detection of the magneto-optical rotation (MOR) effect, where a linearly polarized laser field is interacting with cold $^{87}$Rb atoms in an integrating sphere. The rotation angle of the probe beam's polarization plane is detected in the experiment. The results indicate that the biased magnetic field, the probe light intensity and detuning, and the cold atoms' temperature are key parameters for the MOR effect. This scheme may improve the contrast of the rotation signal and provide an useful approach for high contrast cold atom clocks and magnetometers.
We present a new scheme of compact Rubidium cold-atom clock which performs the diffuse light cooling, the microwave interrogation and the detection of the clock signal in a cylindrical microwave cavity. The diffuse light is produced by the reflection of the laser light at the inner surface of the microwave cavity. The pattern of injected laser beams is specially designed to make most of the cold atoms accumulate in the center of the microwave cavity. The microwave interrogation of cold atoms in the cavity leads to Ramsey fringes whose line-width is 24.5 Hz and the contrast of 95.6% when the free evolution time is 20 ms. The frequency stability of $7.3\times10^{-13}τ^{-1/2}$ has been achieved recently. The scheme of this physical package can largely reduce the complexity of the cold atom clock, and increase the performance of the clock.