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Xiaoyu Zeng

Publications and source records attributed to Xiaoyu Zeng.

31 records · Page 2Linked to original sources
arXiv2020

Vision Skills Needed to Answer Visual Questions

The task of answering questions about images has garnered attention as a practical service for assisting populations with visual impairments as well as a visual Turing test for the artificial intelligence community. Our first aim is to identify the common vision skills needed for both scenarios. To do so, we analyze the need for four vision skills---object recognition, text recognition, color recognition, and counting---on over 27,000 visual questions from two datasets representing both scenarios. We next quantify the difficulty of these skills for both humans and computers on both datasets. Finally, we propose a novel task of predicting what vision skills are needed to answer a question about an image. Our results reveal (mis)matches between aims of real users of such services and the focus of the AI community. We conclude with a discussion about future directions for addressing the visual question answering task.

cs.HC↗
arXiv2018

Ground States of Two-Component Attractive Bose-Einstein Condensates II: Semi-trivial Limit Behavior

As a continuation of [14], we study new pattern formations of ground states $(u_1,u_2)$ for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in $R^2$, where the intraspecies interaction $(-a,-b)$ and the interspecies interaction $-β$ are both attractive, $i.e,$ $a$, $b$ and $β$ are all positive. If $0<b<a^*:=\|w\|^2_2$ and $0<β<a^*$ are fixed, where $w$ is the unique positive solution of $Δw-w+w^3=0$ in $R^2$, the semi-trivial behavior of $(u_1,u_2)$ as $a\nearrow a^*$ is proved in the sense that $u_1$ concentrates at a unique point and while $u_2\equiv 0$ in $R^2$. However, if $0<b<a^*$ and $a^*\leβ<β^*=a^*+\sqrt{(a^*-a)(a^*-b)}$, the refined spike profile and the uniqueness of $(u_1,u_2)$ as $a\nearrow a^*$ are analyzed, where $(u_1,u_2)$ must be unique, $u_1$ concentrates at a unique point, and meanwhile $u_2$ can either blow up or vanish, depending on how $β$ approaches to $a^*$.

math.AP↗
arXiv2017

Cylindrically Symmetric Ground State Solutions for Curl-Curl Equations with Critical Exponent

We study the following nonlinear critical curl-curl equation \begin{equation}\label{eq0.1}\nabla\times \nabla\times U +V(x)U=|U|^{p-2}U+ |U|^4U,\quad x\in \mathbb{R}^3,\end{equation} where $V(x)=V(r, x_3)$ with $r=\sqrt{x_1^2+x_2^2}$ is 1-periodic in $x_3$ direction and belongs to $L^\infty(\R^3)$. When $0\not\in σ(-Δ+\frac{1}{r^2}+V)$ and $p\in(4,6)$, we prove the existence of nontrivial solution for (\ref{eq0.1}), which is indeed a ground state solution in a suitable cylindrically symmetric space. Especially, if $ σ(-Δ+\frac{1}{r^2}+V)>0$, a ground state solution is obtained for any $p\in(2,6)$.

math.AP↗
arXiv2017

Ground States of Two-Component Attractive Bose-Einstein Condensates I: Existence and Uniqueness

We study ground states of two-component Bose-Einstein condensates (BEC) with trapping potentials in $R^2$, where the intraspecies interaction $(-a_1,-a_2)$ and the interspecies interaction $-β$ are both attractive, $i.e,$ $a_1$, $a_2$ and $β$ are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated $L^2$-critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as $β\nearrow β^*=a^*+\sqrt{(a^*-a_1)(a^*-a_2)}$, where $0<a_i<a^*:=\|w\|^2_2$ ($i=1,2$) is fixed and $w$ is the unique positive solution of $Δw-w+w^3=0$ in $R^2$. The semi-trivial limit behavior of ground states is tackled in the companion paper [12].

math.AP↗
arXiv2017

Blow-up behavior of ground states for a nonlinear Schrödinger system with attractive and repulsive interactions

We consider a nonlinear Schrödinger system arising in a two-component Bose-Einstein condensate (BEC) with attractive intraspecies interactions and repulsive interspecies interactions in $\mathbb{R}^2$. We get ground states of this system by solving a constrained minimization problem. For some kinds of trapping potentials, we prove that the minimization problem has a minimizer if and only if the attractive interaction strength $a_i (i=1,2)$ of each component of the BEC system is strictly less than a threshold $a^*$. %attractive intraspecies interactions satisfies $a_i< %a^*= \|Q\|_2^2,\ i=1,\,2$, where $Q$ is the unique positive radial solution of $Δu-u+u^3=0$ in $\mathbb{R}^2$; in contrast, there is no minimizer if either $a_i > a^*$ for $i=1$ or $2$, or $a_1=a_2=a^*$. Furthermore, as $(a_1, a_2)\nearrow (a^*, a^*)$, the asymptotical behavior for the minimizers of the minimization problem is discussed. Our results show that each component of the BEC system concentrates at a global minimum of the associated trapping potential.

math-ph↗
arXiv2017

Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions

The paper is concerned with a system of two coupled time-independent Gross-Pitaevskii equations in $\mathbb{R}^2$, which is used to model two-component Bose-Einstein condensates with both attractive intraspecies and attractive interspecies interactions. This system is essentially an eigenvalue problem of a stationary nonlinear Schrödinger system in $\mathbb{R}^2$, solutions of the problem are obtained by seeking minimizers of the associated variational functional with constrained mass (i.e. $L^2-$norm constaints). Under certain type of trapping potentials $V_i(x)$ ($i=1,2$), the existence, non-existence and uniqueness of this kind of solutions are studied. Moreover, by establishing some delicate energy estimates, we show that each component of the solutions blows up at the same point (i.e., one of the global minima of $V_i(x)$) when the total interaction strength of intraspecies and interspecies goes to a critical value. An optimal blowing up rate for the solutions of the system is also given.

math.AP↗
arXiv2017

Existence and Uniqueness of Normalized Solutions for the Kirchhoff equation

For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent $p$ for its $L^2$-normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations. Secondly, we search for the mountain pass type critical point for the functional on the $L^2$-normalized manifold, and also prove that this type critical point is unique up to translations. Our proof relies only on some simple energy estimates and avoids using the concentration-compactness principles. These conclusions extend some known results in previous papers.

math.AP↗
arXiv2017

Existence and asymptotic behavior for the ground state of quasilinear elliptic equation

In this paper, we are concerned with the existence and asymptotic behavior of minimizers for a minimization problem related to some quasilinear elliptic equations. Firstly, we proved that there exist minimizers when the exponent $q$ equals to the critical case $q^*=2+\frac{4}{N}$, which is different from that of \cite{cjs}. Then, we proved that all minimizers are compact as $q$ tends to the critical case $q^*$ when $a a^*$.

math.AP↗
arXiv2016

Eigenvalue problem for a p-Laplacian equation with trapping potentials

Consider the following eigenvalue problem of p-Laplacian equation \begin{equation}\label{P} -Δ_{p}u+V(x)|u|^{p-2}u=μ|u|^{p-2}u+a| u|^{s-2}u, x\in \mathbb{R}^{n}, \tag{P} \end{equation} where $a\geq0$, $p\in (1,n)$ and $μ\in\mathbb{R}$. $V(x)$ is a trapping type potential, e.g., $\inf\limits_{x \in \mathbb{R}^n}V(x)< \lim\limits_{|x|\rightarrow+\infty}V(x)$. By using constrained variational methods, we proved that there is $a^*>0$, which can be given explicitly, such that problem (\ref{P}) has a ground state $u$ with $\|u\|_{L^p}=1$ for some $μ\in \mathbb{R}$ and all $a\in [0,a^*)$, but (\ref{P}) has no this kind of ground state if $a\geq a^*$. Furthermore, by establishing some delicate energy estimates we show that the global maximum point of the ground states of problem (\ref{P}) approach to one of the global minima of $V(x)$ and blow up if $a\nearrow a^*$. The optimal rate of blowup is obtained for $V(x)$ being a polynomial type potential.

math.AP↗
arXiv2015

Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials

We are interested in the attractive Gross-Pitaevskii (GP) equation in $\R^2$, where the external potential $V(x)$ vanishes on $m$ disjoint bounded domains $Ω_i\subset \R^2\ (i=1,2,\cdots,m)$ and $V(x)\to\infty$ as $|x|\to\infty$, that is, the union of these $Ω_i$ is the bottom of the potential well. By making some delicate estimates on the energy functional of the GP equation, we prove that when the interaction strength $a$ approaches some critical value $a^*$ the ground states concentrate and blow up at the center of the incircle of some $Ω_j$ which has the largest inradius. Moreover, under some further conditions on $V(x)$ we show that the ground states of GP equations are unique and radially symmetric at leat for almost every $a \in (0, a^*)$.

math.AP↗
arXiv2014

Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials

This paper is concerned with the properties of $L^2$-normalized minimizers of the Gross-Pitaevskii (GP) functional for a two-dimensional Bose-Einstein condensate with attractive interaction and ring-shaped potential. By establishing some delicate estimates on the least energy of the GP functional, we prove that symmetry breaking occurs for the minimizers of the GP functional as the interaction strength $a>0$ approaches a critical value $a^*$, each minimizer of the GP functional concentrates to a point on the circular bottom of the potential well and then is non-radially symmetric as $a\nearrow a^*$. However, when $a>0$ is suitably small we prove that the minimizers of the GP functional are unique, and this unique minimizer is radially symmetric.

math.FA↗
arXiv2013

Concentration behavior of standing waves for almost mass critical nonlinear Schrödinger equations

We study the following nonlinear Schrödinger equation $$ iu_t=-Δu+V(x)u-a|u|^qu \quad (t,x)\in \mathbb{R}^1\times \mathbb{R}^2, $$ where $a>0, \ q\in(0,2)$ and $V(x)$ is some type of trapping potentials. For any fixed $a>a^*:= \|Q\|_2^2$, where $Q$ is the unique (up to translations) positive radial solution of $Δu-u+u^3=0$ in $\mathbb{R}^2$, by directly using constrained variational method and energy estimates we present a detailed analysis of the concentration and symmetry breaking of the standing waves for the above equation as $q\nearrow 2$.

math.AP↗