arXiv subjects

Yutian Lei

Publications and source records attributed to Yutian Lei.

At least 19 recordsLinked to original sources

arXiv2026

Distribution solutions of a static dispersion Schrödinger equation

In this paper, we study qualitative properties of distribution solutions of a fourth order equation $$ -Δu(x)+a^2Δ^2u(x)=u^q(x), \quad u(x)>0 \ \ in \ \ \mathbb{R}^3, $$ where $a>0$ and $q>0$. It is the static equation of a mixed dispersion Schrodinger equation, and also the Euler-Lagrange equation satisfied by extremal functions of an embedding inequality. We obtain some Liouville theorems and the corresponding related critical exponents, which imply the best constant of the embedding inequality cannot be attainable. We also obtain some regularity results (involving differentiability, integrability, radial symmetry) and asymptotics at infinity of distribution solutions. Here an equivalent integral equation with the Coulomb potential $|x|^{-1}(1-e^{-|x|/a})$ plays a key role. In addition, we also use the Pohozaev identity in integral form to obtain the Liouville theorem of this integral equation. Such the Pohozaev identity still works to handle the Allen-Cahn-type integral equation.

math.AP↗

arXiv2026

Radial symmetry of positive solutions of an integral system associated with the reversed Stein-Weiss inequality

Whether the solutions of conformal equations in the whole space are radially symmetric is an interesting topic. Chen-Li-Ou proved the radial symmetry for integral systems of the Hardy-Littlewood-Sobolev type and the Stein-Weiss type by the method of moving planes in integral form. In 2015, Dou-Zhu obtained the radial symmetry of extremal functions of the reversed Hardy-Littlewood-Sobolev inequality by the method of moving spheres, and Liu proved the radial symmetry of solutions of the Euler-Lagrange system by the method of moving planes developed by Dou-Guo-Zhu. In this paper, we also use the method of moving planes to prove the radial symmetry of positive solutions of the Euler-Lagrange system satisfied by the extremal functions of the reversed Stein-Weiss inequality established by Chen-Liu-Lu-Tao in 2018.

math.AP↗

arXiv2026

Reversed inequality of the Herbst-type and the related Euler-Lagrange system

In 2008, Beckner (Proc. Amer. Math. Soc. 136(5), 1871-1885) proved two inequalities of the Herbst type, which are the critical forms of the Stein-Weiss inequality. In 2018, Chen et al. (Tran. Amer. Math. Soc. 370(12), 8429-8450) established the reversed Stein-Weiss inequality. In this paper, we are concerned about its critical case and give a reversed Herbst inequality. Namely, $$ \left|\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|x-y|^{α/q'-n}|y|^{α/q'}g(x)h(y)dxdy\right| \geq C_{n,α,p,q'}\|g\|_{L^{q'}(\mathbb{R}^n)}\|h\|_{L^p(\mathbb{R}^n)} $$ holds for any nonnegative functions $g \in L^{q'}(\mathbb{R}^n)$ and $h \in L^p(\mathbb{R}^n)$, where $n\geq 1$, $p, q' \in (0,1)$, $α>n$ satisfying ${1}/{p}+{1}/{q'}-{2α}/(q'n)=1$. Such an inequality is not covered by the reversed Stein-Weiss inequality. Meanwhile, we prove the existence of extremal functions of this inequality. Finally, we study the Euler-Lagrange system satisfied by those extremal functions $$ \left\{\begin{matrix} u(x)=\int_{\mathbb{R}^n}|x-y|^{β-n}v^{-p_2}(y)|y|^βdy, v(x)=\int_{\mathbb{R}^n}|x-y|^{β-n}u^{-p_1}(y)|x|^βdy. \end{matrix}\right. $$ We obtain necessary conditions for the existence of positive solutions, and investigate their integrability and asymptotic behavior when $|x| \to 0$ and $|x| \to \infty$.

math.AP↗

arXiv2025

Sharp criteria for a degenerate diffusion-aggregation system with the intermediate exponent

In this paper, we investigate a multi-dimensional nonlocal degenerate diffusion-aggregation equation with a diffusion exponent $m$ in the intermediate range $\frac{2d}{2d-γ}<m<\frac{d+γ}{d}$, where the nonlocal aggregation term is given by singular potential $|x|^{-γ}$, $0<γ\leq d-2$. Under two different assumptions on the initial data, we establish two sharp criteria (i.e., the critical thresholds in Theorem 1.1 and Theorem 1.2) governing the global existence and finite-time blow-up of solutions. Once the initial free energy is less than a constant that depends on the total mass (or depends on the extremum function of the Hardy-Littlewood-Sobolev inequality), the first criterion depends on the relationship between the $L^{\frac{2d}{2d-γ}}$-norm of initial data and total mass, while the second relies on the relationship between the $L^m$-norm of initial data and extremal function. In the discussion of the second criterion, we do not require $L^\infty(\mathbb{R}^d)$ boundedness of the initial data, which is necessary in reference \cite{B}. Furthermore, with the help of moment estimate, we manage to prove the compactness argument on the whole space by using the Lions-Aubin Lemma. Importantly, we demonstrate that the two initial free energy conditions on which two criteria are based are equivalent. Building on this, we further prove that the two sharp criteria themselves are also equivalent, thereby unifying the classification results obtained from two different approaches.

math.AP↗

arXiv2024

Mining In-distribution Attributes in Outliers for Out-of-distribution Detection

Out-of-distribution (OOD) detection is indispensable for deploying reliable machine learning systems in real-world scenarios. Recent works, using auxiliary outliers in training, have shown good potential. However, they seldom concern the intrinsic correlations between in-distribution (ID) and OOD data. In this work, we discover an obvious correlation that OOD data usually possesses significant ID attributes. These attributes should be factored into the training process, rather than blindly suppressed as in previous approaches. Based on this insight, we propose a structured multi-view-based out-of-distribution detection learning (MVOL) framework, which facilitates rational handling of the intrinsic in-distribution attributes in outliers. We provide theoretical insights on the effectiveness of MVOL for OOD detection. Extensive experiments demonstrate the superiority of our framework to others. MVOL effectively utilizes both auxiliary OOD datasets and even wild datasets with noisy in-distribution data. Code is available at https://github.com/UESTC-nnLab/MVOL.

cs.LG↗

arXiv2024

RT-Grasp: Reasoning Tuning Robotic Grasping via Multi-modal Large Language Model

Recent advances in Large Language Models (LLMs) have showcased their remarkable reasoning capabilities, making them influential across various fields. However, in robotics, their use has primarily been limited to manipulation planning tasks due to their inherent textual output. This paper addresses this limitation by investigating the potential of adopting the reasoning ability of LLMs for generating numerical predictions in robotics tasks, specifically for robotic grasping. We propose Reasoning Tuning, a novel method that integrates a reasoning phase before prediction during training, leveraging the extensive prior knowledge and advanced reasoning abilities of LLMs. This approach enables LLMs, notably with multi-modal capabilities, to generate accurate numerical outputs like grasp poses that are context-aware and adaptable through conversations. Additionally, we present the Reasoning Tuning VLM Grasp dataset, carefully curated to facilitate the adaptation of LLMs to robotic grasping. Extensive validation on both grasping datasets and real-world experiments underscores the adaptability of multi-modal LLMs for numerical prediction tasks in robotics. This not only expands their applicability but also bridges the gap between text-based planning and direct robot control, thereby maximizing the potential of LLMs in robotics.

cs.RO↗

arXiv2024

Reasoning Grasping via Multimodal Large Language Model

Despite significant progress in robotic systems for operation within human-centric environments, existing models still heavily rely on explicit human commands to identify and manipulate specific objects. This limits their effectiveness in environments where understanding and acting on implicit human intentions are crucial. In this study, we introduce a novel task: reasoning grasping, where robots need to generate grasp poses based on indirect verbal instructions or intentions. To accomplish this, we propose an end-to-end reasoning grasping model that integrates a multimodal Large Language Model (LLM) with a vision-based robotic grasping framework. In addition, we present the first reasoning grasping benchmark dataset generated from the GraspNet-1 billion, incorporating implicit instructions for object-level and part-level grasping. Our results show that directly integrating CLIP or LLaVA with the grasp detection model performs poorly on the challenging reasoning grasping tasks, while our proposed model demonstrates significantly enhanced performance both in the reasoning grasping benchmark and real-world experiments.

cs.RO↗

arXiv2024

RLingua: Improving Reinforcement Learning Sample Efficiency in Robotic Manipulations With Large Language Models

Reinforcement learning (RL) has demonstrated its capability in solving various tasks but is notorious for its low sample efficiency. In this paper, we propose RLingua, a framework that can leverage the internal knowledge of large language models (LLMs) to reduce the sample complexity of RL in robotic manipulations. To this end, we first present a method for extracting the prior knowledge of LLMs by prompt engineering so that a preliminary rule-based robot controller for a specific task can be generated in a user-friendly manner. Despite being imperfect, the LLM-generated robot controller is utilized to produce action samples during rollouts with a decaying probability, thereby improving RL's sample efficiency. We employ TD3, the widely-used RL baseline method, and modify the actor loss to regularize the policy learning towards the LLM-generated controller. RLingua also provides a novel method of improving the imperfect LLM-generated robot controllers by RL. We demonstrate that RLingua can significantly reduce the sample complexity of TD3 in four robot tasks of panda_gym and achieve high success rates in 12 sampled sparsely rewarded robot tasks in RLBench, where the standard TD3 fails. Additionally, We validated RLingua's effectiveness in real-world robot experiments through Sim2Real, demonstrating that the learned policies are effectively transferable to real robot tasks. Further details about our work are available at our project website https://rlingua.github.io.

cs.RO↗

arXiv2024

VIHE: Virtual In-Hand Eye Transformer for 3D Robotic Manipulation

In this work, we introduce the Virtual In-Hand Eye Transformer (VIHE), a novel method designed to enhance 3D manipulation capabilities through action-aware view rendering. VIHE autoregressively refines actions in multiple stages by conditioning on rendered views posed from action predictions in the earlier stages. These virtual in-hand views provide a strong inductive bias for effectively recognizing the correct pose for the hand, especially for challenging high-precision tasks such as peg insertion. On 18 manipulation tasks in RLBench simulated environments, VIHE achieves a new state-of-the-art, with a 12% absolute improvement, increasing from 65% to 77% over the existing state-of-the-art model using 100 demonstrations per task. In real-world scenarios, VIHE can learn manipulation tasks with just a handful of demonstrations, highlighting its practical utility. Videos and code implementation can be found at our project site: https://vihe-3d.github.io.

cs.RO↗

arXiv2023

MAC: ModAlity Calibration for Object Detection

The flourishing success of Deep Neural Networks(DNNs) on RGB-input perception tasks has opened unbounded possibilities for non-RGB-input perception tasks, such as object detection from wireless signals, lidar scans, and infrared images. Compared to the matured development pipeline of RGB-input (source modality) models, developing non-RGB-input (target-modality) models from scratch poses excessive challenges in the modality-specific network design/training tricks and labor in the target-modality annotation. In this paper, we propose ModAlity Calibration (MAC), an efficient pipeline for calibrating target-modality inputs to the DNN object detection models developed on the RGB (source) modality. We compose a target-modality-input model by adding a small calibrator module ahead of a source-modality model and introduce MAC training techniques to impose dense supervision on the calibrator. By leveraging (1) prior knowledge synthesized from the source-modality model and (2) paired {target, source} data with zero manual annotations, our target-modality models reach comparable or better metrics than baseline models that require 100% manual annotations. We demonstrate the effectiveness of MAC by composing the WiFi-input, Lidar-input, and Thermal-Infrared-input models upon the pre-trained RGB-input models respectively.

cs.CV↗

arXiv2023

Simultaneous 3D Construction and Imaging of Plant Cells Using Plasmonic Nanoprobe Assisted Multimodal Nonlinear Optical Microscopy

Nonlinear optical (NLO) imaging has emerged as a promising plant cell imaging technique due to its large optical penetration, inherent 3D spatial resolution, and reduced photodamage, meanwhile exogenous nanoprobes are usually needed for non-signal target cell analysis. Here, we report in-vivo, simultaneous 3D labeling and imaging of potato cell structures using plasmonic nanoprobe-assisted multimodal NLO microscopy. Experimental results show that the complete cell structure could be imaged by the combination of second-harmonic generation (SHG) and two-photon luminescence (TPL) when noble metal silver or gold ions are added. In contrast, without noble metal ion solution, no NLO signals from the cell wall could be acquired. The mechanism can be attributed to noble metal nanoprobes with strong nonlinear optical responses formed along the cell walls via a femtosecond laser scan. During the SHG-TPL imaging process, noble metal ions that cross the cell wall could be rapidly reduced to plasmonic nanoparticles by fs laser and selectively anchored onto both sides of the cell wall, thereby leading to simultaneous 3D labeling and imaging of potato cells. Compared with traditional labeling technique that needs in-vitro nanoprobe fabrication and cell labeling, our approach allows for one-step, in-vivo labeling of plant cells, thus providing a rapid, cost-effective way for cellular structure construction and imaging.

physics.optics↗

arXiv2020

A Liouville theorem for an integral equation of the Ginzburg-Landau type

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\overrightarrow{l}+C_*\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-α}}dy. \end{equation*} Here $u: \mathbb{R}^{n} \to \mathbb{R}^{k}$ is a bounded, uniformly continuous and differentiable function with $k \geq 1$ and $1<α<n$, $\overrightarrow{l} \in \mathbb{R}^{k}$ is a constant vector, and $C_*$ is a real constant. If $u$ is the finite energy solution, we prove that $|\overrightarrow{l}| \in \{0,1\}$. Furthermore, we also give a Liouville type theorem (i.e., $u \equiv \overrightarrow{l}$).

math.AP↗

arXiv2020

A Liouville theorem for the fractional Ginzburg-Landau equation

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-α}}dy, \end{equation*} where $u: \mathbb{R}^{n} \to \mathbb{R}^{k}$ with $k \geq 1$ and $1<α<n/2$. We prove that $u \in L^2(\mathbb{R}^n) \Rightarrow u \equiv 0$ on $\mathbb{R}^n$, as long as $u$ is a bounded and differentiable solution.

math.AP↗

arXiv2016

On critical exponents of a $k$-Hessian equation in the whole space

In this paper, we study negative classical solutions and stable solutions of the following $k$-Hessian equation $$ F_k(D^2V)=(-V)^p \quad in~R^n $$ with radial structure, where $n \geq 3$, $1 1$. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to research $k$-Hessian equations without radial structure.

math.AP↗

arXiv2014

Some existence and nonexistence results for a Schrödinger-Poisson type system

In this paper, we study the Schrödinger-Poisson system $$ \left \{ \begin{array}{l} -Δu=\sqrt{p}u^{p-1}v, \quad u>0 \quad in \quad R^n, -Δv=\sqrt{p}u^p, \quad v>0 \quad in \quad R^n \end{array} \right. $$ with $n \geq 3$ and $p>1$. We investigate the existence and the nonexistence of positive classical solutions with the help of an integral system involving the Newton potential $$ \left \{ \begin{array}{l} u(x)=c_1\displaystyle\int_{R^n}\frac{u^{p-1}(y)v(y)dy}{|x-y|^{n-2}}, \quad u>0 \quad in \quad R^n, v(x)=c_2\displaystyle\int_{R^n}\frac{u^p(y)dy}{|x-y|^{n-2}} \quad v>0 \quad in \quad R^n. \end{array} \right. $$ First, the system has no solution when $p\leq \frac{n}{n-2}$. When $p>\frac{n}{n-2}$, the system has a singular solution on $R^n \setminus \{0\}$ with slow asymptotic rate $\frac{2}{p-1}$. When $p<\frac{n+2}{n-2}$, the system has no solution in $L^{\frac{n(p-1)}{2}}(R^n)$. In fact, if the system has solutions in $L^{\frac{n(p-1)}{2}}(R^n)$, then $p=\frac{n+2}{n-2}$ and all the positive classical solutions can be classified as $u(x)=v(x)=c(\frac{t}{t^2+|x-x^*|^2})^{\frac{n-2}{2}}$, where $c,t$ are positive constants. When $p>\frac{n+2}{n-2}$, by the shooting method and the Pohozaev identity, we find another pair of radial solution $(u,v)$ satisfying $u \equiv v$ and decaying with slow rate $\frac{2}{p-1}$.

math.AP↗

arXiv2014

Finite energy solutions and critical conditions of nonlinear equations in $R^n$

This paper is concerned with the critical conditions of nonlinear elliptic equations with weights and the corresponding integral equations with Riesz potentials and Bessel potentials. We show that the equations and some energy functionals are invariant under the scaling transformation if and only if the critical conditions hold. In addition, the Pohozaev identity shows that those critical conditions are the necessary and sufficient conditions for existence of the finite energy positive solutions or weak solutions. Finally, we discuss respectively the existence of the negative solutions of the $k$-Hessian equations in the subcritical case, critical case and supercritical case. Here the Serrin exponent and the critical exponent play key roles.

math.AP↗

arXiv2013

Qualitative properties of positive solutions of quasilinear equations with Hardy terms

In this paper, we are concerned with the quasilinear PDE with weight $$ -div A(x,\nabla u)=|x|^a u^q(x), \quad u>0 \quad \textrm{in} \quad R^n, $$ where $n \geq 3$, $q>p-1$ with $p \in (1,2]$ and $a \in (-n,0]$. The positive weak solution $u$ of the quasilinear PDE is $\mathcal{A}$-superharmonic and satisfies $\inf_{R^n}u=0$. We can introduce an integral equation involving the wolff potential $$ u(x)=R(x) W_{β,p}(|y|^au^q(y))(x), \quad u>0 \quad \textrm{in} \quad R^n, $$ which the positive solution $u$ of the quasilinear PDE satisfies. Here $p \in (1,2]$, $q>p-1$, $β>0$ and $0 \leq -a \frac{(n+a)(p-1)}{n-pβ}$, the positive solution $u$ of the integral equation is bounded and decays with the fast rate $\frac{n-pβ}{p-1}$ if and only if it is integrable (i.e. it belongs to $L^{\frac{n(q-p+1)}{pβ+a}}(R^n)$). On the other hand, if the bounded solution is not integrable and decays with some rate, then the rate must be the slow one $\frac{pβ+a}{q-p+1}$. Thus, all the properties above are still true for the quasilinear PDE. Finally, several qualitative properties for this PDE are discussed.

math.AP↗

arXiv2013

Decay properties of the Hardy-Littlewood-Sobolev systems of the Lane-Emden type

In this paper, we study the asymptotic behavior of positive solutions of the nonlinear differential systems of Lane-Emden type $2k$-order equations $$\{{array}{l} (-Δ)^k u=v^q,u>0 \quad in ~R^n, (-Δ)^k v=u^p,v>0 \quad in ~R^n, {array}. $$ and the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations $$ \{{array}{l} u(x)=\displaystyle\int_{R^n}\frac{v^q(y)dy}{|x-y|^{n-α}},u>0 \quad in ~R^n, v(x)=\displaystyle\int_{R^n}\frac{u^p(y)dy}{|x-y|^{n-α}},u>0 \quad in ~R^n. {array}. $$ Such an integral system is related to the study the extremal functions of the HLS inequality. We point out that the bounded solutions $u,v$ converge to zero either with the fast decay rates or with the slow decay rates when $|x| \to \infty$ under some assumptions. In addition, we also find a criterion to distinguish the fast and the slow decay rates: if $u,v$ are the integrable solutions (i.e. $(u,v) \in L^{r_0}(R^n) \times L^{s_0}(R^n)$), then they decay fast; if the bounded solutions $u,v$ are not the integrable solutions (i.e. $(u,v) \not\in L^{r_0}(R^n) \times L^{s_0}(R^n)$), then they decay almost slowly. Here, for the HLS type system, $r_0=\frac{n(pq-1)}{α(q+1)}$, $s_0=\frac{n(pq-1)}{α(p+1)}$; and for the Lane-Emden type system, $r_0,s_0$ are still the forms above where $α$ is replaced by $2k$.

math.AP↗