arXiv subjects

Lidan Wang

Publications and source records attributed to Lidan Wang.

At least 19 recordsLinked to original sources

arXiv2025

Existence and multiplicity of solutions to discrete fractional logarithmic Kirchhoff equations

In this paper, we study the discrete fractional logarithmic Kirchhoff equation $$ \left(a+b \int_{\mathbb{Z}^d}|\nabla^s u|^{2} d μ\right) (-Δ)^s u+h(x) u=|u|^{p-2}u \log u^{2}, \quad x\in \mathbb{Z}^d, $$ where $a,\,b>0$ and $0 4$; then we verify the existence of ground state sign-changing solutions based on the method of Nehari manifold for $p>6$. Finally, we establish the multiplicity of nontrivial weak solutions.

math.AP↗

arXiv2025

Exploring Fourier Prior and Event Collaboration for Low-Light Image Enhancement

The event camera, benefiting from its high dynamic range and low latency, provides performance gain for low-light image enhancement. Unlike frame-based cameras, it records intensity changes with extremely high temporal resolution, capturing sufficient structure information. Currently, existing event-based methods feed a frame and events directly into a single model without fully exploiting modality-specific advantages, which limits their performance. Therefore, by analyzing the role of each sensing modality, the enhancement pipeline is decoupled into two stages: visibility restoration and structure refinement. In the first stage, we design a visibility restoration network with amplitude-phase entanglement by rethinking the relationship between amplitude and phase components in Fourier space. In the second stage, a fusion strategy with dynamic alignment is proposed to mitigate the spatial mismatch caused by the temporal resolution discrepancy between two sensing modalities, aiming to refine the structure information of the image enhanced by the visibility restoration network. In addition, we utilize spatial-frequency interpolation to simulate negative samples with diverse illumination, noise and artifact degradations, thereby developing a contrastive loss that encourages the model to learn discriminative representations. Experiments demonstrate that the proposed method outperforms state-of-the-art models.

cs.CV↗

arXiv2025

Positive solutions to fractional $p$-Laplacian Choquard equation on lattice graphs

In this paper, we study the fractional $p$-Laplacian Choquard equation $$ (-Δ)_{p}^{s} u+h(x)|u|^{p-2} u=\left(R_α *F(u)\right)f(u) $$ on lattice graphs $\mathbb{Z}^d$, where $s\in(0,1)$, $ p\geq 2$, $α\in(0, d)$ and $R_α$ represents the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under suitable assumptions on the potential function $h$, we first prove the existence of a strictly positive solution by the mountain-pass theorem for the nonlinearity $f$ satisfying some growth conditions. Moreover, if we add some monotonicity condition, we establish the existence of a positive ground state solution by the method of Nehari manifold.

math.AP↗

arXiv2025

Existence and convergence of ground state solutions for Choquard-type systems on lattice graphs

In this paper, we study the $p$-Laplacian system with Choquard-type nonlinearity $$ \begin{cases}-Δ_{p} u+(λa+1)|u|^{p-2} u=\frac{1}γ \left(R_α\ast F(u,v)\right)F_{u}(u, v), \\ -Δ_{p} v+(λb+1)|v|^{p-2} v=\frac{1}γ \left(R_α\ast F(u,v)\right)F_{v}(u, v),\end{cases} $$ on lattice graphs $\mathbb{Z}^N$, where $α\in(0,N),\,p\geq 2,\,γ> \frac{(N+α)p}{2N},\,λ>0$ is a parameter and $R_α$ is the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under some assumptions on the functions $a,\,b$ and $F$, we prove the existence and asymptotic behavior of ground state solutions by the method of Nehari manifold.

math.AP↗

arXiv2025

The eternal solutions of parabolic equations with boundary condition

In this paper, we study the parabolic equations of the form $$ \left\{ \begin{array}{rcll} Lu(y,t) &=& f, \qquad &(y,t)\in Q,\\ u(y,t)&=& 0, \qquad &(y,t)\in \partial Q, \\ u(y,t)&& \hspace{-8mm}\mbox{is uniformly bounded from below}, \qquad &(y,t)\in Q, \end{array} \right. $$ where $Q=Ω\times\mathbb{R}\subset\mathbb{R}^{n+1}$ and $Ω\subset\mathbb{R}^{n}$ is a bounded Lipschitz domain with $0\inΩ$. Here $L$ is a general second order uniformly parabolic differential operator in non-divergence form or divergence form. For $f=0$, we establish the structure of the solution space, which is one dimensional and the solutions in this space grow exponentially at one end and decay exponentially at the other. For $f\neq0$, we show that all solutions can be presented by the solutions corresponding to the homogenous equations($f=0$) and a bounded special solution of the inhomogeneous equations. Our method is based on maximum principle in $Q$ and the Harnack type inequalities.

math.AP↗

arXiv2024

Fractional logarithmic Schrödinger equations on lattice graphs

In this paper, we study the fractional logarithmic Schrödinger equation $$ (-Δ)^{s} u+h(x) u=u \log u^{2} $$ on lattice graphs $\mathbb{Z}^d$, where $s\in (0,1)$. If $h(x)$ is a bounded periodic potential, we prove the existence of ground state solution by mountain pass theorem and Lions lemma. If $h(x)$ is a coercive potential, we show the existence of ground state sign-changing solutions by the method of Nehari manifold.

math.AP↗

arXiv2024

p-Laplacian equations with general Choquard nonlinearity on lattice graphs

In this paper, we study the following $p$-Laplacian equation $$ -Δ_{p} u+h(x)|u|^{p-2} u=\left(R_α *F(u)\right)f(u) $$ on lattice graphs $\mathbb{Z}^N$, where $p\geq 2$, $α\in(0,N)$ are constants and $R_α$ is the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under different assumptions on potential function $h$, we prove the existence of ground state solutions respectively by the methods of Nehari manifold.

math.AP↗

arXiv2024

Solutions to discrete nonlinear Kirchhoff-Choquard equations with power nonlinearity

In this paper, we study the following Kirchhoff-Choquard equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d μ\right) Δu+h(x) u=\left(R_α\ast|u|^{p}\right)|u|^{p-2}u,\quad x\in \mathbb{Z}^3, $$ where $a,\,b>0$, $α\in(0,3)$ are constants and $R_α$ is the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under some suitable assumptions on potential function $h$, for $p>2$, we first establish the existence of ground state solutions based on the Nehari manifold. Subsequently, for $p>4$, we obtain the existence of ground state sign-changing solutions by adopting constrained minimization arguments on the sign-changing Nehari manifold.

math.AP↗

arXiv2024

Dirichlet Heat kernel estimates for a large class of anisotropic Markov processes

Let $Z=(Z^{1}, \ldots, Z^{d})$ be the d-dimensional Lévy {process} where {$Z^i$'s} are independent 1-dimensional Lévy {processes} with identical jumping kernel $ ν^1(r) =r^{-1}ϕ(r)^{-1}$. Here $ϕ$ is {an} increasing function with weakly scaling condition of order $\underline α, \overline α\in (0, 2)$. We consider a symmetric function $J(x,y)$ comparable to \begin{align*} \begin{cases} ν^1(|x^i - y^i|)\qquad&\text{ if $x^i \ne y^i$ for some $i$ and $x^j = y^j$ for all $j \ne i$}\\ 0\qquad&\text{ if $x^i \ne y^i$ for more than one index $i$}. \end{cases} \end{align*} Corresponding to the jumping kernel $J$, there exists an anisotropic Markov process $X$, see \cite{KW22}. In this article, we establish sharp two-sided Dirichlet heat kernel estimates for $X$ in $C^{1,1}$ open set, under certain regularity conditions. As an application of the main results, we derive the Green function estimates.

math.PR↗

arXiv2024

Sign-changing solutions to discrete nonlinear logarithmic Kirchhoff equations

In this paper, we study the discrete logarithmic Kirchhoff equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d μ\right) Δu+(λh(x)+1) u=|u|^{p-2}u \log u^{2}, \quad x\in \mathbb{Z}^3, $$ where $a,b>0, p>6$ and $λ$ is a positive parameter. Under suitable assumptions on $h(x)$, we prove the existence and asymptotic behavior of least energy sign-changing solutions for the equation by the method of Nehari manifold.

math.AP↗

arXiv2024

Solutions to discrete nonlinear Kirchhoff-Choquard equations

In this paper, we study the discrete Kirchhoff-Choquard equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d μ\right) Δu+V(x) u=\left(R_α *F(u)\right)f(u),\quad x\in \mathbb{Z}^3, $$ where $a,\,b>0$ are constants, $R_α$ is the Green's function of the discrete fractional Laplacian with $α\in(0,3)$, which has no singularity but has same asymptotics as the Riesz potential. Under some suitable assumptions on $V$ and $f$, we prove the existence of nontrivial solutions and ground state solutions by variational methods.

math.AP↗

arXiv2023

Solutions to discrete fractional Schödinger equations

In this paper, we study the discrete fractional Schrödinger equation $$ (-Δ)^αu+h(x) u=f(x,u),\quad x\in \mathbb{Z}^d,$$ where $d\in\mathbb{N}^*,\,α\in(0, 1)$ and the nonlocal operator $(-Δ)^α$ is defined by discrete Fourier transform, which differs from the continuous case. Under suitable assumptions on $h$ and $f$, we prove the existence and multiplicity of solutions to this equation by variational method.

math.AP↗

arXiv2022

The ground state solutions of nonlinear Schrödinger equations with Hardy weights on lattice graphs

In this paper, we study the nonlinear Schrödinger equation $$ -Δu+(V(x)- \fracρ{(|x|^2+1)})u=f(x,u) $$ on the lattice graph $\mathbb{Z}^N$ with $N\geq 3$, where $V$ is a bounded periodic potential and $0$ lies in a spectral gap of the Schrödinger operator $-Δ+V$. Under some assumptions on the nonlinearity $f$, we prove the existence and asymptotic behavior of ground state solutions with small $ρ\geq 0$ by the generalized linking theorem.

math.AP↗

arXiv2022

The existence and convergence of solutions for the nonlinear Choquard equations on groups of polynomial growth

In this paper, we study the nonlinear Choquard equation \begin{eqnarray*} Δ^{2}u-Δu+(1+λa(x))u=(R_α\ast|u|^{p})|u|^{p-2}u \end{eqnarray*} on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension $N\geq 2$, where $α\in(0,N),\,p>\frac{N+α}{N},\,λ$ is a positive parameter and $R_α$ stands for the Green's function of the discrete fractional Laplacian, which has same asymptotics as the Riesz potential. Under some assumptions on $a(x)$, we establish the existence and asymptotic behavior of ground state solutions for the nonlinear Choquard equation by the method of Nehari manifold.

math.AP↗

arXiv2020

Open-Domain Question Answering with Pre-Constructed Question Spaces

Open-domain question answering aims at solving the task of locating the answers to user-generated questions in massive collections of documents. There are two families of solutions available: retriever-readers, and knowledge-graph-based approaches. A retriever-reader usually first uses information retrieval methods like TF-IDF to locate some documents or paragraphs that are likely to be relevant to the question, and then feeds the retrieved text to a neural network reader to extract the answer. Alternatively, knowledge graphs can be constructed from the corpus and be queried against to answer user questions. We propose a novel algorithm with a reader-retriever structure that differs from both families. Our reader-retriever first uses an offline reader to read the corpus and generate collections of all answerable questions associated with their answers, and then uses an online retriever to respond to user queries by searching the pre-constructed question spaces for answers that are most likely to be asked in the given way. We further combine retriever-reader and reader-retriever results into one single answer by examining the consistency between the two components. We claim that our algorithm solves some bottlenecks in existing work, and demonstrate that it achieves superior accuracy on real-world datasets.

cs.CL↗

arXiv2020

Learning to Fuse Sentences with Transformers for Summarization

The ability to fuse sentences is highly attractive for summarization systems because it is an essential step to produce succinct abstracts. However, to date, summarizers can fail on fusing sentences. They tend to produce few summary sentences by fusion or generate incorrect fusions that lead the summary to fail to retain the original meaning. In this paper, we explore the ability of Transformers to fuse sentences and propose novel algorithms to enhance their ability to perform sentence fusion by leveraging the knowledge of points of correspondence between sentences. Through extensive experiments, we investigate the effects of different design choices on Transformer's performance. Our findings highlight the importance of modeling points of correspondence between sentences for effective sentence fusion.

cs.CL↗

arXiv2020

Heat kernel bounds for a large class of Markov process with singular jump

Let $Z=(Z^{1}, \ldots, Z^{d})$ be the $d$-dimensional Lévy processes where $Z^{i}$'s are independent $1$-dimensional Lévy processes with jump kernel $J^{ϕ, 1}(u,w) =|u-w|^{-1}ϕ(|u-w|)^{-1}$ for $u, w\in \mathbb R$. Here $ϕ$ is an increasing function with weak scaling condition of order $\underline α, \overline α\in (0, 2)$. Let $J(x,y) \asymp J^ϕ(x,y)$ be the symmetric measurable function where \begin{align*} J^ϕ(x,y):=\begin{cases} J^{ϕ, 1}(x^i, y^i)\qquad&\text{ if $x^i \ne y^i$ for some $i$ and $x^j = y^j$ for all $j \ne i$}\\ 0\qquad&\text{ if $x^i \ne y^i$ for more than one index $i$.} \end{cases} \end{align*} Corresponding to the jump kernel $J$, we show the existence of non-isotropic Markov processes $X:=(X^{1}, \ldots, X^{d})$ and obtain sharp two-sided heat kernel estimates for the transition density functions.

math.PR↗

arXiv2020

Bayesian Optimization for Selecting Efficient Machine Learning Models

The performance of many machine learning models depends on their hyper-parameter settings. Bayesian Optimization has become a successful tool for hyper-parameter optimization of machine learning algorithms, which aims to identify optimal hyper-parameters during an iterative sequential process. However, most of the Bayesian Optimization algorithms are designed to select models for effectiveness only and ignore the important issue of model training efficiency. Given that both model effectiveness and training time are important for real-world applications, models selected for effectiveness may not meet the strict training time requirements necessary to deploy in a production environment. In this work, we present a unified Bayesian Optimization framework for jointly optimizing models for both prediction effectiveness and training efficiency. We propose an objective that captures the tradeoff between these two metrics and demonstrate how we can jointly optimize them in a principled Bayesian Optimization framework. Experiments on model selection for recommendation tasks indicate models selected this way significantly improves model training efficiency while maintaining strong effectiveness as compared to state-of-the-art Bayesian Optimization algorithms.

cs.LG↗