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Desheng Li

Publications and source records attributed to Desheng Li.

17 recordsLinked to original sources

CL-Polyp: A Contrastive Learning-Enhanced Network for Accurate Polyp Segmentation

Accurate segmentation of polyps from colonoscopy images is crucial for the early diagnosis and treatment of colorectal cancer. Most existing deep learning-based polyp segmentation methods adopt an Encoder-Decoder architecture, and some utilize multi-task frameworks that incorporate auxiliary tasks like classification to improve segmentation. However, these methods often need more labeled data and depend on task similarity, potentially limiting generalizability. To address these challenges, we propose CL-Polyp, a contrastive learning-enhanced polyp segmentation network. Our method uses contrastive learning to enhance the encoder's extraction of discriminative features by contrasting positive and negative sample pairs from polyp images. This self-supervised strategy improves visual representation without needing additional annotations. We also introduce two efficient, lightweight modules: the Modified Atrous Spatial Pyramid Pooling (MASPP) module for improved multi-scale feature fusion, and the Channel Concatenate and Element Add (CA) module to merge low-level and upsampled features for {enhanced} boundary reconstruction. Extensive experiments on five benchmark datasets-Kvasir-SEG, CVC-ClinicDB, CVC-ColonDB, CVC-300, and ETIS-show that CL-Polyp consistently surpasses state-of-the-art methods. Specifically, it enhances the IoU metric by 0.011 and 0.020 on the Kvasir-SEG and CVC-ClinicDB datasets, respectively, demonstrating its effectiveness in clinical polyp segmentation.

cs.CV

Global Existence, Regularity, and Dissipativity of Reaction-diffusion Equations with State-dependent Delay and Supercritical Nonlinearities

This work aims to study the initial-boundary value problem of the reaction-diffusion equation $\pa_{t}u-Δu=f(u)+g(u(t-τ(t,u_t)))+h(t,x)$ in a bounded domain with state-dependent delay and supercritical nonlinearities. We establish the global existence and discuss the regularity and dissipativity of the problem under weaker assumptions. In particular, the existence of a global pullback attractor is proved regardless of uniqueness.

math.AP

A Note on the Krein-Rutman Theorem for Sectorial Operators

In this note we present some generalized versions of the Krein-Rutman theorem for sectorial operators. They are formulated in a fashion that can be easily applied to elliptic operators. Another feature of these generalized versions is that they contain some information on the generalized eigenspaces associated with non-principal eigenvalues, which are helpful in the study of the dynamics of evolution equations in ordered Banach spaces.

math.FA

A Remark on Attractor Bifurcation

In this paper we present some local dynamic bifurcation results in terms of invariant sets of nonlinear evolution equations. We show that if the trivial solution is an isolated invariant set of the system at the critical value $λ=λ_0$, then either there exists a one-sided neighborhood $I^-$ of $λ_0$ such that for each $λ\in I^-$, the system bifurcates from the trivial solution to an isolated nonempty compact invariant set $K_λ$ with $0\not\in K_λ$, or there is a one-sided neighborhood $I^+$ of $λ_0$ such that the system undergoes an attractor bifurcation for $λ\in I^+$ from $(0,λ_0)$. Then we give a modified version of the attractor bifurcation theorem. Finally, we consider the classical Swift-Hohenberg equation and illustrate how to apply our results to a concrete evolution equation.

math.DS

On Relative Category and Morse Decompositions for Infinite-Dimensional Dynamical Systems

We employ the relative category to develop relations between the Ważewski pair $(N,E)$ and the Morse decomposition of the maximal invariant set in $\ol{N\sm E}$ for infinite-dimensional dynamical systems. Via these relations, we can detect connecting trajectories between Morse sets and obtain a dynamical-system version of critical point theorem with relative category.

math.DS

Uniform Decay Estimates for Solutions of a Class of Retarded Integral Inequalities

Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities: $$y(t)\leq E(t,τ)||y_τ||+\int_τ^t K_1(t,s)||y_s||ds+\int_t^\infty K_2(t,s)||y_s||ds+ρ, \hspace{0.5cm} t\geqτ\geq 0.$$ As a simple example of application, the retarded scalar functional differential equation $\dot x=-a(t)x+B(t,x_t)$ is considered, and the global asymptotic stability of the equation is proved under weaker conditions. Another example is the ODE system $\dot x=F_0(t,x)+\sum_{i=1}^m F_i(t,x(t-r_i(t)))$ on $R^n$ with superlinear nonlinearities $F_i$ ($0\leq i\leq m$). The existence of a global pullback attractor of the system is established under appropriate dissipation conditions. The third example for application concerns the study of the dynamics of the functional cocycle system $\frac{du}{dt}+Au=F(θ_tp,u_t)$ in a Banach space $X$ with sublinear nonlinearity. In particular, the existence and uniqueness of a nonautonomous stationary solution $Γ$ is obtained under the hyperbolicity assumption on operator $A$ and some additional hypotheses, and the global asymptotic stability of $Γ$ is also addressed.

math.DS

New Schemes for Solving the Principal Eigenvalue Problems of Perron-like Matrices via Polynomial Approximations of Matrix Exponentials

A real square matrix is Perron-like if it has a real eigenvalue $s$, called the principal eigenvalue of the matrix, and $\mbox{Re}\,μ<s$ for any other eigenvalue $μ$. Nonnegative matrices and symmetric ones are typical examples of this class of matrices. The main purpose of this paper is to develop a set of new schemes to compute the principal eigenvalues of Perron-like matrices and the associated generalized eigenspaces by using polynomial approximations of matrix exponentials. Numerical examples show that these schemes are effective in practice.

math.NA

Global Estimates and Regularity of Retarded Parabolic Equations with Fast-growing Nonlinearities

This paper is concerned with global estimates and regularity of solutions for the initial value problem of the retarded parabolic equation $$\frac{\patial u}{\patial t}-Δu=f(x,u)+g(u(x,t-r_1(t)),\cdots,u(x,t-r_m(t)))+h(x,t)$$ in a bounded domain $Ω\subset R^n$ with fast-growing nonlinearities and a dissipative structure, which is associated with the homogeneous Dirichlet boundary condition. Our results reveal some deeper inherent connections between dissipative structures and the regularity of solutions for such problems.

math.DS

Compactly Generated Shape Index Theory and its Application to a Retarded Nonautonomous Parabolic Equation

We establish the compactly generated shape (H-shape) index theory for local semiflows on complete metric spaces via more general shape index pairs, and define the H-shape cohomology index to develop the Morse equations. The main advantages are that the quotient space $N/E$ is not necessarily metrizable for the shape index pair $(N,E)$ and $N\sm E$ need not to be a neighborhood of the compact invariant set. Moreover, in this new theory, the phase space is not required to be separable. We apply H-shape index theory to an abstract retarded nonautonomous parabolic equation to obtain the existence of bounded full solutions.

math.DS

Equilibrium Index of Invariant Sets and Global Static Bifurcation for Nonlinear Evolution Equations

We introduce the notion of equilibrium index for statically isolated invariant sets of the system $u_t+A u=f_λ(u)$ on Banach space $X$ (where $A$ is a sectorial operator with compact resolvent) and present a reduction theorem and an index formula for bifurcating invariant sets near equilibrium points. Then we prove a new global static bifurcation theorem where the crossing number $\mathfrak{m}$ may be even. In particular, in case $\mathfrak{m}=2$, we show that the system undergoes either an attractor/repeller bifurcation, or a global static bifurcation. An illustrating example is also given by considering the bifurcations of the periodic boundary value problem of second-order differential equations.

math.DS

Global Bifurcation of Dynamical Systems and Nonlinear Evolution Equations

We establish new global bifurcation theorems for dynamical systems in terms of local semiflows on complete metric spaces. These theorems are applied to the nonlinear evolution equation $u_t+A u=f_λ(u)$ in a Banach space $X$, where $A$ is a sectorial operator with compact resolvent. Assume that $0$ is always a trivial stationary solution of the equation. We show that the global dynamic bifurcation branch $Γ$ of a bifurcation point $(0,λ_0)$ either meets another bifurcation point $(0,λ_1)$, or is unbounded, completely extending the well-known Rabinowitz Global Bifurcation Theorem on operator equations to nonlinear evolution equations without any restrictions on the crossing number. In the case where $f_λ(u)=λu+f(u)$, due to the {\em nonnegativity} of the Conley index we can even prove a stronger conclusion asserting that only one possibility occurs for $Γ$, that is, $Γ$ is necessarily unbounded. This result can be expected to help us have a deeper understanding of the dynamics of nonlinear evolution equations. As another example of applications of the abstract bifurcation theorems, we also discuss the bifurcation and the existence of nontrivial solutions of the elliptic equation $-Δu=f_λ(u)$ on a bounded domain in $\mathbb{R}^n$ ($n\geq 3$) associated with the homogenous Dirichlet boundary condition. Some new results with global features are obtained.

math.DS

Local and Global Dynamic Bifurcations of Nonlinear Evolution Equations

We present new local and global dynamic bifurcation results for nonlinear evolution equations of the form $u_t+A u=f_λ(u)$ on a Banach space $X$, where $A$ is a sectorial operator, and $λ\in R$ is the bifurcation parameter. Suppose the equation has a trivial solution branch $\{(0,λ):\,\,λ\in R\}$. Denote $Φ_λ$ the local semiflow generated by the initial value problem of the equation. It is shown that if the crossing number $n$ at a bifurcation value $λ=λ_0$ is nonzero and moreover, $S_0=\{0\}$ is an isolated invariant set of $Φ_{λ_0}$, then either there is a one-sided neighborhood $I_1$ of $λ_0$ such that $Φ_λ$ bifurcates a topological sphere $\mathbb{S}^{n-1}$ for each $λ\in I_1\setminus\{λ_0\}$, or there is a two-sided neighborhood $I_2$ of $λ_0$ such that the system $Φ_λ$ bifurcates from the trivial solution an isolated nonempty compact invariant set $K_λ$ with $0\not\in K_λ$ for each $λ\in I_2\setminus\{λ_0\}$. We also prove that the bifurcating invariant set has nontrivial Conley index. Building upon this fact we establish a global dynamical bifurcation theorem. Roughly speaking, we prove that for any given neighborhood $Ω$ of the bifurcation point $(0,λ_0)$, the connected bifurcation branch $Γ$ from $(0,λ_0)$ either meets the boundary $\partialΩ$ of $Ω$, or meets another bifurcation point $(0,λ_1)$. This result extends the well-known Rabinowitz's Global Bifurcation Theorem to the setting of dynamic bifurcations of evolution equations without requiring the crossing number to be odd. As an illustration example, we consider the well-known Cahn-Hilliard equation. Some global features on dynamical bifurcations of the equation are discussed.

math.DS

Attractors of Local Semiflows on Topological Spaces

In this paper we introduce a notion of an attractor for local semiflows on topological spaces, which in some cases seems to be more suitable than the existing ones in the literature. Based on this notion we develop a basic attractor theory on topological spaces under appropriate separation axioms. First, we discuss fundamental properties of attractors such as maximality and stability and establish some existence results. Then, we give a converse Lyapunov theorem. Finally, the Morse decomposition of attractors is also addressed.

math.DS

Linking Theorems of Local Semiflows on Complete Metric Spaces

In this paper we prove some linking theorems and mountain pass type results for dynamical systems in terms of local semiflows on complete metric spaces. Our results provide an alternative approach to detect the existence of compact invariant sets without using the Conley index theory. They can also be applied to variational problems of elliptic equations without verifying the classical P.S. Condition. As an example, we study the resonant problem of the nonautonomous parabolic equation $ u_t-Δu-μu=f(u)+g(x,t) $ on a bounded domain. The existence of a recurrent solution is proved under some Landesman-Laser type conditions by using an appropriate linking theorem of semiflows. Another example is the elliptic equation $-Δu+a(x)u=f(x,u)$ on $R^n$. We prove the existence of positive solutions by applying a mountain pass lemma of semiflows to the parabolic flow of the problem.

math.DS

Morse Theory of Attractors via Lyapunov Functions

This paper is concerned with the Morse theory of attractors for semiflows on complete metric spaces. First, we construct global Morse-Lyapunov functions for Morse decompositions of attractors. Then we extend some well known deformation results in the critical-point theory to Morse-Lyapunov functions which are only continuous. Based on these works, we finally introduce the concept of critical groups for Morse sets and establish Morse inequalities and Morse equations for attractors.

math.DS