arXiv subjects

Qianwei Zhang

Publications and source records attributed to Qianwei Zhang.

5 recordsLinked to original sources

arXiv2026

Gradient estimates and volume doubling for locally finite weighted graphs with $CD\psi(n,-K)$ condition

We study gradient estimates and volume growth on locally finite weighted graphs satisfying the $CD\psi(n,-K)$ condition with $K\geq0$. We establish a variational inequality for the heat semigroup and derive from it a family of Li-Yau type gradient estimates. Furthermore, under suitable assumptions on $\psi$, we establish a uniform heat retention estimate for metric balls. Combined with a heat kernel Harnack inequality obtained from the established gradient estimate, this yields a curvature dependent exponential volume doubling estimate \begin{equation*} V(x,2r)\leq C e^{cKr^2}V(x,r). \end{equation*} When $K=0$, the result reduces to a uniform volume doubling and further implies that the bottom of the spectrum of $-\Delta$ vanishes on infinite graphs.

math.DG↗

arXiv2024

Common revenue allocation in DMUs with two stages based on DEA cross-efficiency and cooperative game

In this paper, we examine two-stage production organizations as decision-making units (DMUs) that can collaborate to form alliances. We present a novel approach to transform a grand coalition of n DMUs with a two-stage structure into 2n single-stage sub-DMUs by extending the vectors of the initial input, intermediate product, and final output, thus creating a 2n*2n DEA cross-efficiency (CREE) matrix. By combining cooperative game theory with CREE and utilizing three cooperative game solution concepts, namely, the nucleolus, the least core and the Shapley value, a characteristic function is developed to account for two types of allocation, i.e., direct allocation and secondary allocation. Moreover, the super-additivity and the core non-emptiness properties are explored. It is found that the sum of the revenue allocated to all DMUs will remain constant at each stage regardless of the allocation manner and the cooperative solution concept selected. To illustrate the efficiency and practicality of the proposed approach, both a numerical example and an empirical application are provided.

cs.GT↗

arXiv2023

Gradient estimates on graphs with the $CDψ(n,-K)$condition

This paper investigates gradient estimates on graphs satisfying the $CDψ(n,-K)$ condition with positive constants $n,K$, and concave $C^{1}$ functions $ψ:(0,+\infty)\rightarrow\mathbb{R}$. Our study focuses on gradient estimates for positive solutions of the heat equation $\partial_{t}u=Δu$. Additionally, the estimate is extended to a heat-type equation $\partial_{t}u=Δu+cu^σ$, where $σ$ is a constant and $c$ is a continuous function defined on $[0,+\infty)$. Furthermore, we utilize these estimates to derive heat kernel bounds and Harnack inequalities.

math.DG↗

arXiv2023

Existence of Solutions to a Class of Kazdan-Warner Equations on Finite Graphs

Let $G=(V, E)$ be a connected finite graph, $h$ be a positive function on $V$ and $λ_{1}(V)$ be the first non-zero eigenvalue of $-Δ$. For any given finite measure $μ$ on $V$, define functionals \begin{eqnarray*} J_{ β}(u)&=&\frac{1}{2}\int_{V}|\nabla u|^{2}d μ-β\log\int_{V}he^{u}d μ, J_{ α,β}(u)&=&\frac{1}{2}\int_{V}\left(|\nabla u|^{2}- αu^{2}\right) d μ-β\log\int_{V}he^{u}d μ\end{eqnarray*} on the functional space $$ {\bf H}= \left\{ u\in{\bf W}^{1,2}(V) \Bigg| \int_{V}u\!\ dμ=0 \right\}. $$ For any $β\in \mathbb{R}$, we show that $J_{ β}(u)$ has a minimizer $u\in{\bf H}$, and then, based on variational principle, the Kazdan-Warner equation $$ Δu=-\frac{βhe^{u}}{\displaystyle{\int_{V}he^{u}d μ}}+\frac{β}{\text{Vol}(V)} $$ has a solution in ${\bf H}$. If $α< λ_{1}(V)$, then for any $β\in \mathbb{R} , J_{ α,β}(u)$ has a minimizer in ${\bf H}$, thus the Kazdan-Warner equation $$ Δu+α\!\ u=-\frac{βhe^{u}}{\displaystyle{\int_{V}he^{u}d μ}}+\frac{β}{\text{Vol}(V)} $$ has a solution in ${\bf H}$. If $α> λ_{1}(V)$, then for any $β\in \mathbb{R}$, $\displaystyle{\inf_{u\in{\bf H}} J_{ α,β}(u) =- \infty}$. When $α=λ_{1}(V)$, the situation becomes complicated: if $β=0$, the corresponding equation is $-Δu=λ_{1}(V)u$ which has a solution in ${\bf H}$ obviously; if $β>0$, then $\displaystyle{\inf_{u\in {\bf H}} J_{α,β}(u) =- \infty}$; if $β<0$, $J_{ α,β}(u)$ has a minimizer in some subspace of ${\bf H}$. Moreover, we consider the same problem where higher eigenvalues are involved.

math.DG↗

arXiv2021

Scalable universal tunable virtual-wavevector spatial frequency shift (TVSFS) super-resolution imaging

Developing a chip-based super-resolution imaging technique with large field-of-view (FOV), deep subwavelength resolution, and compatibility for both fluorescent and non-fluorescent samples is desired for material science, biomedicine, and life researches, etc. Previous on-chip super-resolution methods focus on either fluorescent or non-fluorescent imaging, putting an urgent requirement on the general imaging technique compatible with both of them. Here, we introduce a universal super-resolution imaging method based on tunable virtual-wavevector spatial frequency shift (TVSFS), realizing both labeled and label-free super-resolution imaging on a single delicately fabricated scalable photonic chip. Theoretically, with TVSFS, the diffraction limit of a linear optical system can be overcome, and the resolution can be improved more than three times, which is the limitation for most super-resolution imaging based on spatial frequency engineering. Diffractive units were fabricated on the chip's surface to provide a wavevector-variable evanescent wave illumination and induce tunable deep SFS in the samples' Fourier space. A resolution of λ/4.7 for the label-free sample and λ/7.1 for the labeled sample with a large FOV could be achieved with a CMOS-compatible process on a GaP chip. The large FOV, high-compatibility, and high-integration TVSFS chip may advance the fields like cell engineering, precision inspection in the industry, chemical research, etc.

physics.optics↗