Jesse Ratzkin, Andrejs Treibergs
We resolve a question of Bramson and Griffeath by showing that the expected capture time of four independent Brownian predators pursuing one Brownian prey on a line is finite. Our main tool is an eigenvalue estimate for a particular spherical domain, which we obtain by a coning construction and domain perturbation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We establish a large deviation principle for the process of the largest eigenvalue of an Hermitian Brownian motion. By a contraction principle, we recover the LDP for the largest eigenvalue of a rank one deformation of the GUE.
We investigate the process of eigenvalues of a symmetric matrix-valued process which upper diagonal entries are independent one-dimensional Hölder continuous Gaussian processes of order gamma in (1/2,1). Using the stochastic calculus with respect to the Young's integral we show that these eigenvalues do not collide at any time with probability one. When the matrix process has entries that are fractional Brownian motions with Hurst parameter H in (1/2,1), we find a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding matrix fractional Brownian …
We consider a model of Brownian motion on a bounded open interval with instantaneous jumps. The jumps occur at a spatially dependent rate given by a positive parameter times a continuous function positive on the interval and vanishing on its boundary. At each jump event the process is redistributed uniformly in the interval. We obtain sharp asymptotic bounds on the principal eigenvalue for the generator of the process as the parameter tends to infinity. Our work answers a question posed by Arcusin and Pinsky.
In the present paper, the Karhunen-Lo{è}ve eigenvalues for a sub-fractional Brownian motion are considered in the case of $H>\frac12$. Rigorous large $n$ asymptotics for those eigenvalues are shown, based on functional analysis method. By virtue of these asymptotics, along with some standard large deviations results, asymptotically estimates for the closely related problem of small $L^2$-ball probabilities for a sub-fractional Brownian motion are derived. By the way, asymptotic analysis on the Karhunen-Lo{è}ve eigenvalues for the corresponding "derivative" process is also established.
We study the problem of when a Brownian motion in the unit ball has a positive probability of avoiding a countable collection of spherical obstacles. We give a necessary and sufficient integral condition for such a collection to be avoidable.
We derive a system of stochastic partial differential equations satisfied by the eigenvalues of the symmetric matrix whose entries are the Brownian sheets. We prove that the sequence $\left\{L_{d}(s,t), (s,t)\in[0,S]\times [0,T]\right\}_{d\in\mathbb N}$ of empirical spectral measures of the rescaled matrices is tight on $C([0,S]\times [0,T], \mathcal P(\mathbb R))$ and hence is convergent as $d$ goes to infinity by Wigner's semicircle law. We also obtain PDEs which are satisfied by the high-dimensional limiting measure.
Let $\{B_β(x), x \in \mathbb{S}^N\}$ be a fractional Brownian motion on the $N$-dimensional unit sphere $\mathbb{S}^N$ with Hurst index $β$. We study the excursion probability $\mathbb{P}\{\sup_{x\in T} B_β(x) > u \}$ and obtain the asymptotics as $u\to \infty$, where $T$ can be the entire sphere $\mathbb{S}^N$ or a geodesic disc on $\mathbb{S}^N$.
Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam and van Moerbeke that the process of eigenvalues of two consecutive minors of an Hermitian Brownian motion is a Markov process, whereas if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.
We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R}^{n+\ell}$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R}^{n+\ell}$ allows us to geometrically realize the Wright-Fisher diffusion wit…
In this paper we consider a random potential derived from the Brownian motion. We obtain upper and lower bounds for the expected value of the main eigenvalue of the associated Ruelle operator and for its quenched topological pressure. We also exhibit an isomorphism between the space $C(Ω)$ endowed with its standard norm and a proper closed subspace of the Skorokhod space which is used to obtain a stochastic functional equation for the main eigenvalue and for its associated eigenfunction.
By further developing the generalized $Γ$-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose generator is a step-3 generating hypoelliptic Hörmander's type operator. To prove hypocoercivity in that case, the key point is to show the existence of a convenient Riemannian foliation associated to the diffusion. We will then deduce, under suitable geometric conditions, the co…
We derive a three-term asymptotic expansion for the expected lifetime of Brownian motion and for the torsional rigidity on thin domains in R^n, and a two-term expansion for the maximum (and corresponding maximizer) of the expected lifetime. The approach is similar to that which we used previously to study the eigenvalues of the Dirichlet Laplacian and consists of scaling the domain in one direction and deriving the corresponding asymptotic expansions as the scaling parameter goes to zero. Apart from being dominated by the one-dimensional Brownian motion along the direction of the scaling, we a…
We describe an exact simulation algorithm for the increments of Brownian motion on a sphere of arbitrary dimension, based on the skew-product decomposition of the process with respect to the standard geodesic distance. The radial process is closely related to a Wright-Fisher diffusion, increments of which can be simulated exactly using the recent work of Jenkins & Spanò (2017) [JS17]. The rapid spinning phenomenon of the skew-product decomposition then yields the algorithm for the increments of the process on the sphere.
Persistence probabilities of spherical fractional Brownian motion
本文计算了球面分数布朗运动持续概率的衰减率，发现其与欧几里得情形相似，并作为副产品推广了具有代数端点奇点函数的级数表示系数研究。
Let $D\subset R^d$ be a bounded domain and let $\mathcal P(D)$ denote the space of probability measures on $D$. Consider a Brownian motion in $D$ which is killed at the boundary and which, while alive, jumps instantaneously according to a spatially dependent exponential clock with intensity $γV$ to a new point, according to a distribution $μ\in\mathcal P(D)$. From its new position after the jump, the process repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator $-L_{γ,μ}$, defined by L_{γ,μ}u\equiv -\frac12Δu+γ…
The extreme statistics of some noncolliding Brownian processes
研究非碰撞粒子系统极限理论，揭示布朗运动最大特征值的极限行为、Airy过程极限及非碰撞布朗桥最大路径的Fredholm行列式公式。
We study the spectrum of the kinetic Brownian motion in the space of $d\times d$ Hermitian matrices, $d\geq2$. We show that the eigenvalues stay distinct for all times, and that the process $Λ$ of eigenvalues is a kinetic diffusion (i.e. the pair $(Λ,\dotΛ)$ of $Λ$ and its derivative is Markovian) if and only if $d=2$. In the large scale and large time limit, we show that $Λ$ converges to the usual (Markovian) Dyson Brownian motion under suitable normalisation, regardless of the dimension.
Since the introduction of Dyson's Brownian motion in early 1960's, there have been a lot of developments in the investigation of stochastic processes on the space of Hermitian matrices. Their properties, especially, the properties of their eigenvalues have been studied in great details. In particular, the limiting behaviors of the eigenvalues are found when the dimension of the matrix space tends to infinity, which connects with random matrix theory. This survey reviews a selection of results on the eigenvalues of stochastic processes from the literature of the past three decades. For most rec…
In this article, we study high-dimensional behavior of empirical spectral distributions $\{L_N(t), t\in[0,T]\}$ for a class of $N\times N$ symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H \in(1/2,1)$. For Wigner-type matrices, we obtain almost sure relative compactness of $\{L_N(t), t\in[0,T]\}_{N\in\mathbb N}$ in $C([0,T], \mathbf P(\mathbb R))$ following the approach in \cite{Anderson2010}; for Wishart-type matrices, we obtain tightness of $\{L_N(t), t\in[0,T]\}_…
We establish high probability estimates on the eigenvalue locations of Brownian motion on the $N$-dimensional unitary group, as well as estimates on the number of eigenvalues lying in any interval on the unit circle. These estimates are optimal up to arbitrarily small polynomial factors in $N$. Our results hold at the spectral edges (showing that the extremal eigenvalues are within $\mathcal{O} (N^{-2/3+})$ of the edges of the limiting spectral measure), in the spectral bulk, as well as for times near $4$ at which point the limiting spectral measure forms a cusp. Our methods are dynamical and …
We investigate the "hot--spots" property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the "hot--spots" property for the mixed Dirichlet--Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other
In this article, we consider flat and curved Riemannian symmetric spaces in the complex case and we study their basic integral kernels, in potential and spherical analysis: heat, Newton, Poisson kernels and spherical functions, i.e. the kernel of the spherical Fourier transform. We introduce and exploit a simple new method of construction of these $W$-invariant kernels by alternating sum formulas. We then use the alternating sum representation of these kernels to obtain their asymptotic behavior. We apply our results to the Dyson Brownian Motion on $R^d$.
In this contribution, we extend the methodology proposed in Abry and Didier (2017) to obtain the first joint estimator of the real parts of the Hurst eigenvalues of $n$-variate OFBM. The procedure consists of a wavelet regression on the log-eigenvalues of the sample wavelet spectrum. The estimator is shown to be consistent for any time reversible OFBM and, under stronger assumptions, also asymptotically normal starting from either continuous or discrete time measurements. Simulation studies establish the finite sample effectiveness of the methodology and illustrate its benefits compared to uni…
Spectral Simplicity and Joint Eigenvalue Densities for a Non-Gaussian Brownian Time Change
该研究针对单位正方形上带独立非高斯系数狄利克雷特征函数展开构造的速度测度的布朗时间变换，通过鞅论证、酉变换等方法证明其谱简单性并得到联合特征值密度。
Eigenvalue processes in light of Riemannian submersion and gradient flow of isospectral orbits
本文利用黎曼几何语言，证明了多种特征值过程是紧 Lie 群等距作用诱导的黎曼淹没投影结果，统一处理了不同设定下的特征值过程，并据此推导出特征向量过程和一般 $β>0$ 的 $β$ 过程。
Consider an $n\times n$ Hermitean matrix valued stochastic process $\{H_t\}_{t\geq 0}$ where the matrix elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect. In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the $k\times k$ in the upper left corner of $H_t$. If you project this process to a space-like path it is a determinantal …
Spectral-capacitary bounds for homological recovery from reflected Brownian trajectories
本文研究从单条反射布朗轨迹恢复未知反射域拓扑所需的最优观测时间，提出由谱间隙、布朗容量和特征数量三项组成的界，并证明其极小最优性。
Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$
研究\(\mathrm{GL}(N,\mathbb{C})\)上布朗运动特征值，通过一族非退化扩散过程及对小\(t\)时布朗运动的定量逼近，证明其特征值经验法则几乎必然收敛到确定性概率测度，解决了相关猜想。
A functional limit theorem for the empirical measure-valued process of eigenvalues of a matrix fractional Brownian motion is obtained. It is shown that the limiting measure-valued process is the non-commutative fractional Brownian motion recently introduced by Nourdin and Taqqu. Young and Skorohod stochastic integral techniques and fractional calculus are the main tools used.
We determine the long time behavior and the exact order of the tail probability for the maximal displacement of a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of the associated Schrödinger type operator. To establish our results, we show a sharp and locally uniform growth order of the Feynman-Kac semigroup.
Eigenvalues, eigenvector-overlaps, and regularized Fuglede-Kadison determinant of the non-Hermitian matrix-valued Brownian motion
研究非厄密矩阵值布朗运动的本征值、本征向量重叠及正则化的Fuglede-Kadison行列式，推导耦合本征值过程与本征向量重叠过程的随机微分方程并证明其尺度不变性。
Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian
本文研究d维区域中拉普拉斯算子谱底部与布朗运动退出时间p阶矩乘积的普适上下界，证明整数p时下界精确、p≥1时上界渐近精确，并证明凸对称区域类中存在极值区域，猜想立方体为极值。
Fractional Brownian motion, H-FBM , of index with d-dimensional time is considered in a spherical domain that contains 0 at its boundary. The main result : the log-asymptotics of probability that H-FBM does not exceed a fixed positive level is (H-d)logT(1+o(1)), where T>>1 is radius of the domain.
Using Itô's calculus and the mass optimal transportation theory, we study the generalized Dyson Brownian motion (GDBM) and the associated McKean-Vlasov evolution equation with an external potential $V$. Under suitable condition on $V$, we prove the existence and uniqueness of strong solution to SDE for GDBM. Standard argument shows that the family of the process of empirical measures $L_N(t)$ of GDBM is tight and every accumulative point of $L_N(t)$ in the weak convergence topology is a weak solution of the associated McKean-Vlasov evolution equation, which can be realized as the gradient flow…
We consider a system of noncolliding Brownian motions introduced in our previous paper, in which the noncolliding condition is imposed in a finite time interval $(0,T]$. This is a temporally inhomogeneous diffusion process whose transition probability density depends on a value of $T$, and in the limit $T \to \infty$ it converges to a temporally homogeneous diffusion process called Dyson's model of Brownian motions. It is known that the distribution of particle positions in Dyson's model coincides with that of eigenvalues of a Hermitian matrix-valued process, whose entries are independent Brow…
We examine the non-exit probability of a multidimensional Brownian motion from a growing truncated Weyl chamber. Different regimes are identified according to the growth speed, ranging from polynomial decay over stretched-exponential to exponential decay. Furthermore we derive associated large deviation principles for the empirical measure of the properly rescaled and transformed Brownian motion as the dimension grows to infinity. Our main tool is an explicit eigenvalue expansion for the transition probabilities before exiting the truncated Weyl chamber.
Private Low-Rank Approximation for Covariance Matrices, Dyson Brownian Motion, and Eigenvalue-Gap Bounds for Gaussian Perturbations
针对(ε,δ)-差分隐私下协方差矩阵的秩k逼近问题，提出高斯机制的复变体并结合戴森布朗运动视角分析，改进了弗罗贝尼乌斯范数上界，还证明了扰动矩阵特征值间隙以高概率较大，成果对相关领域有独立价值。
An explicit formula for free multiplicative Brownian motions via spherical functions
通过球函数推导出自由乘法布朗运动的显式公式，并展示其在不同根系中的应用。
We show that the Brydges-Fröhlich-Spencer-Dynkin and the Le Jan's isomorphisms between the Gaussian free fields and the occupation times of symmetric Markov processes generalize to the $β$-Dyson's Brownian motion. For $β\in\{1,2,4\}$ this is a consequence of the Gaussian case, however the relation holds for general $β$. We further raise the question whether there is an analogue of $β$-Dyson's Brownian motion on general electrical networks, interpolating and extrapolating the fields of eigenvalues in matrix-valued Gaussian free fields. In the case $n=2$ we give a simple construction.
We study $n$ non-intersecting Brownian motions, corresponding to the eigenvalues of an $n\times n$ Hermitian Brownian motion. At the boundary of their limit shape we find that only three universal processes can arise: the Pearcey process close to merging points, the Airy line ensemble at edges and a novel determinantal process describing the transition from the Pearcey process to the Airy line ensemble. The three cases are distinguished by a remarkably simple integral condition. Our results hold under very mild assumptions, in particular we do not require any kind of convergence of the initial…
We study the radial parts of the Brownian motions on Kähler and quaternion Kähler manifolds. Thanks to sharp Laplacian comparison theorems, we deduce as a consequence a sharp Cheeger-Yau type lower bound for the heat kernels of such manifolds and also sharp Cheng's type estimates for the Dirichlet eigenvalues of metric balls.
Dyson Brownian Motion and motion by mean curvature
通过将Riemannian流形上Brownian motion的外在构造适应于Hermitian矩阵空间的群轨道几何，构造了$\beta\in (0,\infty]$的Dyson Brownian motion，并揭示了$\beta$无穷大时特征值与群轨道的演化规律。
We give a geometric description of the motion of eigenvalues of a Brownian motion with values in some matrix spaces. In the second part we consider a paper by Polya where he introduced a function close to the Riemann zeta function, which satisfies Riemann hypothesis. We show that each of these two functions can be related to Brownian motion on a symmetric space.
Time changed spherical Brownian motions with longitudinal drifts
本文研究带纵向漂移的二维球面扩散过程的时间变换，通过谱分解法求解时间非局部Kolmogorov方程强解，构造复合过程得到解的随机表示与谱分解，给出低正则初始数据下的极弱解，并证明该过程为反常扩散，改进了已有结果。
In this paper, we consider tridiagonal matrices the eigenvalues of which evolve according to $β$-Dyson Brownian motion. This is the stochastic gradient flow on $\mathbb{R}^n$ given by, for all $1 \leq i \leq n,$ \[ dλ_{i,t} = \sqrt{\frac{2}β}dZ_{i,t} - \biggl( \frac{V'(λ_i)}{2} - \sum_{j: j \neq i} \frac{1}{λ_i - λ_j} \biggr)\,dt \] where $V$ is a constraining potential and $\left\{ Z_{i,t} \right\}_1^n$ are independent standard Brownian motions. This flow is stationary with respect to the distribution \[ ρ^β_N(λ) = \frac{1}{Z^β_N} e^{-\fracβ{2} \left( -\sum_{1 \leq i \neq j \leq N} \log|λ_i -…
We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schrödinger type operator. We first determine their growth rates on the survival event. We then establish the upper deviation for the maximal displacement under the possibility of extinction. Under the non-extinction condition, we further discuss the decay rate of the upper deviation probability and the population growth at the critical phase.
We study the density X(t,x) of one-dimensional super-Brownian motion and find the asymptotic behaviour of P(0<X(t,x)<a) as a approaches 0, as well as the Hausdorff dimension of the boundary of the support of X(t). The answers are in terms of the lead eigenvalue of the Ornstein-Uhlenbeck generator with a particular killing term. This work is motivated in part by questions of pathwise uniqueness for associated stochastic partial differential equations.
Edge homogenization of Dyson Brownian motion and applications
证明了耦合Dyson Brownian运动在谱边缘的均化估计，并应用于量化广义Wigner矩阵特征值分布的边缘普适性，给出了Kolmogorov-Smirnov距离的上下界。
We study the overlaps between right and left eigenvectors for random matrices of the spherical and truncated unitary ensembles. Conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent random variables. This enables us to prove that the scaled diagonal overlaps, conditionally on one eigenvalue, converge in distribution to a heavy-tail limit, namely, the inverse of a $γ_2$ distribution. These results are analogous to what is known for the complex Ginibre ensemble. We also provide formulae for the conditional expectation of diagonal and off-diag…
We derive asymptotics for the probability of the origin to be an extremal point of a random walk in R^n. We show that in order for the probability to be roughly 1/2, the number of steps of the random walk should be between e^{c n / log n}$ and e^{C n log n}. As a result, we attain a bound for the ?pi/2-covering time of a spherical brownian motion.
Given a submersion $ϕ: M \to N$, where $M$ is Riemannian, we construct a stochastic process $X$ on $M$ such that the image $Y:=ϕ(X)$ is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping $π: GL(n) \to GL(n)/O(n)$, whose image is equivalent to the space of $n$-by-$n$ positive definite matrices, $\pdef$, and the said MCF has deterministic image. We are able to compute explicitly the mean curvature (and hence the drift term) of the fibers w.r.t. this map, (i) under diagonalization and (ii) in matrix entries, writing mean curvature as the gra…
We construct an analogue of Dyson Brownian motion in the Siegel half-space H that we term Siegel Brownian motion. Given βin (0,\infty], a stochastic flow for Z_t in H is introduced so that the law of the eigenvalues λ_t of the cross ratio matrix R(Z_t,iI_n) is determined by the Ito differential equation corresponds to stochastic gradient ascent of a function S. S turns out to be the log volume of isospectral orbit in H and can be understood as a Boltzmann entropy. In the limit β=\infty, the group orbits evolve by motion by minus a half times mean curvature.
We find the exponential growth rate of the population outside a ball with time dependent radius for a branching Brownian motion in Euclidean space. We then see that the upper bound of the particle range is determined by the principal eigenvalue of the Schrödinger type operator associated with the branching rate measure and branching mechanism. We assume that the branching rate measure is small enough at infinity, and can be singular with respect to the Lebesgue measure. We finally apply our results to several concrete models.
We investigate the process of eigenvalues of a fractional Wishart process defined as N=B*B, where B is a matrix fractional Brownian motion recently studied by Nualart and Pérez-Abreu. Using stochastic calculus with respect to the Young integral we show that the eigenvalues do not collide at any time with probability one. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter $H\in(1/2,1)$ we derive a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally a f…
We study the trajectorywise blowup behavior of a semilinear partial differential equation that is driven by a mixture of multiplicative Brownian and fractional Brownian motion, modeling different types of random perturbations. The linear operator is supposed to have an eigenfunction of constant sign, and we show its influence, as well as the influence of its eigenvalue and of the other parameters of the equation, on the occurrence of a blowup in finite time of the solution. We give estimates for the probability of finite time blowup and of blowup before a given fixed time. Essential tools are …
Bounding Escape Rates and Approximating Quasi-Stationary Distributions of Brownian Dynamics
本文提出一种基于平方和规划的框架，用于界定布朗动力学生成元的主特征值，从而近似逃逸率和拟平稳分布，并在低维问题上验证了有效性。
We obtain new upper tail probabilities of $m$-times integrated Brownian motions under the uniform norm and the $L^p$ norm. For the uniform norm, Talagrand's approach is used, while for the $L^p$ norm, Zolotare's approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities (large ball probabilities) for general Gaussian random variable in Banach spaces. As applications, explicit bounds are given for the largest eigenvalue of t…
The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix $G+\mathrm{diag}(\mathbf{a})$, where $G$ is the random matrix from the Gaussian Unitary Ensemble (GUE), and $\mathrm{diag}(\mathbf{a})$ is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application is equivalent in distribution to a deterministic shift of the matrix. This result also leads to a new distributional symmetry for a family of reflected Brownian motions with drifts coming from an arith…
This paper provides yet another look at the mixed fractional Brownian motion (fBm), this time, from the spectral perspective. We derive an approximation for the eigenvalues of its covariance operator, asymptotically accurate up to the second order. This in turn allows to compute the exact $L_2$-small ball probabilities, previously known only at logarithmic precision. The obtained expressions show an interesting stratification of scales, which occurs at certain values of the Hurst parameter of the fractional component. Some of them have been previously encountered in other problems involving su…
On the Limit of the Tridiagonal Model for $β$-Dyson Brownian Motion
本文研究了β-戴逊布朗运动下三对角模型的极限，推导了动态β-随机Airy算子的结构，并通过数值验证了β=1,2,4的情况。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。