由于张量数据具有高阶特性和内部结构依赖性, 传统分位回归建模面临诸多挑战. 本文通过引入张量环分解技术, 提出了一种新型张量环分位回归模型, 用以刻画张量值协变量、向量值协变量与标量值响应变量之间的复杂关系. 在低维情景下, 本文证明了估计量的一致性、渐近正态性及有效性; 在高维情景下, 提出了$\ell_1$惩罚估计量, 并建立了相应的估计误差界. 模拟和实证研究结果表明, 该方法在估计与预测方面均具有出色的表现.

本文探讨模型结构基本公理的独立性问题. 通过完备余挠对等方法，本文构造了四个范畴 $\mathcal{A}$ 及其上的态射类三元组 $(\CoFib(\mathcal{A}), \ \Fib(\mathcal{A}), \ \Weq(\mathcal{A}))$, 每个三元组满足 Quillen 的模型结构中的三条公理但违反第四条公理. 这些构造表明了 Quillen 模型结构定义中四条公理的独立性, 即, 没有任何一条公理是其余公理的逻辑推论.

In this paper, we mainly investigate some local and global properties of positive solutions to the degenerate elliptic equation Δp u + a∇ u^p + b u^q <roman>e</roman>^c u= 0 defined on a Riemannian manifold. We establish Cheng-Yau type gradient estimates for positive solutions of this equation. As corollaries, we derive Harnack inequalities and Liouville theorems for such solutions.

In this paper, we prove the Freidlin-Wentzell-type large deviation principle for a general system of fully-coupled slow-fast McKean-Vlasov stochastic dynamics, when the noise intensity δ→0 and the time scale parameter varepsilon(δ) satisfies varepsilon²(δ)/δ→0. The coefficients can depend on the distribution of the slow motion and that of the fast motion. The main techniques are based on the Poisson equation and the weak convergence approach for the large deviation principle.

In this expository paper, we discuss some recent developments on Ricci solitons and ancient solutions in the Ricci flow. We begin with variously typical examples of Ricci solitons and ancient solutions, then focus on some important topics in Ricci flow, such as the rigidity problem of steady gradient Ricci solitons, classification of blow-up solutions, and asymptotic behavior analysis. We also present and discuss some open problems in this area.

In this paper, we introduce the finite-time averaged Lyapunov exponent (FTALE) to characterize chaotic mixing in dynamical systems from a fresh perspective. Different from the traditional finite-time Lyapunov exponent (FTLE), which adopts a single-perturbation-evaluation paradigm and only measures the amplification of an initial disturbance at the final instant, FTALE employs an infinite-perturbation-evaluation protocol. It continuously tracks the average sensitivity of particle trajectories to impulsive perturbations introduced at any instant throughout the entire time window, thereby capturing the full sequence of instabilities experienced along the way. To make the framework practical, we develop two efficient Eulerian algorithms tailored for distinct scenarios and provide rigorous complexity and error estimates. Numerical experiments demonstrate that FTALE, together with its Eulerian algorithms, unveils the intricate chaotic skeleton of the underlying flow robustly and efficiently.

In this paper, we systematically review the intrinsic relationships between the volume of manifolds and Sobolev constants, Ricci curvature, or scalar curvature, summarize the research progress on the rigidity of Sobolev constants on manifolds, and outline the classification results for positive solutions of the critical p-Laplace equation defined on non-compact complete manifolds with non-negative Ricci curvature. Based on the current state of the two major research directions mentioned above, we further propose some key open problems.