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.

首先，我们给出了单位圆盘$D$上一个典型积分的双向估计，该结果推广了1980年Rudin一本专著中的命题1.4.10 (情形$n=1$). 其次，我们应用该积分估计给出了单位圆盘上广义Bergman型算子\ $T_{\lambda,\tau,c,k}$\ 和\ $S_{\lambda,\tau,c,k}$\ 从加权Lebesgue空间\ $L^{p}(D,dA_{\alpha})$\ 到\ $L^{q}(D,dA_{\beta})$\ （$1\leq p\leq q\leq +\infty$, \ $\alpha$\ 和\ $\beta$\ 为实数）有界的充要条件，该结果推广了诸多学者的相关工作.

设$\{X_i\}_{i=1}^n$ 为一族有限维Banach 空间, $(X,\norm{\cdot}_p)$为它们的的$p$-直和空间, $B$为$X$的闭单位球, $\Gamma_m(B)$ 为用$m$个$\gamma B$ 的 平移覆盖 $B$所需的最小正数 $\gamma$. 本文给出了 $\Gamma_m(B)$ 的几个与$p$有关的估计, 并在$X$的维数为3和4时, 分别证明了$\Gamma_8(B) \leq 0.8599$, $\forall p \in [1,\infty]$以及$\Gamma_{16}(B) \leq 0.8486$, $\forall p \in [1,\infty]$.

In this paper, we study the existence of multiple normalized solutions to the following fractional Schrödinger equations:beginalign*begincasesvarepsilon^2s-Δ)^s u+V(x)u=λ u+f(u), displaystyle∫mathbbR^Nu²dx=a²varepsilon^N,endcasesendalign*where 0<s<1, Ngeq2, a, varepsilon>0, λ∈mathbbRis an unknown parameter to appear as a Lagrange multiplier, V is a continuous potential function, and f is a continuous function with L²-subcritical growth. We use Lusternik-Schnirelmann category and the penalization method to prove the existence of multiple normalized solutions, which is related to the topology of the potential V around its minimum.

In this survey, we summarize the recent progress in complex Wiener-Itô chaos decomposition. It mainly includes two aspects: the characterization of the mutual representation between complex and real multiple Wiener-Itô integrals, and the complex fourth moment 定理 and the results of its related convergence rates.

The mechanical foundation of heat is still an important and unresolved 问题 in fundamental physics. In this paper, we systematically study the emergent deterministic descriptions of stochastic motions in the continuous space-time (macroscopic) limit, where a Liouville kinematic equation and a Hamilton-Jacobi equation (HJE) for the evolution of an entropy function are presented. As a characterization of the exponentially small infinitesimal, the HJE provides an energetic description of the kinematic equation. Most importantly, we find that the Legendre transform of the kinetic energy in the Hamiltonian of the HJE equals the macroscopic-limit heat exchange rate in the stochastic thermodynamics, leading to the fact that heat and kinetic energy are dual descriptions for the same phenomenon, based on which we propose the concept of thermokinetic energy in this paper.

We show that the sample entropy production rate of a continuous-time Markov chain can be represented as a compound Poisson integral. Based on this representation, we further prove Birkhoff's ergodic 定理 and the central limit 定理, providing an explicit variance for the sample entropy production rate of countable-state Markov chains.

The continuous emergence of high-dimensional biological big data presents new challenges for mathematical modeling and computation in the fields of probability and statistics, data science, and artificial intelligence. In this paper, we review the research progress in dynamic modeling and computation of snapshot-type high-dimensional data, with time-series single-cell sequencing data as a representative 例. We focus on mathematical models and optimization methods based on optimal transport theory, emphasizing the key role of integrating mathematical models with deep learning for efficient computation in high-dimensional big data. Additionally, we discuss the challenges and opportunities in modeling and computation of complex nonlinear systems, such as inferring cell-cell interactions.