空间形式中具有两个线性相关平均曲率函数的超曲面

在空间形式中, 我们构造了一类泛函, 其临界点包括极小与r 极小超曲面. 给出了临界超曲面的代数、微分和变分刻画. 我们证明了Simons 类不存在定理: 在单位球面中不存在稳定的临界超曲面. 同时证明了Alexandrov 类存在性定理: 在欧氏空间中球面是唯一的稳定的临界超曲面.     

关于有限p-群的正规闭包的一些条件

Berkovich 提出了研究满足下列条件的有限p-群G, 对于G的每一个非正规子群H满足(N₁)exp(H)=exp(H^G); (N₂) |H^G:H|≤ p; (N₃) H^G=HG''. 本文首先研究满足条件(N₃) 的有限p-群,然后讨论满足条件(N₁); (N₂) 和(N₃) 的有限p-群.     

Banach 空间上算子的几种不可约性

本文根据Cowen-Douglas 算子的定义引入两类与强不可约算子有紧密联系的算子类— B_n 算子与B 算子. 说明了在可分Banach 空间上存在B_n 算子与B 算子. 文章详细讨论了几种具有不可约性的算子类之间的关系并得到了一个关系图. 本文还给出这些算子类的一些性质, 包括算子的(拟) 相似不变性等.     

n-Lie 代数的扩张

文章主要研究n-Lie 代数的扩张问题. 首先利用n-Lie 代数的模作n-Lie 代数的T_θ- 扩张与T_θ^*-扩张. 再利用模度量3-Lie 代数,做3-Lie 代数的双扩张. 文章最后利用4- 指标阵构造了m 维3-Lie代数的双扩张.     

关于曲率积分不等式的注记

本文主要研究平面卵形线的曲率积分不等式.利用积分几何中凸集的支持函数以及外平行集的性质,得到了Gage等周不等式与曲率的熵不等式的一个积分几何的简化证明;进一步地,我们得到了一个新的关于曲率积分的不等式.     

广义态空间中Krein-Milman定理的算子类似

讨论在C^*-凸理论下C^*-代数A的广义态空间S_Cⁿ(A)中的Krein-Milman型问题.证明了S_Cⁿ(A)的任意一个BW-紧的C^*-凸子集K都具有一个C^*-端点,而且K是其C^*-端点的C^*-凸包.     

关于可构成实数在Lω1中分布的几个定理

任何一个可构成实数都是在L_ω1中构成的。本文指出,并不是L_ω1前面的每一层都会产生新的实数,并用元数学的方法(或者就是模型的方法)对这样的层作了宏观上的描述。     

泛函Γ-收敛和相应梯度流的关系

给出了泛函序列Γ-收敛会导致相应梯度流解收敛的充分条件 ,部分证明了DeGiorgi猜想 .     

群的F-次正规与F-次反正规链

给定一个子群闭的饱和群系F ,定义群类Fpc　 ,使得G ∈F_pc 当且仅当对于每个子群X ≤G ,存在G的一个F 次正规子群S ,X≤S并且X在S中F 次反正规 .借助F投射子和F覆盖子群 ,给出了F_pc群的特征 .     

Q-curvature flow on 4-manifolds

We formulate an appropriate gradient flow in order to study the evolution of the Q-curvature to a prescribed function on a 4-manifold. For a class of prescribed functions, we show convergence and describe the asymptotic behaviour at infinity.     

A generalized mean field type flow on a closed Riemann surface

Let $(\mathrm{\Sigma },g)$ be a closed Riemann surface. Let ψ, h be two smooth functions on Σ with ${\int }_{\mathrm{\Sigma }}\psi d{v}_g\ne 0$ and $h\ge 0,h\not\equiv 0$. In this paper, using the method of flow due to Cast$\acute{\mathrm{e}}$ras (Pacific J. Math. 276(2015), no. 2, 321–345) and Sun–Zhu (Calc. Var. Partial Differential Equations 60(2021), no. 1, 26), we prove that the solution of the equation $$-{\mathrm{\Delta }}_{g}u=8\pi \left(\frac{h{e}^u}{{\int }_{\mathrm{\Sigma }}h{e}^{u}d{v}_g}-\frac{\psi }{{\int }_{\mathrm{\Sigma }}\psi d{v}_g}\right)$$ exists under given conditions. This gives a new proof of the main results of Zhu (Nonlinear Anal. 169(2018), 38–58).     

Q-curvature flow on 4-manifolds with boundary

Given a compact four-dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero. Using integral methods, we prove global existence and convergence for the Q-curvature flow to a smooth metric conformal to g of prescribed Q-curvature, zero T-curvature and vanishing mean curvature under conformally invariant assumptions.     

The second Sobolev best constant along the Ricci flow

In this work we present some properties satisfied by the second L ²-Riemannian Sobolev best constant along the Ricci flow on compact manifolds of dimensions n ≥ 4. We prove that, along the Ricci flow g(t), the second best constant B ₀(2, g(t)) depends continuously on t and blows-up in finite time. In certain cases, the speed of the explosion is, at least, the same one of the curvature operator. We also show that, on manifolds with positive curvature operator or pointwise 1/4-pinched curvature, one of the situations holds: B ₀(2, g(t)) converges to an explicit constant or extremal functions there exists for t large.      

The asymptotics of blow-up in inviscid Boussinesq flow and related systems

We consider the approach to blow-up in two-dimensional inviscidflows with stagnation-point similitude, in particular a buoyancy-drivenflow resulting from a horizontally quadratic density variationin a horizontally unbounded slab. The blow-up, which is onlypossible because the flow has infinite energy, proceeds by intensificationof the vorticity and density gradient in a layer adjacent tothe upper boundary, while the remainder of the flow tends towardsirrotationality. The governing Boussinesq flow equations arefirst solved numerically, and the results suggest scalings whichare then used in an asymptotic analysis as 0, where is thetime remaining until blow-up. The structure of the asymptoticsolution, involving exponential orders as well as powers andlogarithms of the small parameter, is suggested by the analysisof a simpler related problem for which an exact solution isavailable. The expansion is uniformly valid across the upperboundary layer and the outer region, but there is a layer adjacentto the lower boundary where the flow remains dependent on theinitial conditions and is undetermined by the asymptotics.     

Remarks on Blow-Up Phenomena in p-Laplacian Heat Equation with Inhomogeneous Nonlinearity

We investigate the $p$-Laplace heat equation $u_t-Delta_p u=ζ(t)f(u)$ in a bounded smooth domain. Using differential-inequality arguments, we prove blow-up results under suitable conditions on $ζ, f$, and the initial datum $u_0$. We also give an upper bound for the blow-up time in each case.     

概率论教学笔记四则

本文列举了笔者在概率论教学中处理的四个例子:生日问题、写数问题、猜拳问题、骰子问题这些例子都是日常生活中的司空见惯看似浅显的典型随机现象,通过设计随机试验让学生积极参与,或者通过理论计算分析,让学生体验抽象复杂的理论在日常生活中的应用,感受概率论的魅力,起到了活跃课堂气氛、提高学习兴趣、加强师生互动、加深学生对深奥理论和方法理解的良好效果。     

关于轴对称Navier-Stokes方程正则性的一个注记

建立了一个关于轴对称不可压Navier-Stokes系统的正则性准则.证明了如果局部的轴对称光滑解u满足‖ωr‖_{Lα1((0,T);Lβ1)}+‖ωθ/r‖_{Lα2((0,T);Lβ2)}<∞,其中2/α₁+3/β₁≤1+3/β₁,2/α₂+3/β₂≤2和β₁≥3, β₂>3/2,那么此强解将保持光滑性直至时刻T.