关于经典分析中偏导数和全导数概念的一个歧义

关于首导数处含小参数的偏积分微分方程的差分估计

About estimations of difference for the partial integro-differential equation with small parameter at the leading derivative

本文研究了二阶偏积分微分方程的奇摄动问题。得到了Cauchy问题解的估计。

The partial derivative of Okamoto's functions with respect to the parameter

The differentiability of the one parameter family of Okomoto's functions as functions of $x$ has been analyzed extensively since their introduction in 2005. As an analogue to a similar investigation, in this paper, we consider the partial derivative of Okomoto's functions with respect to the parameter $a$. We place a significant focus on $a = 1/3$ to describe the properties of a nowhere differentiable function $K(x)$ for which the set of points of infinite derivative produces an example of a measure zero set with Hausdorff dimension $1$.

Discreet Solutions for Matrix Systems in Partial Derivatives Hyperbolic and Singular

In this paper we study the construction of a discrete solution for a hyperbolic system of partial differentials of the strongly coupled type. In its construction, the discrete separation of matricial variable method was followed. Two separate equations in differences were obtained: a singular matricial and the other one a Sturm Liouville vectorial problem, which by the superposition principle yield a stable discrete solution.

The equality of mixed partial derivatives under weak differentiability conditions

We review and develop two little known results on the equality of mixed partial derivatives which can be considered the best results so far available in their respective domains. The former, due to Mikusiński and his school, deals with equality at a given point, while the latter, due to Tolstov, concerns equality almost everywhere. Applications to differential geometry and General Relativity are commented.

关于多变量函数的修正 Bernstein-Stancu 多项式的混合偏导数

On mixed partial derivatives of modified Bernstein-Stancu polynomials for functions of several variables

本文研究在最小正则性条件下，利用修正 Bernstein 多项式对多变量函数二阶混合偏导数进行 $L_1$ 范数逼近的问题。

格林函数关于任意两点边界条件的偏导数对参数依赖性的显式公式

An explicit formula of the parameter dependence of de partial derivatives of the Green's functions related to arbitrary two-point boundary conditions

本文推导了格林函数偏导数对参数依赖的显式积分公式，并据此简化了其常符号单调性证明，改进了现有结果，且方法可推广至非局部条件及偏/分数阶微分方程。

关于凸区域上偏导数范数之间的不等式

On inequalities between norms of partial derivatives on convex domains

本文研究凸区域上边界为零的二元凹函数偏导数 $L_p$-范数比值的有界性问题，证明了 $p=1$ 时比值始终有界并给出精确估计，而 $p>1$ 时结论依赖于区域几何。

关于涉及多项式导数的时间尺度上偏动态方程的研究

A study on partial dynamic equation on time scales involving derivatives of polynomials

该研究在二维时间尺度框架下，推导了奇次幂多项式的时间尺度导数与特定点处多项式偏导数的关联恒等式，并借此统一阐释了有限差分、经典导数、q-导数等多种导数算子。

On the derivatives $\partial^{2}P_ν(z)/\partialν^{2}$ and $\partial Q_ν(z)/\partialν$ of the Legendre functions with respect to their degrees

We provide closed-form expressions for the degree-derivatives $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=n}$ and $[\partial Q_ν(z)/\partialν]_{ν=n}$, with $z\in\mathbb{C}$ and $n\in\mathbb{N}_{0}$, where $P_ν(z)$ and $Q_ν(z)$ are the Legendre functions of the first and the second kind, respectively. For $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=n}$, we find that $$\displaystyle\frac{\partial^{2}P_ν(z)}{\partialν^{2}}\bigg|_{ν=n}=-2P_{n}(z)\textrm{Li}_{2}\frac{1-z}{2}+B_{n}(z)\ln\frac{z+1}{2}+C_{n}(z),$$ where $\textrm{Li}_{2}[(1-z)/2]$ is the dilogarithm function, $P_{n}(z)$ is the Legendre polynomia…

Green's Theorem for Generalized Fractional Derivatives

We study three types of generalized partial fractional operators. An extension of Green's theorem, by considering partial fractional derivatives with more general kernels, is proved. New results are obtained, even in the particular case when the generalized operators are reduced to the standard partial fractional derivatives and fractional integrals in the sense of Riemann-Liouville or Caputo.

On a sampling expansion with partial derivatives for functions of several variables

Let $B^p_σ$, $1\le p<\infty$, $σ>0$, denote the space of all $f\in L^p(\mathbb{R})$ such that the Fourier transform of $f$ (in the sense of distributions) vanishes outside $[-σ,σ]$. The classical sampling theorem states that each $f\in B^p_σ$ may be reconstructed exactly from its sample values at equispaced sampling points $\{πm/σ\}_{m\in\mathbb{Z}} $ spaced by $π/σ$. Reconstruction is also possible from sample values at sampling points $\{πθm/σ\}_m $ with certain $1< θ\le 2$ if we know $f(θπm/σ) $ and $f'(θπm/σ)$, $m\in\mathbb{Z}$. In this paper we present sampling series for functions of sev…

偏导数界的 UMD 条件的必要性与精确常数

Bounds for partial derivatives: necessity of UMD and sharp constants

本文证明了不同偏导数间的 $L^p$ 界不等式需要 UMD 条件，并给出了 UMD 常数的定量估计，指出最佳常数 $K$ 等于 UMD 常数的一半，该精确值对标量值函数也是新结论。

On a $L^\infty$ functional derivative estimate relating to the Cauchy problem for scalar semi-linear parabolic partial differential equations with general continuous nonlinearity

In this paper, we consider a $L^\infty$ functional derivative estimate for the first spatial derivative of bounded classical solutions $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ to the Cauchy problem for scalar semi-linear parabolic partial differential equations with a continuous nonlinearity $f:\mathbb{R}\to\mathbb{R}$ and initial data $u_0:\mathbb{R}\to\mathbb{R}$, of the form, \[ \sup_{x\in\mathbb{R}}|u_x (x , t)| \leq \mathcal{F}_t (f,u_0,u) \ \ \ \forall t\in [0,T] . \] Here $\mathcal{F}_t:\mathcal{A}_t\to\mathbb{R}$ is a functional as defined in \textsection 1. We establish that the functi…