| 本文研究一类高阶非线性微分方程的Lyapunov 不等式，是对《Lyapunov-type inequalities for $\psi$-Laplacian equations》有关结论的进一步探讨和推广. |
The obstacle problem for non-coercive equations with lower order term and $L^1-$data
Jun Zheng
| The aim of this paper is to study the obstacle problem associated with an elliptic operator having degenerate coercivity, and with a low order term and $L^1-$data. We prove the existence of an entropy solution to the obstacle problem and show its continuous dependence on the $L^{1}-$data in $W^{1,q}(\Omega)$ with some $q>1$. |
Lyapunov-type inequalities for a class of nonlinear higher order differential equations
Jun Zheng; Haofan Wang
| In this work, we establish several Lyapunov-type inequalities for a class of nonlinear higher order differential equations having a form \begin{align*} (\psi(u^{(m)}(x)))'+\sum_{i=0}^nr_i(x)f_i(u^{(i)}(x))=0, %\ \ \ \ \text{or}\ \ \ \ (\psi(u^{(m)}))^{(m)}+r_i(x)f(u)=0, \end{align*} with anti-periodic boundary conditions, where $m> n\geq 0$ are integers, $\psi$ and $f_i (i=0,1,2,...,n)$ satisfy certain structural conditions such that the considered equations have general nonlinearities. The obtained inequalities are extensions and complements of the existing results in the literature. |
Lyapunov-type inequalities for ψ−Laplacian equations
Zheng, Jun; Guo, Xu
| In this work, we present several Lyapunov-type inequalities for a class of $\psi-$Laplacian equations of the form \begin{align*} (\psi(u'(x)))'+r(x)f(u(x))=0, \end{align*} with Dirichlet boundary conditions, where $\psi$ and $f$ satisfies certain structural conditions with general nonlinearities. We do not require any sub-multiplicative property of $\psi$, and any convexity of $\frac{1}{\psi(t)}$ or $\psi (t)t$ in the establishment of Lyapunov-type inequalities. The obtained inequalities can be seen as extensions and complements of the existing results in the literature. |
| 本文通过严格数学分析找出了逻辑回归过拟合的成因：边界样本的损失贡献比重大且随法向量增长而加速增大、边界样本分布散乱，顺便理清了正则项的作用机理。 利用过拟合机制，本文提出一种反拉方法，既能缓解过拟合，又能减少训练步数，在MNIST数据集上实现加速38.25倍，在CIFAR10数据集上实现加速5.61倍。 |
T-Area-Marker for Scientific Images
陈光; 郭春芳
| Labeled images are one of the most important means of scientific communication and education. However, traditional markers (arrows, lines) are point markers; do not include information about how large the feature is. We designed an efficient marker system for labeling scientific images (electron or light microscopy, CT, MRI, ultrasonography, camera pictures, etc), called the “T Area Marker, (TAM)”. The basic TAM marker looks like a “T”, composed of a line segment and a small tick on one end; it defines an imagined circle that stands on the tickless end and the diameter of the circle is equal to the length of the line segment. Thus the TAM can define an exact area rather than a single point; and the imagined circle does not break the continuity of the image (unlike traditional visible circles, rectangles, etc). A TAM with N ticks (N>1) means the diameter equals to N times the length of TAM. A TAM may also have a tail and/or several tail branches to define translation of the imagined circle, thus define complicated areas. tAreaMarker.py is free software that combines the drawing and reading of TAMs, although in most cases TAMs are easily interpreted without computer assistance. |
Gibbsian representation of knowledge in infinite dimensional random neural networks(IDRNN)
xi guangcheng
| Abstract. In studying of a class of random neural network, some of relative researchers have proposed Markov model of neural network. Wherein Markov property of the neural network is based on “assuming”. To reveal mechanism of generating of Markov property in neural network, it is studied how infinite-dimensional random neural network (IDRNN) forms inner Markov representation of environment information in this paper.Because of equivalence between markov property and Gibbsian our conclusion is that knowledge is eventually expressed by extreme Gibbs probability measure—ergodic Gibbs probability measure in IDRNN. This conclusion is also applicable to quantum mechanical level of IDRNN. Hence one can see “ concept “- “ consciousness” is generated at particle(ion) level in the brain and is experienced at the level of the neurons; We have discussed also ergodicity of IDRNN with random neural potential. |
| 本文通过严格数学分析找出了逻辑回归过拟合的成因：边界样本的损失贡献比重大且随法向量增长而加速增大、边界样本分布散乱，顺便理清了正则项的作用机理。 利用过拟合机制，本文提出一种反拉方法，既能缓解过拟合，又能减少训练步数，在MNIST数据集上实现加速38.25倍，在CIFAR10数据集上实现加速5.61倍。 |
| 本文利用代数和组合的相关理论对超平面配置理论的自由性做出了相关研究，给出圈图所对应的图配置的生成元和导模的具体形式，并以图论的形式对圈图配置的导模及其自由解做出相关的组合解 释. |
Upper bounds for Z$_1$-eigenvalues of generalized Hilbert tensors
孟娟; 宋义生
| In this paper, we introduce the concept of Z$_1$-eigenvalue to infinite dimensional generalized Hilbert tensors (hypermatrix) $\mathcal{H}_\lambda^{\infty}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$ \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}+\lambda},\ \lambda\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=0,1,2,\cdots,n,\cdots, $$ and proved that its $Z_1$-spectral radius is not larger than $\pi$ for $\lambda>\frac{1}{2}$, and is at most $\frac{\pi}{\sin{\lambda\pi}}$ for $\frac{1}{2}\geq \lambda>0$. Besides, the upper bound of $Z_1$-spectral radius of an $m$th-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_\lambda^n$ is obtained also, and such a bound only depends on $n$ and $\lambda$. |