P14原子簇的理论研究

以Ｐ_２、Ｐ_４（Ｔｄ）、Ｐ_８（Ｃ_（２ｖ））及Ｐ_（１０）（Ｃ_（２ｖ））为结构单元设计了Ｐ_２＋Ｐ_４（Ｔ_ｄ）＋Ｐ_８（Ｃ_（２ｖ））→Ｐ_（１４）（Ｃ_ｓ）（Ａ）、Ｐｓ（Ｃ_（２ｖ）＋３Ｐ_２→Ｐ_（１４）（Ｃ_（２ｖ））（Ｉ）（Ｂ）、Ｐ_（１０）（Ｃ_（２ｖ）＋２Ｐ_２→Ｐ１４（Ｃ_（２ｖ））（Ⅱ）（Ｃ），利用从头算Ｇａｕｓｓｉａｎ－９４程序，选择６－３１Ｇ基组，对４种结构单元及Ｐ_（１４）原子簇的３种几何构型Ｃｓ、Ｃ_（２ｖ）（Ⅰ）、Ｃ_（２ｖ）（Ⅱ）进行全优化。相对能量的计算结果表明，Ｐ_（１４）（Ｃ_ｓ）与Ｐ_（１４）（Ｃ_（２ｖ））（Ⅰ）构型稳定。进一步设计（７／２）Ｐ_４（Ｔ_ｄ）→Ｐ_（１４）（Ｄ）及７Ｐ_２→Ｐ_（１４）（Ｅ），其相对能量表明，Ｐ_（１４）（Ｃ_ｓ）构型是稳定的．磷与磷连结单键键长范围为０．２２０～０．２２８ｎｍ，双键键长范围为０．２００～０．２０２ｎｍ，与实验结果大体相当。     

On the uniqueness of overcomplete dictionaries, and a practical way to retrieve them

A full-rank under-determined linear system of equations Ax = b has in general infinitely many possible solutions. In recent years there is a growing interest in the sparsest solution of this equation—the one with the fewest non-zero entries, measured by ∥x∥₀. Such solutions find applications in signal and image processing, where the topic is typically referred to as “sparse representation”. Considering the columns of A as atoms of a dictionary, it is assumed that a given signal b is a linear composition of few such atoms. Recent work established that if the desired solution x is sparse enough, uniqueness of such a result is guaranteed. Also, pursuit algorithms, approximation solvers for the above problem, are guaranteed to succeed in finding this solution.Armed with these recent results, the problem can be reversed, and formed as an implied matrix factorization problem: Given a set of vectors {b_i}, known to emerge from such sparse constructions, Ax_i = b_i, with sufficiently sparse representations x_i, we seek the matrix A. In this paper we present both theoretical and algorithmic studies of this problem. We establish the uniqueness of the dictionary A, depending on the quantity and nature of the set {b_i}, and the sparsity of {x_i}. We also describe a recently developed algorithm, the K-SVD, that practically find the matrix A, in a manner similar to the K-Means algorithm. Finally, we demonstrate this algorithm on several stylized applications in image processing.     

Quantum Fourier Transform-Based Arithmetic Logic Unit on a Quantum Processor

This study proposes and construct a primitive quantum arithmetic logic unit (qALU) based on the quantum Fourier transform (QFT). The qALU is capable of performing arithmetic ADD (addition) and logic NAND gate operations. It designs a scalable quantum circuit and presents the circuits for driving ADD and NAND operations on two-input and four-input quantum channels, respectively. By comparing the required number of quantum gates for serial and parallel architectures in executing arithmetic addition, it evaluates the performance. It also execute the proposed quantum Fourier transform-based qALU design on real quantum processor hardware provided by IBM. The results demonstrate that the proposed circuit can perform arithmetic and logic operations with a high success rate. Furthermore, it discusses in detail the potential implementations of the qALU circuit in the field of computer science, highlighting the possibility of constructing a soft-core processor on a quantum processing unit.     

Ultradistributional boundary values of harmonic functions on the sphere

We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderón and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.     

Hall’s and Kőnig’s theorem in graphs and hypergraphs

We investigate the relation between Hall’s theorem and K?nig’s theorem in graphs and hypergraphs. In particular, we characterize the graphs satisfying a deficiency version of Hall’s theorem, thereby showing that this class strictly contains all K?nig–Egerváry graphs. Furthermore, we give a generalization of Hall’s theorem to normal hypergraphs.     

Problems on matchings and independent sets of a graph

Let $G$ be a finite simple graph. For $X?V\left(G\right)$, the difference of $X$, $d\left(X\right)?\left|X\right|?\left|N\left(X\right)\right|$ where $N\left(X\right)$ is the neighborhood of $X$ and $max\{d\left(X\right):X?V\left(G\right)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d\left(X\right)$ equals the critical difference and $\ker \left(G\right)$ is the intersection of all critical sets. $\mathrm{diadem}\left(G\right)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with$d\left(S\right)>0$ if no proper subset of $S$ has positive difference.A graph $G$ is called a König–Egerváry graph if the sum of its independence number $\alpha \left(G\right)$ and matching number $\mu \left(G\right)$ equals $\left|V\left(G\right)\right|$.In this paper, we prove a conjecture which states that for any graph the number of inclusion minimal independent set $S$ with $d\left(S\right)>0$ is at least the critical difference of the graph.We also give a new short proof of the inequality $\left|\ker \left(G\right)\right|+\left|\mathrm{diadem}\left(G\right)\right|\le 2\alpha \left(G\right)$.A characterization of unicyclic non-König–Egerváry graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $\alpha \left(G\right)?\mu \left(G\right)$, is proved.We also make an observation about $\ker \left(G\right)$ using Edmonds–Gallai Structure Theorem as a concluding remark.     

Null decomposition of trees

Let $T$ be a tree. We show that the null space of the adjacency matrix of $T$ has relevant information about the structure of $T$. We introduce the Null Decomposition of trees, which is a decomposition into two different types of trees: N-trees and S-trees. N-trees are the trees that have a unique maximum (perfect) matching. S-trees are the trees with a unique maximum independent set. We obtain formulas for the independence number and the matching number of a tree using this decomposition. We also show how the number of maximum matchings and the number of maximum independent sets in a tree are related to its null decomposition.     

用在线梯度法训练积单元神经网络的收敛性分析

<正>1引言仅由加和单元构成的传统前向神经网络已经广泛应用于模式识别及函数逼近等领域.但在处理比较复杂的问题时,这种网络往往需要补充大量的隐节点,这样就不可避免地增     

中国经济增长与环境污染的协整关系研究——基于1991-2007年省际面板数据

利用1991-2007中国29个省市的面板数据,对经济增长与环境污染的长期均衡关系进行了研究,对人均实际GDP及代表环境污染的五类污染物排放指标进行了面板单位根检验和面板协整分析。结果表明,在五类污染物排放指标中,只有工业固体废弃物排放量与人均GDP之间符合EKC特征,工业废水排放量随经济增长而逐渐减少,而工业废气排放量与经济增长不存在协整关系。