The construction and approximation of feedforward neural network with hyp erb olic tangent function

In this paper, we discuss some analytic properties of hyperbolic tangent function and estimate some approximation errors of neural network operators with the hyperbolic tangent activation functionFirstly, an equation of partitions of unity for the hyperbolic tangent function is givenThen, two kinds of quasi-interpolation type neural network operators are constructed to approximate univariate and bivariate functions, respectivelyAlso, the errors of the approximation are estimated by means of the modulus of continuity of functionMoreover, for approximated functions with high order derivatives, the approximation errors of the constructed operators are estimated.     

Asymptotically optimal neighbour sum distinguishing colourings of graphs

Consider a simple graph G = (V,E) and its proper edge colouring c with the elements of the set {1,2,…,k}. The colouring c is said to be neighbour sum distinguishing if for every pair of vertices u, v adjacent in G, the sum of colours of the edges incident with u is distinct from the corresponding sum for v. The smallest integer k for which such colouring exists is known as the neighbour sum distinguishing index of a graph and denoted by . The definition of this parameter, which makes sense for graphs containing no isolated edges, immediately implies that , where Δ is the maximum degree of G. On the other hand, it was conjectured by Flandrin et al. that for all those graphs, except for C₅. We prove this bound to be asymptotically correct by showing that . The main idea of our argument relays on a random assignment of the colours, where the choice for every edge is biased by so called attractors, randomly assigned to the vertices. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 776–791, 2015     

On a problem of Hornik

In 1991,Hornik proved that the collection of single hidden layer feedforward neural networks(SLFNs)with continuous,bounded,and non-constant activation functionσis dense in C(K)where K is a compact set in R~s(see Neural Networks,4(2),251-257(1991)).Meanwhile,he pointed out"Whether or not the continuity assumption can entirely be dropped is still an open quite challenging problem".This paper replies in the affirmative to the problem and proves that for bounded and continuous almost everywhere(a.e.)activation functionσon R,the collection of SLFNs is dense in C(K)if and only ifσis un-constant a.e..     

Estimation of rating classes and default probabilities in credit risk models with dependencies

Let Y = m(X) + ε be a regression model with a dichotomous output Y and a one‐step regression function m . In the literature, estimators for the three parameters of m , that is, the breakpoint θ and the levels a and b , are proposed for independent and identically distributed (i.i.d.) observations. We show that these standard estimators also work in a non‐i.i.d. framework, that is, that they are strongly consistent under mild conditions. For that purpose, we use a linear one‐factor model for the input X and a Bernoulli mixture model for the output Y . The estimators for the split point and the risk levels are applied to a problem arising in credit rating systems. In particular, we divide the range of individuals' creditworthiness into two groups. The first group has a higher probability of default and the second group has a lower one. We also stress connections between the standard estimator for the cutoff θ and concepts prevalent in credit risk modeling, for example, receiver operating characteristic. Copyright © 2014 John Wiley & Sons, Ltd.     

Semiclassical matrix model for quantum chaotic transport with time-reversal symmetry

We show that the semiclassical approach to chaotic quantum transport in the presence of time-reversal symmetry can be described by a matrix model. In other words, we construct a matrix integral whose perturbative expansion satisfies the semiclassical diagrammatic rules for the calculation of transport statistics. One of the virtues of this approach is that it leads very naturally to the semiclassical derivation of universal predictions from random matrix theory.     

Orientational order parameter studies on 3.Om and 3O.Om liquid crystals

Orientational order parameter S is evaluated in the nematic phase of six liquid crystal compounds, N-(p-n-propyl benzylidene)-p-n-alkoxy anilines, 3.Om and N-(p-n-propyloxy benzylidene)-p-n-alkoxy anilines, 3O.Om compounds with m = 6, 7 and 8, using different methods. The techniques employed are S from birefringence δn, Haller's approximation from (1?T/T_c) ^β, effective geometry parameter α_g and Vuks’ scaling factor S_C. The values of S obtained using the above methods are compared with one another and with the results on a number of liquid crystals; the liquid crystals favor isotropic Vuks’ method.     

空间关联白噪声影响下小世界神经元网络系统的同步动力学

以电耦合的Terman-Wang小世界神经元网络系统为研究对象, 研究了空间关联白噪声影响下神经元网络系统的同步动力学. 首先将动力学平均场近似理论扩展到受空间关联白噪声影响下的小世界网络系统中, 将描述网络系统动力学演化的2N维随机微分方程简化为11个确定性的矩微分方程. 其次, 基于动力学平均场近似理论所推导的矩方程, 讨论了空间关联噪声、网络结构参数对神经元网络系统同步动力学的关键影响, 发现较大的噪声空间关联系数、耦合强度及节点平均度均对神经元网络系统同步放电具有积极作用. 进一步地, 利用计算机仿真数值模拟原神经元网络系统的同步动力学, 并与基于动力学平均场近似理论所得到的结果进行比较, 发现二者具有较好的一致性.     

二进制信号的混沌压缩测量与重构

二进制信号的压缩感知问题对应超奈奎斯特信号系统中未编码的二进制符号的检测问题, 具有重要的研究意义. 已有的二进制信号压缩测量采用高斯随机矩阵, 信号重构采用经典的l₁最小化方法. 本文利用混沌映射构造基于Cat序列的循环测量矩阵, 并提出一种针对二进制信号的全新的重构算法——平滑函数逼近法. 文章构造的混沌循环测量矩阵兼具确定性和随机性的优点, 能够抵御低信令效率和低信噪比的影响, 取得更好的压缩测量效果. 文章提出的平滑函数逼近法利用非凸函数代替原问题不连续的目标函数, 将组合优化问题转化为具有等式约束的优化问题进行求解. 利用稀疏贝叶斯学习算法进一步修正误差, 得到更准确的重构信号. 在信道含有加性高斯白噪声的条件下对二进制信号进行了压缩测量与重构的数值仿真, 仿真结果表明:基于Cat 序列的循环测量矩阵的压缩测量效果明显优于传统的高斯随机矩阵; 平滑函数逼近法对二进制信号的重构性能明显优于经典的l₁最小化方法.     

Computation of electronic structure and magnetic properties of strongly correlated materials with LDA+DMFT method

In this review, we describe general ideas of the LDA+DMFT method which merges dynamical mean-field theory (DMFT) and density functional theory (in particular the local density approximation (LDA)). Nowadays, the LDA+DMFT computational scheme is the most powerful numerical tool for studying physical properties of real materials and chemical compounds. It incorporates the advantage of DMFT to treat the full range of local dynamical Coulomb correlations and the ability of band methods to describe material-specific band dispersion caused by the lattice periodicity. We briefly discuss underlying physical ideas of LDA+DMFT and its mathematical implementation. Then different algorithms applied to solution of the DMFT impurity problem are briefly described. We then give examples of successful applications of the LDA+DMFT method to study spectral and magnetic properties of recently synthesized compounds like pnictide superconductors as well as classic charge-transfer systems NiO and MnO.     

基于半经典近似研究费米气体的稳定性及顺磁性

基于半经典近似方法,给出重力场和强磁场共存下费米子的能谱.然后由泊松公式导出费米气体的热力学势函数.在此基础上,运用热力学关系式求解低温条件下系统的稳定性及磁化率的解析式,并通过数值模拟分析强磁场背景下重力场对稳定性及顺磁性的影响机制.