Convergence for a family of neural network operators in Orlicz spaces

In this paper, we develop the theory for a family of neural network (NN) operators of the Kantorovich type, in the general setting of Orlicz spaces. In particular, a modular convergence theorem is established. In this way, we study the above family of operators in many instances of useful spaces by a unique general approach. The above NN operators provide a constructive approximation process, in which the coefficients, the weights, and the thresholds of the networks needed in order to approximate a given function f , are known. At the end of the paper, several examples of Orlicz spaces, and of sigmoidal activation functions for which the present theory can be applied, are studied in details.     

A universal result for consecutive random subdivision of polygons

We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with edges, we choose a point at random on each edge, such that the proportions in which these points divide edges are i.i.d. copies of some random variable ξ. These new points form a new (smaller) polygon. By repeatedly implementing this procedure we obtain a sequence of random polygons. The aim of this paper is to show that under very mild non‐degenerateness conditions on ξ, the shapes of these polygons eventually become “flat” The convergence rate to flatness is also investigated; in particular, in the case of triangles (d = 3), we show how to calculate the exact value of the rate of convergence, connected to Lyapunov exponents. Using the theory of products of random matrices our paper greatly generalizes the results of [11] which are achieved mostly by using ad hoc methods. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 341–371, 2017     

Improving the stability and robustness of incomplete symmetric indefinite factorization preconditioners

Sparse symmetric indefinite linear systems of equations arise in numerous practical applications. In many situations, an iterative method is the method of choice but a preconditioner is normally required for it to be effective. In this paper, the focus is on a class of incomplete factorization algorithms that can be used to compute preconditioners for symmetric indefinite systems. A limited memory approach is employed that incorporates a number of new ideas with the goal of improving the stability, robustness, and efficiency of the preconditioner. These include the monitoring of stability as the factorization proceeds and the incorporation of pivot modifications when potential instability is observed. Numerical experiments involving test problems arising from a range of real‐world applications demonstrate the effectiveness of our approach.     

Structure of Fukui matrices

Fukui matrices considered as the generalization of the concept of Fukui densities are decomposed into their pairing and unpairing contributions within the theory of the reduced density matrices. Their algebraic structure become clear from this decomposition providing their relationships with the spin density matrices and the irreducible part of the second‐order reduced density matrix cumulant, that is, the explicit contributions of the many‐body or correlation effects. The uncorrelated state function approximation is a simple way to emphasize the physical meaning of these matrices and represents the appropriate starting point for the treatment of a quasi‐analytical model to denote the occurrence of correlation effects.     

模块式护理计划对痛风患者疼痛及生活质量的影响分析

目的研究分析痛风患者采取模块式护理计划后对其生活质量、疼痛的影响。方法将佛山市第一人民医院风湿免疫内科2015年12月至2016年11月经临床诊断为痛风的住院患者156例随机分为常规组和实验组各78例,常规组采取常规护理措施,实验组采取模块式护理计划,比较两组患者的疼痛评分、疼痛缓解时间、生活质量评分。结果实验组、常规组疼痛缓解时间分别为(4.6±1.8)d、(7.3±2.7)d,(P0.05),护理2周后疼痛评分分别为(1.9±1.1)分、(4.1±2.5)分,(P0.05),生活质量评分分别为(8.2±2.6)分、(4.9±1.5)分,(P0.05)。结论模块式护理计划应用于痛风患者的护理中,能够显著降低患者的疼痛程度,提升其生活质量,缩短疼痛缓解时间,提高护理质量及效率,建议推广使用。     

Construction of nonnegative symmetric matrices with given spectrum

Let σ = (λ₁, … , λ_n) be the spectrum of a nonnegative symmetric matrix A with the Perron eigenvalue λ₁, a diagonal entry c and let τ = (μ₁, … , μ_m) be the spectrum of a nonnegative symmetric matrix B with the Perron eigenvalue μ₁. We show how to construct a nonnegative symmetric matrix C with the spectrum

(λ₁+max{0,μ₁-c},λ₂,…,λ_n,μ₂,…,μ_m).