Pseudo-Semi-Overlap Functions-Based Fuzzy Rough Sets Applied to Image Edge Extraction

As an extension of overlap functions, pseudo-semi-overlap functions are a crucial class of aggregation functions. Therefore, (I, PSO)-fuzzy rough sets are introduced, utilizing pseudo-semi-overlap functions, and further extended for applications in image edge extraction. Firstly, a new clustering function, the pseudo-semi-overlap function, is introduced by eliminating the symmetry and right continuity present in the overlap function. The relaxed nature of this function enhances its applicability in image edge extraction. Secondly, the definitions of (I, PSO)-fuzzy rough sets are provided, using (I, PSO)-fuzzy rough sets, a pair of new fuzzy mathematical morphological operators (IPSOFMM operators) is proposed. Finally, by combining the fuzzy C-means algorithm and IPSOFMM operators, a novel image edge extraction algorithm (FCM-IPSO algorithm) is proposed and implemented. Compared to existing algorithms, the FCM-IPSO algorithm exhibits more image edges and a 73.81% decrease in the noise introduction rate. The outstanding performance of (I, PSO)-fuzzy rough sets in image edge extraction demonstrates their practical application value.

Some properties of monotone set functions defined by Choquet integral

In this paper, some properties of the monotone set function defined by theChoquet integral are discussed. It is shown that several important structural characteristics of theoriginal set function, such as weak null-additivity, strong order continuity, property (s) andpseudomelric generating property, etc., are preserved by the new set function. It is also shown thatC-integrability assumption is inevitable for the preservations of strong order continuous andpseudometric generating property.

SET悬吊训练结合生物反馈技术在卒中患者下肢运动功能训练中的疗效观察

目的:探讨SET悬吊训练结合生物反馈技术在卒中患者下肢运动功能训练中的疗效。方法:选取佛山市南海区第五人民医院2022年1月—2023年12月期间收治的卒中下肢运动功能障碍患者72例为研究对象,分为2组,对照组(36例,实施SET悬吊训练)和实验组(36例,SET悬吊训练结合生物反馈技术),比较两组干预效果。结果:治疗后,实验组患者肌张力低于对照组;治疗后,实验组患者运动功能(FMA)评分、平衡功能(BBS)评分高于对照组(P<0.05)。结论:SET悬吊训练结合生物反馈技术对改善卒中患者下肢运动功能具有积极意义,有效改善患者的神经肌肉控制、降低肌张力、增强平衡和运动功能,进而提高其生活质量,值得推广和实施。

Function S-Rough sets and its applications

Based on S-rough sets（singular rough sets）, this paper presents function S-rough sets （function singular rough sets）and its mathematical structures and features. Function S-rough sets has two forms： function one direction S-rough sets （function one direction singular rough sets） and function two direction S-rough sets （function two direction singular rough sets）. This paper advances the relationship theorem of function S-rough sets and S-rough sets. Function S-rough sets is the general form of S-rough sets, and S-rough sets is the special ease of function S-rough sets. In this paper, applications of function S-rough sets in rough law mining-discovery of system are given. Function S-rough sets is a new research direction of rough sets and rough system.

Function S-rough sets and mining-discovery of rough law in systems

Function S-rough sets （function singular rough sets） is defined on a -function equivalence class [u]. Function S-rough sets is the extension form of S-rough sets. By using the function S-rough sets, this paper gives rough law generation model of a-function equivalence class, discussion on law mining and law discovery in systems, and application of law mining and law discovery in communication system. Function S-rough sets is a new theory and method in law mining research.

Rough Integrals Generated by Function Rough Sets

The concepts of the lower approximation integral,the upper approximation integral and rough integrals are given on the basis of function rough sets.Based on these concepts,the relation of the lower approximation integrals,the relation of the upper approximation integrals,the relation of rough integrals,and the double median theorem of rough integrals are discussed.Rough integrals have finite contraction characteristic and finite extension characteristic.

MEROMORPHIC FUNCTIONS SHARING TWO FINITE SETS

Let S1 = {∞} and S2 = {ω ： Ps（ω） = 0}, Ps（ω） being a uniqueness polynomial under some restricted conditions. Then, for any given nonconstant meromorphic function f, there exist at most finitely many nonconstant meromorphic functions g such that f^-1 （Si） = g^-1 （Si） （i = 1, 2）, where f^-1 （Si） and g^-1 （Si） denote the pull-backs of Si considered as a divisor, namely, the inverse images of Si counted with multiplicities, by f and g respectively.

Function S-rough Sets and Their Law Characteristic

Function S-rough sets(Function Singular rough sets) are defined by R-function equivalence class which has dynamic characteristic, and a function is s law, function S-rough sets have law characteristic. Function S-rough sets has these forms: function one direction S-rough sets, function two direction S-rough sets and dual of function one direction S-rough sets. This paper presents the law characteristic of function one direction S-rough sets and puts forward the theorems of law-chain-attribute and law-belt. Function S-rough sets is s new research direction of the rough sets theory.

A Method of Gene-Function Annotation Based on Variable Precision Rough Sets

It is very important in the field of bioinformatics to apply computer to perform the function annotation for new sequenced bio-sequences. Based on GO database and BLAST program, a novel method for the function annotation of new biological sequences is presented by using the variable-precision rough set theory. The proposed method is applied to the real data in GO database to examine its effectiveness. Numerical results show that the proposed method has better precision, recall-rate and harmonic mean value compared with existing methods.

A Note on Normal Families of Meromorphic Functions and Sharing Set

In the paper,we prove the main result:Let k(≥2)be an integer,and a,b and c be three distinct complex numbers.Let F be a family of functions holomorphic in a domain D in complex plane,all of whose zeros have multiplicity at least k.Suppose that for each f∈F,f(z)and f(k)(z)share the set{a,b,c}.Then F is a normal family in D.

ON UNIQUENESS OF MEROMORPHIC FUNCTIONS WITH SHARED-SETS IN AN ANGULAR DOMAIN

In this article, we deal with the uniqueness problems on meromorphic functions sharing two finite sets in an angular domain instead of the whole plane C. In particular, we investigate the uniqueness for meromorphic functions of infinite order in an angular domain and obtain some results. Moreover, examples show that the conditions in theorems are necessary.

Function S-rough sets and two law forecast

Function S-rough sets has the properties of dynamics, heredity, and memory. Function S-rough sets is penetrated and crossed with the issue of economic law forecast, then a new forecast model based on function S-rough sets namely the two law forecast model is proposed, which includes upper law forecast model and lower law forecast model; and its＇ implement algorithm is given. Finally, the validity of the model is demonstrated by the forecast for region economic development of Hainan Province.

On the Differentiability of Vector Valued Additive Set Functions

The Lebesgue-Nikodym Theorem states that for a Lebesgue measure an additive set function ?which is -absolutely continuous is the integral of a Lebegsue integrable a measurable function;that is, for all measurable sets.?Such a property is not shared by vector valued set functions. We introduce a suitable definition of the integral that will extend the above property to the vector valued case in its full generality. We also discuss a further extension of the Fundamental Theorem of Calculus for additive set functions with values in an infinite dimensional normed space.

Monotone Set-Valued Function Defined by Set-Valued Choquet Integral

Structural characteristics and absolute continuities of monotone set-valued function defined by set- valued Choquet integral are discussed. Similar to the single-valued monotone set function, several important structural characteristics of set-valued function are defined and have been proven the same as those in the original set functions, such as null-additivity, weakly null-additivity, order continuity, strong order continuity and property（S）. A counterexample shows that order continuity and strong order continuity of the original set functions are no longer kept in a monotone set-valued function when Choquet integrably bounded assumption is abandoned. Four kinds of absolute continuities are defined for set-valued function, and all been proven valid with respect to the original set functions.

The Normal Meromorphic Functions Family Concerning Higher Order Derivative and Shared Set by One-Way

Let F be a meromorphic functions family on the unit disc Δ, If for every (the zeros of f is a multiplicity of at least k) and if then and ( ), then F is normal on Δ.

SOME RESULTS ON TOPICAL FUNCTIONS AND UPWARD SETS

The purpose of this paper is to introduce and discuss the concept of topical functions on upward sets. We give characterizations of topical functions in terms of upward sets.

CONTINUITY OF JULIA SET FOR A FAMILY OF ENTIRE FUNCTIONS

Based on the work of McMullen about the continuity of Julia set for rational functions, in this paper, we discuss the continuity of Julia set and its Hausdorff dimension for a family of entire functions which satisfy some conditions.

Normal Families of Meromorphic Functions and Shared Set

In this paper,we study the normality criterion for families of meromorphic functions concerning shared set depending on f∈F.Let F be a family of meromorphic functions in the unit disc A.For each f∈F,all zeros of f have multiplicity at least 2 and there exist nonzero complex numbers b_f,c_f satisfying(i) b_f/c_f is a constant;(ii) min{σ(0,b_f),σ(0,c_f),σ(b_f,c_f)} ≥m for some m > 0;(iii) E_f'(S_f)■ E_f(S_f),where S_f = {b_f,c_f}.Then F is normal in A.At the same time,the corresponding results are also proved.The results in this paper improve and generalize the related results

COMPUTATIONAL COMPLEXITY IN WORST, STOCHASTIC AND AVERAGE CASE SETTING ON FUNCTIONAL APPROXIMATION PROBLEM OF MULTIVARIATE

The order of computational complexity of all bounded linear functional ap proximation problem is determined for the generalized Sobolev class Wp?（Id）, Nikolskii class H|∞k（Id） in the worst （deterministic）, stochastic and average case setting, from which it is concluded that the bounded linear functional approximation problem for the classes Wp?（Id） and H∞k（Id） is intractable in worst case setting, but is tractable with respect to stochastic and average case setting.

Function P-sets and Mining-discovery Theorems of Unknown Information Law

Function P-sets are a set pair which are composed of function internal packetsets S F and function outer packet sets S F or (S F , S F ) are function P-sets. Function P-sets have dynamic characteristics and law characteristics. Based on function P-sets, P-information law discovery theorems which has attribute supplementing-deleting are given. Mining-discovery and mining-discovery theorems of unknown information law are proposed. Function P-sets are a new theory and new method to research dynamic information laws for information systems.