scholarly journals On Uniform Convergence of Sequences and Series of Fuzzy-Valued Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Uğur Kadak ◽  
Hakan Efe
Keyword(s):  
Power Series ◽  
Uniform Convergence ◽  
Membership Function ◽  
Level Sets ◽  
Radius Of Convergence ◽  
Extension Principle ◽  
Membership Functions ◽  
Function Series ◽  
Convergence Of Sequences ◽  
The Relationship

The class of membership functions is restricted to trapezoidal one, as it is general enough and widely used. In the present paper since the utilization of Zadeh’s extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct for a fuzzy-valued function via related trapezoidal membership function. We derive uniform convergence of fuzzy-valued function sequences and series with some illustrated examples. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, we introduce the power series with fuzzy coefficients and define the radius of convergence of power series. Finally, by using the notions of H-differentiation and radius of convergence we examine the relationship between term by term H-differentiation and uniform convergence of fuzzy-valued function series.

scholarly journals On Fourier Series of Fuzzy-Valued Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Uğur Kadak ◽  
Feyzi Başar
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Uniform Convergence ◽  
Level Sets ◽  
Closed Interval ◽  
Complex Form ◽  
Dirichlet Kernel ◽  
Extension Principle ◽  
Science And Engineering ◽  
Interest In Science

Fourier analysis is a powerful tool for many problems, and especially for solving various differential equations of interest in science and engineering. In the present paper since the utilization of Zadeh’s Extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct a fuzzy-valued function on a closed interval via related membership function. We derive uniform convergence of a fuzzy-valued function sequences and series with level sets. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, Fourier series of periodic fuzzy-valued functions is defined and its complex form is given via sine and cosine fuzzy coefficients with an illustrative example. Finally, by using the Dirichlet kernel and its properties, we especially examine the convergence of Fourier series of fuzzy-valued functions at each point of discontinuity, where one-sided limits exist.

Meaning and Measurement

2012 ◽  
Author(s):  
Gary Goertz ◽  
James Mahoney
Keyword(s):  
Membership Function ◽  
Fuzzy Set ◽  
Quantitative Research ◽  
Fundamental Principle ◽  
The Other ◽  
Membership Functions ◽  
Research Paradigm ◽  
Fuzzy Set Analysis ◽  
Quantitative Indicators ◽  
The Relationship

This chapter examines how translation problems are manifested across the qualitative and quantitative cultures for issues related to concepts and measurement. In the quantitative research paradigm, one speaks of variables and indicators. X and Y are normally latent, unobserved variables for which one needs (quantitative) indicators. In practice, quantitative scholars might fuse the variable and the indicator into one entity. Qualitative researchers, on the other hand, tend to use the variable-indicator language which causes a translation problem and does not capture research practices in the qualitative culture. The chapter first considers the notion of “membership function,” which is important in the fuzzy-set approach to concepts, before discussing a fundamental principle of semantic transformations in the qualitative culture: the Principle of Unimportant Variation. It also explains the relationship between scale types and membership functions in fuzzy-set analysis.

The Arzelà–Ascoli theorem by means of ideal convergence

2018 ◽  
Vol 25 (3) ◽  
pp. 475-479
Author(s):  
Emre Taş ◽  
Tugba Yurdakadim
Keyword(s):  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Statistical Convergence ◽  
Real Numbers ◽  
Ideal Convergence ◽  
Convergence Of Sequences ◽  
The Relationship

AbstractIn this paper, using the concept of ideal convergence, which extends the idea of ordinary convergence and statistical convergence, we are concerned with the I-uniform convergence and the I-pointwise convergence of sequences of functions defined on a set of real numbers D. We present the Arzelà–Ascoli theorem by means of ideal convergence and also the relationship between I-equicontinuity and I-continuity for a family of functions.

Entire Functions with Some Derivatives Univalent

1974 ◽  
Vol 26 (1) ◽  
pp. 207-213 ◽  
Author(s):  
S. M. Shah ◽  
S. Y. Trimble
Keyword(s):  
Power Series ◽  
Entire Functions ◽  
Radius Of Convergence ◽  
Open Disc ◽  
Image Position ◽  
Positive Integers ◽  
The Relationship ◽  
Derivatives Of

This paper is a continuation of the author's previous work, [6; 7], on the relationship between the radius of convergence of a power series and the number of derivatives of the power series which are univalent in a given disc.In particular, let D be the open disc centered at 0, and let f be regular there. Suppose that is a strictly-increasing sequence of positive integers such that each f(np) is univalent in D. Let R be the radius of convergence of the power series, centered at 0, that represents f. In [7], we investigated the connection between R and . We showed that, in general

scholarly journals Inner Product of Fuzzy Vectors

2021 ◽  
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Potential Application ◽  
Level Sets ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Decomposition Theorem ◽  
Inner Product ◽  
Extension Principle ◽  
Membership Functions ◽  
Linear Optimization Problems ◽  
Fuzzy Interval

Abstract The inner product of vectors of non-normal fuzzy intervals will be studied in this paper by using the extension principle and the form of decomposition theorem. The membership functions of inner product will be different with respect to these two different methodologies. Since the non-normal fuzzy interval is more general than the normal fuzzy interval, the corresponding membership functions will become more complicated. Therefore, we shall establish their relationship including the equivalence and fuzziness based on the a-level sets. The potential application of inner product of fuzzy vectors is to study the fuzzy linear optimization problems.

scholarly journals Arithmetics of Vectors of Fuzzy Sets

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1614
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Fuzzy Sets ◽  
Level Sets ◽  
Decomposition Theorem ◽  
Extension Principle ◽  
Membership Functions ◽  
Arithmetic Operations ◽  
Mild Conditions ◽  
Different Types ◽  
Scalar Products ◽  
Α Level

The arithmetic operations of fuzzy sets are completely different from the arithmetic operations of vectors of fuzzy sets. In this paper, the arithmetic operations of vectors of fuzzy intervals are studied by using the extension principle and a form of decomposition theorem. These two different methodologies lead to the different types of membership functions. We establish their equivalences under some mild conditions. On the other hand, the α-level sets of addition, difference and scalar products of vectors of fuzzy intervals are also studied, which will be useful for the different usage in applications.

AUTOMATIC DEVELOPMENT OF FUZZY MEMBERSHIP FUNCTIONS ON HEPATITIS PATIENTS DATA USING PARTICLE SWARM OPTIMIZATION (PSO)

2015 ◽  
Vol 77 (22) ◽  
Author(s):  
Candra Dewi ◽  
Ratna Putri P.S ◽  
Indriati Indriati
Keyword(s):  
Particle Swarm Optimization ◽  
Membership Function ◽  
Fuzzy System ◽  
Particle Swarm ◽  
Pso Algorithm ◽  
Membership Functions ◽  
Swarm Optimization ◽  
Global Weight ◽  
The Status ◽  
And Function

Information about the status of disease (prognosis) for patients with hepatitis is important to determine the type of action to stabilize and cure this disease. Among some system, fuzzy system is one of the methods that can be used to obtain this prognosis. In the fuzzification process, the determination of the exact range of membership function will influence the calculation of membership degree and of course will affect the final value of fuzzy system. This range and function can usually be formed using intuition or by using an algorithm. In this paper, Particle Swarm Optimization (PSO) algorithm is implemented to form the triangular membership functions in the case of patients with hepatitis. For testing process, this paper conducts four scenarios to find the best combination of PSO parameter values . Based on the testing it was found that the best parameters to form a membership function range for the hepatitis data is about 0.9, 0.1, 2, 2, 100, 500 for inertia max, inertia min, local ballast constant, global weight constant, the number of particles, and maximum iterations respectively.  

scholarly journals On the size of linearization domains

2008 ◽  
Vol 145 (2) ◽  
pp. 443-456
Author(s):  
XAVIER BUFF ◽  
CARSTEN L. PETERSEN
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Radius Of Convergence ◽  
Holomorphic Map ◽  
Root Of Unity ◽  
Rotation Numbers ◽  
Formal Power

AbstractAssume $f{:}\,U\subset \C\to \C$ is a holomorphic map fixing 0 with derivative λ, where 0 < |λ| ≤ 1. If λ is not a root of unity, there is a formal power series φf(z) = z + ${\cal O}$(z2) such that φf(λ z) = f(φf(z)). This power series is unique and we denote by Rconv(f) ∈ [0,+∞] its radius of convergence. We denote by Rgeom(f) the largest radius r ∈ [0, Rconv(f)] such that φf(D(0,r)) ⊂ U. In this paper, we present new elementary techniques for studying the maps f ↦ Rconv(f) and f ↦ Rgeom(f). Contrary to previous approaches, our techniques do not involve studying the arithmetical properties of rotation numbers.

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