The produced results generalize earlier real ones into the fuzzy setting. Here the high order fuzzy pointwise convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity of the N th order N 1 H-fuzzy derivative of the engaged fuzzy number valued function. At the end we present a related Lp result for fuzzy neural network operators. In Chapter 17 we study the fuzzy random positive linear operators acting on fuzzy random continuous functions. We establish a series of fuzzy random Shisha—Mond type inequalities of Lq -type 1 q 14 1.

We also study the rates of convergence of fuzzy positive linear operators. The aim of Chapter 20 is to present a fuzzy trigonometric Korovkintype approximation theorem by using a matrix summability method. We also study the rates of convergence of fuzzy positive linear operators in trigonometric environment. Finally about Chapter The basic fuzzy wavelet type operators Ak ; Bk ; Ck ; Dk ; k 2 Z were studied in [3], [5], for their pointwise and uniform convergence with rates to the fuzzy unit operator. For prior related and similar study of convergence to the unit of real analogs of these wavelet type operators see [1], section II.

This is done via elegant tight Jackson type inequalities involving the modulus of continuity of the higher order derivative of the engaged real function. Based on these real analysis results in Section 2 we establish the corresponding fuzzy results regarding uniform estimates for the fuzzy differences between the fuzzy wavelet type operators. The last inequalities involve the fuzzy modulus of continuity of the higher order fuzzy derivative of the engaged fuzzy function. The de…ning all these operators real scaling function is not assumed to be orthogonal and is of compact support.

Finally is given a multivariate Hfuzzy Taylor formula. This treatment relies in [10]. For the notion of H-fuzzy derivative see [93] and [53]. First we give some background from Fuzziness, motivation and justi…cation, necessary for the results to follow. In Propositions 2. In Lemmas 1. Then come the main results. Theorem 2. We conclude with Theorem 2. We assume that f i : T! Also we assume that f n , is fuzzy continuous on T. This formula is invalid when s We mention Theorem 2.

C[0; 1] iso- 2. Lemma 2. Here h is a small positive quantity approaching zero. By Lemma 4. And we see that lim D t u h! It follows Proposition 2. Let u 2 RF be …xed. One can do other examples of calculation of H-derivatives of basic fuzzy functions, working as above with power series over appropriate intervals. We mention Lemma 2. RF be fuzzy continuous functions. Then f g is a fuzzy continuous function on a; b. Let xn ; x 2 a; b such that xn! Let U be an open subset of R2 and let f; g : U!

RF be fuzzy continuous jointly in x; y 2 U. Then D f x; y ; g x; y is continuous jointly in x; y. It is similar to [66], p. Let I be an open interval of R and let f; g : I! Let h! Hence 2. The counterpart of the above follows. Proposition 2. Let I be an open interval of R and let f :! Hence h ; i. We need Lemma 2. Let un! Then D un ; vn! In particular D un ; v! Then in D-metric un cn! We notice that D un cn ; u c D un by Lemma 2. Let I be a closed interval in R. Here g : I! Assume that g is strictly increasing. Next follows the multivariate fuzzy chain rule.

Consider f : U! RF a fuzzy continuous function. Assume that fxi : U! Let …rst t 2 a; b. So as t! The theorem basically is proved. A further explanation follows.

## Fuzzy Mathematics: Approximation Theory (Electronic book text)

Then f are continuous functions from U into R, for all r 2 [0; 1]. Then by continuity of f we get D f xm ; f x! Consequently f xm! RF be a fuzzy continuous function. Assume that all H-fuzzy partial derivatives of f up to order m 2 N exist and are fuzzy continuous. We make use of Theorem 5. By Lemma 2. But by basic real analysis, Theorem , p. Let U be an open convex subset of Rn , n 2 N and f : U! Proof of Theorem 2.

RF is fuzzy continuous, also fxi exist and are fuzzy continuous, by Theorem 2. The last is true by Theorem 2. That is both Taylor formulae in that case coincide. And, of course, the fuzzy Taylor formula now can be applied trivially for gz. Furthermore in that case it coincides with the Taylor formula proved earlier for gz. That is 2. At last we give the useful Corollary 2. Let U be an open convex subset of Rn , n 2 N, and f : U! Assume that all the …rst Hfuzzy partial derivatives fxi of f exist and are fuzzy continuous. By Theorem 2. The second part of 2. RF and it has all the properties of f as in Theorem 2.

Clearly here x0 ; z 2 U. This chapter is based on [19]. Theorem 3. Then n 1 1! Let r 2 [0; 1].

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Furthermore by 3. The theorem has been proved. Next we present another multivariate fuzzy Taylor theorem. We need the following multivariate fuzzy chain rule. Here the H-fuzzy partial derivatives are de…ned according to the De…nition 3. We present 54 3. On Fuzzy Taylor Formulae Theorem 3. Explaining formula 3. Proof of Theorem 3. The last is true by Theorem 3 10 of [10] under the assumptions that all H-partial derivatives of f up to order m exist and they are all fuzzy continuous including f itself.

Rm 0; 1 ; where Rm 0; 1 comes from 3. By Theorem 3. That is 3. RF and it has all the properties of f as in Theorem 3. Clearly here we take x0 ; z 2 U. Clearly G t is fuzzy continuous in t; see Lemma 1. That is, there are non-trivial higher order H fuzzy derivatives. We continue with Theorem 3. We …nish with Theorem 3. There are higher order nontrivial H fuzzy partial derivatives. This chapter relies on [13]. The last inequality is sharp, see [5]. Since when A. Fink, see [64]. The author in [5] continued that tradition. Theorem 4. We use Proposition 4. We mention Proposition 4.

Let I be an open interval of R and let f : I! Let f 2 C [a; b]; RF , the space of fuzzy continuous functions, x 2 [a; b] be …xed. Fuzzy Ostrowski Inequalities Optimality of 4. Proposition 4. Inequality 4. Finally we have! Then for x 2 [a; b],! By Propositions 4.

We have that! We conclude with the following Ostrowski type inequality fuzzy generalization in 70 4. Fuzzy Ostrowski Inequalities Theorem 4. Let x 2 [a; b], then by Theorem 4. We prove that g is fuzzy continuous. Let xn 2 [a; b] such that xn! This chapter is based on [31]. It is obvious that pn;k x 0, 8x 2 [0; 1] and pn;0 x , pn;1 x ; : : : ; pn;n x are linearly independent algebraic polynomials of degree n.

Concerning these fuzzy polynomials recently was proved the following G. A fuzzy trigonometric approximation theorem of Weierstrass-type Theorem 5. A function f : R! A generalized fuzzy trigonometric n, will be P polynomial of degree de…ned as a …nite sum of the form Tk x ck , where ck 2 RF and Tk x are trigonometric polynomials of degree n. We need a particular case of the Henstock integral introduced in [52].

De…nition 5. If a; b are not both 0 then Lemma 5. RF ; f is fuzzy continuous and 2 periodic on Rg. Theorem 5. Let us de…ne the fuzzy Jackson-type operators see [66, p. A fuzzy trigonometric approximation theorem of Weierstrass-type F where! By [52, Theorem 2. They can be summarized by the following: Lemma 5. Because u 5. For the last property, by e. Remark 5. The properties ii and iii in Lemma 5. But the properties of D and Lemma 5. The fact that the above Bernstein-type and Jackson-type see the proof of Theorem 5. In case Pk x or Tk x are not positive, by Lemma 5.

The concept of fuzzy polynomials can also be de…ned as in e. Obviously the generalized fuzzy polynomials of Bernstein-type and of Jackson-type mentioned earlier are not of the latter kind. The property iv of Lemma 5. This chapter relies on [41]. RF ; f is 2 periodic and continuous on Rg: In [31], the following Weierstrass-type result is proved. Theorem 6. Jackson-Type Estimates by Generalized Fuzzy Polynomials Other results concerning approximation and interpolation of fuzzy-numbervalued functions can be found in: [11], [59], [57], [81], [84], [72], [80]. But the problems of existence of best approximation by fuzzy polynomials and of convergence of fuzzy Lagrange polynomials, were not discussed much in the fuzzy mathematical literature.

In Section 6. Section 6. As an immediate consequence of Theorem 6. T is a generalized fuzzy trigonometric polynomial of degree n of best approximation for f: Proof. Since by Theorem 6. T is of best approximation, which proves the corollary. If f 2 C2F R ; then by f [ in Corollary 6. F 1 EnK;[A;B] f f; 1 n ; 8n n0 where! In [66, p. But reasoning exactly as in the usual case see [30, p.

F 1 f; [ 1 f; 1 f; [ ; ] ; ]: As a consequence, we obtain the following Jackson-type estimate D Tn x ; f x C! Note that above! F 1 f; n can be replaced by! K;[ 1;1]. Remark 6. From the proof it is easily seen that an interval [A; B] independent of f and n can be constructively determined such that the Jackson kind estimate in Theorem 6. RF ; f continuous on [a; b]g where [a; b] is a compact subinterval of R. If we de…ne the concept of generalized fuzzy algebraic n as in [49], i.

Consequently, we obtain the following Corollary 6. T is a generalized fuzzy algebraic polynomial of degree n , of best approximation for f: Remark 6. By [66, p. For this aim we need the following lemmas. Lemma 6. The result in the above Lemma 6. For the next results we need an embedding theorem. The following lemmas give some approximation properties in Banach spaces. B n continuous. Then 1 ; kRn0 g; x k n! The proof is the same as that of [, Lemma 2], written in the case of functions with values in a Banach space. Thus by 6. The proof is the same as the proof of [56, p.

B; f continuous on I 0 g: Then for …xed x 2 I 0 ; as in [56, p. B1 f; n1 I 0 : The lemma is proved. Then C! B1 Rn j f ; n1 [ 1 ; 1 ] : 4 4 4 4 By Lagrange theorem for functions with values in Banach spaces we obtain It is easy to see that! B1 j f; I0 It is easy to check that! F 1 f; I0 1 n I0 which completes the proof. As an immediate consequence we obtain the following Jackson-type estimate for the error in approximation by generalized fuzzy algebraic polynomials.

Corollary 6. For the best approximation by algebraic polynomials we have EnK;1 C! We can obtain the above results in any interval [a; b] in1 1 stead of 4 ; 4 by mapping this interval in [a; b] through a linear function which maps 14 to a and 14 to b: 6. By Lemma 8. F 1 f; n D Ln ; f which completes the proof. This chapter relies on [18].

Of course pre-existed some Korovkin type set valued literature not related at all to this chapter. The results of Theorems 7. The same real assumptions are kept here in the fuzzy setting, and they are the only assumptions we make along with the very natural and general G. Basic Fuzzy Korovkin Theory realization condition 7. Condition 7. At each step of the chapter we provide an example to justify our method.

We use the following De…nition 7. RF be a fuzzy real number valued function. We de…ne the …rst fuzzy modulus of continuity of f by F! De…nition 7. We mention the very interesting with rates approximation motivating this chapter. Theorem 7. F The last fact comes by the property that! We need to use! Let [a; b] R. Let L be a sequence of positive linear operators from C [a; b] into itself.

We …nally need Lemma 7. Then it holds yj ,0 sup max j f x b a. Then we have r f y r2[0;1] sup max! F Note. For f 2 CF [a; b] we get that f is fuzzy bounded and! We assume that fL bounded. The fuzzy Bernstein operators Bn and the real corresponding ones Bn acting on CF [0; 1] and C [0; 1] , respectively, ful…ll assumption 7.

We present now the Fuzzy Korovkin Theorem. Then D Ln f; f! I unit operator in the fuzzy sense, as n! Use of 7. F Example for Theorem 7. Proof of Theorem 7. Basic Fuzzy Korovkin Theory proving 7. Application 7. Let f 2 CF [0; 1] then by applying 7. This chapter is based on [32]. References [], [18], [24] are the …rst articles dealing with the fuzzy Korovkin matter and inequalities, however [] is very specialized and restrictive though very interesting dealing with fuzzy random variables and positive linearity. The main results here are Theorems 8.

They are simple, basic and very general directly transferring the real trigonometric case, of the convergence with rates of positive linear operators to the unit under trigonometric assumptions, to the fuzzy one. The same real trigonometric G. Fuzzy Trigonometric Korovkin Theory assumptions are kept here in the fuzzy setting and they are the only convergence assumptions we make, along with the general realization condition 8. Condition 8.

At each step of the development of our method we present an example that satis…es our theory. RF be fuzzy real number valued functions. We use the following De…nition 8.

We have a similar obvious de…nition for subsets of R. De…nition 8. RF ; f is fuzzy continuous and 2 -periodic on Rg. We need Lemma 8. Then Qn x is a fuzzy 2 periodic continuous function in x 2 R. We present Lemma 8. RF be a 2 -periodic and fuzzy continuous func F tion, i. Then for all 2 [0; ] we have F! The left hand side inequality is obvious. Case 1. Let f 2 C2 R , then f is fuzzy bounded and fuzzy uni F U formly continuous. F By Proposition 2 of [24] we have that lim! Thus by Lemma 8.

RF be a fuzzy real number valued F r r function. Assume that! Then it holds F! By Corollary Also, 8. Theorem 8. RF be fuzzy continuous function. We are motivated by F De…nition 8. By [66], p. We mention Theorem 8. Based on Theorem 3. By Theorem 8. In fact almost all fuzzy operators de…ned via fuzzy summation or fuzzy integration are fuzzy positive linear operators. Let L ear positive operators, whose common domain K consists of real functions with domain 1; 1. In particular, if L as is often the case, the last inequality 8.

Hence we get with rates in an inequality form, quantitatively, the famous trigonometric Korovkin theorem, see [78]. We assume that fL bounded in n over [a; b] R. Fuzzy Trigonometric Korovkin Theory Proof. We present now the …rst Fuzzy Trigonometric Korovkin theorem. Furthermore assume that L 1, L sin x, L cos x, as n! I fuzzy unit operator, over [a; b], as n!

From 8. Also L are bounded in n over [a; b]. Hence by 8. As an example for Theorem 8. C cos x over R, as n! Then by Theorem 8. That is recon…rming Theorem 8. This chapter relies on [21]. In general and in applications, we prefer to have nice and …t approximations.

## Higher Order Multivariate Fuzzy Approximation by basic Neural Network Operators

That is the approximants, in this case operators, e. This feature is expressed with the property of global smoothness preservation by the approximating operators. In general let L be a fuzzy linear operator acting on a space of fuzzy continuous functions T de…ned on a metric space X; d and taking values in RF , the set of fuzzy real numbers. Fuzzy Global Smoothness Preservation F where!

In the real setting the analogous to the above inequality is true under certain conditions, see [26], [25], [30], [33]. So here we establish related inequalities in the fuzzy setting for various spaces T , by transferring from the real case under a minimal and natural realization assumption on L, see Theorems 9. This assumption is ful…lled by almost all fuzzy operators de…ned via fuzzy summation or fuzzy integration.

We provide also some interesting examples that motivate and ful…ll our general theory. On the way to prove the main results we prove other important side results. We need De…nition 9. Assume F! Then F! Similarly 9. We need B X , or Proposition 9. Clearly then kLk is …nite and kLk kLk. Let [a; b] R and t0 2 [a; b] …xed. Then L Proof. That is an important property motivating our theory next. Also from [26] and [30], p. We also present F Example 9.

Let f 2 C2 R space of 2 -periodic fuzzy continuous functions on R. We de…ne see [66], p. It is known that Kn t 0 being an even trigonometric polynomial of order n. The fuzzy operator Jn is linear, by the linearity of Fuzzy—Riemann integral, see [67]. By Lemma By Theorem Let f 2 C X , we de…ne the …rst modulus of continuity of it by! We would like to mention the following related fundamental result.

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Theorem 9. Let X be a compact metric space, and L : C X! It is known that over [a; b] R we have! Let X; d be a compact metric space. Let L be a fuzzy linear operator acting on CF X. Notice here that all f 2 C X. Then by Proposition 9. We give Example 9. Using Theorem 9. Let open convex subset of a real normed vector space B. Assume for L V; k k. These operators are multivariate fuzzy analogs of earlier studied multivariate real ones.

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The produced results generalize earlier real ones into the fuzzy setting. Here the high order multi-variate fuzzy pointwise convergence with rates to the multivariate fuzzy unit operator is established through multivariate fuzzy inequalities involving the multivariate fuzzy moduli of continuity of the Nth order N 1 H-fuzzy partial derivatives, of the engaged multivariate fuzzy number valued function. Keywords and Phrases: multivariate fuzzy real analysis, multivariate fuzzy neural network operators, high order multivariate fuzzy approximation, multivariate fuzzy modulus of continuity and multivariate Jackson type inequalities.

Estos operadores son analogos difusos multivariados de los reales multi-variados estudiados anteriormente. Los resultados obtenidos generalizan los resultados reales anteriores en el marco difuso. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case, Journal of Mathematical Analysis and Application, Vol. After all, we have a percentage of approvals, a percentage of denials, and a percentage of abstentions. For this situation, special 'intuitive fuzzy' negators, t- and s-norms can be provided.

The concept of IFS has been extended into two major models. The two extensions of IFS are neutrosophic fuzzy sets and Pythagorean fuzzy sets. Neutrosophic fuzzy sets were introduced by Smarandache in This value indicates that the degree of undecidedness that the entity x belongs to the set. The other extension of IFS is what is known as Pythagorean fuzzy sets. Pythagorean fuzzy sets are more flexible than IFS. This is why Yager proposed the concept of Pythagorean fuzzy sets. With these valuations, many-valued logic can be extended to allow for fuzzy premises from which graded conclusions may be drawn.

This extension is sometimes called "fuzzy logic in the narrow sense" as opposed to "fuzzy logic in the wider sense," which originated in the engineering fields of automated control and knowledge engineering , and which encompasses many topics involving fuzzy sets and "approximated reasoning. Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at fuzzy logic. Fuzzy numbers can be likened to the funfair game "guess your weight," where someone guesses the contestant's weight, with closer guesses being more correct, and where the guesser "wins" if he or she guesses near enough to the contestant's weight, with the actual weight being completely correct mapping to 1 by the membership function.

The latter means that fuzzy intervals are normalized fuzzy sets. As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous. However, there are other concepts of fuzzy numbers and intervals as some authors do not insist on convexity. The use of set membership as a key components of category theory can be generalized to fuzzy sets. This approach which initiated in shortly after the introduction of fuzzy set theory [21] led to the development of "Goguen categories" in the 21st century.

Physical interpretation of k is the Boltzmann constant k B. There are many mathematical constructions similar to or more general than fuzzy sets. Since fuzzy sets were introduced in , a lot of new mathematical constructions and theories treating imprecision, inexactness, ambiguity, and uncertainty have been developed. While most of the above can be generally categorized as truth-based extensions to fuzzy sets, bipolar fuzzy set theory presents a philosophically and logically different, equilibrium-based generalization of fuzzy sets. Accordingly it can be seen that A two probability measures are sufficient to describe fuzziness measure theoretically, and B fuzzy sets do conform to field theoretic norms.

It may be noted that in the theory of fuzzy sets it is nowhere mentioned how exactly one should proceed to construct the membership function of a fuzzy number. In the approach mentioned above, how to construct a fuzzy number can be explained. Further, for a fuzzy vector, one has to define a membership surface. In the existing literature, with reference to fuzzy vectors there is no mention of membership surface. In the approach mentioned above, how to construct membership surfaces of fuzzy vectors can be explained. Finally, fuzzy logic would have to be revisited if this approach to define fuzziness is found acceptable.

Bezdek, J. Fuzzy Sets and Systems. From Wikipedia, the free encyclopedia. Main article: Fuzzy set operations. Main article: Fuzzy logic. Main article: Fuzzy number. This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Zadeh "Fuzzy sets". Information and Control 8 3 — Berlin 7, — A recent in-depth analysis of this paper has been provided by Gottwald, S.

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Dubois and H. Prade Fuzzy Sets and Systems. Academic Press, New York. BMC Bioinformatics. Archived from the original on August 5, Vemuri, A.