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Recently, D. Zhang et al. introduced the generalized Choquet integral, extending pseudo-integrals and Choquetlike integrals while exploring their foundational properties. Building on this framework, we introduce the concept of generalized Choquet integrals for triangular fuzzy number (TFN)-valued functions, referred to as TGC-integrals. This work investigates the key properties of TGC-integrals, including monotone non-decreasing convergence theorems and inequalities such as the Fatou type, Jensen type, Minkowski type, and H¨older type inequalities, specifically tailored for TFN-valued functions. Furthermore, we provide illustrative examples that demonstrate practical applications of TGCintegrals, such as TFN-valued Choquet expected utility and pseudo-functional analysis. These results establish a robust theoretical foundation for analyzing TFN-valued functions and highlight their potential for addressing uncertainty and ambiguity in real-world problems. [ABSTRACT FROM AUTHOR]

This study introduces the Choquet integral of fuzzifying functions with respect to a fuzzy measure. To express various phenomena or ambiguous values in many applications may not be enough to show them as a function, in which case a fuzzifying function can be applied to achieve better expressions or flexibility for given function values. To apply Choquet integrals of fuzzifying functions, we consider Choquet integrals of interval-valued functions as an operator which are α-level functions of fuzzifying functions. In this study, we investigate some properties of Choquet integral of fuzzifying functions and present their applications. As part of this, a series of relevant examples and their subsequent applications are provided, along with the fuzzification of two integrands: A probability density function (PDF) and a utility function. [ABSTRACT FROM AUTHOR]

The Bernoulli numbers are among the most interesting and important number sequences in mathematics. They first appeared in the posthumous work "Ars Conjectandi" (1713) by Jacob Bernoulli (1654-1705) in connection with sums of powers of consecutive integers (Bernoulli, 1713; or Smith, 1959). Bernoulli numbers are particularly important in number theory, especially in connection with Fermat's last theorem (Ribenboin, 1979). They also appear in the calculus of finite differences (Norlund, 1924), in combinatorics (Comtet, 1970), and in other fields. Bernoulli polynomials and Bernoulli numbers possess many interesting properties and arise in many areas of mathematics and physics. This paper shows how to derive the sum of the nth powers of positive q-integers up to k-1, and displays the shapes of q-Bernoulli polynomials. (Contains 8 figures and 3 tables.)

We define the twisted -Bernoulli polynomials and the twisted generalized -Bernoulli polynomials attached to of higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain some relationships between twisted -Bernoulli numbers and polynomials and between twisted generalized -Bernoulli numbers and polynomials.

Recently, Kim-Kim-Dolgy-Seo [10] studied the Bernoulli polynomials of the second kind and obtained some interesting identities involving those polynomials by using umbral calculus. Also, Kim-Kim-Kwon-Seo [11] introduced nonlinear differential equations associated with Bernoulli numbers of the second kind. In this paper, we revisit differential equations arising from generating functions of the higher-order Bernoulli numbers of the second kind and give some explicit identities for those numbers. [ABSTRACT FROM AUTHOR]

In this paper, we consider twisted ζ-Euler numbers and polynomials, and discuss p-adic ζ-integral representation of twisted Bernstein polynomials. By using these properties, we derive some identities on the twisted Bernstein polynomials. [ABSTRACT FROM AUTHOR]

Recently the symmetry identities for the generalized twisted Euler polynomials attached to X in complex number field are studied in [7]. These symmetry identities are interesting and valuable. In this paper, we investigate the symmetric properties for the p-adic invariant integral on ℤp. From these symmetric properties, we derive the different proof for the symmetry identities of the generalized twisted Euler polynomials which was treated in [7]. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a systemic study of some families of the Kim's q-Bernstein polynomials. By using double fermionic p-adic integral representation on ᄤp, we give some interesting and new formulae related to q-Euler numbers. [ABSTRACT FROM AUTHOR]

We consider the weighted q-Genocchi numbers and polynomials. From the construction of the weighted q-Genocchi numbers and polynomials, we investigate many interesting identities and relations satisfied by these new numbers and polynomials. [ABSTRACT FROM AUTHOR]

Recently, Kim's work (in press) introduced q-Bernstein polynomials which are different Phillips' q-Bernstein polynomials introduced in the work by Phillips, 1996; 1997. The purpose of this paper is to study some properties of several type Kim's q-Bernstein polynomials to express the p-adic q-integral of these polynomials on ℤp associated with Carlitz's q-Bernoulli numbers and polynomials. Finally, we also derive some relations on the p-adic q-integral of the products of several type Kim's q-Bernstein polynomials and the powers of them on ℤp. [ABSTRACT FROM AUTHOR]

The fermionic p-adic integral (=I-1-integral) is defined by T.Kim in the previous paper, see [2]. In this paper we consider I-1-Fourier transform and investigate some properties which are related to this transform. [ABSTRACT FROM AUTHOR]

In the recent, several mathematician have studied the second kind ordinary Euler numbers which were defined by [this character cannot be represented into ASCII code]. The purpose of this paper is to construct q-expansion of the second kind Euler numbers. From these numbers, we will derive some interesting properties and identities. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new type of q-Bernstein polynomials with fixed weight (α/β). We call them q-Bernstein polynomials with weight (α/β). We establish many interesting identities and relations satisfied by these new polynomials. These formulae are the arithmetical nature. [ABSTRACT FROM AUTHOR]

To determine management guidelines for symptomatic duodenal diverticulum, we reviewed medical records of 26 patients. Complicated duodenal diverticulum was the only possible cause of symptoms-abdominal pain, fever and chills, melena, vomiting-in 18 patients. Ten patients improved with conservative management, and eight patients underwent diverticulectomy with or without various other procedures. Among the eight patients, one patient who had duodenal fistula died of respiratory complications on the second postoperative day. Symptoms recurred in two patients: One had a distal common bile duct (CBD) stricture and underwent choledochojejunostomy. In the other patient a CBD stone developed 3 years later, and choledocholithotomy and choledochojejunostomy were performed. Eight patients had associated gallstone disease as well as the diverticulum. Five of the eight had a history of operation for gallstone disease; four improved with conservative treatment, and one underwent choledochojejunostomy. Two patients were thought to have an innocent diverticulum and underwent cholecystectomy and choledocholithotomy only. One patient underwent diverticulectomy and sphincteroplasty for a CBD stone and pervaterian diverticulum. In conclusion, operations for duodenal diverticulum should be reserved for seriously complicated diverticula, and the surgeon should be aware that pervaterian diverticulum can be a cause of choledocholithiasis.