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Orthopairs (pairs of disjoint sets) have points in common with many approaches to managing vaguness/uncertainty such as fuzzy sets, rough sets, soft sets, etc. Indeed, they are successfully employed to address partial knowledge, consensus, and borderline cases. One of the generalized versions of orthopairs is intuitionistic fuzzy sets which is a well-known theory for researchers interested in fuzzy set theory. To extend the area of application of fuzzy set theory and address more empirical situations, the limitation that the grades of membership and non-membership must be calibrated with the same power should be canceled. To this end, we dedicate this manuscript to introducing a generalized frame for orthopair fuzzy sets called “(m,n)-Fuzzy sets”, which will be an efficient tool to deal with issues that require different importances for the degrees of membership and non-membership and cannot be addressed by the fuzzification tools existing in the published literature. We first establish its fundamental set of operations and investigate its abstract properties that can then be transmitted to the various models they are in connection with. Then, to rank (m,n)-Fuzzy sets, we define the functions of score and accuracy, and formulate aggregation operators to be used with (m,n)-Fuzzy sets. Ultimately, we develop the successful technique “aggregation operators” to handle multi-criteria decision-making problems in the environment of (m,n)-Fuzzy sets. The proposed technique has been illustrated and analyzed via a numerical example.

The purpose of this paper is to introduce and study the structure of p -tuple of ( n , m ) - D -normal operators. This is a generalization of the class of p -tuple of n -normal operators. We consider a generalization of these single variable n - D -normal and ( n , m ) - D -normal operators and explore some of their basic properties.

Abstract A useful tool for expressing fuzziness and uncertainty is the recently developed n,m-rung orthopair fuzzy set (n,m-ROFS). Due to their superior ability to manage uncertain situations compared to theories of q-rung orthopair fuzzy sets, the n,m-rung orthopair fuzzy sets have variety of applications in decision-making in daily life. To deal with ambiguity and unreliability in multi-attribute group decision-making, this study introduces a novel tool called $$k^{n}_{m}$$ k m n -rung picture fuzzy set ( $$k^{n}_{m}$$ k m n -RPFS). The suggested $$k^{n}_{m}$$ k m n -RPFS incorporates all of the benefits of n,m-ROFS and represents both the quantitative and qualitative analyses of the decision-makers. The presented model is a fruitful advancement of the q-rung picture fuzzy set (q-RPFS). Furthermore, numerous of its key notions, including as complement, intersection, and union are explained and illustrated with instances. In many decision-making situations, the main benefit of $$k^{n}_{m}$$ k m n -rung picture fuzzy sets is the ability to express more uncertainty than q-rung picture fuzzy sets. Then, along with their numerous features, we discover the basic set of operations for the $$k^{n}_{m}$$ k m n -rung picture fuzzy sets. Importantly, we present a novel operator, $$k^{n}_{m}$$ k m n -rung picture fuzzy weighted power average ( $$k^{n}_{m}$$ k m n -RPFWPA) over $$k^{n}_{m}$$ k m n -rung picture fuzzy sets, and use it to multi-attribute decision-making issues for evaluating alternatives with $$k^{n}_{m}$$ k m n -rung picture fuzzy information. Additionally, we use this operator to pinpoint the countries with the highest expat living standards and demonstrate how to choose the best option by comparing aggregate findings and applying score values. Finally, we compare the outcomes of the q-RPFEWA, SFWG, PFDWA, SFDWA, and SFWA operators to those of the $$k^{n}_{m}$$ k m n -RPFWPA operator.

In this study, the solvability of a general form of product type of n ‐classes of 2D‐Hadamard–Volterra integral equations in the Banach algebra C ([1, a ] × [1, b ]) is studied and investigated under more general and weaker assumptions. We use a general form of the Petryshyn’s fixed point theorem (F.P.T.) in combination with a suitable measure of noncompactness. This forms a generalization of the Schauder, Darbo, and classical Petryshyn’s F.P.T. The problem under study encompasses various integral equations, as different special cases have been previously investigated in the literature. Finally, we present several illustrative examples to validate our underlying assumptions.

In the present paper, we prove spectral mapping theorem for (𝑚,𝑛)-paranormal operator 𝛵 on a separable Hilbert space, that is, 𝑓(𝜎𝜔(𝛵)) = 𝜎𝜔(𝑓(𝛵)) when 𝑓 is an analytic function on some open neighborhood of 𝜎(𝛵). We also show that for (𝑚,𝑛)-paranormal operator 𝛵, Weyl's theorem holds, that is, 𝜎(𝛵)—𝜎𝜔(𝛵) = 𝜋00(𝛵). Moreover, if 𝛵 is algebraically (𝑚,𝑛)-paranormal, then spectral mapping theorem and Weyl's theorem hold.

A sophisticated inquiry into tourism's social and economic power across the South. In the early 19th century, planter families from South Carolina, Georgia, and eastern North Carolina left their low-country estates during the summer to relocate their households to vacation homes in the mountains of western North Carolina. Those unable to afford the expense of a second home relaxed at the hotels that emerged to meet their needs. This early tourist activity set the stage for tourism to become the region's New South industry. After 1865, the development of railroads and the bugeoning consumer culture led to the expansion of tourism across the whole region. Richard Starnes argues that western North Carolina benefited from the romanticized image of Appalachia in the post-Civil War American consciousness. This image transformed the southern highlands into an exotic travel destination, a place where both climate and culture offered visitors a myriad of diversions. This depiction was futher bolstered by partnerships between state and federal agencies, local boosters, and outside developers to create the atrtactions necessary to lure tourists to the region. As tourism grew, so did the tension between leaders in the industry and local residents. The commodification of regional culture, low-wage tourism jobs, inflated land prices, and negative personal experiences bred no small degree of animosity among mountain residents toward visitors. Starnes's study provides a better understanding of the significant role that tourism played in shaping communities across the South.