Fuzzy Spaces Research Articles

Fuzzy Topological Spaces Generated by Fuzzy Digraphs

In this paper, we defined various fuzzy topological spaces on fuzzy digraph and study interrelation between them. Using adjacency relation on fuzzy vertices and fuzzy edges of fuzzy digraph we defined, namely two types of fuzzy topologies, left(right) fuzzy vertex topology and left(right) fuzzy edge topology on fuzzy digraph, respectively. We have obtained results related to fuzzy topology on fuzzy subdigraph [Formula: see text] of fuzzy digraph [Formula: see text]. Separation axioms [Formula: see text], [Formula: see text] and [Formula: see text] for these fuzzy topological spaces are studied. Some characterization results for fuzzy topological spaces related to isomorphic fuzzy digraphs are given. Further, we have shown that two fuzzy digraphs are isomorphic if and only if their corresponding fuzzy topologies are homeomorphic, this relation on set of fuzzy digraphs forms an equivalence relation. We have also defined and studied fuzzy bitopological spaces on fuzzy digraphs.

Retraction Note: Fuzzy hyperconnected proximity spaces and fuzzy summability over CW complexes. Application of Smirnov fuzzy similarity in video analysis

Retraction Note: Fuzzy hyperconnected proximity spaces and fuzzy summability over CW complexes. Application of Smirnov fuzzy similarity in video analysis

Research on the discrimination and classification of multi-modal fault bearing based on multi-attribute decision-making mechanism

In order to solve the problem of low fault diagnosis rate of friction torque in upright placement, this paper proposes a fault diagnosis method based on multi-attribute decision-making mechanism to accurately evaluate the fault damage degree of bearing friction torque test bench. We detect defects by utilizing non-destructive, non-contact infrared thermal imaging technology to perform data acquisition and testing in six situations, including non-destructive, damaged inner ring, damaged outer ring, marble-defect, lack of lubrication, and damaged inner and outer ring bearings. Then, the infrared data of different bearings are transformed in a standardized form, and the decision data is mapped to the Pythagorean fuzzy information space. The correlation between the decision data are considering, the PA operator and the BM operator are used for data processing to achieve the accurate distinction of different faulty bearings. Finally, qualitative and quantitative methods are used to analyze the decision results, and the effectiveness of the method is explained. The variation of the overall singular value of each object in the decision result of the proposed method is less than 0.002, while the data fluctuation of WA, WG, HM and DHM related decision methods is 0.005, and there is a singular value greater than 0.02. In addition, sensitivity analysis provides a feasible scheme for parameter optimization and optimal parameter determination.

SOME FEATURES OF PAIRWISE α-R0 SPACES IN SUPRA FUZZY BITOPOLOGY

This paper introduces and studies four concepts of supra fuzzy bitopological spaces. We have exhibited that all these four concepts are ‘good extensions’ of the corresponding concepts bitopological spaces and building relationships among them. It has been justified that all the definitions are hereditary, productive, and projective. Furthermore, additional properties of these concepts are studied.

General Trapezoidal-Type Inequalities in Fuzzy Settings

In this study, trapezoidal-type inequalities in fuzzy settings have been investigated. The theory of fuzzy analysis has been discussed in detail. The integration by parts formula of analysis of fuzzy mathematics has been employed to establish an equality. Trapezoidal-type inequality for functions with values in the fuzzy number-valued space is proven by applying the proven equality together with the properties of a metric defined on the set of fuzzy number-valued space and Höler’s inequality. The results proved in this research provide generalizations of the results from earlier existing results in the field of mathematical inequalities. An example is designed by defining a function that has values in fuzzy number-valued space and validated the results numerically using the software Mathematica (latest v. 14.1). The p-levels of the defined fuzzy number-valued mapping have been shown graphically for different values of p∈0,1.

On fuzzy inner products constructed by fuzzy numbers in linear spaces

In this paper a new definition for fuzzy inner product spaces is presented. Just as the set of real numbers can be embedded in a set of fuzzy numbers, a crisp inner product space can be considered as a special case of fuzzy inner product spaces. An example is given to demonstrate that, the new definition is a nontrivial generalization for the crisp inner product spaces. Under certain restrictions, a fuzzy inner product space can become a fuzzy normed space. Based on some elementary properties for the families of semi-inner products of endpoints, the linearly topological structure of new spaces is discussed. Moreover, the orthogonality between two vectors is considered and a fuzzy version of Pythagorean theorem is given.

Quantum scalar field on fuzzy de Sitter space. Part I. Field modes and vacua

We study a scalar field on a noncommutative model of spacetime, the fuzzy de Sitter space, which is based on the algebra of the de Sitter group SO(1, d) and its unitary irreducible representations. We solve the Klein-Gordon equation in d = 2, 4 and show, using a specific choice of coordinates and operator ordering, that all commutative field modes can be promoted to solutions of the fuzzy Klein-Gordon equation. To explore completeness of this set of modes, we specify a Hilbert space representation and study the matrix elements (integral kernels) of a scalar field: in this way the complete set of solutions of the fuzzy Klein-Gordon equation is found. The space of noncommutative solutions has more degrees of freedom than the commutative one, whenever spacetime dimension is d > 2. In four dimensions, the new non-geometric, internal modes are parametrised by S2 × W, where W is a discrete matrix space. Our results pave the way to analysis of quantum field theory on the fuzzy de Sitter space.

Intuitionistic fuzzy normal bi-topological space-approach of intuitionistic fuzzy open sets

This paper provides nine newly proposed notions of intuitionistic fuzzy normal bi-topological spaces (IFNBTS) based on the concept of most explored field fuzzy bi-topological spaces using intuitionistic fuzzy open sets (IFOS). Further, the authors establish implications among the prescribed notions and show that these notions are good extensions of normal and fuzzy normal bi-topological spaces. Finally, the authors study the image and pre-image of IFNBTS, demonstrating that they are also IFNBTS in the sense of IFOS.

A hybrid framework for Detection of Multivariate porphyry Cu Signatures and Anomaly Enhancement: Incorporation of SFA, GMPI, and Grey Wolf Optimization

Geochemical data from stream sediment are commonly employed for regional scale mineral exploration, as they can be utilized to detect geochemical anomalies associated with mineralization processes. However, recognizing efficient multielement geochemical signatures related to target mineralization using stream sediment geochemical data can be challenging due to the intricate nature of mineralization events. To achieve this goal, this article proposes a combination of statistics, logistic functions and machine learning techniques. Initially, the staged factor analysis (SFA) approach was applied to the data to determine the geochemical footprints associated with porphyry copper mineralization in the Varzaghan area. Subsequently, the geochemical mineralization probability index (GMPI) was utilized to facilitate appropriate separation, more precise identification, and the transfer of data into a fuzzy space. By using a combination of SFA and GMPI, it was feasible to enhance the detection of geochemical halos and concealed anomalies. Ultimately, the output of these two approaches was subjected to an improved clustering technique known as GWO-K-means to detect geochemical anomalies. The findings indicated that there is a strong relationship between known mineral occurrences (KMOs) and the identified anomalous areas. The success rate curve demonstrated that the optimized model outperformed the K-means model, proposing that optimization can boost the accuracy in identifying geochemical targets. Overall, the outcomes of this study can be beneficial for proper planning of future exploration activities.

On ((Τ ) ̃^*-N)^η-Fuzzy Soft Quasi Normal Operators

The concept of fuzzy soft sets is considered more versatile and widely applicable than traditional soft set theory, as the latter struggles to handle the fuzziness of problem parameters effectively. In this study, we introduce and describe the fuzzy soft quasinormal operator in a more concise manner. This operator is defined on a fuzzy soft Hilbert space, based on the notion of a fuzzy soft vector space, which has been adapted to an FS-inner product space in this work. We present various characterizations and results related to the fuzzy soft quasinormal operator, demonstrating that it serves as an appropriate example of FS-Hilbert spaces, alongside other related examples. Furthermore, we explore the relationships between these categories. Finally, we provide some fundamental guidelines on this topic.

A novel directional simulation method for estimating failure possibility

Failure possibility plays a crucial role in assessing the safety level of structures under fuzzy uncertainty. However, the traditional fuzzy simulation method suffers from computational inefficiency as it requires a large number of samples for accurate estimation. To address this issue, a directional simulation method is proposed to improve the efficiency of estimating failure possibility. The directional simulation method reformulates the failure possibility estimation into two key steps: the generation of direction samples and the estimation of conditional failure possibility under each direction sample in the polar coordinate system of the standard fuzzy space. To ensure direction uniformity, these direction samples are generated by adopting a good lattice point set based on stratified sampling on the unit hypercube. The conditional failure possibility under each direction sample is estimated by solving the minimum root of a nonlinear equation. The proposed method effectively reduces the dimensionality of the fuzzy input and greatly improves the computational efficiency. To further enhance efficiency, an adaptive Kriging model is embedded into the directional simulation method to reduce the number of performance function evaluations. Four examples are performed to illustrate the accuracy and efficiency of the proposed method. The results highlight the superiority of the directional simulation method over the traditional fuzzy simulation method, offering substantial improvements in computational efficiency while maintaining high estimation accuracy.

(α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces

Abstract In this study, we generalize fuzzy metric-like, non-Archimedean fuzzy metric-like, and all the variants of fuzzy metric spaces. We propose the idea of fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike, respectively. We also propose the idea of ( α , F ) \left(\alpha ,F) -Geraghty-type generalized F F -contraction mappings utilizing fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces. We investigate the presence of unique fixed points using the recently introduced contraction mappings. In order to complement our study, we consider an application to dynamic market equilibrium.

Metrics and geodesics on fuzzy spaces

Abstract We study the fuzzy spaces (as special examples of noncommutative manifolds) with their quasicoherent states in order to find their pertinent metrics. We show that they are naturally endowed with two natural “quantum metrics” which are associated with quantum fluctuations of “paths”. The first one provides the length the mean path whereas the second one provides the average length of the fluctuated paths. Onto the classical manifold associated with the quasicoherent state (manifold of the mean values of the coordinate observables in the state minimising their quantum uncertainties) these two metrics provides two minimising geodesic equations. Moreover, fuzzy spaces being not torsion free, we have also two different autoparallel geodesic equations associated with two different adiabatic regimes in the move of a probe onto the fuzzy space. We apply these mathematical results to quantum gravity in BFSS matrix models, and to the quantum information theory of a controlled qubit submitted to noises of a large quantum environment.

Analysis of Functional Properties of Fuzzy \(\tilde{L}\) \(^p\Omega\) Spaces

This paper proposes a study of fuzzy \(\tilde{L}\)p (Ω) spaces, incorporating functions with triangular fuzzy coefficients. These fuzzy functional spaces provide a better adaptation to fuzzy or imprecise functions. We establish the theoretical foundations of these spaces by examining key functional properties, such as the fuzzy scalar product and the fuzzy norm. To do this, we checked the bilinearity, symmetry, positivity, homogeneity and triangular inequality in a fuzzy environment and in the presence of functions whose coefficients are triangular fuzzy numbers, by the \(\alpha\) -cut Dubois and Prade approach. The aim of this paper is to address observations identified in the existing literature, where some functional properties of \(\tilde{L}\)p (Ω) fuzzy spaces are often addressed in a too general manner, without specifying the types of fuzzy functions. This study aims to provide a more detailed and rigorous analysis, thus enriching mathematical understanding and paving the way for practical applications in diverse fields such as: fuzzy differential equations, artificial intelligence, information processing, and decision making in uncertain environments.

Some New Results on Partial Fuzzy Metric Spaces

In this work, we introduce a different interpretation of the notion of a partial fuzzy metric, which we refer to as a partial fuzzy co-metric. We define a partial fuzzy co-metric from a t-conorm and compare it with partial fuzzy metric, in contrast to the conventional approach to the theory of partial fuzzy metric spaces, which is based on the use of a t-norm. Here, we limit the scope of our analysis to Sedghi's definition of partial fuzzy metrics. Additionally, we proposed and compared the ideas of strong partial fuzzy co-metric spaces and strong partial fuzzy metric spaces. We also presented a few examples of these novel ideas.

Neutrosophic Fuzzy Tribonacci ℐ-Lacunary Statistical Convergent Sequence Spaces

Abstract The aim of this paper is to introduce and investigate some neutrosophic fuzzy tribonacci ℐ-lacunary statistical convergent sequence spaces by utilizing the domain of regular tribonacci matrix A = (ajk ). Morever, we also put forward various algebraic and topological features of these convergent sequence spaces and establish several interesting inclusion relations.

Fixed point results in intuitionistic fuzzy pentagonal controlled metric spaces with applications to dynamic market equilibrium and satellite web coupling.

This manuscript contains several new spaces as the generalizations of fuzzy triple controlled metric space, fuzzy controlled hexagonal metric space, fuzzy pentagonal controlled metric space and intuitionistic fuzzy double controlled metric space. We prove the Banach fixed point theorem in the context of intuitionistic fuzzy pentagonal controlled metric space, which generalizes the previous ones in the existing literature. Further, we provide several non-trivial examples to support the main results. The capacity of intuitionistic fuzzy pentagonal controlled metric spaces to model hesitation, capture dual information, handle imperfect information, and provide a more nuanced representation of uncertainty makes them important in dynamic market equilibrium. In the context of changing market dynamics, these aspects contribute to a more realistic and flexible modelling approach. We present an application to dynamic market equilibrium and solve a boundary value problem for a satellite web coupling.

Mackey convergence and separation in $(L, M)$-fuzzy bornological vector spaces

This paper aims to introduce the concepts of Mackey convergence degree for sequences and separation degree for spaces in $(L, M)$-fuzzy bornological vector spaces. Additionally, the paper presents the concept of bornological closure degree for fuzzy sets. Moreover, the paper discusses various characteristics of these concepts. Furthermore, the paper examines the degree relationships among a Mackey convergence sequence, a separated space, and a bornologically closed fuzzy set. Finally, the paper analyzes the properties of functors $\omega$ and $\iota$ between $M$-fuzzifying bornological vector spaces and $(L, M)$-fuzzy bornological vector spaces in terms of Mackey convergence degree and separation degree.

([formula omitted])-contractive and ([formula omitted])-contractive mapping based fixed point theorems in fuzzy bipolar metric spaces and application to nonlinear Volterra integral equations

In this paper, we introduce some novel concepts within the realm of fuzzy bipolar metric spaces, namely (αη)-contractive type covariant mappings and contravariant mappings, and (βχ)-contractive type covariant mappings. We establish some fixed point theorems that demonstrate both the existence and uniqueness of fixed points for (αη)-contractive type covariant mappings and contravariant mappings, and for (βχ)-contractive type covariant mappings in complete fuzzy bipolar metric spaces utilizing the triangular property. Additionally, to substantiate the findings, some illustrative examples and consequential outcomes are presented. Furthermore, the proven results serve to extend, generalize, and enhance the corresponding outcomes documented in existing literature. A practical application of these findings in the context of non-linear Volterra integral equations is demonstrated, solidifying and reinforcing the credibility of the established results. Overall, this paper contributes to the understanding of fixed point theory in the context of fuzzy bipolar metric spaces and highlights the significance of (αη)-contractive mappings and (βχ)-contractive mappings in this domain.

On Ekeland Variational Principle and Its Applications Through Fuzzy Quasi Metric Spaces with Non-Archimedean t-norm

The aim of this article is to introduce Ekeland variational principle (EVP) and some results in fuzzy quasi metric space (FQMS) under the non-Archimedean \(t\)-norms. In this article the basic topological properties and a partial order relation are defined on FQMS. Utilizing Brézis-Browder principle on a partial order set, we extend the EVP to FQMS also. Moreover, we derive Takahashi’s minimization theorem, which ensures the existence of a solution of an optimal problem without taking the help of compactness and convexity properties on the underlying space. Furthermore, we give an equivalence chain between these two theorems. Finally, two fixed point results namely the Banach fixed point and the Caristi-Kirk fixed point theorems are described extensively.