In this work, we introduce generalizations of fuzzy -quasinormal operators defined on fuzzy Hilbert spaces over fuzzy vector spaces. We study several fundamental properties of these operators and explore specific operations associated with them

Calculus and study of fuzzy dynamic equations for fuzzy vector functions on time scales

Fuzzy matrices play a crucial role in fuzzy logic and fuzzy systems. This paper investigates the problem of supervised learning fuzzy matrices through sample pairs of input–output fuzzy vectors, where the fuzzy matrix inference mechanism is based on the max–min composition method. We propose an optimization approach based on stochastic gradient descent (SGD), which defines an objective function by using the mean squared error and incorporates constraints on the matrix elements (ensuring they take values within the interval [0, 1]). To address the non-smoothness of the max–min composition rule, a modified smoothing function for max–min is employed, ensuring stability during optimization. The experimental results demonstrate that the proposed method achieves high learning accuracy and convergence across multiple randomly generated input–output vector samples.

The Banach fixed‐point theorem, along with a fuzzy number characterized by normality, convexity, upper semicontinuity, and a compactly supported interval to look into the possibility of a solution equation to the fuzzy nonlinear neutral integrodifferential equation of the Sobolev‐type within a fuzzy vector space of n dimensions, is employed in this research. At the end, to demonstrate the practical application of the findings, an example is also presented.

A Note on Aggregation of t-norm-based, Fuzzy Vector Subspaces

Binary convex fuzzy vector spaces over binary vector spaces and their applications

The purpose of this work is to study an approach to solving the problem of fuzzy vector optimization for compiling a diet. It is necessary to develop a daily ration that provides human needs in nutrients and energy and is the best in terms of costs and weight. For this purpose the problem of choosing the best alternative from a given fuzzy set of alternatives is solved, while the quality of the alternative is evaluated using several partial efficiency cri- teria. The goal in the task is fuzzily defined. According to the Zadeh – Bellman idea, a fuzzy solution to a problem is the intersection of a fuzzy goal and a fuzzy set of alternatives. The paper considers the problem of fuzzy two-criteria optimization. Partial criteria that are minimized are the weight and cost of the daily ration. Fuzzy needs for nutrients and kilocalories are determined by fuzzy triangular numbers. For an approximate solution of the problem, an algorithm is proposed according to which a sequence of linear programming problems is solved. The computer program was created that implements the solution of this problem in the Wolfram Matematica package. The following results were obtained: a set of products, their weight and cost were determined for the daily diet. The degree of confidence that the found plan is optimal is determined. On the basis of the proposed mathe- matical model and the method of its solution, a software application allowing to choose food products, enter the necessary restrictions in a user- friendly form, and receive one or more options for the daily ration at the output was created. Such an application will be useful for nutritionists, ath- letes, doctors and other people who are concerned with the problems of healthy eating.

Flag codes are a recent network coding strategy based on linear algebra. Fuzzy vector subspaces extend the notions of classical linear algebra. They can be seen as abstractions of flags to the point that several fuzzy vector subspaces can be identified to the same flag, which naturally induces an equivalence relation on the set of fuzzy vector subspaces. The main contributions of this work are the methodological abstraction of flags and flag codes in terms of fuzzy vector subspaces, as well as the generalisation of three distinct equivalence relations that originated from the fuzzy subgroup theory and study of their connection with flag codes, computing the number of equivalence classes in the discrete case, which represent the number of essentially distinct flags, and a comprehensive analysis of such relations and the properties of the corresponding quotient sets.

The algebraic structures Group, Ring, Field and Vector spaces are important innovations in Mathematics. Most of the theoretical concepts of Mathematics are based on the theorems related to these algebraic structures. Initially many mathematicians developed theorems related to all these algebraic structures. In 20th century most of the researchers introduced the theorems on the algebraic structures with Fuzzy and Intuitionistic fuzzy sets. Recently in 21st century the researchers concentrated on Neutrosophic sets and introduced the algebraic structures like Neutrosophic Group, Neutrosophic Ring, Neutrosophic Field, Neutrosophic Vector spaces and Neutrosophic Linear Transformation. In the current scenario of relating the spaces with the structures, we have introduced the concepts of Neutrosophic topological vector spaces. In this article, the study of Neutrosophic Topological vector spaces has been initiated. Some basic definitions and properties of classical vector spaces are generalized in Neutrosophic environment over a Neutrosophic field with continuous functions. Neutrosophic linear transformations and their properties are also included in Neutrosophic Topological Vector spaces. This article is an extension work of fuzzy and intuitionistic fuzzy vector spaces which were introduced in fuzzy and intuitionistic fuzzy environments. Even though it is an extension work, Neutrosophic Topological Vector space will play an important role in Neural Networks, Image Processing, Machine Learning and Artificial Intelligence Algorithms.

<p>In this paper, the fuzzy counterparts of the collineations defined in classical projective spaces are defined in a 3-dimensional fuzzy projective space derived from a 4-dimensional fuzzy vector space. The properties of fuzzy projective space $ (\lambda, \mathcal{S}) $ left invariant under the fuzzy collineations are characterized depending on the membership degrees of the given fuzzy projective space and also depending on the pointwise invariant of the lines. Moreover, some relations between membership degrees of the fuzzy projective space are presented according to which are of the base point, base line, and base plane invariant under a fuzzy collineation. Specifically, when all membership degrees of $ (\lambda, \mathcal{S}) $ are distinct, the base point, base line, and base plane of $ (\lambda, \mathcal{S}) $ are invariant under the fuzzy collineation $ \bar{f} $. Conversely, if none of the base point, base line, or base plane remain invariant, then the system becomes crisp in $ (\lambda, \mathcal{S}) $. Additionally, some relations between the membership degrees of the fuzzy projective space, concerning the invariance of the base point, base line, and base plane, are presented.</p>

This paper contributes to the broader studies of fuzzy vector metric spaces and fuzzy metric spaces based on order structures beyond the unit interval. It defines the notions of the left (right) order convergence and continuity in non-Arcimedean $\mathcal{L}$-fuzzy vector metric spaces. The notation $\mathcal{M}_E(a,b,s)$ means the nearness between $a$ and $b$ according to any positive vector $s$. This study exemplifies definitions and reaches some well-known results. Moreover, it proposes the concept of $\mathcal{L}$-fuzzy vector metric diameter and studies some of its basic properties. Further, the present paper proves the Cantor intersection theorem and the Baire category theorem via these concepts. Finally, this study discusses the need for further research.

This paper contributes to the ongoing investigation of aggregating algebraic structures, with a particular focus on the aggregation of fuzzy vector spaces. The article is structured into three distinct parts, each addressing a specific aspect of the aggregation process. The first part of the paper explores the self-aggregation of fuzzy vector subspaces. It delves into the intricacies of combining and consolidating fuzzy vector subspaces to obtain a coherent and comprehensive outcome. The second part of the paper centers around the aggregation of similar fuzzy vector subspaces, specifically those belonging to the same equivalence class. This section scrutinizes the challenges and considerations involved in aggregating fuzzy vector subspaces with shared characteristics. The third part of the paper takes a broad perspective, providing an analysis of the aggregation problem of fuzzy vector subspaces from a general standpoint. It examines the fundamental issues, principles, and implications associated with aggregating fuzzy vector subspaces in a comprehensive manner. By elucidating these three key aspects, this paper contributes to the advancement of knowledge in the field of aggregation of algebraic structures, shedding light on the specific domain of fuzzy vector spaces.

In this paper, we used the definition of fuzzy vector spaces and fuzzy subsets in combination in order to define fuzzy codes over a fuzzy vector space. We found that the combined definitions did not satisfy the definition of fuzzy vector spaces over a Galois field F 2 , and gave the conditions in which the combined definitions hold the definition of fuzzy vector spaces in order to define binary fuzzy codes. By defining binary fuzzy codes over fuzzy vector space in relation to the probability of binary symmetric channels (BSC) sending a codeword incorrectly and the weight of error pattern between codewords, we updated properties of Hamming distance of binary fuzzy codes over fuzzy vector space. From the updated properties, we found that the properties of Hamming distance stated over a vector space also satisfy in binary fuzzy codes defined over a fuzzy vector space. Furthermore, we found some interesting results on the decoding of fuzzy codewords sent over a BSC using minimum nonzero Hamming distance of binary fuzzy codes.

Multiplication and division of fuzzy arithmetic have brought out theoretical drawbacks to fuzzy analytic hierarchy process based decision making systems. To cope with the drawbacks, trapezoidal fuzzy additive reciprocal preference matrices (TFARPMs) are utilized to characterize preference information and addition and subtraction of fuzzy arithmetic are applied to fuzzy numbers. This paper introduces formulas to calculate left and right spread indices and imprecision indices of cores and supports for fuzzy elements in a TFARPM. Based on the addition of fuzzy arithmetic, a transitivity equation system with a parameterized trapezoidal fuzzy vector is built and computational formulas are devised to identify values of parameters from imprecision indices of cores and supports of the trapezoidal fuzzy elements in a TFARPM. A fuzzy addition based non-parametric transitivity equation is then established to define additive consistency of TFARPMs. Properties of additively consistent TFARPMs are proposed and an index formula is brought forward to compute additive inconsistency degrees of TFARPMs. A novel approach is presented to generate additively consistent TFARPMs from fuzzy vectors and a new framework is put forward to normalize [0, 1]-valued trapezoidal fuzzy vectors. An absolute deviation based minimization model is developed and converted equivalently into a linear program to acquire normalized trapezoidal fuzzy priority vectors from TFARPMs. A closed-form solution of the minimization model is found to calculate normalized optimal trapezoidal fuzzy priority vectors of additively consistent TFARPMs. Two illustrating examples including a multi-criteria decision making problem are provided to validate the proposed models.

This paper discusses current formulations based on fuzzy-logic control concepts as applied to the removal of impulsive noise from digital images. We also discuss the various principles related to fuzzy-ruled based logic control techniques, aiming at preserving edges and digital image details efficiently. Detailed descriptions of a number of formulations for recently developed fuzzy-rule logic controlled filters are provided, highlighting the merit of each filter. Fuzzy-rule based filtering algorithms may be designed assuming the tailoring of specific functional sub-modules: (a) logical controlled variable selection, (b) the consideration of different methods for the generation of fuzzy rules and membership functions, (c) the integration of the logical rules for detecting and filtering impulse noise from digital images. More specifically, we discuss impulse noise models and window-based filtering using fuzzy inference based on vector directional filters as associated with the filtering of RGB color images and then explain how fuzzy vector fields can be generated using standard operations on fuzzy sets taking into consideration fixed or random valued impulse noise and fuzzy vector partitioning. We also discuss how fuzzy cellular automata may be used for noise removal by adopting a Moore neighbourhood architecture. We also explain the potential merits of adopting a fuzzy rule based deep learning ensemble classifier which is composed of a convolutional neural network (CNN), a recurrent neural networks (RNN), a long short term memory neural network (LSTM) and a gated recurrent unit (GRU) approaches, all within a fuzzy min-max (FMM) ensemble. Fuzzy non-local mean filter approaches are also considered. A comparison of various performance metrics for conventional and fuzzy logic based filters as well as deep learning filters is provided. The algorhitms discussed have the following advantageous properties: high quality of edge preservation, high quality of spatial noise suppression capability especially for complex images, sound properties of noise removal (in cases when both mixed additive and impulse noise are present), and very fast computational implementation.

A rehabilitation framework based on motor imagery induced wheelchair movement using fuzzy vector quantization

This paper discusses a new approximate solution of a class of fully fuzzy linear systems A ˜ x ˜ = b ˜ in which the coefficient matrix A ˜ is a positive fuzzy matrix. The original fuzzy linear systems is extended into simple crisp linear equation using the obtained approximate multiplication of positive fuzzy number and near zero fuzzy number. Two cases are analysed: (a) the unknown vector x ˜ is a near zero fuzzy vector with positive mean value; (b) the unknown vector x ˜ is a near zero fuzzy vector with negative mean value. Two computing models are established and respective expression of the solution to fully fuzzy linear system are derived, and the sufficient condition for the existence of strong fuzzy solution are analyzed correspondingly. Some numerical examples are given to illustrated our proposed method.

The research objective of this article is to train a computer (agent) with market information data so it can learn trading strategies and beat the market index in stock trading without having to make any prediction on market moves. The approach assumes no trading knowledge, so the agent will only learn from conducting trading with historical data. In this work, we address this task by considering Reinforcement Learning (RL) algorithms for stock portfolio management. We first generate a three-dimension fuzzy vector to describe the current trend for each stock. Then the fuzzy terms, along with other stock market features, such as prices, volumes, and technical indicators, were used as the input for five algorithms, including Advantage Actor-Critic, Trust Region Policy Optimization, Proximal Policy Optimization, Actor-Critic Using Kronecker Factored Trust Region, and Deep Deterministic Policy Gradient. An average ensemble method was applied to obtain trading actions. We set SP100 component stocks as the portfolio pool and used 11 years of daily data to train the model and simulate the trading. Our method demonstrated better performance than the two benchmark methods and each individual algorithm without fuzzy extension. In practice, real market traders could use the trained model to make inferences and conduct trading, then retrain the model once in a while since training such models is time0consuming but making inferences is nearly simultaneous.

In Machine learning and pattern recognition, building a better predictive model is one of the key problems in the presence of big or massive data; especially, if that data contains noisy and unrepresentative data samples. These types of samples adversely affect the learning model and may degrade its performance. To alleviate this problem, sometimes, it becomes necessary to sample the data after eliminating unnecessary instances by maintaining the underlying distribution intact. This process is called sampling or instance selection (IS). However, in this process, a substantial computational cost is involved. This paper discusses an uncertainty based optimal sample selection (UBOSS) method which can select a subset of optimal samples efficiently. Our proposed work comprises three main steps; initially, it uses an IS method to identify the patterns of representative and unrepresentative samples from the original data set; then, an uncertainty-based selector is designed to obtain fuzziness (i.e., a type of uncertainty) of those samples using a classifier whose output is a membership or fuzzy vector; this process further utilizes the divide-and-conquer strategy to obtain a subset of representative samples. Experiments are conducted on six datasets to evaluate the performance of the proposed IS method. Results show that our proposed methodology outperforms when compared with the selection performance (i.e., optimum samples) of the baseline methods (i.e., CNN, IB3, and DROP3).

The concept of two-fold algebras is generated by merging some special sets such as fuzzy sets, neutrosophic sets or plithogenic sets with various algebraic structures. The main objective of this research paper is to provide a strict mathematical definition and a general study of two-fold fuzzy vector spaces defined over the real numbers and also two-fold fuzzy vector spaces defined over the complex numbers, as well as two-fold fuzzy algebraic modules defined over commutative rings with unity. The elementary properties and algebraic operations of those structures will be explained through many theorems and mathematical proofs.