The purpose of this research is to interpolate bipolarity into the definition of the vague soft set. This gives a new more applicable, flexible, and generalized extension of the soft set, the fuzzy soft set, or even the vague soft set, which is the bipolar vague soft set. In addition, types of bipolar vague soft sets, as well as some new related concepts and operations are established with examples. Moreover, properties of bipolar vague soft sets including absorption, commutative, associative, distributive, and De Morgan’s laws are discussed in detail. Furthermore, a bipolar vague soft set-designed decision-making algorithm is provided generalizing Roy and Maji method. This allows making more effective decisions to choose the optimal alternative. Finally, an applied problem is introduced with a comparative analysis to illustrate how the proposed algorithm works more successfully than the previous models for problems that contain uncertain ambiguous data.

In this work, we define the fuzzy soft Hilbert space \(\tilde{H}\) based on the definition of the fuzzy soft inner product space \((\tilde{U},\widetilde{<\cdot,\cdot>})\), introduced by Faried {et al.} [{N. Faried, M. S. S. Ali, H. H. Sakr, Appl. Math. Inf. Sci., {\bf 14} (2020), 709--720}], in terms of the fuzzy soft vector \(\tilde{v}_{f_{G(e)}}\). Moreover, we show that \(\mathbb{C}^{n}(A)\), \(\mathbb{R}^{n}(A)\) and \(\ell_{2}(A)\) are suitable examples of fuzzy soft Hilbert spaces. In addition, it is proved that the fuzzy soft orthogonal complement of any non-empty fuzzy soft subset of \(\tilde{H}\) is a fuzzy soft closed fuzzy soft subspace of \(\tilde{H}\) and we study some of the fuzzy soft Hilbert spaces properties and some of the fuzzy soft inner product spaces properties. Furthermore, we introduce the definition of the fuzzy soft orthogonal family and the fuzzy soft orthonormal family and introduce examples satisfying them. Moreover, we present the fuzzy soft Bessel's inequality and the fuzzy soft Parseval's formula in this generalized setting.

This paper presents several nonparametric estimators for the Sharma–Taneja–Mittal entropy measure of a continuous random variable with known support, utilizing spacing, a local linear model, and a kernel function. The properties of these estimators are discussed. Their performance was also examined through real data analysis and Monte Carlo simulations. In the Monte Carlo experiments, the proposed Sharma–Taneja–Mittal entropy estimators were employed to create a test of goodness-of-fit under the standard uniform distribution. The suggested test statistics demonstrate strong performance, as evidenced by a comparison of their power with that of other tests for uniformity. Finally, we examine a classification issue in the recognition of patterns to underscore the significance of these measures.

The complementary dual of entropy has received significant attention in the literature. Due to the emergence of many generalizations and extensions of entropy, the need to generalize the complementary dual of uncertainty arose. This article develops the residual cumulative generalized fractional extropy as a generalization of the residual cumulative complementary dual of entropy. Many properties, including convergence, transformation, bounds, recurrence relations, and connections with other measures, are discussed. Moreover, the proposed measure’s order statistics and stochastic order are examined. Furthermore, the dynamic design of the measure, its properties, and its characterization are considered. Finally, nonparametric estimation via empirical residual cumulative generalized fractional extropy with an application to blood transfusion is performed.

The Molodtsov-initiated soft set theory plays an important role as a powerful mathematical tool for handling uncertainty. As an extension of the soft set, the fuzzy soft set can be seen to be more generic and flexible than utilizing the soft set only that fails to represent problem parameters fuzziness. Through this progress, the fuzzy soft set theory cannot deal with decision-making problems involving multi-attribute sets, bipolarity, or some effective considered parameters. Therefore, the goal of this article is to adapt effectiveness and bipolarity concepts with the multi-fuzzy soft set of order n. One can see that this approach generates a novel, extended, effective decision-making environment that is more applicable than any previously introduced one. In addition, types, concepts, and operations of effective bipolar-valued multi-fuzzy soft sets of dimension n are provided, each with an example. Furthermore, properties like absorption, associative, distributive, commutative, and De Morgan’s laws of those new sets are investigated. Moreover, a decision-making methodology under effective bipolar-valued multi-fuzzy soft settings is established. This technique facilitates reaching the final decision that this student is qualified to take a certain education level, or this patient is suffering from a certain disease, etc. In addition, a case study represented in a medical diagnosis example is discussed in detail to make the proposed algorithm clearer. Applying matrix techniques in this example as well as using MATLAB^®, not only makes it easier and faster in doing calculations, but also gives more accurate, optimal, and effective decisions. Finally, the sensitivity analysis, as well as a comparison with the existing methods, are conducted in detail and are summarized in a chart to show the difference between them and the current one.