Let be a unital ∗‐algebra with the unit I and be the symmetric center of . Assume that contains a nontrivial projection P such that implies A = 0 and implies A = 0. In this paper, we proved that if a map satisfies [ φ ( A ), B ] ∗ = −[ φ ( B ), A ] ∗ for all , then there exists a map such that φ ( A ) = Z A + f ( A ) for all , where . As applications, we obtain the concrete form of nonadditive skew commuting maps on prime ∗‐algebra, factor von Neumann algebras and standard operator algebras.

In real‐time, visually impaired persons face challenging issues in identifying and avoiding obstacles in their path, making it difficult to move around freely and confidently. Object detection technology benefits visually impaired persons and contains the procedure of determining several individual objects from the image and recognizing their place with bounding boxes with class labels. Both exploit typical machine learning (ML) approaches and utilize deep learning (DL) models to execute object detection methods. Recently, several researchers have leveraged DL techniques such as convolutional neural networks (CNNs) for solving object detection problems and presented various recent object detection approaches. The study presents a sine cosine optimization with DL–driven object detector (SCODL–ODC) technique for visually impaired persons. The presented SCODL–ODC technique detects and classifies the objects, which can help them navigate their environment more efficiently and avoid obstacles. To accomplish this, the proposed SCODL–ODC technique employs a YOLOv5 object detector at the initial stage to detect the objects accurately, and the SCO model performs its hyperparameter tuning process. The following step is to classify them using the deep stacked autoencoder (DSAE) model when the objects are detected. Finally, the Harris Hawks optimization (HHO) technique is used to adjust the hyperparameters of the DSAE model. The experimentation outcome of the SCODL–ODC method is examined on the object detection dataset. The investigational validation of the SCODL–ODC method portrayed a superior accuracy value of 99.31% over existing models.

We use the new variant of Szász–Mirakjan operators to construct a generalized version of Szász‐beta type operators and obtain auxiliary lemmas. We present the weighted approximation theorems and, by using Peetre’s K ‐function, the local approximation results of these operators are studied. We also analyze the rate of convergence by utilizing Lipschitz‐type maximal functions and the ordinary modulus of continuity. Moreover, using aforesaid operators, we finally examine the Voronovskaya‐type theorem.

The purpose of this paper is to study the fixed point problem of cyclic mapping in the generalized metric spaces. In an extended b ‐metric space, we introduce a new cyclic mapping, which is called L B 3 S 2 ‐type cyclic mapping into this space, a specific example of L B 3 S 2 ‐type cyclic mapping is shown, and the fixed point theorem for it is established. In addition, we give a concrete example to fully illustrate the correctness of the conclusion. An application to supply chain equilibrium problem is shown. The results of this paper further extend some previous results in generalized metric spaces.

The sum‐connectivity (SC) index of a graph is defined as , where Θ μ denotes the vertex degree of μ in . In this paper, the fourth largest value of SC index for the chemical unicyclic graphs of order n ≥ 7 is determined.

In this article, we study a new subfamily of analytic functions associated with q ‐Janowski function using q ‐Sălăgean operator. We explore certain properties of the functions belonging to this new class which include sufficient condition, inclusion results, and coefficient estimate bounds for Fekete–Szegö functional. Several consequences of main results are also given.

This study presents a detailed investigation into the analytical properties of the generalized Gamma function introduced by Dilcher. We establish a fundamental recurrence relation and derive novel reflection formula for this generalized function, extending the classical identities known for the Euler Gamma function. Further, we analyze the generalized digamma function, the logarithmic derivative of the generalized Gamma function, and obtain a significant difference equation that characterizes its behavior. Explicit expressions for the evaluation of this generalized digamma function at half‐integer arguments are provided. The study also introduces the natural extension to generalized polygamma functions and reports on several special values for both classical and generalized cases. To complement the theoretical analysis, a graphical representation of the generalized digamma function is included, alongside supporting numerical computations that illustrate its properties and validate the derived results. Our findings provide a comprehensive framework that enhances the understanding of these generalized special functions and underscores their potential for broader application in mathematical physics and number theory.

In graph theory, the Randic index ( R ) is a topological graph invariant widely used as a physicochemical descriptor in the mathematical modeling of molecular structures. However, traditional molecular graphs fail to capture the heterogeneity of chemical bonds, since they treat all edges as uniform, ignoring variations in bond lengths and strengths. To overcome this limitation, we adopt a fuzzy graph structure, which provides a more realistic framework for representing molecular interactions. This dual perspective developing rigorous theoretical bounds for the Randic index on fuzzy graph classes and simultaneously applying them to real pharmaceutical compounds ensure both mathematical novelty and practical relevance. We derive analytical bounds for the Randic index over fuzzy versions of complete graphs, paths, stars, and cycles, as well as for graph operations such as Cartesian product, union, join, and composition, confirming the sharpness of the results in each case. To demonstrate the applicability, we compute the Randic index for fuzzy graph representations of Parkinson’s disease drugs (Levodopa, Procyclidine, Trihexyphenidyl, and Apomorphine). The findings indicate that the Randic index, within the fuzzy graph framework, reliably estimates key physicochemical properties such as polarizability, molar refractivity, surface tension, and molar volume, highlighting the strength of combining theoretical results with drug modeling applications.

This paper intends to extend certain special functions from one complex variable to the bicomplex setting. Specifically, we define the k ‐bicomplex Gamma and k ‐bicomplex Beta functions. Subsequently, several properties and formulas pertaining to k ‐bicomplex Gamma, k ‐bicomplex Beta, and k ‐shifted factorial are derived. We also explore applications of the k ‐bicomplex Pochhammer symbol based on these results.

For a given Banach algebra and a continuous endomorphism ω on , we define weakly ω ‐approximately biprojective and weakly ω ‐approximately Helemskii biflat Banach algebras. We then examine the relationship between them and express the correlation between them and ω ‐pseudoamenability. In addition, we show that if ω is not an idempotent continuous endomorphism on , then is weak ω ‐approximately biprojective. Here, represents the free group on two generators, and denotes the Fourier algebra of .