Mathematics

Mathematics is a discipline that forms the foundation of many fields and should be learned gradually, starting from early childhood. However, some subjects can be difficult to learn due to their abstract nature, the need for attention and planning, and math anxiety. Therefore, in this study, a system that contributes to mathematics teaching using computer vision approaches has been developed. In the proposed system, users can write operations directly in their own handwriting on the system interface, learn their results, or test the accuracy of their answers. They can also test themselves with random questions generated by the system. In addition, visual graph generation has been added to the system, ensuring that education is supported with visuals and made enjoyable. Besides the character recognition test, which is applied on public datasets, the system was also tested with images obtained from 22 different users, and successful results were observed. The study utilizes CNN networks for handwritten character detection and self-created image processing algorithms to organize the obtained characters into equations. The system can work with equations that include single and multiple unknowns, trigonometric functions, derivatives, integrals, etc. Operations can be performed, and successful results can be achieved even for users who write in italicized handwriting. Furthermore, equations written within each closed figure on the same page are evaluated locally. This allows multiple problems to be solved on the same page, providing a user-friendly approach. The system can be an assistant for improving performance in mathematics education.

In this paper, we propose three variants of Mann-type inertial forward–backward iterative methods for approximating the minimum-norm solution of the monotone inclusion problem and the fixed points of nonexpansive mappings. In the first two methods, we compute the Mann-type iteration together with the inertial extrapolation and fixed-point iteration in the initiation of the process, while the last method computes only the Mann-type iteration with inertial extrapolation at the start of the process. We establish the strong convergence results for each method with appropriate assumptions and discuss some applications of the presented methods. Finally, we present numerical examples in both finite- and infinite-dimensional Hilbert spaces to demonstrate their efficiency. A comparative analysis with existing methods is also provided.

There is a correspondence between equivalence classes of fuzzy ideals, on a commutative ring, and decreasing gradual ideals. In this paper, we explore how to construct a fuzzy ideal starting from any decreasing gradual ideal $\sigma$. To achieve this, we consider an interior operator, ${\sigma }^d$, and a closure operator, ${\sigma }^e$, and show that the pair is always an F-pair, which defines a fuzzy ideal. Furthermore, this correspondence, and its inverse, preserves sums, intersections and products. This therefore provides an algebraic framework for studying fuzzy ideals. In particular, prime fuzzy ideals and weakly prime fuzzy ideals have their counterparts in the theory of decreasing gradual ideals, offering us a new perspective on these particular objects. One of the main objectives is to characterize fuzzy prime ideals using single fuzzy elements and gradual ideals.

This study investigates the existence of mild solutions to impulsive fractional differential equations involving almost-sectorial operators. Through the application of solution operator techniques, fixed-point theory, and Laplace transform, we demonstrate the existence of a unique mild solution to the considered system.