Preface to the Special Issue “Recent Developments in Theoretical and Applied Mathematics”

This Special Issue, “Recent Developments in Theoretical and Applied Mathematics”, aims to present current trends in pure and applied mathematics to the global mathematical community. This collection explores the application of mathematical concepts to real-world analytical problems across technology and science, while also encouraging creativity in theoretical and applied mathematics. By bridging the gap between abstract structures and practical implementation, this Special Issue highlights how foundational theories drive innovation in fields including engineering, data science, and economics.

Furthermore, it provides a collaborative platform for researchers to exchange their ideas, methodologies and computational techniques. Collating diverse perspectives, the Special Issue seeks to inspire further research and prove the relevance of mathematics in solving complex global problems.

In the first contribution, Adel Lachouri and Muath Awadalla provide a rigorous analysis of a coupled hybrid fractional differential system involving a generalized Hilfer operator under integral and antiperiodic boundary conditions (Contribution 1). Complementing this, in the second study, Gülsüm Ulusoy Ada and Ali Aralwe explore the weighted approximation properties of a family of Szász–Mirakyan-type operators preserving two exponential functions on the unbounded interval $[0,\infty )$ (Contribution 2). In Contribution 3, Das et al. develop a numerical solution to the Kudryashov–Sinelshchikov equation—which is crucial for modeling pressure waves in liquids containing gas bubbles—using the reproducing kernel Hilbert space method to find a numerical solution for the time-fractional form of the equation. In Contribution 4, Ibtsam A. R. Alroily and Brahim Chaourar present a theoretical approach for finite topological spaces regarding the number $T\left(n\right)$ of distinct topologies on n points. Next, in Contribution 5, Abdulrahman A. Sharif and Muath Awadalla investigate a class of Volterra–Fredholm-type implicit neutral fractional integro-differential equations, extending their study to general Banach spaces by employing Darbo’s fixed point theorem combined with the Kuratowski measure of noncompactness. The same authors, in Contribution 6, explore a global approach to climate change which requires climate models that balance computational speed with accurate, long-term physical dynamics. In Contribution 7, Muath Awadalla presents theoretical aspects regarding the fractional Heun operator constructed through a symmetric composition of left and right Riemann–Liouville fractional derivatives, while in Contribution 8, Leonid G. Fel, using polynomial identities for degrees of syzygies, present a different approach to obtaining an affirmative answer to Wilf’s questions in semigroups $S_3$, questions posted in [1]. In Contribution 9, Geonwoo Kim and Junkee Jeon address an infinite-horizon optimal consumption and investment problem for an agent consuming both necessities and luxuries. Operating within a complete financial market that includes risk-free assets and risky stock, they analytically solve the problem using the dual-martingale method. In Contribution 10, Dwivedi et al. develop approximate weak and strong stationary conditions for interval-valued multiobjective optimization problems with vanishing constraints, and in Contribution 11, Suxena et al. present the algebraic structure of generalized quasi-polycyclic codes, which represents a generalization of the right quasi-polycyclic (QPC) and generalized quasi-cyclic (GQC) codes over a finite field $F_q$. By unifying several established code families into a single structure, this approach enables the explicit construction of numerous new linear codes, many of which meet or exceed the previously best-known parameters. Finally, in Contribution 12, Alahmari et al. studied the compactness of the complex Green operator $G_q$ associated with the Kohn Laplacian ${\square }_b$ on boundaries of pseudoconvex domains in Stein manifolds.

The Guest Editors wish to express their deepest gratitude to all the authors for their excellent contributions, which have significantly enriched this Special Issue. We are also profoundly indebted to the anonymous reviewers for their rigorous peer review, insightful comments, and invaluable suggestions that greatly enhanced the quality of each manuscript. In addition, we appreciate the collaboration of the publishing house, the constant support provided by MDPI Assistant Editors, and the exceptional support of the Editorial Office in the successful completion of this project.