SOME PROPERTIES OF η-MADIAL ON d-ALGEBRA

• RACHID BAHLOUL (Bahloul Rachid)

This work presents the concept of η-medial, which is an additional medial generalization. Furthermore, the notions of η-endomorphism and η-derivation are introduced, and some important properties are given. Finally, the condition for injectivity is obtained.

(α, n)-LAPLACE TRANSFORM AND THEIR APPLICATIONS

• KHADIJA EL GOURMATI (Sultan Moulay Slimane University, Beni Mellal, Morocco.)

In this paper, we extend the formulas of both the Laplace and Sumudu transforms to the (α, n)-conformable fractional order, and we derive and analyze several fundamental properties related to these two transforms.

A Frational Extension of AdamW

• Houssam Rachad (Sultan Moulay Slimane University)

Adaptive optimizers such as Adam and AdamW have transformed large-scale machine learning, yet they share a common structural weakness: their memory of past gradients decays exponentially, so gradient information older than roughly $1/(1-\beta_{1})$ iterations rapidly vanishes. In highly non-convex, noisy, or ill-conditioned optimization landscapes, this short-memory mechanism may prevent the optimizer from exploiting long-range trajectory structure. We propose \emph{Adaptive Fractional AdamW} , a new optimizer that replaces the raw stochastic gradient in AdamW's moment estimates with a normalized fractional long-memory correction built on the Gr\"{u}nwald--Letnikov (G--L) operator. Unlike existing fractional optimizers that fix the fractional order $\alpha$ prior to training, FAAdamW continuously adapts $\alpha_t$ based on the current gradient norm: when gradients are large and informative the method behaves similarly to AdamW; when gradients are small or noisy it increases its effective memory depth automatically. We prove that FAAdamW reduces exactly to standard AdamW when the memory length $K=0$, establish a norm-boundedness lemma for the fractional correction, derive a per-step stability bound, and provide a conditional stationary-point convergence result under standard non-convex stochastic assumptions. We also present an approximate-reasoning interpretation of the correction as a convex combination of gradient evidence governed by a power-law credibility kernel, connecting the method to belief-function and possibility-theory frameworks. Experiments on five analytical benchmarks with stochastic noise, accompanied by ablation and sensitivity studies, demonstrate that FAAdamW is competitive with AdamW and other baselines, while providing improved stability and reduced variance in several noisy, ill-conditioned, and multimodal settings.

Analysis of a Diffusive Epidemic Model (SIR) with Crowley–Martin Incidence and Holling Type III Treatment in a Heterogeneous Environment

• Oualid Bahouh (Faculty of Sciences and Techniques, Beni Mellal)

We study a reaction–diffusion SIR model that incorporates nonlinear infection and treatment effects. We prove that the system is well posed by establishing the global existence of positive solutions and deriving uniform bounds. Using variational arguments, we formulate the basic reproduction number R0 and analyze its threshold properties. In particular, we study the local and global stability of the disease-free equilibrium (DFE). Numerical simulations illustrate and support the analytical results, highlighting how diffusion interacts with nonlinear incidence and saturating treatment in epidemic dynamics.

Higher-order conformable derivatives and their applications

• Mohammed Bouziani (Center for Educational Orientation and Planning (COPE), BP 6222, Avenue Zaytoune, Hay Ryad, Rabat, Morocco.)

After presenting some results and introducing a new definition of the higher-order conformable derivative, we discuss how these findings relate to a novel idea: The higher-order conformable Laplace transform and the higher-order conformable Sumudu transform.

Generalization of derivations on lattices

• Faiza Shujat (Taibah University, Madinah, Saudi Arabia)

Over the course of its development, the theory of binary operations has become a foundational component of algebraic structures, providing a rigorous framework for the formulation and analysis of abstract algebraic structures and playing a central role in lattice theory and its numerous applications, where many fundamental notions and properties, including the very definition of a lattice, can be characterized and examined effectively in terms of appropriate binary operations. In this study, we investigate the structural development of generalized γ-derivations on distributive lattices, where γ denotes a lattice homomorphism. We establish the following characterization: let η be a mapping from a distributive lattice to itself. Then η is a generalized γ-derivation and a joinitive mapping if and only if η is a lattice homomorphism. Utilizing this homomorphic framework, we further analyze and derive several fundamental properties and salient characteristics of generalized γ-derivations. From an applications perspective, lattice structures play a central role in various domains, including information theory, information retrieval, and access control systems. Durfee employed techniques from the geometry of numbers to address a range of cryptanalytic problems. Moreover, several public-key cryptosystems have been subjected to cryptanalysis via algebraic methods, with key results obtained through tools originating in the theory of integer lattices.

On $d$-set derivative

• Rania Saadeh (Al-Balqa Applied University, Jordan)

The concept of new d-set derivatives resulting from certain algebraic operations (a law) was introduced. From this concept, a set of results and their applications were derived.

Linear Operators and Fractional Calculus in Banach Spaces Applications to Integral and Control Systems in Biomathematics

• Manal Elzain Mohamed Abdalla (King Khalid University, Saudi Arabia)

In recent years, the study of linear operators and fractional calculus in Banach spaces has gained significant attention due to its fundamental importance in both pure and applied mathematics. Banach spaces, as complete normed vector spaces, provide a general and powerful framework for analyzing a wide variety of problems, particularly those involving infinite-dimensional systems that frequently arise in functional analysis and mathematical modeling. Linear operators are essential tools in this context, as they allow mathematicians to describe transformations between spaces and to investigate the structural and functional properties of mathematical systems. Their applications extend to solving differential and integral equations, which are central to many scientific and engineering disciplines. On the other hand, fractional calculus generalizes classical calculus by introducing derivatives and integrals of non-integer order. This extension has proven to be highly effective in modeling systems with memory, hereditary characteristics, and non-local behavior. The integration of linear operator theory with fractional calculus offers a rich and flexible approach to studying complex systems, particularly in the field of biomathematics. Many biological processes, such as population dynamics, neural responses, and the spread of infectious diseases, exhibit behaviors that cannot be accurately captured using traditional integer-order models. Fractional models provide a more realistic description of these phenomena. This research aims to investigate the theoretical foundations of linear operators and fractional calculus in Banach spaces and to explore their applications in integral equations and control systems within biomathematics. The study seeks to highlight the effectiveness of these mathematical tools in analyzing and solving real-world problems.

Small-Jump Approximation for Infinite-Intensity L´evy Processes

• Driss BOUGGAR (Hassan II University of Casablanca, Morocco.)

We investigate L´evy processes with infinite jump intensity and study the approximation of compensated small jumps by a suitably scaled continuous stochastic process. We investigate the theoretical properties of small-jump approximations and study their convergence behavior in the framework of Lévy-driven stochastic models.