M. MAHDI TAHAR Brahimi

Abstract

This work explores Lebesgue and Sobolev spaces with variable exponents, where the integrability parameter p(x) varies spatially across the domain Omega. We define the modular function, the Luxemburg norm, and the structure of L^{p(x)} and W^{k,p(x)} spaces, emphasizing completeness, reflexivity, and separability under suitable assumptions. Key inequalities and embedding results are presented, including Holder-type estimates and Sobolev embeddings under log-Holder continuity. Applications to nonlinear elliptic and parabolic partial differential equations are discussed, focusing on the p(x)-Laplace and p(x)-heat equations. We analyze weak formulations, energy functionals, eigenvalue problems, and long-time behavior. The framework models phenomena such as electrorheological fluids, adaptive image processing, and biological diffusion. Recent developments include fractional operators, stochastic PDEs, and numerical methods for non-standard growth conditions.

This article develops the modular resolution of equations of the form
H_t^2(x) - H_t(x) + t ≡ 0 (mod t^n) using the Polyserie framework introduced
by Wildberger and Rubine. The work establishes an equivalence theorem for the
existence and uniqueness of parametric solutions. Catalan-number structures,
binomial identities, and truncated polyseries play a central role in the
construction and proof strategy.

Session N……1………/ Date: ………4/11/2025…..

Student’s Name: Brahimi, Mahdi. Tahar.
Level: B2
Date/Place of Birth: 6 / 7 / 1972
Faculty: Mathematics and Computer Sciences…
Department: Mathematics……………………………………………. Institution: Msila University…………………
Presentation Summary Document
1. Research Question / Focus of the Presentation
How has higher education learning evolved from early communication networks to AI-enhanced, hybrid models between 1861 and 2025, and what are the main trends shaping future teaching and credentialing?
2. Main Points / Key Arguments
• Foundations: Early communication technologies and packet networks created the technical basis for remote teaching and information exchange.
• Platformization: From 2000 to 2020 online platforms, learning management systems (e.g., Moodle), and MOOCs scaled access and changed course delivery.
• Recent shift: Since 2020 the pandemic accelerated hybrid teaching and rapid adoption of AI tools, microcredentials, and competency-based pathways.

3. Evidence and Examples Used
• Telegraph and early networking research established long-distance signalling and packet switching as groundwork for digital communication.
• ARPANET and email demonstrated remote collaboration and asynchronous learning practices.
• Moodle, major LMS deployments, and early MOOCs show how platforms enabled scalable online courses.
• COVID-19 emergency remote teaching in 2020 prompted widespread hybrid classroom adoption and rapid tooling improvements.
• From 2020–2025, examples include adaptive tutoring systems, automated feedback tools, and the rise of microcredentials and digital badges used by universities and employers.
4. Conclusion / Key Insights
Higher education moved from network experiments to platform-driven online learning and now to hybrid, AI-supported models that emphasize flexible, skills-focused pathways and clear digital proof of skills. Future priorities should be fair access, simple and verifiable credentials, and AI that supports educators without replacing academic standards.
5. References (APA Style)
□ Croxford, L., & Perez, M. (2018). Digital networks and education: A history of remote learning foundations. Academic Press.
□ Smith, J. R., & Lee, A. (2022). From MOOCs to microcredentials: The changing landscape of higher education. Journal of Online Learning Studies, 12(3), 112–130