Edgard A. Pimentel – författare

Presenting the basics of elliptic PDEs in connection with regularity theory, the book bridges fundamental breakthroughs - such as the Krylov-Safonov and Evans-Krylov results, Caffarelli's regularity theory, and the counterexamples due to Nadirashvili and Vlăduţ - and modern developments, including improved regularity for flat solutions and the partial regularity result. After presenting this general panorama, accounting for the subtleties surrounding C-viscosity and Lp-viscosity solutions, the book examines important models through approximation methods. The analysis continues with the asymptotic approach, based on the recession operator. After that, approximation techniques produce a regularity theory for the Isaacs equation, in Sobolev and Hölder spaces. Although the Isaacs operator lacks convexity, approximation methods are capable of producing Hölder continuity for the Hessian of the solutions by connecting the problem with a Bellman equation. To complete the book, degenerate models are studied and their optimal regularity is described.

Presenting the basics of elliptic PDEs in connection with regularity theory, the book bridges fundamental breakthroughs – such as the Krylov–Safonov and Evans–Krylov results, Caffarelli''s regularity theory, and the counterexamples due to Nadirashvili and Vlăduţ – and modern developments, including improved regularity for flat solutions and the partial regularity result. After presenting this general panorama, accounting for the subtleties surrounding C-viscosity and Lp-viscosity solutions, the book examines important models through approximation methods. The analysis continues with the asymptotic approach, based on the recession operator. After that, approximation techniques produce a regularity theory for the Isaacs equation, in Sobolev and Hölder spaces. Although the Isaacs operator lacks convexity, approximation methods are capable of producing Hölder continuity for the Hessian of the solutions by connecting the problem with a Bellman equation. To complete the book, degenerate models are studied and their optimal regularity is described.

This book is an exploration of differential and algorithmic intelligent game theory, offering a comprehensive look at its methods and applications across diverse fields. As the first installment in a pioneering series, it underscores the expanding influence of game theory in the physical, biological, social, and computational sciences, as well as engineering and technology. Readers will delve into dynamic strategic interactions, exploring continuous, discrete, and quantum strategies. The volume addresses computational challenges in finding equilibria and presents efficient algorithms for computing Nash equilibria. It also highlights the role of game theory in artificial intelligence, examining its use in AI-assisted behavioral and evolutionary game theory, multi-agent systems, and adversary training in generative adversarial networks. This book is essential for researchers, graduate students, and professionals in mathematics, computer science, and economics who are interested in applying game theory to analyze and design strategic systems. Offering a unique interdisciplinary perspective, combining differential game theory's continuous decision-making with algorithmic methodologies, this volume sets the stage for future installments, providing real-world case studies and applications that illustrate the practical use of game theory in dynamic, complex environments

This book is an exploration of differential and algorithmic intelligent game theory, offering a comprehensive look at its methods and applications across diverse fields. As the first installment in a pioneering series, it underscores the expanding influence of game theory in the physical, biological, social, and computational sciences, as well as engineering and technology. Readers will delve into dynamic strategic interactions, exploring continuous, discrete, and quantum strategies. The volume addresses computational challenges in finding equilibria and presents efficient algorithms for computing Nash equilibria. It also highlights the role of game theory in artificial intelligence, examining its use in AI-assisted behavioral and evolutionary game theory, multi-agent systems, and adversary training in generative adversarial networks. This book is essential for researchers, graduate students, and professionals in mathematics, computer science, and economics who are interested in applying game theory to analyze and design strategic systems. Offering a unique interdisciplinary perspective, combining differential game theory's continuous decision-making with algorithmic methodologies, this volume sets the stage for future installments, providing real-world case studies and applications that illustrate the practical use of game theory in dynamic, complex environments

Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields.