Authors: Alexandru Buium (ニューメキシコ大学), Adrian Vasiu (ビンガムトン大学)

DETAILS
EXPLANATIONS (Book Summary)

This advanced research monograph introduces a novel synthesis of arithmetic differential geometry and the classical theory of modular forms. Specifically, the text develops a comprehensive "$\delta$-invariant theory" (arithmetic jet spaces and $\delta$-geometry, pioneered by Buium) tailored to the study of Hecke correspondences on modular curves and related arithmetic varieties.

The authors systematically analyze how arithmetic differential operators interact with Hecke operators, uncovering new arithmetic invariants that remain stable under these correspondences. Designed for researchers and advanced graduate students in algebraic geometry, number theory, and arithmetic geometry, this work provides both the foundational framework and cutting-edge techniques for exploring the deeper structures of $p$-adic modular forms and arithmetic dynamics.

TABLE OF CONTENTS (Expected Chapters)

  1. Foundations of $\delta$-Arithmetic Geometry and Jet Spaces

  2. Review of Classical Hecke Correspondences on Modular Curves

  3. Construction of $\delta$-Invariants for Hecke Operators

  4. Arithmetic Differential Equations and Modular Forms

  5. Overconvergent $\delta$-Modular Forms and $p$-adic Invariants

  6. Direct and Inverse Images under Hecke Correspondences

  7. Local and Global Structure of $\delta$-Invariants

  8. Applications to Arithmetic Dynamics and Diophantine Geometry


    Editors (Authors): Mohamed Ait Mansour, Akhtar Khan, Hassan Riahi

    DETAILS
    EXPLANATIONS (Book Summary)

    This book revolves around minimax and equilibrium problems, optimization problems and set-optimization, optimal control problems, differential equations, and evolution problems. The text sufficiently covers the deterministic aspect as well as problems with the presence of parametric perturbations.

    An important part of the book is devoted to the problems of evolution, population dynamics, and parametric differential equations besides quasi-equilibrium problems. Here, the focus is not only on the quantitative stability of such problems under perturbation effects but also on the convergence over time of trajectories towards solutions of optimization problems through inertial dynamics.

    The book is composed of sixteen chapters written in a simple and constructive style with a view to offering young doctoral students a supporting and highly informative reference on current topics in the considered fields by underlining the key ideas of the main resolution methods with an emphasis on applications to different related fields.

    TABLE OF CONTENTS (Overview of Topics)

    本書は計16章（16 Chapters）で構成されており、主に以下のコアトピックに分かれています。

    1. Minimax and Equilibrium Problems (ミニマックス問題と平衡問題の存在性とアルゴリズム)

    2. Optimization and Set-Optimization (最適化および集合値最適化論)

    3. Optimal Control Problems (最適制御問題と決定論的アプローチ)

    4. Differential Equations and Evolution Problems (微分方程式と進化問題)

    5. Parametric Perturbations and Stability (パラメータ摂動下の定量的安定性解析)

    6. Population Dynamics and Trajectory Convergence (個体群動態および慣性力学による最適解への収束)

    7. Quasi-Equilibrium Systems (準平衡問題とその応用手


      Authors: Farzin Asadi (OSTIM Technical University, Ankara, Turkey)

      • A Problem-Solving Approach to Multi-Variable Calculus: Volume I

      DETAILS

      • Publisher: Springer Nature Switzerland / Springer

      • Publication Date: May – June 2026 (予定)

      • Language: English

      • Format: Hardcover / eBook

      • ISBN-13: 978-3-03221-437-9

      • ISBN-10: 3032214378

      EXPLANATIONS (Book Summary)

      Multi-variable calculus expands the study of change into higher-dimensional spaces, providing the essential language for modeling systems where multiple variables interact simultaneously. It evolves its core tools into the partial derivative and gradient, which map the direction and magnitude of change across surfaces, and the multiple integral, which calculates accumulation over volumes and vector fields. This framework is the vital bridge for students in physics, advanced engineering, and data science who must navigate the interconnected complexities of the real world.

      To truly master the concepts and techniques of calculus, practice is paramount. A Problem-Solving Approach to Multi-Variable Calculus is your essential guide, offering a comprehensive, three-volume set filled with plenty of meticulously solved, step-by-step problems designed to build your skills and deepen your understanding. This first volume empowers you to confidently tackle core multi-variable calculus challenges, transforming complex mathematical hurdles into clear triumphs.

      TABLE OF CONTENTS (Expected Outline for Volume I)

      本書は全3巻構成の第1巻（Volume I）にあたり、多変数微積分の基礎・前半部分（多次元空間の幾何学、ベクトル関数、および偏微分・勾配の基礎など）に焦点を当てた詳細なステップ・バイ・ステップの演習問題で構成されています。

      1. Frontmatter (Preface, Introduction to Higher-Dimensional Spaces)

      2. Vectors and the Geometry of Space (三次元空間の座標系、ベクトル、内積と外積、直線と平面の方程式、二次曲面の演習)

      3. Vector-Valued Functions (ベクトル関数と空間曲線、微分と積分、弧長と曲率、空間運動の軌跡)

      4. Partial Derivatives and Gradients (Introduction) (多変数関数、極限と連続性、偏微分の基礎、勾配ベクトル（Gradient）と方向微分の基本問題演習)

        *******************************************************************************************

        Carlos Cabrelli, Christopher Heil, Ursula Molter, Alexander M. Powell, Sui Tang

        DETAILS

        • Publisher: Birkhäuser / Springer Nature Switzerland

        • Series: Applied and Numerical Harmonic Analysis

        • Publication Date: July 2026 (Scheduled)

        • Language: English

        • Format: Hardcover / eBook

        • Pages: Approx. 418–424 pages

        • ISBN-13: 978-3-03223-232-8

        • ISBN-10: 3032232325

        EXPLANATIONS (Book Summary)

        This special contributed volume is compiled as a mathematical celebration dedicated to Akram Aldroubi on the occasion of his 65th birthday. Bringing together peer-reviewed papers and survey articles by leading experts in the field, the book highlights recent developments, modern techniques, and cutting-edge research across applied harmonic analysis.

        The core focus of the text spans sampling theory, frame theory, wavelet analysis, and their diverse mathematical intersections—including applications to dynamical sampling, signal processing, and structural learning theory. It serves as both an advanced reference for researchers and an insightful text for graduate students looking to explore the active frontiers of mathematical analysis and its implementations in data science.

        TABLE OF CONTENTS (Selected Contributions)

        While the full sequential chapter outline is heavily specialized, the volume features dedicated mathematical papers and comprehensive surveys, including:

        • Foundational Frameworks: Advanced topics in frame theory, Parseval frames, and non-uniform sampling geometries.

        • On Sylvester Equations in Banach Subalgebras – by Qiquan Fang, Chang Eon Shin, and Qiyu Sun.

        • Convolutional Dynamical Sampling and Some New Results – by Longxiu Huang, Martina Newman, and Yuying Xie.

        • Data-Driven and Structural Dynamics: Learning theory for inferring interaction kernels in second-order interacting agent systems, alongside mathematical signal processing paradigms.

          Snehashish Chakraverty, Sandeep Kumar Samota, Reema Gupta

          DETAILS

          • Publisher: Springer Nature Switzerland

          • Series: Synthesis Lectures on Mathematics & Statistics

          • Publication Date: July 2026 (Scheduled)

          • Language: English

          • Format: Hardcover / eBook

          • Pages: Approx. 138 pages

          • ISBN-13: 978-3-03222-573-3

          • ISBN-10: 3032225738

          EXPLANATIONS (Book Summary)

          This book provides a comprehensive overview of cutting-edge machine learning (ML) methods applied to solving computational differential equations. It primarily focuses on modern frameworks such as Physics-Informed Neural Networks (PINNs), Neural Operators, and other emerging ML methodologies designed to handle complex mathematical modeling.

          By bridging traditional computational mathematics with contemporary artificial intelligence, the text serves as a practical guide for utilizing advanced soft computing techniques to resolve differential equations that model real-world uncertainties. It is designed for researchers, data scientists, and advanced students in applied mathematics and engineering who are looking to integrate machine learning into structural dynamics and numerical analysis.

          TABLE OF CONTENTS (Selected Chapters)

          1. Introduction to Differential Equations (DEs) and Machine Learning

          2. Mathematical Preliminaries

          3. Machine Learning Basics for DEs

          4. Physics-Informed Neural Networks (PINNs) and Applications

          5. Advanced Neural Operators for Computational Methods

            Noémie C. Combe, Philippe G. Combe, Hanna K. Nencka

            DETAILS

            • Publisher: Birkhäuser / Springer Nature Switzerland

            • Series: Frontiers in Mathematics

            • Publication Date: June 2026 (Scheduled)

            • Language: English

            • Format: Softcover / eBook

            • Pages: Approx. 228 pages

            • ISBN-13: 978-3-03223-053-9

            • ISBN-10: 3032230535

            EXPLANATIONS (Book Summary)

            This monograph offers a modern exploration of information geometry, bridging classical statistical manifolds with cutting-edge concepts in mathematical physics. The text specifically highlights recent advances in the field and uncovers deep structural connections between informational geometric models and topological field theory.

            Designed as an accessible yet rigorous guide within the Frontiers in Mathematics series, the book serves as an important resource for graduate students and researchers in differential geometry, mathematical statistics, and theoretical physics who wish to understand the geometric formatting of probability spaces and their quantum or topological analogues.

            TABLE OF CONTENTS (Expected Outline)

            1. Foundations of Information Geometry and Statistical Manifolds

            2. Divergence Functions and Dual Affine Connections

            3. Geometric Structures on Spaces of Probability Measures

            4. Introduction to Topological Field Theory for Geometers

            5. Intersections: Information Metrics and Topological Invariants

            6. Recent Advances in Quantum Information Geometry

            7. Applications to Complex Systems and Statistical Mechanics


    Details
    Explanations & Summary

    This textbook provides a rigorous yet highly accessible introduction to foundational quantitative methods, specifically tailored for academic programs and professionals in economics.

    Key themes addressed in the book include:

    ️ Table of Contents (Outline)

    Because this is an upcoming 2026 release, the broad thematic chapters based on the publisher's roadmap include:

    1. Introduction to Set Theory and Mathematical Foundations

    2. Sequences, Series, and Limits

    3. Mathematical Analysis and Functions in Economics

    4. Optimization Theory and Economic Applications (Static and dynamic applications)

    5. Foundations of Probability Theory (Random variables, sample spaces, and uncertainty modeling)

    6. Classical Probability Distributions (Discrete and continuous models applied to markets)

    7. Mathematical Statistics and Statistical Inference

    8. Integrated Case Studies: Decision-Making Under Uncertainty

      Kathrin Hellmuth (Postdoctoral researcher at California Institute of Technology / Caltech)

      On Qualitative Experimental Design for PDE Parameter Identification:
      An Application to the Kinetic Chemotaxis Inverse Problem

      Details

      • Publisher: Springer Nature

      • Publication Date: Expected June 2026

      • Language: English

      • Format: Hardcover (approx. 233 pages)

      • Subject Categories: Mathematics, Mathematical Biology, Differential Equations, Inverse Problems

      Explanations & Abstract

      Experimental design is a foundational task in inverse problems that focuses on planning data collection to meet specific reconstruction goals. Since not all data provides the same value, choices regarding how measurements are taken or how a physical test system is built ultimately determine if the resulting dataset contains useful information for inference.

      This book presents alternative approaches to traditional optimal experimental design optimization frameworks. Rooted in sensitivity and identifiability analysis, these methods bridge the gap between theoretical analysis (input-to-output mapping) and practical, finite data applications. Because they search for a sufficient setup rather than an absolute optimal design, these strategies are qualitative in nature.

      • First Approach: Derives a finite experimental design by relaxing infinite experimental designs used in theoretical uniqueness proofs. This is specifically applied to an inverse problem in mathematical biology: reconstructing the mesoscopic chemotaxis tumbling parameter (which describes the directional changes of bacteria moving toward chemical stimuli) using macroscopic data of bacterial density.

      • Second Approach: Provides a more generally applicable framework assuming a predefined parametric model. It utilizes a matrix sketching algorithm from randomized linear algebra to formulate an importance sampling distribution, ensuring that the design's sensitivity to the target parameter is successfully preserved throughout data down-sampling.

      ️ Table of Contents

      1. Introduction

      2. Inverse Problems

      3. Identifiability Analysis

      4. Experimental Design

      5. The Inverse Problem for Chemotaxis

      6. Structural Identifiability

      7. Theory-based Experimental Design

      8. Numerical Experiments

      9. Experimental Design through Sampling

      10. Discussion

      11. Appendix


    Yao-Lin Jiang

    Professor Yao-Lin Jiang is a Full Professor with the School of Mathematics and Statistics at Xi'an Jiaotong University and is a distinguished Chang Jiang Professor of China. His core research fields include the numerical solution of partial differential equations, model order reduction, and neural network-driven numerical algorithms.

    Model Order Reduction Via Orthogonal Polynomial Approximation

    Details
    Explanations & Abstract

    This monograph presents a comprehensive, deep dive into modern model order reduction (MOR) techniques that are fundamentally rooted in orthogonal polynomials. High-dimensional dynamical models are notoriously computationally expensive to simulate or control; this text provides the mathematical frameworks needed to simplify them into lower-dimensional forms without losing essential systemic characteristics.

    The book covers diverse and complex dynamical structures, including:

    By integrating general and specific continuous polynomials (such as Laguerre and Chebyshev) alongside discrete orthogonal polynomials, the author introduces highly innovative frameworks. Key focus areas include structure-preserving reductions, computational efficiency, and robust algorithm behavior across a spectrum of industrial and mathematical engineering systems.

    ️ Table of Contents