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34925 results in Mathematics (general)

Page 13 of 1747

Essays in Classical Number Theory
  • Yoichi Motohashi
  • Published online:
    12 August 2025
    Print publication:
    28 August 2025
  • Offering a comprehensive introduction to number theory, this is the ideal book both for those who want to learn the subject seriously and independently, or for those already working in number theory who want to deepen their expertise. Readers will be treated to a rich experience, developing the key theoretical ideas while explicitly solving arithmetic problems, with the historical background of analytic and algebraic number theory woven throughout. Topics include methods of solving binomial congruences, a clear account of the quantum factorization of integers, and methods of explicitly representing integers by quadratic forms over integers. In the later parts of the book, the author provides a thorough approach towards composition and genera of quadratic forms, as well as the essentials for detecting bounded gaps between prime numbers that occur infinitely often.

  • 5 - Advanced Miscellany

  • Peter J. Cameron, University of St Andrews, Pierre-Philippe Dechant, University of Leeds, Yang-Hui He, London Institute for Mathematical Sciences, John McKay, Concordia University, Montréal
  • Book:
    ADE
    Published online:
    01 August 2025
    Print publication:
    07 August 2025, pp 122-164

  • Summary

    We treat some more advanced topics: monstrous (and other) moonshine, Monster and E_8, Niemeier lattices, the triangle property, generalized line graphs, quiver representations, cluster algebras, von Neumann algebras, catastrophes, Calabi–Yau, elliptic fibrations.

  • 3 - ADE Classifications

  • Peter J. Cameron, University of St Andrews, Pierre-Philippe Dechant, University of Leeds, Yang-Hui He, London Institute for Mathematical Sciences, John McKay, Concordia University, Montréal
  • Book:
    ADE
    Published online:
    01 August 2025
    Print publication:
    07 August 2025, pp 40-92

  • Summary

    We discuss some areas where the ADE classification arises: polytopes, tessellations, root systems, Coxeter groups, spectra of graphs, binary polyhedral groups, reflections, Clifford algebras, Lie groups and algebras.

ADE
  • Patterns in Mathematics
  • Peter J. Cameron, Pierre-Philippe Dechant, Yang-Hui He, John McKay
  • Published online:
    01 August 2025
    Print publication:
    07 August 2025
  • The ADE diagrams, shown on the cover, constitute one of the most universal and mysterious patterns in all of mathematics. John McKay's remarkable insights unveiled a connection between the 'double covers' of the groups of regular polyhedra, known since ancient Greek times, and the exceptional Lie algebras, recognised since the late nineteenth century. The correspondence involves the ADE diagrams being interpreted in different ways: as quivers associated with the groups and as Dynkin diagrams of root systems of Lie algebras. The ADE diagrams arise in many areas of mathematics, including topics in algebraic geometry, string theory, spectral theory of graphs and cluster algebras. Accessible to students, this book explains these connections with exercises and examples throughout. An excellent introduction for students and researchers wishing to learn more about this unifying principle of mathematics, it also presents standard undergraduate material from a novel perspective.

  • Chapter 10 - Pattern systems and the brain

  • Robert V. Moody, University of Alberta, Ming-Dao Deng
  • Book:
    The Pattern of Change
    Published online:
    17 June 2025
    Print publication:
    03 July 2025, pp 322-355

  • Summary

    One of life’s most fundamental revelations is change. Presenting the fascinating view that pattern is the manifestation of change, this unique book explores the science, mathematics, and philosophy of change and the ways in which they have come to inform our understanding of the world. Through discussions on chance and determinism, symmetry and invariance, information and entropy, quantum theory and paradox, the authors trace the history of science and bridge the gaps between mathematical, physical, and philosophical perspectives. Change as a foundational concept is deeply rooted in ancient Chinese thought, and this perspective is integrated into the narrative throughout, providing philosophical counterpoints to customary Western thought. Ultimately, this is a book about ideas. Intended for a wide audience, not so much as a book of answers, but rather an introduction to new ways of viewing the world.

  • Chapter 15 - Afterword

  • Robert V. Moody, University of Alberta, Ming-Dao Deng
  • Book:
    The Pattern of Change
    Published online:
    17 June 2025
    Print publication:
    03 July 2025, pp 552-561

  • Summary

    One of life’s most fundamental revelations is change. Presenting the fascinating view that pattern is the manifestation of change, this unique book explores the science, mathematics, and philosophy of change and the ways in which they have come to inform our understanding of the world. Through discussions on chance and determinism, symmetry and invariance, information and entropy, quantum theory and paradox, the authors trace the history of science and bridge the gaps between mathematical, physical, and philosophical perspectives. Change as a foundational concept is deeply rooted in ancient Chinese thought, and this perspective is integrated into the narrative throughout, providing philosophical counterpoints to customary Western thought. Ultimately, this is a book about ideas. Intended for a wide audience, not so much as a book of answers, but rather an introduction to new ways of viewing the world.

Page 13 of 1747