Skip to main content Accessibility help
×
  1. Home
  2. Browse subjects
  3. Physics and Astronomy
  4. Statistical Physics
  5. Content listing

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

2192 results in Statistical Physics

Page 21 of 110

  • 7 - Transforming Graphs

  • from Part II - Applications
  • Michael Rudolph
  • Book:
    The Mathematics of Finite Networks
    Published online:
    30 April 2022
    Print publication:
    12 May 2022, pp 291-324

  • Summary

    Navigating across sometimes treacherous waters, we demonstrated in the previous chapter the construction and utilisation of a small selection of graph observables for measuring various properties of finite random graphs. Together with the generation of operator representations for arbitrary graph models, we should now be in possession of a sufficiently equipped toolset with which to further explore and characterise on rigorous algebraical grounds the plethora of graph models in the applied graph-theoretical literature. However, our adventurous journey would not be complete without touching upon another crucial aspect exhibited by many real-world networks - their dynamic nature. In this final chapter, we will explore with one hopefully light-hearted, playful example - the game of chess - how to formulate such dynamical aspects in our operator graph-theoretical language. As we will witness here, the construction of graphs that describe possible moves of chess pieces at any position during a game, and the transformations that lead to changes of such positional chess graphs, pose a formidable challenge for not only computational algorithms.

  • 5 - Generating Graphs

  • from Part II - Applications
  • Michael Rudolph
  • Book:
    The Mathematics of Finite Networks
    Published online:
    30 April 2022
    Print publication:
    12 May 2022, pp 181-237

  • Summary

    The first part of this book led us on a journey from one of the undoubtedly most cherished fields of applied mathematics, classical graph theory, across the ghastly depths of an inherently dynamic formalisation of physical reality in terms of mappings and operators, to an inspired attempt at a fusion of both of these perspectives. With the backing of a conceptional and notational framework at hand, it is now time to put this attempt to a test, as we continue our adventurous journey with an excursion into the endless realm of applications. This second part of our journey will start in this chapter with an exploration of graph generators and their operator graph-theoretical formulation. In this undertaking, we will focus primarily on the generation of random graphs as such models enjoy, in one way or another, widespread and prominent employment throughout almost all fields of science and technology. Only the last section will see the exemplary generation of an exact graph model, the finite square grid graph, as preparation for a closer inspection of an intriguing yet unsolved problem at the very heart of condensed matter physics in the next chapter.

The Principles of Deep Learning Theory
  • An Effective Theory Approach to Understanding Neural Networks
  • Daniel A. Roberts, Sho Yaida
  • With contributions by Boris Hanin
  • Published online:
    05 May 2022
    Print publication:
    26 May 2022
  • This textbook establishes a theoretical framework for understanding deep learning models of practical relevance. With an approach that borrows from theoretical physics, Roberts and Yaida provide clear and pedagogical explanations of how realistic deep neural networks actually work. To make results from the theoretical forefront accessible, the authors eschew the subject's traditional emphasis on intimidating formality without sacrificing accuracy. Straightforward and approachable, this volume balances detailed first-principle derivations of novel results with insight and intuition for theorists and practitioners alike. This self-contained textbook is ideal for students and researchers interested in artificial intelligence with minimal prerequisites of linear algebra, calculus, and informal probability theory, and it can easily fill a semester-long course on deep learning theory. For the first time, the exciting practical advances in modern artificial intelligence capabilities can be matched with a set of effective principles, providing a timeless blueprint for theoretical research in deep learning.

The Mathematics of Finite Networks
  • An Introduction to Operator Graph Theory
  • Michael Rudolph
  • Published online:
    30 April 2022
    Print publication:
    12 May 2022
  • Since the early eighteenth century, the theory of networks and graphs has matured into an indispensable tool for describing countless real-world phenomena. However, the study of large-scale features of a network often requires unrealistic limits, such as taking the network size to infinity or assuming a continuum. These asymptotic and analytic approaches can significantly diverge from real or simulated networks when applied at the finite scales of real-world applications. This book offers an approach to overcoming these limitations by introducing operator graph theory, an exact, non-asymptotic set of tools combining graph theory with operator calculus. The book is intended for mathematicians, physicists, and other scientists interested in discrete finite systems and their graph-theoretical description, and in delineating the abstract algebraic structures that characterise such systems. All the necessary background on graph theory and operator calculus is included for readers to understand the potential applications of operator graph theory.

Page 21 of 110