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5 results

Fractional Sobolev Spaces and Inequalities
  • D. E. Edmunds, W. D. Evans
  • Published online:
    06 October 2022
    Print publication:
    13 October 2022
  • The fractional Sobolev spaces studied in the book were introduced in the 1950s by Aronszajn, Gagliardo and Slobodeckij in an attempt to fill the gaps between the classical Sobolev spaces. They provide a natural home for solutions of a vast, and rapidly growing, number of questions involving differential equations and non-local effects, ranging from financial modelling to ultra-relativistic quantum mechanics, emphasising the need to be familiar with their fundamental properties and associated techniques. Following an account of the most basic properties of the fractional spaces, two celebrated inequalities, those of Hardy and Rellich, are discussed, first in classical format (for which a survey of the very extensive known results is given), and then in fractional versions. This book will be an Ideal resource for researchers and graduate students working on differential operators and boundary value problems.

  • 9 - Ideals and Prime Factorization

  • John Stillwell, University of San Francisco
  • Book:
    Algebraic Number Theory for Beginners
    Published online:
    28 July 2022
    Print publication:
    11 August 2022, pp 189-210

  • Summary

    The remaining step on the road to unique prime ideal factorization is to define "prime ideal" itself. This involves a definition of "division" for ideals. If we also define "product" of ideals, then a prime ideal is one with a property originally discovered by Euclid: if a prime p divides a product ab, then p divides a or p divides b. We then have all the ingredients needed for the definition of Dedekind domain: a Noetherian, integrally closed ring in which all prime ideals are maximal. And the main theorem follows: in a Dedekind domain, each ideal is a unique product of prime ideals.