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91 results

Page 1 of 5

  • Recent Developments in Material Microstructure: a Theory of Coarsening

  • K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer, R. Sharp, S. Ta’asan
  • Journal:
    MRS Online Proceedings Library Archive / Volume 1753 / 2015
    Published online by Cambridge University Press:
    11 June 2015, mrsf14-1753-nn01-01
    Print publication:
    2015

  • Cellular networks are ubiquitous in nature. Most engineered materials are polycrystalline microstructures composed of a myriad of small grains separated by grain boundaries, thus comprising cellular networks. The recently discovered grain boundary character distribution (GBCD) is an empirical distribution of the relative length (in 2D) or area (in 3D) of interface with a given lattice misorientation and normal. During the coarsening, or growth, process, an initially random grain boundary arrangement reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, if the given energy density depends only on lattice misorientation, then the steady state GBCD and the energy are related by a Boltzmann distribution. This is among the simplest non-random distributions, corresponding to independent trials with respect to the energy. Why does such simplicity emerge from such complexity? Here we describe an entropy based theory which suggests that the evolution of the GBCD satisfies a Fokker-Planck Equation, an equation whose stationary state is a Boltzmann distribution.

Commutative Algebra
  • Durham 1981
  • Edited by R. Y. Sharp
  • Published online:
    05 May 2013
    Print publication:
    27 January 1983
  • This book is concerned with the research conducted in the late 1970s and early 1980s in the theory of commutative Neotherian rings. It consists of articles by invited speakers at the Symposium of Commutative Algebra held at the University of Durham in July 1981; these articles are all based on lectures delivered at the Symposium. The purpose of this book is to provide a record of at least some aspects of the Symposium, which several of the world leaders in the field attended. Several articles are included which provide surveys, incorporating historical perspective, details of progress made and indications of possible future lines of investigation. The book will be of interest to scholars of commutative and local algebra.

Local Cohomology
  • An Algebraic Introduction with Geometric Applications
  • 2nd edition
  • M. P. Brodmann, R. Y. Sharp
  • Published online:
    05 December 2012
    Print publication:
    15 November 2012
  • This second edition of a successful graduate text provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, including in multi-graded situations, and provides many illustrations of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Serre's Affineness Criterion, the Lichtenbaum–Hartshorne Vanishing Theorem, Grothendieck's Finiteness Theorem and Faltings' Annihilator Theorem, local duality and canonical modules, the Fulton–Hansen Connectedness Theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. The book is designed for graduate students who have some experience of basic commutative algebra and homological algebra and also experts in commutative algebra and algebraic geometry. Over 300 exercises are interspersed among the text; these range in difficulty from routine to challenging, and hints are provided for some of the more difficult ones.

  • 18 - Applications to reductions of ideals

  • M. P. Brodmann, Universität Zürich, R. Y. Sharp, University of Sheffield
  • Book:
    Local Cohomology
    Published online:
    05 December 2012
    Print publication:
    15 November 2012, pp 388-404

  • Summary

    Graded local cohomology theory has played a substantial rôle in the study of Rees rings and associated graded rings of proper ideals in local rings. We do not have enough space in this book to include all we would like about the applications of local cohomology to this area, and so we have decided to select a small portion of the theory which gives some idea of the flavour. The part we have chosen to present in this chapter concerns links between the theory of reductions of ideals in local rings and the concept of Castelnuovo regularity, discussed in Chapter 16. The highlight will be a theorem of L. T. Hoa which asserts that, if b is a proper ideal in a local ring having infinite residue field, then there exist t0 ∈ ℕ and c ∈ ℕ0 such that, for all t > t0 and every minimal reduction a of bt, the reduction number ra(bt) of bt with respect to a is equal to c. This statement of Hoa's Theorem is satisfyingly simple, and makes no mention of local cohomology, and yet Hoa's proof, which we present towards the end of this chapter, makes significant use of graded local cohomology.

    Throughout this chapter, all graded rings and modules are to be understood to be ℤ-graded, and all polynomial rings R[X1, …, Xt] (and R[T]) over R are to be understood to be (positively) ℤ-graded so that each indeterminate has degree 1 and deg a = 0 for all a ∈ R \ {0}.

  • 20 - Links with sheaf cohomology

  • M. P. Brodmann, Universität Zürich, R. Y. Sharp, University of Sheffield
  • Book:
    Local Cohomology
    Published online:
    05 December 2012
    Print publication:
    15 November 2012, pp 438-479

  • Summary

    In this last chapter we shall develop the links between local cohomology and the cohomology of quasi-coherent sheaves over certain Noetherian schemes. Here we shall assume for the first time that the reader has some basic knowledge about schemes and sheaves: our reference for these topics is Hartshorne's book [30]. The central idea in this chapter is to extend our earlier relations in 2.3.2 and 15.1.3 between ideal transforms and rings of regular functions on varieties to quasi-coherent sheaves over certain Noetherian schemes. We shall be very concerned with a generalization of the ‘Deligne Isomorphism’ (see [30, Chapter III, Exercise 3.7, p. 217]) which links the group of sections (over an open subset) of an induced sheaf on an affine scheme with an ideal transform. More precisely, let denote the sheaf induced by an R-module M on the affine scheme Spec(R), and let U = Spec(R) \ Var(a), where a ⊂ R; then the group of sections Γ (U,) is isomorphic to the ideal transform Da(M). We shall use standard techniques involving negative strongly connected sequences of functors to extend this Deligne Isomorphism, and our generalization of it, to produce the Deligne Correspondence 20.3.11. This correspondence provides connections between higher cohomology groups of induced sheaves on the one hand, and local cohomology modules on the other.

    We shall also examine the case when R is graded in some detail. Here the central result for us is the Serre–Grothendieck Correspondence 20.3.15, which we shall also derive from the Deligne Isomorphism by standard ‘connected sequence’ arguments.

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