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6974 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 144 of 349

  • CPC volume 22 issue 4 Cover and Back matter

  • Journal:
    Combinatorics, Probability and Computing / Volume 22 / Issue 4 / July 2013
    Published online by Cambridge University Press:
    11 June 2013, pp. b1-b3
    • Article
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  • CPC volume 22 issue 4 Cover and Front matter

  • Journal:
    Combinatorics, Probability and Computing / Volume 22 / Issue 4 / July 2013
    Published online by Cambridge University Press:
    11 June 2013, pp. f1-f2
    • Article
      • You have access
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  • Freiman Homomorphisms of Random Subsets of $\mathbb{Z}_{N}$

  • GONZALO FIZ PONTIVEROS
  • Journal:
    Combinatorics, Probability and Computing / Volume 22 / Issue 4 / July 2013
    Published online by Cambridge University Press:
    11 June 2013, pp. 592-611

  • Given a set $A\subset\mathbb{Z}_{N}$, we say that a function $f\colon A \to \mathbb{Z}_{N}$ is a Freiman homomorphism if f(a)+f(b)=f(c)+f(d) whenever a,b,c,d ∈ A satisfy a+b=c+d. This notion was introduced by Freiman in the 1970s, and plays an important role in the field of additive combinatorics. We say that A is linear if the only Freiman homomorphisms are functions of the form f(x) = ax+b.

    Suppose the elements of A are chosen independently at random, each with probability p. We shall look at the following question: For which values of p=p(N) is A linear with high probability as N → ∞? We show that if p=(2logN − ω(N))1/3N−2/3, where ω(N) → ∞ as N → ∞, then A is not linear with high probability, whereas if p=N−1/2+ε for any ε>0 then A is linear with high probability.

  • Lipschitz Functions on Expanders are Typically Flat

  • RON PELED, WOJCIECH SAMOTIJ, AMIR YEHUDAYOFF
  • Journal:
    Combinatorics, Probability and Computing / Volume 22 / Issue 4 / July 2013
    Published online by Cambridge University Press:
    11 June 2013, pp. 566-591

  • This work studies the typical behaviour of random integer-valued Lipschitz functions on expander graphs with sufficiently good expansion. We consider two families of functions: M-Lipschitz functions (functions which change by at most M along edges) and integer-homomorphisms (functions which change by exactly 1 along edges). We prove that such functions typically exhibit very small fluctuations. For instance, we show that a uniformly chosen M-Lipschitz function takes only M+1 values on most of the graph, with a double exponential decay for the probability of taking other values.

Boolean Models and Methods in Mathematics, Computer Science, and Engineering
  • Edited by Yves Crama, Peter L. Hammer
  • Published online:
    05 June 2013
    Print publication:
    28 June 2010
  • This collection of papers presents a series of in-depth examinations of a variety of advanced topics related to Boolean functions and expressions. The chapters are written by some of the most prominent experts in their respective fields and cover topics ranging from algebra and propositional logic to learning theory, cryptography, computational complexity, electrical engineering, and reliability theory. Beyond the diversity of the questions raised and investigated in different chapters, a remarkable feature of the collection is the common thread created by the fundamental language, concepts, models, and tools provided by Boolean theory. Many readers will be surprised to discover the countless links between seemingly remote topics discussed in various chapters of the book. This text will help them draw on such connections to further their understanding of their own scientific discipline and to explore new avenues for research.

Solving Polynomial Equation Systems II
  • Macaulay's Paradigm and Gröbner Technology
  • Teo Mora
  • Published online:
    05 June 2013
    Print publication:
    05 May 2005
  • The second volume of this comprehensive treatise focusses on Buchberger theory and its application to the algorithmic view of commutative algebra. In distinction to other works, the presentation here is based on the intrinsic linear algebra structure of Groebner bases, and thus elementary considerations lead easily to the state-of-the-art in issues of implementation. The same language describes the applications of Groebner technology to the central problems of commutative algebra. The book can be also used as a reference on elementary ideal theory and a source for the state-of-the-art in its algorithmization. Aiming to provide a complete survey on Groebner bases and their applications, the author also includes advanced aspects of Buchberger theory, such as the complexity of the algorithm, Galligo's theorem, the optimality of degrevlex, the Gianni-Kalkbrener theorem, the FGLM algorithm, and so on. Thus it will be essential for all workers in commutative algebra, computational algebra and algebraic geometry.

  • 0 - Introduction

  • Andrew M. Pitts, University of Cambridge
  • Book:
    Nominal Sets
    Published online:
    05 July 2013
    Print publication:
    30 May 2013, pp 1-10

  • Summary

    This is a book about names and symmetry in the part of computer science that has to do with programming languages. Although symmetry plays an important role in many branches of mathematics and physics, its relevance to computer science may not be so clear to the reader. This introduction explains the computer science motivation for a theory of names based upon symmetry and provides a guide to what follows.

    Atomic names

    Names are used in many different ways in computer systems and in the formal languages used to describe and construct them. This book is exclusively concerned with what Needham calls ‘pure names’:

    A pure name is nothing but a bit-pattern that is an identifier, and is only useful for comparing for identity with other such bit-patterns – which includes looking up in tables to find other information. The intended contrast is with names which yield information by examination of the names themselves, whether by reading the text of the name or otherwise. […] like most good things in computer science, pure names help by putting in an extra stage of indirection; but they are not much good for anything else.

    (Needham, 1989, p. 90)

    We prefer to use the adjective ‘atomic’ rather than ‘pure’, because for this kind of name, internal structure is irrelevant; their only relevant attribute is their identity. Although such names may not be much good for anything other than indirection, that one thing is a hugely important and very characteristic aspect of computer science.

Page 144 of 349