665,541 research outputs found
Algebraic entropy for algebraic maps
We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations
Fast Algebraic Attacks and Decomposition of Symmetric Boolean Functions
Algebraic and fast algebraic attacks are power tools to analyze stream
ciphers. A class of symmetric Boolean functions with maximum algebraic immunity
were found vulnerable to fast algebraic attacks at EUROCRYPT'06. Recently, the
notion of AAR (algebraic attack resistant) functions was introduced as a
unified measure of protection against both classical algebraic and fast
algebraic attacks. In this correspondence, we first give a decomposition of
symmetric Boolean functions, then we show that almost all symmetric Boolean
functions, including these functions with good algebraic immunity, behave badly
against fast algebraic attacks, and we also prove that no symmetric Boolean
functions are AAR functions. Besides, we improve the relations between
algebraic degree and algebraic immunity of symmetric Boolean functions.Comment: 13 pages, submitted to IEEE Transactions on Information Theor
Almost algebraic actions of algebraic groups and applications to algebraic representations
Let G be an algebraic group over a complete separable valued field k. We
discuss the dynamics of the G-action on spaces of probability measures on
algebraic G-varieties. We show that the stabilizers of measures are almost
algebraic and the orbits are separated by open invariant sets. We discuss
various applications, including existence results for algebraic representations
of amenable ergodic actions. The latter provides an essential technical step in
the recent generalization of Margulis-Zimmer super-rigidity phenomenon due to
Bader and Furman.Comment: Correction of a small mistake in Proposition 5.
Algebraic models for higher categories
We introduce the notion of algebraic fibrant objects in a general model
category and establish a (combinatorial) model category structure on algebraic
fibrant objects. Based on this construction we propose algebraic Kan complexes
as an algebraic model for oo-groupoids and algebraic quasi-categories as an
algebraic model for (oo,1)-categories. We furthermore give an explicit proof of
the homotopy hypothesis.Comment: 23 pages, minor change
Approximation of complex algebraic numbers by algebraic numbers of bounded degree
We investigate how well complex algebraic numbers can be approximated by
algebraic numbers of degree at most n. We also investigate how well complex
algebraic numbers can be approximated by algebraic integers of degree at most
n+1. It follows from our investigations that for every positive integer n there
are complex algebraic numbers of degree larger than n that are better
approximable by algebraic numbers of degree at most n than almost all complex
numbers. As it turns out, these numbers are more badly approximable by
algebraic integers of degree at most n+1 than almost all complex numbers.Comment: 34 page
Algebraic geometry over algebraic structures II: Foundations
In this paper we introduce elements of algebraic geometry over an arbitrary
algebraic structure. We prove Unification Theorems which gather the description
of coordinate algebras by several ways.Comment: 55 page
The algebraic hyperstructure of elementary particles in physical theory
Algebraic hyperstructures represent a natural extension of classical
algebraic structures. In a classical algebraic structure, the composition of
two elements is an element, while in an algebraic hyperstructure, the
composition of two elements is a set. Algebraic hyperstructure theory has a
multiplicity of applications to other disciplines. The main purpose of this
paper is to provide examples of hyperstructures associated with elementary
particles in physical theory.Comment: 13 page
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