274,405 research outputs found
On the Derivative Imbalance and Ambiguity of Functions
In 2007, Carlet and Ding introduced two parameters, denoted by and
, quantifying respectively the balancedness of general functions
between finite Abelian groups and the (global) balancedness of their
derivatives , (providing an
indicator of the nonlinearity of the functions). These authors studied the
properties and cryptographic significance of these two measures. They provided
for S-boxes inequalities relating the nonlinearity to ,
and obtained in particular an upper bound on the nonlinearity which unifies
Sidelnikov-Chabaud-Vaudenay's bound and the covering radius bound. At the
Workshop WCC 2009 and in its postproceedings in 2011, a further study of these
parameters was made; in particular, the first parameter was applied to the
functions where is affine, providing more nonlinearity parameters.
In 2010, motivated by the study of Costas arrays, two parameters called
ambiguity and deficiency were introduced by Panario \emph{et al.} for
permutations over finite Abelian groups to measure the injectivity and
surjectivity of the derivatives respectively. These authors also studied some
fundamental properties and cryptographic significance of these two measures.
Further studies followed without that the second pair of parameters be compared
to the first one.
In the present paper, we observe that ambiguity is the same parameter as
, up to additive and multiplicative constants (i.e. up to rescaling). We
make the necessary work of comparison and unification of the results on ,
respectively on ambiguity, which have been obtained in the five papers devoted
to these parameters. We generalize some known results to any Abelian groups and
we more importantly derive many new results on these parameters
On algebraic relations between solutions of a generic Painleve equation
We prove that if y" = f(y,y',t,\alpha, \beta,..) is a generic Painleve
equation (i.e. an equation in one of the families PI-PVI but with the complex
parameters \alpha, \beta,.. algebraically independent) then any algebraic
dependence over C(t) between a set of solutions and their derivatives
(y_1,..,y_n,y_1',..,y_n') is witnessed by a pair of solutions and their
derivatives (y_i,y_i',y_j,y_j'). The proof combines work by the Japanese school
on "irreducibility" of the Painleve equations, with the trichomoty theorem for
strongly minimal sets in differentially closed fields.Comment: 23 page
Examples of derivation-based differential calculi related to noncommutative gauge theories
Some derivation-based differential calculi which have been used to construct
models of noncommutative gauge theories are presented and commented. Some
comparisons between them are made.Comment: 22 pages, conference given at the "International Workshop in honour
of Michel Dubois-Violette, Differential Geometry, Noncommutative Geometry,
Homology and Fundamental Interactions". To appear in a special issue of
International Journal of Geometric Methods in Modern Physic
Aspects of p-adic non-linear functional analysis
The article provides an introduction to infinite-dimensional differential
calculus over topological fields and surveys some of its applications, notably
in the areas of infinite-dimensional Lie groups and dynamical systems.Comment: 19 pages; LaTe
Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups
We present a Galois theory of parameterized linear differential equations
where the Galois groups are linear differential algebraic groups, that is,
groups of matrices whose entries are functions of the parameters and satisfy a
set of differential equations with respect to these parameters. We present the
basic constructions and results, give examples, discuss how isomonodromic
families fit into this theory and show how results from the theory of linear
differential algebraic groups may be used to classify systems of second order
linear differential equations
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