210,167 research outputs found
Semifields, relative difference sets, and bent functions
Recently, the interest in semifields has increased due to the discovery of
several new families and progress in the classification problem. Commutative
semifields play an important role since they are equivalent to certain planar
functions (in the case of odd characteristic) and to modified planar functions
in even characteristic. Similarly, commutative semifields are equivalent to
relative difference sets. The goal of this survey is to describe the connection
between these concepts. Moreover, we shall discuss power mappings that are
planar and consider component functions of planar mappings, which may be also
viewed as projections of relative difference sets. It turns out that the
component functions in the even characteristic case are related to negabent
functions as well as to -valued bent functions.Comment: Survey paper for the RICAM workshop "Emerging applications of finite
fields", 09-13 December 2013, Linz, Austria. This article will appear in the
proceedings volume for this workshop, published as part of the "Radon Series
on Computational and Applied Mathematics" by DeGruyte
Determinantal random point fields
The paper contains an exposition of recent as well as old enough results on
determinantal random point fields. We start with some general theorems
including the proofs of the necessary and sufficient condition for the
existence of the determinantal random point field with Hermitian kernel and a
criterion for the weak convergence of its distribution. In the second section
we proceed with the examples of the determinantal random point fields from
Quantum Mechanics, Statistical Mechanics, Random Matrix Theory, Probability
Theory, Representation Theory and Ergodic Theory. In connection with the Theory
of Renewal Processes we characterize all determinantal random point fields in
R^1 and Z^1 with independent identically distributed spacings. In the third
section we study the translation invariant determinantal random point fields
and prove the mixing property of any multiplicity and the absolute continuity
of the spectra. In the fourth (and the last) section we discuss the proofs of
the Central Limit Theorem for the number of particles in the growing box and
the Functional Central Limit Theorem for the empirical distribution function of
spacings.Comment: To appear in the Russian Mathematical Surveys; small misprints are
correcte
Decimation of the Dyson-Ising Ferromagnet
We study the decimation to a sublattice of half the sites, of the
one-dimensional Dyson-Ising ferromagnet with slowly decaying long-range pair
interactions of the form , in the phase transition
region (1< 2, and low temperature). We prove non-Gibbsianness of
the decimated measure at low enough temperatures by exhibiting a point of
essential discontinuity for the finite-volume conditional probabilities of
decimated Gibbs measures. Thus result complements previous work proving
conservation of Gibbsianness for fastly decaying potentials ( > 2) and
provides an example of a "standard" non-Gibbsian result in one dimension, in
the vein of similar resuts in higher dimensions for short-range models. We also
discuss how these measures could fit within a generalized (almost vs. weak)
Gibbsian framework. Moreover we comment on the possibility of similar results
for some other transformations.Comment: 18 pages, some corrections and references added, to appear in
Stoch.Proc.App
Solvability of the cohomological equation for regular vector fields on the plane
We consider planar vector field without zeroes X and study the image of the
associated Lie derivative operator LX acting on the space of smooth functions.
We show that the cokernel of LX is infinite-dimensional as soon as X is not
topologically conjugate to a constant vector field and that, if the topology of
the integral trajectories of X is ``simple enough'' (e.g. if X is polynomial)
then X is transversal to a Hamiltonian foliation. We use this fact to find a
large explicit subalgebra of the image of LX and to build an embedding of R^2
into R^4 which rectifies X. Finally we use this embedding to characterize the
functions in the image of LX.Comment: 21 pages, 2 figure
- …