698,804 research outputs found
On a class of discrete functions
We consider classes of functions which depend in a certain way on their variables. The relation between the number of H-functions of n variables of the k-valued logic and the number of n-dimensional Latin hypercubes of order k is found. We have shown how from an arbitrary Latin hypercube we can "construct" (present in table form) an H-function and vice versa - how every H-function can be represented as a Latin hypercube. We extend the concepts of H-function and Latin hypercube
On some Generalizations of a Class of Discrete Functions
In this paper we examine discrete functions that depend on
their variables in a particular way, namely the H-functions. The results
obtained in this work make the “construction” of these functions possible.
H-functions are generalized, as well as their matrix representation by Latin
hypercubes
Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group
The paper is about methods of discrete Fourier analysis in the context of
Weyl group symmetry. Three families of class functions are defined on the
maximal torus of each compact simply connected semisimple Lie group . Such
functions can always be restricted without loss of information to a fundamental
region of the affine Weyl group. The members of each family satisfy
basic orthogonality relations when integrated over (continuous
orthogonality). It is demonstrated that the functions also satisfy discrete
orthogonality relations when summed up over a finite grid in
(discrete orthogonality), arising as the set of points in
representing the conjugacy classes of elements of a finite Abelian subgroup of
the maximal torus . The characters of the centre of the Lie
group allow one to split functions on into a sum
, where is the order of , and where the component
functions decompose into the series of -, or -, or -functions
from one congruence class only.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems
We present dichotomy theorems regarding the computational complexity of
counting fixed points in boolean (discrete) dynamical systems, i.e., finite
discrete dynamical systems over the domain {0,1}. For a class F of boolean
functions and a class G of graphs, an (F,G)-system is a boolean dynamical
system with local transitions functions lying in F and graphs in G. We show
that, if local transition functions are given by lookup tables, then the
following complexity classification holds: Let F be a class of boolean
functions closed under superposition and let G be a graph class closed under
taking minors. If F contains all min-functions, all max-functions, or all
self-dual and monotone functions, and G contains all planar graphs, then it is
#P-complete to compute the number of fixed points in an (F,G)-system; otherwise
it is computable in polynomial time. We also prove a dichotomy theorem for the
case that local transition functions are given by formulas (over logical
bases). This theorem has a significantly more complicated structure than the
theorem for lookup tables. A corresponding theorem for boolean circuits
coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on
Theoretical Computer Science (ICTCS'2007
Second order Riesz transforms on multiply-connected Lie groups and processes with jumps
We study a class of combinations of second order Riesz transforms on Lie
groups that are multiply connected, composed of a discrete abelian component
and a compact connected component. We prove sharp estimates for these
operators, therefore generalising previous results.
We construct stochastic integrals with jump components adapted to functions
defined on our semi-discrete set. We show that these second order Riesz
transforms applied to a function may be written as conditional expectation of a
simple transformation of a stochastic integral associated with the function.
The analysis shows that Ito integrals for the discrete component must be
written in an augmented discrete tangent plane of dimension twice larger than
expected, and in a suitably chosen discrete coordinate system. Those artifacts
are related to the difficulties that arise due to the discrete component, where
derivatives of functions are no longer local. Previous representations of Riesz
transforms through stochastic integrals in this direction do not consider
discrete components and jump processes
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