78,455 research outputs found
SMT Solving for Functional Programming over Infinite Structures
We develop a simple functional programming language aimed at manipulating
infinite, but first-order definable structures, such as the countably infinite
clique graph or the set of all intervals with rational endpoints. Internally,
such sets are represented by logical formulas that define them, and an external
satisfiability modulo theories (SMT) solver is regularly run by the interpreter
to check their basic properties.
The language is implemented as a Haskell module.Comment: In Proceedings MSFP 2016, arXiv:1604.0038
Antisymmetric Orbit Functions
In the paper, properties of antisymmetric orbit functions are reviewed and
further developed. Antisymmetric orbit functions on the Euclidean space
are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a
Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such
functions are described. These functions are closely related to irreducible
characters of a compact semisimple Lie group of rank . Up to a sign,
values of antisymmetric orbit functions are repeated on copies of the
fundamental domain of the affine Weyl group (determined by the initial Weyl
group) in the entire Euclidean space . Antisymmetric orbit functions are
solutions of the corresponding Laplace equation in , vanishing on the
boundary of the fundamental domain . Antisymmetric orbit functions determine
a so-called antisymmetrized Fourier transform which is closely related to
expansions of central functions in characters of irreducible representations of
the group . They also determine a transform on a finite set of points of
(the discrete antisymmetric orbit function transform). Symmetric and
antisymmetric multivariate exponential, sine and cosine discrete transforms are
given.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
E-Orbit Functions
We review and further develop the theory of -orbit functions. They are
functions on the Euclidean space obtained from the multivariate
exponential function by symmetrization by means of an even part of a
Weyl group , corresponding to a Coxeter-Dynkin diagram. Properties of such
functions are described. They are closely related to symmetric and
antisymmetric orbit functions which are received from exponential functions by
symmetrization and antisymmetrization procedure by means of a Weyl group .
The -orbit functions, determined by integral parameters, are invariant with
respect to even part of the affine Weyl group corresponding
to . The -orbit functions determine a symmetrized Fourier transform,
where these functions serve as a kernel of the transform. They also determine a
transform on a finite set of points of the fundamental domain of the
group (the discrete -orbit function transform).Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Four types of special functions of G_2 and their discretization
Properties of four infinite families of special functions of two real
variables, based on the compact simple Lie group G2, are compared and
described. Two of the four families (called here C- and S-functions) are well
known, whereas the other two (S^L- and S^S-functions) are not found elsewhere
in the literature. It is shown explicitly that all four families have similar
properties. In particular, they are orthogonal when integrated over a finite
region F of the Euclidean space, and they are discretely orthogonal when their
values, sampled at the lattice points F_M \subset F, are added up with a weight
function appropriate for each family. Products of ten types among the four
families of functions, namely CC, CS, SS, SS^L, CS^S, SS^L, SS^S, S^SS^S,
S^LS^S and S^LS^L, are completely decomposable into the finite sum of the
functions. Uncommon arithmetic properties of the functions are pointed out and
questions about numerous other properties are brought forward.Comment: 18 pages, 4 figures, 4 table
Orbit Functions
In the paper, properties of orbit functions are reviewed and further
developed. Orbit functions on the Euclidean space are symmetrized
exponential functions. The symmetrization is fulfilled by a Weyl group
corresponding to a Coxeter-Dynkin diagram. Properties of such functions will be
described. An orbit function is the contribution to an irreducible character of
a compact semisimple Lie group of rank from one of its Weyl group
orbits. It is shown that values of orbit functions are repeated on copies of
the fundamental domain of the affine Weyl group (determined by the initial
Weyl group) in the entire Euclidean space . Orbit functions are solutions
of the corresponding Laplace equation in , satisfying the Neumann
condition on the boundary of . Orbit functions determine a symmetrized
Fourier transform and a transform on a finite set of points.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
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