54,194 research outputs found
On the lexicographic representation of numbers
It is proven that, contrarily to the common belief, the notion of zero is not
necessary for having positional representations of numbers. Namely, for any
positive integer , a positional representation with the symbols for is given that retains all the essential properties of the usual
positional representation of base (over symbols for ).
Moreover, in this zero-free representation, a sequence of symbols identifies
the number that corresponds to the order number that the sequence has in the
ordering where shorter sequences precede the longer ones, and among sequences
of the same length the usual lexicographic ordering of dictionaries is
considered. The main properties of this lexicographic representation are proven
and conversion algorithms between lexicographic and classical positional
representations are given. Zero-free positional representations are relevantt
in the perspective of the history of mathematics, as well as, in the
perspective of emergent computation models, and of unconventional
representations of genomes.Comment: 15 page
Schur Superpolynomials: Combinatorial Definition and Pieri Rule
Schur superpolynomials have been introduced recently as limiting cases of the
Macdonald superpolynomials. It turns out that there are two natural
super-extensions of the Schur polynomials: in the limit and
, corresponding respectively to the Schur
superpolynomials and their dual. However, a direct definition is missing. Here,
we present a conjectural combinatorial definition for both of them, each being
formulated in terms of a distinct extension of semi-standard tableaux. These
two formulations are linked by another conjectural result, the Pieri rule for
the Schur superpolynomials. Indeed, and this is an interesting novelty of the
super case, the successive insertions of rows governed by this Pieri rule do
not generate the tableaux underlying the Schur superpolynomials combinatorial
construction, but rather those pertaining to their dual versions. As an aside,
we present various extensions of the Schur bilinear identity
Preconditioning of Improved and ``Perfect'' Fermion Actions
We construct a locally-lexicographic SSOR preconditioner to accelerate the
parallel iterative solution of linear systems of equations for two improved
discretizations of lattice fermions: the Sheikholeslami-Wohlert scheme where a
non-constant block-diagonal term is added to the Wilson fermion matrix and
renormalization group improved actions which incorporate couplings beyond
nearest neighbors of the lattice fermion fields. In case (i) we find the block
llssor-scheme to be more effective by a factor about 2 than odd-even
preconditioned solvers in terms of convergence rates, at beta=6.0. For type
(ii) actions, we show that our preconditioner accelerates the iterative
solution of a linear system of hypercube fermions by a factor of 3 to 4.Comment: 27 pages, Latex, 17 Figures include
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