54,611 research outputs found
Why Delannoy numbers?
This article is not a research paper, but a little note on the history of
combinatorics: We present here a tentative short biography of Henri Delannoy,
and a survey of his most notable works. This answers to the question raised in
the title, as these works are related to lattice paths enumeration, to the
so-called Delannoy numbers, and were the first general way to solve Ballot-like
problems. These numbers appear in probabilistic game theory, alignments of DNA
sequences, tiling problems, temporal representation models, analysis of
algorithms and combinatorial structures.Comment: Presented to the conference "Lattice Paths Combinatorics and Discrete
Distributions" (Athens, June 5-7, 2002) and to appear in the Journal of
Statistical Planning and Inference
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Analysis of Schr\"odinger operators with inverse square potentials I: regularity results in 3D
Let be a potential on \RR^3 that is smooth everywhere except at a
discrete set \maS of points, where it has singularities of the form
, with for close to and continuous on
\RR^3 with for p \in \maS. Also assume that and
are smooth outside \maS and is smooth in polar coordinates around each
singular point. We either assume that is periodic or that the set \maS is
finite and extends to a smooth function on the radial compactification of
\RR^3 that is bounded outside a compact set containing \maS. In the
periodic case, we let be the periodicity lattice and define \TT :=
\RR^3/ \Lambda. We obtain regularity results in weighted Sobolev space for the
eigenfunctions of the Schr\"odinger-type operator acting on
L^2(\TT), as well as for the induced \vt k--Hamiltonians \Hk obtained by
restricting the action of to Bloch waves. Under some additional
assumptions, we extend these regularity and solvability results to the
non-periodic case. We sketch some applications to approximation of
eigenfunctions and eigenvalues that will be studied in more detail in a second
paper.Comment: 15 pages, to appear in Bull. Math. Soc. Sci. Math. Roumanie, vol. 55
(103), no. 2/201
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