54,169 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
An evaluation of a teaching package constructed using a Webâbased lecture recorder
This paper describes an evaluation of a teaching package used to replace lectures in two closely related university courses on Discrete Mathematics. The package was developed using Audiograph, a Webâbased lecture recorder developed at the University of Surrey. Two groups of subjects were studied: a group of undergraduates, mostly fresh from schools, and a group of postgraduates, mostly with postâuniversity work experience. Although the postgraduates with their greater maturity and experience were significantly more positive in their appraisal than the undergraduates, both groups agreed on the beneficial aspects of being able to work at one's own time and pace, and being able to repeat material at will. It is clear, however, that, in the context investigated, where the lecturer was readily available, such a package can never supplant a human teacher, and that considerable effort needs to be expended in order to integrate the package into a rich learning environment
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
A geometry of information, I: Nerves, posets and differential forms
The main theme of this workshop (Dagstuhl seminar 04351) is `Spatial
Representation: Continuous vs. Discrete'. Spatial representation has two
contrasting but interacting aspects (i) representation of spaces' and (ii)
representation by spaces. In this paper, we will examine two aspects that are
common to both interpretations of the theme, namely nerve constructions and
refinement. Representations change, data changes, spaces change. We will
examine the possibility of a `differential geometry' of spatial representations
of both types, and in the sequel give an algebra of differential forms that has
the potential to handle the dynamical aspect of such a geometry. We will
discuss briefly a conjectured class of spaces, generalising the Cantor set
which would seem ideal as a test-bed for the set of tools we are developing.Comment: 28 pages. A version of this paper appears also on the Dagstuhl
seminar portal http://drops.dagstuhl.de/portals/04351
On Topological Properties of Wireless Sensor Networks under the q-Composite Key Predistribution Scheme with On/Off Channels
The q-composite key predistribution scheme [1] is used prevalently for secure
communications in large-scale wireless sensor networks (WSNs). Prior work
[2]-[4] explores topological properties of WSNs employing the q-composite
scheme for q = 1 with unreliable communication links modeled as independent
on/off channels. In this paper, we investigate topological properties related
to the node degree in WSNs operating under the q-composite scheme and the
on/off channel model. Our results apply to general q and are stronger than
those reported for the node degree in prior work even for the case of q being
1. Specifically, we show that the number of nodes with certain degree
asymptotically converges in distribution to a Poisson random variable, present
the asymptotic probability distribution for the minimum degree of the network,
and establish the asymptotically exact probability for the property that the
minimum degree is at least an arbitrary value. Numerical experiments confirm
the validity of our analytical findings.Comment: Best Student Paper Finalist in IEEE International Symposium on
Information Theory (ISIT) 201
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