302 research outputs found
Die Lehre vom Beruf unter gegenwaertigen Verhaeltnissen
Die Lehre vom Beruf unter gegenwaertigen Verhaeltnissen (The apprenticeship of the profession under present circumstances
An asymptotic bound for secant varieties of Segre varieties
This paper studies the defectivity of secant varieties of Segre varieties. We
prove that there exists an asymptotic lower estimate for the greater
non-defective secant variety (without filling the ambient space) of any given
Segre variety. In particular, we prove that the ratio between the greater
non-defective secant variety of a Segre variety and its expected rank is lower
bounded by a value depending just on the number of factors of the Segre
variety. Moreover, in the final section, we present some results obtained by
explicit computation, proving the non-defectivity of all the secant varieties
of Segre varieties of the shape (P^n)^4, with 1 < n < 11, except at most
\sigma_199((P^8)^4) and \sigma_357((P^10)^4).Comment: 14 page
Reed-Muller codes for random erasures and errors
This paper studies the parameters for which Reed-Muller (RM) codes over
can correct random erasures and random errors with high probability,
and in particular when can they achieve capacity for these two classical
channels. Necessarily, the paper also studies properties of evaluations of
multi-variate polynomials on random sets of inputs.
For erasures, we prove that RM codes achieve capacity both for very high rate
and very low rate regimes. For errors, we prove that RM codes achieve capacity
for very low rate regimes, and for very high rates, we show that they can
uniquely decode at about square root of the number of errors at capacity.
The proofs of these four results are based on different techniques, which we
find interesting in their own right. In particular, we study the following
questions about , the matrix whose rows are truth tables of all
monomials of degree in variables. What is the most (resp. least)
number of random columns in that define a submatrix having full column
rank (resp. full row rank) with high probability? We obtain tight bounds for
very small (resp. very large) degrees , which we use to show that RM codes
achieve capacity for erasures in these regimes.
Our decoding from random errors follows from the following novel reduction.
For every linear code of sufficiently high rate we construct a new code
, also of very high rate, such that for every subset of coordinates, if
can recover from erasures in , then can recover from errors in .
Specializing this to RM codes and using our results for erasures imply our
result on unique decoding of RM codes at high rate.
Finally, two of our capacity achieving results require tight bounds on the
weight distribution of RM codes. We obtain such bounds extending the recent
\cite{KLP} bounds from constant degree to linear degree polynomials
Functional Limit Theorems for Multiparameter Fractional Brownian Motion
We prove a general functional limit theorem for multiparameter fractional
Brownian motion. The functional law of the iterated logarithm, functional
L\'{e}vy's modulus of continuity and many other results are its particular
cases. Applications to approximation theory are discussed.Comment: AMS-LaTeX, 23 page
Conditional Intensity and Gibbsianness of Determinantal Point Processes
The Papangelou intensities of determinantal (or fermion) point processes are
investigated. These exhibit a monotonicity property expressing the repulsive
nature of the interaction, and satisfy a bound implying stochastic domination
by a Poisson point process. We also show that determinantal point processes
satisfy the so-called condition which is a general form of
Gibbsianness. Under a continuity assumption, the Gibbsian conditional
probabilities can be identified explicitly.Comment: revised and extende
Random Cluster Models on the Triangular Lattice
We study percolation and the random cluster model on the triangular lattice
with 3-body interactions. Starting with percolation, we generalize the
star--triangle transformation: We introduce a new parameter (the 3-body term)
and identify configurations on the triangles solely by their connectivity. In
this new setup, necessary and sufficient conditions are found for positive
correlations and this is used to establish regions of percolation and
non-percolation. Next we apply this set of ideas to the random cluster
model: We derive duality relations for the suitable random cluster measures,
prove necessary and sufficient conditions for them to have positive
correlations, and finally prove some rigorous theorems concerning phase
transitions.Comment: 24 pages, 1 figur
Quenched large deviation principle for words in a letter sequence
When we cut an i.i.d. sequence of letters into words according to an
independent renewal process, we obtain an i.i.d. sequence of words. In the
\emph{annealed} large deviation principle (LDP) for the empirical process of
words, the rate function is the specific relative entropy of the observed law
of words w.r.t. the reference law of words. In the present paper we consider
the \emph{quenched} LDP, i.e., we condition on a typical letter sequence. We
focus on the case where the renewal process has an \emph{algebraic} tail. The
rate function turns out to be a sum of two terms, one being the annealed rate
function, the other being proportional to the specific relative entropy of the
observed law of letters w.r.t. the reference law of letters, with the former
being obtained by concatenating the words and randomising the location of the
origin. The proportionality constant equals the tail exponent of the renewal
process. Earlier work by Birkner considered the case where the renewal process
has an exponential tail, in which case the rate function turns out to be the
first term on the set where the second term vanishes and to be infinite
elsewhere. In a companion paper the annealed and the quenched LDP are applied
to the collision local time of transient random walks, and the existence of an
intermediate phase for a class of interacting stochastic systems is
established.Comment: 41 pages, 2 figures. Acronym LDP spelled out in title, main result
strengthened to cover more general "letter" spaces, application to collision
local times removed (this part will become a separate manuscript
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
Concise and Tight Security Analysis of the Bennett-Brassard 1984 Protocol with Finite Key Lengths
We present a tight security analysis of the Bennett-Brassard 1984 protocol
taking into account the finite size effect of key distillation, and achieving
unconditional security. We begin by presenting a concise analysis utilizing the
normal approximation of the hypergeometric function. Then next we show that a
similarly tight bound can also be obtained by a rigorous argument without
relying on any approximation. In particular, for the convenience of
experimentalists who wish to evaluate the security of their QKD systems, we
also give explicit procedures of our key distillation, and also show how to
calculate the secret key rate and the security parameter from a given set of
experimental parameters. Besides the exact values of key rates and security
parameters, we also present how to obtain their rough estimates using the
normal approximation.Comment: 40 pages, 4 figures, revised arguments on security, and detailed
explanaions on how to use theoretical result
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