3,257 research outputs found
Levels of Distribution and the Affine Sieve
This article is an expanded version of the author's lecture in the Basic
Notions Seminar at Harvard, September 2013. Our goal is a brief and
introductory exposition of aspects of two topics in sieve theory which have
received attention recently: (1) the spectacular work of Yitang Zhang, under
the title "Level of Distribution," and (2) the so-called "Affine Sieve,"
introduced by Bourgain-Gamburd-Sarnak.Comment: 34 pages, 2 figure
Synchronizing Automata on Quasi Eulerian Digraph
In 1964 \v{C}ern\'{y} conjectured that each -state synchronizing automaton
posesses a reset word of length at most . From the other side the best
known upper bound on the reset length (minimum length of reset words) is cubic
in . Thus the main problem here is to prove quadratic (in ) upper bounds.
Since 1964, this problem has been solved for few special classes of \sa. One of
this result is due to Kari \cite{Ka03} for automata with Eulerian digraphs. In
this paper we introduce a new approach to prove quadratic upper bounds and
explain it in terms of Markov chains and Perron-Frobenius theories. Using this
approach we obtain a quadratic upper bound for a generalization of Eulerian
automata.Comment: 8 pages, 1 figur
Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables
Let be a a negative discriminant and let vary over a set of
representatives of the integral equivalence classes of integral binary
quadratic forms of discriminant . We prove an asymptotic formula for for the average over of the number of representations of by an
integral positive definite quaternary quadratic form and obtain results on
averages of Fourier coefficients of linear combinations of Siegel theta series.
We also find an asymptotic bound from below on the number of binary forms of
fixed discriminant which are represented by a given quaternary form. In
particular, we can show that for growing a positive proportion of the
binary quadratic forms of discriminant is represented by the given
quaternary quadratic form.Comment: v5: Some typos correcte
A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements
The basic methods of constructing the sets of mutually unbiased bases in the
Hilbert space of an arbitrary finite dimension are discussed and an emerging
link between them is outlined. It is shown that these methods employ a wide
range of important mathematical concepts like, e.g., Fourier transforms, Galois
fields and rings, finite and related projective geometries, and entanglement,
to mention a few. Some applications of the theory to quantum information tasks
are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two
more references adde
On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAs
We tackle the problem of the randomized generation of slowly synchronizing
deterministic automata (DFAs) by generating random primitive sets of matrices.
We show that when the randomized procedure is too simple the exponent of the
generated sets is O(n log n) with high probability, thus the procedure fails to
return DFAs with large reset threshold. We extend this result to random
nondeterministic automata (NDFAs) by showing, in particular, that a uniformly
sampled NDFA has both a 2-directing word and a 3-directing word of length O(n
log n) with high probability. We then present a more involved randomized
algorithm that manages to generate DFAs with large reset threshold and we
finally leverage this finding for exhibiting new families of DFAs with reset
threshold of order .Comment: 31 pages, 9 figures. arXiv admin note: text overlap with
arXiv:1805.0672
Point regular groups of automorphisms of generalised quadrangles
We study the point regular groups of automorphisms of some of the known
generalised quadrangles. In particular we determine all point regular groups of
automorphisms of the thick classical generalised quadrangles. We also construct
point regular groups of automorphisms of the generalised quadrangle of order
obtained by Payne derivation from the classical symplectic
quadrangle . For with we obtain at least two
nonisomorphic groups when and at least three nonisomorphic groups
when or . Our groups include nonabelian 2-groups, groups of exponent 9
and nonspecial -groups. We also enumerate all point regular groups of
automorphisms of some small generalised quadrangles.Comment: some minor changes (including to title) after referee's comment
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