103 research outputs found
A number-theoretic problem concerning pseudo-real Riemann surfaces
Motivated by their research on automorphism groups of pseudo-real Riemann
surfaces, Bujalance, Cirre and Conder have conjectured that there are
infinitely many primes such that has all its prime factors mod~. We use theorems of Landau and Raikov to prove that the number of
integers with only such prime factors is asymptotic to
for a specific constant . Heuristic
arguments, following Hardy and Littlewood, then yield a conjecture that the
number of such primes is asymptotic to
for a constant . The theorem, the conjecture and a similar
conjecture applying the Bateman--Horn Conjecture to other pseudo-real Riemann
surfaces are supported by evidence from extensive computer searches.Comment: 20 page
Counting dependent and independent strings
The paper gives estimations for the sizes of the the following sets: (1) the
set of strings that have a given dependency with a fixed string, (2) the set of
strings that are pairwise \alpha independent, (3) the set of strings that are
mutually \alpha independent. The relevant definitions are as follows: C(x) is
the Kolmogorov complexity of the string x. A string y has \alpha -dependency
with a string x if C(y) - C(y|x) \geq \alpha. A set of strings {x_1, \ldots,
x_t} is pairwise \alpha-independent if for all i different from j, C(x_i) -
C(x_i | x_j) \leq \alpha. A tuple of strings (x_1, \ldots, x_t) is mutually
\alpha-independent if C(x_{\pi(1)} \ldots x_{\pi(t)}) \geq C(x_1) + \ldots +
C(x_t) - \alpha, for every permutation \pi of [t]
Impossibility of independence amplification in Kolmogorov complexity theory
The paper studies randomness extraction from sources with bounded
independence and the issue of independence amplification of sources, using the
framework of Kolmogorov complexity. The dependency of strings and is
, where
denotes the Kolmogorov complexity. It is shown that there exists a
computable Kolmogorov extractor such that, for any two -bit strings with
complexity and dependency , it outputs a string of length
with complexity conditioned by any one of the input
strings. It is proven that the above are the optimal parameters a Kolmogorov
extractor can achieve. It is shown that independence amplification cannot be
effectively realized. Specifically, if (after excluding a trivial case) there
exist computable functions and such that for all -bit strings and with , then
On double Hurwitz numbers in genus 0
We study double Hurwitz numbers in genus zero counting the number of covers
\CP^1\to\CP^1 with two branching points with a given branching behavior. By
the recent result due to Goulden, Jackson and Vakil, these numbers are
piecewise polynomials in the multiplicities of the preimages of the branching
points. We describe the partition of the parameter space into polynomiality
domains, called chambers, and provide an expression for the difference of two
such polynomials for two neighboring chambers. Besides, we provide an explicit
formula for the polynomial in a certain chamber called totally negative, which
enables us to calculate double Hurwitz numbers in any given chamber as the
polynomial for the totally negative chamber plus the sum of the differences
between the neighboring polynomials along a path connecting the totally
negative chamber with the given one.Comment: 17 pages, 3 figure
Upper bounds for the number of orbital topological types of planar polynomial vector fields "modulo limit cycles"
The paper deals with planar polynomial vector fields. We aim to estimate the
number of orbital topological equivalence classes for the fields of degree n.
An evident obstacle for this is the second part of Hilbert's 16th problem. To
circumvent this obstacle we introduce the notion of equivalence modulo limit
cycles. This paper is the continuation of the author's paper in [Mosc. Math. J.
1 (2001), no. 4] where the lower bound of the form 2^{cn^2} has been obtained.
Here we obtain the upper bound of the same form. We also associate an equipped
planar graph to every planar polynomial vector field, this graph is a complete
invariant for orbital topological classification of such fields.Comment: 23 pages, 5 figure
MDL Convergence Speed for Bernoulli Sequences
The Minimum Description Length principle for online sequence
estimation/prediction in a proper learning setup is studied. If the underlying
model class is discrete, then the total expected square loss is a particularly
interesting performance measure: (a) this quantity is finitely bounded,
implying convergence with probability one, and (b) it additionally specifies
the convergence speed. For MDL, in general one can only have loss bounds which
are finite but exponentially larger than those for Bayes mixtures. We show that
this is even the case if the model class contains only Bernoulli distributions.
We derive a new upper bound on the prediction error for countable Bernoulli
classes. This implies a small bound (comparable to the one for Bayes mixtures)
for certain important model classes. We discuss the application to Machine
Learning tasks such as classification and hypothesis testing, and
generalization to countable classes of i.i.d. models.Comment: 28 page
Noncomputability Arising In Dynamical Triangulation Model Of Four-Dimensional Quantum Gravity
Computations in Dynamical Triangulation Models of Four-Dimensional Quantum
Gravity involve weighted averaging over sets of all distinct triangulations of
compact four-dimensional manifolds. In order to be able to perform such
computations one needs an algorithm which for any given and a given compact
four-dimensional manifold constructs all possible triangulations of
with simplices. Our first result is that such algorithm does not
exist. Then we discuss recursion-theoretic limitations of any algorithm
designed to perform approximate calculations of sums over all possible
triangulations of a compact four-dimensional manifold.Comment: 8 Pages, LaTex, PUPT-132
Genus expansion for real Wishart matrices
We present an exact formula for moments and cumulants of several real
compound Wishart matrices in terms of an Euler characteristic expansion,
similar to the genus expansion for complex random matrices. We consider their
asymptotic values in the large matrix limit: as in a genus expansion, the terms
which survive in the large matrix limit are those with the greatest Euler
characteristic, that is, either spheres or collections of spheres. This
topological construction motivates an algebraic expression for the moments and
cumulants in terms of the symmetric group. We examine the combinatorial
properties distinguishing the leading order terms. By considering higher
cumulants, we give a central limit-type theorem for the asymptotic distribution
around the expected value
- …