240 research outputs found
Enumeration of Weighted Plane Trees
In weighted trees, all edges are endowed with positive integral weight. We
enumerate weighted bicolored plane trees according to their weight and number
of edges.Comment: 6 pages, 4 figure
Methods for checking and enforcing physical quality of linear electrical network models
Most CAD tools allow system-level simulation for signal integrity by computing and connecting models together for the various sub-parts. The success of this model derivation depends on the quality of the network parameters. Different errors may seriously affect the quality of the frequency characterization: frequency-dependent measurement errors, errors due to the numerical simulation and/or discretization, etc. When these errors are large, model assembly and simulation becomes difficult and may even fail. This thesis gives an overview of the most significant properties of physically valid network parameters, describes existing methods for checking and enforcing these properties, and presents several new methodologies for checking and enforcing causality. A time domain methodology based on the vector fitting approximation as well as the frequency domain methodologies based on the Kramers-Kronig relations enforcement by numerical integration and Fast Fourier Transform are presented. A new algorithm is developed for a stable recursive convolution after time domain causality enforcement. In addition, global qualities of data for system simulations are discussed: a study of an accurate causal frequency domain interpolation as well as a robust technique for extrapolation to DC is included --Abstract, page iii
Algorithmic Complexity Bounds on Future Prediction Errors
We bound the future loss when predicting any (computably) stochastic sequence
online. Solomonoff finitely bounded the total deviation of his universal
predictor from the true distribution by the algorithmic complexity of
. Here we assume we are at a time and already observed .
We bound the future prediction performance on by a new
variant of algorithmic complexity of given , plus the complexity of the
randomness deficiency of . The new complexity is monotone in its condition
in the sense that this complexity can only decrease if the condition is
prolonged. We also briefly discuss potential generalizations to Bayesian model
classes and to classification problems.Comment: 21 page
Modular Subgroups, Dessins d’Enfants and Elliptic K3 Surfaces
We consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck’s dessins d’enfants. In the particular case of the index 36 subgroups, the corresponding Calabi–Yau threefolds are identified, in analogy with the index 24 cases being associated to K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over P1 as extremal K3 surfaces with six singular fibres. In each case, a representative of the corresponding class of subgroups is identified by specifying a generating set for that representative
Physical Complexity of Symbolic Sequences
A practical measure for the complexity of sequences of symbols (``strings'')
is introduced that is rooted in automata theory but avoids the problems of
Kolmogorov-Chaitin complexity. This physical complexity can be estimated for
ensembles of sequences, for which it reverts to the difference between the
maximal entropy of the ensemble and the actual entropy given the specific
environment within which the sequence is to be interpreted. Thus, the physical
complexity measures the amount of information about the environment that is
coded in the sequence, and is conditional on such an environment. In practice,
an estimate of the complexity of a string can be obtained by counting the
number of loci per string that are fixed in the ensemble, while the volatile
positions represent, again with respect to the environment, randomness. We
apply this measure to tRNA sequence data.Comment: 12 pages LaTeX2e, 3 postscript figures, uses elsart.cls.
Substantially improved and clarified version, includes application to EMBL
tRNA sequence dat
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]
An extension of Chaitin's halting probability \Omega to a measurement operator in an infinite dimensional quantum system
This paper proposes an extension of Chaitin's halting probability \Omega to a
measurement operator in an infinite dimensional quantum system. Chaitin's
\Omega is defined as the probability that the universal self-delimiting Turing
machine U halts, and plays a central role in the development of algorithmic
information theory. In the theory, there are two equivalent ways to define the
program-size complexity H(s) of a given finite binary string s. In the standard
way, H(s) is defined as the length of the shortest input string for U to output
s. In the other way, the so-called universal probability m is introduced first,
and then H(s) is defined as -log_2 m(s) without reference to the concept of
program-size.
Mathematically, the statistics of outcomes in a quantum measurement are
described by a positive operator-valued measure (POVM) in the most general
setting. Based on the theory of computability structures on a Banach space
developed by Pour-El and Richards, we extend the universal probability to an
analogue of POVM in an infinite dimensional quantum system, called a universal
semi-POVM. We also give another characterization of Chaitin's \Omega numbers by
universal probabilities. Then, based on this characterization, we propose to
define an extension of \Omega as a sum of the POVM elements of a universal
semi-POVM. The validity of this definition is discussed.
In what follows, we introduce an operator version \hat{H}(s) of H(s) in a
Hilbert space of infinite dimension using a universal semi-POVM, and study its
properties.Comment: 24 pages, LaTeX2e, no figures, accepted for publication in
Mathematical Logic Quarterly: The title was slightly changed and a section on
an operator-valued algorithmic information theory was adde
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
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