446 research outputs found
Polylogarithms, Multiple Zeta Values and Superstring Amplitudes
A formalism is provided to calculate tree amplitudes in open superstring
theory for any multiplicity at any order in the inverse string tension. We
point out that the underlying world-sheet disk integrals share substantial
properties with color-ordered tree amplitudes in Yang-Mills field theories. In
particular, we closely relate world-sheet integrands of open-string tree
amplitudes to the Kawai-Lewellen-Tye representation of supergravity amplitudes.
This correspondence helps to reduce the singular parts of world-sheet disk
integrals -including their string corrections- to lower-point results. The
remaining regular parts are systematically addressed by polylogarithm
manipulations.Comment: 79 pages, LaTeX; v2: final version to appear in Fortschritte der
Physik; for additional material, see: http://mzv.mpp.mpg.d
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
We consider the calculation of the master integrals of the three-loop massive
banana graph. In the case of equal internal masses, the graph is reduced to
three master integrals which satisfy an irreducible system of three coupled
linear differential equations. The solution of the system requires finding a matrix of homogeneous solutions. We show how the maximal cut can be
used to determine all entries of this matrix in terms of products of elliptic
integrals of first and second kind of suitable arguments. All independent
solutions are found by performing the integration which defines the maximal cut
on different contours. Once the homogeneous solution is known, the
inhomogeneous solution can be obtained by use of Euler's variation of
constants.Comment: 39 pages, 3 figures; Fixed a typo in eq. (6.16
Time-warping invariants of multidimensional time series
In data science, one is often confronted with a time series representing
measurements of some quantity of interest. Usually, as a first step, features
of the time series need to be extracted. These are numerical quantities that
aim to succinctly describe the data and to dampen the influence of noise. In
some applications, these features are also required to satisfy some invariance
properties. In this paper, we concentrate on time-warping invariants. We show
that these correspond to a certain family of iterated sums of the increments of
the time series, known as quasisymmetric functions in the mathematics
literature. We present these invariant features in an algebraic framework, and
we develop some of their basic properties.Comment: 18 pages, 1 figur
On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem
We prove that the number of limit cycles generated by a small
non-conservative perturbation of a Hamiltonian polynomial vector field on the
plane, is bounded by a double exponential of the degree of the fields. This
solves the long-standing tangential Hilbert 16th problem. The proof uses only
the fact that Abelian integrals of a given degree are horizontal sections of a
regular flat meromorphic connection (Gauss-Manin connection) with a
quasiunipotent monodromy group.Comment: Final revisio
Time-warping invariants of multidimensional time series
In data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants.We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties
Time-warping invariants of multidimensional time series
In data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants.We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties
On Buffon Machines and Numbers
The well-know needle experiment of Buffon can be regarded as an analog (i.e.,
continuous) device that stochastically "computes" the number 2/pi ~ 0.63661,
which is the experiment's probability of success. Generalizing the experiment
and simplifying the computational framework, we consider probability
distributions, which can be produced perfectly, from a discrete source of
unbiased coin flips. We describe and analyse a few simple Buffon machines that
generate geometric, Poisson, and logarithmic-series distributions. We provide
human-accessible Buffon machines, which require a dozen coin flips or less, on
average, and produce experiments whose probabilities of success are expressible
in terms of numbers such as, exp(-1), log 2, sqrt(3), cos(1/4), aeta(5).
Generally, we develop a collection of constructions based on simple
probabilistic mechanisms that enable one to design Buffon experiments involving
compositions of exponentials and logarithms, polylogarithms, direct and inverse
trigonometric functions, algebraic and hypergeometric functions, as well as
functions defined by integrals, such as the Gaussian error function.Comment: Largely revised version with references and figures added. 12 pages.
In ACM-SIAM Symposium on Discrete Algorithms (SODA'2011
On the periods of some Feynman integrals
We study the related questions: (i) when Feynman amplitudes in massless
theory evaluate to multiple zeta values, and (ii) when their
underlying motives are mixed Tate. More generally, by considering
configurations of singular hypersurfaces which fiber linearly over each other,
we deduce sufficient geometric and combinatorial criteria on Feynman graphs for
both (i) and (ii) to hold. These criteria hold for some infinite classes of
graphs which essentially contain all cases previously known to physicists.
Calabi-Yau varieties appear at the point where these criteria fail
Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths
We consider differential equations driven by rough paths and study the
regularity of the laws and their long time behavior. In particular, we focus on
the case when the driving noise is a rough path valued fractional Brownian
motion with Hurst parameter . Our contribution
in this work is twofold. First, when the driving vector fields satisfy
H\"{o}rmander's celebrated "Lie bracket condition," we derive explicit
quantitative bounds on the inverse of the Malliavin matrix. En route to this,
we provide a novel "deterministic" version of Norris's lemma for differential
equations driven by rough paths. This result, with the added assumption that
the linearized equation has moments, will then yield that the transition laws
have a smooth density with respect to Lebesgue measure. Our second main result
states that under H\"{o}rmander's condition, the solutions to rough
differential equations driven by fractional Brownian motion with
enjoy a suitable version of the strong Feller
property. Under a standard controllability condition, this implies that they
admit a unique stationary solution that is physical in the sense that it does
not "look into the future."Comment: Published in at http://dx.doi.org/10.1214/12-AOP777 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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