67,475 research outputs found
Some Useful Integral Representations for Information-Theoretic Analyses
This work is an extension of our earlier article, where a well-known integral
representation of the logarithmic function was explored, and was accompanied
with demonstrations of its usefulness in obtaining compact, easily-calculable,
exact formulas for quantities that involve expectations of the logarithm of a
positive random variable. Here, in the same spirit, we derive an exact integral
representation (in one or two dimensions) of the moment of a nonnegative random
variable, or the sum of such independent random variables, where the moment
order is a general positive noninteger real (also known as fractional moments).
The proposed formula is applied to a variety of examples with an
information-theoretic motivation, and it is shown how it facilitates their
numerical evaluations. In particular, when applied to the calculation of a
moment of the sum of a large number, , of nonnegative random variables, it
is clear that integration over one or two dimensions, as suggested by our
proposed integral representation, is significantly easier than the alternative
of integrating over dimensions, as needed in the direct calculation of the
desired moment.Comment: Published in Entropy, vol. 22, no. 6, paper 707, pages 1-29, June
2020. Available at: https://www.mdpi.com/1099-4300/22/6/70
The geometric mean is a Bernstein function
In the paper, the authors establish, by using Cauchy integral formula in the
theory of complex functions, an integral representation for the geometric mean
of positive numbers. From this integral representation, the geometric mean
is proved to be a Bernstein function and a new proof of the well known AG
inequality is provided.Comment: 10 page
Feynman Integrals and Intersection Theory
We introduce the tools of intersection theory to the study of Feynman
integrals, which allows for a new way of projecting integrals onto a basis. In
order to illustrate this technique, we consider the Baikov representation of
maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of
differential forms with logarithmic singularities on the boundaries of the
corresponding integration cycles. We give an algorithm for computing a basis
decomposition of an arbitrary maximal cut using so-called intersection numbers
and describe two alternative ways of computing them. Furthermore, we show how
to obtain Pfaffian systems of differential equations for the basis integrals
using the same technique. All the steps are illustrated on the example of a
two-loop non-planar triangle diagram with a massive loop.Comment: 13 pages, published versio
A stroll along the gamma
We provide the first in-depth study of the "smart path" interpolation between
an arbitrary probability measure and the gamma-
distribution. We propose new explicit representation formulae for the ensuing
process as well as a new notion of relative Fisher information with a gamma
target distribution. We use these results to prove a differential and an
integrated De Bruijn identity which hold under minimal conditions, hereby
extending the classical formulae which follow from Bakry, Emery and Ledoux's
-calculus. Exploiting a specific representation of the "smart path", we
obtain a new proof of the logarithmic Sobolev inequality for the gamma law with
as well as a new type of HSI inequality linking relative
entropy, Stein discrepancy and standardized Fisher information for the gamma
law with .Comment: Typos correcte
Does multifractal theory of turbulence have logarithms in the scaling relations?
The multifractal theory of turbulence uses a saddle-point evaluation in
determining the power-law behaviour of structure functions. Without suitable
precautions, this could lead to the presence of logarithmic corrections,
thereby violating known exact relations such as the four-fifths law. Using the
theory of large deviations applied to the random multiplicative model of
turbulence and calculating subdominant terms, we explain here why such
corrections cannot be present.Comment: 7 pages, 1 figur
Logarithmic Conformal Field Theory - or - How to Compute a Torus Amplitude on the Sphere
We review some aspects of logarithmic conformal field theories which might
shed some light on the geometrical meaning of logarithmic operators. We
consider an approach, put forward by V. Knizhnik, where computation of
correlation functions on higher genus Riemann surfaces can be replaced by
computations on the sphere under certain circumstances. We show that this
proposal naturally leads to logarithmic conformal field theories, when the
additional vertex operator insertions, which simulate the branch points of a
ramified covering of the sphere, are viewed as dynamical objects in the theory.
We study the Seiberg-Witten solution of supersymmetric low energy effective
field theory as an example where physically interesting quantities, the periods
of a meromorphic one-form, can effectively be computed within this conformal
field theory setting. We comment on the relation between correlation functions
computed on the plane, but with insertions of twist fields, and torus vacuum
amplitudes.Comment: LaTeX, 38 pp. 3 figures (provided as eps and as pdf). Contribution to
the Ian Kogan Memorial Volume "From Fields to Strings: Circumnavigating
Theoretical Physics
Relative entropy for compact Riemann surfaces
The relative entropy of the massive free bosonic field theory is studied on
various compact Riemann surfaces as a universal quantity with physical
significance, in particular, for gravitational phenomena. The exact expression
for the sphere is obtained, as well as its asymptotic series for large mass and
its Taylor series for small mass. One can also derive exact expressions for the
torus but not for higher genus. However, the asymptotic behaviour for large
mass can always be established-up to a constant-with heat-kernel methods. It
consists of an asymptotic series determined only by the curvature, hence common
for homogeneous surfaces of genus higher than one, and exponentially vanishing
corrections whose form is determined by the concrete topology. The coefficient
of the logarithmic term in this series gives the conformal anomaly.Comment: 20 pages, LaTeX 2e, 2 PS figures; to appear in Phys. Rev.
3d Modularity
We find and propose an explanation for a large variety of modularity-related
symmetries in problems of 3-manifold topology and physics of 3d
theories where such structures a priori are not manifest. These modular
structures include: mock modular forms, Weil
representations, quantum modular forms, non-semisimple modular tensor
categories, and chiral algebras of logarithmic CFTs.Comment: 119 pages, 10 figures and 20 table
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