7,550 research outputs found
Hessian and concavity of mutual information, differential entropy, and entropy power in linear vector Gaussian channels
Within the framework of linear vector Gaussian channels with arbitrary
signaling, closed-form expressions for the Jacobian of the minimum mean square
error and Fisher information matrices with respect to arbitrary parameters of
the system are calculated in this paper. Capitalizing on prior research where
the minimum mean square error and Fisher information matrices were linked to
information-theoretic quantities through differentiation, closed-form
expressions for the Hessian of the mutual information and the differential
entropy are derived. These expressions are then used to assess the concavity
properties of mutual information and differential entropy under different
channel conditions and also to derive a multivariate version of the entropy
power inequality due to Costa.Comment: 33 pages, 2 figures. A shorter version of this paper is to appear in
IEEE Transactions on Information Theor
Pseudo-Exponential-Type Solutions of Wave Equations Depending on Several Variables
Using matrix identities, we construct explicit pseudo-exponential-type
solutions of linear Dirac, Loewner and Schr\"odinger equations depending on two
variables and of nonlinear wave equations depending on three variables
Towards the n-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure
In this final part of a series of three papers, we will assemble
supersymmetric expressions for one-loop correlators in pure-spinor superspace
that are BRST invariant, local, and single valued. A key driving force in this
construction is the generalization of a so far unnoticed property at tree
level; the correlators have the symmetry structure akin to Lie polynomials.
One-loop correlators up to seven points are presented in a variety of
representations manifesting different subsets of their defining properties.
These expressions are related via identities obeyed by the kinematic
superfields and worldsheet functions spelled out in the first two parts of this
series and reflecting a duality between the two kinds of ingredients.
Interestingly, the expression for the eight-point correlator following from
our method seems to capture correctly all the dependence on the worldsheet
punctures but leaves undetermined the coefficient of the holomorphic Eisenstein
series . By virtue of chiral splitting, closed-string correlators
follow from the double copy of the open-string results.Comment: 77 pages, v2: published versio
Double-Copy Structure of One-Loop Open-String Amplitudes
In this Letter, we provide evidence for a new double-copy structure in
one-loop amplitudes of the open superstring. Their integrands with respect to
the moduli space of genus-one surfaces are cast into a form where
gauge-invariant kinematic factors and certain functions of the punctures --
so-called generalized elliptic integrands -- enter on completely symmetric
footing. In particular, replacing the generalized elliptic integrands by a
second copy of kinematic factors maps one-loop open-string correlators to
gravitational matrix elements of the higher-curvature operator R^4.Comment: 5 pages, v2: modifications in the structure to match published
versio
Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry
Due to the isotropy -dimensional hyperbolic space, there exist a
spherically symmetric fundamental solution for its corresponding
Laplace-Beltrami operator. On the -radius hyperboloid model of
-dimensional hyperbolic geometry with and , we compute
azimuthal Fourier expansions for a fundamental solution of Laplace's equation.
For , we compute a Gegenbauer polynomial expansion in geodesic polar
coordinates for a fundamental solution of Laplace's equation on this
negative-constant sectional curvature Riemannian manifold. In three-dimensions,
an addition theorem for the azimuthal Fourier coefficients of a fundamental
solution for Laplace's equation is obtained through comparison with its
corresponding Gegenbauer expansion.Comment: arXiv admin note: substantial text overlap with arXiv:1201.440
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