1,398 research outputs found
Two remarks on generalized entropy power inequalities
This note contributes to the understanding of generalized entropy power
inequalities. Our main goal is to construct a counter-example regarding
monotonicity and entropy comparison of weighted sums of independent identically
distributed log-concave random variables. We also present a complex analogue of
a recent dependent entropy power inequality of Hao and Jog, and give a very
simple proof.Comment: arXiv:1811.00345 is split into 2 papers, with this being on
Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions
Polar transforms are central operations in the study of polar codes. This
paper examines polar transforms for non-stationary memoryless sources on
possibly infinite source alphabets. This is the first attempt of source
polarization analysis over infinite alphabets. The source alphabet is defined
to be a Polish group, and we handle the Ar{\i}kan-style two-by-two polar
transform based on the group. Defining erasure distributions based on the
normal subgroup structure, we give recursive formulas of the polar transform
for our proposed erasure distributions. As a result, the recursive formulas
lead to concrete examples of multilevel source polarization with countably
infinite levels when the group is locally cyclic. We derive this result via
elementary techniques in lattice theory.Comment: 12 pages, 1 figure, a short version has been accepted by the 2019
IEEE International Symposium on Information Theory (ISIT2019
Closed geodesics on connected sums and 3-manifolds
We study the asymptotics of the number N(t) of geometrically distinct closed
geodesics of a Riemannian or Finsler metric on a connected sum of two compact
manifolds of dimension at least three with non-trivial fundamental groups and
apply this result to the prime decomposition of a three-manifold. In particular
we show that the function N(t) grows at least like the prime numbers on a
compact 3-manifold with infinite fundamental group. It follows that a generic
Riemannian metric on a compact 3-manifold has infinitely many geometrically
distinct closed geodesics. We also consider the case of a connected sum of a
compact manifold with positive first Betti number and a simply-connected
manifold which is not homeomorphic to a sphere.Comment: 15 page
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