1,812 research outputs found
Asymptotic Distribution of Multilevel Channel Polarization for a Certain Class of Erasure Channels
This study examines multilevel channel polarization for a certain class of
erasure channels that the input alphabet size is an arbitrary composite number.
We derive limiting proportions of partially noiseless channels for such a
class. The results of this study are proved by an argument of convergent
sequences, inspired by Alsan and Telatar's simple proof of polarization, and
without martingale convergence theorems for polarization process.Comment: 31 pages; 1 figure; 1 table; a short version of this paper has been
submitted to the 2018 IEEE International Symposium on Information Theory
(ISIT2018
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
Motion of the Tippe Top : Gyroscopic Balance Condition and Stability
We reexamine a very classical problem, the spinning behavior of the tippe top
on a horizontal table. The analysis is made for an eccentric sphere version of
the tippe top, assuming a modified Coulomb law for the sliding friction, which
is a continuous function of the slip velocity at the point of
contact and vanishes at . We study the relevance of the gyroscopic
balance condition (GBC), which was discovered to hold for a rapidly spinning
hard-boiled egg by Moffatt and Shimomura, to the inversion phenomenon of the
tippe top. We introduce a variable so that corresponds to the GBC
and analyze the behavior of . Contrary to the case of the spinning egg,
the GBC for the tippe top is not fulfilled initially. But we find from
simulation that for those tippe tops which will turn over, the GBC will soon be
satisfied approximately. It is shown that the GBC and the geometry lead to the
classification of tippe tops into three groups: The tippe tops of Group I never
flip over however large a spin they are given. Those of Group II show a
complete inversion and the tippe tops of Group III tend to turn over up to a
certain inclination angle such that , when they are
spun sufficiently rapidly. There exist three steady states for the spinning
motion of the tippe top. Giving a new criterion for stability, we examine the
stability of these states in terms of the initial spin velocity . And we
obtain a critical value of the initial spin which is required for the
tippe top of Group II to flip over up to the completely inverted position.Comment: 52 pages, 11 figures, to be published in SIAM Journal on Applied
Dynamical Syste
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