Generalizations of Fano's Inequality for Conditional Information Measures via Majorization Theory

Fano's inequality is one of the most elementary, ubiquitous, and important tools in information theory. Using majorization theory, Fano's inequality is generalized to a broad class of information measures, which contains those of Shannon and R\'{e}nyi. When specialized to these measures, it recovers and generalizes the classical inequalities. Key to the derivation is the construction of an appropriate conditional distribution inducing a desired marginal distribution on a countably infinite alphabet. The construction is based on the infinite-dimensional version of Birkhoff's theorem proven by R\'{e}v\'{e}sz [Acta Math. Hungar. 1962, 3, 188{\textendash}198], and the constraint of maintaining a desired marginal distribution is similar to coupling in probability theory. Using our Fano-type inequalities for Shannon's and R\'{e}nyi's information measures, we also investigate the asymptotic behavior of the sequence of Shannon's and R\'{e}nyi's equivocations when the error probabilities vanish. This asymptotic behavior provides a novel characterization of the asymptotic equipartition property (AEP) via Fano's inequality.Comment: 44 pages, 3 figure

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

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

Development of a rhythmic dance curriculum designed to stimulate "independent, interactive and deep learning"

In this study of a curriculum designed for rhythm-based dance, we consciously considered the stage of the participants’ development and the characteristics of rhythmic play, and practiced using "rhythm" and "expression" as keywords. As a result, the following points became clear. (1) The use of onomatopoeia, such as clapping and "don", which are frequently used on a daily basis, was effective in training the rhythmic feeling in lower grades. (2) Proceeding with the selection of songs, selecting songs with a BPM（Beat Per Minute） of about 100, is effective in training lower- and higher-grade children as it allows them to feel the rhythm of the music. (3) Junior high school students tend to have a higher interest in complex rhythms, and tend to learn the rhythms by imitating sample images from ICT equipment. (4) The “step acquisition” that triggers the movement and the activities that allow the children to express themselves and led to widespread improvisation among the younger children. (5) In higher-grade children, expressions of originality created new perspectives on concrete expressions of space, time, and power. (6) In junior high school students, improvised dance is performed through activities such as choreography, strength, strength, body angle, facial expression, and formation through activities that imitate sample images using ICT equipment. It is thought that it became possible