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Publication

Ergodic Theory Meets Polarization I: A Foundation of Polarization Theory

Rajai Nasser
2015
Conference Papers

Abstract

An open problem in polarization theory is to determine the binary operations that always lead to polarization when they are used in Arıkan style constructions. This paper solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. The characterization is given in terms of a new mathematical framework that we introduce. We show that a binary operation is polarizing if and only if its inverse is strongly ergodic.

Official source

Rajai Nasser
2015
Conference Papers

Abstract

Official source

Related concepts (19)

Binary tree

In computer science, a binary tree is a k-ary tree data structure in which each node has at most two children, which are referred to as the and the . A recursive definition using just set theory notions is that a (non-empty) binary tree is a tuple (L, S, R), where L and R are binary trees or the empty set and S is a singleton set containing the root. Some authors allow the binary tree to be the empty set as well. From a graph theory perspective, binary (and K-ary) trees as defined here are arborescences.

Polarization (physics)

Polarization (also polarisation) is a property of transverse waves which specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string (see image); for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string.

Binary operation

In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary operation on a set is a binary operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, and multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups.

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