We will discuss the possibility to 'prove' mathematical theorems with the aid of digital computers and present different methods to approach this goal. Special focus will be on integer and floating point arithmetic, on computer algebra methods as well as on so-called selfvalidating methods. This note can only cover a very small part of the subject and is intended to stimulate discussions, and possibly reconsideration of one or the other point of view.

View full abstract

Low-density parity-check codes and turbo codes are known as "capacity-approaching codes" since they achieve excellent error performance close to the theoretical limit. They are invented or rediscovered during 1990s and since then have been extensively investigated. This paper presents an overview of LDPC codes, turbo codes, and their iterative decoding.

View full abstract

Belief propagation is a universal method used in many field, such as AI, statistical physical, and error correction codes. It gives the exact inference when a graph is tree, but also a good approximation even if it is loopy. The authors have developed an information geometrical framework to analyze the belief propagation algorithm, which gives a unified view. In this article, the authors show the idea of the belief propagation algorithm, the information geometrical framework, and some results of the analysis.

View full abstract

This paper presents a tutorial introduction to a simple algorithmic solution to the convex optimization problem defined over the fixed point set of nonexpansive mapping in a real Hilbert space. The algorithmic solution was named the hybrid steepest descent method because it is constructed by blending important ideas in the steepest descent method and in the fixed point theory, and generates a sequence converging strongly to the solution of the problem. In this paper, a classical scheme named the projected gradient method is also highlighted for clarifying its limitation and motivations of the above mentioned problem. The remarkable applicability of the method to the broad range of convexly constrained generalized inverse problems is demonstrated based on a recent unified view of the problems.

View full abstract