Yuri I. Manin
Error-correcting codes and phase transitions.
Wed, 10/28/2009 - 16:01 — SomNambulAuthors: Yuri I. Manin, Matilde Marcolli
Subjects: Information Theory
AbstractThe theory of error-correcting codes is concerned with constructing codes
that optimize simultaneously transmission rate and relative minimum distance.
These conflicting requirements determine an asymptotic bound, which is a
continuous curve in the space of parameters. The main goal of this paper is to
relate the asymptotic bound to phase diagrams of quantum statistical mechanical
systems. We first identify the code parameters with Hausdorff and von Neumann
dimensions, by considering fractals consisting of infinite sequences of code
words.Renormalization and Computation II: Time Cut-off and the Halting Problem.
Tue, 08/25/2009 - 22:41 — SomNambulAuthors: Yuri I. Manin
Subjects: Quantum Algebra
AbstractThis is the second installment to the project initiated in [Ma3]. In the
first Part, I argued that both philosophy and technique of the perturbative
renormalization in quantum field theory could be meaningfully transplanted to
the theory of computation, and sketched several contexts supporting this view.In this second part, I address some of the issues raised in [Ma3] and provide
their development in three contexts: a categorification of the algorithmic
computations; time cut--off and Anytime Algorithms; and finally, a Hopf algebra
renormalization of the Halting Problem.- 10 comments
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