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Novel Sequence Discovery by Subtractive Genomics
09:40

Novel Sequence Discovery by Subtractive Genomics

Published on: January 25, 2019

A zoo of computable binary normal sequences.

Steve Pincus1, Burton H Singer

  • 1stevepincus@alum.mit.edu

Proceedings of the National Academy of Sciences of the United States of America
|November 6, 2012
PubMed
Summary
This summary is machine-generated.

This study introduces new algorithms for generating normal numbers, which are candidates for certifiably random sequences. These methods allow precise control over convergence rates and bias, advancing the creation of random number sequences.

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Area of Science:

  • Number Theory
  • Probability Theory
  • Theoretical Computer Science

Background:

  • Historically, generating certifiably random infinite sequences has lacked constructive methods.
  • Normal numbers, exhibiting limiting equidistribution of subblocks, are candidates for random sequences.
  • Previous work provided algorithms for generating normal numbers.

Purpose of the Study:

  • To develop and systematize algorithmic approaches for generating normal numbers.
  • To construct normal numbers with specified rates of convergence and controlled bias.
  • To explore implications for probability theory and randomness metrics.

Main Methods:

  • Developing explicit algorithms for normal number generation.
  • Classifying normal numbers based on deviation from maximal irregularity.
  • Constructing families of biased normal numbers with specified convergence rates.
  • Analyzing convergence properties of singleton and pair blocks.

Main Results:

  • Delineated sets of normal numbers with controllable deviation rates and asymmetry.
  • Explicit construction of a normal number satisfying the Law of the Iterated Logarithm.
  • Families of biased normal numbers with arbitrary specified convergence rates.
  • Construction of a normal sequence with differing convergence rates for singleton and pair blocks.

Conclusions:

  • The developed algorithms offer unprecedented control over the properties of normal numbers.
  • These findings provide new tools for generating and analyzing sequences with near-random properties.
  • The results have significant implications for probability theory and the mathematical understanding of randomness.