In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of classical discrepancy theory, namely distributing points in some space such that they are evenly distributed with respect to some (mostly geometrically defined) subsets. The discrepancy (irregularity) measures how far a given distribution deviates from an ideal one.
Discrepancy theory can be described as the study of inevitable irregularities of distributions, in measure-theoretic and combinatorial settings. Just as Ramsey theory elucidates the impossibility of total disorder, discrepancy theory studies the deviations from total uniformity.
A significant event in the history of discrepancy theory was the 1916 paper of Weyl on the uniform distribution of sequences in the unit interval.[1]
^Weyl, Hermann (1 September 1916). "Über die Gleichverteilung von Zahlen mod. Eins" [About the equal distribution of numbers]. Mathematische Annalen (in German). 77 (3): 313–352. doi:10.1007/BF01475864. ISSN 1432-1807. S2CID 123470919.
^József Beck and Tibor Fiala (1981). ""Integer-making" theorems". Discrete Applied Mathematics. 3 (1): 1–8. doi:10.1016/0166-218x(81)90022-6.
^Joel Spencer (June 1985). "Six Standard Deviations Suffice". Transactions of the American Mathematical Society. 289 (2). Transactions of the American Mathematical Society, Vol. 289, No. 2: 679–706. doi:10.2307/2000258. JSTOR 2000258.
^Spielman, Daniel (11 May 2020). "Using discrepancy theory to improve the design of randomized controlled trials". {{cite journal}}: Cite journal requires |journal= (help)
Further readingedit
Beck, József; Chen, William W. L. (1987). Irregularities of Distribution. New York: Cambridge University Press. ISBN0-521-30792-9.
Matousek, Jiri (1999). Geometric Discrepancy: An Illustrated Guide. Algorithms and combinatorics. Vol. 18. Berlin: Springer. ISBN3-540-65528-X.
January 01, 1970
discrepancy, theory, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, scholar, jstor, januar. This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Discrepancy theory news newspapers books scholar JSTOR January 2018 Learn how and when to remove this message In mathematics discrepancy theory describes the deviation of a situation from the state one would like it to be in It is also called the theory of irregularities of distribution This refers to the theme of classical discrepancy theory namely distributing points in some space such that they are evenly distributed with respect to some mostly geometrically defined subsets The discrepancy irregularity measures how far a given distribution deviates from an ideal one Discrepancy theory can be described as the study of inevitable irregularities of distributions in measure theoretic and combinatorial settings Just as Ramsey theory elucidates the impossibility of total disorder discrepancy theory studies the deviations from total uniformity A significant event in the history of discrepancy theory was the 1916 paper of Weyl on the uniform distribution of sequences in the unit interval 1 Contents 1 Theorems 2 Major open problems 3 Applications 4 See also 5 References 6 Further readingTheorems editDiscrepancy theory is based on the following classic theorems The theorem of van Aardenne Ehrenfest Axis parallel rectangles in the plane Roth Schmidt Discrepancy of half planes Alexander Matousek Arithmetic progressions Roth Sarkozy Beck Matousek amp Spencer Beck Fiala theorem 2 Six Standard Deviations Suffice Spencer 3 Major open problems editThe unsolved problems relating to discrepancy theory include Axis parallel rectangles in dimensions three and higher folklore Komlos conjecture Heilbronn triangle problem on the minimum area of a triangle determined by three points from an n point setApplications editApplications for discrepancy theory include Numerical integration Monte Carlo methods in high dimensions Computational geometry Divide and conquer algorithm Image processing Halftoning Random trial formulation 4 Randomized controlled trialSee also editDiscrepancy of hypergraphsReferences edit Weyl Hermann 1 September 1916 Uber die Gleichverteilung von Zahlen mod Eins About the equal distribution of numbers Mathematische Annalen in German 77 3 313 352 doi 10 1007 BF01475864 ISSN 1432 1807 S2CID 123470919 Jozsef Beck and Tibor Fiala 1981 Integer making theorems Discrete Applied Mathematics 3 1 1 8 doi 10 1016 0166 218x 81 90022 6 Joel Spencer June 1985 Six Standard Deviations Suffice Transactions of the American Mathematical Society 289 2 Transactions of the American Mathematical Society Vol 289 No 2 679 706 doi 10 2307 2000258 JSTOR 2000258 Spielman Daniel 11 May 2020 Using discrepancy theory to improve the design of randomized controlled trials a href Template Cite journal html title Template Cite journal cite journal a Cite journal requires journal help Further reading editBeck Jozsef Chen William W L 1987 Irregularities of Distribution New York Cambridge University Press ISBN 0 521 30792 9 Chazelle Bernard 2000 The Discrepancy Method Randomness and Complexity New York Cambridge University Press ISBN 0 521 77093 9 Matousek Jiri 1999 Geometric Discrepancy An Illustrated Guide Algorithms and combinatorics Vol 18 Berlin Springer ISBN 3 540 65528 X Retrieved from https en wikipedia org w index php title Discrepancy theory amp oldid 1208065716, wikipedia, wiki, book, books, library,
