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{{Short description|Technique of studying linear partial differential equations}}
{{dablink|Not to be confused with the common phrase "algebraic analysis of <nowiki></nowiki>", meaning "the algebraic study of <nowiki></nowiki>"}} {{Hatnote|Not to be confused with the common phrase "algebraic analysis of <nowiki></nowiki>", meaning "the algebraic study of <nowiki></nowiki>"}}
'''Algebraic analysis''' is an area of ] that deals with systems of linear ]s by using ] and ] to study properties and generalizations of functions such as ]s and microfunctions. As a research programme, it was started by ] in 1959.<ref>{{cite article|title=Professor Mikio Sato and Microlocal Analysis|author1=Masaki Kashiwara|author2=Takahiro Kawai|journal=PRIMS|volume=47|issue=1|year=2011|url=http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=47&iss=1&rank=2|doi=10.2977/PRIMS/29|via=EMS-PH}}</ref>
'''Algebraic analysis''' is an area of ] that deals with systems of ] by using ] and ] to study properties and generalizations of ] such as ]s and microfunctions. Semantically, it is the application of algebraic operations on analytic quantities. As a research programme, it was started by the Japanese mathematician ] in 1959.{{sfn|Kashiwara|Kawai|2011|pp=11–17}} This can be seen as an algebraic geometrization of analysis. It derives its meaning from the fact that the differential operator is right-invertible in several function spaces.

It helps in the simplification of the proofs due to an algebraic description of the problem considered.


== Microfunction == == Microfunction ==
{{expand section|date=September 2019}} {{expand section|date=September 2019}}
Let ''M'' be a real-analytic manifold and ''X'' its complexification. (The definition of microfunctions here). Let ''M'' be a ]-] of ] ''n'', and let ''X'' be its complexification. The sheaf of '''microlocal functions''' on ''M'' is given as{{sfn|Kashiwara|Schapira|1990|loc=Definition 11.5.1}}
:<math>\mathcal{H}^n(\mu_M(\mathcal{O}_X) \otimes \mathcal{or}_{M/X})</math>
where
* <math>\mu_M</math> denotes the ],
* <math>\mathcal{or}_{M/X}</math> is the ].<!-- need to give a more gentle definition -->


A microfunction can be used to define a hyper function. By definition, the sheaf of ]s on ''M'' is the restriction of the sheaf of microfunctions to ''M'', in parallel to the fact the sheaf of real-analytic functions on ''M'' is the restriction of the sheaf of holomorphic functions on ''X'' to ''M''. A microfunction can be used to define a Sato's ]. By definition, the sheaf of ]s on ''M'' is the restriction of the sheaf of microfunctions to ''M'', in parallel to the fact the sheaf of ] on ''M'' is the restriction of the sheaf of ]s on ''X'' to ''M''.


==See also== ==See also==
*] * ]
*] * ]
*] * ]
*] * ]
*] * ]
*] * ]
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*] * ]
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*] * ]

== Citations ==
{{Reflist}}

==Sources==
{{refbegin}}
*{{cite journal | title = Professor Mikio Sato and Microlocal Analysis
| last1 = Kashiwara | first1 = Masaki
| last2 = Kawai | first2 = Takahiro
| author1-link = Masaki Kashiwara
| author2-link = Takahiro Kawai
| journal = Publications of the Research Institute for Mathematical Sciences | via = EMS-PH
| year = 2011 | volume = 47 | issue = 1 | pages = 11–17
| url = http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=47&iss=1&rank=2
| doi = 10.2977/PRIMS/29
| doi-access = free
}}
*{{cite book| title = Sheaves on Manifolds
| last1 = Kashiwara | first1 = Masaki
| last2 = Schapira | first2 = Pierre
| author2-link = Pierre Schapira (mathematician)
| year = 1990
| publisher = Springer-Verlag | location = Berlin
| isbn = 3-540-51861-4
}}
{{refend}}


==Further reading== ==Further reading==
* * {{Webarchive|url=https://web.archive.org/web/20120225173659/http://people.math.jussieu.fr/~schapira/mispapers/Masaki.pdf |date=2012-02-25 }}
* *

{{Authority control}}


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{{mathanalysis-stub}} {{mathanalysis-stub}}

Latest revision as of 08:52, 16 August 2023

Technique of studying linear partial differential equations Not to be confused with the common phrase "algebraic analysis of ", meaning "the algebraic study of "

Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunctions. Semantically, it is the application of algebraic operations on analytic quantities. As a research programme, it was started by the Japanese mathematician Mikio Sato in 1959. This can be seen as an algebraic geometrization of analysis. It derives its meaning from the fact that the differential operator is right-invertible in several function spaces.

It helps in the simplification of the proofs due to an algebraic description of the problem considered.

Microfunction

This section needs expansion. You can help by adding to it. (September 2019)

Let M be a real-analytic manifold of dimension n, and let X be its complexification. The sheaf of microlocal functions on M is given as

H n ( μ M ( O X ) ⊗ o r M / X ) {\displaystyle {\mathcal {H}}^{n}(\mu _{M}({\mathcal {O}}_{X})\otimes {\mathcal {or}}_{M/X})}

where

A microfunction can be used to define a Sato's hyperfunction. By definition, the sheaf of Sato's hyperfunctions on M is the restriction of the sheaf of microfunctions to M, in parallel to the fact the sheaf of real-analytic functions on M is the restriction of the sheaf of holomorphic functions on X to M.

See also

Citations

  1. Kashiwara & Kawai 2011, pp. 11–17.
  2. Kashiwara & Schapira 1990, Definition 11.5.1.

Sources

Further reading


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