See π-system in All languages combined, or Wiktionary
Download JSON data for π-system meaning in English (4.6kB)
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{
"ref": "2007, Rabi Bhattacharya, Edward C. Waymire, A Basic Course in Probability Theory, Springer, page 49",
"text": "To see this, first check that #x5C;sigma(X#x5F;0,X#x5F;1,#x5C;dots)#x3D;#x5C;sigma(#x5C;mathcalF#x5F;0), where #x5C;textstyle#x5C;mathcalF#x5F;0#x3A;#x3D;#x5C;bigcup#x5C;infty#x5F;#x7B;k#x3D;0#x7D;#x5C;sigma(X#x5F;0,#x5C;dots,X#x5F;k) is a field and, in particular, a #x5C;boldsymbol#x5C;pi-system.",
"type": "quotation"
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"ref": "2017, Willem Adriaan de Graaf, Computation with Linear Algebraic Groups, Taylor & Francis (CRC Press), page 221",
"text": "We start with a basis of simple roots #x5C;Delta of #x5C;Phi. Then we apply all possible elementary transformations and add the resulting #x5C;boldsymbol#x5C;pi-systems to the list. Of course, if #x5C;Gamma is a #x5C;boldsymbol#x5C;pi-system, and #x5C;Gamma' is a #x5C;boldsymbol#x5C;pi-system obtained from it by an elementary transformation and the diagrams of #x5C;Gamma and #x5C;Gamma' are the same, the root subsystems they span are the same, and therefore we do not add #x5C;Gamma'.",
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"ref": "2021, Jeremy J. Becnel, Tools for Infinite Dimensional Analysis, Taylor & Francis (CRC Press)",
"text": "Clearly the definitions for a #x5C;boldsymbol#x5C;pi-system and a #x5C;lambda-system are both satisfied by a #x5C;sigma-algebra.[…]\nProposition 4.1.8 Let #x5C;Omega be a set and B be a collection of subsets of #x5C;Omega. The collection B is a #x5C;sigma-algebra if and only if B is a #x5C;lambda-system and a #x5C;boldsymbol#x5C;pi-system.",
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"A non-empty collection of subsets of a given set Ώ that is closed under non-empty finite intersections."
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"(set theory, measure theory, probability theory) A non-empty collection of subsets of a given set Ώ that is closed under non-empty finite intersections."
],
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"word": "δ-ring"
},
{
"english": "type of chemical bond",
"word": "π bond"
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"roman": "system involving π bonds",
"word": "π system"
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"word": "π-calculus"
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"code": "fr",
"lang": "French",
"sense": "collection of subsets closed under non-empty finite intersections",
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"word": "π-système"
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"sense": "collection of subsets closed under non-empty finite intersections",
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"word": "π-System"
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"sense": "collection of subsets closed under non-empty finite intersections",
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"word": "π-sistema"
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"ref": "2007, Rabi Bhattacharya, Edward C. Waymire, A Basic Course in Probability Theory, Springer, page 49",
"text": "To see this, first check that #x5C;sigma(X#x5F;0,X#x5F;1,#x5C;dots)#x3D;#x5C;sigma(#x5C;mathcalF#x5F;0), where #x5C;textstyle#x5C;mathcalF#x5F;0#x3A;#x3D;#x5C;bigcup#x5C;infty#x5F;#x7B;k#x3D;0#x7D;#x5C;sigma(X#x5F;0,#x5C;dots,X#x5F;k) is a field and, in particular, a #x5C;boldsymbol#x5C;pi-system.",
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{
"ref": "2017, Willem Adriaan de Graaf, Computation with Linear Algebraic Groups, Taylor & Francis (CRC Press), page 221",
"text": "We start with a basis of simple roots #x5C;Delta of #x5C;Phi. Then we apply all possible elementary transformations and add the resulting #x5C;boldsymbol#x5C;pi-systems to the list. Of course, if #x5C;Gamma is a #x5C;boldsymbol#x5C;pi-system, and #x5C;Gamma' is a #x5C;boldsymbol#x5C;pi-system obtained from it by an elementary transformation and the diagrams of #x5C;Gamma and #x5C;Gamma' are the same, the root subsystems they span are the same, and therefore we do not add #x5C;Gamma'.",
"type": "quotation"
},
{
"ref": "2021, Jeremy J. Becnel, Tools for Infinite Dimensional Analysis, Taylor & Francis (CRC Press)",
"text": "Clearly the definitions for a #x5C;boldsymbol#x5C;pi-system and a #x5C;lambda-system are both satisfied by a #x5C;sigma-algebra.[…]\nProposition 4.1.8 Let #x5C;Omega be a set and B be a collection of subsets of #x5C;Omega. The collection B is a #x5C;sigma-algebra if and only if B is a #x5C;lambda-system and a #x5C;boldsymbol#x5C;pi-system.",
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"(set theory, measure theory, probability theory) A non-empty collection of subsets of a given set Ώ that is closed under non-empty finite intersections."
],
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"sense": "collection of subsets closed under non-empty finite intersections",
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"word": "pi-système"
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"sense": "collection of subsets closed under non-empty finite intersections",
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"word": "π-System"
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{
"code": "it",
"lang": "Italian",
"sense": "collection of subsets closed under non-empty finite intersections",
"tags": [
"masculine"
],
"word": "π-sistema"
},
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"sense": "collection of subsets closed under non-empty finite intersections",
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