See direct product in All languages combined, or Wiktionary
Download JSON data for direct product meaning in English (7.8kB)
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"ref": "1976, Marshall Hall, Jr., The Theory of Groups, 2nd edition, page 40",
"text": "Theorem 3.2.3. A periodic Abelian group is the direct product of its Sylow subgroups, S(p).",
"type": "quotation"
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"text": "A Boolean ring of order 2ⁿ (or dimension n) may be constructed as the direct product of n Boolean rings of dimension one."
}
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"Such a set of tuples formed from two or more rings, forming another ring whose operations arise from the component-wise application of the corresponding original ring operations."
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"(ring theory) Such a set of tuples formed from two or more rings, forming another ring whose operations arise from the component-wise application of the corresponding original ring operations."
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"ref": "1978, K. Itô, An Introduction to Probability Theory, page 53",
"text": "Let us start with the definition of the direct product of two probability measures. Let P#x5F;1 and P#x5F;2 be probability measures on #x5C;Omega#x5F;1 and #x5C;Omega#x5F;2, respectively, and denote #x5C;Omega#x5F;1#x5C;times#x5C;Omega#x5F;2 by #x5C;Omega. A probability measure P on #x5C;Omega with #x5C;mathfrak#x7B;D#x7D;#x5C;left(P#x5C;right)#x3D;#x5C;mathfrak#x7B;D#x7D;#x5C;left(P#x5F;1#x5C;right)#x5C;times#x5C;mathfrak#x7B;D#x7D;#x5C;left(P#x5F;2#x5C;right) is called the direct product of P#x5F;1 and P#x5F;2 (written P#x5F;1#x5C;timesP#x5F;2) if\nP(B_1×B_2)=P(B_1)P(B_2) B_i∈𝔇(P_i);i=1,2.\nThe probability space (Ω,P) is called the direct product of (Ω₁,P_1) and (Ω₂,P_2), written\n(Ω,P)=(Ω₁,P_1)×(Ω₂,P_2).\nFor example, the Lebesgue measure on [0, 1]² is the direct product of that on [0, 1] and itself.",
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"_dis1": "33 0 0 0 0 67",
"english": "synonymous in specific domains",
"word": "direct sum"
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"word": "product ring"
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"_dis1": "22 17 9 20 14 18",
"code": "it",
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"sense": "product of sets — see also Cartesian product",
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"word": "prodotto diretto"
}
],
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"type": "quotation"
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}
],
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"text": "Let us start with the definition of the direct product of two probability measures. Let P#x5F;1 and P#x5F;2 be probability measures on #x5C;Omega#x5F;1 and #x5C;Omega#x5F;2, respectively, and denote #x5C;Omega#x5F;1#x5C;times#x5C;Omega#x5F;2 by #x5C;Omega. A probability measure P on #x5C;Omega with #x5C;mathfrak#x7B;D#x7D;#x5C;left(P#x5C;right)#x3D;#x5C;mathfrak#x7B;D#x7D;#x5C;left(P#x5F;1#x5C;right)#x5C;times#x5C;mathfrak#x7B;D#x7D;#x5C;left(P#x5F;2#x5C;right) is called the direct product of P#x5F;1 and P#x5F;2 (written P#x5F;1#x5C;timesP#x5F;2) if\nP(B_1×B_2)=P(B_1)P(B_2) B_i∈𝔇(P_i);i=1,2.\nThe probability space (Ω,P) is called the direct product of (Ω₁,P_1) and (Ω₂,P_2), written\n(Ω,P)=(Ω₁,P_1)×(Ω₂,P_2).\nFor example, the Lebesgue measure on [0, 1]² is the direct product of that on [0, 1] and itself.",
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This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-05 from the enwiktionary dump dated 2024-05-02 using wiktextract (f4fd8c9 and c9440ce). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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