@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-VR3DQVH1-X
  skos:prefLabel "one-parameter subgroup"@en, "sous-groupe à un paramètre"@fr ;
  a skos:Concept ;
  dc:modified "2023-08-30"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Sous-groupe_%C3%A0_un_param%C3%A8tre>, <https://en.wikipedia.org/wiki/One-parameter_group> ;
  skos:inScheme psr: ;
  skos:altLabel "one-parameter group"@en ;
  skos:definition """Un sous-groupe à un paramètre d'un groupe de Lie réel <i>G</i> est un morphisme de groupes de Lie <i>c</i> : ℝ → <i>G</i>. Plus explicitement, <i>c</i> est une application différentiable vérifiant :  <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\orall s,t\\\\in \\\\mathbb {R} ,c(t+s)=c(t)c(s)}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi mathvariant="normal">∀<!-- ∀ --></mi>
         <mi>s</mi>
         <mo>,</mo>
         <mi>t</mi>
         <mo>∈<!-- ∈ --></mo>
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">R</mi>
         </mrow>
         <mo>,</mo>
         <mi>c</mi>
         <mo stretchy="false">(</mo>
         <mi>t</mi>
         <mo>+</mo>
         <mi>s</mi>
         <mo stretchy="false">)</mo>
         <mo>=</mo>
         <mi>c</mi>
         <mo stretchy="false">(</mo>
         <mi>t</mi>
         <mo stretchy="false">)</mo>
         <mi>c</mi>
         <mo stretchy="false">(</mo>
         <mi>s</mi>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\orall s,t\\\\in \\\\mathbb {R} ,c(t+s)=c(t)c(s)}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c974ea63d4d7f32ebcfccb9acf78d2fed429a25a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.057ex; height:2.843ex;" alt="{\\\\displaystyle \\orall s,t\\\\in \\\\mathbb {R} ,c(t+s)=c(t)c(s)}"></span>.</dd></dl>
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Sous-groupe_%C3%A0_un_param%C3%A8tre">https://fr.wikipedia.org/wiki/Sous-groupe_%C3%A0_un_param%C3%A8tre</a>)"""@fr, """In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\varphi :\\\\mathbb {R} \\
ightarrow G}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>φ<!-- φ --></mi>
         <mo>:</mo>
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">R</mi>
         </mrow>
         <mo stretchy="false">→<!-- → --></mo>
         <mi>G</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\varphi :\\\\mathbb {R} \\
ightarrow G}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c6f5a08e92f887e36333c9af4d90bfeb2c67504" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.576ex; height:2.676ex;" alt="\\\\varphi :{\\\\mathbb  {R}}\\
ightarrow G"></span></dd></dl>
<br/>from the real line <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbb {R} }">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">R</mi>
         </mrow>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} }</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\\\\mathbb {R} "></span> (as an additive group) to some other topological group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle G}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>G</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle G}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="G"></span>. 
         If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\varphi }">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>φ<!-- φ --></mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\varphi }</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="\\\\varphi "></span> is injective then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\varphi (\\\\mathbb {R} )}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>φ<!-- φ --></mi>
         <mo stretchy="false">(</mo>
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">R</mi>
         </mrow>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\varphi (\\\\mathbb {R} )}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b44f20dcb0d5787767cde926d517fe43b053b7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.008ex; height:2.843ex;" alt="\\\\varphi ({\\\\mathbb  {R}})"></span>, the image, will be a subgroup of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle G}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>G</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle G}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="G"></span> that is isomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbb {R} }">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">R</mi>
         </mrow>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} }</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\\\\mathbb {R} "></span> as an additive group.
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/One-parameter_group">https://en.wikipedia.org/wiki/One-parameter_group</a>)"""@en ;
  dc:created "2023-08-30"^^xsd:date ;
  skos:broader psr:-RMQ1RP9W-P .

psr:-RMQ1RP9W-P
  skos:prefLabel "groupe de Lie"@fr, "Lie group"@en ;
  a skos:Concept ;
  skos:narrower psr:-VR3DQVH1-X .