@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:-JN985JTM-R
  skos:prefLabel "distribution"@fr, "distribution"@en ;
  a skos:Concept ;
  skos:narrower psr:-DVQD50Q7-R .

psr: a skos:ConceptScheme .
psr:-DVQD50Q7-R
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:prefLabel "bump function"@en, "fonction C∞ à support compact"@fr ;
  dc:created "2023-08-04"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Bump_function>, <https://fr.wikipedia.org/wiki/Fonction_C%E2%88%9E_%C3%A0_support_compact> ;
  skos:altLabel "test function"@en, "fonction test"@fr ;
  dc:modified "2023-08-08"^^xsd:date ;
  skos:definition """In mathematics, a <b>bump function</b> (also called a <b>test function</b>) is a function <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 f:\\\\mathbb {R} ^{n}\\	o \\\\mathbb {R} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo>:</mo>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="double-struck">R</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">R</mi>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f:\\\\mathbb {R} ^{n}\\	o \\\\mathbb {R} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/306c097f43c91dce633d12cde024948d39e73752" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\\\\displaystyle f:\\\\mathbb {R} ^{n}\\	o \\\\mathbb {R} }"></span> on a Euclidean space <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} ^{n}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="double-struck">R</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} ^{n}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\\\\displaystyle \\\\mathbb {R} ^{n}}"></span> which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported.  The set of all bump functions with domain <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} ^{n}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="double-struck">R</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} ^{n}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\\\\displaystyle \\\\mathbb {R} ^{n}}"></span> forms a vector space, denoted <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 \\\\mathrm {C} _{0}^{\\\\infty }(\\\\mathbb {R} ^{n})}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">C</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="double-struck">R</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathrm {C} _{0}^{\\\\infty }(\\\\mathbb {R} ^{n})}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c42ad5e1bd5583c3f724800d5175f1ad4d8e0f3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.259ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mathrm {C} _{0}^{\\\\infty }(\\\\mathbb {R} ^{n})}"></span> or <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 \\\\mathrm {C} _{\\\\mathrm {c} }^{\\\\infty }(\\\\mathbb {R} ^{n}).}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">C</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">c</mi>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="double-struck">R</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathrm {C} _{\\\\mathrm {c} }^{\\\\infty }(\\\\mathbb {R} ^{n}).}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7f95aa101a636985c636ec114ddd6920598fe7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.906ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mathrm {C} _{\\\\mathrm {c} }^{\\\\infty }(\\\\mathbb {R} ^{n}).}"></span>  The dual space of this space endowed with a suitable topology is the space of distributions. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Bump_function">https://en.wikipedia.org/wiki/Bump_function</a>)"""@en, """En mathématiques, une <b>fonction C<sup>∞</sup> à support compact</b> (également appelée <b>fonction test</b>) est une fonction infiniment dérivable dont le support est compact. 
<br/>Ces fonctions sont au cœur de la théorie des distributions, puisque ces dernières sont construites comme éléments du dual topologique de l'espace des fonctions tests.
<br/>Les fonctions C<sup>∞</sup> à support compact sont également utilisées pour construire des suites régularisantes et des partitions de l'unité de classe C<sup>∞</sup>.
<br/>Si Ω est un ouvert non vide de ℝ<sup><i>n</i></sup>, l'espace des fonctions C<sup>∞</sup> à support compact de Ω dans ℝ est noté <i>C</i><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="vertical-align:baseline">∞</sup><br><sub style="vertical-align:baseline"><i>c</i></sub></span>(Ω), <span class="nowrap"><i>C</i><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="vertical-align:baseline">∞</sup><br><sub style="vertical-align:baseline">0</sub></span>(Ω)</span> ou 𝒟(Ω). 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_C%E2%88%9E_%C3%A0_support_compact">https://fr.wikipedia.org/wiki/Fonction_C%E2%88%9E_%C3%A0_support_compact</a>)"""@fr ;
  skos:broader psr:-JN985JTM-R .