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About: Segre's theorem
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This statement was assumed 1949 by the two Finnish mathematicians and and its proof was published in 1955 by B. Segre. In pappian projective planes of even order greater than four there are ovals which are not conics. In an infinite plane there exist ovals, which are not conics. 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title="Switch to /sparql endpoint"><i class="bi-box-arrow-up-right"></i> Sparql Endpoint </a> </li> </ul> </div> </div> </nav> <div style="margin-bottom: 60px"></div> <!-- /navbar --> <!-- page-header --> <section> <div class="container-xl"> <div class="row"> <div class="col"> <h1 id="title" class="display-6"><b>About:</b> <a href="http://dbpedia.org/resource/Segre's_theorem">Segre's theorem</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> <span class="text-nowrap">An Entity of Type: <a href="javascript:void()">Thing</a>, </span> <span class="text-nowrap">from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, </span> <span class="text-nowrap">within Data Space: <a href="http://dbpedia.org">dbpedia.org</a></span> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> <p class="lead">In projective geometry, Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement: * Any oval in a finite pappian projective plane of odd order is a nondegenerate projective conic section. This statement was assumed 1949 by the two Finnish mathematicians and and its proof was published in 1955 by B. Segre. In pappian projective planes of even order greater than four there are ovals which are not conics. In an infinite plane there exist ovals, which are not conics. In the real plane one just glues a half of a circle and a suitable ellipse smoothly.</p> </div> <div class="col-xs-3 col-sm-2"> <a href="#" class="thumbnail"> <img src="http://commons.wikimedia.org/wiki/Special:FilePath/Oval-def-fin.svg?width=300" alt="thumbnail" class="img-fluid" /> </a> </div> </div> </div> </section> <!-- page-header --> <!-- property-table --> <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract"><small>dbo:</small>abstract</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="de" >Der Satz von Segre, benannt nach dem italienischen Mathematiker Beniamino Segre, ist in der projektiven Geometrie die Aussage: * In einer endlichen pappusschen projektiven Ebene ungerader Ordnung ist jedes Oval ein Kegelschnitt. Die Aussage wurde 1949 von den finnischen Mathematikern G. Järnefelt und P. Kustaanheimo vermutet und ihr Beweis 1955 von B. Segre publiziert. Eine endliche pappussche projektive Ebene kann man sich in inhomogenen Koordinaten wie die reelle projektive Ebene beschrieben denken, nur dass man statt der reellen Zahlen einen endlichen Körper benutzt. Ungerader Ordnung bedeutet, dass ungerade ist. Ein Oval ist eine kreisähnliche Kurve (s. u.): Eine Gerade schneidet höchstens 2-mal und in jedem Punkt gibt es genau eine Tangente. Die Standardbeispiele von Ovalen sind die nicht ausgearteten (projektiven) Kegelschnitte. Der Satz von Segre hat für endliche Ovale eine sehr große Bedeutung, da es im pappusschen ungeraden Fall außer den Kegelschnitten keine weiteren Ovale geben kann. Im Gegensatz zu geraden pappussche Ebenen: Hier gibt es Ovale, die keine Kegelschnitte sind (s. Satz von Qvist). In unendlichen pappusschen Ebenen gibt es Ovale, die keine Kegelschnitte sind. Im Reellen muss man nur einen Halbkreis glatt mit einer geeigneten Halbellipse zusammensetzen. Der Beweis des Satzes für den Nachweis, dass das gegebene Oval ein Kegelschnitt ist, wird mit Hilfe der 3-Punkte-Ausartung des Satzes von Pascal geführt. Dabei wird die für Körper ungerader Ordnung typische Eigenschaft, dass das Produkt aller Elemente, die nicht 0 sind, gleich −1 ist, verwendet.</span><small> (de)</small></span></li> <li><span class="literal"><span property="dbo:abstract" lang="en" >In projective geometry, Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement: * Any oval in a finite pappian projective plane of odd order is a nondegenerate projective conic section. This statement was assumed 1949 by the two Finnish mathematicians and and its proof was published in 1955 by B. Segre. A finite pappian projective plane can be imagined as the projective closure of the real plane (by a line at infinity), where the real numbers are replaced by a finite field K. Odd order means that |K| = n is odd. An oval is a curve similar to a circle (see definition below): any line meets it in at most 2 points and through any point of it there is exactly one tangent. The standard examples are the nondegenerate projective conic sections. In pappian projective planes of even order greater than four there are ovals which are not conics. In an infinite plane there exist ovals, which are not conics. In the real plane one just glues a half of a circle and a suitable ellipse smoothly. The proof of Segre's theorem, shown below, uses the 3-point version of Pascal's theorem and a property of a finite field of odd order, namely, that the product of all the nonzero elements equals -1.</span><small> (en)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/thumbnail"><small>dbo:</small>thumbnail</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" rel="dbo:thumbnail" resource="http://commons.wikimedia.org/wiki/Special:FilePath/Oval-def-fin.svg?width=300" href="http://commons.wikimedia.org/wiki/Special:FilePath/Oval-def-fin.svg?width=300"><small>wiki-commons</small>:Special:FilePath/Oval-def-fin.svg?width=300</a></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageExternalLink"><small>dbo:</small>wikiPageExternalLink</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" 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resource="http://dbpedia.org/resource/Category:Conic_sections" href="http://dbpedia.org/resource/Category:Conic_sections"><small>dbc</small>:Conic_sections</a></span></li> <li><span class="literal"><a class="uri" rel="dct:subject" resource="http://dbpedia.org/resource/Category:Incidence_geometry" href="http://dbpedia.org/resource/Category:Incidence_geometry"><small>dbc</small>:Incidence_geometry</a></span></li> <li><span class="literal"><a class="uri" rel="dct:subject" resource="http://dbpedia.org/resource/Category:Theorems_in_projective_geometry" href="http://dbpedia.org/resource/Category:Theorems_in_projective_geometry"><small>dbc</small>:Theorems_in_projective_geometry</a></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#comment"><small>rdfs:</small>comment</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="de" >Der Satz von Segre, benannt nach dem italienischen Mathematiker Beniamino Segre, ist in der projektiven Geometrie die Aussage: * In einer endlichen pappusschen projektiven Ebene ungerader Ordnung ist jedes Oval ein Kegelschnitt. Die Aussage wurde 1949 von den finnischen Mathematikern G. Järnefelt und P. Kustaanheimo vermutet und ihr Beweis 1955 von B. Segre publiziert.</span><small> (de)</small></span></li> <li><span class="literal"><span property="rdfs:comment" lang="en" >In projective geometry, Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement: * Any oval in a finite pappian projective plane of odd order is a nondegenerate projective conic section. This statement was assumed 1949 by the two Finnish mathematicians and and its proof was published in 1955 by B. Segre. In pappian projective planes of even order greater than four there are ovals which are not conics. In an infinite plane there exist ovals, which are not conics. In the real plane one just glues a half of a circle and a suitable ellipse smoothly.</span><small> (en)</small></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label"><small>rdfs:</small>label</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="de" >Satz von Segre (Projektive Geometrie)</span><small> (de)</small></span></li> <li><span class="literal"><span property="rdfs:label" lang="en" >Segre's theorem</span><small> (en)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#sameAs"><small>owl:</small>sameAs</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" rel="owl:sameAs" resource="http://yago-knowledge.org/resource/Segre's_theorem" href="http://yago-knowledge.org/resource/Segre's_theorem"><small>yago-res</small>:Segre's 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