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Induced topology - Revision history
2024-03-29T05:23:30Z
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80.168.197.71: == See also == * Natural topology
2012-11-22T12:50:52Z
<p>== See also == * <a href="/index.php?title=Natural_topology&action=edit&redlink=1" class="new" title="Natural topology (page does not exist)">Natural topology</a></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:50, 22 November 2012</td>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[File:Ion Barbu.jpg|thumb|200px|Ion Barbu]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''Ion Barbu''' ({{IPA-ro|iˈon ˈbarbu}}; [[pen name]] of '''Dan Barbilian'''; 18 March 1895 &ndash;11 August 1961) was a [[Romania]]n [[mathematician]] and [[poet]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Early life==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Born in [[Câmpulung-Muscel]], [[Argeş County]], he was the son of Constantin Barbilian and Smaranda, born Şoiculescu. He attended the [[Ion Brătianu National College (Pitești)|Ion Brătianu High School]] in [[Piteşti]] and the [[Gheorghe Lazăr National College (Bucharest)|Gheorghe Lazăr High School]] in [[Bucharest]]. During that time, he discovered that he had a talent for mathematics, and started publishing in ''Gazeta Matematică'', one of the most prestigious math publications of that time;{{cn|date=August 2013}} it was also then that he discovered his passion for [[poetry]]. Ion Barbu was known as "one of the greatest Romanian poets of the twentieth century and perhaps the greatest of all" according to Romanian literary critic [[Alexandru Ciorănescu]]</ins>.<ins style="font-weight: bold; text-decoration: none;">{{cn|date=August 2013}} As a poet, he </ins>is <ins style="font-weight: bold; text-decoration: none;">best known for his volume ''Joc secund'' ("Mirrored Play").{{cn|date=August 2013}} </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">He was a student at the [[University of Bucharest]] when [[World War </ins>I<ins style="font-weight: bold; text-decoration: none;">]] caused his studies to be interrupted by military service. He completed his degree in 1921. He then went to [[Göttingen University]] to study [[number theory]] with [[Edmund Landau]] for two years. Returning to Bucharest, he studied with [[Gheorghe Țițeica]], completing his thesis in 1929<ref>Wladimir G. Boskoff, Bogdan Suceavă (2007) "Barbilian spaces: the history of a geometric idea", [</ins>[<ins style="font-weight: bold; text-decoration: none;">Historia Mathematica]] 34(2)</ins>: <ins style="font-weight: bold; text-decoration: none;">221&ndash;224. {{doi|10.1016/j.hm.2006.06.001}} {{MathSciNet|id=2320101}}<</ins>/<ins style="font-weight: bold; text-decoration: none;">ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Apollonian metric==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In 1934, Barbilian published his article<ref>"Einordnung von Lobayschewskys Massenbestimmung in einer gewissen algemeinen Metrik der Jordansche Bereiche", ''Casopis Matematiky a Fysiky'' 64:182,3<</ins>/<ins style="font-weight: bold; text-decoration: none;">ref> describing metrization of a region ''K'', the interior of a [[simple closed curve]] ''J''</ins>. <ins style="font-weight: bold; text-decoration: none;">Let ''x''y denote the [[Euclidean distance]] from ''x'' to ''y''</ins>. <ins style="font-weight: bold; text-decoration: none;">Barbilian’s function for the distance from ''a'' to ''b'' in ''K'' is</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>d(a,b) = \log \underset{p \in J}{\max} (pa/pb) + \log \underset{q \in J}{\max} (qb</ins>/<ins style="font-weight: bold; text-decoration: none;">qa) </ins>.<ins style="font-weight: bold; text-decoration: none;"><</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">At the [[University of Missouri]] in 1938 L.M. Blumenthal wrote ''Distance Geometry. A Study of the Development of Abstract Metrics'',<ref>University of Missouri Studies #13<</ins>/<ins style="font-weight: bold; text-decoration: none;">ref> where he used the term "Barbilian spaces" for [[metric space]]s based on Barbilian's function to obtain their [[metric (mathematics)|metric]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In 1954, P.J. Kelly wrote "Barbilian geometry and the Poincaré model" for the [[American Mathematical Monthly]] (61:311&ndash;19). In distant [[Romania]], Barbilian read [[Mathematical Reviews]] to see his function in use. He answered in 1959 with an article<ref>"Asupra unui principiu de metrizare", ''Studii şi Cercetări Matematice'' 10:69&ndash;116, Academia Republicii Populare Romîne<</ins>/<ins style="font-weight: bold; text-decoration: none;">ref> which described "a very general procedure of metrization through which the positive functions of two points, on certain sets, can be refined to a distance." Besides Blumenthal and Kelly, articles on "Barbilian spaces" have appeared from W.G. Boskoff and P. Souza. Dan Barbilian preferred the term "Apollonian metric space", and articles from A.F. Beardon, Gehring & Hag, P.A. Häströ, and Z. Ibragimov use that term.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Ring geometry==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Barbilian made a contribution </ins>to <ins style="font-weight: bold; text-decoration: none;">the [[Foundations of mathematics#Projective geometry|foundations of geometry]] with his articles in 1940 and 1941 in [[Jahresbericht der Deutschen Mathematiker</ins>-<ins style="font-weight: bold; text-decoration: none;">Vereinigung]] on [[projective plane]]s with coordinates from a [[ring (mathematics)|ring]].<ref>D. Barbilian (1940,1) "Zur Axiomatik der projecktiven ebenen Ringgeometrien" I,II, [[Jahresbericht der Deutschen Mathematiker</ins>-<ins style="font-weight: bold; text-decoration: none;">Vereinigung]] 50:179&ndash;229 {{MathSciNet|id=0003710}}, 51:34&ndash;76, {{MathSciNet|id=0005628}}<</ins>/<ins style="font-weight: bold; text-decoration: none;">ref><ref>T.G. Kvirikashvili (2008) "Projective geometries </ins>over <ins style="font-weight: bold; text-decoration: none;">rings and modular lattices", ''Journal of Mathematical Sciences'' 153(4)</ref> According to Boskoff & Suceava, this work "inspired research in ring geometries, nowadays associated with his, Hjelmslev’s and Klingenberg’s names."</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A more critical stance was taken in 1995 by Ferdinand D Velkamp: </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:A systematic study of projective planes over large classes of associative rings was initiated by D. Barbilian. His very general approach in [1940 and 41] remained rather unsatisfactory, however, his axioms were partly of a geometric nature, partly algebraic as pertaining to </ins>the <ins style="font-weight: bold; text-decoration: none;">ring of coordinates, and there were a number of difficulties which Barbilian could not overcome.<ref></ins>[http://<ins style="font-weight: bold; text-decoration: none;">www</ins>.<ins style="font-weight: bold; text-decoration: none;">sciencedirect</ins>.<ins style="font-weight: bold; text-decoration: none;">com/science/article</ins>/<ins style="font-weight: bold; text-decoration: none;">pii/B9780444883551500219 Geometry over Rings], chapter 19 of ''Handbook </ins>of <ins style="font-weight: bold; text-decoration: none;">Incidence Geometry: Buildings and Foundations'', Francis Buekenhout editor,North</ins>-<ins style="font-weight: bold; text-decoration: none;">Holland, {{MathSciNet|id=1360734}}<</ins>/<ins style="font-weight: bold; text-decoration: none;">ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Nevertheless, in 1989 John R. Faulkner wrote an article "Barbilian Planes"<ref>J.R. Faulkner (1989) "Barbilian Planes", [[Geometriae Dedicata</ins>]<ins style="font-weight: bold; text-decoration: none;">] 30:125&ndash;81 {{MathSciNet|id=1000255}}</ref> that clarified terminology </ins>and <ins style="font-weight: bold; text-decoration: none;">advanced the study. In his introduction he wrote:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:A classical result from projective geometry is that a Desarguesian projective plane is coordinatized by an associative division ring. A Barbilian plane is a geometric structure which extends the notion of a projective plane and thereby allows a coordinate ring which is not necessarily a division ring. There are advantages ...</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In 1942, Barbilian was named professor at the [[University of Bucharest]] (with some help from fellow mathematician [[Grigore Moisil]]<ref>O'Connor, John J; Edmund F. Robertson, </ins>[http://<ins style="font-weight: bold; text-decoration: none;">www-history</ins>.<ins style="font-weight: bold; text-decoration: none;">mcs.st-andrews.ac.uk</ins>/<ins style="font-weight: bold; text-decoration: none;">Biographies</ins>/<ins style="font-weight: bold; text-decoration: none;">Moisil.html "Grigore C. Moisil"</ins>]<ins style="font-weight: bold; text-decoration: none;">, ''[[MacTutor History of Mathematics archive]]''</ref>)</ins>. <ins style="font-weight: bold; text-decoration: none;"> As a mathematician, Barbilian authored 80 research papers and studies. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Political creed==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Barbu was mostly apolitical, with one exception: around 1940 he became a sympathizer </ins>of the <ins style="font-weight: bold; text-decoration: none;">fascist movement [[The Iron Guard]] (hoping to get a professorship if they came </ins>to <ins style="font-weight: bold; text-decoration: none;">power), dedicating </ins>some <ins style="font-weight: bold; text-decoration: none;">poems to one of its leaders, [[Corneliu Zelea Codreanu]]</ins>. In <ins style="font-weight: bold; text-decoration: none;">1940, he also wrote a poem praising [</ins>[<ins style="font-weight: bold; text-decoration: none;">Hitler]].<ref>{{citeweb|url=</ins>http://www.<ins style="font-weight: bold; text-decoration: none;">romaniaculturala</ins>.<ins style="font-weight: bold; text-decoration: none;">ro</ins>/<ins style="font-weight: bold; text-decoration: none;">articol.php?cod=100|title=Căderea poetului|publisher=[[România Literară]]|language=Romanian|accessdate=August 30, 2013}}<</ins>/<ins style="font-weight: bold; text-decoration: none;">ref></ins><<ins style="font-weight: bold; text-decoration: none;">ref</ins>><ins style="font-weight: bold; text-decoration: none;">{{citeweb|url=http://adevarul.ro/cultura/istorie/riga-crypto-drogurile-legionarii-1_50ba02487c42d5a663af8f9a/index.html|title=Riga Crypto, drogurile şi legionarii|publisher=[[Adevarul]]|language=Romanian|accessdate=August 30, 2013}}</ins><<ins style="font-weight: bold; text-decoration: none;">/ref</ins>></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Death==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Ion Barbu died in [[Bucharest]] in 1961, and </ins>is <ins style="font-weight: bold; text-decoration: none;">buried at [[Bellu Cemetery]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==References==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Reflist}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Dan Barbilian, ''Teoria aritmetică a idealelor (în inele necomutative)'', Editura Academiei, Bucharest, Romania, 1956</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Dan Barbilian, ''Grupuri cu operatori: Teoremele de descompunere ale algebrei'', </ins> <ins style="font-weight: bold; text-decoration: none;">Editura Academiei, Bucharest, Romania, 1960</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Sources==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[http://www.<ins style="font-weight: bold; text-decoration: none;">ici</ins>.<ins style="font-weight: bold; text-decoration: none;">ro/romania/en</ins>/<ins style="font-weight: bold; text-decoration: none;">cultura</ins>/<ins style="font-weight: bold; text-decoration: none;">l_barbu.html Short biography on ICI website]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Alexandru Ciorănescu, ''Ion Barbu'', Twayne Publishers, Boston, 1981 ISBN 0-8057-6432-1. Translated in Romanian, Editura Fundaţiei Culturale Române, Bucharest, 1996. ISBN 973-577-042-3</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* P.J. Kelly (1954) "Barbilian geometry and the Poincaré model", [[American Mathematical Monthly]] 61:311&ndash;19.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Authority control|VIAF=66461987}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. --></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| NAME = Barbu, Ion</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| ALTERNATIVE NAMES =</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| SHORT DESCRIPTION = Romanian mathematician and poet</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| DATE OF BIRTH = 1895</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| PLACE OF BIRTH =</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| DATE OF DEATH = 1961</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| PLACE OF DEATH =</ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{DEFAULTSORT:Barbu, Ion}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Romanian avantgarde]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Romanian poets]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Romanian mathematicians]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Romanian scientists]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Romanian inventors]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:20th-century mathematicians]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Members of </ins>the <ins style="font-weight: bold; text-decoration: none;">Romanian Academy elected post-mortem]</ins>]</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Romanian textbook writers]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:University of Bucharest faculty]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Gheorghe Lazăr National College (Bucharest</ins>) <ins style="font-weight: bold; text-decoration: none;">alumni]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:University of Bucharest alumni]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:People from Argeș County]]</ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Pseudonymous mathematicians]]</ins></div></td></tr>
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</table>
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https://en.formulasearchengine.com/index.php?title=Induced_topology&diff=245378&oldid=prev
en>Beroal: initial final topology
2012-07-16T19:17:52Z
<p>initial final topology</p>
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