Reflection Groups, Root Systems, and SingularitiesBojko BakalovDepartment of MathematicsNorth Carolina State UniversityNCSU SUM SeriesJanuary 27, 2010JSFHJSFHJSFHJSFHJSFH
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial" foreground="[51,51,153]" size="16">Reflection Groups</Font></Text-field>JSFHReflection with respect to the line 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 flips the coordinates 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reflection with respect to the plane 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 flips 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 reflection with respect to the plane 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 flips 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Taking compositions we obtain all 6 permutations of the coordinates 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: 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: 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: 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They form the symmetric group 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Group: the composition of two permutations is a permutation, and the inverse of a permutation is a permutation.The symmetric group LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KVEiU0YnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JS1GLDYpUSJuRidGL0YyRjVGOEY6Rj1GLy9GPlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== has 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 elements.JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial" foreground="[51,51,153]" size="16">Root Systems</Font></Text-field>JSFHA plane is determined by the two vectors of length LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYjNictSSNtbkdGJDYlUSIyRicvJSVzaXplR1EjMTRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdRJmZhbHNlRidGO0Y4Rj5GQUY7 perpendicular to it.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEmcGxvdHNGJ0YvRjJGOUY5RjUvJStleGVjdXRhYmxlR0Y9Rjk=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 the 3 planes 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 we obtain 6 vectors:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEmcGxvdHNGJ0YvRjJGOUY5RjUvJStleGVjdXRhYmxlR0Y9Rjk=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 and their inverses.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEmcGxvdHNGJ0YvRjJGOUY5RjUvJStleGVjdXRhYmxlR0Y9Rjk=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIlBGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA=The image of a root vector under one of our reflections is again a root vector. The length of each 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 is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbXNxcnRHRiQ2Iy1GIzYmLUkjbW5HRiQ2JVEiMkYnLyUlc2l6ZUdRIzE0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUnZmFtaWx5R1EmQXJpYWxGJ0Y0RjdGOkY0Rjc=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LlEiLkYnLyUlc2l6ZUdRIzE0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkYvJSdmYW1pbHlHUSZBcmlhbEYnRi8vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHRjcvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGMg== The angles between 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 and 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 are 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The vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2KVEnJiM5NDU7RicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW5HRiQ2KFEiMUYnRjJGOEY7Rj5GQS8lJ2ZhbWlseUdRJkFyaWFsRidGMkZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRidGSkYyRkE= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2KVEnJiM5NDU7RicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW5HRiQ2KFEiM0YnRjJGOEY7Rj5GQUYyL0Y2Rj0vRkJRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUSZBcmlhbEYnRjJGQQ== are orthogonal. This means that we have dot 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 the root systems we assign Dynkin diagramsLUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYoLUkjbW5HRiQ2JVEiMUYnRi8vRjZRJ25vcm1hbEYnRi8vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==: \342\210\230LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYoLUkjbW5HRiQ2JVEiMkYnRi8vRjZRJ25vcm1hbEYnRi8vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==: \342\210\230\342\210\222\342\210\230LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYoLUkjbW5HRiQ2JVEiM0YnRi8vRjZRJ25vcm1hbEYnRi8vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==: \342\210\230\342\210\222\342\210\230\342\210\222\342\210\230All root systems and Dynkin diagrams are: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 \342\210\230\342\210\222\342\210\230\342\210\222\342\213\205\342\213\205\342\213\205\342\210\222\342\210\230 \342\210\230 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: \342\210\230\342\210\222\342\213\205\342\213\205\342\213\205\342\210\222\342\210\230\342\210\222\342\210\230\342\210\222\342\210\230 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 \342\210\230\342\210\222\342\210\230\342\210\222\342\213\205\342\213\205\342\213\205\342\210\222\342\210\230<=\342\210\230 LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KVEiRkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JS1JI21uR0YkNihRIjRGJ0YvRjVGOEY6L0Y+USdub3JtYWxGJ0YvRkcvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==: \342\210\230\342\210\222\342\210\230=>\342\210\230\342\210\222\342\210\230 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\342\210\230\342\211\241>\342\210\230 They correspond to simple Lie algebras.JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial" foreground="[51,51,153]" size="16">Example: The Lie Algebra </Font><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2KVEjc2xGJy8lJXNpemVHUSMxNkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRk8vJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJy9GMFEnaXRhbGljRictSShtZmVuY2VkR0YkNigtRiM2Jy1JI21uR0YkNihRIjNGJ0ZKRlBGU0ZVRi9GSi8lK2V4ZWN1dGFibGVHRk8vRlZRKTJEfklucHV0RidGL0ZKRlBGU0ZVRi8tRkc2I1EhRicvRktRIzE0RidGXW9GX29GLw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2KVEjc2xGJy8lJXNpemVHUSMxNkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRk8vJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJy9GMFEnaXRhbGljRictSShtZmVuY2VkR0YkNigtRiM2Jy1JI21uR0YkNihRIjNGJ0ZKRlBGU0ZVRi9GSi8lK2V4ZWN1dGFibGVHRk8vRlZRKTJEfklucHV0RidGL0ZKRlBGU0ZVRi8tRkc2I1EhRicvRktRIzE0RidGXW9GX29GLw==</Equation></Text-field>JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2KVEjc2xGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRjQvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNigtRiM2Ji1GLDYpUSJuRidGL0YyRjVGOEY6Rj1GLy9GO1EoMkR+TWF0aEYnL0Y+USdub3JtYWxGJ0YvRjVGOEY6RkotRiw2I1EhRidGSEZK is the set of all LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KVEibkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1EnTm9ybWFsRicvRj5RJ25vcm1hbEYn by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KVEibkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1EnTm9ybWFsRicvRj5RJ25vcm1hbEYn matrices with trace LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2KFEiMEYnLyUlc2l6ZUdRIzE0RicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9GOVEnTm9ybWFsRidGOw==, equipped with the 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 (commutator / Lie bracket).It corresponds to the Dynkin diagram 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has a basis: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 for any two roots \316\261 , \316\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 when 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 is a root.
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial" foreground="[51,51,153]" size="16">Equations</Font></Text-field>The quadratic equation 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 has two solutions: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 , 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 the discriminant 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 Note that: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and 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The cubic equation 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 has three distinct real solutions 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precisely when the discriminant 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 It has a multiple root when the discriminant is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2KFEiMEYnLyUlc2l6ZUdRIzE0RicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y7. We have:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZoby1JJW1zdWJHRiQ2Ji1JI21pR0YkNiZRInhGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiVRIjFGJ0YyL0Y5USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEpdGFibGVyZWZGJy1JI21vR0YkNi5RJyZwbHVzO0YnRjJGPy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGTS8lKXN0cmV0Y2h5R0ZNLyUqc3ltbWV0cmljR0ZNLyUobGFyZ2VvcEdGTS8lLm1vdmFibGVsaW1pdHNHRk0vJSdhY2NlbnRHRk0vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0Zmbi1GLDYmRi4tRjw2JVEiMkYnRjJGP0ZBRkRGRy1GLDYmRi4tRjw2JVEiM0YnRjJGP0ZBRkQtRkg2LlEifkYnRjJGP0ZLRk5GUEZSRlRGVkZYL0ZlblEmMC4wZW1GJy9GaG5GZ28tRkg2LlEpJmVxdWFscztGJ0YyRj9GS0ZORlBGUkZURlZGWC9GZW5RLDAuMjc3Nzc3OGVtRicvRmhuRl1wRmNvLUY8NiVGQ0YyRj9GY28tRkg2LlEoJmNvbW1hO0YnRjJGP0ZLL0ZPRjdGUEZSRlRGVkZYRmZvL0ZoblEsMC4zMzMzMzMzZW1GJ0Zjb0Zjb0Zjb0Zjb0Zjb0Zjb0Zjb0Zjb0Zjb0Zjb0YrLUZINi1RMSZJbnZpc2libGVUaW1lcztGJ0Y/RktGTkZQRlJGVEZWRlhGZm9GaG8tRiw2JkYuLUYjNilGW29GMkY1LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lKXJlYWRvbmx5R0ZNLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRjhGQUZERkdGK0ZncC1GLDYmRi4tRiM2KUZgb0YyRjVGXnFGYXFGY3FGOEZBRkRGR0ZpbkZncEZeb0Zjb0Zpb0Zjby1GSDYuUSomdW1pbnVzMDtGJ0YyRj9GS0ZORlBGUkZURlZGWEZaRmduLUYvNiZRImJGJ0YyRjVGOEZjby1GSDYuUSIsRidGMkY/RktGZHBGUEZSRlRGVkZYRmZvRmVwRmNvRmNvRmNvRmNvRmNvRmNvRmNvRmNvRmNvRmNvRmNvRmNvRmNvRmNvRmNvRmNvRmNvRitGZ3BGaW5GZ3BGXm8tRi82I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGW3MvJSpsaW5lYnJlYWtHUSVhdXRvRidGaW9GanEtRi82JlEiY0YnRjJGNUY4RmZyRmFwRmZyRmZyRmZyRmNyRjJGXnFGYXFGPw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY5LUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRitGK0YrRitGKy1JI21uR0YkNihRIjRGJy8lJXNpemVHUSMxNEYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0YvLUYsNjFGLkZKRk1GUEZTRi9GMkY1RjdGOUY7Rj1GP0ZBRkQtSSVtc3VwR0YkNiUtSSNtaUdGJDYpUSJiRidGSi8lJ2l0YWxpY0dGUkZNRlBGUy9GMFEnaXRhbGljRictRkc2KVEiM0YnRkpGTS8lK2V4ZWN1dGFibGVHRlJGUEZTRi8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYsNjJRKCZtaW51cztGJ0ZKRk1GYG9GUEZTRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yMjIyMjIyZW1GJy9GRUZpby1GRzYpUSMyN0YnRkpGTUZgb0ZQRlNGLy1GLDYtUTEmSW52aXNpYmxlVGltZXM7RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1GWTYlLUZmbjYqUSJjRidGSkZpbkZNRmBvRlBGU0Zbby1GRzYpUSIyRidGSkZNRmBvRlBGU0YvRmJvLUYsNjJGLkZKRk1GYG9GUEZTRi9GMkY1RjdGOUY7Rj1GP0ZBRkQtRiw2MlEiPUYnRkpGTUZgb0ZQRlNGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRl9xRmlwRmlwLUZZNiUtSShtZmVuY2VkR0YkNiktRiM2Jy1JJW1zdWJHRiQ2JS1GZm42KlEieEYnRkpGaW5GTUZgb0ZQRlNGW28tRiM2JS1GRzYpUSIxRidGSkZNRmBvRlBGU0YvRkpGLy8lL3N1YnNjcmlwdHNoaWZ0R0Zkb0Zlby1GaXE2JUZbci1GIzYlRmZwRkpGL0ZjckZKRi9GSkZNRmBvRlBGU0YvRmdyRmJvRmlwLUZZNiUtRmRxNiktRiM2J0ZocUZlby1GaXE2JUZbci1GIzYlRl1vRkpGL0ZjckZKRi9GSkZNRmBvRlBGU0YvRmdyRmJvRmlwLUZZNiUtRmRxNiktRiM2J0ZlckZlb0Zfc0ZKRi9GSkZNRmBvRlBGU0YvRmdyRmJvRi8=JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEmcGxvdHNGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOUY1RlpGOQ==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JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial" foreground="[51,51,153]" size="16">Singularities of Surfaces</Font></Text-field>The surface given by the equation 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 has a singularity at the origin 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 an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2KVEiQUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzI1NSwwLDBdRicvJSlyZWFkb25seUdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNilRIm5GJ0YyRjVGOEY7Rj1GQEYyRjVGOEZALyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRicvRjNRIzE0RidGOC8lK2V4ZWN1dGFibGVHRjcvRj5RKTJEfklucHV0RicvRkFRJ25vcm1hbEYn-singularity. There are singularities corresponding to all Dynkin diagrams.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEmcGxvdHNGJ0YvRjJGOUY5RjUvJStleGVjdXRhYmxlR0Y9Rjk=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
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial" foreground="[51,51,153]" size="16">Deforming Singularities</Font></Text-field>After deforming the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2KVEiQUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2KFEiMUYnRjJGOEY7Rj0vRkFRJ25vcm1hbEYnRjJGSS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiNRIUYnL0YzUSMxNEYnRkk= singularity 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 to the surface LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUklbXN1cEdGJDYlLUkjbWlHRiQ2KVEieEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2KFEiMkYnRjJGOEY7Rj0vRkFRJ25vcm1hbEYnRjJGSS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYxUSIrRidGMkY4RjtGPUZJLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZULyUpc3RyZXRjaHlHRlQvJSpzeW1tZXRyaWNHRlQvJShsYXJnZW9wR0ZULyUubW92YWJsZWxpbWl0c0dGVC8lJ2FjY2VudEdGVC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRl1vLUYvNilRImNGJ0YyRjVGOEY7Rj1GQC1GTzYxUSJ+RidGMkY4RjtGPUZJRlJGVUZXRllGZW5GZ25GaW4vRlxvUSYwLjBlbUYnL0Zfb0Znby1GTzYxUSI9RidGMkY4RjtGPUZJRlJGVUZXRllGZW5GZ25GaW4vRlxvUSwwLjI3Nzc3NzhlbUYnL0Zfb0ZdcEZjby1GLDYlLUYvNilRInlGJ0YyRjVGOEY7Rj1GQEZFRktGTi1GLDYlLUYvNilRInpGJ0YyRjVGOEY7Rj1GQEZFRktGMkZJ it becomes non-singular (smooth) for 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 when the discriminant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2MVEiPkYnLyUlc2l6ZUdRIzE0RicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLUkjbW5HRiQ2KFEiMEYnRi9GMkY1RjhGO0YvRjs= .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 deform the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2KVEiQUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2KFEiMkYnRjJGOEY7Rj0vRkFRJ25vcm1hbEYnRjJGSS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiNRIUYnL0YzUSMxNEYnRkk= singularity 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 to the surface 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. Then it is non-singular when the discriminant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2MVEiPkYnLyUlc2l6ZUdRIzE0RicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLUkjbW5HRiQ2KFEiMEYnRi9GMkY1RjhGO0YvRjs= and singular when the discriminant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2MVEiPUYnLyUlc2l6ZUdRIzE0RicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLUkjbW5HRiQ2KFEiMEYnRi9GMkY1RjhGO0YvRjs=.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEmcGxvdHNGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOUY1RlpGOQ==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Here is a deformed the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2KVEiQUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2KFEiM0YnRjJGOEY7Rj0vRkFRJ25vcm1hbEYnRjJGSS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiNRIUYnL0YzUSMxNEYnRkk= singularity with discriminant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2MVEiPUYnLyUlc2l6ZUdRIzE0RicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLUkjbW5HRiQ2KFEiMEYnRi9GMkY1RjhGO0YvRjs=. Notice the resemblance with the Dynkin diagrams \342\210\230 , \342\210\230\342\210\222\342\210\230 , \342\210\230\342\210\222\342\210\230\342\210\222\342\210\230 . This holds in general for types ADE.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEmcGxvdHNGJ0YvRjJGOUY5RjUvJStleGVjdXRhYmxlR0Y9Rjk=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JSFHJSFHJSFHJSFH