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Zeros, moments and determinants Integrability and Randomness in Mathematical Physics and Geometry, April 8 th , 2019 Nina Snaith <latexit

  1. Zeros, moments and determinants Integrability and Randomness in Mathematical Physics and Geometry, April 8 th , 2019 Nina Snaith

  2. <latexit sha1_base64="P9zwSozUBMSdZX7xkgnRBRa7cEQ=">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</latexit> <latexit sha1_base64="P9zwSozUBMSdZX7xkgnRBRa7cEQ=">ADm3icfVLbatAEF1LvSTuzWlfAqWw1GlwKDaScutLS2geGuo+pFAnAa9jVquVvWS1ErurgBH6qP5K3vI3XdnypXLTAaGZc2bmzA7jJ5wp7Tj3Nct+9PjJ043N+rPnL16+amy9vlBxKgntkZjH8srHinImaE8zelVIimOfE4v/ZvTgr+8pVKxWPzSk4QOIjwSLGQEawMNt2q/d1BANeI01C3k0xETGZYST/KMFJZD5DMR5mbZ92m8PdObA/B8r4cBGTINZqmegV8IxDaAFW2x1U2h1V2y26rfV6eJZfLzU+v9otwW7W/kFK9GKaLci0v3orSEHD7xhv9ueJyMqgnLhSLRWO/Bz7DVdveuTb3hvetsKejl+U69Pmw0nY4zNbjuKXTBKWdDxt3KIhJGlGhCcdK9V0n0QOjqhnhNK+jVNEkxs8on3jChxRNcimt5XDwYJYBhL8wkNp+hqRYjpSaRbzIjrMeqyhXgv7h+qsNPg4yJNVUkJlQmHKoY1gcKgyYpETziXEwkczMCskYS0y0OediCW71yevOhdxnY702uefC3XsQHegvegBVxwDE7AGTgHPUCsbeuL9c06s9/Zp/Z3+8cs1aqVNW/AX2b3/gAy1B+l</latexit> <latexit sha1_base64="P9zwSozUBMSdZX7xkgnRBRa7cEQ=">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</latexit> <latexit sha1_base64="P9zwSozUBMSdZX7xkgnRBRa7cEQ=">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</latexit> 0 1 3 5 � 2 K − 1 1 � � � � � � � · · · K − 1 K − 1 K − 1 K − 1 � 2 K 2 4 6 � � � � � � � B C · · · K − 1 K − 1 K − 1 K − 1 B C 3 5 7 � 2 K − 1 � � � � � � � = ( − 1) K +1 2( K 2 ) B C det · · · K − 1 K − 1 K − 1 K − 1 B C . . . . ... B C . . . . B C . . . . @ A � K � K +2 � K +4 � 3 K − 2 � � � � · · · K − 1 K − 1 K − 1 K − 1 Emilia Alvarez (thesis)

  3. <latexit sha1_base64="EI03RG15oRDX9Cua6agoX/RiT4E=">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</latexit> <latexit sha1_base64="EI03RG15oRDX9Cua6agoX/RiT4E=">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</latexit> <latexit sha1_base64="EI03RG15oRDX9Cua6agoX/RiT4E=">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</latexit> <latexit sha1_base64="EI03RG15oRDX9Cua6agoX/RiT4E=">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</latexit> 1 1 1 1   · · · Γ (2 K ) Γ (2 K − 2) Γ (2 K − 4) Γ (2) 1 1 1 1   · · · Γ (2 K − 1) Γ (2 K − 3) Γ (2 K − 5) Γ (1)   . . . . ...   . . . . det . . . .   K × K   1 1 1 1   · · · Γ ( K +2) Γ ( K ) Γ ( K − 2) Γ ( − K +4)   1 1 1 1 · · · Γ ( K ) Γ ( K − 2) Γ ( K − 4) Γ ( − K +2) 1 1 1 1   · · · Γ (2 K − 1) Γ (2 K − 3) Γ (2 K − 5) Γ (1) 1 1 1 1   = K ( K − 1) · · · Γ (2 K − 2) Γ (2 K − 4) Γ (2 K − 6) Γ (0)   det . . . . ...   . . . . 2 K × K . . . .     1 1 1 1 · · · Γ ( K ) Γ ( K − 2) Γ ( K − 4) Γ ( − K +2) Ian Cooper (thesis)

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