Japanese Side Prof. Dr. Yasumasa Baba (The Institute of Statistical Mathematics) • Prof. Dr. Tadashi Imaizumi (Tama University) • Prof. Dr. Akinori Okada (Tama University) • Prof. Dr. Hiroshi Yadohisa (Doshisha University) • Germany Side Prof. Dr. Daniel Baier (Brandenburgische Technische Universität Cottbus) • Prof. Dr. Wolfgang Gaul (Karlsruhe Institut für Technologie) • Prof. Dr. Andreas Geyer-Schulz (Karlsruher Institut für Technologie) • Prof. Dr. Claus Weihs (Universität Dortmund) • 1 i Japanese Side Prof. Dr. Yasumasa Baba (The Institute of Statistical Mathematics) • Prof. Dr. Tadashi Imaizumi (Tama University) • Prof. Dr. Akinori Okada (Tama University) • Prof. Dr. Hiroshi Yadohisa (Doshisha University) • Germany Side Prof. Dr. Daniel Baier (Brandenburgische Technische Universität Cottbus) • Prof. Dr. Wolfgang Gaul (Karlsruhe Institut für Technologie) • Prof. Dr. Andreas Geyer-Schulz (Karlsruher Institut für Technologie) • Prof. Dr. Claus Weihs (Universität Dortmund) • ii 1 Friday March 9 㻣㻦㻡㻜㻙㻝㻤㻦㻜㻜 㻾㼑㼓㼕㼟㼠㼞㼍㼠㼕㼛㼚 㻼㼍㼓㼑 㻤㻦㻟㻜㻙㻤㻦㻠㻡 㻻㼜㼑㼚㼕㼚㼓 㻿㼑㼟㼟㼕㼛㼚㻝㻌㻔㼀㼔㼑㻌㼏㼔㼍㼕㼞㼜㼑㼞㼟㼛㼚㻦㻌㻴㼍㼚㼟㻙㻴㼑㼞㼙㼍㼚㼚㻌㻮㼛㼏㼗㻕 㻯㼛㼙㼜㼍㼞㼕㼟㼛㼚㻌㼛㼒㻌㼠㼣㼛㻌㻰㼕㼟㼠㼞㼕㼎㼡㼠㼕㼛㼚㻌㼂㼍㼘㼡㼑㼐㻌㻰㼕㼟㼟㼕㼙㼕㼘㼍㼞㼕㼠㼕㼑㼟㻌㼍㼚㼐㻌㼕㼠㼟㻌㻭㼜㼜㼘㼕㼏㼍㼠㼕㼛㼚㻌㼒㼛㼞㻌㻿㼥㼙㼎㼛㼘㼕㼏㻌㻯㼘㼡㼟㼠㼑㼞㼕㼚㼓 㻤㻦㻠㻡㻙㻥㻦㻝㻜 㼅㼡㼟㼡㼗㼑㻌㻹㼍㼠㼟㼡㼕㻌㼨㻌㼅㼡㼞㼕㼗㼛㻌㻷㼛㼙㼕㼥㼍㻌㼨㻌㻴㼕㼞㼛㼥㼡㼗㼕㻌㻹㼕㼚㼍㼙㼕㻌㼨㻌㻹㼍㼟㼍㼔㼕㼞㼛㻌㻹㼕㼦㼡㼠㼍㻌㻔㻴㼛㼗㼗㼍㼕㼐㼛㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻝 㻳㼞㼍㼜㼔㻌㻿㼥㼙㼙㼑㼠㼞㼕㼑㼟㻌㼍㼚㼐㻌㼀㼑㼟㼠㼟㻌㼒㼛㼞㻌㻲㼛㼞㼙㼍㼘㻌㻯㼘㼡㼟㼠㼑㼞㻌㻿㼠㼍㼎㼕㼘㼕㼠㼥㻌㼕㼚㻌㻹㼛㼐㼡㼘㼍㼞㻌㻯㼘㼡㼟㼠㼑㼞㼕㼚㼓 㻥㻦㻝㻜㻙㻥㻦㻟㻡 㻤㻦㻠㻡㻙㻝㻜㻦㻞㻡 㻳㼑㼥㼑㼞㻙㻿㼏㼔㼡㼘㼦㻌㻭㼚㼐㼞㼑㼍㼟㻌㼨㻌㻹㼕㼏㼔㼍㼑㼘㻌㻻㼢㼑㼘㼓㾁㼚㼚㼑㻌㼨㻌㻿㼠㼑㼕㼚㻌㻹㼍㼞㼠㼕㼚㻌㻔㻷㻵㼀㻙㻯㼍㼙㼜㼡㼟㻌㻿㾇㼐㻕 㻞 㻲㼛㼘㼐㻌㻯㼔㼍㼚㼓㼑㻌㻯㼘㼍㼟㼟㼕㼒㼕㼑㼞㻌㼒㼛㼞㻌㼠㼔㼑㻌㻭㼚㼍㼘㼥㼟㼕㼟㻌㼛㼒㻌㻳㼑㼚㼑㻌㻱㼤㼜㼞㼑㼟㼟㼕㼛㼚㻌㻼㼞㼛㼒㼕㼘㼑㼟 㻥㻦㻟㻡㻙㻝㻜㻦㻜㻜 㻴㼍㼚㼟㻌㻭㻚㻌㻷㼑㼟㼠㼘㼑㼞㻌㻔㼁㼘㼙㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻟 㻹㼛㼐㼑㼘㻙㻮㼍㼟㼑㼐㻌㻯㼘㼡㼟㼠㼑㼞㼕㼚㼓㻌㻹㼑㼠㼔㼛㼐㼟㻌㼒㼛㼞㻌㼀㼕㼙㼑㻌㻿㼑㼞㼕㼑㼟 㻝㻜㻦㻜㻜㻙㻝㻜㻦㻞㻡 㻴㼍㼚㼟㻙㻴㼑㼞㼙㼍㼚㼚㻌㻮㼛㼏㼗㻌㻔㻾㼃㼀㻴㻌㻭㼍㼏㼔㼑㼚㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻠 iii 㻝㻜㻦㻞㻡㻙㻝㻜㻦㻠㻜 㼀㼑㼍㻌㻮㼞㼑㼍㼗 㻿㼑㼟㼟㼕㼛㼚㻞㻌㻔㼀㼔㼑㻌㼏㼔㼍㼕㼞㼜㼑㼞㼟㼛㼚㻦㻌㻭㼗㼕㼚㼛㼞㼕㻌㻻㼗㼍㼐㼍㻕 㻻㼚㻌㻲㼕㼚㼐㼕㼚㼓㻌㼁㼚㼕㼝㼡㼑㻌㻯㼘㼡㼟㼠㼑㼞㼟㻌㼛㼒㻌㻵㼚㼐㼕㼢㼕㼐㼡㼍㼘㼟㻌㼕㼚㻌㻓㻼㼕㼏㼗㻌㼞㻛㼚㻓㻌㻰㼍㼠㼍㻌㻹㼍㼠㼞㼕㼤 㻝㻜㻦㻠㻜㻙㻝㻝㻦㻜㻡 㼀㼍㼐㼍㼟㼔㼕㻌㻵㼙㼍㼕㼦㼡㼙㻌㻔㼀㼍㼙㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕㻘㻌㻹㼕㼠㼟㼡㼍㼗㼕㻌㻴㼡㼦㼕㼕㻌㼨㻌㼀㼛㼟㼔㼕㼚㼍㼞㼕㻌㻷㼍㼙㼍㼗㼡㼞㼍㻌㻔㻯㼔㼡㼛㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻡 㻭㼓㼓㼘㼛㼙㼑㼞㼍㼠㼕㼢㼑㻌㻯㼘㼡㼟㼠㼑㼞㼕㼚㼓㻌㼁㼟㼕㼚㼓㻌㻭㼟㼟㼥㼙㼑㼠㼞㼕㼏㻌㻹㼑㼍㼟㼡㼞㼑㼟㻌㼣㼕㼠㼔㼛㼡㼠㻌㻾㼑㼢㼑㼞㼟㼍㼘㼟㻌㼕㼚㻌㻰㼑㼚㼐㼞㼛㼓㼞㼍㼙㼟 㻝㻝㻦㻜㻡㻙㻝㻝㻦㻟㻜 㻢 㻝㻜㻦㻠㻜㻌㻙㻌㻝㻞㻦㻞㻜 㻿㼍㼠㼛㼟㼔㼕㻌㼀㼍㼗㼡㼙㼕㻌㼨㻌㻿㼍㼐㼍㼍㼗㼕㻌㻹㼕㼥㼍㼙㼛㼠㼛㻌㻔㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼀㼟㼡㼗㼡㼎㼍㻕 㻱㼤㼍㼙㼕㼚㼍㼠㼕㼛㼚㻌㼛㼒㻌㼠㼔㼑㻌㻺㼑㼏㼑㼟㼟㼕㼠㼥㻌㼛㼒㻌㼁㼟㼕㼚㼓㻌㻻㼚㼑㻙㼙㼛㼐㼑㻌㼀㼔㼞㼑㼑㻙㼣㼍㼥㻌㻭㼟㼥㼙㼙㼑㼠㼞㼕㼏㻌㻹㼛㼐㼑㼘 㻝㻝㻦㻟㻜㻙㻝㻝㻦㻡㻡 㻭㼠㼟㼡㼔㼛㻌㻺㼍㼗㼍㼥㼍㼙㼍㻌㻔㼀㼛㼗㼥㼛㻌㻹㼑㼠㼞㼛㼜㼛㼘㼕㼠㼍㼚㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕㻘㻌㻴㼕㼞㼛㼥㼡㼗㼕㻌㼀㼟㼡㼞㼡㼙㼕㻌㻔㼅㼛㼗㼛㼔㼍㼙㼍㻌㻺㼍㼠㼕㼛㼚㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕㻘 㻣 㻭㼗㼕㼚㼛㼞㼕㻌㻻㼗㼍㼐㼍㻌㻔㼀㼍㼙㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻸㼑㼍㼟㼠㻌㻿㼝㼡㼍㼞㼑㼟㻌㻼㼑㼞㼙㼡㼠㼍㼠㼕㼛㼚㻌㼍㼚㼐㻌㻵㼠㼟㻌㻭㼜㼜㼘㼕㼏㼍㼠㼕㼛㼚㼟㻌㼠㼛㻌㻲㼍㼏㼠㼛㼞㻌㻾㼛㼠㼍㼠㼕㼛㼚㻌㼍㼚㼐㻌㻲㼕㼤㼑㼐㻌㻿㼕㼦㼑㻌㻯㼘㼡㼟㼠㼑㼞㼕㼚㼓 㻝㻝㻦㻡㻡㻙㻝㻞㻦㻞㻜 㻷㼛㼔㼑㼕㻌㻭㼐㼍㼏㼔㼕㻌㻔㻻㼟㼍㼗㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻤 㻝㻞㻦㻞㻜㻌㻙㻌㻝㻟㻦㻟㻜 㻸㼡㼚㼏㼔㻌㻮㼞㼑㼍㼗 㻿㼑㼟㼟㼕㼛㼚㻟㻌㻔㼀㼔㼑㻌㼏㼔㼍㼕㼞㼜㼑㼞㼟㼛㼚㻦㻌㻷㼑㼕㼖㼕㻌㼅㼍㼖㼕㼙㼍㻕 㻼㼍㼓㼑 㼀㼔㼑㻌㻱㼒㼒㼑㼏㼠㼟㻌㼛㼒㻌㻽㻙㼙㼍㼠㼞㼕㼤㻌㻿㼜㼑㼏㼕㼒㼕㼏㼍㼠㼕㼛㼚㻌㼍㼚㼐㻌㻹㼕㼟㼟㼜㼑㼏㼕㼒㼕㼏㼍㼠㼕㼛㼚㻌㼕㼚㻌㻹㼡㼘㼠㼕㼜㼘㼑㻙㼏㼔㼛㼕㼏㼑㻌㻰㻵㻺㻭㻌㻹㼛㼐㼑㼘㼟 㻝㻟㻦㻟㻜㻙㻝㻟㻦㻡㻡 㻷㼛㼗㼑㼚㻌㻻㼦㼍㼗㼕㻌㻔㼀㼔㼑㻌㻵㼚㼟㼠㼕㼠㼡㼠㼕㼛㼚㻌㼛㼒㻌㻿㼠㼍㼠㼕㼟㼠㼕㼏㼍㼘㻌㻹㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟㻕㻘㻌㻵㼗㼗㼛㻌㻷㼍㼣㼍㼔㼍㼟㼔㼕㻌㻔㻶㼍㼜㼍㼚㻌㻲㼛㼡㼚㼐㼍㼠㼕㼛㼚㻕㻘 㻥 㼀㼛㼙㼛㼗㼛㻌㼀㼍㼗㼍㼔㼍㼟㼔㼕㻌㼨㻌㼅㼡㼍㼚㻌㻿㼡㼚㻌㼨㻌㻿㼡㼙㼕㼛㻌㻷㼍㼗㼕㼚㼡㼙㼍㻌㻔㻺㼍㼠㼕㼛㼚㼍㼘㻌㻵㼚㼟㼠㼕㼠㼡㼠㼑㻌㼛㼒㻌㻵㼚㼒㼛㼞㼙㼍㼠㼕㼏㼟㻕 㻭㼡㼠㼛㼙㼍㼠㼕㼏㻌㻾㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㼛㼒㻌㻲㼍㼏㼠㼛㼞㼕㼦㼍㼠㼕㼛㼚㻌㻹㼛㼐㼑㼘㼟 㻝㻟㻦㻡㻡㻙㻝㻠㻦㻞㻜 㻿㼠㼑㼒㼒㼑㼚㻌㻾㼑㼚㼐㼘㼑㻌㻔㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻷㼛㼚㼟㼠㼍㼚㼦㻕 㻝㻜 㻝㻟㻦㻟㻜㻌㻙㻌㻝㻡㻦㻝㻜 㼀㼔㼞㼑㼟㼔㼛㼘㼐㼕㼚㼓㻌㻸㼛㼍㼐㼕㼚㼓㼟㻌㼕㼚㻌㻲㼍㼏㼠㼛㼞㻌㻭㼚㼍㼘㼥㼟㼕㼟 㻝㻠㻦㻞㻜㻙㻝㻠㻦㻠㻡 㼅㼡㼟㼡㼗㼑㻌㻹㼕㼥㼍㼙㼛㼠㼛㻌㻔㻻㼟㼍㼗㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻝㻝 㻭㼟㼟㼑㼟㼟㼙㼑㼚㼠㻌㼛㼒㻌㼠㼔㼑㻌㻾㼑㼘㼍㼠㼕㼛㼚㼟㼔㼕㼜㻌㼎㼑㼠㼣㼑㼑㼚㻌㻺㼍㼠㼕㼢㼑㻌㼀㼔㼛㼞㼍㼏㼕㼏㻌㻭㼛㼞㼠㼕㼏㻌㻯㼡㼞㼢㼍㼠㼡㼞㼑㻌㼍㼚㼐 㻝㻠㻦㻠㻡㻙㻝㻡㻦㻝㻜 㻱㼚㼐㼛㼘㼑㼍㼗㻌㻲㼛㼞㼙㼍㼠㼕㼛㼚㻌㼍㼒㼠㼑㼞㻌㼀㻱㼂㻭㻾㻌㻮㼍㼟㼑㼐㻌㼛㼚㻌㻸㼕㼚㼑㼍㼞㻌㻰㼕㼟㼏㼞㼕㼙㼕㼚㼍㼚㼠㻌㻭㼚㼍㼘㼥㼟㼕㼟 㻝㻞 㻷㼡㼚㼕㼥㼛㼟㼔㼕㻌㻴㼍㼥㼍㼟㼔㼕㻌㼨㻌㻲㼡㼙㼕㼛㻌㻵㼟㼔㼕㼛㼗㼍㻌㻔㻻㼗㼍㼥㼍㼙㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻯㻾㻱㻿㼀㻕㻘㻌㻮㼔㼍㼞㼓㼍㼢㻌㻾㼍㼙㼍㼚㻌㼨㻌㻰㼍㼚㼕㼑㼘㻌㼅㻚㻌㻿㼦㼑㻌㻔㻿㼠㼍㼚㼒㼛㼞㼐㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㻿㼏㼔㼛㼛㼘㻌㼛㼒㻌㻹㼑㼐㼕㼏㼕㼚㼑㻕㻘 㻴㼕㼞㼛㼟㼔㼕㻌㻿㼡㼕㼠㼛㻌㻔㻻㼗㼍㼥㼍㼙㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻯㻾㻱㻿㼀㻕㻘㻌㼀㼍㼗㼡㼥㼍㻌㼁㼑㼐㼍㻌㻔㻿㼠㻚㻌㻸㼡㼗㼑䇻㼟㻌㻵㼚㼠㼑㼞㼚㼍㼠㼕㼛㼚㼍㼘㻌㻴㼛㼟㼜㼕㼠㼍㼘㻘㻌㻯㻾㻱㻿㼀㻕㻘㻌㻷㼛㼖㼕㻌㻷㼡㼞㼕㼔㼍㼞㼍㻌㻔㻻㼗㼍㼥㼍㼙㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻯㻾㻱㻿㼀㻕 㻝㻡㻦㻝㻜㻌㻙㻌㻝㻡㻦㻞㻡 㼀㼑㼍㻌㻮㼞㼑㼍㼗 iv 㻿㼑㼟㼟㼕㼛㼚㻠㻌㻔㼀㼔㼑㻌㼏㼔㼍㼕㼞㼜㼑㼞㼟㼛㼚㻦㻌㻿㼔㼕㼦㼡㼔㼕㼗㼛㻌㻺㼕㼟㼔㼕㼟㼍㼠㼛㻕 㼀㼔㼞㼑㼑㻙㼣㼍㼥㻌㻰㼍㼠㼍㻌㻭㼚㼍㼘㼥㼟㼕㼟㻌㼒㼛㼞㻌㻹㼡㼘㼠㼕㼢㼍㼞㼕㼍㼠㼑㻌㻿㼜㼍㼠㼕㼍㼘㻌㼀㼕㼙㼑㻌㻿㼑㼞㼕㼑㼟 㻝㻡㻦㻞㻡㻙㻝㻡㻦㻡㻜 㻹㼕㼠㼟㼡㼔㼕㼞㼛㻌㼀㼟㼡㼖㼕㻌㻔㻷㼍㼚㼟㼍㼕㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕㻘㻌㼀㼛㼟㼔㼕㼛㻌㻿㼔㼕㼙㼛㼗㼍㼣㼍㻌㻔㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼅㼍㼙㼍㼚㼍㼟㼔㼕㻕 㻝㻟 㼀㼔㼞㼑㼑㻙㻹㼛㼐㼑㻌㻿㼡㼎㼟㼜㼍㼏㼑㻌㻯㼘㼡㼟㼠㼑㼞㼕㼚㼓㻌㼒㼛㼞㻌㻯㼛㼚㼟㼕㼐㼑㼞㼕㼚㼓㻌㻱㼒㼒㼑㼏㼠㼟㻌㼡㼚㼐㼑㼞㻌㻺㼛㼕㼟㼑㻌㼂㼍㼞㼕㼍㼎㼘㼑㼟㻌㼍㼚㼐㻌㻻㼏㼏㼍㼟㼕㼛㼚㼟 㻝㻡㻦㻡㻜㻙㻝㻢㻦㻝㻡 㻷㼑㼚㼟㼡㼗㼑㻌㼀㼍㼚㼕㼛㼗㼍㻌㼨㻌㻴㼕㼞㼛㼟㼔㼕㻌㼅㼍㼐㼛㼔㼕㼟㼍㻌㻔㻰㼛㼟㼔㼕㼟㼔㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻝㻠 㻝㻡㻦㻞㻡㻌㻙㻌㻝㻣㻦㻟㻜 㻯㼘㼍㼟㼟㼕㼒㼕㼏㼍㼠㼕㼛㼚㻘㻌㻯㼘㼡㼟㼠㼑㼞㼕㼚㼓㻌㼍㼚㼐㻌㼂㼕㼟㼡㼍㼘㼕㼟㼍㼠㼕㼛㼚㻌㻮㼍㼟㼑㼐㻌㼛㼚㻌㻰㼡㼍㼘㻌㻿㼏㼍㼘㼕㼚㼓 㻝㻢㻦㻝㻡㻙㻝㻢㻦㻠㻜 㻴㼍㼚㼟㻙㻶㼛㼍㼏㼔㼕㼙㻌㻹㼡㼏㼔㼍㻌㻔㼃㼑㼕㼑㼞㼟㼠㼞㼍㼟㼟㻌㻵㼚㼟㼠㼕㼠㼡㼠㼑㻕 㻝㻡 㻿㼠㼞㼡㼏㼠㼡㼞㼍㼘㻌㻾㼑㼜㼞㼑㼟㼑㼚㼠㼍㼠㼕㼛㼚㻌㼛㼒㻌㻯㼍㼠㼑㼓㼛㼞㼕㼏㼍㼘㻌㻰㼍㼠㼍㻌㼍㼚㼐㻌㻯㼘㼡㼟㼠㼑㼞㻌㻭㼚㼍㼘㼥㼟㼕㼟㻌㼠㼔㼞㼛㼡㼓㼔㻌㻲㼕㼘㼠㼑㼞㼟 㻝㻢㻦㻠㻜㻙㻝㻣㻦㻜㻡 㻿㼔㼕㼦㼡㼔㼕㼗㼛㻌㻺㼕㼟㼔㼕㼟㼍㼠㼛㻌㻔㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼀㼛㼞㼛㼚㼠㼛㻕 㻝㻢 㼂㼍㼞㼕㼍㼎㼘㼑㻌㻿㼑㼘㼑㼏㼠㼕㼛㼚㻌㼕㼚㻌㻷㻙㼙㼑㼍㼚㼟㻌㻯㼘㼡㼟㼠㼑㼞㼕㼚㼓㻌㼢㼕㼍㻌㻾㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㻹㼑㼠㼔㼛㼐 㻝㻣㻦㻜㻡㻙㻝㻣㻦㻟㻜 㼅㼡㼠㼍㼗㼍㻌㻺㼕㼟㼔㼕㼐㼍㻌㻔㻻㼟㼍㼗㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻝㻣 㻝㻥㻦㻜㻜㻌㻙㻞㻝㻦㻜㻜 㻯㼛㼚㼒㼑㼞㼑㼚㼏㼑㻌㻰㼕㼚㼚㼑㼞 Saturday March 10 㻤㻦㻞㻜㻌㻙㻌㻝㻡㻦㻜㻜 㻾㼑㼓㼕㼟㼠㼞㼍㼠㼕㼛㼚 㻼㼍㼓㼑 㻿㼑㼟㼟㼕㼛㼚㻡㻌㻔㼀㼔㼑㻌㼏㼔㼍㼕㼞㼜㼑㼞㼟㼛㼚㻦㻌㼀㼍㼐㼍㼟㼔㼕㻌㻵㼙㼍㼕㼦㼡㼙㼕㻕 㻻㼚㻌㼠㼔㼑㻌㻿㼠㼞㼑㼟㼟㻌㻲㼡㼚㼏㼠㼕㼛㼚㻌㼛㼒㻌㻭㼟㼥㼙㼙㼑㼠㼞㼕㼏㻌㼢㼛㼚㻌㻹㼕㼟㼑㼟㻌㻿㼏㼍㼘㼕㼚㼓 㻥㻦㻜㻜㻙㻥㻦㻞㻡 㻷㼛㼖㼕㼞㼛㻌㻿㼔㼛㼖㼕㼙㼍㻌㻔㼀㼔㼑㻌㻺㼍㼠㼕㼛㼚㼍㼘㻌㻯㼑㼚㼠㼑㼞㻌㼒㼛㼞㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㻱㼚㼠㼞㼍㼚㼏㼑㻌㻱㼤㼍㼙㼕㼚㼍㼠㼕㼛㼚㼟㻕 㻝㻤 㻮㼍㼥㼑㼟㼕㼍㼚㻌㻭㼚㼍㼘㼥㼟㼕㼟㻌㼛㼒㻌㻭㼟㼥㼙㼙㼑㼠㼞㼥㻌㼎㼥㻌㼠㼔㼑㻌㻿㼘㼕㼐㼑㻙㼂㼑㼏㼠㼛㼞㻌㻹㼛㼐㼑㼘 㻥㻦㻞㻡㻙㻥㻦㻡㻜 㻷㼑㼚㼟㼡㼗㼑㻌㻻㼗㼍㼐㼍㻌㻔㻿㼑㼚㼟㼔㼡㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻝㻥 㻥㻦㻜㻜㻙㻝㻜㻦㻠㻜 㻭㻌㻺㼑㼣㻌㻹㼡㼘㼠㼕㼐㼕㼙㼑㼚㼟㼕㼛㼚㼍㼘㻌㻿㼏㼍㼘㼕㼚㼓㻌㻹㼑㼠㼔㼛㼐㼛㼘㼛㼓㼥㻌㼒㼛㼞㻌㻭㼚㼍㼘㼥㼟㼕㼟㻌㼛㼒㻌㻭㼟㼥㼙㼙㼑㼠㼞㼕㼏㻌㻯㼕㼠㼍㼠㼕㼛㼚㻌㻰㼍㼠㼍 㻥㻦㻡㻜㻙㻝㻜㻦㻝㻡 㼕㼚㻌㻿㼏㼕㼑㼚㼠㼕㼒㼕㼏㻌㻼㼡㼎㼘㼕㼏㼍㼠㼕㼛㼚㼟 㻞㻜 㼅㼡㼍㼚㻌㻿㼡㼚㻌㻔㻺㼍㼠㼕㼛㼚㼍㼘㻌㻵㼚㼟㼠㼕㼠㼡㼠㼑㻌㼛㼒㻌㻵㼚㼒㼛㼞㼙㼍㼠㼕㼏㼟㻕㻘㻌㼀㼍㼐㼍㼟㼔㼕㻌㻵㼙㼍㼕㼦㼡㼙㼕㻌㻔㼀㼍㼙㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻭㼟㼥㼙㼙㼑㼠㼞㼕㼏㻌㻹㼡㼘㼠㼕㼐㼕㼙㼑㼚㼟㼕㼛㼚㼍㼘㻌㻿㼏㼍㼘㼕㼚㼓㻌㼣㼕㼠㼔㻌㻳㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㻴㼥㼜㼑㼞㼑㼘㼘㼕㼜㼟㼑㻌㻹㼛㼐㼑㼘 㻝㻜㻦㻝㻡㻙㻝㻜㻦㻠㻜 㼅㼛㼟㼔㼕㼗㼍㼦㼡㻌㼀㼑㼞㼍㼐㼍㻌㻔㻻㼟㼍㼗㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕㻘㻌㻴㼕㼞㼛㼟㼔㼕㻌㼅㼍㼐㼛㼔㼕㼟㼍㻌㻔㻰㼛㼟㼔㼕㼟㼔㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻞㻝 v 㻝㻜㻦㻠㻜㻙㻝㻜㻦㻡㻡 㼀㼑㼍㻌㻮㼞㼑㼍㼗 㻿㼑㼟㼟㼕㼛㼚㻢㻌㻔㼀㼔㼑㻌㼏㼔㼍㼕㼞㼜㼑㼞㼟㼛㼚㻦㻌㻯㼘㼍㼡㼟㻌㼃㼑㼕㼔㼟㻕 㻿㼠㼍㼠㼕㼟㼠㼕㼏㼍㼘㻌㻼㼞㼛㼏㼑㼟㼟㻌㻹㼛㼐㼑㼘㼘㼕㼚㼓㻌㼒㼛㼞㻌㻹㼍㼏㼔㼕㼚㼕㼚㼓㻌㼛㼒㻌㻵㼚㼔㼛㼙㼛㼓㼑㼚㼑㼛㼡㼟㻌㻹㼕㼚㼑㼞㼍㼘㻌㻿㼡㼎㼟㼛㼕㼘 㻝㻜㻦㻡㻡㻙㻝㻝㻦㻞㻜 㻯㼘㼍㼡㼟㻌㼃㼑㼕㼔㼟㻌㻔㼀㼑㼏㼔㼚㼕㼟㼏㼔㼑㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㽯㼠㻌㻰㼛㼞㼠㼙㼡㼚㼐㻕 㻞㻞 㼆㼛㼚㼑㻌㻰㼑㼠㼑㼏㼠㼕㼛㼚㻌㻼㼞㼛㼏㼑㼟㼟㻌㼒㼛㼞㻌㻾㼍㼕㼚㼒㼍㼘㼘㻌㻵㼚㼒㼘㼛㼣㻌㼕㼚㼠㼛㻌㻿㼑㼣㼍㼓㼑㻌㻼㼕㼜㼑 㻝㻝㻦㻞㻜㻙㻝㻝㻦㻠㻡 㻷㼑㼚㻌㼃㼍㼐㼍㻌㼨㻌㼀㼛㼙㼛㼥㼡㼗㼕㻌㻴㼍㼟㼑㼓㼍㼣㼍㻌㻔㻿㼀㻻㻺㻱㻳㻭㼀㻱㻌㻯㼛㻚㻕㻘㻌㻷㼑㼕㼖㼕㻌㼅㼍㼖㼕㼙㼍㻌㻔㻵㼚㼐㼑㼜㼑㼚㼐㼑㼚㼠㻕 㻞㻟 㻝㻜㻦㻡㻡㻌㻙㻌㻝㻞㻦㻟㻡 㼀㼔㼑㻌㼁㼠㼕㼘㼕㼠㼥㻌㼛㼒㻌㻿㼙㼍㼘㼘㼑㼟㼠㻌㻿㼜㼍㼏㼑㻌㻭㼚㼍㼘㼥㼟㼕㼟㼒㼛㼞㻌㼠㼔㼑㻌㻯㼞㼛㼟㼟㻙㻺㼍㼠㼕㼛㼚㼍㼘㻌㻿㼡㼞㼢㼑㼥㻌㻰㼍㼠㼍㻌㻭㼚㼍㼘㼥㼟㼕㼟㻦 㻝㻝㻦㻠㻡㻙㻝㻞㻦㻝㻜 㼀㼔㼑㻌㻿㼠㼞㼡㼏㼠㼡㼞㼑㻌㼛㼒㻌㻾㼑㼘㼕㼓㼕㼛㼟㼕㼠㼥 㻞㻠 㻷㼍㼦㼡㼒㼡㼙㼕㻌㻹㼍㼚㼍㼎㼑㻌㻔㻭㼛㼥㼍㼙㼍㻌㻳㼍㼗㼡㼕㼚㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻭㼚㼍㼘㼥㼟㼕㼟㻌㼛㼒㻌㻯㼔㼍㼚㼓㼑㼟㻌㼛㼒㻌㻮㼞㼍㼚㼐㻌㻯㼍㼠㼑㼓㼛㼞㼕㼑㼟㻌㼁㼟㼕㼚㼓㻌㻼㼡㼞㼏㼔㼍㼟㼑㻌㻴㼕㼟㼠㼛㼞㼥㻌㻰㼍㼠㼍㻌㼍㼚㼐㻌㻱㼕㼓㼑㼚㼢㼍㼘㼡㼑 㻝㻞㻦㻝㻜㻙㻝㻞㻦㻟㻡 㼠㼛㻌㻲㼕㼚㼐㻌㻺㼑㼣㻌㻯㼍㼠㼑㼓㼛㼞㼥 㻞㻡 㼅㼡㼗㼕㻌㼀㼛㼥㼛㼐㼍㻌㻔㼀㼍㼙㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻝㻞㻦㻟㻡㻌㻙㻌㻝㻟㻦㻠㻡 㻸㼡㼚㼏㼔㻌㻮㼞㼑㼍㼗 㻿㼑㼟㼟㼕㼛㼚㻣㻌㻔㼀㼔㼑㻌㼏㼔㼍㼕㼞㼜㼑㼞㼟㼛㼚㻦㻌㼅㼍㼟㼡㼙㼍㼟㼍㻌㻮㼍㼎㼍㻕 㻼㼍㼓㼑 㻭㼚㻌㻭㼡㼠㼛㼙㼍㼠㼕㼏㻌㻱㼤㼠㼞㼍㼏㼠㼕㼛㼚㻌㼛㼒㻌㻭㼏㼍㼐㼑㼙㼕㼍㻙㻵㼚㼐㼡㼟㼠㼞㼥㻌㻯㼛㼘㼘㼍㼎㼛㼞㼍㼠㼕㼢㼑㻌㻾㼑㼟㼑㼍㼞㼏㼔 㻌㼍㼚㼐㻌㻰㼑㼢㼑㼘㼛㼜㼙㼑㼚㼠㻌㻰㼛㼏㼡㼙㼑㼚㼠㼟㻌㼛㼚㻌㼠㼔㼑㻌㼃㼑㼎 㻝㻟㻦㻠㻡㻙㻝㻠㻦㻝㻜 㻷㼑㼕㻌㻷㼡㼞㼍㼗㼍㼣㼍㻌㼨㻌㼅㼡㼍㼚㻌㻿㼡㼚㻌㻔㻺㼍㼠㼕㼛㼚㼍㼘㻌㻵㼚㼟㼠㼕㼠㼡㼠㼑㻌㼛㼒㻌㻵㼚㼒㼛㼞㼙㼍㼠㼕㼏㼟㻕㻘 㻞㻢 㻺㼍㼓㼍㼥㼛㼟㼔㼕㻌㼅㼍㼙㼍㼟㼔㼕㼠㼍㻌㻔㻶㼍㼜㼍㼚㻌㻿㼛㼏㼕㼑㼠㼥㻌㼒㼛㼞㻌㼠㼔㼑㻌㻼㼞㼛㼙㼛㼠㼕㼛㼚㻌㼛㼒㻌㻿㼏㼕㼑㼚㼏㼑㻕㻘 㼅㼍㼟㼡㼙㼍㼟㼍㻌㻮㼍㼎㼍㻌㻔㼀㼔㼑㻌㻵㼚㼟㼠㼕㼠㼡㼠㼑㻌㼛㼒㻌㻿㼠㼍㼠㼕㼟㼠㼕㼏㼍㼘㻌㻹㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟㻕 㻝㻟㻦㻠㻡㻙㻝㻡㻦㻞㻡 㻭㼚㼏㼕㼑㼚㼠㻌㻼㼛㼜㼡㼘㼍㼠㼕㼛㼚㻌㻰㼥㼚㼍㼙㼕㼏㼟㻌㻱㼟㼠㼕㼙㼍㼠㼕㼛㼚㻌㼒㼞㼛㼙㻌㻭㼞㼏㼔㼍㼑㼛㼘㼛㼓㼕㼏㼍㼘㻌㼐㼍㼠㼍㻌㻓㻺㼡㼦㼕㻌㼜㼑㼞㼟㼛㼚㼍㼘㻌㼚㼍㼙㼑㼟㻓 㻝㻠㻦㻝㻜㻙㻝㻠㻦㻟㻡 㻿㼡㼙㼕㼑㻌㼁㼑㼐㼍㻌㻔㼀㼔㼑㻌㻵㼚㼟㼠㼕㼠㼡㼠㼕㼛㼚㻌㼛㼒㻌㻿㼠㼍㼠㼕㼟㼠㼕㼏㼍㼘㻌㻹㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟㻕㻘㻌㻷㼡㼙㼕㻌㻹㼍㼗㼕㼚㼛㻌㻔㻷㼍㼙㼍㼗㼡㼞㼍㻌㼃㼛㼙㼑㼚㻓㼟㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕㻘 㻞㻣 㼅㼛㼟㼔㼕㼍㼗㼕㻌㻵㼠㼛㼔㻌㻔㼀㼔㼑㻌㻵㼚㼟㼠㼕㼠㼡㼠㼕㼛㼚㻌㼛㼒㻌㻿㼠㼍㼠㼕㼟㼠㼕㼏㼍㼘㻌㻹㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟㻕㻘㻌㼀㼍㼗㼍㼟㼔㼕㻌㼀㼟㼡㼏㼔㼕㼥㼍㻌㻔㻺㼍㼠㼕㼛㼚㼍㼘㻌㻳㼞㼍㼐㼡㼍㼠㼑㻌㻵㼚㼟㼕㼠㼡㼠㼑㻌㼒㼛㼞㻌㻼㼛㼘㼕㼏㼥㻌㻿㼠㼡㼐㼕㼑㼟㻕 㻯㼘㼍㼟㼟㼕㼒㼕㼏㼍㼠㼕㼛㼚㻌㼛㼒㻌㻸㼕㼠㼑㼞㼍㼠㼡㼞㼑㻌㼎㼥㻌㻭㼚㼍㼘㼥㼦㼕㼚㼓㻌㻲㼕㼓㼡㼞㼑㻙㻳㼞㼛㼡㼚㼐㻌㻾㼑㼘㼍㼠㼕㼛㼚㼟㼔㼕㼜㻌㼛㼒㻌㻯㼔㼍㼞㼍㼏㼠㼑㼞㼟 㻝㻠㻦㻟㻡㻙㻝㻡㻦㻜㻜 㼀㼑㼠㼟㼡㼥㼍㻌㻹㼍㼠㼟㼡㼕㻌㼨㻌㼅㼡㼗㼕㼛㻙㼜㼑㼓㼕㼛㻌㻳㼡㼚㼖㼕㻌㼨㻌㻱㼡㼓㼑㼚㼕㼛㻙㻿㼏㼔㼚㼑㼕㼐㼑㼞㻌㻷㼕㼠㼍㼙㼡㼞㼍㻌㻔㻷㼛㼎㼑㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻞㻤 㻺㼛㼚㻙㻭㼐㼐㼕㼠㼕㼢㼑㻌㼁㼠㼕㼘㼕㼠㼥㻌㻲㼡㼚㼏㼠㼕㼛㼚㼟㻦㻌㻯㼔㼛㼝㼡㼑㼠㻌㻵㼚㼠㼑㼓㼞㼍㼘㻌㼢㼑㼞㼟㼡㼟㻌㼘㼛㼓㼕㼏㻙㼎㼍㼟㼑㼐㻌㻽㼡㼑㼞㼥㼕㼚㼓 㻝㻡㻦㻜㻜㻙㻝㻡㻦㻞㻡 㻵㼚㼓㼛㻌㻿㼏㼔㼙㼕㼠㼠㻌㻔㻮㼞㼍㼚㼐㼑㼚㼎㼡㼞㼓㼕㼟㼏㼔㼑㻌㼀㼑㼏㼔㼚㼕㼟㼏㼔㼑㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼍㼠㻌㻯㼛㼠㼠㼎㼡㼟㻕 㻞㻥 㻝㻡㻦㻞㻡㻙㻝㻡㻦㻠㻜 㼀㼑㼍㻌㻮㼞㼑㼍㼗 vi 㻿㼑㼟㼟㼕㼛㼚㻤㻌㻔㼀㼔㼑㻌㼏㼔㼍㼕㼞㼜㼑㼞㼟㼛㼚㻦㻌㼃㼛㼘㼒㼓㼍㼚㼓㻌㻳㼍㼡㼘㻕 㻲㼑㼍㼠㼡㼞㼑㻌㻿㼑㼘㼑㼏㼠㼕㼛㼚㻌㼍㼚㼐㻌㻯㼘㼡㼟㼠㼑㼞㼕㼚㼓㻌㼛㼒㻌㻰㼕㼓㼕㼠㼍㼘㻌㻵㼙㼍㼓㼑㼟㻌㼂㼑㼞㼟㼡㼟 㻝㻡㻦㻠㻜㻙㻝㻢㻦㻜㻡 㻽㼡㼑㼟㼠㼕㼛㼚㼚㼍㼕㼞㼑㻌㻮㼍㼟㼑㼐㻌㻳㼞㼛㼡㼜㼕㼚㼓㻌㼛㼒㻌㻯㼛㼚㼟㼡㼙㼑㼞㼟㻦㻌㻭㻌㻯㼛㼙㼜㼍㼞㼕㼟㼛㼚 㻟㻜 㻵㼚㼑㼟㻌㻰㼍㼚㼕㼑㼘㻌㼨㻌㻰㼍㼚㼕㼑㼘㻌㻮㼍㼕㼑㼞㻌㻔㻮㼞㼍㼚㼐㼑㼚㼎㼡㼞㼓㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼀㼑㼏㼔㼚㼛㼘㼛㼓㼥㻌㻯㼛㼠㼠㼎㼡㼟㻕 㻭㼚㼍㼘㼥㼟㼕㼟㻌㼛㼒㻌㻭㼟㼥㼙㼙㼑㼠㼞㼕㼏㻌㻾㼑㼘㼍㼠㼕㼛㼚㼟㼔㼕㼜㼟㻌㻭㼙㼛㼚㼓㻌㻿㼛㼒㼠㻌㻰㼞㼕㼚㼗㻌㻮㼞㼍㼚㼐㼟 㻝㻢㻦㻜㻡㻙㻝㻢㻦㻟㻜 㻝㻡㻦㻠㻜㻌㻙㻌㻝㻣㻦㻞㻜 㻭㼗㼕㼚㼛㼞㼕㻌㻻㼗㼍㼐㼍㻌㻔㼀㼍㼙㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻟㻝 㻴㼛㼣㻌㼠㼛㻌㼡㼟㼑㻌㼃㼕㼘㼘㼕㼚㼓㼚㼑㼟㼟㻙㼠㼛㻙㻼㼍㼥㻌㻰㼍㼠㼍㻌㼒㼛㼞㻌㻼㼞㼛㼐㼡㼏㼠㻌㻮㼡㼚㼐㼘㼕㼚㼓 㻝㻢㻦㻟㻜㻙㻝㻢㻦㻡㻡 㼃㼛㼘㼒㼓㼍㼚㼓㻌㻳㼍㼡㼘㻌㻔㻷㻵㼀㻙㻯㼍㼙㼜㼡㼟㻌㻿㾇㼐㻕 㻟㻞 㻲㼞㼛㼙㻌㻻㼚㼘㼕㼚㼑㻌㻯㼡㼟㼠㼛㼙㼑㼞㻌㻾㼑㼢㼕㼑㼣㼟㻌㼠㼛㻌㻺㼑㼣㻌㻹㼍㼞㼗㼑㼠㼕㼚㼓㻌㻵㼚㼟㼕㼓㼔㼠㼟㻌㻹㼑㼠㼔㼛㼐㼛㼘㼛㼓㼕㼏㼍㼘㻌㻵㼟㼟㼡㼑㼟㻌㼍㼚㼐㻌㻯㼔㼍㼘㼘㼑㼚㼓㼑㼟 㻝㻢㻦㻡㻡㻙㻝㻣㻦㻞㻜 㻾㼑㼕㼚㼔㼛㼘㼐㻌㻰㼑㼏㼗㼑㼞㻌㻔㻮㼕㼑㼘㼑㼒㼑㼘㼐㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻕 㻟㻟 㻝㻣㻦㻞㻜㻌㻙㻌㻝㻣㻦㻟㻜 㻯㼘㼛㼟㼕㼚㼓 vii Comparison of two Distribution Valued Graph Symmetries and Tests for Formal Dissimilarities and its Application for Symbolic Cluster Stability in Modular Clustering Clustering Geyer-Schulz, Andreas1, Ovelgonne, Michael1 and Stein, Martin1 1 2 2 2 Yusuke Matsui , Yuriko Komiya , Hiroyuki Minami and Masahiro Mizuta KIT-Campus S¨ud [email protected] 1 Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan 2 Information Initiative Center, Hokkaido University, Sapporo, Japan Abstract. The analysis of resolution limits in community detection by Fortunato and Barthelemy (2007) shows that modularity as a formal cluster criterion does not Abstract. Symbolic Data Analysis is a new approach for data analysis. We are solve the problem of determining the number of clusters and detecting a proper interested in dissimilarities represented by distributions. Let symbolic objects be cluster structure simultaneously. The counter-examples given are symmetric graphs O1,O2,...,On . We define distribution valued dissimilarities between the objects { } (cyclic graphs and complete cliques). In addition, we show how we can use these Oi and Oj as counter-examples to construct test cases for almost all well-known measures for d (Oi,Oj ) Sij , ≡ comparing clusterings (e.g. the RAND index or mutual information measures) fail where Sij are random variables. with arbitrary large errors. { } We give a comparison rule for ordering Sij as follows. In our contribution we investigate the construction of measures of graph symme- { } 1 tries as tests of formal cluster stability in modular clustering. Our approach applies Sij >Skl Pr(Sij >Skl) > , ⇐⇒ 2 results from graph enumeration and Polya’s combinatorial approach as well as from we put automorphic forms and representations. Pr(Sij >Skl)=Pij,kl. Note that Pij,kl =1 Pkl,ij . Here, we apply the rule to Symbolic Clustering as − follows. References 1. Initialize cls = C1,C2,...,Cn , Ci = Oi , K := n. { } { } 2. Select the pair of clusters (i, j) to be merged such that (i, j) = arg min Pij , .., Fortunaty, Santo and Barthelemy, Marc (2007): Resolution Limit in Community De- i,j n tection. Proceedings of the National