mm-2569 === Subject: Re: Chained arrow notation experts- 9->9->9->9 sketch? Regarding Graham's number,here is a diagram that I've seen on the web- / 1) 3^^^^3 | | 2) 3^^...1)...^^3 [where there are 1) = 3^^^^3 up-arrows] | | 3) 3^^...2)...^^3 [where there are 2) up-arrows] | . 64 levels < . | . | . | 63) 3^^...62)...^^3 | 64) 3^^...63)...^^3 <--- Graham's # (-the process of so and so many arrows is far from being mathematically precise and needs to be reworked;however,it lends itself to being spacially compact. :-) ) Labelling sections and bays one section ________________^_______ | / | 2 bays | ______________^_________ | / | bay 1 bay 2 | ___^___ ______^________ | / / > one section / 1) 3^^^^3 | | 2) 3^..1)..^3 | | 3) 3^..2)..^3 | 64 levels< . | | . | | . | 64) 3^..63)..^3 / where a bay (OL) includes all levels and a section is all bays and their levels. Using this mechanism, the Conway-Guy expression a -> b -> ..... x -> y -> z can be defined in terms of Knuth up-arrows, e.g. 2->3->3->4 8 bays ____________________________^________________________________ / bay 1 bay 2 bay 3 4-7 bay 8 __^__ _________^__________ _____^_______ __^_ _____^______ / / / / / / 1) 2^3 = 8 / 1) 2^3 / / 1) 2^3 | 2) 2^..1)..^3 | 2) 2^..1)..^3| | 2) 2^..1)..^3 8 | 3) 2^..2)..^3 | . | | . levels< . | . | 4 | . | . | . | bays| . 8) 2^..7)..^3 levels< . |here | . = 2->3->8->2 | . | | . = 2->3->2->3 ..2^........^3<.....< . levels| | . ..2^........^3 = 2->3->8->3 bays = 2->3->2->4 ________________________________________________________^_____ / / 1) 2^3 / 1) 2^3 / / 1) 2^3 | . | . | | . levels< . | . | | . | . | . | | . 8) 2^..7)..^3 levels< . | | . | . | | . ..2^........^3<.....< . levels| | . ..2^.......^3 = 2->3->3->4 the above can be called 2 section levels crunch the diagram down to this;dropping bays # 1 and 3 --- 7 bays ___________^_____________ / 1) 2^3 / / 1) 2^3 . | | . . | | . 8)2^..7)..^3<..< . levels | | . ..2^.....^3 = 2->3->2->4 bays ___________________^_____ /1) 2^3 / / 1) 2^3 . | | . . | | . 8)2^..7)..^3<..< . levels| | . ..2^.....^3 = 2->3->3->4 Following and extending the concept--- > another try at 9->9->9->9--- > 387420488 section bays > _______________________________________________^____________________________ _ ________________________________ > / > / 387420488 bays > | _______________________^_____________________ > |/ 1) 9^9=387420489 / / 1) 9^9 > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > | . | | . > | . | | . > |387420489) 9^..387420488)..^9<..< . > | levels| | . > | ..9^........^9 > | bays > /387420488< ______________________________________^______ > | section|/ . > | levels | . / / 387420488 bays > | | . | | _______________________^_____________________ > | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 > | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . > | | . | | . | | . | | . > | | . | | . | |387420489) 9^..387420488)..^9<..< . > | |387420489) 9^..387420488)..^9<..< . | | levels| | . > | | levels | | . | | ..9^........^9 > | ..9^.........^9<...< bays > | section levels | | ___________________________________^_________ > | | |/ . > | | | . > | | | ___________________________________^_________ > | | |/ 1) 9^9 / / 1) 9^9 > | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > 387420488< | | . | | . > levels of| | | . | | . > section | | |387420489) 9^..387420488)..^9<..< . > levels | | | levels | | . > | ..9^........^9 > | section bays > | ____________________________________________________________________________ _ _____________________^_________ > |/ . > | . > | ____________________________________________________________________________ _ _____________________^_________ > |/ / 387420488 bays > | | _______________________^_____________________ > | |/ 1) 9^9=387420489 / / 1) 9^9 > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > | | . | | . > | | . | | . > | |387420489) 9^..387420488)..^9<..< . > | | levels| | . > | | ..9^........^9 > | | bays > 387420488< ______________________________________^______ > section|/ . > levels | . / / 387420488 bays > | . | | _______________________^_____________________ > | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 > |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . > | . | | . | | . | | . > | . | | . | |387420489) 9^..387420488)..^9<..< . > |387420489) 9^..387420488)..^9<..< . | | levels| | . > | levels | | . | | ..9^........^9 > ..9^.........^9<...< bays > section levels | | ___________________________________^_________ > | |/ . > | | . > | | ___________________________________^_________ > | |/ 1) 9^9 / / 1) 9^9 > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > | | . | | . > | | . | | . > | |387420489) 9^..387420488)..^9<..< . > | | levels | | . > ..9^........^9 > levels of > 387420488 bays of section bays section levels > _______________________________________________________________^____________ _ ______________________________________ | > / v > v > 387420488 section bays v > _______________________________________________________________^____________ _ _______________________________ | | v > / | | v > / 387420488 bays | | v > | _______________________^_____________________ | | v > |/ 1) 9^9=387420489 / / 1) 9^9 | | v > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | v > | . | | . | | v > | . | | . | | v > |387420489) 9^..387420488)..^9<..< . | | v > | levels| | . | | v > | ..9^........^9 | | v > | bays | | v > 387420488< ______________________________________^______ | | v > section|/ . | | v > levels | . / / 387420488 bays | | v > | . | | _______________________^_____________________ | | v > | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | v > |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | v > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | v > | . | | . | | . | | . | | v > | . | | . | |387420489) 9^..387420488)..^9<..< . | | v > |387420489) 9^..387420488)..^9<..< . | | levels| | . | | v > | levels | | . | | ..9^........^9 | | v > ..9^.........^9<...< bays | | v > section levels | | ___________________________________^_________ | | v > | |/ . | | v > | | . | | v > | | . | | v > | | ___________________________________^_________ | | v > | |/ 1) 9^9 / / 1) 9^9 | | v > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | v > | | . | | . | | v > | | . | | . | | v > | |387420489) 9^..387420488)..^9<..< . | | v > | | levels | | . | | v > ..9^........^9 | | v > section bays | | v > ____________________________________________________________________________ _ _____________________^_________ | | v > / . | | v > . >....> . | | > ____________________________________________________________________________ _ _____________________^_________ | | > / / 387420488 bays | | > | _______________________^_____________________ | | > |/ 1) 9^9=387420489 / / 1) 9^9 | | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | > | . | | . | | > | . | | . | | > |387420489) 9^..387420488)..^9<..< . | | > | levels| | . | | > | ..9^........^9 | | > | bays | | > 387420488< ______________________________________^______ | | > section|/ . | | > levels | . / / 387420488 bays | | > | . | | _______________________^_____________________ | | > | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | > |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | > | . | | . | | . | | . | | > | . | | . | |387420489) 9^..387420488)..^9<..< . | | > |387420489) 9^..387420488)..^9<..< . | | levels| | . | | > | levels | | . | | ..9^........^9 | | > ..9^.........^9<...< bays | | > section levels | | ___________________________________^_________ | | > | |/ . | | > | | . | | > | | . | | > | | ___________________________________^_________ | | > | |/ 1) 9^9 / / 1) 9^9 | | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | > | | . | | . | | > | | . | | . | | > | |387420489) 9^..387420488)..^9<..< . | | > | | levels | | . | | > ..9^........^9 / / > bays of section bays > ____________________________________________________________________________ _ __________________________^_________________________________________________ _ ____________________________________________________________________________ _ ____ | > / 387420488 section bays | > _______________________________________________________________^____________ _ _______________________________ | > / | > / 387420488 bays | > | _______________________^_____________________ | > |/ 1) 9^9=387420489 / / 1) 9^9 / / 387420488 section bays | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | _______________________________________________________________^____________ _ _______________________________ | > | . | | . | |/ | > | . | | . | | / 387420488 bays | > |387420489) 9^..387420488)..^9<..< . | | | _______________________^_____________________ | > | levels| | . | | |/ 1) 9^9=387420489 / / 1) 9^9 | > | ..9^........^9 | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | bays | | | . | | . | > /387420488< ______________________________________^______ | | | . | | . | > | section|/ . | | |387420489) 9^..387420488)..^9<..< . | > | levels | . / / 387420488 bays | | | levels| | . | > | | . | | _______________________^_____________________ | | | ..9^........^9 | > | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | | bays | > | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | |387420488< ______________________________________^______ | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | section|/ . | > | | . | | . | | . | | . | | levels | . / / 387420488 bays | > | | . | | . | |387420489) 9^..387420488)..^9<..< . | | | . | | _______________________^_____________________ | > | |387420489) 9^..387420488)..^9<..< . | | levels| | . | | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | > | | levels | | . | | ..9^........^9 | | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | ..9^.........^9<...< bays | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | > | section levels | | ___________________________________^_________ | | | . | | . | | . | | . | > | | |/ . | | | . | | . | |387420489) 9^..387420488)..^9<..< . | > | | | . | | |387420489) 9^..387420488)..^9<..< . | | levels| | . | > | | | . | | | levels | | . | | ..9^........^9 | > | | | ___________________________________^_________ | | ..9^.........^9<...< bays | > | | |/ 1) 9^9 / / 1) 9^9 | | section levels | | ___________________________________^_________ | > | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | | |/ . | > 387420488< | | . | | . | | | | . | > levels of| | | . | | . | | | | . | > section | | |387420489) 9^..387420488)..^9<..< . | | | | ___________________________________^_________ | > levels | | | levels | | . | | | |/ 1) 9^9 / / 1) 9^9 | > | ..9^........^9 | | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | section bays | | | | . | | . | > | ____________________________________________________________________________ _ _____________________^_________ | | | | . | | . | > |/ . | | | |387420489) 9^..387420488)..^9<..< . | > | . | | | | levels | | . | > | . | | ..9^........^9 | > | ____________________________________________________________________________ _ _____________________^_________ | | section bays | > |/ / 387420488 bays | | ____________________________________________________________________________ _ _____________________^_________ | > | | _______________________^_____________________ | |/ . | > | |/ 1) 9^9=387420489 / / 1) 9^9 | | . | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | > | | . | | . | | ____________________________________________________________________________ _ _____________________^_________ | > | | . | | . | |/ / 387420488 bays | > | |387420489) 9^..387420488)..^9<..< . | | | _______________________^_____________________ | > | | levels| | . | | |/ 1) 9^9=387420489 / / 1) 9^9 | > | | ..9^........^9 | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | | bays | | | . | | . | > |387420488< ______________________________________^______ | | | . | | . | > | section|/ . | | |387420489) 9^..387420488)..^9<..< . | > levels | . / / 387420488 bays | | | levels| | . | > | . | | _______________________^_____________________ | | | ..9^........^9 | > | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | | bays | > |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | |387420488< ______________________________________^______ | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | section|/ . | > | . | | . | | . | | . | | levels | . / / 387420488 bays | > | . | | . | |387420489) 9^..387420488)..^9<..< . | | | . | | _______________________^_____________________ | > |387420489) 9^..387420488)..^9<..< . | | levels| | . | | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | > | levels | | . | | ..9^........^9 | | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > ..9^.........^9<...< bays | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | > section levels | | ___________________________________^_________ | | | . | | . | | . | | . | > | |/ . | | | . | | . | |387420489) 9^..387420488)..^9<..< . | > | | . | | |387420489) 9^..387420488)..^9<..< . | | levels| | . | > | | . | | | levels | | . | | ..9^........^9 | > | | ___________________________________^_________ | | ..9^.........^9<...< bays | > | |/ 1) 9^9 / / 1) 9^9 | | section levels | | ___________________________________^_________ | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | | |/ . | > | | . | | . | | | | . | > | | . | | . | | | | . | > | |387420489) 9^..387420488)..^9<..< . | | | | ___________________________________^_________ | > | | levels | | . | | | |/ 1) 9^9 / / 1) 9^9 | > ..9^........^9 | | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > levels of section bays<....< | | . | | . | > | | | | . | | . | > | | | |387420489) 9^..387420488)..^9<..< . | > | | | | levels | | . | > ..9^........^9 | > bays of bays > section bays | > ____________________________________________________________________________ _ ____________________________________________________________________________ _ _______________________________________________________________________^____ _ ____ | > / . | > . | > . | > ____________________________________________________________________________ _ ____________________________________________________________________________ _ _______________________________________________________________________^____ _ ____ | > / | 8 > 387420488 section bays > levels > _______________________________________________________________^____________ _ _______________________________ |of bays > / |of sect. > / 387420488 bays |bays > | _______________________^_____________________ | > |/ 1) 9^9=387420489 / / 1) 9^9 / / 387420488 section bays | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | _______________________________________________________________^____________ _ _______________________________ | > | . | | . | |/ | > | . | | . | | / 387420488 bays | > |387420489) 9^..387420488)..^9<..< . | | | _______________________^_____________________ | > | levels| | . | | |/ 1) 9^9=387420489 / / 1) 9^9 | > | ..9^........^9 | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | bays | | | . | | . | > /387420488< ______________________________________^______ | | | . | | . | > | section|/ . | | |387420489) 9^..387420488)..^9<..< . | > | levels | . / / 387420488 bays | | | levels| | . | > | | . | | _______________________^_____________________ | | | ..9^........^9 | > | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | | bays | > | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | |387420488< ______________________________________^______ | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | section|/ . | > | | . | | . | | . | | . | | levels | . / / 387420488 bays | > | | . | | . | |387420489) 9^..387420488)..^9<..< . | | | . | | _______________________^_____________________ | > | |387420489) 9^..387420488)..^9<..< . | | levels| | . | | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | > | | levels | | . | | ..9^........^9 | | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | ..9^.........^9<...< bays | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | > | section levels | | ___________________________________^_________ | | | . | | . | | . | | . | > | | |/ . | | | . | | . | |387420489) 9^..387420488)..^9<..< . | > | | | . | | |387420489) 9^..387420488)..^9<..< . | | levels| | . | > | | | . | | | levels | | . | | ..9^........^9 | > | | | ___________________________________^_________ | | ..9^.........^9<...< bays | > | | |/ 1) 9^9 / / 1) 9^9 | | section levels | | ___________________________________^_________ | > | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | | |/ . | > 387420488< | | . | | . | | | | . | > levels of| | | . | | . | | | | . | > section | | |387420489) 9^..387420488)..^9<..< . | | | | ___________________________________^_________ | > levels | | | levels | | . | | | |/ 1) 9^9 / / 1) 9^9 | > | ..9^........^9 | | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | section bays | | | | . | | . | > | ____________________________________________________________________________ _ _____________________^_________ | | | | . | | . | > |/ . | | | |387420489) 9^..387420488)..^9<..< . | > | . | | | | levels | | . | > | . | | ..9^........^9 | > | ____________________________________________________________________________ _ _____________________^_________ | | section bays | > |/ / 387420488 bays | | ____________________________________________________________________________ _ _____________________^_________ | > | | _______________________^_____________________ | |/ . | > | |/ 1) 9^9=387420489 / / 1) 9^9 | | . | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | > | | . | | . | | ____________________________________________________________________________ _ _____________________^_________ | > | | . | | . | |/ / 387420488 bays | > | |387420489) 9^..387420488)..^9<..< . | | | _______________________^_____________________ | > | | levels| | . | | |/ 1) 9^9=387420489 / / 1) 9^9 | > | | ..9^........^9 | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | | bays | | | . | | . | > |387420488< ______________________________________^______ | | | . | | . | > | section|/ . | | |387420489) 9^..387420488)..^9<..< . | > levels | . / / 387420488 bays | | | levels| | . | > | . | | _______________________^_____________________ | | | ..9^........^9 | > | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | | bays | > |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | |387420488< ______________________________________^______ | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | section|/ . | > | . | | . | | . | | . | | levels | . / / 387420488 bays | > | . | | . | |387420489) 9^..387420488)..^9<..< . | | | . | | _______________________^_____________________ | > |387420489) 9^..387420488)..^9<..< . | | levels| | . | | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | > | levels | | . | | ..9^........^9 | | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > ..9^.........^9<...< bays | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | > section levels | | ___________________________________^_________ | | | . | | . | | . | | . | > | |/ . | | | . | | . | |387420489) 9^..387420488)..^9<..< . | > | | . | | |387420489) 9^..387420488)..^9<..< . | | levels| | . | > | | . | | | levels | | . | | ..9^........^9 | > | | ___________________________________^_________ | | ..9^.........^9<...< bays | > | |/ 1) 9^9 / / 1) 9^9 | | section levels | | ___________________________________^_________ | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | | |/ . | > | | . | | . | | | | . | > | | . | | . | | | | . | > | |387420489) 9^..387420488)..^9<..< . | | | | ___________________________________^_________ | > | | levels | | . | | | |/ 1) 9^9 / / 1) 9^9 | > ..9^........^9 | | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > levels of section bays<....< | | . | | . | > | | | | . | | . | > | | | |387420489) 9^..387420488)..^9<..< . | > | | | | levels | | . | > ..9^........^9 / > | > v > 9->9->9->9 === Subject: Re: Chained arrow notation experts- 9->9->9->9 sketch? Regarding Graham's number,here is a diagram that I've seen on the web- / 1) 3^^^^3 | | 2) 3^^...1)...^^3 [where there are 1) = 3^^^^3 up-arrows] | | 3) 3^^...2)...^^3 [where there are 2) up-arrows] | . 64 levels < . | . | . | 63) 3^^...62)...^^3 | 64) 3^^...63)...^^3 <--- Graham's # (-the process of so and so many arrows is far from being mathematically precise and needs to be reworked;however,it lends itself to being spacially compact. :-) ) Labelling sections and bays one section ________________^_______ | / | 2 bays | ______________^_________ | / | bay 1 bay 2 | ___^___ ______^________ | / / > one section / 1) 3^^^^3 | | 2) 3^..1)..^3 | | 3) 3^..2)..^3 | 64 levels< . | | . | | . | 64) 3^..63)..^3 / where a bay (OL) includes all levels and a section is all bays and their levels. Using this mechanism, the Conway-Guy expression a -> b -> ..... x -> y -> z can be defined in terms of Knuth up-arrows, e.g. 2->3->3->4 8 bays ____________________________^________________________________ / bay 1 bay 2 bay 3 4-7 bay 8 __^__ _________^__________ _____^_______ __^_ _____^______ / / / / / / 1) 2^3 = 8 / 1) 2^3 / / 1) 2^3 | 2) 2^..1)..^3 | 2) 2^..1)..^3| | 2) 2^..1)..^3 8 | 3) 2^..2)..^3 | . | | . levels< . | . | 4 | . | . | . | bays| . 8) 2^..7)..^3 levels< . |here | . = 2->3->8->2 | . | | . = 2->3->2->3 ..2^........^3<.....< . levels| | . ..2^........^3 = 2->3->8->3 bays = 2->3->2->4 ________________________________________________________^_____ / / 1) 2^3 / 1) 2^3 / / 1) 2^3 | . | . | | . levels< . | . | | . | . | . | | . 8) 2^..7)..^3 levels< . | | . | . | | . ..2^........^3<.....< . levels| | . ..2^.......^3 = 2->3->3->4 the above can be called 2 section levels crunch the diagram down to this;dropping bays # 1 and 3 --- 7 bays ___________^_____________ / 1) 2^3 / / 1) 2^3 . | | . . | | . 8)2^..7)..^3<..< . levels | | . ..2^.....^3 = 2->3->2->4 bays ___________________^_____ /1) 2^3 / / 1) 2^3 . | | . . | | . 8)2^..7)..^3<..< . levels| | . ..2^.....^3 = 2->3->3->4 Following and extending the concept--- > another try at 9->9->9->9--- > 387420488 section bays > _______________________________________________^____________________________ _ ________________________________ > / > / 387420488 bays > | _______________________^_____________________ > |/ 1) 9^9=387420489 / / 1) 9^9 > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > | . | | . > | . | | . > |387420489) 9^..387420488)..^9<..< . > | levels| | . > | ..9^........^9 > | bays > /387420488< ______________________________________^______ > | section|/ . > | levels | . / / 387420488 bays > | | . | | _______________________^_____________________ > | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 > | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . > | | . | | . | | . | | . > | | . | | . | |387420489) 9^..387420488)..^9<..< . > | |387420489) 9^..387420488)..^9<..< . | | levels| | . > | | levels | | . | | ..9^........^9 > | ..9^.........^9<...< bays > | section levels | | ___________________________________^_________ > | | |/ . > | | | . > | | | ___________________________________^_________ > | | |/ 1) 9^9 / / 1) 9^9 > | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > 387420488< | | . | | . > levels of| | | . | | . > section | | |387420489) 9^..387420488)..^9<..< . > levels | | | levels | | . > | ..9^........^9 > | section bays > | ____________________________________________________________________________ _ _____________________^_________ > |/ . > | . > | ____________________________________________________________________________ _ _____________________^_________ > |/ / 387420488 bays > | | _______________________^_____________________ > | |/ 1) 9^9=387420489 / / 1) 9^9 > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > | | . | | . > | | . | | . > | |387420489) 9^..387420488)..^9<..< . > | | levels| | . > | | ..9^........^9 > | | bays > 387420488< ______________________________________^______ > section|/ . > levels | . / / 387420488 bays > | . | | _______________________^_____________________ > | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 > |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . > | . | | . | | . | | . > | . | | . | |387420489) 9^..387420488)..^9<..< . > |387420489) 9^..387420488)..^9<..< . | | levels| | . > | levels | | . | | ..9^........^9 > ..9^.........^9<...< bays > section levels | | ___________________________________^_________ > | |/ . > | | . > | | ___________________________________^_________ > | |/ 1) 9^9 / / 1) 9^9 > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 > | | . | | . > | | . | | . > | |387420489) 9^..387420488)..^9<..< . > | | levels | | . > ..9^........^9 > levels of > 387420488 bays of section bays section levels > _______________________________________________________________^____________ _ ______________________________________ | > / v > v > 387420488 section bays v > _______________________________________________________________^____________ _ _______________________________ | | v > / | | v > / 387420488 bays | | v > | _______________________^_____________________ | | v > |/ 1) 9^9=387420489 / / 1) 9^9 | | v > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | v > | . | | . | | v > | . | | . | | v > |387420489) 9^..387420488)..^9<..< . | | v > | levels| | . | | v > | ..9^........^9 | | v > | bays | | v > 387420488< ______________________________________^______ | | v > section|/ . | | v > levels | . / / 387420488 bays | | v > | . | | _______________________^_____________________ | | v > | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | v > |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | v > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | v > | . | | . | | . | | . | | v > | . | | . | |387420489) 9^..387420488)..^9<..< . | | v > |387420489) 9^..387420488)..^9<..< . | | levels| | . | | v > | levels | | . | | ..9^........^9 | | v > ..9^.........^9<...< bays | | v > section levels | | ___________________________________^_________ | | v > | |/ . | | v > | | . | | v > | | . | | v > | | ___________________________________^_________ | | v > | |/ 1) 9^9 / / 1) 9^9 | | v > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | v > | | . | | . | | v > | | . | | . | | v > | |387420489) 9^..387420488)..^9<..< . | | v > | | levels | | . | | v > ..9^........^9 | | v > section bays | | v > ____________________________________________________________________________ _ _____________________^_________ | | v > / . | | v > . >....> . | | > ____________________________________________________________________________ _ _____________________^_________ | | > / / 387420488 bays | | > | _______________________^_____________________ | | > |/ 1) 9^9=387420489 / / 1) 9^9 | | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | > | . | | . | | > | . | | . | | > |387420489) 9^..387420488)..^9<..< . | | > | levels| | . | | > | ..9^........^9 | | > | bays | | > 387420488< ______________________________________^______ | | > section|/ . | | > levels | . / / 387420488 bays | | > | . | | _______________________^_____________________ | | > | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | > |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | > | . | | . | | . | | . | | > | . | | . | |387420489) 9^..387420488)..^9<..< . | | > |387420489) 9^..387420488)..^9<..< . | | levels| | . | | > | levels | | . | | ..9^........^9 | | > ..9^.........^9<...< bays | | > section levels | | ___________________________________^_________ | | > | |/ . | | > | | . | | > | | . | | > | | ___________________________________^_________ | | > | |/ 1) 9^9 / / 1) 9^9 | | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | > | | . | | . | | > | | . | | . | | > | |387420489) 9^..387420488)..^9<..< . | | > | | levels | | . | | > ..9^........^9 / / > bays of section bays > ____________________________________________________________________________ _ __________________________^_________________________________________________ _ ____________________________________________________________________________ _ ____ | > / 387420488 section bays | > _______________________________________________________________^____________ _ _______________________________ | > / | > / 387420488 bays | > | _______________________^_____________________ | > |/ 1) 9^9=387420489 / / 1) 9^9 / / 387420488 section bays | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | _______________________________________________________________^____________ _ _______________________________ | > | . | | . | |/ | > | . | | . | | / 387420488 bays | > |387420489) 9^..387420488)..^9<..< . | | | _______________________^_____________________ | > | levels| | . | | |/ 1) 9^9=387420489 / / 1) 9^9 | > | ..9^........^9 | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | bays | | | . | | . | > /387420488< ______________________________________^______ | | | . | | . | > | section|/ . | | |387420489) 9^..387420488)..^9<..< . | > | levels | . / / 387420488 bays | | | levels| | . | > | | . | | _______________________^_____________________ | | | ..9^........^9 | > | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | | bays | > | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | |387420488< ______________________________________^______ | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | section|/ . | > | | . | | . | | . | | . | | levels | . / / 387420488 bays | > | | . | | . | |387420489) 9^..387420488)..^9<..< . | | | . | | _______________________^_____________________ | > | |387420489) 9^..387420488)..^9<..< . | | levels| | . | | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | > | | levels | | . | | ..9^........^9 | | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | ..9^.........^9<...< bays | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | > | section levels | | ___________________________________^_________ | | | . | | . | | . | | . | > | | |/ . | | | . | | . | |387420489) 9^..387420488)..^9<..< . | > | | | . | | |387420489) 9^..387420488)..^9<..< . | | levels| | . | > | | | . | | | levels | | . | | ..9^........^9 | > | | | ___________________________________^_________ | | ..9^.........^9<...< bays | > | | |/ 1) 9^9 / / 1) 9^9 | | section levels | | ___________________________________^_________ | > | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | | |/ . | > 387420488< | | . | | . | | | | . | > levels of| | | . | | . | | | | . | > section | | |387420489) 9^..387420488)..^9<..< . | | | | ___________________________________^_________ | > levels | | | levels | | . | | | |/ 1) 9^9 / / 1) 9^9 | > | ..9^........^9 | | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | section bays | | | | . | | . | > | ____________________________________________________________________________ _ _____________________^_________ | | | | . | | . | > |/ . | | | |387420489) 9^..387420488)..^9<..< . | > | . | | | | levels | | . | > | . | | ..9^........^9 | > | ____________________________________________________________________________ _ _____________________^_________ | | section bays | > |/ / 387420488 bays | | ____________________________________________________________________________ _ _____________________^_________ | > | | _______________________^_____________________ | |/ . | > | |/ 1) 9^9=387420489 / / 1) 9^9 | | . | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | > | | . | | . | | ____________________________________________________________________________ _ _____________________^_________ | > | | . | | . | |/ / 387420488 bays | > | |387420489) 9^..387420488)..^9<..< . | | | _______________________^_____________________ | > | | levels| | . | | |/ 1) 9^9=387420489 / / 1) 9^9 | > | | ..9^........^9 | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | | bays | | | . | | . | > |387420488< ______________________________________^______ | | | . | | . | > | section|/ . | | |387420489) 9^..387420488)..^9<..< . | > levels | . / / 387420488 bays | | | levels| | . | > | . | | _______________________^_____________________ | | | ..9^........^9 | > | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | | bays | > |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | |387420488< ______________________________________^______ | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | section|/ . | > | . | | . | | . | | . | | levels | . / / 387420488 bays | > | . | | . | |387420489) 9^..387420488)..^9<..< . | | | . | | _______________________^_____________________ | > |387420489) 9^..387420488)..^9<..< . | | levels| | . | | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | > | levels | | . | | ..9^........^9 | | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > ..9^.........^9<...< bays | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | > section levels | | ___________________________________^_________ | | | . | | . | | . | | . | > | |/ . | | | . | | . | |387420489) 9^..387420488)..^9<..< . | > | | . | | |387420489) 9^..387420488)..^9<..< . | | levels| | . | > | | . | | | levels | | . | | ..9^........^9 | > | | ___________________________________^_________ | | ..9^.........^9<...< bays | > | |/ 1) 9^9 / / 1) 9^9 | | section levels | | ___________________________________^_________ | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | | |/ . | > | | . | | . | | | | . | > | | . | | . | | | | . | > | |387420489) 9^..387420488)..^9<..< . | | | | ___________________________________^_________ | > | | levels | | . | | | |/ 1) 9^9 / / 1) 9^9 | > ..9^........^9 | | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > levels of section bays<....< | | . | | . | > | | | | . | | . | > | | | |387420489) 9^..387420488)..^9<..< . | > | | | | levels | | . | > ..9^........^9 | > bays of bays > section bays | > ____________________________________________________________________________ _ ____________________________________________________________________________ _ _______________________________________________________________________^____ _ ____ | > / . | > . | > . | > ____________________________________________________________________________ _ ____________________________________________________________________________ _ _______________________________________________________________________^____ _ ____ | > / | 8 > 387420488 section bays > levels > _______________________________________________________________^____________ _ _______________________________ |of bays > / |of sect. > / 387420488 bays |bays > | _______________________^_____________________ | > |/ 1) 9^9=387420489 / / 1) 9^9 / / 387420488 section bays | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | _______________________________________________________________^____________ _ _______________________________ | > | . | | . | |/ | > | . | | . | | / 387420488 bays | > |387420489) 9^..387420488)..^9<..< . | | | _______________________^_____________________ | > | levels| | . | | |/ 1) 9^9=387420489 / / 1) 9^9 | > | ..9^........^9 | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | bays | | | . | | . | > /387420488< ______________________________________^______ | | | . | | . | > | section|/ . | | |387420489) 9^..387420488)..^9<..< . | > | levels | . / / 387420488 bays | | | levels| | . | > | | . | | _______________________^_____________________ | | | ..9^........^9 | > | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | | bays | > | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | |387420488< ______________________________________^______ | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | section|/ . | > | | . | | . | | . | | . | | levels | . / / 387420488 bays | > | | . | | . | |387420489) 9^..387420488)..^9<..< . | | | . | | _______________________^_____________________ | > | |387420489) 9^..387420488)..^9<..< . | | levels| | . | | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | > | | levels | | . | | ..9^........^9 | | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | ..9^.........^9<...< bays | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | > | section levels | | ___________________________________^_________ | | | . | | . | | . | | . | > | | |/ . | | | . | | . | |387420489) 9^..387420488)..^9<..< . | > | | | . | | |387420489) 9^..387420488)..^9<..< . | | levels| | . | > | | | . | | | levels | | . | | ..9^........^9 | > | | | ___________________________________^_________ | | ..9^.........^9<...< bays | > | | |/ 1) 9^9 / / 1) 9^9 | | section levels | | ___________________________________^_________ | > | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | | |/ . | > 387420488< | | . | | . | | | | . | > levels of| | | . | | . | | | | . | > section | | |387420489) 9^..387420488)..^9<..< . | | | | ___________________________________^_________ | > levels | | | levels | | . | | | |/ 1) 9^9 / / 1) 9^9 | > | ..9^........^9 | | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | section bays | | | | . | | . | > | ____________________________________________________________________________ _ _____________________^_________ | | | | . | | . | > |/ . | | | |387420489) 9^..387420488)..^9<..< . | > | . | | | | levels | | . | > | . | | ..9^........^9 | > | ____________________________________________________________________________ _ _____________________^_________ | | section bays | > |/ / 387420488 bays | | ____________________________________________________________________________ _ _____________________^_________ | > | | _______________________^_____________________ | |/ . | > | |/ 1) 9^9=387420489 / / 1) 9^9 | | . | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | > | | . | | . | | ____________________________________________________________________________ _ _____________________^_________ | > | | . | | . | |/ / 387420488 bays | > | |387420489) 9^..387420488)..^9<..< . | | | _______________________^_____________________ | > | | levels| | . | | |/ 1) 9^9=387420489 / / 1) 9^9 | > | | ..9^........^9 | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > | | bays | | | . | | . | > |387420488< ______________________________________^______ | | | . | | . | > | section|/ . | | |387420489) 9^..387420488)..^9<..< . | > levels | . / / 387420488 bays | | | levels| | . | > | . | | _______________________^_____________________ | | | ..9^........^9 | > | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | | | bays | > |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | |387420488< ______________________________________^______ | > | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | | section|/ . | > | . | | . | | . | | . | | levels | . / / 387420488 bays | > | . | | . | |387420489) 9^..387420488)..^9<..< . | | | . | | _______________________^_____________________ | > |387420489) 9^..387420488)..^9<..< . | | levels| | . | | | ______________________________________^______ | |/ 1) 9^9 / / 1) 9^9 | > | levels | | . | | ..9^........^9 | | |/ 1) 9^9 / / 1) 9^9 | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > ..9^.........^9<...< bays | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | . | | . | > section levels | | ___________________________________^_________ | | | . | | . | | . | | . | > | |/ . | | | . | | . | |387420489) 9^..387420488)..^9<..< . | > | | . | | |387420489) 9^..387420488)..^9<..< . | | levels| | . | > | | . | | | levels | | . | | ..9^........^9 | > | | ___________________________________^_________ | | ..9^.........^9<...< bays | > | |/ 1) 9^9 / / 1) 9^9 | | section levels | | ___________________________________^_________ | > | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | | | |/ . | > | | . | | . | | | | . | > | | . | | . | | | | . | > | |387420489) 9^..387420488)..^9<..< . | | | | ___________________________________^_________ | > | | levels | | . | | | |/ 1) 9^9 / / 1) 9^9 | > ..9^........^9 | | | | 2) 9^..1)..^9 | | 2) 9^..1)..^9 | > levels of section bays<....< | | . | | . | > | | | | . | | . | > | | | |387420489) 9^..387420488)..^9<..< . | > | | | | levels | | . | > ..9^........^9 / > | > v > 9->9->9->9 === Subject: Re: Quick Math Guide to core error issues thank *o*, for small things, like using a vriable consistently -- another milestone for the Ten Year Programme! > Oh yeah, I've taken your advice though as I'm using only m as a > variable in my recent postings as I'm *really* ready to finish things > up. > it's not a big deal. In any event, unlike with Nora Baron, I won't > put quotes around your name. --il duce d'Enron! === Subject: Re: Quick Math Guide to core error issues > > Is it me or can anyone follow what JSH is ranting about? > > David Moran > http://www.crank.net/harris.html Now that's just mean. I think that Sam Wormley is a worm. He's a weak, ineffectual and intellectually deficient person, who unfortunately has delusions of grandeur, so he posts a lot. And yes, you may THINK I have delusions of grandeur, but I'm putting out my research while Sam Wormley is just being an annoying twerp. He's too small to do his own thing, so he attacks others and post links as if knowing a link shows you actually know something. James Harris === Subject: Re: Quick Math Guide to core error issues <3F918BA4.32121731@mchsi.com> Discussion, linux) >> http://www.crank.net/harris.html > Now that's just mean. I think that Sam Wormley is a worm. Please, try to stay within your role. Sam Wormley is a worm comments are reserved for Herc. When you start crossing into Herc's turf, it's difficult for those of us keeping score at home. How would you like it if he started threatening congressional hearings? -- Jesse F. Hughes My proofs are out there. -- James S. Harris === Subject: Re: Quick Math Guide to core error issues > > Is it me or can anyone follow what JSH is ranting about? > > David Moran > > http://www.crank.net/harris.html > Now that's just mean. I think that Sam Wormley is a worm. > He's a weak, ineffectual and intellectually deficient person, who > unfortunately has delusions of grandeur, so he posts a lot. > And yes, you may THINK I have delusions of grandeur, but I'm putting > out my research while Sam Wormley is just being an annoying twerp. > He's too small to do his own thing, so he attacks others and post > links as if knowing a link shows you actually know something. James, when you attack someone personally, why are you surprised when people turn around and attack you personally? === Subject: Re: Quick Math Guide to core error issues > > Is it me or can anyone follow what JSH is ranting about? > > David Moran > > http://www.crank.net/harris.html > > Now that's just mean. I think that Sam Wormley is a worm. > > He's a weak, ineffectual and intellectually deficient person, who > unfortunately has delusions of grandeur, so he posts a lot. > > And yes, you may THINK I have delusions of grandeur, but I'm putting > out my research while Sam Wormley is just being an annoying twerp. > > He's too small to do his own thing, so he attacks others and post > links as if knowing a link shows you actually know something. > > James, when you attack someone personally, why are you surprised when > people turn around and attack you personally? Sam Wormley STARTED IT by posting a flame link in MY THREAD!!! It turns out that Sam Wormley is wormy, and intellectually deficient, convinced that he can post links rather than actually know anything. If he wants to behave like a twerp, and an intellectual weakling, then I have the right to answer. And my answer is that he's deficient morally and intellectually, posts too damn much for what he's saying, and apparently has some delusions of grandeur dependent on posting links versus actually knowing of what he speaks. He's a worm. He's inferior. He's mentally deficient. James Harris === Subject: Re: Explaining math definition problem correction to the correction, and this is the 2nd ed. of B&M: he said that Cusa had a rectification of the circle, but he actually gave a simple argument, showing that it was not even rational, or transendental, as we now say. he said, because the approximations by inscribing/circumscribing polgona with more & more sides, just get further from the ultimate shape of circularity, although the error from pi gets smaller. It's a different species of shape. > another error that it made was: > said that Cusa made an erroneous proof > that the circle was incomeasurable with teh tetragon > the question is, do all cases fall to your peculiar method, or > did you find a counterexample in Object Ring Theory? --les ducs d'Enron! === Subject: Re: Explaining math definition problem > Readers can look at the argument, and see what actually is in it. > Notice how I'll be strongly emphasizing constant terms all the way > down. > P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 > which has a constant term that is 1078. > Well P(x) can also be written out as > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 You can keep on reposting this error, but it only reflects on you. The correct factorization is: P(x)= 7^2(2401 x^3 - 147 x^2 - 3x) (5^3) + 3(1 + 49 x )(5)(7^2) + 7^3 Your factorization is wrong. It does not expand to agree with the original equation defining P(x). Do you ever check your work? Or isn't that necessary with an intellect as advanced as yours? -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: Explaining math definition problem > Readers can look at the argument, and see what actually is in it. > Notice how I'll be strongly emphasizing constant terms all the way > down. > P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 Hmmm...it looks like C. Bond is actually right, and I have a sign problem as it should be -17640 x. Your factorization is wrong. It does not expand to agree with the original equation defining > P(x). Do you ever check your work? Or isn't that necessary with an intellect as advanced as > yours? You are correct that I gave the wrong polynomial as it should be P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078. James Harris === Subject: Re: Explaining math definition problem > You are correct that I gave the wrong polynomial as it should be > P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078. > James Harris Happy to oblige. No doubt you'll be equally grateful for future posts which identify further errors. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: Explaining math definition problem >>I parafrase you and ahve a question at the end: > You don't have my permission to parafrase and I don't find mocking > posts of interest. He wasn't mocking you. English isn't his first language. > Now I've looked over some of your posts and can rather easily explain > what you seem to find significant. > Actually, part of the reason I haven't been terribly worried about > answering you in detail is I've wondered how many people may begin to > doubt algebra and think that maybe math is inconsistent if they can't > figure out what's happening in your examples. > I will say that it's quite simple. > After all, constants *are* constants, and math is consistent, so there > has to be a rational explanation. We all know that. You are treating non-constants as if they behaved like constants, however. Please try to understand what the objections actually are before defending yourself. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Explaining math definition problem >where the a's are roots of >a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). >Notice it *appears* that the constant terms for the three factors are >all 7, which can't be right, as the constant term of P(x) is 1078, so >setting x=0, reveals Why do you keep talking about constant terms? The a_1, a_2, a_3 do not have constant terms; that concept is meaningless in this context. If you mean the value at 0, why don't you say so? If you mean something else, what is the constant term of sqrt(x+1) + 1? Peter van Rossum -- Peter van Rossum, | Universal law of linearity: for all Dept. of Mathematics, New Mexico | f : R -> R and for all x, y in R: State University, Las Cruces, NM, USA. | f(x + y) = f(x) + f(y) === Subject: Re: Explaining math definition problem > I'm an independent researcher, which means that I use my *own* > funding, and my *own* direction to go out and see what knowledge I can > obtain. Some of my research has been in the area of mathematics. Short, straightforward question: Let Q(m) = 7*(m^2*x^2 + 3*x + 7). Then Q(0) = 7*3*x + 7^2 = 7*(3*x + 7). Note that Q(0)/7 = 3*x + 7. This is similar to your polynomial P(m), where you have P(0) = 7^2*(3*x + 7), or P(0)/7^2 = (3*x + 7). Now assume that my polynomial Q(m) is factored in the form Q(m) = (b1*x + 7)*(b2*x + 7). As in your P(m), the coefficients b1 and b2 are going to be functions of m: b1(m) and b2(m). The fact that Q(0) = 7*(3*x + 7) is consistent with b1(0) = 0 and b2(0) = 3. Agreed? So b1(0) is divisible by 7 and b2(0) is relatively prime to 7. This is similar to your polynomial P(m), where a1(0) = a2(0) = 0 and a3(0) = 3, with a1 and a2 being divisible by 7 and a3 being relatively prime to 7. It appears to me that my (quadratic) polynomial Q(m) has all the essential properties that your cubic polynomial P(m) has. Your arguments regarding independence of the constant term, etc., should apply. So, for my polynomial Q, I would like to see your answer to the question: for my polynomial Q(m), for m in general, is b1(m) or b2(m) divisible by 7? Both, only one, or neither ? If the answer differs from what you get with P(m), I wonder if you might explain why? Andrzej === Subject: 7th degree polynomial possible way to solve 7th degree polynomial http://www.geocities.com/jongiff2000/a9_polyroot_index.html === Subject: Re: Wiles' Proof permission for an emailed response. As it happens, I just picked up Fermat's Last Theorem For Amateurs by Paulo Ribenboim, published by Springer. ISBN 0-387-98508-5. The book mostly concentrates on various partial solutions. The epilogue gives a sketch of the proof in a few pages, but doesn't claim to do more than make one excited by the idea. Thomas === Subject: converging partial arithmetic means why is it that lim (x1+...+xn)/n = a, if lim xn = a (lims are to inf)? === Subject: Re: converging partial arithmetic means > why is it that lim (x1+...+xn)/n = a, if lim xn = a (lims are to inf)? The proofs essentially split the xi into two parts, those with abs(xi-a) > eps and those with abs(xi-a) <= eps. For any eps > 0, the first part is negligable (sp?), and the second part is, for the i <= n, as n gets large, almost all the sequence. So, the average value is within 2*eps of a (the 2 takes care of the first part of the sequence). For the product, take logs. Martin Cohen === Subject: Re: converging partial arithmetic means > why is it that lim (x1+...+xn)/n = a, if lim xn = a (lims are to inf)? > even more interesting, why lim (x1*...*xn)^(1/n) = a too? especially since Cesaro theorem. === Subject: Re: converging partial arithmetic means > why is it that lim (x1+...+xn)/n = a, if lim xn = a (lims are to inf)? > even more interesting, why lim (x1*...*xn)^(1/n) = a too? especially since > Cesaro theorem. that seems like a much more general theorem, is there anyway to prove this case directly? === Subject: Re: Multiplying by Five Interesting but not quite. I recall she taught it in the 7th grade. >A long time ago, I vaguely recall an elementary school teacher telling >us some trick for quickly multiplying two numbers (2 digits, I believe) >with one or both having 5 as the second digit. Anyone recall the trick? >Numbers like 35*45 or maybe even 20*35. > Something like this maybe? > Let x=35-5=30 and y=45+5=50 then: > 35*45=(x+5)(y-5)=xy - 5x + 5y -25 = xy +5(y-x) - 25=1500+100-25=1575 === Subject: Re: Multiplying by Five That's an interesting one, but I don't believe what I was think about. WW > A long time ago, I vaguely recall an elementary school teacher telling > us some trick for quickly multiplying two numbers (2 digits, I believe) > with one or both having 5 as the second digit. Anyone recall the trick? > Numbers like 35*45 or maybe even 20*35. > Maybe this isn't what you're thinking of, but there's an > easy-to-use trick for _squaring_ a number ending in 5: multiply > the bit before the 5 by (itself+1), then append 25. > e.g. > 15^2 = 225 (1*2=2) > 25^2 = 625 (2*3*6) > ... > 125^2 = 15625 (12*13=156) > Andrew Taylor > Cambridge UK === Subject: Re: Wiles' Proof Seems to me there is a niche here. Someone needs to write the book, Fermat's Last Theorem for Dummies. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu (who someday will write Stochastic Optimal Control for Dummies) === Subject: Re: Wiles' Proof > Seems to me there is a niche here. Someone needs to write the book, > Fermat's Last Theorem for Dummies. Dr. Herschkorn, for someone with your talent for clarity and for explaining things to dummies like me........ It would only take a few months' sabbatical, probably. Think of the money and compare to academic salary! All the best am === Subject: Re: Wiles' Proof > Seems to me there is a niche here. Someone needs to write the book, > Fermat's Last Theorem for Dummies. Hey! I'm not a dummy! :) But seriously, someone _should_ write a book that outlines the proof or the idea of the proof, teaching some of the necesary math from undergraduate level and up. /David === Subject: Re: Wiles' Proof I suggest that Dummies (tm) hire the team of Magadin and Harris. > But seriously, someone _should_ write a book that outlines the proof or > the idea of the proof, teaching some of the necesary math from > undergraduate level and up. --les ducs d'Enron! http://www.cecaust.com.au/ http://members.tripod.com/~american_almanac/ === Subject: Re: Wiles' Proof > But seriously, someone _should_ write a book that outlines the proof or > the idea of the proof, teaching some of the necesary math from > undergraduate level and up. > /David Some of the books mentioned by earlier posters attempt to do this. But as was also noted, if an author needs to start with undergrad math and keep the book to manageable size (say 300 pages or so), he/she is going to end up being very sketchy. Possibly a more worthwhile project would be a book that explains the proof, to the extent possible, for someone who has at least the basic undergraduate math courses. Those courses would include: (1) linear algebra (2) abstract algebra (groups, rings, fields, including some Galois theory) (3) complex analysis Note that these are the minimal prerequisites for graduate courses such as graduate level algebra (i.e., the material covered in, say, Lang's Algebra, though I don't know that I'd recommend Lang's book as a textbook for self-study!), algebraic geometry, algebraic and analytic number theory, and representation theory. And the material covered in the graduate courses provides the background to _begin_ to get into the details of Wiles's proof. I guess what I'm trying to say is that if someone wants anything beyond a superficial hand-waving approach to Wiles's proof in a book that's less than a couple of thousand pages in length, then they'll probably need to start at least at the point of someone with an undergrad math major degree. So a well-written and worthwhile Wiles's Proof for Dummies book would have as its target audience a rather select group of dummies! (I've heard math majors called various names, but dummy isn't usually one of them.) None of this is to say that it isn't possible to write an entertaining book that gives some vague idea of the proof for someone with very little (i.e., less than an undergrad math major) background. Singh's book attempts to do that, with some success. But reading these posts, I think people want more of the math detail than that. JHS === Subject: Re: Wiles' Proof ... stuff deleted ... > ... So a well-written and worthwhile Wiles's > Proof for Dummies book would have as its target audience a rather > select group of dummies! (I've heard math majors called various names, > but dummy isn't usually one of them.) I'm thinking you haven't paid attention to many of the JSH threads. ... the rest deleted ... > JHS Dale === Subject: Re: Wiles' Proof > Seems to me there is a niche here. Someone needs to write the book, > Fermat's Last Theorem for Dummies. > > Hey! I'm not a dummy! :) > But seriously, someone _should_ write a book that outlines the proof or > the idea of the proof, teaching some of the necesary math from > undergraduate level and up. > /David Gouv.90a, Fernando Q. A marvelous proof. Amer. Math. Monthly 101 (1994), no. 3, 203--222. Here's the review: Recently, Andrew Wiles announced a stunning breakthrough toward proving the conjecture of Shimura and Taniyama that every elliptic curve over the rationals is modular. This announcement has also generated excitement and interest among the general public, because of the connection between the Shimura-Taniyama conjecture and Fermat's last theorem. Intended for an audience of nonspecialists, this paper communicates some of the basic ingredients that go into Wiles' work, and how this work relates to Fermat's last theorem. The author begins by discussing the main actors in the proof: elliptic curves, modular forms, and the Shimura-Taniyama conjecture which postulates a fascinating, mysterious connection between these two notions. He explains the fundamental result of Ribet, based on a construction of Hellegouarch and Frey and work of Serre, which gives a link between the Shimura-Taniyama conjecture and Fermat's last theorem. He then briefly mentions some of the ideas behind Wiles' attack on the Shimura-Taniyama conjecture. Written in a clear, lively and engaging style, this paper is ideal for the cultivated layperson who wishes to be introduced to the fascinating ideas that may lead to a solution of number theory's most famous unsolved problem. === Subject: Re: Wiles' Proof > Gouv.90a, Fernando Q. > A marvelous proof. > Amer. Math. Monthly 101 (1994), no. 3, 203--222. /David === Subject: Re: Wiles' Proof > > Gouv.90a, Fernando Q. > A marvelous proof. > Amer. Math. Monthly 101 (1994), no. 3, 203--222. > Or, Mathematical Association of America sells back issues. Maybe Gouvea has it on his web site. === Subject: Re: Wiles' Proof Sure. > Or, Mathematical Association of America sells back issues. Okay. > Maybe Gouvea has it on his web site. Who? /David === Subject: Re: Wiles' Proof > Maybe Gouvea has it on his web site. > Who? > /David http://www.colby.edu/personal/f/fqgouvea/ === Subject: Re: Wiles' Proof >Sure. >> Or, Mathematical Association of America sells back issues. >Okay. >> Maybe Gouvea has it on his web site. >Who? Lee Rudolph === Subject: How to prove this?!?! Any ideas on this? Prove that: int{[int[(int(f(t) dt, 0,u),0,v]du],0,x} dv= (1/2)*int((x-t)^2*f(t) dt, 0,x) === Subject: Re: How to prove this?!?! > Any ideas on this? > Prove that: > int{[int[(int(f(t) dt, 0,u),0,v]du],0,x} dv= (1/2)*int((x-t)^2*f(t) dt, > 0,x) Try using integration by parts on the right side to get rid of those (x-t)'s. Have a tolerable existence. Eli === Subject: Re: Damped harmonic oscillation I don't think the proposals solves the problem or maybe I don't understand the correctly. The oscillation is caused by a non specific up/down motion of the of the spring, performed by a person. I can measure the up/down distance, speed and acceleration and know all parametres in the formula but C and phi. I gess that both C and phi in some way must be derived from the accelaration. My idea was to in steps of delta t = t[n]-t[n-1] measure the distance, speed and acceleration and then calculate C and phi, when finally should be inserted in d(t) = C*[e^(-t/tau)]*cos(o*t - phi). Torben W. Hansen Denmark === Subject: Structure of Frobenius Complement I have read that Burnside was able to prove that the Frobenius complement of a Frobenius group had the property that any Sylow p-subgroup of the group had a unique subgroup of order p. Therefore, any Sylow p-subgroup of such a group is cyclic unless p=2 and the Sylow 2-subgroup is a generalized quaternion group. Can anyone give me a reference (preferably in English) on how Burnside did this? ---- David === Subject: Re: Structure of Frobenius Complement >I have read that Burnside was able to prove that the Frobenius >complement of a Frobenius group had the property that any Sylow >p-subgroup of the group had a unique subgroup of order p. Therefore, >any Sylow p-subgroup of such a group is cyclic unless p=2 and the >Sylow 2-subgroup is a generalized quaternion group. Can anyone give >me a reference (preferably in English) on how Burnside did this? I don't know how it compares with Burnside's proof, but there is a proof on page 86 of the book John Dixon & Brian Mortimer, Permutation Groups (Springer, 1996) Derek Holt. === Subject: Graduate schools that teach how to teach I am considering applying to grad school again soon. What I am looking for is a school that not only teaches mathematics, but teaches its students how to become teachers. It also would help if the department fully funds all or most of its students, as I have no desire to increase my student loan debt. :-) I am interested in becoming a professor of mathematics at a teaching-oriented school. I'm not saying research is unimportant, but I would rather be known as a great teacher than a great researcher. If anyone has some recommendations as to schools that might meet my criteria, please post or privately email me. If I receive any responses in private email that don't duplicate what's posted, I'll summarize those as well. === Subject: Re: Graduate schools that teach how to teach > I am considering applying to grad school again soon. What I am > looking for is a school that not only teaches mathematics, but teaches > its students how to become teachers. It also would help if the > department fully funds all or most of its students, as I have no > desire to increase my student loan debt. :-) > I am interested in becoming a professor of mathematics at a > teaching-oriented school. I'm not saying research is unimportant, but > I would rather be known as a great teacher than a great researcher. > If anyone has some recommendations as to schools that might meet my > criteria, please post or privately email me. If I receive any > responses in private email that don't duplicate what's posted, I'll > summarize those as well. Look for schools with graduate degrees in the teaching of mathematics. You may be wanting to combine degrees in mathematics with education. I suspect that most schools that offer graduate math programs will have something in line with what you want. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Graduate schools that teach how to teach > I am considering applying to grad school again soon. What I am > looking for is a school that not only teaches mathematics, but teaches > its students how to become teachers. It also would help if the > department fully funds all or most of its students, as I have no > desire to increase my student loan debt. :-) > I am interested in becoming a professor of mathematics at a > teaching-oriented school. I'm not saying research is unimportant, but > I would rather be known as a great teacher than a great researcher. > If anyone has some recommendations as to schools that might meet my > criteria, please post or privately email me. If I receive any > responses in private email that don't duplicate what's posted, I'll > summarize those as well. A collateral question and approach might be to locate teaching-oriented schools and to ask those math departments if they know schools that could meet your needs. David Ames === Subject: prove it n^4+n^2=6 (mod 7) for all n === Subject: Re: prove it > n^4+n^2=6 (mod 7) for all n try it for n=0 -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: prove it > n^4+n^2 == 6 (mod 7) for all n False for integers in 3 of the 7 conguence classes mod 7, namely false for n == 0 mod 7, n == 1 mod 7 and n == 6 mod 7. Could one say that it is just barely more true than false (4 true equivalence classes to 3 false ones)? === Subject: Re: prove it In sci.math, Euler n^4+n^2=6 (mod 7) for all n Counterexample: n = 0 n^4+n^2=0 Counterexample: n = 1 n^4+n^2=2 Works: n = 2 n^4+n^2=16+4=20=6 (mod7) Works: n = 3 n^4+n^2=81+9=90=6 (mod7) Works: n = 4 n^4+n^2=272=6 (mod7) Works: n = 5 n^4+n^2=650=6 (mod7) Counterexample: n = 6 or -1 n^4+n^2=1332=2 (mod7) Well, 4 out of 7 isn't too bad. :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: prove it > n^4+n^2=6 (mod 7) for all n Interesting: then x^4 + x^2 - 6 = 0 would be a degree 4 equation with 7 roots in the field F_7. :-( -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Re: prove it > n^4+n^2=6 (mod 7) for all n Not true. If n=0 (mod 7) then n^4+n^2=0 (mod 7) If n=1 or n=5 (mod 7) then n^4+n^2=2 (mod 7) Have a tolerable existence. Eli === Subject: Re: prove it >> n^4+n^2=6 (mod 7) for all n > Not true. If n=0 (mod 7) then n^4+n^2=0 (mod 7) If n=1 or n=5 (mod > 7) then n^4+n^2=2 (mod 7) > Have a tolerable existence. Eli You meant if n = 1 or n = 6 = -1 (mod 7), n^4 + n^2 = 2 (mod 7). And obviously, if n = 0 (mod 7), also n^4 + n^2. Only for n = 2, 3, 4 or 5, yes, n^4 + n^2 = 6 (mod 7) -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: prove it > You meant if n = 1 or n = 6 = -1 (mod 7), n^4 + n^2 = 2 (mod 7). Have a tolerable existence. Eli === Subject: Re: prove it Euler > n^4+n^2=6 (mod 7) for all n Something's wrong, Leonhard old chap. A fourth-order polynomial can have at most four zeros (not seven) modulo a prime. LH === Subject: Re: Was: Convergence on a space with no topology by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9TDN4X13238; === >Subject: Was: Convergence on a space with no topology >Convergence on a space with no topology got cluttered relatively >fast. In my opinion, it was getting hard for anyone reading the >message for the first time to follow. Therefore, I try to give the >main points of what was said some sort of chronological order. >With all that feedback, you need present revision. >Skip the blah blah and just get to the punch line. Ok, but first, is the weather so damn cold where you are, too? I mean, I almost froze my fingers off a few days ago on the way to the university... >Are you reinventing the Hausdorff metric for sets? >Do you know what the Hausdorff metric for sets is? For time reasons, I was only able to address the second part of your message in my last reply. The Hausdorff metric begins by defining a neighborhood: if A subset X and r > 0, N_r(A) = {y | d(x,y) < r for some x in A) and then procedes to define a distance function h as: h(A,B) = inf { r > 0: A subset N_r(B) and B subset N_r(A) } which has the propeties of a metric if the sets involved are closed /compact. Since I have, in no way, defined an open set or neighborhood - much less a distance function- how can I have reinvented the Hausdorff metric? The only way I currently see to interpret this question properly is: If A_n -> A, does it not follow that lim n -> infty h(A_n, A) where h is the Hausdorff metric and the sets involved are closed / compact, etc.? (I presume this to be valid on a hunch and would appreciate any proofs, as I have not been able to prove it yet.) This because the converse (from lim n -> infty h(A_n, A) follows A_n -> A) certainly does not appear to be true: If A_n -> A then, as was previously pointed out, A_n would converge to all subsets of A, too, were it not for the additional requirement A_n subset A for all n (or something similar like: there exists an m such that A_n subset A forall n > m). The following does appear to be equivilant... 1. A_n -> A 2. A_n subset A forall n and for all p > 0, p in R for all a in A exists q in N for all n > q exists b in A_n: d(a,b) < p A stricter form of convergence: 1. A_n ->* A 2. A_n subset A forall n and for all p > 0, p in R exists q in N for all a in A for all n > q exists b in A_n: d(a,b) < p To me, the last definition is interesting because it follows (trivially) from it that if A_n ->* A, then A_n is a *Cauchy-sequence (if A_n subset A for all n, then we can obviously substitute A_m for A in the above condition for all m > q). Lastly, your question about B*... Why drop it? B* was put in to emphasize that we may not always want to consider P(X), the set of all subsets of X -which may be insanely large-, but some subset P(X) (such as the set of all Cauchy-sequences in X or the set of all single points of X, etc.). C.Dement === Subject: Re: Was: Convergence on a space with no topology by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9TDNO913283; >Skip the blah blah and just get to the punch line. Ok, but first, is the weather so damn cold where you are, too? I mean, I almost froze my fingers off a few days ago on the way to the university... >Are you reinventing the Hausdorff metric for sets? >Do you know what the Hausdorff metric for sets is? For time reasons, I was only able to address the second part of your message in my last reply. The Hausdorff metric begins by defining a neighborhood: if A subset X and r > 0, N_r(A) = {y | d(x,y) < r for some x in A) and then procedes to define a distance function h as: h(A,B) = inf { r > 0: A subset N_r(B) and B subset N_r(A) } which has the propeties of a metric if the sets involved are closed /compact. Since I have, in no way, defined an open set or neighborhood - much less a distance function- how can I have reinvented the Hausdorff metric? The only way I currently see to interpret this question properly is: If A_n -> A, does it not follow that lim n -> infty h(A_n, A)=0 where h is the Hausdorff metric and the sets involved are closed / compact, etc.? (I presume this to be valid on a hunch and would appreciate any proofs, as I have not been able to prove it yet.) This because the converse (from lim n -> infty h(A_n, A)=0 follows A_n -> A) certainly does not appear to be true: If A_n -> A then, as was previously pointed out, A_n would converge to all subsets of A, too, were it not for the additional requirement A_n subset A for all n (or something similar like: there exists an m such that A_n subset A forall n > m). The following does appear to be equivilant... 1. A_n -> A 2. A_n subset A forall n and for all p > 0, p in R for all a in A exists q in N for all n > q exists b in A_n: d(a,b) < p A stricter form of convergence: 1. A_n ->* A 2. A_n subset A forall n and for all p > 0, p in R exists q in N for all a in A for all n > q exists b in A_n: d(a,b) < p To me, the last definition is interesting because it follows (trivially) from it that if A_n ->* A, then A_n is a *Cauchy-sequence (if A_n subset A for all n, then we can obviously substitute A_m for A in the above condition for all m > q). Lastly, your question about B*... Why drop it? B* was put in to emphasize that we may not always want to consider P(X), the set of all subsets of X -which may be insanely large-, but some subset P(X) (such as the set of all Cauchy-sequences in X or the set of all single points of X, etc.). Your message-attacks are getting more and more personal... >How about studying up on the Hausdorff metric? It's a metric for P(S), >based upon the metric for S. Much like what your attempting. So get >smart, instead of just stumbling around, find out what others have already >complished decades ago that have bearing on your attempts. Hhhm... if I get smart, to that mean I become more and more like you? >> If (X,d) is a space whose metric is induced by a norm, >> then X must be a vector space- since a norm is only defined >> on a vector space (nevertheless, sorry for not explicitly stating this). >Baloney, you said metric space and a norm can be defined for a group >making it a topological group or conversely certain metrics for a >topological group can induce or be given norms. So if you want vector >space, say you want a vector space, a normed vector space or even an inner >product space. >> But your counterexample space {0,1} is not a vector space. >> Indeed, if you look at my proof of the proposition, you >> will see that it relies on (t_n a) also being in the space where >> t_n > 0 is a scalar... >It's not? It's a one dimensional vector space over Z_2, that is with >scalars integers modulus 2. Yes, Z_2 is a field, a finite field. As it >has the discrete topology, it can be given a discrete metric such as > d(0,1) = d(1,0) = 1, d(0,0) = d(1,1) = 0. >For a norm, |0| = 0, |1| = 1 will do. Ok you got me, in my short career (as a physics student who has taken the last two years off on fatherly leave) I have never heard of a normed space defined over a finite scalar field. Indeed, sob sob, I have only seen K=R or K=C. Although I feel very aware of the possibility of definining abstract vector spaces over other more general entities. Especially since I have looked at things like this myself- mainly playing around with the quaternions. C. Dement === Subject: Re: Was: Convergence on a space with no topology === Subject: Re: Was: Convergence on a space with no topology >>Skip the blah blah and just get to the punch line. >Ok, but first, is the weather so damn cold where you are, too? >I mean, I almost froze my fingers off a few days ago on >the way to the university... How affable. Portland having wonderful greenhouse warming weather, warm, sunny, some rain and just today tempeture drop 50-40's. Predicted 40-20's. Where's you? BTW, if you'se going @mathform, why such a double bland name? >The Hausdorff metric begins by defining a neighborhood: >if A subset X and r > 0, N_r(A) = {y | d(x,y) < r for some x in A) >and then procedes to define a distance function h as: >h(A,B) = inf { r > 0: A subset N_r(B) and B subset N_r(A) } >which has the propeties of a metric if the sets involved are >closed /compact. I've got, which as I recall is laborously equivalent to yours: dh(A,B) = max(sup{ d(a,B) | a in A }, sup{ d(b,A) | b in B }) metric for nonnul closed bounded subsets of metric space (S,d) >Since I have, in no way, defined an open set or neighborhood >- much less a distance function- how can I have reinvented >the Hausdorff metric? You're using a metric here. >Let (X, d) be a space whose metric is induced by a norm. >Call (A_n) subset B* subset P(x), the set of all subsets of X, >a *Cauchy sequence if (R = reals, N = natural numbers) >For all p > 0, p in R, exists q in N for all n,m > q for all >a in A_n exists b in A_m: d(a,b) < p. >The only way I currently see to interpret this question properly is: >If A_n -> A, does it not follow that >lim n -> infty h(A_n, A)=0 where h is the Hausdorff >metric and the sets involved are closed / compact, etc.? That's how a metric is supposed to work. For compact sets I'd expect dh(A,B) = 0 ==> A = B I recall a set theory definition of A_n -> A like A = /{ { / Aj | j > k } | k in N } = /{ { / Aj | j > k } | k in N } provided both are equal. The limsup, liminf of sets? >(I presume this to be valid on a hunch and would appreciate >any proofs, as I have not been able to prove it yet.) So far you've just an intuitive notion of A_n -> A, nothing defined. >This because the converse (from lim n -> infty h(A_n, A)=0 follows >A_n -> A) certainly does not appear to be true: If A_n -> A then, >as was previously pointed out, A_n would converge to all subsets of >A, too, were it not for the additional requirement A_n subset A for >all n (or something similar like: there exists an m such that >A_n subset A forall n > m). >The following does appear to be equivilant... >1. A_n -> A >2. A_n subset A forall n and for all p > 0, p in R for all a in A >exists q in N for all n > q exists b in A_n: d(a,b) < p Rather limited, that A_n has to approach from below. A dual, approach from above definition, wants to ensue. >A stricter form of convergence: 'uniform convergence'? >1. A_n ->* A >2. A_n subset A forall n and for all p > 0, p in R exists >q in N for all a in A for all n > q exists b in A_n: d(a,b) < p >To me, the last definition is interesting because it >follows (trivially) from it that if A_n ->* A, then A_n is a >*Cauchy-sequence (if A_n subset A for all n, then we can obviously >substitute A_m for A in the above condition for all m > q). >Lastly, your question about B*... >Why drop it? It wasn't doing anything, wasn't given any meaning and did nothing except briefly get in the way of understanding the definition. >B* was put in to emphasize that we may not always want to consider >P(X), the set of all subsets of X -which may be insanely large-, >but some subset P(X) (such as the set of all Cauchy-sequences in X >or the set of all single points of X, etc.). If B* is the be the set of all Cauchy sequences, then it needs not be in the definition of Cauchy sequence. ---- === Subject: Re: Pi Calculation ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9U0ogP16809; >One way to calculate pi is using the formula for ATAN: > atan(x) = x^1/1 - x^3/3 + x^5/5 - x^7/7 + x^9/9 ... I'm working on a targetting project on a robot controlled by a microcontroller. It has approximately 104 steps per revolution. Using integer math only, how might I find atan(), returned as 104-ians instead of radians or degrees? - Kipp === Subject: Re: Pi Calculation ? >One way to calculate pi is using the formula for ATAN: > atan(x) = x^1/1 - x^3/3 + x^5/5 - x^7/7 + x^9/9 ... > I'm working on a targetting project on a robot controlled by a > microcontroller. It has approximately 104 steps per revolution. > Using integer math only, how might I find atan(), returned as > 104-ians instead of radians or degrees? Lookup table with 104 entries. Precalculate the tables at bootup time, then just do a search through the tables when you want to find the angle value. You'll need a little bit of logic to sort out which quadrant you're in. - Randy === Subject: Re: Very interesting problem on Real Analysis David C. Ullrich > You mean this one, that someone reposted just to get it into the > relevant folder in his newsreader? === >Subject: Re: IT IS A POLYNOMIAL >Distribution: world <1998050216380100.MAA06675@ladder01.news.aol.com> Those links gave me trouble, but here it is again: http://www.math.niu.edu/~rusin/known-math/98/f.is.poly I'm still trying to do it with harmonic analysis. Vaguely: By a change of variable we can switch to a function g on the interval [-1,1]. Write g(x)=sum_n k_n H_n(x) all xin [-1,1] where the k_n are real constants and the H_n are the Hermite polynomials, which form a complete orthogonal system for all the continuous functions on [-1,1]. Termwise differentiation is legitimate, and moreover the derivative of H_(n+1) is H_n times a constant, which looks convenient. But I'm not sure this will really go anywhere. Very interesting problem, that's for sure. LH === Subject: Re: Very interesting problem on Real Analysis Originator: grubb@lola >Prove that f is polynomial in every component of V=Union(int(Cn)) and that >UV has no isolated points. >Then show that UV is empty (or else you'll get a contradiction). How do you prove that this is empty? I can see that f has to be a polynomial on some interval. It is also easy to see that it is a polynomial on each of a collection of intervals whose union has nowhere dense complement. Now what? --Dan Grubb === Subject: Re: Very interesting problem on Real Analysis >>Prove that f is polynomial in every component of V=Union(int(Cn)) and that >>UV has no isolated points. >>Then show that UV is empty (or else you'll get a contradiction). >How do you prove that this is empty? I can see that f has to be a >polynomial on some interval. It is also easy to see that it is >a polynomial on each of a collection of intervals whose union >has nowhere dense complement. Now what? He made a later post which was much more complete. See also the posts by Dor and Israel. >--Dan Grubb ************************ David C. Ullrich === Subject: Re: Very interesting problem on Real Analysis Originator: grubb@lola >Prove that f is polynomial in every component of V=Union(int(Cn)) and that >UV has no isolated points. >Then show that UV is empty (or else you'll get a contradiction). >>How do you prove that this is empty? I can see that f has to be a >>polynomial on some interval. It is also easy to see that it is >>a polynomial on each of a collection of intervals whose union >>has nowhere dense complement. Now what? >He made a later post which was much more complete. >See also the posts by Dor and Israel. ---Dan Grubb === Subject: Harmonic oscillation Im still and desperately searching for a function of either velocity or accelaration ( d(t) = f(v) or/and d(t) = f(a)), which expresses the damped oscillation distance d(t) caused by varying velocity or accelaration. I found this formula, which express a mass m in [kg] hanging in a spring k in [N/m], d(t) = C*[e^(-t/tau)]*cos(o*t - phi) where: d(t) = distance in [m] to the time t in [s], C = distance different from neutral position in [m], o = SQRT(kg/m), resonance frequency in [rad/s] tau = time constant for damping oscillation phi = initial phase The oscillation is caused by a non specific up/down motion of the of the spring, performed by a person. I can measure the up/down distance, speed and acceleration and I know all parametres in the formula but C and phi. I gess that both C and phi in some way must be derived from the acceleration. My idea was to measure the distance, speed and acceleration in steps of delta t = t[n]-t[n-1] and then step by step calculate C and phi to be inserted in : d(t) = C*[e^(-t/tau)]*cos(o*t - phi) where where: d(t)=f(t)/k, tau=2*m/b, o=SQRT(k/m) and as far as I know it should be the solution to the differential equation m*d'' + b*d' +k*d = f(t) also mentioned by Anselm Proschniewski in an earlier thread. where: m=mass, b=damping, k=stiffness, f(t)=force, d=displacement, d'=speed, d=acceleration. I'm not sharp in differential-, integral equations, but have some basic knowledge about this and Laplace Transformation. I also know a part of the laws in physics as f=m*a, E=1/2*m*v^2, Hooks law etc, etc... but I can't solve my problem Further more I use a math tool from Texas Instrument called DERIVE 5.06. What do I miss ? Torben W. Hansen Denmark === Subject: Re: Harmonic oscillation > Im still and desperately searching for a function of either velocity or > accelaration ( d(t) = f(v) > or/and d(t) = f(a)), which expresses the damped oscillation distance d(t) > caused by > varying velocity or accelaration. If the frequency is fixed you seem to need a least squares fit. Take the functions d(t)=c0+e^(-t/tau)*(c1*cos(o*t)+c2*cos(o*t)) see, e.g. numerical recipes in C (or your favourite prog. lang.). phi is something like atan2(c1,c2). If freq / damping is to be determined, the parameter optimization gets nonlinear. I assume that you have measured d at several time offsets? hth Klaus > I found this formula, which express a mass m in [kg] hanging in a spring k > in [N/m], > d(t) = C*[e^(-t/tau)]*cos(o*t - phi) > where: > d(t) = distance in [m] to the time t in [s], > C = distance different from neutral position in [m], > o = SQRT(kg/m), resonance frequency in [rad/s] > tau = time constant for damping oscillation > phi = initial phase > The oscillation is caused by a non specific up/down motion of the of the > spring, performed by a person. > I can measure the up/down distance, speed and acceleration and I know all > parametres in the formula but C and phi. > I gess that both C and phi in some way must be derived from the > acceleration. > My idea was to measure the distance, speed and acceleration in steps of > delta t = t[n]-t[n-1] and then step by step calculate C and phi to be > inserted in : > d(t) = C*[e^(-t/tau)]*cos(o*t - phi) where > where: d(t)=f(t)/k, tau=2*m/b, o=SQRT(k/m) > and as far as I know it should be the solution to the differential equation > m*d'' + b*d' +k*d = f(t) also mentioned by Anselm Proschniewski in an > earlier thread. > where: m=mass, b=damping, k=stiffness, f(t)=force, d=displacement, d'=speed, > d=acceleration. > I'm not sharp in differential-, integral equations, but have some basic > knowledge about this and Laplace Transformation. I also know a part of the > laws in physics as > f=m*a, E=1/2*m*v^2, Hooks law etc, etc... but I can't solve my problem > Further more I use a math tool from Texas Instrument called DERIVE 5.06. > What do I miss ? > Torben W. Hansen > Denmark === Subject: Random Number Generator Is it possible to make an 'ideal' random number generator ? By 'ideal' I mean that its output will be always random irrespective of the way in which its output is sampled.. Jean === Subject: Re: Random Number Generator > Is it possible to make an 'ideal' random number generator ? By 'ideal' > I mean that its output will be always random irrespective of the way > in which its output is sampled.. > Jean Early on (the 40s and 50s) there was some talk of embedding a physical process inside the computer to generate random numbers. They were talking about vacuum tubes, but if we think of a radioactive rock the principle would be the same. The rate of radioactive emission would control the generation of random numbers, Bill === Subject: Re: Random Number Generator > Is it possible to make an 'ideal' random number generator ? By 'ideal' > I mean that its output will be always random irrespective of the way > in which its output is sampled.. > Jean A *finite* sequence of numbers will never be truly random. -Michael. === Subject: scaling data Hi all, Could anyone tell me how to (linearly) rescale a data set on the interval [0, 1] to [0, 0.1]? I've forgotten how to do this. More generally, given a data set on the interval [a, b], is there a formula to rescale it to the interval [x, y]? -- _|//_ ( O-O ) ---------------------------o00--(_)--00o------------------------------ Colm G. Connolly | Department of Computer Science | University College Dublin (UCD) | Belfield, Dublin 4 | .83ire / Republic of Ireland | === Subject: Re: scaling data t is in [a,b] t-a is in [0,b-a] (t-a)/(b-a) is in [0,1] (t-a)*(x-y)/(b-a) is in [0,y-x] x+(t-a)*(x-y)/(b-a) is in [x,y] f(t)=x+(t-a)*(x-y)/(b-a) is an affine function with f(a)=x and f(b)=y. Cool -gs- > Hi all, > Could anyone tell me how to (linearly) rescale a data set on the interval > [0, 1] to > [0, 0.1]? I've forgotten how to do this. > More generally, given a data set on the interval [a, b], is there a formula > to rescale it to the interval [x, y]? > -- > _|//_ > ( O-O ) > ---------------------------o00--(_)--00o------------------------------ > Colm G. Connolly | > Department of Computer Science | > University College Dublin (UCD) | > Belfield, Dublin 4 | > .83ire / Republic of Ireland | === Subject: Re: scaling data t in [a, b] t-a in [a, b-a] (t-a)/(b-a) in [0, 1] ((t-a)/(b-a))/(b-a) in [0, y-x] x + ((t-a)/(b-a))/(b-a) [x, y] f(t)=x+(t-a)*(x-y)/(b-a) with f(a)=x and f(b)=y. > t is in [a,b] > t-a is in [0,b-a] > (t-a)/(b-a) is in [0,1] > (t-a)*(x-y)/(b-a) is in [0,y-x] > x+(t-a)*(x-y)/(b-a) is in [x,y] > f(t)=x+(t-a)*(x-y)/(b-a) is an affine function with f(a)=x and f(b)=y. > Cool > -gs- >> Hi all, >> Could anyone tell me how to (linearly) rescale a data set on the interval >> [0, 1] to >> [0, 0.1]? I've forgotten how to do this. >> More generally, given a data set on the interval [a, b], is there a > formula >> to rescale it to the interval [x, y]? >> -- >> _|//_ >> ( O-O ) >> ---------------------------o00--(_)--00o------------------------------ >> Colm G. Connolly | >> Department of Computer Science | >> University College Dublin (UCD) | >> Belfield, Dublin 4 | >> .83ire / Republic of Ireland | -- _|//_ ( O-O ) ---------------------------o00--(_)--00o------------------------------ Colm G. Connolly | Department of Computer Science | University College Dublin (UCD) | Belfield, Dublin 4 | .83ire / Republic of Ireland | === Subject: Fundamental Theorem of Calculus problem Why is the derivative of f(x) = integral (from x to 0) of [cos(xt)/t]dt not equal to -cos(x^2)/x ? Switch the order of integration to make integral negative, and use fund. theorem of calculus : d/dx (integral from a to x of [f(t)dt]) = f(x) John === Subject: Re: Fundamental Theorem of Calculus problem > Why is the derivative of > f(x) = integral (from x to 0) of [cos(xt)/t]dt > not equal to -cos(x^2)/x ? Because: (1) the integrand cos(xt)/t depends on x; (2) the integrand becomes unbounded as t -> 0 and the improper integral diverges. In neither of these situtations does the fundamental theorem of calculus as you have stated it apply. > Switch the order of integration to make integral negative, and use fund. > theorem of calculus : > d/dx (integral from a to x of [f(t)dt]) = f(x) > John -- P.A.C. Smith 'If the Apocalypse comes, beep me.' <*> http://www.srcf.ucam.org/~pas51 === Subject: Difference equation resource -- help Can anyone tell me how to solve a second order difference equation(DE) such as A(n,m)+A(n-1,m)+2*A(n,m-1)-4*A(n-1,m-1)=1 where f(m,n) is known and b,c,d are constant. Or, show me some online resources discussing multi-variable DE's? I suspect it's closely related to partial differential equation(PDE) theories, but *how*? === Subject: Re: Difference equation resource -- help what's f(m,n)? can also often do separation of variables like in PDE. A(m,n)=M(m)*N(n) if bc allow > Can anyone tell me how to solve a second order difference equation(DE) > such as > A(n,m)+A(n-1,m)+2*A(n,m-1)-4*A(n-1,m-1)=1 > where f(m,n) is known and b,c,d are constant. > Or, show me some online resources discussing multi-variable DE's? > I suspect it's closely related to partial differential equation(PDE) > theories, but *how*? === Subject: Re: Difference equation resource -- help > Can anyone tell me how to solve a second order difference equation(DE) > such as > A(n,m)+A(n-1,m)+2*A(n,m-1)-4*A(n-1,m-1)=1 > where f(m,n) is known and b,c,d are constant. > Or, show me some online resources discussing multi-variable DE's? > I suspect it's closely related to partial differential equation(PDE) > theories, but *how*? Since your equation is linear, you should have good chances. Start by considering the homogeneous equation (0 on right hand side). Insert A(n,m) = alpha^n * beta^m, and solve for alpha and beta. Next generate linear combinations of these. You still need to consider boundary conditions, though. -Michael. === Subject: is there a function satisfying this is there a function u: N -> N so that u(i) + u(j) = even if i <> j u(i) + u(j) = odd if i = j ? === Subject: Re: is there a function satisfying this > is there a function > u: N -> N > so that > u(i) + u(j) = even if i <> j > u(i) + u(j) = odd if i = j Let's see... 1=1, so u(1) + u(1) is odd, but this is 2 times u(1), even. No. === Subject: Re: is there a function satisfying this and what if the function is changed into u: N -> R ? === Subject: Re: is there a function satisfying this >and what if the function is changed into >u: N -> R i.e. 2 u(i) is an odd integer, and u(i) + u(j) is an even integer if i <> j. Well, if u(i) = x/2 and u(j) = y/2, then (x+y)/2 is an even integer iff x+y == 0 mod 4. Of the odd integers 2 u(1), 2 u(2) and 2 u(3), at least two are congruent mod 4; if these are 2 u(i) and 2 u(j), then 2 (u(i)+u(j)) is congruent to 2 mod 4, and u(i)+u(j) is odd. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: is there a function satisfying this > and what if the function is changed into > u: N -> R and what is meant by an even real number? === Subject: Re: Applying to Grad school (GRE verbal?) ...................... >In reality, I suspect that what is given the MOST weight in the final >decision is not the GRE test, but rather the coursework taken and the >letters of recommendation. Particularly the latter. As I recall from >my own application experience, the GRE test was more a filter than a >decision maker: in order for your application to be considered, you >should normally score at least this much on each part; and then it was >ignored in favor of letters of rec. and so on. I have seen thousands of applications for graduate students in statistics, and a fair number in mathematics. The problem at this time is that there is NO good information, and that includes the GRE. The grades in the important undergraduate courses, abstract algebra and foundations of analysis, are often not given in a meaningful manner, and it is even the case that the important parts were not taught; teaching to the level of the bodies in the classroom, whether they should have even gotten near that level, has reached this far, and even into graduate courses. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Applying to Grad school (GRE verbal?) >I don't know how many can comment, but I am applying to Math grad school and >was wondering how important the general GRE test is when applying to the top >math schools. In particular, the Verbal section. I am an American student, >and this section still troubles me...The rest of my application is very >good. I expect to get in the low to mid 500's on the verbal...(800 quant >and 4-5 on the writing section out of 6). >John It depends on each department. Each has its own admission criteria. My experience, after many years in the business (30+), is that the verbal part s a better predector of grad. math success than the other section. Another excellent predictor of math success is a foreign language aptitude test !!! === Subject: Re: Applying to Grad school (GRE verbal?) >I don't know how many can comment, but I am applying to Math grad school and >was wondering how important the general GRE test is when applying to the top >math schools. In particular, the Verbal section. I am an American student, >and this section still troubles me...The rest of my application is very >good. I expect to get in the low to mid 500's on the verbal...(800 quant >and 4-5 on the writing section out of 6). >John > It depends on each department. Each has its own admission criteria. > My experience, after many years in the business (30+), > is that the verbal part s a better predector of grad. math success > than the other section. > Another excellent predictor of math success is a foreign language > aptitude test !!! === Subject: Re: Applying to Grad school (GRE verbal?) > I don't know how many can comment, but I am applying to Math grad school and > was wondering how important the general GRE test is when applying to the top > math schools. In particular, the Verbal section. I am an American student, > and this section still troubles me...The rest of my application is very > good. I expect to get in the low to mid 500's on the verbal...(800 quant > and 4-5 on the writing section out of 6). the subject test is key. > John === Subject: Condition on commuting matrices, and Lyapunov equation Do you know what is the characterization of matrices X, Y such that XY=YX? Suppose X is given? What can you say about Y in general? Regarding a version of Lyapunov matrix equation AX=XB, A, B are known to have block partitioned structure A=[a b1; 0 c], B=[a b2; 0 c], can you name an easy way to solve for X, with the constraint X=[I 0; x1 x2] ? Do you just dispose it entry by entry and look at it a system of equations? In general, is there any known result of solving Lyapunov equation with constraint? B. C. === Subject: Re: Condition on commuting matrices, and Lyapunov equation >Do you know what is the characterization of matrices X, Y such that XY=YX? >Suppose X is given? What can you say about Y in general? These are both square matrices of the same size, say n x n. Note that for any invertible n x n matrix S, SXS^(-1) and SYS^(-1) commute iff X and Y do. So wlog we may assume X is in Jordan Canonical Form. Now since (X-rI)^k and Y commute, Y must leave invariant the generalized eigenspaces {x: (X-rI)^k x = 0} for eigenvalues r of X and positive integers k. Consider the sets of indices B_1 = {i_1 ... i_1 + k_1} and B_2 = {i_2 ... i_2 + k_2} corresponding to two (not necessarily distinct) Jordan blocks. If the blocks are for different eigenvalues, then Y_{ij} = 0 for i in B_1 and j in B_2. If they are for the same eigenvalue r, then the (i_1 + j,i_2 + k) entry of Y is 0 if k < j, is equal to the (i_1+j-1,i_2+k-1) entry if 1 <= j <= k <= min(k_1,k_2), and otherwise is arbitrary. >Regarding a version of Lyapunov matrix equation AX=XB, A, B are known to >have block partitioned structure A=[a b1; 0 c], B=[a b2; 0 c], can you name >an easy way to solve for X, with the constraint X=[I 0; x1 x2] ? Do you just >dispose it entry by entry and look at it a system of equations? I get [ b1 x1, b1 x2 - b2 ] AX - XB = [ c x1- x1 a, c x2 - x1 b2 - x2 c ] If b1 is invertible, then x1 = 0, x2 = b1^(-1) b2, and this x2 must commute with c. If b1 is not invertible I guess it will be more complicated. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Condition on commuting matrices, and Lyapunov equation >Do you know what is the characterization of matrices X, Y such that XY=YX? >Suppose X is given? What can you say about Y in general? Well, if X is an nxn matrix, then any polynomial Y=p(X) commutes with X. Under certain conditions, these are the only Y. Maybe the condition is that X has n distinct eigenvalues, or something like that. === Subject: Re: Condition on commuting matrices, and Lyapunov equation >Well, if X is an nxn matrix, then any polynomial Y=p(X) commutes with X. >Under certain conditions, these are the only Y. Maybe the condition is >that X has n distinct eigenvalues, or something like that. I'm pretty sure it's that every Jordan block of X is for a different eigenvalue, or equivalently that each eigenvalue of X has geometric multiplicity 1 (i.e. dim {x: Xx = rx} = 1). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: numerical differentiation of oscillating singal by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9UE4RL05082; >> It might be helpful to explain what went wrong with the methods you >> tried. Is your signal described by an analytic formula, or by a >> series of samples? >Actually, I am processing a signal f(t) and in each step of time, > f(t+dt) = G[D[f(t)]] >here D means the differentiation of f(t) and G is a function to form >D[f(t)] to a new signal. However, f(t) is a signal with a rapidly >oscillating tail. A common algorithm (such as forward difference) to >perform numerical differentiation will lead to wrong result. For this >purpose, I adopt a high-order difference formula. It's weird that the >high-order formula is even worse than the central difference formula. >I also take the fourier method but no help >D[f(t)] = iff(i*w*F(w)); >w denotes for the angular frequency and ifft denotes the inverse >fourier transformation. F(w) is the fourier spectrum of f(t). It looks like what your problem needs is some control theory. If your function G is a linear function of the set of samples, then it can be modeled with a Laplace transform. Differentiation becomes multiplication by s in the Laplace frequency domain. In order to stabilize your system, a compensating filter needs to be added. The goal of the compensation is to ensure that the loop gain drops below unity gain before the phase shift reaches 180 degrees, otherwise the system will oscilate. In a purely sampled system, the Laplace transform can be rewritten using the substitution z = exp(s T), where T is the sample period. This is called the Z transform. If the system is nonlinear, the problem becomes more difficult. If it is locally close to linear, the compensation can be adjusted along the way. If it is highly nonlinear, some kind of phase space method may be needed. It is hard to go into more detail without knowing what your G is. === Subject: Re: Integral of e^x^2 dx ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9UE5AD05173; >could be expressed as some form of error function Something like: -i sqrt(Pi)/2 erf(i z) >> I am an undergrad college student and I ran across a problem that >intrigued me. What is the integral of e^x^2 dx ? I asked a few teachers >and searched the internet for a little while, but to no avail. Does anyone >out there have any idea how to solve this one? Possibly prove it too? I'm >killing myself not being able to figure it out even after a week of >thought!!! >> Kris === Subject: Re: Fibonacci by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9UE61205237; Fibonacci numbers can also be produced using hyperbolic functions: F(even n) = 2/sqrt(5) sinh(n ln(t)) F(odd n) = 2/sqrt(5) cosh(n ln(t)) Swapping the sinh and cosh produces Lucas numbers: L(even n) = 2 cosh(n ln(t)) L(odd n) = 2 sinh(n ln(t)) where t = (1 + sqrt(5))/2 === Subject: Re: minimum foam by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9UE4ox05134; >Does anyone disagree that space fills with irregular tetrahedra >such that each vertex is shared by 20 tetradrahedra? NO? Okay. >Then.... >Place a vertex at the center of each tetrahedron in that maximum >foam and connect the dots. Each vertex has 4 edges connected. Voila! >Space fills with irregular pentagonal dodecahedra. Minimum foam. >Yes? No? >Dick Fischbeck >East Belfast, Maine >Randome, LLC === >Subject: irregular p-dodecahedra >Does anyone know if space fills with irregular pentagonal dodecahedra? >Why or why not? If you go to 4 dimensions, then the tetrahedra can be regular. What you will find is that exactly 600 of them will close to form a polytope known as the 600-cell. The dual is the 120-cell which is made out of 120 regular dodecahedra. If you try to do this with distortion in Euclidean 3-space, you will get a distorted version of the same result. It is analogous to trying to tile the plane with triangles, while allowing only five around each vertex, or tiling with three pentagons around each vertex. === Subject: Help by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9UE5eK05202; Any people can help me on this problem? M is a m-dimensional submanifold of R^n. Show that the Gauss map f: M --> Gr(m,n) is smooth, where Gr(m,n) denotes the set of m-dimensional subspaces of R^n (Grassmannian). libaiyang2000@yahoo.com === Subject: request help for wigner ville distribution by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9UE5Iq05186; hi.. In wvd formula there's formula exp(-2*pi*f*tau). I can't use these formula below exp(i*x)=cos(x)+i*sin(x) and exp(-i*x)=cos(x)-i*sin(x) so I still calculate using exp. In wvd codes that I write. If you dont mind Could you explain to me how exp(-2*pi*f*tau) to calculate in wvd formula This is one thing that confuse me when I try to write WVD codes. could someone give me the codes to callcuate wvd . === Subject: Re: Glenn Lamb's challenge: Prove 0! = 1 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9UE52o05162; i do not under stand what u are saying.give me a mail how you got 0!=1 === Subject: Re: prove it by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9UE6nF05325; >> n^4+n^2=6 (mod 7) for all n >Not true. If n=0 (mod 7) then n^4+n^2=0 (mod 7) If n=1 or n=5 (mod 7) then >n^4+n^2=2 (mod 7) >Have a tolerable existence. Eli Exactly the problem was: (n^5-n)(n^4+n^2-6)=0 (mod 210) for all n === Subject: Re: prove it >> n^4+n^2=6 (mod 7) for all n >Not true. If n=0 (mod 7) then n^4+n^2=0 (mod 7) If n=1 or n=5 (mod 7) then >n^4+n^2=2 (mod 7) >Have a tolerable existence. Eli > Exactly the problem was: > (n^5-n)(n^4+n^2-6)=0 (mod 210) for all n 210 = 2*3*5*7 You must prove that M(n) = (n^5-n)(n^4+n^2-6) is multiple of 2, 3, 5 and 7 for all n. For 2 is obvious, each factor is even for all n, eben or odd. Then M(n) is always multiple of 4. For 3: 0^5 - 0 = 1^5 - 1 = 2^5 - 2 = 0 (mod 3) For 5: n^5 = n (mod 5) by Fermat little theorem. For 7: 0^5 - 0 = 1^5 - 1 = 0 (mod 7). Also 6^5 - 6 = (-1)^5 - (-1) = -1 + 1 = 0 (mod 7) For n = 2, 3, 4 (= - 3) or 5 (= - 2) (mod 7), n^4 + n^2 - 6 = 0 (mod 7), as it has seen in previous post. Actually, you can say that (n^5-n)(n^4+n^2-6) is always multiple of 420. As M(2) = 420, gcd(M(n)) = 420, for all n. -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: Was: Convergence on a space with no topology by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9UE7Sa05420; It was mentioned that: The following does appear to be equivilant... 1. A_n -> A 2. A_n subset A forall n and for all p > 0, p in R for all a in A exists q in N for all n > q exists b in A_n: d(a,b) < p A stricter form of convergence: 1. A_n ->* A 2. A_n subset A forall n and for all p > 0, p in R exists q in N for all a in A for all n > q exists b in A_n: d(a,b) < p A simple example regarding both types of convergence: Let N be the natural numbers (no zero) and N_n = {1,2, ...,n}. It is easy to see that N_n -> N. However, !(N_n ->* N). Proof: We have to check: for all p > 0, p in R exists q in N for all a in N for all n > q exists b in N_n: |a-b| < p This is equivilant to (choose p < 1) exists q in N for all a in N for all n > q exists b in N_n: a = b which, in turn, is equivilant to exists q in N for all a in N for all n > q: a < n + 1 This is not valid for a = q + 2 and n = q + 1, q.e.d. C.Dement === Subject: if n is prime... by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9UE76k05349; show that: a) 3^n+(-2)^n+(-1)^n is multiple of n. for all n prime. b) 5^n-2(3^n)+1 is also multiple of n for all n prime. === Subject: Re: if n is prime... > show that: > a) 3^n+(-2)^n+(-1)^n is multiple of n. for all n prime. > b) 5^n-2(3^n)+1 is also multiple of n for all n prime. === Subject: Re: if n is prime... > show that: > a) 3^n+(-2)^n+(-1)^n is multiple of n. for all n prime. > b) 5^n-2(3^n)+1 is also multiple of n for all n prime. a) If n = 2, it is true. If n is prime > 2, then n is odd and (-1)^n = -1 and (-2)^n = - 2^n. Then, you must to show that 3^n - 2^n = 1 (mod n) (#1) for all odd prime n But the Fermat Little Theorem says that a^n = a (mod n) for all a, if n is prime. Then #1 reduces to 3 - 2 = 1 (mod n). b) As before, 5^n-2(3^n)+1 = 5 - 2*3 + 1 = 0 (mod n) -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: if n is prime... >> Show 3^p + (-2)^p + (-1)^p is a multiple of p, for all p prime. = 3 + -2 + -1 = 0 (mod p) by Fermat's little theorem Below you treat -2 and -1 different from 3. Do you think a^p = a (mod p) fails if a < 0 ? Hint: mod p: a = b => a^n = b^n e.g. p-2 = -2 => (p-2)^n = (-2)^n > If p = 2, it is true. If p is prime > 2, then p is odd > and (-1)^p = -1 and (-2)^p = - 2^p. > Then, you must show (#1) 3^p - 2^p = 1 (mod p) for all odd prime p > But the Fermat Little Theorem says that a^p = a (mod p) for all a, > if p is prime. Then #1 reduces to 3 - 2 = 1 (mod p). -Bill Dubuque === Subject: Re: if n is prime... > show that: > a) 3^n+(-2)^n+(-1)^n is multiple of n. for all n prime. > b) 5^n-2(3^n)+1 is also multiple of n for all n prime. HINT if p is prime a^p mod p = a === Subject: Re: Algebraic Closure by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9UE7J805391; >What is the algebraic closure of a finite field? >I'd be happy just to know the algebraic closure of Z/<2>. Let K be a finite field and L the algebraic closure of K. Then we know that for every integer n there exists exactly one field M within L that has degree n over K. The field M looks like K[X]/pK[X], where p is an arbitrary irreducibel polynomial of degree n. There are algorithms to determine the irreducibel polynomials over the field K (for example Berlekamps algorithm). The field L is the union over all the field M, and these fields form an ascending chain. Every algebraic computation within L typically involves only finitely many elements. So you can perform the calculation in one of the M's, that are under your control. So in a sense one can say that the algebraic closure of K is >known<. === Subject: Re: Delay differential equation >I have a 2nd order DDE >y''(x) + ay'(x) + b(y(x) - y(x-a) = 0 ... which he later amended to y''(x) + ay'(x) + b(y(x) - y(x-c)) = 0 >with 2 bc's >y(0) = 1 and y(x) = 0 when x->Inf >I know y(x) for 0My problem is that to enforce the second bc I need the solution as >x->Inf , and I am looking for a analytic solution so I was wondering if >there is any shortcuts for getting the solution at x=Large without going >through all x If you try y = exp(r t), you'll see that this is a solution iff > (r^2 + a r + b) exp(r a) - b = 0. ... which of course becomes (r^2 + a r + b) exp(r c) - b = 0. > One root of this is r = 0, and > there are probably infinitely many other roots (mostly complex). > To have y -> 0 as x -> infinity, you'll want to use a linear > combination of the solutions for roots with negative real part. Some further thoughts: I'm assuming a, b, c > 0. 1) If a^2 - 2 b >= 0, I'm pretty sure |r^2 + a r + b| >= b for Re(r) >= 0, with equality only for r=0. Since |exp(r c)| >= 1 as well, the only root with Re(r) >= 0 is r = 0. If a^2 - 2 b < 0, you may have to worry about roots in the right half plane. Of course, these correspond to solutions y(t) = exp(r t) that behave badly as t -> infinity. 2) If y is a solution, let U(x) = y'(x) + a y(x) + b int_{x-c}^x y(t) dt. Then U'(x) = y''(x) + a y'(x) + b((y(x) - y(x-c)) = 0, so U is constant. Now for any solution where y(t) and y'(t) -> 0 as t -> infinity, we have U(x) -> 0 as x -> infinity and therefore U(x) = 0. On the other hand, for the constant solution y = 1 we have U = a + b c. So (at least if a^2 - 2 b >= 0) I think that lim_{t -> infinity} y(t) = U/(a+bc) which you can calculate using your known values for 0 <= x <= c. 3) This must all be well-known in the literature on stability of delay-differential equations. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Delay differential equation >>I have a 2nd order DDE >>y''(x) + ay'(x) + b(y(x) - y(x-a) = 0 >If you try y = exp(r t), you'll see that this is a solution iff >(r^2 + a r + b) exp(r a) - b = 0. One root of this is r = 0, and >there are probably infinitely many other roots (mostly complex). >To have y -> 0 as x -> infinity, you'll want to use a linear >combination of the solutions for roots with negative real part. Your answer doesn't mention the LambertW function. In past similar responses, you have evoked it. Does this mean that in this case with the second order derivative, it won't work? I think I will wait for your answer before I explore any further. Reference: http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/LambertW.ps see especially page 7 John Bailey http://home.rochester.rr.com/jbxroads/mailto.html === Subject: Re: Delay differential equation >>I have a 2nd order DDE >>y''(x) + ay'(x) + b(y(x) - y(x-a) = 0 ... and Stefan later admitted that he meant y''(x) + a y'(x) + b (y(x) - y(x-c)) = 0 >If you try y = exp(r t), you'll see that this is a solution iff >(r^2 + a r + b) exp(r a) - b = 0. One root of this is r = 0, and >there are probably infinitely many other roots (mostly complex). >To have y -> 0 as x -> infinity, you'll want to use a linear >combination of the solutions for roots with negative real part. > Your answer doesn't mention the LambertW function. In past similar > responses, you have evoked it. Does this mean that in this case with > the second order derivative, it won't work? I think I will wait for > your answer before I explore any further. I could be wrong, but I don't think that the equation (r^2 + a r + b) exp(r c) - b = 0 can be solved (in general) using LambertW. It can be done in the special case a = 2 sqrt(b), in which case the roots other than r=0 are 2/c W(-ca/4 exp(ca/4)) - a/2 where W is any of the branches of LambertW. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Delay differential equation > I assume this should be > y''(x) + ay'(x) + b(y(x) - y(x-a)) = 0 > but do you really want the two a's to be the same? No they are not the same, my mistake. It should be y''(x) + ay'(x) + b(y(x) - y(x-c)) = 0 >>with 2 bc's >>y(0) = 1 and y(x) = 0 when x->Inf > So actually the boundary condition on the left is y(x) = (some known > function) for 0 <= x as you know because you can solve for each interval in turn. And then > it's not a matter of enforcing the second bc, it's either true or false. > But maybe you only know y(x) for 0 If you try y = exp(r t), you'll see that this is a solution iff > (r^2 + a r + b) exp(r a) - b = 0. One root of this is r = 0, and > there are probably infinitely many other roots (mostly complex). > To have y -> 0 as x -> infinity, you'll want to use a linear > combination of the solutions for roots with negative real part. /Stefan === Subject: Re: Boolean Algebra - Arithmetic Relationship > Many years ago when I was a student of philosophy I read Popper writing > against this Pavlovian idea. The observation (made by the dog) of A is > followed by B does _not_ give rise to the hypothesis A causes B. > Can't remember why. > I would warn you against confusing A causes B with the logician's if > A then B. For the logician, if A then B is defined by the table > A B if A then B > t t t > t f f > f t t > f f t > where t means true and f means false. The ancient Greeks discussed > this and it was taken up again in the late nineteenth century. > I agree that the Pavlovian response is an inference, whereas the > Logician's assertion that A implies B, is necessarily true by > definition. > But isn't this how we learn and develop logical models of our > environment? I don't think we could live without making inferences > such as these from experience. Popper would say that we have (for whatever reason) the theory first and then we test it (or should) against experience. According to Popper the theory always comes before the experience. > Furthermore, isn't all reasoning basically based upon this idea of > inference? > SomeEquation => SomeOtherEquation > SomeTruth => SomeOtherTruth > Other than that, it appears variable subtitution is all that is > necessary. > Of course I may be terribly confused and misinformed, but this seems > to me to be the way things work. > I have a deep respect for the basic ideas of Karl Popper, Thomas Kuhn > and the Logical Positivists camp in general. I like to learn more > about his argument against Pavlovian inferences. You know, don't you, that Popper was opposed to Kuhn and to the logical positivists. For Popper v Pavlov, see the footnote on page 45 of Realism and the Aim of Science and the works referred to there. > -Steve -- G.C. === Subject: Re: Understanding a proof (Was: Re: Wiles' Proof) > I thought that it might be interesting to quote G. H. Hardy here: > There is strictly no such thing as mathematical proof; proofs are what > Littlewood and I call gas, rhetorical flourishes, devices to stimulate > the imaginations of pupils. > Jose Carlos Santos If that quotation is exact, then Hardy contradicts himself, because in his A Mathematician's Apology (Speaking of Euclid's proof of the infinitude of prime numbers), he says:'Two thousand years have no written a wrinkle on it.'And afterwards:'...the proof can be mastered in an hour by any intelligent reader...' It is strange...Why Hardy mixed Littelwood in that nonsense? L.Rodriguez === Subject: Re: Understanding a proof permission for an emailed response. > That makes it a bit difficult for logic students. After all, suppose > you're studying a purely formal proof of, say, one of the De Morgan > laws. If it's already purely formal, you'll never understand it, > since you *can't* right down any more detailed proof. Huh? That doesn't make any sense at all. De Morgan laws are hardly taken as primitive in most systems these days. (I know that they were taken as such in Hofstadter's GEB, but that's a special case indeed.) Typical fashion these days is to use a natural deduction system, which is essentially a formal system for logic which uses no axioms. The most elegant have a pair of rules for each connective, one to introduce it, one to eliminate it. So if I want to prove that ~(P & Q) entails ~P v ~Q following Fitch-style proof in a natural deduction system: 1 | ~(P & Q) +--- 2 | | ~(~P v ~Q) | +--- 3 | | | ~P | | +--- 4 | | | ~P v ~Q [v Intro; 3] 5 | | | [ Intro; 2, 4] | | 6 | | ~~P [~ Intro; 3-5] 7 | | P [~ Elim; 6] | | 8 | | | ~Q | | +--- 9 | | | ~P v ~Q [v Intro; 8] 10 | | | [ Intro; 2, 9] | | 11 | | ~~Q [~ Intro; 8-10] 12 | | Q [~ Elim; 11] 13 | | P & Q [& Intro; 7, 13] 14 | | [ Intro; 1, 13] | 15 | ~~(~P v ~Q) [~ Intro; 2-14] 16 | ~P v ~Q [~ Elim; 15] That gives you some of the flavor. Fitch-style involves the use of subproofs. You can see above the use of v Intro and & Intro, and how they are justified, as well as the rules for introducing ~ and removing ~. (~ intro is the formalization of indirect proof, for example). The & elim rule lets you extract either conjunct. The rule for v elim is proof by cases, you need a subproof for each disjunct, leading from that disjunct to a common conclusion. The usual fashion is not to define connectives in terms of each other, but rather to give further pairs of rules. So implication has two rules, where the intro rule is justified by a subproof, and the elim rule is just modus ponens. Experience in teaching elementary logic to students shows that natural deduction systems are much more easily grasped than axiomatic systems; the latter lead to the question why that axiom and no other, and you are right that if you don't get it, you end up stuck. Whereas the natural deduction systems seem much closer to people's unformed intuitions. Thomas === Subject: Re: Understanding a proof <3f9d06e2$0$69956$edfadb0f@dread12.news.tele.dk> <3f9d8023_5@127.0.0.1> <87k76ql6wu.fsf@phiwumbda.org> <87k76nqmqo.fsf@becket.becket.net> Discussion, linux) >> That makes it a bit difficult for logic students. After all, suppose >> you're studying a purely formal proof of, say, one of the De Morgan >> laws. If it's already purely formal, you'll never understand it, >> since you *can't* right down any more detailed proof. > Huh? That doesn't make any sense at all. De Morgan laws are hardly > taken as primitive in most systems these days. (I know that they were > taken as such in Hofstadter's GEB, but that's a special case > indeed.) I'm aware of that. Before we go further, let me quote the passage to which I was replying. Then we'll use your proof below to make my (not very deep) point -- not very deep, hell, it was just a joke. But I'll run it into the ground by defending it. ,---- | >>What exactly does it mean to understand a proof? | > ... | > To be able to sit down and rewrite the entire proof | > BUT including one more level of detail for every step. `---- [snip introduction to natural deduction] > So if I want to prove that ~(P & Q) entails ~P v ~Q following > Fitch-style proof in a natural deduction system: > 1 | ~(P & Q) > +--- > 2 | | ~(~P v ~Q) > | +--- > 3 | | | ~P > | | +--- > 4 | | | ~P v ~Q [v Intro; 3] > 5 | | | [ Intro; 2, 4] > | | > 6 | | ~~P [~ Intro; 3-5] > 7 | | P [~ Elim; 6] > | | > 8 | | | ~Q > | | +--- > 9 | | | ~P v ~Q [v Intro; 8] > 10 | | | [ Intro; 2, 9] > | | > 11 | | ~~Q [~ Intro; 8-10] > 12 | | Q [~ Elim; 11] > 13 | | P & Q [& Intro; 7, 13] > 14 | | [ Intro; 1, 13] > | > 15 | ~~(~P v ~Q) [~ Intro; 2-14] > 16 | ~P v ~Q [~ Elim; 15] Now, according to Don Taylor's (now revised) hypothesis, in order to claim that I understand your proof, I must be able to rewrite it with more detail. I confess that I cannot add any more detail to your proof, since each statement is either an assumption or the immediate conclusion from a natural deduction rule of inference such that the premises are [blah blah blah]. Therefore, it is apparent that I do not understand your proof (according to Don's suggestion), despite the fact that I can confirm each step and even describe informally the general strategy that leads to the formal proof. Morgan's law, since I think it makes my point that much more obvious. I sure as heck wasn't gonna to write it down and evidently I didn't get my point across without an explicit example. [snip the remaining evaluation of natural deduction, since it's not that relevant to my comments] -- We are happy that you agree that customers need to know that Open Source is legal and stable, and we heartily agree with that sentence of your letter. The others don't seem to make as much sense, but we find the dialogue refreshing. -- Linus Torvalds to Darl McBride === Subject: Re: Understanding a proof permission for an emailed response. Send your questions to ``ASK ZIPPY'', Box 40474, San Francisco, CA 94140, USA > Now, according to Don Taylor's (now revised) hypothesis, in order to > claim that I understand your proof, I must be able to rewrite it with > more detail. I confess that I cannot add any more detail to your > proof, since each statement is either an assumption or the immediate > conclusion from a natural deduction rule of inference such that the > premises are [blah blah blah]. mistake. Glad my proof could help make your point. ;) I might add that perhaps Don Taylor is on to something, even if not quite right. I hesitate to say exactly how to put it. I think there is something about the ability to proffer explanations of steps as a part of understanding. I have become accustomed to thinking there are two different contexts of proving, and something gets missed when they are conflated: Proof as convincement is what mathematicians normally do. The job of a proof here is to convince another (or in some sense, oneself) that a theorem is true. Taking as a harmless simplification that all theorems are conditional sentences, that is a proof is designed to show that the implication in questino is a valid one. A proof works by reducing the implication to a series of smaller arguments, each of which must be valid. The prover's job is to reduce to a series of steps which both prover and provee agree are valid. We sometimes say that the provee must understand the proof in order to judge it. That means that the provee must see each step and know why it is a valid inference. Roughly speaking, that means that the provee must be able, in turn, to proffer a proof for each step to some unspecified third party. In that sense, we say that a provee has understood a proof when she can rewrite it in more detail. A second context for proving is what students do in proving theorems to instructors. The purpose here is not to convince the provee, but rather for the prover to demonstrate their understanding of the proof. Ideally, this works by some kind of back-and-forth method, in which the provee (the instructor) can ask for clarification of any step, demanding a reduction of it to sub-steps, and in that way to verify to his satisfaction the student's understanding. Here we are not concerned with whether a provee understands a proof, but rather whether a prover does. Still, the measure is much the same: the ability to proffer subproofs of the individual steps on demand. As you rightly note, this bottoms out. Or does it? Unlike the average student in a logic class, I *can* give an explanation for why the natural deduction rules for & are what they are. This is in some sense a different order of proof. It's not a more detailed step in a proof--in that sense, Don Taylor is mistaken. But it *is* some kind of justification. My explanation will be a proof-theoretic demonstration that the rule is truth-preserving when considered in the framework of a Fitch-style proof, given the truth-table that I take to be the definition of the meaning of &. I would say that my students do not actually understand *really* why the rules for & are the ones that they are taught until they begin to grok at least the soundness demonstrations for propositional logic. But that isn't a problem: it's perfectly legitimate for them to understand some things and not others. Moreover, they are entitled to use such steps in their own proofs even if they do not understand why we say they are valid steps. Which brings me to a third kind of understanding of proofs. This is what we mean when someone looks at my natural-deduction proof of a De Morgan law and can explain the structure of it. Not merely check the steps (which is of course mechanizable), and not justify why the rules of the system are the right ones, but rather, to explain the overarching structure of the proof. We overlook this kind of understanding sometimes, because in our usual informal proofs most of the energy is spent in communicating the structure. Some teachers are better at it than others, though, and many beginning students come away from a proof understanding every single step (and are able to offer a subproof of each one) and having no clue whatsoever how the whole fits together and how to characterize what is really going on inside the proof. This I think might well be the most important kind of understanding, and I think Don Taylor's notion leaves it out entirely. Anyway, I offer these more sophisticated (I hope) reflections to atone for my presumption in misreading the previous context and giving a pedantic example to folks who hardly needed one. Thomas === Subject: Re: Understanding a proof (Was: Re: Wiles' Proof) > Surely a novice might make the same objections regarding a proof > by an expert, only because the novice regarded some missing step > as essential. But the missing step might be obvious to all experts, > and hence should be omitted on grounds of obscuring the main idea > (assuming that the proof was presented in a forum for experts). So you are saying that determining has he/she understood the proof depends on the *observer*? In other words, a proof can not be judged objectively, but depends on who the judge is, or at least the intended audience, as well as who the author is... I would have expected a proof to be evaluated completely objectively and disjunct from all surroundings, including the author and the audience. Is this just being too idealistic? I mean, I don't like the idea that I present a proof and get critized for not having understood it, only to find the same proof presented by a well-renowned professional mathematician and subsequently accepted without comment. Can we (the readers of sci.math) accept such relativity? On a second note, maybe I am mixing up two different concepts: (1) Validating that a proof is correct, and (2) assessing whether the author has understood the proof. IMVHO, case (1) should be independent of both author and reader. Perhaps case (2) should be independent too, although a well-renowned professional mathematician might feel disrespected for having to explain trivial concepts in excruciating detail :-) -Michael. === Subject: Re: Understanding a proof (Was: Re: Wiles' Proof) I'd have to issue a proviso: if you really believe in fuzzy logic (or, what is the same, the Copenhagen interpretation of QM ... many universes), then maybe that's possible. but we don't know, Why! > the simplest proof taht i know of PT is the lunes one, > Why is the question absurd? --les ducs d'Enron! === Subject: Re: Understanding a proof (Was: Re: Wiles' Proof) > I'd have to issue a proviso: > if you really believe in fuzzy logic (or, what is the same, > the Copenhagen interpretation of QM ... many universes), > then maybe that's possible. but we don't know, Why! I don't understand what you're saying. My point is very simple and real: you have no way of proving things for certain without it ultimately being dependant on the goodwill of humans. /David === Subject: Re: Understanding a proof (Was: Re: Wiles' Proof) > at 12:52 PM, Michael JÁrgensen said: >To me, understanding means to be able to explain to someone else. > There have been cases of fallacious proofs that were accepted for a > long time. People were able to explain the proof to each other, but > the explanations were actually wrong. IMHO they did not understand > the purported proof. > To me, you understand the proof when you understand each step along > the way. That's not as simple as it sounds, since sometimes a step can > depend on very subtle distinctions or unstated assumptions. Perhaps we must reach the conclusion that the only proofs we can be certain we understand are the fallacious ones.... -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Definition of limit(???) > Edgar has pointed out that I didn't read the section carefully > enough - the right definition comes later. > The two definitions I was talking about are _not_ equivalent. > The problem is not sequences versus epsilons, the problem > is whether f(a) should be relevant: > Say f(x) = 0 for x <> 0, f(0) = 1. Then lim_{x->0} f(x) should > be 0, but by one of the definitions above the limit does > not exist. > (At least that's what I thought the problem was - in fact > I missed a tiny bit of notation, the wrong definition > above is a definition of something other than > lim_{x->a} f(x).) In his The Elements of Real Analysis, Robert Bartle defines two kind of limits like y I've never seen anyone else do. If f is defined on a subset D of a real vector space and has values in another real vector space, and a is a limit point of D (actually, Bartle uses the term accumulation point), then he defines: L is the NON-DELETED limit of f at a if, for every neighborhood V of L, there's a neighborhood U of a, depending on V (Bartle is very careful with definitions and likes to emphasize details like this), such that f(x) is in V for every x in U cap D (another point that Bartle emphasizes). And then, Bartle defines the DELETED limit (the usual concept) l of f at a. The only (and important!) difference is at the very end of the definition: ...., such that f(x) is in V for every x in U cap D such that x<>a. And then Bartle, who likes to define limits and continuity in terms of neighborhoods, proves that such definitions are equivalent to the eps-delta ones, the first (non deleted limit) with |x -a| < delta and the second (deleted limit) with, as usual, 0<|x-a| delta. Then Bartle proves that f is continuous at a iff L = l. Though Bartle is a great author, I think these non-deleted and deleted limit things only cause confusion. I see no point in defining such concepts. In my opinion, they add nothing but lenght to the book. (Well, Bartle suggests such concepts make it easier to prove those theorems about continuity and limits of composite functions) In my opinion, defining limits in terms of sequences is a bad idea, at least in the case of metric spaces. I don't think it's natural. Artur === Subject: Re: Definition of limit(???) > Ross seems to think that things will be easiest to follow if he first > talks about convergent sequences and then bases everything > else on them. I'm not sure that that's right but I don't have any > big complaint with it. >>That's the trouble with really understanding the subject. Equivalent >>definitions are, well, equivalent, so why not just go with the flow? This is not really understanding the subject, but merely being able to prove the theorems. One should attempt to teach the concepts, and while I believe convergence is very important, I would make limits of sequences a special case of the general notion of convergence, NOT otherwise as is done by Ross. There are many cases in which I will absolutely not teach the standard definitions as definitions, as they are often highly misleading, and this is even in elementary courses. Many of the usual definitions in probability hide the concepts quite well; it may be that they are easy to verify, but that should be as theorems or as computational procedures, not as definitions. >>We need people versed in pedagogy to teach courses, so they'll look at >>things from an educational point of view. >Fascinating. The fact that I'm not certain his opinion is correct but >I'm also not certain it's wrong means I'm not looking at things from >an educational point of view. Huh. At this time, it seems that those versed in pedagogy have been taught to avoid starting with concepts. Try teaching elementary school teachers, or even high school teachers of mathematics, mathematical concepts. The pedagogy experts put so much memorization and computation in the way that concepts are very hard to get. >Yes, it certainly is true that having people who really understand >a subject teach it is a bad idea. (On what planet, exactly?) If we ever are going to get good education, it will have to be done by those who have not had courses in pedagogy, or who at least reject them utterly. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Definition of limit(???) >>Ross seems to think that things will be easiest to follow if he first >>talks about convergent sequences and then bases everything >>else on them. I'm not sure that that's right but I don't have any >>big complaint with it. >That's the trouble with really understanding the subject. Equivalent >definitions are, well, equivalent, so why not just go with the flow? > This is not really understanding the subject, but > merely being able to prove the theorems. One should > attempt to teach the concepts, and while I believe > convergence is very important, I would make limits of > sequences a special case of the general notion of > convergence, NOT otherwise as is done by Ross. There > are many cases in which I will absolutely not teach the > standard definitions as definitions, as they are often > highly misleading, and this is even in elementary > courses. Many of the usual definitions in probability > hide the concepts quite well; it may be that they are > easy to verify, but that should be as theorems or as > computational procedures, not as definitions. >We need people versed in pedagogy to teach courses, so they'll look at >things from an educational point of view. >>Fascinating. The fact that I'm not certain his opinion is correct but >>I'm also not certain it's wrong means I'm not looking at things from >>an educational point of view. Huh. > At this time, it seems that those versed in pedagogy > have been taught to avoid starting with concepts. Try > teaching elementary school teachers, or even high school > teachers of mathematics, mathematical concepts. The > pedagogy experts put so much memorization and computation > in the way that concepts are very hard to get. The problem, as I see it, is in goals. The goal of people at the elementary/high school level is to produce students capable of crunching some numbers on a test. The goal of mathematicians at the college level (I think) is to promote understanding of underlying relationships and principles. These are contradictory goals. The question becomes: which is the right goal, and if the first goal is incorrect, how do we get it fixed? >>Yes, it certainly is true that having people who really understand >>a subject teach it is a bad idea. (On what planet, exactly?) > If we ever are going to get good education, it will have > to be done by those who have not had courses in pedagogy, > or who at least reject them utterly. Pedagogy needs to dictate the means to the goals, instead of dictating the goals. There are too many people out there who have an idea of how to teach something, without knowing what the goal is. They consistently fail. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Definition of limit(???) >I'm taeching the beginning epsilon-delta course. Looking >ahead in the book (Ross, Elementary Analysis: the Theory >of Calculus) I see the following definition: >Suppose that f : A -> R. Then lim_{x->a} f(x) = L if for >every sequence (x_n) in A such that x -> a we have >f(x_n) -> L. >At first I thought this must be a typo. But it turns out >he means it - later when he shows that this definition >is equivalent to the one in terms of epsilon and delta >the condition is |x - a| < delta, not 0 < |x-a| < delta. >I'm shocked. Is this version of the definition actually >standard in some circles? My impression is that the >definition is inconsistent with what the students are >almost certainly going to see in later courses - is this >correct? (I hate to cause confusion by using a >definition different from what's in the book unless >I have a very good reason, but if this definition is >as rare as it seems to me it is that could be a good >enough reason - hence the question whether it >really is extremely uncommon.) This is valid for every function from a metric space to another metric space. It can also be done more generally, and should, but with convergence of nets instead of sequences. It is very often the only approach to the topology which is easy to work with. There is no problem in showing that the definition is equivalent to the usual one if open sets are used, as they are in general to define limit. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Definition of limit(???) >>I'm taeching the beginning epsilon-delta course. Looking >>ahead in the book (Ross, Elementary Analysis: the Theory >>of Calculus) I see the following definition: >>Suppose that f : A -> R. Then lim_{x->a} f(x) = L if for >>every sequence (x_n) in A such that x -> a we have >>f(x_n) -> L. >>At first I thought this must be a typo. But it turns out >>he means it - later when he shows that this definition >>is equivalent to the one in terms of epsilon and delta >>the condition is |x - a| < delta, not 0 < |x-a| < delta. >>I'm shocked. Is this version of the definition actually >>standard in some circles? My impression is that the >>definition is inconsistent with what the students are >>almost certainly going to see in later courses - is this >>correct? (I hate to cause confusion by using a >>definition different from what's in the book unless >>I have a very good reason, but if this definition is >>as rare as it seems to me it is that could be a good >>enough reason - hence the question whether it >>really is extremely uncommon.) >This is valid for every function from a metric space to >another metric space. >It can also be done more generally, and should, but with >convergence of nets instead of sequences. It is very often >the only approach to the topology which is easy to work >with. There is no problem in showing that the definition >is equivalent to the usual one if open sets are used, as >they are in general to define limit. If you looked more closely, or read some of the thread, you'd see that it is not equivalent to the usual definition of limit. Let f(x) = 0 for x <> 0, f(0) = 1. By the definition above the limit of f(x) as x->0 does not exist. ************************ David C. Ullrich === Subject: Re: Definition of limit(???) <796tpvcthobffrmuapd5nu7d0dhn624194@4ax.com> wouldnt |x-a| < delta, delta > 0 (unless Ross didnt say that delta >was > 0), be meaning the same as 0 < |x-a| < delta. No, because x might equal a. He's concerned with how to define the limit in the case where the function is discontinuous at a point. By one definition, f(x) does not have a limit at x=a, by the other it may but the limit is not f(a). It's not that critical as long as the definition is used consistently; he's worried about the student having to deal with an inequivalent definition in a subsequent course. He may also be concerned with confusing the student when left handed and right handed limits come up. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do === Subject: Re: Definition of limit(???) <796tpvcthobffrmuapd5nu7d0dhn624194@4ax.com> at 12:41 PM, Jonathan Miller said: >That's the trouble with really understanding the subject. Equivalent >definitions are, well, equivalent, so why not just go with the flow? Because they are not equivalent once things get more general. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do === Subject: Re: Definition of limit(???) >I'm taeching the beginning epsilon-delta course. Looking ahead in >the book (Ross, Elementary Analysis: the Theory of Calculus) I see >the following definition: >Suppose that f : A -> R. Then lim_{x->a} f(x) = L if for every >sequence (x_n) in A such that x -> a we have f(x_n) -> L. I can certainly see a pedagogical problem with that, for the students who eventually take Topology. >At first I thought this must be a typo. But it turns out he means it >- later when he shows that this definition >is equivalent to the one in terms of epsilon and delta >the condition is |x - a| < delta, not 0 < |x-a| < delta. Why does that last matter? 0 <= |x-a| and f(x) = f(x), so the proofs go through with either condition. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do === Subject: education (was: Definition of limit(???) > Yes, it certainly is true that having people who really understand > a subject teach it is a bad idea. (On what planet, exactly?) Multiple Choice: Future high-school mathematics instructors should major in: (A) education (B) mathematics (C) English literature (D) fundraising === Subject: Re: education (was: Definition of limit(???) <796tpvcthobffrmuapd5nu7d0dhn624194@4ax.com> >Multiple Choice: >Future high-school mathematics instructors should major in: What do you mean by should? Are you asking about what they need to do in order to get their certification, or about what they need to do in order to 4be able to communicate the subject matter? > (A) education Needed to get certification. May well destroy the ability to teach. > (B) mathematics Only if you actually care whether the students learn anything. > (C) English literature Sounds good to me. Grammar wouldn't hurt either. > (D) fundraising I hope not. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do === Subject: Re: education >>Yes, it certainly is true that having people who really understand >>a subject teach it is a bad idea. (On what planet, exactly?) > Multiple Choice: > Future high-school mathematics instructors should major in: > (A) education > (B) mathematics > (C) English literature > (D) fundraising B. A is useful, but knowing *how* to teaching without knowing *what* to teach will fail. This seems to be a problem with some of our public schools. Note: you can know how to teach without having a degree that proclaims that. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: education > Yes, it certainly is true that having people who really understand > a subject teach it is a bad idea. (On what planet, exactly?) >> Multiple Choice: >> Future high-school mathematics instructors should major in: >> (A) education >> (B) mathematics >> (C) English literature >> (D) fundraising > B. A is useful, but knowing *how* to teaching without knowing *what* > to teach will fail. This seems to be a problem with some of our > public schools. > Note: you can know how to teach without having a degree that proclaims > that. Unfortunately, many do not realize that talent in and knowledge of mathematics (or any other subject) without the ability to teach is just as useless in the classroom. This lack of recognition is particularly prominent at the university level. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: education <796tpvcthobffrmuapd5nu7d0dhn624194@4ax.com> <3f9ebb78_3@newsfeed.slurp.net> at 11:03 PM, Stephen J. Herschkorn said: >Unfortunately, many do not realize that talent in and knowledge of >mathematics (or any other subject) without the ability to teach is >just as useless in the classroom. Unfortunately, many do not realize that a degree in education confers neither an education nor the ability to teach. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do === Subject: Re: education >> Yes, it certainly is true that having people who really understand >> a subject teach it is a bad idea. (On what planet, exactly?) > Multiple Choice: > Future high-school mathematics instructors should major in: > (A) education > (B) mathematics > (C) English literature > (D) fundraising >> B. A is useful, but knowing *how* to teaching without knowing *what* >> to teach will fail. This seems to be a problem with some of our >> public schools. >> Note: you can know how to teach without having a degree that proclaims >> that. >Unfortunately, many do not realize that talent in and knowledge of >mathematics (or any other subject) without the ability to teach is just >as useless in the classroom. This lack of recognition is particularly >prominent at the university level. I am inclined to disagree; there are few who do not have the ability to teach those who are capable of understanding and who are willing to understand the subject. Those who teach in the manner the educationists have messed things up in elementary and high school, by teaching facts, formulas, and methods of calculation, continue the damage done to the students. Give a student a straight cookbook calculus course, and that student will have difficulty in understanding what a continuous function or a derivative is. We have had a posting by a college student who complained that the instruction was not teaching concepts like the concept of inner product of two sequences as the sum of products. Those are not concepts, but formulas, which add nothing to understanding, but detract much. We would do better by giving few formulas and lots of concepts, starting with variable as standing for anything, which belongs in first grade. In the moderately distant past, college students all had the classical Euclidean geometry, which emphasized the understanding and construction of proofs. Now, few have had this, but their geometry course was facts and formulas, and possibly memorization of some proofs. Concepts are not learned by learning how to calculate answers, and this even includes arithmetic. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: education >> Yes, it certainly is true that having people who really understand >> a subject teach it is a bad idea. (On what planet, exactly?) > Multiple Choice: > Future high-school mathematics instructors should major in: > (A) education > (B) mathematics > (C) English literature > (D) fundraising >> B. A is useful, but knowing *how* to teaching without knowing *what* >> to teach will fail. This seems to be a problem with some of our >> public schools. >> Note: you can know how to teach without having a degree that proclaims >> that. > Unfortunately, many do not realize that talent in and knowledge of > mathematics (or any other subject) without the ability to teach is just > as useless in the classroom. This lack of recognition is particularly > prominent at the university level. Unfortunately, it is also present in the high school level with those bearing a degree that would suggest otherwise. Then I (and others) get to pick up the pieces when they hit college. If they get nailed twice with poor teachers, they may never learn math. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: education >>>>> Yes, it certainly is true that having people who really understand >>> a subject teach it is a bad idea. (On what planet, exactly?) >>>>>> Multiple Choice: >> Future high-school mathematics instructors should major in: >> (A) education >> (B) mathematics >> (C) English literature >> (D) fundraising > B. A is useful, but knowing *how* to teaching without knowing *what* > to teach will fail. This seems to be a problem with some of our > public schools. > Note: you can know how to teach without having a degree that proclaims > that. >> Unfortunately, many do not realize that talent in and knowledge of >> mathematics (or any other subject) without the ability to teach is just >> as useless in the classroom. This lack of recognition is particularly >> prominent at the university level. >Unfortunately, it is also present in the high school level with those >bearing a degree that would suggest otherwise. Then I (and others) get >to pick up the pieces when they hit college. If they get nailed twice >with poor teachers, they may never learn math. We're all making excellent points. What I'm curious about is whether anyone agrees with the point that started this: The idea that a deep understanding of mathematics is a _bad_ thing for a math teacher. (That being the point that I was replying to in the quote at the top of this subthread.) ************************ David C. Ullrich === Subject: Re: education > We're all making excellent points. What I'm curious about is whether > anyone agrees with the point that started this: The idea that a deep > understanding of mathematics is a _bad_ thing for a math teacher. > (That being the point that I was replying to in the quote at the > top of this subthread.) I vehemently *disagree* with it. A deep understanding of mathematics is necessary for being a math teacher. If you don't have sufficient depth of understanding, you cannot guide the students towards what is important and what is not. You also cannot see the deeper relationships that may help them understand what they are learning. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: education (was: Definition of limit(???) >> Yes, it certainly is true that having people who really understand >> a subject teach it is a bad idea. (On what planet, exactly?) >Multiple Choice: >Future high-school mathematics instructors should major in: > (A) education > (B) mathematics > (C) English literature > (D) fundraising That strikes me as a fairly tricky question, to tell you the truth. But I don't see the relevance of the question to what you quoted. Ask me an easier one: True or False: Future high-school teachers should have a sound understanding of mathematics, in fact not just the math they're going to be teaching but a little more than that. I know the answer to _that_ one. (And note that I was replying to this: That's the trouble with really understanding the subject, implying that a really understanding the subject is a bad thing. Saying that it's ridiculous to say understanding the subject you're teaching is a bad thing does not imply that understanding the subject is all that's required to teach young children...) ************************ David C. Ullrich === Subject: B-Splines Explaination Hi All, I am interested in coding in java a b-spline fit of a series of 3d points I have generated through an image tracking application I have written for my final year undergrad project. I am not a maths student, just a humble computer science undergrad, therefore I know little about b-splines. I have had b-splines mentioned to me as the solution to my 3d curve fitting problem. However, as mentioned above I'm not maths genius, therefore I am asking people firstly for a link or for somebody to give me a nice simple explaination of what b-splines are all about. Also how I would go about setting up the problem with the use of my 3d points in a computer program. Matrix, recursion ???etc And secondly if anybody knows a java implementation of this, taking into amount i want to specify the number of inputs i use, that would be even better. Adam === Subject: Hard tensor question (kst). 1) Suppose K= |g^uv|, where g^uv is the contravariant metric tensor, and K is it's determinant and is a relative tensor. And then form a covariant tensor K_uv, by K_uv = K*g_uv where g_uv is a the covariant metric, would the determinant, |K_uv| = 1 ? 2) If so, a metric like g_uv = K_uv +A_u*B_v Eq.(kst1) has the determinant (set g = |g_uv|) g = |g_uv| = |K_uv| + |A_u*B_v| therefore, g = 1 + g*A*B where g*A*B = |A_u*B_v|, and A*B=g^uv*A_u*B_v I've assumed g=1/K so far. 3) If so K = 1 - A*B and 1 = K+A*B Certainly the number 1 is invariant, therefore the sum K+A*B is invariant. Thinking along the lines that the covariant derivative of the metric tensor is zero it would be interesting to investigate DK, (D is the absolute derivative with parameter ds, ds^2 = g_uv dx^u dx^v). ie, DK = 0 = DA*B + DB*A and find the asymmetry, DA*B = - DB*A. Eq.(kst2) Dividing this by A and B produces DA/A = - DB/B and then Dlog A = -Dlog B. This is easily integrated, log A = - log B + k (k= the integration constant). expoteniating produces, A =exp(log A) = exp(- log B + k) A = -B*e where e=(+/-)*log k. Coefficient e is from the integration constant k and is necessarily invariant, and a constant that results from k, there seems to be no data available to assign sign. Now lets have some fun by considering two values of e , e =0 and then e=1, (or e=-1). When e=0 Eq.(kst1) becomes g_uv = K_uv and IMO represents an orthogonal space (orthogonal meaning one where Cartesian perpendicular base vectors i,j,k are valid). OTOH, when e=1 then g_uv = K_uv +A_u*B_v Eq.(kst1) is valid, because A_u*B_v is non-zero, and represents a space where the Cartesian base vectors i,j,k are invalid. To recap for clarity, If a relative tensor |K_uv| =1 then e is an invariant constant, that defines departure from Euclidian space. Sofar, this post is mathematical, but it has very good physical equivalence. I've argued in other threads the gravitational constant is in fact a relative invariant of weight 2, and have included this as the quantity K in Eq.(kst1). Secondly, the geometric quantity e that appears within the above is physically the fundamental charge +e or -e. Finally, I wanted to share my current thinking on the relation of the Gravitational constant K, and the fundamental charge e in a general metric. Opinions and corrections welcomed, flames are ok, if I'm an idiot, (I'm an old man senility is a fact of life) take your best shot. Ken S. Tucker === Subject: How did Euler determine Euler's Constant? I know that : Euler's constant = Lim (n-> infinity) [ Sum(i/j) from j=1 to n - ln(n)] but, I can't see how one actually evaluates this relation to get the value of Euler's constant. MB === Subject: Re: How did Euler determine Euler's Constant? > I know that : > Euler's constant = Lim (n-> infinity) [ Sum(i/j) from j=1 to n - ln(n)] > but, I can't see how one actually evaluates this relation to get the value > of Euler's constant. I gather that, to you, to determine a constant is to get its value. Not everyone shares this way of seeing things. In the first place, to this day nobody knows the value of Euler's constant - we may know it to a thousand or a million or a billion decimal places, but that still leaves quite a few decimal places that we don't know. Nor do we know how to express it in terms of pi and e and square root of 2 and suchlike. So I'll assume you're asking, how did Euler evaluate the constant to however many decimals he got? Well, how many decimals did he get? Do you know whether he evaluated it at all? Seems to me you have to answer that one before you even ask the question (that I think) you're asking. But there are a few things Euler might have used, if he did want to evaluate the constant to a few decimals. One of them is called the Euler-Maclaurin formula, so one presumes Euler had some familiarity with it. I recommend looking it up. And if you really want to know about Euler's constant, there's a nice recent book called Gamma, by Havil, which will tell you heaps about it. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Feigenbaum Number What is a good introduction? /David === Subject: Re: Feigenbaum Number > What is a good introduction? > /David Here are few web sites where you can get your feet wet: http://www.fortunecity.com/emachines/e11/86/expmaths.html#Feigenbaum http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Chaos/Chaos.html http://www.stud.ntnu.no/~berland/math/feigenbaum/ His actual papers on this topic are: Quantative Universality for a Class of Non-linear Transformations, J. Stat. Physics, vol. 19, 25-52, 1978. The Universal Metric Properties of Nonlinear Transformations, J. Stat. Physics, vol. 21, 669-706, 1979 FFeigenbaum, M. J., Quantitative Universality for a Class of Nonlinear Transformations, Journal of Statistical Physics 19, 25-52 Feigenbaum, M. J., Quantitative Universality for a Class of Nonlinear Transformations, Journal of Statistical Physics 19, 25-52 (1978) (1978)eigenbaum, M. J., Quantitative Universality for a Class of Nonlinear Transformations, Journal of Statistical Physics 19, 25-52 (1978) === Subject: Re: Feigenbaum Number > His actual papers on this topic are: > Quantative Universality for a Class of Non-linear Transformations, J. > Stat. Physics, vol. 19, 25-52, 1978. > The Universal Metric Properties of Nonlinear Transformations, J. Stat. > Physics, vol. 21, 669-706, 1979 > FFeigenbaum, M. J., Quantitative Universality for a Class of Nonlinear > Transformations, Journal of Statistical Physics 19, 25-52 > Feigenbaum, M. J., Quantitative Universality for a Class of Nonlinear > Transformations, Journal of Statistical Physics 19, 25-52 (1978) > (1978)eigenbaum, M. J., Quantitative Universality for a Class of Nonlinear > Transformations, Journal of Statistical Physics 19, 25-52 (1978) Do you think his supervisor counts these as 5 papers? === Subject: Limsup, Liminf If a sequence of reals {s_n} converges, then the corresponding Limsup equals its Liminf. These 2, however, appear to be independent of the particular ordering of {s_n}. Is it accurate to state, then, that Limsup=Liminf does not necessarily imply the convergence of {s_n}? Any help/explanation appreciated. === Subject: Re: Limsup, Liminf > If a sequence of reals {s_n} converges, then the corresponding Limsup > equals its Liminf. These 2, however, appear to be independent of the > particular ordering of {s_n}. Is it accurate to state, then, that > Limsup=Liminf does not necessarily imply the convergence of {s_n}? > Any help/explanation appreciated. You can rearrange a SERIES and get a different value, not a sequence. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Limsup, Liminf > If a sequence of reals {s_n} converges, then the corresponding Limsup > equals its Liminf. These 2, however, appear to be independent of the > particular ordering of {s_n}. So is convergence. If a sequence of reals converges, so does any rearrangement of the sequence. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Limsup, Liminf >If a sequence of reals {s_n} converges, then the corresponding Limsup >equals its Liminf. These 2, however, appear to be independent of the >particular ordering of {s_n}. Is it accurate to state, then, that >Limsup=Liminf does not necessarily imply the convergence of {s_n}? No, a sequence has a limit if and only if the limsup equals the liminf. Independent of the ordering is a little informal, but the _limit_ of a sequence is _also_ independent of the ordering, in the same sense in which the liminf and limsup are. (Ie, if (n_j) is a permutation of the integers then the limit of s_{n_j} is the same as the limit of s_n.) >Any help/explanation appreciated. ************************ David C. Ullrich === Subject: Re: Limsup, Liminf > If a sequence of reals {s_n} converges, then the corresponding Limsup > equals its Liminf. These 2, however, appear to be independent of the > particular ordering of {s_n}. Is it accurate to state, then, that > Limsup=Liminf does not necessarily imply the convergence of {s_n}? > Any help/explanation appreciated. Since you're working on the real number line, this simplifies things a bit. You can think of the LimSup as the largest limit point in a sequence of real numbers, and similarly, you can think of LimInf as the smallest limit point in a sequence of real numbers. If they are equal, then the sequence has a unique limit point, and hence converges. This holds even when LimSup and LimInf are +/- infinity also (I'll let you think about that). MB === Subject: Re: Homological algebra >Can someone suggest a good book on Homological algebra?? What are you looking for? Does it have to be selfcontained? Does it have to be current? Take a look at Homology by MacLane. I also vaguely recall a Homological Algebra by MacLane and Eilenberg, but perhaps I'm thinking of a different book. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do === Subject: Re: Homological algebra > Take a look at Homology by MacLane. I also vaguely recall a > Homological Algebra by MacLane and Eilenberg, but perhaps I'm > thinking of a different book. i have cartan & eilenberg homological algebra, released as a paperback in the princeton landmarks in math series. rip === Subject: Re: Homological algebra > Can someone suggest a good book on Homological algebra?? I'm not sure where you are starting from, but I have been reading Hilton and Wu's _A_Course_in_Modern_Algebra_. The final chapter introduces homological techniques (Ext and Tor). This book has much to recommend it, but for now let me just say that it is a reasonably thin book that starts out nice and slow--groups--and concentrates basically on concepts which will lead to homological techniques, without much else. For example, there is a chapter on category theory including a discussion of abelian categories and adjoint functors. Projective and injective modules are discussed at length. The major problem with this book is that it is currently being offered by in the Wiley Classic's Library at approximately $100, which I suspect gives them a healthy profit margin. However, copies are available on line. Best wishes, Mike === Subject: Re: Homological algebra >>Can someone suggest a good book on Homological algebra?? > I'm not sure where you are starting from, but I have been > reading Hilton and Wu's _A_Course_in_Modern_Algebra_. The > final chapter introduces homological techniques (Ext and Tor). > This book has much to recommend it, but for now let me just > say that it is a reasonably thin book that starts out nice > and slow--groups--and concentrates basically on > concepts which will lead to homological techniques, without > much else. For example, there is a chapter on category theory > including a discussion of abelian categories and adjoint functors. > Projective and injective modules are discussed at length. > The major problem with this book is that it is currently being > offered by in the Wiley Classic's Library at approximately $100, > which I suspect gives them a healthy profit margin. However, copies > are available on line. > Best wishes, > Mike That REALLY bugs me when publishers take a classic, photo-copy it, and sell it for an outrageous price. My own (and paid for!) example is Hardy and Wright's classic An Introcuction to the Theory of Numbers. I got the paperback edition in the early 1990's for $32.95. The list price is now $61.95. This is ABSURD! Martin Cohen === Subject: Re: Grad school: late entry >I'd like some advice from people who have been in my position; esp. >those who have made it through grad school! Well, everybody is different, so what worked for me may not work for you. Why did you go back to graduate school? Why in Mathematics? If you don't love Mathematics, then possibly you might be better off rethinking your academic goals. I went back to grad school just short of 5 years after leaving college, and was working 7 days a week on top of my studies. I selected classes based on what I was interested in rather than what might be useful after graduation. Those were the best years of my life. >I know graduate school is hard for everyone, but is hard work the >only solution? It's an essential ingredient. >Because at the end of the day, for first year students at least, it >is the grades on the exams and the courses that really matter, >irrespective of mathematical potential. That attitude is part of your problem. It is the material and the insights that matter; the grades are only a way to measure those. If you're not there for the sake of learning then you are going to have problems. Find topics that you want to learn for their own sakes, and work hard on those. If you're not having fun in the process then something is wrong. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do === Subject: Re: Grad school: late entry > Hallo everyone, > I worked for about 3 years after college before getting into grad > school. (I am currently in the first year). And I find math so much > more difficult now, than I ever did; keeping up with the pace of the > courses is turning out to be a nightmare. I have forgotten even some > elementary results; and find myself stuck in a painstakingly slow > iterative process (going back to my old Calc III notes...). My profs > want me to speed up things, but I cannot seem to find an optimal path. > I'd like some advice from people who have been in my position; esp. > those who have made it through grad school! here is some useful advice: ABANDON SHIP. exception: you love math for its own sake. > I know graduate school is hard for everyone, but is hard work the > only solution? Is there a way of working smart instead of hard? > Because at the end of the day, for first year students at least, it is > the grades on the exams and the courses that really matter, > irrespective of mathematical potential. > Frank > P.S.--I am not sure if there is any wisdom in sampling life for a > few years before beginning grad school. === Subject: Submartingales and Gambling problem help To all math/probability scholars, please help with the following game : Consider the following game : Deck of 52 cards. You start out with k dollars. You place a bet on the first hand. The game is that you guess whether the next card will be black or red. The dealer flips over a card and if you're right, you receive whatever you bet, and if you lose, your bet is gone. You keep playing this game until all 52 cards run out. The caveat is that after the first card is gone, you can use that information to your advantage for the next card. So say the first card was red, then obviously you will bet that the next card will be black since you know there are 26 blacks in the deck and 25 reds. So this is a favorable game since if you bet p dollars, you receive p if you win, and lose p if you lose (and on some hands your odds are greater than 50% chance of winning, and the minimum odds of winning any hand are 50% since you are allowed to memorize all of the cards that were already played) My question is, how much would you pay to play this game? In other words, what is the optimal strategy for this game and what is your expected winnings using this strategy? I can put a minimum bound on the game as follows : If you just bet 0 on every hand until the last card, you know what the last card is since you memorized all the previous cards. Then you bet k dollars and you win k dollars (since you know the last card). So at minimum you can expect to win k dollars, so you should pay k dollars to play the game since in the end you gain 2k and end up with k (which is what you started with). Can I use martingale theory to solve this problem? Is this a famous problem? What is the solution? Michael