計劃進修PhD/Research討論區(27) The brain is wider than the sky

一言筆發 2017-6-29 13:33:57

想問下咩叫Offer of Tenure - Associate Professor
:^(

:^(

tenure prof姐係點
:^(
對哩d 一無所知

:^(

數學白痴 2017-6-29 13:34:41

:^(

:^(

:^(

睇咗老婆先生
:^(

:^(

一言筆發 2017-6-29 13:35:36

:^(

:^(

:^(

睇咗老婆先生
:^(

:^(

:^(

數學白痴 2017-6-29 13:36:00

想問下咩叫Offer of Tenure - Associate Professor
:^(

:^(

tenure prof姐係點
:^(
對哩d 一無所知

:^(

:^(

傳教士谷哥 2017-6-29 13:40:28

:^(

:^(

:^(

睇咗老婆先生
:^(

:^(

:^(

:^(

數學白痴 2017-6-29 13:59:54

想問下咩叫Offer of Tenure - Associate Professor
:^(

:^(

tenure prof姐係點
:^(
對哩d 一無所知

:^(

:^(

associate都鐵飯碗
:^(

:^(

婆你呀麼彈彈波 2017-6-29 14:01:50

想問下咩叫Offer of Tenure - Associate Professor
:^(

:^(

tenure prof姐係點
:^(
對哩d 一無所知

:^(

:^(

associate都鐵飯碗
:^(

:^(

:^(

Athanasios 2017-6-29 14:03:41

:^(

數學白痴 2017-6-29 14:05:16

想問下咩叫Offer of Tenure - Associate Professor
:^(

:^(

tenure prof姐係點
:^(
對哩d 一無所知

:^(

:^(

associate都鐵飯碗
:^(

:^(

:^(

:^(

婆你呀麼彈彈波 2017-6-29 14:07:39 http://www.ams.org/journals/bull/2017-54-01/S0273-0979-2016-01552-7/S0273-0979-2016-01552-7.pdf
WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING?
碌落少少已經唔覺得interesting

:^(

數學白痴 2017-6-29 14:10:40

http://www.ams.org/journals/bull/2017-54-01/S0273-0979-2016-01552-7/S0273-0979-2016-01552-7.pdf
WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING?
碌落少少已經唔覺得interesting
:^(

:^(

backspace 2017-6-29 14:21:53

:^(

:^(

:^(

:^(

傳教士谷哥 2017-6-29 14:50:00

:^(

:^(

:^(

:^(

:^(

婆你呀麼彈彈波 2017-6-29 14:51:47

:^(

:^(

:^(

:^(

:^(

:^(

J痕叔叔 2017-6-29 14:59:31 hea左兩日突然間又新post!

夢追人 2017-6-29 15:49:29

:^(

:^(

傳教士谷哥 2017-6-29 18:13:41

:^(

:^(

:^(

我沒有放棄 2017-6-29 18:17:00

:^(

:^(

:^(

:^(

傳教士谷哥 2017-6-29 18:19:42

:^(

:^(

:^(

:^(

:^(

忒修斯之船 2017-6-29 18:33:08

:^(

:^(

:^(

:^(

:^(

數學白痴 2017-6-29 19:28:56

http://www.ams.org/journals/bull/2017-54-01/S0273-0979-2016-01552-7/S0273-0979-2016-01552-7.pdf
WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING?
碌落少少已經唔覺得interesting
:^(

Oseledets's theorem好interesting
:^(

不過篇文真係寫得麻麻地
:^(
冇乜background應該唔太知噏乜
:^(

Lyapunov exponent大致上就係睇一個dynamical system平均嘅"expansion rate"有幾快

例如f(x)=2x mod 1,你當依個map係個dynamical system on circle嘅話，你會發現依個map會將啲嘢「放大」（例如將[0,1/2]變做[0,1]

但如果你睇g(x)=3x mod 1,應該唔難想像g expand得快過f

個rate可以話係取決於個slope有幾大，上面嘅例子個slope係constant,所以你可以好隨便咁講f嘅rate係2，g嘅rate係3

不過咁樣會唔夠好，因為dynamical system睇嘅唔單止係f,仲想睇f(f(...f(x))...)嘅behaviour,所以要睇嘅應該係f^n嘅slope

用上面嘅例子,f^n嘅slope係2^n,g^n嘅slope係3^n,依個sequence唔converge,但take完logarithm然後除以n就好好多，分別係log2同log3

但喺一般情況，個map個slope唔會係constant,但你仍然可以consider log|(h^n)'(x)|,你會發現用chain rule後會變咗 sum log|h'(h^i(x))|,如果h係ergodic依個sum除以n會converge to integrate log|h'| over the space,而依個數字就係Lyapunov exponent,話你知h平均expand得幾快

:^(

婆你呀麼彈彈波 2017-6-29 20:30:52

http://www.ams.org/journals/bull/2017-54-01/S0273-0979-2016-01552-7/S0273-0979-2016-01552-7.pdf
WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING?
碌落少少已經唔覺得interesting
:^(

Oseledets's theorem好interesting
:^(

不過篇文真係寫得麻麻地
:^(
冇乜background應該唔太知噏乜
:^(

Lyapunov exponent大致上就係睇一個dynamical system平均嘅"expansion rate"有幾快

例如f(x)=2x mod 1,你當依個map係個dynamical system on circle嘅話，你會發現依個map會將啲嘢「放大」（例如將[0,1/2]變做[0,1]

但如果你睇g(x)=3x mod 1,應該唔難想像g expand得快過f

個rate可以話係取決於個slope有幾大，上面嘅例子個slope係constant,所以你可以好隨便咁講f嘅rate係2，g嘅rate係3

不過咁樣會唔夠好，因為dynamical system睇嘅唔單止係f,仲想睇f(f(...f(x))...)嘅behaviour,所以要睇嘅應該係f^n嘅slope

用上面嘅例子,f^n嘅slope係2^n,g^n嘅slope係3^n,依個sequence唔converge,但take完logarithm然後除以n就好好多，分別係log2同log3

但喺一般情況，個map個slope唔會係constant,但你仍然可以consider log|(h^n)'(x)|,你會發現用chain rule後會變咗 sum log|h'(h^i(x))|,如果h係ergodic依個sum除以n會converge to integrate log|h'| over the space,而依個數字就係Lyapunov exponent,話你知h平均expand得幾快

:^(

:^(