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

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Oseledets's theorem好interesting

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不過篇文真係寫得麻麻地

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冇乜background應該唔太知噏乜

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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

:^(

:^(

:^(

:^(

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

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如果想讀pg econ, undergrad 個陣可以點equip 自己?

:^(

:^(

學好數學

:^(

如果想讀pg econ, undergrad 個陣可以點equip 自己?

:^(

:^(

學好數學

:^(

除左math, 如果唔識programming 會唔會好蝕底?

:^(

唔識programming 咩都好蝕底

:^(

:^(

:^(

:^(

:^(

:^(

:^(

:^(

:^(

如果想讀pg econ, undergrad 個陣可以點equip 自己?

:^(

:^(

學好數學

:^(

除左math, 如果唔識programming 會唔會好蝕底?

:^(

唔識programming 咩都好蝕底

:^(

點解econ好似特別多人想讀上去

如果想讀pg econ, undergrad 個陣可以點equip 自己?

:^(

:^(

學好數學

:^(

除左math, 如果唔識programming 會唔會好蝕底?

:^(

唔識programming 咩都好蝕底

:^(

睇你想做咩field啦
如果係pure theory嘅，讀多啲數會好啲
如果係macro or applied 果啲就一定要識programming，我果時啲prof/同學 多數用matlab,stata,C++ or python
不過你仲係ug嘅就咩都最好掂少少,我識好多人phd頭幾年都會轉field

點解econ好似特別多人想讀上去

最近特別多

:^(

:^(

:^(

點解econ好似特別多人想讀上去

最近特別多

:^(

:^(

:^(

前幾日經過連登，見到好多學生一車車咁讀上去，有好多隻流眼淚佢地知就快俾人殺...真係好慘！

如果想讀pg econ, undergrad 個陣可以點equip 自己?

:^(

:^(

學好數學

:^(

除左math, 如果唔識programming 會唔會好蝕底?

:^(

唔識programming 咩都好蝕底

:^(

睇你想做咩field啦
如果係pure theory嘅，讀多啲數會好啲
如果係macro or applied 果啲就一定要識programming，我果時啲prof/同學 多數用matlab,stata,C++ or python
不過你仲係ug嘅就咩都最好掂少少,我識好多人phd頭幾年都會轉field

我而家都讀左啲好basic 嘅數，諗住之後會讀埋real analysis, topology or even functional analysis, 咁樣夠唔夠用？
programming 睇黎真係要下啲苦工

:^(