Cayley定理定理說明每個群G都同構於G上的對稱群S_G的子群。書中使用群作用證明。我本來還不清楚為何要使群作用在別的集合上，這個證明使群作用在自己上，我才明白群作用可以幫助我們證明關於群本身，以及關於被作用的數學結構的定理。我在影片中的證明，是按照書中的方法，從群作用的定義開始，再使群作用在自己上，考慮作用的穩定子群和核，證明正規表示是單射。我省略了幾個和定理沒有直接關係的細節。
本影片摘要：In this video, we will prove Cayley's theorem from the point of view of group actions. First we will define group actions and permutation representations. Then we will consider a group acting on itself by left multiplication and use that to prove Cayley's Theorem.
※ This video is largely unedited. You can fast forward anytime.
更多數學表達練習影片→ https://pse.is/MMB9Q
2020年1月1日 星期三
奶奶與我
我小時候都給奶奶照顧，現在奶奶有失智症，只能臥床。小時候的事雖然大部分都忘記，但留下許多照片，媽媽也會告訴我。奶奶給我的印象是，一生很辛苦，但堅強、樂觀、有愛心。這就像一棵樹，一生都只靠自己和太陽的力量生長，但只要有人或動物需要它遮蔭、遮雨，甚至是吃它、砍下它當材料，都不會拒絕。它死時成為土壤中的養分，還是繼續幫助別人。
奶奶讓我的童年可以過得很快樂。我小時候皮膚病很嚴重，體無完膚，很難照顧。奶奶已經七十幾歲，盡心盡力的帶我，我才能慢慢好轉。我嬰兒時頭很扁，奶奶細心調整我的睡姿，讓我的頭變圓。奶奶也教瘦弱的我學會走路。她是理髮高手，我的頭髮都是她剪的。她也很會烹飪，我喜歡吃她煮的菜。
我小時候很調皮，讓奶奶很傷腦筋。有一次我趁奶奶出門，把大門鎖住，她只好從窗戶跳進來。有一次搭電梯，我沒跟好，被電梯載到地下停車場，在那裡哭。我和奶奶到處去玩，度過快樂的童年。我們去過日本、美國、加拿大，也到過台灣各地的公園、博物館和登山步道。
我慢慢長大，也越來越少和奶奶見面。我國小畢業時，拍了一支訪問奶奶的影片。後來，我去加拿大住兩年，每天拍影片問候奶奶。我搬回臺灣時，變成完全沒有聲音。很「剛好」，奶奶也因為失智症，沒有發現我的改變。在她的記憶裡，我可能還是調皮多話的小孩。出國前拍的那支影片，竟成為我和奶奶最後一次的對話。
奶奶失智之後，忘記很多事情，童年受日本教育的事比較記得，所以幫她拍了說童年的影片。她忽然很會唱日本童謠，也會教我五十音。奶奶的狀況隨時間惡化，現在需要臥床和插管。問她幾歲，她會說十八歲。我經過不斷練習，幾個月前開始可以錄影說話，去探望她時播放給她看。幾天前告訴她我考上大學，她很開心，誇我很棒。奶奶還沒忘記如何說話勉勵人。
我覺得奶奶一生像一棵樹，雖然自己很辛苦，沒人幫助，但還是一直幫助、勉勵別人。我會永遠記得，當做我的榜樣。
奶奶讓我的童年可以過得很快樂。我小時候皮膚病很嚴重，體無完膚，很難照顧。奶奶已經七十幾歲，盡心盡力的帶我，我才能慢慢好轉。我嬰兒時頭很扁，奶奶細心調整我的睡姿，讓我的頭變圓。奶奶也教瘦弱的我學會走路。她是理髮高手，我的頭髮都是她剪的。她也很會烹飪，我喜歡吃她煮的菜。
我小時候很調皮，讓奶奶很傷腦筋。有一次我趁奶奶出門，把大門鎖住，她只好從窗戶跳進來。有一次搭電梯，我沒跟好，被電梯載到地下停車場，在那裡哭。我和奶奶到處去玩，度過快樂的童年。我們去過日本、美國、加拿大，也到過台灣各地的公園、博物館和登山步道。
我慢慢長大，也越來越少和奶奶見面。我國小畢業時，拍了一支訪問奶奶的影片。後來，我去加拿大住兩年，每天拍影片問候奶奶。我搬回臺灣時，變成完全沒有聲音。很「剛好」，奶奶也因為失智症，沒有發現我的改變。在她的記憶裡，我可能還是調皮多話的小孩。出國前拍的那支影片，竟成為我和奶奶最後一次的對話。
奶奶失智之後，忘記很多事情，童年受日本教育的事比較記得，所以幫她拍了說童年的影片。她忽然很會唱日本童謠，也會教我五十音。奶奶的狀況隨時間惡化，現在需要臥床和插管。問她幾歲，她會說十八歲。我經過不斷練習，幾個月前開始可以錄影說話，去探望她時播放給她看。幾天前告訴她我考上大學，她很開心，誇我很棒。奶奶還沒忘記如何說話勉勵人。
我覺得奶奶一生像一棵樹，雖然自己很辛苦，沒人幫助，但還是一直幫助、勉勵別人。我會永遠記得，當做我的榜樣。
2019年10月26日 星期六
The First Isomorphism Theorem
我的數學表達練習，這次嘗試邊寫黑板邊講解，一次錄半小時。初步嘗試，請多指教。
本影片摘要：We will start from the definition of cosets and normal subgroups. Then we will construct quotient groups by defining a binary operation on the cosets of a normal subgroup. Then we will show that the kernel of a homomorphism is a normal subgroup. And moreover, every normal subgroup is the kernel of a homomorphism. Finally we will show that G mod kernel of φ is isomorphic to φ of G, completing the proof.
※ This video is mostly unedited. You can fast forward anytime.
更多數學表達練習影片→ https://pse.is/MMB9Q
本影片摘要：We will start from the definition of cosets and normal subgroups. Then we will construct quotient groups by defining a binary operation on the cosets of a normal subgroup. Then we will show that the kernel of a homomorphism is a normal subgroup. And moreover, every normal subgroup is the kernel of a homomorphism. Finally we will show that G mod kernel of φ is isomorphic to φ of G, completing the proof.
※ This video is mostly unedited. You can fast forward anytime.
更多數學表達練習影片→ https://pse.is/MMB9Q
2019年10月20日 星期日
「全球暖化」桌遊
Global Warming
2 to 4 players
Cards with a red stripe are disaster cards. Cards with a yellow stripe are consequence cards. Cards with a green stripe are human activity cards.
Setup
Every player chooses a color. Place a piece of your color on 0 degrees on the temperature scale and another on 10 units on the money scale. Separate the consequence cards from the other cards in the deck. Shuffle the deck. Deal every player 1 consequence card and 3 cards from the deck. The players place their consequence cards on the table. Put any disaster cards back into the deck, and replace them by drawing from the deck. Shuffle the deck again.
Play
1. On your turn, move your pieces on the temperature scale and the money scale according to the human activity cards you have on the table unless you have none. If your temperature reaches -1.0 degrees, it cannot decrease further but can increase. If your money reaches 30 units, it cannot increase further but can decrease. If your temperature equals the temperature on your consequence card or more, you lose the amount of money on the consequence card.
2. Draw 1 card from the deck. If the card is a disaster card, play it to the discard pile immediately. Lose the amount of money and discard the number of cards on the disaster card. If the card is a human activity card, add it to your hand and play 1 card on the table in front of you. Pay the amount of money on the top right corner of the card. You must play 1 card even if you do not have enough money.
3. Each power plant can power 2 cards that need electricity. If a card that needs electricity cannot be powered, the card can still be played but does not affect temperature or money.
4. If there are no cards left in the deck, shuffle the discard pile to make a new deck.
End of game
You lose if you run out of money or if your temperature is 2.0 degrees or more. Discard all of your human activity cards. You win if your temperature is -1.0 degrees and you have 30 units of money or more, or if you are the last person left.
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