好悶 有無人問數

Given a cubic equation f(x), 如果 f(x) 嘅所有turning points 都 above x-axis ，咁我係咪就可以話 f(x) has only one real root ?

Yes.
Proof:
If two turning points, consider table of f'(x),
Suppose (another case similar)
for x<a, f'<0, decreasing to f(a)>0
for a<x<b, f'>0, increasing to f(b)>0
for b<x, f'<0, decreasing to -infinity
the root will be in (b,+inf)

If one turning point, f'(x)=0 has 2 repeated real roots, f(x)=0 has 3 repeated real root.

If no turning point, f'(x)=0 has no real roots. f(x) is injective.

想問
proof
y=2e^x-x^2 is one to one
感激不盡！！

dy/dx = 2(e^x-x)
Claim: e^x-x>0 for all x
Let f(x)=e^x-x
then f'(x)=e^x-1
f'<0 for x<0
f'=0 for x=0
f'>0 for x>0
Hence, f attains minimum at x=0,
thus f(x)>=f(0)=1 for all x
The claim has been proved.

weak topology of normed vector space係咪metrizable?

2c我做到greens轉polar coordinate之後做唔到

int(Pdx+Qdy)=int int (Q_x-P_y) dxdy
但Q_x-P_y係1喎 答案咪個DISK既AREA

留名
樓主可唔可以解釋一下咩係open ball 同點proof 佢係open ball?

General metric space上既Open ball定義如下
B(x,r):={y:d(x:y)<r}
即係所有離x距離少於r既point
要係唔知中心唔知RADIUS既情形下，證明一個set 係Open Ball，我諗都幾難
我諗首先take 2r=sup{d(x,y):x,y in S} 得出radius 先
但centre點搵我仲未諗到

weak topology of normed vector space係咪metrizable?

唔知
我仲weak過weak topology

geometric 問唔問得

oblique cone parametric equation 係點搵？

呢題我又未遇過，你有咩條件比左？
最直接係寫左一般cone條式，再將條式用rotation matrix將d variable rotate

http://mathworld.wolfram.com/Cone.html

呢到有standard(right) cone parametric equation，咁個尖尖就同個底個圓既圓心同軸，而家我想搵個尖尖座標唔同CONE 底個圓心同軸既parametric equation

另外搵到既話，我仲想問cone 個開口有無得控制(依家係由height ⊂ (0,h) 都係2π，我想知有無得[0,2π]咁變

x = aucosv
y = ausinv
z = u,
唔係好明你想控制開口既乜野，理論上，開口大細已經可以由呢三條eqt求出（你條link入面有）透過調整a,u，可以決定高低大細
如果你想中心軸偏離，你其實係想rotate個cone，咁就要multiply一個rotation matrix，即係點?
設(x，y，z)係你rotate前既cone上一點，
設X，Y，Z係你rotate x,y,z後既點，
咁(X,Y,Z)=rotation matrix * (x,y,z)，再將plug返原式。
我舉個例，原本cone向上，我將佢沿x-z plane扭t度
咁對於cone上任意一點，X=xcost-zsint （點計出黎有D煩，想知再問）
你個rotation matrix 第一row就會係[cost 0 -sint]，如此類推
再將個關係eliminate左x,y,z就做到

個開口意思係咁，個圓柱個横切面會慢慢由一個圓變成一個點，咁其實即係一個oblique cone 樣啦，我要用電腦畫呢個圖，唔識點樣整

1) 乜野係random variable?

2) Proof T= Z / (U/k)^(1/2)
Z~N(0,1) & U~chi squared dist. with k degrees of freedom, where T is t distribution

係唔係有人問convolution?

1) 乜野係random variable?

2) Proof T= Z / (U/k)^(1/2)
Z~N(0,1) & U~chi squared dist. with k degrees of freedom, where T is t distribution

1) measurable function

係唔係有人問convolution?

係我問 估唔到個post突然間咁多人

1) 乜野係random variable?

2) Proof T= Z / (U/k)^(1/2)
Z~N(0,1) & U~chi squared dist. with k degrees of freedom, where T is t distribution

1) measurable function

可唔可以具體D講 唔用definition咁解釋

1) 乜野係random variable?

2) Proof T= Z / (U/k)^(1/2)
Z~N(0,1) & U~chi squared dist. with k degrees of freedom, where T is t distribution

1) measurable function

可唔可以具體D講 唔用definition咁解釋

between 有冇人識解釋點解chi-squared/T dist. 既degree of freedom 係n-1 (for one sample case)

1) 乜野係random variable?

2) Proof T= Z / (U/k)^(1/2)
Z~N(0,1) & U~chi squared dist. with k degrees of freedom, where T is t distribution

1) measurable function

可唔可以具體D講 唔用definition咁解釋

可以就咁當佢係一個random number
簡單嚟講，sample space就係個experiment所有random嘅可能性，而random variable就係喺嗰個可能性下所得出嘅一個數字

唔可以當random variable係一個數字
反而上面巴打話 measurable function其實concept上正確 但function唔同number所以一定唔可以咁諗

利申唔完全明白 想要深入d既解釋

1) 乜野係random variable?

2) Proof T= Z / (U/k)^(1/2)
Z~N(0,1) & U~chi squared dist. with k degrees of freedom, where T is t distribution

1) measurable function

可唔可以具體D講 唔用definition咁解釋

可以就咁當佢係一個random number
簡單嚟講，sample space就係個experiment所有random嘅可能性，而random variable就係喺嗰個可能性下所得出嘅一個數字

唔可以當random variable係一個數字
反而上面巴打話 measurable function其實concept上正確 但function唔同number所以一定唔可以咁諗

利申唔完全明白 想要深入d既解釋

f is a measurable function簡單黎講就係preimage of (a, infinity) under f 係一個可以給予"體積"的function

1) 乜野係random variable?

2) Proof T= Z / (U/k)^(1/2)
Z~N(0,1) & U~chi squared dist. with k degrees of freedom, where T is t distribution

1) measurable function

可唔可以具體D講 唔用definition咁解釋

可以就咁當佢係一個random number
簡單嚟講，sample space就係個experiment所有random嘅可能性，而random variable就係喺嗰個可能性下所得出嘅一個數字

唔可以當random variable係一個數字
反而上面巴打話 measurable function其實concept上正確 但function唔同number所以一定唔可以咁諗

利申唔完全明白 想要深入d既解釋

f is a measurable function簡單黎講就係preimage of (a, infinity) under f 係一個可以給予"體積"的function

都係唔明, 如果 f(x) 可以寫成 x^2+1
咁random variable 可以寫成乜?
input 係唔係parameter? 同埋 output係唔係叫做 data?

1) 乜野係random variable?

2) Proof T= Z / (U/k)^(1/2)
Z~N(0,1) & U~chi squared dist. with k degrees of freedom, where T is t distribution

1) measurable function

可唔可以具體D講 唔用definition咁解釋

可以就咁當佢係一個random number
簡單嚟講，sample space就係個experiment所有random嘅可能性，而random variable就係喺嗰個可能性下所得出嘅一個數字

唔可以當random variable係一個數字
反而上面巴打話 measurable function其實concept上正確 但function唔同number所以一定唔可以咁諗

利申唔完全明白 想要深入d既解釋

f is a measurable function簡單黎講就係preimage of (a, infinity) under f 係一個可以給予"體積"的function

都係唔明, 如果 f(x) 可以寫成 x^2+1
咁random variable 可以寫成乜?
input 係唔係parameter? 同埋 output係唔係叫做 data?

你個input咪就係一個outcome (唔記得左叫咩名 lol), 再出一個real number

3以上的自然數n可以滿足X的n平方加上Y的n平方等於Z的n平方的自然數XYZ不存在
請試著證明之

請求樓主解答

3以上的自然數n可以滿足X的n平方加上Y的n平方等於Z的n平方的自然數XYZ不存在
請試著證明之

請求樓主解答

咪派膠

3以上的自然數n可以滿足X的n平方加上Y的n平方等於Z的n平方的自然數XYZ不存在
請試著證明之

請求樓主解答

咪派膠

岩岩先見到樓主個Post
樓主話悶咪出題有深度嘅題目囉
我諗樓主解完之後都唔會覺得悶

3以上的自然數n可以滿足X的n平方加上Y的n平方等於Z的n平方的自然數XYZ不存在
請試著證明之

請求樓主解答

咪派膠

岩岩先見到樓主個Post
樓主話悶咪出題有深度嘅題目囉
我諗樓主解完之後都唔會覺得悶

不如叫佢睇望月篇文好過啦