Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Outer measure: if $A,B\subset\mathbb{R}$ and $|A|<\infty$, then $|B\setminus A|\geq|B|-|A|$. The outer measure is defined as
$|A|=\inf(\{\sum\limits_{i=1}^{\infty}l(I_i)\text{ with $I_1$,$I_2$,... s.t:} \hspace{0,2cm}A\subset\bigcup\limits_{i=1}^{\infty}I_i\}$
where $l$ is the length of an interval in the intuitive s... | Let $(I_i)$ be a sequence of intervals covering $A$ and $(J_l)$ be a sequence of intervals covering $B \setminus A$. Then $(I_i)\cup (J_l)$ covers $B$. Hence $|B| \leq \sum_i l(I_i)+\sum_l l(J_l)$. Taking infimum over all covers $(I_i)$ and $(J_l)$ we get $|B| \leq |A|+|B \setminus A|$. Since $|A|<
\infty$ we can sub... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3448757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Assume an encrypted message is sent through use of exponential cipher. ...such that modulus $p = 2741 \text{ (p is prime)}$ and $e = 11 \text{ (e = exponent)}$
Message: $1315\quad 0611 \quad 0427 \quad 0091 \quad 0520 \quad 0733$
I am required to determine the decryption exponent and determine what it says.
This is my ... | Unfortunately, The Mathematica online help page about ExtendedGCD gives limited information about this;
\begin{align}
in[1]:&\; \{g, \{a, b\}\} = \operatorname{ExtendedGCD}[2, 3]\\
out[1]:&\; \{1,\{-1,1\}\} \quad \text{ //next, test the result}\\
in[2]: &\;2 a + 3 b == g\\
out[2]:&\; \texttt{True}\\
\end{align}
The e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3448909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A Fourier Analysis Question related to fourier transform of $C_c(\mathbb{R}^n)$ functions $f\in L^p(\mathbb{R}^n),1\leq p\leq 2$ and $g\in C_c(\mathbb{R}^n), g$ is not identically $0$, such that $f*g\equiv 0.$ Prove that $f(x)=0$ for almost every $x\in\mathbb{R}^n$.
My strategy is to use that we have $\hat{f}\cdot \hat... | The strategy would be the same as $\widehat g$ would be holomorphic on $\mathbb{C}^n$ and its zero set would be of strictly lower dimension and hence of zero measure. Have in mind that the awkward part here is talking about the Fourier transform of $f \in L^p(\mathbb{R})$ as you would have to prove the result in a dens... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3449042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Isomorphic quotient modules implies equal submodules? Let $R$ be a commutative ring with unity $M$ an $R$-module and $N,L$ submodules of $M$ with $N\subseteq L$.
$$M/N\cong M/L\implies N=L\ ?$$
| No, this is not true. Consider (the $\Bbb Z$-module) $M=\Bbb Z\times\Bbb Z\times \cdots$, and the two submodules $N=\langle(1,0,0,\ldots)\rangle$ and $L=\langle(1,0,0,\ldots), (0,1,0,0,\ldots)\rangle$. Then $M\cong M/N\cong M/L$, but $N$ and $L$ are not only unequal, they aren't even isomorphic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3449170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can I justify this without determining the determinant? I need to justify the following equation is true:
$$
\begin{vmatrix}
a_1+b_1x & a_1x+b_1 & c_1 \\
a_2+b_2x & a_2x+b_2 & c_2 \\
a_3+b_3x & a_3x+b_3 & c_3 \\
\end{vmatrix} = (1-x^2)\cdot\begin{vmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c... | For another solution, note that
$$
\underbrace{\begin{bmatrix}
a_1+b_1x & a_1x+b_1 & c_1 \\
a_2+b_2x & a_2x+b_2 & c_2 \\
a_3+b_3x & a_3x+b_3 & c_3 \\
\end{bmatrix}}_{A}
=
\underbrace{\begin{bmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3 \\
\end{bmatrix}}_{B}
\underbrace{\begin{bm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3449350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 7,
"answer_id": 0
} |
Subgroups of $\mathbb{Z}_2^n$ of order $2^{n-1}$
What are the subgroups of $\mathbb{Z}_2^n$ of size $2^{n-1}$?
I'm fairly convinced that it will be subgroups of the form $\mathbb{Z}_2 \times \dots \times \{0\} \times \dots \times \mathbb{Z}_2$, but I can't seem to know how to prove it.
edit: my guess is wrong. take $... | $\;V:=\left(\Bbb Z_2\right)^n\;$ is an $\;n\,-$ dimensional vector space over the field $\;\Bbb F_2\cong\Bbb Z_2\;$ , and there's a $1$-$1$ correspondence between the subgroups of $\;V\;$ and the subspaces of $\;V\;$ . Since a group $\;H\le V\;$ has $\;2^{n-1}\;$ elements iff $\;\dim H=n-1\;\iff\; H\;$ is a hyperplane ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3449569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
finding inverse of function in ordered pair notation $$f: \mathbb{R} \times \mathbb{R} \mapsto \mathbb{R} \times \mathbb{R} $$
where f is defined as $$f(x,y) =(\text{somethingforx},\text{somethingfory}) $$
I dont want to post the exact question because I would like to get it on my own, but I am having trouble finding t... | The inverse relation of f is
{ ( ( something for x, something for y), (x,y) ) | ((x,y) , (somethingforx, somethingfory) ) belong to f}.
In general the inverse of R is
{ (b,a) | (a,b) belong to R}
Here
a = (x,y)
and
b = ( something for x, something for y)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3449700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
supremum of brownian motion almost surely > 0 Is it true that
\begin{equation}
M_t = \sup_{s \leq t} B_s > 0 \ \ \text{a.s.}
\end{equation}
for all $t>0$? I remember reading this somewhere, but intuitively, can't the Brownian motion B stay below 0 for some time with probability $>0$?
| One way to argue is using Blumenthal's zero one law. Define $A_n=\{B_{1/n}>\frac{1}{\sqrt{n}}\}$, and set $B=\{B_{1/n}>\frac{1}{\sqrt{n}} ~\text{i.o.}\} $
Then,
\begin{align*}
\mathbb{P}(B) &=\mathbb{P}(\limsup_n A_n)\\
&\geq \limsup_n\mathbb{P}( A_n)\\
&= \limsup_n\mathbb{P}(B_{1/n}>\frac{1}{\sqrt{n}})\\
&=\limsup_n\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3449837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Solve complex equation $\left(\frac{8}{z^3}\right) - i = 0$ from $$(a^3 + b^3) = (a+b)(a^2-ab+b^2)$$
I have $${(2/z)}^3 + i^3 =0$$
I have $$\left(\frac{2}{z} + i\right)\left(\left(\frac{2}{z}\right)^2-(2/z)(i)-1)\right) = 0$$
i.e.$ \left(\frac{2}{z}\right)+i = 0 $ or $ \left(\left(\frac{2}{z}\right)^2-(\frac{2}{z})(i)-... | Since $i^{3} = -i$
Then $(\frac{2}{z})^{3} + i^{3} = 0 \iff (\frac{2}{z})^{3} - i = 0 \iff z^{3} = -2^{3}i = -8i $
Then the solutions are $z_{1} = 2i, z_{2} = 2e^{i\frac{2\pi }{3}}i, z_{3} = 2e^{i \frac{4\pi }{3}}i$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3450012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Decrypt 01 09 00 12 12 09 24 10 knowing encryption was doing with c=character $c^5\pmod{29}$ So the solution said in order to find out the original character number($A=0, B=1, C=2$, etc). That you have to find $5x\equiv1\pmod{28}$ so $x\equiv17\pmod{29}$, because $\phi(29)=28$. So the solution becomes $d$=encrypted wor... | I agree that $\phi(29)=28$, as $29$ is prime. The encryption exponent is $5$ and so the decryption exponent is its inverse modulo $28$, which indeed is $17$.
So the decryption function is $x \to x^{17} \pmod{29}$. The inverse of an exponential function is another exponential function, over such a finite ring. The inver... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3450152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Compute the boundary and interior of $\left \{ \left(t,\frac{1}{t}\right) : \frac{1}{4}
Let $\mathbb{R}^2$ be given with $|\cdot |_\infty$.Compute the boundary and interior of $A=\left \{
\left(t,\frac{1}{t}\right) : \frac{1}{4} <t <4\right \}$
I suppose that the interior of $\left \{
\left(t,\frac{1}{t}\right) : \f... | The interior is empty and the boundary (which is also the closure) is $\{(t,\frac 1 t): \frac 1 4 \leq t \leq 4\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3450308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Notation for functional derivative of two variables I have the following functional
\begin{equation}
F_{\varepsilon}\left[\rho\right]\left(t\right):=\int_{0}^{1}\left[\frac{\varepsilon}{2}\left(\frac{d\rho}{dx}\right)^{2}+\frac{1}{4\varepsilon}\left(1-\rho^{2}\right)^{2}\right]dx.
\end{equation}
where $\rho(t,x)$. Call... | This is just $\frac{\delta F_\varepsilon}{\delta\rho}$ with action $F_\varepsilon=\int_0^1Ldx$. It's the usual functional derivative because $\frac{\partial L}{\partial\dot{\rho}}=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3450549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Proving an interesting inequality with square roots Let $a,b,c>0$ be real numbers such that $c \geq a \geq b$ and $a^2 \geq bc$. Show that
$$\frac{\sqrt{a^2b+b^2c}}{a+c}+\frac{\sqrt{b^2c+c^2a}}{b+a}+\frac{\sqrt{c^2a+a^2b}}{c+b} \geq \frac{\sqrt a +\sqrt b +\sqrt c}{2}.$$
I tried to augment each term on the left using... | Even the following inequality is true for any positives $a$, $b$ and $c$:
$$\sum_{cyc}\frac{\sqrt{a^2b+b^2c}}{a+c}\geq\frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt2}.$$
Indeed, let $a=x^2$, $b=y^2$ and $c=z^2$, where $x$, $y$ and $z$ are positives.
Thus, by Holder
$$\sum_{cyc}\frac{\sqrt{a^2b+b^2c}}{a+c}=\sqrt{\frac{\left(\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3450689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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finding a constant $c$ so that $ \hat{f} (m)=0 $ Let be $ c \in (0, 2 \pi) $
and
$$f_c(x):= \begin{cases} \frac{x}{c} , 0 \leq x \leq c \\ \frac{2 \pi -x}{2 \pi -c}, c < x \leq 2 \pi \end{cases} $$
I want to determine $c$,
so that
$\hat{f} (m)=0 $
for $ m \in 7 \mathbb{Z} \backslash \{ 0\} $
it is
$ \hat{f}(m)= \fr... | We have
\begin{align}
\hat{f}(m) &= \frac{1}{2\pi}\int_0^c \frac{x}{c}\mathrm{e}^{-\mathrm{i}mx} dx
+ \frac{1}{2\pi}\int_c^{2\pi} \frac{2\pi - x}{2\pi - c}\mathrm{e}^{-\mathrm{i}mx} dx\\
&= \frac{1}{2\pi (2\pi - c) m^2}\left(1 - \mathrm{e}^{-\mathrm{i}2m\pi}\right)
- \frac{1}{(2\pi - c) cm^2}\left(1 - \mathrm{e}^{-\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3450871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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A class of sequences is of bounded variation Let $\left\{ a_n \right\}$ be a null sequence s.t.
$$\sum_{n=1}^\infty \left( \frac{1}{n}\sum_{k=n}^\infty \vert\Delta a_k\vert^p\right)^\frac{1}{p}<\infty$$
for some $p>1$.
How to prove that $\left\{ a_n \right\}$ must be of bounded variation?
My attempt: It is enough to p... | The argument is:
We wish to prove $n|\Delta a_n|^p \leq \sum_{k=n}^{\infty} |\Delta a_k|^p$.
Now suppose that the right hand side does not contain at least $n$ terms of size $|\Delta a_n|^p$. Then surely $|\Delta a_k|^p \leq |\Delta a_n|^p$ eventually (i.e. of BV). If it does contain at least $n$ terms of this size, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3450978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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A lower bound for $\sum\limits_\text{cyc} \frac{x}{\sqrt{x^2+y^2}}$ Let $x,y,z>0$. Then
$$\sum_\text{cyc} \frac{x}{\sqrt{x^2+y^2}}>1$$
I found a similar inequality in the other direction but I can‘t apply Cauchy-Schwarz here... All I see is by Cauchy-Schwarz,
$$\sum_\text{cyc} \frac{x}{\sqrt{x^2+y^2}}\geq \frac{\sum_\t... | Note that $$\sum_{\rm cyc} \frac x{\sqrt{x^2+y^2}}>\sum_{\rm cyc} \frac x{x+y}$$ $$>\sum_{\rm cyc}\frac x{x+y+z}=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3451106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Does locally smooth imply globally smooth for a function on a manifold? Say we have a function $f$ from a differentiable manifold $M$ to $\mathbb{R}$ such that for all points $m \in M$ there exists a neighborhood $U \ni m$ such that $f | U$ is smooth. Can we conclude that $f$ is smooth?
I'm trying to figure this out gi... | Yes, like on $\mathbb R^n$, smoothness is a local property. A function is (globally) smooth iff its restriction to any open set is a smooth function.
To prove it strictly using your definition, you should let your domain $U$ be $M$, and let $\phi:V\to \tilde V$ be a coordinate map on $M$. Then play around with your loc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3451261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How to prove that $2\sqrt{3}$ is greater than $\pi$ Without calculator, how to prove that $2 \sqrt{3} > \pi$?
The level is baccalauréat grade.
I confirm it's not a school exercise at all, as I left school like 35 years ago.
|
(Credit to David G. Stork for the image).
I am showing the area-based argument explicitly because it seems that the other answers rely on a perimeter-based argument (which I find unconvincing without a rigorous proof). In contrast, it is quite easy to conclude by simple inspection that the circumscribed hexagon has a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3451431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 0
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If $\lim_{x\rightarrow +\infty} \frac{f(x)}{x}=0,$ show there is $x_n\rightarrow\infty$ such that $\lim_{n\rightarrow\infty}f'(x_n)=0.$ I'm working on the problem:
Suppose $f(x)$ is differentiable on $(0,+\infty)$.
If $$\lim_{x\rightarrow +\infty} \frac{f(x)}{x}=0,$$ show there is
$x_n\rightarrow\infty$ such that $$\l... | If the conclusion is not true then there exists $\epsilon >0$ and $M$ such that $|f'(x)| >\epsilon$ for all $x \geq M$. Using the fact that derivatives have IVP we see that we can actually make $f'(x) >\epsilon$ for all $x \geq M$ or $f'(x)<-\epsilon$ for all $x \geq M$. Consider the former case. Note that $f(n+1)-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3451553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Probability measure over bijection map Let $P$ a probability measure over $\mathbb{R}^n$. Let $f: A \rightarrow B$ a bijection from $A$ to $B$ two sets of $\mathbb{R}^n$. Does the following equality could be hold?
$$
P(f(A)) = P(A).
$$
Thanks,
S
| No. We have $f(A)=B$, and since $B$ can have a different measure than $A$ the equality does not hold always.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3451670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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uniform convergent Given f is a differentiable function. Define $$f_n(x)=n\left(f\left(x+\frac{1}{n}\right)-f(x)\right)$$, prove that $f_n$ is uniformly converge to $f'$
I have tried to make these equations:
$n(f(x+\frac{1}{n})-f(x))=\frac{f(x+\frac{1}{n})-f(x)}{1/n}$ and taking limits as n $\to\infty$ but i got stuck ... | This is true only when $f'$ is uniformly continuous.(Use MVT)
For,
By MVT, there exists $\varepsilon_n \in (x,x+1/n)$ such that $$\Big|\frac{f(x+1/n)-f(x)}{1/n}-f'(x)\Big|=|f'(\varepsilon_n)-f'(x)|$$
How to make the RHS small in order to make uniform convergent?
Ans: Using uniform continuity of $f'$
Otherwise, consid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3451815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
If the operator $T$ is defined by $Tf(x)=\int_0^xf(t)\,dt$, show that $Tf \in C[0,1]$
Consider the operator $T$ on $L^2[0,1]$ defined by $Tf(x)=\displaystyle \int_0^xf(t)\,dt.$ Show that $Tf \in C[0,1].$
I have one question before this:
What are the implications between $L^p$ spaces, i.e if $f \in L^p$ does this imp... | Your first question is actually relevant here: the relevant inclusion is that a $L^p$ function on a finite measure space is in $L^r$ for all $r<p$. Thus a $L^2([0,1])$ function is also in $L^1([0,1])$. This is the main ingredient you need, along with the theorem that if $f \in L^1$ then for all $\varepsilon > 0$ there ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3451968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
For simple, connected graph $G$ with minimum degree $\geq k$, if $k\geq 3$, does $G$ always have a cycle of length exactly $k+1$?
Let $G$ be a simple, connected graph such that $\delta(G)\geq k$ (where $\delta(G)$ is the minimum degree). If $k$ is at least $3$, does $G$ always have a cycle of length exactly $k+1$?
P.... | The Petersen graph is another example. It is 3-regular and has no cycle of length less than 5.
An easy construction for even $k$ is $K_{k,k}$. Every vertex has degree $k$ but since the graph is bipartite, there are no odd cycles in the graph (i.e. no cycle with length $k+1$).
For odd $n$, any $2$-connected graph satis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3452094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Homogeneous principal G-set and its group of automorphisms Let $G$ be a group and consider the left operation of $G$ on itself by left translation; this action is simply transitive, i.e., $G$ operates freely and transitively on itself. Then $G$ together with this operation is a left homogeneous principal $G$-set; denot... | Note that $ G^{op}\cong Aut(G_s) $ via $x\mapsto \delta_x$. Indeed, if $f: G_s\to G_s$ is an isomorphism, then for all $g$, $f(g) = f(g\cdot 1) = g\cdot f(1) = gf(1) = \delta_{f(1)}(g)$, so $f\mapsto f(1)$ is the inverse isomorphism.
Now this map $G^{op}\to Aut(E)$ is just the composition of $G^{op}\to Aut(G_s ) \to A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3452248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve a system of linear inequalities? I am working on the following exercise:
Find a solution to the following system or prove that none exists:
\begin{align}
x_1-x_2 &\le 4\\
x_1-x_5 &\le 2\\
x_2-x_4 &\le -6 \\
x_3-x_2 &\le 1 \\
x_4-x_1 &\le 3 \\
x_4-x_3 &\le 5\\
x_4-x_5 &\le 10 \\
x_4-x_3 &\le -4 \\
x_5-x_4 ... | From the first inequality you have $$x_1 \leq 4+x_2.$$
Now from the second $$4+x_2-x_5 \leq 2$$ or $$ x_2-x_5 \leq -2$$ or $$x_2 \leq -2+x_5.$$
Now from the third you get $$ -2+x_5-x_4\leq -6$$ or $$x_5\leq -4+x_4.$$ This can be applied to the last inequality and you get $$-4+x_4-x_4\leq -8$$ and that gives $$-4 \leq -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3452354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\lim_{n \to \infty} k^nn^p= 0$ where $|k| < 1$ and $p>0$ Hello I am working through some problems in a book and came across this question.
Prove that $\lim_{n \to \infty} k^nn^p= 0$ where $|k| < 1$ and $p>0$
I can see why it should be the case (since exponentials grow faster than polynomials) but I don't r... | It suffices to show
\begin{align*}
(|k|^{1/p})^{n}n\rightarrow 0.
\end{align*}
Let $a=|k|^{1/p}<1$, we are to show that $a^{n}n\rightarrow 0$.
Let $a=1/(1+r)$ for $r>0$, then $a^{n}\leq\dfrac{1}{1+nr+n(n-1)r^{2}/2}$, it is now easy to show that
\begin{align*}
\dfrac{n}{1+nr+n(n-1)r^{2}/2}\rightarrow 0.
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3452517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many solutions to this system of linear equations in $\mathbb{Z}_N$? Given distinct non-negative integers $i$ and $j$, and given $a,b\in\mathbb{Z}_N$, is it true that there is at most one $(x,y)\in \mathbb{Z}_N^2$ so that
$$x+iy\equiv a \mod N,$$
$$x+jy\equiv b \mod N?$$
When it is in $\mathbb{Z}\subseteq\mathbb{Q}... | If $j \equiv i$ mod $N$ there may be zero or infinitely many solutions, depending on $a,b.$ But if not we can subtract the two equations and get $(j-i)y \equiv b-a$ mod $N.$ Then provided $j-i$ is invertible mod $N$ we might have a unique solution (again depending on $a,b$ mod $N.$)
Edit: When $j-i$ is invertible we ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3452666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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maximize product of two sines with given precision and for smallest time parameter
Consider the function $f(t)=\text{sin}(\omega_1 t)\text{sin}(\omega_2 t)$, where $\omega_1 \ \text{and}\ \omega_2 \in \mathbb{R} $.
Is there a numerical or analytical solution to the following optimization problem:
find minimal $t$ s... | Although it is not a complete answer, my answer provides some partial solution to your problem (existence).
Depending on $w_1$ and $w_2$, the minimizer may not exist.
Let us first consider the maximum of $f(t) = \sin(w_1t)\sin(w_2t)$.
Since, at the maximum, $\sin(w_1t) = \sin(w_2t)$, without loss of generality, suppost... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3452783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Tap filling Tank .. Time taken?
A tap can fill a tank in 16 hours whereas another
tap can empty the tank it in 8 hours. If in a three
fourth filled tank both the taps are opened, then
how long will it take to empty the tank in this
scenario?
I know that time to fill+empty = (1/16)+(1/8)
How to incorporate thr... | If the tank is full and you open both taps, then it will take $16$ hours to empty it: indeed in first $8$ hours the second tap empties the tank and in the second $8$ hours it empties one more tank that filled the first tap in these $16$ hours. If the tank was $3/4$ in the beginning, it would take $(3/4)\times16=12$ hou... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Definition of $C^k$ boundaries I am reading the book "Partial Differential Equations" of Lawrance c. Evans by myself and started with Appendix part.
At the very beginning of Appendix C, there exists a definition
"We say $\partial U$ is $C^k$ if for each point $x^0\in\partial U$, there exists $r>0$ and a $C^k$ function ... | The condition $$x_n > \gamma(x_1, \ldots, x_n)$$ says that, locally, the boundary can be written as the graph of a function -- it means that the boundary itself is locally the set were $$x_n = \gamma(x_1, \ldots, x_n)$$
This is equivalent to saying it is an $n-1$ dimensional submanifold or a hypersurface -- maybe you ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Name of partial derivatives where the order of differentiation can be reversed. Is there a name given to partial differential equations of the form: $$\frac{\partial{F}}{\partial{x}\partial{y}}=\frac{\partial{F}}{\partial{y}\partial{x}}$$ Not asking for any kind of proof, just specifically wondering if there is a name ... | I suspect that the equality
$$\frac{\partial{F}}{\partial{x}\partial{y}}=\frac{\partial{F}}{\partial{y}\partial{x}},$$
interpreted as a pde, have no special name because it is not commonly interpreted as a pde.
On the other hand, understood as "a condition on" or "a property of" a certain function $F$, the said equalit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solve the equation exponential radical Solve the equation$$31+8\sqrt{15}=(4+\sqrt{15})^x$$for $x$.
I think you could set up a recursion from the coefficients of $(4+\sqrt{15})^{x}$ to the coefficients of $(4+\sqrt{15})^{x+1}$ then find the general formula using the characteristic equation?
It should look like $a_{n+1}=... | If $x = \log_{4+\sqrt {15}} (31+8\sqrt {15})$ is not an acceptable solution (and they should specify that it is not; it satisfies all the requirements of a solution; It exists and it is a unique value and it solves) then I'm not really sure there is anything to do but guess.
We can note for any positive integer $k$ tha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there any finite dimensional algebra that is not isomorphic to some algebra of matrices? Suppose that $\mathcal A$ is a finite dimensional algebra. Is it true that there always exist some isomorphism
\begin{equation}
\phi:\mathcal{A}\rightarrow C\subset M_{K\times K},
\end{equation}
where $C$ is some subalgebra of t... | In general, the answer is no. The octonions are not isomorphic to any matrix algebra because they're not associative.
If the algebra is associative then the answer is yes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3453519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Hom on sequences of integers is determined by values on finite sequences I have found this problem intended for school students (math olympiad) and couldn't solve it myself.
Consider the short exact sequence of abelian groups
$0 \to\bigoplus \limits_{i \in \mathbb{N}} \mathbb{Z}_i \to \prod\limits_{k \in \mathbb{N}} \m... | I know the proof now but I haven't solved it myself.
Note that for a prime $p$ a sequence of the form $(p^na_n)$ is divisible in $X$ by $p^k$ for all $k \in \mathbb{N}$ because all but finitely many of $p^na_n$ are. Therefore for any $f \in Hom(X, \mathbb{Z}), f(p^na_n) $ is divisible by $p^k$ for all $k$ so $f(p^n... | {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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Translation of a union of sets equals the union of a translation of sets. I'm wondering whether or not this is true for any arbitrary set of real numbers. It seems pretty straightforward to me.
Sorry if this is overly pedantic, I've just been scarred by enough measure theory this semester that I've learned everything ... | For any function $f: \mathbb R \to \mathbb R$ and any collection $(S_{\alpha})_{\alpha \in I}$ we have $f(\bigcup S_{\alpha})_{\alpha \in I})=\bigcup_{\alpha \in I} f(S_{\alpha})$. In particular for the function $f(y)=x+y$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Sum of the series $\sum_{n=0}^{\infty} \lfloor n\sqrt{2} \rfloor x^n$? Can we find the sum of the series $\sum_{n=0}^{\infty} \lfloor n\sqrt{2} \rfloor x^n$ explicitly? The question is related to this one where sum is computed if $\sqrt{2}$ is replaced by some rational number $r$.
| This isn't a solution but it does yield a representation that has a nice approximation built into it. Note the series representation
$$\left\lfloor x \right\rfloor =x-\frac{1}{2}+\frac{1}{\pi }\sum\limits_{k=1}^{\infty }{\frac{\sin \left( 2\pi kx \right)}{k}}$$
And so
$${{B}_{1}}\left( \left\{ x \right\} \right)=\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3453954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Probability distribution between two unit vectors using the taxicab metric for distance Suppose we have two positive, real unit vectors $X$ and $Y$ in $\mathbb{R}^n$.
EDIT: As was suggested in a comment, let me describe how $X$ and $Y$ are randomly generated. For both vectors, pick $n$ random values from the uniform di... | As $n \to \infty$, the histogram will become more and more concentrated at $\frac{2}{3}$. For example, $Pr(|d(X,Y)-\frac{2}{3}| > \epsilon)$ goes to $0$ as $n \to \infty$ for any fixed $\epsilon > 0$. By analyzing the proof below, you could say something stronger (e.g. $Pr(|d(X,Y)-\frac{2}{3}| > \frac{1}{\log N})$ goes... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Monotonicity of a partial binomial sum I'm wondering when does the following partial sum monotonically increase or decrease in $n$:
$$f_x(n)=\sum^{\lfloor\frac{n}{2}\rfloor}_{k=0}{n \choose k}x^k(1-x)^{n-k}.$$
In theory, if $x<\frac{1}{2}$, the probability converges to 1. On the other hand, if $x>\frac{1}{2}$, it conve... | After I saw the answer from joriki, I only focused on odd $n$ cases and could show the monotonicity. I'll leave my reasoning here for a future reference. Thanks for your comment, joriki.
Let $n=2m+1$, $m\geq 0$. I will show the sign of $f_p(2m+3)-f_p(2m+1)$ depends on whether $p>\frac{1}{2}$ or $p<\frac{1}{2}$.
Let $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3454283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that no 2 orthogonal matrices satisfy this equation $A^2-B^2=AB$ This question came up in my linear algebra finals, and I couldn't prove it. Could anyone help me?
Question: Prove that no $2$ orthogonal matrices satisfy this condition: $A^2-B^2=AB$
Attempt: Assume $A$ and $B$ are orthogonal. Mutiplying $A^T, B^... | Consider the traces of both sides of $AB^T-A^TB=I$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove the distributive law for maximum and minimum operation Given $A=\{1,2,3,4\}\subseteq \mathbb{N}$ and define the operation on $A$ as below
\begin{eqnarray}
a\oplus b&=& \max(a,b)\\
a\otimes b&=& \min(a,b)
\end{eqnarray}
for all $a,b\in A$.
Prove $(a\oplus b)\otimes c=(a\otimes b)\oplus(a\otimes c)$ for all $a,b,c\... | You have stated the distributive law incorrectly. You want $$(a\oplus b)\otimes c=(a\otimes c)\oplus(b\otimes c)\quad(*)$$
The easiest approach is to look at 6 cases: $a>b>c,a>c>b,b>a>c,b>c>a,c>a>b,c>b>a$.
If you look at $a>b>c$, then $(a\oplus b)\otimes c=a\otimes c=c$. But $(a\otimes b)\oplus(a\otimes c)=b\oplus c=b$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A computation of $Ext^1(M,N)$ (how to derive the commutative diagram for $Ext^1(M,N)$?). I am trying to understand the proof of Proposition 5.6 in the paper on page 17. How to derive the commutative diagram for $Ext^1(M,N)$? Using the projective resolution
:
\begin{align}
\cdots \to \oplus_{v \in V} P_v \overset{D}{\to... | By laziness I'll oversimplify the notation:
$$
M \xrightarrow{D} N \rightarrow L_I \rightarrow 0
$$
Now break this sequence in two pieces, noting that $\Omega L_I = imD$,
$$
M \xrightarrow{f} \Omega L_I \rightarrow 0, \\ 0 \rightarrow \Omega L_I \xrightarrow{g} N \rightarrow L_I \rightarrow 0
$$
We apply $Hom(-, L_J)$ ... | {
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Sequence: $u_n=\sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}}$
Study the following sequence of numbers:
$$\forall n\in\mathbb{N}, u_n=\sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}}$$
I tried to calculate $u_{n+1}-u_n$, but I couldn't simplify the expression.
Plotting the sequence shows arithmetic (or seems to be an) progression.... | Using the Euler-Maclaurin formula we can get more detailed asymptotics, e.g.:
$$ u_n = \left( \sqrt {5}-\sqrt {2} \right) n
+ \frac{\sqrt{5}}{5} + \frac{\sqrt{2}}{4}
+\left(\frac{\sqrt{5}}{300}- \frac{\sqrt{2}}{48} \right) n^{-1} +
\left(-\frac{\sqrt{5}}{10000} + \frac{\sqrt{2}}{1280} \right) n^{-3}
+O \left( {n}^{
-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3454810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Relation between areas of two non-similar triangles with individual proportional sides Triangle $ABC$ has area $k$ and $D$ is the middle point of $BC$.
We have $AP = 2 \cdot AB$, $AQ = 3 \cdot AD$ and $AR = 4 \cdot AC$. What's the area of triangle $PQR$?
I know that the answer is $k$, but I don't know how to prove it. ... |
Let $\angle PAQ = \alpha$ and $\angle RAQ = \beta$.
$$A_{PQR} = A_{APQ} + A_{AQR}- A_{APR}$$
$$=\frac12 AP\cdot AQ\sin\alpha + \frac12 AR\cdot AQ\sin\beta - \frac12 AP\cdot AR\sin(\alpha+\beta)$$
$$=\frac12 ( 2AB \cdot 3AD\sin\alpha )+ \frac12 ( 3AD \cdot 4AC\sin\beta)
- \frac12 ( 2AB \cdot4AC\sin(\alpha+\beta))$$
$$=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What's wrong in my calculation of $\lim\limits_{n \to \infty} \sum\limits_{k=1}^n \arcsin \frac{k}{n^2}$ I have the following limit to find:
$$\lim\limits_{n \to \infty} \sum\limits_{k=1}^n \arcsin \dfrac{k}{n^2}$$
This is what I did:
$$\lim\limits_{n \to \infty} \sum\limits_{k=1}^n \arcsin \dfrac{k}{n^2} = \lim\limits... | As noted by others, there are infinite many summands, one cannot simply distribute the limit operator to them.
The following might be over-killed, but I think it is somehow interesting:
We know that
\begin{align*}
\lim_{x\rightarrow 0}\dfrac{\sin^{-1}x}{x}=1,
\end{align*}
given $\epsilon\in(0,1)$, there is an $N$ such... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $n^n>\left(\dfrac{n+1}{2}\right)^{n+1}$ for all positive integer $n>1$. Question: Prove that $n^n>\left(\dfrac{n+1}{2}\right)^{n+1}$ for all positive integer $n>1$.
I could not understand what should be the initial approach.
| We want to show the $2$s at the bottom are enough to defeat the $+1$s at the top. The effect of the $2$s is easy. Just write
$$ \left(\dfrac{n+1}{2}\right)^{n+1} = \frac{1}{2^{n+1}}(n+1)^{n+1}$$
We'd like a bound like $(n+1)^{n+1} < Cn^n$ for some $C < 2^{n+1}$.
$$(n+1)^{n+1}= (n+1)(n+1)^{n}$$
To bound $(n+1)^n$ use... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3455262",
"timestamp": "2023-03-29T00:00:00",
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What is the period of the $f(x)=\sin x +\sin3x$?
What is the period of the $f(x)=\sin x +\sin3x$?
$f(x)=\sin x+\sin 3x=2\frac{3x+x}{2}\cos\frac{x-3x}{2}=2\sin2x\cos x=4\sin x\cos^2x\\f(x+T)=4\sin(x+T)\cos^2(x+T)=4\sin x\cos^2 x$
how can I deduct this I have no idea
| In general, if $T$ is the period of a function $f(x)$ then the period of the function $f(ax)$ is $\frac{T}{a}$.
Suppose two periodic functions $f_1(x)$ and $f_2(x)$ have periods $T_1$ and $T_2$. Then the period of the function $g(x)=f_1(x)\pm f_2(x)$ is LCM (least common multiple) of $T_1$ and $T_2$ (although, this ce... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is Euler's Formula for Planar Graph Not Working Here?
I have worked out $r(n) = 2^n$, $e(n) = 1 + 3 \times 2^n$, $v(n) = 2\times(2^n - 1) + 4$
The expressions of $r(n)$, $e(n)$, and $v(n)$ are correct and this can be verified with $n = 0, 1, 2, 3\ldots$
But when I calculate $v(n) - e(n) + r(n)$, it does not equal... |
when I calculate $v(n)−e(n)+r(n)$, it does not equal to $2$. What's wrong?
See Euler's formula for planar graphs :
if a finite, connected, planar graph is drawn in the plane without any edge intersections, and $v$ is the number of vertices, $e$ is the number of edges and $f$ is the number of faces (regions bounded b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What are the last two digits of 1^5 + 2^5 + 3^5 + ... +99^5? What are the last two digits of 1^5 + 2^5 + 3^5 + ... +99^5?
My work:
1^5 ends with 1.
2^5 ends with 2.
3^5 ends with 3.
And so on.
Do I simply add the ending digits to get my answer?
| $1^5+99^5=(1+99)(1-99+99^2-99^3+99^4)=100(\text{positive number})$ what can you see??
Also
$2^5+98^5=(100)(2^4-2^3(98)+2^2(98)^2-2(98)^3+98^4)$
My solution is a special case of lab's
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3455707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Intersection of two primary ideals in $\mathbb{Z}[x]$.
Consider $\mathbb{Z}[x]$, and define $I = (x(x^{2}-2),(x^{2}-2)(x^{2}+2))$, $J = (x^{2}-2)\cap(x^{3},2)$. I want to show that $I = J$.
Notice that $I\subset J$ is clear since the generators of $I$ are clearly in $J$. The other direction is less clear to me. It c... | To show the inclusion $J \subset I$ , we proceed as follows. Let $f(x) \in J$ such that $f(x) = \alpha (x^2-2)$, and $f(x) = \beta x^3 + 2 \gamma$, for $\alpha, \beta, \gamma \in \mathbb{Z}[x]$. As you suggested, we want to show that $f(x) \equiv 0 + I$. Observe that $\beta x^3 +2 \gamma + I = 2x \beta + 2 \gamma + I =... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Given a space $X$, construct a CW complex $L(X)$ s.t. they have the same fundamental group This is exercise 1.2.15 in Hatcher's Algebraic topology
Given a space $X$ with basepoint $x_0∈X$, we may construct a CW complex $L(X)$ having a single $0$-cell, a $1$-cell $e^1_γ$ for each loop $γ$ in $X$ based at $x_0$, and a $... | $L(X)$ is attempting to "model" $X$ as a CW complex with the same fundamental group. To do this, we will consider all possible loops in $X$, then specify which ones should be considered homotopic.
*
*As you note, we have a wedge sum of circles, one for each loop in $X$. You can think of each summand as "representin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3456011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the Basis of the Kernel and the Image $R^4 \to R^3$ where $ f(x,y,z,w) = \left[\begin{array}{ccc}2x + z -w\\x +w\\x +z-2w\end{array}\right]$
I started with matrix $ A = $$\left[\begin{array}{ccc}2 & 0 & 1 & -1\\1 & 0 & 0 & 1\\1 & 0 & 1 &-2\end{array}\right]$
and set it to 0 and the result $\left[\begin{array}{... | You are right indeed we can check that
$$\left[\begin{array}{ccc}2 & 0 & 1 & -1\\1 & 0 & 0 & 1\\1 & 0 & 1 &-2\end{array}\right]\left[\begin{array}{c}0\\1\\0\\0\end{array}\right]=\vec 0$$
$$\left[\begin{array}{ccc}2 & 0 & 1 & -1\\1 & 0 & 0 & 1\\1 & 0 & 1 &-2\end{array}\right]\left[\begin{array}{c}-1\\0\\3\\1\end{array}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3456161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finite abelian groups as direct products of proper characteristic subgroups Suppose that $A$ is a finite abelian group. To make things more interesting, assume further that $A$ is an abelian $p$-group for some prime $p$.
Is it ever the case that $A$ can be written as $A = C \times D$, where $C$, $D$ are proper charact... | No. Both $C$ and $D$ are direct sums of cyclic $p$-groups,
$$
\begin{align*}
C &\cong C_{p^{a_1}}\oplus\cdots\oplus C_{p^{a_r}},\quad a_1\leq\cdots\leq a_r\\
D &\cong C_{p^{b_1}}\oplus\cdots \oplus C_{p^{b_t}},\quad b_1\leq\cdots\leq b_t.
\end{align*}
$$
If $a_i=b_j$ for any $i$ and $j$, then you have an automorphism o... | {
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"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Prove by mathematical induction that $(3n+1)7^n -1$ is divisible by $9$ for integral $n>0$ $7^n(3n+1)-1=9m$
$S_k = 7^k(3k+1)-1=9P$
$\Rightarrow 7^k(3k+1) = 9P+1$
$S_{k+1} = 7\cdot7^k(3(k+1)+1)-1$
$= 7\cdot7^k(3k+1+3)-1$
$= 7\cdot7^k(3k+1) +21\cdot7^k -1$
$= 7(9P+1)+21\cdot7^k -1$
$= 63P+7+21\cdot7^k -1$
$= 63P+6+21\cdo... | You're almost finished.
You assume that
$S_k = 7^k(3k+1)-1$ is divisible by 9, and therefore divisible by 3.
$S_k = 7^k(3k)+ 7^k-1$ is divisible by 3.
Therefore $7^k-1$ is divisible by 3.
let $3x = 7^k-1$.
$7^k=3x+1$
Then in your final steps:
$=63P+6+21\cdot7^k$
$=63P+6+21(3x+1)$
$=63P+63x+27$
$=9(7P+7x+3)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3456485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Suppose $ker f = ker g$, show that $\exists \alpha \in \mathbb{C}$ such that $f(p)=\alpha g(p)$, $\forall p \in \mathcal{P}_7 (\mathbb{C})$ Suppose $f$, $g$: $\mathcal{P}_7 (\mathbb{C})\to \mathbb{C}$ and ker $f$ = ker $g$, show that $\exists \alpha \in \mathbb{C}$ such that $f(p)=\alpha g(p)$, $\forall p \in \mathcal{... | If $\mathcal{P}_7(\mathbb{C}) = \ker f \oplus \operatorname{span} \{p\}$ then every $q \in \mathcal{P}_7(\mathbb{C})$ can be uniquely written as $q = r + \lambda p$ where $r \in \ker f$ and $\lambda \in \mathbb{C}$.
Since $r \in \ker f = \ker g$ we have $f(r) = g(r) = 0$ so
$$f(q) = f(r+\lambda p) = f(r) + \lambda f(p)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3456766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A closed smooth manifold cannot admit free involutions if it does not bound. I have seen the following statement: If a closed smooth manifold does not bound, then it cannot admit fixed point free involutions. Here a manifold $M$ bound means there exists a compact manifold $W^{n+1}$ such that $\partial W=M$. I am consid... | Let $M$ be a closed $n$-dimensional manifold and $\tau: M\to M$ a fixed-point free involution. My argument works in any of the standard categories: topological, PL or smooth. Since you are asking about differentiable manifolds, I will work in the smooth category.
Define the quotient manifold $N=M/\tau$. Let $W\to N$ d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3456909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $\frac f g$ is symmetric then what conclusion can we make about $f$ and $g$? Let $K$ be a field (or a commutative ring with identity). Consider $f,g \in K[X_1,X_2, \cdots , X_n]$ with $g \neq 0.$ Suppose that $\frac f g \in K \left (X_1,X_2, \cdots , X_n \right)$ is symmetric i.e. $\frac f g \in \text {Fix}_{S_n} K\... | I have found the answer. It's quite easy. Take for instance $f=X_1^2X_2 \in K[X_1,X_2]$ and $g=X_1 \in K[X_1,X_2].$ Neither $f$ nor $g$ is symmetric. But $\frac f g = X_1X_2 \in K [X_1,X_2] \subseteq K \left (X_1,X_2 \right )$ is indeed symmetric.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3457102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a good upper ground to the number of digits of a factorial n! when I only know the number of digits of $n$? I was coding a function for calculating the factorial of a big number in $C$. Since I'm using a structure where I don´t know directly the value of the number, I need to find the number of digits of the r... | Given a natural number $n$, the number of decimal digits is equal to $d(n)=\lfloor \log_{10}(n)\rfloor +1$. If you only know the value of $d(n)$, then $n$ could be as large as the number consisting of $d(n)$ nines $999...9$, or $10^{d(n)}-1$. This means that the best upper bound we can get on the number of digits of $n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3457397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Graph Theory Proof Problem(Bipartites graphs) I'm struggling to proof something the Professor gave us. I asked several friends and no one knows what to do. Basically is this:
Prove for every simple bipartite Graph with $n \ge 1$ vertex: $$\delta(G)+\Delta(G)\le n.$$
Professor's Tip: Split you answer in two. First pro... | I don't know how to prove the tip, but the theorem is trivial. Suppose the parts are $V_1$ and $V_2$, that $|V_1|=n_1$ and $|V_2|=n_2$, where $n_1+n_2=n$. We may assume that $n_1\geq n_2$.
Every vertex in $V_1$ has degree at most $n_2$, and every vertex in $V_2$ has degree at most $n_1.$ Thus $\Delta(G)\leq n_1$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3457498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What is an example of XOR? I have a doubt regarding the below lines in "deep learning" book.
I don't have a very good math background, I grasp most of the concepts with examples. First they describe this:
Which I understand it as ie $x_1=$ house_size , $x_2=$year_built and $f(x,w)$ shall be the house price.
But I c... | I don't like going to the movies alone. This week, there are 2 tickets for a movie I'd like to see, so I ask 2 friends A and B if they'd like to see it.
If neither A nor B wants to see it, then I wont go.
If A wants to go but B doesn't, or visa versa, then I will go with whoever wants to join me.
If both A and B want t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3457603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
What is the minimum requirement on $f$ such that $\lim_{s\rightarrow 0}\int_0^{\infty}\exp(-st)f(t)dt = \int_0^{\infty}f(t)dt$? What is the minimum requirement on $f$ such that $\lim_{s\rightarrow 0}\int_0^{\infty}\exp(-st)f(t)dt = \int_0^{\infty}f(t)dt$?
I think if I take $f$ to be measurable and $f\in L^1(0, \infty)$... | If $$F(0)=\lim_{T \to \infty} \int_0^T f(t)dt$$ converges then $$ F(s) = \lim_{T \to \infty} \int_0^T f(t)e^{-st}dt=\int_0^\infty (\int_0^T f(t)dt) s e^{-sT}dT$$ converges and is analytic for $\Re(s) > 0$ and
$$\frac{F(s)}{s} = \int_0^\infty (F(0)+o(1)) e^{-sT}dT= \frac{F(0)}{s}+o(\frac{1}{\Re(s)})$$
Thus for $r\in [... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3457773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$2$-connected Eulerian graph that is not Hamiltonian An exercise in Chartand and Zhang asks to find a $2$-connected graph (that is, connected with order at least $3$ and no cut-vertices) that is Eulerian but not Hamiltonian (or prove none exists).
I was wondering whether the graph $K_{2, 4}$ works. I think it does. I h... | Yes, that is a perfectly fine example. It is obvious that a cycle in a bipartite graph must have the vertices alternate between the two parts. Therefore, an unbalanced bipartite graph can never be Hamiltonian.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3457960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Induced image of the fundamental group of a covering space I have been reading Hatcher´s Algebraic Topology, and he wants to prove that if we have a covering space $(E,p)$, with $p(e)=x_0$ then $p_*(\pi_1(E,e)$) consists of the homotopy classes of loops in $X$ starting at $x_0$ such that their lifts are loops in $E$ st... | This is basically just by definition.
Let $\gamma$ be a loop on $x_0$ that represents $p_*([\vartheta])\,\in\pi_1(X,x_0) $ with $[\vartheta] \in\pi_1(E,e)$.
This means $[\gamma] =p_*([\vartheta]) =[p\circ\vartheta]$, that is, $\gamma$ is homotopic to $p\circ\vartheta$, which lifted to $e$ obviously gives $\vartheta$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3458106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculate areas of $0 \leq x \leq \sqrt{y}, 0 \leq y \leq \sqrt{x}, x+y \leq 3 / 4$ I have to calculate the areas of this set.
Could someone explain me how do I have to interprete the following:
\begin{equation}
\left\{(x, y) \in \mathbb{R}^{2}: 0 \leq x \leq \sqrt{y}, 0 \leq y \leq \sqrt{x}, x+y \leq 3 / 4\right\}
\en... | At first i would calculate the intersection points of the curves
$$x=\sqrt{y},y=\sqrt{x},y=\frac{3}{4}-x$$
I have got
$$\frac{A}{2}=\int_{0}^{\frac{3}{4}}x-x^2dx+\int_\frac{3}{8}^{1/2}x-\frac{3}{4}-x^2dx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3458291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Show a line in the form $ax+by=c$? I'm having trouble with a question which states:
Point $M$ has coordinates $(3,5)$. Points $A$ and $B$ lie on the coordinate axes and have coordinates $(0,p)$ and $(q,0)$ so that $AMB$ is a right angle. Show that $5p+3q=34$.
Edit - solved:
I used the gradient $\frac{p-5}{-3}$ $*$ $\f... | We can proceed as follows
*
*the slope A-M is $$\frac{y_A-y_M}{x_A-x_M}=\frac{p-5}{-3}$$
*the slope B-M is $$\frac{y_B-y_M}{x_B-x_M}=\frac{-5}{q-3}$$
and we need
$$\frac{p-5}{-3}\cdot \frac{-5}{q-3}=-1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3458393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $W$ be a vector field of constant length and let $v$ be a vector. Show that the covariant derivative, $\nabla_vW$ and $W$ are orthogonal I know that the covariant derivative of $W$
with respect to $v$ is the tangent vector
$\nabla_vW=W(p+tv)′(0)$
at the point $p$
We want to show that the dot product of $W$ and
$\... | You need Koszul's formula. It tells you that:
$\langle\nabla_XY,Z\rangle = \frac{1}{2}(X\langle Y,Z\rangle+Y\langle X,Z\rangle-Z\langle X,Y\rangle+\langle[X,Y],Z\rangle-\langle[X,Z],Y\rangle-\langle[Y,Z],X\rangle)$
Then we substitute $X=V$, $Y=Z=W$ and we use the fact that $ V\langle W,W\rangle=0$ since $W$ has a cons... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3458534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find the equation of the line $ r $ Find the equation of the line $ r $ that goes through $ (1, -2,3) $, concurrent with the line
$$\begin{cases}
x=2+3t \\
y=1+2t \\
z= -1 \\
\end{cases}$$
and has orthogonal director vector $ (1, -3,1) $
Solution: line r is contained in the plane $(x-1)-3(y+2)+(z-3)=0$
$x-3y+z=10$
Next... | Use the information that you’ve been given directly. The two lines are coincident, so a direction vector of the line that you’re trying to find is $(2+3t,1+2t,-1)-(1,-2,3)$ for some value of $t$. This vector must be orthogonal to $(1,-3,1)$, so set their dot product equal to zero and solve for $t$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3458667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
How to apply cauchy's residue theorem to rational function? I have the following function:
$$f(z)=\frac{(z-1)^2}{z*(z+1)^3)}$$
And I need to find its residue.
The formula that I know for this is the following:
The residue $a_{-1}$ for a pole of order "n" is:
$$a_{-1} =\frac{1}{(n-1)!} \lim_{z \rightarrow z_0} \frac{\... | I get $-1$ for the residue. $$(z+1)^3f(z)= \frac{(z-1)^2}{z}=z-2+\frac1z$$ The second derivative is $2z^{-3}$ and $\lim_{z\to-1}\frac12\cdot2z^{-3}=-1$.
Alternatively, we want the coefficient of the $\frac{1}{z+1}$ term in the Laurent series fo $f(z)$. We have $$\frac{(z-1)^2}{z} = z-2+\frac1z = (z+1)-3-\frac{1}{1-(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3458822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
An example of Sturm-Liouville eigenvalue problem I'm not sure if this has been asked before but I couldn't find it.
I want to solve the Sturm-Liouville eigenvalue problem
$$ u''+\lambda u=0,\ \ u'(0)=u(1)=0.$$
Let the SL differential operator be $Lu=u''.$ I want to find the eigenvalues/functions of $L$ by solving the b... | A simpler approach.
The general solution for the DE $u''+\lambda u = 0$ is
$$
u = c_1\sin(\sqrt \lambda t)+c_2\cos(\sqrt\lambda t)
$$
the boundary conditions are
$$
\cases{u'(0) = \sqrt\lambda c_1\cos(\sqrt\lambda 0)-\sqrt\lambda c_2\sin(\sqrt\lambda 0) = 0\\
u(1) = c_1\sin(\sqrt \lambda)+c_2\cos(\sqrt\lambda) = 0}
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3458926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Uniform convergence of a Cauchy Sequence in a compact implies continuous limit. I need to show that if I have a Cauchy sequence of functions $f_n$.
which I know converges uniformly to $f$ in a compact set $[-r,r]$, then our limit function $f$ is continuous.
The thing is, for my definition of Cauchy sequence, it follow... | If $f_n$ are not continuous then it is not true.
Take $f_n=1_{[-\frac{1}{2},\frac{1}{2}]}+\frac{1}{n}$ on $[-1,1]$
where $1_A$ is the indicator function of the set $A$
You can see in many textbooks and notes that: If the functions $f_n$ are continuous on a compact interval ,then their uniform limit is continuous on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3459093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Limit of $\dfrac{t}{\ln(1+t)}$ without L'Hospital I'm trying to prove: $\lim_{t\rightarrow 0} \dfrac{t}{\ln(1+t)}=1$ without use L'Hospital and derivate, because I still do not define derived from the logarithm, is there any way to prove it by the definition epsilon-delta? So far I have only defined $\log$ as the inver... | $$\lim\limits_{t\to0}\frac{\ln(1+t)}t=\lim\limits_{t\to0}\frac{\ln(1+t)-\ln1}t$$
That's the definition of the derivative of $\frac d{dx}\ln x$ evaluated at $x=1$, which is obviously $\frac11=1$. Therefore,
$$\lim\limits_{t\to0}\frac t{\ln(1+t)}=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3459184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
} |
$f_n$ uniformly convergent on open interval, convergent at endpoints Suppose $\{ f_n \}$ converges uniformly on $(-1,1)$. Suppose also that $f_n(-1)$ and $f_n(1)$ converge. Then $\{ f_n \}$ converges uniformly on $\lbrack -1, 1 \rbrack$.
Attempt:
Suppose $\{ f_n \}$ does not converge uniformly to $f(x)$ on the closed i... | I think you're over complicating matters.
Let $\varepsilon>0$. By uniform convergence, there is an $N$ such that $|f_n(x)-f(x)|<\varepsilon$ for every $x\in (-1,1)$ and every $n\ge N$. Also, since $f_n(1)\to f(1)$, there is some $N'$ such that $|f_n(1)-f(1)|<\varepsilon$ for every $n\ge N'$. Similarly, there is some $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3459325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prove that all the eigenvalues of $A$ are real.(Gershgorin 's Theorem) The question and its answer are given below:
And this is the Greshgorin theorem:
My questions are:
1- I do not understand why a consequence of Greshgorin's theorem is that if a circle is disjoint then it contains 1 eigenvalue. could anyone exp... | That's a very 'legitimate question', but you could only post a question and work from that. I 'll try
$1$. That's the theorem the proof uses continuity argument.
$2$. If a polynomial $P(x)$ has real coefficients. If $P(z)=0$ then taking conjugate $\overline{P(z)}=P(\bar{z})=0$ so both $z$ and $\bar{z}$ are roo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3459558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove that if $f$ is entire and there exists a bounded sequence of distinct real numbers {$a_n$} with given property, show that $f$ is a constant.
The above problem excerpted from Complex variables with application by Silverman, section 8.2.
The first part of the problem can be shown relatively easily by considering $... | Since $f(t) \in \mathbb R$ for all real $t$, there is, by Rolle, some $t_n \in (a_{2n+1}, a_{2n})$ such that $f'(t_n)=0.$
Since $t_n \to 0$, we see that $0$ is an accumulation point of the zeroes of $f'$. This gives that $f'(z)=0$ for all $z.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3459703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding $\lim_{n \to \infty} \left( 1 + 2\int_0^1 \frac{x^n}{x+1} dx \right)^n$
I have to evaluate
$$\lim_{n \to \infty} \left( 1 + 2\int_0^1 \frac{x^n}{x+1} dx \right)^n. $$
My progress: Since $x \in (0, 1)$ we can use the series expansion of $\frac{1}{1+x} = 1-x+x^2-x^3+...$
Evaluating that integral in the parant... | $$|B_n|=|\int_0^1\dfrac{x^n}{1+x} \text{d}x| \leq \int_0^1 x^n\text{d}x=\dfrac{1}{n+1}$$
$$E_n=(1+2B_n)^n=\exp(n\log(1+2B_n))=\exp(n\times(2B_n+O(B_n^2))$$
so $$\lim E_n=\exp(\lim 2nB_n)$$
and $$nB_n=n\int_0^1 \dfrac{x^n}{1+x} \text{d}x=n\int_0^1\dfrac{u}{1+u^{1/n}}u^{(1-n)/n}\times \dfrac{1}{n}\text{d}u=\int_0^1\dfrac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3459797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
pole is well defined on the Riemann surface I want to show a pole is well defined on the Riemann surface.
Let $M$ is a Reimann surface, $f:M \rightarrow \hat{C}$ and $f(p)=\infty$. Suppose $(U,\varphi) $
and $(V, \psi)$ are two charts of M and $p \in V \cap U$.
$f o \varphi^{-1}$ has a pole of orde $t$ at $\varphi^{-1... | Both maps differ by a holomorphic map $\varphi \circ \psi^{-1}$ and the order of a zero (or pole) does not hange under a holomorphic coordinate change.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3459890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can an algebra be morita equivalent to its dg-extension? Say we have a DG algebra $A=\bigoplus_{n\geq 0}A_n$, let $B=A_0$, the 0th degree of $A$. Assume we have that the category of DG-modules over $A$ is equivalent to the category of module over $B$. Does this imply that $A_i=0$ for $i\geq 1$?
| Yes, but rather vacuously so: it implies that $A_i=0$ for all $i$, not just $i\geq 1$! Indeed, the category of modules over a ring has the following property: if $(M_i)$ is an infinite family of nonzero objects, the canonical map $\bigoplus M_i\to\prod M_i$ is not an isomorphism. On the other hand, if $A$ is a nonzer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3460035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to prove that $ -n \int _0 ^1 x^{n-1} \log(1-x)dx$ equals the $n$-th harmonic number? From (Almost) Impossible Integrals, Sums, and Series section 1.3:
$$H_n = -n\int _0 ^1 x^{n-1} \log(1-x)dx$$
The proof of which was appetizingly difficult. I was unable to answer the follow-up challenge question, and do not have... | If you call the RHS $I_n$, then
\begin{align}I_n-I_{n-1}&=\int_0^1((n-1)x^{n-2}-nx^{n-1})\log(1-x)\,dx\\
&=\left[(x^{n-1}-x^n)\log(1-x)\right]_{x=0}^1
+\int_0^1\frac{x^{n-1}-x^n}{1-x}\,dx
\end{align}
on integration by parts.
Then
$$\lim_{x\to1}(x^{n-1}-x^n)\log(1-x)=\lim_{x\to1}(1-x)\log(1-x)=
\lim_{y\to0}y\log y=-\lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3460270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
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Probability that girl who answers the door is the eldest girl? You know that a family has 3 children. You walk up to and knock on the front door of their house. A girl answers the door.
What is the probability that the she is the eldest girl among the children?
Assume that all 3 children are home and equally likely to... | Condition on whether the girl who opened the door is the eldest child overall, the middle child, or the youngest child. Each of these occur with probability $\frac{1}{3}$. We are told this within the problem statement.
The probability she is the eldest girl given that she is the eldest child is clearly $1$.
The proba... | {
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Book suggestion for real analysis help. I am looking for a good book to supplement my class book on analysis.
I am struggling with concepts and I would like a book that can help me learn how to learn analysis.
| When I was learning introductory real analysis, the text that I found the most helpful was Stephen Abbott's Understanding Analysis. It's written both very cleanly and concisely, giving it the advantage of being extremely readable, all without missing the formalities of analysis that are the focus at this level. While i... | {
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Complex number: cube root of i I understand that the way to calculate the cube root of $i$ is to use Euler's formula and divide $\frac{\pi}{2}$ by $3$ and find $\frac{\pi}{6}$ on the complex plane; however, my question is why the following solution doesn't work.
So $(-i)^3 = i$, but why can I not cube root both sides ... | $-i$ is certainly one of the cube roots of $i$ but it is certainly not the only one.
Every complex number (except $0$) has three cube roots.
A quicker way to find these roots is to use the cube roots of unity, which can be written $1, \omega, \omega^2$ and multiply them successively by the root you've already got.
So i... | {
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What is function $\sin$ without $x$? I'm solving a problem set in ODE:
This is the first time in my life that I see such functions $\sin,\cos,\tan$ without argument $x$.
I would like to ask if $\sin,\cos,\tan$ mean $\sin x,\cos x,\tan x$ or they mean something else.
Thank you so much!
| Usually the convention is that if $f:X\to Y$ is a function, then $f(x)\in Y$ is the value at the point $x\in Y$. As you can do arithmetic with function values, you can do arithmetic with functions, applying the operation pointwise. Thus $h=f+g$ is $h(x)=f(x)+g(x)$ for all $x$.
This gives a little complication if you wa... | {
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Apollonian network and 4 coloring Is there a connection between the Apollonian network and the four-color theorem? Are there any attempts to prove this theorem using the Apollonian network?
| As Wikipedia's article on Apollonian networks mentions, "Birkhoff (1930) is an early paper that uses a dual form of Apollonian networks, the planar maps formed by repeatedly placing new regions at the vertices of simpler maps, as a class of examples of planar maps with few colorings." This refers to Birkhoff, On the nu... | {
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Derivative of $\tan^{-1}\dfrac{\sqrt{1+x^2}-1}{x}$ with respect to $\tan^{-1}x$
Derivative of $\tan^{-1}\dfrac{\sqrt{1+x^2}-1}{x}$ with respect to $\tan^{-1}x$
Method 1
$$
w=\tan^{-1}\dfrac{\sqrt{1+x^2}-1}{x}\quad\&\quad z=\tan^{-1}x
$$
Put $\theta=\tan^{-1}x\implies\tan\theta=x$
$$
w=\tan^{-1}\frac{|\sec\theta|-1}{x... | In the first method for $\theta\in(-\pi/2,\pi/2)$ we have $|\cos\theta|=\cos\theta$ therefore only plus sign holds.
Second method seems more effective and clear to me.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Kernel of nonzero linear function in vector space Let $l$ be a nonzero linear function on a vector space $V$, i.e. $l:V\to \mathbb{k}$. Show that $\text{ker}(l)$ is a maximal nontrivial linear subspace in $V$. Also prove that $V/\text{ker}(l)$ has dimension 1.
Remark: It is not necessary that $V$ is finite-dimensional ... | In a vector space of dimension $1$, a maximal subspace is trivial. To show it is nontrivial in dimension greater than one, simply take a one dimensional subspace as a counterexample to the containment condition.
| {
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"source": "stackexchange",
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If a normal subgroup and its factor group are both abelian, then what can be said about the group? The textbook I am using to self-study Abstract Algebra has the following problem.
Prove or disprove: If $H$ is a normal subgroup of $G$ such that $H$ and $G/H$ are abelian, then $G$ is abelian.
My attempt:
Since $H\... | Hint. The symmetric group $S_3$ has a nontrivial normal subgroup.
Rethink your "proof".
| {
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Unique isomorphism between finitely generated $\tau$-structures Let $\mathcal{A}$, $\mathcal{B}$ be two $\tau$-structures, where $\tau$ is a first order language and let $\vec{a} = (a_1, \dots, a_n) \in A^n$ and $\vec{b} = (b_1, \dots, b_n) \in B^n$. We denote $\langle \vec{a}\rangle$ to be the smallest $\tau$-substruc... | Suppose $\xi$ and $\xi'$ are two such isomorphisms. Let $S=\{x:\xi(x)=\xi'(x)\}$. Note that $S$ is a $\tau$-substructure of $\langle \vec{a}\rangle$ since $\xi$ and $\xi'$ are both homomorphisms, and $a_i\in S$ for each $i$ by hypothesis. So by definition of $\langle \vec{a}\rangle$, $S$ must be all of $\langle \vec... | {
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Prove or Disprove - simple inequality conjecture for ratios of real numbers Please exhibit a proof or counterexample for the following claim.
Let $x,y,a,b \in \mathbb{R}$ and $0 < c < 1$. Then $$|x/y - 1| < c \text{ and } |a/b - 1| < c \iff |ay/bx - 1| < c.$$
If the bi-conditional does not hold, does either directi... | counterexample for $\implies$: If $c=1-10^{-10}$, $a/b = 10^{-1}$, and $x/y = 10^{-5}$ then $\frac{ay}{bx} = 10^4$ which is very far from $1$.
counterexample for $\impliedby$: If $a/b = \frac{1}{4}$ and $x/y = \frac{1}{2}$ then $\frac{ay}{bx} = \frac{1}{2}$ which is within $2/3$ of $1$, but $|a/b-1| > 2/3$.
| {
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Help with proof involving Eisenstein's criterion Let $a\in\mathbb {Z}[X]$ and suppose that $2a\in\mathbb {Z}[X]$ is Eisenstein with respect to a prime $p\in\mathbb {Z}$.
How can I prove that $a$ is an Eisenstein polynomial with respect to $p$?
Any help would be greatly appreciated!
| Hint 1 Show that $p \neq 2$. To do this, use $p\nmid 2a_n$.
Hint 2: If $p|2a_k$ and $p \neq 2$ deduce that $p|a_k$.
Hint 3: If $p^2 \nmid 2a_0$ show that $p^2 \nmid a_0$.
| {
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Is $\frac{1}{\alpha} \in \mathbb{Q}[\alpha]$ for irrational $\alpha$? I have been trying to pick up abstract algebra and just attempted an exercise from Landin's An Introduction to Algebraic Structures which asks to prove whether $\frac{1}{\pi} \in \mathbb{Q}[\pi]$, and would like to ask a (slightly) more general quest... | Not, it's not true for all irrational numbers $0 \lt \alpha \lt 1$. To see this, assume it's true so you get
$$\frac{1}{\alpha} = \sum_{i=0}^{n}a_i \alpha^n \tag{1}\label{eq1A}$$
for some set of $a_i \in \mathbb{Q}$. Multiply by $\alpha$ on both sides and then subtract $1$ from both sides to get
$$\sum_{i=0}^{n}a_i \al... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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3D geometry-skew lines, distance of a point from a line A straight line L intersects perpendicularly both the line:
$$\frac{(x+2)}{2} = \frac{(y+6)}{3}=\frac{(z-34)}{-10} $$
and the line:
$$\frac{(x+6)}{4}=\frac{(y-7)}{-3}=\frac{(z-7)}{-2}$$
Then the square of the perpendicular distance of origin from L is
I could fin... | $$L_1: \frac{x+2}{2}=\frac{y+6}{3}=\frac{z-34}{-10}=a$$
On $L_1$, a general point is $\vec p_1=(2a-2,3b-6,-10a+34)$
$$L_2: \frac{x+6}{4}=\frac{x-7}{-3}=\frac{x-7}{-2}=b$$
A generat point on $L_2$ is $\vec p_2\ (4b-6,-3b+7,-2b+7)$
Let the line $L_3$ which is $P_1P_2$, intersects both $L_1$ and $L_2$ orthogonally,
the... | {
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"source": "stackexchange",
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$(a+bi)$ of $\frac{3+4i}{5+6i}$ and $i^{2006}$ and $\frac{1}{\frac{1}{1+i}-1}$ How can one get the Cartesian coordinate form $(a+bi)$ of the following complex numbers?
$$\frac{3+4i}{5+6i}$$
$$i^{2006}$$
$$\frac{1}{\frac{1}{1+i}-1}$$
Regarding $\frac{3+4i}{5+6i}$ I tried expanding it with $\frac{5+6i}{5+6i}$ and got $\f... | For $\frac{3+4i}{5+6i}$ multiply the numerator and denominator by the conjugate of the denominator, $5 - 6i$, what do you observe ?
For $i^{2006}$. If $i^2 = -1$, what can you say about $i^{2006} = (i^2)^{1003} = (-1)^{1003} $ ?
For $\frac{1}{\frac{1}{1+i}-1}$, multiply numerator and denominator by $1+i$, you get $\f... | {
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Can Abel's test and Dirichlet test be used interchangably? I have seen in most cases in series of constants and in series of functions that where Abel's test of convergence applies Dirichlet's test also applies and vice versa. This makes me to think whether the two tests are equivalent in the sense that one is derivab... | The proofs have some commonality but one test is not a corollary of the other.
Suppose the sequence $(a_n)$ is not monotone and $\sum a_n$ converges. It follows that $\sum(1+1/n)^n a_n$ converges by Abel’s test. However, the conditions for Dirichlet’s test are not met since $(1+1/n)^n$ is not monotone decreasing — alth... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finite simplicial complex can be viewed as a subcomplex of a simplex I am reading the proof of the Simplicial Approximation Theorem (2C.1) in Hatcher.
In the first paragraph of the proof, Hatcher says that:
Choose a metric on $K$ that restricts to the standard Euclidean metric on each simplex of $K$. For example, $K$ ... | The key idea is that in a simplicial complex (unlike in, say, a $\Delta$-complex), each simplex is uniquely determined by its vertices (this is part of the definition of a simplicial complex). So, since $K$ has finitely many vertices, say $N - 1$, consider the simplex $\Delta^n$, and identify the $N - 1$ vertices of $\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Convergence of $(1+f(n))^{g(n)}$ when $f(n) \to 0$ and $g(n) \to +\infty$ Let $f : \Bbb N \to [0, \infty[$ and $g : \Bbb N \to [0, \infty[$ be two functions, such that $f(n) \to 0$ and $g(n) \to +\infty$ when $n \to \infty$.
What are some (necessary/sufficient) conditions on $f$ and $g$ for the limit
$$a_{f,g} := \lim_... | We have that
$$(1+f(n))^{g(n)}=e^{g(n)\log(1+f(n))}$$
and since
$$g(n)\log(1+f(n))=g(n)\cdot f(n)\cdot \frac{\log(1+f(n))}{f(n)}$$
with
$$\frac{\log(1+f(n))}{f(n)}\to 1$$
all boils down in $g(n)f(n)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Gradient of $\mathcal{L}(W) = -\frac{n}{2}\left\{d\ln(2\pi)+\ln|C|+\mbox{Tr}(C^{-1}S)\right\}$ w.r.t. $W$ I have the following function $\mathcal{L}(W)$ and I want to find the gradient with respect to $W$, but I'm struggling with the matrix operations and derivations.
$$\mathcal{L}(W) := -\frac{n}{2}\left\{ d\ln(2\pi)... | Use a colon as a convenient product notation for the trace, i.e.
$\;A:B = {\rm Tr}(A^TB)$.
Define the variables
$$\eqalign{
C &= WW^T +\sigma^2I &\implies dC = W\,dW^T+dW\,W^T \\
I &= C^{-1}C &\implies dC^{-1} = -C^{-1}\,dC\,C^{-1} \\
{\cal J} &= -\frac{n}{2}\log(2\pi) &\implies d{\cal J} = 0 \\
}$$
Write the objective... | {
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"source": "stackexchange",
"question_score": "2",
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How would one show that $\textbf{Z}$ is countable? I am trying to show that the set of all integers $\mathbb{Z}$ is indeed countable which would mean that $|\mathbb{N}|=|\mathbb{Z}|$.
This would further imply that I have to find a bijection between the two sets, but I do not really know how to do that.
I have tried wit... | Another possible bijection (likely similar to one you were thinking of) would be:
$$f(x)=\begin{cases}
\frac n2 &\text{$n$ is even}\\
-\frac {n+1}{2}\quad &\text{$n$ is odd}
\end{cases}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3463552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find area bounded by $y=\frac 3x, y=\frac 5x, y=3x, y=6x$ Let S be the area of the region bounded by the curves
$$y=\frac 3x,\>\>\> y=\frac 5x,\>\>\> y=3x,\>\>\> y=6x$$
Need to find $S$.
The coordinates of the vertices of the resulting figure were found. The problem with the transition from a double integral to a repe... | In polar coordinates, the boundaries are
$$r^2=\frac3{\sin\theta\cos\theta},\>\>\>\>\>r^2=\frac5{\sin\theta\cos\theta},
\>\>\>\>\>\tan\theta = 3,\>\>\>\>\>\tan\theta = 6$$
Thus, the area integral is,
$$S=\int_{\theta_1}^{\theta_2}d\theta \int_{r_1}^{r_2}rdr
=\int_{\theta_1}^{\theta_2}\frac{d\theta}{\sin\theta\cos\theta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How my professor derived this CDF? This was an example problem my professor went over in class.
Let $X =$ uniform $(1,4)$ where $Y=(X-2)^2$ Find the CDF.
He went on to derive:
$F_Y(y)=P(Y\leq y) = P((x-2)^2 \leq y) = P(-\sqrt{y}\leq (x-2) \leq \sqrt(y))$
= $P(2 - \sqrt{y} \leq x \leq 2 + \sqrt{y})$
Then he said the ... | The CDF given was simply that of the variable $X$, it's not the CDF for $Y$. Note that since this is a uniform distribution on $(1,4)$ then the density is given by $p_X(x) = \frac{1}{3}$. Then:
$$
F_X(x) \;\; =\;\; P(X\leq x) \;\; =\;\; \int_1^x \frac{1}{3}d\alpha \;\; =\;\; \frac{x-1}{3}.
$$
The values on the other ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Finding sum of series $\sum^{\infty}_{k=0}\frac{(k+1)(k+2)}{2^k}$
Finding sum of series $\displaystyle \sum^{\infty}_{k=0}\frac{(k+1)(k+2)}{2^k}$
what i try
Let $\displaystyle x=\frac{1}{2},$ Then series sum is $\displaystyle \sum^{\infty}_{k=0}(k+1)(k+2)x^k$
$\displaystyle \sum^{\infty}_{k=0}x^k=\frac{1}{1-x}\Righta... | You almost have it. Instead of multiplying the infinite sum by $x$, try multiplying by $x^2$, so you get
$$\sum^{\infty}_{k=0}x^{k+2} = \frac{x^2}{1-x} \tag{1}\label{eq1A}$$
After differentiating twice, you will then get your series sum on the left, i.e., $\sum^{\infty}_{k=0}(k + 1)(k + 2)x^{k}$. You can determine the ... | {
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"url": "https://math.stackexchange.com/questions/3463896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $X=Y=[0,1]$ and $g(x, y)=\max \{x(1-2 y), y(1-2 x)\}$. How to compute $\max_{x \in X} \min_{y \in Y} g(x,y)$? I'm trying to solve a question from last year final exam in game theory:
I've solved question (3a), but it takes me a lot of time to compute because of many possible cases to consider. I think I can not ... | Computing $\alpha$ involves first minimizing $g(x,y)$ with respect to $y$. This implies that we need to set $x(1-2y)=y(1-2x)$. To see this, note the following: If $x(1-2y)>y(1-2x)$, $g(x,y)=x(1-2y)$ and we can increase $y$ to decrease $g(x,y)$. If $y(1-2x)>x(1-2y)$, then $g(x,y)=y(1-2x)$ and we can decrease $y$ to decr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3464059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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