Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
$m + n \sqrt{2}$ is invertible $\iff$ $m^2 - 2n^2 =\pm 1$ I'm having trouble proving that, for $m, n \in \mathbb{Z}$, the existence of a multiplicative inverse for $m + n \sqrt{2}$ implies that $m^2 - 2n^2 = \pm 1$.
The first step, I believe, is to solve for the inverse, which is clearly $\frac{1}{m + n\sqrt{2}}$, pro... | As you said, the only candidate for the inverse is clearly $\frac1{m+n\sqrt{2}}$.
We have
$$\frac1{m+n\sqrt{2}} = \frac{m-n\sqrt{2}}{m^2-2n^2} = \frac{m}{m^2-2n^2} + \frac{-n}{m^2-2n^2}\sqrt{2}$$
This is an element of $\mathbb{Z}[\sqrt{2}]$ if and only if both $\frac{m}{m^2-2n^2}$ and $\frac{-n}{m^2-2n^2}$ are in $\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2864821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Contour integral of $\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx $ with non rectangular contour Is there a way to solve the integral of $$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx $$ for $$a\in (0,1)$$
without using the rectangular region like in this post but still using a contour integral?
Perhaps using a semicir... | (Not using contour integration ; Sorry )
$$I=\int_{-\infty}^{\infty} \frac {e^{ax}}{1+e^x} dx=\int_{-\infty}^{\infty} \frac {e^x\cdot e^{ax}}{e^x+e^{2x}} dx$$
Use the substitution $e^x=t$
$$I=\int_{0}^{\infty} \frac {t^{a-1}}{1+t}dt =B(a,1-a)=\Gamma(a)\Gamma(1-a)=\frac {\pi}{\sin (\pi a)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2864922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is $[0,1]$ an open subset of $[0,1] \cup [2,3]$? Given a metric space $X = [0,1]\cup[2,3]$
I have to show $[0,1]$ is both open and closed in $X$.
This question is also asked in this thread :
Let $X = [0,1] \cup [2,3]$ be a metric space. Why is $[0,1]$ both open and closed?
I understand why $[0,1]$ is closed, but am... | Thanks a lot MichaelBurr and copper.hat.
The key was to think in terms of the relative metric for $X$ and not the metric used for $\mathbb{R}$. The relative metric restricts the definition of the metric for only those points which belong to $[0,1]\cup[2,3]$. In this case, $[0,1/2)$ is also an open ball in [0,1] when X... | {
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"timestamp": "2023-03-29T00:00:00",
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Example of a sequence of functions where the limit cannot be interchanged
Give an example of a sequence of continuous functions $f_n$ on $[0,1]$ with $f_n$ converges pointwise to a continuous function $f$ such that the following relation does't hold:
$$\lim_{n \rightarrow \infty} \lim_{x \rightarrow 0} f_n(x)=\lim_{x... | Since all $f_n$ are continuous we have $\lim_{x\to0} f_n(x) = f_n(0)$ and since $f = \lim_{n\to\infty} f_n$ is also continuous we have $\lim_{x\to 0} f(x) = f(0)$.
So your relation boils down to $\lim_{n\to\infty} f_n(0) = f(0)$, which is true because $f_n \to f$ pointwise.
| {
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"url": "https://math.stackexchange.com/questions/2865092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve: $2^x\Bigl(2^x-1\Bigl) + 2^{x-1}\Bigl(2^{x-1} -1 \Bigl) + .... + 2^{x-99}\Bigl(2^{x-99} - 1\Bigl) = 0$ The question says to find the value of $x$ if, $$2^x\Bigl(2^x-1\Bigl) + 2^{x-1}\Bigl(2^{x-1} -1 \Bigl) + .... + 2^{x-99}\Bigl(2^{x-99} - 1 \Bigl)= 0$$
My approach:
I rewrote the expression as,
$$2^x\Bigl(2^x-1... | \begin{align}
2^x\Bigl(2^x-1\Bigl) + 2^{x-1}\Bigl(2^{x-1} -1 \Bigl) + .... + 2^{x-99}\Bigl(2^{x-99} - 1 \Bigl) &= 0 \\
\sum_{i=0}^{99}2^{x-i}\left(2^{x-i}-1\right) &= 0 \\
\sum_{i=0}^{99}\left[\left(2^{x-i}\right)^2-2^{x-i}\right] &= 0 \\
\sum_{i=0}^{99}\left(2^{x-i}\right)^2-\sum_{i=0}^{99}2^{x-i} &= 0 \\
\sum_{i=0}^{... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What do you call this property involving a function between two complete metric spaces? I have a notion, for which I am not able to find any reference name, as I am not that familiar with these concepts. Please help me by pointing to a definition for the below scenario.
Is there a name for the following property of the... | Let us call the property described in question as Property P. Continuing the observations made in Stefan Böttner's answer we get the following.
Observation. Let $A$ and $B$ be metric spaces and $e\colon A\to B$ be a continuous function. Then $e$ has the Property P if and only if $A$ has no isolated points and $e$ is no... | {
"language": "en",
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Splitting field of a polynomial $f(x) =(x^2-3)(x^2-5)(x^5-1)$ over $\mathbb{Q}$. I was considering the splitting field E of the polynomial $f(x) =(x^2-3)(x^2-5)(x^5-1)$ over $\mathbb{Q}$.
I expected $E=\mathbb{Q}(\sqrt{5},\sqrt{3},\omega)$, where $\omega=e^{\frac{2\pi i}{5}}$.
But I saw a textbook that claimed $E=\ma... | Note that $$\begin{align}\frac{x^5-1}{x-1}&=x^4+x^3+x^2+x+1=x^2\,\left(t^2+t-1\right)
\\&=x^2\,\left(t-\frac{-1+\sqrt{5}}{2}\right)\left(t-\frac{-1-\sqrt{5}}{2}\right)\,,\end{align}$$
where $t:=x+\dfrac{1}{x}$. Therefore, $\sqrt{5}$ is already in the field $\mathbb{Q}(\omega)$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Set $S=\{(x,y,z)| x,y,z \in \mathbb{Z}\}$ is a subset of vector space $\mathbb{R}^3$, how do I show that it is not a subspace of $\mathbb{R}^3$. So I know that set $S=\{(x,y,z)| x,y,z\in \mathbb{Z}\}$ is a subset of vector space $\mathbb{R}^3$.
Specifically, it is worded in our lecture that it is a " subset of $(\math... | Edit: User Randall pointed out that I misread the question. (I assumed S was under some invalid field.)
First lets look at the definition of a subspace:
*
*All products and sums composed of elements within the subspace also are in the subspace.
*All elements in the subspace must be able to be scaled by the vector s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Is there a metric on the reals $\Bbb R$ so that a subset of $\Bbb R$ is open iff its complement is finite? I wanted to know if there is a distance function $d$ on $\Bbb R$ so that a nonempty subset $U$ of $\Bbb R$ is open with respect to $d$ if and only if its complement $\Bbb R$ \ U is finite ?
| Let $d$ be such a metric and $a,b$ any distinct points. Then the open balls around $a$ and $b$ of radius $\frac 12 d(a,b)$ are disjoint proper non-empty open sets, hence at most one of them can be co-finite.
| {
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"timestamp": "2023-03-29T00:00:00",
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Linear Programming optimization with multiple optimal solutions I am trying to solve the following optimization problem using linear programming (deterministic operations research). According to the book, there are multiple optimal solutions, I don't understand why. I'll show you what I have done.
The problem is:
$max ... | If you solve the problem graphically you should solve the objective function $Z$ for $x_2$ as well.
$Z=500x_{1}+300x_{2}$
$Z-500x_{1}=300x_{2}$
$\frac{Z}{300}-\frac53x_1=x_2$
Now you set the level equal to zero, which means that $z=0$ and draw the line. This line goes through the origin and has a slope of $-\frac53... | {
"language": "en",
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Gradient vector $\nabla F = (z_x, z_y, -1)$ is normal to the integral surface? I have an integral surface $z = z(x, y)$.
Writing this integral surface in implicit form, we get
$$F(x, y, z) = z(x, y) - z = 0$$
I am then told that the gradient vector $\nabla F = (z_x, z_y, -1)$ is normal to the integral surface $F(x, y, ... | Let us start with an example.
$$ z=x^2+y^2$$
$$ F(x,y,z)=x^2+y^2-z$$
$$\nabla F = (z_x, z_y, -1)=< 2x,2y,-1>$$
If a point is given, for example $P(1,2,5)$ Then at that point you have two normal vector to the surface.
Upward normal $$< -2x,-2y,1> = <-2,-4,1>$$
Downward normal $$< 2x,2y,-1> = <2,4,-1>$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2865943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Given $t = \tan \frac{\theta}{2}$, show $\sin \theta = \frac{2t}{1+t^2}$ Given $t = \tan \frac{\theta}{2}$, show $\sin \theta = \frac{2t}{1+t^2}$
There are a few ways to approach it, one of the way i encountered is that using the $\tan2\theta$ formula, we get $$\tan\theta = \frac{2t}{1-t^2}$$
By trigonometry, we know t... | No, because $$ \sqrt {x^2} = |x|$$
Thus $$ \sqrt{(t^2+1)^2} = |(t^2+1)|=(t^2+1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2866118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What are the steps involved in solving a quartic polynomial modulo a prime modulus? This:
$$x^4 + 21x^3 + 5x^2 + 7x + 1 \equiv 0 \mod 23$$
Leads to:
$$x = 18 || x =19$$
I know this because of this WolframAlpha example and because a fellow member posted it in a since deleted & related question.
What I don't understand... | Let
$$f(x)=x^4 -2x^3 + 5x^2 + 7x + 1\tag{1}$$
be defined over the finite field $\mathbb{F}_{23}$. Now check for a linear factor by checking for roots over $\mathbb{F}_{23}=\{0,\pm1,\pm2,\pm3,\pm4,\pm5\pm6,\pm7,\pm8,\pm9,\pm10,\pm11\}$. We find $f(-4)=f(-5)=0$, so $(x+4)$ and $(x+5)$ are linear factors. Now factor $f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2866222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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"answer_id": 6
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Cardinality of $A = \varnothing, B = \{ \varnothing \}, C = \{\{\varnothing\}\}$ Given three sets $A = \varnothing, B = \{ \varnothing \}, C = \{\{\varnothing\}\}$ what are the cardinalities of those sets ?
Obviously cardinality of $A$ is $0$ and cardinality of $B$ is $1$, but I am not sure about set $C$, because some ... | $$A = \varnothing, B = \{ \varnothing \}, C = \{\{\varnothing\}\}$$
The way you have it $B$ and $C$ both have cardinality of $1$
My guess is that you wanted $$C = \{ \varnothing ,\{\varnothing\}\}$$
Which has cardinality $2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2866357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Left Kan extension: switching K and F If $F: ⟶ $ and $K: ⟶ ℰ$ are functors, where $$ is small and both $$ and $ℰ$ are cocomplete. How do the left Kan extension of $F$ along $K$ (${\rm Lan}_K(F)$) and the left Kan extension of $K$ along $F$ (${\rm Lan}_F(K)$) relate with one another? (Is it safe to say that they are a... | If $y : A \to[A°,Set]$ is the Yoneda embedding it is a general fact that, given a functor $f : A \to B$, there is the adjunction
$$
Lan_yf \dashv Lan_fy
$$ this is called nerve-realization paradigm.
As you can see e.g. here, asking that the two functors $F,K$ are related in this way is a rather strong request (in fact,... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that $\int_0^1 4 \space\operatorname{li}(x)^3 \space (x-1) \space x^{-3} dx = \zeta(3) $ My mentor tommy1729 wrote $\int_0^1 4 \space \operatorname{li}(x)^3 \space (x-1) \space x^{-3} dx = \zeta(3) $
I wanted to prove it thus I looked at some methods for computing integrals and also representations of $\zeta(3)$ t... | This is not a complete answer, but just a description of two ideas that might help with the evaluation of the integral
$$ I \equiv 4 \int \limits_0^1 \left(\frac{\operatorname{li}(x)}{x}\right)^3 (x-1) \, \mathrm{d} x \, . $$
They are based on methods that can be applied to find the easier integral
$$ J \equiv \int \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2866629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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The conditions for parameterisation I have proved that $\gamma(t) = (1-cost, tant-sint)$ satisfies the equation for the conchoid $(x-1)^2(x^2+y^2)=x^2$. But is there any reason why this is not a parameterisation? How do I have to restrict the parameter $t$ to get a parameterisation for each branch of the curve?
The gra... | Usually, by a plane curve we mean a continuous function $\gamma \colon I \to \mathbb R^2$ where $I \subseteq \mathbb R$ is an interval.
If $\gamma$ is defined by $\gamma(t) = (1 - \cos t, \tan t - \sin t)$, there are two natural choices for the interval:
*
*if we choose $I_1 = \left ( - \frac \pi 2, \frac \pi 2 \rig... | {
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How to show that $\mathbb{Z}_{12} $ is isomorphic to a subgroup of $S_7$? How to show that $\mathbb{Z}_{12}$ is isomorphic to a subgroup of $S_7$?
My attempt: Using Cayley's theorem one can conclude $\mathbb{Z}_{12}$ is isomorphic to a subgroup of $S_{12}$.
Or, if I use Generalised Cayley's theorem I can show that ther... | In this case, as Nicky Hekster pointed out in the top answer, you know the general structure of elements in $S_7$ so finding one of order $12$ is pretty easy.
I want to add that a more general procedure to look for a copy of $\mathbb{Z}_{12}$ in a group $G$ is to consider that $\mathbb{Z}_{12} = \mathbb{Z}_4 \times \ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2866807",
"timestamp": "2023-03-29T00:00:00",
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Is a non-euclidean-norm preserving map necessarily linear? Let $V$ and $W$ be two normed vector spaces and let $f:V \rightarrow W$ be a norm preserving map. I know that if both norms correspond to some inner product then $f$ is necessarily linear, but I can't find the answer for the more general case of normed vector s... | This is the example from J. A. Baker's paper for easy access.
Define $f:\mathbb{R}^2\to\mathbb{R}$ by $$f(x,y)=\begin{cases}y,&\text{ if }0\leq y\leq x\text{ or }x\leq y\leq 0\\x,&\text{ if }0\leq x\leq y\text{ or }y\leq x\leq 0\\0,&\text{ otherwise}\end{cases}$$
Then $f$ satisfies
*
*$f(tx,ty)=tf(x,y)$
*$|f(x,y)-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2866940",
"timestamp": "2023-03-29T00:00:00",
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Minimizing properties of geodesics problem in do Carmo's book I'm DoCarmo's book Riemannian Geometry and in the section with minimizing properties of geodesics it this proposition.
I don't understand why $\langle\frac{\partial f} {\partial r}, \frac{\partial f} {\partial t} \rangle=0$. Can someone fill in the details... | Note that $f(r,t) = \exp(rv(t))$, hence by the chain rule$$
\partial_r f(r_0,t_0)=(d\exp_p)_{r_0v(t_0)}[v(t_0)]
$$
and
$$
\partial _t f(r_0,t_0)=(d\exp_p)_{r_0v(t_0)}[r_0\dot v(t_0)].
$$
Hence
$$
\langle \partial_r f(r_0,t_0)\vert \partial_t f(r_0,t_0)\rangle = \langle (d\exp_p)_{r_0v(t_0)}[v(t_0)]~\vert~ (d\exp_p)_{r... | {
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"timestamp": "2023-03-29T00:00:00",
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Showing the diagonal on $\mathbb{R}^2$ is closed $D=\{{(x,x)|x\in\mathbb{R}}\}$ , I want to show D is closed with the definition:
$A$ is closed if and only if $\mathbb{R}/A$ is open.
So basically what I really want to show is $A=\{{(x,y)\in \mathbb{R}^2|x\neq y\}}$
.
My attemp so far was taking $a=(x,y)\in A$ and an... | Your approach is fine. In fact (as you can see geometrically), there's no need to divide by $2$. In other words, you can take $r=\frac{|x-y|}{\sqrt2}$. I will prove that the open disk $D\bigl((x,y),r\bigr)$ contains no element of $D$, that is, I will prove that if $z\in\mathbb R$, then $\bigl\|(x,y)-(z,z)\bigr\|\geqsla... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Is it always necessary to prove the 'iff' in both directions? I have an exercise in my course, which asks to prove $A \cup B = B \iff A \subseteq B$.
My proof is: Let $A \nsubseteq B$, that is, $\exists a \in A : a \notin B$. Then from the definition follows $a \in A \cup B = B$, in contradiction to the initial asserti... | It appears that you're trying (without making it completely clear) to prove $A\cup B=B \Leftrightarrow A\subseteq B$ by showing that $A\cup B=B$ together with $A\not\subseteq B$ leads to a contradiction.
If you think that is a complete proof, how about this one, by the same principle:
Claim: For any integer $n>2$, $$ ... | {
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GRE combinatorics question about counting the no. of sets questions satisfying a certain requirement. From ETS Major Field Test in Mathematics
A student is given an exam consisting of
8 essay questions divided into 4 groups of
2 questions each. The student is required to
select a set of 6 questions to answer,
... | The student can choose two questions to omit, not both i the same group. There are $8$ options for the first question, and then there are $6$ options left for the second question. Of course the order in which the questions are chosrn doesn't matter, so we get
$$\frac{8\times6}{2}=24$$
options.
Alternatively, the stude... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sum of squared eigenvalues of $A$ equals $\operatorname{tr}(A^2)$? Is the following always true:
$$\sum_i \lambda_i^2 = \operatorname{tr}(A^2)$$
where $\lambda_i$ are the eigenvalues of $A$. If it's not true in general, then under what conditions is it true? Is it always true if $A$ is square and positive semidefinite?... | tr $A^2 $ = tr $AA$ = tr $UDU^{-1}UDU^{-1} $ = tr $UD^2 U^{-1}$ = tr $U^{-1}UD^2 =$ tr $ D^2$ = $\sum \lambda_i^2$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\arg(\overline{z}), \arg(z^2)$ A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.42,43
Exer 3.42 multiple-valued: NO in general.
$$\arg(1+i)=\frac \pi 4 + 2k \pi$$
$$\arg(1-i)=\frac {-\pi} 4 - 2l \pi$$
Choose $k=1$,$l=17$ to arrive at a contradiction.
Exer... | (Partial answer for 3.42 below, 3.43 can be worked out in a similar way.) The following assumes that $z \ne 0$ and $\arg(ab)=\arg(a)+\arg(b)$ was already established for the multi-valued $\arg$.
Exer 3.42 multiple-valued: NO in general.
YES, in general, since $\arg(z\bar z) = \arg(z)+\arg(\bar z)$, but on the other ... | {
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Does there exist an ideal sheaf $\mathcal F$ on some affine scheme $X$ such that $\mathcal F$ is not quasi-coherent? Please give an example such that $X$ is an affine scheme, and there exists an ideal sheaf $\mathcal F$ on $X$ such that $\mathcal F$ is not quasi-coherent.
| It often helps to translate the problem into commutative algebra.
I will define ideal sheaves on a scheme $X$ to be an $\mathcal{O}_X$ module $\mathcal{I}$ such that for all open sets $U\subset X$, $\mathcal{I}(U)$ is an ideal of $\mathcal{O}_X(U)$. So let $\mathcal{I}$ be an ideal sheaf on $X=\operatorname{Spec}A$. Re... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that $f(x,y)=\dfrac{\sin(x^4-y^4)}{x^2+y^2}$ is steady on $\Bbb{R}$ My task is to show that
$$f(x,y) =
\begin{cases}
\dfrac{\sin(x^4-y^4)}{x^2+y^2}, & \text{if $(x,y) \ne (0,0)$} \\[2ex]
0, & \text{else}
\end{cases}$$
is steady on $\Bbb{R}$
As a composition of steady functions, it is enough to show that $f(x,y)$... | $|f(x,y)| \le \frac{|x^4-y^4|}{x^2+y^2} \le \frac{x^4+y^4}{x^2+y^2} \le x^2+y^2$.
Can you proceed ?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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how can I calculate the probability to get triples or better when throwing n 6-sided dice? I've been banging my head on a wall with this question. I'm designing a game and would like to implement a loot system inspired by a game called "Vermintide" where players roll a certain number of dice and gain loot according to ... | Hint
You can compute the probability of not getting triple. The number $f_n$ of possibilities of not getting triples in $n$ throws is:
$$f_1 = \binom{6}{1}\\
f_2 = \binom{6}{2}2!+\binom{6}{1}\\
f_3=\binom{6}{3}3!+\binom{6}{1}\binom{5}{1}\frac{3!}{2!}\\
f_4=\binom{6}{4}4!+\binom{6}{1}\binom{5}{2}\frac{4!}{2!}+\binom{6}{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do I find the minimum of this function? This might seem trivial to some of you, but I can't for the life of me figure out how to solve this.
$$\underset{x}\arg \min (x - b)^T Ax$$
$$x \in \mathbb{R^n}$$
We may assume A to be invertable, but it is not symmetric.
My idea was to calculate the first and second derivat... | You can rewrite it to a standard quadratic program and use corresponding methods as follows:
$(x-b)^T A x = x^T A x - b^T A x = x^T A x - c^T x$
for $c := A^T b$.
Your method can work too but your derivative calculation was wrong, it would be:
$ \frac{d}{dx} (x-b)^T A x = (x-b)^T A + x^T A = 0 \Leftrightarrow 2A^T x = ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Probability that a sum of uniformly distributed random variables is large Problem
Let $\ell_1 \le \ell_2 \le \dots \ell_n$ be nonnegative real numbers, and $S$ a nonnegative real number that is smaller than the sum of the $\ell_i$.
Suppose that for $i = 1, 2, \dots, n$, a number $a_i$ is picked from the interval $[0, \... | It is a little easier to think about $\mathbb P( a_1+\dots+a_n\le S)$, then subtract from $1$.
It turns out that
$$P( a_1+\dots +a_n\le S)=\frac{1}{n!\ell_1\cdots \ell_n}\sum_{I\subseteq \{1,\dots,n\}}(-1)^{|I|}\Big(\Big(S-\sum_{i\in I}\ell_i\Big)^+\Big)^n,$$
where the notation $x^+$ means $\max(x,0)$.
Essentially... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Find all entire functions with $\int_{\Bbb C} |f(z)|\, dz= 1$ I know that an entire function with bounded $L^1$ norm is identically $0$, but I do not know how to attack this problem. Does this contradict the fact I stated about entire functions with bounded $L^1$ norm?
| There are no such functions.
For each $w \in \mathbb C$, we have
$$
|f(w)| \le \int_{\Bbb C} |f(z)|\, dz= 1
$$
Therefore, $f$ is bounded. Since $f$ is entire, $f$ is constant, by Liouville's theorem.
But then we cannot have $\int_{\Bbb C} |f(z)|\, dz= 1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Given some $n ∈ ℤ$ what conditions must $v$ satisfy for $n \left \lfloor {v} \right \rfloor $ = $\left \lfloor {n v} \right \rfloor $ I'm probably overthinking this.
What constraints must you place on $v\in \mathbb R$ : $n \left \lfloor {v} \right \rfloor $ = $\left \lfloor {n v} \right \rfloor $ if $n$ is an arbitrary... | Here is a powerful result (stronger than needed, though), which can be used to deal with this problem. From the theorem below, you would see that $\{x\}\in\left[0,\dfrac1n\right)$ for $x\in\mathbb{R}$ to satisfy $$\lfloor nx\rfloor=n\,\lfloor x\rfloor\,,$$ where $n\in\mathbb{Z}_{>0}$ is fixed. Here, $\{x\}$ is the fr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof that Vitali set is non-measurable. A Vitali set is a subset $V$ of $[0,1]$ such that for every $r\in \mathbb R$ there exists one and only one $v\in V$ for which $v-r \in \mathbb Q$. Equivalently, $V$ contains a single representative of every element of $\mathbb R / \mathbb Q$.
The proof I read is in this short ar... | The sets $V_k$ are disjoint and countable, hence the measure of the union is exactly equal to the sum of measures.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $f$ is proper, lsc, and $\frac{f(x) + f(y)}{2} = f^{**}\left(\frac{x + y}{2}\right) \implies x = y$, is $f$ necessarily convex?
Suppose $X$ is a real Hilbert Space and $f : X \to (-\infty, \infty]$ is a lower semicontinuous, proper function. Further, suppose $f$ satisfies the following, for all $x, y \in \operatorn... | The conjecture seems to be not true. Take $X=\mathbb R$,
$$
f(x)=\sqrt{|x|}.
$$
Then it holds $f^{**}\equiv 0$. Moreover, if
$$
f(x)+f(y) = 2 f^{**}\left(\frac{x+y}2\right)=0,
$$
then necessarily $x=y=0$. But $f$ is not convex.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Prove the following theorem in propositional calculus I have the following Hilbert-style propositional calculus, having the logical connectives $\{\neg,\rightarrow\}$ of arity one and two respectively, and the following axioms:
A1: $A\rightarrow(B\rightarrow A)$
A2: $(A\rightarrow(B\rightarrow C))\rightarrow((A\rightar... | Lemma
1) $\vdash \lnot A \to (A \to B)$ --- Th.4
2) $\vdash (\lnot A \to (A \to B)) \to ((\lnot A \to A) \to (\lnot A \to \lnot B))$ --- Ax.2
3) $\vdash (\lnot A \to A) \to (\lnot A \to \lnot B)$ --- from 1) and 2)
4) $\vdash (\lnot A \to \lnot B) \to (B \to A)$ --- Ax.3
5) $\vdash (\lnot A \to A) \to (B \to A)$ --- f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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An invisible ghost jumping on a regular hexagon Given a regular hexagon and an invisible ghost at one of the vertices of the hexagon (we don’t know which). We have a special gun, that can kill ghosts. In a step we are able to shoot the gun twice (i.e. choose two vertices and see if the ghost is there). After every step... | Brute force exhaustion of possible strategies gives two solutions requiring four turns:
*
*Shoot at $(1,3)$ then $(4,6)$ then $(2,4)$ then $(1,5)$
*Shoot at $(1,3)$ then $(4,6)$ then $(4,6)$ then $(1,3)$
along with reflections and rotations of these basic solutions. There are none requiring three turns.
To see th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2869002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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What is the plot of this implicit function: $|\sin x|^y+|\cos x|^y = 1$ I'm trying to manually plot the following function:
$$
|\sin x|^y+|\cos x|^y = 1
$$
My basic approach for implicit functions is to try to express $y$ in terms of $x$ and plot it, or $x$ in terms of $y$ and then plot the inverse. Sometimes it's cle... | We can see an obvious solution for the contour: if $y = 2$, we have $|\cos x|^2 + |\sin x|^2 = 1$, which is satisfied for all values of $x$. So the line $y = 2$ is part of the solution set.
If $y > 2$, then since $0\leq |\cos x| \leq 1$, we have $|\cos x|^y \leq |\cos x|^2$, with equality iff $|\cos x| = 0$ or $|\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2869092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Gauss elimination. Where did I go wrong?
Gaussian elimination with back sub:
So my starting matrix:
\begin{bmatrix}
1 & -1 & 1 & -1
\\2 & 1 & -3 & 4
\\2 & 0 & 2 & 2
\end{bmatrix}
multiply the 2nd and 3rd row by -1 * (first row):
\begin{bmatrix}
1 & -1 & 1 & -1
\\0 & 3 & -5 & 6
\\0 & 2 & 0 & 4
\end{bmatrix}
then ad... | I'd use a more systematic method:
\begin{align}
\begin{bmatrix}
1 & -1 & 1 & -1\\
2 & 1 & -3 & 4\\
2 & 0 & 2 & 2
\end{bmatrix}
&\to
\begin{bmatrix}
1 & -1 & 1 & -1\\
0 & 3 & -5 & 6\\
0 & 2 & 0 & 4
\end{bmatrix}
&&\begin{aligned} R_2&\gets R_2-2R_1 \\ R_3&\gets R_3-2R_1 \end{aligned}
\\ &\to
\begin{bmatrix}
1 & -1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2869200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Equipotent Sets We know by definition that if a bijection between two sets A and B exists,then A and B are equivalent.The book I was reading took the function f:Z→N
f(x)= -2x,for x<0
2x+1,for x=>0
which is a bijection(it can be easily proven),but I'm a little confused.
How can Z and N be equivalent sets ... | One immediate (though rather dangerous! .. see paragraph below) way to wrap your mind around this is to consider the fact that both sets are of infinite size. Yes, the one is a strict subset of the other, but they are both infinite ... so maybe it's a little easier to digest that way.
Then again, the really surprising ... | {
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Notation of symmetric sum notation When you use the symmetric sum notation, for example, $$\sum_\text{sym}abc+a$$ if there are 3 variables, then does abc count once, 3 times or 6 times?
I am confused about repetitions of the same expression in a symmetric sum notation.
| We have that
$$\sum_\mathrm{sym}Q(x_i)=\sum_\sigma Q(x_{\sigma(i)})$$
for all permutations of $1, \ldots , n$.
Therefore it should be
$$\sum_\text{sym}abc+a=Q(a,b,c)+Q(a,c,b)+Q(b,a,c)+Q(b,c,a)+Q(c,a,b)+Q(c,b,a)=$$
$$=2abc+2a+2abc+2b+2abc+2c=6abc+2(a+b+c)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2869628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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why radians can be converted to reals in calculus? Consider this integral:
$$ \int \sin^2x dx = \frac x2 - \frac {\sin2x}4 + C $$
Note the first term $\frac x2$ is a real as opposed to radian and can, in fact, be substituted with a real number when taking definite integral.
To make the statement more clear, introduce t... | Your distinction between "reals" and "radians" is not a meaningful one. Radians are a unitless measurement, so "$x$ radians" is, in fact, just the real number $x$ (understood in a particular context---that of angles).
Now, $\sin(x)$ and $\sin^\circ(x)$ are both functions from $\mathbb{R}$ to $\mathbb{R}$, but they are... | {
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Evaluate $\int_0^{\frac{\pi}{2}}\frac{\sin x\cos x}{\sin^4x+\cos^4x}dx$
Evaluate
$$
\int_0^{\frac{\pi}{2}}\frac{\sin(x)\cos(x)}{\sin^4(x)+\cos^4(x)}dx
$$
I used the substitution $\sin x =t$, then I got the integral as $$\int_0^1 \frac{t}{2t^4-2t^2+1}dt $$
After that I don't know how to proceed. Please help me with... | Hint:
$$\dfrac{\sin x\cos x}{\sin^4x+\cos^4x}=\dfrac{\tan x\sec^2x}{\tan^4x+1}$$
Set $\tan^2x=y$
OR $$\dfrac{\sin x\cos x}{\sin^4x+\cos^4x}=\dfrac{\cot x\csc^2x}{\cot^4x+1}$$
Set $\cot^2x=u$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving limit of $a^n\to0$ for $|a|<1$ without use of logarithms Prove that $a^n\to0$ as $n\to∞$ for $|a|<1$ without use of logarithms by using properties of the sequence $u_n=|a|^n.$
I've noticed that I should use the subsequence $u_{2n}$, and the fact that $u_{2n}=u_n^2$. However, I don't know where to go from here. ... | By ratio test
$$\frac{|a|^{n+1}}{|a|^n}=|a|<1 \implies |a|^n \to 0$$
then since
$$-|a|^n\le a^n\le |a|^n$$
by squeeze theorem we conclude that
$$a^n \to 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2869873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 6
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Proof by induction on $r$ variables If there is a statement $P(n)$, proof by induction has three steps.
Base case is to show $P(1)$ is true
Induction step is to assume $P(K)$ is true and then to show $P(k+1)$ is true.
If our statement $P(n_1,n_2,n_3,\cdots, n_r)$ involves $r$ variables, then how to prove it by inductio... | Depends on context.
In general it boils down to finding a suitable well order on $\mathbb N^r$.
Then the induction step is proving that $P(n_1,\dots,n_r)$ implies $P(m_1,\dots,m_r)$ where $(m_1,\dots,m_r)$ denotes the successor of $(n_1,\dots,n_r)$.
Sometimes it is possible to do it with induction on $n=n_1+\cdots+n_r... | {
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Munkres-Analysis on Manifolds: Theorem 20.1 I am studying Analysis on Manifolds by Munkres. I have a problem with a proof in section 20:
It states that:
Let $A$ be an $n$ by $n$ matrix. Let $h:\mathbb{R}^n\to \mathbb{R}^n$ be the linear transformation $h(x)=A x$. Let $S$ be a rectifiable set (the boundary of $S=BdS$ h... | As Munkres explains, since $\det A=0$, $h(S)(=T)$ is contained in a vector space of dimension smaller than $n$. But any subset of such a vector space has measure $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2870136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding a set of continuous functions with a certain property I need help finding the set of continuous functions $f : \Bbb R \to \Bbb R$ such that for all $x \in \Bbb R$, the following integral converges:
$$\int_0^1 \frac {f(x+t) - f(x)} {t^2} \ \mathrm dt$$
I am thinking it could be the set of constant functions but ... | Partial answer: if $f$ is differentiable then it is constant
We write $f(x+h) = f(x) + h g(h)$ where $g(h)$ is continuous and $g(0) = f'(x)$.
Then the required integral becomes:
$$\int_0^1 \frac {g(t)} t \ \mathrm dt$$
If WLOG $g(0) > 0$ then there is $\delta > 0$ such that $g(t) > \frac12 g(0)$ for every $0 \le t < \d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2870314",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "12",
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Doubt about how to find a Lipschitz constant I have a doubt about a sentence of my Calculus text.
Let $f: [t_1, t_2]\times \mathbb{R}^n \to \mathbb{R}^n, (t,y)\to f(t,y)$ such that $|\partial_{y_j} f_i|$ is continuous and bounded for every $i,j=1,...n$. Then f has $\sqrt n L$ as Lipschitz constant (with respect to y... | I suspect it is using the fact that $\|x\|_2 \le \|x\|_1 \le \sqrt{n} \|x\|_2$.
Consider the path $x_0=x \to (y_1,x_2,...,x_n) \to (y_1,y_2,x_3,...,x_n) \to \cdots \to x_n=y$, where one component changes at a time.
\begin{eqnarray}
\|f(x,t)-f(y,t)\|_2 &\le& \sum_k \|f_(x_{k+1},t)-f(x_k,t)\|_2 \\
&\le& \sum_k L \|x_{k+1... | {
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What is the probability that each inhabitant of a three-story building lives on a different floor?
There is a multi-apartment building with $3$ stories and $4$ apartments at each story. In each apartment lives one person. Three random inhabitants of this building are standing outside the building. What the probability... | Your understanding of the solution is correct.
To do it without taking the order of selection into account, observe that there are $\binom{12}{3}$ ways to select three of the twelve apartments. The favorable cases are those in which one of the four apartments on each floor is occupied by the inhabitants standing outsi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The product of three consecutive integers is ...? Odd? Divisible by $4$? by $5$? by $6$? by $12$? If i have the product of three consecutive integers:
$n(n+1)(n+2)$, so the result is:
$A)$ Odd
$B)$ Divisible by $4$
$C)$ Divisible by $5$
$D)$ Divisible by $6$
$E)$ Divisible by $12$
My thought was:
$i)$ If we have three ... | Looks good. The multiplication table for three, is $3,6,9,12,15$ etc any consecutive three integers has to include one of these because there are only two integers between them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2870587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to treat a constant when integrating I am wondering what to do when coming across an integral like this where $a$ is a constant:
$$\int^{1000}_a (x-a){1\over1000}dx $$
As far as I can see, it should be ok to do this:
$$ \left.{1\over 1000} \left({{{x^2}\over2} -ax}\right)\right|^{1000}_a$$
But the book I am using ... | The two expressions are different only by a constant. When you take bounds, the constant vanishes, giving the same result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2870710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Integrate :- $\int dx/(\sin(x) + a\sec(x))^2$ Please help me in evaluating this integral
$$
\int \frac{1}{(\sin(x) + a \sec(x))^2}\,dx
$$
I tried by converting $\sec(x)$ to $\cos(x)$ and by solving it became more complicated so guys please guide me further.
| Hint:
$$\dfrac1{(\sin x+a\sec x)^2}=\dfrac1{2(\sin x\cos x+a)^2}+\dfrac{\cos2x}{2(\sin x\cos x+a)^2}$$
The second part is elementary.
$$\dfrac1{(\sin x\cos x+a)^2}=\dfrac{\sec^2x(1+\tan^2x)}{(\tan x+a\tan^2x+a)^2}$$
Choose $\tan x=u$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Isomorphism Clarification and Identification I understand that to claim isomorphism, algebraic properties must be preserved...so....a) closed under multiplication and closed under addition.
However, I am unsure how to apply these condition-testing properties in the context of polynomial/space based questions
For exampl... | The simpler way to find out if two finite dimensional vector spaces are isomorphic, as others said, is to find out what if the dimension of the vector space that you're studying. So we can state a little "theorem"
Two finite dimensional vector spaces $V,W$ are isomorphic iff $$\dim(V)=\dim(W)$$ Here are some referenc... | {
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"timestamp": "2023-03-29T00:00:00",
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Showing that $A=\{a_1,...,a_r\}$ is a closed set Let $A=\{a_1,\dots,a_r\}, a_i \in \mathbb{R}, i=1,...,r$. Show that $A$ is a closed set. As a bit of a beginner, I have written down a proof and I wanted to see if it is good enough/well structured.
So I want to show that $\partial A\subseteq A$, where $\partial A=\{a \i... | It seems to me the introduction of $\partial A$ into the discussion overly complicates things.
I would argue it like this, which to my mind is somewhat simpler:
For each $a_k$, $1 \le k \le r$, the set $\Bbb R \setminus \{ a_k \}$ is open; this is easy to see, since if $p \in \Bbb R \setminus \{a_k\}$, the open inter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2871206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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} |
Finite morphisms of schemes are closed. Let $f : X \rightarrow Y$ be a finite morphism of schemes. I have to show $f$ is closed. I have been able to prove that for any open affine $V= \mathrm{Spec}(B)$ in $Y$, $f : f^{-1}V \rightarrow V$ is a closed morphism. But I am having trouble to extend this globally. I am arguin... | It follows from the following lemma:
Lemma: Let $X$ be a topological space, and let $\{U_{i}\}_{i \in I}$ be an open cover for $X$. Then a subset $C \subset X$ is closed if and only if $C \cap U_{i}$ is closed in $U_{i}$ for each $i \in I$.
(Note that there are no assumptions on the cardinality of the index set $I$!)
P... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Surface integral - cone below plane After several years I suddenly need to brush up on surface integrals. Looking through my old Calculus book I have been attempting to solve some problems, but the following problem has really made me hit a wall, even though it probably is quite easy to solve:
Find $\int \int_S y dS$, ... | The domain of integration is the projection on the $xy$-plane of the intersection between the plane and the cone (an ellipse in 3D space).
Using cylindrical coordinates we get:
$$ x = r\cos \theta\\y=r\sin\theta\\ z =z\\
\sqrt 2 r= r\sin\theta + 1 $$
We can thus determine the expression for the ellipse:
$$ r = \frac{1}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Example of two subgroups $H$ and $K$ of a non-abelian group $G$ such that $HK$ is not a subgroup of $G$? I need to disprove that the set $HK$ is not a subgroup of $G$ if $G$ is non-abelian. Does anyone have a trivial or easy counterexample? I cannot think of one...
| Take $G=S_3,H=\left<(1,2)\right>$ and $K=\left<(2,3)\right>$. Then $H$ and $K$ are subgroups of $G($each containing two elements$)$ and $$HK=\{1,(12),(23),(132)\}$$a set of size $4$. Therefore, $HK$ is not a subgroup of $G$$($by Lagrange's Theorem$)$ since $4$ does not divide $6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2871614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Bayesian Update in the Presence of Noise - Estimating the Ratio of Balls in a Jar There are two jars with red balls and blue balls. Your goal is to estimate the ratio of red to blue for each jar, assuming some initial prior for each jar.
On each iteration, you are handed a ball. You can see its color, and are told wh... | I'll assume that, as specified in a comment, the balls are known to come from either jar with equal probability, independently chosen for each ball.
I take your first paragraph to imply that your initial prior for the ratios factorizes into a product of marginal priors for the individual jars. This factorizability won'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2871696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving a Polynomial Limit where x → 0 Prove
$$\lim \limits_{x\to 0} x^3+x^2+x = 0$$
Note
$|f(x)-L| = |x^3+x^2+x|$
Assume
$\ \ |x-c|<\delta \implies |x| < \delta$
$\implies |x^3+x^2+x|<\delta\cdot|x^2+x+1|$
Assume $|x| < 1 \implies -1 < x < 1 \implies 0 < x+1 < 2$
And I am not sure where to go from there, since I can't... | As an alternative by squeeze theorem assuming $|x|<1$
$$0\le |x^3+x^2+x|=|x||x^2+x+1|\le 3|x| \to 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2871798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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Maximazing multinomial distribution I have an exercise in probability theory, which I can't solve:
There are 3 factories A B C, which produce 3 types of light bulbs. Factory A / B / C makes 40 / 20 / 40 percent of whole light bulbs. The probability of manufacturing first type of light bulb in factory A / B / C is 0.6 /... | Max $P = \dfrac{6!}{x_1!.x_2!.x_3!.x_4!.x_5!.x_6!.x_7!.x_8!.x_9!}.24^{x_1}.12^{x_2}.04^{x_3}.06^{x_4}.08^{x_5}.06^{x_6}.2^{x_7}.08^{x_8}.12^{x_9}$
with
$x_1$ - Type 1 made in Factory A - probability $= .6*.4 = 0.24$
$x_2$ - Type 2 made in Factory A -probability $= .3*.4 = 0.12$
$x_3$ - Type 3 made in Factory A probabi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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if $a^3+b^3 +3(a^2+b^2) - 700(a+b)^2 = 0$ then find the sum of all possible values of $a+b$
If $a+b$ is an positive integer and $a\ge b$ and
$a^3+b^3 +3(a^2+b^2) - 700(a+b)^2 = 0$ then find the sum of all possible values of $a+b$.
I tried a lot to solve it, i came to a step after which i was not able to proceed for... | Here is a start, not a full answer.
Let $s=a+b$ and $p=ab$. Then $3 p (s + 2) = (s - 697) s^2$ as WA tells us.
Then $s+2$ divides $(s - 697) s^2$ and so $s+2$ divides $2796 = 2^2 \cdot 3 \cdot 233$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
compute the summation $\sum_ {n=1}^\infty \frac{2n-1 }{2\cdot4\cdots(2n)}= \text{?}$ compute the summation
$$\sum_ {n=1}^\infty \frac{2n-1 }{2\cdot4\cdots(2n)}= \text{?}$$
My attempts : i take $a_n =\frac{2n-1}{2\cdot4\cdots(2n)}$
Now
\begin{align}
& = \frac{2n}{2\cdot4\cdot6\cdots2n} -\frac{1}{2\cdot4\cdot6\cdots2n}... | Using telescopic approach with$$a_n=\dfrac{1}{2\cdot4\cdot6\cdot\cdots\cdot2n}$$we have $$S=\sum_{n=1}^{\infty}{a_n-a_{n-1}}=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2872093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Polynomial Division: dividing by a double root I have this fairly interesting problem that is based on Polynomial Division and/or factor/remainder theorem.
Determine the values of $a$ and $b$ such that $ax^4 + bx^3 -3$ is divisible by $(x-1)^2$.
This is interesting because the root we are dividing by is a double root... | Method one (compare the coefficients):
$$\begin{eqnarray}ax^4+bx^3-3 &=& (x-1)^2(cx^2+dx+e)\\
&=& (x^2-2x+1)(cx^2+dx+e)\\
&=& cx^4+(d-2c)x^3+(e-2d+c)x^2+(-2e+d)x+e\end{eqnarray}$$
From here we see that:
$$
\begin{eqnarray*}
c&=&a\\
e&=& -3\\
-2e+d&=& 0\implies d=-6\\
e-2d+c&=&0 \implies c=-9\implies a=-9\\
d-2c&=&b\im... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluate $\iint_D x \sin (y -x^2) \,dA$. Let $D$ be the region, in the first quadrant of the $x,y$ plane, bounded by the curves $y = x^2$, $y = x^2+1$, $x+y=1$ and $x+y=2$. Using an appropriate change of variables, compute the integral
$$\iint_D x \sin (y -x^2) \,dA.$$
I've been reviewing for an upcoming test and this ... | An other interesting variable change which can take us much further and show that this integral cannot be evaluated to closed form solution without series expansion, would be to remove away $x$ in the integral. See that we have $x^2$ inside $\sin(\cdot)$ which can make this happen.
Let, $k = x^2$ and keeping $y$ as it ... | {
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A conditional probability involving order statistic Here is the original question:
Let $X_1,X_2,\cdots$ be i.i.d. random variables uniform on $[0,1]$. Let $N$ be the smallest integer such that $X_N$ is smaller than exactly one of its predecessors $X_1,X_2,\cdots,X_{N-1}$. Find the cumulative distribution function of $... | Let $X^{(i)}$ be the $i$-th order statistic from $n$ draws of the uniform distribution. (To be clear, $X^{(1)}$ is the smallest draw while $X^{(n)}$ is the largest. I chose slightly different notation than the linked Wikipedia article to avoid confusion with the notation you've chosen.)
Conditioning on $\{N>n\}$, $N=n+... | {
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Does the limit rule $\lim_{x \rightarrow 0}\frac{\sin x}{x}=1$ apply to $\lim_{x \rightarrow \pi}\frac{\sin\left(x-\pi\right)}{\left(x-\pi\right)}=1$? In my textbook, I was given an example below :
$$\lim_{x \rightarrow \pi}\frac{\sin\left(x-\pi\right)}{\left(x-\pi\right)}=1$$
Previously I was taught that this formula ... | Set $y=x-\pi$. Then $y$ approaches $0$ if and only if $x$ approaches $\pi$. So we may write the following
$$\lim_{x \rightarrow \pi}\frac{\sin\left(x-\pi\right)}{\left(x-\pi\right)}=\lim_{y \rightarrow 0}\frac{\sin y}{y}=1$$
EDIT: One may also see it as a composition of limits.
| {
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"timestamp": "2023-03-29T00:00:00",
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Notation for element of an ordered tuple? When $X$ is a set, we can write:
for all $x\in X$ ...
But say that $X=(a, b, c, ... ,n)$. I.e. an ordered tuple.
Is it standard notation to still say the following?
for all $x\in X$...
It might be confusing because if you interpret it as a sentence in ZFC, then you're not ... | If we insist on rigor here, we can proceed in a slightly ugly way.
If we define a tuple as a function from sets of integers into the target set then we can proceed as follows.
Let
$$
[n] = \{m \in \mathbb{N}: m<n\} = \{0, \ldots, n-1\} = n
$$
Where the last equality follows if the natural numbers are defined as the Von... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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Ito's formula, and the relationship between dt and dB(t) reading Bernt Oksendal Stochastic Differential Equations.
I have just seen Ito's Formula, after this the author then says where $dX(t)^{2}$ is calculated using
$$dt\cdot dt=dt \cdot dB(t) =dB(t)\cdot dt =0$$
and
$$dB(t) \cdot dB(t) = dt$$
On page 45 at the to... | $dt\cdot dt=0$ is shorthand notation for the fact that the quadratic variation of the identity function with itself is $0$.
$dt\cdot dB(t)=0$ is shorthand notation for the fact that the cross variation of Brownian motion with the identity function is $0$.
$dB(t)\cdot dB(t)=dt$ is shorthand notation for the fact that th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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the derivative of $f(x)=|x|^{\frac 52}$ From my point of view answer must be $\frac 52|x|^{\frac 32}$...but answer in my text book is $\frac 52 x |x|^{\frac 12}$....help me to solve it.I have tried it to break the function for positive and negative parts and directly differentiated it but the answer is not matched
| Breaking it up into a piecewise function is fine.
$f(x) = x^{\frac 52}$ (for $x \geq 0$) [$1$]
$f(x) = (-x)^{\frac 52}$ (for $x < 0$) [$2$]
Differentiating them separately,
$f'(x) = \frac 52x^{\frac 32}$(for $x \geq 0$) [$1$]
$f'(x) = -\frac 52(-x)^{\frac 32}$(for $x < 0$) [$2$]
The second part can be rearranged to:
$f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Derivative to function ratio Is there a physical meaning for derivative of the function to function ratio? That is, this quantity,
$$
Q(x) = \frac{1}{f(x)}\frac{df(x)}{dx}
$$
Like for instance, if $f$ is the potential energy, this would be work to potential energy ratio.
Or even what can we say about $Q(x)$, say when ... | If $f(t)$ measures the size of the population, and $t$ is time, then $Q(t)=f'(t)/f(t)$ is the rate of change of the population per capita.
For example, if each individual reproduces at a constant rate $r$, then $f'(t)/f(t)=r$, so we get exponential population growth $f(t)=f(0) \, e^{rt}$.
But if the per capita growth r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question of Automorphism $T$ on Finite group with property that $T(x)=x$ only for $x=e$
Let $G$ be finite group, $T$ be automorphism on $G$ with property that $T(x)=x$ only for $x=e$. Then
1) every $g \in G $ can be written as $g=T(x)x^{-1}$ for some $x\in G$
2) Furthermore if $T^2=\mathrm{Identity}$ then $G$ is ... | Since $G$ is finite, the map $f\colon G\to G$ defined by $x\mapsto f(x):=T(x)x^{-1}$ is surjective (so the property 1) holds) if and only if it is injective. And it is injective, because:
\begin{alignat}{1}
&f(x)=f(y)&&\Longrightarrow \\
&T(x)x^{-1}=T(y)y^{-1}&&\Longrightarrow \\
&T(y^{-1}x)=y^{-1}x&&\Longrightarrow \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2873136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proving a formula for $\pi$ I found a formula for $\pi$ in this paper. However, I could not find any proof of this formula, and I don't know how to approach to it. Is there good explanation for it?
$$ \pi + 3 = \sum_{n=1}^\infty \frac{n2^nn!^2}{(2n)!} $$
| Here's part of an exercise from the book Pi and the AGM by Borwein and
Borwein.
Prove that
$$\frac{2x\sin^{-1}x}{\sqrt{1-x^2}}=\sum_{m=1}^\infty \frac{m!^2(2x)^{2m}}{m(2m)!}.\tag{1}$$
Hint: show that $f=(\sin^{-1}x)/\sqrt{1-x^2}$ satisfies
$(1-x^2)f'=1+xf$.
Granted $(1)$, differentiating and multiplying by $x$ gives
a ... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluate $\int_M(x-y^2+z^3)dS$ Evaluate $\int_M(x-y^2+z^3)dS$ when $M$ is the part of the cylinder $x^2+y^2=a^2$ where $a>0$ which is between the two planes $x-z=0$ and $x+z=0$.
So I did not manage to use green/gauss/stocks, so I tried to solve it as a surface integral.
first to find $\|n\|$ we use the parameterisation... | You have to compute this as a surface integral. Stokes' and Gauss' theorems deal with the flux of certain vector fields across a surface, but here a scalar function (representing, e.g., a temperature) is integrated over $M$.
Your parametrization is fine, and leads to the scalar surface element
$${\rm d}S=|\phi_u\times\... | {
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Runs Application Probability Problem Feller Taken from a problem in Feller's Probability Theory Vol. 1 Chapter 2 Section 5:
Suppose that an observation yielded the following arrangement of empty and occupied seats along a lunch counter: EOEEOEEEOEEEOEOE. There are $11$ runs here, and Feller argues that the probability ... | I think it is possible that Feller's calculator was off by 0.0001, since I am getting the answer to be 0.0577 after rounding. (It is 3/52).
So, how do we go about calculating this? This is a simple counting argument: Assuming a length 16 string with 6 Es and 10 Os, how do we count the probability of getting a string th... | {
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Differential Polynomials(?) Consider an equation of the form:
cy"+cy'+cy
Or something of the form. Essentially, it's a polynomial but instead of powers, there are derivatives. Do these kind of things have a name? Or are they completely useless?
Note: I KNOW what Taylor Polynomials and the like are, but I mean something... | Yes the polynomial associated to $$ ay'' + by' +cy =0$$ is $$P(\lambda )= a \lambda ^2 + b \lambda +c$$ which is called the charateristic polynomial.
This polynomial plays a very important role in finding the solutions to your differential equation.
The genera solution to the differential equation is $$ y=C_1 e^{\la... | {
"language": "en",
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What algorithm do scientific calculators use to calculate Logarithms I have been introduced to numerical analysis and have been researching quite a bit on its applications recently. One specific application would be the scientific calculator.
From the information that I found, computers typically use the Taylor Series... | Modern Computer Arithmetic suggests using an arithmetic-geometric mean algorithm. I'm not sure if this approach is meant for the low amount of precision one typically works with or if its meant for calculation in very high precision.
Another approach is to observe that the Taylor series for $\ln(x)$ is efficient if $x... | {
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"timestamp": "2023-03-29T00:00:00",
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Definition of dependence in probability Here is classical definition and example of dependent events.
"When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event. For example, if you were to draw a two cards from a deck of $52$ cards. If on your first d... | Perhaps the most concise definition of independent events is $P(A\,\text{and}\,B)=P(A)P(B)$. In your example, constants $p,\,q$ exist for which $P(A)=p,\,P(A\,\text{and}\,B)=P(B)=q$, with independence only if $p=1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simple problem on conditional geometric probability The occurrence of the event $A$ is equally likely in every moment of the interval $[0, T]$. The probability of event $A$ occurring at all in this interval is $p$. Given that $A$ hasn't occurred in the interval $[0,t]$ what's the probability that $A$ will occur in $[t,... | We have $$\Pr(A\in (0,T)\mid A\not\in(0,t))=\frac{\Pr(A\in (t,T))}{\Pr(A\not\in(0,t)}.$$
The top probability is $p\frac{T-t}{T}$ and the bottom is $1-p\frac{t}{T}$; rearranging this gives the same answer you have.
| {
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"url": "https://math.stackexchange.com/questions/2873834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Types of undefined for removable discontinuities and vertical asymptotes Consider this rational function:
$$
f(x) = \frac{x^2 - 2x - 24}{x^2 + 10x + 24}
$$
I have been taught that to solve for a removable discontinuity, I find the $x$ values such that both the numerator and denominator are equal to $0$; and to solve fo... | We have that
*
*at $x=-4$ the function is not defined but we don't have any vertical asymptote since for $x\to -4$ we have $f(x) \to -5$
*at $x=-6$ also the function is not defined but we have a vertical asymptote since for $x\to -6$ we have $f(x) \to \pm \infty$
Note that in the case of $x=-4$ we can remove the ... | {
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How to prove $x^2<4 \implies|x|<2$?
How to prove $x^2<4 \implies|x|<2$?
I don't know exactly there is a proof for this or we take this as an axiom. Please help.
What I did so far is, $$x^2-4<0$$
$$(x-2)(x+2)<0$$
$$-2<x<2$$
From this step can we directly say? $|x|<2$
|
What I did so far is, $$x^2-4<0$$
$$(x-2)(x+2)<0$$
Since $\,x^2=|x|^2\,$, you could also factor it as:
$$\left(|x|-2\right)\left(|x|+2\right) \lt 0$$
Given that $\,|x|+2 \gt 0\,$ it follows that $\,|x| - 2\lt 0\,$.
| {
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Mapping and conservation law What is a general map from a point sitting in one dimensional space, to a set of points sitting in two dimensional space, to a set of points sitting in three dimensional space, to a set of points embedded in a torus sitting in four dimensional space? This is what I have but I'm not sure if ... | If I understand correctly you have first a mapping $\mathbb R \stackrel{f}{\longrightarrow} \mathscr P(\mathbb R^2),$ where $\mathscr P$ denotes power set, and then a chain of mappings
$\mathbb R^2 \stackrel{g}{\longrightarrow} \mathbb R^3 \stackrel{\iota} {\hookrightarrow} \mathbb T^4$ acting on each element in $f(t)... | {
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"timestamp": "2023-03-29T00:00:00",
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Finite dimensional subspace is weak star closed I want to show the weak star closed convex hull of a finite set of points is contained in the linear span of those points.
It's enough to show that any finite dimensional subspace $V$ of a Banach space $Z$ is weak star closed in $Z$. Since $V$ is finite dimensional, it is... | This is true not only for arbitrary subspaces of Banach spaces, but in fact for arbitrary subspaces of locally convex spaces. For any locally convex space $X$, the weak and original closures of any convex set $E \subset X$ are the same (see, for instance, Theorem 3.12 here). Since Banach spaces are locally convex and s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2874351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Why is $\epsilon^{p,q}(X):=\Gamma(X,\bigwedge^p\Omega^1_X\otimes\overline{\bigwedge^p\Omega^1_X})$? Here $\Omega^1_X=(T^*X)^{1,0}$, from : this notes we have
$\epsilon^{p,q}(X):=\Gamma(X,\bigwedge^p\Omega^1_X\otimes\overline{\bigwedge^p\Omega^1_X})$
But I wonder why it's tensor not wedge? Shouldn't it be $\epsilon^{p,q... | Let us reduce ourselves to the fiber. If $V$ and $\bar V$ are two different $\Bbb R$-spaces with $W=V\oplus \bar V$, and assume there is an isomorphism over $\Bbb R$, $v\to\bar v$, between the two summands.
We fix a basis $B$ of $V$. Thus also one, $\bar B$, for $\bar V$.
Now let us consider an element in $$\wedge^\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2874460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
what is the difference between $D(g\circ f)(x)$ and $Dg(f(x))$? Let $A$ be an open in $\mathbb R^m$. Let $B$ be open in $\mathbb R^n$. Let $f: A \to \mathbb R^n$ and $g: B \to \mathbb R^p$ where $B = f(A)$. If $f$ is differentiable at $a$ and $g$ is differentiable at $f(a) = b$. Then $D(g\circ f)(a) = Dg(f(a) Df(a)$.... | I think you are missing parentheses, which makes your question hard to parse. Here is how I parse it:
I can not understand what is the difference between $D(g\circ f)(x)$ and $Dg(f(x))$.
Think of it this way:
$D$ is an operator, which "takes" as input a function and outputs another function. Therefore, if $f$ is the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2874622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Euler Lagrange, weird Second Order DE I have found the Euler-Langrange equation of the functional:
$$\int_{a}^{b} y'^2+y^4\,\mathrm dx$$
to be $y'' = 2y^3$
How do I solve this non-linear DE?
| $$y''=2y^3$$
$$2y''y'=4y^3y'$$
$$y^4=y'^2+c_1$$
$$y'=\pm \sqrt{y^4-c_1}$$
$$x=\pm\int \frac{\mathrm dy}{\sqrt{y^4-c_1}}+c_2$$
This is an elliptic integral : http://mathworld.wolfram.com/EllipticIntegral.html
The inverse function $y(x)$ involves the sn Jacobi elliptic function :
http://mathworld.wolfram.com/JacobiEllipt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2874756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
In a village, $90\%$ of people drink Tea, $80\%$ Coffee, $70\%$ Whiskey, $60\%$ Gin. Nobody drinks all four. What percentage of people drinks alcohol?
In a small village $90\%$ of the people drink Tea, $80\%$ Coffee, $70\%$ Whiskey and $60\%$ Gin. Nobody drinks all four beverages. What percentage of people of this vil... | You can use inclusion/exclusion but you might not have enough information. Or then again you might.
The number of people who are in $A$ or $B$ is $A + B - (A\cap B)$ and so if $A+B > 100$ percent we can conclude $A+B - 100\le A\cap B \le \min (A,B)$
So $WHISKEY + GIN - 100 = 70+60 -100 = 30 \le(WHISKEY \cap GIN) \le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2874859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
"answer_count": 7,
"answer_id": 0
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Metric on the unit circle Let $d$ be a metric on the unit circle $S^1$ which defines the usual topology. Let $D = \sup_{x, y \in S^1} d(x,y)$ be the diameter for $S^1$ under this metric.
Is it true that the map $d : S^1 \times S^1 \to [0,D]$ is an open map?
Thank you!
| The map is not always open. Here is a simple counter-example.
The unit circle is homeomorphic to the following figure.
This figure is endowed with the euclidean metric of the plane, that we then transport to a metric on the unit circle using the homeomorphism.
Now consider the two points in red on the figure (or more ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2875111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Weird Notation for Trace of an Endomorphism I am having some difficulty understanding a piece of notation from Riemannian Geometry: and Introduction to Curvature by John M. Lee.
In Section 2 just under equation 2.3 Lee defines the trace operator which lowers the rank of a tensor by 2.
He defines the map:
$$\mathrm... | For fixed $\omega^1, \ldots, \omega^l \in V^*$ and $V_1, \ldots, V_k \in V$ the notation $F(\omega^1,\dots,\omega^l,\bullet,V_1,\dots,V_k,\bullet)$ signifies an element $G \in T_1^1(V)$ such that
$$G(\omega^{l+1}, V_{k+1}) = F(\omega^1, \ldots, \omega^l, \omega^{l+1}, V_1, \ldots, V_k, V_{k+1}).$$
Then $\operatorname{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2875264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Help with Calculus Optimization Problem! We wish to construct a rectangular auditorium with a stage shaped as a semicircle of radius $r$, as shown in the diagram below (white is the stage and green is the seating area). For safety reasons, light strips must be placed on the perimeter of the seating area. If we have $45... | so you are going correctly
you have $$2x+2r+\pi r=45\pi+60$$
so $$2x=45\pi+60-2r-\pi r$$
now you have $$area=2x*r-\frac{\pi r^2}{2}$$
now replace the value of 2x in the area equation from the 1st equation
to obtain the area in a single variable so that you can calculate the derivative w.r.t. r.
$$A=(45\pi+60-2r-\pi r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2875381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Trying to evaluate $\int_{0}^{\infty}\frac{\ln^{s}(t)\ln(1+t)}{t(1+t^2)}\mathrm dt$ $$G(s)=\int_{0}^{\infty}\frac{\ln^{s}(t)\ln(1+t)}{t(1+t^2)}\mathrm dt$$
Trying a substitution of $t=e^x$
$$G(s)=\int_{-\infty}^{\infty}\frac{x^s\ln(1+e^x)}{1+e^{2x}}\mathrm dx$$
I trying to evaluate $G(s)$, but unable to. Can anyone hel... | Assuming $s\in\mathbb{N}$, you may consider that
$$ \int_{0}^{+\infty}\frac{t^{a-1}\log(1+t)}{1+t^2}\,\mathrm dt=\frac{\pi}{4\sin(\pi a)}\left[H_{-1/2-a/4}-H_{-a/4}+2\log(2)\cos\frac{\pi a}{2}+\pi\sin\frac{\pi a}{2}\right] $$
for any $a$ in a neighbourhood of the origin. In order to compute the wanted integral you may ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2875459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Explain the odd/even inequality in the heights of numbers under the Collatz $(3x+1)/2$ transformation? My kid asked me a question and I'm finding it hard to answer: if every number under repeated application of the Collatz transformation1 eventually reaches $1$, then it must be true for both even and for odd numbers. ... | The reason why one would expect more even numbers to appear at a given height rather than odd numbers is the following: the number of even numbers of height $n$ should be equivalent to the number of odd numbers of height$<n$. Why is that? Let the height of odd $g$ be $k<n$. Hence, $g\cdot 2^{n-k}$ will have height $n$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2875576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Find all $n \in \mathbb{N}$ such that ${{2n}\choose{n}} \equiv (-1)^n\pmod{2n+1}$
Find all $n \in \mathbb{N}$ such that $${{2n}\choose{n}} \equiv (-1)^n\pmod{2n+1}.$$
I know that if $2n+1$ were prime number, then
$${{2n}\choose{n}} = \frac{(2n) (2n-1) \cdots (n+1)}{n!} \equiv \frac{(-1)(-2)\cdots(-n)}{n!} = (-1)^n \... | The solutions are those $n \in \mathbb N$ for which $2n+1$ is either prime or a Catalan pseudoprime.
We say that $2n+1$ is a Catalan pseudoprime if it is a composite number and
$$(-1)^n\, C_n \equiv 2 \pmod{2n+1}$$
where $C_n$ is the $n$-th Catalan number, that is,
$$C_n = \frac 1 {n+1} \binom {2n} n.$$
Rewriting the d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2875712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Exercise 2.4.1 of Durret's Probability : Theory and Examples 4th ed- Is this correct? In exercise 2.4.1. of Durret's Probability book, we're looking to construct a sequence of r.v. $X_n\in \{ 0,1 \}$, $X_n\rightarrow 0$ in prob., $N(n)\rightarrow \infty$ a.s., and $X_{N(n)}\rightarrow 1$ a.s.
Here's picture of the solu... | For $2^{n}\leq k <2^{n+1}$ we have $P\{X_k >\epsilon \} =\frac 1 {2^{n}}<\frac 2 k \to 0$ as $k \to \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2875829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Monotonicity of trigonometric function Consider the function
$$f(x)=\frac{\sin(x)}{\sin((2k+1)x)}$$
with $k$ a positive integer. It seems that $f$ is strictly increasing in $[0,\frac{\pi}{2(2k+1)}]$. Is there some easy proof of this monotonicity property that does not invole differentiation (through suitable inequalit... | You have $\sin2x=2\sin x \cos x$ and $\sin 3x=\sin x \cos 2x + \cos x \sin 2x =\sin x (\cos 2 x + 2 \cos^2 x)$, and similar formulas for higher $\sin (kx)$. Then
$$\frac{\sin(3x)}{\sin x} = \cos 2 x + 2\cos^2x
$$
where the RHS is decreasing in $x$ and positive as long as $\cos 2x>0$. That gives you an interval where $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2875945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Proof of dy=f’(x)dx I’ve been wondering about the usage of $dy=f’(x)dx$ in my textbook.
There’s not a single justification of how it is proved and it just states that it is true.
Since $dy/dx$ can’t be assumed as a fraction, I’m guessing there’s more to it than just multiplying by $dx$ on both sides.
Are there any proo... | Well the derivative is given by:
$$\lim_{dx \to 0} \frac{f(x+dx)-f(x)}{dx}=\lim_{dx\to 0} \frac{dy}{dx}$$
By definition the derivative is the rate of change of y with regard to x. That's why RHS stands. As you realise $\frac{dy}{dx}$ is not just a notation but it's mathematically how derivative is been defined. Since $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2876054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Proof Verification: If $A \subset B$ and $B \subset C$, then $ A \cup B \subset C$ I am trying to prove that:
If $A \subset B$ and $B \subset C$, then $ A \cup B \subset C$
My proof is :
Given some $x \in A \cup B$, it is true that either $x \in A$ and/or $x \in B$. IN the case that $x \in A$ it is true that $x \in ... | Another approach is by using Venn diagrams. Draw circles $A$, $B$ and $C$ for three sets such that $A$ is contained in $B$ and $B$ is contained in $C$ (according to given set inclusions). So you have $A$ as the innermost, $B$ in the middle and $C$ as the outermost of them. Now $A\cup B$ is given by the middle circle wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2876194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Implicit Differentiation- Related Rates Suppose that price, $p$ in dollars, and the number of items sold, $x,$ are related by: $p^2 - xp + x^2 = 175.$
If $p$ and $x$ are functions of time, how fast is $p$ changing with respect to time when $p=\$ 10$ and $x$ is increasing by $5$ units per day?
So far I’ve got: $$\frac{d... | Let's see:
\begin{align*}
\frac{d}{dt}[\,p^2-xp+x^2&=175\,] \\
\underbrace{2p\dot{p}}_{\text{Chain}}-\underbrace{(x\dot{p}+\dot{x}p)}_{\text{Product}}+\underbrace{2x\dot{x}}_{\text{Chain}}&=0 \\
\dot{p}(2p-x)&=\dot{x}(p-2x) \\
\dot{p}&=\frac{\dot{x}(p-2x)}{2p-x}.
\end{align*}
The issue is that the problem statement doe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2876293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
how to demonstrate that $\lim_{x\to2}3x^2=12$ using $\gamma$ and $\epsilon$ definition how to demonstrate that $\lim_{x\to2}3x^2=12$ using $\gamma$ and $\epsilon$ definition?
My current steps I have in order from top to bottom:
$|3x^2-12|<\epsilon$ and $0<|x-2|<\gamma$
assume $z=x-2$
$0<|z|<\gamma$ and $3z^2+12|z|<\ep... | Let $|x-2|\lt 1$ , then
$-1<x-2<1$, or $3<x +2<5$.
$|3x^2-12| = 3|x-2||x+2|$.
Let $\epsilon >0$ be given.
Choose $ \delta = \min (1,\epsilon/(15))$.
Then $|x-2| \lt \delta$ implies
$3|x-2||x+2| \lt $
$(3)(5)|x-2| \lt (15)\delta \lt \epsilon$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2876398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Necessary and sufficient condition for convergent series Let $(a_i)_{i \in \mathbb{N}}$ be a sequence of positive reals such that
$$
\limsup_{i \rightarrow \infty} a_i \, i =0.
$$
Is this condition necessary and sufficient for $\sum\limits_{i=1}^\infty a_i < \infty$?
Of course, if $a_i = 1/i$, then the series is infini... | Note that $\sum_{n=2}^\infty\frac1{n\log n}$ diverges (this follows from the integral test), but $\lim_{n\to\infty}n\frac1{n\log n}=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2876467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Two distinct differentials at the same point, with respect to different norms Is it possible for a function $f:E\to F$ (where $E,F$ are normed vector spaces) to admit two distinct differentials in a point $a\in E$ ? Of course one for every pair of norms on $E$ and $F$. It is known that if the two norms on $E$, and the ... | Sure. Let $F$ have infinite dimension and let $x_1,x_2,\dots\in F$ be linearly independent. We can then pick two different norms for which this sequence of linearly independent vectors converges to two different vectors. Say $\|\cdot\|_1$ is a norm on $F$ such that the sequence $(x_n)$ converges to $x$ and let $\|\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2877124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Calculate matrix powering given one outer product: $(x\cdot{y}^T)^k$ It is an exercise on the chapter one of a book. Book: "Matrix Computations 4th edition" by Golub and Van Loan.
It reads: Give an $O(n^2)$ algorithm for computing $C=(x\cdot{y}^T)^k$ where $x$ and $y$ are $n$-vectors.
I did a lot of research because ... | If $k=1$, then $O(n)$ operations are required.
If $k>1$ then $(x y^T)^k = x (y^T x)^{k-1} y^T = (y^T x)^{k-1} xy^T $.
$(y^Tx)^{k-1}$ takes $O(n)$ operations, and there are $n^2$ multiplications to
compute $(y^T x)^{k-1} xy^T $.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2877235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
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