Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Example of a $\mathbb{Z}$-module with exactly three proper submodules? What is an example of a $\mathbb{Z}$-module which has exactly three proper submodules?
| A $\mathbf Z$-module is but an abelian group. For any prime $p$, $\mathbf Z/p^3\mathbf Z$ is an example of an abelian groups with three proper subgroups, which are linearly ordered. These subgroups are
$$0\subsetneq p^2\mathbf Z/p^3\mathbf Z\simeq \mathbf Z/p\mathbf Z\subsetneq p\mathbf Z/p^3\mathbf Z\simeq\mathbf Z/p^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1590175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
find the maximum possible area of $\triangle{ABC}$ Let $ABC$ be of triangle with $\angle BAC = 60^\circ$
. Let $P$ be a point in its interior so that $PA=1, PB=2$ and
$PC=3$. Find the maximum area of triangle $ABC$.
I took reflection of point $P$ about the three sides of triangle and joined them to vertices of triangle... | Let $\theta=\measuredangle PAB$ in the triangle you specify. Then
$\measuredangle PAC=60°-\theta$.
By the law of cosines,
$$2^2=1^2+c^2-2\cdot 1\cdot c\cdot \cos\theta$$
$$3^2=1^2+b^2-2\cdot 1\cdot b\cdot \cos(60°-\theta)$$
Solving those equations for $b$ and $c$,
$$c=\cos\theta+\sqrt{\cos^2\theta+3}$$
$$b=\cos(60°-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1590262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
} |
Simplify $2(\cos^2 x - \sin^2 x)^2 \tan 2x$ Simplify: $$2(\cos^2 x - \sin^2 x)^2 \tan 2x$$ After some sketching, I arrive at: $$2 \cos 2x \sin2x$$
Now according to the answer sheet, I should simplify this further, to arrive at $\sin 4x$. But how do I derive the latter from the former? Where do I start? Hoe do I use my ... | Use $$\cos^2x-\sin^2x=\cos2x$$ and $$\tan A=\dfrac{\sin A}{\cos A}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1590359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Creating a set of vectors Suppose we are given a number k, and we have to construct $2^k$ vectors in $2^k$ dimensional space using only the coordinates 1 and -1 so that all the vectors are orthogonal to each other. How can such a construction be made?
For, example, for k=2, a valid construction might be
1,1,-1,-1
1,1,1... | A matrix $A \in M_n(\mathbb{R})$ is called a Hadamard matrix of order $n$ if the entries of $A$ are $\pm 1$ and the columns of $A$ are mutually orthogonal. Using the notion above, you ask how can one construct a Hadamard matrix of order $2^k$. The basic observation is that if $H$ is a Hadamard matrix of order $n$ then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1590478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$(x^2+1)(y^2+1)(z^2+1) + 8 \geq 2(x+1)(y+1)(z+1)$ The other day I came across this problem:
Let $x$, $y$, $z$ be real
numbers. Prove that
$$(x^2+1)(y^2+1)(z^2+1) + 8 \geq 2(x+1)(y+1)(z+1)$$
The first thought was power mean inequality, more exactly : $AM \leq SM$ ( we noted $AM$ and $SM$ as arithmetic and square ... | For real $x$,we have(or Use Cauchy-Schwarz inequality)
$$x^2+1\ge\dfrac{1}{2}(x+1)^2$$
the same we have
$$y^2+1\ge\dfrac{1}{2}(y+1)^2$$
$$z^2+1\ge\dfrac{1}{2}(z+1)^2$$
so
$$(x^2+1)(y^2+1)(z^2+1)\ge\dfrac{1}{8}[(x+)(y+1)(z+1)]^2$$
Use AM-GM inequality
$$\dfrac{1}{8}[(x+1)(y+1)(z+1)]^2+8\ge 2\sqrt{\dfrac{1}{8}[(x+)(y+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1590588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Adjoint action on Lie algebra su(2) ($A \in SU(2), X \in \mathfrak{su(2)} \Rightarrow AXA^{-1}\in \mathfrak{su(2)}$) I am trying to understand ho $SU(2)/\{\pm I\} \cong SO(3)$ (see: how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$) but i am not sure about the adjoint action. In especially, as I understand, the adjoint actio... | An element $A$ of $\mathfrak{su}(2)$ must satisfy $A=-A^*$ and $tr(A)=0$, because, respectively, of the conditions $AA^*=I$ and $\det(A)=1$ (see Lee's Book proposition 5.38, pg 117, to see how to calculate the tangent space. hint: use $\Phi:SL(2,\mathbb{C})\to SL(2,\mathbb{C})$, given by $\Phi(X)=XX^*$, and so $SU(2)=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1590691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\int_0^{+\infty}(\frac{\arctan t}{t})^2dt$ Is it possible to calculate $$\int_0^{+\infty}\Big(\frac{\arctan t}{t}\Big)^2 dt$$ without using complex analysis?
I found this on a calculus I book and I don't know how to solve it.
I tried to set $t = \tan u$ but it didn't help.
| The answer can be derived from a tailor-made version of Parseval's theorem, as already suggested by Ron Gordon. The Fourier sine series of $x$ over $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ is given by:
$$ x = \sum_{k\geq 1}\frac{(-1)^{k+1}}{k}\,\sin(2k x) \tag{1} $$
and if $k,j\in\mathbb{N}^+$ we have:
$$ \int_{0}^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1590772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Relation between hypergeometric functions? Is there any relations between the following hypergeometric functions?
$$\ _2F_1(1,-a,1-a,\frac{1}{1-z})$$
$$\ _2F_1(1,-a,1-a,{1-z})$$
$$\ _2F_1(1,a,1+a,\frac{1}{1-z})$$
$$\ _2F_1(1,a,1+a,{1-z})$$
| First, you need to use Barnes integral representation
$${}_2F_1(a,b;c;z) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\frac{1}{2\pi i}
\int_{-i\infty}^{+i\infty}\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(-s)}{\Gamma(c+s)}(-z)^sds. $$
The Gauss hypergeometric function ${}_2F_1(a,b;c;z)$ is usually defined by a power series that conve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1590872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Question about creating a volume form for $SL(2,\mathbb{R})$ This problem comes out of R.W.R. Darling (Differential Forms and Connections) ch.8. In the chapter he shows that if $M$ is an $n$-dimensional differential manifold immersed in $\mathbb{R}^{n+k}$, and $\Psi$ is an immersion from $\mathbb{R}^n \rightarrow \math... | First, I would suggest that you practice computing some pullbacks in a more basic setting. For example, if $f(u,v)=(u+v^2,uv,u^3+v^3)=(x,y,z)$ what is $f^*(x\,dy\wedge dz + z\,dx\wedge dy)$? You should learn how to do this without ever writing down a push-forward.
Second, you want to compute $L_{A^{-1}}^*(dx\wedge dy\w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1590959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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When does equality hold for $|x+y| \leq |x|+|y| ?$ This is problem 1-2 in Calculus on Manifolds. Spivak.
He gives two hints.
*
*Examine the proof and
*The answer is not "when y and y are linearly dependent."
This is the proof we are told to examine:
$|x+y|^2 = \sum^n_{i=1}(x^i+y^i)^2=\sum^n_{i=1}{x^i}^2+\sum^n_{... | Your argument is good, but you can go further.
Suppose $\|x+y\|=\|x\|+\|y\|$ (sorry, but I can't use notation with single bars and upper indices) with $x\ne0$; this is equivalent to
$$
\langle x+y,x+y\rangle=\langle x,x\rangle+2\|x\|\,\|y\|+\langle y,y\rangle
$$
that is, expanding the left-hand side and simplifying,
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Baby Rudin exercise 1.6: Is this the proof Rudin expects?
$\bf Exercise\, 1.6$
Fix $b>1.$
Prove that if $m,n,p,q$ are integers, $n>0,q>0$ and $r=m/n=p/q$, then
$$
(b^m)^{1/n}=(b^p)^{1/q}.
$$
I'm not really sure what I can assume and what I can't assume, I think that all I need is $(x^y)^z=x^{yz}$ for integers $y,z$,... | Rudin defined $b^n; n \in \mathbb Z$ as notation to mean $b^n = b\cdot b ..b$. Simple grouping allows you to assume $b^nb^m = b \cdot b...b \cdot b\cdot b....b = b^{n+m}$ and $(b^n)^m = b^{nm}$.
By theorem 1.21 you know that for $b^m$ there exists a unique $d := (b^m)^{1/n}$ such that $d^n = b^m$.
The excercise is to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
$3^{n+1}$ divides $2^{3^n}+1$ Describe all positive integers,n such that $3^{n+1}$divides $2^{3^n}+1$.
I am little confused about what the question asks-if it asks me to find all such positive integers, or if it asks me to prove that for every positive integer n,$3^{n+1}$ divides $2^{3^n}+1$. Kindly clarify this doubt ... | By induction: case $n=0,1$ is obvious, assume the claim for $n \in \mathbb{N}$. Then, $$2^{3^{n+1}} +1 = ((2^{3^{n}}+1)-1)^3 +1 = (2^{3^{n}}+1)^3 -3(2^{3^{n}}+1)^2 +3 (2^{3^{n}}+1)$$ and by the induction hypothesis $ 3^{n+2}$ divides the last two terms. For the first term (call it $a$), induction again gives $3^{n+1}$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 0
} |
Geometric Inequality $(a+b)(b+c)(c+a)(s-a)(s-b)(s-c)\leq (abc)^{2}$ everyone.
$a$,$b$,$c$ are three sides of a triangle.
Prove or disprove the following.
$(a+b)(b+c)(c+a)(a+b-c)(b+c-a)(c+a-b)\leq 8(abc)^{2}$
I know two inequalities.
$8(s-a)(s-b)(s-c)\leq abc~$ , $~(a+b)(b+c)(c+a)\geq 8abc$
But for the above... | Use Heron's formula
$$(a+b-c)(b+c-a)(c+a-b)(a+b+c)=\dfrac{a^2b^2c^2}{R^2}$$
where $R$ be the center of the circumcircle of $\Delta ABC$.
your inequality can write as
$$8R^2\ge\dfrac{(a+b)(b+c)(a+c)}{a+b+c}$$
since
$$9R^2\ge a^2+b^2+c^2$$
it suffices to prove
$$8(a+b+c)(a^2+b^2+c^2)\ge 9(a+b)(b+c)(a+c)\tag{1}$$
use AM-G... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that if $z+\frac1z$ is real, then either $|z|=1$ or $z$ is real.
Prove that if $z+\frac1z$ is real, then either $|z|=1$ or $z$ is real.
Original image
I am not sure whether my proof is sufficient. So far, I have shown that
$$z+\frac1z = \frac{z^2+1}{z}=\frac{|z|+1}{z}$$
However, I don't think the proof enough... | Another way:
\begin{align}
z+\frac{1}{z}&=\rho(\cos\theta+i\sin\theta)+\frac{1}{\rho}(\cos\theta-i\sin\theta)\\
&\left(\rho+\frac{1}{\rho}\right)\cos\theta+i\left(\rho-\frac{1}{\rho}\right)\sin\theta
\end{align}
And
$$
\left(\rho-\frac{1}{\rho}\right)\sin\theta=0\implies \rho=1 \text{ or } \theta=0 \text{ or } \theta=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Denominator in rational gcd of integer polynomials A recent question tells us that even if two polynomials $f,g\in \mathbb Z[X]$ have no common factor as polynomials, their values at integer points may have common factors. That question gives this example:
$$
f=x^3-x^2+3x-1,
\qquad
g=x^3+2,
\qquad
\gcd(f(27),g(27))=31
... | *
*Regarding Question $1$:
I think that $d$ in your post is -in the general case- some divisor of the resultant. However, any divisor $r>1$ of the resultant would do, in the sense that it may be the $\gcd\big(f(n), g(n)\big)$ for some $n$:
for any divisor $r$ of the resultant $Res(f,g)=d$, we can always find some i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Show for $x,y \in \mathbb{X} : |d_A (x) - d_A (y)| \leq d(x,y)$ For $A \subset \mathbb{X}$ non empty and $x \in \mathbb{X}$ define the distance of $x$ to $A$ by $$d_A (x) = \inf \limits _{a \in A} d(x,a)$$
I am trying to show for $$x,y \in \mathbb{X} : |d_A (x) - d_A (y)| \leq d(x,y)$$
This is the proof I have.
I star... | In order to prove that $$\vert d_A (x) - d_A (y) \vert \leq d(x,y),$$ you have to prove the following two inequalities:
$$\begin{cases}
-d(x,y) \le d_A (x) - d_A (y) \\
d_A (x) - d_A (y) \le d(x,y)
\end{cases}.$$ In the question of your post, you proved the second one.
Rearranging only won't be enough to obtain the fir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How to show that the Banach space $\left(C[a,b],\lVert.\rVert_{\scriptsize C[a,b]}\right)$ is not Hilbert space? I want to show that the Banach space $\left(C[a,b],\lVert.\rVert_{\scriptsize C[a,b]}\right)$ is not a Hilbert space. So I should show that it is not an inner product space. Most likely, The Parallelogram E... | Take $$\begin{cases}
f_1(x)=-1+2\frac{x-a}{b-a}\\
f_2(x)=\left\vert \frac{2x - (a+b)}{b-a} \right\vert
\end{cases}$$
You have $\Vert f_1 \Vert = \Vert f_2 \Vert=1$ and $\Vert f_1+f_2 \Vert = \Vert f_1-f_2 \Vert = 2$. Hence the parallelogram law is not satisfied.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Why does $\left\| {\left| A \right|} \right\| \le \left\| {\left| I \right|} \right\|$, for every doubly stochastic matrix $A \in M_n$? Let $\left\| {\left| . \right|} \right\|$ be a unitarily invariant matrix norm on $M_n$.
Why does $\left\| {\left| A \right|} \right\| \le \left\| {\left| I \right|} \right\|$, for eve... | In the spirit of Omnomnomnom's answer, here is an alternative approach that doesn't require Birkhoff theorem.
It is well-known that when $A$ is doubly stochastic, its operator norm is equal to $1$ (because $1\le\rho(A)^2 \le \|A\|_2^2 = \rho(A^TA) \le \|A^TA\|_1 = 1$). So, by unitary invariance of the norm $|||\cdot|||... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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What will be the remainder when $2^{31}$ is divided by $5$? The question is given in the title:
Find the remainder when $2^{31}$ is divided by $5$.
My friend explained me this way:
$2^2$ gives $-1$ remainder.
So, any power of $2^2$ will give $-1$ remainder.
So, $2^{30}$ gives $-1$ remainder.
So, $2^{30}\times 2$ or ... | The salient feature here is that when taking the remainder of a product modulo some integer, it doesn't matter if you first take the remainder or first compute the product. In other words:
$$ab = (a \bmod n)(b \bmod n) \bmod n$$
Thus $2^{30} \times 2 =(2^{30} \bmod 5)(2\bmod 5) = ((-1)^{15} \bmod 5)2 = -2 = 3 \bmod 5$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 2
} |
Solving $\lim_{x\to+\infty}(\sin\sqrt{x+1}-\sin\sqrt{x})$ Do you have any tips on how to solve the limit in the title? Whatever I think of doesn't lead to the solution. I tried using: $\sin{x}-\sin{y}=2\cos{\frac{x+y}{2}}\sin{\frac{x-y}{2}}$ and I got:
$$\lim_{x\to+\infty}\bigg(2\cos{\frac{\sqrt{x+1}+\sqrt{x}}{2}}\sin{... | Notice, $$\lim_{x\to +\infty}\left(2\cos\left(\frac{1}{2(\sqrt{x+1}-\sqrt x)}\right)\sin\left(\frac{ \sqrt{x+1}-\sqrt x}{2}\right) \right)$$
$$=2\lim_{x\to +\infty}\cos\left(\frac{1}{2(\sqrt{x+1}-\sqrt x)}\right)\sin\left(\frac{1}{2(\sqrt{x+1}+\sqrt x)}\right)$$
$$=2\lim_{x\to +\infty}\cos\left(\frac{1}{2(\sqrt{x+1}-\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Inverse Laplace transform of $\tan^{−1}\left(\frac{1}{s}\right)$ I'm studying Laplace transformations, but I don't understand where $-\frac{1}{t}$ comes from. And what is the relationship between the corollary and the example?
| Think of $\tan^{-1} \left( \frac 1 s\right)$ as the antiderivative of $\frac{-1}{s^2+1}$.
Then in your example, $n=1$. This introduces a factor of $1/t$ on the left hand side, and a negative sign on the right hand side.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Determine the set of solutions of $ |x| <2x+3 $
Determine the set of solutions of $ |x| <2x+3 $ where $|x| $ denotes
the absolute value of $x$.
Forgive the banality of the question but when I try to solve this problem given standard techniques I just can't get an answer :
Assuming $x >0$ we have that $x<2x+3$ which... | When we assume $x > 0$ we get $x < 2x + 3$ wich yields $x > -3$. But we assumed $x > 0$. So far our set of solutions is $x > 0$ since it both satisfies $x > 0$ and $x > 3$ (both our conditions). In terms of set theory, one condition is satisfied when $x \in (0,\infty)$ and the other one when $x\in (-3,\infty)$ so both ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1591984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Fourier transform of simple function I'm stuck on this problem
Calculate the Fourier transform of
$$
f(t) = \sin t \, , \, |t|<\pi \\
f(t) = 0 \, , \, |t|≥ \pi
$$
So the start is really simple
$$
\hat{f}(\omega) = \int_{-\pi}^{\pi}\sin(t)e^{-i \omega t}dt = ... = \frac{1}{2i}\left[\frac{e^{-(\omega-1)it}}{-i(\omega -1)... | You should use this identity after you plug your bounds in the antiderivative:
$$\sin \theta = \frac{e^{i\theta}-e^{-i\theta}}{2i}$$
With this, you get
$$\frac{\sin(\pi(w-1))}{i(w-1)} -\frac{\sin(\pi(w+1))}{i(w+1)}$$
Observe
$$\frac{\sin(\pi(w-1))}{i(w-1)} -\frac{\sin(\pi(w+1))}{i(w+1)} = \frac{-\sin(\pi w)}{i(w-1)} -\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1592079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
The real numbers are a field extension of the rationals? In preparing for an upcoming course in field theory I am reading a Wikipedia article on field extensions. It states that the complex numbers are a field extension of the reals. I understand this since $\mathbb R(i) = \{ a + bi : a,b \in \mathbb R\}$.
Then the a... | For $\mathbb{R}$ to be field extension of $\mathbb{Q}$, all we need is that $\mathbb{R}$ is a field containing $\mathbb{Q}$ as a subfield. That's definitely true.
The construction is a bit more delicate and analytic in nature: $\mathbb{R}$ is the completion of $\mathbb{Q}$ and is substantially larger. Since $\mathbb{R}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1592150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
"answer_id": 4
} |
Beginning to Work on the Runge-Kutta Method I'm starting to play around with techniques in the introductory
numerical methods literature, and I am starting to use the
Runge-Kutta method for approximating solutiosn for systems
of first-order diff eq.'s, and I'm getting a little tripped
up on the method. Take the system
... | We are given the system;
$$u_1' = -4u_1 - 2u_2 + \cos t + 4\sin t, u_1(0) = 0 \\ u_2' = 3u_1 + u_2 - 3\sin t, u_2(0) = -1.$$
Here is the setup for the system of differential equations using a $4^{th}-$ Order Runge-Kutta.
f(t, u_1, u_2) = -4 u_1 - 2 u_2 + cos t + 4 sin t
g(t, u_1, u_2) = 3 u_1 + u_2 - 3 sin t
t_0 = 0
u_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1592234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Intrepreting tuples as functions I have been mulling over this for a while now. I am told $\mathbb R^n$ can be interpreted as a set of functions.
Take $\mathbb R^2$, for example I can see how we might interpret it as a set containing all ordered pairs: $<x_1,x_2>$. However I do not understand the notation:
$ \{ f:\{1,... | The left hand side is the set of all functions from $\{1,2\}$ to $\mathbb{R}$. For instance one such function is given by $1\to \pi$, $2\to -7$. This function corresponds to the point $(\pi, -7)$ in $\mathbb{R}^2$.
More generally, the function that sends $1$ to $a$ and $2$ to $b$ (with $a,b\in\mathbb{R}$) correspon... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Gluing together functions on a closed subvariety I'm trying to get an intuition for what sheafification does. I came across a passage from Perrin's algebraic geometry book about closed subvarieties.
If says that if X is an algebraic variety and Y is a closed subvariety, we can inherit a sheaf on Y from X. It suggests t... | EDIT: Incorrect
Your analysis is correct. Letting $U = U_1 \cup U_2$, we do indeed have a regular function on $U$ that is $0$ on $U_1$ and $1$ on $U_2$ (it is locally rational on $U$). There is clearly no regular function on $\mathbb{A}^2 \supseteq Y$ that restricts to our 'weird function'. However, if we sheafify the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1592387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Alternative way to show that a simple group of order $60$ can not have a cyclic subgroup of order $6$
Suppose $G$ is a simple group of order $60$, show that $G$ can not have a subgroup isomorphic to $ \frac {\bf Z}{6 \bf Z}$.
Of course, one way to do this is to note that only simple group of order $60$ is $A_5$. So i... | (1) Suppose $H=C_6$ is a cyclic subgroup of order $6$ in $G$ (simple group of order $60$). Let $z$ be the element of order $2$ in $H$, so that $\langle z\rangle$ is subgroup of order $2$ in $H$.
(2) $C_G(z)=$centralizer of $z$ in $G$; it clearly contains $H$. Also, $\langle z\rangle$ is contained in Sylow-$2$ subgroup ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Test convergence, find $\alpha$ which makes integral converge I'm testing the convergence of this improper integral
$$\int_2^{\infty} x(\ln x)^{\alpha} dx$$
I used the limit comparison test with $\frac{1}{x}$ which is divergent, I found that this integral diverges for all values of $\alpha$.
Am I correct ?
| You are correct.
Observe that the integrand is positive and we have, for all real values of $\alpha$,
$$
\lim_{x \to +\infty}\left(x(\ln x)^{\alpha}\right)=+\infty
$$ thus the initial integral is divergent.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the difference between an impulse response and a transferfunction? An imupulse response, is the output you get when you apply an impulse, like a delta dirac function, to your system (only for LTI?).
By knowing the impulse response you know the system.
The transferfunction relates the input to the output. I.e. t... | For any partially continuous function $f : \mathbb{R} \to \mathbb{R}$, the Dirac delta function has the nice property
$$ f(t) = \int_{-\infty}^\infty f(s) \delta(t - s) ds $$
So, any partially continuous function can be written as a sum of Dirac delta functions. This is particularly useful for LTI systems. If we know t... | {
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"timestamp": "2023-03-29T00:00:00",
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How to compute $\lim\limits_{x \to +\infty} \left(\frac{\left((x-1)^2-x\ln(x)\right)(x)!}{(x+2)!+7^x}\right)$? I have a problem with this limit, I don't know what method to use. I have no idea how to compute it.
Can you explain the method and the steps used?
$$\lim\limits_{x \to +\infty} \left(\frac{\left((x-1)^2-x\ln(... | I would do as following.
The limit equals $$\lim_{x \to \infty} \frac {(x - 1)^2 - x \log x} {x (x + 1) + 7^x/x!}.$$ Note that $(x - 1)^2 - x \log x \sim x^2$, $x (x + 1) \sim x^2$ and $7^x/x! = o (1)$. Using these, we have $$\lim_{x \to \infty} \frac {(x - 1)^2 - x \log x} {x (x + 1) + 7^x/x!} = \lim_{x \to \infty} \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find $\operatorname{lcm}(2n-1,2n+1)$ I'm trying to find the formula for $\operatorname{lcm}(2n-1,2n+1)$ with $n \in \mathbb{Z}$.
Here is my solution but I'm not sure about it.
We know that
$$\operatorname{lcm}(a,b)=\frac{\lvert ab \lvert}{\gcd(a,b)}$$
Now if we substitute we have:
$$\operatorname{lcm}(2n-1,2n+1)=\frac{... | Yes, your work is fine. A slightly different formulation of your study of the greatest common divisor would be to notice that the gcd of two numbers divides their difference, which is $2$. Hence, the gcd is either $1$ or $2$, and the fact that they are both odd implies that it's $1$.
| {
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How many acute triangles can be formed by 100 points in a plane? Given 100 points in the plane, no three of which are on the same line, consider all triangles that have all their vertices chosen from the 100 given points.
Prove that at most 70% of those triangles are acute-angled.
| This is from the 1970 International Mathematical Olympiad. You can find the questions here, and the (rather easy) solution to this question here.
It was one of the homework questions for aspirants to the British team for IMO 1978 in Bucharest. I managed to prove an explicit upper bound on the maximum possible proportio... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$
Let $x_1,x_2,\ldots,x_n > 0$ such that $\dfrac{1}{1+x_1}+\cdots+\dfrac{1}{1+x_n}=1$. Prove the following inequality.
$$\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1)... | Rewrite our inequality in the following form:
$\sum\limits_{i=1}^n\frac{x_i+1}{\sqrt{x_i}}\sum\limits_{i=1}^n\frac{1}{x_i+1}\geq n\sum\limits_{i=1}^n\frac{1}{\sqrt{x_i}}$,
which is Chebyshov.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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On the join of simplicial sets as a dependent product Prop. 3.5 of Joyal notes in quasicategories gives a description of $X\star Y$ as $i_*(X,Y)$, where $i^*\dashv i_*$ is the adjunction
$$
i^*\colon \mathbf{sSet}/\Delta[1] \leftrightarrows \mathbf{sSet}/\partial\Delta[1]
$$
where $i^*$ is "pullback along $\partial\De... | By the Yoneda lemma for the category $\mathbf{\Delta}/[1]$, using the canonical equivalence $(\mathbf{\Delta}/[1])^\wedge = \mathbf{\Delta}^\wedge/\Delta^1$, it suffices to show that the morphism in question induces a bijection
$$ Hom((\Delta^n,f), X \star Y) \to Hom((\Delta^n,f), i_*(X \sqcup Y)) $$
for each $n$, an... | {
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"source": "stackexchange",
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Example of forward difference approaching derivative only up to $O(h)$ According to the article on Wikipedia about finite differences, the forward and backward difference by $h$ of a function $f(x)$ divided by $h$ approach the derivative to order $O(h)$, i.e.
$$\frac {\Delta_h f(x)}{h} -f'(x)= \frac{f(x+h)-f(x)}{h} -f'... | Taylor's theorem gives
$$
f(x+h) = f(x) + h f'(x) + O(h^2)
$$
(for $h \to 0$) and therefore "only"
$$
\frac{f(x+h)-f(x)}{h} -f'(x) = O(h)
$$
A simple example is $f(x) = x^2$, where
$$
\frac{f(x+h)-f(x)}{h} -f'(x) = \frac{(x+h)^2-x^2}{h} - 2x = h
$$
For central differences, the function
$g(h) = f(x+h) - f(x-h)$ has th... | {
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Prove for every odd integer $a$ that $(a^2 + 3)(a^2 + 7) = 32b$ for some integer $b$. I've gotten this far:
$a$ is odd, so $a = 2k + 1$ for some integer $k$.
Then $(a^2 + 3).(a^2 + 7) = [(2k + 1)^2 + 3] [(2k + 1)^2 + 7]$
$= (4k^2 + 4k + 4) (4k^2 + 4k + 8) $
$=16k^4 + 16k^3 + 32k^2 + 16k^3 + 16k^2 + 32k + 16k^2 + 16k ... | Notice
$$16k^4 +48k = 16k(k^3+3).$$
If $k$ is even, we are done. If $k$ is odd, then $k^3 +3$ is even, and we are done.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Need help with $\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$ Please help me to evaluate this integral:
$$\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$$
Using substitution $x=2\arctan t$ it can be transformed to:
$$\int_0^\infty\frac{2}{1+t^2}\arctan^2\left(\frac{2t}{3+t^2}\right)dt$$
Then I t... | A Fourier analytic approach. If $x\in(0,\pi)$,
$$\begin{eqnarray*}\arctan\left(\frac{\sin x}{2+\cos x}\right) &=& \text{Im}\log(2+e^{ix})\\&=&\text{Im}\sum_{n\geq 1}\frac{(-1)^{n+1}}{n 2^n}\,e^{inx}\\&=&\sum_{n\geq 1}\frac{(-1)^{n+1}}{n 2^n}\,\sin(nx),\end{eqnarray*}$$
hence by Parseval's theorem:
$$ \int_{0}^{\pi}\ar... | {
"language": "en",
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What is the difference between an indexed family and a sequence? For indexed family wikipedia states: Formally, an indexed family is the same thing as a mathematical function; a function with domain J and codomain X is equivalent to a family of elements of X indexed by elements of J
For sequence wikipedia states: Forma... | You can index something by the real numbers for example. So for every real number $\alpha$ you might have a value $x_{\alpha}$. But since you cannot enumerate the real numbers you cannot represent that as a sequence. So it's just an indexed set, something more general than a sequence.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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show that the function is in $L^r$
Let $f$ be a measurable function and $1 \le p < r < q < \infty$. If there is a constant $C$ such that $$\mu ( \{ x : |f(x ) | > \lambda \} ) \le \frac { C }{ \lambda ^p + \lambda ^ q} $$ for every $ \lambda > 0$, show that $f \in L ^r $.
I know that if both $f \in L^p $ and $f \in... | Hint: $$\int_X f(x)^r\,d\mu(x) = \int_0^\infty r\lambda ^{r-1} \mu(\{x\in X:f(x)> \lambda\})\,d\lambda.$$
| {
"language": "en",
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number of ways of choosing $r$ points from $n$ points arranged in a circle such that no consecutive points is taken.
number of ways of choosing $r$ points from $n$ ponts arranged in a circle such that no consecutive points is taken.
(I have seen some question on SE related to this. But I am trying to solve it using ... | Consider $n$ things in a circle namely $P_1,P_2 \dots P_n$. Without loss of generality let us assume that $P_1,P_2 \dots P_n$ are all arranged in a circle in a clockwise manner. Now let us start the question by counting the number of favourable cases always involving $P_1$. Clearly $P_2$ and $P_n$ cannot be selected no... | {
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Primitive of $\sin(4x)$ If $\sin(4x)=2\sin(2x)\cos(2x)$ then:
$$\int \sin(4x)=\int 2\sin(2x)\cos(2x)$$
$$\frac{-\cos(4x)}{4}=\frac{-\cos^2(2x)}{2}$$
But considering now $x=2$:
$$\frac{-\cos(4\cdot 2)}{4}\neq\frac{-\cos^2(2\cdot 2)}{2}$$
$$0.036\neq -0.214$$
What's the error that is behind this process ?
Thanks in advan... | $$\frac { -\cos (4x) }{ 4 } +C_{ 1 }=\frac { -\cos ^{ 2 } (2x) }{ 2 } +C_{ 2 }$$
| {
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Projective space and its basis I am trying to solve an exercise from the book "Permutation Groups" by J. Dixon and B. Mortimer.
Later, I asked a similar question about the basis of Affine geometry " Affine geometry and its basis". Similar to the answer of that question, I think that in this case, all of the basis of ... | I don't think either of the two forms of a projective basis that you suggest is quite right.
By definition, such a basis has the form $\{[v_1],\ldots,[v_{d+1}] \}$, subject to the condition that any $d$-element subset of $\{v_1,\ldots,v_{d+1} \}$ is linearly independent (and hence is a basis of $F^d$). This condition ... | {
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"timestamp": "2023-03-29T00:00:00",
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Solving this limit: $\lim\limits_{x\to\infty}\frac{x^{x+1/x}}{(x+1/x)^x}$ $$\lim\limits_{x\to\infty}\frac{x^{x+1/x}}{(x+1/x)^x}$$
I have tried a lot of things, like:
*
*transforming those terms to:
$$\frac{e^{(x+1/x)\ln(x)}}{e^{x\ln(x+1/x)}}$$
*then I tried L'Hôpital's rule but it was just getting more complex
*... | Rewrite the limit as
$$L = \lim_{x \to +\infty} \frac{x^{\frac{x^2 + 1}{x}}}{\left(\frac{x^2 + 1}{x}\right)^x} = \lim_{x \to +\infty} \frac{x^{\frac{2x^2 + 1}{x}}}{(x^2 + 1)^x} = \lim_{x \to +\infty} e^\ell,$$
where
$$\require{cancel}\ell = \frac{2x^2 + 1}x\ln x - x\ln(x^2 + 1) = \cancel{2x\ln x} + \frac1x\ln x - \canc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1593972",
"timestamp": "2023-03-29T00:00:00",
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For nonzeros $A,B,C\in M_n(\mathbb{R})$, $ABC=0$. Show $\operatorname{rank}(A)+\operatorname{rank}(B)+\operatorname{rank}(C)\le 2n$ Let $A,B,C\in M_n(\mathbb{R})$ be nonzero matrices such that $ABC=0$.
How can we prove that $\operatorname{rank}(A)+\operatorname{rank}(B)+\operatorname{rank}(C)\le 2 n$ ?
I can prove this... | $\newcommand{\rk}{\operatorname{rk}}$
$\newcommand{\im}{\operatorname{im}}$You know that $\im(BC)\subseteq \ker A$. Hence, $n= \rk A+\dim\ker A\ge \rk A+\rk(BC)$.
Now, if you consider the restriction of multiplication by $B$ to the subspace $\im C$ and use rank-nullity theorem, you get $\rk(BC)=\rk C-\dim\ker(B|_{\im C... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Two step random experiment: Density of combined uniform and normal distribution imagine a random experiment, where first some number $u$ is drawn uniformly on $[c-\varepsilon,c+\varepsilon]$ for $c>0$ and $0<\varepsilon<c$. Next, a $N(u,\sigma^2)$-distributed random variable $Y$ is generated (meaning that we use the $u... | Density of $Y$:
\begin{align}
f_Y(y) &= \int_{u=c-\epsilon}^{c+\epsilon} f_{Y|U}(y\mid u)f_U(u)\;du \\
&= \int_{u=c-\epsilon}^{c+\epsilon} \dfrac{1}{2\epsilon}\dfrac{1}{\sqrt{2\pi\sigma^2}}\;e^{-\dfrac{1}{2}\left(\dfrac{y-u}{\sigma}\right)^2}\;du. \\
\end{align}
Let $z=(u-y)/\sigma\;$ so that $dz=du/\sigma$. Then,
\beg... | {
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There is a prime between $n$ and $n^2$, without Bertrand Consider the following statement:
For any integer $n>1$ there is a prime number strictly between $n$ and $n^2$.
This problem was given as an (extra) qualification problem for certain workshops (which I unfortunately couldn't attend). There was a requirement to ... | I've come up with a simple proof based on the prime-counting function $\pi(x)$, which I'm pretty sure doesn't depend on Bertrand's Postulate.
First, I will prove a lemma that for every prime $n$, there is another prime $p$ with $n < p < n^2$. I will use this result later to show the general result (i.e. for composite $... | {
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"url": "https://math.stackexchange.com/questions/1594162",
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Prove that if $a,b,$ and $c$ are positive real numbers, then $\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a} \geq ab + bc + ca$.
Prove that if $a,b,$ and $c$ are positive real numbers, then $\dfrac{a^3}{b}+\dfrac{b^3}{c}+\dfrac{c^3}{a} \geq ab + bc + ca$.
I tried AM-GM and it doesn't look like AM-GM or Cauchy-Schwarz work... | It is actually very simple. Use nothing but AM-GM.
$$\frac{a^3}{b} + ab \geq 2a^2$$
$$\frac{b^3}{c} + bc \geq 2b^2$$
$$\frac{c^3}{a} + ac \geq 2c^2$$
$$LHS + (ab+bc+ac) \geq 2(a^2+b^2+c^2) \geq 2(ab + bc +ac)$$
We are done.
| {
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Solve integral $\int{\frac{x^2 + 4}{x^2 + 6x +10}dx}$ Please help me with this integral:
$$\int{\frac{x^2 + 4}{x^2 + 6x +10}}\,dx .$$
I know I must solve it by substitution, but I don't know how exactly.
| Using long division,
$$ \frac{x^2+4}{x^2+6x+10} = 1 -6\frac{x+1}{x^2+6x+10}= 1 -\frac{6x}{x^2+6x+10}-\frac{6}{x^2+6x+10}$$
Integrating $$1 -\frac{6x}{x^2+6x+10}-\frac{6}{x^2+6x+10} $$ should be straight forward
Note that $x^2+6x+10= (x+3)^2 +1$
| {
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Infinite primes proof There is a proof for infinite prime numbers that i don't understand.
right in the middle of the proof:
"since every such $m$ can be written in a unique way as a product of the form:
$\prod_{p\leqslant x}p^{k_p}$. we see that the last sum is equal to: $\prod_{\binom{p\leqslant x}{p\in \mathbb{P}}}... | It's not every such $m$ that can be written as $\prod_{\substack{p\in\mathbb{P},\\ p\leq x}}\sum_{k\geq0}\frac{1}{p}$, it's that the sum of all such $m$ is the same as the product and summation. What your proof is saying are equal is $$\prod_{\substack{p\in\mathbb{P},\\p\leq x}}\sum_{k\geq0}\frac{1}{p}=\sum_{\substack{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1594411",
"timestamp": "2023-03-29T00:00:00",
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Sum of partial derivatives Suppose that
$$\mu_i(x)=x_i \int_0^1 t^{n-1} \rho(tx) dt$$
where $\rho$ is a function on $\mathbb R^n$ and $tx=(tx_1,\dots,tx_n)\in \mathbb R^n$. Show that
$$\sum_{i=1}^n \frac{\partial\mu_i}{\partial x_i}=\rho .$$
This problem looks simple, but I am having difficulty in showing the result.
I... | $$\dfrac{\partial\mu_i}{\partial x_i}=\int_0^1 t^{n-1} \rho(tx) dt + x_i\int_0^1 t^{n} \rho'(tx) dt,$$
$$\sum_{i=1}^n\dfrac{\partial\mu_i}{\partial x_i} = n\int_0^1 t^{n-1} \rho(tx) dt +\sum_{i=1}^n x_i\int_0^1 t^{n} \rho'(tx) dt = \int_0^1 \rho(tx) dt^n + \int_0^1 t^{n} \dfrac{d\rho(tx)}{dt} dt = t^{n} \rho(tx)\biggr|... | {
"language": "en",
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SPNE of a finitely repeated game So I have this little game between two players, played T times without any discouting factor with the following payoff table:
now if T is 2, is there a SPNE where (B,L) is being played first round?
I'm thinking no, since (T,R) is the unique NE and no credible threat can be made when th... | Because the stage game has a unique NE, in any finitely repeated game of the stage game, there is a unique SPNE. In this SPNE, the stage NE is played after every history. This can be easily shown by backward induction as follows.
Suppose horizon is $T$. Let's use $t=0,1,\cdots, T-1$ to denote each period. In period $T-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing $\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}/(1,-1,0)\mathbb{Z}+(0,1,1)\mathbb{Z}+(1,0,-1)\mathbb{Z}\cong \mathbb{Z}$ I have to prove that if $V_K = \{v_0, v_1, v_2\}$ and $K = \{\{v_0\}, \{v_1\}, \{v_2\}, \{v_0, v_1\}, \{v_0, v_2\}, \{v_1, v_2\}\}$ then $H_q(K, \mathbb{Z})\cong \mathbb{Z}$ for $q = 0, 1$.
Alr... | The quotient is not equal to $\Bbb Z$: it is equal to $\Bbb Z/2$. In particular, to correct the comment by p Groups:
$f(1, -1, 0)=0$ implies $a-b=0$, so $a=b$. Similarly, $a-c=0$ implies $a=c$. Finally, $b+c=0$ implies $2a=0$. So the quotient is just $a\Bbb Z/2a\Bbb Z$, for example with representatives $\{(0, 0, 0)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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To find volume of solid of revolution The volume of a solid generated by revolving about the horizontal line y=2 the region bounded by $y^{2}\leq2x$, $x\leq 8$ and $y\geq 2$.
I have figured out the area to be revolved. But I do not know how to do disk method or washer method here.
Thanks you so much
| Your function that describes the solid is $f(x) = \sqrt{2x} - 2$. Integrate the square of this function over $ \sqrt{2x} = 2 \Leftrightarrow x = 2$ to $ x = 8$ to get the volume:
$$V = \pi* \int_{2}^{8} (\sqrt{2x} - 2)^2dx $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1594769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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find minimal polynomial of matrix product $AB$ where $AB=BA$ Let $x^2-1$ and $x^2+1$ be minimal polynomials of $A,B\in M_n(\mathbb{R})$, respectively.
If $AB=BA$, find the minimal polynomial of $AB$.
| Notice that $A^2 = I$ and $B^2 = -I$. Therefore $(AB)^2 = A^2 B^2 = -I$, so the minimal polynom of $AB$ divides $x^2+1$. Because $x^2+1$ is monic and irreducible over $\mathbb{R}$ this already is the minimal polynomial.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How much group theory is required before undertaking an introductory course on Galois Theory? How much knowledge of group theory is needed in order to begin Galois Theory? Which topics are most relevant?
| Most Galois Theory books are self-explanatory, but you need to familiarize yourself with concepts as solvable groups (this relates to equations being solvable by radicals), simple groups. Also Sylow theory helps a lot. In addition, knowledge of rings and fields is necessary. By the way, Ian Stewart's book Galois Theory... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $2$ is subtracted from each root,the results are reciprocals of the original roots.Find the value of $b^2+c^2+bc.$ The equation $x^2+bx+c=0$ has distinct roots .If $2$ is subtracted from each root,the results are reciprocals of the original roots.Find the value of $b^2+c^2+bc.$
Let $\alpha$ and $\beta$ are the root... | So from equation we get quadratic which is $x^2-2x-1$ so roots if this are $1+\sqrt{2},1-\sqrt{2}$ so sum of roots is $-b/1=2,$ and product is $c/1=-1$ so $b^2+c^2+bc=4+1-2=3$ hope its clear.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I draw the skewed normal distribution curve If its of real importance, I am trying to plot the data on gnuplot.
I have the following of some experimental data, obtained by octave:
*
*mean: $\overline{\mu} = 0.6058$
*median: $\tilde{\nu} = 0.6364$
*std: $\sigma_x = 0.1674$
*variance: $\sigma^2 = 0.028$
*... | You can use the skew normal distribution with parameters $(ξ,ω,α)$ which can be estimated from the given data. If we set $δ=\dfrac{α}{\sqrt{1+α^2}}$, then the mean, variance and skewness of the skew normal distribution are given by (see the link)
*
*mean: $ξ+ωδ\sqrt{\dfrac2π}$
*variance: $ω^2\left(1-\dfrac{2δ^2}{π}\... | {
"language": "en",
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Evaluate the definite integral $\frac{105}{19}\int^{\pi/2}_0 \frac{\sin 8x}{\sin x}dx$ Problem :
Determine the value of $$\frac{105}{19}\int^{\pi/2}_0 \frac{\sin 8x}{\sin x}\ \text dx$$
My approach: using $\int^a_0f(x)\ \text dx = \int^a_0 f(a-x)\ \text dx$,
$$
\begin{align}
\frac{105}{19}\int^{\pi/2}_0 \frac{\sin 8... | Method $1$
$1.$ Use the identity
$$2\sin x \sum_{k=1}^{n}\cos(2k-1)x = \sin2nx$$
which can easily be verified by using
$$2\sin x \cos(2k-1)x = \sin 2k x - \sin 2(k-1) x$$
and the telescoping property of the sums.
$2.$ Use the above formula to get
$$\frac{\sin 2nx}{\sin x} = 2 \sum_{k=1}^{n}\cos(2k-1)x$$
$3.$ Integrate ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $f(x)$ is a continous function such that $f(x) +f(\frac{1}{2}+x) =1; \forall x\in [0, \frac{1}{2}]$ then $4\int^1_0 f(x) dx =$ . Problem :
If $f(x)$ is a continous function such that $f(x) +f(\frac{1}{2}+x) =1; \forall x\in [0, \frac{1}{2}]$ then $4\int^1_0 f(x) dx = ?$
My approach :
$$f(x) +f(\frac{1}{2}+x) =1... | \begin{align*}
&4\int_0^1 f \\
=& 4\int_{0}^{0.5} f(t) dt + 4 \int_{0.5}^1 f(t) dt\\
=& 4\int_{0}^{0.5} f(t) dt + 4 \int_0^{0.5} f(x+\frac12) dx \quad\text{($x = t - \frac12$)}\\
=& 4\int_{0}^{0.5} (f(t) + f(t+\frac12)) dt\\
=& 4\int_{0}^{0.5} 1 dt\\
=& 4\cdot 0.5\\
=& 2
\end{align*}
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Maximizing the sum of the products of endpoints of edges in a graph Let $G$ be a graph with vertex set $V=\{v_1,v_2\dots v_n\}$ and edge set $E$. Let $f:V\rightarrow \mathbb [0,\infty)$ be a real valued function such that $\sum\limits_{i=1}^n f(v_i)=A$.
What is the maximum possible value for $\sum\limits_{uv\in E}f(u)f... | I managed to solve it with some help from fractal in AOPS.
For each function $f:V\rightarrow [0,\infty)$ with the desired properties let $pos(f)$ be $\{v\in V|f(v)>0\}$.
Now, consider the set of all such functions in $f$ such that $\sum\limits_{uv\in E}f(u)f(v)$ reaches the maximum-
Take $f$ to be one of functions in t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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In general, when does it hold that $f(\sup(X)) = \sup f(X)$?
Let $f: [-\infty, \infty] \to [-\infty, \infty]$.
What conditions should we impose on $f$ so that the following statement becomes true?
$$\forall \ X \subset [-\infty, \infty], \sup f(X) = f(\sup X)$$
If that doesn't make much sense, then for some function w... | I believe the sufficient and necessary condition for $f$ is that it is nondecreasing, left-continuous (i.e. for all $x_0$, $\lim\limits_{x\rightarrow x_0^-}f(x)=f(x_0)$) and $f(-\infty)=-\infty$. Last condition is necessary for consideration of $X=\varnothing$. First condition is necessary as for $X=\{a,b\},a\leq b$ we... | {
"language": "en",
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How to solve the following quadratic word problem? The total cost of carpeting a rectangular room is given the expression $$6x^2 + 18x$$
This is the multiple choice type question so the given options were set up like this.
The length of the room is_______feet, its width is ____ feet and the cost of carpeting is _____ ... | Steve X is correct that there are infinitely many solutions.
But let's consider what is the "best" solution. It is perfectly natural in a word problem that the unknown $x$ refers to one of the properties of the physical situation (i.e., length or width or price per square foot). Once we factor the polynomial (but s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1595515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is $\tan^{-1}(\infty)$ here equal to $\dfrac{\pi}{2}$? In the question below in the last step of the solution, they seem to be claiming that $\displaystyle \lim_{n \to \infty+}\tan^{-1} \left (\dfrac{n+1}{d} \right) = \dfrac{\pi}{2}$, which I agree with but what about $\dfrac{5\pi}{2}$ etc.? How can they say it mus... | it is small because the individual angles considered are arctangents from lines with positive slopes: the angle between the positive $x$ axis and a line of positive slope $m$ is $\arctan m$ which is between $0$ and $\pi / 2.$
You should have emphasized an earlier line
$$ \arctan \left( \frac{i+1}{d} \right) - \arctan... | {
"language": "en",
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Is the complex square root of $z^2 = \pm z$? Is $\sqrt{z^2} = \pm z$, for $z$ complex?
I think it is, since either $-z$ or $+z$ satisfies the definition
$\sqrt{z^2}= e^{\large \frac{1}{2}\log(z)^2}$
but I just wanted to make sure. It's a bit tricky going from the positive square to the complex square root.
Thanks,
| The notation $\sqrt x$ is usually avoided except when referring to the non-negative square root of a non-negative real number, because otherwise it's ambiguous. In $C$ it is better to refer to "a square root" not "the square root".
You will also come across the phrase "$x$ is an $n$th root of $1$" (when it is underst... | {
"language": "en",
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question about orthogonal vector Let say I have 2 vectors $(1, 0, 0)$ and $(0, 2, 0)$, and I want to find a third vector that is orthogonal to both of them. I can do a cross product and get $(0, 0, 2)$. However, I know there are infinite vector in the following form $(0, 0, x)$ where $x$ $\in$ R that are orthogonal to ... | The vector you get by performing the cross product is the unique vector orthogonal to both of your original vectors that
*
*has a length equal to the magnitude of the area of the parallelogram (actually rectangle in this case) with sides $(1,0,0)$ and $(0,2,0)$ and
*forms a right-handed set with $(1,0,0)$ and $(0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Bounded Holomorphic function on Right half plane. Does there exist a Bounded Holomorphic function defined on Right half plane which have all $\sqrt{n}$ as root for all natural number $n$?
I guess It is a just $0$ function.
But How Could I approach this one?
(I've been trying to use Blascke product.)
Thanks!
| If $f$ is a bounded non-constant holomorphic function on the disc then $\{1-|z| \mid f(z)=0\}$ is summable (see here). The function $$z\mapsto\frac{1+z}{1-z}$$ maps the unit disc onto the right half plane. The pre-image of $\sqrt{n}$ is $$\frac{\sqrt{n}-1}{\sqrt{n}+1} = 1 -\frac2{\sqrt{n}+1}$$
so a bounded function on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1595796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using polar coordinates to find the area of an ellipse Considering an ellipse with the $x$ radius equal to $a$ and the $y$ radius equal to b$:$
I figured that some kind of parameterization might be:
$x=a\cos\theta$
$y=b\sin\theta$
and then polar $r^2$ is just $x^2 + y^2$
But then I tried to come up with some unit of i... | My answer is linked here.
My name is Marco mamello10@gmail.com.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Is every "weakly square" matrix either a $0$ matrix, or a square matrix? Call a matrix $A$ weakly square iff $\mathrm{det}(A^\top A) = \mathrm{det}(A A^\top)$. Then clearly,
*
*every square matrix is weakly square, and
*every zero matrix is weakly square.
Question. Are these the only examples of weakly-square ma... | A non-square matrix $A$ is weakly square if and only if neither $A$ nor $A^T$ has full rank, which is to say iff $\operatorname{rank}(A)<\min\{m,n\}$.
The key to this observation is to note that
$$
\operatorname{rank}(A^TA)=
\operatorname{rank}(A)=
\operatorname{rank}(A^T)=
\operatorname{rank}(AA^T)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1596005",
"timestamp": "2023-03-29T00:00:00",
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Finite element method books I know this question has been asked before; I just want to enquire if anybody has any suggestions to learn how to compute finite element problems, including plenty of examples.
The topics I would like to focus in are as follows:
Introduction to finite elements for 1D and 2D problems covering... | I would suggest Larson–Bengzon - The Finite Element Method: Theory, Implementation and Applications. It contains everything you requested for. They use Matlab as a programming environment.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Identifying a sequence as subset of subspace If I have some sequence $\mathcal A = (a_i)$ of objects $a_i$ (maybe finite, maybe countably infinite) how can I say that those objects all exist in some subspace $S$? Is it correct to say $\mathcal A \subseteq S$? I'm not sure because $\mathcal A$ isn't a set it's a seque... | You are correct in your assertion that $(a_i)\subseteq S$ lacks some formality, but I think it would be understood as intended if you wrote it that way.
Here are some other options:
*
*Use $(a_i) \in S^{\mathbb{N}}$ if you have a sequence in the usual sense.
*Use $(a_i) \in S^N$ if you have a sequence of finite len... | {
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Prove that $\sum_{n=1}^{\infty}\left(\frac{1}{(8n-7)^3}-\frac{1}{(8n-1)^3}\right)=\left(\frac{1}{64}+\frac{3}{128\sqrt{2}}\right)\pi^3$ Prove that $$\sum_{n=1}^{\infty}\left(\frac{1}{(8n-7)^3}-\frac{1}{(8n-1)^3}\right)=\left(\frac{1}{64}+\frac{3}{128\sqrt{2}}\right)\pi^3$$
I don't have an idea about how to start.
| What we have is
$$\frac1{1^3} - \frac1{7^3} + \frac1{9^3} - \frac1{15^3} + \cdots $$
Imagine if we used negative summation indices (e.g., $n=0, -1, -2, \cdots$). Then we would have
$$\frac1{(-7)^3} - \frac1{(-1)^3} + \frac1{(-15)^3} - \frac1{(-9)^3} + \cdots$$
You should see then that the sum is
$$\frac12 \sum_{n=-\in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solve the integral $\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx$ To find the Variance of a Wiener Process, $Var[W(t)]$, I have to compute the integral
$$
Var[W(t)]=\dots=\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx=\dots=t.
$$
I've tried integration by pa... | Hint: Let $a=\dfrac1{2t}.~$ Then we are left with evaluating $\displaystyle\int_{-\infty}^\infty x^2e^{-ax^2}~dx.~$ But the latter can be written as $-\dfrac d{da}\displaystyle\int_{-\infty}^\infty e^{-ax^2}~dx.~$ Can you take it from here ? ;-$)$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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is the vector space of n- forms of an n-manifold equal to the vector space of compactly supported n-forms? Let $\Omega^{n}(M)$ be the real vector space of smooth n-forms of an n-manifold $M$. It is a real vector space of dimension 1. $\Omega^{n}_c(M)$ is the real vector space of compactly supported smooth n-forms on $M... | I think that, for the purposes of your confusion, we can first consider changing "n-form" to "vector field" in order to gain some more concreteness first. The confusion is the same.
Consider the set of vector fields on a manifold $M$, let's call it $V(M)$. It is the set of smooth sections of the tangent bundle $TM$. Wh... | {
"language": "en",
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For all $\omega \neq 0$, $\rho(L_{\omega}) \ge |1 - \omega|$, where $L_{\omega}$ is the SOR matrix Let $A = (a_{ij}) \in M_n(\Bbb C)$ be invertible, such that $a_{ii} \neq 0$ for all $i$. Split $A$ into $D - E - F$, where $D$ is the diagonal of $A$, $E$ is the strict lower triangular part of $-A$ and $F$ is the strict ... | For $\omega \in \Bbb C \setminus \{0\}$ you have :
\begin{gather*}\det\left(L_{\omega}\right ) & =& \det\left(\left(\frac1{\omega} D - E\right)^{-1} \left(\frac{1-\omega}{\omega} D + F\right) \right) \\ & = &\frac{\det\left(\frac{1-\omega}{\omega} D + F\right)}{\det \left(\frac1{\omega} D - E\right) }
\end{gather*}
B... | {
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Evaluate the integral $\int x^{\frac{-4}{3}}(-x^{\frac{2}{3}}+1)^{\frac{1}{2}}\mathrm dx$ $$x^{\frac{-4}{3}}(-x^{\frac{2}{3}}+1)^{\frac{1}{2}}=\frac{\sqrt{(-\sqrt[3]{x^2}+1)}}{\sqrt[3]{x^4}}$$
Is it necessary to simplify the function further? What substitution is useful?
$u=\sqrt[n]{\frac{ax+b}{cx+d}}$ doesn't work.
| Following RecklessReckoner's comment:
We have $\int x^{\frac{-4}{3}}(-x^{\frac{2}{3}}+1)^{\frac{1}{2}}\mathrm dx$.
Now, let $u=x^{1/3}$ and $3u \mathrm du=\mathrm dx$
This gives $3\int u^{-4}(-u^{2}+1)^{\frac{1}{2}}u\mathrm du=3\int u^{-3}(-u^{2}+1)^{\frac{1}{2}}\mathrm du=3\int u^{-3}(1-u^{2})^{\frac{1}{2}}\mathrm du.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\sum_{i = 1}^n \frac{x_i}{i^2} \geq \sum_{i = 1}^n \frac{1}{i}.$
Let $x_1,x_2,\ldots,x_n$ be distinct positive integers. Prove that $$\displaystyle \sum_{i = 1}^n \dfrac{x_i}{i^2} \geq \sum_{i = 1}^n \dfrac{1}{i}.$$
Attempt
I tried using Cauchy-Schwarz and I got that $$(x_1^2+x_2^2+\cdots+x_n^2) \left (\d... | A proof sketch that doesn't use the rearrangement inequality:
Since the integers are distinct it's enough to prove the claim in the case $\{x_1,\dots,x_n\}$ is a permutation on $\{1,\dots,n\}$. For positive integers $\alpha<\beta$, $\gamma < \delta$ you can show $$\alpha/\gamma + \beta/\delta < \beta/\gamma + \alpha/\d... | {
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"source": "stackexchange",
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Why does the monotonicity imply $2^u < 3^v$ if and only if $3^u < 6^v$? In the question and solution below, I am wondering how to #$7$ it says "The monotonicity of $f$" implies that $2^u < 3^v$ if and only if $3^u < 6^v$, $u,v$ being positive integers." How does this even depend on the definition of $f$? And if it does... | Because $f(mn) = f(m)f(n)$, we have:
$$
f(m^k) = f(\underbrace{m m \dots m}_{\text{$k$ times}}) = \underbrace{f(m)f(m)\dots f(m)}_{\text{$k$ times}} = f(m)^k
$$
Also, because $f$ is strictly increasing, we have:
$$
a > b \implies f(a) > f(b)
$$
(indeed that's the definition of strictly increasing).
With these two toget... | {
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"answer_id": 1
} |
Zeroth homotopy group: what exactly is it? What are the elements in the zeroth homotopy group? Also, why does $\pi_0(X)=0$ imply that the space is path-connected?
Thanks for the help. I find that zeroth homotopy groups are rarely discussed in literature, hence having some trouble understanding it. I do understand that ... | Just a slight rephrase: you can consider $\pi_0(X)$ as the quotient set of the set of all points in $X$ where you mod out by the equivalence relation that identifies two points if there is a path between them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1596822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
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A function satisfies the identity $f(x) + 2f\left(\frac1x\right) = 2x+1$ ... find another identity that $f(x)$ satisfies.
A function satisfies the identity $f(x) + 2f\left( \frac{1}{x} \right) = 2x+1$.
By replacing all instances of $x$ with $\frac{1}{x}$, find another identity that $f(x)$ satisfies.
I have absolute... | You have your original identity and new one obtained by substituting $\frac1x$ instead of $x$:
\begin{align}
f(x)+2f\left(\frac1x\right) &= 2x+1 \tag{1}\\
f\left(\frac1x\right)+2f(x) &= \frac2x+1 \tag{2}
\end{align}
Now you want to combine the two equalities to get $f(x)$. If you add to equation $(1)$ the equation $(2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1596943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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} |
Does a connected countable metric space exist? I'm wondering if a connected countable metric space exists.
My intuition is telling me no.
For a space to be connected it must not be the union of 2 or more open
disjoint sets.
For a set $M$ to be countable there must exist an injective function
from $\mathbb{N} \righ... | Fix $x_0 \in X $. Then, the continuous(!) map
$$
\Phi: X \to \Bbb {R}, x \mapsto d (x,x_0)
$$
has an (at most) countable, connected image.
Thus, the image is a whole (nontrivial!, if $X $ has more than one point) interval, in contradiction to being countable.
EDIT: On a related note, this even show's that every connect... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1597111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Existence of maximum and minimum
Let $f:\mathbb{R}_+\rightarrow \mathbb{R}$ be continuous and such that $f(0)=1$ and $lim_{x\rightarrow+\infty}f(x) = 0$.
Prove that $f$ must have a maximum in $\mathbb{R}_+$. What about the minimum?
I started working on that trying to verify Weierstrass theorem on a smaller interval ... | Suppose $f$ doesn't have a maximum.
Pick any non-zero $x_{1} \in \Bbb R_{+}$. Since $[0, x_{1}]$ is compact, and $f$ is continuous, we know $f$ has a maximum $M > 0$ on this interval.
But since $f$ doesn't have a maximum on $\Bbb R_{+}$, we also know there is some $x_{2} \not \in [0,x_{1}]$ with $f(x_{2}) > M$.
B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1597182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is every converging sequence the sum of a constant sequence and a null sequence? Let $a_n$ be any sequence converging to $a$ when $n \to \infty$.
Can you rewrite $a_n$ so that it is the sum of two other sequences? $$a_n=b_n + c_n,$$ with $b_n=b$ for every $n \in \mathbb{N}$ and $c_n\to 0$ as $n\to \infty$.
In other wo... | Yes, you can do that. Simply take $b_n=a,c_n=a_n-a$. By basic properties of limits $$\lim\limits_{n\rightarrow\infty}c_n=\lim\limits_{n\rightarrow\infty}(a_n-a)=(\lim\limits_{n\rightarrow\infty}a_n)-a=a-a=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1597262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 0
} |
Find the summation $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \cdots$ What is the value of the following sum?
$$\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \cdots$$
The possible answers are:
A. $e$
B. $\frac{e}{2}$
C. $\frac{3e}{2}$
D. $1 + \frac{e}{2}$
I tried to expand the options using the series represen... | Clearly the $r^{th}$ numerator is $1+2+3+...+r= \frac{r(r+1)}{2}$ .
And the $r^{th}$ denominator is $r!$.
Thus $$\displaystyle U_r=\frac{\frac{r(r+1)}{2}}{r!}=\frac{r(r+1)}{2r!}$$
Since the degree of the numerator is $2$ , use partial fractions to find $A,B,C$ such that (If you use partial fractions up to $(r-3)!$ , i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1597328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 1
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If $n$ and $m$ are odd integers, show that $ \frac{(nm)^2 -1}8$ is an integer I am trying to solve:
If $n$ and $m$ are odd integers, show that $ \frac{(nm)^2 -1}8$ is an integer.
If I write $n=2k+1$ and $m=2l+1$ I get stuck at
$$\frac{1}{8}(16k^2 l^2 +4(k+l)^2 +8kl(k+l)+4kl+2(k+l))$$
| $$((2k+1)(2l+1))^2-1=16k^2l^2+4k^2+16kl^2+16kl+4k+4l^2+16lk^2+4l.$$
Dropping all the terms with coefficient $16$,
$$4(k^2+k+l^2+l)=4(k(k+1)+l(l+1))$$ must be a multiple of $8$.
With a slightly simpler evaluation:
$$((2k+1)(2l+1))^2-1=(4kl+2k+2l)(4kl+2k+2l+2)=4(2kl+k+l)(2kl+k+l+1).$$
This is a multiple of $8$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1597394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 9,
"answer_id": 2
} |
How do we find the minimum distance of a narrow sense BCH code? I know the designed distance, $d$, is a lower bound for the minimum distance, $d(C)$.
Usually, in the examples I've seen, what we do is find the generator polynomial $g(x)$ of the code, then from $d \le d(C) \le w(g)$, where $w(g)$ is the weight of the cod... | This is a very special code $C$ known as the binary Golay code. To get $d_{min}=7$ you can do the following.
*
*The smallest extension field containing a 23rd root of unity is $GF(2^{11})$. We see this for example by repeatedly applying the Frobenius (i.e. squaring). If $\alpha$ is a 23rd root of unity, its conjugat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1597468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof of -(-v)=v in a vector space The exercise is: Prove that $-(-v)=v$ for every $v \in V$
Proof
Suppose $v \in V$ and $V$ is a vector space.
Then $-(-v) \in V$ as result of the scalar multiplication property and
$-(-v)=-(-1\cdot v)=-1 \cdot(-1\cdot v) =(-1 \cdot-1)\cdot v = 1 \cdot v = v $
The desired result $-(-v... | Your solution seems good, acknowledging the fact that V is a vector space, and therefore satisfies the axioms pertaining to the proof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1597652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
Finding a function $f$ with the minimal $\|f'\|_1$ I was wondering about the following question, which I am sure the answer is known. I couldn't quite find it and I would appreciate if someone could tell me.
Suppose I have a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
$f(-1)=0$, $f(0)=1$, $f(1)=0$, and... | Sketch: Assuming $f\in C^1,$ we have $\int_{-1}^0f' = f(0)-f(1) = 1.$ Hence $\int_{-1}^0|f'| \ge 1.$ The same applies on $[0,1].$ Thus
$$\int_{\mathbb R}|f'| = \int_{-1}^1|f'| \ge 2.$$
Towards finding the minimum value: Let $f$ be any $C^1$ function satisfying the hypotheses that in addition is increasing on $[-1,0]$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1597719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof of a determinant expansion This is equivalent to a result in Prasolov's book on linear algebra whose proof is not clear to me. I need help in understanding why the result is true.
Let $x_1,x_2,\dots,x_n$ be row vectors in $R^n$. Let $e_1,\dots,e_n$ denote the canonical basis row vectors in $R^n.$
Let $M(x_1,\d... | This follows from the generalized laplace expansion of the determinant from the first $k$ rows.
We have,
$$
M(x_1,\dots,x_n) = \sum_{A : |A| = k}\pm S(1,\dots,k;A) S(k+1,\dots,n;A^c)
$$
where for $A \subset \{1,\dots, n\}$ with $|A| = k$, $S(1,\dots,k;A)$ denotes the determinant of the submatrix of the matrix with rows... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1597783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving a set is a basis, having already given a basis Question: If {$u,v$} is a basis for the subspace U, show that {$u+2v,-3v$} is also a basis for U
My attempt:
We must prove that {$u+2v,-3v$} spans U and is linearly independent
We know that given any $w \in U$ there exists $w = a_1(u) + a_2(v)$ where $a_1,a_2 \in ... | First, note that $\dim U=2$ so it suffices to show that $\{u+2\,v,-3\,v\}$ is linearly independent. To do so, note that
$$
\lambda_1(u+2\,v)+\lambda_2(-3\,v)=0
$$
if and only if
$$
\lambda_1u+(2\,\lambda_1-3\,\lambda_2)v=0\tag{1}
$$
But $\{u,v\}$ is linearly independent so (1) holds if and only if
\begin{array}{rcrcrc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1597862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Showing that $\lim_{{x\to 0}}(1+\sin{x})^{\frac{1}{x}} = e$ How do you calculate $\displaystyle \lim_{{x\to 0}}(1+\sin{x})^{\frac{1}{x}}$? I got it from here. It says L'Hopital, but I can't figure out how to apply it as I don't have a denominator. I also tried to rewrite the limit using trig identities:
$\displaystyle ... | my answer:
$\lim_{x\to 0}\left(1+\sin x\right)^{1/x}$
$=\lim_{x\to 0}(\left(1+\sin x\right)^{\frac{1}{\sin x}})^{\frac{\sin x}{x}}$
Note: $\lim_{x\to 0}\frac{\sin x}{x}=1$, so i get
$=(e)^1=e$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1597940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
How to get the complex number out of the polar form How does one get the complex number out of this equation?
$$\Large{c = M e^{j \phi}}$$
I would like to write a function for this in C but I don't see how I can get the real and imaginary parts out of this equation to store it in a C structure.
| $$\mathcal{R}(c)=M\cdot\cos{\phi}$$
$$\mathcal{I}(c)=M\cdot\sin{\phi}$$
Assuming that $j^2=-1$ (Physics notation). If we follow a math notational convention where $i^2=-1$ and $j=e^{{2i\pi\over 3}}$ is a complex root of unity we just replace in the above $\phi$ by $\phi+{2i\pi\over 3}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1598067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Boolean Algebra Product of Sums I have a question to solve the following expression and get it in terms of product of sums
(AB' + A'B)C
And I tried taking the compliment of this
[(AB' + A'B)C]'
[(AB' + A'B)' + C']
[(AB')'.(A'B)' + C']
[(A' + B).(A + B') + C']
[(A' + B + C').(A + B' + C')]
Is this the correct me... | As @coffeemath noted, your answer is actually the complement of the original expression. So this method is not correct.
I think the easiest way is to use the identities $(x + y)' = x'y'$ and $(xy)' = x' + y'$ (known as De Morgan's laws) and $x'' = x$. We have $$AB' + A'B = ((AB')'(A'B)')' = ((A'+B)(A+B'))'=$$
$$=(A'A +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1598262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Can we find $x_{1}, x_{2}, ..., x_{n}$? Consider this.
$$x_{1}+x_{2}+x_{3}+....+x_{n}=a_{1}$$
$$x_{1}^2+x_{2}^2+x_{3}^2+....+x_{n}^2=a_{2}$$
$$x_{1}^4+x_{2}^4+x_{3}^4+....+x_{n}^4=a_{3}$$
$$x_{1}^8+x_{2}^8+x_{3}^8+....+x_{n}^8=a_{4}$$
$$.............................$$
$$x_{1}^{2^{n-1}}+x_{2}^{2^{n-1}}+x_{3}^{2^{n-1}}+.... | If you mean the diophantine-equation tag and the solutions are supposed to be integers, searching will be easy because high powers are spaced far apart. Because of the symmetry you can insist that $x_1 \ge x_2 \ge \dots \ge x_n$. You can focus just on the last equation $x_{1}^{2^{n-1}}+x_{2}^{2^{n-1}}+x_{3}^{2^{n-1}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1598329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How many combinations for a 5 digit code using 3 numbers. Can anyone please help here?
I have inherited a strange looking safe with only numbers 1 2 and 3.
The code to open it is 5 digits and the code uses all three numbers at least once.
Is there some formula I can apply to list all the combinations?
Thanks
| Much simpler is to just [choose numbers] $\times$ [permute them]
3-1-1 of a kind, e.g 22231: $\binom{3}{1,2}\times\frac{5!}{3!} = 60$
2-2-1 of a kind, e.g. 22113: $\binom{3}{2,1}\times\frac{5!}{2!2!} = 90$
yielding the answer of 150
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1598410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Example of using Delta Method Let $\hat p$ be the proportion of successes in $n$ independent Bernoulli trials each having probability $p$ of success.
(a) Compute the expectation of $\hat p (1-\hat p)$.
(b) Compute the approximate mean and variance of $\hat p (1-\hat p)$ using the Delta Method.
For part (a), I can calcu... | Let $$ k = #{success in n independent Bernoulli trials}$$ then we have $$ k \sim B(n,p)$$ and $$ E(\hat{p})=E(\frac{k}{n})=\frac{E(k)}{n}=\frac{np}{n}=p $$ $$Var(\hat{p})=Var(\frac{k}{n})=\frac{Var(k)}{n^{2}}=\frac{np(1-p)}{n^{2}}=\frac{p(1-p)}{n}$$ thus $$E(\hat{p}^{2})=Var(\hat{p})+ E(\hat{p})^{2}=\frac{p-(1-n)p^{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1598503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Valid method to obtain a basis of a topological subspace? Let $(X,\tau)$ be a topological space and $Y \subset X.$ We know that if $\mathcal{B}$ is a basis for $\tau$ and $\tau_{\small{Y}}$ is the subspace topology on $Y$, then we can obtain a basis for $\tau_{\small{Y}}$
by taking the collection $\mathcal{B}_Y$ of int... | No. Take a line in $\mathbb{R}^2$.
Also, your case $[1,2]$ doesn't seem to work for me. It seems neither $1$ nor $2$ would have a neighbourhood.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1598562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Real analysis supremum proof
Let $A$ be a non-empty bounded sub-set of $\mathbb{R}$. Let
$B\subset\mathbb{R}$, given by $$B=\left\{\frac{a_1+2a_2}{2} \,\Bigg|\,a_1,a_2\in
A\right\}$$ Express $\sup B$ in terms of $\sup A$.
My attempt:
Suppose $a_1,a_2\in A$ and $b\in B$.
Then $a_1 \leq \sup (A)$ and $a_2\leq \sup ... | There are Lemmas that would be helpful.
$\sup(A+B) = \sup(A) + \sup(B)$
$\sup(cA) = c\sup(A), c\geq 0$
where A, B are subset of R, $A + B = \{z = a + b| a \in A, b \in B\}$, and c is a non-negative real number
then your questions can be answered: $\sup((A + 2*A)/2) = 3\sup(A)/2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1598667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
solving $\int \cos^2x \sin2x dx$
$$\int \cos^2x \sin2x dx$$
$$\int \cos^2x \sin2x \, dx=\int \left(\frac{1}{2} +\frac{\cos2x}{2} \right) \sin2x \, dx$$
$u=\sin2x$
$du=2\cos2x\,dx$
$$\int \left(\frac{1}{2} +\frac{du}{4}\right)u \, du$$
Is the last step is ok?
| Using the substitution you proposed $u\leadsto\sin2x$ one would get
$$\begin{align}
\int\cos^2x\sin2x\,\mathrm dx&=\int\left(\dfrac12+\dfrac{\cos 2x}2\right)\sin 2x\,\mathrm dx\\
&=\int\dfrac{\sin 2x}2\,\mathrm dx+\int\dfrac{\sin 2x}{4}\underbrace{{2\cos 2x}\,\mathrm dx}_{\displaystyle\mathrm du}\\
&=\int\dfrac{\sin 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1598733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 2
} |
Determining linear independence in $\mathbb{R}^3$ Let $\{\xi_k\}_{k=1}^4$ be a set of vectors in $\mathbb{R}^3$. If $\{\xi_1, \xi_2\}$ and $\{\xi_2, \xi_3, \xi_4\}$ are independent sets, and $\xi_1$ belongs to the span of $\{\xi_2, \xi_4\}$. Show that $\{\xi_k\}_{k=1}^3$ is linearly independent.
Clearly, $\xi_1 = a_1\... | hint
Start with a linear combination of vectors $1,2,3$ and set it equal to $0$ (vector). Now write vector $1$ as a linear combination of vectors $2$ and $4$. This gives you a linear combination of vectors $2,3$ and $4$ equal to the zero vector. Now use the independence of $2,3,4$ to find the coefficients.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1598947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
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