Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What is it called when you have 2 subsets of a set who's elements are based on alternating by a divisor? I came across this concept when I had to figure out a way to alternate spacing in a program I was writing. I'm curious to know if there's a formal term for this concept.
Here's the description.
You have a divisor ... | Here's one way to solve the problem - it's another way to look at @GregMartin 's solution.
Start with the set $S = \{1, \ldots D\}$. Then $A$ contains the numbers you get by adding even multiples of $D$ to the elements of $S$; $B$ contains the sums with odd multiples.
This would be an easy standalone loop in your prog... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2010883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Extension of Scalars and Polynomial Rings I am wondering if the following is true:
Let $A, R$ commutative rings, $\varphi:R\rightarrow A$ a surjective homomorphism, see $A$ as an $R$-algebra. Then
$$ R[X_1,\dots,X_m]\otimes_R A\approx A[X_1,\dots,X_m]$$
I am thinking in applying this to the case of $R\rightarrow R/J$ w... | It is important to know that your special case is the only case, up to isomorphism.
Here is a proof: define $\phi_1:A\to A[x_1, \dots x_n]$ by $a\mapsto a$ and $\phi_2:R[x_1, \dots x_n]\to A[x_1, \dots x_n]$ by $\sum_i a_Ix^I\mapsto \sum_I \varphi(a_I)x^I$. These maps both coincide on $R$, so by the universal property ... | {
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"url": "https://math.stackexchange.com/questions/2010998",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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The total probability of a minimum amount probability-varied choosers choosing one option over another. OK, the title may seem a bit convoluted but here it goes. This is an simplified example of what I am trying to get at. I am really more interested in the method of figuring this out, and how it can be applied to othe... | An exact answer is going to be painful. Let $p_i$ be the probability that person $i$ chooses box B, and these probabilities are independent.
Let $Q=(1-p_1)\cdots(1-p_{10})$.
Then the probability that at least three of them pick $B$ is:
$$1-Q\left[1+\sum_{i=1}^{10} \frac{p_i}{1-p_i} + \sum_{1\leq i<j\leq 10}\frac{p_i}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2011101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Explain how and why the cancellation of $ 6 $’s in $ \dfrac{16}{64} $ to get $ \dfrac{1}{4} $ is a fallacious statement. Based on what we know from elementary and middle school teachers, most of us know that 16/64 correctly equals 1/4 because 16/64 is simplified with a common divisibility of 16. However, there is anoth... | The wrong proof is more of a joke than a serious fallacy:
$$ \frac{16}{64} = \frac{16\llap{/}}{\rlap{/}64} = \frac 14 $$
This joke exploits the notational ambiguity that writing two symbols next to each other can either mean multiplication or -- if the symbols happen to be digits -- be part of the usual decimal notatio... | {
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"url": "https://math.stackexchange.com/questions/2011233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Number of ways to create a password
You are allowed to choose from digits 0 to 9 (inclusive) and 20
letters to make a password with 2 digits and 4 letters. You can repeat
digits and letters in this password. How many different passwords are
possible?
I know that there's $(10^2)(20^4)$ ways of selecting digits a... | There are you need to fill. Two spaces need to be filled with digits, so there are $6! \over 4!$ ways to arrange the two digits in six spaces. There are $4!$ ways to arrange the four letters in the four remaining spaces. There are $10^2$ ways to fill in the digit spots, and there are $20^4$ ways to fill in the letter s... | {
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"url": "https://math.stackexchange.com/questions/2011340",
"timestamp": "2023-03-29T00:00:00",
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Evaluating $\iiint_B (x^2+y^2+z^2)dV$ where $B$ is the ball of radius $5$ centered at the origin. The question asks to use spherical coords. My answer is coming out wrong and symbolab is saying I'm evaluating the integrals correctly so my set up must be wrong.
Since $\rho$ is the distance from the origin to a point on ... | Partition your ball $B$ into spherical shells of radius $\rho$ $(0\leq\rho\leq5)$ and thickness $d\rho$. The volume of such a shell is $dV=4\pi \rho^2\,d\rho$. It follows that
$$\int_B \rho^2\>dV=\int_0^5\rho^2\>4\pi\rho^2\>d\rho=2500\pi\ .$$
| {
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"url": "https://math.stackexchange.com/questions/2011442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Many products can operate if, out of $n$ parts, $k$ of them are working. What is the probability that the product operates?
Many products can operate if, out of $n$ parts, $k$ of them are working. Say $n = 20$ and $k = 17$, $p = 0.05$ is the probability that a part fails, and assume independence. What is the probabili... | "I mean... this question is looking at sample of 20 parts and 17 of them are said to be functioning; "
I think there is some misunderstanding here. The question meant to say that out of 20 parts, if (at least) 17 of them are functioning, then the product is still working.
So if $X$ is the random variable: Number of par... | {
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Complex number equation solving Can somebody help me to solve this equation?
$$(\frac{iz}{2+i})^3=-8$$ ?
I'm translating this into
$(\frac{iz}{2+i})=-2$
But i recon it's wrong ...
| You are in the right way
$$\frac { iz }{ 2+i } =\sqrt [ 3 ]{ 8\cdot \left( -1 \right) } =-2\left( \cos { \frac { k\pi }{ 3 } +i\sin { \frac { k\pi }{ 3 } } } \right) ,k=0,1,2$$ check for instance $k=0$
$$\frac { iz }{ 2+i } =-2\\ iz=-4-2i\\ z=\frac { -4-2i }{ i } =\frac { -4i+2 }{ { i }^{ 2 } } =-2+4i$$
| {
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"url": "https://math.stackexchange.com/questions/2011754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Triangle inequality with spectral norm Let $A$ be a real square matrix, I have to prove that
$$\left \| A \right \|=\sqrt{\lambda_{\text{max}}(A^*A)}$$
defines a norm. I don't know how to prove the triangle inequality. I have already proved that $\|A\|=\|A\|_2=\sup_{\|x\|_2=1} \|Ax\|_2$, but the exercise is to prove wi... | Preliminary answer: This inequality follows from Theorem 3.3.16 in Horn and Johnson (1994) - Topics in Matrix Analysis, and I think they don't use the equality you said in their proof.
Although I don't understand their proof, I am trying to do so since I have this question too (from research). I will update this answer... | {
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Am I correctly using the conditional probability notation in this problem? Problem: Two numbers are chose at random from among the numbers 1 to 10 without replacement.
Goal: Find the probability that the second number chosen is 5.
Official Solution:
$A_{i}:i \in \left\{1,2,...,10\right\}$ is event that first number cho... | You are asked to evaluate the probability that the second number chosen is $5$. This is clearly $P(B)$. There is no condition mentioned.
Yes you are right that you cannot choose a second number without having chosen the first number. But $P(B)$ this is the probability that the second chosen number is $5$ regardless whi... | {
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Projection onto subspace spanned by a single vector Given the vectors $x=(2,2,3)$ and $u_1=(0,3,0)$, vector $v_1$ is the projection of $x$ on the subspace spanned by $u_1$, so $v_1=\alpha u_1$.
Determine the value of $\alpha$.
My attempt
I haven't dealt with a case where the subspace is spanned by only one vector befor... | Hint:
The component of the vector $\mathbf{x}$ in the direction of $\mathbf{u_1}$ is:
$$
\mathbf{v_1}=\frac{\mathbf{x}\cdot \mathbf{u_1}}{|\mathbf{u_1}|^2}\mathbf{u_1}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2012085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$n$ balls numbered from $1$ to $n$
If we consider a box containing $n$ balls numbered from $1$ to $n$. If $455$
is the number of ways to get three balls from the box such that no two
balls are consecutively numbered, then we have to find the value of $n$.
Someone please help me out in this. I am not getting anyth... | As I tend to make mistakes with this kind of problems, I asked my friend Ruby to do some test runs. She is pretty fast at counting and makes no silly mistakes if I explained it well.
We got different numbers, much larger numbers until I realized that order should not matter for a draw, so drawing $1,3,5$ is considered ... | {
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Are eigenspaces and minimal polynomials sufficient for similarity? This question comes out of the conversation in the comments of this answer. The answerer asserts the following:
$\DeclareMathOperator{\rank}{rank}$
Suppose that $A$ and $B$ have the same minimal polynomial and that for all $\lambda \in \Bbb C$, $\rank... | $\DeclareMathOperator{\rank}{rank}$The answer is no. Notably, the minimal polynomial determines the size of the largest blocks in the Jordan form, $\rank(A - \lambda I)$ determines the total number of blocks.
Let $J_k$ denote the $\lambda = 0$ block of size $k$. Consider the matrices
$$
A = J_3 \oplus J_2 \oplus J_2... | {
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Smoothness of functions up to the boundary Let $\Omega$ be an open set in $R^n$ with $C^1$ boundary $\partial \Omega$.
Suppose that $f$ is a $C^1$ function on $\Omega \cup \partial \Omega $, where $C^1$-smoothness of $f(x)$ on $\partial \Omega$ is defined via local $C^1$- diffeomorphism.
If we define a function $g$ on... | No, it's not true, because your definition of being $C^1$ on $\Omega
\cup \partial \Omega$ is bad.
Your definition only guarantees that the restriction of $f$ to the boundary is $C^1$, but that is not good enough: $f$ can be $C^1$ on $\Omega$ and on $\partial \Omega$ without being even continuous on $\Omega
\cup \par... | {
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Prove that a quotient group has an element of order 2.
Let $H$, $K$ be subgroups of $G$, with $H$ being a normal subgroup of $G$. Suppose the cardinality of $H\cap K$ is odd and $K$ has an element $k$ of order $2$. How would you prove that the quotient group $G/H$ has an element of order $2$?
I really don't understa... | If $k\in K$ has order two, this means that $k^2 = e$. $k$ cannot belong to $H$, because $H\cap K$ is a subgroup of $G$, and the order of any element in a group divides the order of the group (Lagrange's theorem). If $H\cap K$ has an odd number of elements, then there can be no element of even order in $H\cap K$. Elemen... | {
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"timestamp": "2023-03-29T00:00:00",
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Let f,g be continous functions from X to Y. Let E be a dense subset of X. Show if $f(x)=g(x) \forall x \in E$, then f(x)=g(x) $\forall x \in X$. I'm trying to prove that if f,g are continuous functions, and if E is a dense subset of X $(\text{or } Cl(E) = X)$ and if $f(x)=g(x) \forall x \in E$ then $f(x)=g(x) \forall x... | A little bit late, but I decided to give an alternate answer without using contradiction or the sequence definition of continuity. Let $p \in X$ and $\epsilon > 0$ be given. If $p \in E$ we are done, otherwise $p$ is a limit point of $E$. By continuity of $f, g$, there exists $\delta_1, \delta_2 >0$ s.t. $\forall x \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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Line and segment relationship in the circle if the larger gear has 30 teeth and the smaller gear has 18, then the gear ratio(larger to smaller) is 5:3. when the larger gear rotates through and of 60°, through what angel measure does the smaller gear rotate?
| The angle between two teeth in the lager gear is $\frac {360°}{30}=12°$ and for smaller gear is $\frac {360°}{18}=20°$. When the larger gear rotates for $60°$ it means $5$ teeth has moved $(\frac {60°}{12°}=5)$. Because these gears are connected then the smaller gear must move for $5$ teeth too, and $5$ teeth in smalle... | {
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Sum of Maclaurin series Find the sum of the infinite series
\begin{equation}
\sum_{n=2}^\infty\frac{7n(n-1)}{3^{n-2}}
\end{equation}
I think it probably has something to do with a known Maclaurin series, but cannot for the life of me see which one.. Any hints would be appreciated!
Edit: Using your hints, I was able to ... | Hint. We have that for $x\not=1$, and $N\geq 2$,
$$\frac{d^2}{dx^2}\left(\frac{1-x^{N+1}}{1-x}\right)=\frac{d^2}{dx^2}\left(\sum_{n=0}^N x^n\right)=\sum_{n=2}^N n(n-1)x^{n-2}.$$
P.S. for the downvoters. I considered a finite sum because it is not so straightforward to say that we can interchange the differentiation and... | {
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"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Calculus exercise (James Stewart book) : Volume of a box I'm learning calculus but I'm very stuck in this exercise.
Exercise
I know that the volume is :
2HL + 2HW + LW (only LW because open top).
Thanks to the exercise we know that:
W = 12 - 2x
L = 20 - 2x
H = x
Consequently I replaced the values:
2x(20 - 2x) + 2x(12 ... | You haven't calculated the volume of the box, but the surface of the cutout.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2013335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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An differential equation$(x^{2}＋y^{2}＋3)\frac{dy}{dx}＝2x(2y-\frac{x^{2}}{y})$ How to solve this ODE $$(x^{2}＋y^{2}＋3)\frac{dy}{dx}＝2x(2y-\frac{x^{2}}{y})$$
I tried to find its integral factor, but failed.
Many thanks for your help.
| $(y^2+x^2+3)(y^2)'=4x(2y^2-x^2)$
$(y^2+x^2+3)(y^2+x^2+3)'=4x(2y^2-x^2)+2x(y^2+x^2+3)$
$(y^2+x^2+3)(y^2+x^2+3)'=10x(y^2+x^2+3)-4x(3x^2+6)$
With $\enspace z:=y^2+x^2+3\enspace$ we get $\enspace zz'=10xz-12x^3-24x$ .
I don't know how to solve this, but Wolfram solves it in a closed form with
http://www.wolframalpha.com... | {
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Show that $f$ can not be expanded into a power series in a neighborhood of $0$. Let $f(x)$ be defined by $$f(x)=\begin{cases}e^{-1/x^2},& \text{ if }x \neq 0,\\0, &\text{ if }x=0.\end{cases}$$
Show that $f$ can not be expanded into a power series in a neighborhood of $0$.
We know that $f$ must be infinitely differenti... | If you try to compute all the derivatives $f'(0)$, $f''(0)$, etc, they are all zero.
So, if $f$ could be expanded into a power series in a neighborhood of 0, it would be the zero function there.
This is a contradiction, as $f$ is not the zero function in any neighborhood of 0.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2013565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Systems of linear equations: solving by addition I have not understood this question. it confused me a lot. I have tried to solve it but the solution was senseless.
$$\begin{cases}
3x + 4y - 5 = 2x + 3y - 1\\
6x - 2y + 2 = 4x - 3y - 5
\end{cases}$$
Can anyone help me with this?
| HINT:
First turn the top and bottom equations into standard form, so you end up with a system that looks like this.
$$\begin{cases}
Ax + By = C\\
Dx + Ey = F
\end{cases}$$
This can be done by combining the $x$ and $y$-terms on the left and moving the constant terms to the right. Then solve this "normal-looking" system... | {
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"timestamp": "2023-03-29T00:00:00",
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Shortest length of line segment that bisects triangle in two equal area A segment of a line $PQ$ with its extremities on $AB$ and $AC$ bisects a triangle $ABC$ with sides $a,b,c$ into two equal areas, then find the shortest length of the segment $PQ$.
I was looking for small hint as how to approach this question? I am ... | Let $AP=:p\geq0$, $AQ=:q\geq0$. Then we have to mnimize
$$f(p,q):=p^2+q^2-2pq\cos\alpha$$
under a condition of the form $g(p,q):=pq-C=0$ for some $C>0$. As $p\to0+$ enforces $q\to \infty$ and therefore $f(p,q)\to \infty$, and vice versa, it follows that the minimum of $f$ is taken for certain $p>0$, $q>0$ found by usin... | {
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"url": "https://math.stackexchange.com/questions/2013828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How to replace 8-sided dice with other dice The exact question is:
You need an 8-sided die for a game. You only have a coin, two four-sided and one 10-sided dice. How can you replace the 8-sided die?
Re-rolls are not allowed.
There are several solutions to this I've been told, I found one, but my solution wasn't one of... | To keep the uniform repartition, here are some proposals :
*
*Roll the 10-face dice, re-roll on 9 and 10.
*Roll the 4-face dice, then flip the coin. Add 4 on heads.
*Roll the 4-face dice and multiply the result by 2. Flip a coin, and substract 1 on heads.
There are probably a number of other solutions, the impo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2013926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 13,
"answer_id": 12
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Fundamental theorem of calculus with improper integral Suppose $f(x,y)$ is continuous everywhere except where $y=x$. Does FTC apply in a scenario where $x$ is given as a boundary? If so, how? Here's a general example.
$$\frac{d}{dx}\int_x^bf(x,y) dy$$
There is a related post here that doesn't answer this question. The ... | I ended up solving my specific problem by introducing a substitution to set the limits of integration to something nicer.
\begin{align}
\frac{d}{dx}\int_x^\infty \frac{g(y)}{\sqrt{y^2-x^2}} dy
& =\frac{d}{dx}\int_0^\infty \frac{g(y)}{\sqrt{u}} \frac{du}{2y}
&\text{ using: } u = y^2 - x^2
\end{align}
Since $u$ is a d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2014055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Result of Cauchy Schwarz inequlaity (homogeneous case) I have shown the Cauchy Schwarz inequality to be $$ \langle v_i(\vec x),v_j(\vec y)\rangle^2 \ \le \ \langle v_i(\vec x),v_i(\vec x)\rangle\langle v_j(\vec y),v_j(\vec y)\rangle.$$
How would I use this to show $$ \langle v_i(\vec x),v_i(\vec y)\rangle\ \le \ \lang... | What you want to prove does not hold in general.
For a counter example, suppose first the vectors have dimension $1$. You want to prove that $(ab)^2\leq a^2 b^2$ implies $(ab)\leq a^2$. However, if $a=1, b=2$, then $2=(ab)>a^2=1$. More generally, for any nonzero vector $x$ and $y=\alpha x$, $\alpha>1$, we have $\langl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2014240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Find all square roots of $x^2+x+4$ in $\mathbb{Z}_5[x]$.
Find all square roots of $x^2+x+4$ in $\mathbb{Z}_5[x]$. Also show that in $\mathbb{Z}_8[x]$ there are infinitely many square roots of $1$.
I know how to find the square roots when in $\mathbb{Z}[x]$ however I keep getting confused with $\mathbb{Z}_5[x]$. From... | Hints:
A square root of $x^2+x+4$ has the form $\pm x+a$. Let's identify:
$$(\pm x+a)^2=x^2\pm 2a x+a^2=x^2+x+4.$$
We have to solve $\;\begin{cases}a^2=4\\\pm2a=1\end{cases}$.
The first equation has solutions $\;a=\pm 2$. Checking the second equation, we find the square roots
$$x-2\enspace\text{or}\enspace -x+2. $$
For... | {
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$M$ and $N$ equal in $K_0$ $\Rightarrow$ $\exists P$ such that $M\oplus P\cong N\oplus P$ Let $\mathcal P(A)$ be the category of finitely generated projective $A$-modules ($A$ is a ring with unity). Then consider the free group $F$ over the isomorphism classes of $\mathcal P(A)$. I will indicate the isomorphism classes... | By the definition of a free abelian group, we have the following observation:
If $[M_1] + \dotsb + [M_r] = [N_1] + \dotsb + [N_s]$, we have $r=s$ and the summands are the same up to permutations. Since the direct sum is commutative, we get an isomorphism.
$$M_1 \oplus \dotsb \oplus M_r \cong N_1 \oplus \dotsb \oplus N_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2014510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Can the Ricci tensor determine a manifold up to homeomorphism? I don't know how to ask this question. Maybe there are many improper statements.
Let $(M, g)$ and $(N, h)$ be two Riemannian manifolds and $f:M \to N$ a bijection such that $\operatorname{Ric}(x)$ is equal to $\operatorname{Ric}(f(x))$ up to a coordinat... | This is not true. For example, not all Ricci-flat manifolds are homeomorphic. More explicitly, for any bijection $f$ between $\mathbb{R}^n$ and $(S^1)^n$, $\operatorname{Ric}(x) = \operatorname{Ric}(f(x)) = 0$, but $\mathbb{R}^n$ and $(S^1)^n$ are not homeomorphic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2014616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Finding domain of variables in joint density for marginal density Let $(X,Y)$ have joint density $f(x,y) = \frac{1}{2}(1+x+y)$ for $0<x<1$ and $0<y<1$.
So the joint density of $X$ and $U=X+Y$ is $f_{X,U}(x,u)=\frac{1}{2}(1+u)$. Now it is simple to get the domain of $X$ since it is provided in the problem description, b... | The domain of $(X,U)$ is a parallelogram in $\mathbf{R}^2$: $X$ is in $(0,1)$, while $U$ is in $(X,X+1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2014730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove equivalence of two Fibonacci procedures by induction? I want to prove equivalence of the following Fibonacci procedures using proof by induction.
*
*version:
$$\text{fib_v1}(n) = \text{fib_v1}(n-1) + \text{fib_v1}(n-2)$$
, with base cases $\text{fib_v1}(0) = 0$ and $\text{fib_v1}(1) = 1$.
*version:
$$\te... | Let me first save some writing by saying $f_1 = fib_{v1}$ and $f_2 = fib_{v2}$
I think you are stuck because the claim you try to prove: $f_2(n,a_0,a_1) = f_1(n)$ is only true for $a_0 = 0$ and $a_1 = 1$. The claim is not true for other values of $a_0$ and $a_1$.
The claim you really want to prove is that for all $n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2015060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In a $\triangle ABC,\angle B=60^{0}\;,$ Then range of $\sin A\sin C$
In a $\triangle ABC,\angle B=60^{0}\;,$ Then range of $\sin A\sin C$
$\bf{My\; Attempt:}$
Using Sin formula:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
So $\displaystyle \frac{a}{\sin A} = \frac{b}{\sin 60^0}\Rightarrow \sin A = \fr... | Clearly $\sin(A)\sin(C)\geq 0$ since $A$ and $C$ are between $0^{\circ}$ and $180^{\circ}$. Let $A$ approach $0^{\circ}$. Then $\sin(A)$ approaches $0$ as well, while $\sin(C)$ is bounded above by $1$. This shows that
$$\sin(A)\sin(C)\rightarrow 0$$
if $A\rightarrow 0^{\circ}$ so that $\sin(A)\sin(C)$ can be as small a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2015166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Complex numbers: $ (\sqrt{3}+ i)^{30} $ to 'a + ib' form How do I rewrite this to 'a + ib' form? The power of 30 is troubling me.
$ (\sqrt{3}+ i)^{30} $
| Hint. Since $|\sqrt{3}+i|=2$, we have that
$$(\sqrt{3}+ i)^{30}=2^{30}\left(\frac{\sqrt{3}}{2}+ \frac{i}{2}\right)^{30}=2^{30}\left(\cos(\pi/6)+ i\sin(\pi/6)\right)^{30}.$$
Then use De Moivre's Formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2015326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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For a normal Subgroup $N$ of $G$, is there an Isomorphism from $G$ to $N\times (G/N)$? For a normal Subgroup $N$ of $G$, is there an Isomorphism from $G$ to $N\times (G/N)$ (the Product with the Quotient)?
[What I thought about: The number of elements is the same, so one could establish a bijective Map. But it is diffi... | No. This is not true at all. Note that if what you think were true every solvable group would be abelian.
For a specific example take $S_3$ and the subgroup $A_3$ which is cyclic of order $3$. The quotient is cyclic of order $2$. So the product is abelian, in fact cyclic of order $6$. But $S_3$ is non-abelian.
It is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2015399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Expected number of distinct items picked from a set, putting back an item in the box every time it is picked I have a box with $N$ items. For $W$ times, I pick an item, label it somehow, and put it back. Items that are picked twice or more are labeled only once.
At the end of the process, what is the expected number of... | First, if you're just looking around online, you should try the coupon collector's problem. In its canonical form, the coupon collector's problem is to calculate the expected number of picks until all the coupons (or items in your box) are labelled. Your problem has the tables turned. You want to calculate the expecte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2015557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Linear elliptic partial diff. operator Let $L$ be the above operator
\begin{equation}
L u = \sum_{i,j=1}^{n} a_{ij}(x) u_{x_ix_j} + \sum_{i=1}^{n} \beta_i u_{x_i} + c(x) u(x)
\end{equation}
Assume furthermore that L is elliptic i.e.
\begin{equation}
0 < \lambda {|\vec{\xi }|}^2 \leq \sum_{i,j=1}^{n} a_{ij}\xi_i\xi_j... | If you already know how to show that $AH$ is negative semi-definite, then just note that
$$\sum_{i,j=1}^n a_{ij}u_{x_ix_j} = \text{Trace}(AH)\leq 0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2015674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Question about Improper Integrals I am trying to find whether the following integral converges or diverges, but I am a bit uncertain about the steps I'm taking. I have to following integral.
$$\int_{-1}^1 \frac{e^x}{e^x-1} \,dx \\$$
Because I know it is discontinuous at x=0, I split the integral
$$\int_{-1}^0 \frac{e^x... | Let us study $$I=\int_0^1\frac{e^x}{e^x-1}dx$$
with the change $t=e^x$,
then $I$ becomes:
$$\int_1^e\frac{dt}{t-1}$$ which diverges
since
$$\lim_{X\to 1^+}\int_X^2\frac{dt}{t-1}=\lim_{X\to1^+}(-\ln(|X-1|))=+\infty.$$
Your integral is therefore divergent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2015834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$x+y+z+4xyz=2$ prove that$ xy+yz+xz\le 1$ Let $x$,$y$,$z$ be non-negative real numbers, for which:
$x + y + z + 4xyz=2$
Prove that:
$xy + yz + zx \le1$.
I am sure this can be done with Cauchy-Schwarz or AM-GM, but I have long forgotten how to use those...Any help is appreciated !
| I don't know about using the inequalities, but this can be solved without any tricks!
\begin{eqnarray}
(1-2x)(1-2y)(1-2z) & \leq & 1 \\
1 - 2(x+y+z) + 4(xy+xz+yz)-8xyz & \leq & 1 \\
1 - 2(x+y+z+4xyz) + 4(xy+xz+yz) & \leq & 1 \\
1 - 4 + 4(xy+xz+yz) & \leq & 1 \\
xy+xz+yz & \leq & 1
\end{eqnarray}
The first inequality ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2015924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why do we define vector spaces over a general $\mathbb{F}$ (field) rather than $\mathbb{C}$ (complex numbers)? In my linear algebra class, many times we define vector spaces over the field $\mathbb{F}$ (i.e. $\mathbb{F}^n$) and then prove things about them. The instructor has defined $\mathbb{F}$ as "either $\mathbb{C}... | There are different fields: Field of rational numbers, algebraic function fields, algebraic number fields, p-adic fields...
The field is the set of scalars for the vector space.
Is the division in $\mathbb{R}$ similar to that in $\mathbb{C}$? Absolute value?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2016019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Does $\sum\frac{4^n(n!)^2}{(2n)!}$ converge? Does $\sum\frac{4^n(n!)^2}{(2n)!}$ converge? The ratio test is inconclusive. I know that $\sum\frac{2^n(n!)^2}{(2n)!}$ converges, if that's of any help.
| Try
$$
a_n = \frac{4^n (n!)^2}{(2n)!}
$$
$$
\frac{a_{n+1}}{a_{n}} = 4 \frac{(n+1)^2}{(2n+1)(2n+2)} > 1
$$
conclude that $a_n$ is monotonically increasing and also $a_n > 0$ so
$$
\lim \limits_{n \to \infty} a_n \ne 0
$$
So
$\sum a_n $ diverge.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2016076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 6
} |
A picture frame measures $14$ cm by $20$ cm. $160$ cm$^2$ of the picture shows inside the frame. Find the width of the frame.
A picture frame measures $14$cm by $20$cm. $160$cm$^2$ of the picture shows inside the frame. Find the width of the frame.
This is the question I was given, word-for-word. Is it asking for the... | You are expected to assume that the picture frame has a constant width. That width creates a border around the picture. Note that the area of the frame is $280 \text{ cm}^2$, which is larger than the picture. You are supposed to find th width of the frame.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2016177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 1
} |
For $f(x)=4x^3+3x^2-x-1$, The Range of values $\frac{f(x_1)-f(x_2)}{x_1-x_2}$ can take is- My Attempt :-
$f'(x)=12x^2+6x-1$ where $f'(x) \ge \frac{-7}{4}$. So (I think) from LMVT we can directly say that
$$\frac{f(b)-f(a)}{b-a} \ge \frac{-7}{4}$$ But the answer Given is $$\frac{f(b)-f(a)}{b-a} > \frac{-7}{4}$$
So my ... | Intuitively, as long as $f'(x)$ attains its minimum at one single point, there won't exist two distinct points where the secant matches that slope.
To formalize, assume $f'(x)$ has a global minimum $m$, then it follows from MVT that:
$$\frac{f(b)-f(a)}{b-a} \ge m \quad \text{for} \;\;\forall a \ne b$$
If $a,b$ points e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2016315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding PDF of $Z \thicksim Q / \sqrt{Q^2 + R^2}$ where $Q, R \thicksim N(0,1)$? I have only done simple convolutions up to this point to find joint PDFs and am lost about how to proceed with finding the PDF of $Z \thicksim \frac{Q}{\sqrt{Q^2 + R^2}}$ where $Q, R \thicksim N(0,1)$. How do I approach finding the PDF of ... | I assume that $Q$ and $R$ are independent. First, $F_Z(z)=0$ for $z\le -1$ and $F_Z(z)=1$ for $z\ge 1$. Also
$$
\mathsf{P}(Z\le z\)=\mathsf{E}\!\left[\mathsf{P}\!\left(Q\le z\sqrt{Q^2+R^2}\mid R\right)\right].
$$
For $z\in (-1,1)$,
$$
\mathsf{P}\!\left(Q\le z\sqrt{Q^2+r^2}\right)=\mathsf{P}\!\left(Q\le z\sqrt{\frac{r^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2016419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finite sum $\sum_{j=0}^{n-1} j^2$ How can I calculate this finite sum? Can someone help me?
$$\sum_{j=0}^{n-1} j^2$$
| I like this explanation:
Each row in the first triangle sums to $j^2$
So the sum of the rows is the number we seek.
Take that triangle and rotate left and right. The sum is of the three trianges, then is 3 times the number we seek. But when we sum it up, we get $(2n+1)$ in every entry in the triange or $(2n+1)$ time... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2016571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 0
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The locus of the point of trisection of all the double ordinates of the parabola $y^2= lx$
The locus of the point of trisection of all the double ordinates of
the parabola $y^2= lx$ is a parabola whose latus rectum is ?
I feel that since the double ordinates are trisected the latus rectum too should be trisected. S... |
Am I going wrong somewhere ?
It seems that you think that the focuses of the both parabolas have the same $x$-coordinate.
The coordinates of the points both on the parabola $y^2=lx$ and on $x=t$ are $(t,\pm\sqrt{lt})$.
Since the double ordinates on $x=t$ are trisected, we get
$$\left(t,\frac{1\cdot\sqrt{lt}+2(-\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2016669",
"timestamp": "2023-03-29T00:00:00",
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"Predicate" vs. "Relation" What's the difference between a predicate and a relation? I read the definition that an $n$-ary predicate on a set $X$ is a function $X^n\to \{\text{true}, \text{false}\}$ where $\{\text{true}, \text{false}\}$ is the set of truth values. Also, it is well-known that an $n$-ary relation is sim... | I like the definition from Wikipedia. It says that a relation is a concept, and a predicate is simply the indicator function of a relation. It’s like "the concept of prime numbers" and "the isPrime function".
Even in your question you say that a predicate is defined on the whole set, but a relation is a subset.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2016749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 2
} |
Square root vs raising to $\frac{1}{2}$ What about the +/- ??? These seem equivalent, yet the raising to 1/2 seems to ignore the +/- aspect of a square root. Is one more valid than the other?
| When you get a little further in math, you'll be told that raising a number to any exponent with a non-integer absolute value is what's called a "multi-valued function." It's just that with a rational exponent with an even denominator it's two of these that are real rather than one as for any other real exponent. Mos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2016845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Sample uniformly from sorted monotonic integer sequences Consider all ${x+x-1 \choose x}$ different integer valued sequences $S$ of length $x$ whose elements are from $\{1,\dots,x\}$ and where $S_i \leq S_{i+1}$.
How can I sample uniformly from this set of sequences?
The simplest possible strategy is to sample eac... | Stepwise Sampling Method
A uniform random selection from the set of such sequences can be constructed stepwise, sampling each element in turn, using an appropriate (often non-uniform) distribution in each step.
Letting $N(m,x)$ denote the number of nondecreasing length-$m$ sequences with elements in $[1..x]$, it follow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2016988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the limit of $(1-\frac2n)^n$ I am trying to find the limit of $$(1-\frac2n)^n$$
I know how $e$ is defined and I am sure the prove will involve substituting a term with $e$ at some point. But I do not really know where to start. I tried rewriting the term, simplifying it, using the binomial theorem, but all that do... | Note that
$$
\left(1-\frac2n\right)^n = \left(1-\frac{1}{n/2}\right)^n = \left(\left(1-\frac{1}{n/2}\right)^{n/2}\right)^2
$$
Now take the limit as $\frac n2 \to \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2017089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Calculate:$\int _{|z|=1}\text{Re z dz}$. For a positively oriented unit circle
Calculate:$\int _{|z|=1}\text{Re z dz}$.
If $z=x+iy\implies \text{Re z}=x$
So $\int _{|z|=1}\text{Re z dz}=\int_{|z|=1} x d(x+iy)=\int _{|z|=1}x dx$.(as dy=0)
$=\frac{x^2}{2}|_0^{2\pi }=2\pi^2$
But the answer is not matching.It is given ... | I thought it might be useful to present a "brute force way forward." Proceeding, we have
$$\begin{align}
\oint_{|z|=1}\text{Re}(z)\,dz&=\color{blue}{\oint_{\sqrt{x^2+y^2}=1}x\,dx}+\color{red}{i\oint_{\sqrt{x^2+y^2}=1}x\,dy}\\\\
&=\color{blue}{\int_{1}^{-1} x\,dx+\int_{-1}^1x\,dx}+\color{red}{i\underbrace{\int_{-1}^1\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2017213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
moving limit through the integral sign Let $F(\alpha) = \int\limits_0^{\pi/2} \ln( \alpha^2 - \sin^2 x) \mathrm{d} x $ where $\alpha>1$
Im tempting to argue that $F(1) = \int\limits_0^{\pi/2} \ln (1 - \sin^2 x) dx = \int\limits_0^{\pi/2} \ln \cos^2 x dx $
But, $\alpha > 1$. Thus, the only way we can do this is if we c... | It’s $\enspace\lim\limits_{\alpha\to 1^+} \ln(\alpha^2-\sin^2 x) =\ln\cos^2 x\enspace$ and we have to proof with
$$\lim\limits_{\alpha\to 1^+}(( \int\limits_0^{t\pi/2}\ln(\alpha^2-\sin^2 x)dx - \int\limits_0^{t\pi/2}\ln(\cos^2 x) dx )|_{t\to 1^-}) =\frac{\pi}{2} \lim\limits_{\alpha\to 1^+} \int\limits_0^1 \ln(1+\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2017297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Sum of elements of order $p$ in $(\mathbb{Z}/2p^2\mathbb{Z})^\times$? I have been working with the group $\mathbb{Z}/2p^2\mathbb{Z}$ and would like someone to correct my reasoning below.
Let $p$ be an odd prime, and consider the multiplicative group $(\mathbb{Z}/2p^2\mathbb{Z})^\times$. This group is cyclic, and so con... | It seems that you are including $1$ in your sum, even though it has order $1$ and not $p$. That's fine though: given an odd prime $p$, define $A$ to be the sum of the elements of order dividing $p$ modulo $p^2$. You are correct that $A\equiv p\pmod{2p^2}$.
It's not hard to show that the elements of order $1$ or $p$ mod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2017527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Confirming that the sequence $a_n = \frac{\sqrt{n}\cos{n}}{\sqrt{n^3}-1}$ converges I'm still trying to totally grasp the differences in proofs for sequences and series. I have a sequence $a_n = \frac{\sqrt{n}\cos{n}}{\sqrt{n^3}-1}$
In order to prove that this sequence converges, would it be correct to state that since... | You can write
\begin{equation}
-\frac{\sqrt{n}}{\sqrt{n^3}-1}\le\frac{\sqrt{n}\cos{n}}{\sqrt{n^3}-1}\le\frac{\sqrt{n}}{\sqrt{n^3}-1}
\end{equation}
and you use the squeeze theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2017694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Find three distinct triples (a, b, c) consisting of rational numbers that satisfy $a^2+b^2+c^2 =1$ and $a+b+c= \pm 1$. Find three distinct triples (a, b, c) consisting of rational numbers that satisfy $a^2+b^2+c^2 =1$ and $a+b+c= \pm 1$.
By distinct it means that $(1, 0, 0)$ is a solution, but $(0, \pm 1, 0)$ counts as... | Here's a start that shows that
any other solutions
would have to have
distinct $a, b, $ and $c$.
In
$a^2+b^2+c^2 =1$
and
$a+b+c= \pm 1$,
if $a=b$,
these become
$2a^2+c^2 = 1,
2a+c = \pm 1$.
Then
$c = -2a\pm 1$,
so
$1
= 2a^2+(-2a\pm 1)^2
=2a^2+4a^2\pm 4a+1
=6a^2\pm 4a+1
$
so
$0 = 6a^2\pm 4a
=2a(3a\pm 2)
$.
Therefore
$a=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2017818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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pigeonhole principle clarification Would the following statement be applicable to the pigeonhole principle? Or can I simply do $100 \times 50 = 5000$?
What is least amount of students in a school to guarantee that there are
at least 100 students from the same state?
| The minimum number to guarantee that would be $99\cdot 50+1=4951$. If you had fewer, say $4950$, then there's the possibility that there are exactly $99$ students from each state. Any one additional student from any state is enough at this point.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2017902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
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Help to understand a comment in Hoffman and Kunze's linear algebra book I'm reading Hoffman and Kunze's Linear Algebra and on page 52, the authors said:
Let $P$ be the $n \times n$ matrix whose $i,j$ entry is the scalar $P_{ij}$, and let $X$ and $X'$ be the coordinate matrices of the vector $\alpha$ in the ordered bas... | Another way to see this is to use the fact that the vector $\alpha$ has unique coordinates with respect to any fixed basis (as discussed on page 50 of the same textbook).
Let $\{ \beta_1,\dots,\beta_n\}$ be an ordered basis for the $n$-dimensional space $V$. We can always express the zero vector as
$$
0 = 0\beta_1 + \d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2017995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that the sum of a set of matrices isn't direct, and that the sum is the whole vector space. Let $U$ be upper triangular matrices in $M_2$ and $L$ lower triangular matrices in $M_2$. Show that their sum isn't direct, and that their sum is the whole vector space.
I have the following definitions:
If $L\cap M=\{0\}$... | I think the answer above is incomplete, and I'd like to complete it. In order for $M_2(\mathbb{K})$ to not be equal to the direct sum, this is, for $M_2(\mathbb{K})$ to differ from $U \bigoplus L$ we need to know the definition of direct sum which says:
each $x$ in $F_1 + ... + F_n$ is written in an unique way as the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2018148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove using induction that $2^{4^n}+5$ is divisible by 21 I have to show, using induction, that $2^{4^n}+5$ is divisible by $21$. It is supposed to be a standard exercise, but no matter what I try, I get to a point where I have to use two more inductions.
For example, here is one of the things I tried:
Assuming that $2... | Note that $2^3 = 1\mod 7$ and hence $2^{3n} = 1 \mod 7$. Now, $4^k -1 = 0\mod 3$ and it follows that $2^{4^k-1} = 1\mod 7$ and $2^{4^k} = 2\mod 7$, Thus
$$2^{4^k}+ 5 = 0 \mod 7 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2018239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Complement of regular language is regular The Question
Let $L$ be a regular language. Prove the $\overline{L}$ is a regular language without using automata.
Explanation
I have an assignment to solve without using automata at all. To solve one of the questions I want to use the following: "If $L_1$, $L_2$ are regular la... | There is also an algebraic characterization of regular languages.
A language $L\subset \Sigma^*$ is regular iff it exists an homomorphism (of monoids) $\phi : \Sigma^*\rightarrow M$ with $M$ a finite monoid and
$$
L=\phi^{-1}(S)
$$
where $S\subset M$. You end using the formula $\phi^{-1}(\bar{S})=\overline{\phi^{-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2018315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Average minimum distance between $N$ points generate i.i.d. uniformly on the shell (sphere) of ball What is the expected minimum euclidean distance between $N$ points uniformly and independently chosen on the shell(sphere) of 3-D ball of radius $R$? Note that the expected minimum distance might be difficult to comput... | The asymptotics is a straightforward (or boring) generalization of my previous answer.
Sketch: Take a point over the unit sphere ($R=1$). Then the "excluded area" $a(D)$ (points that lie at a -euclidean- distance less than $D$ from the point) is $$a(D)=\pi D^2$$
The probability that a "link is free" (distance between p... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given a divergent series $\sum x_n$ with $x_{n} \rightarrow 0$, show there exists a divergent series $\sum y_{n} $ with $ y_n/x_n\rightarrow 0$ The problem is what the title says, with the added requrement that both series should have positive terms.
I ruled out defining $y_{n}$ by $x_{n}$ divided by some function of $... | Slightly more general result: If $\sum x_n$ is a positive divergent series, then there exists a positive divergent series $\sum y_n$ such that $y_n/x_n \to 0.$ (Note that if in addition we have $x_n$ bounded, then $y_n \to 0.$)
Proof: There are integers $1=n_1 < n_2 < \cdots \to \infty$ such that
$$\sum_{n=n_k}^{n_{k+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2018669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof that a continuous function maps connected sets into connected sets I'm trying to prove that, if f is a function from C to C, and its domain, D, is connected, then f(D) is also connected. How would I go about doing this?
The definition of conectedness at play is "S is disconnected iff there exist open disjoint set... | Suppose that $g:f(D)\rightarrow\{0,1\}$ is continues, (where we equip $\{ 0, 1 \}$ with the discrete topology $\big\{ \emptyset, \{ 0 \}, \{ 1 \}, \{ 0, 1 \} \big\}$) then $g \circ f$ is continues. Since $D$ is connected, we deduce that $g\circ f$ is constant. This implies that $g$ is constant since the image of the r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2018848",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "5",
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$I = \int_{0}^\infty t^2 e^{-t^2/2} dt$ Q: Evaluate the integral $I = \int_\limits{0}^\infty t^2 e^{-t^2/2} dt$
Hint, write $I^2$ as the following iterated integral and convert to polar coordinates:
\begin{align*}
I^2 &= \int_\limits{0}^\infty \int_\limits{0}^\infty x^2 e^{-x^2/2} \cdot y^2 e^{-y^2/2} \, dx \, dy \\
... | *
*Why not integrate by parts from the initial integral? One has $$
I=-\frac12\int_0^\infty x \cdot \left(-2xe^{-x^2}
\right)dx=-\frac12\int_0^\infty x \cdot \left(e^{-x^2} \right)'dx $$
then one may use the Gaussian integral to conclude.
*Another path is to differentiate the gaussian identity $$
\int... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find a general form of a sequence and its sum I have a problem to find a general form of the sequence
\begin{align}
- \frac{{n\left( {n - 1} \right)}}{{2\left( {2n - 1} \right)}},\frac{{n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)}}{{2 \cdot 4 \cdot \left( {2n - 1} \right)\left( {2n - 3} \right)}... | Here is a more compact representation as sum formula, most of it was already stated in the comment section.
Since
\begin{align*}
a_n(k)=\frac{(-1)^kn(n-1)\cdots (n-2k+1)}{2\cdot4\cdots (2k)\cdot(2n-1)(2n-3)\cdots(2n-2k+1)}\qquad\qquad 1\leq k\leq n
\end{align*}
We obtain
\begin{align*}
a_n(k)&=(-1)^k\frac{n!}{(n-2... | {
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The set of all equivalence classes from Cauchy sequences is complete Let $C(x)$ be the set of all Cauchy seq. on $X$ and define for $(x_n)_{n \in \mathbb{N}}, (y_n)_{n \in \mathbb{N}}$ the following relation
$$
(x_n)_{n \in \mathbb{N}} \sim (y_n)_{n \in \mathbb{N}} \Leftrightarrow d_X(x_n, y_n) \to 0 \text{ when } n \t... | In order to avoid confusion, I will denote by $x^m$ the $m$th sequence and $x^m_n$ will be the $n$th element of the $m$th sequence: $x^m=(x^m_1,x^m_2,x^m_3,\ldots)$. We need a Cauchy sequence $y=(y_1,y_2,\ldots)$ such that $d(x^n,y)\to0$.
The idea is to take a kind of diagonal: one element from each sequence,
$y_1\in x... | {
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"timestamp": "2023-03-29T00:00:00",
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Is the matrix $A$ symmetric in the quadratic form? Given $x \in \mathbb{R}^n$ and $A \in \mathbb{R}^{n \times n}$ ($A$ is not necessarily symmetric), the quadratic form is written as $x^TAx$, a scaler. We have,
$$x^TAx=(x^TAx)^T=x^TA^Tx$$
that is $x^T(A-A^T)x=0$
Why couldn't conclude $A=A^T$ from $x^T(A-A^T)x=0$, where... | In fact, any quadratic form can be reduced to a symmetric matrix. Suppose $A$ is a non-symmetric matrix. Then
$(A-A^T)/2$ is skew-symmetric since
$$
\left(\frac{A-A^T}{2}\right)^T=\frac{A^T-A}{2}=-\frac{A-A^T}{2}
$$
And for a skew-symmetric matrix it is always true that
$$
x^T\left(\frac{A-A^T}{2}\right)x=0
$$
Since
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2019154",
"timestamp": "2023-03-29T00:00:00",
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Cardinality of multiplicative group of a field $GF(q)/f(x)$ where $f(x)$ is an irreducible polynomial over GF(q) If you have an irreducible polynomial $f(x)$ over $GF(q)$, then the following should be true:
$GF(q)/f(x) \cong GF(q^k)$, where $k = deg(f(x))$.
So the multiplicative group should have $\varphi(q^k)$ elemen... | For what reason should the multiplicative group of $\operatorname{GF}(q^k)$ have $\varphi(q^k)$ elements? In general it is not true that $\operatorname{GF}(q^k)\cong\Bbb{Z}/q^k\Bbb{Z}$. In fact, if $k>1$ then $\Bbb{Z}/q^k\Bbb{Z}$ is not a even field because $q\neq0$ and $q^{k-1}\neq0$ in $\Bbb{Z}/q^k\Bbb{Z}$ but their ... | {
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"source": "stackexchange",
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What is the relation between two row equivalent matrices? My question is very simple.
Suppose $[T]_{\mathfrak B}$ is the matrix of the linear transformation $T:V\to V$ in the basis $\mathfrak B$.
If a matrix $A$ is row equivalent to $[T]_{\mathfrak B}$, what is the relation between these two matrices?
Can we know ... |
Can we know something about the linear transformation $T$ just looking to the matrix $A$?
Yes, we can. For instance, the rank of $A$ is equal to that of $[T]_{\mathcal{B}}$, which would give you the information about where $T$ is invertible or not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2019333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Area of a trapezoid without the height How would I find the area of a non-iscoceles trapezoid and without the height? The trapezoid's bases are $30$ and $40$, and the legs $14$ and $16$. Thanks
| Area of a trapezium without knowing the height:
(a+c) / 4(a-c) * √(a+b-c+d)(a-b-c+d)(a+b-c-d)(-a+b+c+d)
Where a>c and 'a' is parallel to 'c'. 'b' and 'd' are the 'diagonals'.
Therefore,
(40+30) / 4(40-30) * √(40+14-30+16)(40-14-30+16)(40+14-30-16)(-40+14+30+16)
70 / 40 * √(40)(12)(8)(20)
7/4 * 277.128129...
484.974
| {
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"url": "https://math.stackexchange.com/questions/2019434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Are the functions with the map $\sum a_n b^{-n}\mapsto\sum a_n c^{-n}$ continuous?
Let $(a_n)$ a sequence such that $0\le a_n<\min\{b,c\}$ and $b,c> 1$. Then define the function
$$f:A\to[0,1],\quad \sum_{n=1}^\infty a_n b^{-n}\mapsto\sum_{n=1}^\infty a_n c^{-n}$$
where $A$ is the subset of $[0,1]$ where the function i... | For concreteness, consider the case where $b=4, c=5$. Then take any point $\mathbf{a}$ with a terminating quaternary expansion; for instance, $\mathbf {a}=\frac{9}{16}$, which corresponds to the sequence $\langle a_n\rangle = \langle 2, 1, 0, 0, 0, 0, \ldots\rangle$. Then the sequence of sequences $$\begin{align}
\al... | {
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"timestamp": "2023-03-29T00:00:00",
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How to organize math study groups online for long-distance collaboration? How to organize math study groups online for long-distance collaboration? For example, how to organize a study group to:
*
*Effectively go through a textbook,
*Taking notes (not necessarily collaboratively),
*Organizing time, exercises,
*Pu... | From my experience running computational research projects between people in my research group (in the UK) and with our collaborators in Singapore we tend to use a few tools.
Sharelatex.com / Dropbox with Latex files: Writing notes for maths is easiest in latex. I'm not sure if you already use it but if not its a mark ... | {
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"url": "https://math.stackexchange.com/questions/2019683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
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Ratio between inner products on a vector space
Let $V$ be a vector space and let$ \langle,\rangle_1$ and $\ \langle,\rangle_2$ be inner products in $V$ s.t. $\langle,\rangle_1=0 \iff \langle,\rangle_2=0$.
Prove $ \langle v,w\rangle_1=c\langle v,w\rangle_2$ for every $v,w \in V$.
I've been struggling coming up with ... | You are going in the right direction, but there are some unnecessary complications.
There is no need to split the space into a direct sum. Just pick an orthonormal basis $\{v_1,v_2,\ldots,v_N\}$ of $V$ (suppose its dimension is $N$) with respect to $\langle,\rangle_1$. Then this basis is at least an orthogonal basis wi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Guess Who is the apple thief ?? At the end of class, My mathematics teacher gave us an interesting problem Which is as follow:
Out of six boys exactly two were known to have been stealing apples.
Harry said:Charlie and George.
Donald said:Tom and Charlie
James said:Donald and Tom.
George said:Harry and Charlie.
Charli... | Charlie is obviously a good candidate to focus on first, as he is mentioned more times than anyone else.
We can start by assuming he is innocent, as this will obviously impose strong constraints on the other choices, and see if that is possible.
If Charlie is not a thief, that means that two of George, Tom and Harry a... | {
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Let $A$ be a $2\times 2$ squared matrix. Prove that max $tr\ (A)$ s.t. $AA^{T}=I$ has solution.
Let a squared matrix $A=\bigl(\begin{smallmatrix}
x_{11} & x_{12} \\
x_{21} & x_{22}
\end{smallmatrix}\bigr)$. Prove that the following optimization problem:
max $tr\ (A)$ s.t. $AA^{T}=I$
Where $tr$ is the trace of matrix... | I might be missing something but the condition that $AA^T = I$ implies that $A$ is an orthogonal matrix. Thus, the columns (or rows) of $A$ form an orthonormal basis for $\mathbb{R}^2$ (I'm assuming this is over the reals). Thus $x_{ij} \leq 1$ for all $i,j$.
Therefore, $tr(A) \leq 2$. Since $tr(I) = 2$ we have that t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove that $\lim\limits_{n\to\infty} \int_0^1 \cos^n(x)\, dx = 0$ I've got this tasks to prove that:
$\lim\limits_{n\to\infty} \int_0^1 \cos^n(x) \,dx = 0$
I tried to think about a partition ${0,t,1}$, and say that if $t$ is small enough, I can get:
$\lim\limits_{n\to\infty} \int_0^t \cos(x)\, dx = 0$
But then I... | For any $0<\epsilon< 1$
$$ \int_0^1\cos^n xdx=\int_0^{\frac\epsilon2}\cos^nxdx+\int^1_{\frac\epsilon2}\cos^nxdx\le\frac\epsilon2+\cos^n(\frac\epsilon2)<\frac\epsilon2+\frac\epsilon2=\epsilon\text{ as }n\text{ large}. $$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Evaluate a given limit Evaluate the following limit:
$$\lim_{x \rightarrow 4} \left( \frac{1}{13 - 3x} \right) ^ {\tan \frac{\pi x}{8}}$$
I haven't managed to get anything meaningful yet.
Thank you in advance!
| see my nice answer:
assume $x=t+4$ $$\lim_{x \rightarrow 4} \left( \frac{1}{13 - 3x} \right) ^ {\tan \frac{\pi x}{8}}=\lim_{t \to 0} \left( \frac{1}{13 - 3(t+4)} \right) ^ {\tan \frac{\pi (t+4)}{8}}$$
$$=\lim_{t \to 0} \left( \frac{1}{1-3t} \right) ^ {-\cot \frac{\pi t}{8}}$$
$$=\lim_{t \to 0} \left(1-3t \right) ^ {\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2020250",
"timestamp": "2023-03-29T00:00:00",
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Can a sudoku with valid columns and rows be proved valid without evaluating every 3x3 inside it? I'm trying to solve a computer science challenge and have readily been able to validate whether or not the outside dimensions of a sudoku puzzle are valid. However, it doesn't check the validity of the inside squares and wi... | As proved in https://mathoverflow.net/q/129143, it is insufficient in general to additionally check three (or less) of the $3\times3$ blocks.
On the other hand, it is enough to check four blocks: for instance, the three blocks on the diagonal and one more block.
Together with rows and columns, this makes $9+9+4=22$ che... | {
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Prove that $\lim_\limits{n\to\infty }\frac{1}{\log(\log(n))}=0$ How to prove this limit?
$$\lim_{n\to\infty }\frac{1}{\log(\log(n))}=0$$
I thought of something like
$$0 \le \frac{1}{\log(\log(n))} \le \frac{1}{n}$$
Is it alright?
| We have $\log x = \int_1^x {1 \over t} dt $, hence $\log$ is increasing. From the integral we see that $\log (n+1) \ge {1 \over 2} + {1 \over 3} + \cdots + { 1 \over n}$, and since the Harmonic series is divergent, we see that
$\lim_{x \to \infty} \log x = \infty$.
It follows that $\lim_{x \to \infty} \log (\log x) = \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the derivative of $\cos 2 x - 2 \sin x$ So, I've been teaching myself calculus, and I'm very new to all of this, so apologies in advance for what is probably a rather dumb question.
I'm trying to find the derivative of the function $f(x) = \cos 2x - 2 \sin x$.
I'm 99% sure that the derivative of $\cos$ is $-\s... | So, the trick to this one is trigonometric identities:
$$\sin(2x)=2\sin(x)\cos(x)$$
Basically, we end up with
$$-2\cos(x)(1+2\sin(x))=-2\cos(x)-2\underbrace{(2\sin(x)\cos(x))}_{\large\sin(2x)}$$
And notice a small chain rule in the initial problem.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Distance Between 2 Points on a Given function I was trying to find an equation for the distance between 2 points on a function, following (tracing) that function i.e. not the shortest distance (Pythagoras), rather actually tracing the distance of the function from one point to the other.
I reasoned that if I zoomed up ... | As lordoftheshadows commented, what you need is to use the arc length formula which corresponds to your sum for very small $\Delta x$ (or very large $n$).
I suggest you have a look here.
Between two points on the curve $y(x)$, the formula is just $$L=\int_{a}^{b}\sqrt{1+(y'(x))^2}\,dx$$ In the case of $y(x)=x^2$. If n... | {
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Show that if $f$ is lower continuous then $f^{-1}((\alpha,\infty))$ is open Let a function $f:X\to\Bbb R$, where $X$ is a metric space.
Then $f$ is lower continuous if for all $a\in X$ we have that $f(a)\le \liminf f(x_n)$ for every $(x_n)\to a$. Alternatively we can say that $f$ is lower continuous if for all $\epsilo... | Let's use second:
$f$ LSC if for every $a\in X $ and $\epsilon>0$, $\exists \delta$ such that $f(a)-f(x)<\epsilon$ whenever $x\in B_\delta(a)$
For $\alpha\in {\mathbb R}$ let $I_\alpha=f^{-1}((\alpha,\infty))$.
*
*Suppose $f$ is LSC. Fix $\alpha\in \mathbb R$. Then for every $a\in I_\alpha$ and $0<\epsilon <f(a)-\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How do i find the roots of this polynomial equation? The polynomial equation is: $x^4-5x^3+5x^2+5x-6=0$.
How do i simplify this equation so that i can find its roots.
Please, can anyone teach me how to find roots of equations of degree 4 and degree 3.
| Hint. Look first for rational solutions: for a polynomial of degree $n$,
$$a_n x^n+\cdots +a_1x+a_0$$
with integer coefficients, if $p/q\in\mathbb{Q}$ is a solution then $p$ divides $a_0$ and $q$ divides $a_n$.
In your case, a very "lucky" one I would say, try with the divisors of $-6$ (note that $a_n=1$) that is: $\p... | {
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"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Are $\ell_p$ spaces complete under the $q$-norm? $\ell_p$ spaces are complete under the $p$-norm. The question is:
if we change the norm, what can we say about the completness of the space?
For example:
$\ell_1$ (the space of all summable sequences)
is complete or not in the two norm? If it's not complete is there a ... | In general: no.
In the cases that $\mathscr l^p(X)$ or $L^p(X)$ end up being finite dimensional: yes.
For $p<q$ you have $\mathscr l^p(X)\subset \mathscr l^q(X)$ but $\mathscr l^p(X)\neq\mathscr l^q(X)$ if $X$ is not a finite set. If $\mathscr l^p(X)$ were complete with $\|\cdot\|_q$ norm, then it would have to be a cl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2021158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How can I derive what is $1\cdot 2\cdot 3\cdot 4 + 2\cdot 3\cdot 4\cdot 5+ 3\cdot 4\cdot 5\cdot 6+\cdots + (n-3)(n-2)(n-1)(n)$ ?? I'd like to evaluate the series $$1\cdot 2\cdot 3\cdot 4 + 2\cdot 3\cdot 4\cdot 5+ 3\cdot 4\cdot 5\cdot 6+\cdots + (n-3)(n-2)(n-1)(n)$$
Since I am a high school student, I only know how to p... | Using finite calculus we have that
$$\sum k^{\underline 4}\delta k=\frac{k^{\underline 5}}{5}+C$$
where $k^{\underline 4}=k(k-1)(k-2)(k-3)$ is a falling factorial. Then taking limits
$$\sum_{k=m}^nk^{\underline 4}=\sum\nolimits_m^{n+1}k^{\underline 4}\delta k=\frac{k^{\underline 5}}{5}\bigg|_m^{n+1}$$
The standard case... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2021231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 10,
"answer_id": 7
} |
For which values of $1 ≤ p ≤ \infty$ is $\{f_n\} \subset (C[0,1],\lVert\cdot\rVert_p)$ a Cauchy sequence? The question is:
Let $\{f_n\} \subset (C[0,1], \lVert\cdot\rVert_p)$, where $f_n(x)=1-nx$, if $0\leq x \leq \frac1n$, and $0$ otherwise.
For which values of $p$, $1 \leq p \leq \infty$, is $\{f_n\}$ a Cauchy se... | Note that $f_n(x)=0$ if $x≥1/n$ and $|f_n(x)|≤1$ if $x\in[0,1/n]$. For that reason:
$$\int_0^1|f_n(x)|^p\,dx≤\int_0^{1/n}1\,dx=1/n$$
So
$$\|f_n\|_p≤\sqrt[\leftroot{-13}\uproot{2}p\quad ]{1/n}\to0$$
for all $p<\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2021353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Limits to infinity of a factorial function: $\lim_{n\to\infty}\frac{n!}{n^{n/2}}$ How can this limit to infinity be solved? I've tried with d'Alembert but it just keeps coming up with the wrong answer.
$$\lim\limits_{n\to\infty}\frac{n!}{n^{n/2}}$$
I might have a problem in simplifying factorial numbers.
Thank you in ... | $$n!\geq\left (\frac{n}{4}\right )^{3n/4}\\\lim_{n\to\infty}\frac{\left (\frac{n}{4}\right )^{3n/4}}{n^{n/2}}=\lim_{n\to\infty}\frac{n^{n/4}}{4^{3n/4}}=\lim_{n\to \infty}\left(\frac{n}{64}\right)^{n/4}=\infty$$
The inequality comes since $4n!=1\cdots \underbrace{n\cdots 4n}_{3n}\geq \underbrace{n\cdots n}_{3n}=n^{3n}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2021494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 2
} |
Infinite Fibonacci sums $\sum_{n=1}^{\infty} \frac{1}{f_nf_{n+2}}$ - diverge or converge I am currently going through exercises regarding convergence/divergence.
For my previous question I used the ratio test, and managed to get through it all okay (I think). I proved that: $$\sum_{n=1}^{\infty} \frac{n!}{n^n}$$
conve... | Another approach. The $n$th Fibonacci number is about $\varphi^n$, where $\varphi = (1+\sqrt{5})/2$ is the golden mean. Then your sum behaves like $\Sigma (1/\varphi^{2n})$. It's easy to show that converges. Of course you don't get the value of what it converges to, as in @achillehui 's nice answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2021615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Matrix derivative rule for the product of two matrices How to derive the matrix derivative of $AB$ (product of two matrices, where $A \in \mathbb{R}^{m\times n}$ and $B \in \mathbb{R}^{n\times r}$), like
$$\frac{\partial AB}{\partial A}$$
and
$$\frac{\partial AB}{\partial B}$$
| Let $\phi((A,B))$ and note that
$\phi((A+ \alpha, B+ \beta ) ) = \phi((A,B)) + \alpha B + A \beta + \alpha \beta$,
from which we have
$D \phi((A,B)) ((\alpha, \beta)) = \alpha B + A \beta$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2021796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Counting Group Actions from a Finite Group to Itself Given a finite group $G$, how many group homomorphisms $G \to \mathrm{Perm}(G)$ are there? Alternatively, in how many ways can a finite group act on itself?
| Here are some food for thoughts, this is far from an answer as I don't know it.
*
*If $G$ is cyclic, then $G$ can acts on itself through $|G|!$ different ways. Indeed, if $x\in G$ is a generator of $G$, there are as many homomorphisms from $G$ to $S(G)$ as there are elements in $S(G)$, since a group homomorphism fro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2021923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Every automorphism of L:K fixes K? I know that if an monomorphism $L:K$ is an algebraic extension and $\tau: L \rightarrow L$ has the property that fixes $K$, then $\tau$ is 1-1.
Ok then, in the proof of this result we can see that $\tau$ even permutes the roots of $m_\alpha$ ($\alpha$ is an algebraic element).
The que... | Not necessarily. For example, $\mathrm{Aut}(\mathbb{C})$ is uncountable, while precisely two of these automorphisms (the identity map and complex conjugation) fix $\mathbb{R}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2022034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How many abelian groups of order $p^{5}$ are there? I want to find how many abelian groups of order $p^{5}$ there are, up to isomorphism, where $p$ is a prime number. Are there any theorems that can help with this?
| Edit: Adding my answer of your question in comments:
To use the theorem you must factor suppose $\vert G \vert=n$ and factor $n$ into prime factors. Then you must get divisors $d_1,d_2,\dots,d_m$ such that $d_i\mid d_{i+1}$, and $d_1\cdot\dots\cdot d_{m}=n$, and your group can be
$$G\cong \mathbb{Z}_{d_1}\oplus \mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2022131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Almost Complex Structure The question itself is nothing but linear algebra:
Let $\{x_1,\cdots,x_n\}$ be n linearly independent vectors (not necessarily orthogonal) in $\mathbb{R}^{2n}$ and $J^2=-1$ is the almost complex structure, we want to show that $\{x_1,\cdots, x_n, Jx_1,\cdots, Jx_{n}\}$ span $\mathbb{R}^{2n}$.
... | In fact, choose any $x_1 \neq 0$. Then $x_1$ and $Jx_1$ are linearly independent since otherwise $J$ has a real eigenvalue. Next, choose
$$
x_2 \not\in \mathop{\mathrm{span}}\{x_1,Jx_1\}.
$$
Then $x_1$, $x_2$, $Jx_1$, $Jx_2$ are linearly independent since otherwise $V = \mathop{\mathrm{span}}\{x_1,x_2,Jx_1\}$ is inva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2022245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Alternating Series sum up to $99$ terms is $>0.5$
Prove that $$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\cdots \cdots \cdots -\frac{1}{99}+\frac{1}{100}>0.2$$
$\bf{My\; Try::}$ We can write series as $$\frac{1}{2}\bigg(1+\frac{1}{2}+\frac{1}{3}+\cdots \cdots +\frac{1}{50}\bigg)-\bigg(\frac{1}{3}+\frac{1}{5}+\f... | $$\left(\frac12-\frac13\right)+\left(\frac14-\frac15\right)+\cdots+\left(\frac1{98}-\frac1{99}\right)+\frac1{100}>\left(\frac12-\frac13\right)+\left(\frac14-\frac15\right)=0.21\overline 6$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2022420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does there exist a sequence $x_n$ such that $ \sum x_n >0$ but $ \sum x_n ^3 <0 $ where $ |x_n|<1$ Does there exist a sequence $x_n$ such that $ \sum_{n=1}^{\infty} x_n >0$ but $ \sum_{n=1}^{\infty} x_n ^3 <0 $ where $ |x_n|<1$ for all $n$?
Intuitively I would say no, since cubing preserves the sign of each term and si... | Take, for instance, $x_n=-3/5$ if $n$ is divisible by $3$ and $2/5$ otherwise. Then $\sum_{n=1}^\infty x_n=\infty$ while $\sum_{n=1}^\infty x_n^3=-\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2022576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculating $\limsup\limits_{n \to \infty}a_n$ and $\liminf\limits_{n \to \infty}a_n$ with $a_n=\left( \frac{n+(-1)^n}{n} \right)^n$ I need to calculate $\limsup\limits_{n \to \infty} a_n$ and $\liminf\limits_{n \to \infty}a_n$ where $a_{n} = \left( \dfrac{n+(-1)^n}{n}\right)^n, \: n \in \mathbb{N}$
My approach is to c... | Hint. One may recall that
$$
\lim_{n \to \infty} \left(1+\frac{x}n\right)^n=e^x, \qquad x \in \mathbb{R},
$$ and one may observe that
$$
\left(1-\frac1n\right)^n\le a_n \le \left(1+\frac1n\right)^n, \quad n=1,2,\cdots.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2022675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
The number of ways of selecting 6 shoes from 8 pairs of shoes so that exactly 2 pairs of shoes are formed The number of ways of selecting 6 shoes from 8 pairs of shoes so that exactly 2 pairs of shoes are formed?
My try:
Let us first choose $2$ pairs from $8$ pairs. It can be done in $_{8}C_2$ ways.
Suppose I have c... | When unsure, try with a smaller instance of the problem.
Let's count the ways of choosing 2 shoes from 2 pairs so that they are not in the same pair. Your method would say start with 0 full pairs out of 2 and then multiply by $_4C_1$ and $_2C_1$:
$$\binom{2}{0} \binom{4}{1} \binom{2}{1} = 8$$
But we can count directly:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2022861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
how to calculate percentage of chance in a price negotiation I would like to know how to calculate a Percentage of chance in a negotiation.
Here is the problem :
The product has a price of 30€ which represents an acceptance rate of 100%
The minimum price accepted is 27€ which represents an acceptance rate of 50%
How ... | This is guessing the function $r(p)$, while knowing $r(27€)=50 \%$ and $r(30€)=100 \%$.
If there is nothing else, the simplest thing to do is to assume a linear law, doing linear interpolation
$$
r(p) = r_1 (1 - \lambda) + r_2 \lambda \\
\lambda = \frac{p - p_1}{p_2 - p1}
$$
Example:
$p=28€$, then
$$
r(28€) = 50\% \lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2022955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find the range of $f:[0,2\pi) \rightarrow \Bbb R, f(x)=(1+3^{\lfloor\frac{x}{\pi}\rfloor}|\sin x|)^\frac{1}{2}$. I need to check my solution for this problem:
Let $f:[0,2\pi) \rightarrow \Bbb R, f(x)=(1+3^{\lfloor\frac{x}{\pi}\rfloor}|\sin x|)^\frac{1}{2}$.
$1)$ Find the range of $f$.
$2)$ Let $f_1=f\circ f$. Is $f_... | The given function is
$$f(x)=\begin{cases}
\sqrt{(1+\sin x)} & \text{ if } 0 \leq x < \pi \\
\sqrt{(1-3\sin x)} & \text{ if } \pi \leq x < 2\pi
\end{cases}
$$
Now observe that
$f(f0))=f(f(\pi))$, hence the composition function is not bijective.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2023106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Particular solution for $\sin(x)y'(x)+y(x)\cos(x)=-\sin(2x)$. Let the ODE $$y'(x)\sin(x)+y(x)\cos(x)=-\sin(2x).$$
I know that a particular solution is of the form $w(x)=A\cos(2x)+B\sin(2x)$. When I put it in the equation, I get,
$$-2A\sin(2x)\sin(x)+2B\cos(2x)\sin(x)+A\cos(2x)\cos(x)+B\sin(2x)\cos(x)=-\sin(2x)$$
and th... | Note that you can recognize the product rule of differentiation on the left side, $$(\sin(x)y(x))'=\sin(x)y'(x)+\cos(x)y(x),$$ which greatly simplifies the equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2023215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why is $\lim_{x\rightarrow \infty}(1+\frac{a}{bx+c})^{dx+f}=e^{\frac{a\cdot d}{b}}$? I am studying limits through a study book, and I am given this simple rule but without any explanation.
$$\lim_{x\rightarrow \infty}(1+\frac{a}{bx+c})^{dx+f}=e^{\frac{a\cdot d}{b}}$$
Why does this hold true for all values of $a, b, c, ... | Assuming $b\ne 0$ we have that
\begin{align}\lim_{x\rightarrow \infty}\left(1+\frac{a}{bx+c}\right)^{dx+f}&\\=&\lim_{x\rightarrow \infty}\left(1+\frac{a}{bx+c}\right)^{{(bx+c)}\cdot\frac{dx+f}{bx+c}}\\=&\left(\lim_{x\rightarrow \infty}\left(1+\frac{a}{bx+c}\right)^{bx+c}\right)^{\lim_{x\to\infty}\frac{dx+f}{bx+c}}.\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2023338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
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