Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Which side of a 2d curve is a point on? Given a point $Q$ and $2d$ Cubic Bezier Curve: $$P = A(1-t)^3 + 3Bt(1-t)^2 + 3Ct^2(1-t) + Dt^3$$
Is there a way to know which side of the curve the point lies on? I know that the term "side" is a bit strange since there can be a loop, but I'm wondering if answering this question... | I ended up finding a solution to this that I like better than implicitization (I'm a better programmer than I am a mathematician!). I am trying to use this in a computer graphics situation, and had found a paper that talked about how to use graphics hardware to draw a triangle in a specific way to make it so you could... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1288309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Order on eigenvalues on diagonal matrix If the eigenvalues are say $-1$, $-1$ and $2$ for a $3$ x $3$ matrix, then when comes to the diagonal matrix, is it (from top left, to bottom right) $-1$ $-1$ $2$ or $2$ $-1$ $-1$ or $-1$ $2$ $-1$? Does the order matter? How do you know what the order is?
| It does not matter, it just has to be coherent with the order of the columns of $P$ in $A=PDP^{-1}$.
In some applications we prefer to order the eigenvalues but in theory it's not necessary.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1288578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Help for a problem with inscribed triangles If we have a triangle $ABC$ with $AB = 3\sqrt 7$, $AC = 3$, $\angle{ACB} = \pi/3$, $CL$ is the bisector of angle $ACB$, $CL$ lies on line $CD$ and $D$ is a point of the circumcircle of triangle $ABC$, what's the length of $CD$?
Here I attach my solution. The problem is that I... | Your problem is in stating that $\cos \beta = \frac {\sqrt {7}} {14}$. This is incorrect.
Although it is true that $\cos^2 \beta = \frac {7} {196}$, there are two possible values for $\cos \beta$. If you were to make a scale drawing, you would find that AB is over twice as long as Ac, making $\beta$ an obtuse angle.
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1288662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
Finding cubic bezier curve endpoints based on relationship between endpoints and a point on the curve. I have the following information about a bezier curve:
*
*The curve begins at $x=0$ and ends at $x=1$.
*The curve has two control points each at the same height as their closest endpoints, one at $x=.25$ and one ... | Plugging the known point $(x,y)$ into the equation
$$\underbrace{\bigl((1-x)^3+3(1-x)^2x\bigr)}_\alpha a +
\underbrace{\bigl(3(1-x)x^2+x^3\bigr)}_\beta b=y$$
That's the equation of a line in $a,b$-space. You can intersect that with each linear piece of your piecewise linear function. If you have
$$(\alpha a_i+\beta b_i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1288742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Computing line integral using Stokes´theorem Use Stokes´ theorem to show that $$\int_C ydx+zdy+xdz=\pi a^2\sqrt{3}$$ where $C$ is the curve of intersection of the sphere $x^2+y^2+z^2=a^2$ and the plane $x+y+z=0$
My attempt:
By Stokes´ theorem I know that $$\int_S (\nabla \times F) \cdot n \ dS=\int_c F \cdot d\alpha$... | Your parameterization doesn't look right. In particlar, note that if $x a\cos u \sin v$, $y = a\sin u \cos u$, and $z = a\cos v$, then $x^2+y^2+z^2 = a\cos^2u\sin^2v+a^2\sin^2u\cos^2u+a^2\cos^2v$, which does not simplify to $a^2$. One correct parameterization of the sphere would be $\vec{r}(u,v) = (a\cos u \sin v, a\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1288934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Evaluate $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ I'm completely stuck evaluating $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ how would I go about solving this?
| As all terms are positive, we have $$0 \leq \frac{(n!)^2}{(2n)!} = \frac{n!}{2n \cdot \dots \cdot (n+1)} = \prod_{k=1}^n \frac{k}{k+n} \leq \prod_{k=1}^n \frac{1}{2} = \left(\frac{1}{2}\right)^n$$
So then as
$$\lim\limits_{n\rightarrow\infty} \left(\frac{1}{2}\right)^n = 0$$
It follows that
$$\lim\limits_{n\rightarro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Tough Differential equation Can anyone help me solve this question ?
$$ \large{y^{\prime \prime} + y = \tan{t} + e^{3t} -1}$$
I have gotten to a part when I know $r = \pm 1$ and then plugging them into a simple differential equation. I do not know how to the next step.
Thank you very much for all your help.
| Find the complementary solution by solving
\begin{equation*}
y''+y=0.
\end{equation*}
Substitute $y=e^{\lambda t}$ to get
\begin{equation*}
(\lambda ^2+1)e^{\lambda t}=0.
\end{equation*}
Therefore the zeros are $\lambda=i$ or $\lambda =-i.$ The general solution is given by
\begin{equation*}
y=y_1+y_2=c_1e^{it}+\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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undefined angles with arcsin I have this problem but I couldn't solve it. In a paper I'm reading for controlling a device, I need to generate the following angle
$$
\theta = \tan^{-1}\left( \frac{Y_{2} - Y_{1}}{ X_{2} - X_{1}} \right)
$$
where $Y_{2} = 10, Y_{1} = 0, X_{2} = 10$ and $X_{1} = 0$.
Now I need to generat... | When $\theta=0.7854\approx \frac{\pi}{4}$, $\sin{\theta}=\cos{\theta}=\frac{\sqrt{2}}{2}$. Evaluating your definition for $\phi$ at $\frac{\pi}{4}$ comes out to $-1.278$, which is not in the domain of $\sin^{-1}{\theta}$ (because it is not in the range of $\sin{\theta}$). The problem seems to be in the part of your exp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many $s,t,u$ satisfy: $s +2t+3u +\ldots = n$?
Given $n\in \mathbb{N}^+$, what is the possible number of combinations
$s,t,u,\ldots\in\mathbb{N}$, such that: $$s +2t+3u +\ldots = n\quad?$$
Additionally, is there an efficient way to find these combinations
other than an elimination process?
This problem comes... | The number of solutions of the equation equals the coefficient of $z^n$ in the expression
$$
\frac{1}{(1-z)(1-z^2)(1-z^3)\ldots }=\sum_{n}p(n)z^n,
$$
where $p(n)$ is the Euler partition function, so that the number of combinations is $p(n)$.
(Christoph): There is a bijection between the partition function and the d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why the general formula of Taylor series for $\ln(x)$ does not work for $n=0$? I need to find the taylor series for $\ln(x)$ about $a = 2$, and I have find the following solution, but I don't understand why the general formula does not work for $n = 0$.
| The general formula would be $\frac{(-1)^{0+1} (-1)!}{2^0} = -(-1)!$. The factorial of $-1$ is undefined, so the general formula isn't defined for $n=0$. However $f^{(0)}(2) = f(2) =\ln 2$. That's what is meant by "not working".
Note that this is closely related to $\int x^{n-1}\ \mathrm dx = \frac1n x^n$ wich is also ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integral equation involving Planck radiation formula I am stuck in solving the following integral equation:
$$\sigma T^4=\pi\int_{\lambda_0}^{\lambda_1}d\lambda W_{\lambda,T}$$
where:
$$W_{\lambda,T}=\dfrac{C_1}{\lambda^5\left(\exp\left(\frac{C_2}{\lambda T}\right)-1\right)}$$
and $C_1,C_2$ are constant coefficients. F... | Ok as suggested in my comment, transform $b/\lambda=x, d\lambda=-b/x^2$ with $b=C_2/T$. Therefore our integral becomes:
$$
I(C_1,b,\lambda_1,\lambda_0)=\frac{C_1 \pi}{b^4}\underbrace{\int^{b\lambda_1}_{b\lambda_0}\frac{x^3}{1-e^x}}_{J(b\lambda_1,b\lambda_0)}dx
$$
therefore (i will not care about possible convergence is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can $a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-\sqrt{z}\right)^2$ be an integer if $x$, $y$, and $z$ are not squares? Let $\gcd(x,y,z)=1$.Can we find 3 non-perfect squares $x,y,z\in \mathbb{Z},$ such that $a \in \mathbb{Z} \geq 2$
$$a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-... | Let $y=k^2$, $z=r^2$ and $x=c^2$(with $k,r,c$ integers) then the expression becomes: $$(\sqrt {2\cdot (k+r)(c+r)}-k-r)^2$$
Now for $a$ to be an integer $$2(k+r)(c+r)=n^2$$
with $n$ integer.
We put $$k+r=2^{2p+1}\cdot t^s$$
And $$c+r=t^v$$
(or vice versa,with $v$ and $s$ both odd or even).
If $r=1$ $$k=2^{2p+1}\cdot t^s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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"Rationalizing" an equation $$x=\sqrt[3]{p}+\sqrt[3]{q}$$
I'm trying to figure out some way to "rationalize" the previous equation, meaning to rewrite it purely in terms of whole number powers of $p$, $q$, and $x$. It seems quite simple but I've been stuck trying to do it. I'd appreciate anyone's help with this.
| Hint: Use the high school identity:
$$(a+b)(a^2-ab+b^2)=a^3+b^3.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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For which values of $\alpha \in \mathbb{R}$, does the series $\sum_{n=1}^\infty n^\alpha(\sqrt{n+1} - 2 \sqrt{n} + \sqrt{n-1})$ converge? How do I study for which values of $\alpha \in \mathbb{R}$ the following series converges?
(I have some troubles because of the form [$\infty - \infty$] that arises when taking the l... | Hint: $\sqrt{n+1}-2\sqrt{n}+\sqrt{n-1} = \frac{-2}{(\sqrt{n}+\sqrt{n+1})(\sqrt{n-1}+\sqrt{n+1})(\sqrt{n}+\sqrt{n-1})}$
So, this term is for big n approximately $-2n^{-\frac{3}{2}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1290025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
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Solving an integral with trig substitution I'm looking to solve the following integral using substitution:
$$\int \frac{dx}{2-\cos x}$$
Let $z=\tan\frac{x}{2}$
Then $dz=\frac 1 2 \sec^2 \frac x 2\,dx$
$$\sin x=\frac{2z}{z^2+1}$$
$$\cos x =\frac{1-z^2}{z^2+1}$$
$$dx=\frac{2\,dz}{z^2+1}$$
$$\int \frac{dx}{2-\cos x} = \in... | HINT:
$$\int \frac{1}{a^2+x^2}dx=\frac1a \arctan(x/a)+C$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1290118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Identification of a quadrilateral as a trapezoid, rectangle, or square Yesterday I was tutoring a student, and the following question arose (number 76):
My student believed the answer to be J: square. I reasoned with her that the information given only allows us to conclude that the top and bottom sides are parallel, ... | Of course, you are right. Send an email to the teacher with a concrete example, given that (s)he seems to be geometrically challenged. For instance, you could attach the following pictures with the email, which are both drawn to scale. You should also let him/her know that you need $5$ parameters to fix a quadrilateral... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1290291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "210",
"answer_count": 15,
"answer_id": 10
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What will happen if we remove the hypothesis that $V$ is finite-dimensional in this problem Original problem:
Suppose $V$ is finite-dimensional and $S, T,U ∈L(V)$and $STU = I$. Show that T is invertible and that $inv(T) = US$.
I know that it is because of the hypothesis of finite-dimensional that we can take advantage ... | This is definitely false in infinite dimensions.
Let $V$ be an vector space of countably infinite dimension, and pick a basis $\{v_n\}_{n \in \mathbb{N}}$ for $V$.
Let $T$ be the be the map which sends $v_1 \mapsto 0$ and $v_n \mapsto v_n$ for $n\geq 2$.
Let $U$ be the map which sends $v_n \mapsto v_{n+1}$.
Let $S$ be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1290439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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What does $\mathbb{R} \setminus S$ mean? What does $\mathbb{R}\setminus S$ mean? I am not getting it what it actually means. I have found it manywhere in real-analysis like in the definition of boundary points of a set. Can anyone tell me what it means really?
| That symbols means set difference. It is called \setminus in $\TeX.$
If $A$ and $B$ are sets, then $A \setminus B$ is the set of elements in $A$ but not in $B$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1290494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $H$ is normal subgroup of $G$ I have a following question.
Let $p$ be a prime and let $G$ be a group and $H$ be a subgroup of $G$.
$$
G = \left\{
\begin{bmatrix}
a & b \\
0 & 1
\end{bmatrix}
: a,b \in \mathbb{Z}_p, a \neq 0
\right\}$$
and
$$ H = \left\{
\begin{bmatrix}
1 & b \\
0 & 1
\end{bmatrix}
: b \in ... | to follow Solid Snake's suggestion it will help if you know that
$$
\begin{bmatrix}
a & b \\
0 & 1
\end{bmatrix}^{-1} = \begin{bmatrix}
a^{-1} &-ba^{-1}\\
0 & 1
\end{bmatrix}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1290568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$? $$\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$$
I have difficulty to evaluating above integrals.
First I try the substitution $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes integral worse.
Using Mathematica I fou... | Here is an approach.
You may write
$$\begin{align}
\int_0^{\infty}\frac{x^4}{\left(x^4+x^2+1\right)^3}dx
&=\int_0^{\infty}\frac{x^4}{\left(x^2+\dfrac1{x^2}+1\right)^3\,x^6}dx\\\\
&=\int_0^{\infty}\frac{1}{\left(x^2+\dfrac1{x^2}+1\right)^3}\frac{dx}{x^2} \\\\
&=\int_0^{\infty}\frac{1}{\left(x^2+\dfrac1{x^2}+1\right)^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1290669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 2
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If $\forall V\subseteq X$ where $x\in \overline V; f(x) \in \overline{f(V)}$, then $f$ is continous in $x$
Let $f:(X,\tau_X)\to (Y,\tau_Y)$
Prove: If $\forall V\subseteq X$ where $x\in \overline V; f(x) \in \overline{f(V)}$, then $f$ is continous in $x$.
Could someone verify the following proof?
Proof
If $f$ would no... | Same idea, better put:
Suppose that $U$ is a neighbourhood of $f(x)$ such that $f^{-1}[U]$ is not a neighbourhood of $x$, which indeed means that $x \in \overline{X \setminus f^{-1}[U]}$. So by assumption $f(x) \in \overline{f[X \setminus f^{-1}[U]]}$. As $U$ is a neighbourhood of $f(x)$, $U$ intersects $f[X \setminus ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1290753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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False $\Sigma_1$-sentences consistent with PA I'm preparing for an exam and encounter the following exercise in the notes I use.
In the next chapter we shall see that there are $\Sigma_1$-sentences
which are false in $\mathcal{N}$ but consistent with PA. Use this to show that the
following implication does not hol... | The answer you give works, but I think here's what they're looking for: replace $\varphi(x)$ with $\varphi'(x)\equiv\varphi(x)\wedge\forall y(y<x\implies \neg\varphi(y)$). By induction, $\varphi$ has a solution iff $\varphi'$ has a unique solution. This is a trick which is frequently useful and nontrivial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1290857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Solving $e^\frac1x = x$ non-graphically? This question has come up twice in different tests and the instructions always point out that it should be solved using a graphic calculator. Fair enough, the answer is ≈ 1.76322...(goes on forever?).
But how do you approach $e^\frac1x = x$ analytically for that solution? Is th... | Now note that $e^{\frac{1}{x}}>0$ for $x<0$ and note that $x<0$ for $x<0$. Hence we only have to consider $x>0$.
If $e^{\frac{1}{x}}=x$, then $1/x=\ln(x)$ for $x>0$ , which means that $x\ln(x)=1$.
Now write $x\ln(x)=\ln(x)e^{\ln(x)}=1$ and note that Lambert's W function says then that $\ln(x)=W(1)$, hence $x=e^{W(1)}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1290950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
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Derivative of matrix exponential w.r.t. to each element of the matrix I have $x= \exp(At)$ where $A$ is a matrix. I would like to find derivative of $x$ with respect to each element of $A$. Could anyone help with this problem?
| I arrive at the following. The ij element of $e^A$ is $$e^A_{ij} = \sum_{n=0}^\infty \frac{1}{n!} A_i^{k_1} ... A_{k_{n-1}j} ~.$$ Each matrix element can be seen as an independent variable, so the derivative toward $A^{kl}$ is $$e^A_{ij} = \sum_{n=0}^\infty \frac{1}{n!} \sum_{p=0}^{n-1} A^p_{ik} A^{n-1-p}_{lj} ~.$$
Fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1291063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 5,
"answer_id": 4
} |
Trigonometric double angle formulas problem I want to simplify the answer to an equation I had to compute, namely, simplifying $\sin^2 (2y) + \cos^2 (2y)$. I know that $\sin^2 (y) + \cos^2 (y) = 1$ but is there anything like that I can use at all?
| In general $$\sin^2(\color{red}{\rm something}) + \cos^2(\color{red}{\text{the same thing}})=1.$$So your expression is just $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1291261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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quaternions - understanding a formula Quaternions are new for me. I am trying to understand the following formula:
What are:
*
*$\large{q^x}$ ? I don't think it is a power.
*$\large{q^t}$ ? just a transposition of the quaternion $q$?
Do the subscripts next to the $q's$, represent entire rows or columns of the q... | Addressing your questions in order:
*
*$\mathbf q^\times$ is the $3\times3$ matrix that represents the operation of taking the cross product with $\mathbf q$, i.e. $(\mathbf q^\times)\mathbf x=\mathbf q\times\mathbf x$.
*No, $\mathbf q$ is not a quaternion. The quaternion is being denoted by $\bar{\mathbf q}$, and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1291366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving two Recurrence Relations
I encountered this problem in the International baccalaureate Higher Level math book, and my teacher could not help. If anyone could please take the time to look at it, that would be great. I need help with question (b) in the attached picture. All of the relevant information is in th... | Maybe try this:
$$
a_{n+2} = 3a_{n+1}+b_{n+1}=9a_n+3b_n+4a_n-b_n=14a_n+2b_n
\\
2a_{n+1}+8a_n=6a_n+2b_n+8a_n=14a_n+2b_n
$$
They are equal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1291437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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How many 3 letters-long codes can be made by 5 different letters? You have five letters: C, H, E, S, T
How many different codes, consisting of three letters, can be made from the above letters?
I'd say ${5}\choose{3}$ is the correct answer, since the order of the letters doesn't matter. Is this (that the order doesn't ... | Letters can be repeated. The first letter can be chosen from 5 different letters, so can the second and third letter. Thus the answer is $5^{3} = 125$ different codes.
Thank you @Yinon Eliraz!
| {
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"timestamp": "2023-03-29T00:00:00",
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Direct Proof on Divisibilty Using Induction proof makes sense to me and know how to do, but I am having a problem in using a direct proof for practice problem that was given to us.
The problem is:
For all natural numbers $n$, $2n^3 + 6n^2 + 4n$ is divisible by 4.
We are to use direct proof as a way proving it. I have n... | For all $\,n\!:\,$ $\,4\mid 2f(n)$ $\!\iff\! 2\mid f(n)$ $\!\iff\! f(0)\equiv 0\equiv f(1)\pmod{\!2}$ $\!\iff 2\mid f(0),f(1)$
for any polynomial $\,f(x)\,$ with integer coefficients (see also the Parity Root Test).
This applies to $ $ OP $\ f(x) = n^3+3n^2+2n\,$ since $\ 2\mid f(0) = 0,\ $ and $\ 2\mid f(1) = 6$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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find the derivative of the integral Prove that the following integral $F(x)$ is differentiable for every $x \in \mathbb{R}$ and calculate its derivative.
$$F(x) = \int\limits_0^1 e^{|x-y|} \mathrm{d}y$$
I don't know how to get rid of the absolute value in the integral
any ideas?
| Let $\phi_y(x) = e^{|x-y|}$.
The mean value theorem shows that $|e^a-e^b| \le e^{\max(a,b)} |a-b|$, and
also, $||a|-|b|| \le |a-b|$, hence
we have
$|{\phi_y(x+h)-\phi_y(x) \over h}| \le e^{\max(|x-y|,|x+h-y|)} |h|$ for $h \neq 0$.
If $\phi_y(x) = e^{|x-y|}$ then $\phi_y'(x) = e^{|x-y|}$ for $x>y$ and $\phi_y'(x) = -e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I read this triple summation? $\sum_{1\leq i < j < k \leq 4}a_{ijk}$ How do I read this triple summation?
$$\sum_{1\leq i < j < k \leq 4}a_{ijk}$$
The exercise asks me to express it as three sumations and to expand them in the following way:
1) Summing first on $k$, then on $j$ and last on $i$.
2) Summing firs... | You are correct.
For (1), $\displaystyle\sum_{1\leq i<j<k\leq 4}a_{ijk}=\sum_{1\leq i<j< 3}a_{ij3}+\sum_{1\leq i<j<4}a_{ij4}=\sum_{1\leq i<2}a_{i23}+\sum_{1\leq i<2}a_{i24}+\sum_{1\leq i<3}a_{i34}=a_{123}+a_{124}+a_{134}+a_{234}.$
(2) is analogous.
| {
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"timestamp": "2023-03-29T00:00:00",
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Find an example of a sequence not in $l^1$ satisfying certain boundedness conditions. This question is about getting a concrete example for this question on bounded holomorphic functions posed by @user122916 (something that he really expected as explained in the comments).
Give an example of a sequence of complex numb... | An example is $f(z) = \exp {(-\frac{1+z}{1-z})}.$ As $-\frac{1+z}{1-z}$ is a conformal map of the open unit disc $\mathbb {D}$ onto the left half plane, $f$ is bounded and holomorphic in $\mathbb {D}.$ We have $f$ continuous on $\overline {\mathbb {D}} \setminus \{1\}.$ Check that on $\partial \mathbb {D}\setminus \{1\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the semidirect product of normal complementary subgroups a direct product. If $G$ is a group with $K$ and $N$ as normal complementary subgroups of $G$, then we can form $G \cong K \rtimes_\varphi N$ where $\varphi:N \to Aut(K)$ is the usual conjugation. But someone told me that $G \cong K \times N$ also. Is this tru... | This is a consequence of the fact that if two normal subgroups intersect trivially, then elements of the first group commute with elements of the second group, so conjugation does nothing. This is proved by showing the commutator of the two elements is contained in the intersection of the subgroups, which means it is t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $\overline A\cap B = A\cap \overline B = \varnothing$, $A\cup B$ is disconnected. I'm trying to show that if $\overline A\cap B = A\cap \overline B = \varnothing$, $A\cup B$ is disconnected. First of all, I think I have to assume that $A$ and $B$ are nonempty, or else the statement would not be true if I just let $A... | \begin{align*}
U_1 &= \{x : \mathop{\text{dist}}(x,A) < \mathop{\text{dist}}(x,B) \} \\
U_2 &= \{x : \mathop{\text{dist}}(x,A) > \mathop{\text{dist}}(x,B) \}
\end{align*}
(The use of this kind of distance trick is suggested by the fact that the result is not true in general topological spaces; a nice counterexample is ... | {
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Characterizing affine subspaces order-theoretically Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ in a purely order-theoretic manner. Something like so:
Definition.... | Clearly, it suffices to define lines in a order-theoretically way, as $X$ is affine if and only if it contains (it is bigger than) all the lines through its points.
All sets are supposed convex. The order relation is "$X$ is smaller thatn $Y$ iff $X\subset Y$"
Definition: X is polygonal if $\exists$ a convex $Y\supset... | {
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How to prove $\int_0^1 \ln\left(\frac{1+x}{1-x}\right) \frac{dx}{x} = \frac{\pi^2}{4}$? Can anyone suggest the method of computing
$$\int_0^1 \ln\left(\frac{1+x}{1-x}\right) \frac{dx}{x} = \frac{\pi^2}{4}\quad ?$$
My trial is following
first set $t =\frac{1-x}{1+x}$ which gives $x=\frac{1-t}{t+1}$
Then
\begin{ali... | Hint. By the change of variable
$$
t =\frac{1-x}{1+x}
$$ you get
$$
\int_0^1 \ln\left(\frac{1+x}{1-x}\right) \frac{dx}{x}=-2\int_0^1 \frac{\ln t}{1-t^2} dt=-2\sum_{n=0}^{\infty}\int_0^1t^{2n}\:\ln t \:dt=2\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}= \frac{\pi^2}{4}.
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Non-constant entire function-bounded or not? Show that if $f$ is a non-constant entire function,it cannot satisfy the condition: $$f(z)=f(z+1)=f(z+i)$$
My line of argument so far is based on Liouville's theorem that states that every bounded entire function must be a constant.
So I try-to no avail-to show that if $f$ s... | First you can prove that
$$f(z+\mathbb Z(i)) = f(z)$$
so $f$ is determined by it's values on the square $[0, 1+i)$. Entire functions are bounded on precompact sets. QED.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Squaring a trigonometric inequality A very, very basic question.
We know
$$-1 \leq \cos x \leq 1$$
However, if we square all sides we obtain
$$1 \leq \cos^2(x) \leq 1$$
which is only true for some $x$.
The result desired is
$$0 \leq \cos^2(x) \leq 1$$
Which is quite easily obvious anyway.
So, what rule of inequalities... | Squaring does not preserve inequality since it is a 2-to-1 function over $\mathbb{R}$, hence nonmonotone. Mind you a monotone decreasing function would reverse the inequality there would still be a valid inequality pointing the opposite way.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Suppose $A^2B+BA^2=2ABA$.Prove that there exists a positive integer $k$ such that $(AB-BA)^k=0$. Let $A, B \in M_n(\mathbb{C})$ be two $n \times n$ matrices such that
$$A^2B+BA^2=2ABA$$
Prove that there exists a positive integer $k$ such that $(AB-BA)^k=0$.
Here is the source of the problem. Under the comment by Carcu... | From $[A,[A,B]] = 0$, we decuce by induction that
$$ [A^k, B] = kA^{k-1}[A,B] \tag 1 $$
For $k = 0$, (1) is obviuos, if (1) holds for $k-1$, we have
\begin{align*}
[A^k, B] &= A^kB - BA^k\\
&= A^{k-1}AB - BAA^{k-1}\\
&= A^{k-1}[A,B] + A^{k-1}BA - BA^{k-1}A\\
&= A^{k-1}[A,B] + [... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the period of $f(x) = \sin 2x + \cos 3x$ I want to find the period of the function $f(x) = \sin 2x + \cos 3x$. I tried to rewrite it using the double angle formula and addition formula for cosine. However, I did not obtain an easy function. Another idea I had was to calculate the zeros and find the difference b... | Hint
The period of $\sin(2x)$ is $\pi$, and the period of $\cos(3x)$ is $2\pi/3$.
Can you find a point where both will be at the start of a new period?
| {
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Distribution of server utilisations in an M/M/c queuing model with an unusual dispatching discipline I'm studying an M/M/c queuing model with an unusual (?) dispatching discipline:
*
*Servers are numbered 1...c
*The servers have an identical mean service time, exponentially distributed (as usual), which does not va... | This is (an attempt at) a partial answer.
I think that you should be able to decouple the server selection aspect from the composite load aspect, since the servers are identical with exponentially distributed service time $\mu$. That is, one can analyze the number in system (irrespective of distribution across the ser... | {
"language": "en",
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Changing argument into complex in the integral of Bessel multiplied by cosine I got a problem solving the equation below:
$$ \int_0^a J_0\left(b\sqrt{a^2-x^2}\right)\cosh(cx) dx$$
where $J_0$ is the zeroth order of Bessel function of the first kind.
I found the integral expression below on Gradshteyn and Ryzhik's book ... | $$I=\int_{0}^{a}J_0(b\sqrt{a^2-x^2})\cosh(cx)\,dx = a\int_{0}^{1}J_0(ab\sqrt{1-z^2})\cosh(acz)\,dx$$
The trick is now to expand both $J_0$ and $\cosh$ as Taylor series, then to exploit:
$$ \int_{0}^{1}(1-z^2)^{n}z^{2m}\,dz = \frac{\Gamma(n+1)\,\Gamma\left(m+\frac{1}{2}\right)}{2\cdot\Gamma\left(m+n+\frac{3}{2}\right)}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1292854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The convergence of the series $\sum (-1)^n \frac{n}{n+1}$ and the value of its sum This sum seems convergent, but how to find its precise value?
$$\sum\limits_{n=1}^{\infty}{(-1)^{n+1} \frac{n}{n+1}} = \frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+...=-0.3068... $$ Any help would be much appreciated.
| Hint: This series can't converge, since |$a_n| \rightarrow 0$ is not true, because $\frac{n}{n+1}=1$, if $n \rightarrow \infty$(This is a neccessary term).
| {
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Find the ratio of curved surface area of frustum to the cone.
In the figure, there is a cone which is being cut and extracted in three segments having heights $h_1,h_2$ and $h_3$ and the radius of their bases $1$ cm, $2$cm and $3cm$, then
The ratio of the curved surface area of the second largest segment to that of ... | Area is proportional to the square of linear dimension. So the area of the full cone is $k(3^2)$ for some $k$. The area of the second largest segment is $k(2^2) - k(1^2)$, so the ratio is $3:9 = 1:3$. You are right, and the book is wrong.
| {
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How to prove that a straight line is an infinite set of points? From the basic elementary level when we start reading geometry we get this idea developed in us that a straight line is the conjuction of infinite points.but how to prove this? I mean is this an axiom or its provable?
| Take your endpoints, call them $S$ and $E$. Pick some point between $S$ and $E$, let us call it $M$. Then, pick a point between $S$ and $M$. Call it $M_2$. Pick a point between $S$ and $M_2$ ....
If you can keep going like that, it is easy to see that the line must have infinite points... and you can keep going on lik... | {
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Explicit form of this series expansion? I am considering the following series expansion:
$$f(k):=\sum_{n\geq 1} e^{-k n^2}$$
with $k>0$ a fixed parameter. Is there a possibility to either find a closed form expression for $f(k)$? Or at least an upperbound of the type $|f(k)|\leq \frac{C}{k^p}$ for some constant $C$ and... | For $k\ge 1,$ $$\sum_{n=1}^{\infty}e^{-kn^2} \le \sum_{n=1}^{\infty}e^{-kn} = \frac{e^{-k}}{1-e^{-k}}\le \frac{1}{1-e^{-1}}\cdot e^{-k}.$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that $f$ is differentiable at $0$ Let's consider the following function:
$$f(x,y)=\begin{cases} (x^2+y^2)\sin\left(\dfrac{1}{x^2+y^2}\right) & \text{if }x^2+y^2\not=0 \\{}\\ 0 & \text{if }x=y=0 \end{cases}$$
I know that $f_x$ and $f_y$ are not continuous at $0$. How to prove that $f$ is differentiable at $0$?
| Hint:
Change to polar coordinates and show
$$\lim_{r\to 0}\frac{r^2\sin(r^{-2})-0}{r}=0$$
Hint 2: The sine function is bounded.
| {
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Horse racing question probability Been thinking about this for a while.
*
*Horse Campaign length: 10 starts
*Horse Runs this campaign: 5
*Horse will is guaranteed to win 1 in 10 this campaign
Question: what is the Probability of winning at the sixth start if it hasn't won in the first five runs?
20%?
Thanks for... | I assume, that the horse will win only one race. And the probability of the (garanteed) win and a start at a race are equally like for every start/race.
The probability, that the horse has had no starts ($s_0$) is
$\frac{{5 \choose 0}\cdot {5 \choose 5}}{{10 \choose 5}}$ and the probabilty that then the horse will win... | {
"language": "en",
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Computing the value of a series by telescoping cancellations vs. infinite limit of partial sums $$\sum_{m=5}^\infty \frac{3}{m^2+3m+2}$$
Given this problem my first approach was to take the limit of partial sums. To my surprise this didn't work. Many expletives later I realized it was a telescoping series.
My question... | The integral test simply tells you if an infinite sum is convergent, it will not necessarily tell you what the sum converges to. Imagine back to when you first started learning about the Riemann integral. You might remember seeing pictures of a smooth curve with rectangular bars pasted over the curve, approximating the... | {
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"timestamp": "2023-03-29T00:00:00",
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$\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$ $\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$
How do I simplify this limit?
I tried multiplying dividing $\sin(\frac{x}{2^r})$ to use half angle formula but it doesnt give a telescopic which would have simplified it.
| $$\prod_{r=1}^{\infty} \frac{\cos (x/2^r) \sin (x/2^r)}{\sin (x/2^r)} = \prod_{r = 1}^{\infty} \frac{\sin (x/2^{r-1})}{2\sin(x/2^r)} = \lim_{r \to \infty} \frac{\sin x}{2^r \sin (x/2^r)}$$
Can you take it from here? Recall that $\lim_{t \to 0} \frac{\sin
\alpha t}{t} = \alpha$.
| {
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"url": "https://math.stackexchange.com/questions/1293756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Little confusion about connectedness Consider $X=\{(x,\sin(1/x)):0<x<1\}$. Then clearly $X$ is connected , as it is a continuous image of the connected set $(0,1)$.
So, $\overline X$ is also connected , as closure of connected set is connected.
Now if we notice about the set $\overline X$ then
$$\overline X=X\cup B$$
... | The set $B$ is NOT open, so this is not a partition of $\overline X$ in open sets.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove that for any given $c_1,c_2,c_3\in \mathbb{Z}$,the equations set has integral solution. $$ \left\{
\begin{aligned}
c_1 & = a_2b_3-b_2a_3 \\
c_2 & = a_3b_1-b_3a_1 \\
c_3 & = a_1b_2-b_1a_2
\end{aligned}
\right.
$$
$c_1,c_2,c_3\in \mathbb{Z}$ is given,prove that $\exists a_1,a_2,a_3,b_1,b_2,b_3\in\mathbb{Z}$ meet t... | First let us assume that $c_1, c_2, c_3$ are coprime, that is $(c_1, c_2, c_3) = 1$.
Suppose first that there exist integer numbers $u_1, u_2, u_3$ and $v_1, v_2, v_3$ such that
$$
\begin{vmatrix}
c_1 & c_2 & c_3 \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3
\end{vmatrix}
= 1
$$
In that case we can take $a = c \times u$ and... | {
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How write a periodic number as a fraction? What I call as a periodic number is for exemple
$$0.\underbrace{13}_{period}131313...$$ or $$42.\underbrace{465768}_{period}465768465768.$$
So how can we put theses numbers like a integer fractional, i.e. of the form $\frac{a}{b}$ with $a,b\in\mathbb Z$ ?
| Multiplying your number by a suitable power of $10$ we can make some parts the nummber jump to the left of the decimal point leaving identical fractional part. That is $10^mx$ and $x$ have the same fractional part. SO their difference is an integer $a$: That is $a= (10^k-1)x$, this shows $x$ is a rational number.
| {
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Evaluate the double sum $\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2}$ As a follow up of this nice question I am interested in
$$
S_1=\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2}
$$
Furthermore, I would be also very grateful for a solution to
$$
S_2=\sum_{m=1}^{\i... | Numerically, I get
$$
S_1+S_2 = 0.14836252987273216621
$$
which agrees with
$$
\frac{\pi^6}{6480}
$$
Also numerically,
$$
S_1 = 0.074181264936366083104
\\
S_2 = 0.074181264936366083104
$$
are seemingly equal.
| {
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"timestamp": "2023-03-29T00:00:00",
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Solve of the differential equation $y'=-\frac{x}{y}+\frac{y}{x}+1$ I've tried to solve this equation, and in the course of solving any problems. Please help me understand.
$$y'=-\frac{x}{y}+\frac{y}{x}+1$$
Results in a normal form.
$$y'=-\frac{1}{\frac{y}{x}}+\frac{y}{x}+1$$
Make replacement.
$$\frac{y}{x}=U$$
$$y'=U'x... | HINT: we have
$$\frac{du}{1-\frac{1}{u}}=\frac{dx}{x}$$
the term $$\frac{u}{u-1}=\frac{u-1+1}{u-1}=1+\frac{1}{u-1}$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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limit of sin function as it approches $\pi$ In my assignment I have to find the Classification of discontinuities of the following function:
$$f(x)=\frac{\sin^2(x)}{x|x(\pi-x)|}$$
I wanted to look what happens with the value $x=\pi$ because the function doesn't exist in that value.
I have to check if some $L \in \Bbb... | Since $\sin x=\sin (\pi -x)$ then
\begin{eqnarray*}
\lim_{x\rightarrow \pi }\frac{\sin ^{2}(x)}{x\left\vert x(\pi -x)\right\vert
} &=&\lim_{x\rightarrow \pi }\left( \frac{\sin (\pi -x)}{(\pi -x)}\right)
^{2}\frac{\left\vert \pi -x\right\vert }{x\left\vert x\right\vert } \\
&=&\lim_{x\rightarrow \pi }\left( \frac{\sin ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1294358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number of words which can be formed with INSTITUTION such that vowels and consonants are alternate Question:
How many words which can be formed with INSTITUTION such that vowels and consonants are alternate?
My Attempt:
There are total 11 letters in word INSTITUTION. The 6 consonants are {NSTTTN} and the 5 vowels ar... | You do not have the option of either starting with a vowel or starting with a consonant since you must alternate and you have 6 consonants and 5 vowels. It must start with a consonant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1294462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$xf(y)+yf(x)\leq 1$ for all $x,y\in[0,1]$ implies $\int_0^1 f(x) \,dx\leq\frac{\pi}{4}$ I want to show that if $f\colon [0,1]\to\mathbb{R}$ is continuous and
$xf(y)+yf(x)\leq 1$ for all $x,y\in[0,1]$
then we have the following inequality:
$$\int_0^1 f(x) \, dx\leq\frac{\pi}{4}.$$
The $\pi$ on the right hand side sugge... | Let $I = \displaystyle\int_{0}^{1}f(x)\,dx$.
Substituting $x = \sin \theta$ yields $I = \displaystyle\int_{0}^{\pi/2}f(\sin \theta)\cos\theta\,d\theta$.
Substituting $x = \cos \theta$ yields $I = \displaystyle\int_{0}^{\pi/2}f(\cos \theta)\sin\theta\,d\theta$.
Hence, $I = \dfrac{1}{2}\displaystyle\int_{0}^{\pi/2}\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1294575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
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"answer_id": 0
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Upper bound for truncated taylor series A paper claimed the following but I can't figure out why it's true:
For all $1/2> \delta > 0$, $k\le n^{1/2-\delta}$, and $j\le k-1$ where $n$, $j$, and $k$ are positive integers, the following holds: $$\sum_{i=j}^k \frac{(2/n^{2 \delta})^i}{i!} \le \frac{4}{n^{2\delta j}}.$$
Why... | Notes:
*
*If I read the notation of the paper correctly (and the putative inequality is on line 16 of page 19), the authors' claim is weaker than in your question, namely
$$
\sum_{i=j}^{k} \frac{(2/n^{2\delta})^{i}}{i!} \leq \frac{4}{n^{\delta j}}
$$
(n.b. $n^{\delta j}$ in the right-hand denominator rather than $n^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Average distance between two randomly chosen points in unit square (without calculus) Imagine that you choose two random points within a 1 by 1 square. What is the average distance between those two points? Using a random number generator, I'm getting a value of ~0.521402... can anyone explain why I'm getting this va... | Let $(X_i, Y_i)$ for $i=1,2$ be i.i.d. and has uniform distribution on $[0, 1]^2$. Then $|X_1 - X_2|$ has PDF
$$ f(x) = \begin{cases} 2(1-x), & 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases}. $$
This shows that the average distance is
\begin{align*}
\int_{0}^{1}\int_{0}^{1} 4(1-x)(1-y)(x^2 + y^2)^{1/2} \, dxdy
&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1294800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 2,
"answer_id": 0
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Complex numbers modulo integers Is there a "nice" way to think about the quotient group $\mathbb{C} / \mathbb{Z}$?
Bonus points for $\mathbb{C}/2\mathbb{Z}$ (or even $\mathbb{C}/n\mathbb{Z}$ for $n$ an integer) and how it relates to $\mathbb{C} / \mathbb{Z}$.
By "nice" I mean something like:
*
*$\mathbb{R}/\mathbb{Z... | $\mathbb C/\mathbb Z$ is isomorphic to $\mathbb C/ n\mathbb Z$ for any non-zero complex number $n$.
You can show that $\mathbb C/\mathbb Z\cong (\mathbb C\setminus\{0\},\times)$ via the homomorphism $z \to e^{2\pi iz}$.
You can also think of this as $S^1\times\mathbb R$, which is topologically a cylinder.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1294898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Find all solutions in N of the following Diophantine equation $(x^2 − y^2)z − y^3 = 0$
i divide by $z^3$ and look for rational solutions of the equation $A^2 − B^2 − B^3 = 0.$ The point $(A,B) = (0, 0)$ is a singular point, that is any line
through this point will meet the curve twice in $(0, 0)$. Now i wanna use Diop... | At least for a nonvertical line $B=kA$ when plugged into
$$A^2-B^2-B^3=0 \tag{1}$$ gives
$$A^2-k^2A^2-k^3A^3=0,$$
where here the double root corresponds to $A^2$ being a factor. After dividing by that and solving for $A$ one gets $A=(1-k^2)/k^3,$ and so also $B=kA=(1-k^2)/k^2.$ Finally a check reveals these values of ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the mathematical distinction between closed and open sets? If you wanted me to spell out the difference between closed and open sets, the best I could do is to draw you a circle one with dotted circumference the other with continuous circumference. Or I would give you an example with a set of numbers $(1, 2)$ v... | Let's talk about real numbers here, rather than general metric or topological spaces. This way we don't need notions of Cauchy sequences or open balls, and can talk in more familiar terms.
We define that a set $X \subset \mathbb{R}$ is open if for every $x \in X$ there exists some interval $(x-\epsilon,x+\epsilon)$ wit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
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Help With SAT Maths Problem (Percentages and Numbers) I usually solve SAT questions easily and fast, but this one got me thinking for several minutes and I cannot seem to find an answer.
Here it is:
In 1995, Diana read $10$ English and $7$ French books. In 1996, she read twice as many French books as English books. If... | Total English books: $x=10+E$
Total French books: $y=7+F$
Where $E$ and $F$ are English and French books in 1996.
We know $F=2E$
Thus:
Total books: $T=x+y=17+E+F$
We know that $0.6t=y$ that is
$0.6(17+E+F)=7+F\implies 0.6(17+3E)=7+2E$ - solve this for $E$
Which is sufficient to work it out :)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How can I prove the Carmichael theorem I am trying to prove that these two definitions of Carmichael function are equivalent.
I am using this definition of Carmichael function:
$\lambda(n)$ is the smallest integer such that $k^{\lambda(n)}\equiv1 \pmod n\,$ $\forall k$ such that $\gcd(k,n)=1$.
to prove this one:
$... | You need to prove that there are primitive roots $\bmod p^k$ when $p$ is an odd prime. We can do this by induction over $k$
To do this prove the base case using the fact that multiplicative groups of finite fields are cyclic and using $\mathbb Z_p$ is a field.
After this prove it for $k=2$. Take $a$ a primitive root $\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Problem about covering space Let $p:\tilde{X}\to X$ be a covering space, $\tilde{X}$ and $X$ are both path-connected and locally path-connected, if $p(x_1)=p(x_2)=x$, is $p_\ast(\pi_1(\tilde{X},x_1))=p_\ast(\pi_1(\tilde{X},x_2))$ always true?
| Not in general. These subgroups will be conjugate but might not be equal. You can check this by taking a path between your points in $\widetilde{X}$ and projecting it down to a loop in $X$. It is true if $p$ is a regular covering.
Also, you don't need the "locally path connected" assumption for any of these arguments.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1295403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Learning spectral methods in numerical analysis I'm trying to learn the theory about spectral methods without any specific ties to a particular program like MATLAB. I tried to search for some lecture videos but it seems very limited and I'm not sure which books are well suited for self-learning about this topic.
Does a... | I would recommend:
*
*Canuto C. et al. Spectral Methods (springer link).
*Trefethen L. Spectral Methods in Matlab. (This last one guides you through the implementation of the codes in Matlab but it can be easily extended to other programming language).
Hope it helps. Cheers!
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the range of a $4$th-degree function For the function $y=(x-1)(x-2)(x-3)(x-4)$, I see graphically that the range is $\ge-1$. But I cannot find a way to determine the range algebraically?
| Non-calculus Approach/ Completing Square Approach
Let $u=(x-2)(x-3)=x^2-5x+6$
$(x-1)(x-4)=x^2-5x+4=u-2$
So
\begin{align}
y&=(x-1)(x-2)(x-3)(x-4)\\&=u(u-2)\\&=u^2-2u\\&=(u-1)^2-1
\end{align}
As $(u-1)^2\ge0$ for all $u\in{\Bbb{R}}$
$y\ge0-1=-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1295586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Show that a linear transformation $T$ is one-to-one Problem:
Consider the transformation $T : P_1 -> \Bbb R^2$, where $T(p(x)) = (p(0), p(1))$ for every polynomial $p(x) $
in $P_1$. Where $P_1$ is the vector space of all polynomials with degree less than or equal to 1.
Show that T is one-to-one.
My thoughts:
I'm a litt... | Yet another way is with the useful result that a map $L$ between alebraic structures is injective iff the kernel of $L$ is trivial. This means that
$T(p)=0$ iff $p==0$, i.e., iff $p$ is the $0$ polynomial.
Assume then that $T(p)=(p(0),p(1))=(0,0)$. This means, for $p=ax+b$ : $$p(0)=a(0)+b=b=0 $$ ,together with, $$p(1)... | {
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Ratios and percents Mrs. Smith has 80 birds: geese, hens and ducks.
The ratio of geese to hens is 1:3.
60% of the birds are ducks.
How many geese does Mrs. Smith have?
a) 16
b) 8
c) 12
d) 11
I know that if 60% if the birds are ducks, 40% are geese and hens. Now I have to find 40% of 80 so in order to do this I chang... | Let $G$ be the number of geese and $H$ be the number of Hens. You said $G+H=32$. And you are also given $1 \cdot H=3 \cdot G$ or $H=3G$. So $G+3G=32$ I think you can finishing solving that for $G$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1295858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Verify solution to ODE I am given the ODE
$$\left(f''(x)+\frac{f'(x)}{x} \right) \left(1+f'(x)^2 \right) = f'(x)^2f''(x)$$
and I already know that the solution to this ODE is given by
$$f(x)= c \cdot arcosh \left( \frac{r}{c} \right) + d$$
where $|c|<r$ and $d \in \mathbb{R}.$
The problem is I want to show that this ... | $$
\left(f'' +\frac{f'}{x}\right)(1+f'^2)=f'^2f''
$$
multiply by $x$
$$
(xf''+f')(1+f'^2) = \left(\dfrac{d}{dx}xf'\right)(1+f'^2) = xf'^2f''
$$
then we have
$$
\frac{1}{xf'}\left(\dfrac{d}{dx}xf'\right) = \frac{f'f''}{(1+f'^2)}
$$
thus
$$
\dfrac{d}{dx}\ln(xf') = \frac{1}{2}\dfrac{d}{dx}\ln(1+f'^2)
$$
which is
$$
\ln (x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is there a book on proofs with solutions? I am a biochemistry graduate student who works on cancer. I am interested in learning proofs as a personal interest. I use math as a tool, but would like to start building a deeper understanding on my own. I am not taking any course. Hence, I am looking for a book with theory, ... | Well, there are lots and lots of books with solutions.
An answer to your question depends on which part of mathematics you would like to explore. For instance, a series of great ones about real analysis come from Kaczor & Nowak
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1296029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Checking psd-ness of matrix I have the following problem and don't know how to proceed...
I want to check if
\begin{equation}
\frac{1}{2}(B^\top A^\top A + A^\top A B) - \frac{1}{4}B^\top A^\top A A^\top (AA^\top AA^\top)^{-1}AA^\top A B
\end{equation}
is positive semi-definite (psd). We have $A \in \mathbb{R}^{m \time... | I don't finished the problem, but maybe these equalities can to help you.
Note that:
$$B^T A^T A A^T (AA^T AA^T)^{-1}AA^T AB=B^T A^T A A^T \left((A^T)^{-1}A^{-1}(A^T)^{-1}A^{-1}\right) A A^T AB$$
Then,
$$B^T A^T A A^T (AA^T AA^T)^{-1}AA^T AB=B^T A^T AB=(AB)^T AB.$$
And
$$B^T A^T A + A^T AB=(AB)^T A + A^T AB. $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $G$ be an abelian group of order $(p^n)m$, where $p$ is a prime and $m$ is not divide by $p$. Let $P = \{a\in G\mid a^{p^k} = e\text{ for some $k$ depending on $a$}\}$.
Prove that
(a) $P$ is a subgroup of $G$.
(b) $G/P$ has no elements of order $p$.
(c) order of $P=p^n$.
| I like the rest above, so I just wanted to suggest an argument for b).
Suppose there is an $\overline{x} \in G/P$ with order $p$, then $(x+P)^p=P$. But this means $x^p \in P$, so $(x^p)^{p^k}=e$ for some $k$. But this means $x\in P$ and thus $\overline{x}$ is actually the identity in $G/P$. But the identity cannot h... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determinant of block matrix with commuting blocks I know that given a $2N \times 2N$ block matrix with $N \times N$ blocks like
$$\mathbf{S} = \begin{pmatrix}
A & B\\
C & D
\end{pmatrix}$$
we can calculate $$\det(\mathbf{S})=\det(AD-BD^{-1}CD)$$ and so clearly if $D$ and $C$ commute this reduces to $\det(AD-BC)$, whi... | The answer is affirmative. See, e.g., theorem 1 of John Silvester, Determinants of Block Matrices.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "6",
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What is a Galois closure and Galois group? I was reading wikipedia for Galois groups and this term suddenly appears and there is no definition for it.
What is a Galois closure of a field $F$? Does this mean a maximal Galois extension of $F$ so that it merely means a separable closure of $F$?
Secondly, what is a Galois ... | Two points: One, Galois closure is a relative concept, that is not defined for a field, but for a given extension of fields.
Second, it is not something maximal. To the contrary it is something minimal.
Given an extension of fields $F\subset E$ if it is not Galois, then the smallest extension of $F$ that containing $E$... | {
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Modified version of Simpson Rule I'm supposed to use some different version of Simpson's Rule in my Numerical Methods homework to compute some areas, considering the non-uniform spacing case .
Namely, I've got two equal length vectors $x,y$ representing the pairs $(x_i,f(x_i))$ and the components of the $x$ array aren... | If you want to use Simpson's rule with Taylor polynomials and unequally spaced intervals you can look at the derivation in https://www.math.ucla.edu/~yanovsky/Teaching/Math151A/hw6/Numerical_Integration.pdf . (See Section 3, page 2). It doesn't have the derivation, but it has enough to use the derivation within as a gu... | {
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For an infinite cardinal $\kappa$, $\aleph_0 \leq 2^{2^\kappa}$ I'm trying to do a past paper question which states:
$$
\text{For all infinite cardinals $\kappa$, we have } \aleph_0 \leq 2^{2^\kappa}.
$$
I'm supposed to be able to do this without the axiom of choice, but I can't see how. I know that there can be no bi... | HINT: It suffices to find a surjection from $2^\kappa$ onto the natural numbers, now look at cardinalities of finite sets.
You can also show that equality is never possible. But that is besides the point. You are asked to find an injection, not to prove there are no bijections.
| {
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Proof that a degree $4$ polynomial has a minimum
Let $$f(x) = x^4+a_3x^3+a_2x^2+a_1x+a_0.$$ Prove that $f(x)$ has a minimum point in $\Bbb{R}$.
The Extreme Value Theorem implies that the minimum exists in some $[a,b]$, but how do I find the right $a$ and $b$?
| If $x>|a_3|+\sqrt{|a_2|}+\sqrt[3]{|a_1|}+\sqrt[4]{|a_0|}$ then $f(x)>0$.
| {
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Ring with no identity (that has a subring with identity) has zero divisors. Let $L$ be a non-trivial subring with identity of a ring $R$. Prove that if $R$ has no identity, then $R$ has zero divisors.
So I assumed that there $\exists$ $e \in L$, such that $ex=xe=x$, $\forall$ $x\in L$ and there $\exists$ $x' \in R/L$ s... | Actually you can prove something even better: a nonzero idempotent in a rng without nonzero zero divisors is actually an identity for the rng.. This applies to your case since the identity of the subrng would be a nonzero idempotent.
The linked solution proves the contrapositive of your statement very simply, and along... | {
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Symmetry argument and WLOG Suppose that $f:[0,1]\rightarrow \mathbb{R}$ has a continuous derivative and that $\int_0^1 f(x)dx=0$. Prove that for every $\alpha \in (0,1)$,
$$\left|\int_0^{\alpha}f(x)dx\right| \leq \frac{1}{8}\max_{0\leq x\leq 1}|f'(x)|.$$
The solution goes like this: Let $g(x)=\int_0^x f(y)dy$, then $g... | Consider an $\alpha$ that is greater than $\frac{1}{2}$. Then you know that $1-\alpha \leq \frac{1}{2}$ and then we can make the substitution they give you to "pretend" we have the kind of $\alpha$ that we want (Less than or equal to $\frac{1}{2}$).
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $f$ in monotonic In my assignment I have to prove the following:
Let $f$ a continuous function in $\Bbb R$.
Prove the following:
if $|f|$ is monotonic increasing, in R then $f$ is monotonic in R.
Hint: Prove that $f(x)\ne 0$ for every $x \in \Bbb R$
My proof is rather long and I didn't use the fact th... | It's too complicated and some aguments seem to be not justifief: e.g. Why do ypu try to prove $x> 0$ for all $x$? $x$ is the variable, it can take any value. As you noticed, we can't have $f(x=0, because this is the same as $\lvertf(x)\rvert=0$, which would imply that if $x'
From there, as $f$ is continuous, $f(x)$ h... | {
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"answer_id": 0
} |
For which values of $x$ is the following series convergent: $\sum_0^\infty \frac{1}{n^x}\arctan\Bigl(\bigl(\frac{x-4}{x-1}\bigr)^n\Bigr)$ For which values of $x$ is the following series convergent?
$$\sum_{n=1}^{\infty} \frac{1}{n^x}\arctan\Biggl(\biggl(\frac{x-4}{x-1}\biggr)^n\Biggr)$$
| If $x>1$ the series converges absolutely because $\arctan$ is a bounded function:
$$
\Biggl|\frac{1}{n^x}\,\arctan\Bigl(\frac{x-4}{x-1}\Bigr)^n\Biggr|\le\frac{\pi}{2\,n^x}.
$$
If $x<1$ then $(x-4)/(x-1)>1$ and
$$
\lim_{n\to\infty}\arctan\Bigl(\frac{x-4}{x-1}\Bigr)^n=\frac{\pi}{2}.
$$
It follows that the series diverges... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1297194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Residue Calculus - Showing that the quotient of polynomials have integral $0$ in a simple closed contour in a special case. I'm having difficulty understanding the solution to the following problem.
In the solution below, I can't understand why since $b_m\neq 0$, the quotient of these polynomials is represented by a se... | What I'd do: We have $|P(z)/Q(z)|$ on the order of $1/|z|^2$ as $|z| \to \infty.$ By Cauchy, $$\int_C \frac{P(z)}{Q(z)}\,dz = \int_{|z|=R} \frac{P(z)}{Q(z)}\,dz$$ for large $R.$ Use the "M-L" estimate on the latter to see its absolute value is no more than a constant times $(1/R^2)\cdot2\pi R.$ The latter $\to 0$ as $R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1297282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding a nullspace of a matrix - what should I do after finding equations? I am given the following matrix $A$ and I need to find a nullspace of this matrix.
$$A =
\begin{pmatrix}
2&4&12&-6&7 \\
0&0&2&-3&-4 \\
3&6&17&-10&7
\end{pmatrix}$$
I have found a row reduced form of this matrix, which is:
$$A' =
\beg... | here is an almost algorithm to find a basis for the null space of a matrix $A:$
(a) row reduce $A,$
(b) identify the free and pivot b=variables. the variables corresponding to the non pivot columns, here they are $x_2$ and $x_4,$ are called the free variables and the rest are called pivot variables.
(c) you can set on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1297366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
An introduction to algebraic topology from the categorical point of view I'm looking for a modern algebraic topology textbook from a categorical point of view. Basically, I'd like a textbook that uses the language of functors, natural transformations, adjunctions, etc. right from the start.
I'm comfortable with general... | Rotman's An Introduction To Algebraic Topology is a great book that treats the subject from a categorical point of view. Even just browsing the table of contents makes this clear: Chapter 0 begins with a brief review of categories and functors. Natural transformations appear in Chapter 9, followed by group and cogrou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1297513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 3
} |
Visualize $z+\frac{1}{z} \ge 2$ As we know, always
$$z+\frac{1}{z} \ge 2,~~~~~~~~~ z\in \mathbb{R}^+$$
However, is there any geometric way to visualize this equation for some one who is not that expert in math?
I know this question might have various answers.
| Well, what you wrote is clearly not true, it is only true for $x\in (0,\infty)$.
You could just try plotting the function $x\mapsto x+\frac1x$ and see that it is always above $2$.
http://www.wolframalpha.com/input/?i=plot+z+%2B+1%2Fz
Or, you can plot $x\mapsto x$ and $x\mapsto \frac1x$ and try to understand what is hap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1297652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 2
} |
Finding the solutions of $x^2\equiv 9 \pmod {256}$. Find the solutions of $x^2\equiv 9 \pmod {256}$. I try to follow an algorithm shown us in class, but I am having troubles doing so. First I have to check how many solutions there are. Since $9\equiv 1 \pmod {2^{\min\{3,k\}}}$ where k fulfills $256=2^k$, then since $k\... | The answers above explain the idea great. If you need a general Formula
for solving all the questions of this type then check this out -
$x^2 \equiv a \pmod {2^n}$ where a is odd
for n = 1 there is one answer and it is $x \equiv 1$
for $n = 2$ then we separate for 2 cases -
$ a \equiv 1$ then we have two answers $x \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1297717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Interpolation with nonvanishing constraint Let $x_1,x_2,\ldots,x_n$ be distinct complex numbers. Let $y_1,y_2,\ldots,y_n$
be nonzero complex numbers, and let $K$ be a bounded subset of $\mathbb C$. Does there
always exist a polynomial $P$ such that $P(x_i)=y_i$ for every $i$
and $P$ is everywhere nonzero on $K$ ?
This ... | Yes, such a polynomial always exists:
By scaling and replacing $K$ with a superset, we may assume that $K = \overline{\mathbb{D}} = \{ z : \lvert z\rvert \leqslant 1\}$ and that $\lvert x_i\rvert \leqslant 1$ for all $i$.
For each $i$, consider the interpolation polynomial
$$P_i(z) = \prod_{\substack{k=1\\k\neq i}}^n \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1297856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding a Lyapunov function for a given system I need to find a Lyapunov function for $(0,0)$ in the system:
\begin{cases} x' = -2x^4 + y \\ y' = -2x - 2y^6 \end{cases}
Graph built using this tool showed that there should be stability but not asymptotic stability. I'd like to prove that fact by means of Lyapunov functi... | As explained in the comments, even though simulated phase diagrams seem to exhibit cycling trajectories near the origin, they are not conclusive enough to decide whether the origin is stable or not, that is, whether trajectories cycle or spiral outwardly or spiral inwardly. Caution is advised about approximation errors... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1297904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
Having problem in last step on proving by induction $\sum^{2n}_{i=n+1}\frac{1}{i}=\sum^{2n}_{i=1}\frac{(-1)^{1+i}}{i} $ for $n\ge 1$ The question I am asked is to prove by induction $\sum^{2n}_{i=n+1}\frac{1}{i}=\sum^{2n}_{i=1}\frac{(-1)^{1+i}}{i} $ for $n\ge 1$
its easy to prove this holds for $n =1$ that gives $\fra... | Since an answer how to continue your proof was already given, I want to give another proof not using induction (which is in my eyes not the right way to proof this equality).
We have
$$\sum_{i=1}^{2n}\frac{(-1)^{i+1}}{i} \overset{(1)}{=} \sum_{i=1}^{n}\frac{1}{2i-1}-\sum_{i=1}^{n}\frac{1}{2i} = \sum_{i=1}^{n}\frac{1}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1298016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Confusion with Summations I am having a little bit of confusion regarding summations.
I know that $$\sum_{i=m}^n a_i = a_{m}+a_{m+1}+\cdots +a_{n-1}+a_n$$
Here is my confusion. How do we interpret/decompose the following:
$$ \sum_{i=m}^n a_i~~~~,~~~m=0,1,2,3,,,,k~~? $$
| Is it the $m=0,1,2,3,\dots,k$ that's confusing you? That just means there are several sums: $$
\text{$\sum_{i=0}^na_i$, $\ \ \sum_{i=1}^na_i$, $\ \ \sum_{i=2}^na_i$, $\ \ \dots\ $, $\ \sum_{i=k}^na_i$}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1298104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Kernel Principal Component Analysis (PCA) I learn kernel PCA from wikipedia. In this article, the eigen equation is
\begin{equation}
N \lambda \vec{\alpha} = \boldsymbol{K} \vec{\alpha}
\end{equation}
where $\lambda$ is the eigen value, $\vec{\alpha}$ is the eigen vector of $\lambda$ and $\boldsymbol{K}$ is the kern... | Recall for linear PCA $N$ is the number of data points in the set and comes in the eigen decomposition of the covariance matrix $C$, where $C = \frac{1}{N} \sum_{i=1}^N x_i x_i^\top$ ( we look for eigenvectors $v$ such that $\lambda v = C v$). So we simply multiply through by the constant $N$ to bring it through to the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1298200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove result about basis of a linear map with specific properties I am working on the following problem.
Let $V$ be an $n$-dimensional vector space over $K$ and $T: V\to V$ a linear map.
For $k = 1, \ldots, n$ let $x_k \in V \smallsetminus \{0\}$ and $\lambda_k \in K$ be given such that $T(x_k) = \lambda_k x_k \space \... | If $V$ is a vector space of dimension $n$ over $K$, then any $n$ linearly independent vectors in $V$ form a basis of $V$.
Hence $x_1, \ldots, x_n$ span $V$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1298287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What means to complete a pair of vectors $(w, s)$ to an arbitrary basis of $R^d$? I found in an article this : Let $B = (b_1, b_2, . . . , b_d)$ be an orthonormal basis of $R^d$ such that $<b1, b2 >=< w,x >$ (where $< ... >$ denotes linear span). In order to construct $B$, first complete $(w, x)$ to an arbitrary basis ... | To complete a pair of vectors $(w,s)$ to an arbitrary basis of $\mathbb R^d$ means to find an ordered basis of $\mathbb R^d$ such that $w,s$ are the first two vectors of it. Of course there are a lot of differen possible choices. Not at all completely random however.
The third vector $v_3$ should be chosen as not to b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1298361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving $\int_0^{+\infty}\frac{e^{-\alpha x^2} - \cos{\beta x}}{x^2}dx$ I need to find solution of $$\int_0^{+\infty}\frac{e^{-\alpha x^2} - \cos{\beta x}}{x^2}dx$$
I know that Leibniz rule can help but I don't know how to use it.
Could you help me please?
Thank you.
| Comparison with $1/x$ shows that the integral does not converge at $x=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1298427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Understanding the difference between Span and Basis I've been reading a bit around MSE and I've stumbled upon some similar questions as mine. However, most of them do not have a concrete explanation to what I'm looking for.
I understand that the Span of a Vector Space $V$ is the linear combination of all the vectors in... | The span of a finite subset $S$ of a vector space $V$ is the smallest subvector space that contains all vectors in $S$. One shows easily it is the set of all linear combinations of lelements of $S$ with coefficients in the base field (usually $\mathbf R,\mathbf C$ or a finite field).
A basis of the vector space $V$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1298492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 2
} |
Intuition behind independence result The following problem is from Wasserman's $\textit{All Of Statistic}s$. I have worked through the algebra to arrive at the result, but it still seems very strange to me, so I would appreciate any intuition that could be offered.
Let $N$~Poisson($\lambda$). Suppose we toss a coin $N$... | Suppose you're talking about a fair coin, and you're told that 50 tosses (from an unknown number of tosses) are heads. Intuitively, to infer $Y=N-X$ given $X=50$, say, you would need to first infer $N$, which in turn is inferred based on the probability of heads relative to tails (the next best thing you have as you do... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1298596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to find the greatest prime number that is smaller than $x$? I want to find the greatest prime number that is smaller than $x$, where $ x \in N$. I wonder that is there any formula or algorithm to find a prime ?
| As Emilio Novati stated in a comment, the sieve of Eratosthenes will work. The sieve will probably be fast enough for your needs, although potentially faster approaches exist (see Lykos's answer).
I didn't want to bother converting it to pseudocode, so here is a function written in C that returns the greatest prime les... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1298683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Calculate $\lim_{x \to 0} \frac{e^{3x} - 1}{e^{4x} - 1}$ Question:
Calculate
$$\lim_{x \to 0} \frac{e^{3x} - 1}{e^{4x} - 1}$$
using substitution, cancellation, factoring etc. and common standard limits (i.e. not by L'Hôpital's rule).
Attempted solution:
It is not possible to solve by evaluating it directly, since it l... | $$\frac{(e^x)^3-1}{(e^x)^4-1}=\frac{e^x+1}{2(e^{2x}+1)}+\frac{1}{2(e^x+1)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1298780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 4
} |
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