qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,126,835 | <p>Suppose that $f: [0,1] \rightarrow [0,1]$ is continuously differentiable. Further, suppose that $f$ has a fixed point $x_{0} \in (0,1)$ such that $|f'(x_{0})| < 1$. Then there exists an open interval $I$ containing $x_{0}$ such that $\{ f^{n}(x) \}_{n}$ converges to $x_{0}$ for all $x \in I$. </p>
<p>I know t... | Tsemo Aristide | 280,301 | <p>Since $f$ is continuously differentiable, there exists an interval $I=(x_0-c,x_0+c)$ such that for every $x\in I, |f'(x)|<d<1$. The mean value theorem implies that for every $x\in I$, there exists $y\in I$ such that $|f(x)-f(x_0)|=|f(x)-x_0|=|f'(y)||x-x_0|<d|x-x_0|<c$, this implies that $f(x)\in I$ and $... |
725,547 | <p>This is from the Chapter 15 text of Gourieroux and Monfort's Statistics and Econometric Models II:</p>
<p><strong>Set Up</strong>: Suppose that there are 2 possible parameter values $\theta_0$ and $\theta_1$ from which there are 2 density functions $l_{\theta_0}(y)$ and $l_{\theta_1}(y)$ on the data. Define
$$
F(k)... | ah11950 | 136,851 | <p>We have $x=apq+r$, and $r$ is the class of $x$ modulo $b$. </p>
<p>So, looking at this equality modulo $a$, we have $x \equiv r\; (\textrm{mod}\; a)$ since $apq \equiv 0\; (\textrm{mod}\; a)$. This proves the result, since $x \equiv r\; (\textrm{mod}\; b)$. </p>
|
3,389,361 | <p>For what n is this rational,
<span class="math-container">$$\frac{\sqrt{n^2+1}} {\sqrt{2}}$$</span> </p>
<p>So far I have found the integers 1,7,41 and I have found some rational solutions to this as well but I'm looking to get a more general sense. </p>
<ul>
<li>So when is this a rational number?</li>
<li>Are the... | wendy.krieger | 78,024 | <p>This number is rational in the series 1, 7, 41, 239, ...</p>
<p>Each new number is t(n+1)=6 t(n)-t(n-1).</p>
<p>The feature works because 6^2-4 is a double-square, </p>
<p>EDIT:</p>
<p>There are a lot of solutions when it is rational, for example, putting x=23/7 will produce a rational result of 17/7.</p>
<p>Th... |
3,389,361 | <p>For what n is this rational,
<span class="math-container">$$\frac{\sqrt{n^2+1}} {\sqrt{2}}$$</span> </p>
<p>So far I have found the integers 1,7,41 and I have found some rational solutions to this as well but I'm looking to get a more general sense. </p>
<ul>
<li>So when is this a rational number?</li>
<li>Are the... | J. W. Tanner | 615,567 | <p>If <span class="math-container">$\dfrac{\sqrt{n^2+1}} {\sqrt{2}}$</span> is an <em>integer</em> <span class="math-container">$m$</span>, then <span class="math-container">$\color{blue}{2m^2-n^2=1}$</span>. </p>
<p>This is a Pell-type equation, and there are infinitely many solutions,</p>
<p>including <span class=... |
1,282,111 | <p>How to show that the equation $3x^2-2y^2=1$ has infinitely many integer solutions such that $3|x$ ? ( If this can be shown then solutions of $12x^2-8y^2=4$ give infinitely many powerful numbers differing by $4$ )</p>
| Will Jagy | 10,400 | <p>Put another way,
$$ 27 \cdot 3^2 - 2 \cdot 11^2 = 1. $$
Given one $(x,y)$ pair with
$$ 27 x^2 - 2 y^2 = k, $$ we get the next pair, of infinitely many, from
$$ (485x + 132 y, 1782x+485 y). $$</p>
<p>With your $k=1$ you need start with the single seed pair $(3,11)$ to get all $(x,y)$ pairs with $x,y>0.$</p>
... |
787,322 | <p>For a group $[S,*]$ where $S=\{a,b\}$, how come there are $2^4$ binary operations that can be defined on $S$ instead of $2^2$? I can only see $a*a$, $a*b$, $b*a$, and $b*b$, which is $4=2^2$. What other operations can possibly exist? What am I not seeing here?</p>
| Shubham Avasthi | 240,149 | <p>There are $2^2$ elements in the cartesian product of S i.e. $S^2$ in this case and since every element in the set $S^2$ can be related to either $a$ or $b$, therefore the total number of such binary operations is $2^{2^2}=2^4=16$.</p>
|
2,258,147 | <p>I'm trying to see how to go about this problem for my revision. </p>
<blockquote>
<p><strong>Question 16</strong>
<p>$(a)$ If $g(x)=1-3x$, find $g(x-1)$ in terms of $x$
<p>$(b)$ On the axes below sketch the graph of $g(x)$ and $g(x-1)$</p>
</blockquote>
<p>I have a test coming up but I can't remember how yo... | mrnovice | 416,020 | <p>$$f(x) = |\cos x|^3+|\sin x|^3$$</p>
<p>The triangle inequality tells us that $|a|+|b|\geq |a+b|$</p>
<p>So $$f(x) = |\cos x|^3+|\sin x|^3 = |\cos^3x|+|\sin ^3x|\geq|\sin ^3x+\cos^3x|$$</p>
<p>Consider $g(x) = |\sin ^3x+\cos^3x|$</p>
<p>Clearly $g(x) \geq 0$</p>
<p>$\sin ^3 x+\cos^3x=0\implies \tan^3x=-1\impli... |
2,786,648 | <p>Let $f:[0,1]\rightarrow[0,\infty)$ be a continuous function such that $\int_0^1f(x)dx=1$, and let $M=\max f(x)$. Show that:
$$\frac{1}{2M}\le\int_0^1xf(x)dx\le1-\frac{1}{2M}$$</p>
<p>We have $1=\int_0^1f(x)dx\le\int_0^1Mdx=M$, so $M\ge1$. I tried several inequalities, but none of them seem to be working in this cas... | David C. Ullrich | 248,223 | <p>The following may be harder to follow than the excellent accepted answer. My <strong>excuse</strong> for posting it anyway: I hate using integration by parts on something like this, because I never feel like I shows why the inequality is "really" true. If the inequality explicitly involves derivatives, or if it's tr... |
1,212,198 | <blockquote>
<p>Prove or counter-example. For all nonempty sets $A$ and $B$ and for all functions $F$, $F(A-B) = F(A) - F(B)$; if not, what else does $F$ need to have in order to make the equality hold?</p>
</blockquote>
<p>I am pretty lost on this question. I don't feel like its right since it would be a pretty bas... | egreg | 62,967 | <p>Minimal counterexample: $X=\{1,2\}$, $Y=\{0\}$, $f\colon X\to Y$ is the only possible function. If $A=\{1,2\}$ and $B=\{2\}$, then...</p>
|
565,762 | <p>I have to show that : $$T(n) = Θ({n^3})$$</p>
<hr>
<p>We have this recursive function :</p>
<p>$$T(n) = 8T(n/2) + n^2, n>=2$$</p>
<p>also we know that $$T(1) = 1$$</p>
<p>And it says that there is a "replacement method" to do that.</p>
<hr>
<h2>EDIT</h2>
<hr>
<p>If I say $$n = 2^k, k≥1$$</p>
<p>then T(n... | xavierm02 | 10,385 | <p>$g=f\circ \left(x\mapsto ax\right)$</p>
<p>And you know that $f_1,f_2 \in C^\infty\left(\Bbb R\right)\implies f_1\circ f_2 \in C^\infty\left(\Bbb R\right)$</p>
<p>Note that $g(x)\in \Bbb R$. It's a number, as opposed to $g=x\mapsto g(x)$ which is a function.</p>
|
4,130,129 | <p>I was thinking that it might has to be <span class="math-container">$m$</span> and <span class="math-container">$n$</span> coprimes, but I don't have a consolidated idea of how I can prove it. Incidentally, how could I prove that it doesn't work for any integers? (is there any counterexample? I was thinking about <s... | timon92 | 210,525 | <p>The problem is underconstrained. There are several triangles satisfying this condition.</p>
<p>By the sine law we have <span class="math-container">$\dfrac{r_A}{\sin \delta} = \dfrac{r_B}{\sin \gamma}$</span> , hence the condition is equivalent to
<span class="math-container">$$\frac{r_A}{r_B} = \left(\frac{E_B}{E_A... |
384,553 | <p>Any ideas how to solve it?
$$\int\frac{x^4+2x+4}{x^4-1}dx$$
Thanks!</p>
| Ross Millikan | 1,827 | <p>Hint: First, make the integrand into $1+\frac {2x+5}{x^4-1}$ Now apply partial fractions.</p>
|
211,689 | <p>For a (well-behaved) one-dimensional function $f: [-\pi, \pi] \rightarrow \mathbb{R}$, we can use the Fourier series expansion to write
$$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n\sin(nx) \right)$$</p>
<p>For a function of two variables, Wikipedia lists the formula</p>
<p>$$f(x,y) = \su... | miguel747 | 64,953 | <p>@JohnD only details about the coefficients. The correct formula is: </p>
<p>$$c_{n,m} = \frac{\int_{0}^a \int_0^b f(x,y)sin(\frac{n\pi x}{a})sin(\frac{n\pi y}{b})dxdy}{{\int_{0}^a \int_0^b sin^2(\frac{n\pi x}{a})sin^2(\frac{n\pi y}{b})dxdy}{}}$$</p>
<p>the impression that $c_{n,m}$ is $1$. That's not true. Cheers!... |
211,689 | <p>For a (well-behaved) one-dimensional function $f: [-\pi, \pi] \rightarrow \mathbb{R}$, we can use the Fourier series expansion to write
$$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n\sin(nx) \right)$$</p>
<p>For a function of two variables, Wikipedia lists the formula</p>
<p>$$f(x,y) = \su... | patrick_langan | 322,353 | <p>The full real-valued 2D Fourier series is:
$$
\begin{align}
f(x, y) & = \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\alpha_{n,m}cos\left(\frac{2\pi n x}{\lambda_x}\right)cos\left(\frac{2\pi m y}{\lambda_y}\right) \\
& + \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\beta_{n,m}cos\left(\frac{2\pi n x}{\lambda_x}\right)sin\... |
80,078 | <p>Given $f$:</p>
<p>$$
f(x) = \begin{cases}
\frac1{x} - \frac1{e^x-1} & \text{if } x \neq 0 \\
\frac1{2} & \text{if } x = 0
\end{cases}
$$</p>
<p>I have to find $f'(0)$ using the definition of derivative (i.e., limits). I already know how to differentiate and stuff, b... | JavaMan | 6,491 | <p>This is just an exercise in persistence.</p>
<p>First note that</p>
<p>$$
\lim_{h \to 0}\frac{\frac{1}{h} - \frac{1}{e^h - 1} - \frac{1}{2}}{h} = \lim_{h \to 0} \frac{2(e^h - 1)-2h - h(e^h - 1)}{2h^2 (e^h - 1)}
$$</p>
<p>Applying L'Hopital's rule 4 or 5 times you end up with </p>
<p>$$
\lim_{h \to 0}\frac{-1... |
1,543,054 | <p>Every year the teacher write 4 tests with 6 questions,
from a list of 10 different questions, Is it certain that after 8 years, theres 3 different tests with the same 4 questions?</p>
<p>how do i show that with the pigeonhole principle?</p>
| turkeyhundt | 115,823 | <p>Here's a line of thinking to consider. I have no idea if it is correct.</p>
<p>8 years of tests is 32 tests. So we are looking at 32 tests, each with 6 questions. Is it possible for at most only 2 tests have the same set of 4 questions (4-tuples)?</p>
<p>Each test contains ${6\choose4}=15$ 4-tuples. So 32 tests... |
259,388 | <p>Let $A$ be a real $2\times 2$ matrix such that $\det A=1$, show that $\|A\|=\left\|A^{-1}\right\|$.</p>
<p>Any hint would be appreciated, thanks.</p>
<p>EDIT: $\|\cdot\|$ is the operator norm $\|A\|=\max_{\|x\|=1}\|Ax\|$, all vector norms are Euclidean norms.</p>
| Community | -1 | <p>Recall the singular value decomposition of $A$ as $A_{2 \times 2} = U \begin{bmatrix}\Sigma_{11} & 0\\ 0 & \Sigma_{22} \end{bmatrix}V^*$, then $A^{-1} = V^{*^{-1}} \begin{bmatrix}\dfrac1{\Sigma_{11}} & 0\\ 0 & \dfrac1{\Sigma_{22}} \end{bmatrix} U^{-1}$. Since $\det(A) = 1$, we have that $\Sigma_{11} ... |
2,743,099 | <p>So I have two points lets say <code>A(x1,y1)</code> and <code>B(x2,y2)</code>. I want to find a point <code>C</code> (there will be two points) in which if you connect the points you will have an equilateral triangle. I know that if I draw a circle from each point with radius of equal to <code>AB</code> I will find ... | Vasili | 469,083 | <p>Let $r=AB=\sqrt{((x_1-x_2)^2+(y_1-y_2)^2}$. The equation of the circle radius of $AB$ and $A$ as the center will be $(x-x_1)^2+(y-y_1)^2=r^2$. The equation of the circle radius of $AB$ and $B$ as the center will be $(x-x_2)^2+(y-y_2)^2=r^2$. <p>Thus you need to solve $(x-x_1)^2+(y-y_1)^2=(x-x_2)^2+(y-y_2)^2$. Simpli... |
1,805,431 | <blockquote>
<p>The simplest of the logical paradoxes is <em>Russell's paradox</em>, which can be described as follows: </p>
<p>Let $S$ denote the <em>set of all sets that are not elements of themselves</em>. Is $S$ an element of itself? </p>
<ul>
<li><p>Well, if $S$ is an element of $S$, then - by the v... | Mauro ALLEGRANZA | 108,274 | <p>NO: the statement</p>
<blockquote>
<p>"there exists the set $S$ of all sets that are not elements of themselves"</p>
</blockquote>
<p>has been proved to be <em>false</em>.</p>
<p>The "logical analysis" of <a href="http://plato.stanford.edu/entries/russell-paradox/" rel="nofollow">Russell's Paradox</a> starts fr... |
152,626 | <p>Is there any simple way of computing the following sum?</p>
<p>$$\sum_{k=1}^\infty \frac1{k\space k!}$$</p>
| Steven Stadnicki | 785 | <p>First of all, consider the power series for $e^x$, $\displaystyle\sum_{k=0}^{\infty}\frac{x^k}{k!}$. Now subtract off the constant term and divide by $x$: $\displaystyle{\frac{e^x-1}{x} = \sum_{k=1}^{\infty}\frac{x^{k-1}}{k!}}$. Now integrate: $\displaystyle{\int_0^x \frac{e^t-1}{t} dt = \sum_{k=1}^{\infty}\frac{x... |
2,644,610 | <p>How is it possible to prove that:
$$
|e^{ia}-e^{ib}|=2\sin\frac{|a-b|}{2}\leq|a-b|?
$$</p>
<p>Specifically, I'm looking for an analytic technique to show that the equality $|e^{ia}-e^{ib}|=2\sin\frac{|a-b|}{2}$ is correct.</p>
| Jack D'Aurizio | 44,121 | <p>$$\|e^{ia}-e^{ib}\|^2=(e^{ia}-e^{ib})(e^{-ia}-e^{-ib}) = 2-2\cos(a-b) = 4\sin^2\left(\frac{a-b}{2}\right)$$ </p>
<p>Due to the fact that $\sin$ is a Lipschitz-continuous, odd function with a derivative bounded by $1$, by assuming that $|a-b|$ does not exceed $2\pi$ we have:
$$ \|e^{ia}-e^{ib}\|=2\sin\left(\frac{|a-... |
29,143 | <p>In what context should I use $=$ and $\equiv$?</p>
<p>What is the precise difference?</p>
<p>Thanks!</p>
<p>(I wasn't sure what to tag this with, any suggestions?)</p>
| Zev Chonoles | 264 | <p>Use $=$ when you <strong>precisely</strong> mean that the two expressions refer to the same thing. For example, $2=1+1$ is the definition of 2, so the two sides really are the same thing. However, in advanced mathematics, people sometimes blur the use of $=$ to include isomorphic objects, e.g. $\mathbb{Z}$ and $\pi_... |
27,126 | <p>$$e^{\pi i} + 1 = 0$$</p>
<p>I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us? </p>
<p>Best that I can figure out is that it just emphasizes that the various definitions mathematicians have provided for non-intuitive operations (com... | vonjd | 1,047 | <p>Just wanted to include this excellent illustration of Euler's formula<br>(it really deserves to be shown here in its own right and not just as a link in one of the comments):</p>
<p><img src="https://upload.wikimedia.org/wikipedia/commons/e/e3/Euler%27s_Formula_c.png" alt="alt text"></p>
<p>Source: <a href="http:/... |
2,311,583 | <p>$$
\int_{0}^{1}\sqrt{\,1 + x^{4}\,}\,\,\mathrm{d}x
$$ </p>
<p>I used substitution of tanx=z but it was not fruitful. Then i used $ (x-1/x)= z$ and
$(x)^2-1/(x)^2=z $ but no helpful expression was derived.
I also used property $\int_0^a f(a-x)=\int_0^a f(x) $
Please help me out</p>
| pisco | 257,943 | <p>We can do better than hypergeometric function and elliptic integral:
$$\color{blue}{\int_0^1 {\sqrt {1 + {x^4}} dx} = \frac{{\sqrt 2 }}{3} + \frac{{{\Gamma ^2}(\frac{1}{4})}}{{12\sqrt \pi }}}$$</p>
<hr>
<p>Firstly, integration by part gives
$$\int_0^1 {\sqrt {1 + {x^4}} dx} = \sqrt 2 - 2\int_0^1 {\frac{{{x^4}}... |
2,311,583 | <p>$$
\int_{0}^{1}\sqrt{\,1 + x^{4}\,}\,\,\mathrm{d}x
$$ </p>
<p>I used substitution of tanx=z but it was not fruitful. Then i used $ (x-1/x)= z$ and
$(x)^2-1/(x)^2=z $ but no helpful expression was derived.
I also used property $\int_0^a f(a-x)=\int_0^a f(x) $
Please help me out</p>
| Claude Leibovici | 82,404 | <p>For an approximation, you could use a Padé approximant for the integrand. The simplest one would be
$$\frac{3 x^4+4}{x^4+4}=3-\frac{x+2}{x^2+2 x+2}+\frac{x-2}{x^2-2 x+2}$$
$$\int \frac{3 x^4+4}{x^4+4}\,dx=3x+\frac{1}{2} \log \left(\frac{x^2-2 x+2}{x^2+2 x+2}\right)+\tan ^{-1}(1-x)-\tan ^{-1}(1+x)$$ So, using the giv... |
26,651 | <p>Hi, everybody. I'm recently reading W.Bruns and J.Herzog's famous book-Cohen-Macaulay Rings. I personally believe that it would be perfect if the authors provide for readers more concrete examples. After reading the first two sections of this book, I have two questions.</p>
<ol>
<li><p>Given a non-negative integer ... | Hailong Dao | 2,083 | <p>Some interesting examples of Cohen-Macaulay but not Gorenstein rings:</p>
<p>1) Determinantal rings: Let <span class="math-container">$m\geq n\geq r>1$</span> be integers. Take <span class="math-container">$S=k[x_{ij}]$</span> with <span class="math-container">$1\leq i\leq m, 1\leq j\leq n$</span> and <span clas... |
53,073 | <p>Suppose $X$ is a non-explosive diffusion with dynamics</p>
<p>$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$, </p>
<p>where $W$ is a standard Brownian motion. My intuition about $X$ is that if $\mu$ and $\sigma$ are sufficiently nice, then the sample paths of $X$ are in some sense "deformed" sample paths of $W$. Is there a... | Ilya | 11,768 | <ol>
<li><p>If you want to have $X_t$ as a "deformed" $W_t$ - at first I advise to assume $\sigma\neq 0$ a.s. Otherwise you will have some problems (really in such points you may have almost deterministic dynamics).</p></li>
<li><p>If $\mu = 0$ then you can just change the time since all continuous martingales are tim... |
53,073 | <p>Suppose $X$ is a non-explosive diffusion with dynamics</p>
<p>$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$, </p>
<p>where $W$ is a standard Brownian motion. My intuition about $X$ is that if $\mu$ and $\sigma$ are sufficiently nice, then the sample paths of $X$ are in some sense "deformed" sample paths of $W$. Is there a... | Alekk | 1,590 | <p>As other have said, in the one dimensional case at least, you can suppose that the volatility is constant. Then the solution of the SDE is nothing else than a solution of the integral equation
$$X(t) = \int_0^t \mu(X_s) ds + \sigma W_t \qquad \forall t \in [0,T].$$
You can then check that if $\mu(\cdot)$ is a Lipsch... |
53,073 | <p>Suppose $X$ is a non-explosive diffusion with dynamics</p>
<p>$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$, </p>
<p>where $W$ is a standard Brownian motion. My intuition about $X$ is that if $\mu$ and $\sigma$ are sufficiently nice, then the sample paths of $X$ are in some sense "deformed" sample paths of $W$. Is there a... | Jason Swanson | 15,575 | <p>Suppose $\mu$ and $\sigma$ are sufficiently well-behaved so that we may define
$$B_t := - \int_0^t\frac{\mu(X_s)}{\sigma(X_s)}\,ds
+ \int_0^t \frac1{\sigma(X_s)}\,dX_s.$$
This should be the case, for example, if $\mu$ and $\sigma$ are globally Lipschitz with linear growth and $|\sigma|$ is bounded below. We th... |
267,552 | <p>While using <span class="math-container">$\LaTeX$</span>, there are two characters, \Re and \Im, that represent the real and imaginary parts of a complex number. These look like this:</p>
<p><a href="https://i.stack.imgur.com/0MKrr.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0MKrr.png" alt="en... | kirma | 3,056 | <p>Probably the closest are <code>\[GothicCapitalR]</code> and <code>\[GothicCapitalI]</code> which can be also typed as <kbd>Esc</kbd>goR<kbd>Esc</kbd> and <kbd>Esc</kbd>goI<kbd>Esc</kbd>. The <code>TeXForm</code>s of these are <span class="math-container">$\mathfrak{R}$</span> and <span class="math-container">$\mathf... |
561,648 | <p>this is the limit to evaluate: </p>
<p>$$\eqalign{
& \mathop {\lim }\limits_{n \to \infty } \root n \of {{a_1}^n + {a_2}^n + ...{a_k}^n} = \max \{ {a_1}...{a_k}\} \cr
& {a_1}...{a_k} \ge 0 \cr} $$</p>
<p>As far as I understand, $a_1..a_k$ is finite. right?<br>
Suppose it is, I'm clueless about the... | Daniel Fischer | 83,702 | <p>You should apply the polarization identity in the form</p>
<p>$$4(Ax,y) = (A(x+y),x+y) - (A(x-y),x-y) -i(A(x+iy),x+iy) + i(A(x-iy),x-iy).$$</p>
<p>Since you already know $(Az,z) = (z,Az)$ for all $z \in \mathcal{H}$, it is not difficult to deduce $A^\ast = A$ from that.</p>
|
121,645 | <p>I have a (presumably simple) Laplace Transform problem which I'm having trouble with:</p>
<p>$$\mathcal L\big\{t \sinh(4t)\big\} = ?$$</p>
<p>How would I go about solving this? Would you please show working if possible, or alternatively point me in the right direction regarding how to go about solving this?</p>
<... | anon | 11,763 | <ol>
<li>There is a generic formula to write $G(s)=\mathcal{L}\{tf(t)\}(s)$ in terms of $F(s)=\mathcal{L}\{f(t)\}(s)$; seeing it involves integration by parts. Have you covered this rule?</li>
<li>Hyperbolic sine is a difference of exponentials; can you find the Laplace transform of these?</li>
</ol>
|
450,228 | <p>Let $s$ be complex and $\zeta(s)$ the Riemann Zeta function. What is a series expansion for $\zeta(s)$ valid iff $\operatorname{Re}(s)>\frac{1}{2}$ ?</p>
<p>I want a series expansion such that $\zeta(s)=\sum_{n}^{\infty} f(n,s)$ where the $f(n,s)$ are standard functions without irrational constants.</p>
| DonAntonio | 31,254 | <p>Hints: </p>
<p>For Re$(s)>0\,\;,\;\;s\neq 1$ :</p>
<p>$$\zeta(s)=\frac 1{1-2^{1-s}}\sum_{n=1}^\infty \frac {(-1)^{n-1}}{n^s}$$</p>
<p>You must check (1) that the infintie series there converges in the wanted domain and (2) the poles of the factor multiplying the series ad in fact removable (or "fake poles", i... |
3,676,879 | <p>I'm trying to derive a reduction formula for the following integral:</p>
<blockquote>
<p><span class="math-container">$$I_n=\int _0^1 \left(1+x^2\right)^n \mathrm{d}x$$</span></p>
</blockquote>
<p>So far, I have tried applying integration by parts and have reached till:</p>
<p><span class="math-container">$$I_n... | sai-kartik | 736,802 | <p>Continuing the derived integral:
<span class="math-container">$$I_n=(1+x^2)^n\bigg|_0^1-2n\int_0^1x^2\cdot (1+x^2)^{n-1}\mathrm{d}x$$</span>
<span class="math-container">$$\Rightarrow I_n=2^n-2n\int_0^1(x^2+1-1)\cdot (1+x^2)^{n-1}\mathrm{d}x$$</span>
<span class="math-container">$$\Rightarrow I_n=2^n-2n\left[\int_0^... |
2,041,610 | <p>Let's say I have,</p>
<p><img src="https://i.stack.imgur.com/jRw96.jpg" alt="enter image description here"></p>
<p>Now I have to find the angle CBA. Given that we know just 26 given above.</p>
| super saiyan | 395,535 | <p>$\angle CAB=26$ and $AB$ is a diameter hence $\angle ACB=90$.</p>
<p>So $\angle CBA=180-(90+26)=64$.</p>
|
2,018,703 | <p>Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the property that $\lim_{x \rightarrow \infty} f(x)$ and $\lim_{x \rightarrow -\infty} f(x)$ exist and are equal. Prove that $\forall d > 0$ there exists $x_1, x_2 \in \mathbb{R}$ such that $x_1 - x_2 = d$ and $f(x_1) = f(x_2)$.</p>
<p>I a... | Christian Blatter | 1,303 | <p>Since $f$ is not assumed differentiable you cannot use Rolle's theorem; but the IVT is available, and does the job.</p>
<p>After substracting a suitable constant from $f$ (which is irrelevant for the problem at hand) we may assume that $\lim_{x\to-\infty}f(x)= \lim_{x\to\infty}f(x)=0$, and that there is an $a\in{\m... |
58,306 | <p>Let $X$ be a topological space, and let $\mathscr{F}, \mathscr{G}$ be sheaves of sets on $X$. It is well-known that a morphism $\varphi : \mathscr{F} \to \mathscr{G}$ is epic (in the category of sheaves on $X$) if and only if the induced map of stalks $\varphi_P : \mathscr{F}_P \to \mathscr{G}_P$ is surjective for e... | Akhil Mathew | 536 | <p>Ok, here is another answer. Let for $j$ an open immersion, $j_!$ be the "lower shriek" or "extension by zero" functor. Note that $j_!$ is left-adjoint to the functor $j^*$ of restriction from sheaves on $X$ to sheaves on $U$, and that there is a natural transformation $j_!j^* \to \mathrm{Id}$; note also that the sta... |
58,306 | <p>Let $X$ be a topological space, and let $\mathscr{F}, \mathscr{G}$ be sheaves of sets on $X$. It is well-known that a morphism $\varphi : \mathscr{F} \to \mathscr{G}$ is epic (in the category of sheaves on $X$) if and only if the induced map of stalks $\varphi_P : \mathscr{F}_P \to \mathscr{G}_P$ is surjective for e... | Grigory M | 152 | <p>The statement that section maps are not always surjective for surjective map of sheaves is equivalent to non-exactness of the functor of global sections — or equivalently, to non-triviality of sheaf cohomology.</p>
<p>Now it's easy to construct any number of explicit examples. Say, take $X=S^1$, $\mathcal F$ ... |
3,550,162 | <p>if a complex number is prime in Gauss integers, does it follow that its complex conjugate is also prime?</p>
<p>I know in general if a “regular” number divides <span class="math-container">$a+bi$</span>, it also divided <span class="math-container">$a-bi$</span> but can’t show the same for all cause integers. Irred... | Bart Michels | 43,288 | <p>Yes. Complex conjugation defines an automorphism of the ring <span class="math-container">$\mathbb Z[i]$</span>. The image of a prime element by a ring automorphism, is again a prime element.</p>
|
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | user52817 | 1,680 | <p>Bunny rabbit generations: look at the discrete case $P(n)=2^n$ with $P(0)=1$. It is easy to see that $\frac{\Delta P}{\Delta n}=P(n)$ where $\Delta P=P(n+1)-P(n)$ and $\Delta n=1$. Or more generally for $P(n)=b^n$ where $b>1$ we have $\frac{\Delta P}{\Delta n}=cP(n)$ where $c=b-1$.</p>
<p>This discrete case for ... |
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | Benoît Kloeckner | 187 | <p>As mentioned in other answers, the starting point is the definition of the exponential you are using.</p>
<p>One possible definition is as the solution of $f'=f$ taking value $1$ at $0$ (or other equivalent phrasings if you don't want to involve differential equation explicitly) and then there is nothing to explain... |
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | zvonimir šikić | 7,399 | <p>Geometricaly, ln(x) is the area from 1 to x under the graf of y=1/x.
Geometricaly, derivation of the area under f(x) is f(x).
Hence, d(ln(x))/dx = 1/x.
exp(x) is the inverse of ln(x) i.e. if y=ln(x) then x=exp(y).
Hence, if dy/dx=1/x then dx/dy=x i.e. the derivative of exp is exp. </p>
|
1,786,514 | <p>Let $S$ be a set containing $n$ elements and we select two subsets: $A$ and $B$ at random then the probability that $A \cup B$ = S and $A \cap B = \varnothing $ is?</p>
<p>My attempt</p>
<p>Total number of cases= $3^n$ as each element in set $S$ has three option: Go to $A$ or $B$ or to neither of $A$ or $B$</p>
<... | Clement C. | 75,808 | <p><em>Assuming you mean that $A$ and $B$ are picked independently and uniformly at random.</em></p>
<p><strong>Edit:</strong> below, I'm answering two different questions. I guess, after reading yours carefully, that what you intended is the second one.</p>
<p><strong>First interpretation of the question:</strong> "... |
1,776,177 | <p>Given the definite integral:</p>
<p>$$\int_{1}^{2}\left(x\sqrt{x+3}\right)\text{d}x$$</p>
<p>We can make the Power Substitution:
$$\begin{align}
u^2=&&x+3 \\
2u\text{d}u=&&\text {d}x
\end{align}$$</p>
<p>We get the following: (without the limits)</p>
<p>$$\int{\left(\left(u^2-3\right)\times u\tim... | André Nicolas | 6,312 | <p>It does not matter, as long as we are consistent. We can either integrate from $u=2$ to $u=\sqrt{5}$ or from $u=-2$ to $u=-\sqrt{5}$. </p>
|
3,463,970 | <p>I am trying to see if someone can help me understand the isomorphism between <span class="math-container">$V$</span> and <span class="math-container">$V''$</span> a bit more <strong>intuitively</strong>.</p>
<p>I understand that the dual space of <span class="math-container">$V$</span> is the set of linear maps fro... | Pedro | 70,305 | <p><em>How would you define <span class="math-container">$\varphi:V \rightarrow V''$</span> using the "maps to" symbol?</em></p>
<p>We can write
<span class="math-container">$$\begin{aligned}\varphi:V&\longrightarrow V''\\
v&\longmapsto\left( {\begin{aligned}
g_v:V'&\to\mathbb R\\
f&\mapsto f(v)
\end{a... |
3,463,970 | <p>I am trying to see if someone can help me understand the isomorphism between <span class="math-container">$V$</span> and <span class="math-container">$V''$</span> a bit more <strong>intuitively</strong>.</p>
<p>I understand that the dual space of <span class="math-container">$V$</span> is the set of linear maps fro... | Joppy | 431,940 | <p>A shorthand way to write some partially evaluated functions is by leaving a <span class="math-container">$-$</span> sign (pronounced “blank”) in the space of an argument. As an example, if <span class="math-container">$v \in \mathbb{R}^n$</span> and <span class="math-container">$\cdot$</span> is the dot product, we ... |
1,273,353 | <p>Suppose the Roulette table has 37 numbers (European Roulette table). During 37 spins, I always do the same bet: 35 numbers straight (35 chips in 35 different numbers).
Then:</p>
<ol>
<li>the probability of winning the 37 consecutive spins is $(\frac{35}{37})^{37}\approx 0.1279$,</li>
<li>the probability of losing ... | Olivier Oloa | 118,798 | <p>One may recall that, as $n \to +\infty$, by the <a href="https://math.stackexchange.com/questions/947219/equivalent-of-xx1x2-cdotsxn/952122#952122">generalized Stirling formula</a>, we have
$$
a(a+1)(a+2)\cdots(a+n) \sim \frac{n^{n+a}e^{-n}\sqrt{2\pi n} }{\Gamma(a)}
$$
giving $$
p_n = \frac{a(a+1)...(a+n)}{b(b+1)... |
3,418,811 | <p>Let <span class="math-container">$A$</span> be an integral, finitely-generated algebra over some field <span class="math-container">$k$</span>, of dimension <span class="math-container">$\text{dim}(A)\geq2$</span> such that <span class="math-container">$A = \cap_Q A_Q$</span> where <span class="math-container">$Q$</... | awllower | 6,792 | <p>We want to show that <span class="math-container">$\mathcal O_X(U)\hookrightarrow A_Q,\,\forall Q$</span> prime of height <span class="math-container">$1$</span>. We can show this by considering the elements of <span class="math-container">$\mathcal O_X(U)$</span> as follows.</p>
<p>The idea is to find an open cove... |
2,520,301 | <p>So there was an example in my textbook that explained how to show one spanning set is equal to another. However, while I do understand the algebra, I'm not sure why they are allowed to do certain things, or why they do them. </p>
<p>The example:</p>
<p>If x and y are in $R^{n}$ show that span{x, y} = span{x+y, x-y... | Mark Bennet | 2,906 | <p>It may help to look at this from another perspective.</p>
<p>Consider the span of $w$ and $z$ - it certainly contains all the vectors in the span of $\frac {w+z}2, \frac {w-z}2$ (we are allowed to divide by $2$) because elements in this span are clearly linear combinations of $w$ and $z$.</p>
<p>Now define $w=x+y$... |
1,579,579 | <p>I am struggling with showing that for algebraic number $\alpha$, the ring generated by $\mathbb{Q}[\alpha]$ is a field. I understand that to do this, I will have to show that any $r+s\alpha, r,s\in \mathbb{Q}$ has an inverse in $\mathbb{Q}[\alpha]$. I'm lost on how to go about doing this, though. Help? </p>
| lhf | 589 | <p>If $\alpha$ is an algebraic number, then $\mathbb{Q}[\alpha]$ is a finite-dimensional vector space over $\mathbb{Q}$.</p>
<p>The map $x \mapsto \alpha x$ is an injective linear transformation and so is surjective.</p>
<p>This means that $1$ is in the image and so $\alpha$ has an inverse in $\mathbb{Q}[\alpha]$.</p... |
3,252,737 | <p>Show that <span class="math-container">$$\int_{0}^{\pi/2} \sin^3x \cos^2 x \cos7x ~ dx= \frac{1}{60}$$</span>
I could solve this integral by making use of the special expansion <span class="math-container">$$\cos 7x=64 \cos^7 x- 112\cos^5 x + 56 \cos^3 x -7\cos x$$</span> and then using <span class="math-container"... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Use that <span class="math-container">$$\sin(x)^3\cos(x)^2\cos(7x)=\frac{1}{32} (\sin (2 x)-\sin (4 x)-2 \sin (6 x)+2 \sin (8 x)+\sin (10 x)-\sin (12 x))$$</span></p>
|
3,252,737 | <p>Show that <span class="math-container">$$\int_{0}^{\pi/2} \sin^3x \cos^2 x \cos7x ~ dx= \frac{1}{60}$$</span>
I could solve this integral by making use of the special expansion <span class="math-container">$$\cos 7x=64 \cos^7 x- 112\cos^5 x + 56 \cos^3 x -7\cos x$$</span> and then using <span class="math-container"... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p><span class="math-container">$$\sin3x=3\sin x-4\sin^3x,\cos2x=2\cos^2x-1$$</span></p>
<p><span class="math-container">$$\implies(4\sin^3x)(2\cos^2x)=(3\sin x-\sin3x)(1+\cos2x)=3\sin x-\sin3x+3\sin x\cos2x-\sin2x\cos2x$$</span></p>
<p>Use <a href="http://mathworld.wolfram.com/WernerFormulas.html" rel=... |
2,460,003 | <p>I need some help showing that these are equivalent. I made a couple attempts to get this right but so far the following work is as far as I've gotten.</p>
<p>Here is the question in its entirety:</p>
<blockquote>
<p>Let n be a natural number. Give a combinatorial proof of the following:
$\binom{2n+2}{n+1} = \b... | Community | -1 | <p>Left hand side is the coefficient of $x^{n+1}$ in $(1+x)^{2n+2}$
Now,
$$(1+x)^{2n+2} = (1+x)^{2n}(1+x)^2 = (1+x)^{2n}(1+2x+x^2)$$
The coefficient of $x^{n+1}$ on the right hand side is
$$\binom{2n}{n+1} + 2\binom{2n}{n} + \binom{2n}{n-1}$$
It follows that
$$\binom{2n+2}{n+1} = \binom{2n}{n+1} + 2\binom{2n}{n} + \b... |
2,760,982 | <p>Let $n\in\mathbb{Z}$. What are all solutions, for $z\in\mathbb{C}$, of $ z^n=\overline{z} $?</p>
<p>To solve it, I tryed to write the term in Polar-Form and than take the logarithm. Because of the "ln(r)" term I was unable to find a solution.</p>
<p>I also tryed to write $ z^n $ as a Binomial, but this was not hel... | Rhys Hughes | 487,658 | <p>In short, you're asking to solve:
$$r^ne^{ni\theta}=re^{-i\theta}$$
That is: $$r^{n-1}e^{(n+1)i\theta}=1+0i$$
$$\rightarrow r^{n-1}(\cos[(n+1)\theta]+i\sin[(n+1)\theta])=1$$
Thus $$r^{n-1}(1)=1$$, since when $\sin (x)=0, \cos(x)=\pm1$, and it can't be $-1$ here.</p>
<p>$\sin[(n+1)\theta]=0$ when $\cos[(n+1)\theta]=... |
372,045 | <p>I'm sure everyone already thought about this at least one time.
Why matrix multiplication is not defined the way showed below?</p>
<p>$$\left( \begin{array}{ccc}
a_{11} & a_{12} & \ldots \\
a_{21} & a_{22} & \ldots \\
\vdots & \vdots & \ddots
\end{array} \right) \cdot
\left( \begin{array}{cc... | Alex Provost | 59,556 | <p>The matrix multiplication we use is defined that way because it corresponds to the composition of linear maps. Recall that, given a vector space $V$ over $K$ with basis $(e_1,\ldots,e_n)$, and a vector space $W$ over $K$ with basis $(f_1,\ldots,f_m)$, we have a natural isomorphism $\eta:Hom_K(V,W) \rightarrow M_{m\t... |
1,372,985 | <p>On <a href="https://en.wikipedia.org/wiki/Annihilator_(ring_theory)#Properties" rel="nofollow">this Wikipedia article</a>, it says that you can define an $R$-module $M$ as an $R/Ann_R(M)$-module using the action $\overline{r}m:=rm.$ </p>
<p>What does that action actually mean? What is $\overline{r}$?</p>
| Michael Hardy | 11,667 | <p>That is mistaken. Let's look at the probability distribution of $X_2$ given $X_0=6$.</p>
<p>\begin{align}
& \Pr(X_2 = 6\mid X_0=6) \\[10pt]
= {} & \Pr\Big( \underbrace{(X_1 = 1\ \&\ X_2 = 6)\text{ or }(X_1=2\ \&\ X_2=6)\text{ or }\cdots}_{\text{five disjuncts}}\mid X_0=6\Big) \\[10pt]
= {} & \... |
1,184,501 | <p>I need help putting this in $0/0$ or $\infty/\infty$:</p>
<p>$$\lim_{x\to 0}{1\over xe^{x^{-2}}}$$</p>
<p>I've tried every possible combination, and I don't get what I'm missing. Using a graphic calculator, you easily see that the $\lim_{x\to 0}$ of this function is $0$.</p>
| DeepSea | 101,504 | <p><strong>Note</strong>: this answer is based on OP's first post. There have been several revisions of the original post. I think you mean $\displaystyle \lim_{x \to -\infty} \dfrac{1}{xe^{x^2}}$, and for this case is a little more challenging to you since the original limit does not exist. You can for this one write:... |
2,040,678 | <p>I have a confusion regarding the symmetry of the volume in the following question. </p>
<p>Find the volume common to the sphere $x^2+y^2+z^2=16$ and cylinder $x^2+y^2=4y$.</p>
<p>The author used polar coordinates $x=rcos\theta$ snd $y=rsin\theta$ and does something like this:</p>
<p>Required volume $V=4\int_0^{π/... | John | 7,163 | <blockquote>
<p>My point of confusion is that this solid cannot be cut into 4 identical parts</p>
</blockquote>
<p>Sure it can.</p>
<p>The sphere is centered at the origin, and the axis of the cylinder (which has radius $2$) intersects $(0,2)$ in the $x-y$ plane.</p>
<p>Slice it in half with the $z=0$ plane. Then... |
2,040,678 | <p>I have a confusion regarding the symmetry of the volume in the following question. </p>
<p>Find the volume common to the sphere $x^2+y^2+z^2=16$ and cylinder $x^2+y^2=4y$.</p>
<p>The author used polar coordinates $x=rcos\theta$ snd $y=rsin\theta$ and does something like this:</p>
<p>Required volume $V=4\int_0^{π/... | Erik M | 42,176 | <p>A picture is useful here. The solid can indeed be split into four symmetric segments, as shown in the following surface plot.</p>
<p><a href="https://i.stack.imgur.com/ASWk3.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ASWk3.png" alt="enter image description here"></a></p>
|
1,832,512 | <p>How can you find the inverse of $f(x) = 2x^2+8x+13?$ This is what I've tried so far:</p>
<ol>
<li>$y = 2x^2+8x+13$</li>
<li>$x = 2y^2+8y+13$</li>
<li>$x-13 = 2y^2+8y$</li>
<li>$x-13=y(y+8)$</li>
</ol>
<p>This is where I got stuck. To be clear, I want to write $x$ in terms of $y.$ <strong>Credit to Jenna</strong> f... | Tsemo Aristide | 280,301 | <p>You can't find the inverse of a degree 2 polynomial defined on $R$ since it is not injective.</p>
<p>If you want an inverse on invervals, write $2x^2+8x+13=y$, $2x^2+8x+13-y=0$, $\Delta = 64-8(13-y)$ and $x={{-8+\sqrt{8(13-y)}}\over 4}$ or $x={{-8-\sqrt{8(13-y)}}\over 4}$</p>
|
1,158,601 | <p><img src="https://i.stack.imgur.com/P3crI.jpg" alt="enter image description here"></p>
<p>The solution is $x=50^{\circ}$.</p>
<p>How to prove $x=50^{\circ}$ without trigonometry?</p>
| Blue | 409 | <p><img src="https://i.stack.imgur.com/ZJNUn.png" alt="enter image description here"></p>
<p>Find $K$ on $\overline{BD}$ such that $\angle KAB = 10^\circ$. Then $\angle AKB = 160^\circ = 2\cdot\angle ACB$; this makes $K$ the center of the circumcircle of $\triangle ABC$. Thus, $\overline{KB}\cong\overline{KC}$, whereu... |
1,896,546 | <p>I have a set of 3D points (I'll call them "outer" points) and a 2D point ("inner" point).</p>
<p>I need to quickly calculate a "good" third coordinate for the inner point so that it would place the constructed 3D point as "close" to the outer points as possible. "Close" may be defined as a minimum sum of distances ... | lexicore | 140,512 | <p>Here's a solution I came up with so far.</p>
<p>Assume we have a number of more than 3D points $(x_i,y_i,z_i)$ and a 2D point $(x,y)$ we need to calculate $z$ for.</p>
<p>The idea is that the closer $(x,y)$ is to $(x_i,y_i)$, the closer $z$ should be to $z_i$.</p>
<p>So for each of the points $(x_i,y_i,z_i)$ we'l... |
1,031,038 | <p>"The price of a train-ticket is 110 dollars for grown-ups and 90 dollars for children. To a train, 120 tickets were sold for a total of 11640 dollars. How many grown-ups bought a ticket to the train?"</p>
<p>So here's my thought process:</p>
<p>$11640/120 = 97$</p>
<p>So now I have the mean value, but I have no i... | Mike Pierce | 167,197 | <p>We can set up a system of equations from the information provided:
$$
\begin{align}
g + c &= 120 \\
110g + 90c &= 11640
\end{align}
$$
The first equation is concerning the total number of tickest (where $g$ and $c$ represent the number of grown-up tickets and child tickets sold respectively).
The second equa... |
1,031,038 | <p>"The price of a train-ticket is 110 dollars for grown-ups and 90 dollars for children. To a train, 120 tickets were sold for a total of 11640 dollars. How many grown-ups bought a ticket to the train?"</p>
<p>So here's my thought process:</p>
<p>$11640/120 = 97$</p>
<p>So now I have the mean value, but I have no i... | Holden Rohrer | 194,193 | <p>You're going about it the wrong way. Use elimination in a system of equations.</p>
<pre><code> g + c = 120
110g + 90c = 11640
</code></pre>
<p>Then, you can turn the top equation negative, and simply forget about the reality of it.</p>
<pre><code> ... |
193,053 | <p>Assume that </p>
<p>$$T(n) = 2T\left(\frac{n}{2}\right) + \Theta(n \log n)$$</p>
<p>By <a href="http://en.wikipedia.org/wiki/Master_theorem#Generic_form_2" rel="nofollow">Generic form of master theorem</a> with $a = 2$, $b = 2$ and $f(n) = c \, n \log n$, it can easily be proved that $T(n) = \Theta(n \log^2 n)$.<... | zyx | 14,120 | <p>Here you can consider $S(n)=T(n)/cn$ for which </p>
<p>$S(n) = S(n/2) + \log (n)$</p>
<p>$k$ iterations of this give</p>
<p>$S(n) = S(n/{2^k}) + k \log 2 (\log_2 n - \frac{(k-1)}{2})$</p>
<p>Taking $k = \log_2 n + O(1)$ and performing some algebra, the answer is:</p>
<p>$T(n) = (\frac{c \log 2}{2}) n \log_2^2 n... |
924,551 | <p>$$\displaystyle \lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}}$$ </p>
<p>I tried rationalizing the numerator: </p>
<p>$$\lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}} \times \dfrac{(8-\sqrt{x})}{(8-\sqrt{x})}$$ </p>
<p>$$\lim_{x \to \infty}\dfrac{64-16\sqrt{x}+x}{64-x}$$</p>
<p>Is this correct? how do I p... | Adriano | 76,987 | <p>Try dividing each term by $\sqrt{x}$ instead. Intuitively, it is the dominating term as $x$ gets large:
$$
\lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}}
= \lim_{x \to \infty}\dfrac{\frac{8}{\sqrt x}-1}{\frac{8}{\sqrt x}+1}
= \dfrac{0-1}{0+1}
= -1
$$</p>
|
581,497 | <p>Case $1$: $4$ games: Team A wins first $4$ games, team B wins none = $\binom{4}{4}\binom{4}{0}$</p>
<p>Case $2$: $5$ games: Team A wins $4$ games, team B wins one = $\binom{5}{4}\binom{5}{1}-1$...minus $1$ for the possibility of team A winning the first four.</p>
<p>Case $3$: $6$ games: Team A wins $4$ games, team... | André Nicolas | 6,312 | <p>We count the ways in which Team A can win the series, and double the result. To count the ways A can win the series, we make a list like yours.</p>
<p>A wins in $4$: There is $1$ way this can happen.</p>
<p>A wins in $5$: A has to win $3$ of the first $4$, and then win. There are $\binom{4}{3}$ ways this can happe... |
530,301 | <p>There are two poles (lets say poles A and B) $50$ feet apart and the poles are $15$ and $30$ feet tall. There is a wire which runs from the top of pole A to the ground, and then to the top of pole B so that two triangles are made. What is the minimum length of wire needed to set up this configuration? </p>
<p>I tri... | DannyDan | 101,452 | <p>what the diagram is saying that since the wire need to hit the ground, the shortest way from one pole to another is equivalent to the second pole being mirrored underground. Then the shortest distance between two points is a straight line.</p>
<p>Look at the drawing that Kastar made. It is accurate.<br>
You have a ... |
1,107,013 | <p>Suppose that $f$ is a differentiable real function in an open set $E \subset \mathbb{R^n}$, and that $f$ has a local maximum at a point $x \in E$. Prove that $f'(x)=0$</p>
| R Leitch | 208,387 | <p>Your simplification is wrong.
I think you'll find the right hand side of your equation is bounded above and below, and if you draw the graph, it crosses the graph of 2cos(2x) an infinite number of times!</p>
|
731,764 | <p>I understand what the epsilon-delta definition is saying in regards to the distance from a point c and the distance from your limit, but I don't understand how this defines a limit. Any help is appreciated.
Thanks!</p>
| DanielV | 97,045 | <p>To add to what user99680 said with an example-</p>
<p>The delta epsilon definition of a declaration of the form "if P(F, L, X) is true, then L is the limit of F at X". It is not a universal equivalence, so it leaves many cases undefined.</p>
<p>One of the most prominent examples is that of an infinite series. A... |
3,472,358 | <p>For the function <span class="math-container">$f \in L^2 ([-\pi,\pi])$</span> define the map <span class="math-container">$ T: L^2([-\pi, \pi]) \to R $</span> as <span class="math-container">$T(f)=a_1+b_1 $</span> if the Fourier series of <span class="math-container">$f$</span> is of the form </p>
<p><span class="... | José Carlos Santos | 446,262 | <p>Since<span class="math-container">$$a_1=\frac1\pi\int_{-\pi}^\pi f(t)\cos(t)\,\mathrm dt=\frac1\pi\langle f,\cos\rangle,$$</span>the map <span class="math-container">$f\mapsto a_1$</span> is continuous. A similar argument shows that <span class="math-container">$f\mapsto b_1$</span> is continuous too and so your map... |
3,472,358 | <p>For the function <span class="math-container">$f \in L^2 ([-\pi,\pi])$</span> define the map <span class="math-container">$ T: L^2([-\pi, \pi]) \to R $</span> as <span class="math-container">$T(f)=a_1+b_1 $</span> if the Fourier series of <span class="math-container">$f$</span> is of the form </p>
<p><span class="... | Marios Gretsas | 359,315 | <p>By Parseval's Theorem we have that <span class="math-container">$$\sum_{n \in \Bbb{Z}}|\hat{f}(n)|^2=||f||_2^2$$</span> or <span class="math-container">$$\frac{a_0^2}{2}+\sum_{n=1}^{\infty}a_n^2+b_n^2=||f||_2^2$$</span></p>
<p>So <span class="math-container">$|T(f)|^2 \leq 2(|a_1|^2+|b_1|^2) \leq \frac{2a_0^2}{2}+2... |
1,478,103 | <p>I'm looking for examples of subtle errors in reasoning in a mathematical proof. An example of a 'false' proof would be</p>
<blockquote>
<p>Let $a=b>0$. Then $a^2 - b^2 = ab - b^2$. Factoring, we have $(a-b)(a+b) = b(a-b)$, which after cancellation yields $b = a = 2b$ and thus $1=2$. </p>
</blockquote>
<p>Howe... | Soham | 242,402 | <p>$\cos^2x=1-\sin^2x$</p>
<p>$\cos x = (1-\sin^2x)^{\frac{1}{2}}$</p>
<p>$1+\cos x = 1+(1-\sin^2x)^{\frac{1}{2}}.$</p>
<p>$When,x=180^0$</p>
<p>$1-1 = 1+(1-0)^{\frac{1}{2}}$ $as$$,$$[$$cos 180=-1$ $and$ $sin 180=0$$]$</p>
<p>$or,$$0=2$</p>
|
299,304 | <p>I had posted the following problem on <a href="https://math.stackexchange.com/questions/2722893/the-modulus-of-a-polynomial-are-the-same-is-1">stack exchange</a> before.</p>
<blockquote>
<p>Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of t... | Julienne Franz | 124,188 | <p>In [<em>Uber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise</em>, Math. Z., <strong>14</strong>, 1922, 110-148], A. Cohn established the following result:</p>
<blockquote>
<p>Let <span class="math-container">$p_n(z) = \sum_{k=0}^n a_k z^k$</span> be a polynomial of degree <span class="math-cont... |
1,639,156 | <p>Use logical quantifiers to write:
"Everybody loves somebody sometimes" (Where U=all people)
I came up with this but not sure how to type symbols in here.</p>
<p>$$\forall x \in U\,: \exists y\in U: x \text{ loves } y.$$</p>
<p>So... upside down A="For all"
Backwards E for "there exists"
curly little e for "belongs... | Dan Christensen | 3,515 | <p>With apologies to Dino, that sentence <em>is</em> a bit ambiguous. Here is one way to look at it:</p>
<p>$\forall x:[P(x) \implies \exists y: \exists t:[P(y) \land T(t) \land L(x,y,t)]]$</p>
<p>where</p>
<p>$P(x)$ means $x$ is a person</p>
<p>$T(t)$ means $t$ is an instant in time</p>
<p>$L(x,y,t)$ means $x$ lo... |
934,050 | <p>Prove that if $a$ divides $x^n-1$ and $x^m-1$, then $a$ is coprime with $x$.</p>
<p>I think this should be easy but I can't think of a way to do it.</p>
| André Nicolas | 6,312 | <p>Assume that $n\ge 1$. Let $d$ be a common divisor of $a$ and $x$. </p>
<p>(i) Since $a$ divides $x^n-1$, it follows that $d$ divides $x^n-1$. </p>
<p>(ii) Since $d$ divides $x$, it follows that $d$ divides $x^n$. </p>
<p>From (i) and (ii), we conclude that $d$ divides $x^n-(x^n-1)$, and therefore $d$ divides $1$... |
934,050 | <p>Prove that if $a$ divides $x^n-1$ and $x^m-1$, then $a$ is coprime with $x$.</p>
<p>I think this should be easy but I can't think of a way to do it.</p>
| Sheheryar Zaidi | 131,709 | <p>As a matter fact, I fiddled around with the problem and got this: </p>
<p>Let $m>n$ then if $a$ divides $x^m-1$ and $x^n-1$, it divides $(x^m-1)-(x^n-1) = x^m-x^n = x^n(x^{m-n} - 1)$ Now if $a$ divides $x^n$ then clearly they're not coprime. Let's assume it divides $x^{m-n}-1$ instead, then it must divide $(x^{... |
1,365,268 | <p>Part A is in the title, Part B is here:
Is it true that $(k, n+k)= d$ if and only if $(k, n)=d$?</p>
<p>I am still working on the Part A. </p>
<p>What I have so far:</p>
<p>if $(k, n)= 1$ then $1|k$, $1|n$ and $1|(n-k)$</p>
<p>if $(k, n+k)=1$ then $1|k$, $1|n+k$ and $1|((n+k)- k) \to 1|n$</p>
<p>I was under the... | Vlad | 229,317 | <p>I have already posted an answer where I considered asymptotic behavior of inequality as $n\to \infty$.
Below are <em>more general</em> conclusions.</p>
<hr>
<blockquote>
<p>Given $0\le x \le n$, show that $\left(1 + \dfrac{x}{n}\right)^{-n} \le 2^{-x}$.</p>
</blockquote>
<p><strong>First</strong>, note that
... |
4,377,390 | <p>The famous German physicist Walter Schottky (1986-1976), in a publication on "thermal agitation of electricity in conductors" in the 1920ies, calculated the integral <span class="math-container">$\int_{0}^{\infty} \frac{1}{(1-x^2)^2+r^2 x^2}\;dx$</span> to be <span class="math-container">$\frac{2\pi}{r^2}$... | Lai | 732,917 | <p>This is really a math question about an improper integral with answer <span class="math-container">$\dfrac{\pi}{2 r}.$</span> Therefore Mr Schottky’s answer is wrong. I have a wonderful trick to handle the integral for <span class="math-container">$0<r<2$</span>. <span class="math-container">$$
\begin{aligned}... |
3,381,219 | <p><a href="https://i.stack.imgur.com/mM0OF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mM0OF.png" alt="3B) in the picture"></a>What is an example of an infinite intersection of infinite sets is infinite?</p>
<p>I know that the intersection of infinite sets does not need to be infinite. However,... | nicomezi | 316,579 | <p>Let <span class="math-container">$x_k = 2k$</span> and define <span class="math-container">$A_n = \{ \mathbb{N} \setminus \{x_k\}_{k=1}^n \}$</span>.</p>
<p><span class="math-container">$$\bigcap_{n=1}^\infty A_n = \text{set of odd natural integers}$$</span></p>
|
1,866,639 | <p>Let $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ be a function which is Gâteaux differentiable and let $J_f\in\mathbb{R}^{M\times N}$ be its Jacobian matrix.</p>
<p>Is it true that the Gâteaux derivative of $f$ along a direction $v\in\mathbb{R}^N$ is equal to the matrix-vector product $J_f \cdot v$?</p>
| Chill2Macht | 327,486 | <p>As you probably already know, this is true if $f$ is Frechet differentiable.</p>
<p>However, there is no reason why this should be true if $f$ is only Gateaux differentiable but not Frechet differentiable, because then the directional derivatives need not be linearly related to each other, i.e. as is stated in Wiki... |
3,117,260 | <p>So according to the commutative property for multiplication:</p>
<p><span class="math-container">$a \times b = b \times a$</span> </p>
<p>However this does not hold for division</p>
<p><span class="math-container">$a \div b \neq b \div a$</span> </p>
<p>Why is it that in the following case:</p>
<p><span class="... | Rhys Hughes | 487,658 | <p>Because division is the inverse of multiplication, that is: <span class="math-container">$$X \div Y =X\cdot \frac 1Y$$</span></p>
<p>So you have: <span class="math-container">$$56\cdot 100 \cdot \frac18 =56\cdot\frac18\cdot100$$</span>
Which is obvious.</p>
|
50,209 | <p>Over the past few days I have been pondering about this: I enjoy technical things (like programming and stuff) and try to find the patterns and algorithms in everything. My life is number oriented. I'll spend all day working on a programmatic problem. I'll spend however much time is needed to think of an elegant/eff... | Michael Hardy | 11,667 | <p>In recent years teachers in the USA have been under heavy pressure to teach only what can be tested on standardized tests. That has been corrupting the education system to the point where teachers orchestrate widespread cheating on standardized tests to get federal funding for the school.</p>
<p>There's also the f... |
50,209 | <p>Over the past few days I have been pondering about this: I enjoy technical things (like programming and stuff) and try to find the patterns and algorithms in everything. My life is number oriented. I'll spend all day working on a programmatic problem. I'll spend however much time is needed to think of an elegant/eff... | Community | -1 | <p>When I was in middle and high [public] school, I used to carry around a notebook and do all sorts of little calculations and things. One of my proudest moments is when I figured out the 45-45-90 triangle rule (and, of course, in retrospect this is kind of silly but at the time it felt amazing). But (and here is the... |
2,270,346 | <p>I have a pretty simple question here it looks like but I just can't seem to do it. I'd like to be able to do it the easiest way possible. </p>
<blockquote>
<p>Solve $\dot{x}=y$ and $\dot{y}=x$ for $x(t)$ and $y(t)$.</p>
</blockquote>
<p>I need solve these two equations so I can draw a phase plane portrait and s... | Community | -1 | <p>So you want to solve </p>
<p>$$\begin{cases}x' = y \quad (1)\\y' = x\quad (2)\end{cases}$$</p>
<p>where both $x,y$ are functions of $t$.</p>
<p>Differentiate both sides of $(1)$, obtaining:</p>
<p>$x'' = y'$</p>
<p>Then, use $(2)$ to find $x'' = x$ or $x'' - x = 0$</p>
<p>We search the roots of the associated ... |
2,894,807 | <p>Please, does anyone know of a algorithm to compute the integer part $n$ of natural logarithm of an integer $x$?</p>
<p>$$n = \lfloor \ln(x) \rfloor$$</p>
<p>Preferably using integer arithmetic only (akin to <a href="https://en.wikipedia.org/wiki/Integer_square_root#Algorithm_using_Newton's_method" rel="nofollo... | Community | -1 | <p>$$ \ln(2^n a + b) = n \ln(2) + \ln(a) + \ln\left(1 + \frac{b}{2^n a} \right) $$</p>
<p>Assuming $b < 2^n a$, you have</p>
<p>$$ n \ln(2) + \ln(a) \leq \ln(2^n a + b)
< n \ln(2) + \ln(a) + \frac{b}{2^n a} $$</p>
<p>Unless you are in an unlucky situation where $n \ln(2) + \ln(a)$ is extremely close to an inte... |
1,087,107 | <p>In <a href="http://raudys.com/kursas/Options,%20Futures%20and%20Other%20Derivatives%207th%20John%20Hull.pdf" rel="nofollow">Hull (2008, p. 307)</a>, the following equation is found (Eq. 13A.2):</p>
<p>$$E[\max(V-K,0)]=\int_{K}^{\infty} (V-K)g(V)\:dV$$</p>
<p>Where $g(V)$ is the PDF of $V$ and $V,K>0$.</p>
<p>I... | Jason | 130,776 | <p>Let $Z=\max(X,Y)$ and $f(x,y)$ be the joint PDF of $(X,Y)$ (not assuming independence). Then,
\begin{align*}
Pr[Z<z] &= Pr[\max(X,Y)<z] \\
&= Pr[X<z \text{ and } Y<z] \\
&= \int_{-\infty}^z \int_{-\infty}^z f(x,y)\,dx\,dy \\
&= F(z)
\end{align*}</p>
<p>With the CDF function $F(z)$, we ma... |
1,087,107 | <p>In <a href="http://raudys.com/kursas/Options,%20Futures%20and%20Other%20Derivatives%207th%20John%20Hull.pdf" rel="nofollow">Hull (2008, p. 307)</a>, the following equation is found (Eq. 13A.2):</p>
<p>$$E[\max(V-K,0)]=\int_{K}^{\infty} (V-K)g(V)\:dV$$</p>
<p>Where $g(V)$ is the PDF of $V$ and $V,K>0$.</p>
<p>I... | Math1000 | 38,584 | <p>If <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are random variables defined on a probability space <span class="math-container">$(\Omega,\mathcal F,\mathbb P)$</span>, it is not hard to prove that <span class="math-container">$W:=\max\{X,Y\}$</span> is also a random variable. ... |
78,414 | <p>why is $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I am studying for a differential equation test and I seem to always forget \this, and i am just wondering if there is some intuition i'm missing, or is it just one of those things to memorize? and i know this is not very differential equation related, just one of those things ... | Henry | 6,460 | <p>I am with Henning Makholm on this and produce a sketch</p>
<p><img src="https://i.stack.imgur.com/ZIbtL.png" alt="enter image description here"></p>
<p>The slope of the blue line is the red line, and the slope of the red line is the green line. I know the blue line is sine and the red line is cosine; the green li... |
612,468 | <p>I've been using "conditional random variables" as a notation aid with some good success in problem solving. But I've heard people claim that one shouldn't define conditional random variables.</p>
<p>By a conditional random variable for $X$ given $Y$, a "pseudo" random variable $(X|Y)$ with the density function $f_... | Nick | 338,136 | <p>In the paper (<a href="https://www.sciencedirect.com/science/article/abs/pii/S0888327005001512" rel="nofollow noreferrer">Jardin et al., 2006</a>) the next notation was used:</p>
<blockquote>
<p>It is defined as the conditional random variable:
<span class="math-container">$$T - t|T>t, Z(t),$$</span>
where <span ... |
2,906,832 | <p>I want to write $\csc$ and $\tan$ and terms of classical trigonometric functions like $\sin$ and $\cos$. I know about the identity $\sin(x)^2+\cos(x)^2=1$. But I am not sure where to go from here. </p>
| PackSciences | 588,260 | <p>$\csc x = \frac{1}{\sin x}$ and $\tan x = \frac{\sin x}{\cos x}$ by definition.</p>
|
410,411 | <p>Let $f(x) = x^4 + 1 \in \mathbb{Q}[x]$. We can show that if $\alpha$ is a zero of $f(x)$, then the full set of zeros is given by $\{\alpha, -\alpha, i\alpha, -i\alpha\}$. Since $\alpha^2 = \pm i$ we can easily see that $L = \mathbb{Q}(\alpha)$ is the splitting field of $f$ over $\mathbb{Q}$. Since $L$ is the splitti... | DonAntonio | 31,254 | <p>Trying to apply the nice comments below the post:</p>
<p>$$\Bbb Q(\alpha)=\text{Span}_{\Bbb Q}\{\,1\,,\,\alpha\,,\,\alpha^2=i\,,\,\alpha^3=i\alpha\,\}$$</p>
<p>where in fact</p>
<p>$$\alpha=\frac1{\sqrt 2}(1+i)\;,\;\alpha^2=i\;,\;\;\alpha^3=\frac1{\sqrt 2}(-1+i)$$</p>
<p>Let $\,x:=a+b\alpha+ci+d\alpha i\in\Bbb Q... |
4,418,091 | <p>Is <span class="math-container">$\mathbb Q-\mathbb N$</span> dense in <span class="math-container">$\mathbb R$</span>? I believe that the solution is about two relatively prime integers. However, I do not know how to proceed.</p>
| Thomas Bakx | 545,960 | <p>Here is a topological perspective:</p>
<p>If <span class="math-container">$X,Y \subset Z$</span> and <span class="math-container">$X$</span> is discrete in <span class="math-container">$Z$</span> while <span class="math-container">$Y$</span> is dense in <span class="math-container">$Z$</span>, then <span class="math... |
4,418,091 | <p>Is <span class="math-container">$\mathbb Q-\mathbb N$</span> dense in <span class="math-container">$\mathbb R$</span>? I believe that the solution is about two relatively prime integers. However, I do not know how to proceed.</p>
| eyeballfrog | 395,748 | <p>Let's try to find a noninteger rational between any two reals. For any real numbers <span class="math-container">$a <b$</span>, since <span class="math-container">$\mathbb Q$</span> is dense in <span class="math-container">$\mathbb R$</span>, there is <span class="math-container">$p\in(a,b)\cap \mathbb Q$</span>.... |
157,301 | <p>Here is the limit I'm trying to find out:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)}$$</span></p>
<p>Since it is an indeterminate form, I simply applied l'Hopital's Rule and I ended up with:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)} = \... | qoqosz | 31,604 | <p>Use $\lim_{x \to 0} \frac{\sin x}{x} = 1$:</p>
<p>$$\lim_{x \to 0} \frac{x^3}{\tan^3 2x} = \lim_{x\to 0} \left( \frac{(2x)^3}{\sin^3 2x} \cdot \frac{\cos^3 2x}{8} \right) \stackrel{[1 \cdot \frac{1}{8}]}{=} \frac{1}{8}$$</p>
|
3,251,589 | <p><span class="math-container">$$ \lim_{x \to 1} \frac{x^3-1}{x-1}=3
$$</span>
How to prove it using precise definition of limits? While solving it, I get stuck at |(x-1)(x+2)|<ε. I don't know how to take out the inequality for x only as this inequality contains quadratic form.</p>
| José Carlos Santos | 446,262 | <p>Note that<span class="math-container">\begin{align}\frac{x^3-1}{x-1}-3&=x^2+x+1-3\\&=x^2+x-2\\&=x^2-1+x-1\\&=(x-1)(x+1)+x-1.\end{align}</span>Now, take <span class="math-container">$\varepsilon>0$</span>. If <span class="math-container">$\lvert x-1\rvert<1$</span>, then <span class="math-contai... |
729,101 | <p>Given $ n $ points in $ \mathbb{R}^2 $, does there exist a polynomial function of any degree, whose extremas include these $ n $ points?</p>
<p>Given 3 points:
$
P_1 = (0,4), P_2 = (2,2), P_3 = (4,7)
$</p>
<p>And some function $f(x)$, for which the following holds:
$$
P_1, P_2, P_3 \in f(x)
\\
f'(0) = f'(2) = f'(4... | Jason Zimba | 132,296 | <p>Part A: You have to check $S$ against the definition of a subspace. </p>
<p>That means: </p>
<p>(1) Is the zero vector in $S$? </p>
<p>(2) For any two vectors $p$ and $q$ in $S$, is their sum in $S$? </p>
<p>(3) For any vector $p$ in $S$, is $cp$ in $S$ for any scalar $c$?</p>
<p>$S$ is a subspace if and only i... |
4,364,370 | <p>For <span class="math-container">$$\Phi(x) = (2\pi)^{-\frac 12}\int_{-\infty}^x \mathrm{e}^{-t^2/2}\,\mathrm{d}t,$$</span> it is claimed in the proof of Lemma 8.12 of <a href="https://www.hse.ru/data/2016/11/24/1113029206/Concentration%20inequalities.pdf" rel="nofollow noreferrer">this book</a> that we have the asym... | Dominik Kutek | 601,852 | <p>Note that <span class="math-container">$-\ln\left( \Phi(x)\right) \sim \frac{x^2}{2}$</span> as <span class="math-container">$x \to -\infty$</span> is equivalent with <span class="math-container">$\ln \left( 1 - \Phi(t) \right) \sim -\frac{t^2}{2}$</span> as <span class="math-container">$t \to +\infty$</span>, due t... |
1,376,392 | <p>Let $(\Omega,\mathcal{F},\mathbf{P})$ denote a probability space, $(S,\mathcal{M})$ denote a measurable space, and $X : (\Omega,\mathcal{F},\mathbf{P}) \rightarrow (S,\mathcal{M})$ denote a measurable function (thought of as a random variable). Then there is a pushforward measure induced on $(S,\mathcal{M})$ (though... | yoyo | 6,925 | <p>Given a measureable $f:(X,\mathcal{M},\mu)\to(Y,\mathcal{N})$ (a measure space to a measurable space) one can define a measure $\nu=f_{*}\mu$ on $(Y,\mathcal{N})$ by
$$
\nu(B)=\mu(f^{-1}(B)).
$$
Both $f$ and $\mu$ are necessary to define it, so $f_{*}\mu$ seems like good notation to me.</p>
|
3,464,295 | <p>I thought that QR algorithm decomposes a matrix into an orthogonal matrix Q and a upper triangular matrix R using GramSchmidth process for singular matrices but, what is meant by Explicit and Implicit QR algorithms? and how will they help in decomposing a non-singular matrix?</p>
| Manuel Moreno | 66,296 | <p>The classic QR algorithm iteration:</p>
<ol>
<li><p><span class="math-container">$QR = A $</span> ........decomposition</p>
</li>
<li><p><span class="math-container">$A' = RQ $</span></p>
</li>
</ol>
<p>due <span class="math-container">$Q$</span> is orthogonal, is also true:</p>
<ol start="2">
<li><p><span class="ma... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.