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 |
|---|---|---|---|---|
2,491,448 | <p>We roll a die ten times. What's the probability of getting all the six different numbers of the dice?</p>
<p>If $A_i$ is the event of getting at least of the numbers from $i=1$ to $6$. What the problem is asking is $P(A_1 \cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$ . So I guess I will get this probability if I ... | Zhuoran He | 485,692 | <p>The problem can also be solved using classification. Let $n_1,n_2,\ldots,n_6$ be the number of dices showing $1, 2,\ldots,6$, respectively. Define the set</p>
<p>$$A_{n,r}=\{(n_1,\ldots,n_6)_{\,}|_{\,}n_1\geq r,\cdots,n_6\geq r, n_1+\cdots+n_6=n\}.$$</p>
<p>Then the number of ways to let all numbers $1,2,\ldots, 6... |
2,801,162 | <p>I need help to understand how we compute this kind of limit:</p>
<p>$\lim_{(x,y)\rightarrow (0,0)}\ xy\log(\lvert x\rvert+\lvert y\rvert)$</p>
<p>I think we can use the squeeze theorem but I don't know how to bound the function, so I can use the theorem. If I suppose $0 \lt \sqrt{x^2+y^2} \lt 1$ then, but I'm stru... | user | 505,767 | <p><strong>HINT</strong></p>
<p>We have that</p>
<p>$$xy\log(\lvert x\rvert+\lvert y\rvert)=\frac{xy}{|x|+|y|}\cdot (|x|+|y|)\log(\lvert x\rvert+\lvert y\rvert)$$</p>
<p>then recall that as $u\to 0$</p>
<ul>
<li>$u\log u \to 0$</li>
</ul>
<p>and by polar coordinates</p>
<ul>
<li>$\frac{xy}{|x|+|y|}=r\cdot\frac{\c... |
3,669,269 | <p>I have learned that I can compute the moments of a random variable with this formula
<span class="math-container">$$\mu_n=\mbox{E}(X-\mbox{E}X)^n$$</span>
However, for the moment of order <span class="math-container">$1$</span> I can not use this, since I get <span class="math-container">$\mbox{E}(X-\mbox{E}X)=0$</s... | drhab | 75,923 | <p>It is not correct and looks more like a definition of <em>centralized moments</em>. </p>
<p>BTW if it is replaced by a "correct version" then it is actually a <em>definition</em> of moments and not so much a formula that enables you to <em>compute</em> moments. </p>
|
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | John | 20,946 | <p><a href="http://arxiv.org/" rel="nofollow noreferrer"><img src="https://i.imgur.com/W4xr2.png" alt="arXiv.org -- open access to over 700 000 e-prints"></a></p>
|
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | badp | 92 | <p><a href="http://www.lyx.org/" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/A4RsJ.png" alt="You'll wonder how you even managed to write the simplest equations with Word's poor excuse for an equation editor."></a></p>
|
386,002 | <p>I arrived at the following phrase at a material that I'm reading:</p>
<blockquote>
<p>Let $\pi :N'\rightarrow N$ be the blow-up of center $P$. For a given
$a\in\mathcal{O}$ and $P'\in\pi^{-1}(P)$, the <strong>strict transform</strong> of
$a$ in $P'$ is the ideal $str(a;P')$ of $\mathcal{O}_{N',P'}$ generated ... | Andrew | 34,377 | <p>I think the idea is that when $a$ vanishes at $P$ with multiplicity $v_P(a),$ the pullback $\pi^*(a)$ must contain a factor of at least $v_P(a)$ times the equation of the exceptional divisor. The strict transform is basically the inverse image of the vanishing of $a,$ but without the exceptional divisor. So, we get ... |
386,002 | <p>I arrived at the following phrase at a material that I'm reading:</p>
<blockquote>
<p>Let $\pi :N'\rightarrow N$ be the blow-up of center $P$. For a given
$a\in\mathcal{O}$ and $P'\in\pi^{-1}(P)$, the <strong>strict transform</strong> of
$a$ in $P'$ is the ideal $str(a;P')$ of $\mathcal{O}_{N',P'}$ generated ... | Dhruv Ranganathan | 42,999 | <p>This is how I like to think about proper transform. This is somewhat loosely speaking be warned.</p>
<p>Say you are blowing up $Z\subset X$, to get a space $\pi:\hat X\to X$. For a point, say $p$, in the base space, if $p\notin Z$ there is a unique preimage $\hat p$ in $\hat X$ which 'lives above' $p$. On the other... |
1,670,074 | <p>We know that a necessary and sufficient condition for a path-connected, locally path-connected space to have a universal cover is that it is semi-locally simply connected.</p>
<p>Now since $\mathbb R^2\setminus\{0\}$ is such a space, it must have a universal cover. However I can't see what the universal cover of $\... | Daniel Valenzuela | 156,302 | <p>One (of many) possibilities to consider is as for all infinite cyclic covers of nice $X$ is to pull back the universal cover $\mathbb R \to S^1$ of some $f: X \to S^1$. In the smooth setting $f^{-1}(1)$ (let 1 be a regular value) is a 1-codimensional submanifold and the infinite cyclic cover is obtained by cutting a... |
1,820,036 | <p>I'd be thankful if some could explain to me why the second equality is true...
I just can't figure it out. Maybe it's something really simple I am missing?</p>
<blockquote>
<p>$\displaystyle\lim_{\epsilon\to0}\frac{\det(Id+\epsilon H)-\det(Id)}{\epsilon}=\displaystyle\lim_{\epsilon\to0}\frac{1}{\epsilon}\left[\de... | user1551 | 1,551 | <p><em>Late comments.</em></p>
<ol>
<li>This is a special case of <a href="https://en.wikipedia.org/wiki/Jacobi's_formula" rel="nofollow noreferrer">Jacobi's formula</a>.</li>
<li>One way to prove the problem statement using Laplace/cofactor expansion, but easier to understand than other similar approaches and wit... |
2,536,866 | <p>In tensor notation the change of the electromagnetic field tensor by change of inertial reference frames can be done by the following formula :</p>
<p>$$F^{\alpha\beta} = \varLambda^{\alpha}_{\mu}\varLambda^{\beta}_{\nu}F^{\mu\nu}$$</p>
<p>But when this is represented by matrix multiplications it becomes:</p>
<p>... | Integral | 33,688 | <p>It's perfect ok to denote $\mathbf{A}:\mathbb{R} \to \mathbb{R}^{3 \times 2}$ and $\mathbf{B}:\mathbb{R}^2 \to \mathbb{R}^{3 \times 2}$, just as you did.</p>
<p>You can be even more general. Consider some functions $a_{ij}:\mathbb{R}^k \to \mathbb{R}$ for $i = 1 \ldots m, j = 1 \ldots n$. Then you can define the ma... |
4,358,710 | <p>Trying to understand <a href="https://en.wikipedia.org/wiki/Symmetric_rank-one" rel="nofollow noreferrer">symmetric rank one updates</a> and there is this like in the Wikipedia page that says...</p>
<blockquote>
<p>A twice continuously differentiable function <span class="math-container">$x\mapsto f(x)$</span> has a... | 温泽海 | 948,121 | <p>As you see, ratio test gives a ratio of <span class="math-container">$1$</span> by L'Hôpital's rule. What you really need is Cauchy's condensation test.</p>
|
4,358,710 | <p>Trying to understand <a href="https://en.wikipedia.org/wiki/Symmetric_rank-one" rel="nofollow noreferrer">symmetric rank one updates</a> and there is this like in the Wikipedia page that says...</p>
<blockquote>
<p>A twice continuously differentiable function <span class="math-container">$x\mapsto f(x)$</span> has a... | Átila Correia | 953,679 | <p><strong>HINT</strong></p>
<p>Apply the Condensation test</p>
<p><span class="math-container">\begin{align*}
\sum_{n=2}^{\infty}\frac{2^{n}}{2^{n}\ln^{p}(2^{n})} & = \sum_{n=2}^{\infty}\frac{1}{n^{p}\ln^{p}(2)} = \frac{1}{\ln^{p}(2)}\sum_{n=2}^{\infty}\frac{1}{n^{p}}
\end{align*}</span></p>
<p>Can you take it fro... |
244,115 | <p>I create a very large output</p>
<pre><code> D[x^100*E^(2*x^5)*Cos[x^2], {x, 137}]
</code></pre>
<p>I want to assign this to a function as</p>
<pre><code> f[x_]:= {very large output from previous command}
</code></pre>
<p>This allows me to evaluate that output for various values of <span class="math-container"... | Roman | 26,598 | <p>This is what <a href="https://reference.wolfram.com/language/ref/Set.html" rel="nofollow noreferrer">immediate assignments</a> are for:</p>
<pre><code>f[x_] = D[x^100*E^(2*x^5)*Cos[x^2], {x, 137}]
(* long output (can be suppressed with semicolon) *)
f[3.]
(* -8.90666869434476*10^664 *)
</code></pre>
<p>... |
3,768,086 | <p>Show that <span class="math-container">$(X_n)_n$</span> converges in probability to <span class="math-container">$X$</span> if and only if for every continuous function <span class="math-container">$f$</span> with compact support, <span class="math-container">$f(X_n)$</span> converges in probability to <span class="... | Community | -1 | <p>The answer = 1/2</p>
<p>The game has to end by either A winning or B winning</p>
<p>Let's say A wins. He is just as likely to roll a 1 or a 2 on the last roll. Therefore in a game that A wins, probability of an even roll ending the game is 1/2, as 1(odd) and 2(even) are equally likely.</p>
<p>Let's say B wins. He is... |
3,768,086 | <p>Show that <span class="math-container">$(X_n)_n$</span> converges in probability to <span class="math-container">$X$</span> if and only if for every continuous function <span class="math-container">$f$</span> with compact support, <span class="math-container">$f(X_n)$</span> converges in probability to <span class="... | JMoravitz | 179,297 | <p>Working under the assumption that the intended interpretation of the question was merely asking the probability that <span class="math-container">$B$</span> wins (<em>i.e. distinguishing between the term "rounds" as iterating whenever A has a turn and "turns" iterating whenever either A or B has ... |
371,318 | <p>The original problem was to consider how many ways to make a wiring diagram out of $n$ resistors. When I thought about this I realized that if you can only connect in series and shunt. - Then this is the same as dividing an area with $n-1$ horizontal and vertical lines. When each line only divides one of the current... | André Nicolas | 6,312 | <p>In the ordinary calculus, there are <em>no</em> infinitesimals. </p>
<p>Abraham Robinson and others, from the $1950$'s on, developed <em>non-standard analysis</em>, which does have infinitesimals, and also "infinite" number-like objects, that one can work with in ways that are closely analogous to the way we deal w... |
498,694 | <p>So, I'm learning limits right now in calculus class.</p>
<p>When $x$ approaches infinity, what does this expression approach?</p>
<p>$$\frac{(x^x)}{(x!)}$$</p>
<p>Why? Since, the bottom is $x!$, doesn't it mean that the bottom goes to zero faster, therefore the whole thing approaches 0?</p>
| Community | -1 | <p>The limit is $\infty$, think about it this way:</p>
<p>On the top, $$x^x=\underbrace{x\times x\times\cdots\times x}_{x\text{ number of }x's}$$</p>
<p>On the bottom, $$x!=x\times(x-1)\times\cdots\times 1$$</p>
<p>Which is is bigger? Can you tell?</p>
|
498,694 | <p>So, I'm learning limits right now in calculus class.</p>
<p>When $x$ approaches infinity, what does this expression approach?</p>
<p>$$\frac{(x^x)}{(x!)}$$</p>
<p>Why? Since, the bottom is $x!$, doesn't it mean that the bottom goes to zero faster, therefore the whole thing approaches 0?</p>
| New_to_this | 90,774 | <p>There is a theorem, which states that, if the reciprocal approaches zero, then the original expression approches $+\infty$. That might me a strategy in this case.</p>
|
1,132,063 | <p>For $x=(x_j)_{j\in\mathbb N}\in \ell^1$ let</p>
<p>$$\|x\|=\sup_{n\in \mathbb N}\left \Vert \sum_{j=1}^{n}x_j\right\Vert$$</p>
<p>Show that $(\ell^1,\|\cdot\|)$ is a normed space, but it is not complete.</p>
<p>The first part was easy.</p>
<p>Now I try to find a sequence in $\ell^1$ such that it is a cauchy sequ... | tomasz | 30,222 | <p><strong>Hint</strong>: take some sequence which is <em>just</em> outside $\ell^1$ and play around with signs a little.</p>
|
794,912 | <p>I am reviewing Calculus III using <a href="http://www.jiblm.org/downloads/dlitem.aspx?id=82&category=jiblmjournal" rel="nofollow">Mahavier, W. Ted's material</a> and get stuck on one question in chapter 1. Here is the problem:</p>
<p>Assume $\vec{u},\vec{v}\in \mathbb{R}^3$. Find a vector $\vec{x}=(x,y,z)$ so t... | Matt L. | 70,664 | <p><strong>Hint:</strong> compute the cross-product of $\vec{u}$ and $\vec{v}$ and normalize the result (assuming that they are not collinear).</p>
|
221,712 | <p>I have two matrix <code>A</code> and <code>B</code> of equal dimensions see below. In <code>A</code> matrix I have the variables <code>a,b,c,d</code> which have direct correspondence with matrix <code>B</code> element by each row. In other words, for first row <code>{a, b, c, d}</code> we have <code>{2, 9, 6, 7}</co... | Rupesh | 63,381 | <p>I use loop as a last resort in Mathematica and I believe there are other ways to solve this problem. However, I am using loop for this problem as it is quite intuitive. Please, let me know if you have trouble understanding the soln: </p>
<pre><code> A = {{a, b, c, d}, {d, c, b, a}, {a, c, b, d}};
B = {{2, 9,... |
221,712 | <p>I have two matrix <code>A</code> and <code>B</code> of equal dimensions see below. In <code>A</code> matrix I have the variables <code>a,b,c,d</code> which have direct correspondence with matrix <code>B</code> element by each row. In other words, for first row <code>{a, b, c, d}</code> we have <code>{2, 9, 6, 7}</co... | J. M.'s persistent exhaustion | 50 | <p>I didn't immediately give the full answer, in the hope someone would follow up on the hint in my comment. Anyway, the missing piece is to use <code>Ordering[]</code> to rearrange list <code>B</code>, like so:</p>
<pre><code>MapThread[#1[[Ordering[#2]]] &, {B, A}].{{1, -1}, {0, 1}, {0, 0}, {-1, 0}}
{{-5, 7}, ... |
1,768,317 | <p>Show that $\sin(x) > \ln(x+1)$ when $x \in (0,1)$. </p>
<p>I'm expected to use the maclaurin series (taylor series when a=0)</p>
<p>So if i understand it correctly I need to show that: </p>
<p>$$\sin(x) = \lim\limits_{n \rightarrow \infty} \sum_{k=1}^{n} \frac{(-1)^{k-1}}{(2k-1)!} \cdot x^{2k-1} > \lim\limi... | Jack D'Aurizio | 44,121 | <p>We have to prove that:
$$ \forall x\in(0,1),\qquad \int_{0}^{x}\left(\cos t-\frac{1}{t+1}\right)\,dt > 0$$
but $g(t)=(t+1)\cos t\geq (t+1)\left(1-\frac{t^2}{2}\right)$ for any $t\in(0,1)$, hence $g(t)>1$ and the above integral is positive, as wanted.</p>
|
569,103 | <blockquote>
<blockquote>
<p>How can I calculate the first partial derivative $P_{x_i}$ and the second partial derivative $P_{x_i x_i}$ of function:
$$
P(x,y):=\frac{1-\Vert x\rVert^2}{\Vert x-y\rVert^n}, x\in B_1(0)\subset\mathbb{R}^n,y\in S_1(0)?
$$</p>
</blockquote>
</blockquote>
<p>I ask this with reg... | Idris Addou | 192,045 | <p>Let
\begin{eqnarray*}
g(x) &=&2(1+\cos x)\log \sec x-\sin x\{x+\log (\sec x+\tan x)\} \\
&=&\{\sin x-2(1+\cos x)\}\log \cos x-x\sin x-\sin x\log (1+\sin x) \\
&=&\{\sin x-2(1+\cos x)\}A(x)-B(x)-C(x)
\end{eqnarray*}
where
\begin{eqnarray*}
A(x) &=&\log \cos x \\
B(x) &=&x\sin ... |
287,597 | <p>Can anyone explain how why <a href="http://en.wikipedia.org/wiki/Quaternion#Matrix_representations">the matrix representation of the quaternions using real matrices</a> is constructed as such?</p>
| Chen | 329,111 | <p>If It is not a misunderstanding, the question of "why as such (possible others?)" is equivalent to the problem of finding out ALL 4x4 real matrix representations of the Quaternions. The fact is: all those matrix representations are conjugated to each other. In other words, this is the only 4x4 real matrix representa... |
1,023,193 | <p>Proving this formula
$$
\pi^{2}
=\sum_{n\ =\ 0}^{\infty}\left[\,{1 \over \left(\,2n + 1 + a/3\,\right)^{2}}
+{1 \over \left(\, 2n + 1 - a/3\,\right)^{2}}\,\right]
$$
if $a$ an even integer number so that
$$
a \geq 4\quad\mbox{and}\quad{\rm gcd}\left(\,a,3\,\right) = 1
$$</p>
| xpaul | 66,420 | <p>Here is another solution. Noting
$$\sum_{n=-\infty}^\infty f(n)= -\sum_{j=1}^k \operatorname*{Res}_{z=j}\pi \cot (\pi z)f(z) $$
we have
\begin{eqnarray}
&&\sum_{n=0}^{\infty}\left[{1 \over \left(2n + 1 + a/3\right)^{2}}
+{1 \over \left(2n + 1 - a/3\right)^{2}}\right]\\
&=&\sum_{n=-\infty}^{\infty}{1 ... |
917,276 | <p>If $U$ and $V$ are independent identically distributed standard normal, what is the distribution of their difference?</p>
<p>I will present my answer here. I am hoping to know if I am right or wrong.</p>
<p>Using the method of moment generating functions, we have</p>
<p>\begin{align*}
M_{U-V}(t)&=E\left[e^{t... | Jonathan H | 51,744 | <p>The currently upvoted answer is wrong, and the author rejected attempts to edit despite 6 reviewers' approval. So here it is; if one knows the rules about <a href="https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables" rel="noreferrer">the sum</a> and <a href="https://en.wikipedia.org/wiki/Norma... |
1,556,805 | <p>I'm working in a problem that involves the equation
$$
w(z)=\sqrt{1-z^{2}} \,\, .
$$</p>
<p>I already know that there're two branch points in this equation, namely $\pm 1$, so there's a Riemann surface covering the domain of the function where the branch cut is from the $-1$ to $1$, as shown in the figure below.</... | CR Drost | 42,154 | <h1>What are we doing?</h1>
<p>Branching is about <em>multivalued functions</em>, so it's worth stating more explicitly what we're doing. </p>
<p>We can take the map $z\mapsto z^{1/3}$ and expand it about a point $z_0 = r~e^{i\theta}$ to find that it has an analytic continuation which converges on the disc $|z - z_0|... |
277,250 | <p>Let $\mathbb{N}$ be the set of natural numbers and $\beta \mathbb N$ denotes the Stone-Cech compactification of $\mathbb N$. </p>
<p>Is it then true that $\beta \mathbb N\cong \beta \mathbb N \times \beta \mathbb N $ ? </p>
| Taras Banakh | 61,536 | <p>The spaces $\beta \mathbb N$ and $\beta\mathbb N\times \beta\mathbb N$ are not homeomorphic.</p>
<p>To derive a contradiction, assume that $\beta\mathbb N$ and $\beta\mathbb N\times \beta\mathbb N$ are homeomorphic. Since $\beta \mathbb N$ is homeomorphic to $\beta(\mathbb N\times\mathbb N)$, we conclude that there... |
4,379,693 | <p>How to change the order of integration:</p>
<p><span class="math-container">$$\int_{-1}^1dx \int_{1-x^2}^{2-x^2}f(x,y)dy$$</span></p>
<p>I tried to sketch the area and got:</p>
<p><a href="https://i.stack.imgur.com/xxu5T.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xxu5T.png" alt="enter image d... | Henry | 6,460 | <p>My thinking was that</p>
<ul>
<li><p><span class="math-container">$\frac1n\sum (X_i-E[X_i])^2$</span> cannot be more than <span class="math-container">$\frac1n(1-\frac1n)^2+\frac{n-1}n(0-\frac1n)^2 = \frac{n-1}{n^2}$</span> or less than <span class="math-container">$0$</span> if <span class="math-container">$\sum X_... |
3,992,495 | <blockquote>
<p><span class="math-container">$\displaystyle b_1=\left\lbrace\frac{12}{9},\frac{12}{9},2\right\rbrace^T,b_2=\{-18,-18,21\}^T$</span> and <span class="math-container">$\displaystyle v_1=\{-1,-1,2\}^T,v_2=\{3,3,-3\}^T$</span>. <span class="math-container">$b_1 \in \operatorname{Span}\{v_1,v_2\} \text{ and ... | pietro | 851,281 | <p>Indeed you cannot say that <span class="math-container">$Span\{v_1,v_2\} \subseteq Span\{b_1,b_2\}$</span> in general. To see that consider the trivial case where:
<span class="math-container">$v_1=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$</span>, <span class="math-container">$v_2=\begin{pmatrix} 0 \\ 1 \end{pmatrix}$</... |
4,573,600 | <p>I need help to start solving a differential equation</p>
<p><span class="math-container">$$x^2y'+xy=\sqrt{x^2y^2+1}.$$</span></p>
<p>I would divide the equation with <span class="math-container">$x^2.$</span> Then the equation looks like a homogeneous equation, but I get under the square root <span class="math-conta... | Yaser | 932,614 | <p>let <span class="math-container">$z=xy$</span></p>
<p><span class="math-container">$y'=\frac{xz'-z}{x^{2}}$</span></p>
<p><span class="math-container">$xz'=\sqrt{{z^{2}}+1}$</span></p>
<p><span class="math-container">$\displaystyle \int \frac{1}{\sqrt{z^2+1}} \,dz = \displaystyle \int \frac{1}{x} \,dx$</span></p>
<p... |
4,573,600 | <p>I need help to start solving a differential equation</p>
<p><span class="math-container">$$x^2y'+xy=\sqrt{x^2y^2+1}.$$</span></p>
<p>I would divide the equation with <span class="math-container">$x^2.$</span> Then the equation looks like a homogeneous equation, but I get under the square root <span class="math-conta... | Yaser | 932,614 | <p>But this is not the only answer to the equation!!!</p>
<p><span class="math-container">$\operatorname{arcsinh}(z)=\ln(x)+\ln(c)$</span></p>
<p><span class="math-container">$\ln(\sqrt{z^{2}+1}+z)=\ln(cx)$</span></p>
<p><span class="math-container">$\sqrt{z^{2}+1}+z=cx$</span></p>
<p><span class="math-container">$z^{2... |
902,313 | <p>The wikipedia page on clopen sets says "Any clopen set is a union of (possibly infinitely many) connected components." </p>
<p>I thought any topological space is the union of its connected components? Why is this singled out here for clopen sets?</p>
<p>Does it have something to do with it $x\in C$ a clopen subset... | GFR | 64,803 | <p>My attempt at a proof: suppose $S$ is clopen in $X$. If $X$ is connected we are done hence suppose $X=\bigcup_i C_i$ is the partition of $X$ by its connected components. We need to show that for each $i$ $C_i\cap S$ is either void or equal to $C_i$. But since $S$ is clopen, $S\cap C_i$ is both open and closed in $C_... |
459,428 | <p>How does one evaluate a function in the form of
$$\int \ln^nx\space dx$$
My trusty friend Wolfram Alpha is blabbering about $\Gamma$ functions and I am having trouble following. Is there a method for indefinitely integrating such and expression? Or if there isn't a method how would you tackle the problem?</p>
| Daniel Robert-Nicoud | 60,713 | <p>If $\ln^nx$ denotes $\log(x)^n$, then my hint is to try the simple substitution $e^y = x$, giving
$$\int\log(x)^ndx = \int y^ne^ydy$$</p>
|
1,128,414 | <blockquote>
<p>Let $F:C[0,2]\to C[0,2]$ be the map defined by $(F(f))(x)=x^2f(x)$. Show that $F$ is continuous as a function from $(C[0,2],\|\cdot\|_{\sup})$ to $(C[0,2],\|\cdot\|_{2})$.</p>
</blockquote>
<p>I read this solution:</p>
<blockquote>
<p>Let $f\in C[0,2]$. Let $\epsilon>0$. Choose $\delta=\epsilon... | Berci | 41,488 | <p>For a linear transformation $T:V\to V$ and a basis $\beta=(b_1,b_2,\dots)$ of $V$, you can get the $i$th column of the matrix $[T]_\beta$ by calculating the coordinates of $T(b_i)$, taken in the basis $\beta$. </p>
<p>Being diagonal claims $T(b_i)=\lambda_ib_i$ with some $\lambda_i\in\Bbb R$ for each basis element ... |
649,495 | <p>Hello I'm having trouble understanding the factorizing of a polynomial as </p>
<p>$$x^4-4x$$</p>
<p>After that, I turned it into $$x(x^3-8)$$</p>
<p>But I don't quite understand how it's factored (the process) as </p>
<p>$$x(x−2)(x^2+2x+4)$$</p>
<p>Thanks!</p>
| ncmathsadist | 4,154 | <p>Notice that if $f(x) = x^3 - 8$, we can easily see that $f(2)= 0$ This means that $x - 2$ is a divisor of $x^3 - 8$. Do polynomial long division. You an easily generalize to factor $x^3 - a^3$ for a constant $a$.</p>
|
1,378,633 | <p>It seems that some, especially in electrical engineering and musical signal processing, describe that every signal can be represented as a Fourier series.</p>
<p>So this got me thinking about the mathematical proof for such argument.</p>
<p>But even after going through some resources about the Fourier series (whic... | DanielWainfleet | 254,665 | <p>A bounded periodic integrable function F will certainly "have" a Fourier series, but the sum of the series can fail to be equal to F at some points, even if F is continuous. </p>
|
346,432 | <p>I will think of <span class="math-container">$ \mathbb{R}^{n+m}$</span> as <span class="math-container">$\mathbb{R}^n \times \mathbb{R}^m$</span>.</p>
<p>Let <span class="math-container">$ V \subset \mathbb{R}^{n+m}$</span> be open and <span class="math-container">$g:V \to U \subset \mathbb{R}^{n+m} $</span> be a... | Liviu Nicolaescu | 20,302 | <p>Coarea formula will do this for you. It is a "Fubini formula" relating the integral of a function <span class="math-container">$u$</span> on a Riemann manifold <span class="math-container">$(M_0,g_0)$</span> to the integrals along the fibers of a smooth map <span class="math-container">$F:(M_0,g_0)\to (M_1,g_1)$... |
2,099,550 | <p>How to calculate $$\int_{0}^{\pi/4}\sqrt{1+\tan x}\,\mathrm dx$$
My attempt:
Let
$$I=\int_{0}^{\pi/4}\sqrt{1+\tan x}\,\mathrm dx$$
substitute $\sqrt{1+\tan x}=t$,then
$$I=\int_{1}^{\sqrt{2}}\frac{2t^{2}}{t^{4}-2t^{2}+2}\,\mathrm dt=\int_{1}^{\sqrt{2}}\frac{2}{t^{2}-2+\dfrac{2}{t^{2}}}\,\mathrm dt$$
but I got stuck h... | Renascence_5. | 286,825 | <p><strong>Hint:</strong>
\begin{align*}
I&=\int_{1}^{\sqrt{2}}\frac{2t^{2}}{t^{4}-2t^{2}+2}\,\mathrm{d}t\\
&=\int_{1}^{\sqrt{2}}\frac{\sqrt{2}+t^{2}+\left ( t^{2}-\sqrt{2} \right )}{t^{4}-2t^{2}+2}\,\mathrm{d}t \\
&=\int_{1}^{\sqrt{2}}\frac{\displaystyle\frac{\sqrt{2}}{t^{2}}+1}{t^{2}-2+\displaystyle\frac... |
2,099,550 | <p>How to calculate $$\int_{0}^{\pi/4}\sqrt{1+\tan x}\,\mathrm dx$$
My attempt:
Let
$$I=\int_{0}^{\pi/4}\sqrt{1+\tan x}\,\mathrm dx$$
substitute $\sqrt{1+\tan x}=t$,then
$$I=\int_{1}^{\sqrt{2}}\frac{2t^{2}}{t^{4}-2t^{2}+2}\,\mathrm dt=\int_{1}^{\sqrt{2}}\frac{2}{t^{2}-2+\dfrac{2}{t^{2}}}\,\mathrm dt$$
but I got stuck h... | idk | 397,693 | <p><strong>Hint:</strong> Try to use partial fractions
$$\displaystyle\frac{2t^2}{t^4-2t^2+2}=\frac{At+B}{t^2-\sqrt{2+2\sqrt{2}}t+\sqrt{2}}+\frac{Ct+D}{t^2+\sqrt{2+2\sqrt{2}}t+\sqrt{2}}$$</p>
|
264,572 | <p>I am using <code>Table</code> to plot the time steps of a function. However, since there are a lot of decimal places in the x-axis, they are all cramped up :
<a href="https://i.stack.imgur.com/SCSbK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SCSbK.png" alt="enter image description here" /></a... | Michael E2 | 4,999 | <h3>Introduction</h3>
<p><em>[Evolution notice: I forgot to scale the vector field in <code>StreamPlot</code> and <code>VectorPlot</code>; now fixed. Also nonstandard evaluation in Mathematica takes time to find out if you've thought of all the cute things that can be done: <code>%</code> now seems to work, though per... |
2,616,663 | <p>For this proof, after I convert the definite integral into the riemann sum definition, is it just enough to say $\Delta(x)$ = $\frac{b-a}{n}$ and since b = a, the $\Delta(x)$ becomes 0, thus, making everything else equal to 0, since everything else is being multiplied by zero?</p>
| gen-ℤ ready to perish | 347,062 | <p>Why not just stick with this?</p>
<p>$$\begin{align}
\int_a^a f(x)\, dx &= F(x)\bigr|_a^a \\
&= F(a)-F(a) \\
&= 0
\end{align}$$</p>
<p>where $F’(x)=f(x)$.</p>
|
501,660 | <p>In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?</p>
<p>Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.</p>... | Stefan4024 | 67,746 | <p>Use Taylor Series:</p>
<p><span class="math-container">$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$$</span></p>
<p><span class="math-container">$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$... |
4,572,804 | <p>I'm working my way through Murphy's, C<em>-algebras and Operator Theory and I have a question concernig the proof that every C</em>-algebra admits an approximate identity.</p>
<p>Let A be an arbitrary C*-algebra. We denote by <span class="math-container">$\Lambda$</span> the set of all positive elements a in A such ... | Martin Argerami | 22,857 | <p>By construction <span class="math-container">$\|f\|\leq1$</span>.</p>
<p>If <span class="math-container">$t\in K$</span>,
<span class="math-container">$$
|f(t)-\delta g(t)f(t)|=|f(t)|\,|1-\delta |\leq \varepsilon.
$$</span>
If <span class="math-container">$t\not\in K$</span> then <span class="math-container">$|f(t)... |
2,729,617 | <blockquote>
<p>Find the 5th-order Maclaurin polynomial $P_5(x)$ for $f(x) = e^x$.</p>
</blockquote>
<p>I got
$$P_5(x) = 1 + x +\frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} + O(x^6) $$</p>
<p>From this answer, I'm supposed to approximate $f(-1)$, correct to the fifth decimal place. Is it right ... | tobiasbriones | 525,347 | <p>A strictly increasing function: let $$P=\{x_1,...,x_n; x_i<x_{i+1}\}$$ then $f(x_i)<f(x_{i+1})$ for all $x_i,x_{i+1} \in P$.
A monotonic increasing function: let $$P=\{x_1,...,x_n; x_i<x_{i+1}\}$$ then $f(x_i)\leq f(x_{i+1})$ for all $x_i,x_{i+1} \in P$. So a monotonic function can be constant for some inte... |
1,606,363 | <p><strong>A family has two children. What is the probability that both the children are boys given that at least on of them is a boy?</strong></p>
<p>Solution given in my book is
<a href="https://i.stack.imgur.com/H1N3Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/H1N3Z.png" alt="enter image de... | Graham Kemp | 135,106 | <p>You would be correct if $F$ was "the eldest child was a boy", or such information which gives order. Then you could evaluate the probability that the <em>other</em> child is also a boy the way you suggest.</p>
<p>However the <em>actual</em> event is "at least one child is a boy" and that is <em>not</em> the ... |
1,606,363 | <p><strong>A family has two children. What is the probability that both the children are boys given that at least on of them is a boy?</strong></p>
<p>Solution given in my book is
<a href="https://i.stack.imgur.com/H1N3Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/H1N3Z.png" alt="enter image de... | K. Jiang | 302,781 | <p>Always keep in mind that when you solve for a conditional probability, make sure you are using $\frac{P(\text{successful})}{P(\text{total})}.$ You're total ways here is NOT $2$, since there are actually $3$ ways to get at least one boy: BB, BG, and GB. Only one of these is "successful," namely, BB. Since each of the... |
1,606,363 | <p><strong>A family has two children. What is the probability that both the children are boys given that at least on of them is a boy?</strong></p>
<p>Solution given in my book is
<a href="https://i.stack.imgur.com/H1N3Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/H1N3Z.png" alt="enter image de... | fleablood | 280,126 | <p>Ever hear the puzzle "I have two coins; they add up to 30 cents and one of them is not a quarter; what two coins are they". The answer? One is a a nickle and the other is a quarter. Wait? You said one of them was not a quarter! It isn't; it's a nickel and the <em>other</em> one is a quarter.</p>
<p>Believe it o... |
219,014 | <p>I have the list</p>
<pre><code>t1 = {{-1, 0}, {-2, 0}, {-3, 0}, {0, 0}, {-2, 0}, {1, 1}}
</code></pre>
<p>How do I find the position where an element repeats? In this case it would be element <code>{-2, 0}</code> at position 5, because <code>{-2, 0}</code> first came up at postion 2. So the answer would be 5.</p>
... | JimB | 19,758 | <p>This is just an extended comment as @MikeY has the answer.</p>
<p>For your particular example the function <code>SymmetricReduction</code> can help with seeing patterns:</p>
<pre><code>k = 4;
bernoullies = Table[x[i] \[Distributed] BernoulliDistribution[p[i]], {i, k}];
Expectation[Log[1 + Sum[Log[1 + x[i]], {i, k}... |
1,238,292 | <p>This is a homework problem, so please do not give more than hints. I must convert
\begin{align}
\int_0^\sqrt{2}\int_x^\sqrt{4-x^2}\sin\left(x^2+y^2\right)\:dy\:dx\tag{1}
\end{align}
to polar coordinates. This is my attempt:
\begin{align}
\int_{\pi/4}^{\pi/2}\int_{\color{red}{2\cos\left(\theta\right)}}^{\color{red}{2... | Adhvaitha | 228,265 | <p>Below is the region over which you are integrating.</p>
<p><img src="https://i.stack.imgur.com/yiEpj.png" alt="enter image description here"></p>
<p>Can you now fix the limits of $r$ and $\theta$?</p>
|
2,312,968 | <p>If $t=\ln(x)$, $y$ some function of $x$, and $\dfrac{dy}{dx}=e^{-t}\dfrac{dy}{dt}$, why would the second derivative of $y$ with respect to $x$ be:
$$-e^{-t}\frac{dt}{dx}\frac {dy}{dt} + e^{-t}\frac{d^2y}{dt^2}\frac{dt}{dx}?$$</p>
<p>I know this links into the chain rule. I don't have a good intuition for why the fi... | Doug M | 317,162 | <p>b) is easier than a) so I am going to start there.</p>
<p>the function is period with period $2\pi$ </p>
<p>that is $f(x) = f(x+2n\pi)$</p>
<p>if $f(x) = 0$ then $f(2\pi) = 0,f(-2\pi) = 0$ etc.</p>
<p>In the interval $[-4\pi,0]$</p>
<p>$0-4\pi, \frac {4\pi}{3}-4\pi, -2\pi, \frac {4\pi}{3} - 2\pi, 0$</p>
<p>or... |
2,614,920 | <p>Not understanding the concept well, I am trying to determint the pointwise and uniform convergence of the following sequence of function:</p>
<p>$$f_n(x) = \frac{\sin{nx}}{n^3}, x \in \mathbb{R}$$</p>
<p>The only part I understand so far is that I need $\lim_{x\to\infty}{f_n(x)}$ in which I have determined that (i... | Sunny | 304,150 | <p>Certainly, the way you are restricting the map is not a homotopy. Look at this way: if X = S^{2}-{ k points} and k>1, you can enlarge the region of a one deleted point and make it flat which is homotopically equivalent to disk minus k-1 points which deformation retract onto k-1 wedge of circles.</p>
|
508,550 | <p>I read in an article the Euclidean distance formula can be estimated with about 6% relative error with the following formula. Would you please <strong>why</strong> this is true and <strong>where</strong> can I find such estimations? Is it possible to <strong>extend</strong> it for higher dimensions?</p>
<p>$d(a,b) ... | nobody | 449,752 | <p>I will follow the notation in the previous comment.
First, <em>u</em> itself is not a bad estimation. The maximal possible relative error with it is just $\sqrt{2}-1 < 42\% $. If it is generalized for <em>n</em> dimensions, then the error will be $\sqrt{n} - 1$, which means for higher dimensions this simple appro... |
377,393 | <p>Two players play a game. Player 1 goes first, and chooses a number between 1 and 30 (inclusive). Player 2 chooses second; he can't choose Player 1's number. A fair 30-sided die is rolled. The player that chose the number closest to the value of the roll takes that value (say, in dollars) from the other player. Would... | CommonerG | 74,357 | <p>Let $x_1$ and $x_2$ be the player's choices. To maximize his chances of winning, player 2 will select a number a close as possible to player 1's but must choose whether to select a number above or below $x_1$. The expected payoff for choosing above $x_1$ is greater if:</p>
<p>$\frac{1}{30}\sum_{i=x_1+1}^{30}i\geq\f... |
1,986,247 | <p>The nth Catalan number is :
$$C_n = \frac {1} {n+1} \times {2n \choose n}$$
The problem 12-4 of CLRS asks to find :
$$C_n = \frac {4^n} { \sqrt {\pi} n^{3/2}} (1+ O(1/n)) $$
And Stirling's approximation is:
$$n! = \sqrt {2 \pi n} {\left( \frac {n}{e} \right)}^{n} {\left( 1+ \Theta \left(\frac {1} {n}\right) \right... | robjohn | 13,854 | <p>In <a href="https://math.stackexchange.com/a/932509">this answer</a>, it is shown that
$$
\frac{4^n}{\sqrt{\pi(n+\frac13)}}\le\binom{2n}{n}\le\frac{4^n}{\sqrt{\pi(n+\frac14)}}\tag{1}
$$
Using the fact that as $n\to\infty$,
$$
\begin{align}
(n+a)^b
&=n^b\left(1+\frac an\right)^b\\
&=n^b\left(1+O\left(\frac1n\... |
2,880,384 | <p>Look at the following definition.</p>
<p><strong>Definition.</strong> Let $\kappa$ be an infinite cardinal. A theory $T$ is called $\kappa$-stable if for all model $M\models T$ and all $A\subset M$ with $|A|\leq \kappa$ we have $|S_n^M(A)|\leq \kappa$.
A theory $T$ is called stable if it is $\kappa$-stable for some... | Alex Kruckman | 7,062 | <p>I'd like to argue that the counting types definition of stability is actually very "natural". </p>
<p>At the most basic level, model theory is about the semantics of first-order formulas, i.e. definable sets in models. Since first-order logic is built on classical propositional logic, there are Boolean algebras eve... |
4,528,838 | <p>Find the general solution of the equation <span class="math-container">$$x^{(5)} + 2x^{(4)} + 2x^{(3)} + 4x'' + x' + 2x = 100e^{-2t}.$$</span></p>
<p>I don't understand how solve such tasks. I know that I should solve <span class="math-container">$x^{(5)} + 2x^{(4)} + 2x^{(3)} + 4x'' + x' + 2x =0$</span> and then us... | Saeed | 858,459 | <p>Solving these problems has a straightforward step and a somewhat tricky/intuitive step. Here is how we break the problem in two:</p>
<p>We begin with <span class="math-container">$L(x)=f(t)$</span> , where <span class="math-container">$L$</span> is all the differential operations on the function <span class="math-co... |
1,611,560 | <p>Reading through the first half of Baby Rudin again before taking an Analysis class, I came across the assertion that "it is also easy to show that k-cells are convex". </p>
<p>Previously it gave the example of open/closed balls being convex, and the proof is obvious and easy to understand. That being said, and it... | BigbearZzz | 231,327 | <p>Suppose that k-cell means the product of intervals in $\Bbb R^k$. Let $C=\prod_{n=1}^k I_n $, where $I_n$ is an interval for each $n$, be a k-cell. Recall that in $\Bbb R$ $I$ is an interval <em>iff</em> for $a,b\in I$ such that $a<b$, $c\in (a,b)$ implies that $c\in I$.</p>
<p>Now let $x,y\in C$ where $x=(x_1,x... |
3,909,972 | <p>I used Photomath and Microsoft Math to compute an equation, but they gave me two different results (-411 and -411/38) Why did that happen and which is the correct answer?</p>
<p><a href="https://i.stack.imgur.com/egt5A.jpg" rel="nofollow noreferrer">https://i.stack.imgur.com/egt5A.jpg</a>
<a href="https://i.stack.im... | fleablood | 280,126 | <p>Neither attempt are trying to tell you <em>what</em> <span class="math-container">$\frac {16}{19} -\frac {5\times 81}{38} -1$</span> <em>is</em>. There are both trying to tell you <em>why</em> it is not equal to <span class="math-container">$0$</span>. And the both giving you correct reasons why it <em>isn't</em> ... |
2,418,547 | <p>The following is a proof of
$$\frac{\partial(u,v)}{\partial(x,y)}.\frac{\partial(x,y)}{\partial(u,v)} = 1$$
<a href="https://i.stack.imgur.com/fLfRX.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fLfRX.jpg" alt="Jacobian proof"></a></p>
<p>In the above proof, I cannot understand why $\frac{\par... | Mark | 4,460 | <p>Your book's exposition is a little confusing, since it has $x, y, u, v$ being used as both variables and functions.</p>
<p>To make it clearer, I will try to re-derive the result using $x, y, u, v$ as variables and $X, Y, U, V$ as functions. </p>
<p>Suppose we have the functions $U(x, y), V(x, y), X(u, v), Y(u, v)$... |
396,440 | <p>Suppose we have the function $$f(x) = \frac{x}{p} + \frac{b}{q} - x^{\frac{1}{p}}b^{\frac{1}{q}}$$ where $x,b \geq 0 \land p,q > 1 \land \frac{1}{p}+\frac{1}{q} = 1$</p>
<p>I am trying to show that $b$ is the absolute minimum of $f$. </p>
<p>I proceeded as follows:</p>
<p>$$\frac{df(x)}{dx} = \frac{1}{p} - \fr... | robjohn | 13,854 | <p>This is an instance of the <a href="http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means#Weighted_AM.E2.80.93GM_inequality" rel="nofollow">Weighted Arithmetic Mean-Geometric Mean Inequality</a>. It uses the concavity of $\log(x)$ and can be generalized to
$$
\prod_{i=1}^nx_i^{\alpha_i}\le\sum_{i... |
223,582 | <p>Maps $g$ maps $\left\{1,2,3,4,5\right\}$ onto $\left\{11,12,13,14\right\}$ and $g(1)\neq g(2)$. How many g are there.</p>
<p><strong>My answer</strong>:
I transformed the question to a easy-understand way and find out the solution.
Consider there are five children and four seats. Two of them are willing sitting to... | Berci | 41,488 | <p>Draw a <em>Thales</em> circle over the segment $AC$, it will intersect the desired $D$, because $AD\perp DC$:</p>
<ol>
<li>Draw the segment $AC$.</li>
<li>Construct its midpoint $F$.</li>
<li>Draw a circle with origin $F$ and radio $FA(=FC)$.</li>
</ol>
|
4,045,755 | <blockquote>
<p>If <span class="math-container">$p$</span> is a prime then all the non trivial subgroups of <span class="math-container">$G$</span> with <span class="math-container">$\lvert G\rvert=p^2$</span> are cyclic.</p>
</blockquote>
<p>I tried looking online where does this result come from, but could not find a... | Community | -1 | <p>If <span class="math-container">$H\le G$</span> is nontrivial, then <span class="math-container">$|H|=p$</span> (Lagrange). Then <span class="math-container">$1\ne x\in H\Longrightarrow \langle x\rangle=H$</span> (Lagrange, again), so <span class="math-container">$H$</span> is cyclic.</p>
|
2,788,276 | <p>Let$\ f_n (x)=n^2x(1-x)^n$ I need to prove that$\ f_n→0$ in the interval$\ [0,1]$.</p>
<hr>
<p>Let$\ f_n(x) = nx^n$ prove that$\ f_n→0$ in the interval$\ [0,1)$.</p>
<p>For both of these sequences I tried the following:</p>
<p>By taking the function$\ f(x)=0$ we can see that</p>
<p>$$\lim_{n\rightarrow\infty}f_... | Diego | 41,463 | <p>If $P$ is a prime ideal of $R$, then $P\cap (\mathbb{Z}/n\mathbb{Z})=p\mathbb{Z}/n\mathbb{Z}$ for some prime $p$ dividing $n$, and $\mathbb{Z}/p\mathbb{Z}\subseteq R/P$. Let $\bar{f}\in(\mathbb{Z}/p\mathbb{Z})[X]$ be the reduction of $f$. Then $\bar{f}(\bar{r})$ for every $\bar{r}\in R/P$. Since $f$ is monic, $\bar{... |
2,788,276 | <p>Let$\ f_n (x)=n^2x(1-x)^n$ I need to prove that$\ f_n→0$ in the interval$\ [0,1]$.</p>
<hr>
<p>Let$\ f_n(x) = nx^n$ prove that$\ f_n→0$ in the interval$\ [0,1)$.</p>
<p>For both of these sequences I tried the following:</p>
<p>By taking the function$\ f(x)=0$ we can see that</p>
<p>$$\lim_{n\rightarrow\infty}f_... | quasi | 400,434 | <p>Let $R$ be a commutative ring with $1\ne 0$ such that</p>
<ul>
<li>$R$ contains a subring $R_0$ of Krull dimension $0$.$\\[4pt]$
<li>For some monic $f\in R_0[x]$, we have $f(r)=0$, for all $r\in R$.
</ul>
<p><strong>Claim:</strong>$\;R$ has Krull dimension $0$.
<p>
<strong>Proof:</strong>
<p>
Suppose $P$ is a prim... |
1,231,113 | <p>There are plenty of questions about the homology of the connected sum of two $n$-manifolds, but I didn't find an explicit explanation of the computation done in degree $n-1$.
Let's show some examples of such questions:</p>
<p><strong>1)</strong> In <a href="https://math.stackexchange.com/questions/453132/connected-... | Ben Dyer | 164,207 | <p>You need to know a couple of facts. Namely, that the homology of a compact manifold is finitely generated (see <a href="https://math.stackexchange.com/questions/445727/homology-of-compact-manifolds">Homology of a compact manifold is finitely generated</a>) and Corollary 3.28 from Hatcher. Basically this allows us to... |
2,195,739 | <p>For
$$
f(x) = \begin{cases}
x^2 & \text{if $x\in\mathbb{Q}$,} \\[4px]
x^3 & \text{if $x\notin\mathbb{Q}$}
\end{cases}
$$</p>
<p>What I did was examine each of the limits at $0$ of
$\displaystyle\lim_{x\to0} \frac{f(x)-f(a)}{x-a}$ for each case but I am not sure </p>
| dantopa | 206,581 | <p><strong>Yes</strong></p>
<p>Consider the rationale sequences $\delta_{k} = \pm 2^{-k}$. The limit on the left equals the limit on the right.
$$
\lim_{k\to\infty} \frac{f(\pm\delta_{k}) - f(0)}{\mp\delta_{k}} = 0.
$$</p>
<p>Next consider the irrational sequences $\epsilon_{k} = \pm e^{-k}$. Again, the right and l... |
4,362,741 | <p>Let <span class="math-container">$f:(0,\infty)\rightarrow \mathbb{R}$</span> be a real-valued function, such that for some <span class="math-container">$t,C>0$</span>,
<span class="math-container">\begin{equation}
\limsup_{x\rightarrow\infty} f(x+t)\leq C
\end{equation}</span>
Is it also true that <span class="ma... | David Kraemer | 224,759 | <p>We recall that</p>
<p><span class="math-container">$$
\limsup_{x \to \infty} f(x) = L
$$</span></p>
<p>if and only if the following two conditions hold:</p>
<ol>
<li>For every sequence <span class="math-container">$x_n \to \infty$</span>, we have <span class="math-container">$\limsup_{n \to \infty} f(x_n) \leq L$</s... |
1,158,712 | <p>My question is : Given an invertible matrix $A$ ( with complex entries ) , if $A^n$ is normal,is $A$ normal?</p>
<p>This is related to the question : <a href="https://math.stackexchange.com/questions/1158600/if-a-is-an-invertible-n-times-n-complex-matrix-and-some-power-of-a-is-diag">If $A$ is an invertible $n\times... | Najib Idrissi | 10,014 | <p>No, that's not true. Take a symmetry $\sigma$ which isn't unitary, then $\sigma^2 = I$ is normal but $\sigma$ isn't. For a more explicit example, take:
$$A = PSP^{-1} = \begin{pmatrix}
1 & 0 \\
1 & -1
\end{pmatrix},$$
where $P = \begin{pmatrix}1&1\\0&1\end{pmatrix}$ and $S = \begin{pmatrix} 0 & 1... |
3,069,684 | <p>My question goes like this</p>
<p>If 5a+4b+20c=t, then what is the value of t for which the line ax+by+c-1=0 always passes through a fixed point?</p>
<p>I tried but couldn't solve it so I looked at the solution. The solution says that the equation has 2 independent parameters. I get that. If we choose a and b, c a... | user3482749 | 226,174 | <p>Fix your point as <span class="math-container">$(\alpha,\beta)$</span>. Then, since <span class="math-container">$c = \frac{t - 5a - 4b}{20}$</span>, you need to find <span class="math-container">$t$</span> such that for any values of <span class="math-container">$a$</span> and <span class="math-container">$b$</span... |
2,573,458 | <p>Given $n$ prime numbers, $p_1, p_2, p_3,\ldots,p_n$, then $p_1p_2p_3\cdots p_n+1$ is not divisible by any of the primes $p_i, i=1,2,3,\ldots,n.$ I dont understand why. Can somebody give me a hint or an Explanation ? Thanks.</p>
| fleablood | 280,126 | <p>Because if $k>1$ divides $n$ then $k$ can not divide $n+1$. </p>
<p>If $n = m*k$ then then $n + 1 = k(m + \frac 1m)$ and $m + \frac 1m$ is not an integer. </p>
<p>Or to put it another way. If $n = m*k$ then the <em>next</em> multiple of $k$ is $n+k = m*k + k$ which is <em>larger</em> than $n+1 = mk + 1$.</p>... |
3,143,084 | <p>If <span class="math-container">$f : \mathbb{R} \to \mathbb{R}$</span>, we can think of the derivative of <span class="math-container">$f$</span> at a point <span class="math-container">$x$</span>, denoted <span class="math-container">$f'(x)$</span>, as giving the slope of a line tangent to the graph of <span class=... | Xander Henderson | 468,350 | <p>The expression you give is the derivative of the function <span class="math-container">$x \mapsto x^{123}$</span>. There are a few ways that you can attack this, however nearly all of them are ultimately going to rest on an induction argument somewhere along the line. The other answers cite the binomial theorem, w... |
264,770 | <p>If we have a vector in $\mathbb{R}^3$ (or any Euclidian space I suppose), say $v = (-3,-6,-9)$, then:</p>
<ol>
<li>May I always "factor" out a constant from a vector, as in this example like $(-3,-6,-9) = -3(1,2,3) \implies (1,2,3)$ or does the constant always go along with the vector?</li>
<li>If yes on question 1... | T. Eskin | 22,446 | <p>The answer is yes to both. </p>
<ol>
<li><p>If you have a vector $(v_{1},...,v_{n})\in\mathbb{R}^{n}$ and a real number $a\in \mathbb{R}$, then $(av_{1},...,av_{n})=a(v_{1},...,v_{n})$. This is the definition for multiplication of a vector by a real number.</p></li>
<li><p>If $v\in\mathbb{R}^{n}$ and $a\in \mathbb{... |
194,724 | <p>All graphs discussed are finite and simple. The <em>cycle sequence</em> of a graph $G$, denoted $C(G)$, is the nondecreasing sequence of the lengths of all of the cycles in $G$, where cycles are distinguished by the vertices they contain, not by the edges they contain. </p>
<p>For example, $C(K_{3,2})=4,4,4$ and $C... | Gordon Royle | 1,492 | <p>The answer to your question is "Yes". If you type the following into Sage</p>
<pre><code>g1 = Graph("I?`D@bAfg")
g2 = Graph("I?AE@`g~o")
g1.show();
g2.show();
</code></pre>
<p>you get</p>
<p><img src="https://i.stack.imgur.com/RZl8S.png" alt="enter image description here"></p>
<p><img src="https://i.stack.imgur... |
381,177 | <p>I have a problem in which I have to compute the following integral: <span class="math-container">$$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^k y_i=x} e^{-N^2r(\sum_{i=1}^k y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_k,$$</span>
where this notation means that I want to integrate over <span class="math-container">$... | Carlo Beenakker | 11,260 | <p>The desired integral <span class="math-container">$I$</span> can be written as <span class="math-container">$^\ast$</span></p>
<p><span class="math-container">$$I=e^{N^2rx^2/k}J(x),\;\;J(x)=\mathop{\idotsint}\delta\left(x-{\textstyle{\sum_{j=1}^{k}}y_j}\right) e^{-N^2r\sum_{j=1}^{k}y_j^2} dy_1\dots dy_{k}.$$</span>
... |
381,177 | <p>I have a problem in which I have to compute the following integral: <span class="math-container">$$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^k y_i=x} e^{-N^2r(\sum_{i=1}^k y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_k,$$</span>
where this notation means that I want to integrate over <span class="math-container">$... | Iosif Pinelis | 36,721 | <p><span class="math-container">$\newcommand\1{\mathbf1}\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}$</span>Here is yet another, "multivariate calculus" way to treat your integral, with the same result as in my other two answers on this page.</p>
<p>Let
<span class="math-container">$$c:=N^2r\in(0,\inft... |
3,733,757 | <p>I'm proving that given a nonempty set <span class="math-container">$I$</span>, and given a filter <span class="math-container">$F$</span>, there exists an ultrafilter <span class="math-container">$D$</span> on <span class="math-container">$I$</span> such that <span class="math-container">$F \subseteq D$</span>. I us... | FiMePr | 802,801 | <p>Proof by contradiction is indeed a valid method here.
You probably want to use the so-called "finite meet property" : If <span class="math-container">$C \subseteq P(I)$</span>, we say it has the finite meet property when all finite intersections of elements of <span class="math-container">$C$</span> are no... |
665,759 | <p>Let $G$ be an open subset of $R$. </p>
<p>If $0\notin G$, then show that $H=\{xy:x,y\in G\}$ is an open subset of $R$.</p>
<p>Now Since $G$ is open , given $x,y\in R$, $\exists ,r_x,r_y$ such that $B(x,r_x)\subset G$
and $B(y,r_y)\subset G$.Now all we need to do is find a radius $r$ given a point $xy$ in $H$.</p>... | Martín-Blas Pérez Pinilla | 98,199 | <p>Prove first that if $c\ne 0$ the function
$$P_c:{\Bbb R}\longrightarrow{\Bbb R},\qquad P_c(x)=cx$$
is homeomorphism (is obviously continuous and $P_c^{-1}=\cdots$).</p>
<p>Then, $H=\bigcup_{x\in G}P_x(G)$ is a union of open sets.</p>
|
4,182,343 | <p>I am trying to solve the following simple equation:</p>
<p><span class="math-container">$$\frac{df}{dx}=\sin{f}$$</span> This is the "kink" solution to the Sine-Gordon equation. To solve this, I do the following substitution:</p>
<p><span class="math-container">$$f=\tan^{-1}g$$</span></p>
<p>Then we can us... | Claude Leibovici | 82,404 | <p>You could make it faster switching variables
<span class="math-container">$$\frac{df}{dx}=\sin(f)\implies \frac{dx}{df}=\frac 1{\sin(f)}$$</span>
<span class="math-container">$$x+C=\log \left(\tan \left(\frac{f}{2}\right)\right)\implies\tan \left(\frac{f}{2}\right)=C e^x\implies f=2\tan ^{-1}(C e^x)$$</span></p>
|
4,294,860 | <p>Let's say I have a group of n people. Some are left handed and some are right handed. I need to know a random person identity, knowing if he is right or left handed</p>
<p>As conditional probabilty:</p>
<p>Being <span class="math-container">$P(X)$</span> the probability of correctly guessing a person identity.</p>
<... | Lazy | 958,820 | <p>No. Note that <span class="math-container">$P(X)$</span> is a weird notation, you’d expect <span class="math-container">$X$</span> to be a random person and you’d want <span class="math-container">$P(X=x)$</span> for <span class="math-container">$x$</span> some specific person. Then note that <span class="math-conta... |
1,216,619 | <p>Why are the rings $\mathbb{R}$ and $\mathbb{R}[ x ]$ not isomorphic to eachother ?</p>
<p>Think it might have to do with multiplicative inverses but I'm not sure.</p>
| Salomo | 226,957 | <p>You are right: the element "$x$" has no multiplicative inverse, that is there is no polynomial $p(x)$ such that $x\cdot p(x)=1$.</p>
|
137,501 | <p>P. P. Palfy proved that a primitive solvable subgroup of $S_n$ has order bounded by $24^{-1/3} n^{3.24399\dots}$ (in: Pálfy, P. P.
A polynomial bound for the orders of primitive solvable groups.
J. Algebra 77 (1982), no. 1, 127–137. )</p>
<p>This is sharp (and notice that for $n$ prime, the affine group of the lin... | Igor Pak | 4,040 | <p>I suggest you read the following paper by Pálfy. I don't have it handy here and it doesn't seem to be downloadable, but I recall he surveys similar results on the orders of nilpotent permutation groups. </p>
<p>Péter P. Pálfy,
Estimations for the order of various permutation groups,
<em>Contributions to general... |
3,542,573 | <blockquote>
<p>Solve the differential equation <span class="math-container">$$y''-6y'+25y=50t^3-36t^2 -63t +18$$</span></p>
</blockquote>
<p>I tried solving the homogeneous equation using <span class="math-container">$y = vt$</span>, but I didn't go anywhere. </p>
| Michael Rozenberg | 190,319 | <p>It should be <span class="math-container">$$(2,2,2)=2(1,0,0)+2(0,1,1).$$</span>
Can you end it now?</p>
<p>The final result is right.</p>
|
1,504,483 | <p>Where did the angle convention (in mathematics) come from?</p>
<p>One would imagine that a clockwise direction would be more 'natural' (given
sundials & the like, also a magnetic compass dial).</p>
<p>Also, given time and direction conventions, one would imagine that the
zero degree line would be vertical.</p>... | Amir Asghari | 83,875 | <p>This is by no means answering the question as it is. However, It is just to give a historical piece (taken from "<a href="http://www.ingelec.uns.edu.ar/asnl/Materiales/Cap03Extras/Stokes-Katz.pdf" rel="nofollow noreferrer">The history of Stokes' Theorem</a>" written by Katz) that might come handy when thinking of th... |
3,453,483 | <p>I'm reading Serre's <span class="math-container">$\textit{A course in Arithmetic}$</span> where he defines a Dirichlet series to be an infinite sum of the form
<span class="math-container">$$f(z) = \sum\limits_{n=1}^{\infty} a_ne^{-\lambda_nz}
$$</span>
where <span class="math-container">$\lambda_n$</span> is an in... | reuns | 276,986 | <p>The idea is that if the Dirichlet series converges at some <span class="math-container">$z_0$</span> then <span class="math-container">$$\frac{f(z+z_0)}{z} =\sum_{n=1}^\infty a_n e^{-\lambda_n z_0}\frac{e^{-\lambda_n z}}{z}= \sum_{n=1}^\infty a_n e^{-\lambda_n z_0}\int_{\lambda_n}^\infty e^{-tz}dt = \int_{\lambda_0... |
3,258,372 | <p>I'm doing a practice exam questions and am stuck at this question:</p>
<blockquote>
<p>Are there topological spaces X,Y (each with more than one point), such that [0,1] is homeomorphic to X×Y? What if we replace [0,1] with R?</p>
</blockquote>
<p>I'm not even sure how to start tackle it, any help and clues will ... | Pedro | 178,668 | <p>You had the right idea. <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are the image under the projections of <span class="math-container">$X\times Y$</span>, so they must be path connected.</p>
<p>Now on <span class="math-container">$[0,1]$</span> there are many points that aft... |
211,481 | <p>How can I prove that the determinant function satisfying the following properties is unique:</p>
<p>$\det(I)=1$ where $I$ is identity matrix, the function $\det(A)$ is linear in the rows of the matrix and if two adjacent rows of a matrix $A$ are equal, then $\det A=0$.
This is how Artin has stated the properties.I... | Community | -1 | <p>I have thought about your problem for a while now and I think there is a nice slick way to do this. Consider the space $W$ of all multilinear alternating forms $f$ in $k$ - variables</p>
<p>$$f : V \times \ldots \times V \to \Bbb{C}.$$</p>
<p>We claim that there is a canonical isomorphism between $W$ and $(\bigwed... |
211,481 | <p>How can I prove that the determinant function satisfying the following properties is unique:</p>
<p>$\det(I)=1$ where $I$ is identity matrix, the function $\det(A)$ is linear in the rows of the matrix and if two adjacent rows of a matrix $A$ are equal, then $\det A=0$.
This is how Artin has stated the properties.I... | Marc van Leeuwen | 18,880 | <p>A basic fact about linear functions is that they are <em>completely determined</em> by their values on a basis of the vector space. For a multi-linear function this means (repeating this statement for each argument) that they are determined by their values where each argument independently runs through a basis of th... |
211,481 | <p>How can I prove that the determinant function satisfying the following properties is unique:</p>
<p>$\det(I)=1$ where $I$ is identity matrix, the function $\det(A)$ is linear in the rows of the matrix and if two adjacent rows of a matrix $A$ are equal, then $\det A=0$.
This is how Artin has stated the properties.I... | Berci | 41,488 | <p>An alternative way is the <a href="http://en.wikipedia.org/wiki/Gaussian_elimination#Example">Gaussian elimination</a>: for a given $n\times n$ matrix $A$ with rows $r_1,..,r_n$, the following steps are allowed to use, in order to arrive to the identity matrix or one with a zero row (by the linearity, if $A$ has a ... |
31,480 | <p>I'm having difficulty with my math, fractions and up. I used to understand it all, but it's been so long since I've touched the book (I finished it a couple of months ago, picked it up to review everything), I seem to have forgotten it. </p>
<p>The explanations inside of the individual chapters do no good. They nev... | picakhu | 4,728 | <p>The <a href="http://www.khanacademy.org" rel="nofollow">Khan Academy</a> has hundreds of maths videos and practice questions.</p>
|
1,305,481 | <p>Let $C$ denote the circle $|z|=1$ oriented counterclockwise. Show that</p>
<p>i)$\int_Cz^ne^{\frac{1}{z}}dz=\frac{2\pi i}{(n+1)!}$ for $n=0,1,2$</p>
<p>ii)$\int_C e^{z+\frac{1}{z}}dz=2\pi i\sum_{n=0}^\infty\frac{1}{n!(n+1)!}$</p>
<p>I'm stuck in this exercise, because</p>
<p>$$\int_Cz^ne^{\frac{1}{z}}dz=2\pi i *... | Alex M. | 164,025 | <p>For (i):</p>
<p>$z^n \mathbb e ^{1 \over z} = z^n \sum \limits _{k=0} ^\infty {1 \over z^k k!}$. The term containing $1 \over z$ is obtained when $k = n+1$, so its coefficient is $1 \over {(n+1)!}$. The residue is then $2 \pi \mathbb i \over {(n+1)!}$.</p>
<p>For (ii):</p>
<p>$\mathbb e ^{z + {1 \over z}} = \math... |
1,428,905 | <p>I have two functions:</p>
<p>$n!$</p>
<p>$2^{n^{2}}$</p>
<p>What is the difference between the growth of these two? My thought is that $2^{n^2}$ grows much faster than $n!$. </p>
| Victor | 142,550 | <p>Create a sequence $\{a_n\} = \frac{2^{n^2}}{n!}$ and let $n$ get infinitely large. Upon using the ratio test:
$$
\frac {a_{n}}{ a_{n-1}}=\frac{2^{n^2}/n!}{2^{(n-1)^2}/(n-1)!}=\frac{2^{n^2}}{n2^{(n-1)^2}}=\frac{2^{2n-1}}{n}.
$$</p>
<p>What can one say about this?</p>
|
1,428,905 | <p>I have two functions:</p>
<p>$n!$</p>
<p>$2^{n^{2}}$</p>
<p>What is the difference between the growth of these two? My thought is that $2^{n^2}$ grows much faster than $n!$. </p>
| Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>$$\log 2^{n^2}=n^2\log 2$$</p>
<p>and </p>
<p>$$\log n!<n\log n$$</p>
|
1,428,905 | <p>I have two functions:</p>
<p>$n!$</p>
<p>$2^{n^{2}}$</p>
<p>What is the difference between the growth of these two? My thought is that $2^{n^2}$ grows much faster than $n!$. </p>
| Chris Culter | 87,023 | <p>There's a nice combinatorial relationship between the two: $n!$ is the number of $n\times n$ permutation matrices, and $2^{n^2}$ is the number of $n\times n$ binary matrices. Every permutation matrix is a binary matrix, so we immediately have $2^{n^2}\geq n!$.</p>
<p>Moreover, the probability that a random binary m... |
2,406,587 | <p>Isn't the concept of homomorphism and isomorphism in abstract algebra analogous to functions and invertible functions in set theory respectively? That's one way to quickly grasp the concept into the mind?</p>
| mathreadler | 213,607 | <p>To answer the last part about quadratic equation:</p>
<p>An equation about a quadrat, kvadrat : in many languages other than english it is a geometric shape: a rectangle - which has four sides and four corners. Iff we assume linear expressions as it's side and asking for it's area, becomes a quadratic equation:</p>... |
3,020,988 | <p>Here's my attempt at an integral I found on this site.
<span class="math-container">$$\int_0^{2\pi}e^{\cos2x}\cos(\sin2x)\ \mathrm{d}x=2\pi$$</span>
<strong>I'm not asking for a proof, I just want to know where I messed up</strong></p>
<p>Recall that, for all <span class="math-container">$x$</span>,
<span class="ma... | 2'5 9'2 | 11,123 | <p>You cannot substitute <span class="math-container">$u=\sin^2t$</span>. As <span class="math-container">$t$</span> ranges from <span class="math-container">$0$</span> to <span class="math-container">$2\pi$</span>, this is not a one-to-one relationship.</p>
<p>It's like if you subbed <span class="math-container">$u=x... |
3,020,988 | <p>Here's my attempt at an integral I found on this site.
<span class="math-container">$$\int_0^{2\pi}e^{\cos2x}\cos(\sin2x)\ \mathrm{d}x=2\pi$$</span>
<strong>I'm not asking for a proof, I just want to know where I messed up</strong></p>
<p>Recall that, for all <span class="math-container">$x$</span>,
<span class="ma... | David G. Stork | 210,401 | <p>If by <span class="math-container">$\cos \sin (2 x)$</span> you really mean <span class="math-container">$\cos (2 x ) \sin (2 x)$</span>, then the full function looks like</p>
<p><a href="https://i.stack.imgur.com/d7GNZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/d7GNZ.png" alt="enter image d... |
1,269,447 | <p>I consider the space $C^1[a, b]$ of (complex) functions that are at least once differentiable on $[a, b]$. I want to show that</p>
<p>$$||f||_{C^1} := ||f||_\infty + ||f'||_\infty$$</p>
<p>defines a norm on $C^1[a, b]$.</p>
<p>Now it's easy to see that $||f||_{C^1}$ is non-negative, and that it's zero iff f = 0, ... | ant11 | 110,047 | <p>We solve the second system for $y$:</p>
<p>$$y=\frac{-x^2+\sqrt{x}+1}{\sqrt{x}+\sqrt{1-x^2}}$$</p>
<p>and substitute into the first equation and solve for $x$. It is then seen that $(0,1)$ is the only real-valued solution. Computer analysis finds complex solutions where $x$ is the root of a certain $16$ degree pol... |
2,609,252 | <p>like the title said i'm looking for the best way for me(a 15 year old) to go about learning calculus, thank you :)</p>
| fred goodman | 124,085 | <ol>
<li><p>While you are studying calculus (or anything else), study yourself. How do you learn best? What are your strengths, your goals etc.?</p></li>
<li><p>Always ask yourself, "what is the big idea?” Summarize for yourself the steps in any solution and extract the important concepts.</p></li>
</ol>
<p>2a. K... |
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