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 |
|---|---|---|---|---|
1,821,448 | <p>$$\iiint \frac{1}{\sqrt{x^2+y^2+(z-2)^2}}$$</p>
<p>for $x^2+y^2+z^2 = 1$</p>
<p>I've used spherical coordinates, like this:
$x=\rho sin\phi cos\theta$; $y=\rho sin\phi sin\theta$; $z=\rho cos\phi$ and $J=\rho^2 sin\phi$</p>
<p>$$\rho^2 sin^2\phi cos^2\theta + \rho^2 sin^2\phi sin^2\theta +\rho^2 cos^2\phi = 1$$
$... | user5713492 | 316,404 | <p>$$\begin{align}&\int_0^{2\pi}\int_0^{\pi}\int_0^1\frac1{\sqrt{\rho^2-4\rho\cos\phi+4}}\rho^2d\rho\sin\phi\,d\phi\,d\theta\\
&=2\pi\int_0^{\pi}\int_0^1\frac1{\sqrt{\rho^2-4\rho\cos\phi+4}}\rho^2d\rho\sin\phi\,d\phi\\
&=2\pi\int_0^1\left[\frac12\sqrt{\rho^2-4\rho\cos\phi+4}\right]_0^{\pi}\rho\,d\rho\\
&... |
1,821,448 | <p>$$\iiint \frac{1}{\sqrt{x^2+y^2+(z-2)^2}}$$</p>
<p>for $x^2+y^2+z^2 = 1$</p>
<p>I've used spherical coordinates, like this:
$x=\rho sin\phi cos\theta$; $y=\rho sin\phi sin\theta$; $z=\rho cos\phi$ and $J=\rho^2 sin\phi$</p>
<p>$$\rho^2 sin^2\phi cos^2\theta + \rho^2 sin^2\phi sin^2\theta +\rho^2 cos^2\phi = 1$$
$... | Doug M | 317,162 | <p>I was going to suggest the substitution.</p>
<p>$\rho = 2\sin \phi\tan u + 2 \cos \phi$</p>
<p>And that would clean up the radical. But, after thinking about it, I decided that spherical is not the way to go.</p>
<p>Suppose instead you did cylindrical.</p>
<p>$x = r \cos \theta\\
y = r \sin \theta\\
z = z\\
dx\... |
1,009,169 | <p>I have to examine limit of following sequence $\{a_n\}_{ n \ge 1}$ $a_n = \sqrt[n]{\sum_{k=1}^{n}{(2 -\frac{1}{k})^k}}$. We know that $\lim_{n \to \infty} \sqrt[n]{a} = 1$ for $a > 0$ and we know that $\sqrt[n]{\sum_{k=1}^{n}{(2 -\frac{1}{k})^k}} > 0$ but sequence $a_n$ is increasing (I cannot prove it but I c... | Petite Etincelle | 100,564 | <p>Firstly, for any $k_0 \in \mathbb{N}$, we have</p>
<p>$$a_n = \sqrt[n]{\sum_{k=1}^{n}{(2 -\dfrac{1}{k})^k}} \geq \sqrt[n]{\sum_{k=k_0}^{n}{(2 -\dfrac{1}{k_0})^k}} = \left(\dfrac{(2-1/k_0)^{k_0}(1-(2-1/k_0)^{n-k_0+1})}{1/k_0 -1}\right)^{1/n}= \left(\dfrac{(2-1/k_0)^{k_0}((2-1/k_0)^{n-k_0+1}-1)}{1-1/k_0 }\right)^{1/... |
3,440,648 | <blockquote>
<p>Prove that <span class="math-container">$$ \lim_{x \to 0} \frac{ \sqrt{x + \sqrt{ x + \sqrt{x}}} }{\sqrt[8]{x}}= 1$$</span> </p>
</blockquote>
<p>Let <span class="math-container">$x=y^8$</span> with <span class="math-container">$t>0$</span>. Then after we get
<span class="math-container">$$
\lim ... | Martin R | 42,969 | <p>First consider the sequence of every second fraction:
<span class="math-container">$$
\frac{a_{2n+2}}{b_{2n+2}} = \frac{a_{2n}+b_{2n}}{a_{2n}+2b_{2n}} = \frac{\frac{a_{2n}}{b_{2n}} +1}{\frac{a_{2n}}{b_{2n}} + 2} = f(\frac{a_{2n}}{b_{2n}})
$$</span>
where <span class="math-container">$f(x)$</span> is defined as
<sp... |
238,408 | <p>How can I solve this system of equations in Mathematica?</p>
<pre><code>DCp1total = WA1*DCpA0 + WB1*DCpB0
DCp2total = WA2*DCpA0 + WB2*DCpB0
WA1 + WB1 = 1
WA2 + WB2 = 1
DCp1_total = 0.6
DCp2_total = 0.3
DCpA0 = 0.5
DCpB0 = 0.4
</code></pre>
<p>Unkowns: <code>WA1, WA2, WB1</code> and <code>WB2</code>.</p>
| Bob Hanlon | 9,362 | <p>For the revised problem statement:</p>
<pre><code>DCp1Total = 3/5;
DCp2Total = 3/10;
DCpA0 = 1/2;
DCpB0 = 2/5;
eq[1] = DCp1Total == WA1*DCpA0 + WB1*DCpB0 // Simplify;
eq[2] = DCp2Total == WA2*DCpA0 + WB2*DCpB0 // Simplify;
eq[3] = WA1 + WB1 == 1;
eq[4] = WA2 + WB2 == 1;
min = Minimize[Total[(Subtract @@@ (eq /@ Ra... |
3,570,313 | <blockquote>
<p>Prove that there exist infinitely many pairs <span class="math-container">$(a,b)$</span> of relatively prime positive integers
such that <span class="math-container">$\frac{a^2-5}{b}, \frac{b^2-5}{a}$</span> are both positive integers.</p>
</blockquote>
<p>I saw that this problem came from Germany ... | Arthur | 15,500 | <p><strong>Partial answer:</strong> After writing a quick program that checks for solutions, I found, among others, the following pairs that work:
<span class="math-container">$$
4, 11\\
11, 29\\
29, 76\\
76, 199\\
199, 521\\
521, 1364
$$</span>
They seem like a chain of pairs, each constructed from the previous pair i... |
76,147 | <p>Is it possible to define a Borel $\sigma$-algebra on a topological space which is not second countable, i.e. one which does not have a countable base?</p>
<p>I am trying to learn measure theory and my intuition says that this is NOT possible, because if the space is not second countable, then there are "too many" ... | ZulfiqarIII | 8,042 | <p>Yes, it is possible, no problems.</p>
<p>Let $X$ be a space with topology $T$.
Let $\{A_\alpha\}_\alpha$ be any (possibly uncountable) collection of $\sigma$-algebras on $X$.</p>
<p>Consider $\hat A := \bigcap_\alpha A_\alpha$. It's easy to check this again forms a $\sigma$-algebra.</p>
<p>In particular, conside... |
731,769 | <p>I've finished Calculus 1 and I'm going to go into Calculus II next semester, so I've been trying to keep my integration skills up to par. I noticed that in some parts of integration in Calculus 1, there are some integrals that, with a clever <span class="math-container">$u$</span> substitution turn into something b... | N. Bar | 641,014 | <p>Though you may already know this (and this is more of a trick rather than a specific integral), one of my favorite tricks is symmetric integration. All this requires is knowledge of even and odd functions.</p>
<p>Remember: An even function is symmetric about the y-axis; An odd function is symmetric about the origin... |
255,702 | <p>I was surprised that Rationalize will sometimes leave a Real unchanged even if the fractional part of the Real is well above zero.</p>
<pre><code>a=571628.2812`
b=Rationalize[a]
(* 571628.2812 *)
a===b
(* True *)
</code></pre>
<p>I had code blow up because of this, because I explicitly wanted to use only Integer and... | Manuel --Moe-- G | 13,738 | <pre><code>myRationalize[x_Integer] := x;
myRationalize[x_Rational] := x;
myRationalize[0.] := 0;
myRationalize[x_Real] := Block[
{y},
y = Rationalize@x;
If[Head@y === Real, y = Round[x*1000000000.]/1000000000];
y
]
</code></pre>
<pre><code>myRationalize[571628.2812`]
(* 1429070703/2500 *)
myRationalize[(... |
255,702 | <p>I was surprised that Rationalize will sometimes leave a Real unchanged even if the fractional part of the Real is well above zero.</p>
<pre><code>a=571628.2812`
b=Rationalize[a]
(* 571628.2812 *)
a===b
(* True *)
</code></pre>
<p>I had code blow up because of this, because I explicitly wanted to use only Integer and... | Manuel --Moe-- G | 13,738 | <p>Many thanks to the helpful suggestion from Bob Hanlon to just read the documentation (my face is red) and use the second argument for Rationalize.</p>
<pre><code>Rationalize[571628.2812`, 0]
(* 1429070703/2500 *)
Rationalize[(1/3)*1.,0]
(* 1/3 *)
Rationalize[(1/10)*1.,0]
(* 1/10 *)
</code></pre>
<p>Notice that even ... |
2,484,426 | <p>I have the following function:</p>
<p>$$ f(x,y) = \begin{cases}
\dfrac{xy^3}{{x^2+y^2}} & (x,y)\neq(0,0) \\
0 & (x,y)=(0,0) \\
\end{cases} $$</p>
<p>How can I demonstrate the existence of such a limit using epsilon and delta?</p>
| Peter Szilas | 408,605 | <p>$ |f(x,y) - f(0,0)| = $</p>
<p>$|\dfrac{xy^3}{x^2+y^2}| \le \dfrac{|xy|(x^2+y^2)}{(x^2+y^2)} =$</p>
<p>$|xy| \lt (x^2+y^2)$.</p>
<p>Let $\epsilon >0$ be given.</p>
<p>Choose: $\delta= \epsilon^{1/2}$.</p>
<p>Then $|x^2+y^2|^{1/2} \lt \delta$ implies</p>
<p>$|f(x,y)-f(0,0)| \lt $</p>
<p>$(x^2+y^2)\lt \delt... |
3,670,963 | <p>I am trying to calculate the probability of taking from a regular deck of cards (52) 5 cards with 4 different suits, my go: </p>
<p>first we think about picking 'diamond' (just to make it more simple) it is: <span class="math-container">$\frac{13}{52}$</span><br>
second we think about picking 'heart': <span class=... | Christian Fun In Tuscany | 787,332 | <p>I was thinking that the problem could be solved by using the hypergeometric multivariate distribution </p>
<p>P(x,y,z,t) = [ 13Cx 13Cy 13Cz 13Ct ] / 52C5</p>
<p>where x,y,z,t are the random variables associated to the four disjoint sets of the suites</p>
<p>as the only 4 possible configurations of the random vari... |
3,119,058 | <p>I'm given the complex rational expression</p>
<p><span class="math-container">$$\frac{\frac{a}{b} - \frac{b}{a}}{\frac{a+b}{ab}}$$</span></p>
<p>and asked to simplify. The solution provided is <span class="math-container">$a - b$</span>; however I get <span class="math-container">$\frac{a^2 - b^2}{a+b}$</span>.</p... | Marvin | 544,299 | <p>You did nothing wrong. Factor the numerator <span class="math-container">$a^{2} - b^{2}$</span> into the difference the squares:</p>
<p><span class="math-container">$$a^{2} - b^{2} = (a + b)(a - b)$$</span></p>
<p>This will leave with:</p>
<p><span class="math-container">$$\frac{(a+b)(a-b)}{a+b}$$</span></p>
<p>... |
1,361,766 | <p>I am stucked at this combinatorics/probability problem:</p>
<hr>
<p>There are 10 urns, each contains 8 balls numbered 1,2,...,8</p>
<p>If we randomly choose 1 ball from <strong>each</strong> urn, what is the probability that the <strong>total</strong> number of 1's will be <strong>exactly</strong> 3 and the <stro... | drhab | 75,923 | <p>Let $X$ denote the number of $1$'s, let $Y$ denote the number of $4$'s and let let $Z$ denote the number of other numbers. Then:$$P(X=3\wedge Y=2\wedge Z=5)=\frac{10!}{3!2!5!}\left(\frac18\right)^3\left(\frac18\right)^2\left(\frac68\right)^5$$</p>
<p>First a selection concerning: from wich urn are the $1$'s, $4$'s ... |
106,985 | <h2>Background</h2>
<p>I am currently working on the homology of some moduli space and there exists a much simpler chain complex with the same homology.
It is a quotient of a bisimplicial complex by a subcomplex. One could say that some faces are degenerate, thus are zero.
Unfortunately, it's too hard to compute the h... | Vidit Nanda | 18,263 | <p>I realize that this is an old question and it is likely that the OP has moved on, but let me summarize the state-of-the-art as it exists now.</p>
<hr>
<p>The given sizes (18 million by 15 million) are pretty hopeless if you are planning to resort to matrix methods. To put things in perspective, latest version of <... |
3,687,299 | <p>I have a stupid question about continuity in higher dimensions.</p>
<p>There are maps, for example, <span class="math-container">$f(x,y)=\frac{xy^2}{x^2+y^4}$</span>, when <span class="math-container">$(x,y)\neq (0,0)$</span> and <span class="math-container">$f(x,y)=(0,0)$</span> when <span class="math-container">$... | GReyes | 633,848 | <p>The reason is that the function is approaching zero along each straight line at a different speed, depending on the line. So, for <span class="math-container">$f$</span> to be, say, less than <span class="math-container">$1/10$</span> along one of the lines, you need to be within a distance <span class="math-contain... |
1,936,712 | <p>Suppose I have a die with sides $1,2,3,4,5,5$. What is the probability of getting any order of $1,2,3,4,5$ in $5$ rolls?</p>
<p>Would I be right in saying you can't use combinations to try and calculate this because the probability of each outcome isn't equally likely?</p>
| Mike F | 6,608 | <p>I would try thinking like this:</p>
<ul>
<li>What is the probability of rolling 1,2,3,4,5 in <em>exactly that order</em>?</li>
<li>What about rolling them in some other order, say 2,1,3,4,5. Is the probability still the same?</li>
<li>How many different orderings are there?</li>
<li>The probability of rolling any o... |
132,354 | <p>I need to integrate an equation of the form </p>
<p>$f''(x) g_1+f'(x) g_2=a g_3$</p>
<p>where $a$ is a constant and $g_1,g_2,g_3$ are functions of $x$
whose values on the nodes of integration are known. The boundary conditions are
$f(x=0)=0$, $f(x=1)=f_0$, $f_0$ is a constant. $g_1,g_2,g_3$ are arrays of data, a... | Alexey Popkov | 280 | <p>According to the <a href="http://reference.wolfram.com/language/ref/DefaultStyleDefinitions.html" rel="noreferrer">Documentation</a>,</p>
<blockquote>
<p><code>DefaultStyleDefinitions</code> is a global option that specifies the default stylesheet for all new notebooks.</p>
<p>The stylesheet must be located in a dir... |
2,604,548 | <p>I am trying to solve some <strong>ratio tests</strong>. I came across to my notes and i can't figure out how is that equation simplified.
Well, how is $\frac{\left(2\left(n+1\right)\right)!}{\left(n+1\right)!}\cdot \frac{n!}{\left(2n\right)!}$ simplified to $\frac{\left(2n+1\right)\left(2n+2\right)}{n+1}$</p>
| Community | -1 | <p>Note that: $$(2(n+1))!=(2n+2)!$$ $$=(2n)!\times (2n+1)\times (2n+2)$$ giving us: $$\frac{(2(n+1))!}{(2n)!}\times \frac{n!} {(n+1)!} $$ $$= \frac{(2n)!\times (2n+1) \times (2n+2)}{(2n)!}\times \frac{n!} {n! \times (n+1)}$$ $$=\,? $$</p>
|
4,695 | <p>Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose geometric fibers are (smooth) K3 surfaces, endowed with polarization of some fixed degree.) Are there only finitely man... | user1640 | 1,640 | <p>One classical trick that may be useful for reduction to the abelian scheme case is the Kuga-Satake construction which takes a weight 2 Hodge structure of K3 type to a weight 1 Hodge structure built out of its Clifford algebra. Although I've never read it, there is a paper of Rizov (<a href="http://arxiv.org/abs/mat... |
4,174,404 | <p>So I'm trying to find conditions on <span class="math-container">$p$</span> and <span class="math-container">$k$</span> under which <span class="math-container">$1+p^k$</span> is a power of <span class="math-container">$2$</span>, i.e.,
<span class="math-container">$$1+p^k=2^\ell.$$</span></p>
<p>Let's first note so... | Mastrem | 253,433 | <p>By <a href="https://en.wikipedia.org/wiki/Catalan%27s_conjecture" rel="nofollow noreferrer">Mihăilescu's theorem</a> there are no solutions with <span class="math-container">$k\ge 2$</span>. For <span class="math-container">$k=1$</span>, we obtain that <span class="math-container">$p=2^\ell-1$</span>. Such primes <s... |
1,459,824 | <p><a href="https://i.stack.imgur.com/J229G.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/J229G.png" alt="Picture:"></a></p>
<p>I first thought I'd just have to do cases, i.e. if $m,n$ are even, $m=2l, n=2k$, where $k, l\in \Bbb Z$. But even in this case, alone, I wind up with $4kl(16l - 16k) = 6... | Tintarn | 197,823 | <p>Alright, let's see how to do this.</p>
<p>First, it clearly suffices to prove that the term is always divisible by 2, 3 and 5 to prove that it's divisible by $30=2 \cdot 3 \cdot 5$.</p>
<p>So why is it divisible by 2?</p>
<p>Assume that it's not i.e. it's odd. Then clearly all factors must be odd.
In particular, ... |
1,598,555 | <p>How to solve such questions in which we are given $N$ distinct numbers i.e $1$ to $N$ and also given certain restrictions like $(x,y) , (x,z) , (z,x) , (z,a) , (b,c)$ and so on should never occur simultaneously.</p>
<p>For example:</p>
<p>$N=4$ and $(1,2) , (2,3) , (3,4)$ should never come together.</p>
<p><stro... | JMoravitz | 179,297 | <p>The easiest in my opinion would be to use Inclusion-Exclusion.</p>
<p>For your example of permutations of $\{1,2,3,4\}$ avoiding substrings $(1,2),(2,3),(3,4)$, let $U$ be the set of all possible permutations of $\{1,2,3,4\}$.</p>
<p>Let $A_{(1,2)}$ be the set of permutations which <strong>do</strong> contain a su... |
3,426,659 | <p>Find the domain and range of <span class="math-container">$f(x)=\sin^{-1}(\sqrt{x^2+x+1})$</span></p>
<p><strong>Finding the domain:</strong></p>
<p><span class="math-container">$$x^2+x+1>=0 \text { it is always true }$$</span></p>
<p><span class="math-container">$$-1<=\sqrt{x^2+x+1}<=1$$</span>
<span cl... | Z Ahmed | 671,540 | <p>The range of <span class="math-container">$f(x)=sin^{-1} x$</span> is <span class="math-container">$[-\pi/2,\pi/2]$</span> over the domain <span class="math-container">$[-1,1]$</span>.
So here range the range of the the given function is <span class="math-container">$[\pi/3,\pi/2].$</span></p>
|
3,426,659 | <p>Find the domain and range of <span class="math-container">$f(x)=\sin^{-1}(\sqrt{x^2+x+1})$</span></p>
<p><strong>Finding the domain:</strong></p>
<p><span class="math-container">$$x^2+x+1>=0 \text { it is always true }$$</span></p>
<p><span class="math-container">$$-1<=\sqrt{x^2+x+1}<=1$$</span>
<span cl... | Mohammad Riazi-Kermani | 514,496 | <p>The domain of <span class="math-container">$$f(x)=\sin^{-1}(\sqrt{x^2+x+1})$$</span> is where you get <span class="math-container">$$0\le x^2+x+1\le 1$$</span> which is the interval <span class="math-container">$[-1,0]$</span></p>
<p>Note that <span class="math-container">$f(-1/2)=\pi/3$</span> is the minimum value... |
2,886,238 | <p>While I was considering random walk on $S^1$, I needed to compute the following.</p>
<p>$$\lim_{m\rightarrow\infty}\sum_{k=1}^m \left(\cos{\frac{2\pi k}{2m+1}}\right)^{(2m+1)^2}$$</p>
<p>I guessed it should converges to trace of heat kernel on $S^1$. So I tried to use Mathematica. Mathematica says, it converges to... | Szeto | 512,032 | <p>In response to the general case:</p>
<blockquote>
<p>Claim:</p>
<p>If $|f_m(k)|\le g(x)$ where $g$ is an integrable function, and $\lim_{m\to\infty}f_m(x)=f(x)$ then $$\lim_{m\to\infty}\sum^m_{k=1}f_m(k)=\sum^\infty_{k=1}f(k)$$</p>
</blockquote>
<p>Proof:</p>
<p>Define a ‘step function’:
$$
H_t(x)=
\begin{... |
896,270 | <p>Is the collection of all finite sets a set or a proper class? If it's a set how to prove otherwise alos how to prove? And finite set means sets which are in 1-to-1 correspondence with <span class="math-container">$\{1,2,3,...,n\}$</span> where <span class="math-container">$n$</span> is a natural number.</p>
| drhab | 75,923 | <p>Let <span class="math-container">$U$</span> denote the universe of all sets. </p>
<p>If <span class="math-container">$F=\left\{ a\mid a\text{ finite}\right\} $</span>
is indeed a set then <span class="math-container">$S=\left\{ \left\{ x\right\} \mid x\in U\right\} $</span>
is a subset of <span class="math-containe... |
159,654 | <p>Is there a closed form expression for the expectation of $\log(x+a)$ (with $a>0$, the case $a=0$ is obvious) when X follows a beta distribution?</p>
| Carlo Beenakker | 11,260 | <p>closed form --- yes, simple --- no:</p>
<p>$$I(\alpha,\beta;a)=\int_0^1 dx\; \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}\ln(x+a)=$$</p>
<p>$$\frac{\alpha}{a(\alpha+\beta)} \; _3{F}_2\left(1,1,\alpha+1;2,\alpha+\beta+1;-1/a\right)+\ln a$$</p>
<p>a few special cases:</p>
<p... |
72,588 | <p>Imagine I have a point-like Brownian particle, with diffusion constant $D$, and I place it at some initial coordinate in a cage of known geometry. Assuming the volume $V$ of the cage is "everywhere" accessible, how long does it take the particle to lose its "memory" of the initialization coordinate and achieve a un... | Alice | 16,937 | <p>Seems like the timescale should depend on the number and sizes of spherical balls it takes to cover the entire space $V$. For example if your space has a narrow neck connecting two balloons then it would take a while for your particle to diffuse from the balloon where it started through the neck to the other side.<... |
1,000,836 | <p><strong>Consider a regular tetrahedron with vertices $A,B,C,D$. A bug starts crawling from $A$. The bug moves from one vertex to the other along the edges continuously until it reaches $D$, where there is glue and hence once the bug reaches $D$, it gets stuck. The bug can move to any one vertex from any other vertex... | Pradhan | 166,800 | <p>Using symmetry, you can cut down the number of variables a bit more and extend this to the case of a complete graph with $n$ vertices - the tetrahedron is the case with $n=4$. Assume that the bug starts from vertex $curr$ and that the designated end vertex is $glue(\neq curr)$. Let $p_{v}$ and $p_{other}$ be the pro... |
4,013,573 | <p>I think this set is uncountable, but I don't really know how to prove this. Maybe somehow find an injection from <span class="math-container">$\mathcal{P}( \mathbb{N} \times \mathbb{N}) \rightarrow A$</span>?</p>
| jjagmath | 571,433 | <p>For any <span class="math-container">$X\subseteq \mathbb N$</span>, the relation <span class="math-container">$R_X = \{(x,x) | x \in X\}$</span> is anti-symmetric.</p>
|
699,347 | <p>At p. 107 of the book <em>Cohen-Macaulay Rings</em> by Bruns and Herzog, the authors write </p>
<blockquote>
<p>"any module of finite projective dimension (over a Gorenstein ring $R$) has finite injective dimension as well, simply because $R$ has finite injective dimension by definition." </p>
</blockquote>
<p>C... | Dietrich Burde | 83,966 | <p>Because a Noetherian ring $R$ has finite injective dimension if and only all R-modules with finite projective dimension also have finite injective dimension. See the proof of this on page $8$, and also the Theorem below there in <a href="http://www.math.hawaii.edu/~lee/homolog/Goren.pdf" rel="nofollow">http://www.ma... |
37,643 | <p>I have a set of data that measurements of temperature vs time (date).
It is easy to realize when the sensor went wrong as one might see a blip on the data.
How to teach to Mathematica to:
1) Detect the "Blip"
2) Delete this Blip
3) Replace the corrupted point for one derived as the mean of the previous data (before ... | Dr. belisarius | 193 | <p>Filtering by the "second difference" and removing the two most prominent outliers. The rest of the data remain unchanged:</p>
<pre><code>dd = Ordering@Differences@Abs@Differences[l[[All, 2]]];
Show@{DateListPlot[l, PlotRange -> All, PlotStyle -> Large, Joined -> True],
DateListPlot[ReplacePart[l, Th... |
1,232,871 | <p>A closed subset of a locally connected space is locally connected?</p>
<p>A closed subset of a locally path-connected space is locally path-connected?</p>
<p>Is there any counter-example?</p>
| bof | 111,012 | <p>$\mathbb R$ is locally connected and locally path-connected. The Cantor set is a closed subset of $\mathbb R$ which is not locally connected at any of its points. For a simpler example, take $\{0\}\cup\{\frac1n:n\in\mathbb N\}$; however this example is locally connected at some of its points. </p>
|
78,301 | <p>What can be said about the rate of growth of $f(n)$, defined by
$$f(n) = \min_{|V(G)|=n} \left[ \chi(G) + \chi(\bar{G}) \right],$$ where the minimum is taken over all graphs $G$ on $n$ vertices.</p>
<p>Two observations.</p>
<p>(1) Either $G$ or $\bar{G}$ contains a clique on roughly $\log{n}$ vertices by Ramsey ... | GeoffDS | 8,671 | <p>Although there is already an accepted answer, I answer to give a bit more information, for what I have to say would not fit in a comment.</p>
<p>This is the Nordhaus-Gaddum Theorem:</p>
<blockquote>
<p>If $G$ is a graph of order $n$, then</p>
<ol>
<li>$2 \sqrt{n} \leq \chi(G) + \chi(\bar{G}) \leq n+1$</li... |
618,984 | <p>Given $2$ prime numbers,$ p$ and $q$, that are both not even, and $3$ doesn't divide $p-1$ or $q-1$, and
$n=pq$, how many elements in $Z^*_n$ exists that has $b$ such that $b^3\equiv a\pmod n$ .</p>
<p>I know that the answer is $\phi(n)$ - that is, $(p-1)*(q-1)$, but I can't find out how to prove it :( </p>
| benh | 115,596 | <p>Let $a \in \Bbb (Z/n \Bbb Z)^\times$. We show that there is $k$ such that we can choose $b=a^k$, so $a^{3k}\equiv a \mod n$.</p>
<p>By Fermat's Little Theorem, we have $a^{\varphi(n)}\equiv 1 \mod n$.</p>
<p>As $3 \nmid (p-1)(q-1) = \varphi(n)$, we know $\gcd(3,\varphi(n)) = 1$. So by the Euclidean Algorithm you ... |
654,688 | <p>Why $\log|x|$ is a locally integrable function and $1/|x|$ is not? Id know how their graphs look like but I don't know what is the exact difference causing local integrability of the first one.</p>
| BaronVT | 39,526 | <p>First: there are two obstacles to <strong>global</strong> integrability here: behavior at $0$ and behavior at $\infty$. The function $1/|x|$ fails in both regards, that is</p>
<p>$$
\int_0^b 1/|x| \,dx = \infty
$$</p>
<p>for any $b$, and </p>
<p>$$
\int_a^\infty 1/|x|\,dx = \infty
$$</p>
<p>for any $a$.</p>
<p>... |
3,286,415 | <p>Solve for the angle <span class="math-container">$x$</span> in the right triangle without trigonometry.</p>
<p><a href="https://i.stack.imgur.com/swWdP.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/swWdP.jpg" alt="enter image description here"></a></p>
<p>I don't know how to find the angle <sp... | farruhota | 425,072 | <p>Alternatively, from the Sine theorem:
<span class="math-container">$$\frac{PQ}{\sin x}=\frac{BQ}{\sin \angle BPQ} \Rightarrow\\
\sin \angle BPQ=\frac{BQ\sin x}{PQ}=\frac{CQ}{PQ}=\frac12 \Rightarrow \angle BPQ=30^\circ \Rightarrow \angle PAQ=15^\circ \Rightarrow \\
2x=90-\angle PAQ=75^\circ \Rightarrow x=37.5^\circ.$... |
1,701,470 | <p>Could someone please point out resources or provide the solution with detailed steps to calculate the following limits?</p>
<p>If possible, please provide solutions with or without using L'Hopital's rule.</p>
<p>$$
\underset{x\rightarrow\infty}{\lim}\left[\frac{e^{x}}{e^{x^{2}}}\right]
$$</p>
<p>Thanks,</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>Hint:
$$\frac{e^{x}}{e^{x^2}} = e^{x-x^2}$$</p>
|
100,577 | <p>We know that the intersection of a smooth cubic surface in $\mathbb{P}^3$ with a plane is a cubic, a line+a conic or three disjoint lines. Can this intersection consist of a conic and its tangent line?</p>
| Will Sawin | 18,060 | <p><a href="https://en.wikipedia.org/wiki/Preadditive_category" rel="nofollow noreferrer">Preadditive categories</a> are probably the most general context. They have endomorphism rings, since one can both add and multiply morphisms. Most examples of preadditive categories are abelian categories, such as abelian groups,... |
100,577 | <p>We know that the intersection of a smooth cubic surface in $\mathbb{P}^3$ with a plane is a cubic, a line+a conic or three disjoint lines. Can this intersection consist of a conic and its tangent line?</p>
| Qiaochu Yuan | 290 | <p>First, I am not really sure what you mean by "it is hard to come across a general theory of ring actions." This is <strong>precisely</strong> module theory! If <span class="math-container">$f : R \to S$</span> is any ring homomorphism whatsoever, then composition with the natural map <span class="math-container">$S ... |
3,893,117 | <p>Claim: <span class="math-container">$2^{n+1} >n^{2}$</span> for any positive integer n.</p>
<p>My proof:</p>
<p><span class="math-container">$n = 1 \longrightarrow$</span></p>
<p><span class="math-container">$2^{1+1} > 1^{2} \iff 4 > 1$</span></p>
<p>Check.</p>
<p><span class="math-container">$n=2\longright... | marty cohen | 13,079 | <p>Playing algebraically.</p>
<p><span class="math-container">$\begin{array}\\
s(n)
&=\sum_{k=0}^{n}\left(\sum_{l=0}^{k}\binom{n}{k}\binom{k}{l} l\right)\\
&=\sum_{k=0}^{n}\left(\sum_{l=0}^{k}\dfrac{n!k!}{k!(n-k)!l!(k-l)!}l\right)\\
&=\sum_{k=0}^{n}\left(\sum_{l=0}^{k}\dfrac{n!l}{(n-k)!l!(k-l)!}\right)\\
&a... |
77,306 | <p>I'm not entirely sure what I want to ask here, so please bear with me!</p>
<p>I think the explanation they give us in school for how finding the antiderivative by substitution works is: $$\int f(g(t))g'(t)dt=(Fg)(t)=F(g(t))=F(u)=\int f(u)du$$</p>
<p>But I never really understood the equality $F(u)=\int f(u)du$... | Riley E | 5,330 | <p>If I understand correctly, the issue is due to unfortunately sloppy (standard) notation. Let's try tracing substitution through for a <em>definite</em> integral.
Suppose we're looking at
$$ \int_a^b f(g(t)) g'(t) dt. $$
If we make the substitution $u=g(t)$, then $du=g'(t)dt$. However, there's more to do be... |
3,554,502 | <blockquote>
<p>Two players, A and B, play the following backgammon variant.</p>
<p>A series of steps is labelled <span class="math-container">$-2$</span>, <span class="math-container">$-1$</span>, <span class="math-container">$0$</span>, <span class="math-container">$1$</span> and <span class="math-container">$... | joriki | 6,622 | <p>The analysis in quarague’s answer isn’t correct because it only takes the possibility of doubling once into account, whereas future doubling opportunities in fact increase the expected payoff.</p>
<p>Denote by <span class="math-container">$x_k$</span> the expected value of the game for <span class="math... |
1,915,097 | <p>For five years I tried to understand how Fourier transform works. Read a lot of articles, but nobody could explain it in simple terms. Two weeks ago I stumbled upon the video about a 100 years old machine that calculates Fourier series mechanically: <a href="https://www.youtube.com/watch?v=NAsM30MAHLg">https://www.y... | Sandipan Dey | 370,330 | <p>As per the <code>python</code> implementation <a href="https://pythonnumericalmethods.berkeley.edu/notebooks/chapter24.03-Fast-Fourier-Transform.html" rel="nofollow noreferrer">here</a> of the 1D Cooley-Tukey FFT algorithm (with the assumption that the length of the time-domain input signal <span class="math-contain... |
2,838,874 | <p>Consider $$φ(x, y , z) = x^2 + 2y^2 + 4z^2 −xy −2xz −3yz$$
find the coordinate transformation (translation or rotation) to eliminate $xy$, $xz$ and $yz$.
In $\mathbb R²$, with conic sections, I would do this with
$$\begin{pmatrix}x \\ y\end{pmatrix}=\begin{pmatrix}\cos\alpha &-\sin\alpha\\\sin\alpha & \cos\... | José Carlos Santos | 446,262 | <p>Suppose that $10X^6-15X^2+7=P(X)Q(X)$, where $P(X)$ and $Q(X)$ are non-constant polynmials, whose degrees are $m$ and $n$ respectively. Then $m+n=6$ and$$X^6P\left(\frac1X\right)Q\left(\frac1X\right)=7X^6-15X^4+10.\tag1$$But $X^6P\left(\frac1X\right)Q\left(\frac1X\right)=X^mP\left(\frac1X\right)X^nQ\left(\frac1X\rig... |
1,925,176 | <p>For the question "The corners of a triangle are the points $P(4,7)$, $Q(-2,5)$ and $R(3, -10)$. Find the length of each side of $\triangle PQR$, giving your answers in terms of surds".
I have the answer to the first part as : $|PQ| = 2\sqrt{10}$, $|PR| = \sqrt{290}$, $|RQ| = 5\sqrt{10}$.</p>
<p>The second part asks... | JMP | 210,189 | <p>As $|PR|$ is the maximum length, the right-angle, if any, must be $\angle PQR$.</p>
<p>The gradient of $QR$ is $\dfrac{-2-3}{5-(-10)}=\dfrac{-5}{15}=\dfrac{-1}3$.</p>
<p>The gradient of $QP$ is $\dfrac{-2-4}{5-7}=\dfrac{-6}{-2}=3$.</p>
<p>Two lines are perpedicular iff $m _1=\dfrac{-1}{m_2}$, where $m_1,m_2$ are ... |
1,158,652 | <p>$$f(x) = \frac{1}{1-2\cos^2x}$$</p>
<p>The result of $f'(x)$ should be equals</p>
<p>$$f'(x) = \frac{-4\cos x\sin x}{(1-2\cos^2x)^2}$$</p>
<p>I'm trying to do it in this way but my result is wrong.</p>
<p>$$f'(x) = \frac {1'(1-2\cos x)-1(1-2\cos^2x)'}{(1-2\cos^2x)^2} =
\frac {1-2\cos^2x-(1-(2\cos^2x)')}{(1-2\c... | Uncountable | 215,630 | <p>The derivative of the function that is constantly $1$ should be zero, not $1$.</p>
|
1,158,652 | <p>$$f(x) = \frac{1}{1-2\cos^2x}$$</p>
<p>The result of $f'(x)$ should be equals</p>
<p>$$f'(x) = \frac{-4\cos x\sin x}{(1-2\cos^2x)^2}$$</p>
<p>I'm trying to do it in this way but my result is wrong.</p>
<p>$$f'(x) = \frac {1'(1-2\cos x)-1(1-2\cos^2x)'}{(1-2\cos^2x)^2} =
\frac {1-2\cos^2x-(1-(2\cos^2x)')}{(1-2\c... | creative | 166,713 | <p>$f(x)=\dfrac{1}{1-2\cos^2 x}=-\dfrac{1}{2\cos^2 x-1}=-\dfrac{1}{\cos 2x}=-\sec 2x$ </p>
<p>$\therefore f'(x)=-2\sec(2x)\tan(2x) \quad\blacksquare$ </p>
<p>Check the list of formulas for finding derivatives of trigonometric functions <a href="http://www.math.com/tables/derivatives/more/trig.htm" rel="nofollow">here... |
1,652,107 | <p>Since the limit $\frac{\sin(x)}{x}=1$ for $x \rightarrow 0$, I wondered about the infinite product:</p>
<p>$$\prod^{\infty}_{n=1} n \sin \left( \frac{1}{n} \right)=\sin(1) \cdot 2 \sin\left( \frac{1}{2} \right) \cdot 3 \sin\left( \frac{1}{3} \right) \dots$$</p>
<p>By numerical experiment in Mathematica it seems to... | choco_addicted | 310,026 | <p>I suppose that you know
$$
\lim_{x\to 0}\frac{\sin x}{x}=1,\qquad \lim_{x\to 0}\frac{1-\cos x}{x^2}=\frac{1}{2}.
$$
Then,
\begin{align}
\lim_{x\to 0}\frac{1-\cos x \sqrt{\cos 2x}}{x^2}&= \lim_{x\to 0}\frac{1-\cos x \sqrt{\cos 2x}}{x^2}\cdot \frac{1+\cos x \sqrt{\cos 2x}}{1+\cos x \sqrt{\cos 2x}}\\
&=\lim_{x\... |
321,417 | <p>I was wondering: does every quadratic matrix define an endomorphism? </p>
<p>Sure the map defined by the quadratic matrix goes from a vector space of $n$ dimensions to a vector space of $n$ dimensions. But why are these vector spaces necessarily the same?</p>
| Julien | 38,053 | <p>Edit: Another decomposition which involves an upper triangular matrix (therefore a lower triangular one if you apply it to the adjoint first) is the <a href="http://en.wikipedia.org/wiki/QR_decomposition" rel="nofollow">QR decomposition</a>, which is based on <a href="http://en.wikipedia.org/wiki/Gram-Schmidt_orthog... |
321,417 | <p>I was wondering: does every quadratic matrix define an endomorphism? </p>
<p>Sure the map defined by the quadratic matrix goes from a vector space of $n$ dimensions to a vector space of $n$ dimensions. But why are these vector spaces necessarily the same?</p>
| imj | 65,139 | <p>Edit : Did not see the edit above, my answer provides comparaison between Choleski and QR:</p>
<p>Once you have $A= ( P D^{1/2} P^t) (P D^{1/2} P^t)^t$ you can perform $QR$ decomposition (using Householder for instance) on $B=(PD^{1/2}P^t)^t=QR$ Then since $A=B^tB$, you get :</p>
<p>$$A = R^tQ^tQR = R^tR$$ and thu... |
2,283,981 | <p>When presented with an equation, say, $y=5x^3+7x^2+4x+9$, you can write on the second line, $\frac{dy}{dx}=15x^2+14x+4$. Similarly, $f(x)=5x^3+7x^2+4x+9$ and $f'(x)=15x^2+14x+4$. But is there a way to write "the derivative of $5x^3+7x^2+4x+9$ is $15x^2+14x+4$" in just one line? </p>
<p>What should l write, $\frac{d... | MPW | 113,214 | <p>Just as the symbol $$\int(\cdots)\;dx$$ denotes the antiderivative of something (the expression where the "$\cdots$" is), so the symbol $$\frac{d}{dx}(\cdots)$$ denotes the derivative of something (again, the expression where the "$\cdots$" is).</p>
<p>For example, you would have $$\frac{d}{dx}(13x^2-27x+1) = 26x-2... |
3,138,158 | <p>How can I show that if the cube power of a matrix is the null matrix, then the matrix itself is singular?</p>
| YiFan | 496,634 | <p><strong>Hint:</strong> Note that <span class="math-container">$\det(AB)=\det(A)\det(B)$</span>, so <span class="math-container">$\det(A^3)=(\det(A))^3$</span>. What does <span class="math-container">$(\det A)^3=0$</span> tell you?</p>
|
508,522 | <p>Take a list of the first $n$ primes $P_n=\{2,3,5,7,11,\ldots\}$ and convert the sequence into a binary string</p>
<p>$$S_n = 101110111\ldots$$</p>
<p>Compress the string with your favorite compression algorithm (the details don't matter) and let the size of the string after compression be $C(S_n)$. Compare this to... | marty cohen | 13,079 | <p>It turns out that
storage of primes can be compressed
by an arbitrary amount,
though the storage needed
is exponential in the compression factor.</p>
<p>Note:
This has been corrected and been made more precise.</p>
<p>Let $p_n$ be the $n$-th prime
(with $p_1 = 2$),
and let
$P_n$ be the product of the first $n$ pri... |
870,937 | <p>Is there a function almost everywhere $0$ on $\mathbb{R}$ whose graph is dense in $\mathbb{R^2}$?</p>
<p>How to establish such strange funciton?</p>
| Rene Schipperus | 149,912 | <p>Yes. Define a function $f:\mathbb{Q} \rightarrow \mathbb{Q}$ such that
for every $q \in \mathbb{Q}$ and every interval $I$ there is $p\in I$ such that $f(p)=q$.
This is actually easy to achieve, enumerate pairs $(I,q)$ here $I$ is an interval with rational endpoints (there are countably many) amd $q$ is rational. N... |
19,124 | <p>To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and "identify opposite sides". We can imagine gluing together one pair of opposite sides to get a cylinder, and then gluing together opposite ends of the cylinder to get a torus. (Pro... | Community | -1 | <p>I don't think it's necessary or a good idea to invoke a specific game, because any given game will be one that only a small fraction of your students have played. Just say, "You know how in some video games, if you go off one side of the screen, you reappear on the other side? Say you have a game where this hap... |
4,116,827 | <p>Consider <span class="math-container">$x:\mathbb{R}_+ \to \mathbb{R}_+$</span>, where <span class="math-container">$x\not\in\mathcal{L}^1$</span>, i.e.,
<span class="math-container">$$\int_0^\infty x(s)ds = \infty.$$</span>
Consider the convolution
<span class="math-container">$$h(t) = \int_0^t e^{-(t-s)}x(s)ds.$$</... | Henry Lee | 541,220 | <p>you could rewrite:
<span class="math-container">$$h(t)=e^{-t}\int_0^t e^sx(s)ds$$</span>
Now consider the conditions which are required for:
<span class="math-container">$$t\to\infty,h\to0$$</span></p>
<p>it looks like the integral, and so the integrand, would have to diverge slower than <span class="math-container"... |
1,330,752 | <p>Please excuse the simplicity of this question, but I am very new to groups and fields. I only seek an simplistic / intuitive expalnation, and confirmation / refutation re whether I am on the right track. I confess I am a little averse to reading about them from the ground up, as my motivation is quickly deteriorated... | Zach Stone | 38,565 | <p>While that's actually a pretty clever idea, it's not what ideal means. First of all, Ideals need to be subrings, so if $16$ and $-4$ are in the ideal, then so must $16+(-4)=12$. Secondly, Ideals are meant to be "closed under multiplication." The idea is to generalize the notion of "has a factor of n." So, for any nu... |
3,315,113 | <p>Are there some basic criteria by which to check whether a polytope <span class="math-container">$P\subset\Bbb R^n$</span> is a <span class="math-container">$01$</span>-polytope, that is, can be rotated and scaled to have vertices in <span class="math-container">$\{0,1\}^n$</span>, maybe even by using higher-dimensio... | M. Winter | 415,941 | <p>I found two further criteria that helped me to eliminate a lot of candidates from my list of symmetric polytopes.</p>
<ul>
<li><p>Assuming a sufficiently symmetric polytope <span class="math-container">$P$</span> with barycenter in the origin, we can ask for the cosines <span class="math-container">$\cos\sphericala... |
833,017 | <blockquote>
<p>What is the maximum size of an antichain of $[n]:=\{1,2,3,\dots,n\}$ (say $\mathcal{A}$) such that $\mid A\cap B\mid \ge k$ where
$A,B\in \mathcal{A}$ and $1\le k\le n-1$?</p>
</blockquote>
<p>By antichain, I mean an antichain of the poset $(2^{[n]},\subseteq)$ where $2^{[n]}$ is the power set of $... | Ben Grossmann | 81,360 | <p>Not sure how to answer fully, but this provides a decent lower-bound:</p>
<p>Set $k = \lfloor n/2 \rfloor$.
Split $[n]$ into $A = \{1,\dots,k\}$ and $B = \{k + 1,\dots,2k\}$. Label these elements $a_1,\dots,a_k$ and $b_1,\dots,b_k$ respectively.</p>
<p>Consider any $S \subset \{1,\dots,k\}$. Define the set $f(S)... |
1,056,194 | <p>With the regular trig functions, if I ever end up with something like $\operatorname{trig}_1(\operatorname{arctrig}_2(f(x))$, where $\text{trig}_1$ and $\text{trig}_2$ are two arbitrary trigonometric functions, I can draw a right triangle to find a formula for this that doesn't involve any trigonmetric functions.</p... | Najib Idrissi | 10,014 | <p>Hyperbolic functions satisfy the fundamental identity
$$\cosh(x)^2 - \sinh(x)^2 = 1.$$</p>
<p>This identity comes from the interpretation of hyperbolic functions in terms of hyperbolic triangles, cf. <a href="https://en.wikipedia.org/wiki/Hyperbolic_function" rel="nofollow">Wikipedia</a> for example.</p>
<p>So now... |
1,854,963 | <p>Let $\overline{\mathbb{F}}_2$ be an algebraic closure of $\mathbb{F}_2$, and let $\alpha\in\overline{\mathbb{F}}_2$ be such that $\alpha^2+\alpha+1=0$. Prove that if a polynomial $P$ is irreducible in $\mathbb{F}_2$ but $P$ is reducible in $\mathbb{F}_2[\alpha]$, the deg $P$ is even.</p>
| Lubin | 17,760 | <p>Here’s a shorter argument:<br>
Your $\alpha$ is a cube root of unity, and $\Bbb F(\alpha)=\Bbb F_4$. The extension os Galois of course, and the nontrivial automorphism exchanges $\alpha$ with $\bar\alpha=\alpha^2=\alpha+1$.<br>
A monic polynomial $g\in\Bbb F_4[X]$ either has coefficients involving $\alpha$ or it doe... |
1,517,202 | <p>Question: Find $\frac{dy}{dx}$ implicity if $x=\tan(xy)$.</p>
<p>So far, I have used implicit differentiation on both sides of the equation and ended up with two different answers. The first one was a bit more work than the second one. </p>
<p>My first answer is $\frac{dy}{dx}=\frac{\cos^2{xy}(1-y\sec^2{xy})}{x}$.... | Andrew D. Hwang | 86,418 | <p>Your answers are identical:
$$
\frac{1}{\sec^{2}(xy)} = \cos^{2}(xy).
$$</p>
|
81,079 | <p>Some time ago I met a forester. He told that there are only larches and spruces in his forest.
He also said that there are exactly $10$ spruces at the distance of exactly 1 km from each larch. Next, he argued that there are more larches than spruces in his forest. Can he be right?</p>
| TonyK | 1,508 | <p>Not an answer, but I think some top-level clarification is needed here. From the comments, it seems that the following interpretation is probably what was intended: </p>
<ol>
<li>For each larch, the number of spruces that are exactly 1km from it is exactly 10.</li>
<li>There are more larches than spruces.</li>
</o... |
1,400,987 | <blockquote>
<p>Given $x\in \left(0; \frac\pi2\right)$. Prove that $$\sin x>\frac{2x}{\pi}$$</p>
</blockquote>
<p>This is my try:</p>
<p>Let $y=\sin x-\frac{2x}\pi\implies y ' = \cos x - \frac2\pi\implies y ''=-\sin x <0; \forall x\in \left(0; \frac\pi2\right)$
$\implies \left\{\begin{matrix} y '(x)> y '(\... | Empty | 174,970 | <p>Consider the function , $f(x)=\frac{\sin x}{x}$ in $(0,\pi/2)$. </p>
<p>Then , $$f'(x)=\frac{x\cos x-\sin x}{x^2}<0$$[<strong>Using the relation $x<\tan x$ in $(0,\pi/2)$</strong>]</p>
<p>So, $f$ is monotone decreasing in $(0,\pi/2)$.</p>
<p>So, $x<\pi/2\implies f(x)>f(\pi/2)\implies \sin x>\frac{... |
1,313,348 | <p>I am trying to figure out how $p$-adic numbers work and currently am having trouble wrapping my head around how they work, so I made a pun! HAH!</p>
<p>Jokes aside, I am working on this question</p>
<blockquote>
<p>Show that the sequence $(3,34,334,3334,.....)$ is equal to $2/3$ in $\hat{\mathbb{Z}}_5$</p>
</blo... | Joel Cohen | 10,553 | <p>Here, the notation is in the usual base $10$, so the number $a_n = 33\ldots34$ simply denotes</p>
<p>$$a_n = 3 \times 10^n + 3 \times 10^{n-1} + \ldots + 3 \times 10 + 4 = 1 + 3\sum_{k = 0}^n 10^k = 1 + \frac{10^{n+1}-1}{3}$$</p>
<p>Now, in the $5$-adic topology, the sequence $(10^n)_{n \ge 0}$ goes to $0$ at infi... |
437,960 | <p>I can't find any logic equivalence or inference rules on this. Personally, I feel that $\neg P \implies \neg T$ and $P \implies \neg T$ would mean that it follows that $\neg T$ is true regardless, and I should be able to use that fact as such in my next step. Is this proper reasoning, though?</p>
| Adriano | 76,987 | <p>Here's another way:
$$ \begin{array}{c|c|c}
\# & \text{Statement} & \text{Reason}\\
\hline
(1) & T & \text{Assumption} \\
(2) & \neg P \implies \neg T & \text{Given} \\
(3) & T \implies P & \text{Contrapositive of (2)} \\
(4) & P & \text{(1), (3), and Modus Ponens} \\
(5) &... |
3,372,482 | <p>Looking at some of the more common pythagorean triples, I noticed a trend that there are usually two numbers which differ by only one or two, with more much smaller or larger number in comparison. For example:</p>
<p><span class="math-container">$$5,12,13$$</span>
<span class="math-container">$$8,15,17$$</span>
<sp... | Mark Bennet | 2,906 | <p>A primitive Pythagorean triple is of the form <span class="math-container">$$m^2-n^2, 2mn, m^2+n^2$$</span> where <span class="math-container">$m\gt n$</span>, <span class="math-container">$m,n$</span> have no common prime factor and one of <span class="math-container">$m,n$</span> is even.</p>
<p>If <span class="m... |
2,797,106 | <p>If $C$ is a fixed non-empty set, which one of the following options imply $A=B$?</p>
<p>1) $A - C = B - C$</p>
<p>2) $A \cup C = B \cup C$</p>
<p>3) $A \cap C = B \cap C$</p>
<p>4) $A \Delta C = B \Delta C$</p>
<p>I'm not sure why all of them are not valid option.</p>
<p>The problem I'm having is with premise.... | hmakholm left over Monica | 14,366 | <blockquote>
<p>The problem I'm having is with premise. If $A=B$
then it would imply that all of the options are true. Can you maybe expand that a bit more on how to reason the problem?</p>
</blockquote>
<p>You seem to have misunderstood what the problem is. Taking part (b) as an example, you are right that
$$A=B ... |
2,797,106 | <p>If $C$ is a fixed non-empty set, which one of the following options imply $A=B$?</p>
<p>1) $A - C = B - C$</p>
<p>2) $A \cup C = B \cup C$</p>
<p>3) $A \cap C = B \cap C$</p>
<p>4) $A \Delta C = B \Delta C$</p>
<p>I'm not sure why all of them are not valid option.</p>
<p>The problem I'm having is with premise.... | Ross Millikan | 1,827 | <p>Your mistake is having the implication backwards. You are correct that $A=B$ implies each of them, but that isn't what is being asked. You are being asked which of them implies $A=B$. For (2), for example, there could be an element that is in $A$ and $C$, so is in both unions, but is not in $B$. That makes $A \n... |
2,716,545 | <p>Suppose that a point with co-ordinates $(X, Y)$ is chosen uniformly at random from the triangle $\{(x,y)\in\mathbb R^2 :0\leq x\leq1\,,0\leq y\leq\min(x,1−x)\}$. Find the density of the distribution of $X$.</p>
<p>I am struggling for when $x>1/2$.</p>
| BCLC | 140,308 | <p>Proof of</p>
<blockquote>
<p>In the case when $\Omega$ consists of finitely many elements and $P[{ω}] >
0$ for every $ω \in \Omega$, then for every probability measure $Q$ we have $Q << P$.</p>
</blockquote>
<p>For any $A \in F$, we need to show that if $P(A)=0$ then $Q(A)=0$. So, for what $A$'s do we h... |
2,669,054 | <p>Question: What is the area of the region enclosed by the curves:</p>
<p>$$2y = 4\sqrt{x},\quad y = 3,\quad \text{and} \quad 2y + 2x = 6.
$$
I have tried calculate all the definite integrals but I am not sure which curve I am supposed to subtract and which one is supposed to come first. And also, I am a little confu... | Community | -1 | <p>$y=2\sqrt{x}$</p>
<p>$y=3$</p>
<p>$y=-x+3$</p>
<p>The intersection point(s) of $y=3$ and $y=2\sqrt{x}$ are:</p>
<p>$x=\left(\dfrac 32\right)^2=\dfrac 94$.</p>
<p>The intersection point(s) of $y=-x+3$ and $y=2\sqrt{x}$ are:</p>
<p>$x^2-6x+9=4x$</p>
<p>$x^2-10x+9=0$</p>
<p>$x=9,1$</p>
<p>$x=9$ is extraneous, ... |
353,576 | <p>I am looking at the proof of Bishop's Theorem on pages 122 and 123 of Rudin's Functional Analysis. The following quote is from the the last two sentences of the proof on pg. 123.</p>
<p>"Every continuous linear functional on $C(S)$ that annihilates $A$ also annihilates g. Hence $g\in A$ by the Hahn-Banach Separat... | Leo Spencer | 66,064 | <p>By the HBST, since $\left\{ f\right\}$ is compact, $A$ is closed, and $C(X)$ is locally convex, there exists a $\Lambda\in {C(X)}^*$ such that $\Lambda(f)\not\in \Lambda(A)$. Since $A$ is a subspace of $C(X)$, the image of $A$ is $\left\{ 0\right\}$ or $\mathbb{C}$. However, any $\Lambda$ that annihilates $A$ must... |
266,228 | <p>I'm new to use the amazing map function in Mathematica, and I've found it really elegant to program in simple lines of code. However, I'd like to perform more elegant stuff. I've found this question, but its not what I'm trying to achieve. <a href="https://mathematica.stackexchange.com/questions/8147/how-do-i-use-m... | user1066 | 106 | <p><em>One</em> way would be to use <code>Inner</code>:</p>
<pre><code>Inner[Times,vector,{1,1,1},Tan[#2/#1]&]
(* {0.328928, 0.0877953, -1.11698, 0.163536, 8.47242, -1.30076, -0.876783, -0.445995,
3.10856, 13.781, -0.0637223} *)
</code></pre>
<p>To join the result to <code>vector</code>, creating a new colu... |
4,182,765 | <p>Claim: <span class="math-container">$3\mid n^4-n^2$</span> for all <span class="math-container">$n \in \mathbb{Z^+}$</span>, <span class="math-container">$n\ge2$</span>.</p>
<p>What I've done so far:</p>
<p>Base case: Let <span class="math-container">$n=2$</span>. Then <span class="math-container">$\exists k \in \m... | Henry | 6,460 | <p>If you want to continue this, then</p>
<p>you found <span class="math-container">$(m+1)^4 - (m+1)^2 = (m^4-m^2)+4m^3+6m^2+2m$</span></p>
<p>and this is <span class="math-container">$(m^4-m^2)+3(m^3+2m^2+m) +(m^3-m)$</span></p>
<p>so you have reduced the problem to showing <span class="math-container">$3|(n^3-n)$</sp... |
1,174,463 | <p><span class="math-container">$$\int x^3 \cos(2x^2-5)dx$$</span></p>
<p>I tried using the substitution method but to no avail.</p>
<p>I also tried integrating by parts but that didn't work either.</p>
| DeepSea | 101,504 | <p><strong>hint</strong>: $u = 2x^2 - 5$. Can you continue?</p>
|
1,174,463 | <p><span class="math-container">$$\int x^3 \cos(2x^2-5)dx$$</span></p>
<p>I tried using the substitution method but to no avail.</p>
<p>I also tried integrating by parts but that didn't work either.</p>
| Kishalay Sarkar | 691,776 | <p>Put <span class="math-container">$u=2x^2-5$</span>,then the integral equals <span class="math-container">$\frac{5}{8}\sin(2x^2-5)+\frac{1}{8}(2x^2-5)\sin(2x^2-5)+\frac{1}{8}\cos(2x^2-5)+C$</span>.</p>
|
2,778,914 | <p>In the book of <em>Foundations of Geometry</em> by Hilbert, at page 2, it is stated that</p>
<blockquote>
<p>I, 1. Two distinct points A and B always completely determine a
straight line a. We write AB = a or BA = a.</p>
<p>I, 2. Any two distinct points of a straight line completely determine
that line; ... | Intelligenti pauca | 255,730 | <p>If $AB=a$ and $AC=a$, with $B\ne C$, then you know from Axiom 1 that points $BC$ lie on some line: Axiom 2 states that such line is still $a$. </p>
<p>So I think the words added after "that is" are a way to better explain the meaning of this axiom, while your $AB=AC=BC=a$ would be a mere repetition of the first par... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | Tony Huynh | 2,233 | <p>I do not really know the answer, but I am inclined to agree with Brendan McKay's answer which I will just paraphrase as <em>in the long term, none.</em></p>
<p>As a counterpoint though, I do not think this will mark the end of human mathematicians since I regard mathematics as a fundamentally <em>human endeavor</em... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | Yochay Jerby | 112,792 | <p>You might be interested in the recent following great piece from Quanta magazine <a href="https://www.quantamagazine.org/in-computers-we-trust-20130222" rel="nofollow noreferrer">In Computers We Trust?</a> touching exactly on that subject.</p>
<p><em>Just a side remark — there is a difference between "how to pr... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | Fallen Apart | 62,635 | <blockquote>
<p>What advantage humans have over computers in mathematics?</p>
</blockquote>
<p>There is an active research in machine learning world, more specifically in NLP (natural language processing) that tries to answer contrary, and even more general question:</p>
<blockquote>
<p>What computers can do, that we p... |
1,867,078 | <p>The primitive element theorem is a basic result about field extensions. I was wondering whether there are nice geometric ways to visualize it or think about it. Since field spectra are singletons, it has to be about the non-trivial automorphisms of points (I think), and I don't know how to think about it.</p>
| Lubin | 17,760 | <p>Here’s a geometric argument that has nothing to do with algebraic geometry. You may find it insufficiently rigorous, but the idea is certainly sound.</p>
<p>Consider a separable extension $K\supset k$. One consequence of separability is that there are only finitely many intermediate fields $E$, $K\supset E\supset k... |
256,723 | <p>When are Noetherian conditions on a scheme genuinely essential in algebraic geometry? </p>
<p>I am under the impression that most of the time these conditions are imposed for expositional clarity and to simplify the commutative algebra involved (for example throughout Hartshorne and FGA), and that they can be remo... | Georges Elencwajg | 450 | <p>Here are some examples illustrating the genuine necessity of noetherian assumptions: </p>
<p><strong>1) Every scheme with just one point is the spectrum of a local artinian ring?</strong><br>
This is true for every noetherian one point scheme and false for every non-noetherian one point scheme.<br>
A non-noether... |
76,832 | <p>I have a complex <code>Manipulate-</code>based demonstration containing a few sliders and a Setterbar. </p>
<p>I would like to make a movie out of it, avi, or whatever else. The movie will then be used for a journal publication as a Supporting material, so the quality should be good enough. </p>
<p>When the movie... | Karsten 7. | 18,476 | <p>You can define a function that creates <code>Manipulate</code>s with a "fake" <code>SetterBar</code> and a specific <code>AutorunSequencing</code></p>
<pre><code>m[k_, seq_] := Manipulate[
Plot[{Sin[a*x] + b*Cos[3 a*x], k*x}, {x, 0, Pi}, ImageSize -> 400],
{a, 1, 2}, {b, 0, 1}, Grid@{{"k", SetterBar[k, {0, .5... |
3,333,655 | <p>Mount Kea is a dormant volcano on the island of Hawaii, and also the highest seamount in the world as measured from its base on the oceanic plate. At an unknown distance along with the oceanic plate from the peak of Mauna Kea, the angle of elevation to the peak is approximately <span class="math-container">$45.29^\c... | Rick | 449,129 | <p><span class="math-container">$$\lim\limits_{h \to 0} \frac{\ln(x+h)-\ln(x)}{h} = \lim\limits_{h \to 0}\frac{1}{h} \ln\left(1+\frac{h}{x}\right) = \lim\limits_{h \to 0} \ln\left(1+\frac{h}{x} \right)^{\frac{1}{h}}$$</span></p>
<p>We use the limit formula: <span class="math-container">$$\lim\limits_{h \to 0}(1+a)^{\f... |
2,551,410 | <p>Cardinals have a well-defined arithmetic, but I don't have enough background knowledge to check if this notion makes sense.</p>
| Alejandro Nasif Salum | 481,187 | <p>I think you can follow the usual ideas, but if it were possible to have an extension field, say $\mathbb{R}(\kappa),$ then all such fields would be isomorphic for all infinite cardinals $\kappa$, since these are all idempotent ($\kappa^n=\kappa$ for all finite $n$). Anyway, although my intuition fears the answer mig... |
2,051,843 | <p>I am analyzing the below puzzle presented <a href="https://books.google.ie/books?id=8EDCAgAAQBAJ&pg=PA37&lpg=PA37&dq=in%20an%20election,%20two%20candidates,%20Albert%20and%20Benjamin&source=bl&ots=GGih2A4OTC&sig=had3_VHldWIeszXN1HtMpG31YnE&hl=en&sa=X&ved=0ahUKEwjghIXlk-jQAhVYF8AKH... | Samar Holkar | 747,067 | <p>Adding this to Jeremy's answer. </p>
<p>Lets us suppose in a sequence m A's and n B's, such that m > n, the first time their values are equal, we have x A's and x B's, which means at 2xth position. </p>
<p>Before 2x votes, either A will be more or B. But using the Principle of Symmetric outcomes, we can declare th... |
1,458,055 | <p>What i want to prove is following integral
\begin{align}
\int_{-\infty}^{\infty} e^{-x^2} \cos(x^2) dx=\frac{1}{2} \sqrt{\left(1+\sqrt{2}\right) \pi }
\end{align}
can you give some explicit method to obtain this result?</p>
| Eclipse Sun | 119,490 | <p><strong>Hint</strong>: Try to compute
$$\left(\int_{-\infty}^{\infty} e^{-x^2} \cos(x^2) dx\right)\times\left(\int_{-\infty}^{\infty} e^{-y^2} \cos(y^2) dy\right)$$
by polar coordinates. Use the formula $2\cos x\cos y= \cos(x-y)+\cos(x+y)$ to break it into two parts and conclude one of them is zero.</p>
|
395,416 | <blockquote>
<p>Consider the sets dened by the real solutions of the inequalities $$A=\{(x,y):x^2+y^4\le 1\}$$ and $$B=\{(x,y):x^4+y^6\le 1\}$$Then which is true $A$ is subset of $B$ or $B$ is subset of $A$. </p>
</blockquote>
<p>This is <a href="http://www.isical.ac.in/~deanweb/sylques.pdf" rel="nofollow">Question ... | V-X | 64,349 | <p>The set A looks more like circle, the B looks more like a squre... and the almost-circle lies inside of almost-square. try that on your own...</p>
<p>A may be expressed as $y\in (-(1-x^2)^\frac{1}{4},(1-x^2)^\frac{1}{4})$ </p>
<p>B may be expressed as $y\in (-(1-x^4)^\frac{1}{6},(1-x^4)^\frac{1}{6})$ </p>
<p>You ... |
395,416 | <blockquote>
<p>Consider the sets dened by the real solutions of the inequalities $$A=\{(x,y):x^2+y^4\le 1\}$$ and $$B=\{(x,y):x^4+y^6\le 1\}$$Then which is true $A$ is subset of $B$ or $B$ is subset of $A$. </p>
</blockquote>
<p>This is <a href="http://www.isical.ac.in/~deanweb/sylques.pdf" rel="nofollow">Question ... | Mikasa | 8,581 | <p>Hint: Try the following points: $$(0.5,0.5),~~(0.8,0.8),~~\{(0.7,0.9),(0.7,0.8)\}$$ I know this is an strange way based on trial and error method. I did it just because your question is a multiple choices.</p>
|
1,963,572 | <p>Is it possible that a function takes in an equation and pops out another equation by modifying the first one like f(2x+2=0)=3x+3=1. </p>
| Simon | 377,049 | <p>Well, in computeralgebra you obviously do that. i.e. in software like maple/mathematica/sage you can type "simplify(3*x-x=5)" and get $x=5/2$ as a result. But that is more about programming than pure mathematics. The problem is that you have to define what an "equation" actually is. The obvious answer is something l... |
2,158,796 | <p>Let $K = \mathbb{Q}(\alpha)$ where $\alpha^3 - 50\alpha- 10= 0$.</p>
<p>Prove that $\{1, \alpha, \alpha^{2}\}$ is an integral basis of $\mathcal O_{K}$.</p>
<p>I know that the minimal polynomial is</p>
<blockquote>
<p>$$m_\alpha(x)=x^3 - 50x -10$$</p>
</blockquote>
<p>but I'm not sure where to even begin. I ha... | Adam Hughes | 58,831 | <p>Eisenstein easily verifies the polynomial is irreducible. Next, consider that the discriminant of the polynomial $x^3+ax+b$ is $-4a^3-27b^2=497300$ which is $100$ times a prime. So if we can show that $2$ and $5$ are ramified, we will see that this is an integral basis as otherwise the true discriminant would differ... |
3,332,999 | <p>Why are there <span class="math-container">$2^n-1$</span> terms in the inclusion-exclusion formula of <span class="math-container">$n$</span> sets?</p>
<p>An example of what I mean by inclusion-exclusion formula is this:</p>
<p>There are three sets (i.e. <span class="math-container">$n$</span> <span class="math-co... | Bernard | 202,857 | <p>This is because you have to take these sets <span class="math-container">$1$</span> by <span class="math-container">$1$</span>, then <span class="math-container">$2$</span> by <span class="math-container">$2$</span>, &c. and ultimately <span class="math-container">$n$</span> by <span class="math-container">$n$</... |
1,465,138 | <p>I know how to prove that the derivative of $\ln(x)$ is ${1\over x}$ using the definition $f'(x) = {f(x+h) - f(x) \over h}$ but I have ran into trouble proving that the derivative of $\ln(f(x))$ is ${f'(x) \over f(x)}$. Is there a way to prove this strictly using the definition of the derivative?</p>
| mathreadler | 213,607 | <p>The critical points are where partial derivatives are 0, right?</p>
<p><strong>Hint:</strong></p>
<p>So we have a product of numbers. A product which is equal to 0 is when <em>at least one</em> of the factors is 0. Also we can use that the exponential function is never $0$ for any input.</p>
|
79,012 | <p>Let
$$f(z)=\frac{1}{(z-1)(z-2)}$$
and let
$$R_1=\Bigl\{z\Bigm| 1<|z|<2\Bigr\}\quad\text{ and }\quad R_2=\Bigl\{z\Bigm| |z|>2\Bigr\}.$$</p>
<p>How do you find the Laurent series convergent on $R_1$? Also how do you do it for $R_2$?</p>
<p>I'm having serious trouble with this as I can't see how to expand ... | Matthew Towers | 5,316 | <p>Most problems like this can be done with two tools only: partial fractions, and the result $1/(1-w) = 1+w+w^2+\cdots$ for $|w|<1$. So first split your function $f$ into $1/(z-2)-1/(z-1)$. I will show you how to cope with one of these factors, $1/(z-2)$. Writing this as $\frac{-1}{2} \frac{1}{1-z/2}$ is temptin... |
79,012 | <p>Let
$$f(z)=\frac{1}{(z-1)(z-2)}$$
and let
$$R_1=\Bigl\{z\Bigm| 1<|z|<2\Bigr\}\quad\text{ and }\quad R_2=\Bigl\{z\Bigm| |z|>2\Bigr\}.$$</p>
<p>How do you find the Laurent series convergent on $R_1$? Also how do you do it for $R_2$?</p>
<p>I'm having serious trouble with this as I can't see how to expand ... | Américo Tavares | 752 | <p>The function $f(z)$ can be expanded into two partial fractions
$$
f(z):=\frac{1}{\left( z-1\right) \left( z-2\right) }=\frac{1}{z-2}-\frac{1}{z-1}.
$$</p>
<p>We now expand each partial fraction into a geometric series. On $R_{2}$
these series are
$$
\begin{eqnarray*}
\frac{1}{z-2} &=&\frac{1}{z\left( 1-2/z\... |
1,480,844 | <p>Let ${a_n}$ be a sequence of real numbers. The backwards differences of this
sequence are defined recursively:</p>
<p>The first difference $∇a_n$ is an new sequence defined by:$∇a_{n} = a_{n} - a_{n-1} $</p>
<p>The $(k + 1)$st difference $∇^{k+1}a_n $ is $∇^{k+1}a_n = ∇^{k}a_{n} - ∇^{k}a_{n-1} $ Find $∇a_{n}$ and... | Brian M. Scott | 12,042 | <p>For the first problem you know that $\nabla a_n=2$ for all $n$, so $\nabla a_n-\nabla a_{n-1}=2-2=0$.</p>
<p>In the second you made an algebra error: $\nabla a_n=n^2-(n-1)^2=n^2-n^2+2n-1=2n-1$, not $-2n+1$. Note that this is an error that you should have caught: $-2n+1$ is clearly negative for $n\ge 1$, but $n^2-(n... |
910,844 | <p>I need help evaluating this with contour integration
$$
\int_{0}^{1}{\ln\left(\,x\,\right)\over
\,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x
$$
I am not sure as to how to work with the branch cuts of both
$\ln\left(\,x\,\right)$ and $\sqrt{1 - x^{2}}$</p>
<p>Second part is to evaluate
$$ \int_{0}^1 \frac{\s... | dustin | 78,317 | <p>As one of the comments suggested, we do want to obtain $\ln(\sin(\theta))$. Then we will solve the problem using complex analysis.</p>
<p>Let $x = \sin(\theta)$ and $dx = \cos(\theta)d\theta$. Therefore, $0\mapsto 0$ and $1\mapsto\pi/2$. Thus, the integral is
$$
\int_0^{\pi/2}\ln(\sin(\theta))d\theta = \frac{1}{2}\... |
1,171,733 | <p>Let $\{x_n\}$ be a sequence in $\mathbb{R}$ where $\forall n\in\mathbb{N}:x_n\neq 0$ and it converges to some $x\neq 0$. If the sequence is NOT monotone is it ever true that $\frac{1}{x_n}\rightarrow\frac{1}{x}$? If so what other conditions (if any) are needed and how would you show it. </p>
<p>Thanks in advance a... | Timbuc | 118,527 | <p>Suppose <span class="math-container">$x{}_n\to x\neq 0$</span> and <span class="math-container">$x{}_n\neq 0$</span> (enough to assume <span class="math-container">$x{}_n\to x\neq 0$</span>, then <span class="math-container">$x{}_n\neq 0$</span> for almost all <span class="math-container">$n\in\Bbb N$</span> - i.e. ... |
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