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 |
|---|---|---|---|---|
1,705,481 | <blockquote>
<p>$$\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2}$$</p>
</blockquote>
<p>I have tried the comparison test with $\frac{1}{n}$ and got $0$ with $\frac{1}{n^2}$ I got $\infty$</p>
<p>What should I try?</p>
| Marco Cantarini | 171,547 | <p>Fix $0 < \alpha<1 $. If $n
$ is sufficiently large, say $n\geq N$, holds $$\log\left(n\right)\leq n^{\alpha}\tag{1}
$$ hence $$\sum_{n\geq1}\frac{\log\left(n\right)}{n^{2}}\leq C + \sum_{n\geq N}\frac{1}{n^{2-\alpha}}<\infty.$$ Maybe it's interesting to note that, if $s>1
$ the Riemann zeta function i... |
45,163 | <p>I would like to get reccomendations for a text on "advanced" vector analysis. By "advanced", I mean that the discussion should take place in the context of Riemannian manifolds and should provide coordinate-free definitions of divergence, curl, etc. I would like something that has rigorous theory but also plenty of ... | amWhy | 9,003 | <p>Se<a href="http://www.archive.org/details/117714283" rel="nofollow">e Willard Gibbs, archive text</a> for an old text on Vector analysis, also referenced in Wikipedia, available free, and downloadable...At the very least, it should be of historical importance?</p>
<p>Most of my search returned Janich's text as a re... |
3,980,845 | <p>Given <span class="math-container">$X,Y$</span> i.i.d where <span class="math-container">$\mathbb{P}(X>x)=e^{-x}$</span> for <span class="math-container">$x\geq0$</span> and <span class="math-container">$\mathbb{P}(X>x)=1$</span> for all <span class="math-container">$x<0$</span>
and <span class="math-contai... | D F | 501,035 | <p><span class="math-container">$$E[V|X] = E[X|X, Y\ge X]P(Y\ge X|X) + E[Y|X, Y<X]P(Y<X|X) = X\int_{X}^{\infty}e^{-y}dy + \int_{0}^Xye^{-y}dy = Xe^{-X} + 1 - e^{-X}(1+X) = 1 - e^{-X}$$</span>
Which means that <span class="math-container">$E[V|X]\sim U(0, 1)$</span> since <span class="math-container">$F_{X}(X)$</s... |
246,606 | <p>I have matrix:</p>
<p>$$
A = \begin{bmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 3 & 3 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 2
\end{bmatrix}
$$</p>
<p>And I want to calculate $\det{A}$, so I have written:</p>
<p>$$
\begin{array}{|cccc|ccc}
1 & 2 & 3 & 4 & 1 & 2 ... | user1551 | 1,551 | <p>The others have pointed out what's wrong with your solution. Let's calculate the determinant now:
\begin{align*}
\det \begin{bmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 3 & 3 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 2
\end{bmatrix} &\stackrel{r1 - \frac12(r2+r3+r4)}{=}
\det \begin{bm... |
553,431 | <p>In the <a href="http://demonstrations.wolfram.com/NoFourInPlaneProblem/" rel="nofollow noreferrer">No-Four-In-Plane problem</a>, points are selected so that no four of them are coplanar.</p>
<p>Eight points can be picked from a <span class="math-container">$3\times3\times3$</span> space in a unique way.</p>
<p>Can 1... | Oleg567 | 47,993 | <p>On $4\times 4 \times 4$:</p>
<p>maximal number of points for $4\times 4 \times 4$ grid is $10$.</p>
<p>As I checked, there are no way to build $11$ points in a $4 \times 4 \times 4$ grid (ignoring rotating, reflecting) with <strong>No-Four-In-Plane</strong> rule.</p>
<p>And there are $232$ ways to build $10$ such... |
3,503,999 | <p>Consider the function <span class="math-container">$$f(x):=\frac{x-x_0}{\Vert x-x_0 \Vert^2} + \frac{x-x_1}{\Vert x-x_1 \Vert^2}$$</span></p>
<p>for two fixed <span class="math-container">$x_0,x_1 \in \mathbb R^2$</span> and <span class="math-container">$x \in \mathbb R^2$</span> as well. </p>
<p>Does anybody know... | PierreCarre | 639,238 | <p>Well, you can just compute <span class="math-container">$g(x)$</span>... I'll denote <span class="math-container">$x_0,x_1$</span> by <span class="math-container">$u,v$</span>.</p>
<p><span class="math-container">$$
f(x) = \frac{x-u}{\|x-u\|^2}+\frac{x-v}{\|x-v\|^2}=\left(\frac{x_1-u_1}{(x_1-u_1)^2+(x_2-u_2)^2}+\fr... |
3,503,999 | <p>Consider the function <span class="math-container">$$f(x):=\frac{x-x_0}{\Vert x-x_0 \Vert^2} + \frac{x-x_1}{\Vert x-x_1 \Vert^2}$$</span></p>
<p>for two fixed <span class="math-container">$x_0,x_1 \in \mathbb R^2$</span> and <span class="math-container">$x \in \mathbb R^2$</span> as well. </p>
<p>Does anybody know... | Calvin Khor | 80,734 | <p>(Edit) RIP bounty, but here's the (correct) solution computed via Sympy/Jupyter.</p>
<p>setting <span class="math-container">$x_1=0$</span> and relabelling <span class="math-container">$x_0$</span> as <span class="math-container">$(x_0,y_0)$</span>, <span class="math-container">$x$</span> as <span class="math-conta... |
3,503,999 | <p>Consider the function <span class="math-container">$$f(x):=\frac{x-x_0}{\Vert x-x_0 \Vert^2} + \frac{x-x_1}{\Vert x-x_1 \Vert^2}$$</span></p>
<p>for two fixed <span class="math-container">$x_0,x_1 \in \mathbb R^2$</span> and <span class="math-container">$x \in \mathbb R^2$</span> as well. </p>
<p>Does anybody know... | Christian Blatter | 1,303 | <p>If you choose <span class="math-container">${\bf x}_0$</span> and <span class="math-container">${\bf x}_1$</span> at <span class="math-container">$(\pm a,0)$</span> of the <span class="math-container">$(x,y)$</span>-plane you have
<span class="math-container">$$f(x,y)={(x+a,y)\over(x+a)^2+y^2}+{(x-a,y)\over(x-a)^2+y... |
3,676,911 | <p>I'm trying to understand what is the Hessian matrix of <span class="math-container">$f\colon\mathbb{R}^{n}\to\mathbb{R}$</span>
defined by <span class="math-container">$f\left(x\right)=\left\langle Ax,x\right\rangle \cdot\left\langle Bx,x\right\rangle $</span>
where <span class="math-container">$A,B$</span> are syme... | greg | 357,854 | <p>Your function is the product of the following scalar functions
<span class="math-container">$$\eqalign{
\alpha &= x^TAx \quad\implies d\alpha = (2Ax)^Tdx \\
\beta &= x^TBx \quad\implies d\beta = (2Bx)^Tdx \\
f &= \alpha\beta \\
}$$</span>
Calculate the differential and the gradient of <span class="mat... |
2,709,809 | <blockquote>
<p>Let $Q$ be the square with corners $0$,$1$,$i$,$1+i$ and $R$ the rectangle with corners $0$,$2$,$i$,$2+i$. Prove there's no conformal map $Q$ to $R$ (on the interiors) that extends to a surjective homeomorphism on the closure, and takes corners to corners.</p>
</blockquote>
<p>This is from an old qua... | ts375_zk26 | 204,508 | <p>Suppose that there is a conformal map $w=f(z):Q \to R$ satisfying the described conditions.
We may assume that $f(0)=0, f(1)=2,f(i)=i$ and $f(1+i)=2+i$ for corners correspondence.
Let $C_y=\{x+iy: 0\le x\le 1\}$ $(0\le y\le 1)$ be a horizontal segment joining two points $iy$ and $1+iy$ in $Q$. Its image $f(C_y)$ is... |
2,709,809 | <blockquote>
<p>Let $Q$ be the square with corners $0$,$1$,$i$,$1+i$ and $R$ the rectangle with corners $0$,$2$,$i$,$2+i$. Prove there's no conformal map $Q$ to $R$ (on the interiors) that extends to a surjective homeomorphism on the closure, and takes corners to corners.</p>
</blockquote>
<p>This is from an old qua... | Dap | 467,147 | <p>Here's another argument that might be interesting. Apply the Schwarz reflection principle to extend along an edge of $Q$ to a map from a rectangle twice as big as $Q$ to a rectangle twice as big as $P.$ Continue extending in this way to extend to an automorphism of the whole complex plane. This must be a Möbius tran... |
3,500,418 | <p>I am fining the pointwise limit of the function <span class="math-container">$f_n(x) = \frac{x^n}{3-x^n}$</span> for <span class="math-container">$x ∈ [0,1]$</span> and <span class="math-container">$n ∈ N$</span></p>
<p>In order to do this I first divided through by <span class="math-container">$x^n$</span>, yieldi... | José Carlos Santos | 446,262 | <p>Since:</p>
<ul>
<li><span class="math-container">$\cos(x)=1$</span> if and anly if <span class="math-container">$x=2k\pi$</span> for some <span class="math-container">$k\in\mathbb Z$</span>;</li>
<li><span class="math-container">$(\forall x\in\mathbb R):\cos(x)\in[-1,1]$</span>,</li>
</ul>
<p>you have <span class=... |
39,424 | <p>I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?</p>
| Micah Milinovich | 3,659 | <p>If it is a course elementary number theory, look at "Elementary Number Theory" by Dudley.</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/048646931X" rel="nofollow">http://www.amazon.com/Elementary-Number-Theory-Underwood-Dudley/dp/048646931X</a></p>
|
39,424 | <p>I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?</p>
| Romeo | 7,867 | <p>For an introductory undergrad course I'd say the book to use by a long-shot is Kenneth Rosen's<a href="http://rads.stackoverflow.com/amzn/click/0321500318" rel="nofollow"> Elementary Number Theory and its Applications</a></p>
<p>The theory is all there, but it's placed nicely in a context appropriate for a mixed ba... |
3,949,580 | <p>Is it possible to set this integral up without using substitution?</p>
<p><span class="math-container">$$\iint_D e^{x+y} \,\mathrm{d}x\,\mathrm{d}y\,,$$</span> where</p>
<p><span class="math-container">$$D = \left\{-1\le x+y \le 1, -1 \le -x + y \le 1\right\}$$</span></p>
<p>The answer is: <span class="math-containe... | MrCool690000 | 862,505 | <p>You still won't know how many balls are inside the jar until you look inside the jar. You can't add two numbers when you only know one of the numbers...</p>
|
2,788,015 | <p>I'm trying to solve an exercise that says</p>
<blockquote>
<p>Show that a locally compact space is $\sigma$-compact if and only if is separable.</p>
</blockquote>
<p>Here locally compact means that also is Hausdorff. I had shown that separability imply $\sigma$-compactness but I'm stuck in the other direction.</... | spaceisdarkgreen | 397,125 | <p>Not sure if this is part of what you're wondering about, but will fill in the proof Henno omitted (slightly too long for a comment). </p>
<p>Let $\kappa >|\mathbb R|,$ $U$ and $U’$ be disjoint, open proper subsets of $I=[0,1],$ and for $\alpha<\beta<\kappa$ define $U_{\alpha,\beta} \subseteq I^\kappa$ to b... |
3,232,341 | <p>How would I show this? I know a directed graph with no cycles has at least one node of outdegree zero (because a graph where every node has outdegree one contains a cycle), but do not know where to go from here.</p>
| Markoff Chainz | 377,313 | <p>Suppose that there exists a graph with no cycles and there are no nodes of indegree <span class="math-container">$0$</span>. Then each node has indegree <span class="math-container">$1$</span> or higher. Pick any node, since its indegree is <span class="math-container">$1$</span> or higher we can go to its parent no... |
215,333 | <p>There are many symbols for understanding internet-related properties: <code>$NetworkConnected</code>, <code>PingTime</code>, <code>NetworkPacketTrace</code>, <code>NetworkPacketRecording</code>, etc.</p>
<p>But is there any convenient way of testing your network's upload speed from within Mathematica?</p>
| Carl Lange | 57,593 | <p>The easiest method I can think of to get an estimate is to upload a file to a server and measure how long it takes. Should give a reasonable guess, but like all of these things, it can only be a guess. In this example latency to WRI's servers may add a lot.</p>
<pre><code>file = "mytestfile";
time = AbsoluteTiming... |
3,426,756 | <p>From a point <span class="math-container">$O$</span> on the circle <span class="math-container">$x^2+y^2=d^2$</span>, tangents <span class="math-container">$OP$</span> and <span class="math-container">$OQ$</span> are drawn to the ellipse <span class="math-container">$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$</span>, <span ... | Arctic Char | 629,362 | <p>This may not be the solution you are looking for:</p>
<p>Let <span class="math-container">$\mathcal C=\{x^2+y^2=1\}$</span> be the unit circle. Let <span class="math-container">$O' = (\alpha, \beta)$</span> be any point outside of this circle. Let <span class="math-container">$O'P'$</span> and <span class="math-con... |
1,338,980 | <p>Suppose you have a set of data $\{x_i\}$ and $\{y_i\}$ with $i=0,\dots,N$. In order to find two parameters $a,b$ such that the line
$$
y=ax+b,
$$
give the best linear fit, one proceed minimizing the quantity
$$
\sum_i^N[y_i-ax_i-b]^2
$$
with respect to $a,b$ obtaining well know results. </p>
<p>Imagine now to desi... | Claude Leibovici | 82,404 | <p>The model being intrisically nonlinear wih respect to its parameter, you will need nonlinear regression.</p>
<p>However, the problem is to provide good estimates. One way to do it is to rewrite the model as $$y=a e^{qx}+b$$ with $q=\log(p)$. Now, a hint already provided by Yves Daoust <a href="https://math.stackexc... |
1,338,832 | <p>Assume we have a group consisting of both women and men. (In my example it is 67 women and 43 men but that is not important.) The women are indistinguishable and the men are also indistinguishable.</p>
<p>In how many ways can we pick a subgroup consisting of $n$ women and $n$ men, i.e., the same number of women and... | Ofir Schnabel | 140,778 | <p>you asking is how many vectors of order $2n$ are there when in any cordinat there is a man or a woman such that their number is equal. Your answer is $$\frac{(2n)!}{(n!)^2}.$$ Therefore, for $3$ you are wrong.</p>
|
1,030,335 | <blockquote>
<p>Let <span class="math-container">$n$</span> and <span class="math-container">$r$</span> be positive integers with <span class="math-container">$n \ge r$</span>. Prove that:</p>
<p><span class="math-container">$$\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}.$$</span></p>
</bloc... | Michael Hardy | 11,667 | <p>Assign numbers $1,2,3,\ldots,n,n+1$ to $n+1$ objects. The number of ways to choose $r+1$ of them is $\dbinom{n+1}{r+1}$.</p>
<p><em>Either</em> you choose the very last one and $r$ others bearing lower numbers (the number of ways to do that is $\dbinom n r$),</p>
<p><em>or</em> you choose the one just before the ... |
19,261 | <p>Every simple graph $G$ can be represented ("drawn") by numbers in the following way:</p>
<ol>
<li><p>Assign to each vertex $v_i$ a number $n_i$ such that all $n_i$, $n_j$ are coprime whenever $i\neq j$. Let $V$ be the set of numbers thus assigned. <br/></p></li>
<li><p>Assign to each maximal clique $C_j$ a unique p... | Sebastian | 4,572 | <p>I think you should read something about the Ricci flow and Perelmann s work (for 3mfs), or Seiberg Witten/Yang-Mills theory (for 4-mfs). These theories give you very deep results in topology. But the hole theory is geometric.</p>
|
4,539,739 | <p>Here is the curve <span class="math-container">$y=2^{n-1}\prod\limits_{k=0}^n \left(x-\cos{\frac{k\pi}{n}}\right)$</span>, shown with example <span class="math-container">$n=8$</span>, together with the unit circle centred at the origin.</p>
<p><a href="https://i.stack.imgur.com/mBNbY.png" rel="noreferrer"><img src=... | metamorphy | 543,769 | <p>This question is closely related to computing the discriminant of Chebyshev polynomials of the second kind, found in literature (e.g., see G. Szegő <a href="https://books.google.com/books?id=3hcW8HBh7gsC" rel="nofollow noreferrer"><em>Orthogonal polynomials</em></a>, theorem <span class="math-container">$6.71$</span... |
3,313,272 | <p><strong>Inscribe in a given cone, the height of which is equal to the radius of the base, a cylinder whose volume is a maximum.</strong> </p>
<p>I'm stuck. The answer key says the cylinder's height should be <span class="math-container">$\frac23$</span> the radius of the base of the cone, but the answer I'm getting... | Sharky Kesa | 398,185 | <p>Let the cone's radius be <span class="math-container">$r$</span>, and suppose the radius of the cylinder was <span class="math-container">$x$</span>. Then the height of the cylinder can be determined to be <span class="math-container">$r-x$</span> using similar triangles in a triangular cross-section of the cone thr... |
3,313,272 | <p><strong>Inscribe in a given cone, the height of which is equal to the radius of the base, a cylinder whose volume is a maximum.</strong> </p>
<p>I'm stuck. The answer key says the cylinder's height should be <span class="math-container">$\frac23$</span> the radius of the base of the cone, but the answer I'm getting... | j.wood | 476,276 | <p>With the help of Sharky Kesa's answer I figured out the issue. I solved for the cylinder's <em>hight</em> and misunderstood the answer key, which gives its <em>radius</em>. The cylinder of maximum volume has radius and height that are <span class="math-container">$\frac23$</span> and <span class="math-container">$\f... |
1,443,441 | <blockquote>
<p>If <span class="math-container">$\frac{x^2+y^2}{x+y}=4$</span>,then all possible values of <span class="math-container">$(x-y)$</span> are given by<br></p>
<p><span class="math-container">$(A)\left[-2\sqrt2,2\sqrt2\right]\hspace{1cm}(B)\left\{-4,4\right\}\hspace{1cm}(C)\left[-4,4\right]\hspace{1cm}(D)\l... | juantheron | 14,311 | <p>Given $$\displaystyle \frac{x^2+y^2}{x+y} = 4\Rightarrow x^2+y^2 = 4x+4y$$</p>
<p>So we get $$x^2-4x+4+y^2-4y+4 = 8\Rightarrow (x-2)^2+(y-2)^2 = (2\sqrt{2})^2$$</p>
<p>Now Put $$x-2 = 2\sqrt{2}\cos \phi\Rightarrow x = 2+2\sqrt{2}\cos \phi$$</p>
<p>and $$y-2 = 2\sqrt{2}\sin \phi\Rightarrow y = 2+2\sqrt{2}\sin \phi... |
1,443,441 | <blockquote>
<p>If <span class="math-container">$\frac{x^2+y^2}{x+y}=4$</span>,then all possible values of <span class="math-container">$(x-y)$</span> are given by<br></p>
<p><span class="math-container">$(A)\left[-2\sqrt2,2\sqrt2\right]\hspace{1cm}(B)\left\{-4,4\right\}\hspace{1cm}(C)\left[-4,4\right]\hspace{1cm}(D)\l... | juantheron | 14,311 | <p>Given $$\displaystyle \frac{x^2+y^2}{x+y} = 4\Rightarrow x^2+y^2 = 4x+4y$$</p>
<p>So we get $$x^2-4x+4+y^2-4y+4 = 8\Rightarrow (x-2)^2+(y-2)^2 = 8$$</p>
<p>Now we can write $$x-y = (x-2)-(y-2) = \left[(x-2)+(2-y)\right]$$</p>
<p>Now Using $\bf{Cauchy\; Schwartz\; Inequality}$</p>
<p>$$\displaystyle \left[(x-2)^2... |
1,187,713 | <p>How would I go about proving that if $a_n$ is a real sequence such that $\lim_{n\to\infty}|a_n|=0$, then there exists a subsequence of $a_n$, which we call $a_{n_k}$, such that $\sum_{k=1}^\infty a_{n_k}$ is convergent.</p>
<p>I think that I can choose terms $a_{n_k}$ such that they are terms of a geometric series,... | graydad | 166,967 | <p>Start with a series you know is convergent. A geometric series will work as you have guessed. I'll use the series $\sum_{n=1}^\infty \frac{1}{n^2}$ for my example. Choose each $a_{n_{k}}$ such that $\left|a_{n_{k}}\right|\leq \frac{1}{n_k^2}$ for some integer $n_k$. You will need to prove that you can find infinitel... |
332,760 | <blockquote>
<p>For an odd prime, prove that a primitive root of $p^2$ is also a primitive root of $p^n$ for $n>1$. </p>
</blockquote>
<p>I have proved the other way round that any primitive root of $p^n$ is also a primitive root of $p$ but I have not been able to solve this one. I have tried the usual things th... | Ivan Loh | 61,044 | <p>Let $g$ be a primitive root $\pmod{p^2}$. Then $p|(g^{p-1}-1)$ by Fermat's little theorem and $p^2 \nmid (g^{p-1}-1)$ since $g$ is a primitive root. Thus by <a href="http://www.artofproblemsolving.com/Resources/Papers/LTE.pdf">Lifting the Exponent Lemma</a>, $p^{n-1}\|((g^{p-1})^{p^{n-2}}-1)$ and $p^n\|((g^{p-1})^{p... |
332,760 | <blockquote>
<p>For an odd prime, prove that a primitive root of $p^2$ is also a primitive root of $p^n$ for $n>1$. </p>
</blockquote>
<p>I have proved the other way round that any primitive root of $p^n$ is also a primitive root of $p$ but I have not been able to solve this one. I have tried the usual things th... | lab bhattacharjee | 33,337 | <p>We know from <a href="https://math.stackexchange.com/questions/227199/order-of-numbers-modulo-p2/229918#229918">here</a>, if $ord_pa=d, ord_{(p^2)}a= d$ or $pd$</p>
<p>If $a$ is a primitive root $\pmod {p^2}, ord_{(p^2)}a=\phi(p^2)=p(p-1)$ </p>
<p>Then $ord_pa$ can be $p-1$ or $p(p-1)$</p>
<p>But as $ord_pa<p,... |
3,386,999 | <p>How can I ind the values of <span class="math-container">$n\in \mathbb{N}$</span> that make the fraction <span class="math-container">$\frac{2n^{7}+1}{3n^{3}+2}$</span> reducible ?</p>
<p>I don't know any ideas or hints how I solve this question.</p>
<p>I think we must be writte <span class="math-container">$2n^{7}... | Will Jagy | 10,400 | <p>The extended euclidean algorithm for gcd (of polynomials with rational coefficients) also tells us that
<span class="math-container">$$ \left( 2 x^{7} + 1 \right) \left( { 1728 x^{2} - 1944 x + 2187 } \right) - \left( 3 x^{3} + 2 \right) \left( { 1152 x^{6} - 1296 x^{5} + 1458 x^{4} - 768 x^{3} ... |
3,388,457 | <p>I made an equation
<span class="math-container">$$(100b+40+a)-(100a+40+b)=99$$</span> simplified that to <span class="math-container">$b-a=1$</span> , but do not know where to go from there.</p>
| The Demonix _ Hermit | 704,739 | <p>Since <span class="math-container">$a4b$</span> is divisible by <span class="math-container">$9$</span> , we have <span class="math-container">$$a+b=5 \text { or } a+b = 14$$</span></p>
<p>Since by reversing the number, we get a bigger number , we conclude <span class="math-container">$a\lt b$</span></p>
<p>All po... |
3,631,042 | <p>Probably, <span class="math-container">$y = x^2$</span> plots a parabola only given certain assumptions that structure a cartesian coordinate plane, and it does not plot a parabola in e.g. the polar coordinate plane.</p>
<p>Now, why exactly does a parabola share an equation with the area of a square? 'Why' here is ... | Jesus is Lord | 187,128 | <p><code>y = x * x</code> says <code>To compute the number y, multiply the number x with itself.</code></p>
<p>If you're talking about length, that corresponds to area of a square.</p>
<p>If you're talking about real numbers, you get a parabola curve.</p>
<p>If you're talking about complex numbers or some other set,... |
3,631,042 | <p>Probably, <span class="math-container">$y = x^2$</span> plots a parabola only given certain assumptions that structure a cartesian coordinate plane, and it does not plot a parabola in e.g. the polar coordinate plane.</p>
<p>Now, why exactly does a parabola share an equation with the area of a square? 'Why' here is ... | Sam Cassidy | 339,509 | <p>If you take the graph of <span class="math-container">$y = x$</span>, the region under the graph between <span class="math-container">$0$</span> and <span class="math-container">$t$</span> is half of a square of side length <span class="math-container">$t$</span>, and <span class="math-container">$\int_0^t x \, \mat... |
3,360,914 | <p>Let <span class="math-container">$A$</span>, <span class="math-container">$B$</span> and <span class="math-container">$C$</span> be symmetric, positive semi-definite matrices. Is it true that
<span class="math-container">$$ \|(A + C)^{1/2} - (B + C)^{1/2}\| \leq \|A^{1/2} - B^{1/2}\|,$$</span>
in either the 2 or Fro... | Conrad | 298,272 | <p>Roughly speaking the sum behaves (for large <span class="math-container">$m,n$</span>) like <span class="math-container">$\sum_{m,n \ne 0}\frac{1}{m^2+n^2}$</span> and that is divergent since the sum say in m for fixed <span class="math-container">$n$</span> is about <span class="math-container">$\frac{1}{n}$</span>... |
2,020,128 | <p>For $r$ is a real number, I can write $r \in \mathbb{R}$.</p>
<p>For $\varepsilon$ is an infinitesimal, I'd like to write something like $\varepsilon \in something$ Is there a symbol for "the set of infinitesimals"? Or alternatively, a commonly used abbreviation for "infinitesimal"?</p>
<p>For $H$ is an infinite (... | mhwombat | 3,483 | <p>After more research, I have concluded that, as Bye_world suggests, there is no standard notation for the set of infinitesimals. Here are some of the notations I have seen used:</p>
<p>$\mathcal{I}$,
$N$,
$\mathbb{N}$,
$\Delta$.</p>
<p>Also, for "$x$ is an infinitesimal", I have seen the notation $x \approx 0$.</p>... |
4,637,565 | <p>I am thinking of positive sequences whose sum is infinite but whose sum of squares is not?</p>
<p>One representative sequence is <span class="math-container">$$x[n] = \frac{a}{n+b},$$</span> where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are given real numbers such that <sp... | Balaji sb | 213,498 | <p>As the answer from Gareth Ma points out, look for sequences in <span class="math-container">$\ell^2 \setminus \ell^1$</span>. This will be your set of sequences.</p>
<p>Concretely speaking, take <span class="math-container">$\{a_n : a_n \geq 0\}$</span> such that <span class="math-container">$\sum_n a_n < \infty$... |
2,405,205 | <p>The Wikipedia article on <a href="https://en.wikipedia.org/wiki/Fraction_(mathematics)#Complex_fractions" rel="nofollow noreferrer">Fractions</a> says:</p>
<blockquote>
<p>If, in a complex fraction, there is no unique way to tell which fraction lines takes precedence, then this expression is improperly formed, be... | Community | -1 | <p>it's only because of the way it interprets it potentially. The reason the order of operations is needed is to stop ambiguous answers. in this case with parentheses added around the divisions you can make it equal 1, or ${5\over(10*20*40)}= {1\over1600} $,etc. some arithmetic without implied parentheses by order of o... |
4,319,590 | <blockquote>
<p>Let <span class="math-container">$H$</span> be a subgroup of <span class="math-container">$G$</span> and <span class="math-container">$x,y \in G$</span>. Show that <span class="math-container">$x(Hy)=(xH)y.$</span></p>
</blockquote>
<p>I have that <span class="math-container">$Hy=\{hy \mid h \in H\}$</s... | Shaun | 104,041 | <p>You're correct.</p>
<hr />
<p>This can be done in a few lines:</p>
<p><span class="math-container">$$\begin{align}
x(Hy)&=\{xh'\mid h'\in Hy\}\\
&=\{x(hy)\mid h\in H\}\\
&=\{(xh)y\mid h\in H\}\\
&=\{h''y\mid h''\in xH\}\\
&=(xH)y.
\end{align}$$</span></p>
|
2,030,547 | <p>The following expression came up in a proof I was reading, where it is said "It is easily shown: $$\lim_{x\to\infty} x(1-\frac{\ln (x-1)}{\ln x})=0."$$</p>
<p>Unfortunately I'm not having an easy time showing it. I guess it should come down to showing that the ratio $\frac{\ln (x-1)}{\ln x}$ converges to 1 superlin... | Community | -1 | <p>We might define a linear transformation $\theta : V \to V^{**}$ by the equation</p>
<p>$$ \theta(v)(f) = f(v) $$</p>
<p>The notation here is recursive; we are defining the linear transformation $\theta$ by specifying its value at every $v \in V$. In turn, we define the linear functional $\theta(v) \in V^{**}$ by s... |
133,711 | <p>I am trying to show that $$\int_{-\pi}^{\pi}e^{\alpha \cos t}\sin(\alpha \sin t)dt=0$$</p>
<p>Where $\alpha$ is a real constant.</p>
<hr>
<p>I found the problem while studying a particular question in this room,<a href="https://math.stackexchange.com/questions/124868/evaluate-int-c-frace-alpha-zzdz-where-alpha-in... | joriki | 6,622 | <p>This is false. In the interior of the interval of integration, the value of the inner sine is in $(0,1]$. For sufficiently small $\alpha$, that means the value of the outer sine is positive, so since $\mathrm e^\alpha$ is also positive, the integral is positive.</p>
<p>[<em>Edit in response to the change in the que... |
2,359,621 | <p>Consider $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ where</p>
<p>$$f(x,y):=\begin{cases}
\frac{x^3}{x^2+y^2} & \textit{ if } (x,y)\neq (0,0) \\
0 & \textit{ if } (x,y)= (0,0)
\end{cases} $$</p>
<p>If one wants to show the continuity of $f$, I mainly want to show that </p>
<p>$$ \lim\limits_... | Alekos Robotis | 252,284 | <p>The formal definition is as follows: given a function of $n$ real variables (here $n=2$): $f(x_1,\ldots, x_n),$ we say that
$$\lim_{(x_1,\ldots, x_n)\to (p_1,\ldots, p_n)}f(x_1,\ldots,x_n)=L$$
if for every $\epsilon>0$, there exists a $\delta$ sufficiently small that $$ \lvert (x_1,\ldots, x_n)-(p_1,\ldots, p_n)... |
2,359,621 | <p>Consider $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ where</p>
<p>$$f(x,y):=\begin{cases}
\frac{x^3}{x^2+y^2} & \textit{ if } (x,y)\neq (0,0) \\
0 & \textit{ if } (x,y)= (0,0)
\end{cases} $$</p>
<p>If one wants to show the continuity of $f$, I mainly want to show that </p>
<p>$$ \lim\limits_... | Lonidard | 206,444 | <p><strong>TLDR</strong>:You can intuitively think of it as $\lim_{||(x,y||\to 0}$. </p>
<p>This becomes more clear and formal switching to polar coordinates; if you write
$$f(x,y)=f(r\cos \theta, r\sin \theta)$$
we say that
$$\lim_{(x,y)\to (x_0,y_0)}f(x,y)=L$$
if for every $\theta \in [0,2\pi)$ we have
$$\lim_{r\t... |
2,289,935 | <p>Please help, I don't know how to go with this. So far I've done this :</p>
<p>if $c_1v + c_2w + c_3(v\times w) = 0$ , then $c_1,c_2,c_3$ must be $0$, and $0$ must be the only solution.</p>
| marty cohen | 13,079 | <p>I would use the fact that
$0
= v\cdot (v\times w)
= w\cdot (v\times w)
$.</p>
<p>If
$v\times w
= av+bw$
then
$0
=v\cdot(v\times w)
= v\cdot(av+bw)
=a|v|^2+b(v\cdot w)
$
and
$0
=w\cdot(v\times w)
= w\cdot(av+bw)
=av\cdot w+b|w|^2
$
which implies
$a = b = 0$.</p>
|
2,289,935 | <p>Please help, I don't know how to go with this. So far I've done this :</p>
<p>if $c_1v + c_2w + c_3(v\times w) = 0$ , then $c_1,c_2,c_3$ must be $0$, and $0$ must be the only solution.</p>
| copper.hat | 27,978 | <p>By (one) definition, $a \times b$ is the unique vector satisfying
$\langle x, a \times b \rangle = \det \begin{bmatrix} x & a & b \end{bmatrix} $, and in particular we have $\|a \times b \|^2 = \det \begin{bmatrix} a \times b & a & b \end{bmatrix}$.</p>
<p>Hence we see that $a \times b = 0$ <strong>... |
829,449 | <p>I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, it is true. Yet, when we consider x(x+1) and X^2 + X, we can say that the x is the same for = 1. However, we call this... | mweiss | 124,095 | <p>A few thoughts:</p>
<ul>
<li>Resolving vectors into components is all about trigonometric ratios, which in turn are all about similar triangles.</li>
<li>Inverse-square laws (gravity, electrical force) can be interpreted geometrically in terms of the surface area of a sphere.</li>
<li>Planetary orbits and ellipses.... |
829,449 | <p>I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, it is true. Yet, when we consider x(x+1) and X^2 + X, we can say that the x is the same for = 1. However, we call this... | cesaruliana | 128,809 | <p>People already gave some good ideas, but I'll pitch in what could be a more systematic approach</p>
<p>Without calculus is somewhat difficult to study physics after Newton. Fortunately a lot was known before him. An account of this can be seen in Rene Dugas' "<a href="http://rads.stackoverflow.com/amzn/click/048665... |
1,649,053 | <p>In figure $AD\perp DE$ and $BE\perp ED$.$C$ is mid point of $AB$.How to prove that $$CD=CE$$<a href="https://i.stack.imgur.com/ZtAA0.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZtAA0.png" alt="enter image description here"></a></p>
| Narasimham | 95,860 | <p>Draw $GH$ parallel to $DE$. GAC equals to the alternate one CBH because cutting by parallels,GCA to its vertically opposite HCB, given $ CA= CB, $ so congruent.</p>
<p><a href="https://i.stack.imgur.com/nj2BO.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nj2BO.png" alt="QnTrap"></a></p>
|
4,273,026 | <p>Let <span class="math-container">$\Omega\subset\mathbb{R}^n$</span> be a bounded open set, <span class="math-container">$n\geq 2$</span>. For <span class="math-container">$r>0$</span>, denote by <span class="math-container">$B_r(x_0)=\{x\in\mathbb{R}^n:|x-x_0|<r\}$</span> whose closure is a proper subset of <s... | Math | 471,409 | <p>@ user378654 the below limit
<span class="math-container">$$
\lim_{a\to\infty} a(1-(\frac{1}{2})^\frac{1}{a})^n=0,
$$</span>
but I hope you have used the above limit to be non-zero to give a lower bound in your argument, which is independent of <span class="math-container">$a$</span>.</p>
<p>The above limit is zero,... |
3,553,975 | <p>I fear that this is a stupid question, but I want to have a go anyway. </p>
<p>Let <span class="math-container">$k$</span> be a field, and let <span class="math-container">$f(x,y)$</span> be an irreducible homogeneous quadratic polynomial in <span class="math-container">$k[x,y]$</span>. </p>
<p><em>Question</em>: ... | Henno Brandsma | 4,280 | <p>I would call a function that obeys <span class="math-container">$x < y \to f(x) < f(y)$</span> for all <span class="math-container">$x,y$</span>, a <em>strictly increasing</em> function.</p>
|
2,659,781 | <p>I saw a problem yesterday, which can be easily be solved if we are using fractions. But the problem is for the 4th grade children, and I don't know how to solved this using what they what learned.</p>
<p>I tried solved it using the graphic method ( segments ). Here's the problem:</p>
<p>A team of workers has to fi... | user326210 | 326,210 | <p>You can work backwards starting from the last day:</p>
<ul>
<li>The work took place over days 1, 2, and 3. They built the complete road starting from nothing.</li>
<li>They built some amount of the road on day 1. </li>
<li>They had to build the rest of it on days 2 and 3. We can think about days 2 and 3 collectivel... |
3,027,925 | <p>Just for my own understanding of how exactly integration works, are these steps correct:</p>
<p><span class="math-container">$$\begin{align}\int x\,d(x^2) \qquad &\implies x^2 = u \\ & \implies x= \sqrt{u}\end{align}$$</span> </p>
<p>Thus, it becomes <span class="math-container">$$\int\sqrt{u}\,du = \frac... | zero | 532,480 | <p>Yes, except for an integration on the LHS of the last line. Also make sure to keep track of your limits. You would get the same expression when you use <span class="math-container">$d(x^2) = 2x dx$</span></p>
|
2,741,832 | <p>When one first learns measure theory, it is a small novelty to find out that
$$\bigcup_{n=0}^\infty B_{\epsilon/2^n}(r_n)$$
is not all of $\mathbb{R}$, where $\{r_n\}$ is an enumeration of the rationals and $\epsilon$ is an arbitrary positive number (notice this fact is equally impressive if $\epsilon$ is small or l... | Eric Wofsey | 86,856 | <p>Yes, this is possible. Your proposed construction of $(c_n)$ works with no difficulty. To construct $(d_n)$, the easiest thing to do is just pick one point that you want to not be covered. So fix some irrational number $\alpha$. We would like to just let $d_n=|\alpha-r_n|$. Then $\alpha$ will not be in any $B_{... |
4,066,601 | <p>The question is</p>
<blockquote>
<p>Find all solutions <span class="math-container">$z\in \mathbb C$</span> for the following equation: <span class="math-container">$z^2 +3\bar{z} -2=0$</span></p>
</blockquote>
<p>I have attempted numerous methods of approaching this question, from trying to substitute <span class="... | Michael Hoppe | 93,935 | <p>No need for real and imaginary parts here.</p>
<p>From <span class="math-container">$z^2 +3\bar z = 2$</span> we have <span class="math-container">$\overline{z^2 +3\bar z} = \bar 2$</span>. Now <span class="math-container">$\overline{z^2 +3\bar z}= \bar z^2+3z$</span>.
Hence we have
<span class="math-container">$$... |
634,890 | <blockquote>
<p><strong>Moderator Notice</strong>: I am unilaterally closing this question for three reasons. </p>
<ol>
<li>The discussion here has turned too chatty and not suitable for the MSE framework. </li>
<li>Given the recent pre-print of <a href="http://arxiv.org/abs/1402.0290" rel="noreferrer">T. Ta... | Stephen Montgomery-Smith | 22,016 | <p>OK, I spent an afternoon getting help with Russian, and I think I understand a lot more.</p>
<p>So first he actually proves a rather abstract theorem (Theorem 2), and strong solutions of the Navier-Stokes is merely a corollary. He shows the existence of solutions satisfying certain bounds to
$$ \dot u + Au + B(u,u... |
634,890 | <blockquote>
<p><strong>Moderator Notice</strong>: I am unilaterally closing this question for three reasons. </p>
<ol>
<li>The discussion here has turned too chatty and not suitable for the MSE framework. </li>
<li>Given the recent pre-print of <a href="http://arxiv.org/abs/1402.0290" rel="noreferrer">T. Ta... | nick kave | 125,228 | <p>On the Spanish site
<a href="http://francis.naukas.com/2014/01/18/la-demostracion-de-otelbaev-del-problema-del-milenio-de-navier-stokes/#comment-21031" rel="nofollow">http://francis.naukas.com/2014/01/18/la-demostracion-de-otelbaev-del-problema-del-milenio-de-navier-stokes/#comment-21031</a>
the following info appea... |
137,794 | <p>I'm plotting the electric field of a charged ring based a solution from Jackson's <em>Electrodynamics</em>. </p>
<p><em>Mathematica</em> handles <code>VectorPlot3D</code> and <code>SliceVectorPlot3D</code> for the field without a hitch, and <code>SliceContourPlot3D</code> of the field magnitude as well. </p>
<p>Ho... | Jack LaVigne | 10,917 | <p>Using your definitions one can determine that the source of the problem is when <code>x</code> and <code>y</code> are very close to zero.</p>
<pre><code>contourList = Partition[
Flatten[
Table[{x, y, z, Norm[dΦc]}, {x, -0.35, 0.35, 0.01},
{y, -0.35, 0.35, 0.01},
{z, -0.35, 0.35, 0.01}
]
],... |
4,496,815 | <blockquote>
<p>For <span class="math-container">$n, m \in \mathbb{N}, m \leq n$</span>, let <span class="math-container">$P(n, m)$</span> denote the number of permutations of length <span class="math-container">$n$</span> for which <span class="math-container">$m$</span> is the first number whose position is left unch... | Drew Brady | 503,984 | <p>This is an immediate consequence of AM-GM.</p>
<p>First, <span class="math-container">$u + 1/u \geq 2$</span> whenever <span class="math-container">$u > 0$</span>.</p>
<p>Lower bound: by inequality above,
<span class="math-container">$\mu \geq {2^2}^2 = 16$</span>, and this is attained with <span class="math-cont... |
3,104,706 | <p>Let <span class="math-container">$T$</span> be the left shift operator on <span class="math-container">$B(l^{2}(\mathbb{N}))$</span>. How to see that von Neumann algebra generated by <span class="math-container">$T$</span> is <span class="math-container">$B(l^{2}(\mathbb{N}))$</span>?</p>
| David Hill | 145,687 | <p>If <span class="math-container">$X$</span> is infinite, then there is an infinite subset <span class="math-container">$Y=\{y_1, y_2, \ldots\}\subset X$</span>. </p>
<ol>
<li><p>Define <span class="math-container">$g:X\to X$</span> so that <span class="math-container">$g(x)=x$</span> if <span class="math-container">... |
747,949 | <p>There is a complex serie: $f(t_n)=\alpha_n+\beta_n i$, for $n = 1,...,N$,$t_n,\alpha_n$ and $\beta_n$ are known.When we have know that $f(t)$ has the following form:
$$f(t)=Ae^{-iBt}$$
with unknown amplitude $A$ and unknown phase $B$, how to estimate the parameters $A$ and $B$ by using a numerical optimization metho... | Claude Leibovici | 82,404 | <p>As written by Martín-Blas Pérez Pinilla, let us suppose that you want to find the optimum values of parameters $A$ and $B$ which minimize the objective function $$\Phi(A,B)=\sum _{n=1}^N (\alpha_n-A\cos (Bt_n))^2+(\beta_n+A\sin (Bt_n))^2=\sum _{n=1}^N r_n$$ Now, since you want the objective function to be minimum, w... |
70,429 | <p>For a $n$-dim smooth projective complex algebraic variety $X$, we can form the complex line bundle $\Omega^n$ of holomorphic $n$-form on $X$. Let $K_X$ be the divisor class of $\Omega^n$, then $K_X$ is called the canonical class of $X$.</p>
<p><strong>Question</strong>: Is homology class of $K_X$ in $H_{2n-2}(X)$ ... | Francesco Polizzi | 7,460 | <p>For the question about <em>homeomorphisms</em> the answer is <em>no</em>, even if $X$ and $X'$ are algebraic surfaces. </p>
<p>In fact, in his paper [Orientation reversing homeomorphisms in surface geography, Math. Ann. 292 (1992)], D. Kotschick proves the following result:</p>
<blockquote>
<p><strong>Theorem.</... |
1,027,486 | <p>How do I integrate this?</p>
<p>$$\int_0^{2\pi}\frac{dx}{2+\cos{x}}, x\in\mathbb{R}$$</p>
<p>I know the substitution method from real analysis, $t=\tan{\frac{x}{2}}$, but since this problem is in a set of problems about complex integration, I thought there must be another (easier?) way.</p>
<p>I tried computing t... | Math-fun | 195,344 | <p>Here is an elementary treatment: </p>
<p>First note that $\displaystyle2+\cos x=\frac{3+\tan ^2 \frac{x}{2}}{1+\tan^2 \frac{x}{2}}$. Also note that for $\displaystyle f(x)=\frac{3+\tan ^2 \frac{x}{2}}{1+\tan^2 \frac{x}{2}}$, it holds that $\displaystyle f(x+\pi)=f(x)$ for $0<x<\pi$. Therefore
$$\begin{align}\... |
244,769 | <p>I am DMing a game of DnD and one of my players is really into fear effects, which is cool, but the effect of having monsters suffer from the "panicked" condition gets tedious to render via dice rolls.</p>
<p>The rule is, on the battle grid the monster will run for 1 square in a random direction, then from ... | Ulrich Neumann | 53,677 | <p>Try <code>FindGeometricTransform</code> to describe the transformation <code>{x,y}<->{u,v}</code>.</p>
<p>Therefore it's necessary to know the three points <code>A,B,C</code> (I added <code>Buv</code> ).</p>
<pre><code>{Axy, Bxy, Cxy} = {{0.2, 0.8}, {0.1, 0.15}, {0.8, 0.25}}
{Auv, Buv, Cuv} = {{0, 75}, {0, -4... |
3,858,414 | <p>I need help solving this task, if anyone had a similar problem it would help me.</p>
<p>The task is:</p>
<p>Calculate using the rule <span class="math-container">$\lim\limits_{x\to \infty}\left(1+\frac{1}{x}\right)^x=\large e $</span>:</p>
<p><span class="math-container">$\lim_{x\to0}\left(\frac{1+\mathrm{tg}\: x}{... | Shaun | 104,041 | <p>It suffices to prove that, for all <span class="math-container">$x,y\in A$</span>, if <span class="math-container">$[x]\cap[y]\neq \varnothing$</span>, then <span class="math-container">$[x]=[y]$</span>.</p>
<p>Suppose <span class="math-container">$x,y$</span> are arbitrary in <span class="math-container">$A$</span>... |
4,531,652 | <p>In my school book, I read this theorem</p>
<blockquote>
<p>Let <span class="math-container">$n>0$</span> is an odd natural number (or an odd positive integer), then the equation <span class="math-container">$$x^n=a$$</span> has exactly one real root.</p>
</blockquote>
<p>But, the book doesn't provide a proof, onl... | Suzu Hirose | 190,784 | <p>If <span class="math-container">$n$</span> is even then <span class="math-container">$x^n=a$</span> has two real roots, <span class="math-container">$x=\pm\sqrt[n]{a}$</span>, since <span class="math-container">$x^n=\left(x^{(n/2)}\right)^2$</span> is always positive, but solutions are restricted to <span class="mat... |
664 | <p>Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large monochromatic subgraphs?</p>
<p>That is, can we explicitly construct (edge) 2-colourings on graphs of size $c^k$, for some... | Mike | 1,579 | <p>A question I have here is what do you mean by "explicit"? </p>
<p>Personally, I like the definition that a construction is explicit if it can be constructed in polynomial time (due to Alon? Wigderson??). Given that we are talking about exponentials in n here, this gets (slightly) complicated, but we'll say the cont... |
664 | <p>Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large monochromatic subgraphs?</p>
<p>That is, can we explicitly construct (edge) 2-colourings on graphs of size $c^k$, for some... | Gil Kalai | 1,532 | <p>Finding explicit constructions for Ramsey graphs is a central problem in extremal combinatorics. Indeed, computational complexity gives a way to formalize this problem. Asking for a graph which can be constructed in polynomial time is a fairly good definition although sometimes the definition is taken as having a lo... |
232,777 | <p>Let $F$ be an ordered field.</p>
<p>What is the least ordinal $\alpha$ such that there is no order-embedding of $\alpha$ into any bounded interval of $F$?</p>
| Fedor Petrov | 4,312 | <p>This is similar to your proof but without induction.</p>
<p>We prove that there are at least 3 such sets. For $r=k$ this is clear, so assume that $k>r$. Consider our $\binom{k}{r}+1$ $r$-sets. Call an element $v\in V$ appropriate if $v$ belongs to at most $\binom{k-1}{r-1}$ our sets. Then there exist at least $\... |
2,566,803 | <p>Let $A,B,C$ be sets such that $f:A\to B$ is a function.</p>
<p>Let $F: C^B \to C^A$ be a function, such that $F(k)=k\circ f$.</p>
<p>Prove/disprove that if $f$ is surjective then $F$ is surjective.</p>
<p>I tried to prove it: If $f$ is surjective so for every $b\in B$ there is $a\in A$ so $f(a)=b$, but what now?... | Andres Mejia | 297,998 | <p>Leet $A=\{1,2\}$, $C=\{a,b\}$ and $B=\{1\}$. Consider the function $f:A \to 1$, which is constant and surjective.</p>
<p>Now, consider $g \in C^A$ given by $1 \mapsto a$ and $2 \mapsto b$. Clearly, there is no $h:B \to C$ so that $h \circ f=g$</p>
<hr>
<p>Suppose that $f$ is <em>injective</em>.
Then let $g: A \to... |
196,902 | <p>Hello fellow Ace Users.</p>
<p>Currently I'm working on a project to implement Peridynamics.
This is a discretization technique in the fashion of a meshless particle method.
AceGen/AceFEM provides the feature of arbitrary nodes per element which suits my need perfectly as such a peridynamic particle interacts with ... | Pinti | 42,046 | <p>I have no experience with meshless methods, but I will try to answer/comment on your questions.</p>
<ol>
<li><p>In AceFEM assembling of global matrices and vectors ("Tangent and residual" subroutine) is parallelized. Solving the linear system is also parallelized (Intel MKL PARDISO). "Tasks" subroutine is not paral... |
2,128,182 | <p>I've been looking for a definition of game in game theory. I'd like to know if there is a definition shorter than that of Neumann and Morgenstern in <em>Theory of Games and Economic Behavior</em> and not so vague like "interactive decision problem" or "situation of conflict, or any other kind of interaction". I've s... | Hector | 318,351 | <p>I know this question already has an accepted answer, but games are usually defined depending on their form and their information structure. Therefore, the definition of a normal form game is different from that one of extensive form of incomplete information (for example). I usually define a game in normal form (its... |
53,185 | <p>Let us consider a noncompact Kähler manifold with vanishing scalar curvature but nonzero Ricci tensor. I'm wondering what can it tell us about the manifold. The example (coming from physics) has the following Kähler form</p>
<p><span class="math-container">$$K = \bar{X} X + \bar{Y} Y + \log(\bar{X} X + \bar{Y} Y)$$<... | Peter Koroteev | 5,550 | <p>Partly the answer for 4 manifolds (2d complex manifolds) is given in the paper by C Lebrun ``Counter-examples to the generalized positive action conjecture'' <a href="https://projecteuclid.org/euclid.cmp/1104162166" rel="nofollow noreferrer">paper</a>. The author considers vanishing scalar curvature and derives the ... |
488,141 | <p>\begin{align*}A=\left(\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \\\end{array}\right);\end{align*}</p>
<p>The eigenvalues are $1$, I know one of the eigenvectors is $(1,0,0,0)$, Is that all?</p>
<p>The mathematica gives, why ... | Bill Kleinhans | 73,675 | <p>But more simply, if $\sqrt3$ is an element of $Q[\sqrt[4]2]$, then $Q[\sqrt2,\sqrt3]$ is a subfield of $Q[\sqrt[4]2]$. However, since both are order 4, they must be isometric. But the first is a splitting field and the second is not.</p>
|
2,823,758 | <p>I was learning the definition of continuous as:</p>
<blockquote>
<p>$f\colon X\to Y$ is continuous if $f^{-1}(U)$ is open for every open $U\subseteq Y$</p>
</blockquote>
<p>For me this translates to the following implication:</p>
<blockquote>
<p>IF $U \subseteq Y$ is open THEN $f^{-1}(U)$ is open</p>
</blockq... | Benjamin Dickman | 37,122 | <p>Perhaps the following paper would be of interest to you:</p>
<blockquote>
<p>Velleman, D. J. (1997). Characterizing continuity. <em>The American Mathematical Monthly, 104</em>(4), 318-322. <a href="https://www.jstor.org/stable/2974580" rel="nofollow noreferrer"><strong>Link</strong></a>.</p>
</blockquote>
<p>Her... |
109,734 | <p>I am trying to do this homework problem and I have no idea how to approach it. I have tried many methods, all resulting in failure. I went to the books website and it offers no help. I am trying to find the derivative of the function
$$y=\cot^2(\sin \theta)$$</p>
<p>I could be incorrect but a trig function squared ... | Arturo Magidin | 742 | <p>Indeed, $\cot^2(a)$ means
$$\left(\cot (a)\right)^2.$$</p>
<p>You need to apply the Chain Rule <em>twice</em>: first, to deal with the square: set $g(u)=u^2$ as your "outside function", and $u=f(\theta) = \cot(\sin(\theta))$ as your inside function. Since $g'(u) = 2u$, then
$$\frac{d}{d\theta}\cot^2(\sin(\theta... |
4,268,962 | <blockquote>
<p>Check whether <span class="math-container">$y=\ln (xy)$</span> is an answer of the following differential equation or not</p>
<p><span class="math-container">$$(xy-x)y''+xy'^2+yy'-2y'=0$$</span></p>
</blockquote>
<p>First I tried to solve the equation,</p>
<p><span class="math-container">$$x(yy''-y''+y'... | Math Lover | 801,574 | <p>We have <span class="math-container">$x y'=\frac y{y-1}$</span> and <span class="math-container">$(xy')' = - \frac{y'}{(y-1)^2}$</span></p>
<p>DE is <span class="math-container">$(xy-x)y''+xy'^2+yy'-2y'=0$</span></p>
<p>Rearranging LHS we get,</p>
<p><span class="math-container">$(xy-x)y''+xy'^2+yy'-2y'$</span></p>
... |
4,268,962 | <blockquote>
<p>Check whether <span class="math-container">$y=\ln (xy)$</span> is an answer of the following differential equation or not</p>
<p><span class="math-container">$$(xy-x)y''+xy'^2+yy'-2y'=0$$</span></p>
</blockquote>
<p>First I tried to solve the equation,</p>
<p><span class="math-container">$$x(yy''-y''+y'... | Etemon | 717,650 | <p>Continuing my first approach:</p>
<p><span class="math-container">$$(xy-x)y''+xy'^2+yy'-2y'=0$$</span>
<span class="math-container">$$x(yy''+y'^2)-xy''+yy'-2y'=0$$</span>
<span class="math-container">$$x(yy')'+(x)'(yy')-xy''-2y'=0$$</span>
<span class="math-container">$$(xyy')'-xy''-y'-y'=0$$</span>
<span class="mat... |
2,615,185 | <p>The title is not complete, since it would be too long. Consider the following statement:</p>
<blockquote>
<p>Let $U \subset \mathbb{R}^n$ be open, connected and such that its one-point compactification is a manifold. Then, this compactification must be (homeomorphic to) the sphere $S^n$.</p>
</blockquote>
<p>Is ... | Nick A. | 412,202 | <p>I can imagine an ellementary approach only for the special case of $\mathbb{R^2} $ and $\mathbb{R}$.</p>
<p>For $\mathbb{R^2}$:We know that all the compact surfaces arise from adding to the sphere a finite amount of handles or Mobius-strips. In any case, if you remove a point from a compact surface which is somet... |
2,402,410 | <p>I defined the "function":</p>
<p>$$f(t)=t \delta(t)$$</p>
<p>I know that Dirac "function" is undefined at $t=0$ (see <a href="http://web.mit.edu/2.14/www/Handouts/Convolution.pdf" rel="nofollow noreferrer">http://web.mit.edu/2.14/www/Handouts/Convolution.pdf</a>).</p>
<p>In Wolfram I get $0 \delta(0)=0$ (<a href=... | Cauchy | 360,858 | <p>Look at $\delta$ as distribution: $\langle \delta, f \rangle = f(0)$. Then $\langle t \delta(t), f(t) \rangle = \langle \delta (t), t f(t) \rangle = (tf(t))\mid_{t = 0} = 0 \cdot f(0) = 0$ (see multiplication of distribution by smooth functions).</p>
|
1,297,319 | <p>integration equation </p>
<p>$$\int_{0}^{1/8} \frac{4}{\sqrt{(1-4x^2)}} \,dx$$</p>
<p>my work </p>
<p>$t= \sqrt{(1-4x^2)} $</p>
<p>$dt = -4x/\sqrt{(1-4x^2)} dx $</p>
<p>stuck here also </p>
| Anurag A | 68,092 | <p>Use the substitution $2x=\sin \theta$. Then $\frac{d}{d\theta}x=\frac{1}{2}\cos \theta $ and the integral becomes
$$\int_{0}^{1/8} \frac{4}{\sqrt{(1-4x^2)}} \,dx = \int_{0}^{\arcsin\left(\frac{1}{4}\right)} 2\,d\theta$$</p>
|
827,740 | <p>This is a new integral that I propose to evaluate in closed form:
$$ {\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$$
where $\Re$ denotes the real part and $\log (z)$ denotes the principal value of the logarithm defined for $z \neq 0$ by
$$ \log (z) = \ln |z| + i \mathrm{Arg}z, \qu... | Hakim | 85,969 | <p>I don't think a closed form exists after computing that integral numerically in Mathematica, and looking up in the <a href="http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html" rel="nofollow"><em>Inverse Symbolic Calculator</em></a>. It is approximately equal to: $-0.10617124113817\ldots$ If you need more digits jus... |
1,018,248 | <p>Let $X:=(X_t)_{t\geq0}$ be a Lévy process with triple $(b,A,\nu)$. Is there any known relation between the "distribution" of its jumps and the Lévy measure $\nu$? E.g. can we express something like $\mathbb{P}[X$ has $n$ jumps in $[0,1]]$ or $\mathbb{P}[X$ has a jump of absolute value $>u$ in $[0,1]]$ for some $... | binkyhorse | 18,357 | <p>Sample path properties are discussed e.g. in Sato's <em>Lévy processes and infinitely divisible distributions</em>, Section 21. For example, the following results are given there:</p>
<ul>
<li>Sample functions of $X$ are a.s. continuous if and only if $\nu=0$.</li>
<li>Sample functions of $X$ are a.s. piecewise con... |
269,665 | <p>Is Klein bottle an algebraic variety? I guess no, but how to prove. How about other unorientable mainfolds? </p>
<p>If we change to Zariski topology, which mainfold can be an algebraic variety? </p>
| Community | -1 | <p>In the introduction (second page) of <a href="http://www.mathematik.uni-bielefeld.de/documenta/vol-12/17.pdf">this paper</a> of Biswas and Huisman, it is explained that any non-orientable compact topological surface $X$ is real algebraic (<em>i.e.</em> there exists a real smooth algebraic surface $S$ whose real poin... |
1,133,544 | <p>I've been struggling to show that $\mathrm{SL}_2(\mathbb{R})$ is a normal subgroup of $\mathrm{GL}_2(\mathbb{R})$. I already proved that $\mathrm{SL}_2(\mathbb{R})\leq\mathrm{GL}_2(\mathbb{R})$ (not shown). Now I want to show that
$$
A\cdot \mathrm{SL}_2(\mathbb{R})=\mathrm{SL}_2(\mathbb{R})\cdot A
$$
for every $A\... | Mister Benjamin Dover | 196,215 | <p>Hint: Can you write $SL_2$ as a kernel? You certainly know some multiplicative maps from linear algebra. (From my experience, the easiest way to show that some subgroup is normal is to exhibit it as a kernel of a homomorphism.)</p>
|
1,133,544 | <p>I've been struggling to show that $\mathrm{SL}_2(\mathbb{R})$ is a normal subgroup of $\mathrm{GL}_2(\mathbb{R})$. I already proved that $\mathrm{SL}_2(\mathbb{R})\leq\mathrm{GL}_2(\mathbb{R})$ (not shown). Now I want to show that
$$
A\cdot \mathrm{SL}_2(\mathbb{R})=\mathrm{SL}_2(\mathbb{R})\cdot A
$$
for every $A\... | Brian Rushton | 51,970 | <p>Hint: determinants are multiplicative, and real numbers commute.</p>
|
1,133,544 | <p>I've been struggling to show that $\mathrm{SL}_2(\mathbb{R})$ is a normal subgroup of $\mathrm{GL}_2(\mathbb{R})$. I already proved that $\mathrm{SL}_2(\mathbb{R})\leq\mathrm{GL}_2(\mathbb{R})$ (not shown). Now I want to show that
$$
A\cdot \mathrm{SL}_2(\mathbb{R})=\mathrm{SL}_2(\mathbb{R})\cdot A
$$
for every $A\... | Ivo Terek | 118,056 | <p><strong>Hint:</strong> If $A \in {\rm SL}(n, \Bbb R)$ and $G \in {\rm GL}(n, \Bbb R)$, you want to prove that $G^{-1}AG \in {\rm SL}(n, \Bbb R)$. But: $$\det(G^{-1}AG) = \det(G^{-1})\det A\, \det G.$$</p>
|
2,825,789 | <p>I struggle to understand the following theorem (not the proof, I can't even validate it to be true). Note: I don't have a math background.</p>
<blockquote>
<p>If S is not the empty set, then (f : T → V) is injective if and only if Hom(S, f) is injective.</p>
<p>Hom(S, f) : Hom(S, T) → Hom(T, V)</p>
</blockquote>
<p>... | Kaj Hansen | 138,538 | <blockquote>
<p><strong>Theorem</strong>: A continuous function <span class="math-container">$f: [a,b] \rightarrow \mathbb{R}$</span> is Riemann integrable.</p>
</blockquote>
<p><em>Proof:</em></p>
<p>Let <span class="math-container">$f: [a,b] \rightarrow \mathbb{R}$</span> be a continuous function. Any function t... |
353,087 | <p>Solve the interior Dirichlet Problem</p>
<p>$$(r^2u_r)_r+\dfrac{1}{\sin\phi}(\sin\phi~u_\phi)_\phi+\dfrac{1}{\sin^2\phi}u_{\theta\theta}=0\,, \,\,\,\,\,\,\, 0<r<1 $$</p>
<p>where $u(1,\phi)=\cos3\phi$</p>
| Ron Gordon | 53,268 | <p>You are really just solving Laplace's equation </p>
<p>$$\Delta u = 0$$</p>
<p>in the interior of the unit sphere, with a boundary condition that is independent of $\theta$. The solution to this problem is well known:</p>
<p>$$u(r,\phi,\theta) = \sum_{n=0}^{\infty} a_n r^n \, P_n(\cos{\phi})$$</p>
<p>where $P_n... |
4,531,939 | <p>I know that <span class="math-container">$$p(a|b)=\frac{p(a, b)}{p(b)}$$</span>
And I also know <span class="math-container">$$p(a, b) = p(a)p(b)$$</span></p>
<p>So, algebraically, it all seems to me that <span class="math-container">$$p(a|b)=p(a)$$</span>
I know something is wrong with this situation that I'm think... | BatMath | 1,038,433 | <p>To give a more detailed explanation -- For two events <span class="math-container">$a$</span> and <span class="math-container">$b$</span>, the quantity <span class="math-container">$p(a\vert b)$</span> is called the <em>probability of <span class="math-container">$a$</span> given <span class="math-container">$b$</sp... |
4,531,939 | <p>I know that <span class="math-container">$$p(a|b)=\frac{p(a, b)}{p(b)}$$</span>
And I also know <span class="math-container">$$p(a, b) = p(a)p(b)$$</span></p>
<p>So, algebraically, it all seems to me that <span class="math-container">$$p(a|b)=p(a)$$</span>
I know something is wrong with this situation that I'm think... | Suzu Hirose | 190,784 | <p>For example if eye colour is the gene, let <span class="math-container">$p(B)$</span> be the probability that the child's eye colour is blue, and <span class="math-container">$P(B|MB)$</span> be the probability that the child's eye colour is blue given that the mother's eye colour is blue.</p>
<p>Suppose the probabi... |
1,618,373 | <p>Prove that $S_4$ cannot be generated by $(1 3),(1234)$</p>
<p>I have checked some combinations between $(13),(1234)$ and found out that those combinations cannot generated 3-cycles.</p>
<p>Updated idea:<br>
Let $A=\{\{1,3\},\{2,4\}\}$<br>
Note that $(13)A=A,(1234)A=A$<br>
Hence, $\sigma A=A,\forall\sigma\in \langl... | Ennar | 122,131 | <p>If we denote $a = (1234)$ and $b = (13)$, one can easily check that $a^4 = e$, $b^2 = e$ and $ab = ba^{-1}$ which are precisely relations that define dihedral group $D_4$. Thus, subgroup generated by $a$ and $b$ in $S_4$ is isomorphic to quotient of $D_4$, and thus it's order is less or equal than $8$. Since $|S_4| ... |
139,575 | <p>I use Magma to calculate the L-value, yields</p>
<p>E:=EllipticCurve([1, -1, 1, -1, 0]);
E;
Evaluate(LSeries(E),1),RealPeriod(E),Evaluate(LSeries(E),1)/RealPeriod(E);</p>
<p>Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - x over Rational Field
0.386769938387780043302394751243 3.09415950710224034641915800995
0.... | Joe Silverman | 11,926 | <p>Tim's answer is great, but I want to mention one other place where people have lost a power of 2. The canonical height is often defined relative to the divisor (O), as I do in my books. So it is given by
$$ \hat h(P) = \frac12 \lim_{n\to\infty} 4^{-n} h\bigl(x([2^n]P)\bigr). $$
Here the $\frac12$ is inserted becaus... |
1,119,027 | <p>I'm trying to learn Bayes's formula, and am coming up with some poker problems to learn this.</p>
<p>My problem is as following: given a $H4,H5$ ($4$ of hearts, $5$ of hearts) hand, what are the odds that I'll hit a straight flush?</p>
<p>My reasoning is like this:</p>
<p>$$\Pr(\text{straight flush}|H4H5) = (\Pr(... | Tahir Imanov | 208,078 | <p>What are you asking is what is the probability of drawing two cards, without putting them back.
Probability of drawing first card is 1/52.
Now, there are 51 cards left.
Therefore the probability of drawing second card is 1/51.
Therefore PR(A & B) = PR(A)*PR(B).</p>
|
1,186,825 | <p>Prove $$\lim_{n\to\infty}\int_0^1 \left(\cos{\frac{1}{x}} \right)^n\mathrm dx=0$$</p>
<p>I tried, but failed. Any help will be appreciated.</p>
<p>At most points $(\cos 1/x)^n\to 0$, but how can I prove that the integral tends to zero clearly and convincingly?</p>
| Jack D'Aurizio | 44,121 | <p>$$I_n=\int_{0}^{1}\cos^n\frac{1}{x}\,dx = \int_{1}^{+\infty}\frac{\cos^n x}{x^2}\,dx=\sum_{n\geq 0}\int_{1+2n\pi}^{1+(2n+2)\pi}\frac{\cos^n x}{x^2}\,dx$$
hence:
$$ I_n = \frac{1}{4\pi^2}\int_{1}^{1+2\pi}\psi'\left(\frac{x}{2\pi}\right)\cos^n x\,dx$$
and by Cauchy-Schwarz inequality:
$$ |I_n| \leq \frac{1}{4\pi^2}\sq... |
2,048,054 | <p>I need to find signed distance from the point to the intersection of 2 hyperplanes. I was quite sure that this is something that every mathematician do twice a week :) But not found any good solution or explanation for same problem.</p>
<p>In my case the hyperplanes is defined as $y = w'*x + x_0$, but it is ok to d... | Henno Brandsma | 4,280 | <p>If $X$ is locally connected, then every connected component $C$ of $X$ is open (and closed). For any space $X$, the connected components form a disjoint cover of $X$ (every point is in a component, and two components are disoint, or their union would be striclty larger, contradicting their maximality). Clearly, a di... |
1,410,163 | <p>Show that the limit of the function, $f(x,y)=\frac{xy^2}{x^2+y^4}$, does not exist when $(x,y) \to (0,0)$.</p>
<p>I had attempted to prove this by approaching $(0, 0)$ from $y = mx$, assuming $m = -1$ and $m = 1$. The result was $f(y, -y) = \frac{y}{1+y^2}$ and $f(y, y) = \frac{y}{1+y^2}$ as the limits which are ob... | tattwamasi amrutam | 90,328 | <p>Suppose that $A \subset B$. Let $ x \in B^c$. Then $x \not\in B$. Then $x \not \in A$. Thus $x \in A^c$. </p>
<p>Similarly assuming that $B^c \subset A^c$. Let $x \in A$. Then $x \not \in A^c$. Thus $x \not \in B^c$. Hence $x \in B$</p>
|
1,704,410 | <p>If we have two groups <span class="math-container">$G,H$</span> the construction of the direct product is quite natural. If we think about the most natural way to make the Cartesian product <span class="math-container">$G\times H$</span> into a group it is certainly by defining the multiplication</p>
<p><span class... | Michael Burr | 86,421 | <p>It is nice to think about $D_4$ as a semidirect product. Namely, $D_4=\langle \sigma,\tau:\sigma^4=\tau^2=1,\tau\sigma=\sigma^{-1}\tau\rangle$. You can see the automorphism because $\sigma$ and $\tau$ do not commute, but the automorphism ($x\mapsto x^{-1}$) tells you how to move the $\tau$ past the $\sigma$.</p>
... |
1,704,410 | <p>If we have two groups <span class="math-container">$G,H$</span> the construction of the direct product is quite natural. If we think about the most natural way to make the Cartesian product <span class="math-container">$G\times H$</span> into a group it is certainly by defining the multiplication</p>
<p><span class... | Mariano Suárez-Álvarez | 274 | <p>You are looking at this in the wrong way.</p>
<p>The main reason for which we define the direct product of groups is that we like describing/understanding the structure of groups and we noticed that many groups are, well, direct products.</p>
<p>Now not all groups are direct products. For example, the dihedral gro... |
4,243,344 | <blockquote>
<p><span class="math-container">${43}$</span> equally strong sportsmen take part in a ski race; 18 of
them belong to club <span class="math-container">${A}$</span>, 10 to club and 15 to club <span class="math-container">${C}$</span>. What is the
average place for (a) the best participant from club <span c... | Especially Lime | 341,019 | <p>The position of the highest placed person from <span class="math-container">$B$</span> (call this <span class="math-container">$X$</span>) can be anywhere from <span class="math-container">$1$</span> to <span class="math-container">$34$</span>, but these are not equally likely. For <span class="math-container">$X=1$... |
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