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 |
|---|---|---|---|---|
263,983 | <p>Is there a method to measure <strong>theoretic time load</strong> (hope +theoretic memory load) of an evaluation?</p>
<p>They should not change - <strong>always generate same value after pressing Enter key</strong>(=evaluation).</p>
<p>Hope they are even same in different environment(=different computers).</p>
<p>Fo... | John Doty | 27,989 | <p>For the sort of computer that you need for <em>Mathematica</em>, it is not possible. Modern high-speed processors use all kinds of shortcuts to speed execution, and these shortcuts depend on the history of everything that has gone on in the computer in its recent execution trajectory, not just <em>Mathematica</em>. ... |
135,308 | <p>Let $G$ be a compact Lie group and $M$ be a smooth manifold on which $G$ acts smoothly and effectively. Then the orbit space $M/G$ admits an Whitney stratification as follows
$$M/G=\bigsqcup_{H<G}(M_{(H)}/G).$$
Where $H$ is a closed subgroup of $G$, and $M_{(H)}$ is the set of the points in $M$ such that the isot... | Oliver Straser | 32,972 | <p>Maybe I add something to Peter Michor's answer. The slice theorem says, that for $y\in M_{(H)}$ with $G_y=H$ there exists a $G$-invariant open neighbourhood $U_y$, such that
$$ U_y\cong G\times_H V_y$$
and in particular $$M_{(H)}\cap U_y= G\times_H V_y^H$$
where $V_y= T_y M/T_y (G\cdot y)$. So $M_{(H)}$ can only... |
306,744 | <p>So if the definition of continuity is: $\forall$ $\epsilon \gt 0$ $\exists$ $\delta \gt 0:|x-t|\lt \delta \implies |f(x)-f(t)|\lt \epsilon$. However, I get confused when I think of it this way because it's first talking about the $\epsilon$ and then it talks of the $\delta$ condition. Would it be equivalent to say: ... | genepeer | 50,955 | <p>It's already been shown how your definition fails but I'll try to explain why it is the way it is. What the definition tries to get at is basically "You can get as close as you want to the limit." $\epsilon$ represents the any closeness to the limit you want to achieve. and the $\delta$ tells you how to achieve it. ... |
103,317 | <p>I have a 3D Plot with 3 separate functions for <code>z</code> in terms of <code>x</code> and <code>y</code>. I would like to convert this to a RegionPlot with the <code>max</code> of the three functions plotted for the variables <code>x</code> and <code>y</code>.</p>
<p>Essentially, this would be the top view of th... | Jason B. | 9,490 | <p>Here are three example functions,</p>
<pre><code>f1 = Exp[-x^2 - y^2];
f2 = .5 Sin[x - y];
f3 = .002 x^2 + .07 y^2;
</code></pre>
<p>And here is a top-down view of the 3D plot,</p>
<pre><code>Plot3D[{f1, f2, f3}, {x, -6, 6}, {y, -4, 4}, PlotPoints -> 100,
ViewCenter -> {0.5, 0.5, 0.5}, ViewPoint -> {0,... |
2,491,881 | <p>I'm trying to factorise $(4 + 3i)z^2 + 26iz + (-4+3i)$.</p>
<p>I tried to use the quadratic formula to factorise $(4 + 3i)z^2 + 26iz + (-4+3i)$, where $b = 26i$, $a = (4 + 3i)$, $c = (-4+3i)$. This gives us the roots $z = \dfrac{-4i - 12}{25}$ and $z =-4i - 3$.</p>
<p>But $\left( z + \dfrac{4i + 12}{25} \right) \l... | John Griffin | 466,397 | <p>Your proof is fine with the slight modification I noted in the comments.</p>
<p>You need to show that $X\cup Y$ is compact. This means that you need to show that every sequence in $X\cup Y$ has a convergent subsequence, so you fix $(x_n)$ in $X\cup Y$. The goal is to now find a subsequence of $(x_n)$ which converge... |
2,971,980 | <p>Show that if <span class="math-container">$0<b<1$</span> it follows that
<span class="math-container">$$\lim_{n\to\infty}b^n=0$$</span>
I have no idea how to express <span class="math-container">$N$</span> in terms of <span class="math-container">$\varepsilon$</span>. I tried using logarithms but I don't see h... | Peter Szilas | 408,605 | <p>Set <span class="math-container">$b=\dfrac{1}{1+x}$</span> , <span class="math-container">$x >0.$</span></p>
<p>Note: <span class="math-container">$ (1+x)^n \gt 1+nx.$</span></p>
<p><span class="math-container">$0<b^n =\dfrac{1}{(1+x)^n} \lt \dfrac{1}{1+nx}\lt$</span></p>
<p><span class="math-container">$(1... |
2,971,980 | <p>Show that if <span class="math-container">$0<b<1$</span> it follows that
<span class="math-container">$$\lim_{n\to\infty}b^n=0$$</span>
I have no idea how to express <span class="math-container">$N$</span> in terms of <span class="math-container">$\varepsilon$</span>. I tried using logarithms but I don't see h... | with-forest | 519,097 | <p>From <span class="math-container">$0<b<1$</span>, we get</p>
<p><span class="math-container">$$0< (n+1)b^n < 1+b+\cdots+b^n = \frac{1-b^{n+1}}{1-b} < \frac{1}{1-b}.$$</span> </p>
<p>Hence, </p>
<p><span class="math-container">$$0< b^n < \frac{1}{n+1} \frac{1}{1-b}.$$</span></p>
<p>And as <sp... |
1,104,163 | <p>I have come up with the equation in the form $${{dy}\over dx} = axe^{by}$$, where a and b are arbitrary real numbers, for a project I am working on. I want to be able to find its integral and differentiation if possible. Does anyone know of a possible solution for $y$ and/or ${d^2 y}\over {dx^2}$?</p>
| Dave L. Renfro | 13,130 | <p>This is a separable differential equation, as is the case for any equation that can be put into the form $\frac{dy}{dx} = f(x)g(y)$ for some functions $f$ and $g.$ In your case, divide both sides by $e^{by}$ and multiply both sides by $dx,$ then integrate.</p>
|
1,104,163 | <p>I have come up with the equation in the form $${{dy}\over dx} = axe^{by}$$, where a and b are arbitrary real numbers, for a project I am working on. I want to be able to find its integral and differentiation if possible. Does anyone know of a possible solution for $y$ and/or ${d^2 y}\over {dx^2}$?</p>
| abel | 9,252 | <p>separate the variables like $$ e^{-by}dy = axdx$$ on integrating, you get $$ -\dfrac{1}{b}e^{-by} = \dfrac{1}{2}ax^2 - \dfrac{C}{2b}$$ which can be solved for $y = \dfrac{1}{b}\ln\left( \dfrac{2}{C - ab x^2}\right)$</p>
<p>for $\dfrac{d^2y}{dx^2},$ differencing $$e^{-by}\dfrac{dy}{dx} = ax$$ gives you
$$e^{-by}\df... |
3,220,135 | <p>How many integer numbers, <span class="math-container">$x$</span>, verify that the following</p>
<p><span class="math-container">\begin{equation*}
\frac{x^3+2x^2+9}{x^2+4x+5}
\end{equation*}</span></p>
<p>is an integer?</p>
<p>I managed to do:</p>
<p><span class="math-container">\begin{equation*}
\frac{x^3+2x^2+... | zwim | 399,263 | <p><em>Here is another method that works well when Diophantine quadratics <span class="math-container">$ax^2+bx+c=0$</span> are involved, it consists in saying that since a solution should exists then <span class="math-container">$\exists \delta\in\mathbb Z\mid b^2-4ac=\delta^2$</span>.</em></p>
<p><br></p>
<p>In thi... |
140,615 | <p>A rectangular page is to have a printed area of 62 square inches. If the border is to be 1 inch wide on top and bottom and only 1/2 inch wide on each side find the dimensions of the page that will use the least amount of paper</p>
<p>Can someone explain how to do this?</p>
<p>I started with:</p>
<p>$$A = (x + 2)(... | David Mitra | 18,986 | <p>If the dimensions of the printed area are $x$ and $y$, where $y$ is the dimension with the $1/2$ inch borders (the "width"),
then the printed area is $$\tag{1}62= x y.$$ You want to minimize the area of the entire page, which is
$$\tag{2}A=(x+2)(y+1).$$
We want $A$ expressed in terms of one variable only; so solve... |
3,343,550 | <p>Show that if <span class="math-container">$(a,15)=1$</span>, then <span class="math-container">$a^4\equiv1 \mod 15$</span>, so that we do not have primitive roots of <span class="math-container">$15$</span>
Please help me with this problem.</p>
| Community | -1 | <p>It is a good idea which you will be able to use in similar questions to consider mod 3 and mod 5 separately.
You will then find that Fermat's Little Theorem will save you a lot of work.</p>
|
1,526,882 | <p>Let $$r(x,y)=\begin{cases} y &\mbox{ if } 0\leq y\leq x \\ x &\mbox{ if } x\leq y\leq 1\end{cases}$$</p>
<p>Show that $v(x)=\int_0^1r(x,y)f(y) \ dy$ satisfies $-v''(x)=f(x)$, where $0\leq x \leq 1$ and $f$ is continuous. </p>
<p>How can I take the second derivative of this? When I try to do it I feel like... | Samrat Mukhopadhyay | 83,973 | <p>Note that $$v(x)=\int_{0}^ x yf(y) dy+\int_{x}^1 xf(y) dy\\\implies v'(x)=xf(x)+\int_{x}^1 f(y) dy-xf(x)\\\implies v''(x)=-f(x)$$ The differentiation results follow from <a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus" rel="nofollow">fundamental theorem of calculus</a>.</p>
|
2,043,690 | <p>I'm in the process of studying for an exam and this just popped into my head. Sorry if it's a dumb question</p>
| Joffan | 206,402 | <p>Indeed they are the same, although I'd write them either without brackets or with brackets around the whole expression. </p>
<p>I really wanted to add that that using the $(-1 \bmod 13)$ representation can be very useful, especially when try to find the result of a multiplication. For a simple example, $12\times 7 ... |
1,258,623 | <p>Let $a, b, c, d, e$ be distinct positive integers such that $a^4 + b^4 = c^4 + d^4 = e^5$. Show that $ac + bd$ is a composite number.</p>
| Einar Rødland | 37,974 | <p>Assume that $p=ac+bd$ is prime. Without loss of generality, we may assume that $a>c,d$ from which follows that $b<c,d$. Also, $a,b,c,d<p$, and so must all be relatively prime with $p$.</p>
<p>Computing modulo $p$, we have $ac \equiv -bd$ which leads to $(ac)^4\equiv(bd)^4$. Exploiting $a^4+b^4=c^4+d^4$ we... |
2,893,388 | <p>My textbook is confusing me a little. Here is a worked example from my textbook:</p>
<blockquote>
<p>Line <span class="math-container">$l$</span> has the equation <span class="math-container">$\begin{pmatrix}3\\ -1\\ 0\end{pmatrix}+\lambda \begin{pmatrix}1\\ -1\\ 1\end{pmatrix}$</span> and point <span class="math-c... | Henno Brandsma | 4,280 | <p>If a space $X$ is <em>hereditarily Lindelöf</em> and $\mathcal{B}$ is any base for the space $X$, then any open set $O$ can be written as a countable union of elements from $\mathcal{B}$.
A space can be hereditarily Lindelöf without there being a countable base for $X$, an example of this is the Sorgenfrey line (or... |
1,507,519 | <p>How do I solve this question?</p>
<p><a href="https://i.stack.imgur.com/drfiA.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/drfiA.jpg" alt="Question"></a></p>
<p>I tried using the quadratic formula on the question equation and got</p>
<p>$x_1 = 0.25 +1.089724..i = \ln r$</p>
<p>$x_2 = 0.25 -... | Mythomorphic | 152,277 | <p>Actually the key of solving this question is to make use of the <strong>RELATION OF ROOTS</strong>. </p>
<p><strong>Recall:</strong> </p>
<p>If $\forall a,b,c\in\Bbb{R}$, given quadratic of $x$, $ax^2+bx+c=0$, </p>
<p>which can be converted to the form of $x^2+\frac bax+\frac ca=0$, </p>
<p>we say that $-\frac ... |
3,152,532 | <p>What is meaning of <span class="math-container">$f^2(x)$</span> ? There seems to be confusion in its interpretation.</p>
<p>Is <span class="math-container">$f^2(x)$</span> same as <span class="math-container">$(f(x))^2$</span> or <span class="math-container">$f\circ f$</span>?</p>
| Community | -1 | <p>It can be both, but unless you are in a context where function iteration makes sense, it is more likely to be the square.</p>
<hr>
<p>The confusion is a little worse with the exponent <span class="math-container">$-1$</span>: <span class="math-container">$\sin^{-1}(x)$</span> is more likely to be the arc sine than... |
4,369,480 | <p>In's commonly said that in VAE, we use reparameterization trick because "we can't backpropagate through stochastic node"
<a href="https://i.stack.imgur.com/ED72y.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ED72y.png" alt="enter image description here" /></a></p>
<p>It makes sense fro... | etal | 389,777 | <p>I think your maximum likelihood equation is not correct. In particular, if I understand your setting correctly <span class="math-container">$X_i$</span> is your data, which does not depend on <span class="math-container">$u$</span>. Their log-likelihood does depend on <span class="math-container">$u$</span> since fo... |
2,985,256 | <blockquote>
<p>Let there are three points <span class="math-container">$(2,5,-3),(5,3,-3),(-2,-3,5)$</span> through which a plane passes. What is the equation of the plane in Cartesian form?</p>
</blockquote>
<p>I know how to find it in using vector form by computing the cross product to get the normal vector and p... | sree_ranjani | 492,385 | <p>After using Cramer's rule, <span class="math-container">$a=-2,b=-3,c=-4$</span>.</p>
|
3,291,190 | <p>Compute</p>
<p><span class="math-container">$$\text{const} \left( \frac{1}{(1-z_1z_2)(1-z_1^2z_2)(1-z_1)(1-z_2)z_1^{3t}z_2^{2t}} \right) $$</span> </p>
<p>The answer should be a polynomial in <span class="math-container">$t$</span>. This is exercise 2.38 from Beck's Computing the Continuous Discretely, and the cor... | amd | 265,466 | <p>I don’t know of any real short cuts for attacking this. I think you just need to bash it out one variable at a time. </p>
<p>Starting from <a href="https://math.stackexchange.com/a/3291206/265466">gt6989b’s answer</a>, we’re trying to find <span class="math-container">$[x^{3t}y^{2t}] F(x,y)$</span> with <span clas... |
4,316,771 | <p>I'm asked to compute <span class="math-container">$$\sum_{k=-3}^{10} 2k^4$$</span>
I looked up for<a href="https://en.m.wikipedia.org/wiki/Bernoulli_number" rel="nofollow noreferrer">Bernoulli Number</a> on Wikipedia and found a general formula for that. But my teacher has asked me to evaluate this by breaking the s... | miracle173 | 11,206 | <p>Assume we have no electronic or mechanical device that assists us in doing the calculations and we are short of time and paper so that we want to avoid excessive calculations by pencil and paper and our mental arithmetic skills are rather limited than I would propose the following way to calculate the value:</p>
<p>... |
740,294 | <p>Let $y_n$ satisfy the nonlinear difference equation:</p>
<p>$$(n+1)y_n=(2n)y_{n-1}+n.$$</p>
<p>Let $u_n=(n+1) y_n$. Show that</p>
<p>$$u_n= 2u_{n-1}+n.$$</p>
<p>Solve the linear difference equation for $u_n$. Hence find $y_n$ subject to the initial condition $y_0=4$.</p>
<p>I have showed that $u_n=2u_{n-1}+n$, ... | Yiyuan Lee | 104,919 | <p>Firstly, divide throughout by $2^n$ to get:</p>
<p>$$\frac{u_n}{2^n} = \frac{u_{n-1}}{2^{n-1}} + \frac{n}{2^n}$$</p>
<p>Rearrange to get</p>
<p>$$\frac{u_n}{2^n} - \frac{u_{n-1}}{2^{n-1}} = \frac{n}{2^n}$$</p>
<p>Sum from $1$ to $n$:</p>
<p>$$\sum_{i=1}^n \frac{u_i}{2^i} - \frac{u_{i-1}}{2^{i-1}} = \sum_{i=1}^n... |
747,816 | <p>1) Can a non-square matrix have eigenvalues? Why?</p>
<p>2) True or false: If the characteristic polynomial of a matrix A is p($\lambda$)=$\lambda$^2+1, then A is invertible.
Thank you!</p>
| David | 119,775 | <p><strong>Hint</strong>. (1) Try it! For example, can you solve
$$\pmatrix{1&2&3\cr4&5&6\cr}\pmatrix{v_1\cr v_2\cr v_3\cr}
=\lambda\pmatrix{v_1\cr v_2\cr v_3\cr}\ ?$$</p>
<p>(2) You should know</p>
<ul>
<li>the connection between whether or not $A$ is invertible and $\det(A)$;</li>
<li>the connec... |
3,390,407 | <p>If <span class="math-container">$t>0,t^2, t+\frac{1}{t},t+t^2,\frac{1}{t}+\frac{1}{t^2}$</span> are all irrational number,
<span class="math-container">$$a_n=n+\left \lfloor \frac{n}{t} \right \rfloor+\left \lfloor \frac{n}{t^2} \right \rfloor,\\
b_n=n+\left \lfloor \frac{n}{t} \right \rfloor +\left \lfloor nt \r... | user90369 | 332,823 | <p><em>Often it's crucial how a question is formulated to understand it well. My following considerations involve a functional view. It's a proof for the generalisation.</em></p>
<p><span class="math-container">$k\in\mathbb{N}~$</span> fixed.</p>
<p><span class="math-container">$t_i\in\mathbb{R}^+~$</span> and <span ... |
983,849 | <p>Given</p>
<ul>
<li>$G$ be a finite group</li>
<li>$X$ is a subset of group $G$</li>
<li>$|X| > \frac{|G|}{2}$</li>
</ul>
<p>I noticed that any element in $G$ can be expressed as the product of 2 elements in $X$. Is there a valid way to prove this？</p>
<p>If the third condition was $|X| = \frac{|G|}{2}$ instead... | David | 119,775 | <p>Answer for the additional question: the statement need not be true if $|X|=|G|/2$.</p>
<p>For example, let $G$ be the cyclic group of order $2$ and $X$ consist of the non-identity element.</p>
<p>A (slightly) more general example: let $G$ be the cyclic group of order $2n$, let $g$ be a generator, and let
$$X=\{g^{... |
1,148,043 | <p>Is $2\sqrt{12}$ or $4\sqrt{3}$ a better representation? Also, for $\sqrt{675}$, is $3\sqrt{75}$ or $15\sqrt{3}$ considered more simplified? Why is one more simplified than the others?</p>
| Asinomás | 33,907 | <p>I would say it should be $4\sqrt 3$ or $\sqrt {48}$ . </p>
<p>Why should it be $\sqrt{48}$? You can get a very easy approximation of it just by knowing $\sqrt{49}=7$.</p>
<p>Why should it be $4\sqrt{3}$? You can calculate just by knowing $\sqrt{3}$. Which most mathematicians should know up to at least two places. ... |
398,371 | <p>How to calculate $$\lim_{t\rightarrow1^+}\frac{\sin(\pi t)}{\sqrt{1+\cos(\pi t)}}$$? I've tried to use L'Hospital, but then I'll get</p>
<p>$$\lim_{t\rightarrow1^+}\frac{\pi\cos(\pi t)}{\frac{-\pi\sin(\pi t)}{2\sqrt{1+\cos(\pi t)}}}=\lim_{t\rightarrow1^+}\frac{2\pi\cos(\pi t)\sqrt{1+\cos(\pi t)}}{-\pi\sin(\pi t)}$$... | Bill Kleinhans | 73,675 | <p>In the original expression, use the half angle equation in the denominator, and the double angle equation in the numerator. Then $ cos(\pi t/2) $ cancels, and the original expression equals minus $\sqrt 2 sin(\pi t/2) $</p>
|
3,958,142 | <p>The following problem is in Mathematical Methods in physical science CH8 miscellaneous problems
<span class="math-container">$$
(2x-y\sin(2x))dx = (\sin^2x-2y)dy
$$</span>
it isnt an exact equation the only difference is <span class="math-container">$-$</span> sign
because it not an exact equation I tried to rearran... | J.G. | 56,861 | <p>Since <span class="math-container">$d[x^2+y^2-y\sin^2x]=(2x-y\sin2x)dx+(2y-\sin^2x)dy=0$</span>, the solution is <span class="math-container">$x^2+y^2-y\sin^2x=c$</span>.</p>
|
3,958,142 | <p>The following problem is in Mathematical Methods in physical science CH8 miscellaneous problems
<span class="math-container">$$
(2x-y\sin(2x))dx = (\sin^2x-2y)dy
$$</span>
it isnt an exact equation the only difference is <span class="math-container">$-$</span> sign
because it not an exact equation I tried to rearran... | user577215664 | 475,762 | <p><span class="math-container">$$(2x-y\sin(2x))dx = (\sin^2x-2y)dy$$</span>
<span class="math-container">$$(2x-y\sin(2x))= (\sin^2x-2y)y'$$</span>
<span class="math-container">$$2x+2yy'=y'\sin^2 x +y \sin (2x)$$</span>
<span class="math-container">$$2x +(y^2)'=(y \sin^2 x)'$$</span>
Integrate.</p>
|
855,329 | <p>$$
\mbox{Question: Evaluate}\quad
\tan^{2}\left(\pi \over 16\right) + \tan^{2}\left(2\pi \over 16\right) + \tan^{2}\left(3\pi \over 16\right) + \cdots + \tan^{2}\left(7\pi \over 16\right)
$$</p>
<p>What I did:
Well I know that $\tan^{2}\left(7\pi/16\right)$ is the same as
$\cot^{2}\left(\pi/16\right)$. Thus this wi... | user8268 | 8,268 | <p>The 15 numbers $\tan k\pi/16$, $-7\leq k\leq7$, are the roots of the polynomial (of degree 15)
$(1+ix)^{16}-(1-ix)^{16}$ (using the geometric interpretation of complex numbers and their multiplication). The polynomial is of the form $-32ixp(x^2)$, where $p(t)=t^7-35t^6+\dots$ is a polynomial of degree 7, the roots o... |
855,329 | <p>$$
\mbox{Question: Evaluate}\quad
\tan^{2}\left(\pi \over 16\right) + \tan^{2}\left(2\pi \over 16\right) + \tan^{2}\left(3\pi \over 16\right) + \cdots + \tan^{2}\left(7\pi \over 16\right)
$$</p>
<p>What I did:
Well I know that $\tan^{2}\left(7\pi/16\right)$ is the same as
$\cot^{2}\left(\pi/16\right)$. Thus this wi... | lab bhattacharjee | 33,337 | <p>Like <a href="https://math.stackexchange.com/questions/718346/find-sum-limits-k-112-tan-frack-pi13-cdot-tan-frac3k-pi13/720170#720170">Find $\sum\limits_{k=1}^{12}\tan \frac{k\pi}{13}\cdot \tan \frac{3k\pi}{13}$</a>,</p>
<p>$$\tan16\theta=\frac{\binom{16}1t-\binom{16}3t^3+\cdots+\binom{16}{13}t^{13}-\binom{16}{15}t... |
500,579 | <p>Is there a formula for the coefficients of $x^n$ for </p>
<p>$$
\prod_{i=1}^N(x+x_i)
$$</p>
<p>in terms of $x_i$?</p>
| bubba | 31,744 | <p>Another approach that might be easier to program:</p>
<p>Let's call your function $f(x)$. We know it's a polynomial of degree $n$, so it can be written in the form $p(x) = \sum_{i=0}^n a_ix^i$. Choose $n+1$ values $z_0, \ldots, z_n$. By equating values of $f$ and $p$ at $x = z_0, \ldots, z_n$, we get a system of $n... |
2,775,087 | <blockquote>
<p>Without using a calculator, what is the sum of digits of the numbers from $1$ to $10^n$?</p>
</blockquote>
<p>Now, I'm familiar with the idea of pairing the numbers as follows:</p>
<p>$$\langle 0, 10^n -1 \rangle,\, \langle 1, 10^n - 2\rangle , \dotsc$$</p>
<p>The sum of digits of each pair is $9n$... | Christian Blatter | 1,303 | <p>From $0000$ till $9999$ we have $10\,000$ four digit words. The $40\,000$ digits have an average value of $4.5$ each. Add $1$ for the extra number $10\,000$ and obtain $180\,001$.</p>
|
1,344,883 | <p>I had an idea that passes by declaring a new type of <strong>computer variable</strong> (like Integer, Double, etc.) <strong>that represents a statistical probability distribution (PDF)</strong>, for that I would need to define the basic operations; sum, multiplication, inverse and negation.</p>
<p>The problem is t... | Tolaso | 203,035 | <p><strong>There is no closed solution</strong>. However note that:</p>
<p>$$e^x = \frac{1}{x}$$</p>
<p>has one real root. To see this you can consider it as a function , differentiate , determine the range etc. Therefore we have a root. </p>
<p>This root is a famous constant denoted as $\Omega$. Its approximate val... |
482,596 | <p>How do I evaluate $ sin(20)$ exactly? [in degrees]</p>
<p>I derived the relationship between $sin(x) $ and $sin(3x)$ where $x = sin(x)$ and $ y = sin(3x)$</p>
<p><a href="http://www.wolframalpha.com/input/?i=-4x%5E3+%2B+3x+%3D+y%2C+solve+for+x" rel="nofollow">http://www.wolframalpha.com/input/?i=-4x%5E3+%2B+3x+%3... | André Nicolas | 6,312 | <p>If $7s+11t=1$, then $7(s+11k)+11(t-7k)=1$ for all $k$. So there are infinitely many choices.</p>
<p>As for <em>finding</em> one pair $(s,t)$ that work, though there are general procedures, for such small numbers it is easier to go by experimentation. Note that we can take $s=8$ and $t=-5$.</p>
<p><strong>Remark:</... |
3,215,556 | <p>I have a question in my paper, Express 4225 as the product of its prime factors in index notation. That was easy to answer, but the next question is express the square root of 42250000 using prime factorisation. Apparently there is a way to use my answer in the first question to do the second, but how do I?</p>
| Arthur | 15,500 | <p>By way of
<span class="math-container">$$
42\,250\,000 = 4225\cdot 10\,000
$$</span>
the two numbers have a lot of factorisation in common.</p>
|
324,219 | <p>Given an urn with $M$ unique balls, how many times do I need to draw with replacement before the probability that I have seen each ball at least once is greater than $\epsilon$?</p>
| Coiacy | 62,460 | <p>Order the $M$ balls form $1$ to $M$ and suppose you have drawn $n$ balls with replacement. Among the $n$ balls, the number of ball$1$ is $x_1$, the number of ball$2$ is $x_2,\ \cdots$, the number of ball$M$ is $x_M$, then
$$
x_1+x_2+\cdots +x_M=n
$$
you want to see each ball at least once, that is the restiction:
$... |
41,183 | <p>Is it true that any manifold homotopy equivalent to a k-dimensional CW-complex admits a proper Morse function with critical points all of index <= k? I believe this is not true, so I would like to see a counterexample.</p>
| Ryan Budney | 1,465 | <p>Take a contractible $3$-manifold which is not homeomorphic to $\mathbb R^3$ -- like the Whitehead manifold. </p>
<p>If such a Morse function existed on the Whitehead manifold, it would be a Morse function with only one critical point, the minimum, and therefore the Whitehead manifold would be an open 3-ball. The p... |
1,752,506 | <p>Question: $ \sqrt{x^2 + 1} + \frac{8}{\sqrt{x^2 + 1}} = \sqrt{x^2 + 9}$</p>
<p>My solution: $(x^2 + 1) + 8 = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$</p>
<p>$=> (x^2 + 9) = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$</p>
<p>$=> (x^2 + 9) - \sqrt{x^2 + 9} \sqrt{x^2 + 1} = 0$</p>
<p>$=> \sqrt{x^2 + 9} (\sqrt{x^2 + 9} - \sqrt{x^... | brainst | 309,526 | <p>It would indeed , because then you wont have $x=\pm 3i$ as the root because that would make the denominator zero , which is the greatest offense one can do in algebra!! or more broadly in Mathematics (:P)</p>
|
1,187,142 | <p>Let a(n) be an arithmetic sequence and, as usual let $A(n)=a(1) + a(2) + \dots + a(n).$ If $A(5) = 50$ and $A(20) = 650$, find $A(15).$</p>
<p>I'm not so sure how to solve that.
So </p>
| Jr Antalan | 207,778 | <p>Let $A(n)$ be an arithmetic sequence, then: </p>
<p>$A(n)=\frac{n}{2}(2a+(n-1)d)$</p>
<p>where $a$ is $A(1)$ and $d$ is the common difference.</p>
<p>Using the formula above we yield the system of 2 equations below:</p>
<p>$\frac{5}{2}(2a+4d)=50$</p>
<p>$\frac{20}{2}(2a+19d)=650$</p>
<p>Now with this two equat... |
2,066,501 | <p>If Q = \begin{bmatrix}1&1/√6&1/3√3\\1&-1/√6&-1/3√3\\0 & √2/√3 & -1/3√3\end{bmatrix}</p>
<p>and R = \begin{bmatrix}2√2&1/√2&1/√2\\0&4/√6&1/√6\\0&0&1/3√3\end{bmatrix}.</p>
<p>Is A invertible? (no computation required) Is the system Ax=b solvable for each b in $R^3$ (gi... | TheGeekGreek | 359,887 | <p><strong>Hint.</strong> Consider the determinant of $A$. I assume you mean the $QR$-decomposition of $A$, i.e. $A = QR$.</p>
|
1,470,039 | <p>This problem will deal with the group <span class="math-container">$G = D_4 \times S_3$</span>.</p>
<p>How many elements of each order do the groups have <span class="math-container">$D_4$</span> and <span class="math-container">$S_3$</span> have? Using this info, determine how
many elements of each order <span cl... | Groups | 28,017 | <p>Elements of order $2$ will come from produc of an element of order $\leq 2$ in $D_4$ and an element of order $\leq 2$ in $S_3$ (not both taken identity). Count them.</p>
<p>Elements of order $4$ will be product of elements of order $4$ inside $D_4$, with elements of order $\leq 2$ in $S_3$ (count them).</p>
<p>Ele... |
1,470,039 | <p>This problem will deal with the group <span class="math-container">$G = D_4 \times S_3$</span>.</p>
<p>How many elements of each order do the groups have <span class="math-container">$D_4$</span> and <span class="math-container">$S_3$</span> have? Using this info, determine how
many elements of each order <span cl... | Nicky Hekster | 9,605 | <p>Hint: if <span class="math-container">$x \in G, y \in H$</span>, then in <span class="math-container">$G \times H$</span>, the <span class="math-container">$ord((x,y))=lcm(ord(x),ord(y))$</span></p>
|
411,875 | <p>This is an exam question I encountered while studying for my exam for our topology course:</p>
<blockquote>
<p>Give two continuous maps from $S^1$ to $S^1$ which are not homotopic. (Of course, provide a proof as well.)</p>
</blockquote>
<p>The only continuous maps from $S^1$ to $S^1$ I can think of are rotations... | Karl Kroningfeld | 67,848 | <p>The identity is not homotopic to a constant map; otherwise, $S^1$ would be contractible, which would imply $\pi_1(S^1)=0$.</p>
|
3,733,428 | <p>I would like to understand why <span class="math-container">$\mathbb{R}^2\setminus \mathbb{Q}^2$</span> endowed with the subspace topology is not a topological manifold. It seems to me it is Hausdorff and second countable. So I am wondering. Why is it not locally Euclidian?</p>
| Alessandro Codenotti | 136,041 | <p>Any neighbourhood of any point in <span class="math-container">$\Bbb R^2\setminus\Bbb Q^2$</span> is not simply connected, so no it cannot be homeomorphic to <span class="math-container">$\Bbb R^n$</span>.</p>
|
1,232,143 | <p>If
$$f(x)=\sum_{n=0}^\infty\frac{x^n}{n!}, x\in\mathbb{R}$$
and
$$
g(x) = 1 + \int_0^x f(t) \,dt
$$</p>
<p>prove that $g(x)=f(x)$ for all $x\in\mathbb{R}$
and prove that $f$ is differentiable on $\mathbb{R}$ as well as show that $f'(x)=f(x)$ for all $x\in\mathbb{R}$.</p>
<p>I know that $f$ is continuous on $\math... | Paramanand Singh | 72,031 | <p>The problem is trivial if we use uniform convergence because then we can do term by term differentiation and get $f'(x) = f(x)$ and by definition of $g(x)$ we get $g'(x) = f(x)$ so that $h'(x) = 0$ where $h(x) = f(x) - g(x)$. Then $h(x)$ is a constant and $$h(x) = h(0) = f(0) - g(0) = 1 - 1 = 0$$ and therefore $f(x)... |
1,655,884 | <blockquote>
<p>How many integer-sided right triangles exist whose sides are combinations of the form $\displaystyle \binom{x}{2},\displaystyle \binom{y}{2},\displaystyle \binom{z}{2}$?</p>
</blockquote>
<p><strong>Attempt:</strong></p>
<p>This seems like a hard question, since I can't even think of one example to ... | vonbrand | 43,946 | <p><strong>Hint:</strong> Start the other way around, by the formula to generate all <a href="https://en.wikipedia.org/wiki/Pythagorean_triple" rel="nofollow">Pythagorean triples</a>.</p>
|
901,357 | <p>Let there be $T:R^3 \rightarrow R^3$
<br>
$T(0,-1,1)=(3,3,3)$
<br>
$T(1,0,-1)=(0,1,1)$
<br>
$T(1,1,0)=(1,2,-1)$</p>
<p>Is (1,2,3) is the only image of the vector $(1, \frac{-7}{9}, \frac{-8}{9})$?</p>
<p>I have thought to create a matrix $[T]^T_E$*$[T]^E_C$=$[T]^T_C$
so I will have a matrix that does the transform... | Adam Hughes | 58,831 | <p>You want to write</p>
<p>$${1\over 1-z}={1\over 1-5i-(z-5i)}$$</p>
<p>Then this is just</p>
<p>$${1\over 1-5i}\left({1\over 1-{z-5i\over 1-5i}}\right)$$</p>
<p>Which, by geometric series, instantly gives</p>
<p>$${1\over 1-5i}\sum_n {(z-5i)^n\over (1-5i)^n}=\sum_n{(z-5i)^n\over (1-5i)^{n+1}}$$</p>
<p>and moreo... |
869,358 | <p>This is a seemingly simple induction question that has me confused about perhaps my understanding of how to apply induction</p>
<p>the question;</p>
<p>$$\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\ \le\ 2-\frac{1}{n},\ \forall\ n \ge1.$$</p>
<p>this true for $n=1$, so assume the expression is true for $n\le k$. which pr... | Martin Sleziak | 8,297 | <p>Despite the fact that such solution is given in other posts (for example in this question about similar infinite series: <a href="https://math.stackexchange.com/questions/1000642/proof-that-sum-1-infty-frac1n2-2">Proof that $\sum_{1}^{\infty} \frac{1}{n^2} <2$</a> it is used as an auxiliary result in some answers... |
3,007,785 | <p><span class="math-container">$\lim\limits_{x\to 0}\frac{e^x-\sqrt{1+2x+2x^2}}{x+\tan (x)-\sin (2x)}$</span> I know how to count this limit with the help of l'Hopital rule. But it is very awful, because I need 3 times derivate it. So, there is very difficult calculations. I have the answer <span class="math-container... | user | 505,767 | <p><strong>HINT</strong></p>
<p>By Taylor's series we have that</p>
<p><span class="math-container">$$\frac{e^x-\sqrt{1+2x+2x^2}}{x+\tan (x)-\sin (2x)}=\frac{\overbrace{1+x+\frac12 x^2+\frac16x^3+o(x^3)}^{\color{red}{e^x}}-(\overbrace{1+x+\frac12x^2-\frac12 x^3+o(x^3)}^{\color{red}{\sqrt{1+2x+2x^2}}})}{x+\underbrace{... |
2,874,840 | <blockquote>
<p>If $P\left(A\right)=0.8\:$ and $P\left(B\right)=0.4$, find the maximum and minimum values of $\:P(A|B)$.</p>
</blockquote>
<p>My textbook says the answer is $0.5$ to $1$. But I think the answer should be $0$ to $1$.</p>
<p>The textbook claims $P(A∩B)$ is $0.2$ when $P(A'∩B')=0$</p>
<p>I think that ... | Clarinetist | 81,560 | <p>I think a visualization helps most with understanding these probability bounds.</p>
<p>To begin,
$$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} = \dfrac{P(A \cap B)}{0.4}\text{.}$$
To finish this problem, we need to find bounds for $P(A \cap B)$.
Let's draw a Venn diagram for these two events.
<a href="https://i.stack... |
786,329 | <p>$$f(x,y)=\begin{cases}
\frac{xy}{x^2+y^2}, \text{ if } x^2+y^2\neq 0
\\
0, \text{ if } x^2+y^2=0
\end{cases}$$</p>
<p>Is it continuous at $(0,0)$?</p>
| Jlamprong | 105,847 | <p>No, since along line $y=x$
$$
\lim_{x,y\to0}f(x,y)=\frac12\neq0=f(0,0).
$$</p>
|
2,317,496 | <p><strong>Definition:</strong> An ordinal number $\alpha$ is called a <strong><em>limit ordinal number</em></strong> if there is no ordinal number immediately preceding $\alpha$. </p>
<p>Now my lecture notes say that $\omega, 2\omega, \omega^2, \omega^\omega$ are limit ordinal numbers whereas $\omega+3,2^\omega+5$ ar... | Asaf Karagila | 622 | <p>There can be several answers, depending on what you mean by "identify".</p>
<p>The simplest answer would be to look at the <a href="https://en.wikipedia.org/wiki/Ordinal_arithmetic#Cantor_normal_form" rel="noreferrer">Cantor normal form</a> of $\alpha$, and see if it has any finite ordinal there. If the answer is n... |
2,484,855 | <p>How can I prove that these three sets have no common values:</p>
<ul>
<li>A: {prime numbers} </li>
<li>B: {Fibonacci numbers}</li>
<li>C: {8|n+1} </li>
</ul>
<blockquote>
<ul>
<li>C: for example 15: 15+1 = 16 => 8|16 <= 16/8 = 2</li>
<li>C: for example 23: 23+1 = 24 => 8|24 <= 24/8 = 3</li>
</ul>
... | Barry Cipra | 86,747 | <p>Consider the Fibonacci numbers mod $8$:</p>
<p>$$1,1,2,3,5,0,5,5,2,7,1,0,\ldots$$</p>
<p>We see that $F_m\equiv7$ mod $8$ if and only if $m\equiv10$ mod $12$, which implies $m$ is even (and greater than $4$). But $F_n\mid F_{2n}$ for all $n$, so $F_m$ cannot be a prime if $8\mid F_m+1$.</p>
<p>Remark: The theor... |
2,733,142 | <p>This question has been asked a <a href="https://math.stackexchange.com/questions/207029/a-b2-for-which-matrix-a">few</a> <a href="https://math.stackexchange.com/questions/583442/square-root-of-nilpotent-matrix">times</a>. In the former case I noticed that there was some argument trending towards using the Jordan fo... | Will Jagy | 10,400 | <p>trying 3 by 3</p>
<p>$$
\left(
\begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i
\end{array}
\right)^2 =
\left(
\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{array}
\right) \; ,
$$
then
$$
\left(
\begin{array}{ccc}
a^2 + bd+cg & ab+be+ch & ... |
2,727,237 | <p>$$
\begin{matrix}
1 & 0 & -2 \\
0 & 1 & 1 \\
0 & 0 & 0 \\
\end{matrix}
$$</p>
<p>I am told that the span of vectors equal $R^m$ where $m$ is the rows which has a pivot in it. So when describing the span of the above vectors, is it correct it saying that they don't span $... | P Vanchinathan | 28,915 | <p>The point is the distinction between vectors and their representation as linear combination with respect to a basis. Orthogonality as a concept is a condition <em>on the given two vectors</em>. Of course this can be translated into a condition on their representation. But that translated condition will vary dependi... |
2,145,098 | <p>This was part of a three part question where I was supposed to prove two sets have equal cardinality by finding bijections. I've created a bijection $f: \Bbb Z \Rightarrow 2\Bbb Z$ by $f(x)=2x$. I've created a bijection $g: (0,1) \Rightarrow (4,50)$ by $g(x)=46x+4$. I think those are both correct. My last question i... | fleablood | 280,126 | <p>It's counter intuitive but push 0 to $a_1$. And push $1$ to $a_2$. Then create an infinite sequence $a_i $ and push $a_i $ to $a_{i+3}$. Meanwhile for any $x $ not in the sequence, map $x $ to $x $. </p>
<p>Example: Let $a_n = 1/n$. Let $f (0)=1/2$. Let $f (1)= 1/3$. Let $f (1/n)=\frac 1 {n+2} $. If $x \ne ... |
1,642,427 | <p>If the stem of a mushroom is modeled as a right circular cylinder with diameter $1$, height $2$, its cap modeled as a hemisphere of radius $a$ the mushroom has axial symmetry, is of uniform density,and its center of mass lies at center of plane where the cap and stem join, then find $a$.</p>
<p>I really need help.<... | Wojowu | 127,263 | <p>The problem with this argument, and many other alike, is that the arc length of a curve is not a continuous function of a curve, which means that if sequence of curves $\gamma_i$ converges to a curve $\gamma$ (notion of convergence of functions like that can be made formal, but let me not do that here), then there i... |
2,071,828 | <p>Find the splitting field of $x^6-2x^4-8x^2+16$ over $\mathbb {F}_3$ and list the intermediate fields between the base camp and the splitting field.</p>
| lhf | 589 | <p>$x^6-2x^4-8x^2+16 = x^6+x^4+x^2+1$ in $\mathbb {F}_3[x]$.</p>
<p>$x^6+x^4+x^2+1 = \dfrac{x^8-1}{x^2-1} = (x^2 + 1) (x^4 + 1)$</p>
<p>Therefore, the splitting field of $x^6-2x^4-8x^2+16$ is the same as the splitting field of $x^4 + 1=(x^2 + x + 2) (x^2 + 2 x + 2)$, which is $\mathbb {F}_9$.</p>
|
46,726 | <p>In many proofs I see that some variable is "fixed" and/or "arbitrary". Sometimes I see only one of them and I miss a clear guideline for it. Could somebody point me to a reliable source (best a well-known standard book) which explains, when and how to use both in proofs?</p>
<p>EDIT: A little add-on to the question... | ItsNotObvious | 9,450 | <p>When you encounter the term "arbitrary", it usually just means that a given statement is specified for any element from a given set of elments. For example, if I say, let $x$ be an arbitrary element of the interval $[0, 1]$ I just mean that $x$ can take on any value within that interval. </p>
<p>The term <em>fixed<... |
2,046,492 | <p>Let's think of two events $1$ and $2$.</p>
<p>Both events happen randomly $n_1$/$n_2$-times during a given time $T$ and last for a time of $t_1$/$t_2$.</p>
<p><strong>What is probability $P$, that both events happen simultaneously at some moment?</strong></p>
<hr>
<hr>
<p><em>EXAMPLE 1:</em></p>
<p>$T = 60$ mi... | leonbloy | 312 | <p>[Not an answer, but some pointers]</p>
<p>In statistical physics parlance, you have a pair of independent <a href="http://www.sklogwiki.org/SklogWiki/index.php/1-dimensional_hard_rods" rel="nofollow noreferrer">1D hard-rods</a> (or Tonk gas). In the limit $L\to \infty$ the "canonical ensemble" (fixed volumen and nu... |
1,630,074 | <p>My brother needs help coming up with a formula for a problem that I already did but failed to write out the formula for. </p>
<p>The problem is:</p>
<p>Consider a circle with the point (5,4) and a radius of 3.
Determine if a vertical line segment with the points (3,5) being one end of the segment and (3,1) being t... | Plutoro | 108,709 | <p>The problem arises when applying $H_{t}$ to the point $(-1,0,0,...)$. We get $$H_t(-1,0,0,...)=\frac{(2t-1,0,0,...)}{\Vert (2t-1,0,0,...)\Vert}=\begin{cases} (-1,0,0,...) & t<1/2\\
(1,0,0,...) & t>1/2\\
\text{undefined} & t=1/2.\end{cases}$$
This is not continuous. We do not need a two step descrip... |
376,530 | <p>Let <span class="math-container">$G$</span> be a graph which does not contain a simple cycle <span class="math-container">$v_1\ldots v_k$</span> and two "crossing" chords <span class="math-container">$v_iv_j$</span> and <span class="math-container">$v_pv_q$</span>, <span class="math-container">$i<p<j... | David Wood | 25,980 | <p>Thomassen and Toft [<a href="https://doi.org/10.1016/S0095-8956(81)80025-1" rel="noreferrer">JCTB 31(2):199-224, 1981</a>] showed that any graph with minimum degree at least 3 contains a cycle with two crossing chords from neighbouring vertices on the cycle. The <span class="math-container">$2n-3$</span> upper bound... |
33,743 | <p>I have a lot of sum questions right now ... could someone give me the convergence of, and/or formula for, $\sum_{n=2}^{\infty} \frac{1}{n^k}$ when $k$ is a fixed integer greater than or equal to 2? Thanks!!</p>
<p>P.S. If there's a good way to google or look up answers to these kinds of simple questions ... I'd lo... | Qiaochu Yuan | 232 | <p>Yes, this sum converges, and yes, you can prove this using the integral test. For $k$ even the exact value (plus $1$) turns out to be a rational multiple of $\pi^k$ (see, for example, <a href="http://en.wikipedia.org/wiki/Bernoulli_number#Asymptotic_approximation" rel="nofollow">Wikipedia</a>), but for odd $k$ the e... |
2,990,642 | <p><span class="math-container">$\lim_{n\to \infty}(0.9999+\frac{1}{n})^n$</span></p>
<p>Using Binomial theorem:</p>
<p><span class="math-container">$(0.9999+\frac{1}{n})^n={n \choose 0}*0.9999^n+{n \choose 1}*0.9999^{n-1}*\frac{1}{n}+{n \choose 2}*0.9999^{n-2}*(\frac{1}{n})^2+...+{n \choose n-1}*0.9999*(\frac{1}{n})... | user | 505,767 | <p><strong>HINT</strong></p>
<p>We have that</p>
<p><span class="math-container">$$\left(0.9999+\frac{1}{n}\right)^n=(0.9999)^n\left(1+\frac{\frac{1}{0.9999}}{n}\right)^n$$</span></p>
|
1,651,922 | <p>I'm solving this equation:</p>
<p>$$\sin(3x) = 0$$</p>
<p>The angle is equal to 0, therefore:</p>
<p>$$3x=0+2k\pi \space\vee\space3x= (\pi-0)+2k\pi$$ </p>
<p>$$x = \frac {2}{3}k\pi \space \vee \space x = \frac {\pi + 2k\pi}{3}$$</p>
<p>Though, the answer is </p>
<p>$$x = k\frac {\pi}{3}$$</p>
<p>It looks like... | fosho | 166,258 | <p>All we need is for $3x$ to be an integer multiple of $\pi$. In other words</p>
<p>$$3x = k\pi \Rightarrow x = \frac{k\pi}{3}$$</p>
|
1,651,922 | <p>I'm solving this equation:</p>
<p>$$\sin(3x) = 0$$</p>
<p>The angle is equal to 0, therefore:</p>
<p>$$3x=0+2k\pi \space\vee\space3x= (\pi-0)+2k\pi$$ </p>
<p>$$x = \frac {2}{3}k\pi \space \vee \space x = \frac {\pi + 2k\pi}{3}$$</p>
<p>Though, the answer is </p>
<p>$$x = k\frac {\pi}{3}$$</p>
<p>It looks like... | StackTD | 159,845 | <p>You're not missing anything: sometimes it's possible to efficiently combine sets of solutions.</p>
<p>Notice that for $\sin x=0$, the solutions $x=0+2k\pi$ ($0$ and then 'adding full circles') $\vee \; x=\pi+2k\pi$ ($\pi$ and then 'adding full circles') can be combined as $x=k\pi$ ($0$ and 'adding <strong>half</str... |
708,633 | <p>My question is as in the title: is there an example of a (unital but not necessarily commutative) ring $R$ and a left $R$-module $M$ with nonzero submodule $N$, such that $M \simeq M/N$?</p>
<p>What if $M$ and $N$ are finitely-generated? What if $M$ is free? My intuition is that if $N$ is a submodule of $R^n$, then... | Mariano Suárez-Álvarez | 274 | <p>One keyword which should bring up many useful results: Leavitt álgebras.</p>
|
1,672,882 | <p>Why we always see extended operations, like arbitrary unions, products, etc. in different parts of mathematics in the form of extensions of finite ones upon arbitrary <em>sets</em> (called index set)? Why never use an index-<em>class</em> for a proper class?</p>
<p>EDIT:
Of course I think this a reason, for example... | Martín-Blas Pérez Pinilla | 98,199 | <p>The greatest operation that you will see is the power of the universe $V^A$ with $A$ set. This is a proper class, but the <em>elements</em> are sets: functions $f:A\longrightarrow V$. In the case $\text{set}^{\text{proper class}}$ the possible "elements" are proper classes.</p>
|
159,510 | <p>I have data points of the form <code>{x,y,z}</code>:</p>
<pre><code>s = {{0, 1, 2}, {3, 4, 5}, {6, 7, 8}}
</code></pre>
<p>And I need to remove <em>entire data points</em> based on a condition. For example, let's say that I want to remove any data points where <code>z > 6</code>. The result should be this:</p>
... | Jack LaVigne | 10,917 | <p>I copied the 10 lines in your question and pasted them into a text file called <strong><em>out.txt</em></strong>.</p>
<pre><code>data = Import["D:\\at_work\\mathematica\\stack_exchange\out.txt",
"text"]
</code></pre>
<p>The basic problem with the way you were working is that what appears to be numbers are strin... |
159,510 | <p>I have data points of the form <code>{x,y,z}</code>:</p>
<pre><code>s = {{0, 1, 2}, {3, 4, 5}, {6, 7, 8}}
</code></pre>
<p>And I need to remove <em>entire data points</em> based on a condition. For example, let's say that I want to remove any data points where <code>z > 6</code>. The result should be this:</p>
... | george2079 | 2,079 | <p>another approach.</p>
<pre><code>ImportString[
StringReplace[
Import["test.txt", "Text"], {"[", "]"} -> ""], "CSV"]
</code></pre>
|
1,129,052 | <p>How many 3-digits numbers possess the following property: </p>
<blockquote>
<p>After subtracting $297$ from such a number, we get a $3$-digit number consisting of the same digits in the reverse order.</p>
</blockquote>
| Pp.. | 203,995 | <p>A $3$-digit number is of the form $$a_0+10a_1+100a_2\ \ \ \ \ \ \ \ \text{with }\ \ \ \ \ \ 0\leq a_0,a_1\leq9,\ \ \ \ \ \ 1\leq a_2\leq9$$
Then the condition to impose is $$a_0+10a_1+100a_2-297=a_2+10a_1+100a_0\ \ \ \ \ \text{ and }\ \ \ \ a_0\neq0$$
Therefore
$$99a_2-99a_0=297$$</p>
<p>or equivalently $$a_2-a_0... |
1,129,052 | <p>How many 3-digits numbers possess the following property: </p>
<blockquote>
<p>After subtracting $297$ from such a number, we get a $3$-digit number consisting of the same digits in the reverse order.</p>
</blockquote>
| Community | -1 | <p>Following the perfect explanation given by Arian, there are a total of 70 solutions -- 60 of them are <em>true</em> solutions (> 390):</p>
<p>$$ \left[
\begin{array}{cccccccccc}
300&310&320&330&340&350&360&370&380&390\\
\hline
401&411&421&431&441&a... |
4,285,448 | <p>For training I have decided to solve this limit of a succession</p>
<p><span class="math-container">$$\lim\limits_{n\to \infty }\left(3^{n+1}-3^{\sqrt{n^2-1}}\right).$$</span></p>
<p><strong>My first attempt</strong>:</p>
<p><span class="math-container">\begin{split}
\lim_{n\to \infty }\left(3^{n+1}-3^{\sqrt{n^2-1}}... | Matsmir | 685,805 | <p>The obvious way to obtain <span class="math-container">$\infty$</span> as the limit:</p>
<p><span class="math-container">$$
3^{n+1} - 3^{\sqrt{n^2 - 1}} \ge 3^{n+1} - 3^{n} = 2(3^n) \rightarrow \infty.
$$</span>
Your second attempt seems correct but is more complicated.</p>
|
3,902,501 | <p>Over <span class="math-container">$\mathbb R$</span>, the only linear maps are those of the form <span class="math-container">$ax$</span>.</p>
<p>If we discuss rational functions over <span class="math-container">$\mathbb R$</span>, this extra structure would allow us to describe a wider variety of linear maps.</p>
... | GEdgar | 442 | <p>A example of a linear functional on <span class="math-container">$\mathbb R(X)$</span> ... write it in partial fraction form, and take the coefficient of <span class="math-container">$1/(X-3)^2$</span>. (Of course <span class="math-container">$0$</span> if that terms does not appear.)</p>
|
2,697,251 | <p>In my c++ application I have a 3d plane, with a point above it. The plane is rotated, and the point is at 0,0,0. </p>
<p>In the image below, the plane is represented by the triangle side B - C. (S3)</p>
<p>A is the point that is at zero.
B is the mid-point of the plane.</p>
<p>What I need to calculate is the dis... | user | 505,767 | <p>Yes we can, indeed recall that</p>
<ul>
<li>$S_3=S_2 \cos B$</li>
<li>$S_1=S_2 \sin B$</li>
</ul>
<p><a href="https://i.stack.imgur.com/5WWvx.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5WWvx.gif" alt="enter image description here"></a></p>
|
2,697,251 | <p>In my c++ application I have a 3d plane, with a point above it. The plane is rotated, and the point is at 0,0,0. </p>
<p>In the image below, the plane is represented by the triangle side B - C. (S3)</p>
<p>A is the point that is at zero.
B is the mid-point of the plane.</p>
<p>What I need to calculate is the dis... | Mohammad Riazi-Kermani | 514,496 | <p>Yes, the right triangle is uniquely determined by its hypotenuse and an angle $\alpha.$</p>
<p>The other two sides are $$ BC=AB cos( \alpha) \\AC= AB sin( \alpha)$$ </p>
|
2,624,272 | <p>It is well known that, in order to track constant reference values with zero asymptotic error, in the presence of uncertainty, an integrator is required in the open loop (either by the plant or the controller itself).</p>
<p>Lets for simplicity assume the plant $P$ has no integral components.</p>
<p>In order to tr... | Kwin van der Veen | 76,466 | <p>The reason why this can be done in the first place can be derived from the <a href="https://en.wikipedia.org/wiki/Final_value_theorem" rel="nofollow noreferrer">final value theorem</a>. For this one needs the Laplace transform of the tracking error, which can be obtained by multiplying the sensitivity transfer funct... |
1,558,665 | <p>I am looking for examples of finitely generated solvable groups that are not polycyclic. In <a href="http://groupprops.subwiki.org/wiki/Finitely_generated_and_solvable_not_implies_polycyclic#Some_examples_based_on_the_general_construction_and_otherwise" rel="nofollow">Wikipedia</a> Baumslag-Solitar group $BS(1,2)$ i... | anomaly | 156,999 | <p>The Baumslag-Solitar group $BS(1, 2) = \mathbb{Z} \ltimes \mathbb{Z}[\frac{1}{2}]$ under the isomorphism $a \to (0, 1), b \to (1, 0)$, where the generator $g$ of the left term acts by $g.1 = 2$. But $\mathbb{Z}[\frac{1}{2}]\subset BS(1, 2)$ is clearly not finitely generated, so $BS(1, 2)$ is not polycyclic.</p>
|
3,702,309 | <p>The question reads,
Find all differentiable functions <span class="math-container">$f$</span> such that <span class="math-container">$$f(x+y)=f(x)+f(y)+x^2y$$</span> for all <span class="math-container">$x,y \in \mathbb{R} $</span>. The function <span class="math-container">$f$</span> also satisfies <span class="ma... | Kavi Rama Murthy | 142,385 | <p>There cannot be any function satisfying this equation for the simple reason that <span class="math-container">$f(x+y)-f(x)-f(y)$</span> does not change if you interchange <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. So if such a function exists we must have <span class="math-c... |
1,099,214 | <p>I'm having trouble calculating some probabilities for a simple card game I'm into </p>
<p>1) So say we have a deck of 13 cards total. You draw 3 cards so there are 10 cards left in the deck. Say there is 1 special card in the deck that we want to figure out the probability of getting it in the hand. How do we figur... | BCLC | 140,308 | <p>Looks right.</p>
<p>"I was thinking I might have to multiply by 3! because there are 3! ways to order the 3 cards in the hand " --> No need. Order doesn't matter in this case.</p>
<p>"Now let's say in the deck there is that 1 special card, and also 3 copies of a different type of card that we want. How would we ca... |
687,897 | <p>I need help with a differential equation, the trouble is I don't think it's separable and I have tried and failed to apply the method of characteristics to figure it out. z is also bound between zero and one.</p>
<p>$$
\frac{\partial u}{\partial x}+z\frac{\partial u}{\partial y}-Cz\frac{\partial u}{\partial z}=0
$$... | user127096 | 127,096 | <p>Parametrized by $x$, characteristic curves have equation
$$x=x,\quad y = y_0+\frac{z_0}{C}(1-e^{-Cx}),\quad z = z_0 e^{-Cx} \tag{1}$$
where $(0,y_0,z_0)$ is the point of crossing the plane $x=0$. To check this, differentiate (1) with respect to $x$:
$$x'=1,\quad y' =z_0 e^{-Cx} = z ,\quad z' = -C z_0 e^{-Cx} = -Cz... |
238,673 | <p><a href="http://vihart.com/doodling/" rel="nofollow">Vi Hart's doodling videos</a> and a 4 year old son interested in mazes has made me wonder:</p>
<p>What are some interesting mathematical "doodling" diversions/games that satisfy the following criteria:</p>
<p>1) They are "solitaire" games, i.e. require only one ... | Egor Skriptunoff | 50,643 | <p>@Karolis Juodelė<br>
... or emulate <a href="http://en.wikipedia.org/wiki/Busy_beaver" rel="nofollow">4-state, 2-symbol busy beaver</a> :-)</p>
|
1,038,060 | <p>Can anyone help me with this question: I know it before, but I have tried to solve it myself and didnt succeed.
what is the regular expression for this language: L=all words that have 00 or 11 but not both.</p>
<p>Thank you!</p>
| student | 379,844 | <p>Assume that $A_n$ is a simple group for $n\neq 4$. If $H$ is any subgroup of index $2$ in $S_n$, then $H$ must be normal in $S_n$ (because the only non-trivial coset in $S_n/H$ must be the set-theoretic complement of $H$, and must therefore be both the right and left coset obtained by multiplying on the right, or th... |
136,808 | <p>Let $A$ be a nonempty subset of $\omega$, the set of natural numbers.
I want to prove this statement:</p>
<blockquote>
<p>If $\bigcup A=A$ then $n\in A \implies n^+\in A$.</p>
</blockquote>
<p>Help...</p>
| lesnikow | 30,013 | <p>Oh yes, this can be pretty confusing! Here is how I think of it:</p>
<p>$\bigcup A = A$ really means $\bigcup A \subseteq A$ and $\bigcup A \supseteq A$.</p>
<p>Now assume $a \in A$. From $A \subseteq \bigcup A$, we get $a \in \bigcup A$,
which, looking at the definition of $\bigcup A$, means that $a$ is an elemen... |
2,220,105 | <p>Hatcher makes the following definition in exercise 7 on page 258. </p>
<p>For a map $f: M \rightarrow N $ between connected, closed, orientable $n$-manifolds with fundamental classes $[M]$ and $[N]$, the <em>degree</em> of $f$ is defined to be the integer $d$ such that $f_{*}[M]=d[N]$ (so the sign of the degree dep... | Doug M | 317,162 | <p>This sort of sorting problem is called pancake sorting. Bill Gates made important contributions to the pancake sorting problem. Then he dropped out of school. He did all right for himself, though.</p>
<p>Base case:</p>
<p>$n = 2$</p>
<p>Either the sequence is already sorted, or it needs to be inverted and can ... |
1,745,832 | <p>It is stated in Kolmogorov & Fomin's <em>Introductory Real Analysis</em> that every normal space is Hausdorff. I cannot seem to find an explanation for this anywhere, and don't see why this is obviously true... since it is not necessarily the case in an arbitrary topological space that a singleton is closed. </... | DanielWainfleet | 254,665 | <p>It is taken as part of the def'n of a normal space that it has the $T_1$ property, and it follows that a normal space is $T_2$. Engelking, in General Topology, calls a space in which disjoint closed subsets are completely separated a pseudo-normal space, as the $T_1$ property may fail. The simplest examples of a non... |
991,530 | <p>I am trying to prove the following:
For any positive real number $x>0$, there exist a positive integer N such that $x>1/N>0$. </p>
<p>What I know is:
since x is a real number, it can be expressed as the formal limit of a Cauchy sequence
$$x:=LIM (b_n)_{n=1}^\inf$$
since $x>0$, the sequence $(b_n)$ is b... | Community | -1 | <p><strong>Hint:</strong>
$$x>0$$ There will be infinitely many positive integers $A$ such that $$A>\frac{1}{x}.$$</p>
|
1,485,463 | <p>At a school function, the ratio of teachers to students is $5:18$. The ratio of female students to male students is $7:2$. If the ratio of the female teachers to female students is $1:7$, find the ratio of the male teachers to male students. </p>
| eyqs | 280,815 | <p>Let's say that there are $T_f$ female teachers, $T_m$ male teachers, $S_f$ female students, and $S_m$ male students. You are given that,</p>
<p>$$
(T_f + T_m) / (S_f + S_m) = 5/18\\
S_f / S_m = 7/2\\
T_f / S_f = 1/7
$$</p>
<p>And what you want to find is,</p>
<p>$$
T_m / S_m
$$</p>
<p>You can solve this now by s... |
113,960 | <p>When trying to compute the (Serre-generalized) intersection number of two varieties at a closed point, I came to a need to compute the following $\operatorname{Tor}$:</p>
<blockquote>
<p>Let $k$ be an algebrically closed field, $A=k[x_1,x_2,x_3,x_4]$ and $\mathfrak m=(x_1,x_2,x_3,x_4)$. Let $M = k[x_1,x_2,x_3,x_4... | Mariano Suárez-Álvarez | 274 | <p>You can do this computation without thinking much, really.</p>
<p>The two generators of the ideal which defines $N$ form a regular sequence, so you can use a Koszul complex to projectively resolve $N$. Tensor it with $M$ and just compute the homology of the resulting complex.</p>
|
1,412,091 | <p>The typewriter sequence is an example of a sequence which converges to zero in measure but does not converge to zero a.e.</p>
<p>Could someone explain why it does not converge to zero a.e.?</p>
<blockquote>
<p><span class="math-container">$f_n(x) = \mathbb 1_{\left[\frac{n-2^k}{2^k}, \frac{n-2^k+1}{2^k}\right]} \tex... | Ben Grossmann | 81,360 | <p>Note that at any choice of $x$ and for any integer $N$, there is an $n>N$ with $f_n(x)=1$. So, the numerical sequence $f_n(x)$ cannot converge to $0$.</p>
<p>Note, however, that we can certainly select a <em>subsequence</em> of this sequence of functions that converges pointwise a.e.</p>
|
3,839,244 | <p>Is <span class="math-container">$\{3\}$</span> a subset of <span class="math-container">$\{\{1\},\{1,2\},\{1,2,3\}\}$</span>?</p>
<p>If the set contained <span class="math-container">$\{3\}$</span> plain and simply I would know but does the element <span class="math-container">$\{1,2,3\}$</span> include <span class=... | user | 505,767 | <p>No it is not, we have that <span class="math-container">$\{1\}\in \{\{1\},\{1,2\},\{1,2,3\}\}$</span> but <span class="math-container">$\{3\}$</span> is not an element of the set, what is true is that <span class="math-container">$3\in\{1,2,3\}$</span>.</p>
|
1,935,320 | <p>If $a^2-b^2=2$ then what is the least possible value of: \begin{vmatrix} 1+a^2-b^2 & 2ab &-2b\\ 2ab & 1-a^2+b^2&2a\\2b&-2a&1-a^2-b^2 \end{vmatrix}</p>
<p>I tried to express the determinant as a product of two determinants but could not do so. Seeing no way out, I tried expanding it but that ... | MJD | 25,554 | <p>Do you know that 2×2 matrices correspond to linear transformations of the plane, and that composing the transformations multiplies the corresponding matrices?</p>
<p>Can you think of a linear transformation of the plane which, repeated three times, is the identity transformation?</p>
<p>Mouse over for hint:</p>
<... |
599,656 | <p>If the objects of a category are algebraic structures in their own right, this often places additional structure on the homsets. Is there somewhere I can learn more about this general idea?</p>
<hr>
<p><strong>Example 1.</strong> Let $\cal M$ denote the category of magmas and <strong>functions</strong>, not necess... | mainak | 114,270 | <p>obviously 8 is one of the solutions with no doubts at all.</p>
|
2,101,894 | <p>As part of a larger proof, I need to use the fact that for a homomorphism $\phi : A \rightarrow B$, where $A$ is cyclic and $a \in A$ its generator, $\phi(a^{-n}) = \phi(a^n)^{-1}$. All I came up with is that $\phi(a^{-n}) = \phi((a^n)^{-1})$ and then I hit a wall. Is it true in general that $\phi(a^{nm}) = \phi(a^n... | Mustafa | 400,050 | <p>because $\phi$ is homomorphism group then
$\phi ( (a^n)^{-1}))=\phi((a.a...a)^{-1})=\phi(a^{-1}...a^{-1})=\phi(a^{-1})...\phi(a^{-1})= \phi(a)^{-1}...\phi(a)^{-1}=(\phi(a)...\phi(a))^{-1}=(\phi(a^n))^{-1} $</p>
|
2,992,477 | <p>Can u help me prove that every simple module is cyclic? Ive already proved that a cyclic module is isomorphic to A quotiented by I ( I being a left ideal of A) using homomorphisms, but i cant seem to make any advance in proving this.Thx in advance.</p>
| sandoh | 1,083,786 | <p>Let M be a simple module. Then the only submodules of M are the trivial submodule and M itself.
Suppose M is not cyclic, that is, it is not generated by any of its elements. Because M is a module, we know it is an additive subgroup and so the sets generated by each of its non-zero elements will form a subgroup of M ... |
548,563 | <p>Calculate $$\sum \limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$$</p>
<p>I use software to complete the series is $\frac{2}{27} \left(18+\sqrt{3} \pi \right)$</p>
<p>I have no idea about it. :|</p>
| Ron Gordon | 53,268 | <p>Consider the function</p>
<p>$$f(x) = \frac{\arcsin{x}}{\sqrt{1-x^2}}$$</p>
<p>$f(x)$ has a Maclurin expansion as follows:</p>
<p>$$f(x) = \sum_{n=0}^{\infty} \frac{2^{2 n}}{\displaystyle (2 n+1) \binom{2 n}{n}} x^{2 n+1}$$</p>
<p>Differentiating, we get</p>
<p>$$f'(x) = \frac{x \, \arcsin{x}}{(1-x^2)^{3/2}} + ... |
3,450,692 | <p>How to solve this equation?</p>
<p><span class="math-container">$$
\frac{dy}{dx}=\frac{x^{2}+y^{2}}{2xy}
$$</span></p>
| emonHR | 565,609 | <p><span class="math-container">$
\frac{dy}{dx}=\frac{x^{2}\left(1+\frac{y^2}{x^{2}}\right)}{2xy}=\frac{1+\left(\frac{y}{x}\right)^2}{2\frac{y}{x}}
$</span> Can you go from here<span class="math-container">$?$</span></p>
|
2,238,830 | <p>I have just come across "Foundations of Mathematics". Wikipedia describes it as </p>
<blockquote>
<p>Foundations of mathematics can be conceived as the study of the basic
mathematical concepts (number, geometrical figure, set, function,
etc.) and how they form hierarchies of more complex structures and
conc... | J.G. | 56,861 | <p>In ancient Greece, Euclid proposed an idea mathematicians have liked (in principle) ever since: have assumptions called "axioms" we take for granted, then prove "theorems" from them. In theory, you could write a list of all the axioms of modern mathematics, then take it from there. Euclid had a few axioms of geometr... |
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