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 |
|---|---|---|---|---|
1,100,845 | <p>Let $z=f(x,y)$ and let r and $\theta$ be polar coordinates in the x-y plane. Recall that $x=r \cos \theta$ and $y=r \sin \theta$. Express $\frac{\partial^{2}z}{\partial r^{2}}$ in terms of r, $\theta$, and the partial derivative of f.</p>
<p>Letting </p>
<p>$$
g(x,y,\theta)=\frac{\partial z}{\partial r}
=\frac{\... | pre-kidney | 34,662 | <p>$$
\frac{y^2}{b^2}-\frac{x^2}{a^2}=1\implies \frac{y^2}{b^2}=1+\frac{x^2}{a^2}\implies y^2=b^2\left(1+\frac{x^2}{a^2}\right)
$$</p>
|
1,100,845 | <p>Let $z=f(x,y)$ and let r and $\theta$ be polar coordinates in the x-y plane. Recall that $x=r \cos \theta$ and $y=r \sin \theta$. Express $\frac{\partial^{2}z}{\partial r^{2}}$ in terms of r, $\theta$, and the partial derivative of f.</p>
<p>Letting </p>
<p>$$
g(x,y,\theta)=\frac{\partial z}{\partial r}
=\frac{\... | Teoc | 190,244 | <p>$$\begin{align}
\large\frac{y^2}{b^2}-\frac{x^2}{a^2}&=1\\
\large y^2a^2-x^2b^2&=a^2b^2\\
\large y^2a^2&=x^2b^2+a^2b^2\\
\large y^2&=\frac{x^2b^2}{a^2}+b^2\\
\large y&=\pm\sqrt{\frac{x^2b^2}{a^2}+b^2}\\
\large y&=\pm\sqrt{b^2(\frac{x^2}{a^2} +1)}\\
\end{align}$$</p>
|
2,071,977 | <p>I have a matrix $M$ and a vector $x$. I would like to compute the vector $y= M^n x$.</p>
<p>I found the eigenvalues $λ_k$ of $M$ and the left and right eigenvectors, properly normalized. If M was symmetric, I would now project $x$ on each eigenvector, multiply by $λ_k^n$, and then recombine them. That would give me... | Florian | 185,854 | <p>Is $M$ diagonalizable? If so, you can write $M$ as $M=P \cdot \Lambda \cdot P^{-1}$, where $P$ is a matrix containing the eigenvectors of $M$ and $\Lambda$ is a diagonal matrix containing the eigenvalues $\lambda_k$. In this case, we have that $M^n = P \cdot \Lambda^n \cdot P^{-1}$. Based on this result you can do w... |
678,599 | <p>Here is a problem from Lang's Algebra,chapter 12 and problem 12,</p>
<p>a)Let f(x) be a polynomial of degree 1 in $\mathbb Z[x]$. Show that the values $f(a)$ for $a \in \mathbb Z$ are divisible by infinitely many primes.</p>
<p>b) Let F be a finite extension of $\mathbb Q$ . Show that there are infintely many prim... | Karolis Juodelė | 30,701 | <p>Say $f(n)$ is only ever divisible by primes $p_i$, $i \in \{1\dots K\}$. Then $f(n) = \prod p_i^{c_i}$ for some $c_i \in 0 \cup \mathbb N$. You should be able to show that the values of the product on the right are much more sparse than the values of $f$ and thus cannot be equal.</p>
|
364,321 | <p>Is it possible to nest big-oh notation with omega-notation? I came across this here, while doing calculations on an exercise: </p>
<p>$$
f(x) \in O(\Omega(\log x))
$$</p>
<p>I'm really unsure on how to properly read this. Either it means, that $f(x)$ is unbounded, as $\Omega(\log x)$ is anything growing as fast or... | Alex | 38,873 | <p>You are probably right. We can write
$$
g(x)= \Omega(\log x) \geq c_1 \log x
$$
hence
$$
\lim_{x \to \infty}\frac{f(x)}{g(x)} \leq \frac{1}{c_1} \lim_{x \to \infty} \frac{f(x)}{\log x}\leq \frac{c_{2}}{c_1}
$$
hence $f(x) = O(\log x)$</p>
|
2,268,261 | <p>if you are a Londoner you are familiar with the situation: When you rent a bike a machine will produce a $5$-number code made up of $\{1,2,3\}$. My friend said "oh, I had this number before", but I said it is very unlikely as there are plenty of combinations; when he asked how many, I got stuck. </p>
<p>So here is ... | The Dead Legend | 433,379 | <p>If it's the total number of combinations you are asking, then It's simply filling 5 places with 10 numbers? which amounts to ${10}^5$ cases.</p>
<p><strong>IF THAT'S WHAT YOU ARE ASKING</strong></p>
|
2,268,261 | <p>if you are a Londoner you are familiar with the situation: When you rent a bike a machine will produce a $5$-number code made up of $\{1,2,3\}$. My friend said "oh, I had this number before", but I said it is very unlikely as there are plenty of combinations; when he asked how many, I got stuck. </p>
<p>So here is ... | Peter | 242,929 | <p>3 way to pick the first number, 3 ways to pick the second number and so on which equates to $$3^5 =243$$
After only 18 visits, the chances are more likely than not, that you will get a number you have had before.</p>
|
794,268 | <p>Binomial theorem is a very popular theorem that:
$$(x + y) ^ n = \sum_{i=0}^n {n \choose i}x^i y^{n-i}$$</p>
<p>I am looking for any papers (the newer the better) where I can find any informations about asymptotic approximation of the binomial theorem, especially for the case when I have $(1 + y)^n$.
<a href="http... | Alex | 38,873 | <p>The asymptotic approximation you have is just binomial expansion up to the 4th term. What happens as $n \to \infty$ depends on $y$. If $y$ is a constant larger than $0$, the sum diverges, if $-1<y<0$ it converges to 0. If $y= \frac{a}{n}$, the limit is of course $e^a$. Hence, no single answer exists. </p>
|
1,345,008 | <p>In general how do you prove inequalities using calculus, I believe it is using maxima or minima right? For example</p>
<blockquote>
<p>$$a^2b+b^2c+c^2a \le 3, \qquad a,b,c \ge 0,\quad a+b+c=3.$$</p>
</blockquote>
<p>How would you use calculus, just a sketch, I am interested in the method itself. </p>
| DeepSea | 101,504 | <p>$f(a,b) = a^2b + b^2(3-a-b) + (3-a-b)^2a = a^2b + 3b^2-ab^2-b^3+9a+a^3+ab^2-6a^2-6ab+2a^2b=3a^2b+3b^2-b^3+9a+a^3-6a^2-6ab$. Thus: $f_a = 6ab+9+3a^2-12a-6b = 0 = f_b = 3a^2+6b-3b^2-6a\Rightarrow 3(a^2-b^2) = 6(a-b)\Rightarrow (a-b)(a+b-2)=0$. There are $2$ cases:</p>
<p><strong>Case</strong> $1$: $a = b \Rightarrow ... |
381,771 | <p>Let <span class="math-container">$p$</span> be a prime. How many solutions <span class="math-container">$(x, y)$</span> are there to the equation <span class="math-container">$x^2 + 3y^2 \equiv 1 \pmod{p}$</span>? Here <span class="math-container">$x, y \in \{0, 1, \ldots p-1\}$</span>. This paper (<a href="https://... | GH from MO | 11,919 | <p>For any for any nonzero residues <span class="math-container">$a,b,c\not\equiv 0\pmod{p}$</span>, the number of solutions of the congruence <span class="math-container">$ax^2+by^2\equiv c\pmod{p}$</span> equals <span class="math-container">$p-\left(\frac{-ab}{p}\right)$</span>. For the proof, see my post <a href="ht... |
392,066 | <p>I know the title is not descriptive enough, but I don't know how else to say it. I don't know why I can't use the Binomial distribution to get the result I'm looking for. The teacher solved it long ago with a more straightforward method, but I tried today to solve the same problem using Binomial distribution and the... | Calvin Lin | 54,563 | <p>1) Yes. Take the Eulerian circuit, and delete that edge. You get a eulerian path.</p>
<p>2) No. Use the classification that a graph is Eulerian if and only if every vertex has even degree.</p>
|
392,066 | <p>I know the title is not descriptive enough, but I don't know how else to say it. I don't know why I can't use the Binomial distribution to get the result I'm looking for. The teacher solved it long ago with a more straightforward method, but I tried today to solve the same problem using Binomial distribution and the... | Douglas S. Stones | 139 | <p>It is important that the definition of an <em>Eulerian graph</em> here includes the property "is connected". Without this property, we can have the following situation where $G_1$ is the blue graph and $G_2$ is the red graph, and we add in the orange edge:</p>
<p><img src="https://i.stack.imgur.com/0wYx3.png" alt=... |
2,431,679 | <p>First I denote $x=\arctan{2}$ and $y=\arcsin{\frac{1}{\sqrt{10}}}$ and then use the addition formula for sine:</p>
<p>$$\sin{(x-y)}=\sin{x}\cos{y}-\cos{x}\sin{y}=\sin{x}\cos{y}-\cos{x}\cdot \frac{1}{\sqrt{10}}.$$</p>
<p>Now I use the fact that $\cos{y}=1-\sin^2{y}$ which gives</p>
<p>$$\cos{y}=1-\sin^2{\left(\arc... | George Coote | 445,167 | <p>Was going to leave a comment on GAVD's answer, but it's not difficult to do with </p>
<p>$$\sin(\arctan(x)) = \frac x {\sqrt{x^2 + 1}}$$</p>
<p>Which you can prove using, </p>
<p>$$1 + \cot^2(\arctan(x)) = \csc^2(\arctan(x))$$</p>
<p>By the Pythagorean identity. From which, </p>
<p>$$1 + \frac 1 {x^2} = \frac{1... |
2,179,243 | <p><strong>Question:</strong> Is there more than one "appropriate" way to generalize the orthogonal group of $\mathbb{R}^n$?</p>
<p>The two competing ways which come to mind for me are:</p>
<ol>
<li><p>The group of all isometries with the origin as a fixed point, for any (finite-dimensional) vector space $V$ with any... | mathlove | 78,967 | <p>Your answer is correct.</p>
<blockquote>
<p>Is there an easier way to find the answer, please suggest ?</p>
</blockquote>
<p>How about the following way?</p>
<p>You already have
$$4x^2+35x+4=0$$
from which we get $$x+\frac 1x=\frac{-35}{4}\tag1$$</p>
<p>Now dividing top and bottom of
$$\frac{(1+x)^5}{1+x^5}$$
... |
2,257,878 | <p>We have to prove that the equation $\displaystyle \frac{x^5}{5!} +\frac{x^4}{4!} +\frac{x^3}{3!}+\frac{x^2}{2!} +x+1=0$ have exactly one real root .</p>
<p>My sir told me it is just an application of derivative .</p>
<p>But I could not understand what he mean by that . Can anybody please explain me .</p>
| egreg | 62,967 | <p>Denote by $F_n$ the $n$-th Fibonacci number
$$
F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n
$$
and consider the series
$$
F=\sum_{n\ge0}F_nx^n
$$
in the <a href="https://en.wikipedia.org/wiki/Formal_power_series" rel="noreferrer">ring of formal power series with coefficients in $\mathbb{R}$</a>. This ring has the p... |
1,009,169 | <p>I have to examine limit of following sequence $\{a_n\}_{ n \ge 1}$ $a_n = \sqrt[n]{\sum_{k=1}^{n}{(2 -\frac{1}{k})^k}}$. We know that $\lim_{n \to \infty} \sqrt[n]{a} = 1$ for $a > 0$ and we know that $\sqrt[n]{\sum_{k=1}^{n}{(2 -\frac{1}{k})^k}} > 0$ but sequence $a_n$ is increasing (I cannot prove it but I c... | mvggz | 167,171 | <p>I think you can get your way around this one using limited expansion.</p>
<p>Here I use : $ln(1-u) = -u -\frac{u^2}{2} +o(u^2)$ ; and $\exp(u) = 1+ u +o(u)$ , when $ u \rightarrow 0$</p>
<p>Let : $u_k = (2-\frac{1}{k})^k = 2^k*(1-\frac{1}{2k})^k = 2^k*\exp{[k*ln(1-\frac{1}{2k})]} = 2^k*\exp[-\frac{1}{2} -\frac{1}{... |
1,440,571 | <p>There are two events, $P(A) = .45$ & $P(B) = .65$. Also, $P( A \cap B) = .25$.</p>
<p>How can I get $P (B \cap A^c)$ ?</p>
<p>Any help will be appreciated. Thanks!</p>
| Harish Chandra Rajpoot | 210,295 | <p>Notice, $$P(B\cap A^{c})=P(B)-P(B\cap A)$$
Since, $P(B\cap A)=P(A\cap B)$, hence
$$P(B\cap A^{c})=P(B)-P(A\cap B)$$
Setting the corresponding values $P(B)=0.65$ & $P(A\cap B)=0.25$
$$P(B\cap A^{c})=0.65-0.25=\color{red}{0.40}$$</p>
|
238,408 | <p>How can I solve this system of equations in Mathematica?</p>
<pre><code>DCp1total = WA1*DCpA0 + WB1*DCpB0
DCp2total = WA2*DCpA0 + WB2*DCpB0
WA1 + WB1 = 1
WA2 + WB2 = 1
DCp1_total = 0.6
DCp2_total = 0.3
DCpA0 = 0.5
DCpB0 = 0.4
</code></pre>
<p>Unkowns: <code>WA1, WA2, WB1</code> and <code>WB2</code>.</p>
| Nasser | 70 | <p>There is no solution. But you can use least squares approximation</p>
<p>ps. do not use <code>_</code> in variable names in Mathematica as <code>_</code> is meant be used for pattern and you will get in trouble if you do that.</p>
<pre><code>(*from clean kernel*)
DCpA0 = 1/2;
DCpB0 = 4/10;
WA0 = 7/10;
WB0 = 3/10;
D... |
25,573 | <p>Let's generate some noisy data with uneven noise.</p>
<pre><code> data = Table[{x,Exp[-(x-2)^2] + Exp[-(x+2)^2]*RandomReal[{0.5, 1.5}]},
{x, RandomReal[{-4, 4}, 500]}];
</code></pre>
<p><img src="https://i.stack.imgur.com/VrQVD.png" alt="enter image description here"></p>
<p>Let's fit it with a nonl... | Rod | 5,369 | <p>I believe there are some big misunderstandings in the question...</p>
<p><strong>First:</strong> usually error terms are <strong><em>additive</strong>, <strong>not multiplicative</em></strong>. </p>
<p>There are the so-called <em>multiplicative error models</em> (MEM), but I don't think it's the case here, because... |
2,484,426 | <p>I have the following function:</p>
<p>$$ f(x,y) = \begin{cases}
\dfrac{xy^3}{{x^2+y^2}} & (x,y)\neq(0,0) \\
0 & (x,y)=(0,0) \\
\end{cases} $$</p>
<p>How can I demonstrate the existence of such a limit using epsilon and delta?</p>
| G Tony Jacobs | 92,129 | <p>Presumably, the limit you want to check is $$\lim_{(x,y)\to(0,0)} f(x) = \lim_{(x,y)\to(0,0)} \frac{xy^3}{x^2+y^2}$$</p>
<p>There are some useful inequalities that help with questions like this. One comes from noting the following. Since $(x+y)^2\ge 0$, and $(x-y)^2\ge 0$, we can expand and rearrange these to concl... |
151,516 | <p>I am trying to find this by using integration by parts but I am not sure how to do it.</p>
<p>$$\int_0^{\pi/2} (\sin x)^7 (\cos x)^5 \mathrm{d} x$$</p>
<p>I tried rewriting as </p>
<p>$$\int_0^{\pi/2} \sin x\cdot\ (\sin x)^6\cdot\ (\cos x)^5 \mathrm{d} x = \int_0^{\pi/2}\sin x(1-\ (\cos x)^3)\cdot\ (\cos x)^5 \m... | Valentin | 31,877 | <p><a href="http://en.wikipedia.org/wiki/Beta_function" rel="nofollow">Beta</a> version:</p>
<p>$$\begin{aligned}\int_{0}^{\frac{\pi}{2}}\left(\sin x\right)^{7}\left(\cos x\right)^{5}dx & =\frac{1}{2}\int_{0}^{\frac{\pi}{2}}\left(\sin x\right)^{6}\left(\cos x\right)^{4}2\sin x\cos xdx\\
& =\frac{1}{2}\int_{0}... |
2,257,778 | <p>I read this statement: </p>
<blockquote>
<p>In a monoid, if an element has two distinct left inverses, it cannot
have a right inverse, and hence cannot have a two-sided inverse.</p>
</blockquote>
<p>I'm wondering if there's an easy example that I could use to illustrate this fact. </p>
| Derek Holt | 2,820 | <p>For a set $S$, let $M(S)$ be the monoid of all maps $S \to S$ under composition. If $S$ is infinite then there are maps that are injective but not surjective, and they have infinitely many left inverses but no right inverse. Similarly, maps that are surjective but not injective have infinitely many right inverses bu... |
2,257,778 | <p>I read this statement: </p>
<blockquote>
<p>In a monoid, if an element has two distinct left inverses, it cannot
have a right inverse, and hence cannot have a two-sided inverse.</p>
</blockquote>
<p>I'm wondering if there's an easy example that I could use to illustrate this fact. </p>
| quasi | 400,434 | <p>Let $M$ be the monoid of all functions from $\mathbb{R}$ to $\mathbb{R}$, with the operation of composition.
<p>
Consider the functions $f,g,h \in M$ defined by
\begin{align*}
f(x) &= e^x\\[4pt]
g(x) &=
\begin{cases}
\ln{x}&\text{if}\;x > 0\\[4pt]
0&\text{otherwise}\\[4pt]
\end{cases}\\[4pt]
h(x)... |
177,181 | <p>I have a object which is similar to the shape of irregular rock like this <img src="https://i.stack.imgur.com/Zz55K.gif" alt="enter image description here"></p>
<p>I would like to find the volume of this. How to do it?</p>
<p>If I have to find the volume, what are the things I would need. eg., If it is cylindrical... | Lenar Hoyt | 15,616 | <p>There is a very simple and elegant algorithm for calculating the volume of an arbitrary, closed mesh described in <a href="http://amp.ece.cmu.edu/Publication/Cha/icip01_Cha.pdf" rel="nofollow">this paper</a> (<em>Zhang, Cha, and Tsuhan Chen. "Efficient feature extraction for 2D/3D objects in mesh representation." Im... |
624,881 | <p>I have set of (x,y) points which can be connected to form a graph, my goal is to detect dynamic parts of this graph. by dynamic I mean ranges where the values are not stable but they are changing by going up/down and forming different graphs than a straigh y = C line.</p>
<p>I tried to look at the slopes graph and ... | John Hughes | 114,036 | <p>To be sure I'm answering what you're asking:</p>
<p>Your $x_i$ and $y_i$ values might be, say, dates and temperatures on those days, but you only get to read the thermometer now and then, so that (calling Jan 1 = 1, Jan 2 = 2, Feb 1 = 32, etc.), you have data like</p>
<p>(1, 12)</p>
<p>(2, 11)</p>
<p>(15, 13)</... |
3,119,058 | <p>I'm given the complex rational expression</p>
<p><span class="math-container">$$\frac{\frac{a}{b} - \frac{b}{a}}{\frac{a+b}{ab}}$$</span></p>
<p>and asked to simplify. The solution provided is <span class="math-container">$a - b$</span>; however I get <span class="math-container">$\frac{a^2 - b^2}{a+b}$</span>.</p... | Andrei | 331,661 | <p><span class="math-container">$$a^2-b^2=(a-b)(a+b)$$</span> If <span class="math-container">$a+b\ne 0$</span>, you can divide both the numerator and denominator by it.</p>
|
2,148,582 | <p>In how many ways can I obtain $4$ as a sum of : $0.125$, $0.25$, $0.5$, $1$, $2$, $4$ with repetitions of values?
The order of the sum is important, for example $(1, 1, 2)$ is different from $(1, 2, 1)$ and different from $(2, 1, 1)$,</p>
| M. Winter | 415,941 | <p>$4=32\times 0.125$, so write down
$$4 = \underbrace{0.125+\cdots +0.125}_{32 \text{ times}}.$$
Now it is just a matter of in how many appropriate ways you can group the $0.125$ summands to your actual summands. E.g. you can group it like that:
$$
4 =
\underbrace{0.125+\cdots +0.125}_{4 \text{ times }\rightarrow\, 0... |
1,231,829 | <blockquote>
<p>Find two solutions in $\mathbb{R}$ for the Riccati equation $$y'+y+y^2 = 2$$
Which satisfy the following initial conditions:</p>
<p>a) $y = b$ when $x=0$, $-2 \leq b < 1$</p>
<p>b) $y = b$ when $x=0$, $b \geq 1$ or $b < -2$</p>
</blockquote>
<p><strong>What I tried to do:</strong>
F... | Jack D'Aurizio | 44,121 | <p>Your ODE is separable. It can be written as:
$$\frac{y'}{2-y-y^2}=1\tag{1} $$
and by integrating both sides of $(1)$ we get:
$$\log\left(\frac{2+y}{1-y}\right)=3t+C \tag{2}$$
hence:
$$ y(t) = \frac{K e^{3t}-2}{K e^{3t}+1}\tag{3} $$
where the value of $K$ depends on the initial conditions.</p>
|
259,382 | <p>it seems first easy to me, but now i am tossing my head against wall not being able to solve the problem. i need to check for convergence of this sequence below. i dont know how to start although it seems to be very easy one</p>
<p>$\lim_{n \to \infty} i^{3n} = help = ?$ </p>
<p>i need help here. do i have to work... | Mikasa | 8,581 | <p>If $i=\sqrt{-1}$ then the sequence $(i^{3n})_1^{\infty}$ is divergent.</p>
|
585 | <p>I would like to give the built-in function <code>PlusMinus</code>, which is undefined by default, the intuitive meaning: considering all possible combinations of $+$ and $-$.</p>
<p>For simple expressions, this can be implemented as follows:</p>
<pre><code>PlusMinus[a_,b_] := Flatten[{a+b,a-b}]
a \[PlusMinus] b \[... | David | 103 | <p>I managed to throw together sort of a solution. Note however that <strong>this is not what I am looking for</strong> (since I'm defining a helper function, whereas I would like the symbol itself to behave like this), the question remains open. However, I thought this might be useful for others nevertheless.</p>
<p>... |
1,717,989 | <p>This question came to mind when I saw another question involving modular-arithmetic division.</p>
<p>In $\mathbb{Z}/10\mathbb{Z}$ we can divide 4 by 7 by identifying 3 as a multiplicative inverse of 7 and multiplying 4 by that inverse, thus $4\cdot3=2$. This does not work for $4/8$ because 8 does not have a multip... | A. Thomas Yerger | 112,357 | <p>Well if $n$ is of the form $2q+1$, then $n+1 = 2q+2 = 2(q+1)$ is also even, so either way one of them works.</p>
<p>In general, with these kinds of problems, just follow your nose. You computed one case, then I just computed the other, and it turns out that everything worked out.</p>
|
1,598,555 | <p>How to solve such questions in which we are given $N$ distinct numbers i.e $1$ to $N$ and also given certain restrictions like $(x,y) , (x,z) , (z,x) , (z,a) , (b,c)$ and so on should never occur simultaneously.</p>
<p>For example:</p>
<p>$N=4$ and $(1,2) , (2,3) , (3,4)$ should never come together.</p>
<p><stro... | Ross Millikan | 1,827 | <p>There is no general way, especially because you have not said what sort of restrictions might be applied. You can either count the permutations directly or subtract the disallowed ones from the total. Interlocking restrictions like your example (where $2,3$ are in more than one restriction) can make it a bunch of ... |
2,886,238 | <p>While I was considering random walk on $S^1$, I needed to compute the following.</p>
<p>$$\lim_{m\rightarrow\infty}\sum_{k=1}^m \left(\cos{\frac{2\pi k}{2m+1}}\right)^{(2m+1)^2}$$</p>
<p>I guessed it should converges to trace of heat kernel on $S^1$. So I tried to use Mathematica. Mathematica says, it converges to... | Antonio Vargas | 5,531 | <p>A sketch:</p>
<p>By exploiting symmetry of the sum, we can rewrite it as</p>
<p>$$
\begin{align}
&\sum_{k=1}^m \left(\cos{\frac{2\pi k}{2m+1}}\right)^{(2m+1)^2} \\
&\qquad = \sum_{k=1}^{m/2} \left(\cos{\frac{\pi 2k}{2m+1}}\right)^{(2m+1)^2} - \sum_{k=1}^{m/2} \left(\cos{\frac{\pi (2k-1)}{2m+1}}\right)^{(2m... |
896,270 | <p>Is the collection of all finite sets a set or a proper class? If it's a set how to prove otherwise alos how to prove? And finite set means sets which are in 1-to-1 correspondence with <span class="math-container">$\{1,2,3,...,n\}$</span> where <span class="math-container">$n$</span> is a natural number.</p>
| Clive Newstead | 19,542 | <p><strong>Hint:</strong> If $x$ is a set then $\{ x \}$ is a finite set.</p>
|
695,015 | <p>If $A$ and $B$ are subspaces of $\mathbb R^n$. Is it possible to find a basis for $\mathbb R^n$ that contains a basis for $A$ and $B$?</p>
<p>It has been suggested to me that we define a basis for $A\cap B$ and then use that to define basises $A$ and $B$. I would like to understand why this approach is taken and ... | Christiaan Hattingh | 90,019 | <p>it is most certainly possible...and the approach suggested is the way to go. Basically you have a subspace \begin{equation} A+B=\{a+b:a \in A \text{ and } b \in B\}\end{equation} known as the sum of of $A$ and $B$; this is the smallest subspace containing $A$ and $B$. Also, the subspace $A \cap B$ is a subspace. So ... |
695,015 | <p>If $A$ and $B$ are subspaces of $\mathbb R^n$. Is it possible to find a basis for $\mathbb R^n$ that contains a basis for $A$ and $B$?</p>
<p>It has been suggested to me that we define a basis for $A\cap B$ and then use that to define basises $A$ and $B$. I would like to understand why this approach is taken and ... | John Hughes | 114,036 | <p>Here's a general approach to certain classes of basis problems in finite-dimensional spaces (e.g., $R^n$). You work with two main facts. </p>
<p>i. If $S = \{ v_1, \ldots, v_k\}$ is a dependent set of vectors, then (a) at least one of the $v$s, say $v_i$, can be written as a linear combination of the others, and (b... |
903,875 | <p>I got the following problem:</p>
<p>Evaluate the following limits or show that it does not exist: </p>
<p>$$\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$$</p>
<p>and</p>
<p>$$\lim_{(x,y)\to(\infty,8)}\left(1+\frac{1}{3x}\right)^\frac{x^2}{x+y}$$</p>
<p>I tried for an hour and half evaluating each of th... | Gardel | 18,036 | <p>Alternative solutions:</p>
<p>Consider the limit on all lines passing through the origin, i.e.,
$y=ax$. $$\lim_{(x,y)\to (\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}=\lim_{x\to\infty}\frac{x\left[2-a\right]}{x^2\left[1-a+a^2\right]}=\lim_{x\to\infty}\frac{\left[2-a\right]}{x\left[1-a+a^2\right]}=0, $$ independently of ... |
150,856 | <p>From the result discussed in <a href="https://mathoverflow.net/questions/75049/does-the-inverse-function-theorem-hold-for-everywhere-differentiable-maps">Does the inverse function theorem hold for everywhere differentiable maps?</a> (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obta... | Liangpan Li | 160,595 | <p>The answer to the first question is YES. There were two proofs of this result, one by Jean Saint Raymond (Mathematika, 2002), the other by L. Hurwicz and M. K. Richter (Discussion paper no. 279, Dept. of Economics, University of Minnesota, 1994).</p>
|
1,967,312 | <p>Say we have disjoint sets $A$ and $B$ where $A = \{1\}$ and $B = \{2\}$. </p>
<p>Thus the power set of $A$ is $\{\emptyset,1\}$ and the power set of $B$ is $\{\emptyset,2\}$. </p>
<p>I know the intersection set of $A$ and $B$ is $\emptyset$ </p>
<p>My question is this: Given the above information, does that mean ... | ziggurism | 16,490 | <p>No, the intersection of P(A) and P(B) is the set containing the empty set, which is itself not an empty set since it contains a set. </p>
<p>$\mathcal{P}(A) \cap \mathcal{P}(B) = \{\emptyset\}\neq \emptyset$</p>
|
49,453 | <p>$\mathrm{DISCLAIMER~:~}$I am not interested in working with compactly generated spaces.</p>
<p>This post is related to this one : <a href="https://math.stackexchange.com/questions/20462/exponential-law-for-based-spaces">Exponential Law for based spaces</a>. I learned about the exponential law for topological spaces... | Henno Brandsma | 4,280 | <p>As to Question 2, let $\Sigma$ be the composition map $\mathrm{Map}(X,Y)\times\mathrm{Map}(Y,Z)\rightarrow\mathrm{Map}(X,Z)$.</p>
<p>All functions spaces have the compact-open topology, with as a subbase all sets of the form
$\mathrm{M}(C,U) = \{ f: f[C] \subset U \}$, where $C$ is a compact subset of the domain, ... |
1,000,836 | <p><strong>Consider a regular tetrahedron with vertices $A,B,C,D$. A bug starts crawling from $A$. The bug moves from one vertex to the other along the edges continuously until it reaches $D$, where there is glue and hence once the bug reaches $D$, it gets stuck. The bug can move to any one vertex from any other vertex... | JimmyK4542 | 155,509 | <p>Let $p_A$ be the probability that the bug reaches $D$ from $B$ given that it is currently at $A$. </p>
<p>Let $p_B$ be the probability that the bug reaches $D$ from $B$ given that it is currently at $B$. </p>
<p>Let $p_C$ be the probability that the bug reaches $D$ from $B$ given that it is currently at $C$. </p>
... |
4,013,573 | <p>I think this set is uncountable, but I don't really know how to prove this. Maybe somehow find an injection from <span class="math-container">$\mathcal{P}( \mathbb{N} \times \mathbb{N}) \rightarrow A$</span>?</p>
| radekzak | 876,772 | <p>For any <span class="math-container">$S \subset \mathbb{N}$</span> take <span class="math-container">$\mathcal{R}_S = \bigcup_{s \in S} \{(s,s+1)\}$</span>. It is clear that <span class="math-container">$\mathcal{R}$</span> is anti-symmetric, as it's a subset of anti-symmetric relation <span class="math-container">$... |
807,560 | <p>The function is
$$ f(n) = \sum_{i=1}^{n} \frac{1}{2i-1}$$
How can I compute for example $f(20)$ or $f(50)$ without using a calculator. I want to have an approximation</p>
| Alex | 38,873 | <p>It's a harmonic sum with odd terms:
$$
S_n = 1 + \frac{1}{3} + \ldots \frac{1}{2n-1} = 1 +\frac{1}{2}+\frac{1}{3} + \ldots \frac{1}{2n} - \frac{1}{2}-\frac{1}{4} -\ldots \frac{1}{2n} = H_{2n}-\frac{1}{2}H_n
$$
and each $H_{k} \sim \log k +\gamma$.</p>
<p>EDIT: Since $H_{2n} \sim \log 2n + \gamma = \log 2 + \log n... |
893,406 | <p>I have been given this example 2.0013−2.0005=0.0008. The destructive cancellation is the large common value, here the 2, that disappears. </p>
| TonyK | 1,508 | <p>Suppose $x$ and $y$ are known to five significant figures: $x \approx 2.0013$ and $y \approx 2.0005$. This means that $2.00125 \le x \le 2.00135$ and $2.00045 \le y \le 2.00055$ (some of the $\le$ signs can be replaced by $<$ signs, but exactly which ones depends on the rounding mode used).</p>
<p>But then if $z... |
538,444 | <p><img src="https://i.stack.imgur.com/mBZOe.jpg" alt="enter image description here"></p>
<p>I'm studying and I'm not that sure how to answer this question. </p>
<p>Is $97.1%$ $=$ $650,000,000$?</p>
<p>I was going to find $2.9%$ of $650,000,000$ however this would be wrong as I would finding our the annum of the $20... | Dan Rust | 29,059 | <p>Relevant facts: $$|\mathbb{Z}_n|=n\\ |G\times G'|=|G|\cdot |G'| \\ [G\: :\: H]\cdot|H|=|G|\quad\mbox{(Lagrange's Theorem)}$$</p>
|
1,232,871 | <p>A closed subset of a locally connected space is locally connected?</p>
<p>A closed subset of a locally path-connected space is locally path-connected?</p>
<p>Is there any counter-example?</p>
| Crostul | 160,300 | <p>$\Bbb{R}^2$ is locally path connected and locally connected. There is a standard example of a closed subset $X \subset\Bbb{R}^2$ which is neither locally connected, nor locally path-connected: see <a href="http://en.wikipedia.org/wiki/Topologist%27s_sine_curve" rel="nofollow">http://en.wikipedia.org/wiki/Topologist%... |
618,984 | <p>Given $2$ prime numbers,$ p$ and $q$, that are both not even, and $3$ doesn't divide $p-1$ or $q-1$, and
$n=pq$, how many elements in $Z^*_n$ exists that has $b$ such that $b^3\equiv a\pmod n$ .</p>
<p>I know that the answer is $\phi(n)$ - that is, $(p-1)*(q-1)$, but I can't find out how to prove it :( </p>
| Bill Dubuque | 242 | <p>$\rm x\mapsto x^3$ is onto, being $1$-$1\!:$ $\, \begin{eqnarray}\rm a^3 &=&\rm b^3\\\Rightarrow\rm (a/b)^3 &=& 1\end{eqnarray}\!$ so $\rm\,a/b\,$ has order $\,n\mid 3,\phi,\,$ so $\,n\mid (3,\phi) = 1\ \ $ <strong>QED</strong></p>
|
2,874,316 | <p>I started off with:</p>
<p>$$((A\lor B)\land ((B\leftrightarrow A)\rightarrow C))\lor(C\rightarrow \lnot A)$$</p>
<p>And I've managed to get to:</p>
<p>$$((A\lor B) \land (((B\land \lnot A) \lor (A \land \lnot B)) \lor C)) \lor (\lnot C \lor \lnot A)$$</p>
<p>How do I finish and get this into CNF? </p>
| cdipaolo | 173,871 | <p>To be clear about what is hinted at in the comments,
$$
f^{-1}(y) = -y\,W(-\tfrac{1}{y})
$$
where $W$ is the <a href="https://en.wikipedia.org/wiki/Lambert_W_function" rel="nofollow noreferrer">Lambert-$W$ function</a>. (You can verify this in Mathematica.) If you wanted an asymptotic expansion check the wiki articl... |
3,098,290 | <p>For <span class="math-container">$a,b,c>0 $</span> and <span class="math-container">$a+b+c=ab+bc+ca$</span> . Prove or disprove that : <span class="math-container">$\sqrt{24ab+25}+\sqrt{24bc+25}+\sqrt{24ca+25}\geq 21$</span></p>
<p>I checked in very many cases. Example :<span class="math-container">$c=1,
a=2,b=\... | Hai Smit | 537,207 | <p>I consider function <span class="math-container">$ f(x)=\sqrt{24x^2+25}$</span>
And <span class="math-container">$f’(x)=\frac{24x}{\sqrt{24x^2+25}}$</span>,
<span class="math-container">$f’’(x)=\frac{600}{(24x^2+25)\sqrt{24x^2+25}}>0$</span>
So <span class="math-container">$f(x)+f(y)+f(z)\geq 3f(\frac{x+y+z}{3})... |
607,403 | <p>Find,</p>
<p>$$x^2y^2-2xy+x=2$$</p>
<p>find the slope and the equation of the tangent line at $(2,0)$ by implicit differentiation.</p>
<hr>
<p>I have gotten this question wrong on a quiz and exam. I'm trying to understand it for my final. Part of the problem is that my text book says find $d/dx$ and then ignore ... | Alec | 114,173 | <p>Not sure what you are doing, but this is my take. Hope it helps.
$$
\frac{d}{dx}\left(x^2y^2\right) - \frac{d}{dx}\left(2xy\right) + \frac{d}{dx}\left(x\right) = 0
$$
Then, remember that $\frac{d}{dx}\left(y^2\right)=\frac{d}{dy}\left(y^2\right)\frac{dy}{dx} = 2yy'$ (chain rule). Then, using product rule and chain r... |
1,915,097 | <p>For five years I tried to understand how Fourier transform works. Read a lot of articles, but nobody could explain it in simple terms. Two weeks ago I stumbled upon the video about a 100 years old machine that calculates Fourier series mechanically: <a href="https://www.youtube.com/watch?v=NAsM30MAHLg">https://www.y... | Matthew Hampsey | 46,080 | <p>The FFT is just an algorithm for computing the discrete Fourier transform (DFT). It turns out that the DFT matrix is highly symmetric (due to the symmetry and periodicity properties of $e^{ix}$). The FFT is just a matrix factorisation of the DFT into a series of sparse matrices. You can multiply the sparse matrices ... |
1,008,170 | <p>We know that if $x_i^2 = -1 \mod n$ we are able to find more solutions of the form, $x^2 = -y^2 \mod n$ </p>
<blockquote>
<p><strong>Simple Proof:</strong></p>
<p>Let $x_i$ be the initial solution to $x_i^2+1 \equiv 0 \mod n$</p>
<p>$y^2x_i^2+y^2 \equiv 0 \mod n$</p>
<p>$x^2 = -y^2 \mod n$, $x = x_... | Przemysław Scherwentke | 72,361 | <p>Let us consider the equation
$
y^3+py+q=0
$.
The theory of its solution tells, that we should consider
$$
\Delta=\left(\frac{p}3\right)^3+\left(\frac{q}2\right)^2.
$$
If $\Delta>0$, then
$$
y=\sqrt[3]{-\frac{q}2-\sqrt{\Delta}}+\sqrt[3]{-\frac{q}2+\sqrt{\Delta}}
$$
is unique real solution.</p>
<p>If you know th... |
2,510,496 | <p>Let $\mathcal{H}$ be a complex Hilbert space. Let $T\in \mathcal{B}(\mathcal{H})$ and let $A\in \mathcal{B}(\mathcal{H})^+$ (i.e. $A^*=A$ and $\langle Ax\;| \;x\rangle \geq0,\;\forall x\in \mathcal{H}$).</p>
<blockquote>
<p>Assume that $AT=TA$. Why $A^{1/2}T=TA^{1/2}$?</p>
</blockquote>
<p>Thank you</p>
| Frederik vom Ende | 495,075 | <p>Looking at the construction of the square-root operator, the commutation property in question is a somewhat simple consequence. The following argument is taken from Section 104 of the book <em>"Functional Analysis"</em> by Riesz & Nagy (1990, Dover publications).</p>
<p>Let $A\in\mathcal B(\mathcal H)^+$ be giv... |
1,925,176 | <p>For the question "The corners of a triangle are the points $P(4,7)$, $Q(-2,5)$ and $R(3, -10)$. Find the length of each side of $\triangle PQR$, giving your answers in terms of surds".
I have the answer to the first part as : $|PQ| = 2\sqrt{10}$, $|PR| = \sqrt{290}$, $|RQ| = 5\sqrt{10}$.</p>
<p>The second part asks... | JDF | 314,709 | <p>To verify that $PQR$ contains a right angle, you can use Pythagoras' Theorem, which says that in a right angled triangle, if the side lengths are $a\leq b\leq c$, then $a^2+b^2=c^2$.</p>
<p>Note that $(2\sqrt{10})^2+(5\sqrt{10})^2 = 40 + 250 = 290 = (\sqrt{290})^2$</p>
|
1,158,652 | <p>$$f(x) = \frac{1}{1-2\cos^2x}$$</p>
<p>The result of $f'(x)$ should be equals</p>
<p>$$f'(x) = \frac{-4\cos x\sin x}{(1-2\cos^2x)^2}$$</p>
<p>I'm trying to do it in this way but my result is wrong.</p>
<p>$$f'(x) = \frac {1'(1-2\cos x)-1(1-2\cos^2x)'}{(1-2\cos^2x)^2} =
\frac {1-2\cos^2x-(1-(2\cos^2x)')}{(1-2\c... | SA-255525 | 135,922 | <p>$f(x) = \frac{1}{1-2cos^2x}$</p>
<p>The result of $f'(x)$ should be equal $f'(x) = \frac{-4cosxsinx}{(1-2cos^2x)^2}$</p>
<p>$$
f'(x) =
\frac {0-1(1-2cos^2x)'} {(1-2cos^2x)^2} = </p>
<p>\frac {-4sinxcosx} {(1-2cos^2x)^2}
$$</p>
|
993,051 | <p>Just by looking at the function:
$$y(t) = \tan(t)+t$$
I can immediately see that there is a root at $t=0$, though after graphing it I can see many more roots and I can calculate them using computer aided programs; i.e. the next root is at $t=2.0288$.</p>
<p>What I want to know is: what methods I can use to solve t... | Claude Leibovici | 82,404 | <p>As you know, equations which mix polynomials and trigonometric functions do not have explicit solutions and numerical methods should be used for solving them.</p>
<p>As you noticed from the graph, the solutions are closer and closer to the vertical asymptotes and there are an infinite number say $$t_n=(2n-1)\frac{\... |
426,661 | <p>Can someone please explain these answers? I have reviewed the slides and read about properties of equality but I still don't understand how to apply it to these sets.</p>
<p>For each the following relations on the set of integers list all that apply
(Reflexive, Symmetric, Antisymmetric, Transitive, or none): </p>
... | Nick Peterson | 81,839 | <p>Let's review what each property means. Your question deals with relations on the integers; so, let's say that we are working with a relation $R\subseteq\mathbb{Z}\times\mathbb{Z}$.</p>
<h2>Symmetric:</h2>
<p>$R$ is called <em>symmetric</em> if for all $(a,b)\in R$, we also have $(b, a)\in R$. </p>
<p>For instanc... |
1,663,411 | <blockquote>
<p>Let $I=({ X }^{ 2 },2X)$ ideal of $\mathbb Z[X]$ generated by ${ X }^{ 2}$ and $2X$. Show that $I$ is not primary.</p>
</blockquote>
<p>I tried to find $$\sqrt { I } =\sqrt { ({ X }^{ 2 })+(2X) } =\sqrt { \sqrt { { (X }^{ 2 }) } +\sqrt { (2X) } } =\sqrt { \sqrt { { (X) }^{ 2 } } +\sqrt { (2X) } } =... | Saleh Hamdan | 104,972 | <p>If there was no universally agreed upon definition, you would use whatever metric is most suitable for a particular subject you're studying. For example, if you are working with $L^{1}(\Omega)$ space, you would use its corresponding metric. </p>
<p>But I should point out that the uniform metric $d_{u}$ as it is def... |
1,663,411 | <blockquote>
<p>Let $I=({ X }^{ 2 },2X)$ ideal of $\mathbb Z[X]$ generated by ${ X }^{ 2}$ and $2X$. Show that $I$ is not primary.</p>
</blockquote>
<p>I tried to find $$\sqrt { I } =\sqrt { ({ X }^{ 2 })+(2X) } =\sqrt { \sqrt { { (X }^{ 2 }) } +\sqrt { (2X) } } =\sqrt { \sqrt { { (X) }^{ 2 } } +\sqrt { (2X) } } =... | quid | 85,306 | <p>The first definition is (implicitly) for functions with image the real (or possibly complex) numbers, where the absolute value makes sense. </p>
<p>In this case the most common way to define the distance of two values is by saying $d(s,t)= |s-t|$. </p>
<p>More generally, if one has a normed spaced, like $\mathbb{R... |
172,439 | <p>Is there an analytic approximation to the inequality:</p>
<p>$$\sum_{i=1}^{n} |x_i| \leq \delta ? $$</p>
<p>I would like to replace the above inequality with a smooth inequality that is "valid" in the sense that if the approximate smooth inequality is satisfied then the original inequality will also be satisfied. ... | Ravi | 36,224 | <p>I found that $ |x| $ can be approximated by the smooth function $ g(x) = x \cdot \mathrm{erf}(\frac{x}{\sigma}) $. We can get accurate approximation by letting $ \sigma $ go to $ 0 $.</p>
|
2,173,289 | <p>I want to calculate </p>
<p>$$\int_0^\infty \frac{\ln x}{x^2+4}\,dx$$</p>
<p>using the Residue Theorem. The contour I want to use is almost the upper half circle of radius $R$, but going around $0$ following a half-circle of radius $\epsilon$, and using the branch cut at $3\pi/2$. The picture is below. Let's call ... | Marco Cantarini | 171,547 | <p>It is more useful the <a href="https://www.google.it/search?hl=it&site=imghp&tbm=isch&source=hp&biw=1366&bih=662&q=keyhole+contour&oq=keyhole+con&gs_l=img.3.0.0i30k1j0i8i30k1.96.2391.0.3366.11.11.0.0.0.0.181.1295.0j9.9.0....0...1ac.1.64.img..2.9.1290...0j0i10k1.PoivChLol4Y#imgrc=b9MQh... |
4,443,170 | <p>Let <span class="math-container">$I$</span> be an uncountable set and let <span class="math-container">$2^I=\{0, 1\}^I$</span> be the set of all functions from <span class="math-container">$I$</span> into <span class="math-container">$\{0, 1\}$</span>. Consider the product measure <span class="math-container">$\mu$<... | zwim | 399,263 | <p>We can reduce the gcd by successively removing <span class="math-container">$an+b$</span> where the <span class="math-container">$a$</span> is the smallest, i.e. <span class="math-container">$$\gcd(u,v)=\gcd(u-v,v)$$</span></p>
<p><span class="math-container">$\begin{align}\gcd(8n-7,3n+1)
&=\gcd(5n-8,3n+1)&\... |
2,931,740 | <p>Find the first three non-zero terms in the asymptotic expansion for each of the three roots of the cubic equation <span class="math-container">$$\epsilon x^3+x^2-1=0, |\epsilon|<<1.$$</span> </p>
<p>I found the expansions <span class="math-container">$1-\frac{\epsilon}{2}+\frac{5}{8}\epsilon^2$</span> and <sp... | skbmoore | 321,120 | <p>Maybe you want to experience the joy of tedious calculation, but I make my computer slave do this for me and reward him with extra time to bounce virtual polygons across the screen.</p>
<p>Simply calculate the cubic roots explicitly and expand them in a power series. In Mathematica the code looks like:</p>
<p>roo... |
3,908,915 | <p>I was reading Lecture 7 of Afternotes on Numerical Analysis by G. W. Stewart, and he mentions without proof the following result:</p>
<p>If <span class="math-container">$n \epsilon_M \leq 0.1$</span> and <span class="math-container">$\epsilon_i \leq \epsilon_M (i = 1,2,...,n)$</span>, then</p>
<p><span class="math-c... | DinosaurEgg | 535,606 | <p>The answer above covers the computational part of the question more than adequately but it does not answer how to prove that a bound like that can be rigorously obtained.</p>
<p>Here's an outline of a proof using some basic calculus.</p>
<ul>
<li><strong>Lemma 1</strong>: The function <span class="math-container">$f... |
3,315,113 | <p>Are there some basic criteria by which to check whether a polytope <span class="math-container">$P\subset\Bbb R^n$</span> is a <span class="math-container">$01$</span>-polytope, that is, can be rotated and scaled to have vertices in <span class="math-container">$\{0,1\}^n$</span>, maybe even by using higher-dimensio... | Dr. Richard Klitzing | 518,676 | <p>When the 24-cell would be a 01-polytope, then its equatorial section, the cuboctahedron, would be too. And, iterating that argument, when the cuboctahedron would be a 01-polytope, then its equatorial section, the regular hexagon, would be too. Sure, the hexagon well can be inscribed into a cube by means of a single ... |
372,135 | <p>Question: Let $P = 27+108x+90x^2-80x^3-60x^4+48x^5-8x^6$, find $\sqrt[3]{P}$.</p>
<p>Just wondering if there is a general way for dealing with this sort of question, I was able to figure how to find $\sqrt{P}$, but $\sqrt[3]{P}$ is beyond me. Long division is used in an example, unfortunately it lacks justification... | lab bhattacharjee | 33,337 | <p>Let $(3+ax-2x^2)^3=27+108x+90x^2-80x^3-60x^4+48x^5-8x^6$</p>
<p>Now use <a href="http://en.wikipedia.org/wiki/Multinomial_theorem#Example" rel="nofollow">this</a> and compare the coefficients of the different powers of $x$ to find the unique value of $a$ such that all the equation formed by comparing the coeffic... |
183,123 | <p>By using Cantor's Theorem: if We have a Set A, with n members the Cardinality of the Power Set is $P(A)=2^n$.
There is Formula to calculate n Power Sets $P(...P(P(A))...)$ of a set A with m elements without doing the simple iteration $(((2^2)^2)..)^2^m$ n times?
Many thanks</p>
| Brian M. Scott | 12,042 | <p>The correct formula is $$\large{2^{\left(2^{\left(2^{\dots(2^n)}\right)}\right)}}\;.$$ That is, you start with $2^n$ for the cardinality of $\wp(A)$ when $|A|=n$, then get $2^{2^n}=2^{(2^n)}$ for the cardinality of $\wp(wp(A))$, and so on. I know of no simpler form for this <a href="http://en.wikipedia.org/wiki/Tetr... |
340,701 | <p>I'm thinking about the following problem. Assume that one is given a sequence of binary data $(x_i)_i\in\{0,1\}^n$ and that wants to determine whether or not this sequence has been generated by a Bernoulli-like random mechanism, such as coin tosses.</p>
<p>My first idea was to use classical statistical test theory,... | Wintermute | 67,388 | <p>You might try something like testing the independence of events. If you are looking at something Bernoulli then data points must be independent. You might just check to see if $P(A \cap B) = P(A)P(B)$ fails for two data points in which case you are not Bernoulli.</p>
|
3,912,109 | <p>Let <span class="math-container">$T$</span> be a complete first-order theory over a <em>countable</em> language <span class="math-container">$L$</span>, such that <span class="math-container">$T$</span> has an infinite model.</p>
<p>It is well-known that we can construct an elementary chain of size-<span class="math... | Tryst with Freedom | 688,539 | <h3>Bisection method</h3>
<p><span class="math-container">$$ f= 3x^2 + 2 \sin x -2$$</span></p>
<p>We can use the bisection method for finding the root, the first is to guess an interval. I will start with an interval <span class="math-container">$[0,1]$</span>.</p>
Note that:
<p><span class="math-container">$$ f(1) = ... |
2,235,645 | <p>I am struggling with solving sum of this alternate series:</p>
<p>$$
\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)}\
$$</p>
<p>I know that:</p>
<p>$$
\log(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \cdot x^n\
$$</p>
<p>But It seems that I can't find a way to get to this form.
Thanks.</p>
| haqnatural | 247,767 | <p>It is well-known fact that</p>
<blockquote>
<p>$${ \sum _{ n=1 }^{ \infty }{ \frac { (-1)^{ n+1 } }{ n } } =\ln { 2 } . }$$</p>
</blockquote>
<p>So
$$\sum _{ n=1 }^{ \infty } \frac { (-1)^{ n+1 } }{ n(n+1) } =\sum _{ n=1 }^{ \infty } \left( \frac { (-1)^{ n+1 } }{ n } -\frac { (-1)^{ n+1 } }{ n+1 } \right)... |
252,257 | <p>I need to show that the DE $y'=y^{\alpha}$, where $\alpha$ is a constant with $0<\alpha<1$, has infinitely many solutions passing through the point $(0,0)$.
Also I need four of such solutions.
Thank you for your help!
Klara</p>
| Artem | 29,547 | <p>Here how you can proceed. First note that $y=0$ is a solution. Next, you use integration to get
$$
\int_0^y\frac{dy}{y^{\alpha}}=t+C_1.
$$
Note that on the left hand side you have an improper integral which converges if $0<\alpha<1$ -- this is the reason the solutions are not unique at the point $(0,0)$. Assum... |
1,590,619 | <p>I must find the sum $$\sum_\limits{n=1}^{\infty}\frac1{n(2n+1)}$$ I have already tried partial fractions, but obtained an indeterminate form of infinity minus infinity. Perhaps completing the square would help?</p>
| Gyumin Roh | 292,972 | <p>We have $$\sum_{n=1}^{\infty} \frac{1}{n(2n+1)}=\sum_{n=0}^{\infty} \frac{1}{2(n+1)(n+\frac{3}{2})} = \sum_{n=0}^{\infty} \frac{1}{n+1}-\frac{1}{n+\frac{3}{2}}=\psi(\frac{3}{2})+\gamma = 2+\psi(\frac{1}{2})+\gamma = 2-\ln 4$$</p>
|
1,757,914 | <p>My instructor proved the central limit theorem using the characteristic function. I think the proof is a standard one because I found basically the same proof in wikipedia.</p>
<p>So for i.i.d. ${X_1, X_2,\cdots, X_n}$ with $E[X_i]=\mu$ and $\text{Var}(X_i)=\sigma^2<+\infty$, he then defined
$$Y_i=\frac{X_i-\mu}... | Axel Johansson | 597,416 | <p>It's definitely a way of seeing it, from my own experience it very much depends on how you define the mobius strip in the <span class="math-container">$3$</span>-d plane.</p>
<p>Personally I set the midsection of the strip in the <span class="math-container">$XY$</span>-plane with a <span class="math-container">$Z$... |
3,339,979 | <p>The problem is as follows:</p>
<p>"Prove that if the limit <span class="math-container">$f(x)$</span> as <span class="math-container">$x$</span> tends to <span class="math-container">$a$</span> is <span class="math-container">$L$</span>, then there exists an <span class="math-container">$m\gt0$</span> and <span cla... | John Hughes | 114,036 | <p>You can slightly improve your choice of <span class="math-container">$M$</span> by making it very explicit. Pick
<span class="math-container">$$
M = 2 + |L|.
$$</span>
Then
<span class="math-container">$$
L \le |L| = M - 2 < M
$$</span>
and
<span class="math-container">$$
-L \le |L| = M - 2 < M.
$$</span></p... |
1,802,892 | <p>If $ Q(x)= x^2-5x+1 $ , find $ \dfrac {Q(5+h)-Q(5)}{h} $</p>
<p>can someone show the steps to reach the answer of $h+5$?</p>
<p>I got it down to $\frac{h^2+5h-2}{h}$</p>
| DeepSea | 101,504 | <p><strong>hint</strong>: $Q(5+h) = (5+h)^2 - 5(5+h) +1=...$</p>
|
1,850,977 | <p>I am creating "children" in script and would like to calculate the number of children before launching the script. I have "width" the number of child nodes and "depth" the depth of the nodes.</p>
<ul>
<li>5 wide and 1 deep = 5</li>
<li>5 wide and 2 deep = 30 = (5+25)</li>
<li>5 wide and 3 deep = 155 = (5+25+125)</l... | barak manos | 131,263 | <p><strong>First, show that this is true for $n=1$:</strong></p>
<p>$3^{2+3}+40-27=64\cdot4$</p>
<p><strong>Second, assume that this is true for $n$:</strong></p>
<p>$3^{2n+3}+40n-27=64k$</p>
<p><strong>Third, prove that this is true for $n+1$:</strong></p>
<p>$3^{2(n+1)+3}+40(n+1)-27=$</p>
<p>$9\cdot(\color\red{... |
1,850,977 | <p>I am creating "children" in script and would like to calculate the number of children before launching the script. I have "width" the number of child nodes and "depth" the depth of the nodes.</p>
<ul>
<li>5 wide and 1 deep = 5</li>
<li>5 wide and 2 deep = 30 = (5+25)</li>
<li>5 wide and 3 deep = 155 = (5+25+125)</l... | Bill Dubuque | 242 | <p>Many inductive proofs of this and similar divisibilities boil down to computing the first two terms of the Binomial Theorem <span class="math-container">$\,\color{#0a0}{\rm BT}.\,$</span> However, this innate structure is often (greatly) obfuscated by details of the special case. Let's bring this structure to the fo... |
3,730,376 | <p>Let <span class="math-container">$A$</span> be a non empty set and <span class="math-container">$X$</span> be the set of all partitions of <span class="math-container">$A$</span>. If <span class="math-container">$T \subseteq X$</span>, I want to write the set of intersections of sets in the partitions of <span class... | Royce Pacibe | 801,693 | <p><em>This answer assumes that <span class="math-container">$T$</span> is already defined as <span class="math-container">$T = \left\{T_{i} \mid i \in I\right\}$</span> for some index set <span class="math-container">$I$</span>.</em></p>
<p>Take some <span class="math-container">$B \in S$</span>. By definition of <spa... |
3,730,376 | <p>Let <span class="math-container">$A$</span> be a non empty set and <span class="math-container">$X$</span> be the set of all partitions of <span class="math-container">$A$</span>. If <span class="math-container">$T \subseteq X$</span>, I want to write the set of intersections of sets in the partitions of <span class... | Asaf Karagila | 622 | <p>This is slightly easier to write if you think about the actual cells in the partitions. Ultimately, I suppose, the goal is to find the coarsest partition that refines all the partitions in your collection.</p>
<p>So given <span class="math-container">$a\in A$</span>, what will be the cell containing <span class="mat... |
266,228 | <p>I'm new to use the amazing map function in Mathematica, and I've found it really elegant to program in simple lines of code. However, I'd like to perform more elegant stuff. I've found this question, but its not what I'm trying to achieve. <a href="https://mathematica.stackexchange.com/questions/8147/how-do-i-use-m... | lericr | 84,894 | <p>Since you're looking to learn more about functional style, I'll elaborate rather than just provide an answer.</p>
<p>We start with our table:</p>
<pre><code>vector = Table[{RandomReal[], RandomReal[], RandomReal[]}, {n, 0, 10}]
</code></pre>
<p>We could either re-structure the table first, or deal with selecting whi... |
4,539,866 | <blockquote>
<p>If <span class="math-container">$a+b+c=1$</span> and <span class="math-container">$a, b, c\geq0$</span>, <br />
what is the maximum value of <span class="math-container">$\frac{1}{a^2-4a+9}+\frac{1}{b^2-4b+9}+\frac{1}{c^2-4c+9}$</span> ?</p>
</blockquote>
<p>I found its answer by using CAS Calculator. ... | Suzu Hirose | 190,784 | <p>This is something of a brute force method.</p>
<p>Let <span class="math-container">$f(a,b,c)=\sum_{a,b,c}1/ (a^2-4a+9)$</span> and <span class="math-container">$g(a,b,c)=a+b+c-1$</span> then calculate <span class="math-container">$\nabla f-\lambda \nabla g=0$</span> to find
<span class="math-container">$$
{2a-4\over... |
1,976,538 | <p>How do you know your answer to a probability question is correct without checking the solutions?</p>
<p>For example, when you have an equation you can get a graphing calculator, create the graph for the first part of the equation, the graph for the second and then get the intersection to make sure your calculations... | Graham Kemp | 135,106 | <blockquote>
<p>I doubt I will be allowed to do that during an exam </p>
</blockquote>
<p>If you are having doubts with answering a question of this simplicity in an exam, you may not pass the exam <em>due to nervousness</em>. Have a little more confidence in yourself.</p>
<p>This sort of problem has just tw... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | Mikhail Katz | 28,128 | <p>Human mathematical creativity is not reducible to computation. An illustration of this is that human beings would have an easier time figuring out Bourbaki's <em>Set theory</em>. As A. Mathias showed, Bourbaki's definition of "1" requires billions of characters. The definition of "2" is probably inaccessible to m... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | Brendan McKay | 9,025 | <p>The day will come when not only will computers be doing good mathematics, but they will be doing mathematics beyond the ability of (non-enhanced) humans to understand. Denying it is understandable, but ultimately as short-sighted as it was to deny computers could ever win at Go.</p>
<p>This is not as depressing as... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | logicute | 85,130 | <p>It boils down to creativity. What we truly admire in mathematical achievements are the creative leaps when creating new theories (category theory, calculus), not mathematical prowess, which is a tool. Good problem solving is also highly regarded, but it's also admired for the creative ideas it takes while doing the ... |
222,114 | <p>It is known (via Kotschwar's uniqueness of backwards Ricci flows) that the isometry group of a Riemannian metric remains unchanged under the Ricci flow. But, one can easily observe that it can change at the limit. For example, one can perturb a sphere $S^2$ slightly so that it has no symmetry. As long as the Ricci f... | Otis Chodosh | 1,540 | <p>The answer is yes (I'm assuming you are asking about closed manifolds, non-compactness allows for all sort of crazy things to happen, you can check out the work of Topping and collaborators). </p>
<p>Kotschwar's backwards uniqueness proves that the isometry group cannot <em>increase</em> under the Ricci flow in fin... |
2,643,778 | <p>I can intuitively realize why this is true. All the points that are on the line $y=-x$ are in the set. How can I prove this with sequences?</p>
| Pedro | 178,668 | <p><strong>Hint</strong>: inequalities are preserved by limits.</p>
|
2,643,778 | <p>I can intuitively realize why this is true. All the points that are on the line $y=-x$ are in the set. How can I prove this with sequences?</p>
| Peter Szilas | 408,605 | <p>Correct me if wrong.</p>
<p>$A:=${$(x,y)\in \mathbb{R^2}: x+y \ge 0$}.</p>
<p>Let $a=(x,y)$ be an accumulation point of the set $A$.</p>
<p>There exists a sequence $(x_n,y_n) \in A$ </p>
<p>such that $\lim_{n \rightarrow \infty} (x_n,y_n)= (x,y).$</p>
<p>Since $(x_n,y_n) \in A$: </p>
<p>$x_n+y_n \ge 0.$</p>
... |
1,894,926 | <p>I know no formal system containing arithmetic can prove itself consistent. But is it possible to add some <em>ungrounded</em> axiom in the sense described here <a href="https://en.wikipedia.org/wiki/Liar_paradox#Saul_Kripke" rel="nofollow">https://en.wikipedia.org/wiki/Liar_paradox#Saul_Kripke</a> such that the new ... | Noah Schweber | 28,111 | <p>Leaving aside a number of issues (which also applied to your previous question on illogicality) around foundational matters, there is arguably a sense in which the answer to your question is yes: if we pass to an appropriate theory which can talk about the set of all natural numbers - for example, $ZFC$! - then such... |
1,696,143 | <p>Prove that:</p>
<p>$$\int_0^B e^\left(x^2\right) \;dx \sim \frac{e^\left(B^2\right)}{2B}$$ where $$B \to \infty$$</p>
<p>We should prove the equivalence of the given integral to the expression in the right part. </p>
<p>It's obvious, that the integral in the given form cannot be expressed through known indefinite... | Ron Gordon | 53,268 | <p>Integrate by parts:</p>
<p>$$\begin{align}\int_0^B dx \, e^{x^2} &= \underbrace{\int_0^1 dx \, e^{x^2}}_{C} + \int_1^B dx \, e^{x^2}\\ &= C + \int_1^B dx \,x \frac{e^{x^2}}{x} \\ &= C + \frac12 \left [\frac{e^{x^2}}{x} \right ]_1^B + \frac12 \int_1^B dx \, \frac{e^{x^2}}{x^2}\\ &= \underbrace{C - \... |
4,076,273 | <p>Given a one dimensional random walk where I have <span class="math-container">$p$</span> probability to go forward and <span class="math-container">$q$</span> probability to go backwards.</p>
<p>I have to prove that, starting from the origin, the probability of reaching <span class="math-container">$a>0$</span> b... | Ian | 83,396 | <p>So first you have to go right, so that introduces a factor of <span class="math-container">$p$</span>. After that, you need to hit <span class="math-container">$a$</span> before hitting <span class="math-container">$0$</span> starting at <span class="math-container">$1$</span>. There is a standard way to compute the... |
4,512,690 | <p>I am having problems in evaluating the following Cauchy principal value</p>
<p><span class="math-container">$$\int_{-\infty}^{+\infty} \frac{\cos(3x)}{x-1}d x$$</span></p>
<p>I know it is supposed to show my work, but I got barely nothing. I tried with using <span class="math-container">$\cos(x) = (z + z^{-1})/2$</s... | Jack D'Aurizio | 44,121 | <p>An even shorter solution: for any <span class="math-container">$n\in\mathbb{N}^+$</span> we have <span class="math-container">$\int_{-\infty}^{+\infty}\frac{\sin(nx)}{x}\,dx=\pi$</span> and</p>
<p><span class="math-container">$$\begin{eqnarray*} \text{PV}\int_{\mathbb{R}}\frac{\cos(3x)}{x-1}\,dx &\stackrel{x\map... |
1,458,055 | <p>What i want to prove is following integral
\begin{align}
\int_{-\infty}^{\infty} e^{-x^2} \cos(x^2) dx=\frac{1}{2} \sqrt{\left(1+\sqrt{2}\right) \pi }
\end{align}
can you give some explicit method to obtain this result?</p>
| Jameson | 591,805 | <p>Let your integral be labelled
$$I_1=\int_{-\infty}^{\infty} e^{-x^2}\cdot \cos(x^2)\space dx,$$
and a second integral
$$I_2=\int_{-\infty}^{\infty} e^{-x^2}\cdot\sin(x^2)\space dx.$$
It follows that
$$I_1-i\cdot I_2=\int_{-\infty}^{\infty} e^{-x^2}\cdot(\cos(x^2)-i\cdot \sin(x^2))\space dx.$$
Apply Euler's formula i... |
304,646 | <p>I am asked to solve the following system of equations using Cramer's Rule</p>
<p>$$
A^2 + B = x\\
AB = 1-x^2
$$</p>
<p>where $A$ and $B$ are 2 unknowns that are to be solved for some given $x$. ie I'm expecting the solution to be $A = a(x)$ and $B = b(x)$ where $a(x)$ and $b(x)$ is some function of $x$.</p>
<p>I... | Kaster | 49,333 | <p>Cramer's rule is for linear system of equations only. So, no, you can't apply it for the system above.</p>
<h2>Update</h2>
<p>Also, what's $x$ in your equations? Because if it's actually a variable, then your system is a functional equations. If it's just a number, never mind then.</p>
|
395,416 | <blockquote>
<p>Consider the sets dened by the real solutions of the inequalities $$A=\{(x,y):x^2+y^4\le 1\}$$ and $$B=\{(x,y):x^4+y^6\le 1\}$$Then which is true $A$ is subset of $B$ or $B$ is subset of $A$. </p>
</blockquote>
<p>This is <a href="http://www.isical.ac.in/~deanweb/sylques.pdf" rel="nofollow">Question ... | DonAntonio | 31,254 | <p>Hints:</p>
<p>$$(x,y)\in A\implies x^2+y^4\le 1\implies |x|,|y|\le 1$$</p>
<p>so</p>
<p>$$ x^4+y^6\le x^2+y^4\implies \ldots\ldots$$</p>
|
30,983 | <p>Firstly, are there always nontrivial real solutions to the sysytem
of equations, $A_{1}x^{5}+B_{1}y^{5}+C_{1}z^{5}=0$ and $A_{2}x+B_{2}y+C_{2}z=0$,
for real numbers $A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, $B_{2}$,
and $C_{2}$? [Answered]</p>
<p>Secondly, are there always nontrivial real solutions to the sysytem
of equa... | Andy B | 4,542 | <p>A very good reference for this is the recent book of De Loera, Rambau and Stantos called "Triangulations:
Structures for Algorithms and Applications". They have a whole chapter on regular triangulations. I don't know if the book is in print yet. It's possible that careful googling could reveal drafts of the book tha... |
207,871 | <p>I am writing a code to display longitudinal waves.
I started with the following, filling the slots using the Range function. </p>
<pre><code> l = {Blue, Disk[{3 Sin[(5) t + 0.1 #] + 1.4 #, 1}, 0.5]} & /@
Range[25];
Animate[Evaluate[Graphics[l, PlotRange -> {{-4, 44}, All}]], {t, 0,
20. \[Pi]}, Anim... | Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
nRow = 10;
nCol = 25;
l = {Blue,
Disk[{3 Sin[5 t + 0.1 #[[1]]] + 1.4 #[[1]], 1.5 #[[2]]},
0.5]} & /@ Flatten[Array[{#1, #2} &, {nCol, nRow}], 1];
Animate[Evaluate[Graphics[l, PlotRange -> {{-4, 44}, All}]], {t, 0,
20. π}, AnimationRate -> .3]
</code></pre>
... |
3,043,144 | <p><strong>Problem</strong>:</p>
<p>Let <span class="math-container">$A$</span> be some linear operator such that:
<span class="math-container">$$ (A^{2006}-I)^{2006}-I=0. $$</span>
Determine all real numbers that can be eigenvalues of operator <span class="math-container">$A$</span>.</p>
<p><strong>Question</strong>: ... | Ross Millikan | 1,827 | <p>If <span class="math-container">$x$</span> is an eigenvalue it must satisfy <span class="math-container">$(x^{2006}-1)^{2006}-1=0$</span>, so <span class="math-container">$x^{2006}-1$</span> is a <span class="math-container">$2006^{th}$</span> root of <span class="math-container">$1$</span>. The only reals that sat... |
4,470,820 | <p>I shall find the integral by Feynman’s Technique Integration on a particular integral</p>
<p><span class="math-container">$\displaystyle I(a)=\int_{0}^{\pi} \ln (a \cos x+1) d x,\tag*{} $</span>
where <span class="math-container">$-1\leq a \leq 1.$</span>
<span class="math-container">$\displaystyle \begin{aligned}I^... | Frank W | 552,735 | <p>Another option as opposed to using an infinite sequence and piecewise integration is to use a Weierstrass substitution and then a double integral.</p>
<p>Take <span class="math-container">$x\mapsto\tan\left(\tfrac x2\right)$</span> such that <span class="math-container">$\cos x\mapsto(1-x^2)/(1+x^2)$</span>. The int... |
2,961,323 | <p>In transport theory in plasma physics, there's an important integral called the Coulomb logarithm, which relates to the scattering cross section off the Yukawa potential. It can be written as
<span class="math-container">$$
\ell(\Lambda) = \int_0^\infty \cos^2\left(\int_0^{u^*}\left[1 - 2\frac{u}{\xi}\exp\left(-\fra... | Diger | 427,553 | <p>I want to build on what Yuriy already established. First of all let me emphasize that I think at no order in the small <span class="math-container">$t$</span> approximation you will get a finite integral. This is because the equation <span class="math-container">$$1-wv^2-2v \exp\left({-\frac{t}{v}}\right)=0 \tag{0}$... |
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