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 |
|---|---|---|---|---|
1,517,502 | <p>Point P$(x, y)$ moves in such a way that its distance from the point $(3, 5)$ is proportional to its distance from the point $(-2, 4)$. Find the locus of P if the origin is a point on the locus.</p>
<p><strong>Answer</strong>:</p>
<p>$$(x-3)^2 + (y-5)^2 = (x+2)^2 + (y-4)^2$$
or, $$10x+2y-14=0$$
or, $$5x+y-7=0$$</... | Community | -1 | <p>$$(x-3)^2 + (y-5)^2 = \lambda((x+2)^2 + (y-4)^2).$$</p>
<p>We express that the curve passes through the orgin:</p>
<p>$$(-3)^2 + (-5)^2 = \lambda((+2)^2 + (-4)^2),$$</p>
<p>hence
$$\lambda=\frac{17}{10}.$$</p>
|
2,249,841 | <p>Let $a_n$ denote the number of those permutations $\sigma$ on $\{1,2,3....,n\}$ such that $\sigma$ is a product of exactly two disjoint cycles. Then </p>
<ol>
<li><p>$a_5 = 50$</p></li>
<li><p>$a_4 = 14$</p></li>
<li><p>$a_5 = 40$</p></li>
<li><p>$a_4 = 11$</p></li>
</ol>
<p>I tried specifically for $a_5$ and $a... | Marko Riedel | 44,883 | <p>I would like to present the connection to Stirling numbers since it
has not been pointed out. For the first interpretation where the
cycles may be singletons we get the species
$\mathfrak{P}_{=2}(\mathfrak{C}(\mathcal{Z}))$ which yields per
generating function</p>
<p>$$n! ... |
51,096 | <p>Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?</p>
| Qiaochu Yuan | 232 | <p>Sure. For example, let $A_n$ be the natural numbers with exactly $n$ ones in their binary expansion.</p>
<p>Alternately, pick your favorite way of decomposing $\mathbb{N}$ into two disjoint infinite subsets $A, B$, and pick a bijection $f : B \to \mathbb{N}$. Then $f(B)$ can be decomposed into two disjoint subsets ... |
184,361 | <p>I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:</p>
<blockquote>
<p><strong>DEFINITION</strong> Given $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x < z+1$$ and we d... | Community | -1 | <p>Hint: Set $\{ x \} = x - [x]$. To prove $[nx] = \displaystyle\sum_{k = 0}^{n - 1} [x + \frac{k}{n}]$, consider the cases $\frac{k - 1}{n} \leq \{ x\} < \frac{k}{n}$ for $k = 1,2,\ldots,n$ separately.</p>
<p>The idea is that we want to see when exactly $[x + \frac{k}{n}] = [x] + 1$ starts to hold as $k$ grows.</p... |
1,116,445 | <p>I am trying to understand <a href="http://en.wikipedia.org/wiki/Diophantine_equation" rel="nofollow">Diophantine equation</a> article in wiki. They say that in the given equation:</p>
<p>$$ax + by = c$$</p>
<p>There will be such integers $x,y$ <strong>if and only if</strong> $c$ is a multiplier of greatest common... | GPerez | 118,574 | <p>It does in fact verify the statement, because $$\mathrm{gcd}(3,2) = \mathrm{gcd}(3,4) = 1$$</p>
|
1,116,445 | <p>I am trying to understand <a href="http://en.wikipedia.org/wiki/Diophantine_equation" rel="nofollow">Diophantine equation</a> article in wiki. They say that in the given equation:</p>
<p>$$ax + by = c$$</p>
<p>There will be such integers $x,y$ <strong>if and only if</strong> $c$ is a multiplier of greatest common... | MJD | 25,554 | <p>Here you have $a=3, b=2$. The greatest common divisor of $3$ and $2$ is $1$. $17$ is a multiple of $1$, so there is a solution to the equation, namely that $x=3, y=4$.</p>
|
1,906,146 | <p>Can the following expression be further simplified: $$a^{(\log_ab)^2}?$$</p>
<p>I know for example that $$a^{\log_ab^2}=b^2.$$</p>
| Jan Eerland | 226,665 | <p>Use:</p>
<ul>
<li>$$\log_x(y)=\frac{\ln(y)}{\ln(x)}$$</li>
<li>$$\exp\left[\ln\left(x\right)\right]=e^{\ln(x)}=x$$</li>
</ul>
<hr>
<p>So, we get:</p>
<p>$$a^{\left(\log_a(b)\right)^2}=a^{\left(\frac{\ln(b)}{\ln(a)}\right)^2}=a^{\frac{\ln^2(b)}{\ln^2(a)}}=\exp\left[\frac{\ln^2(b)}{\ln(a)}\right]$$</p>
|
1,906,146 | <p>Can the following expression be further simplified: $$a^{(\log_ab)^2}?$$</p>
<p>I know for example that $$a^{\log_ab^2}=b^2.$$</p>
| mfl | 148,513 | <p>$$a^{(\log_ab)^2}=a^{\log_a b\cdot \log_a b}=(a^{\log_a b})^{\log_a b}=b^{\log_a b}=b^{\frac{1}{\log_b a}}.$$</p>
|
1,636,207 | <p>I understand the basics of Cartesian products, but I'm not sure how to handle a set inside of a set like $C = \{\{1,2\}\}$. Do I simply include the set as an element, or do I break it down?</p>
<p>If I use it as an element I think it would be something like this:</p>
<p>$$\{(1,\{1,2\}), (2,\{1,2\})\}$$</p>
<p>If... | mm8511 | 180,904 | <p>You are correct. Even if a set has sets as elements, you still treat "each element" separately.</p>
<p><a href="https://i.stack.imgur.com/FAsbG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FAsbG.png" alt="mathematica output"></a></p>
|
1,333,994 | <p>We have a function $f: \mathbb{R} \to \mathbb{R}$ defined as</p>
<p>$$\begin{cases} x; \ \ x \notin \mathbb{Q} \\ \frac{m}{2n+1}; \ \ x=\frac{m}{n}, m\in \mathbb{Z}, n \in \mathbb{N} \ \ \ \text{$m$ and $n$ are coprimes} \end{cases}.$$</p>
<p>Find where $f$ is continuous</p>
| user84413 | 84,413 | <p>Let $ a=\log_{3}\frac{1}{2}$, so $\;a<0$ and $\;\displaystyle\log_{\frac{1}{2}}3=\frac{\log_{3}3}{\log_3\frac{1}{2}}=\frac{1}{a}$.</p>
<p>Then $(a+1)^2>0\implies a^2+1>-2a\implies a+\frac{1}{a}<-2\;\;$ (since $a<0$)</p>
|
4,636,101 | <p>Given a curve <span class="math-container">$y = x^3-x^4$</span>, how can I find the equation of the line in the form <span class="math-container">$y=mx+b$</span> that is tangent to only two distinct points on the curve?</p>
<p>The problem given is part of the Madas Special Paper Set. This paper set, seems to not hav... | Jan-Magnus Økland | 28,956 | <p><span class="math-container">$(x,y)=(t,t^3-t^4),$</span> has <a href="https://en.wikipedia.org/wiki/Dual_curve" rel="nofollow noreferrer">dual</a> <span class="math-container">$(X,Y)=(-\frac{y'}{xy'-x'y},\frac{x'}{xy'-x'y})=(-\frac{3t^2-4t^3}{t(3t^2-4t^3)-(t^3-t^4)},\frac1{t(3t^2-4t^3)-(t^3-t^4)})$</span> and implic... |
2,188,965 | <p>Can someone explain to me how this step done? I got a different answer than what the solution said.</p>
<p>Simplify $x(y+z)(\bar{x} + y)(\bar{y} + x + z)$</p>
<p>what the solution got </p>
<p>$x(y+z)(\bar{x} + y)(\bar{y} + x + z)$ = $x(y + z\bar{x})(\bar{y} + x + z)$ (Using distrubitive)</p>
<p>What I got</p>
<... | Bram28 | 256,001 | <p>Put the following equivalence into your boolean algebra toolkit:</p>
<p><strong>Absorption</strong></p>
<p>$x +xy = x$</p>
<p>Using Absorption twice in one step we get:</p>
<p>$y\bar{x}+y+z\bar{x}+zy = y+z\bar{x}$</p>
<p>Done!</p>
|
1,431,289 | <p>Find the average rate of change of $2x^3 - 5x$ on the interval $[1,3]$.</p>
<p>I'm really confused about this problem. I keep ending up with the answer $12$, but the answer key says otherwise. Someone please help! Thanks!</p>
| DirkGently | 88,378 | <p>The total change is $h(3)-h(1)=52$. The length of the interval is $2$. So the average rate of change is $52/2=26$.</p>
<p><strong>Update:</strong> You have apparently changed the function in the question from $h(x)=2x^3-5$ to $h(x)=2x^3-5x$. The answer for the new function is $\frac {h(3)-h(1)}2=21$.</p>
|
3,209,237 | <p>The proof of the CRT goes as follows:<br>
Given the number <span class="math-container">$x \epsilon Z_m$</span>, <span class="math-container">$m=m_1m_2...m_k$</span>
<span class="math-container">$$M_k = m/m_k$$</span>
construct:
<span class="math-container">$$ x = a_1M_1y_1+a_2M_2y_2+...+a_nM_ny_n$$</span>
where <sp... | Bernard | 202,857 | <p>I think <span class="math-container">$x$</span> is calculated in <span class="math-container">$\mathbf Z$</span>, using representatives of <span class="math-container">$M_k^{-1}\bmod m_k$</span>. Note that the congruence class of <span class="math-container">$x$</span> <span class="math-container">$\bmod m$</span> ... |
3,953,674 | <p>Here is a common argument used to prove that the sum of an infinite geometric series is <span class="math-container">$\frac{a}{1-r}$</span> (where <span class="math-container">$a$</span> is the first term and <span class="math-container">$r$</span> is the common ratio):
<span class="math-container">\begin{align}
S &... | Gauge_name | 813,708 | <p>Your argument is rigorous if you know that your series converges. Namely you first should know that the series converges and afterwards you may use that method to compute its limit. Prior to know that the series converges the method is wrong since, if the series diverges, in the first step you are subtracting two in... |
127,412 | <p>How to take a 3 random given name?</p>
<p>I tried:</p>
<p><a href="https://i.stack.imgur.com/BUYIT.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BUYIT.png" alt="enter image description here"></a></p>
| Greg Hurst | 4,346 | <p>IMO the best way is <a href="http://reference.wolfram.com/language/ref/RandomEntity.html" rel="nofollow noreferrer"><code>RandomEntity</code></a> as <a href="https://mathematica.stackexchange.com/users/731/c-e">C.E.</a> points out in the comments:</p>
<pre><code>RandomEntity["GivenName", 3]
</code></pre>
<p><a hre... |
127,412 | <p>How to take a 3 random given name?</p>
<p>I tried:</p>
<p><a href="https://i.stack.imgur.com/BUYIT.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BUYIT.png" alt="enter image description here"></a></p>
| Anton Antonov | 34,008 | <p>Another approach is to use the Wolfram Function Repository function <a href="https://resources.wolframcloud.com/FunctionRepository/resources/RandomPetName/" rel="nofollow noreferrer"><code>RandomPetName</code></a>:</p>
<pre class="lang-mathematica prettyprint-override"><code>SeedRandom[33];
ResourceFunction["Ra... |
514,517 | <p>So this is what my book states:</p>
<p>Random variables $X,Y, and Z$ are said to form a Markov chain in that order denoted $X\rightarrow Y \rightarrow Z$ if and only if:</p>
<p>$p(x,y,z)=p(x)p(y|x)p(z|y) $</p>
<p>That's great and all but that doesn't give any intuition as to what a Markov chain is or what it impl... | Balbichi | 24,690 | <p>Hint: $f:[a,b]\to \mathbb{R}$ be continuous, such that $f(a)f(b)<0$. Then there exists $c\in(a,b)$ such that $f(c)=0$.</p>
|
1,480,511 | <p>I have included a screenshot of the problem I am working on for context. T is a transformation. What is meant by Tf? Is it equivalent to T(f) or does it mean T times f?</p>
<p><a href="https://i.stack.imgur.com/LtRS1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LtRS1.png" alt="enter image desc... | vonbrand | 43,946 | <p>Use the <a href="https://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow">rational root theorem</a>: If the polynomial $a_n x^n + \dotsb + a_0$ with integer coefficients, such that $a_n \ne 0$ and $a_0 \ne 0$ has a rational zero $u / v$, then $u$ divides $a_0$ and $v$ divides $a_n$.</p>
<p><strong>Proof:... |
3,660,101 | <p>I want to determine if the series <span class="math-container">$ \sum_{n=2}^{\infty}\frac{\left(-1\right)^{n}}{\left(-1\right)^{n}+n} $</span> converge/diverge. the sequence in the denominator is not monotinic, so I cant use Dirichlet's or Abel's tests. My intuition is that this series converge, becuase its looks cl... | marty cohen | 13,079 | <p>Let
<span class="math-container">$s(m)
=\sum_{n=2}^{m}\dfrac{\left(-1\right)^{n}}{\left(-1\right)^{n}+n}
$</span>.
The terms go to zero,
so it enough to show that
<span class="math-container">$s(2m+1)$</span>
converges.</p>
<p><span class="math-container">$\begin{array}\\
s(2m+1)
&=\sum_{n=2}^{2m+1}\dfrac{\left... |
3,660,101 | <p>I want to determine if the series <span class="math-container">$ \sum_{n=2}^{\infty}\frac{\left(-1\right)^{n}}{\left(-1\right)^{n}+n} $</span> converge/diverge. the sequence in the denominator is not monotinic, so I cant use Dirichlet's or Abel's tests. My intuition is that this series converge, becuase its looks cl... | P. Lawrence | 545,558 | <p>The series <span class="math-container">$\sum_{n=2}^{\infty}\frac{(-1)^n}{n}$</span> converges by the alternating series test. <span class="math-container">$$\text{Your given series }-\sum_{n=2}^{\infty}\frac{(-1)^n}{n}=-\sum_{n=2}^{\infty}\frac{1}{n((-1)^n+n)}$$</span>The series <span class="math-container">$$\sum_... |
2,451,469 | <p>I have a system of differential equations:</p>
<p>$$x(t)'' + a \cdot x(t)' = j(t)$$
$$j(t)' = -b \cdot j(t) - x(t)' + u(t)$$</p>
<p>The task is: Substitute $v(t) = x(t)'$ into the system and rewrite the system as 3 coupled linear
differential equations of the same form (with
$y(t) = x(t)$ the solution sought... | Koto | 355,087 | <p>You just need to reduce the order of the first equation. If $v=x'$, then $v'=x''$ and the first equation can be written as the system $$v'=j-av$$ $$x'=v$$ so adding $j'=-bj-v+u$, you get a system in the form $X'=AX+b$, where $X=(x,v,j)^T$ and $$A=\begin{bmatrix}0&1&0\\0 &-a&1\\0 & -1&-b\end{b... |
3,460 | <p>I asked the question "<a href="https://mathoverflow.net/questions/284824/averaging-2-omegan-over-a-region">Averaging $2^{\omega(n)}$ over a region</a>" because this is a necessary step in a research paper I am writing. The answer is detailed and does exactly what I need, and it would be convenient to directly cite t... | Andy Putman | 317 | <p>I think you should just reproduce the argument in your paper while attributing it to the user in question (with a link to the question). As long as you give the correct attribution, there is nothing academically dishonest about this.</p>
<p>I had to do this in one of my papers; see the top of page 20 of <a href="h... |
3,002,874 | <p>I found this limit in a book, without any explanation:</p>
<p><span class="math-container">$$\lim_{n\to\infty}\left(\sum_{k=0}^{n-1}(\zeta(2)-H_{k,2})-H_n\right)=1$$</span></p>
<p>where <span class="math-container">$H_{k,2}:=\sum_{j=1}^k\frac1{j^2}$</span>. However Im unable to find the value of this limit from my... | Jack D'Aurizio | 44,121 | <p>Let's see:</p>
<p><span class="math-container">$$\begin{eqnarray*} \sum_{k=0}^{n-1}\left(\zeta(2)-H_k^{(2)}\right) &=& \zeta(2)+\sum_{k=1}^{n-1}\left(\zeta(2)-H_k^{(2)}\right)\\&\stackrel{\text{SBP}}{=}&\zeta(2)+(n-1)(\zeta(2)-H_{n-1}^{(2)})+\sum_{k=1}^{n-2}\frac{k}{(k+1)^2}\\&=&\zeta(2)+(n-... |
1,960,169 | <p><a href="http://puu.sh/rCwCy/c78a9ef78a.png" rel="nofollow noreferrer">Asymptote http://puu.sh/rCwCy/c78a9ef78a.png</a></p>
<p>Well my thinking was if the asymptote is at x = 4, it will reach as close to 4 as possible but will never reach 4, meaning it's not defined at 4. </p>
| Frank | 332,250 | <p>The vertex can be found by plugging $x$ with $-\frac {b}{2a}$ give the form $ax^2+bx+c=0$.</p>
<p>So with your example $x^2+4x-5=0$, we have $$a=1\\b=4\\c=-5\tag{1}$$
So $-\frac {b}{2a}=-\frac {4}{2}=-2$. Plugging that into the quadratic gives $$f(-2)=4-8-5=-9\tag{2}$$
Therefore, the vertex is $(-2,-9)$.</p>
|
1,904,553 | <blockquote>
<p>$$\displaystyle\lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$$</p>
</blockquote>
<p>I have this limit to be calculated. Since the first term takes the form $\frac 00$, I apply the L'Hospital rule. But after that all the terms are taking the form $\frac 10$. S... | zhw. | 228,045 | <p>Hint: $e^{-u} = 1-u+u^2/2 + O(u^3).$</p>
|
4,600,131 | <blockquote>
<p>If <span class="math-container">$$f(x)=\binom{n}{1}(x-1)^2-\binom{n}{2}(x-2)^2+\cdots+(-1)^{n-1}\binom{n}{n}(x-n)^2$$</span>
Find the value of <span class="math-container">$$\int_0^1f(x)dx$$</span></p>
</blockquote>
<p>I rewrote this into a compact form.
<span class="math-container">$$\sum_{k=1}^n\binom... | Sangchul Lee | 9,340 | <p><strong>Solution 1.</strong> Here is another approach. Consider the shift operator <span class="math-container">$\Delta$</span> defined for functions on <span class="math-container">$\mathbb{R}$</span> by</p>
<p><span class="math-container">$$ \Delta f(x) = f(x-1). $$</span></p>
<p>Then</p>
<p><span class="math-cont... |
4,600,131 | <blockquote>
<p>If <span class="math-container">$$f(x)=\binom{n}{1}(x-1)^2-\binom{n}{2}(x-2)^2+\cdots+(-1)^{n-1}\binom{n}{n}(x-n)^2$$</span>
Find the value of <span class="math-container">$$\int_0^1f(x)dx$$</span></p>
</blockquote>
<p>I rewrote this into a compact form.
<span class="math-container">$$\sum_{k=1}^n\binom... | Alexander Burstein | 499,816 | <p>We wish to evaluate the sum <span class="math-container">$$\sum_{k=0}^{n}(-1)^k\binom{n}{k}(x-k)^2=x^2-\sum_{k=1}^{n}(-1)^{k-1}\binom{n}{k}(x-k)^2.$$</span></p>
<p>Consider an <span class="math-container">$x\times x$</span> board, where <span class="math-container">$x\ge n$</span> is an integer, in which we want to ... |
178,473 | <p>The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there is a function $f:\kappa\to S$ such that for every $\alpha<\kappa$, $f\upharpoonright\alpha R f(\alpha)$. The axiom ... | Asaf Karagila | 7,206 | <p>The idea is to mimic the permutation models as given in Jech. One can then ask, "Well, in Jech he chooses some set of objects in the full universe, and shows it has a support. But in forcing we don't have a simple access to names like that, since they might not be "sufficiently determined" for us to c... |
178,473 | <p>The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there is a function $f:\kappa\to S$ such that for every $\alpha<\kappa$, $f\upharpoonright\alpha R f(\alpha)$. The axiom ... | Lorenzo | 141,146 | <p>In the comments of Asaf' answer I explain why there is a problem in his proof of Theorem 1. In this answer I try to correct them by slightly modifying his argument. I'll keep the same notation of his answer.</p>
<hr />
<h2>Theorem I</h2>
<blockquote>
<p>Let <span class="math-container">$\kappa$</span> be a successor... |
4,600,992 | <p>I have two sequences of random variables <span class="math-container">$\{ X_n\}$</span> and <span class="math-container">$\{Y_n \}$</span>. I know that
<span class="math-container">$X_n \to^d D, Y_n \to^d D$</span>. Can I conclude that <span class="math-container">$X_n - Y_n \to^p 0$</span>?</p>
<p>If I cannot, what... | donaastor | 251,847 | <p>This is a way you could formaly express your intuition:
<span class="math-container">$$f(x)=x\int_1^x\Big(\frac{1}{t}+\sum_{n=1}^\infty\frac{t^{n-1}}{n!}\Big)dt-e^x=x\int_1^x\frac{1}{t}dt+x\int_1^x\sum_{n=1}^\infty\frac{t^{n-1}}{n!}dt-e^x=$$</span>
<span class="math-container">$$=x\ln x+x\int_1^x\sum_{n=1}^\infty\fr... |
78,243 | <p>A positive integer $n$ is said to be <em>happy</em> if the sequence
$$n, s(n), s(s(n)), s(s(s(n))), \ldots$$
eventually reaches 1, where $s(n)$ denotes the sum of the squared digits of $n$.</p>
<p>For example, 7 is happy because the orbit of 7 under this mapping reaches 1.
$$7 \to 49 \to 97 \to 130 \to 10 \to 1$$
B... | David Moews | 17,657 | <p>I started working on this question after it was posted to MathOverflow and found bounds similar to those found by Justin Gilmer: upper asymptotic density of the happy numbers 0.1962 or greater, lower asymptotic density no more than 0.1217. However, I was also able to prove that the upper asymptotic density of the h... |
118,545 | <p>I gather that the question whether the Bruck-Chowla-Ryser condition was sufficient used to top the list, but now that that's settled - what is considered the most interesting open question?</p>
| Chris Godsil | 1,266 | <p>Fix $\lambda>1$. Are there infinitely many symmetric $(v,k,\lambda)$ designs?
(The Hadamard conjectures would be at the top of my list though.)</p>
|
2,072,666 | <p>I have a set as : <b> {∀x ∃y P(x, y), ∀x¬P(x, x)}. </b>. In order to satisfy this set I know that there should exist an interpretation <b> I </b> such that it should satisfy all the elements in the set. For instance my interpretation for x is 3 and for y is 4. Should I apply the same numbers (3,4) to ∀x¬P(x, x) as ... | Clayton | 43,239 | <p><strong>Hint:</strong> The function $f(x)=1/x$ is continuous, so $\lim_{n\to\infty}f(x_n)=f(\lim_{n\to\infty} x_n)$ as long as the sequence does not tend to zero.</p>
|
607,862 | <p>Let $f$ be a continuous function. What is the maximum of $\int_0^1 fg$ among all continuous functions $g$ with $\int_0^1 |g| = 1$?</p>
| ncmathsadist | 4,154 | <p>Put $M = \|f\|_\infty$. Note that
$$\int_0^1 f(x) g(x)\, dx \le \|f\|_\infty \|g\|_1 = M.$$
Let $\epsilon > 0$. Then choose a point $x$ so $|f(x)| = M$. Wlog, we may assume
$f(x) = M$. Choose an interval $I$ so that $f\ge M - \epsilon$ on $I$. </p>
<p>Define $g$ to be $1/|I|$ on $I$ and $0$ off $[f > 0]$... |
332,993 | <p>How do I approach the problem?</p>
<blockquote>
<p>Q: Let $ \displaystyle z_{n+1} = \frac{1}{2} \left( z_n + \frac{1}{z_n} \right)$ where $ n = 0, 1, 2, \ldots $ and $\frac{-\pi}{2} < \arg (z_0) < \frac{\pi}{2} $. Prove that $\lim_{n\to \infty} z_n = 1$.</p>
</blockquote>
| Michael Hardy | 11,667 | <p>Let $f(z) = \frac12\left(z+\frac1z\right)$. Clearly $f(1)=1$ and $f'(1)=0$. So suppose $z=1+\Delta z$. Then
$$
f(z) = 1 +f'(1)\,\Delta z + \text{higher-degree terms in $\Delta z$},
$$
so $f(z)$ is closer to $1$ than $z$ is. You have an attractive fixed point.</p>
<p><b>Later edit:</b></p>
<p>Or put it this way... |
613,961 | <p>I got the following problem:</p>
<p>Let $V$ be a real vector space and let $q: V \to \mathbb R$ be a real quadratic form,<br/> Prove that if the set $L = \{v \in V | q(v) \ge 0\}$ forms a subspace of $V$
then q is definite (meaning $q$ is positive definite, positive semidefinite, negative definite or negative semi... | Jeremy Daniel | 115,164 | <p>Suppose by contradiction that there exists $x$ and $y$ such that $q(x) < 0 < q(y)$. Then for any real $ \lambda$, $q(x + \lambda y) = q(x) + \lambda^2 q(y) + 2\lambda B(x,y)$ where $B$ is the bilinear form associated to $q$. So when $\lambda$ is large, $q(x + \lambda y)$ is positive, so $x + \lambda y \in L$. ... |
666,503 | <p>How to isolate $x$ in this equation: $px+(\frac{b}{a})px=m$</p>
<blockquote>
<p>Blockquote</p>
</blockquote>
<p>And get $\frac{a}{a+b}*\frac{m}{p}$</p>
| Eleven-Eleven | 61,030 | <p>$$px+\left(\frac{b}{a}\right)px=m$$
$$\left(1+\frac{b}{a}\right)px=m$$
$$px=\frac{m}{1+\frac{b}{a}}=\frac{am}{a+b}$$
$$x=\frac{am}{p(a+b)}$$</p>
|
405,449 | <blockquote>
<p>Is the polynomial $x^{105} - 9$ reducible over $\mathbb{Z}$?</p>
</blockquote>
<p>This exercise I received on a test, and I didn't resolve it. I would be curious in any demonstration with explanations. Thanks!</p>
| Zev Chonoles | 264 | <p><strong>Hint:</strong> Make a <a href="https://en.wikipedia.org/wiki/Newton_polygon" rel="nofollow noreferrer">Newton polygon</a> for the prime $p=3$. Use the corollary at the top of page 2 in <a href="http://www.math.umn.edu/~garrett/m/number_theory/newton_polygon.pdf" rel="nofollow noreferrer">these notes by Paul ... |
228,437 | <p>The ODE in question: <code>y'' + 3y' + 2y = 8t + 8</code></p>
<p>But I get something like this for my solution:</p>
<p><a href="https://i.stack.imgur.com/6QFoV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6QFoV.png" alt="enter image description here" /></a></p>
<p>I also tried getting the solut... | Steffen Jaeschke | 61,643 | <p>"The ODE in question: y'' + 3y' + 2y = 8t + 8"</p>
<p>is a linear inhomogeneous ordinary differential equation with real constant coefficients. The inhomogeneity is a linear polynomial with constant real coefficients.</p>
<p>Solution:</p>
<pre><code>DSolve[y''[t] + 3 y'[t] + 2 y[t] == 8 t + 8, y, t]
(*{{y ... |
315,235 | <p>I am learning about vector-valued differential forms, including forms taking values in a Lie algebra. <a href="http://en.wikipedia.org/wiki/Vector-valued_differential_form#Lie_algebra-valued_forms">On Wikipedia</a> there is some explanation about these Lie algebra-valued forms, including the definition of the operat... | Olivier Bégassat | 11,258 | <p>A $\frak g$-valued differential form is , as far as I know, just a section $\alpha$ of the tensor product of the exterior power of the cotangent bundle $\Lambda^{\bullet}T^*M$ of some manifold $M$ with the trivial vector bundle $M\times\frak{g}$. As such, locally over some chart domain $U$, $\alpha$ can be cast in t... |
9,416 | <p>Say I pass 512 samples into my FFT</p>
<p>My microphone spits out data at 10KHz, so this represents 1/20s.</p>
<p>(So the lowest frequency FFT would pick up would be 40Hz).</p>
<p>The FFT will return an array of 512 frequency bins
- bin 0: [0 - 40Hz)
- bin 1: [40 - 80Hz)
etc</p>
<p>So if my original sound con... | Ben Voigt | 4,923 | <p>Remember that the FFT is circular. Inputs which contain an integer number of cycles will come out clean as a single point, in the corresponding bin. Those which do not, act as if they are multiplied by a rectangular pulse in the time domain, which creates convolution by a sinc function in the frequency domain. Si... |
9,416 | <p>Say I pass 512 samples into my FFT</p>
<p>My microphone spits out data at 10KHz, so this represents 1/20s.</p>
<p>(So the lowest frequency FFT would pick up would be 40Hz).</p>
<p>The FFT will return an array of 512 frequency bins
- bin 0: [0 - 40Hz)
- bin 1: [40 - 80Hz)
etc</p>
<p>So if my original sound con... | Sebastian Reichelt | 1,386 | <p>I just noticed this old question and thought I'd expand on J.M.'s comment (that is, what I think he/she was hinting at).</p>
<p>First of all, a small remark on the "bins" you talk about: The frequencies associated with the coefficients should be thought of as lying in the center of their "bin," and you're also off ... |
55,232 | <p>I'm looking for a concise way to show this:
$$\sum_{n=1}^{\infty}\frac{n}{10^n} = \sum_{n=1}^{\infty}\left(\left(\sum_{m=0}^{\infty}{10^{-m}}\right){10^{-n}}\right)$$
With this goal in mind:
$$\sum_{n=1}^{\infty}\left(\left(\sum_{m=0}^{\infty}{10^{-m}}\right){10^{-n}}\right) =
\sum_{n=1}^{\infty}\left(\left(\frac{... | anon | 11,763 | <p>Personally, I'd go the following route:
$$\sum_{n=1}^\infty\frac{n}{10^n}=\left(\sum_{n=1}^\infty\frac{1}{10^n}\right)+\left(\sum_{n=2}^\infty\frac{1}{10^n}\right)+\left(\sum_{n=3}^\infty\frac{1}{10^n}\right)+\cdots$$
$$=\left(\sum_{n=1}^\infty\frac{1}{10^n}\right)+\frac{1}{10}\left(\sum_{n=1}^\infty\frac{1}{10^n}\r... |
1,532,202 | <p>I want to find out $$\mathcal{L^{-1}}\{\frac{e^{-\sqrt{s+2}}}{s}\}$$
How do you find the inverse Laplace? </p>
<p>thanks</p>
| hbp | 131,476 | <p>Thank you for the interesting question. Here is a rather brute force solution, which may add a few steps to <a href="https://math.stackexchange.com/users/226665/jan-eerland">Jan Erland</a>'s solution.</p>
<p>First, let us recall that, if
$$
\mathcal L(f) = \int_0^\infty f(t) \, e^{-st} \, dt = F(s)
$$
Then
$$
\ma... |
587,198 | <p>I am having problems with this question, it would be wonderful if someone can help.</p>
<p>Given that $f(x)= x^2 + x - 3$</p>
<p>1) Find $f(x + h)$</p>
<p>2) Then express $f(x+h)-f(x)$ in its simplest form</p>
<p>3) Deduce $\lim\limits_{h->0}\dfrac{f(x+h)-f(x)}{h}$</p>
<p>Thanks for the help, i was stuck on ... | Mufasa | 49,003 | <p>if $2a^2=b^2$ it means $b$ must be even (because only an even number squared leads to an even number). So let $b=2m$ - this leads to:$$2a^2=(2m)^2=4m^2$$$$\therefore a^2=2m^2$$and hence $a$ must be even (for same reasons as above).</p>
<p>Thus both $a$ and $b$ must be even.</p>
|
906,332 | <blockquote>
<p>Prove $\ell_{ki}\ell_{kj}=\delta_{ij}$</p>
</blockquote>
<p>where $\{\hat{\mathbf{e}}_i\}$ and $\{\hat{\mathbf{e}}_i'\}$ are sets of orthonormal basis vectors for $i\in\{1,2,3\}$, $\ell$'s are the direction cosines such that $\ell_{ij}=\cos{(\hat{\mathbf{e}}_i',\hat{\mathbf{e}}_j)}=\hat{\mathbf{e}}_i... | BeaumontTaz | 147,480 | <p>Yeah, I needed to rewrite everything and start on a new piece of paper before I realized the EDITed reverse transformation was relevant.</p>
<p>The proof is as follows:
$$\begin{align*}
\ell_{ki}\ell_{kj}&=(\hat{\mathbf{e}}_k'\cdot\hat{\mathbf{e}}_i)\ell_{kj}\\
&=(\ell_{kj}\hat{\mathbf{e}}_k')\cdot\hat{\mat... |
906,332 | <blockquote>
<p>Prove $\ell_{ki}\ell_{kj}=\delta_{ij}$</p>
</blockquote>
<p>where $\{\hat{\mathbf{e}}_i\}$ and $\{\hat{\mathbf{e}}_i'\}$ are sets of orthonormal basis vectors for $i\in\{1,2,3\}$, $\ell$'s are the direction cosines such that $\ell_{ij}=\cos{(\hat{\mathbf{e}}_i',\hat{\mathbf{e}}_j)}=\hat{\mathbf{e}}_i... | user_of_math | 161,022 | <p>There is a simpler proof. Since you are transforming from one set of orthonormal Cartesian coordinates to another, your change of basis matrix $[l]$ is orthogonal (its transpose is also its inverse).</p>
<p>Thus, $[l][l]^T = [l]^T[l]=[I]$.</p>
<p>Clearly then, $l_{kj}l_{ki}$ = $l^T_{ik}l_{kj} = \delta_{ij}$, since... |
104,170 | <p>I am trying to solve a fundamental problem in analytical convective heat transfer: laminar free convection flow and heat transfer from a flat plate parallel to the direction of the generating body force.</p>
<p><strong>Brief History of the problem</strong></p>
<p>Effectively: a flat plate is vertical and parallel ... | Michael E2 | 4,999 | <p>The problem is with the default starting initial conditions used by the shooting method in <code>NDSolve</code>. The shooting method is where <code>FindRoot</code> is being used internally, so the OP's error message is a strong hint that this is the problem. Getting convergence in a nonlinear system can depend grea... |
4,546,415 | <p>Say <span class="math-container">$I = \mathbb{N} \setminus \{0, 1\}$</span> and</p>
<p><span class="math-container">$$A(n) = \left\{x \in \mathbb{R}\,\middle|\, −1−
\frac{1}{n}
< x \leq
\frac{1}{n}
\text{ or } 1−
\frac{1}{n} \leq x < 2−
\frac{1}{n}
\right\}$$</span>
with <span class="math-container">$n \in I$<... | Gino | 1,102,186 | <p>Actually, you can express your derivative in a more compact form.</p>
<p>Since</p>
<p><span class="math-container">$v(x)=A(x)(A(x)^{-1}v(x))$</span> (Eq. 1)</p>
<p>differentiating (Eq.1) w.r.t. <span class="math-container">$x$</span>, gives:</p>
<p><span class="math-container">$\frac{d}{dx}(v(x))=(\frac{d}{dx}A(x))A... |
2,158,369 | <p>prove that :
$$a,b>0\\,0<x<\pi/2$$
$$a\sqrt{\sin x}+b\sqrt{\cos x}≤(a^{4/3}+b^{4/3})^{3/4}$$
my try :</p>
<p>$$a\sqrt{\sin x}+b\sqrt{\cos x}=a(\sqrt{\sin x}+\frac{a}{b}\sqrt{\cos x})$$</p>
<p>$$\frac{a}{b}=\tan y$$</p>
<p>$$a\sqrt{\sin x}+b\sqrt{\cos x}=a(\sqrt{\sin x}+\tan y\sqrt{\cos x})$$</p>
<p>$$\... | Léreau | 351,999 | <p>Let <span class="math-container">$n := \vert V \vert$</span>.
Since <span class="math-container">$G$</span> is connected, there is a walk <span class="math-container">$\gamma$</span> passing through all <span class="math-container">$n$</span> points of <span class="math-container">$G$</span> and having at least leng... |
928,826 | <p>I have a function </p>
<p>$$f(x)=\frac{2x^2 - x - 1}{x^2 + 3x + 2}$$</p>
<p>from the interval $[0,\infty)$</p>
<p>The limit of this function is $2$. Is the range then simply from $f(0)$ to $2$, and if yes, would I write it as $[f(0],2]$ or $[f(0),2)$, i.e open brackets or closed? </p>
<p>Also, would i first nee... | amWhy | 9,003 | <p>You need the open bracket at $2$. And since $f(0) = -\frac 12$, the range of $f$ is given by $$\left[-\frac{1}{2}, 2\right)$$</p>
<p>Yes, since $f$ is monotonically increasing, $f(0) = -\frac 12$ is its greatest lower bound. So all you need here is to prove (establish) that it is monotonically increasing. Furthermo... |
2,189,818 | <p>I am having trouble finding the natural parameterization of these curves:</p>
<blockquote>
<p>$$\alpha(t)=\left(\sin^2\left(\frac{t}{\sqrt{2}}\right),\frac{1}{2}\sin \left(t\sqrt{2}\right), \left(\frac{t}{\sqrt{2}}\right)\right)$$</p>
</blockquote>
<p>The thing is when finding $$\|\alpha'(t)\|=\sqrt{\frac{3}{2}\... | Old Peter | 340,536 | <p>$$106202791239577=9996044^2+2506371^2$$</p>
<p>The target number is, as I expect you know, a Pythagorean prime <a href="https://en.wikipedia.org/wiki/Pythagorean_prime" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Pythagorean_prime</a>
As such, it has only one way to be expressed as the sum of two square... |
337,252 | <p>I'm guessing that the free group on an empty set is either the trivial group or isn't defined.
Some clarification would be appreciated.</p>
| Michael Hardy | 11,667 | <p>My guess is that would be a group with one element. There should be results about what direct products of groups have to do with sets of generators. What happens when you take the <s>direct</s> free product of a no generators and hence no relations, with a group with some other specified set of generators and rela... |
337,252 | <p>I'm guessing that the free group on an empty set is either the trivial group or isn't defined.
Some clarification would be appreciated.</p>
| Alexander Gruber | 12,952 | <p><em>Short version:</em> It's the trivial group. The only element is the empty word.</p>
<p><em>Long version:</em> To elaborate, we write the set of words in $A$ as $\mathfrak{W}_A$ (considered as the monoid generated by $A\cup A^{-1}$ under concatenation) and the free group on $A$ as $\mathfrak{F}_A$. $$\newcomma... |
337,252 | <p>I'm guessing that the free group on an empty set is either the trivial group or isn't defined.
Some clarification would be appreciated.</p>
| Martin Brandenburg | 1,650 | <p>Left adjoints preserve colimits, in particular the empty colimits, i.e. initial objects. In particular, the free group functor takes the initial set, i.e. $\emptyset$, to the initial group, i.e. the trivial group. Another example: The polynomial ring without variables is the base ring.</p>
<p>A must read: <a href="... |
3,362,916 | <p>I'm trying to graph <span class="math-container">$|x+y|+|x-y|=4$</span>. I rewrote the expression as follows to get a function that resembles the direction of unit vectors at <span class="math-container">$\pi/4$</span> to the horizontal axis (take it to be <span class="math-container">$x$</span>)<span class="math-co... | David K | 139,123 | <p>I think you dismiss the solution by cases too quickly.
Actually you only need to do one case, and the rest is developed by symmetry.</p>
<p>The easiest case is when <span class="math-container">$x+y\geq 0$</span> and <span class="math-container">$x - y \geq 0,$</span>
equivalently <span class="math-container">$x\ge... |
17,975 | <p>How to systematically classify Mathematica expressions? I can think of using <code>Head[]</code>, <code>Depth[]</code>, <code>Length[]</code>, and some special pattern based on the problems at hand. What other key words, or functions should I consider?</p>
<h2>Update</h2>
<p>I mostly want to group symbols by how... | s.s.o | 840 | <p>Did you considered looking at <a href="http://www.wolframscience.com/nksonline/toc.html" rel="nofollow">the book from S. Wolfram</a> a new kind of science. There he discuses some main <strong>principals and rules</strong> applied to mathematica in particular Chapter 11: The Notion of Computation and further chapters... |
17,975 | <p>How to systematically classify Mathematica expressions? I can think of using <code>Head[]</code>, <code>Depth[]</code>, <code>Length[]</code>, and some special pattern based on the problems at hand. What other key words, or functions should I consider?</p>
<h2>Update</h2>
<p>I mostly want to group symbols by how... | jVincent | 1,194 | <p>In your updated example, you would find the "classification" using your scheme by replacing the lowest level elements with a pattern based on their heads:</p>
<pre><code>classify[expression_] := Map[Blank[Head[#]] &, expression, {-1}]
</code></pre>
<p>Then you can apply this on template examples of the pattern... |
1,386,307 | <p>If you consider that you have a coin, head or tails, and let's say tails equals winning the lottery. If I participate in one such event, I may not get tails. It's roughly 50%. But if a hundred people are standing with a coin and I or them get to flip it, my chances of having gotten a tail after these ten attempts... | ignoramus | 155,096 | <p>Your chances of winning the lottery <strong>does</strong> increase if you participate in more lotteries. Say you particpate in the lottery where you have a 1 in 12 million chance of winning 1000 times. Then the probability that you don't win a single time is $$\biggl(1-\frac{1}{12000000}\biggr)^{1000} \approx 99.992... |
1,176,938 | <p>How do you show that $-1+(x-4)(x-3)(x-2)(x-1)$ is irreducible in $\mathbb{Q}$?</p>
<p>I don't think you can use the eisenstein criterion here</p>
| TorsionSquid | 202,777 | <p>Using the Gauss lemma as suggested, note that $1,2,3,4$ are clearly not roots of $p(x)$. Also, when $x\leq 0$ or $x\geq 5$ we have $p(x)\geq -1+24>0$. So there are no integer roots. So $p$ is irreducible over $\mathbb{Z}$ and hence $\mathbb{Q}$.</p>
|
2,869,442 | <blockquote>
<p>Check whether the series
$$\sum_{n=1}^{\infty}\int_0^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^2}\ dx$$
is convergent.</p>
</blockquote>
<p>I tried to sandwich the function by $\dfrac{1}{1+x^2}$ and $\dfrac{x}{1+x^2}$ , but this did not help at all.
Any other way of approaching?</p>
| Doug M | 317,162 | <p>When $x\in [0,1], \frac {\sqrt x}{2} \le\frac {\sqrt x}{1+x^2} \le \sqrt x$</p>
<p>Which means that $\frac 12\int_0^\frac1n \sqrt x\ dx \le \int_0^\frac1n \frac {\sqrt x}{1+x^2} \ dx \le \int_0^\frac1n \sqrt x \ dx$</p>
<p>if $\sum_\limits{n=1}^{\infty} \int_0^\frac1n \sqrt x \ dx $ converges then $\sum_\limits{n... |
2,225,650 | <p>Given that $\vec{a}$ and $\vec{b}$ are two non-zero vector. The two vectors form 4 resultant vectors such that $\vec{a} + 3\vec{b}$ and $2\vec{a} - 3\vec{b}$ are perpendicular, $\vec{a} - 4\vec{b}$ and $\vec{a} + 2\vec{b}$ are perpendicular. How can I find the angle between $\vec{a}$ and $\vec{b}$?</p>
<p>The answe... | user26872 | 26,872 | <p>Consider the function $\rho(\phi) = 1-d + d\tanh^2 c\phi$.
This has a dent of depth $d$ at $\phi=0$.
(There are many other possible functions $\rho(\phi)$.
For example, something of the form
$\rho(\phi) = (1-d+c^2\phi^2)/(1+c^2\phi^2)$
should also work well.)
The parameter $c$ is roughly the inverse angular widt... |
2,225,650 | <p>Given that $\vec{a}$ and $\vec{b}$ are two non-zero vector. The two vectors form 4 resultant vectors such that $\vec{a} + 3\vec{b}$ and $2\vec{a} - 3\vec{b}$ are perpendicular, $\vec{a} - 4\vec{b}$ and $\vec{a} + 2\vec{b}$ are perpendicular. How can I find the angle between $\vec{a}$ and $\vec{b}$?</p>
<p>The answe... | anderstood | 36,578 | <p>You have not been very precise on what you were exactly looking for, but this should serve as a good basis.</p>
<p>The idea is to use a <a href="https://upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gaussian_2d.png/300px-Gaussian_2d.png" rel="nofollow noreferrer">Gaussian</a> to make the dent. The number of den... |
2,291,310 | <p>I'm seeking an alternative proof of this result:</p>
<blockquote>
<p>Given $\triangle ABC$ with right angle at $A$. Point $I$ is the intersection of the three angle lines. (That is, $I$ is the incenter of $\triangle ABC$.) Prove that
$$|CI|^2=\frac12\left(\left(\;|BC|-|AB|\;\right)^2+|AC|^2\right)$$</p>
</bloc... | Blue | 409 | <p><a href="https://i.stack.imgur.com/1gMfOm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1gMfOm.png" alt="enter image description here"></a></p>
<p>Let the circle about $B$ through $C$ meet the extension of $\overline{AB}$ at the point $C^\prime$. By symmetry and a little angle chasing in isosce... |
2,919,841 | <blockquote>
<p><span class="math-container">$$\Large\bigcup\limits_{k\in\bigcup\limits_{i\in I}J_i}A_k=\bigcup\limits_{i\in I}\bigg(\bigcup\limits_{k\in J_i}A_k\bigg)$$</span></p>
</blockquote>
<hr />
<p><strong>My attempt:</strong></p>
<p><span class="math-container">$\large x\in\bigcup\limits_{k\in\bigcup\limits_{i\... | Peter Szilas | 408,605 | <p>$c>1$. $c^n= \exp (n\log c)$, where $\log c >0.$</p>
<p>$\dfrac{n^a}{\exp (n\log c)}$;</p>
<p>Set $b:=\dfrac{a}{\log c} >0$.</p>
<p>$(\dfrac{n^b}{\exp n})^{\log c}.$</p>
<p>Take the limit $n \rightarrow \infty$.</p>
|
2,231,487 | <p>In [Mathematical Logic] by Chiswell and Hodges, within the context of natural deduction and the language of propositions LP (basically like <a href="http://www.cs.cornell.edu/courses/cs3110/2011sp/lectures/lec13-logic/logic.htm" rel="nofollow noreferrer">here</a>) it is asked to show, by counter-example that a certa... | jadn | 70,766 | <p>$p_0$ could be: 'it is raining'. Then, while we can claim without any assumptions that '(it is raining) or (it is not raining)' is correct, it is clear that we cannot from that claim derive a claim, using no assumptions, that '(it is raining)' is correct, similarly for '(it is not raining)'.</p>
|
1,750,104 | <p>I've had this question in my exam, which most of my batch mates couldn't solve it.The question by the way is the Laplace Transform inverse of </p>
<p>$$\frac{\ln s}{(s+1)^2}$$</p>
<p>A Hint was also given, which includes the Laplace Transform of ln t.</p>
| Ashok Saini | 539,858 | <p>$$x(t) \rightleftharpoons X(s)$$</p>
<p>$$tx(t) \rightleftharpoons -\frac{dX(s)}{ds}$$</p>
<p>$$x_1(t) \rightleftharpoons ln(s)$$</p>
<p>$$tx_1(t) \rightleftharpoons -\frac{1}{s}$$</p>
<p>$$tx_1(t) = -u(t)$$</p>
<p>$$x_1(t) = -\frac{u(t)}{t}$$</p>
<p>$$x_2(t) \rightleftharpoons \frac{1}{(s+2)^2}$$</p>
<p>$$e^... |
19,285 | <p>Is anyone aware of Mathematica use/implementation of <a href="http://en.wikipedia.org/wiki/Random_forest">Random Forest</a> algorithm?</p>
| Seth Chandler | 5,775 | <p>I very much enjoy Dan's approach in part because it is so simple both in concept and implementation. I'm taking the liberty here of suggesting a few arguable improvements to his terrific code. For makeForest (a) the data is in the same format as is used in functions such as LinearModelFit (a simple array instead of ... |
3,053,386 | <p>This might be a very basic question for some of you. Indeed in <span class="math-container">$\textbf Z$</span>, it's very easy. For example, <span class="math-container">$\textbf Z / \langle 2 \rangle$</span> consists of <span class="math-container">$\langle 2 \rangle$</span> and <span class="math-container">$\langl... | nguyen quang do | 300,700 | <p>I'm afraid there is no general approach if not going through the ring of integers of the number field. Recall that for a number field <span class="math-container">$K$</span> of degree <span class="math-container">$n$</span> over <span class="math-container">$\mathbf Q$</span>, the norm <span class="math-container">... |
496,011 | <p>Suppose that $ p_n(t) $ is the probability of finding n particle at a time t. And the dynamics of the particle is described by this equation : </p>
<p>$$ \frac{d}{dt} p_n(t) = \lambda \Delta p_n(t) $$</p>
<p>Defining one - dimensional lattice translation operator $ E_m = e^{mk} $ with $ km - mk = 1 $ and $\Delta ... | Felix Marin | 85,343 | <p>\begin{align}
&\int x^{3}\,\sqrt{x^{2} + 1\,}\,{\rm d}x
=
\int x^{2}\,{\rm d}\left[{1 \over 3}\,\left(x^{2} + 1\right)^{3/2}\right]
\\[3mm]&=
x^{2}\,{1 \over 3}\left(x^{2} + 1\right)^{3/2}
-
\int{1 \over 3}\left(x^{2} + 1\right)^{3/2}
\,{\rm d}\left(x^{2} + 1\right)
\\[3mm]&=
{1 \over 3}\,x^{2}\left(x^{2... |
37,804 | <p>I'm trying to gain some intuition for the usefullness of the spectral theory for bounded self adjoint operators. I work in PDE and any interesting applications/examples I've ever encountered are concerning <em>compact operators</em> and <em>unbounded operators</em>. Here I have the examples of $-\Delta$, the laplaci... | user36539 | 36,539 | <p>Pseudodifferential operators are bounded (Theorem of Calderon-Vaillancourt) in contrast to Differential operators which are unbounded (they are defined on a dense subspace of $L^2$). We can try to use the spectral to obtain explicitely the spectral function $E(\lambda)$. </p>
<p>Another example is given by the pape... |
2,201,085 | <p>Let
$$x_{1},x_{2},x_{3},x_{5},x_{6}\ge 0$$ such that
$$x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}=1$$
Find the maximum of the value of
$$\sum_{i=1}^{6}x_{i}\;x_{i+1}\;x_{i+2}\;x_{i+3}$$
where
$$x_{7}=x_{1},\quad x_{8}=x_{2},\quad x_{9}=x_{3}\,.$$</p>
| Michael Rozenberg | 190,319 | <p>For $x_i=\frac{1}{6}$ we get $\frac{1}{216}$.</p>
<p>We'll prove that it's a maximal value.</p>
<p>Indeed, let $x_1=\min\{x_i\}$, $x_2=x_1+a$, $x_3=x_1+b$, $x_4=x_1+c$, $x_5=x_1+d$ and $x_6=x_1+e$.</p>
<p>Hence, $a$, $b$, $c$, $d$ and $e$ are non-negatives and we need to prove that:
$$216\sum_{i=1}^6x_ix_{i+1}x_{... |
203,114 | <p>If we have a pair of coordinates <span class="math-container">$(x,y)$</span>, let's say</p>
<pre><code>pt = {1,2}
</code></pre>
<p>then we can easily rotate the coordinates, by an angle <span class="math-container">$\theta$</span>, by using the rotation matrix</p>
<pre><code>R = {{Cos[\[Theta]], -Sin[\[Theta]]}, ... | Henrik Schumacher | 38,178 | <p>Maybe this way?</p>
<pre><code>data = {{1}, {-0.3, 1}, {2, -0.2}, {2}, {-2, 1}, {4, -2}, {3}, {1, 1}, {-0.2, -0.3}};
R = {{Cos[\[Theta]], -Sin[\[Theta]]}, {Sin[\[Theta]], Cos[\[Theta]]}};
data2 = data;
data2[[2 ;; ;; 3]] = data[[2 ;; ;; 3]].Transpose[R];
data2[[3 ;; ;; 3]] = data[[3 ;; ;; 3]].Transpose[R];
</code><... |
1,348,127 | <p>I'm struggling with this problem, because I'm not sure how to integrate $1/\ln(x)$</p>
<blockquote>
<p>Suppose that you have the following information about a function
$F(x)$:</p>
<p>$$F(0)=1, F(1)=2, F(2)=5$$ $$F'(x)=\frac1{\ln(x)}$$</p>
<p>Using the Fundamental Theorem of Calculus evaluate $$\int_0... | Daniel Fischer | 83,702 | <p>Since $\cos$ is an even function, you have in fact a telescoping series:</p>
<p>\begin{align}
\sum_{n = 1}^N \sin x\sin (nx) &= \frac{1}{2}\sum_{n = 1}^N \bigl(\cos\bigl((n-1)x\bigr) - \cos \bigl((n+1)x\bigr)\bigr)\\
&= \frac{1}{2}\bigl( 1 + \cos x - \cos (Nx) - \cos \bigl((N+1)x\bigr)\bigr).
\end{align}</p... |
3,794,507 | <p>On <span class="math-container">$(5),$</span> <span class="math-container">$(6),$</span> and <span class="math-container">$(7),$</span> what's the difference between <span class="math-container">$S^2$</span> and <span class="math-container">$\sigma_x^2$</span>?</p>
<p>Also, why does:</p>
<p><span class="math-contain... | Wim Nevelsteen | 799,896 | <p><span class="math-container">$\sigma$</span> is the population standard deviation of the random variable <span class="math-container">$X$</span>.</p>
<p><span class="math-container">$X_i$</span> represents the value of the i-th sample. If you have <span class="math-container">$n$</span> different such samples, the s... |
96,377 | <p>I have a polynomial with the coefficients of {a1, b1, b2}</p>
<pre><code>x = 1/8 (a1^4 E^(4 I τ ω) - 2 a1^2 E^( 2 I τ ω) (b2 Sqrt[1 - t] + b1 Sqrt[t])^2 + (b2 Sqrt[ 1 - t] + b1 Sqrt[t])^4);
</code></pre>
<p><a href="https://i.stack.imgur.com/6W2Ob.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com... | Jason B. | 9,490 | <p>I think this would do what you are asking for:</p>
<pre><code>Normal[Series[x, {a1, 0, 4}, {b1, 0, 4}, {b2, 0, 4}]]
</code></pre>
<p><a href="https://i.stack.imgur.com/If7gs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/If7gs.png" alt="enter image description here"></a></p>
<p>or in copyable ... |
222,480 | <p>How many $10$-digit numbers have two digits $1$, two digits $2$, three digits $3$, three digits $4$ so that between the two digits $1$ it has at least <strong>other two digits</strong> and between two digits $2$ it has at least <strong>other two digits</strong> (not necessarily distinct)? Thanks!</p>
| Community | -1 | <p>The motivation behind Frobenius method is to seek a power series solution to ordinary differential equations.</p>
<p>Let $y(x) = \displaystyle \sum_{n=0}^{\infty} a_n x^n$. Then we get that $$y'(x) = \sum_{n=0}^{\infty} na_n x^{n-1}$$
$$3xy'(x) = \sum_{n=0}^{\infty} 3na_n x^{n}$$
$$y''(x) = \sum_{n=0}^{\infty} n(n-... |
395,618 | <p>If n squares are randomly removed from a $p \ \cdot \ q$ chessboard, what will be the expected number of pieces the chessboard is divided into? </p>
<p>Can anybody please provide how can I approach the problem? There are numerous cases and when I go through case consideration it becomes extremely complex.</p>
| user78391 | 78,391 | <p>2 = = 2 piece
3 = (3+1)/2 = 2 piece
4 = = 2 piece
5 = (5+1)/2 = 3 piece
6 = 3 piece
7 = (7+1)/2 = 4 piece
8 = = 4 piece</p>
<p>Series = (n+1)/2 pieces</p>
<p>Regards,
Yuvaraj</p>
|
33,389 | <p>Consider Schrödinger's <em>time-independent</em> equation
$$
-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi.
$$
In typical examples, the potential $V(x)$ has discontinuities, called <em>potential jumps</em>.</p>
<p>Outside these discontinuities of the potential, the wave function is required to be twice differentiable... | José Figueroa-O'Farrill | 394 | <p>To answer your first question:</p>
<p>Actually the assumption is <em>not</em> that the wave function and its derivative are continuous. That follows from the Schrödinger equation once you make the assumption that the probability amplitude $\langle \psi|\psi\rangle$ remains finite. That is the physical assumption.... |
33,389 | <p>Consider Schrödinger's <em>time-independent</em> equation
$$
-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi.
$$
In typical examples, the potential $V(x)$ has discontinuities, called <em>potential jumps</em>.</p>
<p>Outside these discontinuities of the potential, the wave function is required to be twice differentiable... | Jiahao Chen | 1,674 | <p>Here is a tangential response to your first question: sometimes these discontinuities do have physical significance and are not just issues of mathematical trickery surrounding pathological cases. Wavefunctions for molecular Hamiltonians become pointy where the atomic nuclei lie, which indicate places where the 1/r ... |
4,551,407 | <p>Here's the question:<br />
If we have m loaves of bread and want to divide them between n people equally what is the minimum number of cuts we should make?<br />
example:<br />
3 loaves of bread and 15 people the answer is 12 cuts.<br />
6 loaves of bread and 10 people the answer is 8 cuts.</p>
<p>for example 1, I f... | Kevin Dietrich | 1,103,878 | <p>If you say that <span class="math-container">$y = f(x)$</span> you can solve it like this (homogeneous):</p>
<p><em>step 2: Assume a solution to this Euler-Cauchy equation will be proportional to for some constant <span class="math-container">$λ$</span>. Substitute <span class="math-container">$y(x) := x^{λ}$</span>... |
4,551,407 | <p>Here's the question:<br />
If we have m loaves of bread and want to divide them between n people equally what is the minimum number of cuts we should make?<br />
example:<br />
3 loaves of bread and 15 people the answer is 12 cuts.<br />
6 loaves of bread and 10 people the answer is 8 cuts.</p>
<p>for example 1, I f... | user | 1,053,451 | <p>Treat it as if it's a quadratic and you're searching for roots (this is not true, it's a second order ODE, but you'll notice that the following method feels familiar to you:)</p>
<p>Rewrite it like this:
<span class="math-container">$$ x^2y^{(2)}+4xy^{(1)}+2y-f(x)=0. $$</span></p>
<p>Notice how this is similar to <s... |
2,432,213 | <p>I am having a really hard time understanding this problem. I know that for uniqueness we need that the derivative is continuous and that the partial derivative is continuous. I also know that the lipschitz condition gives continuity. I can't figure out what to do with this problem though. </p>
<p><a href="https://i... | H. H. Rugh | 355,946 | <p>The problem has some inherent unboundedness in the way it is posed. You need to get rid of that in order to pursue. Some suggestions:</p>
<p>Pick first $r_0>0$ arbitrary and note that by compactness + continuity, $g([0,r_0])=[a,b]$ is a compact subset of $(-\infty,r_0]$. Therefore,
$$ R = \sup \left\{ |\phi(u)| ... |
4,073,821 | <p>If <span class="math-container">$\sum a_n$</span> converges, then does <span class="math-container">$\sum |a_n|$</span> converge as well? This is the same as "absolute convergence", where if <span class="math-container">$\sum a_n$</span> converges, then <span class="math-container">$\sum|a_n|$</span> might... | Community | -1 | <p>Assume a series of positive terms <span class="math-container">$a_n$</span>. The corresponding alternating series <span class="math-container">$a_1-a_2+a_3-\cdots$</span> can be seen as the series with terms <span class="math-container">$b_1=a_1-a_2,b_2=a_3-a_4,b_3=a_5-a_6,\cdots$</span> As those terms are differenc... |
2,614,127 | <p>I have been trying to show the statement below using the $AC$ but I am starting to think that it is not strong enough to do it. </p>
<p><strong>Context:</strong> Let $\Gamma$ be an uncountable linearly ordered set with a smallest element (not necessarily well-ordered). </p>
<p>For each $\alpha\in\Gamma$, let $C_\a... | Asaf Karagila | 622 | <p>The answer is negative. This is not provable, even when assuming the axiom of choice. Even under the assumption that $\Gamma$ is a well-ordered set.</p>
<p>Let $T$ be a tree of height $\omega_1$ without a branch (either an Aronszajn tree, assuming choice; or any counterexample to $\sf DC_{\omega_1}$ otherwise).</p>... |
19,373 | <p>I posted this question earlier today on the Mathematics site (<a href="https://math.stackexchange.com/q/3988907/96384">https://math.stackexchange.com/q/3988907/96384</a>), but was advised it would be better here.</p>
<p>I had a heated argument with someone online who claimed to be a school mathematics teacher of man... | practical man | 15,302 | <p><em>IF</em> the horse ride were 22.00000000 miles then the other person would be right.</p>
<p>Else if it were 22 miles then you should round the answer to zero decimal places.</p>
<p>Some people are illogically pedantic without any rational reason for what they promulgate.</p>
|
747,519 | <p><img src="https://i.stack.imgur.com/jYzfz.png" alt="enter image description here"></p>
<p>I tried this problem on my own, but got 1 out of 5. Now we are supposed to find someone to help us. Here is what I did:</p>
<p>Let $f:[a,b] \rightarrow \mathbb{R}$ be continuous on a closed interval $I$ with $a,b \in I$, $a \... | Patrick Stevens | 259,262 | <p>Another way of showing "closed", because it's useful to be able to switch between the various definitions of these concepts: recall that continuous functions preserve the convergence of sequences, and that a closed set is precisely one which contains all its limit points.</p>
<p>Let $[f(c_i)]$ be a sequence in $f([... |
613,105 | <p>I was observing some nice examples of equalities containing the numbers $1,2,3$ like $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi$ and $\log 1+\log 2+ \log 3=\log (1+2+3)$. I found out this only happens because $1+2+3=1*2*3=6$.<br> I wanted to find other examples in small numbers, but I failed. How can we find all of the s... | Chris Taylor | 4,873 | <p>If $a=0$ then you require $b+c=0$ and hence $b=c=0$.</p>
<p>Note that you can assume $a\leq b \leq c$. If $a, b, c \geq 2$ then $abc \geq 4c > c + b + a$. Hence at least one of $a,b,c$ is equal to $1$.</p>
<p>Wlog assume $a=1$, and look for solutions to $b+c+1 = bc$. If $b,c\geq 3$ then $bc \geq 3c > b + c +... |
794,389 | <p>Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2$ squares, with sides $1, 2, 3\ldots n^2$ ($n$ odd). Which seems like it would work with ANY odd $n$. It's so simple, surely it's well-known, but I haven't seen it in my (brief) web travels. I ha... | TonyK | 1,508 | <p>There is a big difference between "arbitrarily large" and "infinite". This example shows the difference clearly:</p>
<blockquote>
<p>For any positive integer $n$, there exists a strictly decreasing
sequence of positive integers of length $n$.</p>
</blockquote>
<p>True, obviously. But this is false:</p>
<block... |
1,815,662 | <blockquote>
<p>Prove that $X^4+X^3+X^2+X+1$ is irreducible in $\mathbb{Q}[X]$, but it has two different irreducible factors in $\mathbb{R}[X]$.</p>
</blockquote>
<p>I've tried to use the cyclotomic polynomial as:
$$X^5-1=(X-1)(X^4+X^3+X^2+X+1)$$</p>
<p>So I have that my polynomial is
$$\frac{X^5-1}{X-1}$$ and now... | Community | -1 | <p>Every polynomial splits completely over the complexes. There are only three possibitilies:</p>
<ul>
<li>There are four real roots</li>
<li>There are two real roots and one pair of complex conjugate roots</li>
<li>There are two pairs of complex conjugate roots</li>
</ul>
<p>The roots of the polynomial are fifth roo... |
1,815,662 | <blockquote>
<p>Prove that $X^4+X^3+X^2+X+1$ is irreducible in $\mathbb{Q}[X]$, but it has two different irreducible factors in $\mathbb{R}[X]$.</p>
</blockquote>
<p>I've tried to use the cyclotomic polynomial as:
$$X^5-1=(X-1)(X^4+X^3+X^2+X+1)$$</p>
<p>So I have that my polynomial is
$$\frac{X^5-1}{X-1}$$ and now... | Jyrki Lahtonen | 11,619 | <p>A different route to the factorization over the reals (obviously the end result is same as in Egreg's post, but I give the factors explicitly).</p>
<p>Let $p(x)$ be your polynomial. By a direct calculation we see that
$$
(x^2+\frac x2+1)^2=x^4+x^3+\frac94x^2+x+1=p(x)+\frac54 x^2.
$$
This calculation is aided by pal... |
167,575 | <p>I have 6 sets of 4D points. Here is an example of one set :</p>
<pre><code>{{30., 5., 111.925, 113.569}, {30., 7.5, 114.7, 158.286}, {30., 10., 115.625, 206.023},
{30., 12.5, 115.625, 257.528}, {30., 15., 117.475, 294.663}, {30., 17.5, 119.325, 328.03},
{30., 20., 121.175, 357.982}, {30., 22.5, 122.1, 393.646}, {... | OkkesDulgerci | 23,291 | <p>Just join all your data and use J.M's code.</p>
<pre><code>data = {{30., 5., 111.925, 113.569}, {30., 7.5, 114.7, 158.286}, {30.,
10., 115.625, 206.023}, {30., 12.5, 115.625, 257.528}, {30., 15.,
117.475, 294.663}, {30., 17.5, 119.325, 328.03}, {30., 20.,
121.175, 357.982}, {30., 22.5, 122.1, 393.646... |
3,443,082 | <p>Find all the positive integral solutions of, <span class="math-container">$\tan^{-1}x+\cos^{-1}\dfrac{y}{\sqrt{y^2+1}}=\sin^{-1}\dfrac{3}{\sqrt{10}}$</span></p>
<p>Assuming <span class="math-container">$x\ge1,y\ge1$</span> as we have to find positive integral solutions of <span class="math-container">$(x,y)$</span>... | lab bhattacharjee | 33,337 | <p>Another way</p>
<p>As we need <span class="math-container">$x,y>0$</span></p>
<p>If <span class="math-container">$x/y<1$</span></p>
<p><span class="math-container">$$\tan^{-1}x+\tan^{-1}(1/y)=\tan^{-1}\dfrac{xy+1}{y-x}$$</span></p>
<p><span class="math-container">$$xy+1=3(y-x)$$</span></p>
<p><span class=... |
3,167,571 | <p>Let consider a square <span class="math-container">$10\times 10$</span> and write in the every unit square the numbers from <span class="math-container">$1$</span> to <span class="math-container">$100$</span> such that every two consecutive numbers are in squares which has a... | nonuser | 463,553 | <p>Well you could try with Am-Gm inequality:</p>
<p><span class="math-container">$$ 5\sqrt[5]{p_1p_2...p_5x_1(x_2-x_1)...(x_5-x_4)}\leq 1$$</span></p>
<p>So <span class="math-container">$$p_1p_2...p_5 \leq {1\over 5^5x_1(x_2-x_1)...(x_5-x_4)}$$</span></p>
<p>and this value is achivable if <span class="math-container... |
65,059 | <p>I have two points ($P_1$ & $P_2$) with their coordinates given in two different frames of reference ($A$ & $B$). Given these, what I'd like to do is derive the transformation to be able to transform any point $P$ ssfrom one to the other.</p>
<p>There is no third point, but there <em>is</em> an extra constra... | Benjol | 16,163 | <p>OK, I've been thinking about this (like an engineer, not a mathematician), and here's my (half-baked) take:</p>
<p>I take Frame A, and translate it (TA) such that it's origin is at P1, then rotate it (RA) around Z and Y such that P2 is on the X axis: this gives me a new Frame A'.</p>
<p>I do the <em>same</em> thin... |
3,340,093 | <p>Is the following statement true?</p>
<blockquote>
<p>Two real numbers a and b are equal iff for every ε > 0, |a − b| < ε.</p>
</blockquote>
<p>I got that if a and b are equal then |a-b|=0 which is less than ε.
But I'm not sure if the converse also holds.</p>
| Community | -1 | <p>This is outright proven in <a href="https://rads.stackoverflow.com/amzn/click/com/1493927116" rel="nofollow noreferrer" rel="nofollow noreferrer"><em>Understanding Analysis</em> (2016 2 edn)</a>. pp 9 - 10.</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/q2dvi.jpg" rel="nofollow noreferrer"><img src="https... |
1,199,746 | <p>Let $f : (−\infty,0) → \mathbb{R}$ be the function given by $f(x) = \frac{x}{|x|}$. Use the $\epsilon -\delta$ definition of a $\lim\limits_{x \to 0^-} f(x) = -1.$</p>
<p>Workings:</p>
<p>Informal Thinking:
We want $|f(x) - L| < \epsilon$</p>
<p>$\left|\frac{x}{|x|} - -1\right| < \epsilon$</p>
<p>$\left|\f... | graydad | 166,967 | <p>Since you are approaching $x$ from the left, that means $x<0$. As such, we have $|x| = -x$. So, when you get to your third line we have $$\left|\frac{x + |x|}{|x|} \right|=\left|\frac{x + (-x)}{|x|} \right| = 0 < \varepsilon$$ The inequality clearly holds for all $x<0$ and for all $\varepsilon>0$. Can yo... |
3,700,938 | <p>I know that the equations are equivalent by doing the math with the same value for x, but I don't understand the rules for changing orders or operations.<br>
When it is not the first addition or subtraction happening in the equation, parentheses make the addition subtraction and vice versa? Are there any other rules... | Prime Mover | 466,895 | <p>Let's simplify it, and suppose you have <span class="math-container">$x^2 - x - 1$</span>.</p>
<p>Examine what you are doing. You start with <span class="math-container">$x^2$</span>.</p>
<p>Then you subtract <span class="math-container">$x$</span> from <span class="math-container">$x^2$</span>.</p>
<p>Then you s... |
1,319,767 | <p>If we know that $\frac{2^n}{n!}>0$ for every $n\in \mathbb{N}$ and $$\frac{2^n}{n!}=\frac{2}{1}\frac{2}{2}...\frac{2}{n}$$ how to bound this sequence above?</p>
| Atvin | 215,617 | <p><strong>Hint:</strong> <a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow">Stirling's approximation</a></p>
|
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