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 |
|---|---|---|---|---|
1,489,078 | <p>Let $P(S)$ denotes the power set of set $S$. Which of the following is always true$?$</p>
<ol>
<li>$P(P(S)) = P(S)$</li>
<li>$P(S) ∩ P(P(S)) = \{ Ø \}$</li>
<li>$P(S) ∩ S = P(S)$</li>
<li>$S ∉ P(S)$</li>
</ol>
<hr>
<p>I try to explain $:$
If $S$ is the set $\{x, y, z \},$ then the subsets of S are:</p>
<p>${}$ (... | hmakholm left over Monica | 14,366 | <p>No, $\{\}$ and $\{\{\}\}$ are two different sets. You can see that by noting that there's something that is an element of one but not of the other:</p>
<p>$$ \{\}\notin \{\} \qquad\text{but}\qquad \{\}\in\{\{\}\} $$</p>
<p>$\{\}$ is a set that <em>has no elements</em>. It is the same set that is also notated $\var... |
3,624,953 | <p>Let <span class="math-container">$(X_t)_{t \in [0, T]}$</span> be a continuous stochastic process with paths which are a.s. continuous, the underlying space of which is irrelevant but is well defined. Let <span class="math-container">$a$</span> be a constant
Define two stopping times
<span class="math-container">$$\... | Surb | 154,545 | <p>Yes. Indeed, let <span class="math-container">$\omega \in \{X_{\tau_1}>a\}$</span>. In particular, <span class="math-container">$X_{\tau_1(\omega )-h}\leq a$</span>. Therefore, <span class="math-container">$$\lim_{h\to 0}X_{\tau_1(\omega )-h}=X_{\tau_1^-(\omega )}\leq a<X_{\tau_1(\omega )}.$$</span></p>
<p>Si... |
92,613 | <p>This is a question in Pinter's <em>A Book of Abstract Algebra</em>.</p>
<blockquote>
<p>Let $S=\{g\in G\mid \operatorname{ord}(g)=p\}$. Prove the order of $S$ is a multiple of $p-1$.</p>
</blockquote>
<p>In his solution Pinter says $a \in S$ implies that $a$ generates a subgroup with $p-1$ elements. Shouldn't th... | Dan Petersen | 677 | <p>No, you are right, every element $a \in S$ generates a subgroup with $p$ elements. However, only $p-1$ of those will lie in the set $S$, which I guess is what Pinter means.</p>
|
1,123,247 | <p>let $a_n$ be a sequence of real numbers such that the series $\sum |a_n|^2$ is convergent.Find range of $p$ such that the series $\sum |a_n|^p$ is convergent.</p>
<p>My try:</p>
<p>To show the series it is necessary to show that sequence of partial sums of $\sum |a_n|^p$ is bounded.</p>
<p>Consider $S_n=|a_1|^p+|... | Cm7F7Bb | 23,249 | <p>Suppose that $\ell_q$ is the space of real sequences $x=\{x_n\}$ such that $\sum|x_n|^q<\infty$ with the norm
$$
\|x\|_q=\biggl(\sum|x_n|^q\biggr)^{1/q}
$$
for $q\ge1$. We have that $\|x\|_p\le\|x\|_q$ for $1\le q\le p<\infty$ (see <a href="https://math.stackexchange.com/questions/69125/inequality-between-ellp... |
102,932 | <p>This is a naive question but I hope that the answers will be educational. When is it the case that a finitely presented group $G$ admits a faithful $2$-dimensional complex representation, e.g. an embedding into $\text{GL}_2(\mathbb{C})$? (I am mostly interested in sufficient conditions.) </p>
<p>I think I can figur... | Geoff Robinson | 14,450 | <p>I'm not sure whether it is worth remarking on, but if $G$ is any irreducible subgroup of ${\rm GL}(2,\mathbb{C}),$ its non-central Abelian subgroups have some very restrictive properties:
If $A$ is a non-central Abelian subgroup of $G$, then $C_{G}(A)$ is also Abelian and $[N_{G}(A):C_{G}(A)] \leq 2.$</p>
|
3,029,519 | <p>I have a variable, <span class="math-container">$X$</span>, which is normally distributed</p>
<p><span class="math-container">$$X \sim \mathcal{N}(\mu, \sigma^2)$$</span></p>
<p>I also have an event, <span class="math-container">$A$</span>, which measures the probability that <span class="math-container">$X$</span... | lab bhattacharjee | 33,337 | <p>I think it is simpler to utilize binomial identities:</p>
<p><span class="math-container">$$\dfrac{\binom nk}{\binom{k+3}k}=\dfrac{n!3!}{(n-k)!(k+3)!}=\dfrac6{(n+1)(n+2)(n+3)}\binom{n+3}{k+3}$$</span></p>
<p><span class="math-container">$$\sum_{k=0}^n\dfrac{\binom nk}{\binom{k+3}k}=\dfrac6{(-1)^3(n+1)(n+2)(n+3)}\s... |
2,263,520 | <p>I'm trying to understand how to simplify summations. My text says that: $$\sum_{i=1}^{\frac{n}{2}} \sum_{j=i}^{n-i} \sum_{k=1}^{j} 1 = \frac{n^3}{8}$$</p>
<p>But does not explain how to get to the right-hand side.</p>
<p>I think the above nested summation evaluates to $\sum_{i=1}^{n/2} \left[\sum_{j=i}^{n-i} j = i... | Ng Chung Tak | 299,599 | <p>\begin{align}
\sum_{k=1}^{j} 1 &= j \\
\sum_{j=i}^{n-i} j
&= \frac{n(n-2i+1)}{2} \\
S &= \sum_{i=1}^{n/2} \frac{n(n-2i+1)}{2} \\
&= \frac{n(n+1)}{2} \sum_{i=1}^{n/2} 1-
n \sum_{i=1}^{n/2} i
\end{align}</p>
<ul>
<li><p>If $n=2m$,
\begin{align}
S &= \frac{2m(2m+1)}{2} \times m-2m\... |
2,263,520 | <p>I'm trying to understand how to simplify summations. My text says that: $$\sum_{i=1}^{\frac{n}{2}} \sum_{j=i}^{n-i} \sum_{k=1}^{j} 1 = \frac{n^3}{8}$$</p>
<p>But does not explain how to get to the right-hand side.</p>
<p>I think the above nested summation evaluates to $\sum_{i=1}^{n/2} \left[\sum_{j=i}^{n-i} j = i... | Hypergeometricx | 168,053 | <p>Exploit the symmetry.</p>
<p>Put $n=2m$ (assuming $n$ is even):</p>
<p>$$\begin{align}
S
=\sum_{i=1}^{\frac n2}\sum_{j=i}^{n-i}\sum_{k=1}^j 1
=\sum_{i=1}^{\frac n2}\sum_{j=i}^{n-i}j
&=\sum_{i=1}^m\sum_{j=i}^{2m-i}j&&(n=2m)\tag{1}\\
&=\sum_{i=1}^m\sum_{j=i}^{2m-i}(2m-j)&&(j\leftarrow 2m-j)\t... |
1,818,557 | <p>I will be teaching some "topology" to high school students. I was wondering how to explain to such a school student that on a sphere the shortest path between 2 points is given by a great circle?</p>
<p>Also, how to explain that if they lived on a sphere they would have no notion of "above" or "below"? I cannot fin... | Narasimham | 95,860 | <p>There are two ways I tried with students. </p>
<p>Case 1. Equator of ball</p>
<p>On a plastic ball toy carefully tie a string around any great circle, ( use a smal cellulose tape/tab if needed, to prevent side slippage ,) for exactly one rotation. Make the string taut by pulling in opposite directions. The ball w... |
1,818,557 | <p>I will be teaching some "topology" to high school students. I was wondering how to explain to such a school student that on a sphere the shortest path between 2 points is given by a great circle?</p>
<p>Also, how to explain that if they lived on a sphere they would have no notion of "above" or "below"? I cannot fin... | zyx | 14,120 | <p>Set up lines of latitude and longitude on the sphere so that the two points, call them $A$ and $B$, are on the same meridian.</p>
<p>Then any motion from $A$ to $B$ can be "tracked" along the meridian by taking a moving point on the meridian that has the same latitude as the point moving from $A$ to $B$. The moti... |
1,812,525 | <p>Let $T:H \to H$ be defined as $Tx=\sum_{n=1}^{\infty} \lambda_n \langle x,\varphi _n \rangle \varphi _n$, given that $\{\varphi _n\}_{n=1}^\infty$ is an orthonormal sequence (not necessarily a basis) and $\{\lambda_n\}_{n=1}^\infty$ is a sequence of numbers (which may be complex if the Hilbert space is complex).</p... | walkar | 98,077 | <p>HINT: You need to use the fact that $\phi_n$ is orthonormal. That is, $\left\langle \phi_n,\phi_m \right\rangle = \left\lbrace \begin{array}{cc}
0 & n \neq m \\
1 & n = m
\end{array} \right.$. Start by supposing that $f \in \ker(T)$ so that $T(f) = \sum_{n=1}^\infty \lambda_n \left\langle f,\phi_n \right\ra... |
702,506 | <p>We have an exam in $3$ hours and I need help how to solve such trigonometric equations for intervals.</p>
<p>How to solve</p>
<p>$$\sin x - \cos x = -1$$</p>
<p>for the interval $(0, 2\pi)$.</p>
| Community | -1 | <p>$$\sin x-\cos x=-1\\
\implies (\sin x-\cos x)^2=\sin^2x+\cos^2x-2\sin x\cos x=1+\sin(2x)=1$$
So,
$$\sin(2x)=0$$
Therefore,
$$x=3\pi/2,\mbox{ as }x\in(0,2\pi)$$</p>
|
702,506 | <p>We have an exam in $3$ hours and I need help how to solve such trigonometric equations for intervals.</p>
<p>How to solve</p>
<p>$$\sin x - \cos x = -1$$</p>
<p>for the interval $(0, 2\pi)$.</p>
| lab bhattacharjee | 33,337 | <p><strong>Method</strong> $\#1$ </p>
<p>Avoid squaring which immediately introduces <a href="http://en.wikipedia.org/wiki/Extraneous_and_missing_solutions" rel="nofollow">extraneous roots</a> which demand exclusion </p>
<p>We have $\displaystyle\sin x-\cos x=-1$</p>
<p>$$\iff\sin x=-(1-\cos x)\iff2\sin\frac x2\cos\... |
2,380,880 | <p>The following question was asked in an exam:<br>
Consider the problem:<br>
Maximize $2y_1+3y_2+5y_3+4y_4$<br>
subject to<br>
$y_1+y_2\leq 1,$ $y_2+y_3\leq 1,$<br>
$y_4+y_1\leq 1,$ $y_3+y_4\leq 1$ and $y_i\geq 0$ for i=1,2,3,4.<br>
Then the optimum value is<br>
1. equal to 8<br>
2. between 8 and 9<br>
3. greater tha... | Michael Hartley | 96,763 | <p>You could let $z_1=y_1+y_2$, $z_2=y_2+y_3$ and $z_3=y_3+y_4$. Then the problem becomes:</p>
<p>Minimise $2z_1+z_2+4z_3$ subject to $z_1\leq 1$, $z_2\leq 1$, $z_3\leq 1$, $z_1-z_2+z_3\leq 1$, </p>
<p>Then, by inspection $z_1=z_2=z_3=1$ satisfies the constraints and attains the maximum. </p>
|
198,298 | <p>Is there a tool for editing the spelling dictionary? Now that <em>Mathematica</em> 12 does spellchecking on the fly, I've been using it. And of course, as I was wondering how I might delete a word accidentally added to the dictionary, I accidentally added a misspelled word to the dictionary. I'm sure I can hunt it ... | Alexey Popkov | 280 | <p><a href="https://mathematica.stackexchange.com/a/198308/280">Answer by ciao</a> shows a way to edit user-supplied list of correct words via GUI. The same can be done via setting <code>$FrontEnd</code> option:</p>
<pre><code>Options[$FrontEnd, SpellingDictionaries]
</code></pre>
<blockquote>
<pre><code>{SpellingDic... |
387,365 | <p>I have a physics project, and I have to develop an argument, but am not allowed to use phrases like "From the graph you can tell..."</p>
<p>How can the nature of the graph be determined manually, e.g. finding that it is expontential rather than quadratic?</p>
| Dale M | 55,635 | <p>I assume by disjoint squares you mean squares made with 2 cuts, each parallel to an axis and that the leftover rectangles are thrown away.</p>
<p>A PDF uniform on $x$ when $y=y_0$ and 0 otherwise would give Alice nothing.</p>
<p>In light of @RossMillikan comment</p>
<p>$$P(x,y)=\begin{cases}
\frac{1}{5}&[0,0]... |
387,365 | <p>I have a physics project, and I have to develop an argument, but am not allowed to use phrases like "From the graph you can tell..."</p>
<p>How can the nature of the graph be determined manually, e.g. finding that it is expontential rather than quadratic?</p>
| Erel Segal-Halevi | 29,780 | <p>I think Alice can always assure herself at least $1 \over 4$ cdf, in the following way.</p>
<p>First, in each of the 4 corners, mark a square that contains $1 \over 4$ cdf. Since the pdf is finite, it is always possible to construct such a square, by starting from the corner and increasing the square gradually, unt... |
2,783,129 | <p>How many different value of x from 0° to 180° for the equation $(2\sin x-1)(\cos x+1) = 0$?</p>
<p>The solution shows that one of these is true:</p>
<p>$\sin x = \frac12$ and thus $x = 30^\circ$ or $120^\circ$ </p>
<p>$\cos x = -1$ and thus $x = 180^\circ$</p>
<p><strong>Question:</strong> Inserting the $\arcsin... | Christopher Marley | 510,133 | <p>If $ab = 0$, then either $a=0$ or $b=0$.<br>
Given $(2\sin x - 1)(\cos x + 1) = 0$, we can separate these and solve separately.</p>
<p>So... $\ \ \ \ \ 2\sin x -1 = 0$<br>
$\Rightarrow \sin x = \frac12 \Rightarrow x = 30^\circ, 150^\circ$<br>
We get those two values because $\sin x$ is symmetrical across the $y$-ax... |
1,217,771 | <p>Let $x,y \in R$.
If $0 \leq y < x$ for all $x > 0$, then $y=0$.</p>
<p>Proof by contradiction: </p>
<p>Assume the opposite that is; "If $0 \leq y < x$ for all $x > 0$, then $y\neq0$".
Subtract $x$ from each part of the inequality to get,
$0-x \leq y-x < 0$
Th... | Olivier Oloa | 118,798 | <p>You may just observe that, as $x\to \infty$:
$$
\frac{1+\sqrt{x}}{\sqrt{x+\sqrt{x}}+\sqrt{x-1}}\sim \frac{\sqrt{x}\left(1+\frac1{\sqrt{x}}\right)}{\sqrt{x}\left(\sqrt{1+\frac{1}{\sqrt{x}}}\right)+\sqrt{x}\left(\sqrt{1-\frac{1}{x}}\right)}\sim \frac12.
$$</p>
|
1,217,771 | <p>Let $x,y \in R$.
If $0 \leq y < x$ for all $x > 0$, then $y=0$.</p>
<p>Proof by contradiction: </p>
<p>Assume the opposite that is; "If $0 \leq y < x$ for all $x > 0$, then $y\neq0$".
Subtract $x$ from each part of the inequality to get,
$0-x \leq y-x < 0$
Th... | davidlowryduda | 9,754 | <p>Let's start from your last line:
$$\begin{align}
\lim_{x\to+\infty}\frac{1+\sqrt{x}}{\sqrt{x+\sqrt{x}}+\sqrt{x-1}} &= \lim \frac{\sqrt x}{\sqrt x} \frac{\frac{1}{\sqrt x} + 1}{\sqrt{1 + \frac{1}{\sqrt x}} + \sqrt{1 - \frac{1}{ x}}} \\
&= \frac{1}{1 + 1} = \frac{1}{2}
\end{align}$$
where we note that everywhe... |
138,866 | <p>I have data in a csv file. The first row has labels, and the first column, too.</p>
<pre><code>Datos = Import["C:\\Users\\jodom\\Desktop\\Data.csv"]
</code></pre>
<p>Tha data in the csv file is that:</p>
<pre><code>{{"No", "Vol", "Vel"}, {1, 500, 45}, {2, 700, 67}, {3, 350, 87}, {4,
123, 23}, {5, 587, 45}, {6, 4... | Ali Hashmi | 27,331 | <pre><code>data = {{"No", "Vol", "Vel"}, {1, 500, 45}, {2, 700, 67}, {3, 350,
87}, {4, 123, 23}, {5, 587, 45}, {6, 435, 89}, {7, 896, 65}, {8, 125, 45},
{9, 476, 27}, {10, 987, 80}}
dataset = Dataset[Association@MapThread[(#1 -> <|"Vol" -> #2, "Vel" -> #3 |> &),
Transpose@Rest@data]]
dataset[1,... |
4,231,580 | <p>I have been working on th integral
<span class="math-container">$$\int_0^\infty \frac{\sin x}{1+x^2} dx$$</span>
for a short while now trying substitutions and even Laplace transforms and other stuff but gave up. I looked to Wolfram Alpha but how did it get the series expansion of the constant that it did (see <a hr... | robjohn | 13,854 | <p><span class="math-container">$$\newcommand{\Re}{\operatorname{Re}}\newcommand{\Im}{\operatorname{Im}}\newcommand{\Ei}{\operatorname{Ei}}\newcommand{\PV}{\operatorname{PV}}
\begin{align}
\int_0^\infty\frac{\sin(x)}{1+x^2}\,\mathrm{d}x
&=\Im\left(\int_0^\infty\frac{e^{ix}}{1+x^2}\,\mathrm{d}x\right)\tag1\\
&=\... |
196,002 | <p>The paper <a href="http://www.sciencedirect.com/science/article/pii/S002212369690110X" rel="nofollow noreferrer">Lattices of Intermediate Subfactors</a> of Y. Watatani, received on December 1994, finishes by: </p>
<blockquote>
<p><strong>Prop. 6.2.</strong> $ \ $ Any finite lattice with at most five elements ca... | Sergei | 49,208 | <p>Some results for differences $A_n-G_n, G_n-H_n$ and so after summing up for $A_n-H_n$ you may find in the book: Classical and new inequalities in analysis by D. S. Mitrinovic; J. E. Pecaric; A. M. Fink on pages 25,39. </p>
|
163,153 | <p>How do i find the absolute maximum and absolute minimum values of f on this given interval.</p>
<p>$f(x) = 6x^3 − 9x^2 − 36x + 7, \
[−2, 3]$</p>
| Roman Chokler | 38,328 | <p>I am not sure on how to prove this.
I myself have heard about other properties on periodicity.
A particularly interisting one is as follows:
Label a each primary bulb with a fraction $\frac pq$ where the bulb has period $q$ and the $p$th spoke the going counterclockwise from the main spoke is the biggest spoke of th... |
1,990,638 | <p>This is a purely affine variety. Define $F_1=y-x^2, F_2=z-xy, F_3=xz-y^2$. </p>
<p>It is clear that $(x,x^2,x^3)$ is a solution to $F_1=F_2=F_3=0$. Since projective twisted cubic is not intersection of two quadractics which contains an extra line, I would not expect this to be the case in affine as well. However, $... | Ted | 15,012 | <blockquote>
<p>Since projective twisted cubic is not intersection of two quadratics
which contains an extra line, I would not expect this to be the case in
affine as well.</p>
</blockquote>
<p>This is not correct. When you go from projective to the affine space, you remove some points. Claims like "projecti... |
313,254 | <p>I know this question has been asked before on MO and MSE (<a href="https://mathoverflow.net/questions/59605/reference-in-riemann-surfaces">here</a>, <a href="https://math.stackexchange.com/questions/407004/good-book-for-riemann-surfaces">here</a>, <a href="https://math.stackexchange.com/questions/1839673/books-on-ri... | Chris Judge | 7,120 | <p>For a hyperbolic perspective: Buser's "geometry and spectrum of compact riemann surfaces" is a nice book. </p>
|
234,668 | <p>I have an equation in $x$ and I would like to determine if it has any solutions modulo a large prime $p$. Suppose $p$ is large enough that I can factor numbers up to $p$, but I cannot test all values up to $p$. (Actually, so far, I have been doing just that -- but I'd like to avoid this as it takes a long time. If y... | Cameron Buie | 28,900 | <p>This is false, as written. If $k$ is even, then so is $kn$, regardless of whether $n$ is odd or not.</p>
<p>Now, suppose that $k,n$ are <em>both</em> odd. Then $k=2j+1$ and $n=2m+1$ for some integers $j,m$. Therefore $kn=2(\text{stuff})+1$. (I leave the determination of "stuff" to you, though I will tell you it's a... |
854,079 | <blockquote>
<p>Let $|A|=\mathcal c, \ |B|=\aleph_0, \ A\cap B=\emptyset,$ Prove that $ |A\cup B|=\mathcal c$</p>
</blockquote>
<p>So $|A\cup B|=|A|+|B|$ but this just leads to cardinal arithmetic which I don't think is the right approach. </p>
<p>Maybe embedding could work ? There has to be a 1-1 function from $A$... | martini | 15,379 | <p>I will suppose $a_n \ge 0$, as otherwise the statement is wrong.</p>
<p>By induction, we have $a_n < k^na_0$, hence $0 \le a_n \le k^n a_0$. As $0 \to 0$ and $k^n a_0 \to 0$ (we have $k \in (0,1)$), by the squeeze theorem $a_n \to 0$.</p>
|
854,079 | <blockquote>
<p>Let $|A|=\mathcal c, \ |B|=\aleph_0, \ A\cap B=\emptyset,$ Prove that $ |A\cup B|=\mathcal c$</p>
</blockquote>
<p>So $|A\cup B|=|A|+|B|$ but this just leads to cardinal arithmetic which I don't think is the right approach. </p>
<p>Maybe embedding could work ? There has to be a 1-1 function from $A$... | user157745 | 157,745 | <p>we have a theorem if a sequence is bounded and descend this sequence ins convergence and this condition is true for your question .and i think we have $a_n\geq 0 $</p>
|
4,116,732 | <p>If we have a topological category and the underlying category forgetting the topological structure, are the nerves same. They should be, is what my guess is from the definition of nerve of a category. Then, I have some facts which leads to contradiction (which should not be, thus I am missing something). The facts a... | Connor Malin | 574,354 | <p>No, these two nerves will not be the same. For example, let <span class="math-container">$G$</span> be a topological group. We can form its classifying category <span class="math-container">$BG$</span>, and this is a topological category. If <span class="math-container">$G^\partial$</span> denotes the discrete group... |
2,682,093 | <blockquote>
<p>Let $\circ$ be an inequality.</p>
<p>Prove $|x| \circ a \equiv -a \circ x \circ a$.</p>
</blockquote>
<p>If $x$ is positive, then $|x| \circ a = x \circ a$.</p>
<p>If $x$ is negative, then $|x| \circ a = x \circ a$ ?</p>
| farruhota | 425,072 | <p>Note $\circ$ can be either $<$ or $>$.</p>
<p>If $x>0$, then $|x|\circ a \equiv x\circ a$.</p>
<p>If $x<0$, then $ |x|\circ a \equiv -x\circ a$.</p>
|
2,741,511 | <p><a href="https://i.stack.imgur.com/eh6nF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/eh6nF.png" alt="enter image description here"></a></p>
<ul>
<li>How long does it take this object to arrive at the point $M$? Assume that $|KL| =|LM|$. </li>
</ul>
<p><strong>Options:</strong> </p>
<p>a) $... | Takahiro Waki | 268,226 | <p>Use <a href="http://www.mathsmutt.co.uk/files/newt.htm" rel="nofollow noreferrer">Newton's equations of motion</a></p>
<p>$mg\sinθ=ma$ </p>
<p>$⇒a=g\sinθ$</p>
<p>$v^2=2g\sinθ*h/\sinθ$ (velocity on point L).</p>
<p>$v=g\sinθt_1$</p>
<p>$t_1=\dfrac{\sqrt{2gh}}{g\sinθ}$</p>
<p>$t=t_1+t_2=\dfrac{\sqrt{2gh}}{g\sinθ... |
2,741,511 | <p><a href="https://i.stack.imgur.com/eh6nF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/eh6nF.png" alt="enter image description here"></a></p>
<ul>
<li>How long does it take this object to arrive at the point $M$? Assume that $|KL| =|LM|$. </li>
</ul>
<p><strong>Options:</strong> </p>
<p>a) $... | Peter Szilas | 408,605 | <p>1)$ |KL|:=s.$</p>
<p>Note : $h/s= \sin \theta$; or </p>
<p>$s= h/(\sin \theta)$.</p>
<p>Constant $a=g \sin\theta$ down the slope.</p>
<p>A) Final speed at $L$:</p>
<p>$v_f = at$ , where $t$ is the time taken down the slope, and</p>
<p>$s= (1/2)at^2$, or</p>
<p>$t_1:= t =((2s)/a)^{1/2}$; and</p>
<p>$v_f = a(... |
701,176 | <p>A function $f$ is differentiable over its domain and has the following properties:</p>
<ol>
<li><p>$\displaystyle f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$</p></li>
<li><p>$\lim_{h \to 0} f(h) = 0$</p></li>
<li><p>$\lim_{h \to 0} f(h)/h = 1$</p></li>
</ol>
<p>i) Show that $f(0)=0$</p>
<p>ii) show that $f'(x)=1+[f(x)]^... | Barry Cipra | 86,747 | <p>All you need to do is show that the equation $a^3-6a^2-2=0$ has no positive integer solution. If it did, then you'd have $a^2(a-6)=2$. But $a^2\mid2$ implies $a=1$, in which case $a^2(a-6)=-5\not=2$.</p>
|
235,319 | <p>The problem states: Suppose $f'(b) = M$ and $M <0$. Find $\delta>0$ so that if $x\in (b-\delta, b)$, then $f(x) > f(b).$</p>
<p>This intuitively makes sense, but I am not exactly sure how to find $\delta$. I greatly appreciate any help I can receive. </p>
| Robert Israel | 8,508 | <p>You could use dynamical programming. Let $a(m,n)$ be the number of $m$-digit nonnegative integers with each digit 0 to 7 and the sum of digits is $n$. Then $a(1,n) = 1$ for $0 \le n \le 7$, $0$ otherwise, and
$$a(m+1,n) = \sum_{d=0}^{\min(n,7)} a(m,n-d)$$</p>
<p>EDIT: That solution allows leading zeros. If lead... |
183,039 | <p>I'm pretty weak in the field of mathematics, but a strong programmer. I am looking for a mathematical solution that, given two points on a line will give me a curve between them, including those two points within the curve itself.</p>
<p>For instance, if I have a set of points { (0, 3) (1,10) } I'd like a mathemati... | RougeSegwayUser | 15,870 | <p>You also might want to check out Lagrange interpolation. The idea is that you can create a polynomial that goes through the given points, provided that the points arn't lined up vertically. Here's the link to the wikipedia article.</p>
<p><a href="http://en.wikipedia.org/wiki/Lagrange_polynomial" rel="nofollow">htt... |
1,335,446 | <p>Ordinary sphere in $\mathbb{R}^3$ is two-dimensional object (2-sphere), i.e. it requires at least two coordinates to define point on a surface. As I notice, however, there is a catch. <br/>
If we use spherical coordinates,$(\phi,\theta)$, then there are two points where $\phi$ is not defined (polar regions). If we ... | Andrew D. Hwang | 86,418 | <p>First, there is no way to cover the sphere with a single coordinate system. If there were, the sphere would be diffeomorphic to an open set in the plane, which it isn't. (Generally, a manifold of positive dimension can be covered by a single "coordinate chart" only if it is diffeomorphic to some open subset of a Euc... |
18,224 | <p>I need to generate a very large sparse block matrix, with blocks consisting only of ones along the diagonal. I have tried several ways of doing this, but I seem to always run out of memory.</p>
<p>The fastest way of doing this that I've come up with so far is as follows:</p>
<p>(typically, I will need <code>n</cod... | b.gates.you.know.what | 134 | <p>Maybe this ?</p>
<pre><code>SparseArray[{Band[{1, 1}, Dimensions[tmp] {m, m}] -> {tmp}}, Dimensions[tmp] {m, m}]
</code></pre>
|
18,224 | <p>I need to generate a very large sparse block matrix, with blocks consisting only of ones along the diagonal. I have tried several ways of doing this, but I seem to always run out of memory.</p>
<p>The fastest way of doing this that I've come up with so far is as follows:</p>
<p>(typically, I will need <code>n</cod... | Mr.Wizard | 121 | <p>I find that this performs equivalently to b.gatessucks' code while being significantly cleaner.</p>
<pre><code>n = 3;
m = 2;
SparseArray[Band[{1, 1}] -> SparseArray[{}, {m, n, n}, 1]]
</code></pre>
<p>$\left(
\begin{array}{cccccc}
1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0... |
2,002,672 | <p>I am ok with finding a power series for $\tanh^{-1}{z}$ since I can just see that since $\tanh iz = i\tan z$, then $\tanh^{-1}z = \frac1i \tan^{-1}{iz}$ and use the power series for $\tan$ to get:</p>
<p>$$\tanh^{-1}{z} = \sum^\infty_{k=0} \frac{z^{2k+1}}{2k+1}$$</p>
<p>But then I am asked to deduce that:</p>
<p>... | Martin Argerami | 22,857 | <p>For the functions you mention you can estimate the error in the Taylor polynomial. For cases like $\sin x$ and $\cos x$, it is easy to show that the error term goes to zero for each fixed $x\in\mathbb R$. This makes the function equal to its Taylor series everywhere, and thus analytic. </p>
<p>If you can bound the ... |
2,388,613 | <p>1) a right cone has a surface area 12 m${^2}$ and radius 1.3 m</p>
<p><strong>here is my answers:</strong></p>
<p>1) $s$ = 1.28</p>
<p><a href="https://i.stack.imgur.com/a8ovb.jpg" rel="nofollow noreferrer">(photo of how i got my answer)</a></p>
<p><strong>textbook answer:</strong> 1.64 m</p>
<p>how did they ge... | Cye Waldman | 424,641 | <p>Your error is in thinking that they want the vertical height of the cone. For a cone, $s$ usually represents the slant height. So the first thing you want to is to express the total surface are as</p>
<p>$$S=\pi r s+\pi r^2$$</p>
<p>where $\pi r s$ is the lateral area. Then</p>
<p>$$
s=\frac{S-\pi r^2}{\pi r}\app... |
87,271 | <p>For a category $C$, let $C-Set$ denote the category of set-valued functors $\delta\colon C\to Set$. Given categories $C$ and $D$, and a functor $F\colon C\to D$, composition with $F$ yields a functor that I'll denote by $$\Delta_F\colon D-Set\longrightarrow C-Set.$$ The functor $\Delta_F$ has both a left adjoint, wh... | Karol Szumiło | 12,547 | <p>I've been wondering about the same thing recently and here's my best guess. Let $\widetilde C$ denote the Cauchy completion of $C$. if the associated functor $\widetilde F : \widetilde C \to \widetilde D$ induces a bijection of the sets isomorphism classes of objects in $\widetilde C$ and $\widetilde D$, then $\Delt... |
109,037 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/107336/why-doesnt-dx-n-x-n1-rightarrow-0-as-n-rightarrow-infty-imply-x-n">Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?</a> </p>
</blockquote>
<p>my questio... | Dave L. Renfro | 13,130 | <p>Your intuition and argument are correct as far as they go, but note that you've only shown that the Cauchy condition is at least as strong, not strictly stronger. You can show "strictly stronger" by finding a counterexample, or by finding a property that all Cauchy sequences have and the weaker formulation doesn't h... |
31,992 | <p>I am attempting to generate partially transparent images for PNG <code>Export</code>, but seem to run to the following issue. If I <code>Rasterize</code> a simple piece of <code>Graphics</code> with <code>Background -> None</code> (transparent) it looks worse:</p>
<pre><code>rings = Image[
Rasterize[
Gr... | Michael Hale | 5,568 | <p>Oops. I was too tired to fully verify the expected output before I wrote my comment. Here is a simple hack to make the image you want though.</p>
<pre><code>disk[c_, r_] :=
Polygon@Table[c + r {Cos@a, Sin@a}, {a, 0, 2 Pi, Pi/50}]
Image[Rasterize[
Graphics[{Black, disk[{0, 0}, 1], White,
disk[{1, 1}/78, 1 -... |
417,131 | <p>Does limit $\frac{4xy}{\sqrt{x^2+y^2}}$ as $(x,y) \to (0,0)$ exist or not?</p>
<p>To remove the root, I squared the whole equation and I get
limit $\frac{16x^2y^2}{\sqrt{x^2+y^2}}$. Then I dont know how to continue working it. Can anyone help to do this questions for me please.</p>
<p>Thank you for your effort in ... | A.S | 24,829 | <p>Remember that $|\sin(z)| < z$ for $z \ne 0$. So $0 \le \sin^2(z) < z^2$. Set $z = x-y$. Your denominator satisfies $|z| = |x-y| \le |x| + |y|$ by the triangle inequality.</p>
<p>$$0 \le \lim_{(x,y) \to (0,0)} \frac{\sin^2(x-y)}{|x|+|y|} \le \lim_{(x,y) \to (0,0)} \frac{z^2}{|z|} = \lim_{(x,y) \to (0,0)} |z... |
3,435,286 | <p>Suppose that <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are random variables with joint density function</p>
<p><span class="math-container">$$f_{X,Y}(x,y)=\begin{cases} 8xy \space \text{ for } 0<y<x<1\\ 0 \space\space\space\space\space\text{ otherwise.}\end{cases}$... | J.G. | 56,861 | <p>With <span class="math-container">$c:=\cos x$</span>, <span class="math-container">$f=2c^2+c-1$</span>, so find the extrema of this function on <span class="math-container">$[-1, 1]$</span>. <strong>Do not</strong> assume each extremum is at an endpoint of this range. The quadratic has its global minimum at <span cl... |
1,132,922 | <p>Let $f$ and $g$ be two differentiable functions s.t $ f '(x) \le g '(x) $ for all $ x\lt 1$ and $ f '(x) \ge g'(x) $ for all $ x\gt 1$ then</p>
<ol>
<li><p>If $f(1) \ge g(1)$, then $f(x)\ge g(x)$ for all $x$</p></li>
<li><p>If $f(1) \le g(1)$, then $f(x)\le g(x)$ for all $x$</p></li>
<li><p>$f(1) \le g(1)$</p></li>... | Subhra Mazumdar | 329,168 | <p>Let $f(x)=(x-1)^{2}$ and $g(x)=(x-1)$. Then $f'(x)=2(x-1)$ and $g'(x)=1$. This clearly satisfies the constraint $f '(x) \ge g'(x)$ for all $x>1$ and $f '(x) \le g'(x)$ for all $x<1$. $f(1)=0$ and $g(1)=0$, but for any value of $x , f (x) > g(x)$. So option (1)</p>
|
706,514 | <p>I know that the fundamental group of homeomorphic spaces are isomorphic. Is the converse true? I mean, can we say the two spaces with isomorphic fundamental groups are homeomorphic? </p>
| Brian Fitzpatrick | 56,960 | <p>No. In fact, there are spaces that are not <em>homotopy equivalent</em> (isomorphic in the homotopy category of topological spaces) but have the same fundamental group. Take $X=S^2$ and $Y=\{\operatorname{pt}\}$.</p>
|
706,514 | <p>I know that the fundamental group of homeomorphic spaces are isomorphic. Is the converse true? I mean, can we say the two spaces with isomorphic fundamental groups are homeomorphic? </p>
| Moishe Kohan | 84,907 | <p>This is an addendum to Jason DeVito's answer; it concerns the current status of Borel conjecture: </p>
<p><em>If two closed aspherical (i.e., with contractible universal covers) manifolds $M, N$ have isomorphic fundamental groups then they are homeomorphic.</em> </p>
<p>The strongest to date result towards Borel c... |
209,044 | <p>Please help me deal with this kind of question about ODEs.
My codes are as follows</p>
<pre><code>m = 100;
a = D[x[t], {t, 2}];
t1up = 2 x''[t] + 1/2 (490 + 34 x''[t] + 2 (490 + 50 x''[t]));
t1down = 490 + 53 x''[t];
t1 = Piecewise[{{t1up, x'[t] >= 0}, {t1down, x'[t] < 0}}]
equa00 = t1 == m*a
t0 = 50;
s1 = ND... | xzczd | 1,871 | <p>Another solution is to use <a href="https://mathematica.stackexchange.com/a/133530/1871"><code>Simplify`PWToUnitStep</code></a>:</p>
<pre><code>s1 = NDSolve[{equa00 // Simplify`PWToUnitStep, x[0] == 1, x'[0] == 1}, x, {t, 0, 50}]
</code></pre>
|
2,936,236 | <p>I have two 2D disks, <span class="math-container">$(C_1, r_1)$</span> (blue) and <span class="math-container">$(C_2, r_2)$</span> (red), where the blue disk somehow overlaps the red disk (the figure below shows one example). I’m interested in finding the positive distance along the <span class="math-container">$y$<... | amd | 265,466 | <p>Find the point(s) on the vertical line through the center of the blue disk that are at a distance equal to the sum of the two disks’ radii from the red disk’s center. That’s where the center of the blue disk should be for it to be tangent to the red. Looks like a job for the Pythagorean theorem.</p>
|
2,936,236 | <p>I have two 2D disks, <span class="math-container">$(C_1, r_1)$</span> (blue) and <span class="math-container">$(C_2, r_2)$</span> (red), where the blue disk somehow overlaps the red disk (the figure below shows one example). I’m interested in finding the positive distance along the <span class="math-container">$y$<... | Narasimham | 95,860 | <p>Shift the green circle upwards until increasing distance between circle centers just equals</p>
<p><span class="math-container">$$(C_1 C_2) =(r_1+r_2).$$</span></p>
|
2,279,660 | <p>On $C^1([0,1],\mathbb{R})$ I have two norms
\begin{align*}
N_1(f)&=|f(0)|+\|f'\|_{\infty}\\
N_2(f)&=\|f\|_{\infty}+\|f'\|_{\infty}.
\end{align*}</p>
<p>We have always that $N_1(f)\leq N_2(f)$ but what about the other inequality please?</p>
<p>And please how to see If $N_1$ and $N_2$ are equivalent to $\le... | zhw. | 228,045 | <p>Hint for the first question: By the mean value theorem, $f(x) = f(0) + f(x)-f(0) = f(0) + f'(c_x)x.$</p>
<p>Hint for the second question: Consider the functions $x^n,n=1,2,\dots $</p>
|
249,597 | <p>I am suppose to find all the solutions to this problem, I think some theorem states that there can only be as many solutions to the problem as the highest degree. I know that calculus reinforces this so I know that</p>
<p>$2x^2 + 4x + 1 = 0$</p>
<p>Can have at most two solutions. In calculus this is proven by the ... | Tengu | 37,026 | <p>If you don't want to use the formula, you can do the following.(Actually this is how the formula is derived though. I used to use this method when I didn't memorize the formula.)</p>
<p>$2x^2+4x+1=0$</p>
<p>$x^2+2x+1/2=0$</p>
<p>$(x+1)^2-1/2=0$</p>
<p>$(x+1)^2=1/2$</p>
<p>$x+1=+1/\sqrt 2$ or $-1/\sqrt 2$</p>
|
249,597 | <p>I am suppose to find all the solutions to this problem, I think some theorem states that there can only be as many solutions to the problem as the highest degree. I know that calculus reinforces this so I know that</p>
<p>$2x^2 + 4x + 1 = 0$</p>
<p>Can have at most two solutions. In calculus this is proven by the ... | P.K. | 34,397 | <p>The <a href="http://en.wikipedia.org/wiki/Quadratic_equation#Quadratic_formula" rel="nofollow">quadratic formula</a> works for all quadratic equations. For any equation [in the form] $ax^2 + bx + c = 0$, this is true:$$x = {-b \pm \sqrt{b^2 - 4ac} \over 2a}$$</p>
|
996,103 | <p>Suppose that $A$, $B$, and $C$ are sets. Prove that $(A\cap B)\times C =(A\times C)\cap(B\times C)$. Prove the statement both ways or use only if and only if statements.</p>
| marty cohen | 13,079 | <p>Here is a way
to show convergence
without finding the sum.</p>
<p>If $x > 0$,
$\ln(1+x)
=\int_1^{1+x} \frac{dt}{t}
$.
Therefore
(the $1+x$ in the denominator
of the next integral
is intentional)
$\int_1^{1+x} \frac{dt}{1+x}
< \ln(1+x)
< \int_1^{1+x} \frac{dt}{1}
$
or
$\frac{x}{1+x}
< \ln(1+x)
< x
$.<... |
475,221 | <p>I am working on problem #25 of <em>Linear Algebra and its Applications</em> and the question asks:</p>
<blockquote>
<p>Find an equation involving $g$, $h$, and $k$ that makes this augmented matrix correspond to a consistent system:
$$\left(\begin{array}{ccc|c}
1& -4& 7& g \\
0& 3& -5& h ... | Community | -1 | <p><strong>Hint</strong></p>
<ul>
<li>$$\frac{y^2}{y^2+d^2}=\frac{y^2+d^2-d^2}{y^2+d^2}=1-\frac{d^2}{y^2+d^2}$$</li>
<li>$$\frac{d^2}{y^2+d^2}=\frac{1}{(y/d)^2+1}=\frac{1}{t^2+1}$$</li>
<li>$$\int\frac{dt}{t^2+1}=\arctan t+C$$</li>
</ul>
|
4,095,361 | <p>Given <span class="math-container">$f : [0, 1] \rightarrow \mathbb{R}$</span> be a bounded function which is continuous on <span class="math-container">$[0,1]$</span> except at <span class="math-container">$1/2$</span>. Let <span class="math-container">$\alpha(x) = x^2$</span>. No further description of function <s... | tommik | 791,458 | <p>your distribution is not absolutely continuous thus your expectation is wrong. Your random variable has a positive probability mass in <span class="math-container">$x=5/6$</span></p>
<p>An easy way to calculate its expectation is the following purple area <span class="math-container">$=35/72$</span></p>
<p><a href="... |
820,490 | <p>I am studying pre-calculus mathematics at the moment, and I need help in verifying if $\sin (\theta)$ and $\cos (\theta)$ are functions? I want to demonstrate that for any angle $\theta$ that there is only one associated value of $\sin (\theta)$ and $\cos (\theta)$. How do I go about showing this?</p>
| Christian Blatter | 1,303 | <p>Usually one begins studying the $\sin$ function in connection with right triangles. Here the notion of angle is not questioned, so I shall stick with this intuitive concept. It follows that we have $\sin \theta$ defined for $0\leq\theta\leq{\pi\over2}$.</p>
<p>As a next step one considers polar angles in the $(x,y)... |
549,254 | <blockquote>
<p>$$\frac{1}{3}=.33\bar{3}$$ </p>
</blockquote>
<p>is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ratio? </p>
<p>I understand that $\pi$ is irrational because it extends infinitely <em>... | Khosrotash | 104,171 | <p>Let <span class="math-container">$$a=0.333333333333333...$$</span></p>
<p>Then multiply by <span class="math-container">$10$</span></p>
<p><span class="math-container">$$10a=3.333333333333333...$$</span></p>
<p>Substract <span class="math-container">$a$</span> from both sides</p>
<p><span class="math-container">$$10... |
1,336,209 | <p>I need to understand very good how the properties of this formula</p>
<p>$\frac{4}{\pi} = \frac{5}{4} + \sum_{N \geq 1} \left[ 2^{-12N + 1} \times(42N + 5)\times {\binom {2N-1} {N}}^3 \right] $</p>
<p>Taken from the paper "Radian Reduction for Trigonometric Function" (Hanek Payne Algorithm)</p>
<p>Some remarkable... | Ant | 66,711 | <p>The Stolz Cesaro theorem yields $$\lim_{n \to \infty} \frac{(n+1)^4}{(n+1)^k-n^k}$$</p>
<p>The degree of the numerator is $4$ while the degree of the denominator is $k-1$. So if $k \le 4$, the limit is $\infty$.
If $k \ge 6$, the limit is $0$. </p>
<p>If $k = 5$, you only need to compute the coefficient of $n^4$... |
1,336,209 | <p>I need to understand very good how the properties of this formula</p>
<p>$\frac{4}{\pi} = \frac{5}{4} + \sum_{N \geq 1} \left[ 2^{-12N + 1} \times(42N + 5)\times {\binom {2N-1} {N}}^3 \right] $</p>
<p>Taken from the paper "Radian Reduction for Trigonometric Function" (Hanek Payne Algorithm)</p>
<p>Some remarkable... | user26486 | 107,671 | <p>You can prove $1^4+2^4+3^4+\cdots+n^4=\frac{1}{5}n^5+\frac{1}{2}n^4+\frac{1}{3}n^3-\frac{1}{30}n$ using simple induction.</p>
<p>$$\lim_{n\to \infty} \frac{1^4+2^4+3^4+\ldots+n^4}{n^k}$$</p>
<p>$$=\lim_{n\to \infty} \frac{\frac{1}{5}n^5+\frac{1}{2}n^4+\frac{1}{3}n^3-\frac{1}{30}n}{n^k}$$</p>
<p>If $k<5$, the l... |
3,542,374 | <p>I have the matrix of stacked constraints</p>
<p><span class="math-container">$$\begin{bmatrix} x_1^2 & x_1y_1 & y_1^2 & x_1 & y_1 & 1 \\ x_2^2 & x_2y_2 & y_2^2 & x_2 & y_2 & 1 \\ x_1^2 & x_3y_3 & y_3^2 & x_3 & y_3 & 1 \\ x_4^2 & x_4y_4 & y_4^2 &... | user8675309 | 735,806 | <p><strong>a linear algebra (+ Kronecker products) proof of the rank of your interpolation matrix</strong></p>
<p><span class="math-container">$\begin{bmatrix} x_1^2 & x_1y_1 & y_1^2 & x_1 & y_1 & 1 \\ x_2^2 & x_2y_2 & y_2^2 & x_2 & y_2 & 1 \\ x_3^2 & x_3y_3 & y_3^2 &... |
544,673 | <p>let $\alpha$ be a cycle of length $s$, say $\alpha = (a_1, a_2, \ldots, a_s)$</p>
<p>Prove $\alpha^2$ is a cycle if and only if $s$ is odd.</p>
<p>Let me start off by saying I am in my 5th week of Group Theory. I often have trouble getting these problems started. This is my first proof based course. </p>
<p>I ... | Dan Shved | 47,560 | <p>Here is my somewhat crude illustration of what happens when you multiply a cycle with itself.</p>
<p>If the cycle has odd length, it just changes the order in which vertices are visited. The pentagon on the left is the graph of a $5$-cycle. The star on the right is the graph of that cycle multiplied with itself.</p... |
66,480 | <p>I have the following problem:</p>
<blockquote>
<p>Given $P(A)=0.2$, $P(B)=0.4$, $P(C)=0.8$, $P(D)=0.5$, find $P(A\cup B\cup C\cup D)$</p>
</blockquote>
<p>And the final answer should be 0.952</p>
<p>I know how to find the union of two and three elements (for 2, its: $A+B-AB$), but the formula becomes clumsy aft... | Dilip Sarwate | 15,941 | <p>The question you pose cannot be solved; all that can be said is that
$$
\max\{P(A), P(B), P(C), P(D)\} = 0.8 \leq P(A \cup B \cup C \cup D) \leq 1
$$
However, the answer $0.952$ that you give corresponds to the case when $A$, $B$, $C$, and $D$ are <em>independent</em> events. Did you leave out this important piece ... |
4,270,857 | <p>Solve by separation of variables
<span class="math-container">$$\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}$$</span>
given intitial conditions:</p>
<p><span class="math-container">$$\frac{\partial u}{\partial x}=0 \text{ at } x=a \text{ and } x=-a, \forall t\geq 0;$$</span>
<span class="math-con... | Community | -1 | <p>First, for <span class="math-container">$1\le i\le n+1$</span>,
<span class="math-container">$$
\mathsf{P}(x_i=a)=\frac{n!}{(n+1)!}=\frac{1}{n+1},
$$</span>
and for <span class="math-container">$i\ne j$</span> and <span class="math-container">$a\ne b$</span>,
<span class="math-container">$$
\mathsf{P}(x_i=a,x_j=b)=\... |
1,557,686 | <p>I have a definition of a <a href="https://en.wikipedia.org/wiki/Hypergeometric_distribution" rel="nofollow">Hypergeometric distribution</a> as follows:</p>
<blockquote>
<p>Definition: the Hypergeometric distribution is a discrete probability distribution that describes the probability of $k$ successes in $n$ draw... | Asker | 201,024 | <p>According to your book, if you have $N$ different things that can be picked, $K$ is the number of things from those $N$ things which would be considered "successes". $X$ is then the number of successes from those $K$ successes <em>that are actually picked.</em> </p>
<p>The number of trials is not $K$. The number of... |
4,249,191 | <p>I need to find chromatic number of graph <span class="math-container">$G$</span> obtained from <span class="math-container">$K_{2019}$</span> , by removing one Hamiltonian cycle from it. I already did this example for even <span class="math-container">$n$</span>. Now if <span class="math-container">$V(G)=\{1,2,...,n... | Empy2 | 81,790 | <p>Suppose a colouring with 1009 colours is possible.<br />
Take 1008 consecutive even points. They have 1008 different colours.<br />
There is a gap of 5 between the ends. Either of the middle two can complete a set of 1009, so they are the same colour.<br />
Likewise, all adjacent points are the same colour, which ... |
4,249,191 | <p>I need to find chromatic number of graph <span class="math-container">$G$</span> obtained from <span class="math-container">$K_{2019}$</span> , by removing one Hamiltonian cycle from it. I already did this example for even <span class="math-container">$n$</span>. Now if <span class="math-container">$V(G)=\{1,2,...,n... | kabenyuk | 528,593 | <p>Let the vertices of the complete graph <span class="math-container">$K_n$</span> be located at the vertices of a regular <span class="math-container">$n$</span>-gon. Then the edges of <span class="math-container">$K_n$</span> are all possible diagonals of <span class="math-container">$n$</span>-gon and all its sides... |
1,704,707 | <p>The function $f: ℝ → ℝ$ defined by $f(x) = x^{3}$ is onto because for any real number $r$, we have that $\sqrt[3]r$ is a real number and $f(\sqrt[3]r)=r$. Consider the same function defined on the integers $g: ℤ → ℤ$ by $g(n) = n^3.$ Explain why $g$ is not onto $ℤ$ and give one integer that $g$ cannot output. </p>
... | KGD | 322,022 | <p>Note that each $f_k$ is 1-lipschitz. In particular, this implies that, for each $\varepsilon > 0$, you can write $(|x-y| < \varepsilon) \Rightarrow (|f_k(x)-f_k(y)| < \varepsilon)$ for each $ k \ge 1$, therefore the family is equicontinuous.</p>
|
1,704,707 | <p>The function $f: ℝ → ℝ$ defined by $f(x) = x^{3}$ is onto because for any real number $r$, we have that $\sqrt[3]r$ is a real number and $f(\sqrt[3]r)=r$. Consider the same function defined on the integers $g: ℤ → ℤ$ by $g(n) = n^3.$ Explain why $g$ is not onto $ℤ$ and give one integer that $g$ cannot output. </p>
... | Lutz Lehmann | 115,115 | <p>Note that with $k=0$ this sequence contains its own limit. This is the second most trivial situation for a compact set, the most trivial being a set of finitely many points.</p>
|
3,271,917 | <p><span class="math-container">$$ \sum_{n=1}^\infty (-1)^{n+1} \left( \frac{1.4.7\dots .(3n-2)}{2.3.8\dots .(3n-1)} \right)^2 $$</span></p>
<p>I have done the<span class="math-container">$ \sum_{n=1}^\infty \left( \frac{1.4.7\dots .(3n-2)}{2.3.8\dots .(3n-1)} \right)^2 $</span> part , and showed it divergent using G... | Jack D'Aurizio | 44,121 | <p>It is enough to prove that <span class="math-container">$\frac{\Gamma(n+1/3)}{\Gamma(n+2/3)}$</span> is decreasing to zero, then invoke Leibiz' criterion. On the other hand</p>
<p><span class="math-container">$$\frac{\Gamma(n+1/3)}{\Gamma(n+2/3)}=\frac{1}{\Gamma(1/3)}B(n+1/3,1/3)=\frac{3}{\Gamma(1/3)}\int_{0}^{1}\c... |
71,354 | <p>I've been reading Moroianu's Kahler geometry notes and found a unattributed quote that says that if the Kahler properties hold, then
"a long list of miracles occur"</p>
<p>I am guessing that this quote belongs to Kahler himself, but I can't back this up. Does anyone know?</p>
| Igor Rivin | 11,142 | <p>I believe the answer is yes, see:</p>
<p><a href="http://books.google.com/books?id=u6WFVmoHxFkC&pg=PA740&dq=Kahler+%2B+%22a+long+list+of+miracles%22" rel="nofollow">http://books.google.com/books?id=u6WFVmoHxFkC&pg=PA740&dq=Kahler+%2B+%22a+long+list+of+miracles%22</a></p>
|
1,443,166 | <blockquote>
<p>If the proposition ¬p→v is true, then the truth value of the
proposition ¬p∨(p→q), where ¬ is negation, ∨ is inclusive OR and → is
implication, is</p>
<ol>
<li>True</li>
<li>False</li>
<li>Multiple Values</li>
<li>Cannot be determined</li>
</ol>
</blockquote>
<hr>
<p><strong>I try... | Eric Auld | 76,333 | <p>I don't see what $v$ has to do with the question. It seems that the statement is true if $\neg p$, and false if $p \land \neg q$.</p>
|
76,457 | <p>I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane.</p>
<p>How do I determine if a point $(x,y)$ is within the area bounded by the ellipse? </p>
| user3731622 | 341,117 | <p>1) Consider the point as a vector</p>
<p>$$
p=\begin{bmatrix}
x \\
y \\
\end{bmatrix}
$$</p>
<p>2) Consider the center of the ellipse as</p>
<p>$$
c=\begin{bmatrix}
h \\
k \\
\end{bmatrix}
$$</p>
<p>3) Subtract the center of the ellipse
$$
p_{cent... |
898,082 | <p>I have the answer for this, but my teacher hadn't taught the whole "when cosine is an even, the value of $-\arccos (-0.7)$ is a solution too." </p>
<p>Please: </p>
<p>-tell me when a $\cos$/$\sin$ function is even/odd </p>
<p>-what happens if its odd? </p>
<p>-how to use the "$±\arccos(-0.7) + 2kπ$" ( don't unde... | chaffdog | 168,595 | <p>As others have said, a function $f(x)$ is called even if $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$.</p>
<p>We add $2k\pi$ to the answer because both sine and cosine are periodic of period $2\pi$. To see this, you can think of sine and cosine as the $x$ and $y$ values respectively of a point on the circumference of... |
944,965 | <p>Question: whats the order of the element a=33 in Z60 (under modular addition)?</p>
<p>Answer: $\langle 33 \rangle= \{33,6,39,12,45,18,51,24,57,30,3,36,9,42,15,48,21,54,27,0,\}$. Therefore, $\text{order}(33)=20$</p>
<p>Whats the inverse of $33$ in $\mathbb{Z}_{60}$ (under modular addition)?</p>
<p>Im struggling wi... | Adriano | 76,987 | <p><strong>Hint:</strong> From your previous work, we know that:
$$
0 = \underbrace{33 + 33 + \cdots + 33}_{20 \text{ times}} = 33 + \underbrace{33 + \cdots + 33}_{19 \text{ times}}
$$
Thus, we know that the inverse of $33$ is:
$$
\underbrace{33 + \cdots + 33}_{19 \text{ times}}
$$
Incidentally, this is just $60 - 33$.... |
3,552,779 | <p>In a 3D world, given a box <code>B</code>, a pivot <code>P</code>, and a direction vector <code>V</code>, how can I find out how much to rotate at <code>P</code> such that <code>V</code> points towards an arbitrary point <code>A</code>?</p>
<p>Problem source:</p>
<p>I am a software developer that come across the n... | user170231 | 170,231 | <p>Without too much simplification, the substitution you cite yields</p>
<p><span class="math-container">$$\int_2^3\frac9{(x-2)^{1/4}}\,\mathrm dx=36\int_0^1\frac{u^3}u\,\mathrm du$$</span></p>
<p>which you certainly welcome to treat as an improper integral,</p>
<p><span class="math-container">$$36\left(\frac13-\lim... |
2,012,080 | <p>Consider the PDE for $v(x,y)$:</p>
<blockquote>
<p>$v_{xx} − v_{xy} − 6v_{yy} − 5v_{x} − 10v_{y} + 25 = 0$</p>
</blockquote>
<p>Taking $\xi = y-2x$ and $\eta=y+3x$ that I found from $b^2-4ac$ I have transformed the equation to the form</p>
<p>$v_{\eta\xi} + v_{\eta}=1$</p>
<p>I have been posed the question:</p... | Aleksas Domarkas | 562,074 | <p>Solve PDE
$$v_{xx} − v_{xy} − 6v_{yy} − 5v_{x} − 10v_{y} + 25 = 0 $$</p>
<ul>
<li>$D_x^2-D_xD_y-6D_y^2-5D_x-10D_y=
\left( {D_x}-3 {D_y}-5\right) \, \left( {D_x}+2 {D_y}\right)$ </li>
<li>General solution of $\quad v_x+2v_y=0\quad$ is $\quad v_1=f(y-2x)$</li>
<li>General solution of $\quad v_x-3v_y-5v= 0\quad$ is $\... |
165,385 | <p>I am generating a list of 1's, 2's, and 3's with different probabilities for each number. I then convert this list into three binarized lists, giving a list of the locations of each digit in the original list. There must be a more efficient way to do this? Perhaps changing the way I make the original list?</p>
<pre... | Carl Woll | 45,431 | <p>There are many possibilities, but I like using <a href="http://reference.wolfram.com/language/ref/Clip" rel="nofollow noreferrer"><code>Clip</code></a>:</p>
<pre><code>Unitize@Clip[list,{1,1},{0,0}]
Unitize@Clip[list,{2,2},{0,0}]
Unitize@Clip[list,{3,3},{0,0}]
</code></pre>
<blockquote>
<p>{0, 0, 0, 1, 0, 0, 0, ... |
165,385 | <p>I am generating a list of 1's, 2's, and 3's with different probabilities for each number. I then convert this list into three binarized lists, giving a list of the locations of each digit in the original list. There must be a more efficient way to do this? Perhaps changing the way I make the original list?</p>
<pre... | Alucard | 18,859 | <pre><code>list1e = Boole[# == 1] & /@ list;
list2e = Boole[# == 2] & /@ list;
list3e = Boole[# == 3] & /@ list;
</code></pre>
<p>another option is using Piecewise. For example:</p>
<pre><code>fun[x_] := Piecewise[{{1, x == 1}}, 0]
SetAttributes[fun, Listable]
list1c = fun[list];
</code></pre>
<p>or the ... |
141,128 | <p>I have a complicated function for <code>StreamPlot</code> to call that includes some error checking and the use of <code>Message</code>. Somehow the messages are not displayed when the function is called from <code>StreamPlot</code>. Here's a toy example:</p>
<p>Define the function & message:</p>
<pre><code>... | Michael E2 | 4,999 | <p>You need to reset <code>$Messages</code>.</p>
<pre><code>f[x_] := Module[{},
If[x > 1, Print["print: ", x];
Block[{$Messages = Streams["stdout"]},
If[! ListQ@$MessageList, $MessageList = {}];
Message[f::msg, x]]];
x];
</code></pre>
<p><code>$MessageList</code> should be a list, which if is... |
289,757 | <p>I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied field is CFD. Even though every mathematician would say I have not heard a lot of math, for myself I would say that I... | TakeS | 38,349 | <p>In my job as a (thus far) applied computer science researcher, I can sympathize with you in that I am not utilizing as much math directly from my courses into my job. Sure, I have to know some basic probability, but much of my job consists of reading research papers where most math consists of the standard operation... |
2,466,022 | <p>In Real and Complex Analysis, 3rd Edition, Walter Rudin advances the following:</p>
<p><a href="https://i.stack.imgur.com/XEVwK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XEVwK.png" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/dTwbg.png" rel="nofollow noreferrer... | Sunyam | 463,614 | <p>Another way to see is a proof by contradiction. Assume there exists a complex number z for which $e^z=0$.Then $|e^z|=0$. Which implies for this z, $e^{\Re{z}}=0$. Which is not true for any real number $\Re{z}$. Hence proved.</p>
|
2,447,850 | <p>So I have to prove 2 things:</p>
<ol>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R, x>0$. </p></li>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R$. </p></li>
</ol>
<p>For #1, I kn... | Zestylemonzi | 270,448 | <p>Hint: Given $x$, take $N \in \mathbb{N}$ such that $N>x$. Then, for $n >N$,
$$\frac{x^n}{n!} = \frac{x^N}{N!} \cdot \frac{x^{n-N}}{(n!/N!)} \le \frac{x^N}{N!} \cdot \left(\frac{x}{N+1}\right)^{n-N} .$$</p>
|
2,447,850 | <p>So I have to prove 2 things:</p>
<ol>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R, x>0$. </p></li>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R$. </p></li>
</ol>
<p>For #1, I kn... | Michael Hardy | 11,667 | <p>$$
\frac {x^n}{x!} = \frac{\overbrace{x\cdot x\cdot x\cdots\cdots x}^{\Large n \text{ factors}}}{\underbrace{1\cdot2\cdot3\cdots\cdots n}_{\Large n \text{ factors}}}
$$
When $n$ gets to be more than twice as big as $x,$ then multiplying by $\dfrac x n$ reduces the thing to less than half what it was. Then $n$ keeps ... |
128,651 | <p>Let $M$ be a smooth manifold (maybe compact, if that helps). Denote by $\operatorname{Diff}(M)$ the group of diffeomorphisms $M\to M$ and by $R(M)$ the space of Riemannian metrics on $M$. We obtain a canonical group action
$$ R(M) \times \operatorname{Diff}(M) \to R(M), (g,F) \mapsto F^*g, $$
where $F^*g$ denotes th... | Holonomia | 73,953 | <p>The group of C^1 diffeomorphism does not act transitively on the space of Riemannian metrics on a compact manifold. For example, two circles of different radius have different diameters and by pullback we can "copy" the metric of one of them on the other. Doing so we get two metrics on a circle with different diamet... |
1,515,900 | <p>In many books and papers on analysis I met this equality without proof:</p>
<p>$$\sup \limits_{t\in[a,b]}f(t)-\inf \limits_{t\in[a,b]}f(t)=\sup \limits_{t,s\in[a,b]}|f(t)-f(s)|$$</p>
<p>Can anyone show strict and nice proof of that equality?</p>
<p>I would really grateful foe your help!</p>
| robjohn | 13,854 | <p>$$
\begin{align}
\sup_{t\in[a,b]}f(t)-\inf_{s\in[a,b]}f(s)
&=\sup_{t\in[a,b]}f(t)+\sup_{s\in[a,b]}(-f(s))\\
&=\sup_{s,t\in[a,b]}\big(f(t)+(-f(s))\big)\\
&=\sup_{s,t\in[a,b]}\big(f(t)-f(s)\big)\\
&=\sup_{s,t\in[a,b]}\left|f(t)-f(s)\right|
\end{align}
$$
The last equality follows since $\big\{f(t)-f(s)... |
1,051,357 | <p>$$
\begin{bmatrix}
-1&&2&&-3&&4\\
5&&0&&2&&-2\\
2&&1&&1&&2\\
0&&0&&3&&-2\\
\end{bmatrix}
$$</p>
<p>I wanted to confirm that if I use $(w, x,$$ y, z)$ would the image of the vector be:</p>
<pre><code>-w + 2x - 3y + 4z
5w... | NicNic8 | 24,205 | <p>Your statement is correct, however, it may be able to be simplified.</p>
<p>One thing you could do is determine the reduced row echelon form (rref) of your matrix. Then the image (or column space) of your matrix is the column space of the rref of your matrix.</p>
|
830,599 | <p>The function $f$ is defined as follows:
$$f(x):=\sum_{j=1}^{\infty} \frac{x^j}{j!} e^{-x}$$</p>
<p>It's easy to see that $f(0)=0$. But I am interested in the value
$$\lim_{x \rightarrow 0^+} f(x).$$</p>
<p>Even <a href="https://www.wolframalpha.com/input/?i=lim_%28x-%3E0%29+%28sum_%28i%3D1%29%5Einfinity+x%5Ej%2F%... | zuriel | 147,911 | <p>Notice that $e^x=\displaystyle\sum_{j=0}^{\infty}\displaystyle\frac{x^j}{j!}$. </p>
<p>So $\displaystyle\sum_{j=1}^{\infty} \frac{x^j}{j!} e^{-x}=\displaystyle\sum_{j=0}^{\infty} \frac{x^j}{j!} e^{-x}-e^{-x}=1-e^{-x}$</p>
|
1,371,549 | <p>Differentiate the Function : $y=\log_2(e^{-x} \cos(\pi x))$</p>
<p>Here is my work. What I have I done wrong?
<a href="https://i.stack.imgur.com/w9RSN.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/w9RSN.jpg" alt="enter image description here"></a></p>
| mathlove | 78,967 | <p>$$\frac{-\sin(\pi x)\cdot \pi+\cos(\pi x)\cdot (-1)}{e^x(e^{-x}\cos(\pi x))\cdot \ln 2}$$
is correct, <strong>but the last expression you wrote is not correct</strong>. You have a mistake when you simplify the expression (you miss minus signs. Also, note that $e^x\cdot e^{-x}=1$). The correct answer will be
$$\frac{... |
2,135,811 | <p>$ \frac{6x}{x-2} - \sqrt{\frac{12x}{x-2}} - 2\sqrt[4]{\frac{12x}{x-2}}>0 $</p>
<p>I've tried putting $t= \frac{6x}{x-2} $ and play algebraically, using square of sum, but still no luck, any help?</p>
| Michael Rozenberg | 190,319 | <p>Let $\sqrt[4]{\frac{12x}{x-2}}=t$. Hence, $t\geq0$ and
$$\frac{t^4}{2}-t^2-2t>0$$ or
$$t(t^3-2t-4)>0$$ or
$$t(t^3-2t^2+2t^2-4t+2t-2)>0$$ or
$$t(t-2)(t^2+2t+2)>0$$ or
$$t(t-2)((t+1)^2+1)>0$$ or
$$t(t-2)>0$$ and since $t\geq0$, we obtain $t>2$ or
$$\frac{12x}{x-2}>16$$ or
$$\frac{3x}{x-2}-4&g... |
4,279,409 | <p>Consider the ring <span class="math-container">$\mathbb{R}[x]$</span>, determine the ideal <span class="math-container">$$(x^3-5x^2+6x-2,x^4-4x^3+3x^2-4x+2)$$</span></p>
<p>Is it principal? Is it prime? Is it maximal?</p>
<p>Attempt:</p>
<p>We have that <span class="math-container">$x^3-5x^2+6x-2=0$</span> in this i... | user7427029 | 513,618 | <p>I would try something named »coefficient comparison«. For example, for <span class="math-container">$jh + 1 = 0$</span>, we get <span class="math-container">$1 = \dots x^{jh + 1}$</span></p>
|
4,279,409 | <p>Consider the ring <span class="math-container">$\mathbb{R}[x]$</span>, determine the ideal <span class="math-container">$$(x^3-5x^2+6x-2,x^4-4x^3+3x^2-4x+2)$$</span></p>
<p>Is it principal? Is it prime? Is it maximal?</p>
<p>Attempt:</p>
<p>We have that <span class="math-container">$x^3-5x^2+6x-2=0$</span> in this i... | B. Goddard | 362,009 | <p>If you expand <span class="math-container">$(a+bx^h)^4$</span> you get</p>
<p><span class="math-container">$$a^4 + 4a^3bx^h + 6 a^2b^2x^{2h} + 4ab^3x^{3h} + b^4x^{4h}.$$</span></p>
<p>So your equation is</p>
<p><span class="math-container">$$-1 + x+ ax(a+bx^h)^4 = 0.$$</span></p>
|
516,244 | <p>My professor gave us this example on her notes:</p>
<p>$$\sum_{n = 1}^\infty \left(\frac{3}{n(n+3)}+\frac{1}{2^n}\right)$$</p>
<p>So I know we're supposed to find the partial fraction, which ends up being</p>
<p>$$\left(\frac{3}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}=
\frac{1}{n}-\frac{1}{n+3}\right)$$</p>
<p>So ba... | Shobhit | 79,894 | <p>if $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ are the vertices of a triangle then its area is given by:-</p>
<p>$$\frac{
1}{2}|(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2))|$$</p>
|
516,244 | <p>My professor gave us this example on her notes:</p>
<p>$$\sum_{n = 1}^\infty \left(\frac{3}{n(n+3)}+\frac{1}{2^n}\right)$$</p>
<p>So I know we're supposed to find the partial fraction, which ends up being</p>
<p>$$\left(\frac{3}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}=
\frac{1}{n}-\frac{1}{n+3}\right)$$</p>
<p>So ba... | Beedassy Lekraj | 243,296 | <p>The area $A$ of the triangle two of whose vertices lie on the axes, with coordinates $(a, 0)$, $(0, b)$, and a third vertex $(c, d)$ is obtained from previous formula by a mere horizontal axis shift of -a units as
$$A = \frac{|-ad + b(a - c)|}{2}$$</p>
|
801,680 | <p>Is it true that every point in a space equipped with the Hausdorff topology is closed?</p>
<p>I am assuming this is true but am having a difficult time proving it. Fix $a \in X$ where $X$ is equipped with the Hausdorff topology. If I can show that $X \setminus \{a\}$ is open then I am finished.</p>
<p>I know that ... | SomeOne | 87,286 | <p><strong>Hint:</strong> we could use the fact: Intersections of closed sets is a closed set, in order to construct singleton <span class="math-container">$\{x\}$</span>,if we can show that in Hausdorff then automatically it will be a closed set.</p>
<ol>
<li><p>For any <span class="math-container">$y \in X \ \&... |
1,743,465 | <p>If $a=(1,2,3,4,5)$ is an example of a vector in $\mathbb R^5$, what could be an example of a vector in $\mathbb C^5$ ? is it $a=(1,2,3,4,i5)$ ?</p>
<p>Also, $x=a+ib$ is $2$ dimensional, can a complex number be one dimensional? like when $a=0$ or $b=0$, but if $b=0$ then it is a real number, so can we say that all... | EHH | 133,303 | <p>A vector in $\mathbb{P}^n$ is anything of the form $(x_1,\ldots,x_n)$ for $x_i \in \mathbb{P}$, so you $(1,2,3,4,5i)$ is indeed in $\mathbb{C}^5$ and so is any $(x_1,\ldots,x_n)$ with all $x_i \in \mathbb{C}$ (remember $\mathbb{R}$ is a subset of $\mathbb{C}$).</p>
<p>Dimensionality then depends on your field of sc... |
1,022,485 | <p>I'm working on a homework problem for my discrete math class, and I'm stuck. (Note: I made a post about this earlier, but I read the problem incorrectly, thus the work was wrong, so I deleted the post.)</p>
<p>Prove by mathematical induction that for every integer n $\ge$ 0, $12\mid8^{2n+1}+2^{4n+2}$</p>
<p>I sta... | Przemysław Scherwentke | 72,361 | <p>If you already have $F(n)=12k$ for some $k\in\mathbb{N}$ and
$$
F(n+1)=8^{2(n+1)+1}+2^{4(n+1)+2}=8^{2n+3}+2^{4n+6},
$$
then
\begin{align}
8^{2n+3}+2^{4n+6}
&=64\cdot8^{2n+1}+16\cdot2^{4n+2}\\
&=12(5\cdot8^{2n+1}+2^{4n+2})+4(8^{2n+1}+2^{4n+2})\\
&=12(5\cdot8^{2n+1}+2^{4n+2}+4k),
\end{align}
which is div... |
1,573,947 | <p>I'm trying to see the relationship between the sample variance equation </p>
<p>$\sum(X_i- \bar X)^2/(n-1)$ and the variance estimate, $\bar X(1-\bar X),$ in case of binary samples. </p>
<p>I wonder if the outputs are the same, or if not, what is the relationship between the two??</p>
<p>I'm trying to prove their... | zyx | 14,120 | <p>The first quantity is the standard variance estimator that is unbiased for i.i.d samples from any distribution.</p>
<p>The second quantity is a simplified formula (the simplification being valid only for 0-1 binary data) for calculating exactly, not estimating, the variance of the sample.</p>
<p>Using the second i... |
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