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 |
|---|---|---|---|---|
1,739,052 | <p>After years of mathematics, I am struggling with this simple question. </p>
<p>If we have 3 r.v. $X,Y,Z$ and we have $X$ independent to $Y$ and to $Z$, then do we have that $X$ is also independent to $YZ$ ?</p>
<p>At first sight, I thought that if $X$ is independent to $Y$ and $Z$, it is also independent to the si... | sanketalekar | 327,455 | <p><strong>Short answer: No, $X \perp Y , X \perp Z$ doesn't imply $X \perp YZ$</strong></p>
<p>Let's say you do an experiment where you choose a number randomly from:</p>
<p>{1,2,3,4,6,7,8,9}</p>
<p>Let X = 1 if:</p>
<p>(Your chosen number is even AND less than five) OR (your chosen number is odd AND greater than ... |
1,146,294 | <p>How would one prove that $$(n+1)^{n-1}<n^n \ \forall n>1$$</p>
<p>I have tried several methods such as induction.</p>
| JMoravitz | 179,297 | <p>For completeness' sake, here is a proof coming from combinatorics. The terminology and original proof can be attributed to Konheim and Weiss (1966) and the proof reproduced here by Pollak. I first encountered the problem in Stanley's textbook Enumerative Combinatorics. See also <a href="https://www.math.tamu.edu/... |
3,702,263 | <p>How am I supposed to find the blow-up time of this ODE solution?
<span class="math-container">$$y'=e^x + y^2 \qquad y(0)=0$$</span></p>
<p>The fact that it blows up it's granted by the fact that <span class="math-container">$y' \geq y^2$</span> which solution explodes. But how to estimate the time of explosion?
I s... | AVK | 362,247 | <p>Since <span class="math-container">$\forall x>0 \;\;y'(x)>0$</span>, there exists an inverse function <span class="math-container">$x(y)$</span> defined on <span class="math-container">$[0,+\infty)$</span>. Its derivative is
<span class="math-container">$$\tag{1}
x'(y)=\frac1{e^{x(y)}+y^2}.
$$</span>
Integrati... |
3,702,263 | <p>How am I supposed to find the blow-up time of this ODE solution?
<span class="math-container">$$y'=e^x + y^2 \qquad y(0)=0$$</span></p>
<p>The fact that it blows up it's granted by the fact that <span class="math-container">$y' \geq y^2$</span> which solution explodes. But how to estimate the time of explosion?
I s... | Jack D'Aurizio | 44,121 | <p>If we set <span class="math-container">$y(x)=e^{x/2}f(e^{x/2})$</span> and <span class="math-container">$e^{x/2}=t$</span> we are left with</p>
<p><span class="math-container">$$ e^{-x/2}f(e^{x/2})+f'(e^{x/2}) = 2 + 2\,f(e^{x/2})^2 $$</span>
<span class="math-container">$$ \frac{f(t)}{t}+f'(t) = 2 + 2 f(t)^2,\qqua... |
314,856 | <p>Does anybody know how to prove this identity?</p>
<p>$$\int_0^\infty \prod_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx=\frac{\sqrt{\pi}}{2}\frac{\Gamma \left(a+\frac{1}{2}\right)\Gamma(b+1)\Gamma \left(b-a+\frac{1}{2}\right)}{\Gamma(a)\Gamma \left(b+\frac{1}{2}\right)\Gamma(b-a+1)}$$</p>
... | Marko Riedel | 44,883 | <p>To conclude this we actually do the simplification in terms of the $\Gamma$ function. We have
$$\prod_{k=0}^{b-a-1}\frac{1}{2a+2k+1} =
\frac{1}{2^{b-a}} \frac{\Gamma(a + 1/2)}{\Gamma(b+1/2)}.$$
Furthermore $$\prod_{k=0}^{b-a} (a+k) = \frac{\Gamma(b+1)}{\Gamma(a)}.$$
Finally, $$ \frac{1}{2^{b-a}} {2(b-a) - 1 \choose... |
1,058,019 | <p>Evaluate the below integral:
$$
\int_{0}^{\infty}{x^{\alpha - 1} \over 1 + x}\,{\rm d}x
$$
How to start ?.</p>
| Community | -1 | <ol>
<li><p>Translate the rotation center to the origin: $x'=x-x_c, y'=y-y_c$.</p></li>
<li><p>Rotate around the origin: $x''=(x'-y')/\sqrt2, y''=(x'+y')/\sqrt2$.</p></li>
<li><p>Translate back from the origin: $x'''=x''+x_c,y'''=y''+y_c$.</p></li>
</ol>
|
2,072,729 | <p>Given that $n\in \mathbb{N}$.</p>
<p>I know that it converges to $1$ if $ \alpha=3$ and to $0$ if $\left | \alpha \right |< 3$ intuitively but I am not able to convince myself algebraically. </p>
<p>I tried writing it as $e^{2^{n}ln\left ( \frac{\alpha }{3} \right )}$ which tells me that my exponent needs to c... | hamam_Abdallah | 369,188 | <p>like $\lim_{n\to+\infty}q^n$, we have</p>
<p>$$\lim_{n\to+\infty}\left(\frac{\alpha}{3}\right)^{\large 2^n}$$</p>
<ul>
<li><p>$=+\infty$ if $|\alpha|>3$</p></li>
<li><p>$=1$ if $ |\alpha|=3$</p></li>
<li><p>$=0$ if $ |\alpha|<3$</p></li>
</ul>
|
3,197,362 | <p>this is a proof by contradiction
let y and z be least upper bounds of a set A, such that y != z
so, according to a theorem, L - ε < x,for all x in A. where L is the least upper bound and ε is a positive real number.
so my proof goes like this
according to that theorem, we have</p>
<p>1) y - ε < x, for all ... | Divide1918 | 706,588 | <p>The least upper bound of a set S is some x such that:
<span class="math-container">$\forall{s \in S}, x \ge s.$</span>
<span class="math-container">$\forall {x'\lt x}, x'$</span> is not an upper bound.</p>
<p>For a set S with upper bound <span class="math-container">$B$</span>:
Define a sequence of nested intervals... |
3,819,639 | <p>Integral: <span class="math-container">$J=\int_0^1 \frac{x}{1+x^8}dx$</span></p>
<p>Consider the following assertions:</p>
<p><span class="math-container">$I:J> \frac{1}{4}$</span> and <span class="math-container">$II:J< \frac{\pi}{8}$</span></p>
<p>A. Both are true</p>
<p>B. Only <span class="math-container">... | Community | -1 | <p><span class="math-container">$$\int_0^1\frac x{1+x^8}dx=\frac12\int_0^1\frac 1{1+z^4}dz>\frac12\int_0^1\frac 1{1+z^2}dz=\frac\pi8>\frac14.$$</span></p>
<p>(By WA, <span class="math-container">$J\approx0.433486$</span>.)</p>
|
64,265 | <p>I've been using a DateList plot to visualise property information but I don't think it's the best way display my data. My data is formatted as {time (hours), property} where property is an integer between 1 and 20</p>
<pre><code>data = {{0, 0}, {0.2187, 3}, {0.25, 1}, {0.3715, 15}, {0.868,
1}, {1.261, 15}, {1.4595... | Mr.Wizard | 121 | <p>The terse <a href="http://reference.wolfram.com/language/ref/Graphics.html" rel="nofollow noreferrer"><code>Graphics</code></a> primitives approach:</p>
<pre><code>Graphics[{
EdgeForm[{Black, Thick}],
{ColorData[39][#2], Rectangle[{0, 0}, {#, 1}]} & @@@ Reverse[data]
},
Axes -> {True, False}
]
</code>... |
15,351 | <p>For example, in MATLAB, a panel is available where one can see straightaway which variables are used and their dimension sizes. Is such a feature available in <em>Mathematica</em>? I really find it hard to scroll up and down to see where things are in <em>Mathematica</em>; I just want to see at a glance what's been ... | Fabian | 46,396 | <p>Here is my try:</p>
<pre><code>wscount = Length[Notebooks["Currently Defined Variables"]];
While[wscount > 0,
NotebookClose[Notebooks["Currently Defined Variables"][[1]]];
wscount = wscount - 1;
];
CreateWindow[
PaletteNotebook[
Dynamic[Grid[
Prepend[Select[
With[{expr = ToExpression@#},... |
2,397,837 | <p>Please help! I don't remember what happened last night but I woke up this morning in a prison cell with nothing but my pile of game theory lecture notes. The guard came in and says they have another rational person in the cell next door, and we both have to play a one-shot nonzero-sum matrix game without communicat... | Slepecky Mamut | 180,179 | <p>There exist a several solution concepts, for this situation, but don´t exist any "universal", valid for all types of games. Operation is limited to situations where everyone is guided by the same principle.
Simple examples of principles.</p>
<ol>
<li>If unique NE from set of NE is Pareto dominant, use it (that mean... |
250,426 | <p>I want to factorize any quadratic expressions into two complex-valued linear expressions.</p>
<p>My effort below</p>
<pre><code>a := 1;(*needed*)
p := 2;(*needed*)
q := 3;(*needed*)
f[x_] := a (x - p)^2 + q;(*needed*)
AA := Coefficient[f[x], x^2];
BB := Coefficient[f[x], x];
CC := f[0];
DD = BB^2 - 4 AA CC;
EE = Tim... | Ulrich Neumann | 53,677 | <p>substituting and inverse-substituting</p>
<pre><code>((I*x + a) (I*x + b) /. {a -> I ai, b -> I bi} // Factor)
/. {ai -> -I a, bi -> I b}
(*-((-I a + x) (I b + x))*)
</code></pre>
<p>works too</p>
|
12,544 | <blockquote>
<p>Can <span class="math-container">$n!$</span> be a perfect square when <span class="math-container">$n$</span> is an integer greater than <span class="math-container">$1$</span>?</p>
</blockquote>
<p>Clearly, when <span class="math-container">$n$</span> is prime, <span class="math-container">$n!$</span> ... | Gordon Sail | 219,381 | <p>Hopefully this is a little more intuitive (although quite a bit longer) than the other answers up here. </p>
<p>Let's begin by stating a simple fact : (1) when factored into its prime factorization, any perfect square will have an even number of each prime factor.</p>
<p>If $n$ is a prime number, then $n$ will not... |
1,915,560 | <p><a href="https://math.dartmouth.edu/archive/m105f13/public_html/m105f13notes1.pdf" rel="nofollow">These notes on harmonic sum</a> present the following inequality:</p>
<p>$$\frac{1}{n} < \int_{n-1}^n \frac{dt}{t}$$ for $n≥2$.</p>
<p>How can this be shown? By induction and by evaluation the integral?</p>
| gowrath | 255,605 | <p>See if this makes it any clearer. The red line is the function $f(x) = \frac{1}{x}$ and the red shaded area is the $$\int_{n-1}^{n}{\frac{1}{x}dx}$$ whereas the blue rectangle is a rectangle of $\text{width} = 1$ and $\text{height} =\frac{1}{n}$ and thus it's area = $1 \over n$. Can you see why the proposition is tr... |
1,915,560 | <p><a href="https://math.dartmouth.edu/archive/m105f13/public_html/m105f13notes1.pdf" rel="nofollow">These notes on harmonic sum</a> present the following inequality:</p>
<p>$$\frac{1}{n} < \int_{n-1}^n \frac{dt}{t}$$ for $n≥2$.</p>
<p>How can this be shown? By induction and by evaluation the integral?</p>
| abnry | 34,692 | <p>By the mean value theorem, there is an $m \in (n-1, n)$ such that
$$\int_{n-1}^n \frac{1}{t} dt = \log(n)-\log(n-1) = \frac{1}{m} (n-(n-1)) = \frac{1}{m}.$$
Because $1/t$ is decreasing,
$$\frac{1}{n} < \frac{1}{m} = \int_{n-1}^n \frac{1}{t} dt.$$</p>
<p>Note: It is not necessary to find the antiderivative. Also... |
3,677,964 | <p>High school student here.
I'm trying to find the maximum of this function:
<span class="math-container">$$f(x)=\frac{2x-1}{2-x}.$$</span>
where <span class="math-container">$0 \leq x \leq 1$</span>.
The standard process would involve finding the values of <span class="math-container">$x$</span> such that <span clas... | Sahiba Arora | 266,110 | <p>If <span class="math-container">$z=a+ib,$</span> then <span class="math-container">$\Re z=a, \Im z=b.$</span> So you need to write <span class="math-container">$i^{1/4}$</span> in the form of <span class="math-container">$a+ib$</span> to find its real and imaginary parts.</p>
<p>Note that <span class="math-containe... |
138,801 | <p>ImageSize works fine as an option for Show when used in a call.</p>
<pre><code>Show[image,ImageSize->100]
</code></pre>
<p><a href="https://i.stack.imgur.com/rFJEK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rFJEK.png" alt="enter image description here"></a></p>
<p>But SetOptions does no... | Mr.Wizard | 121 | <h1>Pane</h1>
<p>It appears that someone along the way took out my <a href="https://mathematica.stackexchange.com/a/11987/121">carefully placed</a> <a href="http://reference.wolfram.com/language/ref/Pane.html" rel="nofollow noreferrer"><code>Pane</code></a> which prevented downsizing of graphics. If you just want the... |
484,273 | <p>$$\int_0^1\frac{ \arcsin x}{x}\,\mathrm dx$$</p>
<p>I was looking in my calculus text by chance when I saw this example , the solution is written also but it uses very tricky methods for me ! I wonder If there is a nice way to find this integral.</p>
<p>The idea of the solution in the text is in brief , Assume $y... | Ahmed Hejazi | 690,585 | <p>Let <span class="math-container">$$ y=sin^{-1}x\ \ \Rightarrow x= \sin y,\ \ dx= \cos y dy,$$</span>
then
<span class="math-container">$$
I=\int_{0}^{1}\frac{sin^{-1}x}{x}dx=\int_{0}^{\frac{\pi }{2}}\frac{y}{siny}.cosy=\int_{0}^{\frac{\pi }{2}}\frac{y}{tany}dy.$$</span>
Let
<span class="math-container">$$I(a)=\int_{... |
630,614 | <p>Let $n\ge 2$ be an integer. The symmetric group $S_3$ acts on the set $M_n$ of polynomials in $\mathbb{C}[x_1,x_2,x_3]$ whose monomials are of the form $x_1^{a_1}x_2^{a_2}x_3^{a_3}$ with $0\le a_i\le n$ in the obvious way.</p>
<p>Is there a simple way to describe how $M_n$ is decomposed as a sum of the three irredu... | Leox | 97,339 | <p>Let $V=\langle x_1,x_2,x_3 \rangle$ be the natural representation of $S_3.$
Then
$$
M_n={\rm Sym}^0(V) \oplus {\rm Sym}(V) \oplus \cdots \oplus {\rm Sym}^{3n}(V),
$$
where ${\rm Sym}^k(V)$ is the symmetric power of the vector space $V.$ All that you need is to decompose ${\rm Sym}^k(V)$ into irreducible represe... |
3,365,426 | <p>This is a question about a remark someone said to me without giving much precision. </p>
<p>Suppose you have two nice spaces <span class="math-container">$X,Y$</span> and are trying to build a map <span class="math-container">$X\to Y$</span> with certain nice properties. Suppose for simplicity (no pun intended) tha... | Qiaochu Yuan | 232 | <p>I'll write <span class="math-container">$B^n A$</span> for <span class="math-container">$K(A, n)$</span>. Given that there exists a lift, the space of lifts is the space of homotopy sections of the homotopy pullback of the bundle <span class="math-container">$Y_{n+1} \to Y_n$</span> to <span class="math-container">$... |
8,193 | <p><strong>NB. Some answers appear to be for a question I did not ask, namely, "Why is standardized testing bad?" Indeed, these answers tend to underscore the premise of my actual question, which can be found above.</strong></p>
<p>As a foreigner who has spent some time in the US, it seems to me that in the U... | Gerhard Paseman | 3,468 | <p>Not an answer, but perhaps a clue.</p>
<p>Richard Feynman in one of his books (I think the title began "Surely You're Joking...", or it might have began "What Do You Care..."; hopefully someone else will recall the correct and full title) had a couple of stories regarding foreign high-level education (Brazil? physi... |
2,929,238 | <p>Recently I have come arcross the following fraction</p>
<blockquote>
<p><span class="math-container">$$\dfrac{z^2}{(2z^2+3)((s^2+1)z^2+1)}$$</span></p>
</blockquote>
<p>Hence I have encountered this fraction within a task of integration I want to do a partial decomposition. First of all I rewrote it as following... | M. Winter | 415,941 | <p><strong>Hint:</strong> There are two equivalence classes: <span class="math-container">$\{0\}$</span> and <span class="math-container">$\Bbb R^n\setminus \{0\}$</span>.</p>
|
3,942,070 | <p>Prove that <span class="math-container">$\mathbb{Z}_8$</span> is not an internal direct product of two proper subgroups.</p>
<p>I'm really struggling to get the wheels turning on this one. I know what it means for group to be an internal product of two subgroups, I'm just not sure how to get started in showing that ... | Hagen von Eitzen | 39,174 | <p>The orders of the factors would have to be <span class="math-container">$2$</span> and <span class="math-container">$4$</span>. Then the product has no elements of order <span class="math-container">$>4$</span>, but <span class="math-container">$\Bbb Z_8$</span> has.</p>
|
2,136,734 | <p>When we divide $n^5+5$ by $n+5$, we get a remainder $-620$, i.e.,
$n^5+5=K(n+5)-620$ , now how to proceed further?</p>
| JHF | 50,427 | <p>This is what comes to my mind, but it doesn't address your question exactly. If $\pi_2 X = \pi_2 Y = 0$, then you can take $F$ to be the homotopy fiber of the map $X \vee Y \to X \times Y$. Then you will have a short exact sequence $$0 \to \pi_1 F \to \pi_1 (X \vee Y) \to \pi_1(X \times Y).$$ In general, you will... |
3,057,874 | <blockquote>
<p>The following formula shall be proved by induction:
<span class="math-container">$$F(m+n) = F(m-1) \cdot F(n) + F(m) \cdot F(n+1)$$</span>
Where <span class="math-container">$F(i), i \in \mathbb{N}_0$</span> is the Fibonacci sequence defined as:
<span class="math-container">$F(0) = 0$</span>, <s... | ODF | 397,960 | <p>You can get away with inducting on just <span class="math-container">$n$</span>:</p>
<p><span class="math-container">$$ \begin{align} F(m + n) & = F(m + (n - 1)) + F(m + (n -2))\\
& = F(m-1)F(n-1) + F(m)F(n) + F(m-1)F(n-2) + F(m)F(n-1) \\
& = F(m-1)[F(n-1) + F(n-2)] + F(m)[F(n)+F(n-1)] \\
& = F(m-1... |
3,909,307 | <p>I'm having trouble with this question. I tried to make a relation between x, y, and z using the points given, and I came up with <span class="math-container">$x=z=-y+1$</span>, which led to trying to solve the integral</p>
<p><span class="math-container">$$\int_{0}^{1}(y^2+y)\sqrt{1+(2y+1)^2}dy$$</span></p>
<p>But I... | paulinho | 474,578 | <p>Here is another (slightly) different way to use the fact that the adjugate is a rank <span class="math-container">$1$</span> symmetric matrix. If it has rank one and furthermore there does not exist some <span class="math-container">$k$</span> such that the determinant of the <span class="math-container">$(k, k)$</s... |
2,956,158 | <blockquote>
<p>Given <span class="math-container">$z = \cos (\theta) + i \sin (\theta)$</span>,
prove <span class="math-container">$\dfrac{z^{2}-1}{z^{2}+1} = i \tan(\theta)$</span></p>
</blockquote>
<p>I know <span class="math-container">$|z|=1$</span> so its locus is a circle of radius <span class="math-contai... | lab bhattacharjee | 33,337 | <p><span class="math-container">$\dfrac{z^2-1}{z^2+1}$</span> reminds me of <a href="https://math.stackexchange.com/questions/577231/what-are-the-names-in-english-for-alterando-invertendo-componendo-and-dividend">Componendo and Dividendo</a>?</p>
<p><span class="math-container">$\dfrac{z^2}1=(\cos\theta+i\sin\theta)^2... |
2,967,291 | <p><a href="https://i.stack.imgur.com/VA5AP.png" rel="nofollow noreferrer">plot</a></p>
<p>I am trying to prove an inequality connecting the function <span class="math-container">$f(x) = \log(1+x^2)$</span> and the absolute value of its derivative i.e <span class="math-container">$ |f'(x)| =\big|\frac{(2x)}{(1+x^2)}... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$ log (1+x^{2}) \leq x^{2}$</span> so all you need is <span class="math-container">$x^{2} \leq \frac {2|x|} {1+x^{2}}$</span> which is true if <span class="math-container">$|x| \leq 1$</span>. We also get <span class="math-container">$ log (1+x^{2}) \leq |f'(x)|^{2}$</span> by almost th... |
3,421,455 | <p>I'm not sure if I'm supposed to use integration by substitution here, but here's the question:</p>
<p><span class="math-container">$$\int ^{10}_{0}f\left( x\right) dx=25$$</span>
Find the value of
<span class="math-container">$$\int ^{e^{2}}_{1}\dfrac {f\left( 5\times \ln \left( x\right) \right) }{x}dx$$</span>
Do ... | L-- | 517,726 | <p>Putting <span class="math-container">$u=\ln(x)$</span>, the limit will now vary from <span class="math-container">$0$</span> to <span class="math-container">$2$</span>.
<span class="math-container">$$ \int_{1}^{e^2} \frac{f(5 \ln(x))}{x}dx= \int_{0}^{2}f(5u)du$$</span>.
Again if I put <span class="math-container">$5... |
1,563,105 | <p>I have a equation of motion for a forced pendulum show below
$$
{d^2\theta\over dt^2} = -{g\over L}\sin\theta + C\cos\theta\sin(Dt)
$$
$L=10$ cm, $C=2\ \hbox{s}^{-2}$ and $D=5\ \hbox{s}^{-1}$.</p>
<p>I am trying to make this equation dimensionless by setting the follow equations</p>
<p>$$\omega^2 = g/L,\quad ... | hmakholm left over Monica | 14,366 | <p>Suppose you have a representation of $x_1\lor x_2\lor x_3$ as 2CNF clauses, possibly involving with hidden variables. (This would "represent" the three-way disjunction in the sense that the 2CNF is satisfiable together with any consistent combination of $(\neg)x_i$ that contains at least one positive $x_i$, and is n... |
4,461,144 | <p>Consider set <span class="math-container">$S = \{2^0, 2^1, 2^2, 2^3, 2^4, 2^5, \dots, 2^{2003}, 2^{2004}\}$</span> and <span class="math-container">$\log2 = 0.3010$</span>. Find the number of elements in the set <span class="math-container">$S$</span> whose most significant digit is 4.</p>
<p>It is also known that t... | paw88789 | 147,810 | <p>The leading digits of powers of an integer greater than <span class="math-container">$1$</span> will follow Benford's law quite closely (especially in the long run).</p>
<p>According to Benford's law, for appropriate data sets, the proportion of items with leading digit <span class="math-container">$4$</span> should... |
1,399,601 | <p>Could you help me with proving:</p>
<p>Let $f$ be an analytic function defined in on upper half plane(UHP). Suppose that $|f(z)|<1$ for all $z$ in UHP. Prove that for every $z$ in UHP</p>
<p>$$
|f'(z)|\leq{\frac{1}{2\operatorname{Im} z}} .
$$</p>
<p>I guess that I need to use Cauchy's estimate but I am not sur... | Paolo Leonetti | 45,736 | <p><strong>Partial answer:</strong> By Cauchy estimate
$$
|f^\prime(z)| \le \frac{1}{R}\max_{t \in \partial D_R(z)}|f(t)|<\frac{1}{R}.
$$
holds for all $R>0$ for which $f$ is holomorphic in a neighborhood of the closed disk $\overline{D_R(z)}$. It implies that
$$
|f^\prime(z)| \le \frac{1}{|z|}.
$$
Now, if $|z|\g... |
2,368,453 | <p>I am trying to derive a proof of the associative property of addition of complex numbers using only the properties of real numbers.</p>
<p>I found the following answer but was hoping someone can explain why it is correct, since I am not satisfied with it (From <a href="https://math.stackexchange.com/questions/10786... | fleablood | 280,126 | <p>The person who did that solution just made a mess of it. S/he seems to be believing we can treat the imaginary $bi$ and real components $a$ and the imaginary unit $i$ as real summands and rearrange them and distribute them. We can. But that has to be proven.</p>
<p>A better proof:</p>
<p>$z_1 + (z_2 + z_3) = \{... |
20,768 | <p>After a long period a post is deleted, is it still possible to edit the deleted post?</p>
| robjohn | 13,854 | <p>In <a href="https://meta.stackexchange.com/a/5222">this FAQ answer</a>, it says</p>
<blockquote>
<p>Self-deleted posts can be viewed and undeleted by their original authors. However, self-deleted questions cannot be edited by their authors unless undeleted first. Self-deleted answers can be edited by the author w... |
330,710 | <p>Input: a set of $n$ points in general position in $\mathbb{R}^2$.</p>
<p>Output: the pair of points whose slope has the largest magnitude.</p>
<p>Time constraint: $O(n \log n)$ or better.</p>
<p>Please don't spoil the answer for me - I'm just stuck and looking for a nudge in the right direction. Thanks!</p>
| Michael Biro | 29,356 | <p>Hint:</p>
<p>Try recursion. Split the points into two halves, find the slopes for each half, then you just need to figure out the max slope defined by two separated point sets.</p>
|
1,000,025 | <p>Well the question is the title. I tried to grab at some straws and computed the Minkowski bound. I found 19,01... It gives me 8 primes to look at. I get</p>
<p>$2R = (2, 1 + \sqrt{223})^2 = P_{2}^{2}$</p>
<p>$3R = (3, 1 + \sqrt{223})(3, 1 - \sqrt{223}) = P_{3}Q_{3}$</p>
<p>$11R = (11, 5 + \sqrt{223})(11, 5 - \sqr... | Cam McLeman | 20,762 | <p>Hints/relevant information: </p>
<p>1) If the ideal class group has order 3, then it can't have any elements of order 2. That tells you something about your factorization of $2R$.</p>
<p>2) One of your ideal classes is always the principal ideal class.</p>
<p>3) That leaves two ideal classes, $A$ and $B$. T... |
2,766,332 | <blockquote>
<p>If $a,b$ are elements of a group and $a^2=e, b^6=e, ab=b^4a$, then find the order of $ab$ and express ${(ab)}^{-1}$ in terms of $a^mb^n$ and $b^ma^n$ </p>
</blockquote>
<p>I could find the order of $ab$ to be 6 but struggling to find ${(ab)}^{-1}$ in terms of $a^mb^n$ and $b^ma^n$.</p>
<p>Please hel... | orangeskid | 168,051 | <p>We have
$$b = a b^4 a = (a b a)^4= (b^4)^4=b^{16}= b^6 \cdot b^6 \cdot b^4 = b^4$$
so $b^3=e$ and $ab= b a$. From here it is easy to see that the order of $ab$ is $\le 6$ (it is in fact the product of the orders of $a$, $b$, these being relatively prime). Moreover, $(ab)^{-1}=a^{-1}b^{-1}=a b^2= b^2 a$.</p>
|
155,359 | <p>I tried to compute the 40-th iteration by FindRoot but without result.
Please If any body solve that, I would to thank him. This question has a
nonlinear system. And the solution for n=1 is solved. But there is problem
when n=2,3, ...40.</p>
<p>The following code for computing the four unknowns for the n... | yode | 21,532 | <p>More cleaner. Just make a note here.</p>
<pre><code>ImportString[
RunProcess[$SystemShell, "StandardOutput", "dir\n"], "Text",
CharacterEncoding -> "CP936"]
</code></pre>
|
1,048,045 | <blockquote>
<p>$$\int_0^{\frac{\pi}{2}}\arctan\left(\sin x\right)dx$$</p>
</blockquote>
<p>I try to solve it, but failed. Who can help me to find it?</p>
<p>I encountered this integral when trying to solve $\displaystyle{\int_0^\pi\frac{x\cos(x)}{1+\sin^2(x)}\,dx}$.</p>
| David H | 55,051 | <p>Using the integral definition of the arctangent function, we may write $$\arctan{\left(\sin{x}\right)}=\int_{0}^{1}\mathrm{d}y\,\frac{\sin{x}}{1+y^2\sin^2{x}},$$</p>
<p>thus, transforming the integral into a double integral. Changing the order of integration, we find:</p>
<p>$$\begin{align}
\mathcal{I}
&=\int_... |
264,609 | <p>Question on theory: If $\lim_{x\to a^+}F(x)=L$ and $\lim_{x\to a^-}F(x)$ doesn't exist, then does the $\lim_{x\to a}F(x)$ exists and is $L$? </p>
<p>Thanks. </p>
| coffeemath | 30,316 | <p>If the function $F(x)$ has domain say $[a,b]$ where $b>a$, so that $F(x)$ is not even <em>defined</em> for $x<a$, and if $\lim_{x \to a^+}F(x)=L$, then using some definitions of limit, the limit exists. Sometimes limit is defined to mean that, if $x$ approaches $a$ such that $x$ stays in the domain of $F(x)$, ... |
222,863 | <p>Computing $\displaystyle \sum_{k\ge2}k(1-p)^{k-2}$, $p\in ]0,\space1[$</p>
<p><a href="http://www.wolframalpha.com/input/?i=sum%28k%2a%281-p%29%5E%28k-2%29,%202,%20infinity%29" rel="nofollow">WolframAlpha</a> says it is $\cfrac {p+1}{p^2}$ but I couldn't get that value but anyway here is what I did:</p>
<p>$$\disp... | Brian M. Scott | 12,042 | <p>You shifted the index when you should not have done so:</p>
<p>$$\begin{align*}
\sum_{k\ge 2}k(1-p)^{k-1}&=-\left(\sum_{k\ge 2}(1-p)^k\right)'\\
&=-\left(\frac{(1-p)^2}p\right)'\\
&=\frac{(1-p)^2+2p(1-p)}{p^2}\\
&=\frac{1-p^2}{p^2}\\
&=\frac{(1-p)(1+p)}{p^2}\;.
\end{align*}$$</p>
<p>You had $-\... |
916,120 | <p>What is negation of <strong>All birds can fly.</strong></p>
<p>The question seems bit funny but i don't know which of the following two sentences is correct:</p>
<ol>
<li>Some birds can not fly</li>
<li>There is at least one bird which can not fly.</li>
</ol>
<p>Both the sentence seems almost logically same. But ... | amWhy | 9,003 | <p>$B(x):$ x is a bird. </p>
<p>$F(x):$ x can fly.</p>
<p>All birds can fly: $$\forall x(B(x)\rightarrow F(x))$$</p>
<p>Negation of the above $$\lnot \forall x(B(x)\rightarrow F(x))\equiv \exists x \Big(\lnot\big(\lnot B(x) \lor F(x)\big)\Big) \equiv \exists x (B(x) \land \lnot F(x)$$</p>
<p>You can certainly trans... |
1,432,429 | <p>Problem to finish the question: If $n > 4$ is compound then $(n-1)!\equiv 0\pmod n$.
If $n = a\cdot b$ there is no problem, once $a, b$ are factors of $(n-1)!$. The problem is when $ n = p^2$. I know that once $p > 4$ then $p^2 \ge 3$. But, how can I justify that $p^2$ is a factor of $(n-1)!$?</p>
<p>Thanks ... | porridgemathematics | 270,179 | <p>Let $n = p^2$, clearly $p$ must be contained in $(n-1)! = (n-1)(n-2)\cdots 1.$</p>
<p>So $p|(n-1)!$ . Note that $2p < n$, why? Suppose $2p > n$, then $p > {n\over 2}$, so that $p^2 > \frac{n^2}{4}$, but remember you have $n>4$, if $p^2>\frac{n^2}{4}$, then $p^2>n^2$, a contradiction. So $2p &l... |
648,809 | <p>I have a matrix $A$ given and I want to find the matrix $B$ which is closest to $A$ in the frobenius norm and is positiv definite. $B$ does not need to be symmetric.</p>
<p>I found a lot of solutions if the input matrix $A$ is symmetric. Are they any for a non-symmetric matrix $A$? Is it possible to rewrite the pro... | Michael Grant | 52,878 | <p>A real, square matrix $B$ is positive definite iff $v^TBv> 0$ for all $v\neq 0$. But
$$v^TBv = \tfrac{1}{2}(v^TBv+v^TB^Tv) = \tfrac{1}{2}v^T(B+B^T)v.$$
It follows then that $B$ is positive definite iff $B+B^T$ is positive definite. Therefore, your model becomes
$$\begin{array}{ll}
\text{minimize} & \|A-B\|_F ... |
18,280 | <p>More specifically, is it true that a representation of $\dim < p+1$ of the algebraic group $SL_2(\mathbb{F}_p)$ is always completely reducible? (of course above this dimension there are non completely reducible examples)</p>
<p>More general results that might help in this direction are also welcome.</p>
<p>Than... | Jim Humphreys | 4,231 | <p>The essential work in this direction was published from 1994 on by J.-P. Serre
and J.C. Jantzen, concerning both algebraic groups and related finite groups
of Lie type. Related papers by R. Guralnick and G.J. McNinch followed. There are uniform dimension bounds for complete reducibility, stricter in rank 1. For... |
4,508,474 | <p>I would like to solve the ODE
<span class="math-container">$$\left(x-k_1\right)\frac{dY}{dx} + \frac{1}{a}x^2\frac{d^2 Y}{d x^2}-Y+k_2=0$$</span>
with boundary conditions <span class="math-container">$Y\left(k_3\right)=0$</span> and <span class="math-container">$\frac{d Y}{d x}\left(\infty\right)=1-b$</span> (meanin... | Oscar Lanzi | 248,217 | <p>One answer has to do with <a href="https://en.wikipedia.org/wiki/Conformal_map" rel="noreferrer">con-formal mapping</a>. Basically you map the points in a plane in such a way that angles between tangents to curves at their point of intersection are preserved, except perhaps at isolated points. The number of points o... |
4,508,474 | <p>I would like to solve the ODE
<span class="math-container">$$\left(x-k_1\right)\frac{dY}{dx} + \frac{1}{a}x^2\frac{d^2 Y}{d x^2}-Y+k_2=0$$</span>
with boundary conditions <span class="math-container">$Y\left(k_3\right)=0$</span> and <span class="math-container">$\frac{d Y}{d x}\left(\infty\right)=1-b$</span> (meanin... | gnasher729 | 137,175 | <p>A circle in the plane can be described by three parameters: The centre (x, y; 2 parameters) and the radius. As a result, you can determine a circle when you are given three points on the circle. (That is of course something you will have to prove. But the centre has equal distance from any two points; and given thre... |
847 | <p>Is there any mathematical significance to the fact that the law of cosines...</p>
<p>$$
\cos(\textrm{angle between }a\textrm{ and }b) = \frac{a^2 + b^2 - c^2}{2ab}
$$</p>
<p>... for an impossible triangle yields a cosine $< -1$ (when $c > a+b$), or $> 1$ (when $c < \left|a-b\right|$)</p>
<p>For exampl... | Qiaochu Yuan | 232 | <p>One can prove the triangle inequality in any abstract inner product space, such as a Hilbert space; it is a consequence of the Cauchy-Schwarz inequality $\langle a, b \rangle^2 \le ||a||^2 ||b||^2$. In order for the triangle inequality to fail, the Cauchy-Schwarz inequality has to fail, and in order for Cauchy-Schw... |
2,140,607 | <blockquote>
<p>Find $$\lim_{n \to \infty} \int_{1}^{n}\frac{nx^{1/2}}{1+nx^2}dx$$</p>
</blockquote>
<p>I have tried tackling this problem using the DCT but I am not quite sure if I have the right answer.
To begin with, $$\int_{1}^{n}\frac{nx^{1/2}}{1+nx^2}dx=\int_{1}^{\infty}\frac{nx^{1/2}}{1+nx^2}1_{[1,n]}dx$$</p>... | Jack D'Aurizio | 44,121 | <p>$$\lim_{n\to +\infty}\int_{1}^{n}\frac{n\sqrt{x}}{1+nx^2}\,dx = \lim_{n\to +\infty}\left(\int_{1}^{+\infty}\frac{\sqrt{x}}{\frac{1}{n}+x^2}\,dx-\int_{n}^{+\infty}\frac{\sqrt{x}}{\frac{1}{n}+x^2}\right)$$
where
$$ \left|\int_{n}^{+\infty}\frac{\sqrt{x}}{\frac{1}{n}+x^2}\right|\leq \int_{n}^{+\infty}x^{-3/2}\,dx=\frac... |
485,190 | <p>I am trying to prove that every neighborhood of a boundary bound contains a point in interior and $X \setminus A$ where $A$ is the set in consideration. I am given the following definitions</p>
<p>(1) $(X,\mathcal{T})$ is a topological space if $X,\emptyset \in \mathcal{T}$, any arbitrary union of open sets in $\ma... | Martin Argerami | 22,857 | <p>Using the definitions straightforwardly:</p>
<p>Let $x\in\partial A$, and let $B$ be a neighbourhood of $x$. Suppose that $A\cap B=\emptyset$. This implies that $x$ is an interior point of $X\setminus A$: indeed, $A\subset X\setminus B$, which is a closed set; as $x\not\in X\setminus B$, this shows that $x\not\in\b... |
3,187,680 | <p>We have differential equation:</p>
<p><span class="math-container">$$(2 x y^2 + 2 y) + (2 y x^2 + 2 x) y' = 0$$</span> </p>
<p>We can easily check that it is a complete differential equation and with solving methods of Complete differential equation I found that it is:</p>
<p><span class="math-container">$$\frac{... | E.H.E | 187,799 | <p><span class="math-container">$$x^2y^2 + 2xy = C$$</span>
<span class="math-container">$$x^2y^2 + 2xy +1= C+1$$</span>
<span class="math-container">$$(xy+1)^2=C+1$$</span>
<span class="math-container">$$xy+1=\mp \sqrt{C+1}$$</span>
<span class="math-container">$$xy=-1\mp \sqrt{C+1}$$</span>
or
<span class="math-conta... |
3,187,680 | <p>We have differential equation:</p>
<p><span class="math-container">$$(2 x y^2 + 2 y) + (2 y x^2 + 2 x) y' = 0$$</span> </p>
<p>We can easily check that it is a complete differential equation and with solving methods of Complete differential equation I found that it is:</p>
<p><span class="math-container">$$\frac{... | Mohammad Riazi-Kermani | 514,496 | <p>Both solutions are correct.</p>
<p>Note that if <span class="math-container">$xy=C$</span>, then <span class="math-container">$$xy(xy+2)=C(C+2)=K$$</span> and If <span class="math-container">$xy(xy+2)=K$</span> you can solve for <span class="math-container">$xy$</span> and get <span class="math-container">$xy=C$</s... |
1,270,802 | <p>I just finished an exam, it has the following question: Where is the point on the plane $3x + 5y + z = 18$ has the shortest distance to $(0,0,0)$?</p>
<p>I found this question similar: <a href="https://math.stackexchange.com/questions/355460/find-the-point-on-the-plane-2x-y-2z-20-nearest-the-origin">Find the point ... | Asinomás | 33,907 | <p>$6526-8437=-1911$. You can type it into google to check.</p>
<p>What I do when I have to solve this sort of problem is swap the order of the numbers so the big number is on top.</p>
<p>So I calculate $8437-6526=1911$. Then I multiply it by minus one to get $6526-8437$, which is what I wanted.</p>
<p>This works b... |
2,529,616 | <p>$ M= \left[ {
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 1\\
0 & 1 & 1 & 1 & 0\\
0 & 1 & 1 & 1 & 0\\
0 & 1 & 1 & 1 & 0\\
1 & 0 & 0 & 0 & 1\\
\end{array} } \right] $</p>
<p>What is the product of positive eigen values for the above matrix... | Ben Grossmann | 81,360 | <p>Notably, the matrix $M$ has rank $2$, which is to say that it will have $2$ non-zero eigenvalues. Both of these eigenvalues will be positive, since the matrix is positive semidefinite. </p>
<p>Let $\lambda_i$ denote the eigenvalues of $M$. We can note that
$$
\operatorname{tr}(M^2) = \sum_{i}\lambda_i^2\\
\operat... |
3,764,846 | <blockquote>
<p>A martingale <span class="math-container">$\{X_n\}$</span> is bounded in <span class="math-container">$L^2$</span> by definition if <span class="math-container">$\sup\limits_nEX_n^2<\infty$</span>. Show that a martingale <span class="math-container">$\{X_n\}$</span> is bounded in <span class="math-co... | Kavi Rama Murthy | 142,385 | <p>Hint: It is easy to verify from the definition of a martingale that <span class="math-container">$(X_{n+1}-X_n)$</span> is an orthogonal sequence. The result now follows from general Hilbert space Theory: <span class="math-container">$\sum\limits_{n=1}^{N}\|X_{n+1}-X_n\|^{2}=\|x_N\|^{2}-\|x_1\|^{2}$</span> which is ... |
1,690 | <p>I like to ask true-false questions on exams, because I feel that they can be an efficient way to assess students' understanding of concepts and ability to apply them to somewhat unfamiliar situations. In general, I'm very happy with true-false questions, but there is one annoyance that I have never figured out how ... | vonbrand | 123 | <p>Instead of "true/false" it might be better to have mutiple choice questions. Around here the physics people are fond of them, but ask for explanation of the result. AFAIU, only questions with the correct choice are graded. Not my choice at all, but somebody you might consider.</p>
<p>I prefer to give one question ... |
1,690 | <p>I like to ask true-false questions on exams, because I feel that they can be an efficient way to assess students' understanding of concepts and ability to apply them to somewhat unfamiliar situations. In general, I'm very happy with true-false questions, but there is one annoyance that I have never figured out how ... | Dave L Renfro | 745 | <p>I rarely asked true/false questions, but when I did I usually made them worth $2$ or $3$ points each (not $1$ point each), and incorrect answers with a somewhat appropriate and at least a somewhat correct explanation could earn one or two points back. In fact, if it was clear from their explanation that they acciden... |
44,623 | <p>For a bilinear function $T$, it can be shown that $\lVert T(x,y)\rVert\leq C \lVert x\rVert \lVert y \rVert$</p>
<p>I saw some books say a bilinear function T is Lipschitz with Lipschitz constant $C$ given the above inequality holds. </p>
<p>Now I'm confused because a function T is Lipschitz if $\lVert T(\alpha)-T... | Steven Stadnicki | 785 | <p>While it's not quite meaningless to average the direction of the wind, it's ill-defined in many cases (imagine that you have a perfect east wind and a perfect west wind and you want to find the direction of their average!), and probably not really what you're after. Assuming that you have velocity data for the wind... |
44,623 | <p>For a bilinear function $T$, it can be shown that $\lVert T(x,y)\rVert\leq C \lVert x\rVert \lVert y \rVert$</p>
<p>I saw some books say a bilinear function T is Lipschitz with Lipschitz constant $C$ given the above inequality holds. </p>
<p>Now I'm confused because a function T is Lipschitz if $\lVert T(\alpha)-T... | Chris Taylor | 4,873 | <p>Averaging the sines of the inputs and taking the inverse sine of the result can't give you a sensible answer, since the range of the inverse sine function is $180^\circ$ rather than $360^\circ$.</p>
<p>You probably want to take the magnitude of the wind into account. For example, if you have a wind blowing to the e... |
44,623 | <p>For a bilinear function $T$, it can be shown that $\lVert T(x,y)\rVert\leq C \lVert x\rVert \lVert y \rVert$</p>
<p>I saw some books say a bilinear function T is Lipschitz with Lipschitz constant $C$ given the above inequality holds. </p>
<p>Now I'm confused because a function T is Lipschitz if $\lVert T(\alpha)-T... | heltonbiker | 27,435 | <p>Question is old, but I found this in Wikipedia, which might be the "average of sin" stuff you wrote in your question:</p>
<blockquote>
<p>"A simple way to calculate the mean of a series of angles (in the
interval [0°, 360°)) is to calculate the mean of the cosines and sines
of each angle, and obtain the angle... |
3,249,064 | <p>I've read the question: "Why does the derivative of sine only work for radians?" and I can follow the derivation for the derivative of sine when measured in degrees, but the result confuses me. </p>
<p>Does this mean the derivative of the sine changes values when measured in different units? </p>
<p>For example, w... | GReyes | 633,848 | <p>If you call <span class="math-container">$\bar\sin$</span> the function that associates to the angle measured in degrees its sine, you clearly have
<span class="math-container">$$
\bar\sin x=\sin(\pi x/180).
$$</span>
where <span class="math-container">$\sin$</span> is the usual sine function. Then, by the chain rul... |
1,803,843 | <p>Most characterizations of pointwise continuous functions defined on an interval rely on "local" properties. That is, a function is continuous at $x_0 \in I$ if it satisfies some property (epsilon-delta, sequential, oscillation, etc); a function is continuous on an interval if it is continuous at all $x \in I$. </p>
... | jdods | 212,426 | <p>This:
$$( \bullet ) \hspace{-16px}\color{red}{\left(\begin{matrix}\quad\quad \\ \quad\end{matrix}\right)} \quad\quad\text{ bigger $\epsilon$ }$$
where $\color{red}{\left(\begin{matrix}\quad\quad \\ \quad\quad\end{matrix}\right)}$ is the set you want a limit point of and $( \bullet )$ is an epsilon neighborho... |
3,378,754 | <p>if <span class="math-container">$A=1$</span>, <span class="math-container">$y \sim N(1,\sigma^2)$</span></p>
<p>if <span class="math-container">$A=2$</span>, <span class="math-container">$y \sim N(2,\sigma^2)$</span></p>
<p><span class="math-container">$Pr(A=1)=0.5$</span></p>
<p><span class="math-container">$Pr(... | gt6989b | 16,192 | <p><strong>HINT</strong>
Great job arguing the statement was false.</p>
<p>However, I think the intent was to show <span class="math-container">$f(A+B) = f(A)+f(B)$</span> which works out well. </p>
|
128,219 | <p>There is a function <code>FindSequenceFunction</code> in Mathematica, that can identify a sequence of integers based on a few first elements. But what if I have a set of finite sequences <code>sec[n]</code> that grows with <code>n</code>? For example:</p>
<pre><code>sec[0]={1}
sec[1]={1, 1}
sec[2]={1, 6, 1}
sec[3]=... | Michael E2 | 4,999 | <p><code>FindSequenceFunction</code> can find the each individual sequence <code>sec[k]</code> in terms of a recursion with polynomial coefficients:</p>
<pre><code>FindSequenceFunction[PadRight[sec[5], 20]]
(*
DifferenceRoot[
Function[{\[FormalY], \[FormalN]}, {(-66 + 23 \[FormalN] -
2 \[FormalN]^2) \[... |
292,594 | <p>Today I have encounter an integral:</p>
<p>$$\int_0^{\infty}\left[\frac{1}{3}\frac{\sin x}{x}+\cdots+\frac{1\times4\times\cdots\times(3n-2)}{3^nn!}\left(\frac{\sin x}{x}\right)^n+\cdots\right]\text{d}x$$</p>
<p>since $$\int_0^{\infty}\frac{\sin x}{x}=\frac{\pi}{2}$$</p>
<p>so I want to estimate $$\sum_{n=1}^{\inf... | André Nicolas | 6,312 | <p>Informal solution: Take the logarithm of the product. We have $\log(1-t)=-t+O(1/t^2)$. Summing, we get
$$-\frac{2}{3}\left(1+ \frac{1}{2}+\cdots+\frac{1}{n}\right)+O(1/n)=-\frac{2}{3}H_n+O(1/n),$$
where $H_n$ is the $n$-th harmonic number. </p>
<p>But the difference between $H_n$ and $\log n$ is bounded.
It follo... |
225,351 | <p>In one of the eight Thurston geometries there is the geometry of the universal cover of $SL(2, \mathbb{R})$. But from the algebraic point of view $PSL(2,\mathbb{R})$ is sufficient for building 3-manifolds i.e. we let the group act on itself and then quotient out a discrete subgroup of it. This is what we do for th... | Mikhail Katz | 28,128 | <p>Your question "why pass to the universal cover?" is really a topological question that has little to do with the special case of solvgeometry, so it may be helpful to point out that you wouldn't want to study flat manifolds by starting with the tori, but rather you start with euclidean spaces to get more examples as... |
96,720 | <p>Informally, Löb's theorem (<a href="http://en.wikipedia.org/wiki/L%25C3%25B6b%2527s_theorem" rel="noreferrer">Wikipedia</a>, <a href="http://planetmath.org/?op=getobj&from=objects&id=9381" rel="noreferrer">PlanetMath</a>) shows that:</p>
<blockquote>
<p>a mathematical system cannot assert its own soundnes... | Dan Piponi | 1,233 | <p>Löb's theorem can be used to show that there exist equilibria in games like prisoner's dilemma when the participants are computer programs that can read each other's source code.</p>
<p>If player 1 can show that player 2 will cooperate when given player 1's source code, and vice versa, then we can have an equilibri... |
260,857 | <p>I have the following code</p>
<pre><code>rand = {1, 2, 3};
el = EdgeList[CompleteGraph[5]]
g = CompleteGraph[5,
EdgeLabels -> Table[el[[i]] -> RandomChoice[rand], {i, Length[el]}]]
</code></pre>
<p>What I want is to get the labels of each edge of my graph in a list. Is there a way to do it?</p>
| Bob Hanlon | 9,362 | <pre><code>SeedRandom[1234];
rand = {1, 2, 3};
el = EdgeList[CompleteGraph[5]];
g = CompleteGraph[5,
VertexLabels -> "Name",
EdgeLabels -> Table[el[[i]] -> RandomChoice[rand], {i, Length[el]}]]
</code></pre>
<p><a href="https://i.stack.imgur.com/A60Vd.png" rel="nofollow noreferrer"><img src="ht... |
2,023,400 | <p><span class="math-container">$\textbf{Question:}$</span> Find a basis for the vector space of all <span class="math-container">$2\times 2$</span> matrices that commute with <span class="math-container">$\begin{bmatrix}3&2\\4&1\end{bmatrix}$</span>, which is the matrix <span class="math-container">$B$</span>.... | copper.hat | 27,978 | <p>Here is a way of finding one basis:</p>
<p>Let $L(A) = AB-BA$, then $A$ commutes with $B$ <strong>iff</strong> $A \in \ker L$. Using
a standard basis, find the null space of $L$ and use this to determine a
basis of $\ker L$.</p>
<p>This can be simplified a little since $B$ has a full set of eigenvectors.</p>
<p>... |
14,486 | <h2>Speculation and background</h2>
<p>Let <span class="math-container">$\mathcal{C}:=\mathrm{CRing}^\text{op}_\text{Zariski}$</span>, the affine Zariski site. Consider the category of sheaves, <span class="math-container">$\operatorname{Sh}(\mathcal{C})$</span>.</p>
<p>According to <a href="https://ncatlab.org/nlab/s... | LRG | 781 | <p><span class="math-container">$\newcommand\Aff{\mathrm{Aff}}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Hom{Hom}\newcommand\Sh{\mathrm{Sh}}$</span>A morphism of sheaves <span class="math-container">$f: X \to Y$</span> in the fpqc topology on <span class="math-container">$\Aff$</span> [covers are <em>finite</e... |
65,923 | <p>The <a href="http://en.wikipedia.org/wiki/Parity_of_a_permutation">sign of a permutation</a> $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula</p>
<p>$${\rm sgn}(\sigma) = (-1)^m$$</p>
<p>where $m$ is the number ... | supermartingale | 242,606 | <p>$sgn(\sigma)=\prod_{i<j}\frac{\sigma(i)-\sigma(j)}{i-j}$ is quadratic, but straightforward to code.</p>
|
395,118 | <p>In a city of over $1000000$ residents, $14\%$ of the residents are senior citizens. In a simple random sample of $1200$ residents, there is about a $95\%$ chance that the percent of senior citizens is in the interval [pick the best option; even if you can provide a sharper answer than you see in the choices, please ... | Stahl | 62,500 | <p>I think you mean $\ln(x/y) = \ln x - \ln y$. If you've already shown that $\ln x^r = r\ln x$ (for all $r\in\Bbb R$), $\ln(1/y) = \ln y^{-1} = -\ln y$ follows from there, so you're good.</p>
|
2,788,500 | <p><strong>Prove that $9\vert F_{n+24}$ iff $9\vert F_n$.</strong></p>
<p>This can be proved if in some way we can establish $F_{n+24}\equiv F_{n}(mod 9)$.</p>
<p>It is given in hint to use identity $F_{m+n}=F_{m-1}F_{n}+F_{m}F_{n+1}$.</p>
<p>Using the above identity for $F_{n+24}$,we get $F_{n+24}=F_{n-1}F_{24}+F_... | Prajwal Kansakar | 49,781 | <p>Use $$F_{n+24}=F_{(n+12)+12}=F_{n+11}F_{12}+F_{n+12}F_{13}$$ and the fact that $9| F_{12}$ but $9$ does not divide $F_{13}$ to show that $9|F_{n+24}$ iff $9|F_{n+12}$. Then use similar identity to prove that $9|F_{n+12}$ iff $9|F_n$.</p>
|
1,115,732 | <p>I'm trying to do the following probability question involving, I think, the ''amended'' multiplication rule:</p>
<p>A Jar contains 3 red and 5 black balls. What is the probability of drawing
2 red balls simultaneously ?</p>
<p>I used the formula - $P(A \cap B) = P(A)\cdot P(B\mid A)$</p>
<p>I.E. P(Red ball and th... | Alex R. | 22,064 | <p>Yep. $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $</p>
|
1,115,732 | <p>I'm trying to do the following probability question involving, I think, the ''amended'' multiplication rule:</p>
<p>A Jar contains 3 red and 5 black balls. What is the probability of drawing
2 red balls simultaneously ?</p>
<p>I used the formula - $P(A \cap B) = P(A)\cdot P(B\mid A)$</p>
<p>I.E. P(Red ball and th... | Graham Kemp | 135,106 | <p>$\color{green}{\checkmark}$ Yes. You have it. Your equation is appropriate, $P(R_1\cap R_2) = P(R_1)P(R_2\mid R_1)$, and you measured the terms correctly.</p>
<p>Another way would be to use combinations: ways to select 2 of 3 red balls out of the ways to select any 2 of all 8 balls. Which affirms your conditional... |
4,626,092 | <p>Question:
If the wire of length is cut into three pieces then the probability that the three pieces form a triangle is ____</p>
<p>My approach is as follow</p>
<p>For a triangle to exist the sum of two sides should be greater than the third side</p>
<p>Hence <span class="math-container">$x + y + z = \ell \Rightar... | Prem | 464,087 | <p><strong>Solution 1 :</strong></p>
<p>Let <span class="math-container">$x$</span> be the first cut , then <span class="math-container">$y$</span> be the second cut.<br />
Then <span class="math-container">$z$</span> is remaining.</p>
<p>This will not form a triangle when <span class="math-container">$z$</span> is lon... |
2,737,342 | <p>How do I evaluate this integral
$$\int_{0}^{1}\mathrm dx\ln^2(1+\sqrt{x})\ln(1-\sqrt{x})?$$</p>
<p>Enforcing $x=\tan^2(y)$</p>
<p>$$\int_{0}^{\pi/4}\mathrm dy\sec^2y\tan y\ln^2(1+\tan y)\ln(1-\tan y)$$</p>
<p>Enforcing $v=1+\tan y$</p>
<p>$$\int_{1}^{1+\pi/4}\mathrm dv (v-1)\ln^2(v)\ln(2-v)\tag1$$</p>
<p>$$(1)... | Jack D'Aurizio | 44,121 | <p>We are interested in
$$\mathcal{J}= \int_{0}^{1}2x\log^2(1+x)\log(1-x)\,dx$$
and by integration by parts this boils down to computing
$$ \int_{0}^{1}(1+x)\log^2(1+x)\,dx = \frac{3}{4}+2\log^2(2)-2\log(2)$$
$$ \mathcal{K} = \int_{0}^{1}(1-x)\log(1-x)\log(1+x)\,dx$$
and $\mathcal{K}$ can be tackled by enforcing the su... |
856,237 | <p>How can be proven the following inequality?
$$\forall{x\in\mathbb{R}},\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$$
Thanks</p>
| newzad | 76,526 | <p>Hint:</p>
<p>$\sin(2x)+x\sin^2x \leq 1+x$ and $(x-2)^2\geq0$</p>
|
3,559,942 | <p>I am trying to solve the limit:
<span class="math-container">$$\lim_{x\to\infty}x^\frac{5}{3}\left(\left(x+\sin\left(\frac{1}{x}\right)\right)^\frac{1}{3}-x^\frac{1}{3}\right)$$</span></p>
<p>I was trying to find a way to bring it into a fraction form to apply L'Hospital's rule, and I tried using
<span class="math-... | Matthew Leingang | 2,785 | <p>First I would factor <span class="math-container">$x^{1/3}$</span> out of the second term:
<span class="math-container">$$
x^{5/3}\left(\left(x+\sin\left(\frac{1}{x}\right)\right)^\frac{1}{3}-x^\frac{1}{3}\right)
= x^{2}\left(\left(1+ \frac{1}{x}\sin\left(\frac{1}{x}\right)\right)^{1/3}-1\right)
$$</span>
Then I wo... |
3,559,942 | <p>I am trying to solve the limit:
<span class="math-container">$$\lim_{x\to\infty}x^\frac{5}{3}\left(\left(x+\sin\left(\frac{1}{x}\right)\right)^\frac{1}{3}-x^\frac{1}{3}\right)$$</span></p>
<p>I was trying to find a way to bring it into a fraction form to apply L'Hospital's rule, and I tried using
<span class="math-... | Luca Goldoni Ph.D. | 264,269 | <p>You have that</p>
<p><span class="math-container">$$
\begin{gathered}
\mathop {\lim }\limits_{x \to + \infty } x^{\frac{5}
{3}} \left[ {\left( {x + \sin \left( {\frac{1}
{x}} \right)} \right)^{\frac{1}
{3}} - x^{\frac{1}
{3}} } \right] = \hfill \\
\hfill \\
\mathop {\lim }\limits_{x \to + \infty } x^{\fr... |
15,413 | <p>I see that there's a control I can click to hide or "minimize" version 9.0's "suggestion bar". Is there a keyboard shortcut to do this?</p>
| Charlie Joey Hadley | 1,952 | <p>As already said by everyone else, there isn't a keyboard shortcut I'm aware of and don't currently know how to assign one.</p>
<p>But, I use the following when showing people the interface</p>
<pre><code>Column[{Button["Predictive Interface Off", dummy1 = 2; SetOptions[EvaluationNotebook[],
ShowPredictiveInterface... |
1,292,500 | <p>A very, very basic question.</p>
<p>We know
$$-1 \leq \cos x \leq 1$$
However, if we square all sides we obtain
$$1 \leq \cos^2(x) \leq 1$$
which is only true for some $x$.</p>
<p>The result desired is
$$0 \leq \cos^2(x) \leq 1$$
Which is quite easily obvious anyway. </p>
<p>So, what rule of inequalities am I for... | kodlu | 66,512 | <p>Squaring does not preserve inequality since it is a 2-to-1 function over $\mathbb{R}$, hence nonmonotone. Mind you a monotone decreasing function would reverse the inequality there would still be a valid inequality pointing the opposite way.</p>
|
1,292,500 | <p>A very, very basic question.</p>
<p>We know
$$-1 \leq \cos x \leq 1$$
However, if we square all sides we obtain
$$1 \leq \cos^2(x) \leq 1$$
which is only true for some $x$.</p>
<p>The result desired is
$$0 \leq \cos^2(x) \leq 1$$
Which is quite easily obvious anyway. </p>
<p>So, what rule of inequalities am I for... | CiaPan | 152,299 | <p>The square function is not growing, so it does not preserve the inequality direction! $$-5 < -2 < -1$$ but $$(-5)^2 > (-2)^2 > (-1)^2$$</p>
<p>Additionally the square function is not monotonic, so it does not preserve inequalities at all! $$-1 < 1$$ but $$(-1)^2 = (1)^2$$</p>
<p>Squaring preserves i... |
53,798 | <p>Let $f(i),i\in \mathbb N\, $ be a sequence of real or complex numbers then for natural numbers $m,n$ and $r$ holds sum transformation</p>
<p>$$\sum_{i=0}^{mn+r}f(i)=\sum_{i=0}^{r}f(mn+i)+\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}f(mj+i).$$</p>
<p>This identity can be proved by induction by $r$. I am looking for an alternati... | Shai Covo | 2,810 | <p>The result is evident using
$$
\sum\limits_{i = 0}^{m - 1} {\sum\limits_{j = 0}^{n - 1} {f(mj + i)} } = \sum\limits_{j = 0}^{n - 1} {\sum\limits_{i = 0}^{m - 1} {f(mj + i)} }.
$$</p>
|
546,505 | <p>About Goldbach conjecture, (that Every even integer greater than 2 can be expressed as the sum of two primes) and if algorithms exist to solve the Halting Problem, then algorithm that determine Goldbach conjecture is true or false is exist? please explain</p>
| michaelmross | 194,864 | <p>There's a lot of muddled thinking about the Goldbach Conjecture. There's only one reason that it can fail, and that's if you run out of primes. </p>
<p>The conjecture fails if no primes exist for a huge gap, close to $n$ and $2n$. It's been proved there must be a $p$ within that interval. We can easily show, algori... |
155,453 | <p>How can I use a list of variables (possibly subscripted) as an <a href="http://reference.wolfram.com/language/ref/AxesLabel.html" rel="noreferrer"><code>AxesLabel</code></a> without showing the braces.</p>
<p>For example, </p>
<pre><code>Plot[{x, x^2}, {x, 0, 1}, AxesLabel -> {x, {Subscript[y, 1], Subscript[y, ... | eldo | 14,254 | <pre><code>Plot[{x, x^2}, {x, 0, 1},
AxesLabel -> {x, Row[{Subscript[y, 1], " ", Subscript[y, 2]}]}]
</code></pre>
<p><a href="https://i.stack.imgur.com/CUrJV.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/CUrJV.jpg" alt="enter image description here"></a></p>
|
2,537,344 | <p>I'm trying to find an expression for $\prod_{j=1}^n\left(j-\frac{1}{\sqrt{j}}+1\right)$ in terms of n. I have tried finding $\int_1^n\ln\left(x-\frac{1}{\sqrt{x}}+1\right)\,dx$ by parts but end up with a big mess. Any help would be appreciated!</p>
| Claude Leibovici | 82,404 | <p>As you wrote, the calculation of $$I_n=\int_1^n\ln\left(x-\frac{1}{\sqrt{x}}+1\right)\,dx$$ is not the most pleasant one.</p>
<p>Using, as you did, integration by parts, we have
$$\int\ln\left(x-\frac{1}{\sqrt{x}}+1\right)\,dx=x \log \left(x-\frac{1}{\sqrt{x}}+1\right)-\int\frac{2 x^{3/2}+1}{2 \left(\sqrt{x} (x+1)-... |
1,397,036 | <p>A bag labeled $A$ contains $4$ red balls and $7$ green balls.
Another bag $B$ contains $6$ red and $5$ green balls.</p>
<p>A ball is transferred from bag $A$ to bag $B$, after which a ball is drawn from $B$.</p>
<p>Find the probability that it is a red ball?</p>
<p>To be honest I have no idea how to approach the ... | drhab | 75,923 | <p><strong>Hint</strong>:</p>
<p>$$P\left(E\right)=P\left(E\mid R\right)P\left(R\right)+P\left(E\mid G\right)P\left(G\right)$$</p>
<p>Here $R$ is the event that the transferred ball is red, $G$ is the event that the transferred ball is green and $E$ denotes the event that the ball taken out bag $B$ is red.</p>
<p>Se... |
2,217,630 | <p>I need to find how many <em>real</em> roots this polynomial has and prove there existence. I was wondering if my logic and thought process was correct.</p>
<blockquote>
<p>Determine the number of <em>real</em> roots and prove it for $x^3 - 3x + 2$</p>
</blockquote>
<p>First, note that $f'(x) = 3x^2 - 3$ and so <... | Dietrich Burde | 83,966 | <p>Since we have
$$
x^3-3x+2=(x-1)^2(x+2),
$$
we have three real roots $1,1,-2$. Here we count with multiplicities (which is standard for many results in geometry and other areas).</p>
|
130,804 | <p>I have this question just out of curiosity.</p>
<p>If X is a scheme, then a morphism $f: X \rightarrow X$ can be the identity on the underlying topological space of X, but not the identity on the structure sheaf. For example, f can be the Frobenius morphism.</p>
<p>Does someone know an example of such a morphism w... | Loïc Teyssier | 24,309 | <p>I'll finaly settle for "a cutout compact set" for want of a better term. But I think this word expresses well the "finitely many" (connected components, non-smooth points) side of the object, which is what I wanted to highlight.</p>
|
618,665 | <p>Show that $\sqrt{13}$ is an irrational number.</p>
<p>How to direct proof that number is irrational number. So what is the first step..... </p>
| Bill Dubuque | 242 | <p>As mentioned, one can quickly prove the irrationality of square roots using the Rational Root Test, or <em>uniqueness of prime factorizations</em>, or other closely related propeerties such as Euclid's Lemma or <a href="http://en.wikipedia.org/wiki/B%C3%A9zout's_identity" rel="nofollow noreferrer">Bezout's gcd i... |
4,265,066 | <p>For the purpose of CG and animation, I'm looking for a function thats tends to 1 when x tends to +Infinity, and have a tangent of 1 when x = 0.</p>
<p>I found that function:</p>
<p><span class="math-container">$f\left(x\right)=2\frac{\left(\frac{2x}{p}+1\right)^{p}}{\left(\frac{2x}{p}+1\right)^{p}+1}-1$</span></p>
<... | SION | 8,413 | <p>I think I have a solution to the problem.</p>
<p>Let us first recall the following recursive definition of pfaffian from Page 116 of Schubert varieties and degeneracy loci, Fulton-Pragacz. Let <span class="math-container">$\mathrm{pf}^{i j}(A)$</span> denote the pfaffian of the skew-symmetric matrix obtained from <s... |
98,698 | <p>I have the time domain signal
$$
u_o(t) = u(t)e^{-t/\tau}\eta(t) + \sigma(t)
$$
where $\tau$ is known, $\eta$ is non-Gaussian noise, and $\sigma$ is Gaussian noise. The distribution of $\eta(t)$ is known, but only numerically. I also have prior knowledge that $u(t)$ is a sum of a small number of sinusoids. How can I... | ffuu | 24,202 | <p><a href="http://scicomp.stackexchange.com">http://scicomp.stackexchange.com</a></p>
|
3,285,447 | <blockquote>
<p>Given that <span class="math-container">$a$</span> is any vector in a vector space <span class="math-container">$V$</span>, show that the set <span class="math-container">$\{xa : x \in \mathbb{R}\}$</span> of all scalar multiples of <span class="math-container">$a$</span> is a subspace of <span class=... | Community | -1 | <p>1) clearly the set is non empty</p>
<p>2) let <span class="math-container">$xa,ya$</span> be in the set, then <span class="math-container">$xa+ya=(x+y)a$</span>. Since the addition of two real numbers is a real number, <span class="math-container">$(x+y)a$</span> is in the set.</p>
<p>3) let A be any real scalar. ... |
225,953 | <p>I would like to calculate the maximum number of polynomial terms given a certain number of variables and a certain degree. eg. given that the number of variables is 2 and the degree is 3, the maximum number of terms is 9:
<span class="math-container">$$x_1^3 + x_1^2 x_2 + x_1 x_2^2 + x_2^3+ x_1^2 +x_1 x_2 + x_2^2 + ... | Joe Silverman | 11,926 | <p>This really isn't a research level question. But here's a nice trick for getting the answer without doing the sum as in Wolfgang's answer. It's easier to ask for the number of distinct monomials of exact degree $n$ in $k+1$ variables $x_0,\ldots,x_k$. Then you can set $x_0=1$ if you want monomials of degree at most ... |
3,300,954 | <p>How to prove P(A ∩ B) ≤ P(A) using probability theory?</p>
<p>I understand this when drawn on a Venn Diagram but am unsure how it translates to a formal proof. </p>
| A.G. | 115,996 | <p>Solving numerically should be quite easy with newton's method to get a zero of
<span class="math-container">$$
f(t)=e^{30\,t}\,30\,t-.3=0\Rightarrow t\approx 0.00789184.
$$</span>
<a href="https://i.stack.imgur.com/st3wk.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/st3wk.png" alt="enter image... |
3,434,463 | <p>Euler's identity <span class="math-container">$e^{i\pi}+1=0$</span> has always fascinated me, and at the same time freaks me out a bit. Like, they are two <em>very</em> fundamental constants which seem to have absolutely nothing in common, but still there mysteriously is an immediate mathematical connection between ... | eyeballfrog | 395,748 | <p>It's just a natural consequence of the connection between complex exponentials and rotations. That is, <span class="math-container">$e^{i\theta}$</span> is a rotation through angle <span class="math-container">$\theta$</span> in radians. Since <span class="math-container">$-1$</span> is a half-turn and the angle of ... |
78,279 | <p>In another question (<a href="https://mathematica.stackexchange.com/questions/77229/using-timeseriesforecast-for-forecasting-the-traffic-growth/78275">Using TimeSeriesForecast for forecasting the traffic growth</a>) I asked to use the TimeSeriesForecast on <a href="https://docs.google.com/spreadsheets/d/1dmv_C2_J7uG... | xzczd | 1,871 | <h1>Update</h1>
<p>As of <em>v12</em>, it's possible to parallelize when using <code>NDSolve</code> to solve the problem, check <a href="https://mathematica.stackexchange.com/q/208784/1871">this post</a> for more information. (Sadly the <code>gamma</code> matrix in the question is deleted so I can't test. )</p>
<hr />
... |
1,725,343 | <p>I found the quadratic approximation as $9 + \frac{1}{2}(-9x^2 - 9y^2)$</p>
<p>The problem is that the triple derivatives all end up 0 at (0,0), so I get that the error approximation is 0. Wolfram alpha calculates the triple derivatives having sin(y) or sin(x) in them making them 0. I know you are supposed to plug i... | Anonymous | 327,815 | <p>$$
g(x) = \frac{1}{(8+x)} = \sum_{n=0}^{\infty} \frac{(-1)^nx^n}{8^{n+1}}
$$
Taking the derivative in both sides: (n=0 is constant)
$$
g'(x) = \frac{-1}{(8+x)^2} = \sum_{n=1}^{\infty} \frac{n(-1)^nx^{n-1}}{8^{n+1}}
$$
$$
-f(x) = \frac{-1}{(8+x)^2} = \sum_{n=0}^{\infty} \frac{(n+1)(-1)^{n+1}x^{n}}{8^{n+2}}
$$
$$
f(... |
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