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 |
|---|---|---|---|---|
20,773 | <p><strong>Background</strong></p>
<p>The students in my country are supposed to be able to work with and answer questions about functions at the age of around 15. This is asserted in the standard mathematics curriculum for middle school students.</p>
<p>Personally, I think the definition of a function is extremely abs... | Matthew Daly | 12,619 | <p>Functions are far broader and more applicable than you give them credit for. Consider the following:</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th style="text-align: left;">Country or state</th>
<th style="text-align: left;">Capital</th>
<th style="text-align: left;">Elevation (in met... |
20,773 | <p><strong>Background</strong></p>
<p>The students in my country are supposed to be able to work with and answer questions about functions at the age of around 15. This is asserted in the standard mathematics curriculum for middle school students.</p>
<p>Personally, I think the definition of a function is extremely abs... | Michael Bächtold | 590 | <p>Most of pre 1900 mathematics can be done without the modern function concept. Hints that this was actually the case can be found in this hsm question <a href="https://hsm.stackexchange.com/q/6104/3462">Who first considered the <span class="math-container">$f$</span> in <span class="math-container">$f(x)$</span> as a... |
1,058,019 | <p>Evaluate the below integral:
$$
\int_{0}^{\infty}{x^{\alpha - 1} \over 1 + x}\,{\rm d}x
$$
How to start ?.</p>
| gt6989b | 16,192 | <ol>
<li>Translate so that you are rotating about the origin. In your case, subtract (2,2) from both what you are rotating and what you are rotating about.</li>
<li>Perform the rotation about the origin.</li>
<li>Add the original translation back.</li>
</ol>
|
1,058,019 | <p>Evaluate the below integral:
$$
\int_{0}^{\infty}{x^{\alpha - 1} \over 1 + x}\,{\rm d}x
$$
How to start ?.</p>
| paw88789 | 147,810 | <p>This can also be achieved nicely with complex numbers, using the fact that in multiplying complex numbers you multiply the lengths and add the angles. This means that to rotate a complex number $z$ by an angle $\theta$ about the origin you take $z\cdot(\cos\theta+i\sin\theta)$</p>
<p>To rotate $z$ by an angle $\th... |
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... | Fernando Revilla | 401,424 | <p>The key is that if $\lvert q \rvert<1,$ then $\lim q^n=0.$ If $q=0,$ the result is trivial. If $\left | q\right |\neq 0$ and $\epsilon>0,$ then: $$\left | q^n-0 \right |<\epsilon\Leftrightarrow \left | q\right |^n<\epsilon \Leftrightarrow n\log \left | q\right | <\log \epsilon.\quad (*)$$ As $0<\le... |
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... | Alex M. | 164,025 | <p>If $\alpha = 3$, then your sequence is just the constant sequence $1$, so it converges.</p>
<p>If $\alpha > 3$ then, since the exponential function $x \mapsto a^x$ increases when $a>1$, we have that</p>
<p>$$\left( \frac \alpha 3 \right) ^{2^n} \ge \left( \frac \alpha 3 \right) ^n$$</p>
<p>and since $\left(... |
846,341 | <p>My question is to sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1\}$ in the complex plane. </p>
<p>I substituted $z$ for $a+bi$, but did not get anywhere:</p>
<p>$\left|\frac{a+(b-1)i}{a+(b+1)i}\right|<1\\
\left|\frac{(a+(b-1)i)(a-(b+1)i)}{a^2+(b+1)^2}\right|<1\\
\left|\frac{a^2-(ab+a... | Hans Lundmark | 1,242 | <p>Rewrite the inequality as $|z-i|<|z+i|$. Then think geometrically in terms of distances from the point $z$ to the points $\pm i$.</p>
|
846,341 | <p>My question is to sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1\}$ in the complex plane. </p>
<p>I substituted $z$ for $a+bi$, but did not get anywhere:</p>
<p>$\left|\frac{a+(b-1)i}{a+(b+1)i}\right|<1\\
\left|\frac{(a+(b-1)i)(a-(b+1)i)}{a^2+(b+1)^2}\right|<1\\
\left|\frac{a^2-(ab+a... | Eli Elizirov | 76,037 | <p>$|z-i|<|z+i| \implies |x+(y-1)i|<|x+(y+1)i| \implies x^2+y^2-2y+1<x^2+y^2+2y+1\implies y>0$ </p>
<p>is just an algebraic way to show what Hans Lundmark said</p>
|
2,936,329 | <p>Let <span class="math-container">$(X,d)$</span> be a metric space. If <span class="math-container">$a_n$</span> is a sequence in <span class="math-container">$X$</span> and <span class="math-container">$a\in X$</span> such that <span class="math-container">$\frac{d(a_n,a)}{1+d(a_n,a)} \to 0$</span>, then <span class... | DeepSea | 101,504 | <p>Put <span class="math-container">$x_n = d(a_n,a)\implies 1-\dfrac{1}{1+x_n}\to 0\implies 1+x_n \to 1\implies x_n \to 0\implies a_n \to a$</span> as <span class="math-container">$n \to \infty$</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... | Dr. belisarius | 193 | <pre><code>Row[{Column[{
Plot[Interpolation[data, InterpolationOrder -> 0][x], {x, 0, Max[data[[All, 1]]]},
PlotRange -> All, AspectRatio -> 1/4, ImageSize -> 600, AxesOrigin -> {0, -1},
ColorFunction -> "Rainbow", PlotStyle -> Thick],
BarChart[Thread[Differences@data[[All, 1]]... |
1,710,929 | <p>If $3$ people are dealt $3$ cards from a standard deck, determine the probability that none of them is dealt three of a kind?</p>
<p>Here is my attempt:</p>
<p>The total number of hands is
$${_{52}\mathsf C}_3\times{_{49}\mathsf C}_3\times{_{46}\mathsf C}_3=7.75262759\times 10^{12}.$$
The number of ways we can de... | joriki | 6,622 | <p>I haven't checked every detail, but it looks good, and the probability isn't too high. Three of a kind in three cards is quite unlikely – two players getting one is so unlikely that you can get a very good estimate just by calculating the probability that one player gets one and multiplying it by $3$:</p>
<p>... |
1,799,710 | <p>The PDE $$\frac{\partial ^{2}u}{\partial x^{2}}+2\frac{\partial^{2}u}{\partial x\partial y}+\frac{\partial^{2}u}{\partial y^{2}}=x$$ Has</p>
<p>$1.$ Only one particular integral.</p>
<p>$2.$ a particular integral which is linear in x and .</p>
<p>$3.$a particular integral which is a quadratic polynomial is x and ... | Doug M | 317,162 | <p>Here is a trick.</p>
<p>Let $B' = \begin{pmatrix} 1&1&1\\0&1&1\\0&0&1\end{pmatrix}$ This is the basis of B' in terms of the basis of B.</p>
<p>To change the basis of T.</p>
<p>$[T]_B = [B'^{-1}TB']_{B'}$ </p>
|
3,913,856 | <p>we have analytically calculated distance between the centers of a big circle and a small circle in <strong>mm</strong>.</p>
<p><a href="https://i.stack.imgur.com/7WJEv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7WJEv.png" alt="enter image description here" /></a></p>
<p>We also have coordinat... | Christian Blatter | 1,303 | <p>I'm considering the problem on top of your question. There is no "consecutive" involved, and "repetition" does not occur in sets.</p>
<p>We have <span class="math-container">$x_1+x_2+x_3=0$</span> modulo <span class="math-container">$3$</span> <strong>iff</strong> either all <span class="math-con... |
2,668,468 | <p>So I've got the region $R$ like in the image below, and need to find the double integral $$\iint\limits_{R}\frac{1}{4}\sqrt{2x^2+2y^2}dA$$ over that region, given $a=8$ and $c=5$.</p>
<p><a href="https://i.stack.imgur.com/ahhZp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ahhZp.png" alt="enter... | Claude Leibovici | 82,404 | <p>Almost as dxiv commented, write
$$A=\sum_{n=1}^\infty n^2 z^n=\sum_{n=1}^\infty (n(n-1)+n) z^n=z^2\sum_{n=1}^\infty n(n-1) z^{n-2}+z\sum_{n=1}^\infty n z^{n-1} $$ that is to say
$$A=z^2 \left(\sum_{n=1}^\infty z^{n} \right)''+z \left(\sum_{n=1}^\infty z^{n} \right)'$$</p>
|
90,987 | <p>I have coin, and want to get 2 heads exactly. I will throw it until this condition is met. </p>
<p>What is expected number of tries for this condition? </p>
<p>I know that it would be $$\sum\limits_{n=2}^\infty P(X=n)n=0.5^n \cdot n\cdot(n-1)$$
however I don't have an idea how to solve that sum because we didn't l... | André Nicolas | 6,312 | <p>Your analysis is right. The probability that $X=n$ is indeed $(n-1)(0.5)^n$. This is because we need to have exactly $1$ head in the first $n-1$ tosses (probability $(n-1)(0.5)^{n-1}$) and then a head (probability $0.5$). So the required expectation is
$$\sum_{n=2}^\infty (n)(n-1)(0.5)^n. \qquad (\ast) $$</p>
<p... |
194,745 | <p>I'm trying to create a program which converts latitude/longitude (wgs84) to UPS (Universal Polar stereographic) coordinates, and then UPS(x, y) to WGS 84. I mean Given UPS(x, y), be able to compute latitude/longitude and vice versa.</p>
<p>Is there any easy way to do this?</p>
| Carl Lange | 57,593 | <p>You can solve this problem by using a <a href="http://epsg.io/trans?x=50&y=17&z=0&s_srs=4326&t_srs=5514" rel="nofollow noreferrer">web API</a> to do arbitrary transformations (in this way you can convert to and from any spatial reference system, and are not limited to the systems that WL supports nat... |
926,581 | <p>I find the <a href="https://en.wikipedia.org/wiki/Surreal_number" rel="nofollow noreferrer">surreal numbers</a> very interesting. I have tried my best to work through John Conway's <em>On Numbers and Games</em> and teach myself from some excellent <a href="http://www.tondering.dk/claus/sur16.pdf" rel="nofollow noref... | gamma | 88,524 | <p>The surreal numbers contain the real numbers (as well as infinite and infinitesimal numbers).</p>
<p>Both $0.999\dots$ and $1$ are real numbers.</p>
<p>In the real numbers $1 = 0.999\dots$ and so it must also be true in the surreal numbers.</p>
<p>As Bryan correctly points out in his answer, surreal numbers which... |
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... | Vivek Kaushik | 169,367 | <p>Let $$I=\int_{0}^{1} \frac{\arcsin(x)}{x} dx.$$ Put $x=\sin(\theta), dx=\cos(\theta) d\theta$ so that $$I=\int_{0}^{\frac{\pi}{2}}\frac{\theta}{\tan(\theta)} d\theta.$$ We will show this representation of $I$ is indeed equal to a double integral that we can easily evaluate in two ways. </p>
<p>Consider $$J=\int_{0... |
2,001,449 | <p>Where in the analytic hierarchy is the theory of all true sentences in ZFC? In higher-order ZFC? In ZFC plus large cardinal axioms?</p>
<p>Edit: It seems that this is ill-defined. Why is this ill-defined for ZFC, but true for weaker theories like Peano arithmetic and higher-order arithmetic?</p>
| Mitchell Spector | 350,214 | <p>Let $\Gamma$ be the set of sentences of ZFC that happen to be true. I doubt that you can determine much about this set, but you can see that it's consistent with ZFC that $\Gamma$ is as complicated as you want.</p>
<p>For instance, using Easton forcing, you can find a model $M$ of ZFC in which $2^\kappa$ has any v... |
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... | Maxime Ramzi | 408,637 | <p>Let me write another answer to try to understand this from another point of view and using things I learned recently. </p>
<p>Let me assume that <span class="math-container">$X$</span> is simply-connected and very nice. </p>
<p>We're back at our stage where we have <span class="math-container">$X\to Y_n$</span> an... |
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... | Maxime Ramzi | 408,637 | <p>Here's a more lowbrow point of view, with, I think, no gap - as opposed to my previous attempt at an answer . </p>
<p>The idea is that instead of separating uniqueness and existence, we consider the two similarly, by simply using a relative version of the obstruction to lifting. </p>
<p>So suppose we have a map <s... |
24,524 | <p>Consider two (distinct) octahedral diagrams i.e. diagrams mentioned in the octahedron axioms of triangulated categories (with four 'commutative triangular faces' and four 'distinguished triangular faces'). Is it true than one can extend to a morphism of such diagrams:
1. any morphism of one of the 'commutative faces... | Greg Stevenson | 310 | <p>It is true in a Heller triangulated category aka $\infty$-triangulated category (although strictly speaking one only needs a 3-triangulation for octahedra) that any morphism between the bases of octahedra (by which I mean the 3 objects and two composable morphisms from which the octahedron is built) extends to a mor... |
24,524 | <p>Consider two (distinct) octahedral diagrams i.e. diagrams mentioned in the octahedron axioms of triangulated categories (with four 'commutative triangular faces' and four 'distinguished triangular faces'). Is it true than one can extend to a morphism of such diagrams:
1. any morphism of one of the 'commutative faces... | Matthias Künzer | 9,300 | <p>The identity of commutative triangles does not lift to a morphism of Verdier octahedra in general, for it may happen that there exist two mutually nonisomorphic Verdier octahedra on the same commutative triangle.
Cf. <a href="http://www.math.rwth-aachen.de/~kuenzer/counterexample.pdf" rel="nofollow">http://www.math.... |
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... | JTP - Apologise to Monica | 64 | <p>A video clip <a href="https://youtu.be/J6lyURyVz7k" rel="nofollow">Last Week Tonight with John Oliver: Standardized Testing</a> offers an 18 minute discussion on the issues. Issues raised -</p>
<ul>
<li>Too many standardized tests, an average of 113 over the course of one's education through high school.</li>
<li>T... |
1,768,142 | <p>Calculate the line integral
$$
\rm I=\int_{C}\mathbf{v}\cdot d\mathbf{r}
\tag{01}
$$
where
$$
\mathbf{v}\left(x,y\right)=y\mathbf{i}+\left(-x\right)\mathbf{j}
\tag{02}
$$</p>
<p>and $C$ is the semicircle of radius $2$ centred at the origin from $(0,2)$ to $(0,-2)$ to the negative x axis (left half-plane).</p>
<blo... | Frobenius | 321,266 | <p><a href="https://i.stack.imgur.com/GxyfJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GxyfJ.png" alt="enter image description here"></a>
Since the problem is a special case, the answer is given in Figure without any integration.</p>
<p>If the curve (semicircle) lies to the positive $x$ as in a... |
3,988,540 | <p>Does the series <span class="math-container">$\sum_{n=0}^\infty\frac{4^n}{n^3+9^n}$</span> converge or diverge?</p>
<p>So far, I've divided each term by <span class="math-container">$9^n$</span> to get <span class="math-container">$\frac{(4/9)^n}{n^3/9^n + 1}$</span> and tried to apply the ratio test, but that didn'... | Community | -1 | <p><strong>Hint</strong></p>
<p>This is a series of positive terms bounded above by a geometric progression with ratio <span class="math-container">$\frac{4}{9}$</span>.</p>
<p>Alternately, use the Ratio Test. Note that <span class="math-container">$n^3/9^n$</span> tends to <span class="math-container">$0$</span>.</p>
... |
2,417,506 | <blockquote>
<p>Find the radius of convergence of $\displaystyle{\sum_{n=0}^{\infty}} {(n!)^3 \over (3n)!}z^{3n} \ ?$</p>
</blockquote>
<p>I applied Cauchy-Hadamard test and the result is coming $0$ (radius of convergence). To obtain the limit I also used <a href="http://priti2212.blogspot.in/2013/05/cauchys-theorem... | Ahmad | 411,780 | <p>By the ratio test the sum converges if $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} = r$ and $|r| <1$ , special cases are when $|r|=1$ and have to be checked separately.</p>
<p>$a_n = \frac{(n !)^3}{(3n)!} z^{3n}$ so $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} =\frac{((n+1)!)^3 z^{3 n+3}}{\frac{(3 (n+1... |
2,417,506 | <blockquote>
<p>Find the radius of convergence of $\displaystyle{\sum_{n=0}^{\infty}} {(n!)^3 \over (3n)!}z^{3n} \ ?$</p>
</blockquote>
<p>I applied Cauchy-Hadamard test and the result is coming $0$ (radius of convergence). To obtain the limit I also used <a href="http://priti2212.blogspot.in/2013/05/cauchys-theorem... | Bernard | 202,857 | <p><strong>Hint:</strong>
First find the radius of convergence of the series $\;\displaystyle\sum_{n=0}^{\infty} {(n!)^3 \over (3n)!}z^{n}$.</p>
<p>For this apply <em>Cauchy-Hadamard's formula</em>, combined with <em>Stirling's formula</em>:
\begin{align}
\biggl(\frac{(n!)^3}{(3n)!}\biggr)^{\!\tfrac1n}\sim_\infty&... |
42,913 | <p>The following algorithm decides if a number $n>0$ is a totient or a nontotient:</p>
<pre><code>if n = 1
return true
if n is odd
return false
for k in n..n^2
if φ(k) = n
return true
return false
</code></pre>
<p>This is very slow; even using a sieve it takes $n^2$ steps to decide that $n$ is nontotient... | Jack Schmidt | 583 | <p>On <a href="https://mathoverflow.net/questions/17072/the-finite-subgroups-of-sun">mathoverflow</a> a similar question was asked. You can take the list for SU3 and mod out by the centers to get the list for PGL3. Several references for SU3 are given (in more than one answer), and some discussion of SU(n) is also gi... |
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... | Ross Millikan | 1,827 | <p>We have <span class="math-container">$\log 4=2 \log 2 \approx 0.602$</span> and <span class="math-container">$\log 5 = \log 10 - \log 2 \approx 0.699$</span>. From Benford's law we expect about <span class="math-container">$0.699-0.602=0.097$</span> of the numbers to start with <span class="math-container">$4$</spa... |
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... | Brian Tung | 224,454 | <p>To expand on how the solution you included in your question works: Consider the powers of <span class="math-container">$2$</span> with <span class="math-container">$k$</span> digits. There are five possible sequences of first digits of these <span class="math-container">$k$</span>-digit powers of <span class="math-c... |
1,243,159 | <p>I've recently been studying matrices and have encountered a rather intriguing question which has quite frankly stumped me. </p>
<p>Find the $3\times3$ matrix which represents a rotation clockwise through $43°$ about the point $(\frac{1}{2},1+\frac{8}{10})$</p>
<p>For example: if the rotation angle is $66°$ then th... | Raskolnikov | 3,567 | <p>I assume you know how to represent a rotation around the origin $(0,0)$ as a matrix, which I'll henceforth shall call $R$. Now, if we want to rotate about a different point $(x_0,y_0)$, we will first translate all the points of the plane back by $(x_0,y_0)$ so that or rotation center now coincides with the origin. T... |
1,911,037 | <p>So I realized that I have to prove it with the fact that $(x-y)^2+2xy=x^2+y^2$ </p>
<p>So $\frac{(x+y)^2}{xy}+2=\frac{x}{y}+\frac{y}{x}$ $\Leftrightarrow$ $\frac{(x+y)^2}{xy}=\frac{x}{y}+\frac{y}{x}-2$ </p>
<p>Due to the fact that $(x+y)^2$ is a square, it will be positive </p>
<p>$x>0$ and $y>0$ so $xy>... | jaseem | 82,068 | <p>$$\left( \frac { \sqrt { x } }{ \sqrt { y } } -\frac { \sqrt { y } }{ \sqrt { x } } \right) ^{ 2 }\ge 0$$
$$\Rightarrow\frac{ { x } }{ { y } }+\frac { { y } }{ { x } } \ge 2$$</p>
|
720,924 | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| namsap | 111,080 | <p>In Calculus, sine and cosine are defined via the exponential function, meaning that
$ \cos x = \mathrm{Re}\{e^{ix}\}$ and $ \sin x = \mathrm{Im}\{e^{ix}\}$ and as you know, $e^0 = e^{i2\pi} = 1$ which means that $360^{\circ}$ which is the full circle corresponds to $2\pi$. For further reference see <a href="http:/... |
720,924 | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| Ben Grossmann | 81,360 | <p>It really comes down to the following limit:
$$
\lim_{x\to 0} \frac{\sin(x)}{x} = 1
$$
Or in other words, "$\sin x \approx x$ for small $x$". As a consequence, we have
$$
\frac{d}{dx}\sin x = \cos x, \qquad
\frac{d}{dx}\cos x = -\sin x
$$
For any other choice of angular unit, these derivatives require some sort of ... |
720,924 | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| Manuel Ocaña | 887,095 | <p>You can't remember any proof because 'we require radians in calculus' is not a logically well formulated statement, and only those have proofs.
The problem with using degrees in your calculations is that when you integrate or differentiate a function you probably left many logical gaps, which you fill with geometric... |
3,145,832 | <p>I am given real values <span class="math-container">$p, s, t, u$</span> and wish to find unknown values <span class="math-container">$r, v$</span>. As shown in the diagram below, <span class="math-container">$p$</span> and <span class="math-container">$s$</span> are radii of two given circles, with centers at <spa... | amd | 265,466 | <p>The locus of the centers of the circles tangent to the two fixed circles is a hyperbola (that degenerates to a double line when <span class="math-container">$p=s$</span>) with center at <span class="math-container">$C=\left(0,(t-p)/2\right)$</span> and asymptotes perpendicular to the common external tangent lines to... |
2,143,227 | <p>I wish to show whether or not the sequences $f_n(x)=\chi_{[n,n+1]}$ is tight</p>
<p>By definition, Let $(X,m,\mu)$be a measure space, the sequence ${f_n}$ is called tight over $X$ if $\forall \epsilon>0$ $ \exists X_0 \subset X $,with $\mu (X_0)<\infty$ , such that $\forall n$ $\int_{X-X_0}|f_n|d\mu <\e... | grand_chat | 215,011 | <p>Roughly speaking, a sequence of functions is tight if you can find a set $X_0$ with finite measure that contains 'nearly all' of the mass of every function. </p>
<p>We prove your sequence of functions is not tight using a contraposition argument. Suppose $X_0$ is a set such that $\int_{X-X_0}|f_n|d\mu <\epsilon... |
63,589 | <p>A big-picture question: what "physical properties" of a graph, and in particular of a bipartite graph, are encoded by its largest eigenvalue? If $U$ and $V$ are the partite sets of the graph, with the corresponding degree sequences $d_U$ and $d_V$, then it is easy to see that the largest eigenvalue $\lambda_{\max}$ ... | Mahmoud El Chamie | 25,025 | <p>The largest eigenvalue $\Lambda (A)$ of the adjacency matrix $A$ of a general graph satisfies the following inequality:</p>
<p>$\max \ ( d_{av},\sqrt{d_{max}} ) \le \Lambda (A) \le d_{max}$ , </p>
<p>where $d_{av}$ is the average degree of nodes in the graph and $d_{max}$ is the largest degree. </p>
<p>The proof ... |
361,747 | <p>Let me discuss two possible constructions of Gaussian measures on infinite dimensional spaces. Consider the Hilbert space <span class="math-container">$l^{2}(\mathbb{Z}^{d}) := \{\psi: \mathbb{Z}^{d}\to \mathbb{R}: \hspace{0.1cm} \sum_{x\in \mathbb{Z}^{d}}|\psi(x)|^{2}<\infty\}$</span> with inner product <span cl... | user69642 | 69,642 | <p>I think that what you are looking for is the link between the white noise measure <span class="math-container">$\mu_C$</span> and the isonormal process indexed by <span class="math-container">$\ell^2(\mathbb{Z}^d)$</span> with covariance structure given by <span class="math-container">$C$</span>. The white noise mea... |
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 ... | user236182 | 236,182 | <p>More generally: define $\upsilon_p(n!)=k\iff \left(p^k\mid n!,\, p^{k+1}\nmid n!\right)$</p>
<p>(this is standard notation; see <a href="http://www.taharut.org/imo/LTE.pdf" rel="nofollow">this paper</a>). </p>
<p>$\lfloor x\rfloor$ is the <a href="https://en.wikipedia.org/wiki/Floor_and_ceiling_functions" rel="nof... |
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... | George McNinch | 4,653 | <p>Jim Humphreys gave already the answer, but I thought I'd try to clarify
the question of "algebraic vs. non-algebraic" representations. </p>
<p>If G is any reductive algebraic group in char. p (say, over an alg. closure k of F_p),
the result in the paper of Jantzen mentioned in Jim Humphreys' answer shows that
any ... |
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$$</span>
with boundary conditions <span class="math-container">$Y\left(k_3\right)=0$</span> and <span class="math-container">$\frac{d Y}{d x}\left(\infty\right)=1-b$</span> (meanin... | emacs drives me nuts | 746,312 | <p>Extending on <em>Mockingbird</em>'s excellent answer, it's nice and instructive to calculate the <strong>4 intersection points of two concentric circles of different radius</strong>.</p>
<p>Without loss of generality, let the circles be centered at the origin, so that their defining equations are</p>
<p><span class=... |
347,494 | <p>I have a question regarding differential forms.</p>
<p>Let $\omega = dx_1\wedge dx_2$. What would $d\omega$ equal? Would it be 0?</p>
| Henry T. Horton | 24,934 | <p>Recall that
$$d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^{\deg \alpha} \alpha \wedge d\beta.$$</p>
<p>Then
\begin{align*}
d\omega & = d(dx_1 \wedge dx_2) \\
& = ddx_1 \wedge dx_2 - dx_1 \wedge ddx_2 \\
& = 0 - 0 \\
& = 0,
\end{align*}
since $d^2 = 0$.</p>
|
2,885,374 | <blockquote>
<p>Suppose, to build a box (a rectangular solid) of fixed volume and square base, the cost per square inch of the base and top is twice that of the four sides. Choose dimensions for the box that minimize the cost of building it, if the volume is (say) $2000$ cubic inches.</p>
</blockquote>
<p>Here is w... | Xander Henderson | 468,350 | <p>You are off to a good start. If $C(x)$ represents the cost of building a box with a square base of side length $x$, then (assuming that your computations are correct–I did not check them)
$$ C(x)
= m(2x^2) + n(4xh)
= 4nx^2 + \frac{8000 n}{x}
= 4n \left(x^2 + \frac{2000}{x} \right). $$
This tends to infinity both as... |
2,281,184 | <p>I was going through theorem 1.21 from Rudin (without looking at the proof) and I wanted to show that the supremum of $E = \{ y : y > 0, y^n < x \}$ must be the element that satisfies $y^n = x$. But for me to even attempt that next step, I need to show that $E$ is non-empty (then I need to show its bounded) so ... | copper.hat | 27,978 | <p>Assuming that $x>0$, note that there is some $m \in \mathbb{N}$ such that ${1 \over m} < x$. Then ${1 \over m^n} \le {1 \over m}$ and
so ${1 \over m}$ is in the set.</p>
|
485,190 | <p>I am trying to prove that every neighborhood of a boundary bound contains a point in interior and $X \setminus A$ where $A$ is the set in consideration. I am given the following definitions</p>
<p>(1) $(X,\mathcal{T})$ is a topological space if $X,\emptyset \in \mathcal{T}$, any arbitrary union of open sets in $\ma... | Brian M. Scott | 12,042 | <p>Observe that if $x$ is in the boundary of $A$, then $x$ is not in the interior or the exterior of $A$. Since $x\notin\operatorname{int}A$, $x$ has no open nbhd contained entirely in $A$. But that just means that if $U$ is an open nbhd of $x$, then $U\nsubseteq A$, and hence $U\cap(X\setminus A)\ne\varnothing$. A sim... |
1,270,802 | <p>I just finished an exam, it has the following question: Where is the point on the plane $3x + 5y + z = 18$ has the shortest distance to $(0,0,0)$?</p>
<p>I found this question similar: <a href="https://math.stackexchange.com/questions/355460/find-the-point-on-the-plane-2x-y-2z-20-nearest-the-origin">Find the point ... | David | 119,775 | <p>Here is why you got the wrong answer. Part way through the calculation you will have
$$\eqalign{6526&\cr -\ 8437&\cr -\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-&\cr ?089&\cr}$$
The next subtraction will give you $-2$ in place of the question mark. But note that the "places" in your answer are all positiv... |
1,690 | <p>I like to ask true-false questions on exams, because I feel that they can be an efficient way to assess students' understanding of concepts and ability to apply them to somewhat unfamiliar situations. In general, I'm very happy with true-false questions, but there is one annoyance that I have never figured out how ... | Benjamin Dickman | 262 | <p>Your question is in fact three questions.</p>
<blockquote>
<ol>
<li><p>Is there a good solution?</p>
</li>
<li><p>Do you ask your students to give explanations for true-false questions?</p>
</li>
<li><p>How do you deal with the issues raised above?</p>
</li>
</ol>
</blockquote>
<p>My answer for 1 is that I would <st... |
1,690 | <p>I like to ask true-false questions on exams, because I feel that they can be an efficient way to assess students' understanding of concepts and ability to apply them to somewhat unfamiliar situations. In general, I'm very happy with true-false questions, but there is one annoyance that I have never figured out how ... | Linear | 521 | <p>I recall one test I had as an undergraduate for a Linear Algebra course. It was, bar none, the most educational test I'd ever taken. It was a set of 50 questions, of which we only had to answer 30. Every single one was a conjecture, which we had to answer T/F and either formally prove or provide a counterexample for... |
4,631,283 | <p>A question in my textbook says to evaluate <span class="math-container">$\displaystyle \int \frac{1}{\sqrt{x^2-a^2}}~dx$</span> where <span class="math-container">$a \gt 0$</span>. I know how to solve the integral using trig substitution but what i do not understand is why is <span class="math-container">$a \gt 0$<... | aschepler | 2,236 | <p>Start with</p>
<p><span class="math-container">$$ \begin{align*} x &= (R + r \cos(6 \pi \theta)) \cos(2 \pi \theta) \\
y &= (R + r \cos(6 \pi \theta)) \sin(2 \pi \theta) \end{align*} $$</span></p>
<p>And we want an equation involving <span class="math-container">$x,y,r,R$</span> but eliminating <span class="... |
415,795 | <p>I am currently doing exercises from graph theory and i came across this one that i can't solve. Could anyone give me some hints how to do it?</p>
<p>Prove that for every graph G of order $n$ these inequalities are true:
$$2 \sqrt{n} \le \chi(G)+\chi(\overline G) \le n+1$$</p>
| Community | -1 | <p>To prove that $\chi(G)+\chi(\overline G)\ge2\sqrt n$: Suppose $G$ and $\overline G$ have been colored with $\chi(G)$ and $\chi(\overline G)$ colors, respectively. Map each vertex to an ordered pair of colors, namely, $v\mapsto ($color of $v$ in $G,$ color of $v$ in $\overline G)$. Note that this map is injective, sh... |
3,996,442 | <blockquote>
<p>If <span class="math-container">$P(x)= \sum_{n=0}^{\infty} a_{n} x^{n}$</span>. It is known that <span class="math-container">$P$</span> satisfies: <span class="math-container">$$P^{\prime}(x)=2xP(x)$$</span> <span class="math-container">$\\$</span> for all <span class="math-container">$ x\in \mathbb{R}... | Simon | 808,494 | <p>It is true that if two power series have the same coefficients then they are equal. More is true actually. If two power series are equal on some interval then their coefficients must be equal.</p>
<p>Assuming that the series has a positive radius of convergence <span class="math-container">$R$</span>, we can differe... |
1,883,139 | <p>A small petroleum company owns two refineries. Refinery 1 costs \$20,000 per day to
operate, and it can produce 400 barrels of high-grade oil, 300 barrels of medium-grade oil,
and 200 barrels of low-grade oil each day. Refinery 2 is newer and more modern. It costs \$25,000 per day to operate, and it can produce 300 ... | Zubzub | 349,735 | <p>Let $x$ be the number of days refinery $1$ is run and $y$ the number of days refinery $2$ is run.</p>
<p>Your function to optimize is :
$$f(x,y) = 20000x + 25000y$$
Your constraints are :
$$
400x + 300y \geq 25000 \\
300x + 400y \geq 27000 \\
200x + 500y \geq 30000
$$
(Note we cannot use strict equality because it... |
1,883,139 | <p>A small petroleum company owns two refineries. Refinery 1 costs \$20,000 per day to
operate, and it can produce 400 barrels of high-grade oil, 300 barrels of medium-grade oil,
and 200 barrels of low-grade oil each day. Refinery 2 is newer and more modern. It costs \$25,000 per day to operate, and it can produce 300 ... | dtldarek | 26,306 | <p><strong>Hint:</strong></p>
<p>The following diagram depicts what is happening. Yet, please, calculate the exact solution yourself.</p>
<p><a href="https://i.stack.imgur.com/BNaeV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BNaeV.png" alt="diagram"></a></p>
<p>I hope this helps $\ddot\smile$... |
604,359 | <p>My experience with non commutative rings is limited to 2 by 2 matrices and the quaternions. The first of which is not a domain, and the latter is a division ring. I'm looking for an example of a domain that is not a division ring. </p>
<p>Invertible matrices do not produce an example, as they must be division rings... | Daniel Miller | 68,076 | <p>nik's example of $\mathbb H[X]$, where $\mathbb H$ is the quaternions is a good one. The set of invertible matrices is not a ring at all (it is not closed under
addition). Another example is the <a href="http://en.wikipedia.org/wiki/Weyl_algebra" rel="nofollow">Weyl algebra</a>
$$
W_1 = k\langle x,y\rangle / (xy... |
2,197,576 | <p>$(p\land q)\rightarrow r$ and $(p\rightarrow r)\lor (q\rightarrow r)$</p>
<p>Have to try prove if they are logically equivalent or not using the laws listed below and also if need to use negation and implication laws. I was going to use associative law and then distributive but I wasn't sure how to get rid of the "... | Bram28 | 256,001 | <p>With the laws that you provide you will not ba able to prove their equivalence. You need an equivalence involving implications. here is the one that is typically used:</p>
<p>Implication: $p \rightarrow q \equiv \neg p \lor q$</p>
<p>Use it as follows:</p>
<p>$(p \land q) \rightarrow r \equiv$ (implication)</p>
... |
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... | MarianD | 393,259 | <p>Compare graphs of the two functions - the <span class="math-container">$\color{green} {green}$</span> one is for <span class="math-container">$\,\color{green}{y=\sin(x)}\ $</span> for <span class="math-container">$\color{green}x$</span> in <strong>radians</strong>, the <span class="math-container">$\color{red} {red}... |
466,271 | <blockquote>
<p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span>, and <span class="math-container">$d$</span> be the lengths of the sides of a quadrilateral. Show that
<span class="math-container">$$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\... | user21820 | 21,820 | <p>WLOG $a = \max(a,b,c,d)$</p>
<p>Let $t = \frac{1}{2} (b+c+d-a) \ge 0$ because $a$,$b$,$c$,$d$ are sides of a quadrilateral</p>
<p>Then decreasing $(a,b,c,d)$ simultaneously by $t$ reduces the desired expression</p>
<p>And $a-t = (b-t)+(c-t)+(d-t)$</p>
<p>Thus it suffices to minimize the expression when $a=b+c+d$... |
466,271 | <blockquote>
<p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span>, and <span class="math-container">$d$</span> be the lengths of the sides of a quadrilateral. Show that
<span class="math-container">$$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\... | Karl Marx | 145,571 | <p>use the ptolemy inequality !</p>
<p>$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$\Longrightarrow$</p>
<p>$2(ab^3+bc^3+cd^3+da^3)\ge$$b^2\cdot{a}(b+c)+c^2\cdot{b}(c+d)+d^2\cdot{c}(a+d)+a^2\cdot{d}(a+b)$$\Longrightarrow$</p>
<p>$2(ab^3+bc^3+cd^3+da^3)\ge$$b^2\cdot{a}l_{1}+c^2\cdot{b}l_{2}+d^2\cdot{c}l_{1}+a^2\cdo... |
466,271 | <blockquote>
<p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span>, and <span class="math-container">$d$</span> be the lengths of the sides of a quadrilateral. Show that
<span class="math-container">$$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\... | River Li | 584,414 | <p>WLOG, assume that <span class="math-container">$a = \max(a, b, c, d)$</span>.
Let <span class="math-container">$x = a- b, \ y = a-c, \ z = a-d$</span>. Then <span class="math-container">$x, y, z \ge 0$</span>.
Let <span class="math-container">$w = b+c+d - a$</span>. Then <span class="math-container">$w > 0$</spa... |
260,857 | <p>I have the following code</p>
<pre><code>rand = {1, 2, 3};
el = EdgeList[CompleteGraph[5]]
g = CompleteGraph[5,
EdgeLabels -> Table[el[[i]] -> RandomChoice[rand], {i, Length[el]}]]
</code></pre>
<p>What I want is to get the labels of each edge of my graph in a list. Is there a way to do it?</p>
| kglr | 125 | <p>You can use <a href="https://reference.wolfram.com/language/ref/PropertyValue.html" rel="nofollow noreferrer"><code>PropertyValue</code></a> (and/or <a href="https://reference.wolfram.com/language/ref/AnnotationValue.html" rel="nofollow noreferrer"><code>AnnotationValue</code></a> in versions 12.1+) as follows:</p>
... |
1,192,434 | <p>I have a problem with integrating of fraction
$$
\int \frac{x}{x^2 + 7 + \sqrt{x^2 + 7}}
$$
I have tried to rewrite it as $\int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{x^4 + 13x^2 + 42} = \int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{(x^2 + 6)(x^2 + 7)}$ and then find some partial fractions, but it wasn't succesful.</p>
| kobe | 190,421 | <p>Using the factorization $1 - x^{2^{k+1}} = (1 - x^{2^k})(1 + x^{2^k})$ we decompose</p>
<p>\begin{align}&\frac{x}{1 - x^2} + \frac{x^2}{1 - x^4} + \cdots + \frac{x^{2^n}}{1 - x^{2^n}}\\
&= \left(\frac{1}{1 - x} - \frac{1}{1 - x^2}\right) + \left(\frac{1}{1 - x^2} - \frac{1}{1 - x^4}\right) + \cdots + \left(... |
514,702 | <p>I know that this is true and is used to prove that $\mathbb{Q}$ is not a discrete metric space, but I can't figure out, why is it true ?</p>
| Anupam | 84,126 | <p>Any subset of $\mathbb{Q}$ is open if it is of the type $G\cap \mathbb{Q}$, where $G$ is open in $\mathbb{R}$. If you choose any singleton $\{a\}$ in $\mathbb{Q}$, then it can not be written in the above form.</p>
|
315,697 | <p>Let <span class="math-container">$X$</span> be an irreducible normal projective scheme over <span class="math-container">$\mathbb{C}$</span>. Let <span class="math-container">$U$</span> be the open subscheme of smooth points of <span class="math-container">$X$</span>. Consider the closed subscheme <span class="math-... | Piotr Achinger | 3,847 | <p>In fact, quite the opposite tends to be true. Mumford [1] showed that for <span class="math-container">$(X,0)$</span> the germ of a normal surface singularity (over <span class="math-container">$\mathbf{C}$</span>), <span class="math-container">$U=X\setminus 0$</span>, one has <span class="math-container">$\pi_1(U)=... |
2,824,411 | <p>My first thought was successful: <span class="math-container">$x^4+x^2=x^2(x^2+1)$</span> and <span class="math-container">$x^3+x^2+1=x^2(x+1)+1$</span> so it is its own inverse because <span class="math-container">$(x^2(x+1)+1)^2\equiv x^4(x+1)^2+1\equiv x^4(x^2+1)+1\equiv1.$</span></p>
<p>The given solution claims... | dan_fulea | 550,003 | <p>Note that we pick the ten cards without "putting them back", so the space $\Omega$ is the set of all subsets with ten elements of $\Omega_0=\{1,2,\dots,52\}$. This is not the cartesian product $\Omega_0^{\times 10}$. (Which allows repetitions, and also knows the order the ten cards came to the hand.)</p>
<p>Our con... |
2,824,411 | <p>My first thought was successful: <span class="math-container">$x^4+x^2=x^2(x^2+1)$</span> and <span class="math-container">$x^3+x^2+1=x^2(x+1)+1$</span> so it is its own inverse because <span class="math-container">$(x^2(x+1)+1)^2\equiv x^4(x+1)^2+1\equiv x^4(x^2+1)+1\equiv1.$</span></p>
<p>The given solution claims... | Steve Kass | 60,500 | <p>The chance that A♣️ is in your hand is $10/52$. Given that, the chance the A♦️ also is is then $9/51$, etc, so the answer is $\dfrac{10\cdot9\cdot8\cdot7}{52\cdot51\cdot50\cdot49}$.</p>
|
2,824,411 | <p>My first thought was successful: <span class="math-container">$x^4+x^2=x^2(x^2+1)$</span> and <span class="math-container">$x^3+x^2+1=x^2(x+1)+1$</span> so it is its own inverse because <span class="math-container">$(x^2(x+1)+1)^2\equiv x^4(x+1)^2+1\equiv x^4(x^2+1)+1\equiv1.$</span></p>
<p>The given solution claims... | BruceET | 221,800 | <p><strong>Comment:</strong> You have several correct answers already--<em>including your own!</em> I just wanted to show
the connection with the <strong><em>hypergeometric distribution</em></strong>.</p>
<p>The number $X$ of Aces among ten cards chosen at random without replacement
from a 52 card deck has a hypergeom... |
1,793,375 | <p>Question: </p>
<p><a href="https://i.stack.imgur.com/TbjfY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TbjfY.png" alt="enter image description here"></a></p>
<p>My work for parts a and b:
<a href="https://i.stack.imgur.com/C3CgR.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.... | Sarah | 314,963 | <p>First calculate out Cov(X,Y) using $Cov(X,Y)=\sum(X-\mu_X)(Y-\mu_Y)f(X,Y)$ where f(X,Y) is the correspondig pdf.<br/>
Then use the formula of correlation coefficient: $cor(X,Y)=\frac{Cov(X,Y)}{\mu_X\mu_Y}$<br/>You can take a look at this example: <a href="https://onlinecourses.science.psu.edu/stat414/book/export/htm... |
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>.... | Emilio Novati | 187,568 | <p>As you noted, the matrix $B$ is diagonalizable, and we have:
$$
B=\begin{bmatrix}
3 & 2\\
4 & 1
\end{bmatrix}=SDS^{-1}=
\begin{bmatrix}
-1 & 1\\
2 & 1
\end{bmatrix}
\begin{bmatrix}
-1 & 0\\
0 & 5
\end{bmatrix}
\begin{bmatrix}
-1/3 & 1/3\\
2/3 & 1/3
\end{bmatrix}
$$</p>
<p>A matrix $A... |
14,486 | <h2>Speculation and background</h2>
<p>Let <span class="math-container">$\mathcal{C}:=\mathrm{CRing}^\text{op}_\text{Zariski}$</span>, the affine Zariski site. Consider the category of sheaves, <span class="math-container">$\operatorname{Sh}(\mathcal{C})$</span>.</p>
<p>According to <a href="https://ncatlab.org/nlab/s... | Daniel Bergh | 1,084 | <p>I'm not sure if this is what you are after, but when I started to look at Grothendieck-topologies I thought of being an open immersion as a topological property; somehow it should be possible to recover all open immersions from the topology, and if we changed the topology to the etalé site, the same method should gi... |
65,923 | <p>The <a href="http://en.wikipedia.org/wiki/Parity_of_a_permutation">sign of a permutation</a> $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula</p>
<p>$${\rm sgn}(\sigma) = (-1)^m$$</p>
<p>where $m$ is the number ... | Derek O'Connor | 14,701 | <p>If $c_e(n)$ is the number of even-length cycles in a permutation $p$ of length $n$, then one of the formulas for the sign of a permutation $p$ is $\text{sgn}(p) = (-1)^{c_e(n)}$.</p>
<p>Here is an $O(n)$ Matlab function that computes the sign of a permutation vector $p(1:n)$ by traversing each cycle of $p$ and (imp... |
3,790,726 | <p><span class="math-container">$\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$</span></p>
<hr />
<p>By maths calculator it results 1.
I calculate and results <span class="math-container">$\sqrt{-\frac{1}{2}}$</span>.</p>
<p><span class="math-container">$\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$</span>... | Sage Stark | 745,622 | <p>There's something called the "Trivial Inequality" that states <span class="math-container">$$x^2 \geq 0$$</span> for all <span class="math-container">$x\in \mathbb{R}$</span>. So <span class="math-container">$$\left( -\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}.$$</span></p>
|
3,790,726 | <p><span class="math-container">$\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$</span></p>
<hr />
<p>By maths calculator it results 1.
I calculate and results <span class="math-container">$\sqrt{-\frac{1}{2}}$</span>.</p>
<p><span class="math-container">$\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$</span>... | Bachamohamed | 804,312 | <p>Hint:<span class="math-container">$(-\sqrt{3})^2=(-1)^2(3)^{{\frac{1}{2}}2}=3$</span></p>
|
395,118 | <p>In a city of over $1000000$ residents, $14\%$ of the residents are senior citizens. In a simple random sample of $1200$ residents, there is about a $95\%$ chance that the percent of senior citizens is in the interval [pick the best option; even if you can provide a sharper answer than you see in the choices, please ... | Clement C. | 75,808 | <p>You can do the change of variable $u=\frac{1}{t}$: for $x > 0$,
$$
\ln\frac{1}{x} = \int_{1}^{\frac{1}{x}} \frac{dt}{t} = \int_{1}^{x} \frac{-du}{u^2}u = -\int_{1}^{x} \frac{du}{u} = -\ln x
$$</p>
|
2,788,500 | <p><strong>Prove that $9\vert F_{n+24}$ iff $9\vert F_n$.</strong></p>
<p>This can be proved if in some way we can establish $F_{n+24}\equiv F_{n}(mod 9)$.</p>
<p>It is given in hint to use identity $F_{m+n}=F_{m-1}F_{n}+F_{m}F_{n+1}$.</p>
<p>Using the above identity for $F_{n+24}$,we get $F_{n+24}=F_{n-1}F_{24}+F_... | lhf | 589 | <p>Just note that $F_{24} \equiv 0 \bmod 9$ and $F_{25} \equiv 1 \bmod 9$, and so $F_{n+24} = F_{n-1}F_{24}+F_{n}F_{25} \equiv F_{n} \bmod 9$.
In particular, $9 \mid F_{n+24}$ iff $9 \mid F_n$.</p>
<p>Actually, we have $9 \mid F_{n+12}$ iff $9 \mid F_n$ because $F_{12} \equiv 0 \bmod 9$ and $F_{13} \equiv -1 \bmod 9$ ... |
2,912,881 | <p>In an $n\times n$ board ($n\geq 3$), how many colors do we need so that we can color the cells such that no three consecutive cells (horizontal, vertical, or diagonal) are of the same color?</p>
<p>With three colors we can do it, using the pattern</p>
<p>$$131$$
$$232$$
$$312$$</p>
<p>and repeating it as necessar... | Marlo | 74,183 | <p>The answer to your question are <a href="http://mathworld.wolfram.com/StaircaseWalk.html" rel="nofollow noreferrer">staircase walks</a> that are related to <a href="http://mathworld.wolfram.com/DyckPath.html" rel="nofollow noreferrer">Dyck paths.</a></p>
<p>In general there are $\binom{m+n}{n}$ such paths. In your ... |
543,599 | <p>I have a question that is:</p>
<p>$a_1 = 1, a_{k+1} = (k+1)+a_k$</p>
<p>compute $a_8$</p>
<p>I suspect $a_8 = 46$</p>
<blockquote>
<p>from:
$${a_2 = 1 + 1 + 1 = 3}$$
$${a_3 = 2 + 1 + 3 = 6}$$
$$...$$</p>
</blockquote>
<p>Am I computing this correctly?</p>
<p>Thanks</p>
| riboch | 94,101 | <p>It looks like you are on the right track, but you made a mistake somewhere: </p>
<p>$$
\begin{array}{rcl}
a_{2}&=&(1+1)+1=3\\
a_{3}&=&(2+1)+3=6\\
a_{4}&=&(3+1)+6=10\\
a_{5}&=&(4+1)+10=15\\
a_{6}&=&(5+1)+15=21\\
a_{7}&=&(6+1)+21=28\\
a_{8}&=&(7+1)+28=36\\
\end... |
4,442,117 | <p>This is the question I am trying to solve. "Suppose that n items are being tested simultaneously, the items are independent, and the length of life of each item has the exponential distribution with parameter β. Determine the expected length of time until three items have failed"</p>
<p>I approached the pr... | trula | 697,983 | <p>You seem confused about curvature it is not "the global angular rate of change of of the tangent unit vector", but it is the differential change in the case of the circle it happens that it is also the global change.</p>
|
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>$$(1)... | Franklin Pezzuti Dyer | 438,055 | <p><strong>MY "ANSWER":</strong> I can get you started on the right track using the following two identities:</p>
<p>$$\bbox[lightgray,15px]
{
\int_0^1 x^n\ln(1-x)dx=-\frac{H_{n+1}}{n+1}
}$$</p>
<p>$$\bbox[lightgray,15px]
{
\frac{1}{2}\ln^2(1-x)=\sum_{n=1}^\infty \frac{H_n x^{n+1}}{n+1}
}$$</p>
<p>From these, we hav... |
1,794,459 | <p>We have set $S=\{ (a,b) | a,b \in \mathbb{Q}\}$</p>
<p>And we know that $(a,b)\sim \mathbb{R}$ , so $k((a,b))=c$. And $\mathbb Q \sim \mathbb N$, so $k(\mathbb Q)=\aleph_0$.</p>
<p>I don't know how to put all informations together.</p>
<p>I was also thinking that $S \subset \mathbb R$ and than we know $k(S) \le k... | user133281 | 133,281 | <p><strong>Hint:</strong> there exists an injective map $f: S \to \mathbb{Q}^2$. Since $\mathbb{Q}^2$ is countable, $S$ is at most countable.</p>
|
2,148,389 | <p>Consider the equation $x^2 - 5x + 6 = 0$. By factorising I get $(x-3)(x-2) = 0$. Which means it represents a pair of straight lines, namely $x-2 =0 $ and $x- 3 = 0$, but when I plot $x^2 - 5x + 6 = 0$, I get a parabola, not a pair of straight lines. Why?</p>
<p>Plotting: <code>x^2 - 5x + 6 = 0</code> </p>
<p>at <... | CiaPan | 152,299 | <p>You're asking about two different things in your question and in its title.</p>
<p>As for the title:</p>
<blockquote>
<p><strong>Why does "$x^2 - 5x + 6 = 0$", which is the same as "$(x-3)(x-2) = 0$", represent a parabola?</strong></p>
</blockquote>
<p>the answer is: <strong>it doesn't</strong>.</p>
<p>The equ... |
3,965,967 | <p>Assume that customers arrive to a bus stop according to a homogenous Poisson process with rate <span class="math-container">$\alpha$</span> and that the arrival process of buses is an independent renewal process with interarrival distribution <span class="math-container">$F$</span>. <strong>What is the long run perc... | TOMILO87 | 117,933 | <p>First recall the renewal reward theorem:</p>
<p><span class="math-container">$E[R(t)]/t \rightarrow E[R]/E[T]$</span>,</p>
<p>where <span class="math-container">$E[R]$</span> is the expected reward during a cycle and <span class="math-container">$E[T]$</span> is the expected length of a cycle. We denote <span class=... |
1,817,300 | <p>How to integrate $\frac {\cos (7x)-\cos (8x)}{1+2\cos (5x)} $ ? </p>
<p>All I could do is apply difference of cosines formula in numerator.After that I'm stuck.Can somebody please help me?</p>
| Qwerty | 290,058 | <p>The rule is to multiply above and below by the sin(5x) (here).</p>
<p>$${\sin(5x) (\cos (7x)-\cos (8x))\over \sin(5x)+\sin(10x)}$$
$$={\sin(5x)2\sin(15x/2)\sin(x/2)\over 2\sin(15x/2)\cos(5x/2)}$$
$$=2\sin(5x/2)\sin(x/2)$$</p>
<p>Now its easy integration right?</p>
|
1,817,300 | <p>How to integrate $\frac {\cos (7x)-\cos (8x)}{1+2\cos (5x)} $ ? </p>
<p>All I could do is apply difference of cosines formula in numerator.After that I'm stuck.Can somebody please help me?</p>
| juantheron | 14,311 | <p>Let $$I = \int\frac{\cos 7x-\cos 8x}{1+2\cos 5x}dx = \int\frac{(\cos 7x+\cos 3x)-(\cos 8x+\cos 2x)-\cos 3x+\cos 2x}{1+2\cos 5x}dx$$</p>
<p>$$\bullet
\displaystyle \cos C+\cos D = 2\cos \left(\frac{C+D}{2}\right)\cos \left(\frac{C-D}{2}\right)$$</p>
<p>So $$I = \int\frac{2\cos 5x\cos 2x+\cos 2x-2\cos 5x\cos3x-\cos... |
330,236 | <p>I'm dutch and I'm not sure if I translated this right. If there are some more dutchies here, how could I translate: "volledig stelsel van representanten"</p>
<p>Let $H$ be a subgroup of of $G$ and consider the left cosets of $H$. Pick from every distinct left coset one element, and put them in the set $R$. Show tha... | Gerry Myerson | 8,269 | <p>EDIT: OP has edited question so what follows is no longer relevant. </p>
<p>I'm not sure what that little circle is, but I don't see why $G_3\circ G_4$ should be a subset of $G_3\cap G_4$. </p>
|
1,223,909 | <p>The following is a use of eisenstein criterion that i have taken out from my lecture note.</p>
<p>$f(x, y) = x^4 +x^3y^2 +x^2y^3 +y$ is irreducible in Q[x, y]. This can be proved by treating Q[x,y] as (Q[y])[x] and applying the Eisenstein criterion with p = y.</p>
<p>However, I can't understand why i can apply eis... | Andreas Caranti | 58,401 | <p>$\mathbf{Q}[y]/(y) \cong \mathbf{Q}$ is a field, so that $y$ is prime.</p>
|
2,281,161 | <p>Munkres say it is of the form $\{x\}\times(a,b) $, but for me it just the intervals of the form $(a,b)×(c,d)$. Can anyone explain?</p>
| DanielWainfleet | 254,665 | <p>Let $(u,v)_p$ denote an ordered pair, to distinguish it from an open real interval. With respect to the lexicographic (dictionary) order $<_L$ on $\mathbb R^2$ we have $$\{x\}\times (a,b)=\{y\in \mathbb R^2 :\; (x,a)_p<_Ly<_L(x,b)_p \}$$ which is an open interval in the $<_L$ order.</p>
<p>Observe that ... |
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>
| ulead86 | 13,661 | <p>You can try it this way:</p>
<p>A number is irrational, if you can not find a finite continued fraction.</p>
<p>Now try to write $\sqrt{13}$ as continued fraction and you'll see, its periodic:</p>
<p>$$\sqrt{13}=[3;\overline{1,1,1,1,6}]$$</p>
<p>Hope that is what your you looking for.</p>
|
49,679 | <p>I'd like to know whether the following statement is true or not.</p>
<p>Let $T_1, T_2\in \mathbb{C}^{n\times n}$ be upper triangular matrices. If there exists a nonsingular matrix $P$ such that $T_1=PT_2P^{-1}$, then there is a nonsingular uppper triangular matrix $T$ such that $T_1=TT_2T^{-1}$. </p>
| Denis Serre | 8,799 | <p>It is <strong>false</strong> for the following obvious reason. The diagonal elements of a triangular matrix are its eigenvalue. If they are pairwise distinct, the matrix is similar to its diagonal. </p>
<p>Assume now that two upper triangular matrices have the same diagonal elements, pairwise distinct, but not in t... |
3,305,140 | <p>So, I'm going to take an enumerative combinatorics class this upcoming semester. I began reading about it and came across and interesting example, but I am not sure how they arrive at their final answer. The example is in in the image I included. I don't know how they determined the equations for <span class="math... | Cesareo | 397,348 | <p>Hint.</p>
<p><span class="math-container">$$
(t u)' = u^2
$$</span></p>
<p>now making <span class="math-container">$ v = t u$</span></p>
<p><span class="math-container">$$
v' = \frac{v^2}{t^2}\Rightarrow \frac{dv}{v^2} = \frac{dt}{t^2}
$$</span></p>
<p>or integrating</p>
<p><span class="math-container">$$
\frac... |
563,712 | <p>A corollary at page 91 of the book Group Theory I by M. Suzuki is as follows:</p>
<p>Let $A$ be an abelian subgroup of a $p$-group $G$. If $A$ is maximal among abelian normal subgroups of $G$, then $A$ satisfies $C_G(A)=A$. In particular, $A$ is maximal among abelian subgroups of $G$. </p>
<p>I am confused with t... | Beginner | 15,847 | <p>A maximal abelian normal subgroup need not be maximal abelian subgroup. Consider $SL(2,p)$, $p>3$. The only normal subgroups of this group are, the center, $1$ and the whole group. We see that the <strong>center is "maximal abelian normal subgroup", but it is not "maximal abelian subgroup"</strong> since consider... |
152,912 | <p>I am reading probability theory and I have a question. The Bochner-Minlos theorem roughly says that if we have $E \subset H \subset E^*$ where $H$ is a Hilbert space, then there is a unique measure on $E^*$ satisfying some nice properties. </p>
<p>On the other hand, the Wiener measure is supported on the space of... | Abdelmalek Abdesselam | 7,410 | <p>Yes. Wiener measure can be arrived at using the Bochner-Minlos Theorem in at least two ways.</p>
<ol>
<li>One can consider the bilinear form on $S(\mathbb{R})$
$$
C(f,g)=\int_{\mathbb{R}}\ \overline{\widehat{f}(\xi)} \widehat{g}(\xi)\ \frac{d\xi}{2\pi}
$$
then the BM Theorem applied to the characteristic function
... |
1,604,204 | <blockquote>
<p><span class="math-container">$a,b,c$</span> are in A.P ; <span class="math-container">$p,q,r$</span> are in H.P. And <span class="math-container">$ap,bq,cr $</span> are in G.P. Then what is the value of <span class="math-container">${p\over r} + {r\over p}\ \ ?$</span></p>
<p><span class="math-container... | lab bhattacharjee | 33,337 | <p>Let $a=b-d,c=b+d$</p>
<p>and $p=\dfrac1{x-D}, q=\dfrac1x, r=\dfrac1{x+D}$</p>
<p>$ap\cdot cr=(bq)^2\implies\dfrac{b^2}{x^2}=\dfrac{b-d}{x-D}\dfrac{b+d}{x+D}=\dfrac{b^2-d^2}{x^2-D^2}=\dfrac{d^2}{D^2}=k$(say)</p>
<p>$\implies b^2=kx^2,d^2=kD^2$</p>
<p>$\dfrac rp+\dfrac pr=\dfrac{x-D}{x+D}+\dfrac{x+D}{x-D}=\dfrac{2... |
301,198 | <p>I am new to linear algebra and I have a doubt that : in 2D coordinate system is a line which is at 45 degree <strong>NOT</strong> passing through the origin a subspace of the vector space comprising the whole 2D plane i.e. $ \mathbb{R}^2 $ ?
let $V = \{ (x,y) \in \mathbb{R}^2 \}$. and $W = \{ (x,y) \in \mathbb{R}^... | Marc van Leeuwen | 18,880 | <p>A subspace $S$ of a vector space must always contain the zero vector, because (it is not allowed to be the empty set, and) (i) it must be closed under subtraction: if $x\in S$ then $x-x=\vec 0\in S$, or alternatively (ii) it must be closed under scalar multiplication if $x\in S$ then $0x=\vec 0\in S$.</p>
|
2,522,291 | <p>I have two matrices where $A = $ \begin{bmatrix}1&-2\\-1&3\end{bmatrix} and $B =$ \begin{bmatrix}0&-1\\2&-2\end{bmatrix}. The question asks to Solve for the matrix C (i.e. find matrix C):
$$(((AB^T)^{-1})^TB)^T= 2C + B$$</p>
<p>I have done the calculations by following the formula and get \begin{bma... | AlkaKadri | 258,038 | <p>There are a few very useful identities that you should swear on your life to never forget. Ready?
$$(AB)^T = B^T A^T$$
$$(AB)^{-1} = B^{-1} A^{-1}$$
$$(A^{-1})^T = (A^T)^{-1}$$
In summary, when we "distribute" the "transpose" or "inverse" over a product, we need to reverse the order of the product. We can also inter... |
4,096,885 | <p>For example if I have two sets <em>A</em> and <em>B</em>, where I take the Cartesian Product of both, does it matter if I perform the operation in this order <em>A</em>x<em>B</em> or whether I perform the operation in this order or the order <em>B</em>x<em>A</em>, or does the order not even matter? I am enquiring be... | Geoffrey Trang | 684,071 | <p>For any two sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, there is a canonical bijection between <span class="math-container">$A \times B$</span> and <span class="math-container">$B \times A$</span>, but they are not the same set, unless <span class="math-container">$A$</s... |
4,096,885 | <p>For example if I have two sets <em>A</em> and <em>B</em>, where I take the Cartesian Product of both, does it matter if I perform the operation in this order <em>A</em>x<em>B</em> or whether I perform the operation in this order or the order <em>B</em>x<em>A</em>, or does the order not even matter? I am enquiring be... | Samuele Monitto | 740,138 | <p>They are in general different sets, so it does matter, <span class="math-container">$A\times B$</span> will have different elements compared to <span class="math-container">$B\times A$</span> . A simple example can be one where you take <span class="math-container">$A=\{0,1\}$</span> and <span class="math-container"... |
1,446,797 | <p>A sequence is non-decreasing if $k_1 \leq k_2 \leq k_3$.</p>
<p>Now <strong>I need to find the number of non-decreasing sequences of length-$n$ sequences from $\{1,2,....m\}$</strong></p>
<p>I basically see it as sum of the numbers of strictly increasing sequences plus other sequences.</p>
<p>The number of stric... | Probability-Stats-Optimisation | 561,429 | <p>I would also <a href="https://math.stackexchange.com/questions/1396896/number-of-non-decreasing-functions">link it to this problem</a>. It is, in effect, counting the number of non-decreasing functions from a set <span class="math-container">$A$</span> such that <span class="math-container">$\|A\|=r$</span> to a set... |
2,243,598 | <p>Consider this integral $(1)$</p>
<blockquote>
<p>$$\int_{0}^{\infty}\color{red}{{\gamma+\ln x\over e^x}}\cdot{1-\cos x\over x}\,\mathrm dx={1\over 2}\cdot{\pi-\ln 4\over 4}\cdot{\pi+\ln 4\over 4}\tag1$$</p>
</blockquote>
<p>Recall a well-known integral for $\gamma$:</p>
<p>$$\int_{0}^{\infty}e^{-x}\ln x\,\mathr... | Jack D'Aurizio | 44,121 | <p>The Laplace transform of $1-\cos x$ is $\frac{1}{s(1+s^2)}$ and the inverse Laplace transform of $\frac{1}{x e^{x}}$ is the characteristic function of $(1,+\infty)$, hence
$$ \int_{0}^{+\infty}\frac{1-\cos x}{x e^x}\,dx = \int_{1}^{+\infty}\frac{ds}{s(1+s^2)}=\frac{1}{2}\log(2).$$
You may apply the same technique to... |
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