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 |
|---|---|---|---|---|
321,916 | <p>In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also adds a layer of complexity). Why do we do it this way? </p>
<p>The first question is to what extent are the notions d... | Piotr Hajlasz | 121,665 | <p>Actually, in the following book the Lebesgue integral is defined the way you suggested:</p>
<p><strong>Pugh, C. C.</strong> <A HREF="https://link.springer.com/book/10.1007%2F978-3-319-17771-7" rel="noreferrer"><em>Real mathematical analysis</em></A>.
Second edition. Undergraduate Texts in Mathematics. Springer, C... |
1,251,537 | <p>$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically?</p>
<hr>
<p>Using integration by parts I got the form:
$\int_a^bg(x)f(x)-g'(x)F(x)=0$. Where $F'(x)=f(x)$.</p>
| Kitegi | 120,267 | <p>By the Stone Weierstrass Theorem, we can show that if $f$ is a continuous function on $[a,b]$ and $$\forall n \in \Bbb N:\ \int_a^b f(x)x^ndx=0$$
then $f=0$</p>
<p>Since we're only allowed to use functions that vanish in $a$ and $b$, then we have $$\forall n \in \Bbb N:\ \int_a^b f(x)(x-a)(b-x)x^ndx=0$$
So $x\right... |
1,251,537 | <p>$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically?</p>
<hr>
<p>Using integration by parts I got the form:
$\int_a^bg(x)f(x)-g'(x)F(x)=0$. Where $F'(x)=f(x)$.</p>
| TonyK | 1,508 | <p>In particular, putting $g(x)=(x-a)(b-x)f(x)$, we have $\int_a^b{(x-a)(b-x)f(x)^2dx}=0$. The integrand is non-negative in $[a,b]$, so $f = 0$ almost everywhere. As $f$ is continuous, $f$ must be identically zero.</p>
|
2,669,524 | <p>I am reading <strong>Algebraic Geometry</strong>, Vol 1, <em>Kenji Ueno</em>. My problem is that
$$k\left[ x,y,t\right]/\left(xy-t\right)\otimes_{k\left[t\right]}k\left[t\right]/\left(t-a\right) \simeq k\left[x,y\right]/\left(xy-a\right) $$
where $k$ is a field and $a$ is an element in $k$. I don't understand how i... | Allawonder | 145,126 | <p>Contrary to what you might find in many contemporary textbooks (especially if they're elementary), you can well think of an infinitesimal as an entity and manipulate it according to its own rules (this has been rigorously set down in the last century in the so-called nonstandard calculus, which in fact I think is th... |
976,910 | <p>i'm having a small issue with a certain question. </p>
<p>Given a parametric equation of a plane $x=5-2a-3b$, $y=3-4a+2b$, $z=7-6a-2b$, find a point $P$ on the plane so that the position vector of $P$ is perpendicular to the plane.</p>
<p>How would you go about this for a parametric equation? I think I could conve... | layman | 131,740 | <p>What can we deduce from $K$ not having any limit points? </p>
<p>Well, that means if we take $x \in X - K$, then we can find some $\epsilon > 0$ such that $B(x, \epsilon) \cap K = \emptyset$ (otherwise, $x$ would be a limit point of $K$). </p>
<p>But $B(x, \epsilon) \cap K = \emptyset \implies B(x, \epsilon) ... |
746,180 | <p>I'm working through Stephen Abbott's wonderful <em>Understanding Analysis</em> in preparation for entering a math undergrad degree this fall. A personal note about me: Friends and family tell me I tend to be periphrastic; if there's a long-winded, inelegant way of explaining myself, I'll find it. As I work through A... | John Joy | 140,156 | <p>$$\int \frac{1}{x^2 \sqrt{x^2+4}}dx = \int \frac{1}{8(\frac{x}{2})^2 \sqrt{(\frac{x}{2})^2+1}}dx= \int \frac{1}{8tan^2(tan^{-1}(\frac{x}{2})) \sqrt{tan^2(tan^{-1}(\frac{x}{2}))+1}}dx$$</p>
<p>$$=\int \frac{1}{8tan^2(tan^{-1}(\frac{x}{2})) \sqrt{tan^2(tan^{-1}(\frac{x}{2}))+1}}dx\frac{\frac{d(tan^{-1}(\frac{x}{2}))}... |
148,420 | <p>A simple concept but I've not been able to solve it. I'm trying to create a stack of 2D plots in 3D space using Mathematica 9. <strong>This is not a parametric plot</strong>, but I'm creating it from an array of vectors (imported
.csv file). The ListPlot3D function creates a filled mesh but what I want is this typ... | yohbs | 367 | <p>You can also use <a href="http://reference.wolfram.com/language/ref/ParametricPlot3D.html" rel="nofollow noreferrer"><code>ParametricPlot3D</code></a>:</p>
<pre><code>f[x_, y_] := Exp[-x^2 - y^2/(4 + x^2/4) + x y];
ParametricPlot3D[Table[{x, y, f[x, y]}, {y, -5, 5}], {x, -5, 5},
PlotRange -> Al... |
138,203 | <p>Assume I export some data into a file like:</p>
<pre><code>data = {{1, 2, 3}, {4, 5, 6}};
Export["test.h5",data,{"Datasets","/h1"}];
</code></pre>
<p>How can I append <code>{7, 8, 9}</code> to the "test.h5" (by directly writing in the test.h5) such that the results for </p>
<pre><code>Import["test.h5", data, {"Da... | Albert Retey | 169 | <p>Extending an existing dataset is not possible with the standard Mathematica <code>Export</code>, at least not with any version up to 11.0.1. What does work is to add additional datasets one by one like this:</p>
<pre><code>filename = FileNameJoin[{$HomeDirectory, "Desktop", "tst.h5"}]
Export[filename, {{1, 2, 3}},... |
138,203 | <p>Assume I export some data into a file like:</p>
<pre><code>data = {{1, 2, 3}, {4, 5, 6}};
Export["test.h5",data,{"Datasets","/h1"}];
</code></pre>
<p>How can I append <code>{7, 8, 9}</code> to the "test.h5" (by directly writing in the test.h5) such that the results for </p>
<pre><code>Import["test.h5", data, {"Da... | Thunderbird | 67,254 | <p>It seems like it has changed in Version 12.0.0.0</p>
<p>"Append" -> True</p>
<p>is now implemented as</p>
<p>OverwriteTarget->"Append"</p>
|
3,162,464 | <p>I need help making an OGF for <span class="math-container">$1 + x^i + x^{2i}+...+x^{ki}$</span>. I already know how to verify that <span class="math-container">$1 +x +x^2+...+x^k$</span> can be written by <span class="math-container">$({1-x^{k+1}})/({1-x})$</span>. I'm wondering if there is any correlation between t... | PrincessEev | 597,568 | <p>Notice that</p>
<p><span class="math-container">$$1 + x^i + x^{2i} + x^{3i} + ... + x^{ki} = 1 + (x^i)^1 + (x^i)^2 + (x^i)^3 + ... + (x^i)^k$$</span></p>
<p>This is a finite geometric series with ratio <span class="math-container">$x^i$</span>, and thus</p>
<p><span class="math-container">$$1 + x^i + x^{2i} + x^{... |
858,952 | <p>related to <a href="https://math.stackexchange.com/questions/830599/one-sided-limit-lim-x-rightarrow-0-fx-where-wolfram-alpha-does-not-hel">this question</a>:</p>
<p>Is there an easy closed-form term for</p>
<p>$$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$</p>
<p>thus when the sum starts at a constant $k$ instead... | draks ... | 19,341 | <p>$$
\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x}=
\left(\sum_{j=k}^{\infty} \frac{x^j}{j!}+\sum_{j=0}^{k-1} \frac{x^j}{j!}-\sum_{j=0}^{k-1} \frac{x^j}{j!}\right)e^{-x}\\
=\left(e^x-\sum_{j=0}^{k-1} \frac{x^j}{j!}\right)e^{-x}=1-e^{-x}\sum_{j=0}^{k-1} \frac{x^j}{j!}
$$</p>
|
956,235 | <p>This may be a little low-brow for this forum, but I'm trying to figure out what the common base number set is between two other sets of numbers. Here's the situation: I have received quotes from two vendors for a list of products that they sell, and the prices they have quoted are:</p>
<pre><code> Vend... | Leucippus | 148,155 | <p>Given $s(t) = 32 + 112 t - 16 t^{2}$ then $v(t)$, being the derivative of $s(t)$, is $v(t) = 112 - 32 t$. If $v(t) = 0$ then $t = 7/2$. Now, $s(7/2) = 32 + 56 \cdot 7 - 4 \cdot 49 = 228$. </p>
<p>To find the velocity at impact solve for $s(t) = 0$. This yields $16 t^{2} - 112 t - 32 = 0$, or $t^{2} - 7 t - 2 = 0$. ... |
748,325 | <p>In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen), argues as follows: Let $\sigma$ be the first singular value of A, and $v_{1}$ the corresponding singular vector. Let $w$ be another linearly independent vector such that $||Aw||=\sigma$, and construc... | epi163sqrt | 132,007 | <p><em>Note:</em> Please note that this answer was initially incorrect. Thanks to <a href="https://math.stackexchange.com/users/40119/littleO">littleO</a> who draw attention to my mistake. The essential argument at the end is now based on the answer already stated by <a href="https://math.stackexchange.com/users/15381/... |
2,900,294 | <p>I tried this and I only got $\sin( 53^\circ)= \sin( 127^\circ).$ How do I find the equal value in cosine or tangent? Please help me out. Thank you!</p>
| paulplusx | 578,155 | <p>Use:</p>
<p>$sin(90^\circ + \theta )= cos(\theta)$</p>
<p>$sin(90^\circ - \theta )= cos(\theta)$.</p>
<p>So:</p>
<p>$sin(53^\circ)=sin(90^\circ - 37^\circ )= cos(37^\circ)$.</p>
<p>It would better if you learn more about them here <a href="https://www.khanacademy.org/math/trigonometry/trigonometry-right-triangl... |
1,043,094 | <p>I have to find the limit of following</p>
<p><span class="math-container">$$\lim_{x \to 0}\left(\frac{1}{x} - \frac{1}{x^2}\right)$$</span></p>
<p>I have no idea how to start this one off.
How would I do it?</p>
<p>Do I just substitute the <span class="math-container">$0$</span>? It doesn't look that easy and sim... | Aaron Meyerowitz | 84,560 | <p>For $f(x)=\frac{1}{x}-\frac{1}{x^2}$, if you want to know what $\lim_{x \rightarrow 0}f(0)$ is, try looking at $f(\frac{1}{10}),f(\frac{-3}{100})$ and selected similar values. That will start you off and give you an idea what the answer might be (or why the book says that the answer is what it is.) Once you have dec... |
1,043,094 | <p>I have to find the limit of following</p>
<p><span class="math-container">$$\lim_{x \to 0}\left(\frac{1}{x} - \frac{1}{x^2}\right)$$</span></p>
<p>I have no idea how to start this one off.
How would I do it?</p>
<p>Do I just substitute the <span class="math-container">$0$</span>? It doesn't look that easy and sim... | Milo Brandt | 174,927 | <p>A useful thing to do would be to make the substitution $u=\frac{1}x$. Then, this becomes
$$\lim_{u\rightarrow\infty}u-u^2$$
(or the analogous limit to $-\infty$) but $u$ grows much more slowly than $u^2$, the expression in the limit must decrease without bound - in particular, since $u^2>2u$ if $u>2$, we get t... |
1,043,094 | <p>I have to find the limit of following</p>
<p><span class="math-container">$$\lim_{x \to 0}\left(\frac{1}{x} - \frac{1}{x^2}\right)$$</span></p>
<p>I have no idea how to start this one off.
How would I do it?</p>
<p>Do I just substitute the <span class="math-container">$0$</span>? It doesn't look that easy and sim... | egreg | 62,967 | <p>As written, the limit is in the so-called “indeterminate form $\infty-\infty$”), so we want to rewrite it in another way to start with:
$$
\frac{1}{x}-\frac{1}{x^2}=\frac{x-1}{x^2}
$$
It's not restrictive to work under the assumption that $-1<x<1$; thus $|x|<1$ and $|x|^2<|x|$, that is to say
$$
\frac{1}... |
864,237 | <p>Let's take a short exact sequence of groups
$$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$
I understand what it says: the image of each homomorphism is the kernel of the next one, so the one between $A$ and $B$ is injective and the one between $B$ and $C$ is surjective. I get it. But other than being a so... | Hagen von Eitzen | 39,174 | <p>It's getting more interesting as soon the diagrams get more involved. For example: If the following diagram with exact rows commutes, and the outer columns are both mono/epi/iso-morphism, then the middle one is also a mono/epi/iso-morphism:
$$
\begin{matrix}
0\to &A&\to &B&\to &C&\to 0\\
&am... |
1,725,084 | <p>I am currently trying to practice the technique of transfinite induction with the following problem: </p>
<p>Suppose that $X$ is a non-empty subset of an ordinal $\alpha$, so that $X$ is well-ordered by $\in$. Show that $\text{type}(X; \in) \leq \alpha$. </p>
<hr>
<p>My approach thus far: </p>
<p>Let $\beta = \t... | hmakholm left over Monica | 14,366 | <p>Things will go a bit smoother if you strengthen the induction hypothesis to include "... and $f(X\cap\gamma)$ is an initial segment of $\beta$".</p>
<p>If $\gamma\notin X$ then there's nothing new to prove. So assume that $\gamma\in X$.</p>
<p>Now, for $f$ to be an order isomorphism, $f(\gamma)$ has to be the smal... |
949,512 | <p>How do mathematicians define inner product on a vector space. </p>
<p>For example: $a = (x_1,x_2)$ & $ b =(y_1,y_2) $ in $ \mathbb{R}^2.$ </p>
<p>Define $\langle a,b\rangle= x_1y_1-x_2y_1-x_1y_2+4x_2y_2$. It's an inner product.</p>
<p>But how does one motivate this inner product? I think there is some sort of... | rych | 73,934 | <p>Inner product $\langle u,v\rangle$ doesn't depend on the choice of basis. In the given basis $\langle u,v \rangle =x^TMy$.</p>
<p>Every real symmetric matrix is orthogonally diagonalizable: $M=Q^TDQ$, and in the new, "rotated", coordinates $\langle u,v\rangle=x^TMy=x^TQ^TDQy=(Qx)^TD(Qy)=\tilde{x}^TD\tilde{y}$.</p>
... |
576,271 | <p>I integrated to get $\frac{3}{1-x}$, turned it into a power series $3x^n$, and differentiated to get the series $3nx^{n-1}$ which is incorrect.</p>
| Ross Millikan | 1,827 | <p><a href="http://www.wolframalpha.com/input/?i=series%203/%281-x%29%5E2" rel="nofollow">Wolfram</a> agrees with you that $\frac 3{(1-x)^2}=\sum_{n=1}^{\infty} 3nx^{n-1}$</p>
|
576,271 | <p>I integrated to get $\frac{3}{1-x}$, turned it into a power series $3x^n$, and differentiated to get the series $3nx^{n-1}$ which is incorrect.</p>
| Sammy Black | 6,509 | <p>Shift the index:
$$
\frac{3}{(1 - x)^2} = \sum_{n = 1}^{\infty} 3nx^{n - 1} = \sum_{m = 0}^{\infty} 3(m + 1)x^m.
$$</p>
<p>(Here we've let $m = n - 1$, so $n = m + 1$.)</p>
|
2,218,924 | <p>$ \displaystyle \lim_{n\to \infty} \sum_{k=1}^n \frac{k^4}{n^4}=$ ?</p>
<p>I found it difficult to tranform it into the integral form by the definition of Riemann sum, which is a way to solve similar problems.</p>
| Community | -1 | <p>There is probably a typo.</p>
<p>By the Riemannian summation,</p>
<p>$$\lim_{n\to\infty}\frac1n\sum_{k=1}^n\frac{k^p}{n^p}=\int_0^1 x^pdx=\frac1{p+1}.$$</p>
|
1,370,576 | <p>I am working on a trigonometry question at the moment and am very stuck. I have looked through all the tips to solving it and I cant seem to come up with the right answer. The problem is </p>
<blockquote>
<p>What is exact value of<br>
$$\cot \left(\frac{7\pi}{6}\right)? $$</p>
</blockquote>
| Emilio Novati | 187,568 | <p>Hint: $\cot x=\dfrac{\cos x}{\sin x}$ and</p>
<p>$$
\cos \left(\dfrac{7 \pi}{6}\right)=\cos \left(\pi+\dfrac{\pi}{6}\right)
$$</p>
<p>$$
\sin \left(\dfrac{7 \pi}{6}\right)=\sin \left(\pi+\dfrac{\pi}{6}\right)
$$</p>
<p>now you can use sum formulas or reduction to the first quadrant.</p>
|
274,961 | <p>I want to calculate the determinant along the last slice of a 3-dimensional array. So for I do this by slow the <code>Table</code> command. I know that for time reasons I should use <code>Map</code> or <code>Apply</code>, however couldn't successful solve the problem.</p>
<pre><code>m = 2; n = 3; o = 10;
SeedRandom[... | userrandrand | 86,543 | <p><strong>Update</strong></p>
<p>I was curious to see what the speed would be if one used the neural network framework to do the computation. It turns out that it is faster than using <code>Map</code> and the computation is particularly fast with a GPU. However, error accumulates with the number of components</p>
<p>T... |
1,052,180 | <p>I need to find connected graph $G = (V, E), |V| \geq 3$ such that every power of his adjacency matrix contains zeroes.</p>
<p>I know that that graph will be path and adjacency matrix for even and odd powers would look like this (lets say for $|V| = 3$):</p>
<p>$M=
\left[ {\begin{array}{ccccc}
0 & 1 & ... | ml0105 | 135,298 | <p>The crux of Robert Israel's hint is that a bipartite graph has no cycle of odd length. So a walk follows a sequence of edges, where we can repeat vertices. Now suppose $G$ is bipartite and we can walk from $v_{i} \to v_{j}$, where $v_{i}, v_{j}$ are in the same partition. The proof is essentially by algorithm. We be... |
4,330,991 | <p>I do understand that if:</p>
<p><span class="math-container">$a=b \Rightarrow a^2 = b^2 $</span></p>
<p>But clearly, the graph representing these two equations won't be the same. So, (correct me if I'm wrong) this would suggest that if you square both sides of the equation, you essentially get a different set of ans... | Lee Mosher | 26,501 | <p>If <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are general real numbers then the implication <span class="math-container">$a=b \implies a^2 = b^2$</span> is not reversible (because, of course, <span class="math-container">$(-a)^2 = a^2$</span>).</p>
<p>However, if <span class=... |
4,140,956 | <p>I'm trying to determine the order of the pole in the complex expression</p>
<p><span class="math-container">$$f(z)=\frac{1}{(1-\cos(z))^2}$$</span></p>
<p>I have determined the pole to be <span class="math-container">$z=2\pi n, n\in \mathbb{Z}$</span>.</p>
<p>However, when I use the equation <span class="math-conta... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>Note that <span class="math-container">$1-\cos(z)=2\sin^2(z/2)$</span>. Thus, <span class="math-container">$\left(1-\cos(z)\right)^2=4\sin^4(z/2)$</span></p>
<p>Can you proceed now?</p>
|
156,376 | <p>I understand that when we are doing indefinite integrals on the real line, we have $\int f(x) dx = g(x) + C$, where $C$ is some constant of integration. </p>
<p>If I do an integral from $\int f(x) dx$ on $[0,x]$, then is this considered a definite integral? Can I just leave out the constant of integration now? I am... | Hunter | 120,472 | <p>A definite integral is nothing different from an indefinite integral but the constant, that was eliminated during the differentiation, has some definite value. For instance in indefinite integrals we have to write a C that represents all constants after the integration has been done. So in definite integrals this C... |
1,714,278 | <p>Given the sequence $ y_{k}=2^k\tan(\frac{\pi}{2^k})$ for k=2,3,.. prove that $ y_{k} $ is recursively produced by the algorithm:
$$ y_{k+1}=2^{2k+1}\frac{\sqrt{1+(2^{-k}y_{k})^2}-1}{y_{k}} $$ for k=2,3,...</p>
<p>I used the identity $ {\tan^2({a})}=\frac{1-\cos{(2a)}}{1+\cos{(2a)}}$ but I couldn't get it right. An... | Jean Marie | 305,862 | <p>Let us have a heuristic proof, i.e., try to understand how the recursive formula has been found, in order that, at the end, we can say "I understand how they have had the idea (and how simple it was)".</p>
<p>The formula that is to be established deals with tangents; thus, let us stay with tangents by centering our... |
335,651 | <p>I'm having trouble proving $$\left(\frac{\sin(\frac{n\theta}{2})}{\sin(\frac{\theta}{2})}\right)^2=\left|\sum_{k=1}^{|n|}e^{ik\theta}\right|^2$$ where $n\in\mathbb{Z}$ and $\theta\in\mathbb{R}$. Can anyone suggest a hint?</p>
| Community | -1 | <p>Try this-
We know the Euler's Formula-$e^{ix} = \cos x + i\sin x \ $-<a href="http://en.wikipedia.org/wiki/Euler%27s_formula" rel="nofollow">http://en.wikipedia.org/wiki/Euler%27s_formula</a>.</p>
<p>Link this to summation of sines in as arithmetic progression. Try Proving this!
From wiki:-
Sum of sines and cosines... |
1,073,628 | <p>I am trying to find generating functions which will give me a power logarithm. </p>
<p>I am trying to find generating sums in the form</p>
<p>$$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$</p>
<p>or </p>
<p>$$\sum_{n=1}^{\infty} a_n\,x^n = \frac{\log^2(x)}{x}.$$</p>
<p>Something, which will return ... | Jack D'Aurizio | 44,121 | <p>We have:
$$-\log(1-x)=\sum_{n\geq 1}\frac{x^n}{n}$$
for any $x$ such that $|x|<1$, hence:
$$-\frac{\log(1-x)}{1-x}=\sum_{n\geq 1}H_n x^n.$$
and since $\frac{d}{dx}\log^2(1-x) = -2\frac{\log(1-x)}{1-x}$ we have:
$$\log^2(1-x) = 2\sum_{n\geq 1}\frac{H_n}{n+1}x^{n+1}\tag{1}.$$
Since, by partial summation:
$$\sum_{n=... |
2,300,382 | <p>I cannot think of a non-constructible algebraic number of degree $4$ over $\Bbb Q$ so far. I wish if I can find such an example. Could some one tell me some such numbers with justification? Also I would like to know the track of working out such an example. Any help or reference would be appreciate. Thanks in advanc... | sharding4 | 254,075 | <p>Take an $S_4$ extension which is the splitting field of a quartic polynmial, say $f(x)=x^4-4x+2$ with splitting field $K$. If the roots of $f(x)$ were constructible, then all the elements of $K$ would be constructible. For $G$ a Sylow $2$-subgroup of $S_4$, the fixed field of $G$, $K^{G}$ has odd degree over $\Bbb... |
789,407 | <p>If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. </p>
<p><strong>Edit:</strong> I forgot to mention in the question that $a$, $b$, and $c$ are natural numbers. Sorry for the inconvenience.<br>
<strong>Edit 2:</strong> As Hagen von Eitzen said abo... | Hagen von Eitzen | 39,174 | <p>The discrimimnat $D=b^2-4ac$ must be positive to ensure two distinct real roots.
(If double root is not forbidden, we have $4x^2-4x+1$ with double root at $\frac12$ and $abc=16$).
Next, we must have $f(1)>0$, i.e. $$a+c>b.$$
For naturals $a,c$ we also have $ a+c\le 1+ac$ and conclude $$\tag1b\le ac.$$
If $b\le... |
3,386,371 | <p>Find the explicit form of
<span class="math-container">$$
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}.
$$</span></p>
<p>Let <span class="math-container">$S(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}$</span>. It has radius of convergence <span class="math-container">$1$</span>.</p>
<p>Let <span ... | Who am I | 687,026 | <p>Hint :</p>
<p>For large factorials you can use stirling's formula
<span class="math-container">$$n! \approx (\sqrt{2\pi}) n^{n+0.5}e^{-n}$$</span></p>
<p>But if you want to be more accurate you can use Ramanujan's factorial formula</p>
<p><span class="math-container">$$n! \approx \sqrt{\pi}\left(\frac{n}{e}\right... |
2,120,539 | <p>Find the points of local maximum and minimun of the function:
$$f(x)=\sin^{-1}(2x\sqrt{1-x^2})~~~~;~~x\in (-1,1)$$
I know
$$f'(x)=-\frac{2}{\sqrt{1-x^2}}$$</p>
<p>How to find the local maximum and minimum? I have drawn the fig and seen the points of local maximum and minimum. But how to find then analytically?
<a h... | lab bhattacharjee | 33,337 | <p>As suggested in the comment using my answer here in <a href="https://math.stackexchange.com/questions/1764431/solving-arcsin-left2x-sqrt1-x2-right-2-arcsin-x">Solving $\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$</a>,</p>
<p>$$
\arcsin(2x\sqrt{1-x^2}) =\begin{cases}2\arcsin x
\;\;;-\dfrac1{\sqrt2}\le x\le \d... |
2,174,340 | <p>Given the function $$F(X,Y,Z) = \alpha^TXYZ$$ in which $X, Y, Z $ are matrices of size $n \times n$ and $\alpha$ is a vector of size $n \times 1$, how to compute the derivative of $F$ with respect to $Y$?</p>
<p>Actually I found some related questions but did not help.</p>
<p>Edit: if the function is of the form: ... | Rodrigo de Azevedo | 339,790 | <p>Let </p>
<p>$$\rm f (X, Y, Z) := a^{\top} X Y Z$$</p>
<p>Hence,</p>
<p>$$\frac{\mathrm f (\mathrm X, \mathrm Y + h \mathrm V, \mathrm Z) - \mathrm f (\mathrm X, \mathrm Y, \mathrm Z)}{h} = \rm a^{\top} X V Z$$</p>
<p>Vectorizing,</p>
<p>$$\rm \mbox{vec} (a^{\top} X V Z) = \left( \color{blue}{Z^{\top} \otimes a^... |
849,433 | <blockquote>
<p>We have subspaces in $\mathbb R^4: $ </p>
<p>$w_1= \operatorname{sp} \left\{
\begin{pmatrix} 1\\ 1 \\ 0 \\1 \end{pmatrix} ,
\begin{pmatrix} 1\\ 0 \\ 2 \\0 \end{pmatrix},
\begin{pmatrix} 0\\ 2 \\ 1 \\1 \end{pmatrix} \right\}$,
$w_2= \operatorname{sp} \left\{
\begin{pmatrix} 1\\ 1 \\ 1 \\1 ... | whosleon | 117,747 | <p><strong><em>Hint</em></strong>: the intersection of these two spans is NOT empty. What you need to do is find a new spanning set for $w_2$ that contains some of the vectors from the spanning set for $w_1$. The common vectors will span the intersection. Now that you have a basis for $w_1\cap w_2$, you can extend it t... |
1,661,244 | <p>If $R$ is a comutative ring with identity ring and $K$ is an ideal from it, let $R'=R/K$ and $I$ an ideal of $R$ satisfy $K\subseteq I$ and $I'$ is the coresponding ideal of $R'$ (we knew that correspondence theorem gives a certain one-to-one corespondence between the set of ideals of $R$ containing $K$ and the set ... | martini | 15,379 | <p>Note that we have by the third isomorphism theorem
$$ R/I \cong R/K\bigm/I' $$
hence $R/I$ is a domain iff $(R/K)/I'$ is, therefore $I$ is prime iff $I'$ is.</p>
|
114,147 | <p>I have two lists let say</p>
<pre><code>listF = {{7, 2}, {2, 6}, {8, 1}, {1, 7}, {11, 8}, {6, 11}};
</code></pre>
<p>and </p>
<pre><code>newD = {{{2, 7}, {7, 9}, {9, 2}}, {{7, 2}, {2, 6}, {6, 7}}, {{7,
2}, {2, 6}, {6, 7}}, {{11, 6}, {6, 2}, {2, 11}}, {{8, 1}, {1,
7}, {7, 8}}, {{11, 1}, {1, 8}, {8, 11}}, {{1,... | Sumit | 8,070 | <pre><code>Table[Select[newD[[i]],
Complement[{#}, listF] != {} && Complement[{Reverse@#}, listF] != {} &]
, {i, Length[newD]}]
</code></pre>
<blockquote>
<p>{{{7, 9}, {9, 2}}, {{6, 7}}, {{6, 7}}, {{2, 11}}, {{7, 8}}, {{11,
1}}, {{1, 5}, {5, 7}}, {{7, 8}}, {{11, 1}}, {{8, 6}}, {{2,
11}}, {... |
114,147 | <p>I have two lists let say</p>
<pre><code>listF = {{7, 2}, {2, 6}, {8, 1}, {1, 7}, {11, 8}, {6, 11}};
</code></pre>
<p>and </p>
<pre><code>newD = {{{2, 7}, {7, 9}, {9, 2}}, {{7, 2}, {2, 6}, {6, 7}}, {{7,
2}, {2, 6}, {6, 7}}, {{11, 6}, {6, 2}, {2, 11}}, {{8, 1}, {1,
7}, {7, 8}}, {{11, 1}, {1, 8}, {8, 11}}, {{1,... | kglr | 125 | <pre><code>foo[x_] := Sequence[x, Reverse@x];
DeleteCases[newD, Alternatives @@ (foo /@ listF), 2]
</code></pre>
<blockquote>
<p>{{{7, 9}, {9, 2}}, {{6, 7}}, {{6, 7}}, {{2, 11}}, {{7, 8}}, {{11, 1}},<br>
{{1, 5}, {5, 7}}, {{7, 8}}, {{11, 1}}, {{8, 6}}, {{2, 11}}, {{8, 6}}}</p>
</blockquote>
<pre><code>fun = ##... |
587,077 | <p>Given any prime $p$. Prove that $(p-1)! \equiv -1 \pmod p$.</p>
<p>How to prove this?</p>
| Asinomás | 33,907 | <p>This is known as Wilson's theorem: (not completely, but Wilson's theorem is an if and only if while this is only an if)</p>
<p><a href="https://en.wikipedia.org/wiki/Wilson%27s_theorem" rel="nofollow">https://en.wikipedia.org/wiki/Wilson%27s_theorem</a></p>
<p>The idea is that (p-1)! is the product of an element o... |
315,551 | <p>So I'm going over my practice midterms (which all seem to have solutions like this one), </p>
<p><img src="https://i.stack.imgur.com/fC8Gu.png" alt="Image"></p>
<p>Can anyone help clarify this for me? I understand that you multiply by the reciprocal to get to line two. But after that I'm completely lost, I don't u... | Aeolian | 58,941 | <p>This is easier to see if we work backward:</p>
<p>$$\begin{align*}(x-(x+h))(x+x+h) & = (x-(x+h))(x+(x+h)) \\
& = x^2 -(x+h)^2 \\
& = x^2 + 1 -(x+h)^2 - 1\\
& = (x^2 + 1) -[(x+h)^2 + 1]
\end{align*}
$$</p>
<p>And, as others have pointed out, the trick on the first line here is to see a difference o... |
1,063,352 | <p>$A$ and $B$ are sets and $\mathcal{F}$ is a family of sets. I'm trying to prove that</p>
<p>$\bigcap_{A \in \mathcal{F}}(B \cup A) \subseteq B \cup (\cap \mathcal{F})$</p>
<p>I start with "Let $x$ be arbitrary and let $x \in \bigcap_{A \in \mathcal{F}}(B \cup A)$, which means that $\forall C \in \mathcal{F}(x \in... | Raymond Manzoni | 21,783 | <p>Your series is a <a href="http://mathworld.wolfram.com/LambertSeries.html" rel="nofollow">Lambert series</a> and in the case of constant coefficients $a_n$ it may be rewritten as a <a href="http://en.wikipedia.org/wiki/Theta_function" rel="nofollow">classical Jacobi theta function</a> (for $z=0$) : $$\theta_3(z,q):=... |
3,661,474 | <p><span class="math-container">$ h:R^{N+1} \to [0 , \infty)$</span> , <span class="math-container">$ h $</span> is measurable</p>
<p><span class="math-container">$ g:R^{N+1} \to [0 , \infty)$</span> , <span class="math-container">$ g $</span> is measurable</p>
<p><span class="math-container">$x,y \in R^N$</span></p... | Yuri Negometyanov | 297,350 | <p><strong>HINT</strong></p>
<p>This answer does not contain a strict proof, only some details of the full solution.</p>
<p>Let
<span class="math-container">$$F(x) = f(x,x^2),\quad G(x,y)=H\left(\dfrac{1+i}2x+\dfrac{1-i}2y,\dfrac{1-i}2x +\dfrac{1+i}2y\right)\tag1$$</span>
then
<span class="math-container">$$F(x)F(y) ... |
3,153,821 | <p>I'm trying to analyse a game of Mastermind and am having trouble quantifying the amount of possible game states. I know that a code has <span class="math-container">$\text{# of colors}^{\text{# of pegs per guess}}$</span> combinations (in my case that would be <span class="math-container">$6^4=1296$</span>). However... | Peter Taylor | 5,676 | <p>Other way round : <span class="math-container">$\text{# of combinations per row}^{\text{# of rows}}$</span></p>
<p>If you sum over 0 to 10 rows you get a geometric series, giving a total of <span class="math-container">$$\frac{\text{# of combinations per row}^{1+\text{# of rows}}-1} {\text{# of combinations per row... |
4,253,160 | <p>I was recently taught that a subset W is a subspace of V if and only if:</p>
<ol>
<li>W is non-empty.</li>
<li>W is closed under vector addition.</li>
<li>W is closed under scalar multiplication.</li>
</ol>
<p>So we only need to prove 3 out of the 10 vector space axioms; why is this? Is it because it's redundant to ... | Pirate Prentice | 965,950 | <p>As <span class="math-container">$x\rightarrow 0$</span>, the outputs of <span class="math-container">$\cos(1/x)$</span> will oscillate wildly between <span class="math-container">$-1$</span> and <span class="math-container">$1$</span>. But when multiplied by <span class="math-container">$\lvert \sin(x)\rvert$</span>... |
4,253,160 | <p>I was recently taught that a subset W is a subspace of V if and only if:</p>
<ol>
<li>W is non-empty.</li>
<li>W is closed under vector addition.</li>
<li>W is closed under scalar multiplication.</li>
</ol>
<p>So we only need to prove 3 out of the 10 vector space axioms; why is this? Is it because it's redundant to ... | CiaPan | 152,299 | <p>Be careful! <em>'Anything multiplied by zero is zero'</em> is certainly true, but the rule does not hold in limits. One term <em>convergent</em> to zero is not the same as one term <em>equal</em> zero. As a counterexample consider <span class="math-container">$\lim\limits_{x\to 0} \left(x\cdot\frac 1x\right)$</span>... |
2,120,194 | <blockquote>
<p>Let $K_1$ and $K_2$ be two disjoint compact sets in a metric space $(X,d).$ Show that $$a = \inf_{x_1 \in K_1, x_2 \in K_2} d(x_1, x_2) > 0.$$
Moreover, show that there are $x \in K_1$ and $y \in K_2$ such that $a = d(x,y)$.</p>
</blockquote>
<p>For the first part, suppose to the contrary that $... | Ennar | 122,131 | <p>Your attempt is indeed fruitful. If you define $R=\Bbb Q[x]/I$, where $I=(x^2-2)$, then $$(x+I)^2 = x^2+I = (x^2+I) - (x^2-2+I) =2+I$$ so you have solution to $x^2=2$. Note that $\Bbb Q$ is embedded in $R$ via $q\mapsto q + I$ and that this is "sort of cheating". We literally defined $R$ so it will have this root. <... |
4,386,952 | <p>Informally, mathematicians treat Integers like a subset of rational numbers.</p>
<p>But according to the standard, formal construction of <span class="math-container">$\mathbb{Q}$</span>, <span class="math-container">$\mathbb{Q}$</span> is an equivalence class over <span class="math-container">$\mathbb{Z} \times \ma... | Átila Correia | 953,679 | <p>What does it mean that <span class="math-container">$fg$</span> is differentiable?</p>
<p>Well, this means the following limit must exist:
<span class="math-container">\begin{align*}
\lim_{x\to a}\frac{f(x)g(x) - f(a)g(a)}{x - a}
\end{align*}</span></p>
<p>Based on the given assumptions, we arrive at the relation:
<... |
39,466 | <p>I could not solve this problem:</p>
<blockquote>
<p>Prove that for a non-Archimedian field $K$ with completion $L$, $$\left\{|x|\in\mathbb R \mid x\in K\right\} =\left\{|x|\in\mathbb R \mid x\in L\right\}$$</p>
</blockquote>
<p>I considered a Cauchy sequence in $K$ with norms having limit $l$, but I could not co... | lhf | 589 | <p>Use the comparison test on $\sum |a_kx^k|$.</p>
|
529,886 | <p>In the context of learning about comparison theorem, using integrals to determine convergence and learning about exponential series (That's what $n^p$ is called right?).</p>
| Jack D'Aurizio | 44,121 | <p>Set $AB=x$ and $EB=y$. You have $BC=AF\sin\left(\pi/3+\hat{EAB}\right)$, so
$$ BC^2 = \frac{3x^2+y^2+2\sqrt{3}xy}{4}.$$
Now consider $N$ as the projection of $M$ on $AB$. We have:
$$ BM^2 = MN^2+NB^2 = AM^2 \sin^2\left(\pi/3+\hat{EAB}\right)+\left(x-AM \cos\left(\pi/3+\hat{EAB}\right)\right)^2.$$
Since $AM=\frac{AF... |
2,791,087 | <p>I have the following density function:
$$f_{x, y}(x, y) = \begin{cases}2 & 0\leq x\leq y \leq 1\\ 0 & \text{otherwise}\end{cases}$$</p>
<p>We know that $\operatorname{cov}(X,Y) = E[(Y - EY)(X - EX)]$, therefore we need to calculate E[X] and E[Y]. </p>
<p>$$f_x(x)=\int_x^1 2\,\mathrm dy = \big[2y\big]_x^1 =... | Graham Kemp | 135,106 | <blockquote>
<p>However, the provided solution states that $E[X]=1/3$. Have I done a mistake or is the solution wrong?</p>
</blockquote>
<p>Yes. $f_X(x)=\text{correct stuff}={[2y]}_{y=x}^{y=1}\mathbf 1_{x\in(0;1)}=(2-\color{crimson}2x)\mathbf 1_{x\in(0;1)}$ $$\mathsf E(X)=\int_0^1 x(2-\color{crimson}2x)\mathsf d x ... |
1,261,977 | <p>I tried doing this the same way you would find the Fourier transform for $1/(1+x^2)$ but I guess I'm having some trouble dealing with the 2x on top and I could really use some help here.</p>
| Mark Viola | 218,419 | <p>Hint: Taking the derivative with respect to $k$ of $$F(k)=\int_{-\infty}^{\infty}\frac{1}{1+x^2}e^{ikx}dx$$</p>
<p>yields</p>
<p>$$F'(k)=i\int_{-\infty}^{\infty}\frac{x}{1+x^2}e^{ikx}dx$$</p>
<p>Thus, the Fourier Transform of $\frac{2x}{1+x^2}$ is $-2i$ times the derivative with respect to $k$ of the Fourier Tran... |
1,261,977 | <p>I tried doing this the same way you would find the Fourier transform for $1/(1+x^2)$ but I guess I'm having some trouble dealing with the 2x on top and I could really use some help here.</p>
| Disintegrating By Parts | 112,478 | <p>Your function is square integrable. So the Fourier transform will be square integrable, and expressed as the Cauchy principal value
<span class="math-container">$$
\lim_{R\rightarrow\infty}\frac{1}{\sqrt{2\pi}}\int_{-R}^{R}e^{-isx}\frac{2x}{1+x^{2}}dx \\
= \lim_{R\rightarrow\infty}\frac{1}{\sqrt{2\pi}}\int_{-R... |
1,348,099 | <p>We know that cross product gives a vector that is orthogonal to other two vectors. Let this vector denoted by $$|\vec{v} \times \vec{u}| = \vec{n}$$
Then $$\vec{n}\cdot \vec{u} = 0 $$ Everything okay up to here. Then how we choose a vector from two possible orthogonal vectors, $$\vec{n}$$ or $$\vec{-n}$$ Why followi... | Michael Burr | 86,421 | <p>Consider
$$
\int\frac{x}{mg+kx}dx.
$$
Add and subtract $mg/k$ to the numerator to get:
$$
\int\frac{x+mg/k-mg/k}{mg+kx}dx=\int\frac{x+mg/k}{mg+kx}dx-\int\frac{mg/k}{mg+kx}dx.
$$
Factoring out $\frac{1}{k}$ in the first integral and using a $u$-substitution of $u=mg+kx$ (so $du=kdx$) on the second integral, we get
$$... |
2,344,758 | <p>Quoting from Wikipedia article on Euler's totient function theorem :---</p>
<blockquote>
<p>In general, when reducing a power of <span class="math-container">$a$</span> modulo <span class="math-container">$n$</span> (where <span class="math-container">$a$</span> and <span class="math-container">$n$</span> are coprim... | lhf | 589 | <p>This is false when $a$ and $n$ are not coprime.</p>
<p>Take for instance $a=2$ and $n=4$. Then:</p>
<p>$3 \equiv 1 \bmod \phi(2)$ but $2^3 \equiv 0 \not\equiv 2 = 2^1 \bmod 4$.</p>
<p>On the other hand, if $(x \equiv y \bmod \phi(n) \implies a^x \equiv a^y \bmod n)$, then take $x=\phi(n)$ and $y=0$ to get Euler's... |
367,497 | <p>Let $\{f_n\}$ be a sequence of $L^1(\mathbb R)$ functions converging a.e. to zero.
Does
$$
\lim_{n\to \infty} \int_{\mathbb R} \sin(f_n(x)) dx = 0?
$$</p>
<p>I think the answer is no, but I can't find a counterexample.</p>
| 23rd | 46,120 | <p>You are correct. For example, consider $f_n=\chi_{[n, n+1]}$.</p>
|
2,917,848 | <p>Given a real square matrix <span class="math-container">$A$</span> and a vector <span class="math-container">$v$</span>, Krylov subspaces are given by:
<span class="math-container">$$\mathcal K_n(A,v) := \text{span}(v, Av, \cdots A^{n-1} v)$$</span>
These spaces are known to help solve numerical linear algebra probl... | Ross Millikan | 1,827 | <p>The limit does not exist. If you approach along a path with $h=0$ the quantity is always zero. If you approach along $h=k$ with $t=0$ the quantity is $\sqrt {\frac z2}$</p>
|
1,701,260 | <p>My textbook states the following:<br>
i) If $ f : [a,b] \rightarrow \mathbb{R} $ is bounded and is continuous at all but finitely many points of $[a,b]$, then it is integrable on $[a,b]$.<br>
ii) Any increasing or decreasing function on $[a,b]$ is integrable on $[a,b]$.</p>
<p>The proof for (i) is clear to me. I fo... | Sangchul Lee | 9,340 | <p>Let $P = \{ a = t_0 < \cdots < t_n = b \}$ be <em>any</em> partition on $[a, b]$. If $f : [a, b] \to \Bbb{R}$ is monotone increasing, then the upper Riemann sum is</p>
<p>$$ U(P, f) = \sum_{i=1}^{n} \sup_{t \in [t_{i-1}, t_i]} f(t) (t_i - t_{i-1}) = \sum_{i=1}^{n} f(t_i) (t_i - t_{i-1}) $$</p>
<p>and likewis... |
1,701,260 | <p>My textbook states the following:<br>
i) If $ f : [a,b] \rightarrow \mathbb{R} $ is bounded and is continuous at all but finitely many points of $[a,b]$, then it is integrable on $[a,b]$.<br>
ii) Any increasing or decreasing function on $[a,b]$ is integrable on $[a,b]$.</p>
<p>The proof for (i) is clear to me. I fo... | Eugene Zhang | 215,082 | <p><strong>Hint:</strong> Use the fact that monotone function can have at most countable number of discontinuity points. So the Lebesgue measure of discontinuity points is zero and thus Riemann integrable. </p>
|
3,913,032 | <blockquote>
<p><strong>Problem.</strong> Let <span class="math-container">$A$</span> be a non-singular <span class="math-container">$n\times n$</span> matrix and let <span class="math-container">$\Gamma=[\Gamma_1\quad\Gamma_2]$</span> be an <span class="math-container">$n\times n$</span> orthogonal matrix where <span ... | Sort of Damocles | 478,044 | <p>As noted in the comments, the radius of the cylinder is just <span class="math-container">$r$</span>. Otherwise your work appears to be correct, but you've made things harder on yourself than necessary. Note that <span class="math-container">$r^3$</span> appears in both formulas, so once you have solved for <span cl... |
105,868 | <p>Let $f(x)$ be a continuous probability distribution in the plane. It is obvious that if $X$ and $X'$ are two independent random samples from $f$, then $\mathbf{E}(\|X - X'\|) \leq 2 \mathbf{E}(\|X\|)$ by the triangle inequality. Can this upper bound be made tighter if we assume that $f$ is rotationally symmetric a... | Pablo Shmerkin | 11,009 | <p>Under the assumption that $f$ is radial, the angle between $X$ and $X'$ is uniformly distributed in $[0,\pi]$ (since the signed angle that each of $X$ and $X'$ makes with the $x$-axis is uniformly distributed by rotational invariance, hence so is their difference). Thus the distribution of $\|X-X'\|$ is the same as ... |
105,868 | <p>Let $f(x)$ be a continuous probability distribution in the plane. It is obvious that if $X$ and $X'$ are two independent random samples from $f$, then $\mathbf{E}(\|X - X'\|) \leq 2 \mathbf{E}(\|X\|)$ by the triangle inequality. Can this upper bound be made tighter if we assume that $f$ is rotationally symmetric a... | Arthur B | 8,737 | <p>Using the pareto distribution $f(x) = \frac{\alpha}{x^{\alpha+1}}$ ($x > 1$) , the ratio $\frac{E(||X-X'||)}{E(||X||)}$ approaches a $2$ as $\alpha$ tends to 1. </p>
<p>To find such a distribution, consider that all else equal, you want to maximize the difference $||X||-||X'||$ since the angle between the two is... |
1,737,055 | <p>What happens if some blind person want to study math? Is there some "braille alphabet" for mathematical symbols? Are there math books, at least for undergraduate students, written for blind people?</p>
| JM97 | 301,287 | <p>Reading and writing mathematics is fundamentally different than reading and writing text. While Braille is adequate for the representation of text, it is not up to the task of representing mathematics. The two basic reasons for this are:</p>
<p><strong>Linearity</strong></p>
<p>Text is linear in nature while mathe... |
1,737,055 | <p>What happens if some blind person want to study math? Is there some "braille alphabet" for mathematical symbols? Are there math books, at least for undergraduate students, written for blind people?</p>
| MathematicianByMistake | 237,785 | <p>There exists a certain variation-or rather "enrichment"-of the Braille Alphabet, named Nemeth Braille, after its' creator, Abraham Nemeth, which is also using the standard six-dot Braille cells to create mathematical symbols.</p>
<p>I am not sure on whether it is exhaustive-that is, if <em>all</em> mathmematical ex... |
55,435 | <p>I've recently become interested in the elementary theory of groups due to Sela and Myasnikov-Kharlampovich's work with free groups. I'd like a good introduction to the field of the elementary theory of groups, and in particular I'd like a reference to contain examples of group properties that cannot be read from a ... | Jeremy Macdonald | 10,673 | <p>You might try Champetier and Guirardel's <a href="http://arxiv.org/abs/math/0401042" rel="nofollow">Limit groups as limits of free groups</a>.
It has a short section (section 5) on elementary and universal theory, though perhaps none of the "non-examples" you're looking for. It is, however, a pleasure to read and i... |
230,154 | <p><strong>Question.</strong> Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback is always a weak equivalence). Or is there a close statement that I can't remember (browsing nLab did... | David White | 11,540 | <p>You asked if you can check it for less than the collection of all weak equivalences. In A.5 of Motivic Symmetric Spectra, Jardine proves a general right properness result from Corollary A.4, which is the statement that weak equivalences with fibrant codomain are preserved by pullback along a fibration. So that appea... |
23,181 | <p>I have n sectors, enumerated 0 to n-1 counterclockwise. The boundaries between these sectors are infinite branches (n of them).
The sectors live in the complex plane, and for n even,
sector 0 and n/2 are bisected by the real axis, and the sectors are evenly spaced.</p>
<p>These branches meet at certain points, call... | Justin | 6,014 | <blockquote>
<p>but the order of the long branches
should be the same, and you cannot
rotate the tree.</p>
</blockquote>
<p>So 'order' has nothing to do with the geometrical length, right? It is the depth of the tree that you are talking about?</p>
<p>It seems that the identity of a junction is its angular ord... |
3,174,339 | <p>Let <span class="math-container">$M$</span> be a <span class="math-container">$C^{\infty}$</span> manifold. Let <span class="math-container">$U$</span> be an open subset of <span class="math-container">$M$</span>. Now take a closed subset (with respect to the subspace topology on <span class="math-container">$U$</sp... | SOFe | 314,846 | <p>It is not true over <em>multiple</em> Euclidean spaces.</p>
<p>Just consider the natural projection <span class="math-container">$f: \mathbb R \to \mathbb R^2$</span> where <span class="math-container">$f(x) = \left[\begin{matrix}x \\ 0\end{matrix}\right]$</span>. <span class="math-container">$f(S)$</span> is never ... |
3,032,258 | <p>Assume 5 out of 100 units are defective. We pick 3 out of the 100 units at random. </p>
<p>What is the probability that exactly one unit is defective?</p>
<hr>
<p>My answer would be </p>
<p><span class="math-container">$P(\text{Defect}=1) = P(\text{Defect})\times P(\text{Not defect})\times P(\text{Not defect}) =... | BelowAverageIntelligence | 441,199 | <p>Your answer should be <span class="math-container">$$\frac{\binom{5}{1}\binom{95}{2}}{\binom{100}{3}}$$</span> Since we want the total number of ways to choose 3 meeting the criteria over the total number of ways to choose 3 out of the 100.</p>
|
163,672 | <p>Is there a characterization of boolean functions $f:\{-1,1\}^n \longrightarrow \{-1,1\}$,
so that $\mathbf{Inf_i}[f]=\frac{1} {2}$, for all $1\leq i\leq n$? Is it known how many such functions there are? </p>
| Bjørn Kjos-Hanssen | 4,600 | <p>We are looking at functions where <em>each variable has a 50% chance of mattering</em>.
Let <span class="math-container">$c_n$</span> be the number of such functions.
I'll just prove that <span class="math-container">$4^n\le c_n$</span> (of course <span class="math-container">$c_n\le 2^{2^n}$</span>) for <span class... |
163,672 | <p>Is there a characterization of boolean functions $f:\{-1,1\}^n \longrightarrow \{-1,1\}$,
so that $\mathbf{Inf_i}[f]=\frac{1} {2}$, for all $1\leq i\leq n$? Is it known how many such functions there are? </p>
| KEW | 30,582 | <p>For a random function $f : \{-1,1\} \rightarrow \{-1,1\}$, we have that $\mathbf{Pr}[\mathbf{Inf}_i(f) = \frac12]$ is roughly $2^{-n/2}$; it's the probability that a binomial random variable with $2^{n-1}$ trials with success probability $\frac12$ has exactly $2^{n-2}$ successes. These events for $i = 1,2,\ldots,n$... |
162,293 | <p>Consider a "curve" defined by a list of points in finite dimension (here, four):</p>
<pre><code> pts = Table[{Cos[t], 0, Sin[2 t], Sin[t]}, {t, Subdivide[0, 1, 99]}]
</code></pre>
<p>I used known functions to generate <code>pts</code> but of course I am not supposed to know the parametric equation of the curve the... | David G. Stork | 9,735 | <p>For ${\bf x}(t) = \{ x(t), y(t), z(t) \}$ in $\mathbb{R}^3$ the curvature is:</p>
<p>$$ \kappa = {\sqrt{ (z^{\prime\prime}y^\prime + y^{\prime\prime} z^\prime)^2 + (x^{\prime\prime} z^\prime + z^{\prime\prime} x^\prime)^2 + (y^{\prime\prime} x^\prime + x^{\prime\prime} y^\prime)^2} \over ((x^\prime)^2 + (y^\prime)^... |
244,333 | <p>Consider this equation : </p>
<p><span class="math-container">$$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$</span></p>
<p>where <span class="math-container">$t$</span> varies from <span class="math-container">$0$</span> to <span class="math-container">$T$</span> , and <span class="math-container... | Américo Tavares | 752 | <p><em>Comment to your attempt</em>: instead of reasoning based on the initial positions and conditions of the dog and cat, I advise to sketch a <em>generic intermediate position</em> of both. I tried this approach in the sketch below. </p>
<p>I have not checked <a href="https://math.stackexchange.com/a/245537/752">... |
127,322 | <p>Being a new member, I am not yet sure whether my question will be taken as a research level question (and thus, appropriate for MO). However, I have seen similar questions on MO, couple of which led me asking mine, and I seem to not be able to find many resources except discussion on FOM and MO. So, any references t... | Joel David Hamkins | 1,946 | <p>With regard to your sub-question,</p>
<blockquote>
<blockquote>
<p>Now imagine a universe where $\text{Con}(\text{ZFC})$ holds but all the models of $\text{ZFC}$ are $\omega$-nonstandard and believe $\neg \text{Con}(\text{ZFC})$. I do not know whether this scenario is even possible...</p>
</blockquote>
</b... |
947,191 | <p>Show that $\sum _{n=1 } ^{\infty } (n \pi + \pi/2)^{-1 } $ diverges.</p>
<p>Both the root test and the ratio test is inconclusive. Can you suggest a series for the series comparison test?</p>
<p>Thanks in advance!</p>
| Mathronaut | 53,265 | <p>$\displaystyle\frac{1}{n\cdot \pi+\frac{\pi}{2}}\ge\frac{1}{n\cdot\pi+\pi} =\frac{1}{\pi}\cdot\frac{1}{n+1}$, and since $\displaystyle\sum\frac{1}{n+1}$ diverges thus $$\sum\displaystyle\frac{1}{n\cdot \pi+\frac{\pi}{2}}$$ diverges!</p>
|
3,224,455 | <p>I derived the volume of a cone using two approaches and compared the results.</p>
<p>First I integrated a circle of radius <span class="math-container">$r$</span> over the height <span class="math-container">$h$</span> to get the expression: <span class="math-container">$$V_1=\frac{1}{3}\pi r^2 h$$</span></p>
<p>T... | Ethan Bolker | 72,858 | <p>This is not a "result". <span class="math-container">$\infty$</span> is not a number. If your argument were correct you could use it this way:</p>
<p>For all positive integers <span class="math-container">$n$</span>
<span class="math-container">$$
n \times \frac{1}{n} = 1.
$$</span>
Then taking the limit as <span c... |
3,224,455 | <p>I derived the volume of a cone using two approaches and compared the results.</p>
<p>First I integrated a circle of radius <span class="math-container">$r$</span> over the height <span class="math-container">$h$</span> to get the expression: <span class="math-container">$$V_1=\frac{1}{3}\pi r^2 h$$</span></p>
<p>T... | Community | -1 | <p>Generally speaking, <span class="math-container">$\infty\cdot0$</span> has no precise value (not counting that it is a mathematically "illegal" expression).</p>
<p>For more rigor, you can work with limits and write</p>
<p><span class="math-container">$$\infty\cdot0=\lim_{n\to \infty}f(n)g(n)$$</span> where <span c... |
760,032 | <p>Consider the integral
\begin{equation}
I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt
\end{equation}
show that
\begin{equation}
I(x)= \frac{2x}{\pi} +O(x^{3})
\end{equation}
as $x\rightarrow0$.</p>
<p>=> I Have used the expansion of McLaurin series of $I(x)$ but did not work.
please help me.</p>
| Jason | 130,776 | <p>Just appeal to the Taylor expansion of $I(x)$ directly; clearly $I(0)=0$. Now,
$$ I'(x) = \frac{1}{\pi}\int_0^\pi \cos(x\sin{t})\sin{t}\,dt. $$
So,
$$ I'(0) = \frac{1}{\pi} \int_0^\pi \sin{t}\,dt = \frac{2}{\pi}. $$
Also,
$$ I''(x) = -\frac{1}{\pi}\int_0^\pi \sin(x\sin{t})\sin^2{t}\,dt. $$
So, $I''(0)=0$.</p>
<p>Th... |
2,512,137 | <blockquote>
<p>A social worker has 77 days to make his visits. He wants to make at least one visit a day, and has 133 visits to make. Is there a period of consecutive days in which he makes a.) 21 b.) 23 visits? Why?</p>
</blockquote>
<p>a.) Set $a_i$ to be the number of visits up to and including day $i$, for $i =... | Joffan | 206,402 | <p>Consider the cumulative number of visits $a_i$ after day $i$, taken $\bmod 21$. We can also include $a_0=0$ here. Then by the generalized pigeonhole principle there is some residue class with at least $\lceil 78/21\rceil =4$ values in it. Furthermore, if there are no classes with $5$ or more values then there are a... |
3,506,316 | <p>I am trying to evaluate this limit:</p>
<p><span class="math-container">$$\lim_{x\to0^{+}}(x-\sin x)^{\frac{1}{\log x}}$$</span></p>
<p>It's a <span class="math-container">$0^0$</span> intedeterminate form, and I am unsure how to deal with it. I have a feeling that if I could turn it to a form where L'Hopital's ru... | Ryan Shesler | 585,375 | <p>Try converting into an indeterminant product using exponentiation by <span class="math-container">$e$</span>:</p>
<p><span class="math-container">$$\lim_{x \to 0^+} (x - \sin x)^{\frac{1}{\log x}} = \lim_{x \to 0^+} e^{\ln(x - \sin x)(\frac{1}{\log x})}$$</span> and you can go from here using L'Hopital</p>
|
1,532,275 | <p>The kernel of a monoid homomorphism $f : M \to M'$ is the submonoid $\{m \in M : f(m)=1\}$. (This should not be confused with the kernel pair, which is often also named the kernel.)</p>
<p><em>Question.</em> Which submonoids $N$ of a given monoid $M$ arise as the kernel of a monoid homomorphism? (If necessary, let ... | slader.com | 290,016 | <p>$$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{ k (k + 1) (k + 2) \cdots (k + m)
}=\lim_{n\to \infty}\dfrac{1}{m}\sum_{k=1}^n \frac{(k+m)-(k)}{ k (k + 1) (k + 2) \cdots (k + m)
}$$</p>
<p>$$=\lim_{n\to \infty}\dfrac{1}{m}\sum_{k=1}^n \frac{1}{ k (k + 1) (k + 2) \cdots (k + m-1)
}-\frac{1}{ (k + 1) (k + 2) \cdot... |
1,532,275 | <p>The kernel of a monoid homomorphism $f : M \to M'$ is the submonoid $\{m \in M : f(m)=1\}$. (This should not be confused with the kernel pair, which is often also named the kernel.)</p>
<p><em>Question.</em> Which submonoids $N$ of a given monoid $M$ arise as the kernel of a monoid homomorphism? (If necessary, let ... | Jan Eerland | 226,665 | <p>HINT:</p>
<p>$$\lim_{n\to \infty}\sum_{k=1}^n\frac{1}{k(k+1)(k+2)\cdots(k+m)}=$$
$$\lim_{n\to \infty}\sum_{k=1}^n\frac{1}{\prod_{n=0}^{m} (k+n)}=$$
$$\lim_{n\to \infty}\sum_{k=1}^n\frac{1}{\frac{\Gamma(k+m+1)}{\Gamma(k)}}=$$
$$\lim_{n\to \infty}\sum_{k=1}^n\frac{\Gamma(k)}{\Gamma(k+m+1)}=$$
$$\lim_{n\to \infty}\fra... |
3,244,193 | <p>Here's what I did: <span class="math-container">$$\lim_{n\rightarrow +\infty}(2+3^n)^{\frac{1}{2n}}=\lim_{n\rightarrow +\infty}e^{\frac{1}{2n}\ln(2+3^n)}$$</span>
What should I do next in order to solve it?</p>
| Parcly Taxel | 357,390 | <p>In the natural logarithm, <span class="math-container">$3^n$</span> dominates <span class="math-container">$2$</span>, so we can take it out:
<span class="math-container">$$=\lim_{n\to\infty}e^{\ln(3^n)/(2n)}=\lim_{n\to\infty}e^{(n\ln3)/(2n)}=\lim_{n\to\infty}e^{(\ln3)/2}=\sqrt3$$</span></p>
|
3,244,193 | <p>Here's what I did: <span class="math-container">$$\lim_{n\rightarrow +\infty}(2+3^n)^{\frac{1}{2n}}=\lim_{n\rightarrow +\infty}e^{\frac{1}{2n}\ln(2+3^n)}$$</span>
What should I do next in order to solve it?</p>
| Peter Szilas | 408,605 | <p><span class="math-container">$3^{(1/2)}((2/3^n)+1)^{(1/(2n))}$</span>.</p>
<p>For <span class="math-container">$n$</span> large enough: <span class="math-container">$2/(3^n) <1$</span>:</p>
<p><span class="math-container">$\small { √3 \lt √3(2/3^n+1)^{(1/(2n))} \lt √3(√2)^{(1/n)}}$</span>.</p>
<p>Squeeze.</p>... |
4,246,726 | <p>For the system of linear equations <span class="math-container">$Ax = b$</span> with <span class="math-container">$b =\begin{bmatrix}
4\\
6\\
10\\
14
\end{bmatrix}\\
$</span>. The set of solutions is given by- <span class="math-container">$\left... | Ben Grossmann | 81,360 | <p>What you have computed is really <span class="math-container">$\frac{\delta \operatorname{vec}(Y^TY)}{\delta \operatorname{vec}(Y)}$</span>; I will assume this is what you're really after. I will also assume that you are using the <a href="https://en.wikipedia.org/wiki/Row-_and_column-major_order" rel="nofollow nore... |
72,613 | <p>Given a list or string, how do I get a list of all (contiguous) sublists/substrings? The order is not important.</p>
<p>Example for lists:</p>
<pre><code>list = {1, 2, 3};
sublists[list]
(* {{}, {}, {}, {}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}} *)
</code></pre>
<p>Example for strings:</p>
<pre><code>string =... | Martin Ender | 2,305 | <p><a href="https://en.wikipedia.org/wiki/There%27s_more_than_one_way_to_do_it" rel="nofollow noreferrer">TMTOWTDI</a> applies to both of these problems. Below I present an overview of various approaches I've come across, followed by timing data obtained in 10.4 on Windows 10 (the timing code is available as well, so y... |
72,613 | <p>Given a list or string, how do I get a list of all (contiguous) sublists/substrings? The order is not important.</p>
<p>Example for lists:</p>
<pre><code>list = {1, 2, 3};
sublists[list]
(* {{}, {}, {}, {}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}} *)
</code></pre>
<p>Example for strings:</p>
<pre><code>string =... | Kuba | 5,478 | <pre><code>f[l_List] := Take[l, #] & /@ Subsets[Range[Length[l]], {1, 2}];
f[s_String] := StringJoin @@@ f[Characters[s]]
</code></pre>
<hr>
<pre><code>string = "abcabcabcabcabcabcabcabcabcabcabcabcabcabcabcabcab" <>
"cabcabcabcabcabcabcabcabcabcabcabcabcabcabcabcabca";
a = f[string]; // Timing
</code... |
432,811 | <p>I'm trying to solve $$\operatorname{Arg}(z-2) - \operatorname{Arg}(z+2) = \frac{\pi}{6}$$ for $z \in \mathbb{C}$.</p>
<p>I know that
$$\operatorname{Arg} z_1 - \operatorname{Arg} z_2 = \operatorname{Arg} \frac{z_1}{z_2},$$
but that's only valid when $\operatorname{Arg} z_1 - \operatorname{Arg} z_2 \in (-\pi,\pi]$, ... | lab bhattacharjee | 33,337 | <p>Using <a href="http://en.wikipedia.org/wiki/Atan2#Definition_and_computation" rel="nofollow">this</a> and <a href="http://en.wikipedia.org/wiki/Argument_%28complex_analysis%29#Notation" rel="nofollow">this</a>,</p>
<p>if $z=x+iy,$</p>
<p>Case $1:$ If $x>2,\text{Arg}(z-2)=\arctan \frac y{x-2}$ and $\text{Arg}(z+... |
29,766 | <p>I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only bet just to google for them?</p>
| Willie Wong | 3,948 | <p>As a counter-point to my somewhat flippant previous answer (which only really applies if one is a specialist in the field), if you are looking at a field in which you are not as much a specialist in, I suggest reading the articles from the <a href="http://www.ams.org/publications/journals/journalsframework/aboutbull... |
174,165 | <p>I have Maths test tomorrow and was just doing my revision when I came across these two questions. Would anyone please give me a nudge in the right direction?</p>
<p>$1)$ If $x$ is real and $$y=\frac{x^2+4x-17}{2(x-3)},$$ show that $|y-5|\geq2$ </p>
<p>$2)$ If $a>0$, $b>0$, prove that $$\left(a+\frac1b\right)... | vanna | 30,573 | <p>Using identities $\sin(x)^2 = \frac{1-\cos(2x)}{2}$ and $\cos(x)^2 = \frac{1+\cos(2x)}{2}$ we get</p>
<p>$$ \left(\sin\left(\frac{n\pi}{2}\right)\right)^2 = \frac{1 - \cos(n\pi)}{2} = \frac{1 - (-1)^n}{2} $$
$$ \left(\cos\left(\frac{n\pi}{2}\right)\right)^2 = \frac{1 + \cos(n\pi)}{2} = \frac{1 + (-1)^n}{2}$$</p>
<... |
93,621 | <p>As we know, most of the spectral sequences are doubly graded. However, this "doubly graded" condition is not a part of the formal definition of spectral sequence. Is there any useful triply (quadruply, quintuply, etc.) graded spectral sequences? If not, is there a hope that some meaningful work can be done with this... | Peter May | 14,447 | <p>Well, there is an eponymous spectral sequence in my thesis (still never published
in full, but there is an announcement, stuff about it in Ravenel's book, and papers by
Tangora and others). Quite generally, take a connected graded algebra $A$ over a field $k$, filter it for example by the powers of its augmentatio... |
93,621 | <p>As we know, most of the spectral sequences are doubly graded. However, this "doubly graded" condition is not a part of the formal definition of spectral sequence. Is there any useful triply (quadruply, quintuply, etc.) graded spectral sequences? If not, is there a hope that some meaningful work can be done with this... | Neil Strickland | 10,366 | <p>As others have said, there are plenty of examples of spectral sequences that have a third grading such that each $d_r$ preserves the grading up to a shift depending only on $r$. This is all fairly straightforward.</p>
<p>However, there are some more interesting questions along the same lines. When Ravenel was try... |
93,621 | <p>As we know, most of the spectral sequences are doubly graded. However, this "doubly graded" condition is not a part of the formal definition of spectral sequence. Is there any useful triply (quadruply, quintuply, etc.) graded spectral sequences? If not, is there a hope that some meaningful work can be done with this... | Liviu Nicolaescu | 20,302 | <p>J. L. Verdier dissertation (written in the 60s), pre-Ravenel's book, covers multiple-graded complexes. It was reprinted "recently"</p>
<blockquote>
<p>Des catégories dérivées des catégories abéliennes. (French. French summary) [On derived categories of abelian categories]
With a preface by Luc Illusie. Edi... |
873,434 | <p>Let's say I want to find the product of $1,2,3, \dots, 10$. Do I need to do $1 \cdot 2 \cdot 3 \cdot \dots \cdot 10$ manually or is there an easier way to do it?</p>
<p>Something like the sumation of $1$ to $n$ which gives $\frac{n(n+1)}{2}$.</p>
<p>I tried to search but couldn't find a way to do it directly. </... | lhf | 589 | <p>The fact that there is a special notation for the factorial suggests that there is no simpler formula for it other than the definition.</p>
<p>Note that there is no special notation for $\sum_{i=1}^n i$, since it can be written as $n(n+1)/2$. (I'm not sure $\binom{n+1}{2}$ counts as special notation in this context... |
2,356,905 | <p>I was researching about the Newton-Raphson method and came across <a href="http://www.sosmath.com/calculus/diff/der07/der07.html" rel="nofollow noreferrer">http://www.sosmath.com/calculus/diff/der07/der07.html</a>. On the third line of the page near the end of the line it told us to consider $2/x_1$:</p>
<blockquot... | Trevor Gunn | 437,127 | <p>If $x > \sqrt{2}$ then $\dfrac1x < \dfrac{1}{\sqrt{2}}$. See the problem? $\dfrac{1}{\sqrt{2}}$ is not $\sqrt{2}$ but $\dfrac{2}{\sqrt{2}}$ is.</p>
|
2,356,905 | <p>I was researching about the Newton-Raphson method and came across <a href="http://www.sosmath.com/calculus/diff/der07/der07.html" rel="nofollow noreferrer">http://www.sosmath.com/calculus/diff/der07/der07.html</a>. On the third line of the page near the end of the line it told us to consider $2/x_1$:</p>
<blockquot... | browngreen | 321,445 | <p>Since $x_1$ is close to $\sqrt2$, taking $2\over x_1$ will get you another number close to $\sqrt2$ but on the other side from $x_1$. This way you can take the average of $x_1$ and the new number to approximate $\sqrt2$.</p>
|
1,682,341 | <p>While looking at another question on this site about constructable numbers I started wondering. If you can take a countable number of steps (possibly infinite) can you draw an interval of a length corresponding to a computable number?</p>
<p>More strictly if I have a unit interval, a straight edge, a compass, a fi... | Ross Millikan | 1,827 | <p>You can certainly define a (possibly infinite) sequence of segments whose total length has as its limit any computable number. You can compute the binary expansion of the number. The integer part is easy, just add up the proper number of $1$'s. Then add on $\frac 1{2^n}$ if the $n^{\text{th}}$ bit of the expansio... |
387,295 | <p>I need to find $$\underset{n \to \infty}{\lim} \underset{x\in [0,1]}{\sup} \left| \frac{x+x^{2}}{1+n+x} \right|.$$ How to show that supremum will be at the point $x=1$?</p>
| Community | -1 | <p>First let us find the supremum of $f_n(x) = \dfrac{x+x^2}{1+n+x}$. We have
$$f_n'(x) = \dfrac{(1+n+x)(1+2x) - (x+x^2)}{(1+n+x)^2} = \dfrac{x(2n+x+2)+n+1}{(1+n+x)^2} > 0 \,\,\,\, \forall x \in [0,1]$$
Hence, $f_n(x)$ is an increasing function in the interval $[0,1]$. Hence, the supremum is attained at $x=1$. We he... |
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