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 |
|---|---|---|---|---|
3,639,192 | <p>In their article on the <a href="https://en.wikipedia.org/wiki/Brauer_group#Galois_cohomology" rel="nofollow noreferrer">Brauer group</a> Wikipedia writes:</p>
<blockquote>
<p>Since all central simple algebras over a field <span class="math-container">$K$</span> become isomorphic to the matrix algebra over a sepa... | Awnon Bhowmik | 515,900 | <p><span class="math-container">$$\begin{align}9^x+15^x&=25^x\\\left(\dfrac35\right)^{2x}+\left(\dfrac35\right)^x&=1\\y^2+y&=1\qquad\boxed{\text{Let }y=\left(\dfrac35\right)^x}\\y^2+y-1&=0\\y&=\dfrac{-1\pm\sqrt{5}}2\end{align}$$</span></p>
<p><span class="math-container">$$\begin{equation}\begin{sp... |
47,246 | <p>I got some text scraps with this <em>structure</em></p>
<pre><code>"Focal Plane: 198' Active Aid to Navigation: Yes *Latitude: 35.250 N *Longitude: -75.529 W"
</code></pre>
<p>But some of them lack of parts like this</p>
<pre><code>"Focal Plane: 198' Active Aid to Navigation: Yes *Longitude: -75.529 W"
</cod... | Murta | 2,266 | <p>As requested, here is one approach using pure <code>RegularExpression</code>.</p>
<p>Another difference is the use of <code>Internal`StringToDouble</code> that is faster then ToExpression (velocity test <a href="https://mathematica.stackexchange.com/a/19631/2266">here</a>).</p>
<pre><code>getFields[str_String]:=Mo... |
3,065,572 | <p>Function <span class="math-container">$f: (0, \infty) \to \mathbb{R}$</span> is continuous. For every positive <span class="math-container">$x$</span> we have <span class="math-container">$\lim\limits_{n\to\infty}f\left(\frac{x}{n}\right)=0$</span>. Prove that <span class="math-container">$\lim\limits_{x \to 0}f(x)=... | Community | -1 | <p>What you're trying to prove is the transitive relation of subsets. The statement is logically equivalent to if A<span class="math-container">$\nsubseteq$</span>C then A<span class="math-container">$\subseteq$</span>B or B<span class="math-container">$\nsubseteq$</span>C. So assume A<span class="math-container">$\ns... |
113,349 | <p>I have a lot of old <em>Mathematica</em> version 2.2 <code>.ma</code> files that simply crash the current (10.4.1) version of <em>Mathematica</em> if I try to open them.</p>
<p>Is there some way to convert them to <code>.nb</code> notebook files?</p>
<p>Is 5.2 the latest version that will do the conversion? And if... | larry | 38,059 | <p>How to save Older Mathematica Files With The .ma Extension.</p>
<p>In order to use them I needed to convert some older (1995) mathematica files I found in a CD in a used book so that they would run on 11.1. They had extension ".ma". </p>
<p>To do this I did the following.
1) Download the Mathematica CDF program. T... |
361,171 | <p>If I am about to fabricate a bracelet and I can select $24$ pearls out of a total of $500$ pearls ($300$ white, $150$ red and $50$ green) how many possible bracelets can I create?</p>
<p>The (official) solution to this question is
$$ \frac{500!}{476!} = 3.4\cdot10^{64} $$
But how can it be that the colors of the p... | timidpueo | 54,230 | <p>Yeah it seems like the official answer is assuming each pearl is distinct, in which case the fact that it provided colors is meaningless. On top of that even if you assume each pearl is distinct, the official answer is wrong. It would be right if the question was asking about putting pearls in a line instead of a ... |
361,171 | <p>If I am about to fabricate a bracelet and I can select $24$ pearls out of a total of $500$ pearls ($300$ white, $150$ red and $50$ green) how many possible bracelets can I create?</p>
<p>The (official) solution to this question is
$$ \frac{500!}{476!} = 3.4\cdot10^{64} $$
But how can it be that the colors of the p... | Marko Riedel | 44,883 | <p>The first answer is excellent. I would just like to remark that if we are calculating the cycle index of $D_{24}$ anyway, then why not use the Polya enumeration theorem (PET)?</p>
<p>We have by table lookup and/or basic reasoning that for the dihedral group the cycle index is given by
$$ Z(D_n) = \frac{1}{2} Z(C_n)... |
3,697,062 | <p>How many solutions are there for this equation <span class="math-container">$$x_1+x_2+x_3+x_4 = 49\,,$$</span> where <span class="math-container">$x_i,\;i= 1,2,3,4$</span> is a non negative integer such that:<span class="math-container">$$ 1\le x_1\le 8,\;3\le x_2\le 9,\;10\le x_3\le 20,\ 0\le x_4\,?$$</span></p>
| G Cab | 317,234 | <p>We have that
<span class="math-container">$$
\eqalign{
& \left\{ \matrix{
1 \le x_{\,1} \le 8 \hfill \cr
3 \le x_{\,2} \le 9 \hfill \cr
10 \le x_{\,3} \le 20 \hfill \cr
0 \le x_{\,4} \left( { \le 49} \right) \hfill \cr
x_{\,1} + x_{\,2} + x_{\,3} + x_{\,4} = 49 \hfill \cr} \right.\quad \... |
1,560,539 | <p>I'm trying to show that the partial sums of $\log(j) = n\log(n) - n + \text{O}(\log(n))$</p>
<p>I know that $$\int_1^n\log(x)dx = n\log(n) - n + 1$$</p>
<p>so that this number is pretty close to what I want.</p>
<p>Now I look at the difference between sum and integral of log:</p>
<p>$$\sum_{j=1}^n \log(j) - \int... | Marco Cantarini | 171,547 | <p>It is a direct application of <a href="https://en.wikipedia.org/wiki/Abel%27s_summation_formula" rel="nofollow">Abel's summation</a>. We have $$\sum_{k\leq n}\log\left(k\right)=\sum_{k\leq n}1\cdot\log\left(k\right)=n\log\left(n\right)-\int_{1}^{n}\frac{\left\lfloor t\right\rfloor }{t}dt
$$ where $\left\lfloor t\ri... |
234,477 | <p>In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension?</p>
<p>Is it algebraic? Probably not, right? Transcendental? Normal? Finitely generated? Separable?</p>
<p>For instance, I was th... | Mark S. | 26,369 | <p>I don't think it would be enough to adjoin one infinitestimal. To construct the hyperreals via an ultrapower requires the Axiom of Choice, but you could certainly adjoin one infinitesimal without Choice.</p>
|
234,477 | <p>In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension?</p>
<p>Is it algebraic? Probably not, right? Transcendental? Normal? Finitely generated? Separable?</p>
<p>For instance, I was th... | Mikhail Katz | 72,694 | <p>A small correction to the formulation of the original question: the continuum hypothesis is not needed to construct a hyperreal field extension of the reals. The alternative simpler field extension you are proposing is related to the Levi-Civita field, which was of intense interest to Abraham Robinson. However, it... |
2,861,293 | <p>I found this statement with the proof:</p>
<p><a href="https://i.stack.imgur.com/bGRiZ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bGRiZ.jpg" alt="enter image description here" /></a></p>
<p>But I don't understand the proof. Where is the contradiction? We have a nonempty interval <span class="... | uniquesolution | 265,735 | <p>This is essentially Paul Frost's answer, in a slighly less verbose form.</p>
<p>The range $f(I)$ contains the interval $(f(a),f(b))$, so every point in the interval $(f(a),f(b))$ is of the form $f(x)$ for some $x\in I$. On the other hand, only one endpoint of the non-empty interval $J$ is a value of $f$ - because $... |
97,318 | <p>I use this code but it doesn't work.</p>
<pre><code>ZZ = {
{1, 2, 3, 4},
{5, 6, 7, 8},
{9, 10, 11, 12},
{13, 14, 15, 16}
} ;
ZZ // MatrixForm
K = ConstantArray[0, {4, 4, 4}];
K // MatrixForm
For[ i = 1, i = 4, i++,
For [j = 1, j = 4, j++,
K[[i, j, 1]] = ZZ[[i, 1]] ;
K[[i... | Chris Degnen | 363 | <p>Needs <code><=</code> to satisfy the condition.</p>
<pre><code>For[i = 1, i <= 4, i++,
For[j = 1, j <= 4, j++,
K[[i, j, 1]] = ZZ[[i, 1]];
K[[i, j, 2]] = ZZ[[i, 2]];
K[[i, j, 3]] = ZZ[[i, 3]];
K[[i, j, 4]] = ZZ[[i, 4]];]]
</code></pre>
<p>Also <code>K2 = Array[ZZ[[{#, #, #, #}]] &, 4]</code><... |
3,213,854 | <p>From the definition of a inverse standpoint (<span class="math-container">$f^{-1}(f(x))=f(f^{-1}(x))=x$</span>), why does interchanging variables (<span class="math-container">$x$</span> and <span class="math-container">$y$</span>) work to find the inverse? It seems logical to me but I cannot come up with a coherent... | StAKmod | 640,345 | <p>From your definition, suppose that <span class="math-container">$(x,y)$</span> is on that function, we have</p>
<p><span class="math-container">$x=f^{-1}(f(x))=f^{-1}(y)$</span>.</p>
<p>Now like you said, if we change <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, we obtain <s... |
3,829,377 | <p>We know trace of a matrix is the sum of the eigenvalues of the given matrix. Suppose we know the characteristics polynomial of the matrix, is there any result which gives us the sum of the positive eigenvalues of the matrix?</p>
<p>Note that I need the sum of only the positive eigenvalues...not all eigenvalues.</p>
| Disintegrating By Parts | 112,478 | <p>Let <span class="math-container">$C$</span> be a large positively-oriented square contour with the left side of the contour along the imaginary axis. Then
<span class="math-container">$$
p(\lambda)=(\lambda-\lambda_1)^{r_1}(\lambda-\lambda_2)^{r_2}\cdots(\lambda-\lambda_n)^{r_n} \\
\frac{p'(\la... |
1,575,107 | <p>$$L^{-1}\frac{4s}{(s-6)^{3}}$$
$$4L^{-1}\frac{s}{s^{3}}|s=s-6$$
$$4L^{-1}\frac{1}{s^{2}}|s=s-6$$
$$4L^{-1}\frac{1!}{s^{1+1}}|s=s-6$$
$$4te^{6t}$$</p>
<p>Is this correct? symbolab and Wolfram are giving me different answers...</p>
| mrf | 19,440 | <p>If you don't know residues, you can rewrite your function as
$$
\frac{4s}{(s-6)^3} = \frac{4(s-6)}{(s-6)^3} + \frac{24}{(s-6)^3} =
\frac{4}{(s-6)^2} + \frac{24}{(s-6)^3}
$$
and use the "translation rule" to invert the two terms separately.</p>
|
2,482,868 | <p>I am trying to find</p>
<p>$$ \lim\limits_{x\to \infty }[( 1+x^{p+1})^{\frac1{p+1}}-(1+x^p)^{\frac1p}],$$</p>
<p>where $p>0$. I have tried to factor out as</p>
<p>$$(1+x^{p+1})^{\frac1{p+1}}- \left( 1+x^{p}\right)^{\frac{1}{p}} =x\left(1+\frac{1}{x^{p+1}}\right)^{\frac{1}{p+1}}- x\left(1+\frac{1}{x^{p}}\right... | Guy Fsone | 385,707 | <p><strong>Thanks this answer's here</strong> : <a href="https://math.stackexchange.com/questions/2482681/find-lim-n-to-infty-sqrt3n31-sqrtn21#comment5129295_2482681">Find $\lim_{n \to \infty } \sqrt[3]{n^3+1} - \sqrt{n^2+1}$</a></p>
<p>Consider
$$
f(x)=[( 1+x^{p+1})^{\frac{1}{p+1}}-(1+x^p)^{\frac1p}]
$$
Then
$$
f'(x)... |
685,642 | <p>I am trying to find a basis for the set of all $n \times n$ matrices with trace $0$. I know that part of that basis will be matrices with $1$ in only one entry and $0$ for all others for entries outside the diagonal, as they are not relevant.</p>
<p>I don't understand though how to generalize for the entries on th... | Sammy Black | 6,509 | <p>The <em>matrix unit</em> <span class="math-container">$E_{ij}$</span> is the matrix with <span class="math-container">$1$</span> in the <span class="math-container">$(i, j)$</span>-entry and <span class="math-container">$0$</span> everywhere else. A basis for your space consists is
<span class="math-container">$$
\... |
815,739 | <p>Let $f :\mathbb R\to \mathbb R$ be a continuous function such that $f(x+1)=f(x) , \forall x\in \mathbb R$ i.e. $f$ is of period $1$ , then how to prove that $f$ is uniformly continuous on $\mathbb R$ ?</p>
| Hanul Jeon | 53,976 | <p>In my personal opinion, if $\Bbb{R}^{1/2}$ and $\Bbb{R}^{-2}$ is exists meaningfully, then we can expect that these two objects satisfies following properties:</p>
<ul>
<li>$\Bbb{R}^2\times \Bbb{R}^{-2} \cong \{0\}$</li>
<li>$(\Bbb{R}^{1/3})^3 \cong \Bbb{R}$</li>
</ul>
<p>However, the cardinality of $\Bbb{R}^2\tim... |
815,739 | <p>Let $f :\mathbb R\to \mathbb R$ be a continuous function such that $f(x+1)=f(x) , \forall x\in \mathbb R$ i.e. $f$ is of period $1$ , then how to prove that $f$ is uniformly continuous on $\mathbb R$ ?</p>
| coolpapa | 151,988 | <p>None that I know of. </p>
<p>You could <strike>stretch things a bit and</strike> say that $\mathbb{R}^{0}$ is the $0$-dimensional vector space over $\mathbb{R}$. However, raising something to the one-third power should mean a cube root in SOME sense - to make sense with the definition of $\mathbb{R}^{3}$, we would ... |
1,116,435 | <p>How do I get the value of </p>
<p>$$\lim_{x \rightarrow \frac{1}{4} \pi } \frac{\tan x-\cot x}{x-\frac{1}{4} \pi }?$$ </p>
<p>I need the steps without using L'hospital.</p>
| idm | 167,226 | <p>$$...=\underbrace{\frac{1}{\tan x}}_{\to 1}\underbrace{\frac{\tan^2x-1}{(x-\frac{\pi}{4})}}_{ \to \frac{d}{dx}\tan^2(x)\big|_{x=\pi/4}}$$</p>
|
1,437,073 | <p>Ok guys, I have to solve this ODE</p>
<p>$$
\frac{d^2y}{dx^2}=f(x), \quad
x>0,\quad y\left(0\right) = 0, \quad
\left.\frac{dy}{dx}\right\lvert_{x=0}=0
$$
The solution I should get is in the form of
$$y\left(x\right)=\int_0^x k\left(t\right)\, dt $$
Moreover, I should tell what the function $\,k\left(t\right... | Vlad | 229,317 | <p><strong>HINT</strong>: Use the (Leibniz) <a href="https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign" rel="nofollow">formula for differentiation under the integral with variable limits</a></p>
<p>$$
\frac{d}{dx}\left(
\int_{a\left(x\right)}^{b\left(x\right)} f\left(x,t\right)\, dt \right)
= f\b... |
4,341,297 | <p>Evaluate the given expression <span class="math-container">$$\sqrt[n]{\dfrac{20}{2^{2n+4}+2^{2n+2}}}$$</span> The given answer is <span class="math-container">$\dfrac{1}{4}$</span>. My attempt:
<span class="math-container">$$\sqrt[n]{\dfrac{20}{2^{2n+4}+2^{2n+2}}}=\sqrt[n]{\dfrac{20}{2^{2n}\cdot2^4+2^{2n}\cdot2}}\\=... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$2^{2n+4}+2^{2n+2}=2^{2n}(2^4+2^2)=20\times4^n$</span>,<span class="math-container">$$\sqrt[n]{\frac{20}{2^{2n+4}+2^{2n+2}}}=\sqrt[n]{\frac{20}{20\times4^n}}=\frac14.$$</span></p>
|
700,673 | <p>I saw this notation many times, but I don't understand why the $y$ variable is missing in the first term of the first equation below.</p>
<p>$$
\frac{\mathrm{d}y(x)}{\mathrm{d}x} = f(x,y)
$$</p>
<p>It just mean:</p>
<p>$$
\frac{\mathrm{d}y(x,y)}{\mathrm{d}x} = f(x,y)
$$</p>
<p>?</p>
<hr>
<h2>Tell me if I'm wro... | Git Gud | 55,235 | <p>The notation $\dfrac{\mathrm dy(x)}{\mathrm dx}$ is short for $\dfrac{\mathrm dy}{\mathrm dx}(x)$ or $y'(x)$, if you prefer.</p>
<p>In this context, the equality $\dfrac{\mathrm{d}y(x)}{\mathrm{d}x} = f(x,y)$ should be read as $\dfrac{\mathrm dy}{\mathrm dx}(x)=f(x,y(x))$.</p>
<p>As for your last example, you got ... |
3,218,662 | <p>Let <span class="math-container">$T: X \to Y$</span> be a linear operator between normed Banach spaces <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>.
The definition of the operator norm
<span class="math-container">$$
\| T \|
:= \sup_{x \neq 0} \frac{\| Tx \|}{\| x \|}
$$</span>... | operatorerror | 210,391 | <p>This fails in finite dimensions. </p>
<p>Take
<span class="math-container">$$
T=\begin{pmatrix} 2&0\\0&3 \end{pmatrix}
$$</span>
So <span class="math-container">$||T||=3$</span> and
<span class="math-container">$$
T^{-1}=\begin{pmatrix} 1/2&0\\0&1/3 \end{pmatrix}
$$</span>
and <span class="math-c... |
1,590,817 | <p>I was hoping someone could explain to me how to prove a sequence is Cauchy. I've been given two definitions of a Cauchy sequence:</p>
<p>$\forall \epsilon > 0, \exists N \in \mathbb{N}$ such that $n,m> N$ $\Rightarrow |a_n - a_m| ≤ \epsilon$</p>
<p>and equivalently $\forall \epsilon > 0, \exists N \in \ma... | Pedro M. | 21,628 | <p>For the particular example you chose, it is very easy to show directly that it converges to zero, because
$$\sqrt{n+1} - \sqrt{n} = \frac{1}{\sqrt{n+1} + \sqrt{n}}.$$
Nevertheless, this same identity allows you to show that it is Cauchy, since
$$\left|\frac{1}{\sqrt{m+1} + \sqrt{m}} - \frac{1}{\sqrt{n+1} + \sqrt{n}}... |
677,859 | <p>$f(x)= f(x+1)+3$ and $f(2)= 5$, determine the value of $f(8)$.</p>
<p>I don't understand how $f(x)$ can equal $f(x+1)+3$</p>
| Ashot | 51,491 | <p>It means that $f(x+1) = f(x) - 3$. Now you can find values of $f(x)$ consequently.</p>
|
203,827 | <p>Suppose I have the following lists: </p>
<pre><code>prod = {{"x1", {"a", "b", "c", "d"}}, {"x2", {"e", "f",
"g"}}, {"x3", {"h", "i", "j", "k", "l"}}, {"x4", {"m",
"n"}}, {"x5", {"o", "p", "q", "r"}}}
</code></pre>
<p>and </p>
<pre><code>sub = {{"m", "n"}, {"o", "p", "r", "q"}, {"g", "f", "e"}};
</code><... | C. E. | 731 | <p>A rule-based approach:</p>
<pre><code>patt = List /@ OrderlessPatternSequence @@@ Alternatives @@ sub;
Cases[prod, {x_, y : patt} :> {x, y}]
</code></pre>
<blockquote>
<p><code>{{"x2", {"e", "f", "g"}}, {"x4", {"m", "n"}}, {"x5", {"o", "p", "q", "r"}}}</code></p>
</blockquote>
<p>And a functional approach:<... |
1,997,513 | <p>I spend so much time for proving this triangle and i still don't know. </p>
<p>Question :</p>
<p>Given Triangle ABC, AD and BE are altitudes of the triangle. Prove that Triangle DEC similarity with triangle ABC</p>
| arberavdullahu | 377,862 | <p>Since $AD=AC\cdot cos C$ and $BE=BC\cdot cos C$ we have that
$$\frac{AD}{BE}=\frac{AC}{BE}$$
And since the angle that form those two pairs are equal ($\angle C$) the result follows</p>
|
115,385 | <p>I heard that computation results can be very sensitive to choice of random number generator. </p>
<ol>
<li><p>I wonder whether it is relevant to program own Mersenne-Twister or other pseudo-random routines to get a good number generator. Also, I don't see why I should not trust native or library generators as rando... | Yuval Filmus | 1,277 | <p>Let's take your points one by one.</p>
<ol>
<li><p>Is it relevant to program your own PRNG? Not really. Someone already did it for you. There is no more reason to program a PRNG than it is to write your own hash function; that is, only if you're not satisfies with the performance of the library function. In the cas... |
90,712 | <p>How many <em>unique</em> pairs of integers between $1$ and $100$ (inclusive) have a sum that is even? The solution I got was</p>
<p>$${100 \choose 1}{99 \choose 49}$$</p>
<p>I don't have a way to verify it, but I figured you pick one card from the 100, then you can pick 49 of the other cards (if the first card is ... | Arturo Magidin | 742 | <p>This doesn't make much sense: $\binom{100}{1}\binom{99}{49}$ counts the number of ways of picking one out of 100 possibilities <em>and</em> 49 out of 99 possibilities. That is not what you want: you don't want to pick 49 <em>other</em> cards, you just want to pick one out of the 49 that are still "in play".</p>
<p>... |
28,456 | <p>I built a <code>Graph</code> based on the permutations of city's connections from :</p>
<pre><code>largUSCities =
Select[CityData[{All, "USA"}], CityData[#, "Population"] > 600000 &];
uScityCoords = CityData[#, "Coordinates"] & /@ largUSCities;
Graph[#[[1]] -> #[[2]] & /@ Permutations[largUSCitie... | Simon Woods | 862 | <p>I think what you are after is a Delaunay triangulation of the city coordinates. For example:</p>
<pre><code>Graphics`Mesh`MeshInit[];
Graph[
Range[Length[uScityCoords]],
UndirectedEdge @@@ Delaunay[Reverse[uScityCoords, 2]]["Edges"],
VertexCoordinates -> Reverse[uScityCoords, 2],
VertexStyle -> Red,
Pr... |
1,723,718 | <p>Knowing $f(x,y) = 2x^2 +3y^2 -7x +15y$, one simply proves $$|f(x,y)|\leq 5(x^2+y^2)+22 \sqrt{x^2 + y^2}$$ How can I use this info to compute
$$ \lim_{(x,y)\to(0,0)} \frac{f(x,y) - 2(x^2+y^2)^{1/4}}{(x^2+y^2)^{1/4}}\;\;\; ?$$</p>
<p>Thanks!</p>
| Anonymous | 327,815 | <p>Approach limit with $y = mx$
then
$$
\lim_{(x,y)\to(0,0)} \frac{f(x,y) - 2(x^2+y^2)^{1/4}}{(x^2+y^2)^{1/4}} =
$$
$$
\lim_{x\to0} \frac{x(2x +3m^2x -7 +15m)}{x^{1/2}(1+m^2)^{1/4}} - 2 =
$$
$$
\lim_{x\to0} \frac{x^{1/2}( -7 +15m)}{(1+m^2)^{1/4}} - 2 = -2
$$</p>
|
984,852 | <p>Let $p$ be a univariate polynomial over a field $F$, and let $K$ be an extension of $F$. </p>
<p>If $p(x) = 0$ for all $x \in F$, does this imply that $p(x) = 0$ for all $x \in K$? How about if $p$ is multivariate?</p>
<p>For context, I'm trying to understand if doing Schwartz-Zippel-style arithmetic circuit ident... | ajotatxe | 132,456 | <p>No. Take $p(X)=X(X+1)$, $F=\Bbb F_2$, $K=\Bbb F_4$.</p>
<p>Indeed, let $\beta\in\Bbb F_4-\Bbb F_2$. We know that $\beta^2+\beta=1$, that is, $p(\beta)=1$. And $p(0)=p(1)=0$.</p>
|
3,540,045 | <p>The definite integral, <span class="math-container">$$\int_0^{\pi}3\sin^2t\cos^4t\:dt$$</span></p>
<p><strong>My question</strong>: for the trigonometric integral above the answer is <span class="math-container">$\frac{3\pi}{16}$</span>. What I want to know is how can I compute these integrals easily. Is there more... | Vedant Chourey | 638,765 | <p><strong>Note</strong> :-The solution is specifically for the problem in question</p>
<p>The given function is <strong>even</strong></p>
<p>So let <span class="math-container">$$I = 3\int_0^{\pi} \sin^{2}t \cos^{4}t dt$$</span>
And it is equal to
<span class="math-container">$$I = 6\int_0^{\pi/2} \sin^{2}t \cos^... |
3,130,059 | <p>I have faced this differential problem: <span class="math-container">$(y'(x))^3 = 1/x^4$</span>. </p>
<p>From the fundamental theorem of algebra i know there exist 3 solutions <span class="math-container">$y_1$</span>, <span class="math-container">$y_2$</span>, <span class="math-container">$y_3$</span>, but formall... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Write <span class="math-container">$$\frac{dy}{dx}=x^{-4/3}$$</span>
Use the factorization of <span class="math-container">$$a^3-b^3=(a-b)(a^2+ab+b^2)$$</span>
This is
<span class="math-container">$$(y'(x)-x^{-4/3})(y'(x)^2+y'(x)x^{-4/3}+x^{-8/3})$$</span></p>
|
3,130,059 | <p>I have faced this differential problem: <span class="math-container">$(y'(x))^3 = 1/x^4$</span>. </p>
<p>From the fundamental theorem of algebra i know there exist 3 solutions <span class="math-container">$y_1$</span>, <span class="math-container">$y_2$</span>, <span class="math-container">$y_3$</span>, but formall... | user247327 | 247,327 | <p>Every number has three cube roots. You only used one of them.</p>
<p>In particular, if a is real number then the three cube roots are <span class="math-container">$a^{1/3}$</span>, a real number, <span class="math-container">$a^{1/3}(cos(2\pi/3)+ isin(2\pi/3))$</span>, and <span class="math-container">$a^{1/3}(cos... |
1,950,077 | <blockquote>
<p>A standard deck of cards consists of 26 red cards (hearts and diamonds) and 26 black cards (clubs and spades). Suppose you shuffle such a deck and draw three cards at random without replacement. Let <span class="math-container">$A_i =$</span> the event that the <span class="math-container">$i$</span>t... | copper.hat | 27,978 | <p>Each permutation of the deck is equally likely. When you select the three cards, each one is equally likely to be red or black from which the probablity
follows (${1 \over 2}$).</p>
<p>Hence $pA_1 p A_3 = {1 \over 4}$, however $p (A_1 \text{ and } A_3) = { \binom{26}{2}\over \binom{52}{2}} = {25 \cdot 26 \over 51 \... |
2,275,679 | <p>Originally, I want to show that
$$
\frac{\sqrt{a \cdot b + \frac{b}{a}x^2}\arctan \left(\frac{c}{\sqrt{a \cdot b + \frac{b}{a}x^2}}\right)}{\sqrt{a \cdot b}\arctan \left(\frac{c}{\sqrt{a \cdot b}}\right)} \geq 1 \ \ \text{for} \ \ x, a,b,c > 0 \ .
$$
To do so, I figured it is sufficient to show that
$$
f(x) = ... | Cye Waldman | 424,641 | <p>You are quite correct in your supposition that there should be some symmetry in the surface area and volume of a body of revolution. Your only error is that in the surface are we do not want the area under $ydx$, but rather along $yds$.</p>
<p>These ideas are expressed quite clearly in Pappus's Centroid Theorems:</... |
175,971 | <p>Let's $F$ be a field. What is $\operatorname{Spec}(F)$? I know that $\operatorname{Spec}(R)$ for ring $R$ is the set of prime ideals of $R$. But field doesn't have any non-trivial ideals.</p>
<p>Thanks a lot!</p>
| LinAlgMan | 49,785 | <p>I know that the spectrum of a field in the Zariski/commutative-algebra/algebraic-geometry meaning is simply $\{ (0) \}$ but it is too trivial, so I <strong>suspect</strong> there is another meaning for this term applied to fields. In a paper by <em>Yuval Flicker et al.</em> there is a related term: "real spectrum of... |
3,669,700 | <blockquote>
<p>Let <span class="math-container">$f(x)$</span> be a monic, cubic polynomial with <span class="math-container">$f(0)=-2$</span> and <span class="math-container">$f(1)=−5$</span>. If the sum of all solutions to <span class="math-container">$f(x+1)=0$</span> and to <span class="math-container">$f\big(\fr... | Sarvesh Ravichandran Iyer | 316,409 | <p>If <span class="math-container">$\alpha$</span> is a root of <span class="math-container">$f(x)$</span> then <span class="math-container">$\alpha - 1$</span> is a root of <span class="math-container">$f(x+1)$</span> (and vice-versa). So the sum of roots of <span class="math-container">$f(x+1)$</span> is the sum of r... |
9,010 | <p>I hear people use these words relatively interchangeably. I'd believe that any differentiable manifold can also be made into a variety (which data, if I understand correctly, implicitly includes an ambient space?), but it's unclear to me whether the only non-varietable manifolds should be those that don't admit smo... | Matt | 2,000 | <p>A variety is usually defined to be the zero set of some polynomials. I don't know much about this, but I think people usually call manifolds that can be realized as varieties "algebraic manifolds". Take for example the polynomial $x^2+y^2-1$, then the zero set of this is $x^2+y^2=1$, or the circle $S^1$. So that is ... |
2,557,520 | <p>PS: Before posting it, one tried to grasp <a href="https://math.stackexchange.com/questions/1996141/if-ffx-x2-x1-what-is-f0">this</a> (although didn't understand mfl's answer completely either).</p>
<p>I was shown the way to solve it: if I set $$\frac{3x-2}{2}=0 \Rightarrow x=\frac{2}{3} \Rightarrow x^2-x-1=-\frac{... | Dr. Sonnhard Graubner | 175,066 | <p>substituting $$t=\frac{3x-2}{3}$$ then we get
$$x=\frac{2t+2}{3}$$ then we get
$$f(t)=\left(\frac{2t+2}{3}\right)^2-\frac{2+2t}{3}-1$$ then we get ...?</p>
|
50,547 | <p>let $X$ be a $n$-manifold. let $A=\{(x,y,z) \, |\,x=y\}$. I want to see if $A$ is a submanifold of $X^3$.</p>
<p>Consider the map $\Delta\times 1:X\times X\rightarrow X \times X\times X;\, (x,y)\mapsto (x,x,y)$. </p>
<p>If $U_x$ and $U_y$ are neighborhoods of $x$ and $y$ in $X$ then $\Delta\times 1 (U_x\times U_y)... | Marko Riedel | 44,883 | <p>Suppose we are interested in
$$I = \int_0^\infty \frac{x}{1+x^4} dx$$</p>
<p>and evaluate it by integrating
$$f(z) = \frac{z}{1+z^4}$$</p>
<p>around a pizze slice contour with the horizontal side $\Gamma_1$ of
the slice on the positive real axis and the slanted side $\Gamma_3$
parameterized by $z= \exp(2\pi... |
2,610,501 | <p>A five digit number has to be formed by using the digits $1,2,3,4$ and $5$ without repetition such that the even digits occupy odd places. Find the sum of all such possible numbers.</p>
<p>This question came in my test where you literally get $2$ minutes to solve one problem. I want to how to solve this problem mor... | Travis Willse | 155,629 | <p><strong>Hint</strong> If you subtract a number satisfying the criterion from $66666$, you get a different number satisfying the criterion.</p>
|
2,991,719 | <p>How to prove/disprove</p>
<blockquote>
<p>If <span class="math-container">$f : (0, \infty) \to \mathbb{R}^n$</span> is continuous on <span class="math-container">$[a, \infty) , \forall a>0$</span> then <span class="math-container">$f$</span> is continuous on <span class="math-container">$(0, \infty)$</span></p... | user | 505,767 | <p>Yes that's correct, indeed by the <a href="https://en.wikipedia.org/wiki/Determinant#Properties_of_the_determinant" rel="nofollow noreferrer"><strong>properties of the determinant</strong></a> we have that</p>
<p><span class="math-container">$$\det\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatr... |
1,440,106 | <p>I am currently studying how to prove the Fibonacci Identity by Simple Induction, shown <a href="http://mathforum.org/library/drmath/view/52718.html">here</a>, however I do not understand how $-(-1)^n$ becomes $(-1)^{n+1}$. Can anybody explain to me the logic behind this?</p>
| MrYouMath | 262,304 | <p>$-(-1)^n=(-1)^1(-1)^n=(-1)^{n+1}$</p>
|
1,440,106 | <p>I am currently studying how to prove the Fibonacci Identity by Simple Induction, shown <a href="http://mathforum.org/library/drmath/view/52718.html">here</a>, however I do not understand how $-(-1)^n$ becomes $(-1)^{n+1}$. Can anybody explain to me the logic behind this?</p>
| Uri Goren | 203,575 | <p>$$-x=(-1)x$$
Now substitute $x=(-1)^n$
$$-(-1)^n=(-1)(-1)^n=(-1)^{n+1}$$</p>
|
3,066,913 | <p>Given a linear operator <span class="math-container">$T$</span> on a finite-dimensional vector space <span class="math-container">$V$</span>, satisfying <span class="math-container">$T^2 = T$</span> answer the following:</p>
<p>(a) Using the dimension theorem, show that <span class="math-container">$N(T) \bigoplus ... | user3482749 | 226,174 | <p>You've rather overcomplicated the issue: you know that <span class="math-container">$T^{-1}S^{-1}$</span> is a linear map that exists (since <span class="math-container">$S$</span> and <span class="math-container">$T$</span> have inverses), so all you need to is check that it's an inverse for ST. As a hint: this res... |
804,532 | <p>I a working my way through some old exam papers but have come up with a problem. One question on sequences and induction goes:</p>
<p>A sequence of integers $x_1, x_2,\cdots, x_k,\cdots$ is defind recursively as follows: $x_1 = 2$ and $x_{k+1} = 5x_k,$ for $k \geq 1.$</p>
<p>i) calculate $x_2, x_3, x_4$</p>
<p>ii... | vonbrand | 43,946 | <p>Induction is aplicable in many cases, not just sums. In this case you have a expression for $x_n$, prove (base) that it is valid for $n = 1$ and (induction) if it is valid for $n$ it is valid for $n + 1$.</p>
|
1,593,282 | <p>Say we have the function $f:A \rightarrow B$ which is pictured below.<a href="https://i.stack.imgur.com/WfCxs.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/WfCxs.jpg" alt="enter image description here"></a></p>
<p>This function is not bijective, so the inverse function $f^{-1}: B \rightarrow A$... | Alex M. | 164,025 | <p>No, but it does make sense to say that $f^{-1} (\{d\}) = \{6\}$ (which reads as: the preimage of the set $\{d\}$ is $\{6\}$). This is standard mathematical notation. This also allows you to say that $f^{-1} (\{c\}) = \emptyset$, so it's a very convenient notation.</p>
<p>Note that with this notation, you must now v... |
3,336,135 | <p>I was looking at sequences <span class="math-container">$x_n=\{n\alpha\}$</span> with <span class="math-container">$n=1,2,\cdots$</span> and <span class="math-container">$\alpha\in [0, 1]$</span> an irrational number. Such sequences are known to be equidistributed, so they get arbitrarily close to any number <span c... | Daniel Castle | 398,608 | <p>For the case of <span class="math-container">$\frac{\sqrt{2}}{2}$</span>, we look to find <span class="math-container">$k\in\mathbb{N}$</span> such that <span class="math-container">$\frac{\sqrt{2}}{2}k\pmod 1$</span> is as large as possible.</p>
<p>First, consider the sequence of "best possible" rational over-esti... |
3,098,245 | <p><span class="math-container">$$ \frac{d}{dx}e^x =\frac{d}{dx} \sum_{n=0}^{ \infty} \frac{x^n}{n!}$$</span>
<span class="math-container">$$ \sum_{n=0}^{ \infty} \frac{nx^{n-1}}{n!}$$</span>
<span class="math-container">$$ \sum_{n=0}^{ \infty} \frac{x^{n-1}}{(n-1)!}$$</span>
This isn't as straightforward as I thought ... | Theo Bendit | 248,286 | <p>Well, first of all, careful with your first term, since you are dividing by <span class="math-container">$(-1)!$</span>. The first term is constant and differentiates to <span class="math-container">$0$</span>, so your sum should start from <span class="math-container">$n = 1$</span>.</p>
<p>Then, just make a subst... |
3,596,514 | <p>Let <span class="math-container">$0\le x \le 1$</span>
find the maximum value of <span class="math-container">$x(9\sqrt{1+x^2}+13\sqrt{1-x^2})$</span></p>
<p>I try to use am-gm inequality to solve this because it's similar to CMIMC2020 team prob.12 but i don't know how to do next. </p>
| Matteo | 686,644 | <p>You can compute the first derivate of <span class="math-container">$f(x)$</span> which is:
<span class="math-container">$$f'(x)=x\left (\frac{9x}{\sqrt{x^2+1}}-\frac{13x}{\sqrt{1-x^2}}\right )+9\sqrt{x^2+1}+13\sqrt{1-x^2}$$</span>
Now you have to study <span class="math-container">$f'(x)=0$</span> and so:
<span clas... |
738,122 | <p>I have been looking at Kolmogorov's book "Introductory Real Analysis" and have become stuck at the problem 4a) on page 301.</p>
<p>In this problem we are given $f$ a nonnegative and integrable function on $A$, a set of finite measure. We are asked to show that</p>
<p>$\int_A f(x) d \mu \ge 0$</p>
<p>However, how... | AlexR | 86,940 | <p>Hint: First show the result for non-negative simple functions. Then use that you can find $f_n \ge 0$ simple with $f_n \nearrow f$ uniformly to see that
$$\int_A f d\mu = \lim_{n\to\infty} \underbrace{\int_A f_n d\mu}_{\ge 0} \ge 0$$
For simple functions, the result is fairly easy to prove but depends slightly on th... |
352,305 | <p>For <span class="math-container">$\mathcal{S}$</span> the <span class="math-container">$(\infty,1)$</span>-category of spaces <a href="https://mathoverflow.net/questions/239383/the-homotopy-category-is-not-complete-nor-cocomplete">its homotopy category <span class="math-container">$h\mathcal{S}$</span> does not have... | user152131 | 152,131 | <p>No, for a diagram <span class="math-container">$X: I \to \mathcal{S} \to h\mathcal{S}$</span> the colimit in <span class="math-container">$h\mathcal{S}$</span> would satisfy <span class="math-container">$[\mathrm{colim} X(i),Y] \cong \lim [X(i),Y]$</span> where brackets denote morphisms in <span class="math-containe... |
352,305 | <p>For <span class="math-container">$\mathcal{S}$</span> the <span class="math-container">$(\infty,1)$</span>-category of spaces <a href="https://mathoverflow.net/questions/239383/the-homotopy-category-is-not-complete-nor-cocomplete">its homotopy category <span class="math-container">$h\mathcal{S}$</span> does not have... | Tim Campion | 2,362 | <p>Ironically, I was wondering something similar earlier this week (the irony is that I was sitting next to the OP while doing my wondering). Here's another reason why this can't be the case. In general, if <span class="math-container">$C\subseteq D$</span> is a full subcategory and every object in <span class="math-co... |
744,787 | <p>I need the equation of the line passing through a given point $A(2,3,1)$ perpendicular to the given line
$$
\frac{x+1}{2}= \frac{y}{-1} = \frac{z-2}{3}.
$$</p>
<p>I think there must bee some kind of rule to do this, but I can't find it anyway.</p>
| JJacquelin | 108,514 | <p>Your wording (x+1)/2= y/-1 = (z-2)/3 looks strange. Why not (x+1)/2= -y = (z-2)/3 ? Is there not a typing mistake ? If it's OK, an analylical method is shown below :</p>
<p><img src="https://i.stack.imgur.com/0dOWt.jpg" alt="enter image description here"></p>
|
285,719 | <p>Why is $(-3)^4 =81$ and $-3^4 =-81 $?This might be the most stupidest question that you might have encountered,but unfortunately i'am unable to understand this.</p>
| Artem | 28,379 | <p>By definition</p>
<p>$(-3)^4 = (-3)\cdot(-3)\cdot(-3)\cdot(-3) = 9\cdot 9 = 81$</p>
<p>$-3^4 = -(3^4) = -(3\cdot 3\cdot 3\cdot 3) = - 81$</p>
|
645,579 | <p>Let $n$ be a positive integer. Find a general expression for $$\int x^n\cos(x)~dx$$ None of the standard integration techniques or the standard tricks I've seen for difficult integrals seem to apply to this one. I guess it is some type of reduction, but how to get a closed form?</p>
| Peter | 82,961 | <p>Use repeated integration by parts. Take the antiderivate of the cos(x) or sin(x)-function and differentiate the polynomial term. The degree decreases in each step.</p>
|
645,579 | <p>Let $n$ be a positive integer. Find a general expression for $$\int x^n\cos(x)~dx$$ None of the standard integration techniques or the standard tricks I've seen for difficult integrals seem to apply to this one. I guess it is some type of reduction, but how to get a closed form?</p>
| doraemonpaul | 30,938 | <p><a href="http://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions#Integrands_involving_only_cosine" rel="nofollow">http://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions#Integrands_involving_only_cosine</a> can find the reduction formula and also the final result.</p>
|
182,101 | <p>With respect to assignments/definitions, when is it appropriate to use $\equiv$ as in </p>
<blockquote>
<p>$$M \equiv \max\{b_1, b_2, \dots, b_n\}$$</p>
</blockquote>
<p>which I encountered in my analysis textbook as opposed to the "colon equals" sign, where this example is taken from Terence Tao's <a href="http... | Carl Mummert | 630 | <p>It's entirely up to the whim of the author. Other symbols that can mean the same thing are $\triangleq$ and $=_{def}$. I think that only a minority of authors use any special notation, however; the majority just use a regular equals sign. </p>
|
182,101 | <p>With respect to assignments/definitions, when is it appropriate to use $\equiv$ as in </p>
<blockquote>
<p>$$M \equiv \max\{b_1, b_2, \dots, b_n\}$$</p>
</blockquote>
<p>which I encountered in my analysis textbook as opposed to the "colon equals" sign, where this example is taken from Terence Tao's <a href="http... | nullUser | 17,459 | <p>The notation $x:= y$ is preferred as $\equiv$ has another meaning in modular arithmetic (though it is almost always clear from context as to which is meant). However, there is one big advantage to using the $:=$. That is, it is not graphically symmetric and hence allows for strings such as
$$
y:= f(x) \leq g(x) =: ... |
2,909,626 | <p>I'm having trouble remembering how to solve when you have an equation such as </p>
<p>$$0.3=(1-0.63x)^{5.26}$$</p>
<p>any help would be appreciated.</p>
| David C. Ullrich | 248,223 | <p>The sentence you say you're concerned about has more or less nothing to do with "<em>extension</em> by continuity". A careful justification (assuming my guesses about the meaning of the notation are correct):</p>
<p>Say $S\subset L^p$ is the set of simple functions. Define $\Lambda:L^p\to\Bbb R$ by $$\Lambda g=Fg-... |
1,597,247 | <p>Give the continued fraction expansion of two real numbers $a,b \in \mathbb R$, is there an "easy" way to get the continued fraction expansion of $a+b$ or $a\cdot b$?</p>
<p>If $a,b$ are rational it is easy as you can easily conver the back to the 'rational' form, add or multiply and then conver them back to continu... | Gerry Myerson | 8,269 | <p>Gosper found efficient ways to do arithmetic with continued fractions (without converting them to ordinary fractions or decimals). <a href="http://perl.plover.com/yak/cftalk/" rel="noreferrer">Here</a> is a page with links to Gosper's work, but also with an exposition of Gosper's methods. </p>
<p>See also this olde... |
3,535,316 | <p><span class="math-container">$$\int_{0}^{\pi}e^{x}\cos^{3}(x)dx$$</span></p>
<p>I tried to solve it by parts.I took <span class="math-container">$f(x)=\cos^{3}(x)$</span> so <span class="math-container">$f'(x)=-3\cos^{2}x\sin x$</span> and <span class="math-container">$g'(x)=e^{x}$</span> and I got"</p>
<p><spa... | Community | -1 | <p>Your sum is:</p>
<p><span class="math-container">$$\sum_{n\ge1}^{}\frac{n}{\left(n-1\right)!}$$</span>
Setting <span class="math-container">$n-1 \mapsto n$</span> gives:</p>
<p><span class="math-container">$$=\sum_{n\ge0}^{}\frac{n+1}{n!}=1+\sum_{n\ge1}^{}\frac{n+1}{n!}=1+\sum_{n\ge1}^{}\frac{1}{n!}+\sum_{n\ge1}^{... |
3,166,419 | <blockquote>
<p>A fair coin is tossed three times in succession. If at least one of
the tosses has resulted in Heads, what is the probability that at
least one of the tosses resulted in Tails?</p>
</blockquote>
<p>My argument and answer: The coin was flipped thrice, and one of them was heads. So we have two unkn... | LuuBluum | 321,835 | <p>Well, let's look at the problem. It's asking the odds of flipping any tail given that you flipped at least one head, or <span class="math-container">$P(T>0|X>0)$</span> Using the definition of conditional probability, we can say that <span class="math-container">$P(T>0|X>0)=P(T>0,X>0)/P(X>0)$</... |
3,090,879 | <p>Is there prime number of the form <span class="math-container">$1+11+111+1111+11111+...$</span>. I've checked it up to first 2000 repunits, but i found none. If <span class="math-container">$R_1=1$</span>, <span class="math-container">$R_2=1+11$</span>, <span class="math-container">$R_3=1+11+111$</span>, <span cla... | Roman | 632,893 | <p>The first one is at <span class="math-container">$n=2497$</span>, you just missed it!</p>
<pre><code>Do[If[PrimeQ[(10^(n+1)-9n-10)/81], Print[n]], {n, 10^4}]
(* 2497 *)
(* 3301 *)
...
</code></pre>
<p>The direct formula for the <span class="math-container">$n^{\text{th}}$</span> term is from</p>
<pre><code>Sum[10... |
3,090,879 | <p>Is there prime number of the form <span class="math-container">$1+11+111+1111+11111+...$</span>. I've checked it up to first 2000 repunits, but i found none. If <span class="math-container">$R_1=1$</span>, <span class="math-container">$R_2=1+11$</span>, <span class="math-container">$R_3=1+11+111$</span>, <span cla... | DanaJ | 117,584 | <p>To give some more examples of solutions, using Roman's direct formula for conciseness (alternately you can do something like <code>r = 10*r+1; N += r</code> in the loop).</p>
<p>Pari/GP (test is AES BPSW):</p>
<pre><code>? for(n=1,6000,ispseudoprime((10^(n+1)-9*n-10)/81) && print(n))
2497
3301
time = 2min,... |
3,148,094 | <p>In class, my professor computed the density of a gamma random variable by taking the derivative of its CDF, but he skipped many steps. I am trying to go through the derivation carefully but cannot reproduce his final result.</p>
<p>Let <span class="math-container">$k$</span> be the shape and <span class="math-conta... | Peter Szilas | 408,605 | <p>Set <span class="math-container">$A=$</span>{<span class="math-container">$x \in \mathbb{R} | x^2 \le 9$</span> and <span class="math-container">$x^2>4$</span>}.</p>
<p>1) <span class="math-container">$x^2 \le 9$</span> , i.e. <span class="math-container">$-3 \le x \le 3$</span>.</p>
<p>2) <span class="math-con... |
241,210 | <p>I am confused with the concept of topology base. Which are the properties a base has to have?</p>
<p>Having the next two examples for $X=\{a,b,c\}$:</p>
<p>1) $(X,\mathcal{T})$ is a topological space where $\mathcal{T}=\{\emptyset,X,\{a\},\{b\},\{a,b\}\}$. Which is the general procedure to follow in order to get a... | Rudy the Reindeer | 5,798 | <p>A base is a collection $B$ of sets such that every set in the topology can be written as a union of sets in $B$.</p>
<p>1) Can you write every set in $T$ as a union of $\{a\}$ and/or $\{b\}$?</p>
<p>2) Produce all the unions then you'll see what the topology is.</p>
|
417,356 | <p>I'd like to show that the function,</p>
<p>$$f(x) = \begin{cases} 0 & x \not\in \mathbb{Q}\\ 1 & x \in \mathbb{Q} \end{cases}$$</p>
<p>is discontinuous via using the sequence-limit-criterion:</p>
<p>And I'd like to show:</p>
<p>"There is a sequence $(a_n)$ that contains only rational numbers and converge... | Tim | 74,128 | <p>Hint:</p>
<p>Let $k_n$ be the largest integer such that $k_n^2<n^2$. Then $\dfrac {k_n}n\to\sqrt 2$ as $n\to\infty$.</p>
|
417,356 | <p>I'd like to show that the function,</p>
<p>$$f(x) = \begin{cases} 0 & x \not\in \mathbb{Q}\\ 1 & x \in \mathbb{Q} \end{cases}$$</p>
<p>is discontinuous via using the sequence-limit-criterion:</p>
<p>And I'd like to show:</p>
<p>"There is a sequence $(a_n)$ that contains only rational numbers and converge... | DonAntonio | 31,254 | <p>Choose for example</p>
<p>$$x_0:=3\;,\;x_1:=3.1\;,\;....,x_n=3.141592...a_n\;,\;a_n:=\text{the n-th decimal digit of }\;\;\pi$$</p>
<p>Now, the above isn't very nice since one usually does <em>not</em> know what that $\,a_n\,$ is, in particular for big values of $\,n\,$, yet it is possible to calculate as many dig... |
417,356 | <p>I'd like to show that the function,</p>
<p>$$f(x) = \begin{cases} 0 & x \not\in \mathbb{Q}\\ 1 & x \in \mathbb{Q} \end{cases}$$</p>
<p>is discontinuous via using the sequence-limit-criterion:</p>
<p>And I'd like to show:</p>
<p>"There is a sequence $(a_n)$ that contains only rational numbers and converge... | Hagen von Eitzen | 39,174 | <p>Let $a_n=\frac 1n\lfloor nx_0\rfloor$ (where $\lfloor x\rfloor$ denotes "greatest integer $\le x$"). Then $\mathbb Q\ni a_n\to x_0$. If $x_0$ is irrational, this shows that $f$ is not continuous at $x_0$.
If $x_0$ is rational, consider $a_n=\frac 1n\lfloor n(x_0-\sqrt 2)\rfloor+\sqrt 2$.</p>
|
1,557,039 | <h2>Background</h2>
<p>I am a software engineer and I have been picking up combinatorics as I go along. I am going through a combinatorics book for self study and this chapter is absolutely destroying me. Sadly, I confess it makes little sense to me. I don't care if I look stupid, I want to understand how to solve the... | Dr. Sonnhard Graubner | 175,066 | <p>HINT: we have $3^{2(n+1)}=3^{2n}\cdot 3^2$</p>
|
1,583,747 | <p>I just started a course on queue theory, yet equations are given for granted without any demonstrations, which is very frustrating... Thus</p>
<ol>
<li>Why is the mean number of people in a queue system following an $M/M/1$ system</li>
</ol>
<p>$$E(L)=\frac{\rho}{1-\rho}$$</p>
<p>with $\rho=\frac{\lambda}{\mu}$ w... | Jan van der Vegt | 298,403 | <p>The $M/M/1$ queue can be modelled as a so called birth/death process. Since the arrivals are a Poisson process (the interarrival times are exponential) and the departures (service times) are exponentially distributed this will lead to some nice properties.</p>
<p><strong>The birth/death process</strong></p>
<p>If ... |
1,278,442 | <p>I was wondering if the singular homology functor preserve injectivity and surjectivity? I've been trying to figure out a proof or counterexample for ages now and I just can't.</p>
<p>This came up when I was looking at the reduced homology $H_p(S^{n},S^{n-1})$. To calculate it, I have looked at the canonical injecti... | Alex Fok | 223,498 | <p>Your example already answers your question. $\iota_*: H_*(S^{n-1})\to H_*(S^n)$ is not injective. Also, $\pi: S^2\to\mathbb{RP}^2$ is surjective but the induced map on homology is not.</p>
|
3,424,720 | <p>I want to calculate the above limit. Using sage math, I already know that the solution is going to be <span class="math-container">$-\sin(\alpha)$</span>, however, I fail to see how to get to this conclusion.</p>
<h2>My ideas</h2>
<p>I've tried transforming the term in such a way that the limit is easier to find:
... | Vasili | 469,083 | <p>Use that <span class="math-container">$\cos(\alpha + x) - \cos(\alpha)=-2\sin(\alpha+\frac{x}{2})\sin(\frac{x}{2})$</span> to get <span class="math-container">$$\lim_{x \rightarrow 0} -\sin(\alpha+\frac{x}{2})\frac{\sin\frac{x}{2}}{\frac{x}{2}}$$</span></p>
|
2,113,387 | <p><strong>An Ordered Primitive Pythagorean Triple</strong> $(a,b,c)$ is one in which $a \le b \le c$ are coprime and $a^2+b^2 = c^2$.</p>
<p>$f(n) = |\{(a,b,c)~|~ a^2+b^2=c^2,a\le b\le c,~c \le n\}|$.</p>
<p>Function $f(n)$ defines the number of all distinct Ordered Primitive Pythagorean Triple $(a,b,c)$ with $c \le... | Michael Lugo | 173 | <p>Wolfram MathWorld, in its <a href="http://mathworld.wolfram.com/PythagoreanTriple.html" rel="nofollow noreferrer">article on Pythagorean triples,</a> says,</p>
<blockquote>
<p>Lehmer (1900) proved that the number of primitive solutions with
hypotenuse less than N satisfies</p>
<p>$$\lim_{n \to \infty} {\De... |
2,113,387 | <p><strong>An Ordered Primitive Pythagorean Triple</strong> $(a,b,c)$ is one in which $a \le b \le c$ are coprime and $a^2+b^2 = c^2$.</p>
<p>$f(n) = |\{(a,b,c)~|~ a^2+b^2=c^2,a\le b\le c,~c \le n\}|$.</p>
<p>Function $f(n)$ defines the number of all distinct Ordered Primitive Pythagorean Triple $(a,b,c)$ with $c \le... | Thomas Andrews | 7,933 | <p>A loose heuristic argument uses the sort-of-result:</p>
<blockquote>
<p>Picking two random integer, the probability that they are relatively prime is $\frac{6}{\pi^2}$</p>
</blockquote>
<p>This is a "sort-of" result because we are actually talking about density, not probability.</p>
<p>Every primitive triple ca... |
3,237,242 | <p>I have the following problem:</p>
<p>I need to prove that given the following integral</p>
<p><span class="math-container">$\int_{0}^{1}{c(k,l)x^k(1-x)^l}dx = 1$</span>,</p>
<p>we the constant <span class="math-container">$c(k,l) = (k+l+1) {{k+l}\choose{k}} = \frac{(k+l+1)!}{k!l!}$</span>,</p>
<p>with the use of... | Seewoo Lee | 350,772 | <p>In some sense, I think we don't need two-dimensional induction here. Assume that you showed the recurrence relation
<span class="math-container">$$
c(k, l) = \frac{k+1}{l}c(k+1, l-1)
$$</span>
by using the integration by parts. Also, we know base case: <span class="math-container">$c(k, 0) = c(0, k) = k+1$</span>. T... |
109,961 | <p>Suppose $x^2\equiv x\pmod p$ where $p$ is a prime, then is it generally true that $x^2\equiv x\pmod {p^n}$ for any natural number $n$? And are they the only solutions?</p>
| Zev Chonoles | 264 | <p>No, it is not generally true. For example,
$$6^2=36\equiv 6\bmod 5$$
but
$$6^2=36\not\equiv 6\bmod 25.$$</p>
<hr>
<p>Suppose we are given an integer $x$. Note that, for any prime power $p^n$,
$$x^2\equiv x\pmod {p^n}\iff p^n\mid x(x-1)\iff p^n\mid x\;\; \text{ or }\;\;p^n\mid (x-1)$$
because $x$ and $x-1$ are re... |
621,109 | <p>I need to find all the numbers that are coprime to a given $N$ and less than $N$.
Note that $N$ can be as large as $10^9.$ For example, numbers coprime to $5$ are $1,2,3,4$.</p>
<p>I want an efficient algorithm to do it. Can anyone help? </p>
| user44197 | 117,158 | <p>I assume you know how to implement sets. We will only add numbers $<N$ to the set so any bit representation is okay.</p>
<p>If you know that the primes dividing $N$ then the following works:</p>
<pre><code>S={} // Empty sets
for p in primes dividing $N$
add p,2p,3p... to S
end for
</code></pre>
<p>If you do... |
4,222,110 | <p>I am following course on topology that is kind of lack luster (not made for mathematicians). The course starts off with predicate logic and axiomatic set theory (ZFC). Now, I reached a point where the author defined the partition of unity and used the set of all continuous functions between 2 sets. But at the starts... | Spectre | 799,646 | <p>Well, as per your request, I'd add an answer so the comments section is clear (sorry again for the inconvenience).</p>
<p>Notice that you can rearrange the equation as <span class="math-container">$p = (y + 3x)(y-3x)$</span>.
The least bound for <span class="math-container">$p$</span> is <span class="math-container"... |
4,222,110 | <p>I am following course on topology that is kind of lack luster (not made for mathematicians). The course starts off with predicate logic and axiomatic set theory (ZFC). Now, I reached a point where the author defined the partition of unity and used the set of all continuous functions between 2 sets. But at the starts... | Servaes | 30,382 | <p><strong>Solution:</strong> Compute the remainder of <span class="math-container">$p$</span> modulo <span class="math-container">$36$</span>.</p>
<ol>
<li>If <span class="math-container">$p\equiv2,3,5,6,8,10,11,12,14,15,17,18,20,21,22,23,24,26,29,30,32,33,34,35\pmod{36}$</span>, then there are no solutions.</li>
<li>... |
1,042,375 | <p><strong>Question:</strong></p>
<blockquote>
<p>Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Prove that $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$.</p>
</blockquote>
<p>At first I thought this was quite obvious, but then I came up with a counterexample. What if $f(x... | Nishant | 100,692 | <p>You are asked to show that the preimage of $(0, \infty)$ is open. But since $f$ is continuous, the preimage of an open set is open; $(0, \infty)$ is open, therefore its preimage is also open.</p>
|
320,228 | <p>Today, in my lesson, I was introduced to partial derivatives. One of the things that confuses me is the notation. I hope that I am wrong and hope the community can contribute to my learning. In single-variable calculus, we know that, given a function $y =f(x)$, the derivative of $y$ is denoted as $\frac {dy}{dx}$. I... | Community | -1 | <p>A nicer notion is that of the differential:</p>
<p>$$ \text{If} \qquad z = 5x + 3y \qquad \text{then} \qquad dz = 5\, dx + 3\,dy $$</p>
<p>Then if you decide to hold $y$ constant, that makes $dy = 0$, and you have $dz = 5 \, dx$.</p>
<hr>
<p>Another notation that works well with function notation is that if we... |
3,927,845 | <p>n is a natural number. Prove <span class="math-container">$6^n \geq n3^n$</span> holds for every natural number.</p>
<p><span class="math-container">$n = 1:$</span></p>
<p><span class="math-container">$$6 \geq 3 $$</span></p>
<p><span class="math-container">$n \rightarrow n + 1:$</span></p>
<p><span class="math-cont... | Raffaele | 83,382 | <p>Suppose <span class="math-container">$6^n\ge n\cdot 3^n$</span> is true for <span class="math-container">$n$</span> and let's prove it for <span class="math-container">$(n+1)$</span>.</p>
<p><span class="math-container">$6^{n+1}=6\cdot 6^n\ge 6\left(n\cdot 3^n\right)=2\cdot 3\left(n\cdot 3^n\right)=2 n\cdot 3^{n+1}\... |
59,965 | <p>If I have a function $f(x,y)$, is it always true that I can write it as a product of single-variable functions, $u(x)v(y)$?</p>
<p>Thanks.</p>
| anon | 11,763 | <p>If a multivariable function $f(x_1,\dots,x_n)$ is separable then the <a href="http://en.wikipedia.org/wiki/Hessian_matrix" rel="nofollow">Hessian</a> $H\ln f$ is diagonal, which isn't usually the case. Moreover, if $f$ decomposes into $g_1(x_1)\cdots g_n(x_n)$, then</p>
<p>$$\int \frac{\partial\ln f}{\partial x_i} ... |
59,965 | <p>If I have a function $f(x,y)$, is it always true that I can write it as a product of single-variable functions, $u(x)v(y)$?</p>
<p>Thanks.</p>
| Qiaochu Yuan | 232 | <p>It <strong>is</strong> true that linear combinations of such functions are dense. Depending on exactly what space of functions you're working with, this should follow from the locally compact form of the <a href="http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem#Locally_compact_version" rel="nofollow">S... |
2,496,114 | <p>Show that $ a \equiv 1 \pmod{2^3 } \Rightarrow a^{2^{3-2}} \equiv 1 \pmod{2^3} $</p>
<p>Show that$ a \equiv 1 \pmod{2^4 } \Rightarrow a^{2^{4-2}} \equiv 1 \pmod{2^4} $</p>
<p><strong>Answer:</strong></p>
<p>$ a \equiv 1 \pmod{2^3} \\ \Rightarrow a^2 \equiv 1 \pmod{2^3} \\ \Rightarrow a^{2^{3-2}}=a^{2^1} \equiv... | Gabriel Romon | 66,096 | <p>Since $$ \frac{\sin(x)}{x} = \int_{0}^{1} \cos(tx) dt$$</p>
<p>with proper justifications (differentiation under the integral sign) you may derive </p>
<p>$$\left(\frac{\sin(x)}{x}\right)^{(n)} = \int_{0}^{1} t^n \cos(tx+n\pi/2) dt$$ which immediately yields the wanted estimate.</p>
|
1,173,002 | <p>Do I have to use the diagonalization of A?</p>
| Laars Helenius | 112,790 | <p>There are 49 $2\times 2$ boards.</p>
<p>A single domino can cover 6 $2\times 2$ boards.</p>
<p>Thus you need at least 9 dominoes.</p>
<p>On the other hand we can cover your squares claim with 11 dominoes using a greedy type algorithm. I can deploy a maximum of 6 dominoes that cover 6 $2\times 2$ squares per domin... |
449,296 | <p>I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if mathematicians choose the axioms then we are inventing our own maths, surely that is what it is but as it has been pra... | Carl Mummert | 630 | <p>The axioms of mathematics come, in many cases, from the mathematical objects that are being axiomatized.</p>
<p><strong>Example 1: Euclidean Geometry</strong> Many ancient cultures needed to work with geometric ideas in order to build things. Euclid famously wrote a system of axioms for plane geometry based on his ... |
238,970 | <p>Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the next in sequence of iteration functions. (The first three being addition, multiplication, and exponentiation, while ... | Simply Beautiful Art | 272,831 | <p>Disclaimer: My proofs of the following facts may be faulty, but numerical computations suggest otherwise.</p>
<p>The factorial can be uniquely extended by enforcing the condition that it grows at a certain asymptotic rate, namely that <span class="math-container">$(n+x)!\sim n^x\cdot n!$</span> as <span class="math... |
563,927 | <p>Show that $\mathbb{R}$
is not a simple extension of $\mathbb{Q}$
as follow:</p>
<p>a. $\mathbb{Q}$
is countable.</p>
<p>b. Any simple extension of a countable field is countable.</p>
<p>c. $\mathbb{R}$
is not countable.</p>
<p>I 've done a. and c. Can anyone help me a hint to prove b.?</p>
| DonAntonio | 31,254 | <p>To simplify things let us divide in two cases:</p>
<p><strong>The extension of $\;\Bbb Q\;$ is algebraic:</strong> Then we have $\;\Bbb Q(\alpha)/\Bbb Q\;$ of degree $\;n\;$ , and every element in the extension is of the form $\;a_0+a_1\alpha+\ldots+a_{n-1}\alpha^{n-1}\;,\;\;a_i\in\Bbb Q\;$ ...</p>
<p><strong>The... |
1,072,639 | <p>A function is <b>bijective</b> if it is both <b>surjective</b> and <strong>injective</strong>. Is there a term for when a function is both <strong>not surjective</strong> and <strong>not injective</strong>?</p>
| Mister Benjamin Dover | 196,215 | <p>Of course not. This would be a useless notion (so there is no need for such terminology). Most functions are not surjective or injective (in the sense that if you take a "random" one, then you cannot expect it to be surjective or injective), mind you. We are mostly interested in the special class of mappings called ... |
4,057,255 | <p>Suppose we have 10 items that we will randomly place into 6 bins, each with equal probability. I want you to determine the probability that we will do this in such a way that no bin is empty. For the analytical solution, you might find it easiest to think of the problem in terms of six events Ai, i = 1, . . . , 6 wh... | Math Lover | 801,574 | <p>It can be written as,</p>
<p><span class="math-container">$\displaystyle \small 1 - 6 \cdot \bigg(\frac{5}{6}\bigg)^{10} + 15 \cdot \bigg(\frac{4}{6}\bigg)^{10} - 20 \cdot \bigg(\frac{3}{6}\bigg)^{10} + 15 \cdot \bigg(\frac{2}{6}\bigg)^{10} - 6 \cdot \bigg(\frac{1}{6}\bigg)^{10}$</span></p>
<p><span class="math-cont... |
3,827,449 | <p>For example, <span class="math-container">$(1,1,1)$</span> is such a point. The sphere must contain all points that satisfy the condition.</p>
<p>So, I've been milling over this question on and off for the past few days and just can't seem to figure it out. I think I would have to use the distance formula from some ... | JonathanZ supports MonicaC | 275,313 | <p>The standard way to do this is</p>
<ul>
<li><p>let <span class="math-container">$(x, y, z)$</span> be an arbitrary point on your surface,</p>
</li>
<li><p>using your geometry toolkit write the formulas for the various distances you're interested in,</p>
</li>
<li><p>set the appropriate quantities equal to each other... |
3,359,436 | <p>So I remember as a child when I was taught: <span class="math-container">$ . \bar9 =1 $</span>
The proof was taught as:</p>
<p><span class="math-container">$$x = 0.\bar{9} \\
10x = 9.\bar{9} \\
10x - x = 9.\bar{9} - 0.\bar{9} \\
9x = 9 \\
x = 1 \\
\therefore 0.\bar{9} = 1$$</span></p>
<p>I was found the whole thin... | 5xum | 112,884 | <p>A sum of logs is easily simplified as the log of the product, i.e.</p>
<p><span class="math-container">$$\sum_{i=1}^n \log a_n = \log\left(\prod_{i=1}^n a_n\right).$$</span></p>
<p>There is no such conversion for the log of a sum.</p>
|
3,359,436 | <p>So I remember as a child when I was taught: <span class="math-container">$ . \bar9 =1 $</span>
The proof was taught as:</p>
<p><span class="math-container">$$x = 0.\bar{9} \\
10x = 9.\bar{9} \\
10x - x = 9.\bar{9} - 0.\bar{9} \\
9x = 9 \\
x = 1 \\
\therefore 0.\bar{9} = 1$$</span></p>
<p>I was found the whole thin... | Henry | 6,460 | <p>Logarithms are strictly increasing functions and assuming that the logarithm is a real number, i.e. the sum is positive (or at least non-negative, as it should be if this is a sum of probabilities or densities or likelihoods or similar), </p>
<p>then the maximum of the logarithm of a sum is equal to the logarithm o... |
2,995,495 | <p>I'm trying to prove that, for every <span class="math-container">$x \geq 1$</span>:</p>
<p><span class="math-container">$$\left|\arctan (x)-\frac{π}{4}-\frac{(x-1)}{2}\right| \leq \frac{(x-1)^2}{2}.$$</span> </p>
<p>I could do it graphically on <span class="math-container">$\Bbb R$</span>, but how to make a formal... | user | 505,767 | <p>We have that for <span class="math-container">$x\ge 1$</span></p>
<p><span class="math-container">$$\arctan x-\frac{π}{4}-\frac{(x-1)}{2}\le 0 \implies \left|\arctan x-\frac{π}{4}-\frac{(x-1)}{2}\right| =\frac{π}{4}+\frac{(x-1)}{2}-\arctan x$$</span></p>
<p>then consider</p>
<p><span class="math-container">$$f(x)... |
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