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 |
|---|---|---|---|---|
279,238 | <p>I want to create a regular polygon from the initial two points <span class="math-container">$A$</span>, <span class="math-container">$B$</span> and number of vertices <span class="math-container">$n$</span>,<br />
<a href="https://i.stack.imgur.com/dKr9A.png" rel="noreferrer"><img src="https://i.stack.imgur.com/dKr9... | yode | 21,532 | <p>I will recommend <code>ResourceFunction["PolygonFromBase"]</code> here:</p>
<pre><code>base = {{0, 1}, {2, 3}};
Graphics[{Line[base], Table[{RandomColor[],
Line[ResourceFunction["PolygonFromBase"][base, i]]}, {i, 3, 10}]}]
</code></pre>
<p><a href="https://i.stack.imgur.com/3wSl2.png" rel="n... |
4,112,011 | <p>Let <span class="math-container">$f:X\rightarrow Y$</span> be a map and let <span class="math-container">$T'$</span> be a topology on <span class="math-container">$Y$</span>.</p>
<p>Show that <span class="math-container">$$T=\{U\subseteq X \mid \exists V\in T' \text{ with }U=f^{-1}(V)\} $$</span> is a topology on <s... | Henno Brandsma | 4,280 | <p><span class="math-container">$\emptyset = f^{-1}[\emptyset]$</span> and as <span class="math-container">$\emptyset \in T'$</span> we have by definition <span class="math-container">$\emptyset \in T$</span>. Same for <span class="math-container">$X=f^{-1}[Y]$</span> and <span class="math-container">$Y \in T'$</span>.... |
3,486,480 | <p>Find all positive integers <span class="math-container">$n$</span> which are not perfect squares and such that the decimal representation of <span class="math-container">$\sqrt{n}$</span> has the following property: among the first six digits there are at least five equal.</p>
<p>For example <span class="math-conta... | Oscar Lanzi | 248,217 | <p>Numbers consisting of all <span class="math-container">$1$</span> digits in base <span class="math-container">$10$</span> and even number of these digits will have square roots beginning with as many <span class="math-container">$3$</span>'s, thus <span class="math-container">$\sqrt{11}=3.the..., \sqrt{1111}=33.33..... |
3,486,480 | <p>Find all positive integers <span class="math-container">$n$</span> which are not perfect squares and such that the decimal representation of <span class="math-container">$\sqrt{n}$</span> has the following property: among the first six digits there are at least five equal.</p>
<p>For example <span class="math-conta... | Vepir | 318,073 | <p>Up to <span class="math-container">$n\le 10^6$</span>, there are <span class="math-container">$538$</span> non-squares whose square root has <span class="math-container">$5$</span> out of <span class="math-container">$6$</span> (decimal) digits equal:</p>
<p><a href="https://i.stack.imgur.com/K2z9R.png" rel="norefe... |
301,038 | <p>If you want to show that a sequence $(a_{n})$ in $\mathbb{R}$ is convergent, when is it sufficient to show that there is a number $b\in\mathbb{R}$ such that
$$ \liminf a_{n} \geq b \geq \limsup a_{n}$$</p>
<p>In particular, I have a situation where my sequence is bounded and I wanted to use this approach, but I'm n... | Ittay Weiss | 30,953 | <p>It is always sufficient to know such a $b$ exists. Since $\liminf a_n \le \limsup a_n$ always holds, if a $b$ as in the question exists then in particular holds that $\limsup a_n \le \liminf a_n$. But then the equality $\limsup a_n = \liminf a_n $ holds. Now, a sequence converges if, and only if, its limsup is equal... |
2,504,737 | <p>During one of my daily exercises, I was looking for properties of the elements of Cartan calculus. I stumbled on Wikipedia's page about interior products <a href="https://en.wikipedia.org/wiki/Interior_product" rel="nofollow noreferrer">(here)</a>, and I've noticed a property that sounds very useful:
$$
\iota_{[X,Y]... | Anthony Carapetis | 28,513 | <p>Be careful - the interior product places vectors in the first slot, but repeated applications should be read right-to-left, as function composition always is. That is, by definition we have $(i_X d \alpha)(Y) = d \alpha(X,Y);$ so applying $i_Y$ to the 1-form $i_X d \alpha$ yields this same result.</p>
|
4,435,497 | <p>I have to prove the following:</p>
<p>Let <span class="math-container">$p$</span> be a prime and denote <span class="math-container">$\mathbb{Z}_p=\mathbb{Z}/p\mathbb{Z}$</span>. Show that <span class="math-container">$\mathbb{Z}_p[x]/(x^2+x+1)$</span> is not a field if and only if <span class="math-container">$p=3$... | David Lui | 445,002 | <p>By the quadratic formula, <span class="math-container">$x^2 + x + 1$</span> has roots <span class="math-container">$\frac{-1 \pm \sqrt{-3}}{2}$</span>.</p>
<p>Therefore, we consider when <span class="math-container">$-3$</span> has a square root mod <span class="math-container">$p$</span>. If it does, then <span cla... |
281,386 | <p>I have asked this question on stackexchange and have not received any answers or comments after 2 days of it being there.</p>
<p>I read somewhere that the following statement is correct. A proof or any hint as to how to prove it would be helpful.</p>
<p>Let $G$ be a connected reductive group defined over $k$(mayno... | anon | 114,662 | <p>After passing to a finite separable extension of the base field, we may suppose that the reductive group is split. After passing to a finite covering, we may suppose that it is the product of a simply connected semisimple group $G$ and a torus. The torus presents no problem. After passing to a finite separable exten... |
362,944 | <blockquote>
<p>Consider a Brownian bridge <span class="math-container">$B: [0,1]\to \mathbb{R}$</span> with <span class="math-container">$B(0)=B(1)=0$</span>. Let <span class="math-container">$M[0, 1/2]=\max_{x\in[0,1/2]}B(x)$</span>. How to prove that
<span class="math-container">$$\mathbb{P}(M[0, 1/2]\geq s)\le... | Iosif Pinelis | 36,721 | <p>Let <span class="math-container">$B_t:=B(t)$</span>. For <span class="math-container">$t\in[0,1]$</span>, we can write
<span class="math-container">$$B_t=W_t-tW_1,$$</span>
where <span class="math-container">$W$</span> is a standard Wiener process. We have to show that
<span class="math-container">$$P(M_{1/2}\ge s... |
1,037,632 | <p>How to find the last 2 digits of $2014^{2001}$? What about the last 2 digits of $9^{(9^{16})}$?</p>
| Krishnamurari | 195,422 | <p>By Euler's Theorem</p>
<p>$2014^{\phi(25)}=2014^{20}\equiv 1\pmod{25}\implies2014^{2000}\equiv 1\pmod{25}\implies2014^{2001}\equiv 2014\pmod{25}\implies2014^{2001}\equiv 14\pmod{25}$</p>
<p>Obviously $2014^{2001}\equiv 0\pmod{4}$</p>
<p>Let $2014^{2001}=25n+14\implies25n+14\equiv 0\pmod{4}\implies n\equiv 2\pmod{... |
1,761,273 | <p>Good morning. I have a problem with this:</p>
<p>Find the maximum and minimum distances from the origin to the curve* <span class="math-container">$$g\left(x,y\right)=5x^{2}+6xy+5y^{2}$$</span></p>
<p>I have done this:</p>
<p>Function to optimize:<span class="math-container">$f\left(x,y\right)=x^{2}+y^{2}$</span></p... | Rob | 350,976 | <p>Have you tried a software like Mathematica? There are four solutions to your system: $x=y=Sqrt[5/2]$ and the negative of that root (both with $λ=5/8$), and $x=-y=Sqrt[10]$ and the negative of that root (both with $λ=5/2$).</p>
|
1,831,250 | <p>I have come to conclusion that the most efficient and thorough way to prove whether or not a limit exists in three dimensions is to use polar coordinates. </p>
<p>$lim_{x,y \to (0,0)} \frac{x^3+y^3}{x^2+y^2}$</p>
<p>Substituting for polar coordinates:</p>
<p>$lim_{r \to 0^+}$ $\frac{r^3(cos^3 \Theta + sin^3 \Thet... | Mathemagician1234 | 7,012 | <p>Well,first of all, you really mean in 2 dimensions since the limit is defined by the domain,which lies in $\mathbb R^2$. The surface of the graph of the function lies in $\mathbb R^3$,but that's not relevant for the limit</p>
<p>Second, while r in this case <strong>is</strong> independent of $\theta$, I'm really n... |
4,271,166 | <p>I have been studying probability and there are many results regarding the sum of random variables.</p>
<p>For example the sum of iid bernouli random variables is a binomial distributed
The sum of Geometric random variables is Negative binomial distributed
Sum of exponential random variables is Gamma distributed. Al... | G Cab | 317,234 | <p>A list of interrelation among distributions based on sum (like the examples you cited) would be much interesting. However I never came across a paper reporting it.<br />
Nor I saw any list reporting together pdf and characteristic function. That could be of some help, since the CF of the sum of random variables is t... |
152,115 | <p>I would like to open the file "myfile.cdf" or "myfile.nb". One or the other will be in the directory but not both. Because I don't know which, I'm trying to use <code>StringExpression</code> to be general about opening the file. Here is the code below. It does't seem to work. Ideas?</p>
<pre><code>NotebookOpen... | Chris Degnen | 363 | <pre><code>NotebookOpen[First@FileNames[NotebookDirectory[] <> "myfile.*"]]
</code></pre>
|
152,115 | <p>I would like to open the file "myfile.cdf" or "myfile.nb". One or the other will be in the directory but not both. Because I don't know which, I'm trying to use <code>StringExpression</code> to be general about opening the file. Here is the code below. It does't seem to work. Ideas?</p>
<pre><code>NotebookOpen... | Itai Seggev | 4,848 | <p>StringExpression, like all patterns, are template which can be matched against. They don't automagically expand to match eveything--you do some function to do that. I'd write your code like this</p>
<pre><code>NotebookOpen[First[FileNames[NotebookDirectory[] <> "test." <> # & /@ {"cdf", "pdf"}]]]
... |
152,115 | <p>I would like to open the file "myfile.cdf" or "myfile.nb". One or the other will be in the directory but not both. Because I don't know which, I'm trying to use <code>StringExpression</code> to be general about opening the file. Here is the code below. It does't seem to work. Ideas?</p>
<pre><code>NotebookOpen... | m_goldberg | 3,066 | <p>This seems to be reasonably robust. You can even use wildcard * in the name string, but in that case you will open the file which is 1st match in the list returned by <code>FileNames</code>.</p>
<pre><code>openNBorCDF[name_String] :=
Module[{pathNames, file},
pathNames =
FileNames[{name <> ".n... |
1,540,635 | <p>Given $$\min\limits_x \|x\|_2^2$$ $$\text{s.t.} Ax = b$$ <strong>show $x^* = A^T(AA^T)^{-1}b$ where $A \in \mathbb{R}^{m \times n}, m < n$</strong></p>
<p>This is projection $x$ onto the hyperplane $Ax - b = 0$</p>
<p>Multiplying $A^T$ on both sides</p>
<p>$A^TAx - A^Tb = 0$</p>
<p>So we have $\langle A^TAx ... | Omran Kouba | 140,450 | <p>You need the hypothesis that $A$ has rank $m$ so that $AA^T$ is invertible.</p>
<p>Since $x^*=A^T(AA^T)^{-1}b$ we have $Ax^*=b$, so for every $x$ such that $Ax=b$ we have $x-x^*\in {\rm ker}A$ and clearly $x^*\in {\rm Im}A^T$, but it is well-known that ${\rm Im}A^T=({\rm ker}A)^\bot$. Thus, by Pythagoras theorem
... |
4,148,930 | <p>Let's consider the function <span class="math-container">$\log x$</span>; how can I prove that it is a transcendental function on the function field of rational functions, i.e. that a polynomial in two variables <span class="math-container">$p(x,y)$</span> such that <span class="math-container">$p(x,\log x)=0$</span... | Conrad | 298,272 | <p>A direct way to do this is to assume there is such a <span class="math-container">$p(x,\log x)=0, x\in \mathbb R_+$</span>
Since <span class="math-container">$\log$</span> takes infinitely many values, <span class="math-container">$p$</span> cannot be a polynomial in <span class="math-container">$\log x$</span> only... |
2,111,841 | <p>A geodesic is a line representing the shortest route between two points on a sphere, for example on the Earth treated here as a perfect sphere.
Two points on Earth having the same latitude can be also connected with the line being a part of a circle for selected constant latitude.
Differences between these two lines... | Larry B. | 364,722 | <p>Assume the sphere is of radius $1$, and we have our constant lattitude $\phi$ for two coordinates $(\phi, -\lambda_0), (\phi, \lambda_0) \in (-\pi/2,\pi/2)\times [-\pi,\pi]$. Note that the longitudes are positive and negative of a constant value. We can do this because we choose to measure the longitude angle $\lamb... |
3,173,125 | <p>Let <span class="math-container">$f(x) = x^3 + a x^2 + b x + c$</span> and <span class="math-container">$g(x) = x^3 + b x^2 + c x + a\,$</span> where <span class="math-container">$a, b, c$</span> are integers and <span class="math-container">$c\neq 0\,$</span>. Suppose that the following conditions hold:</p>
<ol>
... | Servaes | 30,382 | <p>Let <span class="math-container">$u$</span>, <span class="math-container">$v$</span> and <span class="math-container">$w$</span> be the roots of <span class="math-container">$f$</span>, so that <span class="math-container">$u^2$</span>, <span class="math-container">$v^2$</span> and <span class="math-container">$w^2$... |
1,614,111 | <p>Is there a sequence of positive real numbers $x_1,\ldots,x_n$ for which
$$
\sum_{1\leq i,j\leq n}\left[\frac{\sqrt{ij}}{2}-\min(i,j)\right]x_ix_j> 0?
$$</p>
| River Li | 584,414 | <p>Alternative solution:</p>
<p>There exists such <span class="math-container">$n$</span> and a sequence of positive real numbers <span class="math-container">$x_1, x_2, \cdots, x_n$</span>.</p>
<p>Let <span class="math-container">$A$</span> denote the matrix whose <span class="math-container">$(i,j)$</span>-th entry i... |
1,125,891 | <p>Let $U=<\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & -1 \\ 0 & 1 & 0\end{pmatrix}, \begin{pmatrix}2 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}>$ and $V= <\begin{pmatrix}2& 0 & 0 \\ 0 & 2 & 0 \end{pmatrix}, \begin{pmatrix}... | obataku | 54,050 | <p>Since $u_i$ generate $U$ it follows every element of $U$ can be written in the form $\sum c_iu_i$. As Bernard noted, $f$ acts such that $u_1\mapsto v_1+v_2,u_2\mapsto2(v_1+v_2),u_3\mapsto\frac12(v_1+v_2)$. It follows that $\operatorname{im}(f)$ is generated by $v_1+v_2$ alone.</p>
<p>Now, since we know the action o... |
204,087 | <p>What would be an isomorphism between
$\mathcal{F}(S;V / U)$ and $\mathcal{F}(S;V)/ \mathcal{F}(S;U)$, where
$S$ is a set, $V$ a vector space and $U$ a subspace of $V$. $\mathcal{F}(A,B)$ denotes the set of functions from $A$ to $B$ and $V/U$ is the quotient space. </p>
| Jair Taylor | 28,545 | <p>There is a nice formula for the partial sum of a geometric series:</p>
<p>$\sum_{i=0}^{n-1}{x^i} = \frac{1 - x^n}{1-x}$</p>
<p>So your expression will simplify if you let $\lambda_i = c^i$ for some choice of $c$. Since you want the $\lambda_i$'s to be increasing, you could choose $\lambda_i = 2^i$. Then you get:... |
184,575 | <p>I have a large jagged list, that is each sub-list has a different length. I would like to <code>Flatten</code> this list for <code>Histogram</code> purposes, but it seems to be taking an inordinate amount of time and memory</p>
<pre><code>jaggedList=Table[RandomReal[1,RandomSample[Range[400000,800000],1]],{n,100}]... | kglr | 125 | <p><code>Apply</code>ing <code>Join</code> is much faster than <code>Flatten</code>:</p>
<pre><code>SeedRandom[1]
jaggedList = Table[RandomReal[1, RandomSample[Range[400000, 800000], 1]], {n, 100}];
fullFlatten = Flatten@jaggedList; // AbsoluteTiming // First
</code></pre>
<blockquote>
<p>8.2375848</p>
</blockquot... |
2,337,357 | <p><a href="https://i.stack.imgur.com/iWuCv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/iWuCv.png" alt="enter image description here"></a></p>
<p>The derivative of $\frac1x$ is $\frac{-1}{x^2}$.</p>
<p>How do I find the $c$ if there is no zero in the derivative of the function?</p>
<p>I starte... | User8128 | 307,205 | <p>You're very close. Once the first one is rolled, there is a $(1-1/20)$ chance it is not replicated by the second die. Once the second has also been rolled, assuming it did not replicate the first, there is a $(1-2/20)$ that the third die replicates neither of the original two. Thus the probability is $$1 - (1-1/20)... |
2,337,357 | <p><a href="https://i.stack.imgur.com/iWuCv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/iWuCv.png" alt="enter image description here"></a></p>
<p>The derivative of $\frac1x$ is $\frac{-1}{x^2}$.</p>
<p>How do I find the $c$ if there is no zero in the derivative of the function?</p>
<p>I starte... | Sri-Amirthan Theivendran | 302,692 | <p>Consider the complement namely all three dice are distinct. There are $20\times19\times18$ possible such outcomes out of $20^3$ possible outcomes. Hence the probability is
$$
1-\frac{20\times19\times18}{20^3}=1-\frac{20}{20}\times\frac{19}{20}\times\frac{18}{20}.
$$</p>
|
2,607,333 | <p>Let $L$ be the splitting field of $x^3+2x+1$. I want to know how 59 splits in $L$. I calculated the discriminant of $\mathbb{Z}[\alpha]$ to be $-59$, (where $\alpha$ is a root of the polynomial), which is squarefree therefore $\mathcal{O}_K = \mathbb{Z}$ (because $d(\mathbb{Z}[\alpha]) = (\mathcal{O}_K:\mathbb{Z}[\a... | Eric Wofsey | 86,856 | <p>I assume you want to require $C_i\neq \emptyset$ for all $i$ since otherwise this is trivial. It suffices to show that if $A$ is any topological space with more than $n$ components then it can be written as a union of $n+1$ disjoint nonempty clopen subsets (since then those subsets will be separated in $X$). The s... |
299,140 | <p>Is there a closed form sum of </p>
<p>$\sum_{k=0}^{\infty} \frac{x^k}{(k!)^2}$</p>
<p>It is trivial to show that it is less than $e^x$ but is there a tighter bound?</p>
<p>Thanks</p>
| Iosif Pinelis | 36,721 | <p>$\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\renewcommand{\th}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand... |
229,748 | <p>I have an integral given by <code>intS[zs,c]</code> and I want to plot the expression <code>functionS</code> (which contains <code>intS[zs,c]</code>) with respect to another integral function <code>inta[zs,c]</code> and <code>zs</code>. However, the values of <code>inta[zs,c]</code> are extremely small compared to <... | cvgmt | 72,111 | <p>We can speed up the code by use <code>Newton-Leibniz Formula</code> instead of <code>NIntegrate</code> since all the functions have symbole expression say <code>Hypergeometric2F1</code></p>
<pre><code>Clear["`*"];
d = 4;(*dimensions*)
inta[zs_, c_] =
Subtract @@ (Integrate[((c zs^(d + 1))/(2 d)) (y^((1 -... |
660,034 | <p>I wondered if all decimal expansions of $\frac{1}{n}$ could be thought of in such a way, but clearly for $n=6$,</p>
<p>$$.12+.0024+.000048+.00000096+.0000000192+...\neq.1\bar{6}$$</p>
<p>Why does it work for 7 but not 6? Is there only one such number per base, <em>i.e.</em> 7 in base 10? If so what is the general ... | Henry | 6,460 | <p>The expansion in your title formula is effectively $\displaystyle \sum_1^\infty 7 \times \left(\frac{2}{100}\right)^n = 7 \times \frac {1}{49} = \frac {1}{7}$</p>
<p>Simpler similar ones might be $\displaystyle \sum_1^\infty 2 \times \left(\frac{2}{10}\right)^n = 2 \times \frac {1}{4} = \frac {1}{2}$, i.e. $0.4 + 0... |
660,034 | <p>I wondered if all decimal expansions of $\frac{1}{n}$ could be thought of in such a way, but clearly for $n=6$,</p>
<p>$$.12+.0024+.000048+.00000096+.0000000192+...\neq.1\bar{6}$$</p>
<p>Why does it work for 7 but not 6? Is there only one such number per base, <em>i.e.</em> 7 in base 10? If so what is the general ... | lab bhattacharjee | 33,337 | <p>Please rectify me as I could not find following expression explicitly</p>
<p>$$\frac17=\frac{14}{98}=\frac{14}{100\left(1-\frac2{100}\right)}=\frac{14}{100}\left(1-0.02\right)^{-1}$$</p>
<p>$$=0.14[1+0.02+(0.02)^2+(0.02)^3+\cdots]$$</p>
|
199,842 | <p>I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below:</p>
<p>The area of a square is like a line, the height (one dimension, length) placed several times next to each other up to the square all the way until the square length thus we have heig... | KeithS | 23,565 | <p>The problem with that reasoning is that when showing a rectangle's area by dividing the rectangle's length and height into units, the length and height units form squares. Rotating the radius around the circle by a set number of degrees to divide the circle does not produce squares; these slices are more like triang... |
199,842 | <p>I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below:</p>
<p>The area of a square is like a line, the height (one dimension, length) placed several times next to each other up to the square all the way until the square length thus we have heig... | cloudfeet | 45,136 | <p>OK - so the heart of your intuition is linking the area with the length of line required to fill in that area. It's like shading in a picture using a ballpoint pen.</p>
<p>Take your example with your square: at the end, when you've drawn all your lines, you've shaded the entire area. So if you know how much ink yo... |
134,815 | <p>Assume I have a shuffled deck of cards (52 cards, all normal, no jokers) I'd like to record the order in my computer in such a way that the <em>ordering</em> requires the least bits (I'm not counting look up tables ect as part of the deal, just the ordering itself. </p>
<p>For example, I could record a set of strin... | carlop | 29,324 | <p>There are $52!$ possible ordering of the 52 cards. Each bit can store 2 value, so you need to find the smallest $n$ for which $2^n\geq52!$, $n\geq log_2(52!)$, that is $226$.</p>
<p>The algorithm for rebuild the ordering from the number is the one suggested by Ross Millikan.</p>
|
662,637 | <p>I have a quick question</p>
<p>If S is an orthonormal set of vectors in Rn, then S is a basis for the subspace it spans. I am not sure if this is true or not based off the definition of orthonormal basis which is a subspace w of Rn is a basis for w that is also an orthogonal set.</p>
| Berci | 41,488 | <blockquote>
<p>If $S$ is an orthonormal set of vectors in $\Bbb R^n$, then, <strong>yes</strong>, $S$ is a basis for the subspace it spans.</p>
</blockquote>
<p>Being a basis in the spanned subspace is equivalent to being <em>linearly independent</em>. So this claim means exactly that $S$ is then linearly independe... |
1,105,787 | <p>I was given an excercise in my calculus class that i don't really understand, the problem says : Find the area limited by the curves $$ y = \frac{x+4}{x^2+1} ,\space x = -2 ,\space x = 3,\space y = 0 $$</p>
<p>I don't really know what approach to follow here, my guess would be to solve it using riemann sums or mayb... | Zubin Mukerjee | 111,946 | <p>Let $u = x^2+1$. The derivative of $u$ with respect to $x$ is $2x$. If $A$ is the area we want, then</p>
<p>\begin{align}
A= \int_{-2}^{3} \frac{x+4}{x^2+1}\,\mathrm{d}x&=\frac{1}{2}\left(\int_{-2}^{3} \frac{2x}{x^2+1} \,\mathrm{d}x\right)+4\left(\int_{-2}^{3} \frac{1}{x^2+1}\,\mathrm{d}x\right) \\\\
&=\fra... |
4,318,842 | <p>Thanks for your time.</p>
<p>I am studying Euclidean Geometry through a text book and they have a corollary that afirms that given <span class="math-container">$r \parallel s$</span>, if t intersects r then it must intersect s, although the proof they give is not much convinceable. It says: "If one line can int... | MarioPrix | 998,117 | <p>Yes, your proof is correct.</p>
<p>This is another alternative:</p>
<p>Let <span class="math-container">$m_l$</span> indicates the gradient of the line <span class="math-container">$l$</span>. Then <span class="math-container">$m_r=m_s$</span>.</p>
<p>Additionally, since <span class="math-container">$t$</span> inter... |
3,836,662 | <p>I understand basic group theory. I would say that I've seen most of the standard stuff up to, say, the quotient group.</p>
<p>I feel like I've seen in more than one place the suggestion that group theory is the study of symmetries, or actions that leave something (approximately) unchanged. Unfortunately I can only f... | markvs | 454,915 | <p>The best and standard way to prove it is to consider the map from <span class="math-container">$H\times K$</span> to <span class="math-container">$HK$</span> which sends <span class="math-container">$(h,k)$</span> to <span class="math-container">$hk$</span> and look at the equivalence classes of pairs mapped to the ... |
3,220,273 | <p>It seems the definition of a parallelogram is locked to quadrilaterals for some reason. Is there a reason for this? Why couldn't a parallelogram (given the way the word seems rather than as a mathematical/geometric construct) contain greater than two pairs of parallel sides? In a hexagon for example, all six sides a... | Mark Fischler | 150,362 | <p>I'm going to propose, out of the blue, terms like "hexaparallelogram", "octaparallelogram", and so forth.</p>
<p>I'm wondering whether, for more than <span class="math-container">$4$</span> sides, you would like your definition of hexaparallelogram to be restricted to having 3 pairs of parallel and pairwise equal s... |
2,940,306 | <p>I understand that it would be n! permutations for the given amount of elements, but I am not sure calculate it with these parameters.</p>
| Paul Childs | 584,354 | <p>3 for the first</p>
<p>3 for the last</p>
<p>(n-2)! for the rest.</p>
<p>Multiply them together.</p>
|
4,041,595 | <p>I've been having lectures in group theory with Hungerford's book. We were presented with the following theorem:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/pkL0f.png" rel="noreferrer"><img src="https://i.stack.imgur.com/pkL0f.png" alt="enter image description here" /></a></p>
</blockquote>
<p>And then wit... | Arturo Magidin | 742 | <p>Let’s say you have a group you are interested in. One way to try to understand the group is to stare at it intently until you see interesting things about it.</p>
<p>But the big insight of abstract algebra in general (and of Category Theory) is that you can also learn a lot about the group by considering <em>morphis... |
3,443,672 | <p>The equation is
<span class="math-container">$$\tan\frac{5\pi}{6} \cos x=1-\sin x$$</span>
<span class="math-container">$$\sin\frac{5\pi}{6} \cos x=\cos\frac{5\pi}{6}-\cos\frac{5\pi}{6} \sin x$$</span>
<span class="math-container">$$\sin\left(\frac{5\pi}{6}+x\right)=\cos \frac{5\pi}{6}$$</span> which looks weird to... | Quanto | 686,284 | <p>Note <span class="math-container">$\tan\frac{5\pi}{12}=2+\sqrt3$</span> to rewrite <span class="math-container">$(2+\sqrt 3)\cos x=1-\sin x$</span> as</p>
<p><span class="math-container">$$\sin\frac{5\pi}{12}\cos x+\cos\frac{5\pi}{12}\sin x=\cos\frac{5\pi}{12}$$</span></p>
<p>or</p>
<p><span class="math-container... |
1,827,612 | <p>I have to simplify $\sum_{i,j}\left[{n}\atop{i+j}\right]\binom{i+j}{i}$. I looks like we have $n$ children and we have to answer how many times we can arrange them into circles and color some of circles red.</p>
| RRL | 148,510 | <p>For the first lemma, since $\inf f(x) + \inf g(x) \leqslant f(x) + g(x)$ we have the inequality</p>
<p>$$\inf_{x \in [x_{i-1},x_i]} f(x) + \inf_{x \in [x_{i-1},x_i]} g(x) \leq \inf_{x \in [x_{i-1},x_i]} [f(x) + g(x)],$$</p>
<p>where $[x_{i-1},x_i]$ is a subinterval of some partition $P$.</p>
<p>This implies, fo... |
532,785 | <p>Let $f:[0, 1] \rightarrow \mathbb{R}$ a continuous function. If $a>0$, show that:</p>
<p>$$ f(0)\ln(\frac{b}{a})=\lim_{\epsilon\rightarrow 0}\int_{\epsilon a}^{\epsilon b} \frac{f(x)}{x}dx$$</p>
<p>Tried using Riemann sum, but did not succeed.</p>
| DonAntonio | 31,254 | <p>Another approach using , as you did, the bisectrix of the right angle: if we call $\;M\;$ the vertex where the right angle is, and the upper one in $\;N\;$ and the rightmost one is $\;Q\;$ (next time do <strong>call names</strong> to all the vertices and intersection points! Whithout that geometry exercises become c... |
4,644,847 | <p>Is there a formal fallacy to describe lack of an observation is not proof that it does not exist, or lack of an occurrence is not proof that it can never happen?</p>
| Bram28 | 256,001 | <p>This is the <a href="https://fallacyinlogic.com/appeal-to-ignorance/" rel="nofollow noreferrer">Appeal to Ignorance fallacy</a>: absence of evidence is not evidence of absence. It is not considered a formal fallacy though.</p>
|
2,013,650 | <p>I am trying to prove that
$$n^{n/2}<n!,\text{ for } n\ge2.$$
I can't really figure it out. </p>
| Najib Idrissi | 10,014 | <p>Your "functors" aren't actually functors. I'll stick to standard notations because "$\operatorname{Hom}_k(V,V)$" is too easy to mistake for the space of $k$-linears maps $V \to V$ for some field $k$... I'll also let the base field $\mathbb{R}$ be implied throughout.</p>
<p>Let's do the simplest case, $k=1$. Then yo... |
3,648,315 | <p><a href="https://i.stack.imgur.com/IKnlD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IKnlD.png" alt="enter image description here"></a></p>
<p>I have an item, let's say sword. </p>
<p>As described above, <br/>
If my sword is at state 1, I can try upgrading it. There are 3 possibilities. <br/... | Will Jagy | 10,400 | <p><span class="math-container">$$ (2x+1)(2y+1) = 167 $$</span>
which is prime, so we get
<span class="math-container">$$ (0,83),(83,0),(-84, -1),(-1,-84), $$</span></p>
|
2,698,555 | <p><strong>Question</strong></p>
<p>Find all the rational values of $x$ at which $y=\sqrt{x^2+x+3}$</p>
<p><strong>My attempt</strong></p>
<p>Since we only have to find the rational values of $x$ and $y$, we can assume that
$$ x \in Q$$
$$ y \in Q$$
$$ y-x \in Q $$
Let$$ d = y-x$$
$$d=\sqrt{x^2+x+3}-x$$
$$d+x=\sqrt... | Michael Hoppe | 93,935 | <p>Apply the carpet theorem, see <a href="https://www.cut-the-knot.org/Curriculum/Geometry/CarpetsInSquare.shtml" rel="noreferrer">https://www.cut-the-knot.org/Curriculum/Geometry/CarpetsInSquare.shtml</a>. Consider the upper eighth of the big circle and the left halve circle. Since their areas are equal, the areas w... |
2,983,231 | <p>I'm working out of Salamon's <em>Measure and Integration</em> in preparation for studying probability, and I came across the following exercise.</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be uncountable and let <span class="math-container">$\mathcal{A} \subset 2^X$</span> be the set of all su... | tomasz | 30,222 | <p><strong>Hint</strong>: If <span class="math-container">$(A_n)_n$</span> is a family of pairwise disjoint, countable or cocountable sets, then at most one of them is cocountable.</p>
|
569,484 | <p>A and B are events in a sample space with $p(A) > 0$ and $p(B) > 0$. Write $p(A|B)$ for the conditional probability of $A$ given that $B$ has occurred.</p>
<p>1) If $p(A|B) < p(A)$, show that $p(B|A) < p(B)$ </p>
<p>2) Show that $p(A|B) ≥ \frac{p(A)+p(B)-1}{p(B)}$</p>
<p>For the first part:
I subbed... | Community | -1 | <ol>
<li><p>If $P(A/B)<P(A)$ then $\frac{P(A\cap B)}{P(B)}<P(A)$ then $P(A\cap B)<P(B)P(A)$ and dividing by $P(A)$ to get $\frac{P(A\cap B)}{P(A)}<P(B)$. Thus, $P(B/A)<P(B)$.</p></li>
<li><p>Since $P(A\cup B)\leq 1$ then $P(A)+P(B)-(A\cap B)\leq 1$, this implies $P(A)+P(B)-1\leq (A\cap B)$. Dividing by ... |
35,375 | <p>Good morning,
today I have read that "number theory is nothing but the study of $\mathrm{Gal}(\mathbb{\bar{Q}}/\mathbb{Q})$", here <a href="http://www.math.uconn.edu/~alozano/elliptic/finding%20points.pdf" rel="nofollow">http://www.math.uconn.edu/~alozano/elliptic/finding%20points.pdf</a>
can anyone give a very naiv... | G.. | 14,043 | <p>The object $\mathrm{Gal}(\mathbb{\bar{Q}}/\mathbb{Q})$ is not just an abstract group; it has a natural and intricate topology on it as well. Then one would consider various naturally occurring continuous actions of this topological group on various modules. Two examples are: The absolute Galois group of a number fie... |
128,533 | <p>For the description, I have a simplified problem like the following:</p>
<pre><code>MapAt[f[1, #1], {a, b, c, d}, #2] & @@@ {{1, 2}, {3, 4}}
</code></pre>
<p>will give</p>
<blockquote>
<pre><code>{{a, f[1, 1][b], c, d}, {a, b, c, f[1, 3][d]}}
</code></pre>
</blockquote>
<p>But actually <code>{a, f[1, 1][b], ... | kglr | 125 | <pre><code>ClearAll[f1]
f1[func_, lst_] := Module[{funcs = func[1, #] & @@@ #, pos = #2 & @@@ #},
ReplacePart[lst, Thread[pos -> MapThread[# @ #2 &, {funcs, lst[[pos]]}]]]] &;
f1[f, {a, b, c, d}]@{{1, 2}, {3, 4}}
</code></pre>
<blockquote>
<p><code>{a, f[1, 1][b], c, f[1, 3][d]}</code></p>
... |
1,955,505 | <blockquote>
<p>$\sum_{n=0}^\infty \frac{n^2+3n+2}{4^n} = \frac{128}{27}$ Given hint: $(n^2+3n+2) = (n+2)(n+1)$</p>
</blockquote>
<p><strong>I've tried</strong> converting the series to a geometric one but failed with that approach and don't know other methods for normal series that help determine the actual conver... | user246336 | 246,336 | <p>Here is another approach. Let us define the function </p>
<p>$$\phi(x):=\sum_{n=0}^\infty (n+1)(n+2) x^n.$$</p>
<p>If we differentiate we obtain</p>
<p>$$\phi'(x):=\sum_{n=0}^\infty n(n+1)(n+2) x^{n-1}=\sum_{n=0}^\infty(n+1)(n+2)(n+3)x^n.$$</p>
<p>Now, </p>
<p>$$\phi'(x)-3\phi(x)=\sum_{n=0}^\infty n(n+1)(n+2)x^... |
499,496 | <p>We have the diff. eq:</p>
<p>$$ y' = \dfrac{2y}{t}$$</p>
<p>I tried to do the following:</p>
<p>$$ y' = \dfrac{1}{t} \cdot 3y$$</p>
<p>$$3y = \dfrac{1}{t} \cdot y'$$</p>
<p>$$ \dfrac{3}{2} y^2 = \ln(t)$$</p>
<p>Here I stopped because I noticed the answer had to be $c \cdot t^3$. I don't understand how to get t... | Did | 6,179 | <p>$$\left(\frac{y}{t^\color{red}{\alpha}}\right)'=\frac{y'}{t^\color{red}{\alpha}}-\frac{\color{red}{\alpha}}{t^{\color{red}{\alpha}+1}}y=\frac1{t^\color{red}{\alpha}}\left(y'-\frac{\color{red}{\alpha}y}t\right)$$</p>
|
1,495,885 | <p>I'm trying to integrate this but I think the function can't be integrated? Just wanted to check, and see if anyone is able to find the answer (I used integration by parts but it doesn't work). Thanks in advance; the function I need to integrate is $$\int\frac{x}{x^5+2}dx$$</p>
| Michael Hardy | 11,667 | <p>Probably you'd want to use the calculus of residues to do this.</p>
<p>But below I do it using first-year calculus methods.</p>
<p>The cumbersome part may be the algebra, and that's what I concentrate on here.
\begin{align}
x^5 + 2 & = \left( x+\sqrt[5]{2} \right) \underbrace{\left( x - \sqrt[5]{2} e^{i\pi/5}\... |
3,080,566 | <p>please tell me how I can solve the following equation. </p>
<p><span class="math-container">$$z^3+\frac{(\sqrt2+\sqrt2i)^7}{i^{11}(-6+2\sqrt3i)^{13}}=0$$</span></p>
<p>What formula should I use? If possible, tell me how to solve this equation or write where I can find a formula for solving such an equation. I sear... | Bernard | 202,857 | <p><strong>Hint</strong></p>
<p>You have to use the exponential form of complex numbers: set <span class="math-container">$Z=r\mathrm e^{i\theta}$</span> and observe that, for instance,
<span class="math-container">$$\sqrt 2+\sqrt 2i=\sqrt 2(1+i)=\sqrt 2\cdot \sqrt2\mathrm e^{i\tfrac\pi 4}=2\,\mathrm e^{i\tfrac\pi 4},... |
4,371,244 | <blockquote>
<p>Given an independent set of vectors <span class="math-container">$\left\{v_1, v_2, v_3,v_4\right\}$</span> , then is <span class="math-container">$\left\{v_1+v_2+v_3, v_1+2v_2, 2v_3-v_2\right\}$</span> dependent or independent?</p>
</blockquote>
<p>I've tried this:
<span class="math-container">$$\left\{... | amitava | 566,278 | <p>Let <span class="math-container">$x(v_1+v_2+v_3)+y(v_1+2v_2)+z(2v_3-v_2) = 0$</span></p>
<p>we get
<span class="math-container">$(x+y)v_1+(x+2y-z)v_2+(x+2z)v_3=0$</span>, so</p>
<p><span class="math-container">$x+y = 0,x+2y-z=0,x+2z=0$</span></p>
<p>When we solve this we get <span class="math-container">$x=0,y=0,z=0... |
4,371,244 | <blockquote>
<p>Given an independent set of vectors <span class="math-container">$\left\{v_1, v_2, v_3,v_4\right\}$</span> , then is <span class="math-container">$\left\{v_1+v_2+v_3, v_1+2v_2, 2v_3-v_2\right\}$</span> dependent or independent?</p>
</blockquote>
<p>I've tried this:
<span class="math-container">$$\left\{... | Leox | 97,339 | <p>The rank of the matrix
<span class="math-container">$$
\left[ \begin {array}{cccc} 1&1&1&0\\ 1&2&0&0
\\ 0&-1&2&0\end {array} \right]
$$</span>
is <span class="math-container">$3$</span>, so the vectors are independent.</p>
|
3,113,850 | <blockquote>
<p><span class="math-container">$$f(x)=\sum_{n=1}^{\infty}{\frac{x^{n-1}}{n}},$$</span></p>
<p>Prove that <span class="math-container">$f(x)+f(1-x)+\log(x)\log(1-x)=\frac{{\pi}^2}{6}$</span></p>
</blockquote>
<p>In my mind though,I think that this is related to Basel problem<span class="math-container">$\l... | Arthur | 15,500 | <p>If <span class="math-container">$A$</span> is an <span class="math-container">$n\times n$</span> matrix with <span class="math-container">$n>2$</span>, and all entries are equal, then the adjoint (adjugate) matrix will be zero.</p>
<p>A matrix with a <span class="math-container">$0$</span> adjugate matrix cannot... |
3,161,874 | <p>Find the vector equation of the plane which contains points <span class="math-container">$A(0,1,1)$</span>, <span class="math-container">$B(-1,2,1)$</span> and <span class="math-container">$C(2,0,2)$</span>. </p>
<p>I did this by first finding <span class="math-container">$AB$</span> and <span class="math-container... | Arthur | 15,500 | <p>It has made infinitely many trips. Every trip will be shorter than the last, but the fly will always reach the wall before the car, so it will always have room for one more trip. And one more. And one more.</p>
<p>The total length the fly flies is 50km, as the car crashes into the wall exactly 30 minutes after the ... |
4,627,133 | <p>We have <span class="math-container">$A$</span> <span class="math-container">$(3×3)$</span> matrix with real entries. We know that A is orthogonal and <span class="math-container">$\operatorname{trace}(A)>1$</span>. Show that matrix <span class="math-container">$A+I_{3}$</span> is invertible.</p>
<p>We can see th... | Ben Grossmann | 81,360 | <p>One quick approach using eigenvalues: suppose that <span class="math-container">$A + I$</span> is not invertible. It follows that <span class="math-container">$A$</span> has <span class="math-container">$-1$</span> as an eigenvalue. On the other hand, because <span class="math-container">$A$</span> is orthogonal, al... |
87,319 | <p>How might I show that there's no metric on the space of measurable functions on $([0,1],\mathrm{Lebesgue})$ such that a sequence of functions converges a.e. iff the sequence converges in the metric?</p>
| Davide Giraudo | 9,849 | <p>In fact, the almost converge doesn't correspond to any topology. We use the following fact: </p>
<blockquote>
<p>Let $(X,\mathcal T)$ a topological space. A sequence $\{x_n\}$ converges to $x$ on $(X,\mathcal T)$ if and only if for all subsequence $\{x_{n_k}\}$ we can extract a converging subsequence to $x$. </p>... |
301,106 | <p>In the game connect four with a $7 \times 6$ grid like in the image below, how many game situations can occur?</p>
<p><strong>Rules</strong>:</p>
<blockquote>
<p>Connect Four [...] is a two-player game in which the players first
choose a color and then take turns dropping colored discs from the top
into a se... | Ivan Loh | 61,044 | <p>The number of possible Connect-Four game situations after $n$ plies ($n$ turns) is tabulated at <a href="https://oeis.org/A212693">OEISA212693</a>. The total is 4531985219092. More in-depth explanation can be found at the links provided by the OEIS site. (E.g. <a href="http://homepages.cwi.nl/~tromp/c4/c4.html">John... |
301,106 | <p>In the game connect four with a $7 \times 6$ grid like in the image below, how many game situations can occur?</p>
<p><strong>Rules</strong>:</p>
<blockquote>
<p>Connect Four [...] is a two-player game in which the players first
choose a color and then take turns dropping colored discs from the top
into a se... | Kieleth | 379,373 | <p>I think that:</p>
<p>"Every field of the grid can have three states: Empty, red or yellow disc. Hence, we can have (...) game situations at maximum." </p>
<p>is not correct.</p>
<p>The reason is that by just capturing that each cell can have 3 states, we´re allowing "floating" discs in the board, and the game rul... |
203,212 | <p>Q1: When a question asking me to show a set consists of countably many point, do it mean infinite countably many or both infinite or finite case. </p>
<p>Q2:Also, to show that a set consist of countably many point, is it enough to show that the set is countable or i have to show somethings more?</p>
| André Nicolas | 6,312 | <p>One of the official definitions of countable set is a set $A$ such that there is a bijection between $A$ and a <em>subset</em> of the natural numbers. That definition includes the finite sets.</p>
<p>If a set is countable but not finite, it should be called <em>countably infinite</em>.</p>
<p>But people often slo... |
11,882 | <p>Prove that $\lim\limits_{x \to 2} \frac{x^{2}-2x+9}{x+1}$ using an epsilon delta proof.</p>
<p>So I have most of the work done. I choose $\delta = min{\frac{1}{2}, y}$,<br>
$f(x)$ factors out to $\frac{|x-3||x-2|}{|x+1|}$
But $|x-3| \lt \frac{3}{2}$ for $\delta = \frac{1}{2}$ and also $|x+1| > 5/2$ (I'll spar... | Arturo Magidin | 742 | <p><strong>First:</strong> You <strong>don't</strong> choose $\epsilon$! </p>
<p>In an $\epsilon$-$\delta$ proof, you begin by taking an <em>arbitrary</em> $\epsilon$; you then need to show that it is possible to choose a $\delta$ that "works" for <em>that</em> $\epsilon$. So you are going at things <em>exactly</em> b... |
1,105,056 | <p>There's something I've never understood about polynomials.</p>
<p>Suppose $p(x) \in \mathbb{R}[x]$ is a real polynomial. Then obviously,</p>
<p>$$(x-a) \mid p(x)\, \longrightarrow\, p(a) = 0.$$</p>
<p>The converse of this statement was used throughout high school, but I never really understood why it was true. I ... | fiverules | 102,465 | <p>if $(x-a)$ does not divide $p(x)$, then every linear factor of $p(x)$ is not equal to $x-a$, that is, $p(a)$ is product of nonzero (complex) numbers, therefore $p(a) \neq 0$</p>
|
1,105,056 | <p>There's something I've never understood about polynomials.</p>
<p>Suppose $p(x) \in \mathbb{R}[x]$ is a real polynomial. Then obviously,</p>
<p>$$(x-a) \mid p(x)\, \longrightarrow\, p(a) = 0.$$</p>
<p>The converse of this statement was used throughout high school, but I never really understood why it was true. I ... | Mark Bennet | 2,906 | <p>The proof using the division algorithm is a good one. Here is a more direct approach.</p>
<p>Note that $x^r-a^r=(x-a)(x^{r-1}+ax^{r-2}+a^2x^{r-3} \dots +a^{r-1})=(x-a)p_r(x)$ where $(x-a)$ is a factor.</p>
<p>Let $p(x)=c_rx^r+c_{r-1}x^{r-1}+\dots +c_0$. We are given $p(a)=0$ so $$p(x)=p(x)-p(a)=c_r(x^r-a^r)+c_{r-1... |
2,755,785 | <p>I recently came across a problem in which I had to show that a discrete random variable $X$ has two modes $m_1$ and $m_2$. The information given was that $$\frac{P(X=n)}{P(X=n-1)} = \frac{0.9(n-1)}{n-3}$$</p>
<p>I need to show that this distribution has two modes, $m_1$ and $m_2$. My initial thoughts were to try an... | The Integrator | 538,397 | <p>The given polynomial is $n^3+pn^2+qn+p=0$ $\qquad$</p>
<p>NOTE: im using $n$ as the variable to avoid confusion with it being the root</p>
<p><a href="https://en.wikipedia.org/wiki/Vieta's_formulas" rel="nofollow noreferrer">Vieta's Formula </a> gives us that $x+y+z= -p$</p>
<p>$ xy+yz+zx = q$</p>
<p>$xyz ... |
1,613,857 | <p>I'm reading <a href="https://en.wikipedia.org/wiki/The_Music_of_the_Primes" rel="nofollow">The Music of the Primes</a> by du Sautoy and I've come across a section that I'm having difficulty understanding:</p>
<blockquote>
<p>Euler fed imaginary numbers into the function $2^x$. To his surprise, out came waves whic... | SolUmbrae | 303,202 | <p>I don't have much to say but I had a complex analysis class and as far as I can tell that wouldn't be of any help.</p>
<p>To understand $2^x \cdot (\cos(y\ln 2) + i \sin(y\ln 2)$ as a pure-note-wave, which is of the form $A\cdot(\cos(\omega \cdot t + \phi_0)$ with so-called amplitude $A$, frequency $\omega$, phase-... |
199,235 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/9505/xy-yx-for-integers-x-and-y">$x^y = y^x$ for integers $x$ and $y$</a> </p>
</blockquote>
<p>Determine the number of solutions of the equation
$n^m = m^n$
where both m and n are integers.</p>
| Lubin | 17,760 | <p>This is a nice problem for a calculus class, to describe all pairs $(x,\xi)$ of positive real numbers with $x\ne\xi$ and $x^\xi=\xi^x$. From $\xi\log x=x\log\xi$ you get $(\log x)/x=(\log\xi)/\xi$, in other words, you’re looking for horizontal lines that intersect the graph of $f(x)=(\log x)/x$ twice. Since the func... |
4,482,600 | <p>Prove
<span class="math-container">$$\int_{0}^{\frac{\pi}{2}} \sqrt[n]{\tan x} \,dx = \frac{\pi}{2} \sec \left(\frac{\pi}{2n}\right)$$</span></p>
<p>for all natural numbers <span class="math-container">$n \ge 2$</span>.</p>
<p>There are several answers (<a href="https://math.stackexchange.com/questions/4385116/how-t... | Lai | 732,917 | <p><span class="math-container">\begin{aligned}
\int_{0}^{\frac{\pi}{2}} \sqrt[n]{\tan x} d x&=\int_{0}^{\frac{\pi}{2}} \sin ^{\frac{1}{n}} x \cos ^{-\frac{1}{n}} x d x\\
&= \int_{0}^{\frac{\pi}{2}} \sin ^{2\left(\frac{1}{2 n}+\frac{1}{2}\right)-1} x \cos^ {2\left(-\frac{1}{2n}+\frac{1}{2}\right)-1 }x d x\\
&am... |
77,485 | <p>Find the smallest integer $n$ such that $$\left(1-\frac{n}{365}\right)^n < \frac{1}{2}.$$</p>
<p>I cannot use a calculator, and I do not know where to begin.</p>
| Did | 6,179 | <p>To find an integer $n$ such that this holds, rewrite the inequality as $1-(n/365)<\exp(-(\ln2)/n)$ and use the fact that $\exp(-x)>1-x$ for every $x$. Then $n$ will do as soon as
$1-(n/365)<1-(\ln2/n)$, that is, $n^2>365\cdot\ln2$. Since $\ln2<.7$, one knows that $365\cdot\ln2<365\cdot.7=255.5<2... |
2,053,255 | <p>Let $f_{X,Y}(x,y)=\frac{1}{8}$ for $-2<x<2$, $0<y<2$. Find $f(z)$ where $Z=X+Y$. </p>
<p>I am having difficulty solving the above problem. My attempt at a solution relies on a convolution formula, which says that the $pdf$ of $Z=X+Y$, given the joint $pdf$ $f(x,y)$ is given by $$f_Z(z)= \int_{s+t=z} f(s... | BruceET | 221,800 | <p>The joint distribution of $(X,Y)$ is uniform on a rectangle with
corners at $(-2,0)$ and $(2,2),$ and with area 8. So just by geometry, you should
be able to find the CDF of $S = X + Y.$ For example, draw the rectangle
and the line $x + y = 1.5.$ Then what is the value of $F_S(s) = P(S \le s = 1.5)?$
(Consider thre... |
3,672,948 | <p>Consider the series
<span class="math-container">$$
S = \sum_{n=1}^\infty e^{-n^2x}
$$</span>
then I have to argue for that <span class="math-container">$S$</span> is convergent if and only if <span class="math-container">$x>0$</span>.</p>
<p>As this is if and only if I think I have to assume first that S is con... | Gary | 83,800 | <p>If <span class="math-container">$x>0$</span>, then (since <span class="math-container">$0<e^{-x}<1$</span>)
<span class="math-container">$$
\sum\limits_{n = 1}^\infty {e^{ - n^2 x} } \le \sum\limits_{n = 1}^\infty {e^{ - nx} } = \frac{{e^{ - x} }}{{1 - e^{ - x} }} = \frac{1}{{e^x - 1}} < + \infty .... |
997,116 | <p>If one root of the equation $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ is common and $a \ne b$ then:</p>
<p>The options are as follows:
$$\begin{array}{ll}
(A)\quad& a + b = 0\\
(B)& a + b = -1\\
(C)& a - b = 1\\
(D)& a + b = 1
\end{array}$$</p>
<p>Idk how to solve this, please help me.</p>
| ajotatxe | 132,456 | <p>Let $c$ be the common root. Then
$$c^2+ac+b=0$$
$$c^2+bc+a=0$$
so $ac+b=bc+a$ or $(a-b)c=a-b$. Since $a\neq b$, $c$ must be $1$.</p>
<p>Now we have $a+b=-1$.</p>
|
1,618,952 | <p>How to prove this exercise?</p>
<p>Let $A$ be a $n \times n$ diagonal matrix with characterist polynominal</p>
<p>$$(x-c_1)^{d_1}...(x-c_k)^{d_k}$$</p>
<p>where $c_i$ are distinct. Let $V$ be the space of $n \times n$ matrixes $B$ such that $AB=BA$. Prove that dimension of $V$ is $d_{1}^{2}+...+d_{k}^{2}$.</p>
| Brian M. Scott | 12,042 | <p>HINT: Starting by writing out what the hypotheses mean is good. A good next step is to do the same kind of expansion of what you’re trying to prove: $\lim_{n\to\infty}\sum_{k=1}^nf(k)=a$ means that </p>
<blockquote>
<p>for each $\epsilon>0$ there is an $m_\epsilon\in\Bbb N$ such that $$\left|a-\sum_{k=1}^nf(k)... |
4,320,437 | <p>I was thinking about linear tramsformations and i came up with this example:
<span class="math-container">$$f:\mathbb{R}^n \to \mathbb{C}^n\\
f(x)=ix$$</span>
for this example, domain and co-domain are not defined over the same field and all linear transformations that i encountered by now had domain and co-domain ... | Daniel Wohlrath | 842,628 | <p><span class="math-container">$\mathbb{R}^n$</span> is by default just a set of points. The notation is often abbreviated when talking about vector spaces, so <span class="math-container">$\mathbb{R}^n$</span> can also be considered as a vector space (most often over the field of reals). Actually, ever field is a vec... |
743,465 | <p>Suppose $l,t\in[0,1]$ and $l+t\leq1$ I want to prove $1+l+t>6lt$. When $t=0$ or $l=0$, it is trivial, so I started with $l,t\neq0$ but I couldn't reach anywhere. I don't have time to write in detail what I have already tried, but I tried to manipulate $(l-t)^2$ mostly. Anyway, if anyone help me with the proof tha... | David | 119,775 | <p>If $l+t=a$ then $lt\le\frac{1}{4}a^2$ and we have
$$6lt-(l+t)\le{\textstyle\frac{3}{2}}a^2-a={\textstyle\frac{3}{2}}a(a-{\textstyle\frac{2}{3}})\ .$$
By sketching a graph it is easy to see that for $0\le a\le1$ the right hand side is at most $\frac{1}{2}$.</p>
|
743,465 | <p>Suppose $l,t\in[0,1]$ and $l+t\leq1$ I want to prove $1+l+t>6lt$. When $t=0$ or $l=0$, it is trivial, so I started with $l,t\neq0$ but I couldn't reach anywhere. I don't have time to write in detail what I have already tried, but I tried to manipulate $(l-t)^2$ mostly. Anyway, if anyone help me with the proof tha... | Macavity | 58,320 | <p>By AM-GM, $4lt \le (l+t)^2\le 1$. </p>
<p>Similarly, $2 lt \le 2\sqrt{lt} \le l+t$. Add these... and note equality is not possible for all the inequalities used simultaneously.</p>
|
1,973,724 | <p>Now this was the explanation on how to solve a prove in my book : </p>
<p>We will prove that the two sets complement (A∩B) and complement(A) ∪ complement(B) are equal by showing that each is a subset of the other. </p>
<p>First,we will show that complement (A∩B) ⊆ complement(A) ∪ complement(B). We do this by show... | fleablood | 280,126 | <p>Because $(A\cap B)^c =\{a| a \not \in A\cap B\} $. Now to further understand and describe what it means to <em>not</em> be in the intersection it helps to know the definition of intersection. </p>
<p>$(A\cap B)^c =\{a| a \not \in A\cap B\}=\{a\not \in both A and B\}=\{a|\lnot(a\in A \land a \in B)\}=\{a|\lnot (a\i... |
2,006,927 | <p>I want to prove that $\cap_{i=1}^{\infty}\left(A_i \cap \left(\cup_{j=1}^{\infty}B_j\right)\right)=\cup_{j=1}^{\infty}\left(\left(\cap_{i=1}^{\infty}A_i\right) \cap B_j\right)$ for sets $A_i,B_j$ and natural numbers $i,j$. If an element $x$ belongs to the left hand side, then $x\in A_1$ and $x\in$ some of $B_j$ and ... | MarnixKlooster ReinstateMonica | 11,994 | <p>Here is a way to systematically calculate which elements $\;x\;$ are in both sides of this equality, just by expanding the definitions and simplifying.$
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\op}[1]{\\ #1 \quad & \quad \unicode{x201c}}
\newcommand{\hints}[1]{\mbox{#1} \\ \quad & \quad \ph... |
293,371 | <p>This is part of a homework assignment for a real analysis course taught out of "Baby Rudin." Just looking for a push in the right direction, not a full-blown solution. We are to suppose that $f(x)f(y)=f(x+y)$ for all real x and y, and that f is continuous and not zero. The first part of this question let me assum... | Community | -1 | <p>First note that $f(x) > 0$, for all $x \in \mathbb{R}$. This can be seen from the fact that
$$f(x) = f\left(\dfrac{x}2 + \dfrac{x}2\right) = f \left(\dfrac{x}2\right)^2$$ Further, you can eliminate the case $f(x) = 0$, since this would mean $f \equiv 0$.</p>
<p>One way to go about is as follows.</p>
<p>$1$. Pro... |
98,317 | <p>This comes from Artin Second Edition, page 219. Artin defined $G = \langle x,y\mid x^3, y^3, yxyxy\rangle$, and uses the Todd-Coxeter Algorithm to show that the subgroup $H = \langle y\rangle$ has index 1, and therefore $G = H$ is the cyclic group of order 3.</p>
<p>That being the case, $x$ cannot be either $y$ or ... | djechlin | 79,767 | <p>I computed the powers of $z = xy$:</p>
<p>$xy$, $y^2$, $x$, $y$, $xy^2$, $x^2$, $xy$, $y^2$, $1$.</p>
<p>This in particular gives $z^7 = z$ so $z^6 = x^2 = 1$, and $x^3 = x^2 = 1$ gives $x = 1$.</p>
<p>All you need from this though is</p>
<p>$z^2 = y^2$<br>
$z^3 = x$</p>
<p>So $1 = y^6 = (z^2)^3 = (z^3)^2 = x^... |
573,484 | <p>Prove that if $A \setminus B = \emptyset$, then $A \subseteq B$.</p>
<p>The Venn Diagram helped me to visualize what I'm trying to show (thanks @GA316), but the book asks for a written proof (step by step) by contradiction. Sorry if I wasn't more specific at first, is just that I've had many troubles in the past w... | The Chaz 2.0 | 7,850 | <p>Just for grins, you could prove the (logically equivalent) <strong>contrapositive</strong>, viz. </p>
<blockquote>
<p>If $A$ is <em>not</em> a subset of $B$, then [$A$ "toss" $B$] is nonempty.</p>
</blockquote>
<p>If $A$ is not a subset of $B$, then there is some element $x \in A$ that is <em>not</em> in $B$... |
1,756,448 | <p>Suppose the function, $f$, is differentiable at $x = 1$. $$\lim_{h\rightarrow\ 0}\frac{f(1+h)}{h} = 5$$</p>
<p>Find a) $f(1)$ and, b) $f'(1)$. </p>
<p>I know b) (well at least I think it can) can be found by the definition of the derivative, i.e. </p>
<p>$$\lim_{h\rightarrow\ 0}\frac{f(1+h)-f(1)}{h} $$</p>
<p>Th... | drhab | 75,923 | <p>Hint:</p>
<p>if $f$ is differentiable at $1$ then it is continuous at $1$. For $h\neq 0$ we have:</p>
<p>$$f(1+h)=h\times\frac{f(1+h)}{h}$$</p>
<p>What equality will arise if $h\to0$?</p>
|
1,646,460 | <p>I was thinking about the following integral if I could solve it without using trigonometric formulas. If there is no other way to solve it, could you please explain me why do we replace $x$ with $2\sqrt 2 \sin(t)$? I'm really confused about these types of integrals.</p>
<p>$$\int \sqrt{8 - x^2} dx$$</p>
| Harish Chandra Rajpoot | 210,295 | <p>Let<br>
$$I=\int\sqrt{8-x^2}\ dx\tag 1$$
using integration by parts,<br>
$$I=\sqrt{8-x^2}\int 1\ dx-\int \left(\frac{-2x}{2\sqrt{8-x^2}}\right)\cdot x\ dx$$
$$I=\sqrt{8-x^2}(x)-\int \frac{(8-x^2)-8}{\sqrt{8-x^2}} \ dx$$
$$I=x\sqrt{8-x^2}-\int \left(\sqrt{8-x^2}-\frac{8}{\sqrt{8-x^2}} \right)\ dx$$
$$I=x\sqrt{8-x^2}-... |
176,613 | <p>Working on Harmonic numbers, I found this very interesting recurrence relation :
$$
H_n = \frac{n+1}{n-1} \sum_{k=1}^{n-1}\left(\frac{2}{k+1}-\frac{1}{1+n-k}\right)H_k
,\quad \forall\ n\in\mathbb{N},n>1$$
My proof of this is quite long and complicated, so I was wondering if someone knows an elegant or concise one... | joriki | 6,622 | <p>It helps to visualize the terms in a square of products $1/(ij)$ with $i$ and $j$ running for $1$ to $n$. The sum over the first term contains all products with $i\ne j$ exactly once, whereas the sum over the second term roughly corresponds to the upper left half of the square, but with the left-most column, which a... |
176,613 | <p>Working on Harmonic numbers, I found this very interesting recurrence relation :
$$
H_n = \frac{n+1}{n-1} \sum_{k=1}^{n-1}\left(\frac{2}{k+1}-\frac{1}{1+n-k}\right)H_k
,\quad \forall\ n\in\mathbb{N},n>1$$
My proof of this is quite long and complicated, so I was wondering if someone knows an elegant or concise one... | dbot | 90,699 | <p>This page helped me prove the following related result, so I thought I'd share.</p>
<p>To begin,</p>
<p>$$
\sum_{i=1}^{n}\left ( \frac{1}{x_i} + \frac{1}{x_{n+1-i}}\right ) = 2 \sum_{i=1}^{n}\left ( \frac{1}{x_i} \right )
$$</p>
<p>$$
=\sum_{i=1}^{n}\left ( \frac{x_i+x_{n+1-i}}{x_i x_{n+1-i}} \right )
$$</p>
<p>... |
318,299 | <blockquote>
<p>Let <span class="math-container">$U$</span> be an open set in <span class="math-container">$\mathbb R$</span>. Then <span class="math-container">$U$</span> is a countable union of disjoint intervals. </p>
</blockquote>
<p>This question has probably been asked. However, I am not interested in just get... | Trevor Wilson | 39,378 | <p>Let $U$ be an open subset of $\mathbb{R}$. Let $P$ be the poset consisting of collections $\mathcal{A}$ of disjoint open intervals where we say $\mathcal{A} \le \mathcal{A}'$ if each of the sets in $\mathcal{A}$ is a subset of some open interval in $\mathcal{A}'$. Every chain $C$ in this poset has an upper bound, ... |
318,299 | <blockquote>
<p>Let <span class="math-container">$U$</span> be an open set in <span class="math-container">$\mathbb R$</span>. Then <span class="math-container">$U$</span> is a countable union of disjoint intervals. </p>
</blockquote>
<p>This question has probably been asked. However, I am not interested in just get... | sachin srivastava | 215,804 | <p>Let $G$ be a nonempty open set in $\mathbb{R}$. Write $a\sim b$ if the closed interval $[a, b]$ or $[b, a]$ if $b<a$, lies in $G$.This is an equivalence relation, in particular $a\sim a$ since $\{a\}$ is itself a closed interval. $G$ is therefore the union of disjoint equivalence classes. </p>
<p>Let $C(a)$ be t... |
318,299 | <blockquote>
<p>Let <span class="math-container">$U$</span> be an open set in <span class="math-container">$\mathbb R$</span>. Then <span class="math-container">$U$</span> is a countable union of disjoint intervals. </p>
</blockquote>
<p>This question has probably been asked. However, I am not interested in just get... | Alex | 560,714 | <p>Tell me if this has any potential... I understand the equivalence relation proof. However, since R is separable, we know that there exists a countable base. This means that there exists a family of open subsets of R such that every open set is equal to the union of a subcollection of this family.</p>
<p>But then by ... |
2,056,209 | <p>We know that the union of countably many countable sets is countable.
What can se say about the union of infinitely many countable sets and its cardinality?
Thanks :)</p>
| Michael Hardy | 11,667 | <p>The set $\mathbb N \times \mathbb R$ is the set of all ordered pairs $(n,x)$ where $n\in \{1,2,3,\ldots\}$ and $x$ is a real number. The set of pairs $\{(1,x), (2,x), (3,x),\ldots\}$ where $x$ is <em>just one</em> real number is countably infinite, so the union $\displaystyle \bigcup_{x\,\in\,\mathbb R} \{(1,x), (2,... |
768,925 | <p><img src="https://i.stack.imgur.com/SHRqU.png" alt="enter image description here"></p>
<p>This is quite a tricky question for me, but this is how far I got:<img src="https://i.stack.imgur.com/UV0ou.jpg" alt="enter image description here"></p>
<p>My drawing may not be precise, but I do know the points of tangency. ... | Jonas Granholm | 134,205 | <p>This construction occurs three times:
<img src="https://i.stack.imgur.com/gwcMM.png" alt="A square and a circle inscribed in a right triangle"></p>
|
1,824,638 | <p>The figure shows a piece of string tied to a circle with a radius of one unit. The string is just long enough to reach the opposite side of the circle. Find the area of the region, not including the circle itself that is traced out when the string is unwound counterclockwise and continues counterclockwise until it r... | Doug M | 317,162 | <p>There are 3 parts to the calculation.
The string unwinds until it points straight up.
A semi-circle.
The string winds back onto the wheel.
Part 1 and part 3 trace out equal area.</p>
<p>Part 1:
Let $(\cos\theta,\sin\theta)$ be the point where the string is starting to pull away from the wheel.</p>
<p>let (x,y) be ... |
1,714,053 | <p>How can I prove that if $\sum\limits_{n\in\Bbb N} a_n^2 $ converges then also $\sum\limits_{n\in\Bbb N} \frac{\lvert a_n\rvert}{n}$ converges?</p>
<p>Thanks ! </p>
| DonAntonio | 31,254 | <p>By Cauchy-Schwarz$$\sum_{n=1}^\infty\frac{|a_n|}n\le\left(\sum_{n=1}^\infty a_n^2\right)^{1/2}\cdot\left(\sum_{n=1}^\infty\frac1{n^2}\right)^{1/2}$$</p>
|
29,926 | <p>Is there an algorithm which will allow me to find an isomorphism between two graphs if I have their adjacency lists?</p>
| Aaron | 9,863 | <p><B>Short answer:</B> using logs won't really help. Instead, you should multiply $p$ by something large so that matlab can deal with it, and then divide out in the end.</p>
<p><B>Long answer:</B>There is a $\log$ for matrices, but it doesn't behave quite the same as $\log$ for numbers, and so it's not quite suitabl... |
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | Zev Chonoles | 1,916 | <p>Atiyah + Macdonald, <em>Introduction to Commutative Algebra</em>.</p>
|
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | Jim Conant | 9,417 | <p>"Rational Homotopy Theory and Differential Forms." by Griffiths and Morgan. </p>
|
2,747,074 | <p>I've seen multiple ways on how to solve it online (and most likely the majority are wrong), but I don't know how to solve this fully so that I could get the full grade during my exam.</p>
<p>The question is as follows:</p>
<blockquote>
<p>Prove that if $\sum a_n$ is convergent with $a_n > 0$ for all $n$,
th... | user284331 | 284,331 | <p>Then for large $n$, $a_{n}<1$, then $a_{n}^{2}\leq a_{n}$ for all such $n$ and hence $\displaystyle\sum_{n\geq N}a_{n}^{2}\leq\sum_{n\geq N}a_{n}<\infty$.</p>
|
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