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 |
|---|---|---|---|---|
3,368,130 | <p>Hi I'm having a lot of trouble with the following problem, I tried using the equation dQ/dt = 3Q/2006-3t but it doesn't work.</p>
<blockquote>
<p>A tank contains 90 kg of salt and 2000 L of water. Pure water enters a tank at the rate 6 L/min. The solution is mixed and drains from the tank at the rate 3 L/min. How... | Milo Brandt | 174,927 | <p>It seems like there are two steps here: that certain sums of holomorphic functions remain holomorphic and that certain functions deserve to be called "extensions" of others.</p>
<p>The first part can be captured as follows</p>
<blockquote>
<p>Let <span class="math-container">$f_1,f_2,f_3,\ldots$</span> be a sequ... |
23,285 | <p>Related to <a href="https://mathematica.stackexchange.com/questions/1964/delete-the-current-notebook-cell-using-the-keyboard">Delete the current notebook cell using the keyboard</a>.
Difference is that here I want to delete the whole group of cells, including the original Input cell and all Print, Output, Message et... | Vladimir | 5,981 | <p>Here is the code that should be added to <code>KeyEventTranslations.tr</code>. To learn more about editing this, and similar, system files look at <a href="https://mathematica.stackexchange.com/questions/16165/how-can-i-set-a-keyboard-shortcut-to-run-a-command/16174#16174">this question</a> and its related.</p>
<pr... |
414,642 | <p><strong>Background: Evaluation of</strong> <span class="math-container">$\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$</span></p>
<p>We can prove using the Beta-Function identity that</p>
<p><span class="math-container">$$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma \left(\lambda-\frac{1}{2} \ri... | xpaul | 66,420 | <p>I hope it is not too late. Define
\begin{eqnarray}
I(a)=\int_0^\infty\frac{\log(1+ax^4)}{(1+x^2)^2}dx.
\end{eqnarray}
Then
\begin{eqnarray}
I'(a)&=&\int_0^\infty \frac{x^4}{(1+ax^4)(1+x^2)^2}dx\\
&=&\frac{1}{(1+a)^2}\int_0^\infty\left(-\frac{2}{1+x^2}+\frac{1+a}{(1+x^2)^2}+\frac{1-a+2ax^2}{1+a x^4}\r... |
414,642 | <p><strong>Background: Evaluation of</strong> <span class="math-container">$\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$</span></p>
<p>We can prove using the Beta-Function identity that</p>
<p><span class="math-container">$$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma \left(\lambda-\frac{1}{2} \ri... | Lai | 732,917 | <p>Letting <span class="math-container">$x\mapsto \frac{1}{x}$</span> reduces the power of the denominator and changes the integral into
<span class="math-container">$$
\begin{aligned}
\int_0^{\infty} \frac{\ln \left(1+x^n\right)}{\left(1+x^2\right)^2} d x & =\int_0^{\infty} \frac{x^2 \ln \left(1+x^n\right)}{\left(... |
2,714,660 | <p>I've got a rather elemental doubt about what "mapping a subspace homeomorphically onto its image" means, but I'd like to be sure what is being implied.</p>
<p>Let $f:X\to Y$ a continuous injective map between topological spaces. Let $U\subseteq X$ be an open subspace. Then we could say in some context that $f$ maps... | Masacroso | 173,262 | <p>Just observe that</p>
<p>$$\int_{-1}^1 d[\arctan(1/x)]=\arctan(1)-\lim_{x\to0^+}\arctan(1/x)+\lim_{x\to 0^-}\arctan(1/x)-\arctan(-1)$$</p>
<p>That is: the above is an improper integral.</p>
|
3,519,681 | <p>so I want to prove both sides of this identity</p>
<p><span class="math-container">$$
\sin x \cos y + \cos x \sin y - \sin x = \cos x \sin y - 2\sin x \sin^2\left(\frac{1}{2} y\right)
$$</span></p>
<p>I've already proved it by manipulating the right side. However, when I try to prove it by manipulating the left, ... | an4s | 533,556 | <p><strong>Hint:</strong></p>
<p>From <a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Half-angle_formulae" rel="nofollow noreferrer">here</a>,</p>
<p><span class="math-container">$$\cos \theta = 1 - 2\sin^2\left(\frac\theta2\right)$$</span></p>
<p>Also, you should change your working to</p>
... |
3,519,681 | <p>so I want to prove both sides of this identity</p>
<p><span class="math-container">$$
\sin x \cos y + \cos x \sin y - \sin x = \cos x \sin y - 2\sin x \sin^2\left(\frac{1}{2} y\right)
$$</span></p>
<p>I've already proved it by manipulating the right side. However, when I try to prove it by manipulating the left, ... | Mohammad Riazi-Kermani | 514,496 | <p><span class="math-container">$$\sin x \cos y + \cos x \sin y - \sin x = \cos x \sin y +\sin x (\cos y -1)$$</span></p>
<p>In order to get the above expression identical to the <span class="math-container">$RHS$</span> you need to have <span class="math-container">$$(\cos y -1) = -2\sin ^2 (y/2)$$</span></p>
<p>No... |
1,864,161 | <p>The main question is :</p>
<p>$z^x=x$, $z^y=y$, $y^y=x$</p>
<p>Find $z$, $y$, $x$.</p>
<p>My method : </p>
<p>I first attempted to get two equation for the unknowns $x$ and $y$.</p>
<p>We can happily write :</p>
<p>$z=x^{1/x}$ and $z=y^{1/y}$</p>
<p>Thus we get,
$x^{1/x}=y^{1/y}$
Which is,
$x^y=y^x$.</p>
<p>... | Simply Beautiful Art | 272,831 | <p>And to be different, the third solution is $x=y=z=-1$, by old-fashioned trial and error.</p>
|
1,225,277 | <p>I tried to prove that any subfield of $\mathbb{C}$ must contain every rational number by contradiction. </p>
<p>Proof:</p>
<p>Let $\mathbb{F}$ be any subfield of $\mathbb{C}$. Thus, $\mathbb{F}$ is itself a field under the usual operations of addition and multiplication of complex numbers. For $\mathb... | Dan | 229,636 | <p>As other comments has pointed out, $\mathbb{Q}$ is the smallest subfield containing $\mathbb{Z}$. This is a special case of the more general fact that the fraction field $Frac(R)$ is the smallest field containing the ring $R$. To get $Frac(R)$, we take the following set (very intuitive): $\left \{\frac{m}{n} \mid m,... |
696,996 | <p>I have the equation $y = \sqrt{3x^2 + 1}$, and I need to prove that there will be infinitely many integer solutions. I saw possible solutions with things like Pell's equation, but I did not fully understand these methods. It would be awesome if someone could help me out with simpler methods.</p>
<p>Thanks,</p>
<p>... | r9m | 129,017 | <p>If, $y^2-3x^2=1$ write it as $(y-\sqrt3x)(y+\sqrt3x)=1$ and cube both sides to obtain</p>
<p>$(y^3-3\sqrt3x^3-3\sqrt3xy(y-\sqrt3x))(y^3+3\sqrt3x^3+3\sqrt3xy(y+\sqrt3x))=1$</p>
<p>multiply and simplify to get $(y^3+3x^2y)^2-3(3y^2x+3x^3)^2=1$</p>
<p>So, once you find a solution $(x_0,y_0)$ you can iteratively gene... |
696,996 | <p>I have the equation $y = \sqrt{3x^2 + 1}$, and I need to prove that there will be infinitely many integer solutions. I saw possible solutions with things like Pell's equation, but I did not fully understand these methods. It would be awesome if someone could help me out with simpler methods.</p>
<p>Thanks,</p>
<p>... | Robert Israel | 8,508 | <p>$y^2 - 3 x^2 = 1$ is a Pell's equation, so you can't ask for something "simpler" than that. </p>
<p>The basic idea is this. Consider a Pell's equation $y^2 - d x^2 = 1$ (where $d$ is a positive integer that is squarefree (i.e. not divisible by the square
of an integer $> 1$).
First find the smallest solution in... |
1,133,066 | <p>Is it possible to create uniformly distributed real pseudo random numbers
<br>
$x_1,x_2$, and $y_1,y_2,y_3\in$ $[0,1]$, subject to the following constraints:</p>
<p>$$x_1^2+x_2^2=1$$</p>
<p>$$y_1^2+y_2^2+y_3^2=1$$</p>
<p>I tried to use sines and cosines but that does not work;
the conventional approach in creatin... | drhab | 75,923 | <p>If $x_1^2+x_2^2=1$ and $x_1$ is uniformly distributed over $[0,1]$ then the distribution of $x_2^2=1-x_1^2$ is <em>determined</em>. It is <em>not</em> the same distribution of $z^2$ where $z$ is uniformly distributed over $[0,1]$. So the answer to your "Is it possible..." is: no if it comes to two random numbers. I... |
1,133,066 | <p>Is it possible to create uniformly distributed real pseudo random numbers
<br>
$x_1,x_2$, and $y_1,y_2,y_3\in$ $[0,1]$, subject to the following constraints:</p>
<p>$$x_1^2+x_2^2=1$$</p>
<p>$$y_1^2+y_2^2+y_3^2=1$$</p>
<p>I tried to use sines and cosines but that does not work;
the conventional approach in creatin... | Community | -1 | <p>In 2D, using polar coordinates $x_1=\cos 2\pi u,x_2=\sin 2\pi u$, where $u$ is uniform in $[0,1]$.</p>
<p>In 3D, using spherical coordinates $y_1=2\sqrt{v-v^2}\cos 2\pi u, y_2=2\sqrt{v-v^2}\sin 2\pi u, y_3=2v-1$, where $v$ s uniform in $[0,1]$.</p>
|
1,657,176 | <p>"Let $\ f: A\to B$ and $\ g:B\to C$ be functions. Prove that if $g\circ f$ is surjective and $g$ is injective, then $f$ is surjective."</p>
| DeepSea | 101,504 | <p>Let $b \in B$, and consider $g(b)$. We have $g(b) \in C$, so call it $c = g(b)$, and since $(g\circ f)$ is onto, there exists an $a \in A$ such that $(g\circ f)(a) = c$, and rewrite this as $g(f(a)) = c = g(b)$, and since $g$ is one-to-one, $b = f(a)$, so $f$ is onto.</p>
|
267,507 | <p>Figure shows a rectangle $ABCD$ and an isosceles triangle $\triangle DEC$.</p>
<p>$AD=BC=z$;$AB=DC=y$;$DE=CE=x$</p>
<p>One solution is as follows.</p>
<p>We know that the pythagoras theorem holds for a right triangle $\triangle EBC$.</p>
<p>In the right triangle, $\triangle EBC$.</p>
<p>$z^2+(\frac{y}{2})^{2}=x... | davidlowryduda | 9,754 | <p>Step 1: Assume the Pythagorean theorem for right triangles.</p>
<p>Step 2: Realize that an isosceles right triangle is a right triangle.</p>
<p>Step 3: Conclude that the Pythagorean theorem for isosceles right triangles is a special case (and therefore free from) the Pythagorean theorem for right triangles.</p>
<... |
886 | <p>Okay so <a href="https://math.stackexchange.com/questions/835/optimal-strategy-for-deal-or-no-deal">this question</a> reminded me of one my brother asked me a while back about the hit day-time novelty-worn-off-now snoozathon <a href="http://en.wikipedia.org/wiki/Deal_or_no_deal" rel="noreferrer">Deal or no deal</a>.... | Andre | 73,907 | <p>very interesting thread, Im sorry I found it so late.</p>
<p>In my opinion swapping boxes matters, but the difference is really small one. I am trying to solve the problem this way: when in play only 2 boxes left, 1 million $ not revealed,why it is still unopened?</p>
<p>on 1/22 propability its unopened, because I... |
4,088,439 | <p>First of all, the following question with it's answers are the main reason I put this question, because as I will show the question wording is bad.
<a href="https://math.stackexchange.com/questions/1896238/how-to-calculate-curvature-of-earth-per-surface-kilometer">How to calculate curvature of Earth per surface kilo... | Ripi2 | 688,039 | <p>I will use the same image as in your link.</p>
<p>Horizontal distance (straight): <span class="math-container">$H$</span>
<br>Arc length (round distance): <span class="math-container">$s$</span>
<br>Earth radius; <span class="math-container">$R=6371$</span> km
<br>"Drop" (vertical): <span class="math-conta... |
4,088,439 | <p>First of all, the following question with it's answers are the main reason I put this question, because as I will show the question wording is bad.
<a href="https://math.stackexchange.com/questions/1896238/how-to-calculate-curvature-of-earth-per-surface-kilometer">How to calculate curvature of Earth per surface kilo... | Dan | 910,203 | <p>Using lonza's answer I also created a LibreOffice Calc (excel maybe also works) formula (because I also formulated the need for an excel formula at the end of my question):</p>
<pre><code>=6371-6371*COS(s/6371)
</code></pre>
<p>where you can change "s" for a cell that will contain the distance.</p>
|
1,634,265 | <p>I have some trouble in how to evaluate this integral:
$$
\int_{0}^{\pi}\theta\ln\left(\tan\left(\theta \over 2\right)\right)
\,\mathrm{d}\theta
$$
I think it maybe has another form
$$
\int_{0}^{\pi}\theta\ln\left(\tan\left(\theta \over 2\right)\right)
\,\mathrm{d}\theta
=
\sum_{n=1}^{\infty}{1 \over n^{2}}
\left[\ps... | Renascence_5. | 286,825 | <p>Obviously we have
$$\int_{0}^{\pi }\theta \ln\tan\frac{\theta }{2}\mathrm{d}\theta =4\int_{0}^{\pi /2}x\ln \tan x\mathrm{d}x$$
then use the definition of <strong>Lobachevskiy Function</strong>(You can see this in <em>table of integrals,series,and products,Eighth Edition</em> by Ryzhik,page 900)
$$\mathrm{L}\left ( x... |
1,634,265 | <p>I have some trouble in how to evaluate this integral:
$$
\int_{0}^{\pi}\theta\ln\left(\tan\left(\theta \over 2\right)\right)
\,\mathrm{d}\theta
$$
I think it maybe has another form
$$
\int_{0}^{\pi}\theta\ln\left(\tan\left(\theta \over 2\right)\right)
\,\mathrm{d}\theta
=
\sum_{n=1}^{\infty}{1 \over n^{2}}
\left[\ps... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
989,043 | <p>I'm trying to prove that, for a finite set $A$, if the map $f: A\rightarrow A$ is a surjection, then it's an injection. I've looked at this post: <a href="https://math.stackexchange.com/questions/63072/surjectivity-implies-injectivity">Surjectivity implies injectivity</a> but the arguments there seem very round-abo... | Hagen von Eitzen | 39,174 | <p>Let $f\colon A\to A$ be a surjective map. Let
$$ \mathcal A=\{\,X\subseteq A\mid f|_X\colon X\to A\text{ is injective}\,\}.$$
Then $\mathcal A$ is not empty because $f|_\emptyset$ is injective.
Let $B\in\mathcal A$ be maximal (how?). Then $f|_B$ is onto as otherwise we'd find $b\in A$ with $f(b)\notin f(B)$ and the... |
2,864,538 | <p>Show that $$\sum_{n=1}^\infty 2^{2n}\sin^4\frac a{2^n}=a^2-\sin^2a$$</p>
<p>I am studying for an exam and I bumped into this question. It's really bothering me because I really don't have any clue what to do. Does it have anything to do with the Cauchy condensation? Can somebody help me I would really appreciate it... | J.Dane | 544,749 | <p>Using the hint that @Clayton gave this is what you get if you take the partial sum </p>
<p>$$2^2\sin^2\frac a{2}-\sin^2a+2^4\sin^2\frac a{2^2}-2^2\sin^2\frac a{2}+ \cdots+ 2^{2n}\sin^2\frac a{2^n}-2^{2n-2}\sin^2\frac a{2^{n-1}}=
2^{2n}\sin^2\frac a{2^n}-\sin^2a$$</p>
<p>when you take the limit $$\lim_... |
2,293,731 | <blockquote>
<p>Given $x$ and $y$ are positive real numbers less than 9, what is the probability that the sum of them is greater than 3</p>
</blockquote>
<p>I thought that these are integers and did $\frac{_6C_2}{_8C_2}$ but then I realized that they are all the positive real numbers between $0$ and $9$, not just wh... | AlgorithmsX | 355,874 | <p>Your best bet is a geometric proof. Find the area of the region $0\leq x \leq9$ and $0\leq x \leq9$, which is $81$. Then find the area within that region such that $x+y\geq3$.</p>
<p>This is the region you're looking for.</p>
<p><a href="https://i.stack.imgur.com/6M4xY.gif" rel="nofollow noreferrer"><img src="http... |
2,548,680 | <p>How would one go around proving the convergence of: $ \sum \frac{(-1)^n}{n+(-1)^n} $. I'm fairly certain, that this series converges, but it doesn't do so absolutely $\left( \frac{1}{n+(-1)^n} \approx \frac{1}{n}\right)$. Leibniz criterion of convergence can't be used, because $\left( \frac{1}{n+(-1)^n} \right) $ i... | Sangchul Lee | 9,340 | <p>Notice that</p>
<p>$$ \frac{(-1)^n}{n+(-1)^n}
= \frac{(-1)^n}{n} - \frac{1}{n(n+(-1)^n)}.$$</p>
<p>Now</p>
<ul>
<li>$\sum \frac{(-1)^n}{n}$ converges by alternating series test, and </li>
<li>$ \sum \frac{1}{n(n+(-1)^n)}$ converges by comparison with $\sum \frac{1}{n^2} < \infty$.</li>
</ul>
<p>Therefore thei... |
967,084 | <p>I am asked to find all equilibrium solutions to this system of differential equations:</p>
<p>$$\begin{cases} x ' = x^2 + y^2 - 1 \\ y'= x^2 - y^2 \end{cases} $$</p>
<p>and to determine if they are stable, asymptotically stable or unstable.</p>
<p>I do not know how to proceed, I tried switching to polar coordinat... | Amzoti | 38,839 | <p>We are given:</p>
<p>$$\begin{cases} x ' = x^2 + y^2 - 1 \\ y'= x^2 - y^2 \end{cases} $$</p>
<p>As @RobertLewis has pointed out, we find the equilibrium points $(x,y)$ at the points where we have $x' = y' = 0$. We have</p>
<p>$x^2 + y^2 -1 =0\, \tag{3}$</p>
<p>$x^2 - y^2 = 0. \tag{4}$</p>
<p>From $(4)$, we have... |
186,453 | <p>I want to get <code>x^2 + y^2</code> by evaluating <code>Abs[a + b I]^2</code>. Unfortunately, Mathematica just reproduces the same <code>Abs[a + b I]^2</code>. </p>
| bill s | 1,783 | <p>Another way is to be explicit about the domain of the variables, that is, that x and y are real valued:</p>
<pre><code>FullSimplify[Abs[(x + y I)]^2, Assumptions -> {x, y} \[Element] Reals]
x^2 + y^2
</code></pre>
|
1,387,180 | <p>Given a selfadjoint (maybe unbounded) operator $T$ on a Hilbert space $H$, I want to calculate the adjoint of $\lambda I - T$ for a $\lambda \in \Bbb C$.</p>
<hr>
<p>I am tempted to argue as follows:</p>
<p>According to Lemma 4.26 of <a href="https://people.math.ethz.ch/~kowalski/spectral-theory.pdf" rel="nofollo... | Community | -1 | <p>Let $K$ be an algebraically closed field of characteristic $0$, $L$ be the Lie algebra $M_n(K),[.,.]$, $ad(L)=\{ad_x:y\rightarrow xy-yx=[x,y]|x\in L\}$, $der(L)$ be the set of derivations over $L$. </p>
<p>Prop 0. $L$ has no proper ideal. (well-known)</p>
<p>Prop 1. $ad(L)$ is an ideal of $der(L)$. </p>
<p>Proof.... |
1,882,325 | <p>I am having trouble in understanding and finding "the three-piece Bezier curve forming a triangle with vertices (1,2), (3,4) and (5,1)".</p>
<p>My understanding so far is that I need 4 points to form a cubic Bezier curve, but the textbooks excersice solution insists it is a third-degree (cubic) solution despite onl... | tired | 101,233 | <p>By the Euler Mac-Laurin formula,</p>
<p>$$
S(k)=\lim_{N\rightarrow \infty}\sum_{n=1}^{N} e^{-n^2/k}\sim\lim_{N\rightarrow \infty}\int_1^{N} e^{-n^2/k}dn
$$</p>
<p>it is now not not too difficult to show that the integral asymtotically (large $k$) equals</p>
<p>$$
\frac{\sqrt{\pi k}}{2}-1+\mathcal{O}(k^{-1}) \qqua... |
1,882,325 | <p>I am having trouble in understanding and finding "the three-piece Bezier curve forming a triangle with vertices (1,2), (3,4) and (5,1)".</p>
<p>My understanding so far is that I need 4 points to form a cubic Bezier curve, but the textbooks excersice solution insists it is a third-degree (cubic) solution despite onl... | Refaat Mohamed | 233,155 | <p>Here is a solution using Riemann sum
your limit $$\lim \limits_{k\to \infty }\frac{2}{\sqrt{k} } \sum \limits^{k}_{n=1}e^{\frac{-n^{2}}{k} }$$
Now let $k=u^2$ we get the limit$$\lim \limits_{u\to \infty }\frac{2}{u} \sum \limits^{u^{2}}_{n=1}e^{-\left( \frac{n}{u} \right) ^{2}}=2\int \limits^{\infty }_{0}e^{-y^{2}}... |
2,987,055 | <blockquote>
<p>Let <span class="math-container">$\ V $</span> be a linear space over <span class="math-container">$\ F $</span>, <span class="math-container">$\ \dim V = n $</span> and <span class="math-container">$\ T_1,T_2 : V \rightarrow F $</span> are linear transofrmations. <span class="math-container">$\ T_1 ,... | Jack D'Aurizio | 44,121 | <p>As an alternative approach, the fact that <span class="math-container">$\{(x,y)\in\mathbb{R}^2:x^2-xy+y^2-3x=0\}$</span> is an ellipse follows from the fact that the discriminant of <span class="math-container">$x^2-xy+y^2$</span> is negative. Such ellipse goes through <span class="math-container">$(0;0),(3;0),(3;3)... |
493,928 | <p>I'm having trouble proving this combinatorially. I can't work the $2^n$ into my understanding. </p>
<p>$$\sum_{t=0}^n {2n \choose t + n} {t + n \choose n} = 2^n{2n \choose n}$$
I'm thinking of it as taking $2n$ objects, splitting them into groups of size $t + n$ where $t$ are of one type and $n$ of another, and the... | Hagen von Eitzen | 39,174 | <p>The left hand side counts the ways to pick $n$ or more out of $2n$ objects and then pick $n$ out of these. You might colour those final $n$ ones red, those $0\le t\le n$ ones picked in the first but not the second step blue.</p>
<p>The right hand side counts the ways to pick $n$ out of $2n$ objects to be painted re... |
3,768,027 | <p>We know that a closed subset of a complete metric space is complete.
But I want to find a closed subset <span class="math-container">$A$</span> of an incomplete metric space <span class="math-container">$(X,d)$</span> such that <span class="math-container">$A$</span> is not complete.</p>
| ir7 | 26,651 | <p><em>Hint:</em></p>
<p>For finite measures, use <a href="https://en.wikipedia.org/wiki/Convergence_in_measure" rel="nofollow noreferrer">this wiki fact</a>: A sequence <span class="math-container">$X_n$</span> converges to <span class="math-container">$X$</span> in measure if and only if for any subsequence <span cla... |
3,768,027 | <p>We know that a closed subset of a complete metric space is complete.
But I want to find a closed subset <span class="math-container">$A$</span> of an incomplete metric space <span class="math-container">$(X,d)$</span> such that <span class="math-container">$A$</span> is not complete.</p>
| Oliver Díaz | 121,671 | <p>This statement can be obtained as a consequence of the following Lemma:</p>
<blockquote>
<p><strong>Lemma</strong>: If <span class="math-container">$\nu\ll\mu$</span> and <span class="math-container">$\nu$</span> is finite (as is in your case) then for any <span class="math-container">$\varepsilon>0$</span>, ther... |
1,650,371 | <p>What is the formula to make a semicircle graph that is continuous? By continuous I mean like a sine or cos graph but shaped like semicircles one after the other. Thanks</p>
| MPW | 113,214 | <p>I believe the function $f:\mathbb R\to\mathbb R$ with</p>
<p>$$f(x)=(-1)^{\lfloor x/2\rfloor} \sqrt{1-(x-2\lfloor x/2\rfloor -1)^2}$$</p>
<p>will do the trick. Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.</p>
<p><strong>Explanation:</strong></p>
<p><strong>(1) The semicircle:<... |
2,496,950 | <p>This is probably a silly question for anyone "in the know", so please forgive me.</p>
<p>Consider the "group" given $G = \langle x,y : x^3=y^3=1,xy=yx^2\rangle$.</p>
<p>I got the elements to be $G = \{1,x,x^2,y,yx,yx^2,y^2,y^2x,y^2x^2\}$, meaning that $|G| = 9$.</p>
<p>I calculated the multiplication table (see b... | R.C.Cowsik | 293,582 | <p>Any group of order $P$ or $p^2$ is Abelian. So, $xy = yx^2$ implies, post computation that order is 3 or 9, that $x = 1$</p>
|
800,637 | <p>Buenos Dias, Ciao, Hello!</p>
<p>My fellow math stack users, I will try to solve this integral
$$
\int_0^1 x(1-x)(2-x) e^{-(1-x)^2}\ln(1-x)\,dx
$$
I did this $u=1-x$
$$
-\int_0^1 (u-1)u(u+1)e^{-u^2}\ln u \, du=\\
-\int_0^1 (u^2-u)(u+1)e^{-u^2}\ln u \, du=\\
-\int_0^1 (u^3-u)e^{-u^2}\ln u \, du=\\
-\int_0^1 u^3e^{-u... | Start wearing purple | 73,025 | <p>You can integrate by parts to kill the logarithms. Namely, since
\begin{align}
\frac{d}{du}\left(\frac{u^2e^{-u^2}}{2}\right)=(u-u^3)e^{-u^2},
\end{align}
the integral reduces to
$$-\int_0^1\frac{u^2e^{-u^2}}{2}d\ln u=-\frac12\int_0^1 ue^{-u^2}du=\frac14 e^{-u^2}\biggl|_0^1=\frac{e^{-1}-1}4 .$$</p>
|
173,092 | <p>I'm trying to build a Poincaré map of the following differential equation, </p>
<pre><code>r'[ϕ]==(Sqrt[e (1 + ((-1 + c) M)/(2 - (3 M)/r[ϕ])) - (1 + ((-1 + c) M)/(
2 - (3 M)/r[ϕ])) (k + l^2/r[ϕ]^2) (1 - (2 M)/
r[ϕ])] r[ϕ]^2)/l
</code></pre>
<p>I set the Mathematica's example to my problem, and my code is the fol... | Akku14 | 34,287 | <p>It seems, MMA can't do Integration of squared interpolation functions.</p>
<p>Generate a new interpolationfunction of the squared function values.</p>
<pre><code>Plot[(v[x])^2, {x, -5, 5}, PlotRange -> All]
</code></pre>
<p><a href="https://i.stack.imgur.com/jjkuV.jpg" rel="nofollow noreferrer"><img src="https... |
468,650 | <p>In other words, if $F$ is a proper, closed subset of $[0,1]$, must $\mu(F) < 1$? What about in $[0,1]^n$? </p>
<p>My (frequently misguided) intuition wants to say "no", but I'm not sure how to approach a proof. Suggestions/hints/counterexamples welcome.</p>
| Ben Grossmann | 81,360 | <p><strong>Hint:</strong> any proper closed subset of $[0,1]$ can be obtained by subtracting a non-empty open set.</p>
|
1,091,873 | <p>If I have a function $ u(x,t) = p(x+ct) + q(x-ct) $ (which is the d'Alembert solution to the $1D$-wave equation), I can make the substitutions </p>
<p>$$ \xi(x,t) = x + ct\\
\eta(x,t) = x - ct $$</p>
<p>So I am now left with </p>
<p>$$ u(x,t) = p(\xi) + q(\eta) $$</p>
<p>I could then take that a step further and... | Andreas Blass | 48,510 | <p>Once you've defined $u$ by the equation $u(x,t)=p(x+ct)+q(x-ct)$, you have this same equation for any other quantities that you might substitute for $x$ and $t$. In particular, $u(\xi,\eta)=p(\xi+x\eta)+q(\xi-c\eta)$. So you're quite right that you cannot claim $p(\xi)+q(\eta)=u(\xi,\eta)$. (If you made that claim... |
4,570,552 | <p>This question was asked in my quiz of Ring theory and I was not able to prove it.</p>
<blockquote>
<p>Question: Show that if <span class="math-container">$I$</span> is not prime, then there exists ideal <span class="math-container">$I_1$</span> and <span class="math-container">$I_2$</span> such that <span class="mat... | kathy4k | 333,911 | <p>Suppose we can find <span class="math-container">$I_1,I_2$</span> properly containing <span class="math-container">$I$</span> with <span class="math-container">$I_1I_2\subset I$</span>. So <span class="math-container">$I$</span> cannot be the whole ring. Let's prove that <span class="math-container">$I$</span> canno... |
248,385 | <p>For example, dividing by 2, means we cut something in two.</p>
<p>But dividing by 0.5, can only be explained with multiplying something by 2.</p>
<p>So, is there a "physical" explanation of dividing by 0.5? Is it "I divide by an entity that internally multiplies' or something as so bizarre?</p>
| Pedro | 23,350 | <p>I once heard: "If you feed a kid $Y$ grams of chocolate he would have a density of $X$ $\rm gr/cm^3$ of chocolate in his blood. But if you feed <strong>half</strong> a boy chocolate he will have $2X$ $\rm gr/cm^3$ of chocolate in his blood". </p>
|
248,385 | <p>For example, dividing by 2, means we cut something in two.</p>
<p>But dividing by 0.5, can only be explained with multiplying something by 2.</p>
<p>So, is there a "physical" explanation of dividing by 0.5? Is it "I divide by an entity that internally multiplies' or something as so bizarre?</p>
| user21820 | 21,820 | <p>[This answer is essentially identical to <a href="https://matheducators.stackexchange.com/a/7868">https://matheducators.stackexchange.com/a/7868</a>, which I think deserves to be on Math SE as well.]</p>
<p><strong>What do beginners know?</strong></p>
<p>Multiplication and division of fractions is, in the United S... |
3,049,627 | <p>What is command to obtain Brauer Character in GAP? In magma, it is like </p>
<pre><code>> G := Sym(5);
> I := AbsolutelyIrreducibleModules(G, GF(3));
> [BrauerCharacter(i): i in I];
[
( 1, 1, 1, 0, 1, 1, 0 ),
( 1, -1, 1, 0, -1, 1, 0 ),
( 4, 2, 0, 0, 0, -1, 0 ),
( 4, -2, 0, 0, 0, -1, 0 ),
... | jumpjack | 139,449 | <h2>Short answer</h2>
<p><span class="math-container">$$
t_{65mph} = \frac{m}{\delta}\ln\left(\frac{(27.8-v_1)^{\alpha} * (27.8-v_2)^{\beta} * (27.8-v_3)^{\gamma}}{(-v_1)^\alpha * (-v_2)^\beta * (-v_3)^\gamma }\right)
$$</span></p>
<ul>
<li><span class="math-container">$\alpha = v_1(v_2-v_3)$</span></li>
<li><span c... |
1,261,572 | <p>If $\alpha$ and $\beta$ are the zeros of the polynomial $p(s)=3s^2-6s+4$, find the value of $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+2(\frac{1}{\alpha}+\frac{1}{\beta})+3\alpha\beta$</p>
<p>By calculating, I am finding the answer as $9$, but, the book has the answer as $8$. I did a cross check and my answer seems... | Claude Leibovici | 82,404 | <p>Rewrite $$\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+2(\frac 1{\alpha}+\frac 1{\beta})+3{\alpha}{\beta}=\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+2\frac{\alpha+\beta}{\alpha\beta}+3{\alpha\beta}=\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+2\frac SP+3P$$ where $S$ and $P$ are the sum and the product of the roots.</p>
<... |
2,467,246 | <p>In a quiz there are $12$ questions and $100$ participants. For every question, at least $56$ participants knows the correct answer. How can I show that there are three participants who knows answers to all of the questions?</p>
| Martigan | 146,393 | <p>I think something is missing from your proposition. </p>
<p>EDIT: I think what you meant in fact is that, there exists a team of three people that have the answer to every question, AS A TEAM, and not three people knowing the answer of all question...</p>
<p>let'st ake an example fitting your statement (at least 5... |
1,115,006 | <p>I already tried to putting the square root like this: $\sqrt{\frac{x}{5 + x}}$ but I dont know what to do next.</p>
<p>$$\int \frac{\sqrt{x}}{\sqrt{5+x}}dx$$</p>
| Timbuc | 118,527 | <p>$$\frac x{5+x}=u^2\implies x=\frac{5u^2}{1-u^2}\implies dx=\frac{10u\,du}{(1-u^2)^2}\implies$$</p>
<p>$$\int\sqrt\frac x{5+x}\,dx=\int\frac{10u^2}{(1-u^2)^2}du$$</p>
<p>and now you have the integral of a rational function. Do, for example, partial fractions, or whatever.</p>
|
18,753 | <p>Consider a compact manifold <em>M</em>. For a vector field <em>X</em> on <em>M</em>, let $\phi_X$ denote the diffeomorphism of <em>M</em> given by the time 1 flow of <em>X</em>.</p>
<p>If <em>X</em> and <em>Y</em> are two vector fields, is $\phi_X \circ \phi_Y$ necessarily of the form $\phi_Z$ for some vector field... | Charlie Frohman | 4,304 | <p>The answer to the first question, is $X\circ Y$ a vector field if $X$ and $Y$ are, is in general no. The reason is that the operator does not satisfy the Leibniz rule. One of the cool things about the Lie bracket of vector fields is that the problematic terms vanish and it is a vector field. </p>
<p>The question a... |
18,753 | <p>Consider a compact manifold <em>M</em>. For a vector field <em>X</em> on <em>M</em>, let $\phi_X$ denote the diffeomorphism of <em>M</em> given by the time 1 flow of <em>X</em>.</p>
<p>If <em>X</em> and <em>Y</em> are two vector fields, is $\phi_X \circ \phi_Y$ necessarily of the form $\phi_Z$ for some vector field... | valeri | 1,988 | <p>Does it make any sense - if we consider vector fields as derivations of the algebra of smooth functions on M? Then each \phi_X is an automorphism of this algebra and there is an asymptotic formula for its (i.e., \phi_X) action. Product of two such asymptotic representations give as. formula for superposition \phi_X ... |
693,761 | <p>Prove that the maximum connected components in a graph is $V+1-\left\lceil\frac{1+\sqrt{1+8E}}{2}\right\rceil$</p>
<p>I came up with this intuitively but was not able to prove it. The idea is that a graph with the maximum number of connected components would have individual nodes and 1 clique connected with at most... | Flowers | 93,842 | <p>Assume there is a graph with the maximum number of connected components that has multiple components with more than one node. This graph can be reduced to having only one component with more than one node, without changing the number of edges or connected components (by taking every node with degree>0 and connecting... |
1,156,658 | <p>In Discrete math I remember learning that "a function is a relation that is both 1 to 1 and onto."</p>
<p>Every time I try to look this up I can't find this definition of "function", all I can find is that "a function that is 1 to 1 and onto is a bijective function".</p>
<p>Have I misremembered this, or does it va... | hmakholm left over Monica | 14,366 | <p>The usual (set-theoretic) definition is that a <strong>function</strong> $A\to B$ is a subset of $A\times B$ such that for every $a\in A$ there is exactly one $b\in B$ such that $(a,b)$ is in the function.</p>
<p>In contrast, a <strong>relation</strong> between $A$ and $B$ is just an arbitrary subset of $A\times B$... |
3,197,274 | <p>Given the bottom left coordinates of a square and the side length of the square, how do you find out the top right coordinates of the square?</p>
<p>Example: <span class="math-container">$(-10,-2)$</span> is the bottom left coordinates of the square. And the side length of the square is <span class="math-container">... | Satyajit Ghosh | 291,039 | <p>I am assuming your are asking for some kind of general formula. So, here we go, let's assume that <span class="math-container">$(x_a,y_a)$</span> is the below left corner of your square and length of side is L, then the formula for co-ordinate of above right corner will be,</p>
<p><span class="math-container">$=({x... |
763,665 | <p>Let $\{f_{n}\}$ be continuous functions on $[0,1]$ which converge to a measurable function $f$ in the following sense: $$\lim_{n \to \infty} \int_{0}^{1} |f_{n}(x) - f(x)|^{2} dx \to 0$$</p>
<p>Show $f$ is continuous.</p>
<p>I'm still new at this, but my understanding is that the hypothesis is that the sequence $\... | user2566092 | 87,313 | <p>Just take $f_n$ to be bumps around $x=1/2$ that are basically $0$ everywhere and then go up to $1$ in a small interval around $1/2$, where the widths of the intervals goes to $0$ as $n \to \infty$. Then your sequence of functions will converge in norm to the function which is $0$ everywhere except $1/2$ and $1$ at $... |
506,556 | <p><strong>Problem:</strong> </p>
<blockquote>
<p>Find the number of points which lies inside the triangle : $(0,0) , (0,21); (20,0) $</p>
</blockquote>
<p>Approach : </p>
<p>Let us take point $A = (0,0)$, $B = (0,21)$, and $C = (20,0)$.</p>
<p>Since the figure formed by joining these points $ABC$ is a triangle ... | hjpotter92 | 27,741 | <h2>Hint</h2>
<p>Take the equations:
$$
\begin{align}
0 \le x & \le 20 \\
0 \le y &\le 21 \\
\dfrac{x}{20} + \dfrac{y}{21} &\le 1 \\
x, \ y \in \left\{0,\ 1,\ 2,\ 3,\ \cdots \right\}
\end{align}
$$</p>
|
1,874,363 | <p>For example, how would you respond to this question?</p>
<blockquote>
<p>The earnings of one-hundred workers in a company are normally distributed. If the mean of this data set is 24 and the standard deviation is 4, find an approximate value for the range.</p>
</blockquote>
| Ant | 66,711 | <p>This is pretty straightforward using probabilistic methods: What you're looking for is </p>
<p>$$E[\max_{1 \le i \le n} X_i] = \int_{[0,1]^n} \max x_i \ \ dx_1\dots dx_n$$</p>
<p>where $X_i$ are iid uniformly distribuited on $[0,1]$.</p>
<p>Now call $Y = \max_i X_i$; its distirbution function is given by </p>
<p... |
137,306 | <p>Suppose I have a list that looks like the following (Town names and total distance to that town)</p>
<pre><code>resupply = {
{"Coleman", 0},
{"Highwood House", 106},
{"Canmore", 229},
{"Exshaw", 245},
{"Ghost Station", 288},
{"MountainAire", 370},
{"Nordegg", 552},
{"Robb", 672},
{"Hinton", 720}
... | Mr.Wizard | 121 | <p>The Terse Way</p>
<pre><code>{-#, #2 "km"} & @@@ Differences[resupply] // TraditionalForm
</code></pre>
<p>$\left(
\begin{array}{cc}
\text{Coleman}-\text{Highwood House} & 106 \text{ km} \\
\text{Highwood House}-\text{Canmore} & 123 \text{ km} \\
\text{Canmore}-\text{Exshaw} & 16 \text{ km} \\
... |
369 | <p>This question has been kicking around in the back of my head for a couple of years, but the impetus to post it now came from reading the related question at <a href="https://matheducators.stackexchange.com/questions/244/when-did-the-american-school-systems-progression-of-math-classes-take-its-curre">When did the Ame... | Richard | 3,493 | <p>As SMSG books are public domain, I have found some more PDFs. Thanks mweiss for checking it is the right document.
<a href="http://babel.hathitrust.org/cgi/pt?id=mdp.39015017374235;view=1up;seq=7;skin=mobile" rel="noreferrer">http://babel.hathitrust.org/cgi/pt?id=mdp.39015017374235;view=1up;seq=7;skin=mobile</a></p... |
153,578 | <p>I currently have $$\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$$</p>
<p>Now, I know that when you have square root instead of a cubic root it's easy. You just multiply by the conjugate and simplify afterwards. If it were a sqrt I know that the limit is 1/2. I know that the result here is 1/3 but I can't seem to get ther... | Community | -1 | <p>I think you have multiplied incorrectly. I suppose you want to find $$\lim_{h \to 0 } \frac{(h+1)^{1/3} -1}{h}$$</p>
<p>Note that this is nothing but the derivative of the function $f(x) = (x+1)^{1/3}$ at $x =0$. So your $f'(x) = \frac{1}{3} \cdot (x+1)^{-2/3}$. Now it's easy to see that $f'(0)=\frac{1}{3}$ and so ... |
153,578 | <p>I currently have $$\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$$</p>
<p>Now, I know that when you have square root instead of a cubic root it's easy. You just multiply by the conjugate and simplify afterwards. If it were a sqrt I know that the limit is 1/2. I know that the result here is 1/3 but I can't seem to get ther... | anon | 11,763 | <p>The problem is that $(\sqrt[3]{h+1}-1)(\sqrt[3]{h+1}+1)^2$ is <strong>not</strong> the factorization of $(h+1)-1$, which is the end result you desire. Rather, for cubes, $u^3-1=(u-1)(u^2+u+1)$ is the correct "level higher" version of $u^2-1=(u-1)(u+1)$. Indeed, in general we have $u^n-1=(u-1)(u^{n-1}+\cdots+1)$, whi... |
3,205,242 | <p>Find the number of real roots of the equation <span class="math-container">$54x^4-36x^3+18x^2-6x+1=0$</span></p>
<p>I entered the equation in desmos.com and no roots were lying below or x=0 lines , hence all roots are imaginary.</p>
<p>Using Descartes rule for f(x) 4 sign change occurs , hence 4,2 or zero positive... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Prove that <span class="math-container">$$54x^4-36x^3+18x^2-6x+1>0$$</span> for all real <span class="math-container">$$x$$</span></p>
|
2,929,878 | <blockquote>
<p>Solve the Cauchy problem <span class="math-container">$$(x − y − 1)u_x + (y − x − u + 1)u_y = u,$$</span>
if <span class="math-container">$u=1$</span> on <span class="math-container">$x^2+(y+1)^2=1.$</span></p>
</blockquote>
<p><em>Attempt.</em> <span class="math-container">$$\frac{dx}{x-y-1}=\frac... | user577215664 | 475,762 | <p>For the second constant of integration
<span class="math-container">$$\frac{dx}{x-y-1}=\frac{dy}{y-x-z+1}=\frac{dz}{z}$$</span>
<span class="math-container">$$\frac{d(x+y)}{x+y-k}=\frac{dy}{2y-k+1}$$</span>
<span class="math-container">$$z^2=C({2y + {1-k}})$$</span>
Where <span class="math-container">$k=x+y+z$</span... |
2,383,260 | <p>I am confused about the concept of identity element via ,If (S,*) is an algebraic structure where S={A: A is a 2x2 matrix having all entries a} and * is matrix multiplication (binary operation) then this set have identity element E= [ 1/2 1/2
1/2 1/2] .why this is so it should be usual i... | Lundborg | 227,259 | <p>The identity element $E$ should fulfill that for all $A \in S$</p>
<p>$$
A \cdot E = E \cdot A = A
$$</p>
<p>Now with $E$ being </p>
<p>$$
E = \begin{pmatrix}
\frac{1}{2} & \frac{1}{2}\\
\frac{1}{2} & \frac{1}{2}
\end{pmatrix}
$$</p>
<p>We get for all $A \in S$</p>
<p>$$
A \cdot E = \begin{pmatrix}
a &a... |
2,383,260 | <p>I am confused about the concept of identity element via ,If (S,*) is an algebraic structure where S={A: A is a 2x2 matrix having all entries a} and * is matrix multiplication (binary operation) then this set have identity element E= [ 1/2 1/2
1/2 1/2] .why this is so it should be usual i... | Mark Bennet | 2,906 | <p>Within the set you have chosen the matrix with entries $a\in R$ (where $R$ is the system of numbers where $a$ lives - could be integers or reals etc) corresponds to multiplication by $2a$, so with $a=\frac 12$ you have the identity for the set of matrices you have defined with the operation you have specified. </p>
... |
2,940,217 | <p>From my topology notes...</p>
<p>Let <span class="math-container">$X$</span> be a simply ordered set with the least upper bound property with
respect to the order topology. Then each closed interval is compact.</p>
<p>Proof. First step toward showing <span class="math-container">$[a, b]$</span> is compact: Assumin... | William Elliot | 426,203 | <p>Assume X is the reals. Cover X with<br>
A = { [a,x), (x,b] : a < x < b }.<br>
More than two sets of A will cover [x,y].</p>
<p>Could he be trying to prove [a,b] compact by using<br>
Alexander's subbase theorem, the easiest method? </p>
<p>A space is compact iff every covering by<br>
subbase sets has a fini... |
656,207 | <p>The question is:</p>
<p>What can you say about the set M if sup M = inf M.</p>
<p>I know that supremum is the lowest upper bound and infumum is the biggest lower bound. But I cant figure out what you can say about the set M. And we only had one lession about supremum and infumum.</p>
| Arthur | 15,500 | <p>You know that for <em>any</em> $x \in M$, we have $\inf M \leq x \leq \sup M$, by definition of upper and lower bounds (not necessarily <em>least</em> upper or <em>greatest</em> lower bound). Now, if $\inf M = \sup M$, what does that say about the possible values of $x$?</p>
|
656,207 | <p>The question is:</p>
<p>What can you say about the set M if sup M = inf M.</p>
<p>I know that supremum is the lowest upper bound and infumum is the biggest lower bound. But I cant figure out what you can say about the set M. And we only had one lession about supremum and infumum.</p>
| Hagen von Eitzen | 39,174 | <p>Hint: Assume $a,b\in M$ with $a<b$. What can you say about $\inf M$ compared to $a$? What about $\sup M $ compared to $b$? What does that tell about $\inf M$ compared to $\sup M$?</p>
|
430,126 | <p>I've asked two years ago a post on Mathematics Stack Exchange, were provided two excellent answers. I'm asking on MathOverflow in the hope that some professor can to expand/improve (if it is possible) these results answering my question. The post has the same title and identifier <a href="https://math.stackexchange.... | Wlod AA | 110,389 | <p>Paper by L.D.Lehmer, "On a problem by Stormer", pp.57-79, Illinois J.Math. (received July 25, 1962; that's all the info about the paper that I have + a reprint) is related and should be certainly useful. It contains complete (extensive) tables of integers <span class="math-container">$\ N\ $</span> such th... |
667,162 | <p>Have the $f(x)=x^2$ maximize and minimize $f$ over $[0,1]$. What happens in terms of the maximum and minimum of $f$ if it is changed to $[0, 1)$?</p>
| lsp | 64,509 | <p>$f(x)$ is increasing in the domain $[0,1]$(How? Think). So minimum occurs at $0$ and maximum occurs at $1$.</p>
|
667,162 | <p>Have the $f(x)=x^2$ maximize and minimize $f$ over $[0,1]$. What happens in terms of the maximum and minimum of $f$ if it is changed to $[0, 1)$?</p>
| Ivy_End | 115,835 | <ol>
<li><p>For $x\in\left[0,1\right]$, we have $f\left(x\right)_{min}=f\left(0\right)=0,f\left(x\right)_{max}=f\left(1\right)=1$. For $f^{'}\left(x\right)=2x>0$ in the domain $\left[0,1\right]$.</p></li>
<li><p>For $x\in\left[0,1\right)$, $f\left(x\right)$ have the minimum but do not have a maximum, which minimum i... |
3,610,173 | <p>Is there a linear endomorphism of <span class="math-container">$\mathbb{R}^3$</span> that is the square root of <span class="math-container">$-Id$</span>? If not is there one for <span class="math-container">$\mathbb{R}^n$</span> where n is odd?</p>
| Tsemo Aristide | 280,301 | <p>No, an an endomorphism <span class="math-container">$A$</span>, of <span class="math-container">$\mathbb{R}^3$</span> has a real eigenvalue since it characteristic polynomial has degree 3, if <span class="math-container">$A(u)=cu, c\in\mathbb{R}, A^2(u)=c^2u=-u$</span> implies that <span class="math-container">$c^2=... |
2,903,478 | <p>I really cant underatand what is exactly difference. By their it can be verified that undefined has no value at all but inderminant form we can able to evaluate its limit by various way(such as L'Hospital's rule) whether it exists. </p>
| Villa | 588,918 | <p>Undefined is something that is not and will never be defined. e.g dividing by zero. Is just an expression that has no sense and no possible value. On the other side, indeterminate is an expression that you can't know its value at simple sight, but it can be anything (even undefined). Some indeterminate forms are $0/... |
57,163 | <p>$M^k \le 2^r < M^{k+1}$</p>
<p>where $M>1 , k>0$ for some $r$.</p>
<p>It simply tells you that there exists a $2^r$ between $M^k$ and $M^{k+1}$. for example:</p>
<p>if $M=3$, $k=1$ then
$$M^k = 3, \quad M^{k+1} = 9$$
and there exists $4$ and $8$ in between $3$ and $9$. i.e., $2^2$ and $2^3$</p>
<hr>
... | Patrick Da Silva | 10,704 | <p>The simplest way to state this is using the natural logarithm : the inequality
$$
M^k \le 2^r <M^{k+1}
$$
is equivalent to
$$
k \log M \le r \log 2 < (k+1) \log M
$$
and again equivalent to
$$
k \left( \frac{\log M}{\log 2} \right) \le r < (k+1) \left( \frac{ \log M}{ \log 2} \right).
$$
which clearl... |
2,199,650 | <p>I'm utterly stuck with no where to go. The assignment is to complete the indirect proof. I'm stuck on the following step, and have no clue how to proceed. Where do I go? Also, pardon the poor formatting. I have no clue how properly format here. I can't space it like I'd like, or use the symbol I'd like.</p>
<p>$ p ... | Mauro ALLEGRANZA | 108,274 | <p>Full proof, to avoid misunderstanding:</p>
<p>1) $p \to t$ --- 1st premise</p>
<p>2) $s \to (r \lor ¬t)$ --- 2nd premise</p>
<p>3) $q \land s$ --- 3rd premise</p>
<p>4) $¬[(¬p \land q) \lor r]$ --- assumed [b]</p>
<p>5) $s$ --- from 3) by Simplification</p>
<p>6) $q$ --- from 3) by Simplification</p>
<p>7) $r... |
243,368 | <p>Asking <a href="https://mathoverflow.net/questions/241946/on-the-transitivity-of-the-action-of-the-unitary-group">this</a> question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now I am c... | Venkataramana | 23,291 | <p>First, let us prove that a non-compact subgroup $G$ of $SL_n(\mathbb {R})$ which contains $SO(n)$ is all of $SL_n(\mathbb{R})$. If the connected component $G^0$ is compact, and $SO(n)$ is maximal compact in $SL_n(\mathbb{R})$ , it follows that $G^0=SO(n)$. But it is easy to show that $SO(n)$ it its own normalizer in... |
243,368 | <p>Asking <a href="https://mathoverflow.net/questions/241946/on-the-transitivity-of-the-action-of-the-unitary-group">this</a> question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now I am c... | Thomas | 81,169 | <p>In fact the subgroup generated by $SO(n)$ and $tSO(n)t^{-1}$ is certainly connected by arc (as the union of images of arc-connected spaces $SO(n)^k, k\in \bf N$ by the obvious product map) So it is an immersed Lie subgroup (Yamabe). But the argument of Venkataramana shows that there is no Lie algebra between $so(n)$... |
234,036 | <p>I am trying to generate this type of graphic:
<a href="https://i.stack.imgur.com/KRci4.png" rel="noreferrer"><img src="https://i.stack.imgur.com/KRci4.png" alt="ε-machine graph" /></a></p>
<p>see Fig.1a from <a href="https://arxiv.org/abs/0905.3587" rel="noreferrer">https://arxiv.org/abs/0905.3587</a>.</p>
<p>This p... | Daniel Huber | 46,318 | <p>Something like this:</p>
<pre><code>T = {{{0, p, 0}, {1, 0, 0}, {q, 0, 0}}, {{0, 0, 1 - p}, {0, 0,
0}, {1 - q, 0, 0}}}; vars = {x1, x2, x3};
vertexLabels = {1, 2, 3}
edg1 = Outer[Coefficient[#1, #2] &, T[[1]].vars, vars];
edg2 = Outer[Coefficient[#1, #2] &, T[[2]].vars, vars];
edgs = Reap[
Do[If[edg... |
42,814 | <h3><a href="http://en.wikipedia.org/wiki/Recursion#The_recursion_theorem" rel="noreferrer">The recursion theorem</a></h3>
<blockquote>
<p>In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set <span class="math-container">$X$</span>, <span class="math-container">$a \in X$</... | Zhen Lin | 5,191 | <p>In the form you have stated, you can't get the factorial function without a little bit of extra work. Nonetheless, it can be done.</p>
<p>Let $X = \mathbb{N} \times \mathbb{N}$ be the set of ordered pairs of natural numbers. Set $f(x, y) = (x + 1, (x + 1) y)$. Set $F(0) = (0, 1)$. Then, by the recursion theorem, th... |
3,375,527 | <blockquote>
<p>What is the maximum possible value of <span class="math-container">$x$</span> such that expression <span class="math-container">$3^x - 2^n $</span> results in a prime,
where <span class="math-container">$n$</span> is the maximum value such that <span class="math-container">$2^n<3^x$</span> and ... | sirous | 346,566 | <p>COMMENT</p>
<p>Some of these type of primes may have a relation with Merssene and Fermat numbers:</p>
<p><span class="math-container">$$N=3^x-2^n=3^x-2-2^n+2=3^x-2-2(2^n-1)$$</span></p>
<p>Where <span class="math-container">$M=2^n-1$</span> is Merssene Number. Also:</p>
<p><span class="math-container">$$N=3^x-2^... |
1,819,864 | <p>How can we find the value of $ 3\sin(\frac12\arccos\frac19)+ 4\cos(\frac12\arccos\frac18)$ ?
Substituting A = $\arccos\frac19$ </p>
<p>My approach to this question..
I tried to use the formula $\cos A = \cos^2\frac A2\ - \sin^2\frac A2 $
but it led to a negative value under radical.. Any hints?</p>
| almagest | 172,006 | <p>Use the formulae: $1-\cos A=2\sin^2\frac{A}{2},1+\cos x=2\cos^2\frac{A}{2}$.</p>
<p>So $\frac{8}{9}=2\sin^2\frac{1}{2}A$ and hence $\sin\frac{1}{2}A=\frac{2}{3}$ (note that the angle is obviously in the first quadrant, so we want the positive sign).</p>
<p>Similarly, $\cos\frac{1}{2}A=\frac{\sqrt5}{3}$, so $3\sin\... |
3,780,932 | <p>I have a completely filled in sudoku board. If the sum of all digits in each row, in each column, and in each block (the typical sudoku constraints) are all 45, what is the probability that the board is a valid sudoku solution? (part 2: is checking this condition, and an additional one that each digit 1..9 occurs ex... | 53x15 | 734,148 | <p>Leaving aside part 1, which looks like a difficult question, for part 2 checking these conditions is not enough. For example, take any completed Sudoku board where one box matches this pattern:</p>
<p><span class="math-container">$
\begin{array}{|c|c|c|}
\hline
1 & 4 & a \\ \hline
2 & 3 & b \\ \... |
2,445,569 | <blockquote>
<p>Consider the limit
$$ \lim_{n \to \infty} \int_0^n e^{-px} \cos(ax)\, dx.$$</p>
<p>The exercise is asking about the values of $p$ and $a$ which the limit above exists.</p>
<p>a) exists for infinity values of $p$</p>
<p>b) exists for a finite values of $a$</p>
<p>c) exists for ev... | Arthur | 15,500 | <p>$\in$, which has the telling $\TeX$-command <code>\in</code>, is used to show that something is an element of some set. In your case, $a\in (0,4)$ means that $a$ is some member of the set $(0,4)$.</p>
<p>So what is $(0,4)$? It is the set of numbers between $0$ and $4$, not counting $0$ or $4$ themselves. This is ca... |
497,118 | <p>find all the set of values for k such that $-2x^3+24x+8=k$ has more than one solution.</p>
<p>This problem can only be solved using basic Calculus 1 knowledge.</p>
<p>I thought about setting the equation to zero and then computing the x value that can then be rewritten in terms of k but I am not sure that is the c... | Glen O | 67,842 | <p>Hint: For the function to have more than one solution, it must have two distinct, real turning points. For what values of $k$ do those turning points occur on opposite sides of the $x$ axis?</p>
<p>(where we're taking the function to be $y=-2x^3+24x+8-k$)</p>
|
682,170 | <p>Determine the largest 3-digit prime factor of ${2000 \choose 1000}$. </p>
<p>I could not approach the problem at all. I have no idea how to try the problem. Please help.</p>
| Calvin Lin | 54,563 | <p><strong>Hint:</strong> Show that if $p$ is a prime such that $667 \leq p \leq 1000$, then $ p$ is not a factor of $ {2000 \choose 1000}$. Use the fact that $667 \times 3 > 2000$.</p>
<p><strong>Hint:</strong> Try the next largest prime number $p^*$. Does it work?</p>
|
236,235 | <p>Prove that among any 100 integers there are always two whose difference is divisible by 99. How can I prove this?</p>
| André Nicolas | 6,312 | <p>Let $r_i$ be the remainder when the $i$-th integer in our list is divided by $99$.</p>
<p>There are only $99$ conceivable remainders, the numbers $0$ to $98$. </p>
<p>Since there are $100$ integers in our list, and only $99$ conceivable remainders, there must be two different numbers in our list which have the sam... |
3,496,882 | <p><span class="math-container">$ABC$</span> is a triangle with <span class="math-container">$AB < AC$</span>.
The tangent to the circumcircle at <span class="math-container">$A$</span> meets line <span class="math-container">$BC$</span> at <span class="math-container">$P$</span>. Points <span class="math-container... | Kavi Rama Murthy | 142,385 | <p>Given any sequence <span class="math-container">$b_n$</span> of positive numbers increasing to <span class="math-container">$\infty$</span> there is always a series <span class="math-container">$\sum a_n$</span> of positive terms such that <span class="math-container">$a_1+a_2+\cdots+a_n=b_n$</span> for all <span ... |
3,446,716 | <p>I have a problem where, when trying to solve it, I use some parts which would prove that a k-connected graph contains a k-connected bipartite subgraph. I am fairly sure that this can not be correct, does someone see how my approach is wrong or is this implication true?</p>
<p>The proof for the statement would look ... | oshill | 651,041 | <p>Cliques. Clique sums of cliques. Take two copies of <span class="math-container">$K_{10}$</span> and sum them over <span class="math-container">$K_{8}$</span> and analyze. This is a collision of loved areas in research, what are the maximal properties we can attain from bipartite subgraphs? and given connectivity co... |
3,446,716 | <p>I have a problem where, when trying to solve it, I use some parts which would prove that a k-connected graph contains a k-connected bipartite subgraph. I am fairly sure that this can not be correct, does someone see how my approach is wrong or is this implication true?</p>
<p>The proof for the statement would look ... | Mike | 544,150 | <p>There is a much simple counterexample though. Take any odd-length cycle [such as a 3-cycle]--which is 2 connected. What must every bipartite subgraph be. </p>
|
2,167,441 | <p>i need some help here.</p>
<p>Let $A \in \mathbb{R}$ and $x$ be a limit point of the closure of $A$.</p>
<p>Let $(x_{n})$ be a sequence in closure of $A$ such that $x_{n}\neq x, \forall n\in \mathbb{N}$ and $x_{n}$ converges to $x$.</p>
<p>Define the sequence $(y_{n})$ as the following:</p>
<p>If $x_{n} \in A$, ... | Saradamani | 415,693 | <p>Let $e^{x}-2 = z$</p>
<p>$d(e^{x}-2)/dx = e^x$</p>
<p>Therefore $e^x dx$ = $dz$</p>
<p>Then $∫e^x/(e^x-2) dx = ∫dz/z = \ln(z)+c=\ln(e^x-2)+C$ [where c is the constant of integration]</p>
|
258,806 | <p>It was pretty hard finding a short and precise title. Here is my problem:</p>
<p>The equation $$\bigg|\int_\gamma f(z)\text{d}z\bigg|\le\int_\gamma\big|f(z)||\text{d}z|$$ holds if $f$ is integrable (where $\gamma$ is a differentiable path $[0,1]\rightarrow\mathbb{C}$).
However, in what cases does "$=$" apply? Some ... | 23rd | 46,120 | <p>It suffices to consider the following reduced question: if $g:[0,1]\to\mathbb{C}$ is Riemann/Lebesgue integrable, in the following inequality
$$\left|\int_0^1g(t)dt\right|\le \int_0^1|g(t)|dt,\tag{1}$$
when does "$=$" holds?</p>
<p>To answer this question, let us recall how to deduce $(1)$. Since if $g$ is Rieman... |
2,118,676 | <p>Let $u,v,w$ be 3 pairwise coprime integers. Then $$u^3+v^3+3^{5}w^3=2\cdot3^{2}uvw$$ has no non-trivial solutions. How can I prove this? </p>
<p>I have tried to consider many individual cases such as $uvw>0$,$uvw<0$, $max(u,v,w)=u$ etc. a pretty tedious approach. I am certain there must be simpler ones. Any h... | User8976 | 98,414 | <p>Hint: one of the primes must be $19$.</p>
<p>Now lets say that $p = 19$, then $qr = 19 + q + r \implies (q-1)(r-1)=20$</p>
<p>There are only finitely many choices to factorize $20$ and you can check explicitly which pair satisfies the condition.</p>
|
964,951 | <p>Show that for any strictly increasing function $f:[0,1]\to[0,1]$ there is a fixed point such that $f(x)=x$. ( The function isn't necessarily continuous) .</p>
<p>Any ideas ?</p>
| Cameron Buie | 28,900 | <p>If $f(x)=x$ for some $x<1,$ then we're done. Suppose not. Note that $0<f(0).$ (Why?) Since $f$ is monotone non-decreasing on $[0,1]$, then for every $x_0\in(0,1],$ we have that $\lim_{x\to x_0^-}f(x)$ exists--namely, it is the least upper bound of the set $\{f(x):0<x<x_0\}$--and moreover $$\lim_{x\to x_0... |
4,186,471 | <p>I'm given the series:
<span class="math-container">$$\sum_{n=1}^{+\infty}(n^2+n+1)x^{n-1}$$</span>
which is easy to see converges iff <span class="math-container">$|x|<1$</span>. What I'm missing is the idea behind the computation of the sum. I tried considering the series
<span class="math-container">$$\sum_{n=1... | Claude Leibovici | 82,404 | <p>A small trick for this kind of summation
<span class="math-container">$$\sum_{n=1}^{+\infty}(n^2+n+1)x^{n-1}$$</span>
<span class="math-container">$$n^2=n(n-1)+n\implies n^2+n+1=n(n-1)+2n+1$$</span>
<span class="math-container">$$\sum_{n=1}^{+\infty}(n^2+n+1)x^{n-1}=\sum_{n=1}^{+\infty}n(n-1)x^{n-1}+2\sum_{n=1}^{+\i... |
4,186,471 | <p>I'm given the series:
<span class="math-container">$$\sum_{n=1}^{+\infty}(n^2+n+1)x^{n-1}$$</span>
which is easy to see converges iff <span class="math-container">$|x|<1$</span>. What I'm missing is the idea behind the computation of the sum. I tried considering the series
<span class="math-container">$$\sum_{n=1... | yngabl | 111,900 | <p>As a comment. It's a general method and your series can be obtained as
<span class="math-container">$$
\big((xD_x)^2+(xD_x)+1\big)f(x)
$$</span>
where <span class="math-container">$f(x)$</span> is the geometric serie and <span class="math-container">$D_x$</span> is the derivative wrt <span class="math-container">$x$... |
1,439,210 | <p>upon the quest of understanding spherical harmonics, I came to the a saying "functions defined on the surface of a sphere". </p>
<p><a href="https://en.wikipedia.org/wiki/Spherical_harmonics" rel="nofollow">https://en.wikipedia.org/wiki/Spherical_harmonics</a></p>
<p>Now what does it mean by "functions defined on ... | porridgemathematics | 270,179 | <p>When you mention </p>
<blockquote>
<p>"draw a curve on the surface of a sphere",</p>
</blockquote>
<p>I do not think this is what the article is suggesting.</p>
<p>When you draw a curve on a surface of a sphere, you are defining a vector valued function $r(t) = x(t)i + y(t)j + z(t)k$ where $i, j , k$ are simpl... |
2,462,448 | <blockquote>
<p>Find all integers $m,n$ such that $$(562-5(m+n))(82-5(m-n))=98^2.$$</p>
</blockquote>
<p>What is a simpler method to tackle this? I know the results but my approach is kind of lengthy. I considered every possible factor of $98^2$ to get the result. Thanks.</p>
| kimi Tanaka | 331,356 | <p>Let $a$ be $562 - 5(m + n)$ $=$ $a$. Let $b$ be $82 - 5(m - n)$ $=$ $b$.</p>
<p>Now, you check if the last digit of $a$ has $2$ or $-8$ because it has to be $(a - 562)$ % $5$ $=$ $0$ where $m, n$ $\in$ $Z$. Therefore you should check just $2$ or $-28$, $-98$, $1372$, $4802$ are one of the factor from the combinatio... |
3,232,057 | <p><span class="math-container">$$\lim_{n \to \infty }n\int_{0}^{1}\frac{x^{n}}{1+x+x^{n}}dx$$</span></p>
<p>I factorize <span class="math-container">$x^n$</span> then I tried with <span class="math-container">$u=1+x^{1-n}$</span> but I didn't get too far. Also I tried to get something from <span class="math-container... | robjohn | 13,854 | <p><span class="math-container">$$
\begin{align}
\lim_{n\to\infty}n\int_0^1\frac{x^n}{1+x+x^n}\,\mathrm{d}x
&=\lim_{n\to\infty}\int_0^1\frac{x^{1/n}}{1+x^{1/n}+x}\,\mathrm{d}x\tag1\\
&=\int_0^1\frac1{2+x}\,\mathrm{d}x\tag2\\[3pt]
&=\log\left(\frac32\right)\tag3
\end{align}
$$</span>
Explanation:<br>
<span c... |
18,932 | <p>I'm trying to solve an ODE with two independent variables (a cannon firing from a cliff incorporating wind resistance dependent on velocity). I've tried the following for the x-component:</p>
<pre><code>NDSolve[{x''[t, θ] == -0.2*x'[t, θ]/2.30, x'[0, θ] == 10.8*Cos[θ], x[0, θ] == 0},
x[t, θ], t]
</code></pre>
<p... | b.gates.you.know.what | 134 | <p>If :</p>
<pre><code>sol = DSolve[{D[x[t, th], {t, 2}] == -0.2*D[x[t, th], t]/2.30,
Derivative[1, 0][x][0, th] == 10.8*Cos[th], x[0, th] == 0}, x[t, th], t];
</code></pre>
<p>then you can use the solution as :</p>
<pre><code>Plot3D[x[t, th] /. sol, {t, 0, 10}, {th, -Pi, Pi}]
</code></pre>
<p><img src="https://i... |
104,698 | <p>I have a 36 inch piece of wood (Length) I will placing 6 coat hanger hooks along the length of the wood. The first hook will be placed 1 inch down the length from one end and the last hook will be placed 1 inch from the other end of this 36 inch piece of wood. Question how to find the equal distance between the ot... | Bob Terrell | 15,643 | <p>It is not clear whether the question is about arithmetic, or about a practical method to
be used in construction. I think it is about arithmetic, in which case Peejay's answer is
exactly right. But I will answer it as a woodworker too, as this also contains mathematical
ideas. Don't measure anything. Use "dividers... |
4,620,730 | <blockquote>
<p>What is the real part of this complex number?
<span class="math-container">$$\frac{e^{(n+1)i\theta}-1}{e^{i\theta}-1}$$</span></p>
</blockquote>
<p>I am trying to times conjugate of denominator which will be <span class="math-container">$$\frac{e^{(n+1)i\theta}-1}{e^{i\theta}-1}\cdot\frac{e^{-i\theta}-1... | Bob Dobbs | 221,315 | <p>The real part of <span class="math-container">$\large\frac{a+bi}{c+di}$</span> is <span class="math-container">$\large\frac{ac+bd}{c^2+d^2}$</span> so the real part of <span class="math-container">$\large\frac{e^{(n+1)i\theta}-1}{e^{i\theta}-1}=\frac{\cos(n+1)\theta-1+i\sin(n+1)\theta}{\cos\theta-1+i\sin\theta}$</sp... |
3,009,033 | <p>Help me to compute this limit with L'Hospital's rule.</p>
<p><span class="math-container">$$\lim_{x\to 0} (\sin x)^x$$</span></p>
<p>Thanks in advance.</p>
| user | 505,767 | <p><strong>HINT</strong></p>
<p>We need <span class="math-container">$x>0$</span>, then use</p>
<p><span class="math-container">$$ (\sin x)^x=e^{x \log \sin x}=e^{\frac{\log \sin x}{\frac1x}}$$</span></p>
|
4,432,171 | <p>The lim is about "when x approachs a, then y approachs L". Then, shouldn't the epsilon and delta be like "For all delta, no matter how small the delta is, you can always find an epsilon that makes ε < f(x)-L < ε"? But, the conventional explanation says like "for all epsilon, you find d... | Eric Duminil | 386,794 | <p>"Epsilon-Delta" is supposed to be a <em>challenge</em>, in order to prove that <span class="math-container">$\lim\limits_{x\to a}f(x)=L$</span>.</p>
<h2>Your proposal</h2>
<blockquote>
<p>For all delta, no matter how small the delta is, you can always find
an epsilon that makes -ε < f(x)-L < ε</p>
</... |
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