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 |
|---|---|---|---|---|
28,361 | <p>My motivation is to understand the following situation: Given absolutely and almost simple algebraic group $G$ defined over a number field $k$ and a finite valuation $v$ on $k$, when $G(k_v)$ can be compact (with respect to the $p$-adic topology)?</p>
<p>I more or less understand that if $G=SL_1(D)$ where $D$ is a ... | Charles Matthews | 6,153 | <p>See <a href="http://eom.springer.de/a/a012530.htm" rel="nofollow">http://eom.springer.de/a/a012530.htm</a> and the concept of anisotropic group. As it says there, the classification involved is pretty much the classification of semisimple groups.</p>
|
28,361 | <p>My motivation is to understand the following situation: Given absolutely and almost simple algebraic group $G$ defined over a number field $k$ and a finite valuation $v$ on $k$, when $G(k_v)$ can be compact (with respect to the $p$-adic topology)?</p>
<p>I more or less understand that if $G=SL_1(D)$ where $D$ is a ... | Boyarsky | 6,773 | <p>Here are some remarks on the answers of Charles Matthews, Kevin Buzzard, and Victor Protsak. For justification of Kevin Buzzard's claim, see G. Prasad, "An elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits" from Bull. Soc. Math. France 110 (1982), pp. 197--202, for an incredibly elegant ... |
3,129,029 | <blockquote>
<p>The tournament involves <span class="math-container">$2k$</span> tennis players they play the tournament, each played with each exactly once. What is the minimum number of rounds you need to play to find 3 such that everyone plays with everyone?</p>
</blockquote>
<p>In each round, <span class="math-c... | Mark Fischler | 150,362 | <p>To prove that <span class="math-container">$k$</span> rounds does not force a triangle, consider the following set of matchesamong <span class="math-container">$2k$</span> players:</p>
<p>Divide the players <span class="math-container">$P_i: i = 1 \ldots 2k$</span> into equi-numerous groups <span class="math-contai... |
891,203 | <p>I am wondering about this: Assume you have a class $[f] \in \pi_1(B,b_0)$ and a covering map $p:E \rightarrow B$.
Now, I know that if you take any two paths $g,h \in f$ that are homotopic and they have liftings $\hat{g} , \hat{h}$, then if $E$ is not simply connected it might happen that the liftings are not closed... | Jack D'Aurizio | 44,121 | <p>Let $S$ be the symmetric of $Q$ wrt $R$. Since $\frac{PQ}{TR}=2$, $S$ is the simmetric of $P$ wrt $T$, too. This gives that the intersection of $PR$ and $QT$ is the centroid of the triangle $PQS$, and since the medians cut themselves in thirds, the conclusion.</p>
|
3,429,229 | <p>Let <span class="math-container">$u_n = u_n(t,x)$</span> be a sequence of functions, <span class="math-container">$u_n : (0, \infty) \times \mathbb{R}^N \rightarrow \mathbb{R}$</span>, such that <span class="math-container">$u_n(t)$</span> converges to a function <span class="math-container">$u(t)$</span> in the <sp... | Calvin Khor | 80,734 | <p>You can drop the <span class="math-container">$t$</span>, its not playing any role. Then you can use the following interpolation inequality, which is not hard to prove using Hölder's inequality: </p>
<blockquote>
<p><strong>Lebesgue Interpolation.</strong> Let <span class="math-container">$p_0,p_1\in[1,\infty]$</... |
1,914 | <p><a href="https://matheducators.stackexchange.com/a/1528/42">This answer</a> describes an analogy between finite state machines and mazes. This allows for some playful exercises, like </p>
<blockquote>
<p>Draw a representation of a word accepted by the following automaton...</p>
</blockquote>
<p>which sums up to ... | Community | -1 | <p>The game of <a href="http://en.wikipedia.org/wiki/Nim" rel="nofollow">Nim</a> is often used as introduction to binary numbers and $\mathbf{Z}\ /\ 2\mathbf{Z}$.</p>
<p>It uses the curious addition on $\mathbf{N}$ defined by $a+b=c$ iff $a_i+b_i=c_i \pmod 2$ for all $i$, where $a_i$ is the $i^\text{th}$ digit in the ... |
2,981,450 | <blockquote>
<p>Prove that there do not exist natural <span class="math-container">$n$</span> such that <span class="math-container">$(1+i)^n=1$</span>.</p>
</blockquote>
<p>I try to prove with the binomial and proving by induction but it isn't working</p>
<p><img src="https://i.stack.imgur.com/erGAb.png" alt="try ... | Parcly Taxel | 357,390 | <p><span class="math-container">$$(1+i)^0=1$$</span></p>
<blockquote class="spoiler">
<p>However, it is quite simple to show that 0 is the <em>only</em> integer exponent that solves the equation, e.g. by considering the polar form; the magnitude of <span class="math-container">$(1+i)^n$</span> is <span class="math-c... |
2,981,450 | <blockquote>
<p>Prove that there do not exist natural <span class="math-container">$n$</span> such that <span class="math-container">$(1+i)^n=1$</span>.</p>
</blockquote>
<p>I try to prove with the binomial and proving by induction but it isn't working</p>
<p><img src="https://i.stack.imgur.com/erGAb.png" alt="try ... | drhab | 75,923 | <p>Shortcut:</p>
<p>If indeed <span class="math-container">$z^n=1$</span> then also <span class="math-container">$|z|^n=|z^n|=1$</span>.</p>
|
1,631 | <p>Using <code>MouseAppearance</code> one can change the cursor image when passing over an expression.</p>
<p>Is it possible to change the cursor image for the entire notebook front end (not just one expression within it)?</p>
| cormullion | 61 | <p>The ImageApply function applies any suitable Mathematica function to every pixel in an image. You just have to specify the transformation you want to make. I recently asked this: <a href="https://mathematica.stackexchange.com/questions/207/image-levels-how-to-alter-exposure-of-dark-and-light-areas">Image levels: how... |
4,169,953 | <p>Suppose we have a function <span class="math-container">$f$</span> on the real line and <span class="math-container">$f > 0$</span> on some interval <span class="math-container">$[a,b)$</span> and <span class="math-container">$f(b) = 0$</span>. Assume also that <span class="math-container">$f'<0$</span> on <sp... | Andrew D. Hwang | 86,418 | <p>The mean value theorem is your friend any time you want to pass from information about a derivative <em>on an interval</em> to information about the function. Here, you're assuming <span class="math-container">$f$</span> is differentiable on <span class="math-container">$[b, c]$</span>. The mean value theorem guaran... |
83,882 | <p>Suppose I have a quadratic polynomial in two variables <code>x</code> and <code>y</code> in which the squares with respect to <code>x</code> and <code>y</code> have already been completed:</p>
<pre><code> q = -72 + 9 (-2 + x)^2 + 4 (3 + y)^2 ;
</code></pre>
<p>How might I extract the "constant" part <code>-72</cod... | murray | 148 | <p>OK, I think this is easier than I thought:</p>
<pre><code> c = First[List @@ q]
(* -72 *)
</code></pre>
<p>Is there a yet easier way, or some alternate ways?</p>
<p>(Sorry, I had not intended to post something I could so quickly answer myself!)</p>
|
1,019,833 | <p>Let $x \in A \cap B$. Suppose we have $U \subset A,V \subset B$ such that $ x \in U$ and $x \in V$. So, $x \in U \cap V$. Further, suppose there exists a set $G \subset U \cap V $ so that $x \in G$. Does it follow that </p>
<p>$$ x \in G \subset U \cap V \subset A \cap B $$</p>
<p>?? </p>
| Crostul | 160,300 | <p>Yes. $x \in G \subset U \cap V$ is your assumption, while $U \cap V \subset A \cap B$ is true because $\forall y \in U \cap V$
$$y \in U \Rightarrow y \in A$$
$$y \in V \Rightarrow y \in B$$
hence $y \in A \cap B$</p>
|
73,693 | <p>I have a function $\vec{F}_i(t)$, which is unknown, but I do know it's mean
$\langle \vec{F}_i(t) \rangle = \vec{0}$ and it's variance
$\langle \vec{F}_i(t) \cdot \vec{F}_j(t') \rangle = 2 k_B T \gamma \delta_{ij} \delta(t-t')$. </p>
<p>I am having a long expression and now want to evaluate it's mean - which mean... | Alexei Boulbitch | 788 | <p>The first replacement is easy:</p>
<pre><code>F[t]*g[x]*F[k] /. u_*F[t_] F[x_] -> DiracDelta[t - x]*u
(* DiracDelta[k - t] g[x] *)
</code></pre>
<p>The second one is a bit more tricky because of the integral involved. In such a case I make all substitutions first, and integrate later:</p>
<pre><code> (f1[... |
2,112,161 | <p>This is a question out of "Precalculus: A Prelude to Calculus" second edition by Sheldon Axler. on page 19 problem number 54.</p>
<p>The problem is Explain why $(a−b)^2 = a^2 −b^2 $ if and only if $b = 0$ or $b = a$.</p>
<p>So I started by expanding $(a−b)^2$ to $(a−b)^2 = (a-b)(a-b) = a^2 -2ab +b^2$. To Prove tha... | Community | -1 | <blockquote>
<p>which resulted in $a^2 - 2a(0) + 0^2 = 2a$</p>
</blockquote>
<p>$a^2 - 2a(0) + 0^2 = a^2$, not $2a$.</p>
<blockquote>
<p>or if I do not substitute the $b^2$ I end up with $a^2 + b^2$.</p>
</blockquote>
<p>Why would you not substitute the $b^2$? If you're substituting $b=0$ then you need to do it... |
4,633,165 | <p>There are closed forms for <span class="math-container">$\int_{0}^{\pi/2}\ln(\sin x)\,\mathrm dx\,$</span> and <span class="math-container">$\,\int_{0}^{\pi/2}\ln^2(\sin x)\,\mathrm dx\,$</span> but I can’t seem to find a closed form for <span class="math-container">$$\int_{0}^{\pi/2}\ln^3(\sin x)\,\mathrm dx\;.$$</... | Lai | 732,917 | <p>Noting that <span class="math-container">$$
\int_0^{\frac{\pi}{2}} \ln ^3(\sin x) d x=\left.\frac{\partial^3}{\partial a^3} I(a)\right|_{a=0}
$$</span>
where
<span class="math-container">$$
I(a)=\int_0^{\frac{\pi}{2}} \sin ^a x d x=\frac{1}{2}B\left (\frac{a+1}{2},\frac{1}{2} \right)$$</span></p>
<p><span class="mat... |
3,744,077 | <p>When a group of people need to decide a winner or leader between them, one approach would be that a random hidden integer is chosen with uniform distribution on <span class="math-container">$\{0, 1, ..., n\}$</span> and all <span class="math-container">$p$</span> participants publicly choose a number.</p>
<p>Then, t... | irchans | 372,582 | <p>Just to make things a bit easier, assume the initial random number is a random real number between 0 and 1. If there are <span class="math-container">$p$</span> players, then I think one Nash Equilibrium strategy is for the <span class="math-container">$i$</span>th player to choose the number <span class="math-cont... |
3,035,549 | <p>I'm trying to prove the following inequality: </p>
<p><span class="math-container">$$\sin^{-1}(1)\geq\int_0^b1/\sqrt{1-x^2}\,dx +(1-b)\pi/2$$</span></p>
<p>for every <span class="math-container">$b \in [0,1)$</span>. </p>
<p>I'm given <span class="math-container">$\sin^{-1}(1) = \pi/2$</span> and <span class="mat... | Shubham Johri | 551,962 | <p>The inequality you seek to prove is not true. Put <span class="math-container">$b=1/4$</span>.</p>
<p>However, <span class="math-container">$\sin^{-1}b\le b\pi/2,\ \forall b\in[0,1)$</span></p>
<p>Let <span class="math-container">$f(x)=x\pi/2-\sin^{-1}x, f(0)=f(1)=0$</span></p>
<p><span class="math-container">$f'... |
2,313,895 | <p>I have a doubt about the equivalence between Fourier Transform and Laplace Transform. </p>
<p>It was told me that if I have a function such that:</p>
<ul>
<li>$f(t)=0$ if t<0</li>
<li>$f\in L^1(R) \bigcap L^2(R)$</li>
</ul>
<p>I can define </p>
<p>$F[f(t)]=\int_{0}^{\infty}f(t) e^{-j\omega t}dt$</p>
<p>$L[f(... | Paul | 202,111 | <p>You will always have an infinite (indeed, uncountable) number of invertible matrices along the segment between $A$ and $B$. Determinant is a continuous function of the entries of a matrix. So in some neighborhood of both $A$ and $B$ you are guaranteed invertibility, since the determinant of $\epsilon A + (1-\epsil... |
1,873,702 | <p>My question is very simple. Suppose we have a polynomial defined as follows:
$$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots+a_0 $$
where all of the $a_n$'s are all real and positive. Is there something that we can say about the roots of $p(x)$? Can we say the roots of $p(x)$ all contain negative real parts?</p>
<p>Thank... | Zestylemonzi | 270,448 | <p>You certainly can not say that the real part of any root is negative.
Consider
$$x^3 +24\sqrt3 = 0.$$
This has ($\sqrt3 + 3i)$ as a root.</p>
|
1,873,702 | <p>My question is very simple. Suppose we have a polynomial defined as follows:
$$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots+a_0 $$
where all of the $a_n$'s are all real and positive. Is there something that we can say about the roots of $p(x)$? Can we say the roots of $p(x)$ all contain negative real parts?</p>
<p>Thank... | Nitin Uniyal | 246,221 | <p>The answer is yes, if and only if you are interested in real roots. You can verify it by using Descarte's rule of signs. But in the field of complex numbers, <span class="math-container">$x^2+1=0$</span> is the simplest one for contradiction.</p>
<blockquote>
<p><strong>ERRATUM (April 18,2021)-</strong> Construct a ... |
1,873,702 | <p>My question is very simple. Suppose we have a polynomial defined as follows:
$$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots+a_0 $$
where all of the $a_n$'s are all real and positive. Is there something that we can say about the roots of $p(x)$? Can we say the roots of $p(x)$ all contain negative real parts?</p>
<p>Thank... | Pietro Paparella | 414,530 | <blockquote>
<p>Can we say the roots of p(x) all contain negative real parts?</p>
</blockquote>
<p>As pointed out in another answer, this is not true (in general).</p>
<p>However, there is an <em>exlusion</em> sector for such polynomials. Clearly, such polynomials can not have positive roots; however, Aaron Melman rece... |
513,822 | <p>Show an example where neither $\lim\limits_{x\to c} f(x)$ or $\lim\limits_{x\to c} g(x)$ exists but $\lim\limits_{x\to c} f(x)g(x)$ exists.</p>
<p>Sorry if this seems elementary, I have just started my degree...</p>
<p>Thanks in advance.</p>
| kedrigern | 97,299 | <p>For example </p>
<p><span class="math-container">$f(x) =
\begin{cases}
0 & \text{if $x\in\mathbb{Q}$} \\
1 & \text{if $x\notin\mathbb{Q}$}
\end{cases}$</span></p>
<p><span class="math-container">$g(x) =
\begin{cases}
1 & \text{if $x\in\mathbb{Q}$} \\
0 & \text{if $x\notin\mathbb{Q}$}
\end{cases}$... |
2,035,987 | <p>$$e^{x^2}=e^{18x} \cdot 1/e^{80}$$</p>
<p>I dropped the e since it was the same base and solved:</p>
<p>$$x^2=18x-80$$</p>
<p>I then moved $ 18x$ and $80$ to the other side and got:
$$x^2-18x+80=0$$</p>
<p>The roots are $ 8 $ and $ 10 $ but its not possible with the signs I got. Where did I go wrong with the sig... | Arnaldo | 391,612 | <p>$$e^{x^2}=e^{18x} \cdot 1/e^{80} \Rightarrow x^2=18x+(-80) \Rightarrow x^2-18x+80=0$$
what give us $x=10$ or $x=8$</p>
|
498,674 | <p>I'm trying to interpret the very last sentence. He says nothing about $a_{1}$, so in the case of $a_{1}=0$ all the other coefficients can not be zero, which would imply linearly independence. The case where $a_{1}\neq 0$ all the other coefficients can not be zero because, if they were, we would have $a_{1}v_{1}=0$ w... | Tom Oldfield | 45,760 | <p>What you've said is perfectly correct. A slightly quicker/neater/different way to see it (without splitting into cases) might be the following:</p>
<p>Suppose instead that all of $a_2,\dots,a_m$ are equal to zero. Then $a_1v_1=0$. But $v_1\not=0$ so it must be that $a_1=0$, contradicting our original assumption tha... |
4,631,361 | <p>Let <span class="math-container">$a, b, x, y>0$</span>, <span class="math-container">$a+b \ge x+y$</span> and <span class="math-container">$ab \le xy$</span> then <span class="math-container">$a^n+b^n \ge x^n+y^n$</span> where <span class="math-container">$n=2$</span>.</p>
<blockquote>
<blockquote>
<p><em>Is it t... | marty cohen | 13,079 | <p>Let
<span class="math-container">$d_n
= a^n+b^n
$</span>
and let <span class="math-container">$a+b=d$</span>.</p>
<p><span class="math-container">$\begin{array}\\
d_n(a+b)
&=(a^n+b^n)(a+b)\\
&=a^{n+1}+ba^n+ab^n+b^{n+1}\\
&=a^{n+1}+b^{n+1}+ab(a^{n-1}+b^{n-1})\\
&=d_{n+1}+abd_{n-1}\\
\text{so}\\
d_{n+1... |
744,594 | <p>Let $x\in(0,1)$ be any and let $0<a<b<\frac{1}{2}$, I need to show that $$1-(1-x^a)(1-x^{1-a})>1-(1-x^b)(1-x^{1-b}).$$ Any suggestions? References? In practice I need to solve a more general case where I have $1-\prod_{i=1}^I (1-x^{a_i})$ and the $a_i$ are more spread-out than the $b_i$ and $\sum a_i =\s... | Community | -1 | <p>Break the integral in two : </p>
<p>$$ \int \frac{ x^2 }{x^2 + 1} dx + \int \frac{ \arctan x }{1 + x^2} dx $$</p>
<p>To solve the second one : Notice $d( \arctan x ) = \frac{ dx}{1 + x^2} $ So</p>
<p>$$ \int \frac{ \arctan x }{1 + x^2} dx = \int \arctan x\, d(\arctan x ) = (\arctan x)^2/2 + K$$</p>
<p>As for th... |
2,386,079 | <p>I have a question concerning the composition of morphisms in the category $ \textbf{Rel} \ $. </p>
<p>First, for categories generally, it is frequently stated that if $ f \colon A \to B $ and $ g \colon B \to C $ , i.e., if cod(f) = dom (g), the morphisms must compose in order to fit the definition of a category.</... | Chris Culter | 87,023 | <p>For all $A$ and $B$, the null set is indeed a subset of $A\times B$, so it fits the definition of a morphism $A\to B$. What else is there to say?</p>
|
2,386,079 | <p>I have a question concerning the composition of morphisms in the category $ \textbf{Rel} \ $. </p>
<p>First, for categories generally, it is frequently stated that if $ f \colon A \to B $ and $ g \colon B \to C $ , i.e., if cod(f) = dom (g), the morphisms must compose in order to fit the definition of a category.</... | Community | -1 | <p>The composite $S \circ R$ only exists when the codomain of $R$ is the domain<sup>1</sup> of $S$, where the words "composite", "domain", and "codomain" all refer to the category structure of <strong>Rel</strong>.</p>
<p>This does not preclude there being some other operation that extends the category composition ope... |
1,683,977 | <p>How should I compute the derivative of $e^{x\sin x}$ ?
I am a student of class 11, so can you explain me how to do this without high level mathematics ( I know first principles )
I know that derivative of $e^x$ is $e^x$, but I cannot understand what to do with that $\sin x$?</p>
| choco_addicted | 310,026 | <p>Remind the chain rule
$$(f(g(x)))'=g'(x)f'(g(x))$$
and product rule:
$$(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)$$
You can use follwing formulas:
$$
(e^x)'=e^x\text{ (you already know)}\\
(\sin x)'=\cos x\\
(x\sin x)'=\sin x + x\cos x
$$
and so
$$
(e^{x\sin x})'=(\sin x+x\cos x)e^{x\sin x}.
$$
<a href="https://math.stackexcha... |
1,683,977 | <p>How should I compute the derivative of $e^{x\sin x}$ ?
I am a student of class 11, so can you explain me how to do this without high level mathematics ( I know first principles )
I know that derivative of $e^x$ is $e^x$, but I cannot understand what to do with that $\sin x$?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>it is the chain and the power rule:
$${e^{x\sin(x)}}'=e^{x\sin(x)}(\sin(x)+x\cos(x))$$</p>
|
68,173 | <p>I am working on tool for merging smaller textures into one larger for use on Android app.</p>
<p>I have $n$ rectangles of given size $(w_k, h_k)$, where $k=1,\ldots,n$ and I need to position them within master rectangle of size $2^l \times 2^m$, where $l, m \leq 9$ so none overlapping occur and $2^l \times 2^m$ has... | Listing | 3,123 | <p>I guess <a href="http://www.codeproject.com/KB/web-image/rectanglepacker.aspx" rel="nofollow">this</a>
should be exactly what you need. Basically it is an implementation of the approach presented in <a href="http://www.aaai.org/Papers/ICAPS/2003/ICAPS03-029.pdf" rel="nofollow">this</a> paper by Richard E. Korf.
To ... |
107,972 | <p>The <code>Sign</code> function works in a very straightforward manner:</p>
<pre><code>Sign[8]
Sign[-8]
Sign[0]
</code></pre>
<blockquote>
<pre><code>1
-1
0
</code></pre>
</blockquote>
<p>Unfortunately, in my code I need <code>Sign[0]</code> to equal 1 or negative 1. I tried this but get a funny error:</p>
<pre><... | bill s | 1,783 | <p>You can define your own <code>sign</code>. If you want Sign[0] to be randomly 1 or -1,</p>
<pre><code>sign[x_] := If[Sign[x] == 0, RandomChoice[{-1, 1}], Sign[x]]
{sign[8], sign[-8], sign[0], sign[0], sign[0]}
{1, -1, 1, -1, -1}
</code></pre>
<p>though you could just choose one of the two.</p>
|
107,972 | <p>The <code>Sign</code> function works in a very straightforward manner:</p>
<pre><code>Sign[8]
Sign[-8]
Sign[0]
</code></pre>
<blockquote>
<pre><code>1
-1
0
</code></pre>
</blockquote>
<p>Unfortunately, in my code I need <code>Sign[0]</code> to equal 1 or negative 1. I tried this but get a funny error:</p>
<pre><... | MarcoB | 27,951 | <p>Define your own: </p>
<pre><code>mysign[x_] := Piecewise[{{1, x >= 0}, {-1, True}}]
mysign[-2] (* Out: -1 *)
mysign[2] (* Out: 1 *)
mysign[0] (* Out: 1 *)
</code></pre>
|
4,320,453 | <p>In the context of Lebesgue Integrals, I have came across <span class="math-container">$L^2$</span> as the set of measurable functions <span class="math-container">$f:[a,b] \rightarrow \mathbb{C}$</span> that have Lebesgue integrable squares - that is <span class="math-container">$x \to |f(x)|^2$</span> is Lebesgue ... | 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... |
4,320,453 | <p>In the context of Lebesgue Integrals, I have came across <span class="math-container">$L^2$</span> as the set of measurable functions <span class="math-container">$f:[a,b] \rightarrow \mathbb{C}$</span> that have Lebesgue integrable squares - that is <span class="math-container">$x \to |f(x)|^2$</span> is Lebesgue ... | Elliot Yu | 165,060 | <p>Linear transformations are always defined with a single underlying field <span class="math-container">$K$</span>, so that both the domain and the codomain are <span class="math-container">$K$</span>-vector spaces. This is because for linearity to make sense, we need to have the same notion of scalar multiplication i... |
4,145,772 | <p>I asked a question before, and was directed <a href="https://math.stackexchange.com/questions/3447575/why-is-the-tangent-of-an-angle-called-that/3447586#3447586">here</a> instead. :^( Hopefully, someone reading on this thread might still be able to help answer my question.</p>
<p>I am trying to understand how the w... | Thomas Andrews | 7,933 | <p>It is true now that the tangent is now taught as the slope of a radius.</p>
<p>But that is not the historical understanding of things.</p>
<p>“Slope” is a concept from coordinate (aka Cartesian) geometry.</p>
<p>Trigonometric functions were studied long before coordinate geometry. All six functions were used by the ... |
26,723 | <p>I have a large table (7000 rows × 17 columns) of terse textual data.
In many of the columns, empty entries have been replaced with "." as a marker.
Working from the top of the columns downward, I want to replace each successive "." with the data value (non-".") from above until the next data value is found; at which... | J. M.'s persistent exhaustion | 50 | <p>Maybe there's a more efficient approach, but here's what I came up with:</p>
<pre><code>data = {{"foo", "foo", "blah", "blah", "foo", "blah", "argh"},
{"pfft", ".", ".", ".", ".", ".", "foo"},
{"foo", ".", "foo", ".", "pfft", "blah", "."},
{".", ".", "pfft", ".", ".", ".", "blah"},
{... |
26,723 | <p>I have a large table (7000 rows × 17 columns) of terse textual data.
In many of the columns, empty entries have been replaced with "." as a marker.
Working from the top of the columns downward, I want to replace each successive "." with the data value (non-".") from above until the next data value is found; at which... | Silvia | 17 | <p>Caution: This answer is just for fun. <code>ReplaceRepeated</code> is <em>constantly</em> slow.</p>
<p>Using 0x4A4D's <code>data</code>:</p>
<pre><code>#//.{pre___,a_,b:Longest["."..],post___}:>{pre,a,Sequence@@({b}/."."->a),post}&/@
(data\[Transpose])\[Transpose]
</code></pre>
<p>Or,</p>
<pre><code>... |
397,830 | <blockquote>
<p>If $a,b \in \mathbb{Z}$ and odd, show $8 \mid (a^2-b^2)$.</p>
</blockquote>
<p>Let $a=2k+1$ and $b=2j+1$. I tried to get $8\mid (a^2-b^2)$ in to some equivalent form involving congruences and I started with
$$a^2\equiv b^2 \mod{8} \Rightarrow 4k^2+4k \equiv 4j^2+4j \mod{8}$$
$$\Rightarrow k^2+k-j^2-j... | Inceptio | 63,477 | <p><strong>Hint:</strong></p>
<blockquote>
<p>$$a^2 \equiv b^2 \equiv 1 \pmod8$$</p>
</blockquote>
<p>When both $a$ and $b$ are odd.</p>
<p>$(2k+1)^2=4k^2+1+4k=4k(k+1)+1$, here either $k$ or $k+1$ is even.</p>
<p>Therefore $4k(k+1) \equiv 0 \pmod 8$</p>
|
397,830 | <blockquote>
<p>If $a,b \in \mathbb{Z}$ and odd, show $8 \mid (a^2-b^2)$.</p>
</blockquote>
<p>Let $a=2k+1$ and $b=2j+1$. I tried to get $8\mid (a^2-b^2)$ in to some equivalent form involving congruences and I started with
$$a^2\equiv b^2 \mod{8} \Rightarrow 4k^2+4k \equiv 4j^2+4j \mod{8}$$
$$\Rightarrow k^2+k-j^2-j... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$k^2-j^2+k-j=(k-j)(k+j+1)$</p>
<p>As $(k+j+1)-(k-j)=2j+1$ which is odd, they must be of opposite parity, exactly one of them must be divisible by $2$ </p>
<p><strong>Method 2:</strong></p>
<p>If $a,b$ are odd, observe that one of $(a-b),(a+b)$ is divisible by $4,$ the other by $2$</p>
<p><strong>Me... |
3,725,762 | <p>Prove that <span class="math-container">$ \sum_{i=1}^{N} a_i \leq \sqrt{N \sum_{i=1}^{N}a_i^2} $</span>. Well i choose <span class="math-container">$u=(1,\ldots,1)$</span> and <span class="math-container">$v=(a_1,\ldots,a_N)$</span> whit <span class="math-container">$a_i$</span> positive and. Apply <span class="mat... | Ricky Tensor | 583,074 | <p>So we already know that <span class="math-container">$\sum_i a_i = u\circ v$</span>. You then have <span class="math-container">$|u| = \sqrt{u\circ u} = \sqrt{N}$</span>. And <span class="math-container">$|v| = \sqrt{v\circ v} = \sqrt{\sum_i a_i^2}$</span>. Thus, <span class="math-container">$\sqrt{N\sum_i a_i^2} = ... |
4,445,607 | <blockquote>
<p>Evaluate the following limit: <span class="math-container">$$\lim_{(x,y) \to (0,0)}\frac{-xy}{x^2 + y^2}$$</span></p>
</blockquote>
<hr />
<p><strong>My try:</strong></p>
<p>Using polar substitution - putting <span class="math-container">$x =r\cos(\theta)$</span>, <span class="math-container">$y = r\sin... | copper.hat | 27,978 | <p>Let <span class="math-container">$(Ff)(x) = f(-x)$</span>. Note that <span class="math-container">$g \in G$</span> <strong>iff</strong> <span class="math-container">$Fg =g$</span>.</p>
<p>Note that we can write <span class="math-container">$f$</span> in its odd & even parts with
<span class="math-container">$f =... |
82,734 | <p>Can anyone please recommend some good reading on the geometry of linear groups and their actions?</p>
<p>An example of the kind of question I am interested in: Explicitly describe a fundamental domain for the action of $GL_2(\mathbb{Z})$ on $GL_2(\mathbb{R})$, and compute the volume of the quotient. </p>
<p>I'm fa... | Asaf | 8,857 | <p>You can try to take a look in Dave Witte's book about arithmetic groups here - <a href="http://people.uleth.ca/~dave.morris/books/IntroArithGroups.html" rel="nofollow noreferrer">http://people.uleth.ca/~dave.morris/books/IntroArithGroups.html</a>
He also presents in his site a dynamical approach to this subject (due... |
1,368,310 | <p>Suppose that a large lot with 10000 manufactured items has 30 percent defective items and 70 percent nondefective. You choose a subset of 10 items to test.
(a) What is the probability that at most 1 of the 10 test items is defective?
(b) Approximate the previous answer using the binomial distribution.</p>
<p>I am ... | Conrado Costa | 226,425 | <p>The binomial distribution is only an approximation</p>
<p>More explicitly</p>
<p>$P(\text{at most 1 defective }) = P(\text{ no defective item}) + P(\text{1 defective item}) \\= \frac{7000}{10000}\frac{6999}{9999}\ldots \frac{6991}{9991} + 10 \frac{3000}{10000}\frac{7000}{9999}\frac{6999}{9998} \ldots \frac{6992}{... |
931,951 | <p>How can I find all solutions of $Ax = 0$ in parametric vector form where A is row equivalent to the matrix </p>
<p>$\begin{pmatrix}
-1&-4&0&-4\\2&-8&0&8
\end{pmatrix}$</p>
| M.S.E | 153,974 | <p>I think what your asking is to find the null space.</p>
<p>So follow these steps-</p>
<p>put your matrix to reduced row echleon form and then find the basis.</p>
<p>The span of the basis is the null space (all the solutions to Ax= 0)</p>
<p>Since you want to give it in a parametric vector form, it is the same th... |
2,326,050 | <p>This is a question from RMO 2015.</p>
<p>Show that there are infinitely many triples (x,y,z) of integers such that $x^3+y^4=z^{31}.$</p>
<p>This is how I did my proof:
Suppose $z=0,$ which is possible because $0$ is an integer. Then $x^3+y^4=0 \Rightarrow y^4=-x^3.$ Now, suppose $y$ is a perfect cube such that it ... | pisco | 257,943 | <p>Here is a way to obtain <strong>positive</strong> integer solutions.</p>
<p>Let $n$ be any positive integer such that
$$3\mid n \quad \quad 4\mid n \quad \quad 31 \mid (n+1)$$
There are infinitely many such $n$ by Chinese Reminader Theorem,
then $x^3 = 2^n$, $y^4 = 2^n$, $z^{31} = 2^{n+1}$ satisfies $x^3+y^4=z^{31}... |
716,364 | <p>I am just beginning to study fields and for whatever reason am finding their presentation to be completely baffling - moreso than I think anything I have ever studied. I am reading out of chapter 21 of this free book: <a href="http://abstract.ups.edu/download.html" rel="nofollow">http://abstract.ups.edu/download.htm... | Macavity | 58,320 | <p>When $y=x$, you have $g = -3$. So the form of the linear function is $g(x, y) = a(y-x)-3$. Now you also have noted $g(-2, 2)= 9$. Solve for $a$.</p>
|
128,044 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/9188/is-mathbbq-sqrt2-cong-mathbbq-sqrt3">Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?</a> </p>
</blockquote>
<p>Consider the field extension $\mathbb{Q}(\sqrt2)$. I want to show that $\sqrt5 \not... | lhf | 589 | <p>Squaring both sides of $\sqrt5 = a + b\sqrt2$ will imply that $\sqrt2$ is rational, which it is not.</p>
<p>(Actually, you also need to consider the case $b=0$, which would imply $\sqrt5$ is rational, and the case $a=0$, which would imply $\sqrt{\dfrac52}$ is rational.)</p>
|
128,044 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/9188/is-mathbbq-sqrt2-cong-mathbbq-sqrt3">Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?</a> </p>
</blockquote>
<p>Consider the field extension $\mathbb{Q}(\sqrt2)$. I want to show that $\sqrt5 \not... | Bill Dubuque | 242 | <p><strong>HINT</strong> $\ $ Since $\sqrt{2},\sqrt{5},\sqrt{10}\not\in\mathbb Q,\:$ it is an immediate consequence of this</p>
<p><strong>LEMMA</strong> $\rm\ \ [K(\sqrt{a},\sqrt{b}) : K] = 4\ $ if $\rm\ \sqrt{a},\ \sqrt{b},\ \sqrt{a\:b}\ $ all are not in $\rm\:K\:$ and $\rm\: 2 \ne 0\:$ in $\rm\:K\:.$</p>
<p><str... |
71,070 | <p>Anyone recall a structure determined by a 3rd order partial derivative?
not the general nth order of recent Baranovsky</p>
| Denis Serre | 8,799 | <p>The <a href="http://en.wikipedia.org/wiki/Schwarzian_derivative">Schwarzian derivative</a> is third-order and plays an important role in the geometry of the projective line.</p>
|
116,727 | <p>I was thinking a bit about isometric embeddings into Hilbert spaces and got the following idea.</p>
<p>First, as we recall, many vector spaces over the reals are isomorphic to $\mathbb{R}^{\alpha}$ for some cardinal number $\alpha$. [EDIT: As Carl Mummert remarked below, not every vector space is of this form, as I... | jw_ | 671,015 | <p>For the non-openness:</p>
<p>Open intervals are a basis of the standard topology for R.</p>
<p>A topoloogy T equals the collection of all unions of elments of its basis B, then each open set is a union of elements from basis B, then each open set at least contains one open intervals as an element of a basis, then ... |
2,161,420 | <p>I'm thinking about how to compute $\int_{\frac{3}{2n}}^1 nx^{n-1}(x-\frac{3}{2n})^n dx$ or give limsup of it as n tends to infinity? It seems integrate by part may work, but it is complicated. I try to use dominated convergence theorem, since the function converges to $nx^{2n-1}$, but it seems there is not a uniform... | poweierstrass | 358,056 | <p>Here is an evaluation of the integral.</p>
<p>Let $z=\frac{2n}{3}x$</p>
<p>\begin{equation}
\int\limits_{3/(2n)}^{1} n x^{n-1} \left(x - \frac{3}{2n} \right)^n dx
= n\left(\frac{3}{2n} \right)^{2n} (-1)^n \int\limits_{1}^{2n/3} z^{n-1} (1-z)^n dz
\end{equation}</p>
<p>\begin{align}
I(n) &= \int\limits_{1}^{2n... |
1,399,402 | <p>Let $f:[0,\infty [\to[0,\infty [$ continuous such that $$\lim_{t\to\infty }\frac{f(t)}{t}=\ell\in[0,1).$$</p>
<p>Prove that $f$ has a fixed point, i.e. there is an $x\geq 0$ such that $f(x)=x$. </p>
<p>I don't really know how to solve this problem. My first intension was to use Brouwer, but it's only useable on a ... | Siminore | 29,672 | <p>I am adding the requirement that $f$ be continuous, otherwise the conclusion is trivially false.</p>
<p>Hint: draw $y=x$ and $y=\ell x$, where $0 \leq \ell < 1$. If $f(0)=0$, there is nothing to prove. If $f(0)>0$, then try to convince yourself that sooner or later $f(x)$ must lie below $y=x$, and conclude by... |
3,375,280 | <p>Let A and B two bonded sets from <span class="math-container">$R$</span> i want to prove that the set <span class="math-container">$AB=\{ab,a\in A,b\in B\}$</span> is bounded </p>
<p>Let <span class="math-container">$c\in AB$</span> then there exists <span class="math-container">$a\in A$</span> and <span class="mat... | Kavi Rama Murthy | 142,385 | <p>The inequalities you have written are not true in general: Take <span class="math-container">$A=\{0,1\}, B=\{-1\}$</span> and <span class="math-container">$c=0$</span>. </p>
<p>If <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are bounded then there exist <span class="math-conta... |
931 | <p>What do moderators do? Is it just a matter of dealing with posts flagged for moderator attention, or are there other things too?</p>
| Scott Morrison | 3 | <p>Sometimes we have to write to some one who is misusing the site in some way, either with good intentions (e.g. over-eager editing of old posts), through ignorance (particularly new users), or simply because they are misbehaving. Very often there's first a discussion on the moderator mailing list. Sometimes we also s... |
931 | <p>What do moderators do? Is it just a matter of dealing with posts flagged for moderator attention, or are there other things too?</p>
| Scott Morrison | 3 | <p>We try to spend enough time on meta to ensure that questions requiring our input get answered properly. Often this requires looking up some statistics from the moderator tools, helping write a Data Explorer or API query, looking at a deleted post, or merely remembering some ancient thread on <a href="http://tea.math... |
2,660,193 | <p>$6÷2(1+2)=$?</p>
<p>$6÷2\times3=9$</p>
<p>We need to solve first the parentheses or we can distribute this way:</p>
<p>$6÷(2\times1+2\times2)=$?</p>
<p>$6÷(2+4)=$?</p>
<p>$6÷6=1$</p>
<p>What is the correct? $9$ or $1$?</p>
| Mr Pie | 477,343 | <p>Go <a href="https://m.youtube.com/watch?v=URcUvFIUIhQ" rel="nofollow noreferrer">here</a> to answer your question. Because answers that just provide links are discouraged — and for good reason too — I will give you some insight on how it approaches the problem. The link takes you to a YouTube video which is entirely... |
2,145,822 | <p>I need help to find the derivative of $(9x^6+4x^3)^4 $</p>
<p>I already tried the chain rule but I got the wrong answer and I do not know what I did wrong. </p>
<p>This is what I got:</p>
<p>$$4(9x^6+4x^3)^3 \times 54x^5+12x^2$$</p>
| Futurologist | 357,211 | <p>Here is a way to get the function $h$. The idea is that one can define the smooth map $F \, : \, \mathbb{R}^3 \, \to \, \mathbb{R}^2$ by
\begin{align}
s &= f(x)\\
t &= g(x)
\end{align} where $x = (x_1, x_2, x_3) \in \mathbb{R}^3 $. The restriction $$\nabla f(x) \times \nabla g(x) = 0$$ for all $x \in \mathb... |
3,517,528 | <p>Let <span class="math-container">$X$</span> be the the space of polynomials on <span class="math-container">$[0, 1]$</span> and let <span class="math-container">$\|p\| = \max|p'(x)|$</span>, where <span class="math-container">$p'$</span> is the derivative of p. Is <span class="math-container">$\| . \|$</span> a norm... | Severin Schraven | 331,816 | <p>No, it is not. What is <span class="math-container">$\Vert 1\Vert$</span>?</p>
|
2,604,608 | <p>Need help with a question in my book. How do I prove this is continuous.
question -
Show that f(x) = 2a + 3b is continuous where a and b are constants</p>
| zoli | 203,663 | <p>The question regarding the expected value of the red marbles initially in the bag cannot be answered without an assumption about their distribution. However if we assume that there are at least <span class="math-container">$27$</span> red marbles then the most probable number of red marbles given the result is indep... |
1,167,386 | <p>I have been having trouble understanding the translation of a graph. I understand the 'rule in the sense that the '$+$' shifts to the left and the '$-$' to the right when dealing with something like e.g. $f(x + 2) = f(x)$.</p>
<p>In the book I am using it has written that if $g(x) = f(x-c)$, where $c > 0$ then t... | Ken | 169,838 | <p>You are correct in supposing that if you had some function $f(x)$, then if you let $c > 0$ and tried to plot $f(x-c)$, you would get the graph of $f(x)$ shifted $c$ to the right. So if $g(x) = f(x-c)$, then $g$ is the graph that results from shifting $f$ $c$ to the right.</p>
<p>The part where your confusion is ... |
4,519,955 | <p>By differentiation I got the derivative
<span class="math-container">$$f'(x)=\frac{64\sin^3(x)-27\cos^3(x)}{\sin^2(x)\cos^2(x)}$$</span>
and then got the zero of derivative
<span class="math-container">$$x=\arctan(\frac{3}{4})$$</span>
insert x to f(x)=y get the "minimum" value
<span class="math-container"... | Suneesh Jacob | 511,393 | <p><span class="math-container">$$\sin{\left(y\right)}=\frac{2x}{1+x^2}$$</span></p>
<p><span class="math-container">$$\frac{d}{dx}\left(\sin{\left(y\right)}\right)=\frac{d}{dx}\left(\frac{2x}{1+x^2}\right)$$</span></p>
<p><span class="math-container">$$\cos{\left(y\right)}\frac{dy}{dx}=2x\frac{d}{dx}\left(\frac{1}{1+x... |
1,889,002 | <p>I have been going through and doing some (non-assessed) homework questions, but am getting really stuck on conditional probability. The following problem is one that I simply cannot get my head around.</p>
<p>Question: Die A has four red and two blue faces, and die B has two red and four blue faces. One of the dice... | drhab | 75,923 | <p>Hints:</p>
<p>Let $R_{i}$ denote the event that throw $i$ results in red. Let
$A$ denote the event that $A$ is selected for use. Let $B$ denote
the event that $B$ is selected for use. You are looking for $\Pr\left(R_{3}\mid R_{1}\cap R_{2}\right)$</p>
<ul>
<li><p>$\Pr\left(R_{3}\mid R_{1}\cap R_{2}\right)=\frac{\P... |
3,222,926 | <p>This is the equation:</p>
<p><span class="math-container">$$\sin\theta=0.8\theta$$</span> </p>
| b00n heT | 119,285 | <p>Using Taylor's series up to the third term one gets the approximate polynomial equation:
<span class="math-container">$$\theta-\frac{\theta^3}{6}+\frac{\theta^5}{120}=\frac{4}{5}\theta$$</span>
from which we obtain
<span class="math-container">$$\theta\cdot \left(24-20\cdot \theta^2+\theta^4\right)=0$$</span>
which ... |
3,245,841 | <blockquote>
<p>Let <span class="math-container">$a,b$</span> be two positive numbers. Prove or disprove the statement:</p>
<p>If <span class="math-container">$a+b \leq \frac{1}{2}$</span>, then <span class="math-container">$\dfrac{1-a}{a} \dfrac{1-b}{b} \geq 1$</span>.</p>
</blockquote>
<p>True. Assume <span c... | Community | -1 | <p><span class="math-container">$$\frac{1- a}{a}\times \frac{1- b}{b}> \frac{(\,a+ b\,)- a}{a}\times \frac{(\,a+ b\,)- b}{b}= 1$$</span></p>
<p><span class="math-container">$$\because\,1> a+ b$$</span></p>
|
916,688 | <p>Suppose I have two sets A and B:</p>
<p>$$ A = \lbrace 2k-1 : k \in \mathbb{Z}\rbrace$$
$$ B = \lbrace 2l+1 : l \in \mathbb{Z}\rbrace$$ </p>
<p>I need to prove that A = B. </p>
<p>I know that to prove equality between two sets I need to prove both:</p>
<p>$$ A \subseteq B $$</p>
<p>and</p>
<p>$$ A \supseteq B... | Michael Shi | 172,676 | <p>Your approach is correct, but just to help give you an example in writing out your mathematical argument to give you some confidence, I would write it like this:</p>
<p>Suppose $x \in A$. Then there exists a $k \in \mathbb{Z}$ such that $x = 2k - 1 = 2l + 1$ where $l = k - 1$ (following the hint above/what you wrot... |
682,156 | <p>I have come across two contradicting definitions of characteristics function (CHF). In wikipedia CHF is defined as the inverse Fourier transform (FT) of probability density function (PDF) and at some places (e.g. <a href="http://www.math.nus.edu.sg/~matsr/ProbI/Lecture6.pdf" rel="nofollow">http://www.math.nus.edu.sg... | Did | 6,179 | <p>$$\varphi_X(t)=E(\mathrm e^{\mathrm i\langle t,X\rangle})=\int\mathrm e^{\mathrm i\langle t,x\rangle}\,\mathrm dP_X(x)$$</p>
|
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | Akhil Mathew | 344 | <p>I think <em>Functional Analysis</em> by Riesz-Nagy fits the criteria quite well; there are copious footnotes (I read it a while ago and that is probably the part of it that I remember best!).</p>
|
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | Emerton | 2,874 | <p>Naimark's <I> Normed rings </I> has something of the scholarly flavour being discussed. It is extremely complete in its exposition, with extensive references and notes, and many asides and elabolrations which (at least in the English translation that I studied) were typeset in a very small font (the kind that you m... |
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | Timothy Chow | 3,106 | <p><i>Éléments de mathématique</i> by Nicolas Bourbaki.</p>
|
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | Timothy Chow | 3,106 | <p>The <i><a href="http://eom.springer.de/" rel="nofollow">Encyclopaedia of Mathematics</a></i>.</p>
|
892,926 | <p>I'm reading a bit about Nelson's version of nonstandard analysis and in the notes it is said that [$n$ is standard]$\implies$[$n+1$ is standard]. Right after that it is mentioned that an inductive proof is impossible (using the fact $0$ is standard as the base) because it would be based on external sets. I have two ... | Community | -1 | <p>Your argument that $n+1$ is standard if $n$ is seems correct; I would suggest a different approach: show that evaluating a standard function at a standard value produces a standard value. Then this problem would follow by using $f(x) = x+1$.</p>
<p>For your second question, are you <em>sure</em> every natural numbe... |
3,165,019 | <p><a href="https://i.stack.imgur.com/T0WSV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/T0WSV.png" alt="enter image description here" /></a></p>
<p>I converted both functions into polar form and got <span class="math-container">$z = r$</span> and <span class="math-container">$z = r^2 - r\cos\thet... | fleablood | 280,126 | <p>This is easy and more general than you think.</p>
<p>If <span class="math-container">$f:A \to A$</span> is one to one then <span class="math-container">$f:A\to f(A)\subseteq A$</span> is onto and a bijection. If <span class="math-container">$f:A \to A$</span> is <em>not</em> onto then <span class="math-container">... |
1,370,561 | <p>Consider a function $f \in C([-1,1],\mathbb{R})$ and suppose that </p>
<p>$$\int_{-1}^{1} f(t)t^{2n}dt=0$$</p>
<p>for all $n \in \mathbb{N}_0$. I want to show that under this assumption the function $f$ has to be odd, that is $f(x)=-f(-x)$ for all $x \in [-1,1]$.</p>
<p>Let $A$ denote the set $span\{t^{2n}:n \in ... | Martin Argerami | 22,857 | <p>If you let $B=\text{span}\,\{t^{2n+1}:\ n\in\mathbb N_0\}$, then $\overline A$ and $\overline B$ are two orthogonal subspaces that span $L^2[-1,1]$. As $f$ is orthogonal to $B$ it should lie in $A$. So $f$ is an $L^2$-limit of linear combinations of odd powers of $t$. </p>
|
2,461,551 | <p>How can I prove by induction that $3^n ≥ 1 + 2^n$ for every $n\in\mathbb{N}$?</p>
| EM90 | 308,117 | <p>For $n=1$ you just get $3\geq 2+1$, which is clearly true.
Assume now that the thesis holds for a fixed $n\in\mathbb{N}$, i.e. assume that $3^n\geq 1+2^n$. We shall prove that it holds for $n+1$, too, i.e.
$$
3^{n+1}\geq 1+2^{n+1},
$$
which can be written as
$$
3\cdot 3^n\geq 1+2\cdot2^{n}.
$$
Now, notice that $3\c... |
2,461,551 | <p>How can I prove by induction that $3^n ≥ 1 + 2^n$ for every $n\in\mathbb{N}$?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>we have to prove that $$3^{n+1}\geq 2^{n+1}+1$$ if we multiply $$3^n\geq 2^n+1$$ by $3$ we obtain
$$3^{n+1}\geq 2^n\cdot 3+3$$ and we must show that $$2^n\cdot 3+3\geq 2^{n+1}+1$$
and this is true since
$$2^n\cdot 3+3=2^{n+1}+2^n+3\geq 2^{n+1}+1$$ which is true,. </p>
|
3,177,017 | <p>I am trying to find a good upper bound estimate for the expression <span class="math-container">$ab^3$</span>, where <span class="math-container">$a,b\in\mathbb R$</span>, and it should be of the form <span class="math-container">$ab^3\leq c (a^2+b^2)^2, c\in\mathbb R.$</span></p>
<p>(The reason for the latter is, ... | Macavity | 58,320 | <p><em>Another way</em>: As this is homogeneous, we may set <span class="math-container">$a^2+b^2=1$</span>, whence it is maximising <span class="math-container">$\sin t \cos^3t$</span>, which is easy to see happens when say <span class="math-container">$t=\pi/6$</span>.</p>
|
652,960 | <p>I am lost on this one. I'm still new to ring theory, as we're only a couple weeks into the course, but it's already well over my head. I know that $R$ is an integral domain, so the additive and multiplicative identities are not equal, and if both $a$ and $b$ are nonzero, then their product will be nonzero. So $u$ ca... | Bill Dubuque | 242 | <p>$\begin{eqnarray}{\bf Hint} & \quad\ \ \ aR &=&\, bR\\
\iff &b\in aR,\!\!\!&&\!\!\!\!\ a\in bR\\
\iff&\quad a\overset{\large \times\ u}\rightarrow &b& \overset{\large\times\ v}\rightarrow a\\
\iff&au = b\!\!\!\!\!\!&,&\!\! auv =a \\
\iff& au=b\!\!\!\!\!\!&,&a... |
75,777 | <p>I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is uniformly distributed on the unit interval. With "elementary" I mean that it does not make use of complex analysis in partic... | coudy | 6,129 | <p>An elementary proof was given by Alberto Zorzi in </p>
<p><em><a href="https://doi.org/10.1007/s00283-014-9505-x" rel="nofollow noreferrer">An Elementary Proof for the Equidistribution Theorem</a></em><br>
<em>The Mathematical Intelligencer
September 2015, Volume 37, Issue 3, pp 1–2</em> </p>
<p>Unfortunately the... |
152,949 | <p>I noticed that a factorization over algebraic fields is useless in Mathematica. Here is the example over the field containing I*Sqrt[3]:</p>
<pre><code>Pol=4 (3 I Sqrt[3] (-12 + 6 x - 4 x^2 + x^3) y^3 z +
9 (12 - 12 x + 12 x^2 - 6 x^3 + x^4) y^4 z^2 +
I Sqrt[3] (8 + x^3) y z^3 + (4 - 2 x + x^2)^2 z^6 -
3 y^2 (4 ... | Daniel Lichtblau | 51 | <p>Here is a much cleaner method than what I first posted. It's basically (Barry) Trager's method, and it is a cousin to the code posted by @chyaong.</p>
<pre><code>algpoly =
4 (3 I Sqrt[3] (-12 + 6 x - 4 x^2 + x^3) y^3 z +
9 (12 - 12 x + 12 x^2 - 6 x^3 + x^4) y^4 z^2 +
I Sqrt[3] (8 + x^3) y z^3 + (4 - ... |
1,074,882 | <p>Given a list of $n$ distinct items, where a smaller item behind a larger item is obscured, if you can see $x$ items from one end, and $y$ from the other, how many ways can the items be arranged?<br/></p>
<p>Firstly, I noticed that for $x+y-1\ge n$ there are $0$ ways. Then I tried to find how many elements can be on... | Asinomás | 33,907 | <p>We solve a simpler problem first. How many ways are there to arrange $n$ objects so that $k$ are visible when seen from the left?
Denote this number by $f_k(n)$, then clearly $f_k(k)=1$ and $f_1(n)=(n-1)!$ since exactly one of the objects can be covered.</p>
<p>After this we obtain $f_{k+1}(n+1)=f_k(n)+(n)f_{k+1}(... |
1,074,882 | <p>Given a list of $n$ distinct items, where a smaller item behind a larger item is obscured, if you can see $x$ items from one end, and $y$ from the other, how many ways can the items be arranged?<br/></p>
<p>Firstly, I noticed that for $x+y-1\ge n$ there are $0$ ways. Then I tried to find how many elements can be on... | cromulent | 295,986 | <p>The previous answer is almost correct. You have to multiply to each element of the summation the number of ways to split all the items (except the tallest) into left and right parts. With this modification the formula becomes: $$g_{j,i}(n)=\sum_{k=1}^n\left(\frac{n - 1}{k - 1}\right)S^{(j-1)}_{k-1}S^{(i-1)}_{n-k}$$<... |
1,129,894 | <p>The ratio of adult tickets to student tickets for the play was 4:5. If the sum of the adult tickets and one half of the student tickets is 260, how many adult tickets were sold?</p>
<p>The choices are as follow:</p>
<p>80</p>
<p>100</p>
<p>160</p>
<p>200</p>
<p>None of the above</p>
<p>I tried to solve this a... | coffeemath | 30,316 | <p>The span of a set of vectors $K$ in a vector space $V$ is the collection of linear combinations of them, which means finite sums $a_1v_1+...+a_rv_r$ where the $a_k$ are any scalars and each $v_k \in K.$ For your question $K$ is the set $\{0\}$ having only one vector in it, so the "sums" just mentioned have only one ... |
3,920,939 | <p>Find all <span class="math-container">$z$</span> such that <span class="math-container">$\sqrt{5z+5} - \sqrt{3 - 3z} - 2\sqrt{z} = 0$</span>.</p>
<p>After much trial and error, I was able to rearrange the above equation
as <span class="math-container">$80z^2 - 112z-4 = 0$</span>.</p>
<p><span class="math-container">... | dan_fulea | 550,003 | <p>In order to have a clean complete solution, the "steps in between" have to be mentioned, and moreover it should be clear if we work with <span class="math-container">$\Rightarrow$</span> (only), or if we work with equivalences <em>all the time</em>.</p>
<p>For instance, if we take the initial equation, and... |
2,312,805 | <p>So I'm trying to come up with an answer to this question for hours now. I don't know what I'm doing wrong and none of the calculators on the internet couldn't help so I figured I should ask people.</p>
<p>What have I done so far:</p>
<p>$\frac{(\sin 20^\circ + \cos 20^\circ)^2}{\cos 40^\circ} = \frac{(\frac{2\sin(... | Donald Splutterwit | 404,247 | <p>Notice that <span class="math-container">$\cos 40 =\sin 50$</span> and <span class="math-container">$ \sin 40 = \cos 50$</span>. So we have
<span class="math-container">\begin{eqnarray*}
\frac{1+\sin(40)}{ \cos 40} = \frac{1+\cos(50)}{\sin 50}
\end{eqnarray*}</span>
Now use the double angle formulea <span class="mat... |
2,312,805 | <p>So I'm trying to come up with an answer to this question for hours now. I don't know what I'm doing wrong and none of the calculators on the internet couldn't help so I figured I should ask people.</p>
<p>What have I done so far:</p>
<p>$\frac{(\sin 20^\circ + \cos 20^\circ)^2}{\cos 40^\circ} = \frac{(\frac{2\sin(... | lab bhattacharjee | 33,337 | <p><strong>Generalization</strong>:</p>
<p>$$\dfrac{(\cos x+\sin x)^2}{\cos2x}=\dfrac{(\cos x+\sin x)^2}{\cos^2x-\sin^2x}=\dfrac{\cos x+\sin x}{\cos x-\sin x}$$ provided $\cos x+\sin x\ne0\iff\tan x\ne-1$</p>
<p>Now $$\dfrac{\cos x+\sin x}{\cos x-\sin x}=\dfrac{1+\tan x}{1-\tan x}=\tan\left(45^\circ+x\right)$$</p>
... |
328,984 | <p>I am interested in mathematical systems where first order induction (IND) fails. One example is ring theory + $\forall x(Sx=x+1)$. Non-commutative rings are models of this theory. Induction proves $\forall x\forall y(xy=yx)$ which is false in these models.</p>
<p>Consider the theory $ZF + IND + 0=\{\} + \forall x(S... | A.S | 24,829 | <p>To answer your question, yes, this does prove that "$ZF$ plus induction" is inconsistent, although I find the phrasing somewhat odd.</p>
<p>The induction "axiom" essentially states that the universe in question is an inductive set. As you have stated it here, $A \vDash INF$ it can be reduced to the statement $A \s... |
4,563,516 | <h1>Question</h1>
<p>Evaluate <span class="math-container">$$\int_0^1 \cos^{-1} x\ dx$$</span> by first finding the value of <span class="math-container">$$\frac{d}{dx}(x\cos^{-1} x).$$</span></p>
<h1>My Working</h1>
<p>As the question said to evaluate <span class="math-container">$$\frac{d}{dx}(x\cos^{-1} x),$$</span>... | Lai | 732,917 | <p>We may use the trigonometric substitution <span class="math-container">$y=\cos^{-1} x$</span> to transform the integral into
<span class="math-container">$$
\begin{aligned}
\int_0^1 \cos ^{-1} x d x &=\int_0^{\frac{\pi}{2}} y \sin y d y \\
&=-\int_0^{\frac{\pi}{2}} y d(\cos y) \\
&=-[y \cos y]_0^{\frac{\... |
1,217,654 | <p>I took the Laplace transform and solved for $Y$ which resulted in $Y=\frac{1+e^{-s}}{s^2-4}$. I began to break up the problem separating the result into two equations but the fact that there is a $1$ added to the $e^{-s}$ is messing me up. The answer I got was $y(t)= \frac{1}{8} u_1(t)e^{-2t}$ but I'm not sure if th... | graydad | 166,967 | <p>Let $a,b \in \Bbb{R}$ such that $a = n+x$ and $b = m+y$ where $x,y \in [0,1)$ and $n,m \in \Bbb{Z}$. Then $$\lfloor a\rfloor+\lfloor b\rfloor = \lfloor n+x\rfloor+\lfloor m+y\rfloor \\ = n+m$$ </p>
<p><strong>Case 1:</strong> $x+y\geq 1$. Then $x+y = 1+z$ for $z \in [0,1)$ and $\lfloor a+b\rfloor = n+m+1$, demonst... |
16,342 | <p>Why do you need logarithms?
In what situations do you use them?</p>
| NebulousReveal | 2,548 | <p>See <a href="https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1112&context=tme" rel="nofollow noreferrer">Chopping Logs: A Look at the History and Uses of Logarithms</a>.</p>
|
119,027 | <p>I am trying to plot the equations that describe the Cartesian locations of Bi-Cylinder coordinates. The equations are as follows:</p>
<pre><code>x==(1.5 Sinh[n])/(Cosh[n]-Cos[p])
y==(1.5 Sin[p])/(Cosh[n]-Cos[p])
</code></pre>
<p>I keep getting the following error </p>
<blockquote>
<p>Options expected(instead... | Jason B. | 9,490 | <p>You want some kind of <code>DateHistogramList</code> function that works like <a href="http://reference.wolfram.com/language/ref/HistogramList.html" rel="nofollow noreferrer"><code>HistogramList</code></a> does, but I don't see an option for it. Without that, you could just extract the coordinates for the tops of t... |
119,027 | <p>I am trying to plot the equations that describe the Cartesian locations of Bi-Cylinder coordinates. The equations are as follows:</p>
<pre><code>x==(1.5 Sinh[n])/(Cosh[n]-Cos[p])
y==(1.5 Sin[p])/(Cosh[n]-Cos[p])
</code></pre>
<p>I keep getting the following error </p>
<blockquote>
<p>Options expected(instead... | kglr | 125 | <p>You can use the third argument of <code>DateHistogram</code> to extract bins and heights and use them in <code>DateListPlot</code> as <code>Epilog</code>:</p>
<pre><code>DateHistogram[data, "Day", (((dl = Transpose[{Mean/@#, #2}]); #2)&),
Epilog -> First[DateListPlot[dl, PlotStyle -> Directive[Thick, R... |
245,083 | <p>I'm new to integral calculus, I started literally 15 minutes ago, and I need help with this question:</p>
<p>$$\int \dfrac{\ln(x)^2}{x} dx $$</p>
<p>My first step was:</p>
<p>$$\int \dfrac{1}{x}\ln(x)^2 dx $$</p>
<p>However, what to do next, how to solve this using the reverse chain rule? </p>
| Pragabhava | 19,532 | <p>Taking $u = \log x$, then $du = \frac{dx}{x}$, hence
$$
\int \frac{\log(x)^2}{x} dx = \int u^2 du
$$</p>
<p>Can you finish it?</p>
|
368,763 | <p>A part two, you could say, of <a href="https://math.stackexchange.com/questions/368343/j2-1-but-j-neq-1-what-is-j">my previous question</a>.</p>
<p>I was watching a Numberphile video about the favorite number of some mathematicians, and at one point, the creator of ViHart said the following -</p>
<blockquote>
<p... | Qiaochu Yuan | 232 | <p>Don't think of $\ast$ as a number to begin with; that will only confuse you. $\ast$ is a mathematical object called a <a href="http://en.wikipedia.org/wiki/Combinatorial_game_theory" rel="nofollow">combinatorial game</a>. An example of a combinatorial game you may have played before is <a href="http://en.wikipedia.o... |
1,835,414 | <p>What is the fastest method to find which number is bigger manually?</p>
<p>$\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ or $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $</p>
| Barry Cipra | 86,747 | <p>Using the approximation $\sqrt3\approx1.7$, one can do a bit of mental arithmetic and see that</p>
<p>$${3\sqrt3-4\over7-2\sqrt3}\approx{1.1\over3.6}\lt1\qquad\text{while}\qquad{3\sqrt3-8\over1-2\sqrt3}\approx{-2.9\over-2.4}={2.9\over2.4}\gt1$$</p>
<p>This only works, of course, because the two numbers are not at ... |
2,015,223 | <p>Let $f : A \to A$ and $g : A \to A$ be bijections. Show that $g \circ f$ is a bijection and find its inverse in terms of $f^{-1}$ and $g^{-1}$?</p>
| Will Jagy | 10,400 | <p>Your discriminant, for $x^2 - k x y + y^2,$ is
$$ \Delta = k^2 - 4. $$ The smallest solution, in positive integers, to
$$ \tau^2 - \Delta \sigma^2 = 4 $$
is $\tau = k, \sigma = 1.$ You have the obvious $-1$ automorphism given by interchanging $x,y.$ The matrix generating the oriented automorphisms is
$$
\left(
\be... |
211,379 | <p>The numbers 7, 8, 9, apart from being part of a really lame math joke, also have a unique property. Consecutively, they are a prime number, followed by the cube of a prime, followed by the square of a prime. Firstly, does this occurence happen with any other triplet of consecutive numbers? More importantly, is there... | JACKY88 | 42,605 | <p>Suppose the first prime is $x$ and $x+1=y^3$ where $y$ is also a prime. Then $y$ has be an odd number which implies $x$ is even. So $x$ cannot be a prime. Contradiction.</p>
|
325,582 | <p>I am looking for examples of constructions for transfinite towers <span class="math-container">$(X_{\alpha})_{\alpha}$</span> generated by structures <span class="math-container">$X$</span> where the problem of determining whether the tower <span class="math-container">$(X_{\alpha})_{\alpha}$</span> stops growing is... | Wojowu | 30,186 | <p>Consider the following construction of sets of ordinals.</p>
<ul>
<li><p><span class="math-container">$X_0=\{0\}$</span>,</p></li>
<li><p><span class="math-container">$X_{\alpha+1}=$</span> the closure of <span class="math-container">$X_{\alpha}$</span> under <span class="math-container">$\gamma\mapsto\gamma+1$</sp... |
846,020 | <blockquote>
<p>Let $(\mathfrak a_i)$ be an infinite family of ideals in commutative ring $R$. Is $\bigcap\limits_{i=1}^\infty \mathfrak a_i$ not defined?</p>
</blockquote>
<p>I am trying to understand Zariski topology. Here, $V(\bigcap_i \mathfrak a_i)= \bigcup\limits_{i} V(\mathfrak a_i)$. </p>
<p>If $\bigcap\lim... | Keenan Kidwell | 628 | <p>First of all, the question of whether or not "$\bigcap_i\mathfrak a_i$ is defined" doesn't quite makes sense. Each $\mathfrak a_i$ is a set, and intersections of (arbitrary sets of) sets always makes sense as a set, and since each $\mathfrak a_i$ is a subset of $R$, so is the intersection. The right question to ask ... |
742,577 | <p>It's been a while since I've had to do math and I've been stuck around a problem for two good hours. I hate asking questions but I can't figure it out. </p>
<p>I have the following problem: </p>
<p>Find the zero/domain of </p>
<p>$$ f(x) = \frac{9x^3-4x}{(x-3)(x^2-2x+1)} $$</p>
<p>So far, I've been able to find ... | MJD | 25,554 | <p>Each die roll generates $2.58$ bits of entropy, and a sequence of die rolls can generate a uniformly distributed random real number in $[0,1]$ to any desired degree of precision. (For example, consider the die rolls to be a sequence of base-6 digits, where a roll of 6 represents a <code>0</code> digit.) </p>
<p>Gen... |
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