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 |
|---|---|---|---|---|
291,729 | <p>How to show that $\large 3^{3^{3^3}}$ is larger than a googol ($\large 10^{100}$) but smaller than googoplex ($\large 10^{10^{100}}$).</p>
<p>Thanks much in advance!!!</p>
| Community | -1 | <p>$$3^{3^{3^3}} > 3^{300} > 10^{100}$$ since $$3^{3^3} > 3^7 = 3 \cdot (3^3)^2 > 3 \cdot 10^2 = 300$$ since $$3^3 > 7$$</p>
<hr>
<p>$$3^{3^{3^3}} < 10^{3^{3^3}} < 10^{10^{100}}$$ since $$3^{3^3} < 3^{100} < 10^{100}$$ since $$3^3 < 100$$</p>
|
1,954,470 | <p>The question defines $x \in \mathbb{R}$ where x>0 and a sequence of integers with $a_0: [x], a_1=[10^1(x-a_0)]$ until $a_n=[10^n(x-(a_0+10^{-1}a_1+ ... + 10^{-n}a_{n-1}))]$. I want to prove that $0 \leq a_n \leq 9$ for each $n \in \mathbb{N}$.</p>
<p>I am completely stumped at what to do. I feel like the Archimdea... | Mick | 42,351 | <p>I can only show you the logic.</p>
<p>Note that all square bracket [.] means the least integer function.</p>
<p>Let 75.2609 be an instance of x. Then, $a_0 = [x] = 75$. Its action is equivalent to stripping off the integral part.</p>
<p>$x – a_0$ is to get the decimal part (i.e. .2609).</p>
<p>$10^1(x – a_0)$ is... |
206,227 | <p>I was given the following problem:</p>
<p>Let $V_1, V_2, \dots$ be an infinite sequence of Boolean variables. For each natural number $n$, define a proposition $F_n$ according to the following rules: </p>
<p>$$\begin{align*}
F_0 &= \text{False}\\
F_n &= (F_{n-1} \ne V_n)\;.
\end{align*}$$</p>
<p>Use induc... | copper.hat | 27,978 | <p>First, notice that for any boolean value $v$, we have $(\text{False} \neq v) = v$, and $(\text{True} \neq v) = \neg v$.</p>
<p>The base case for the induction is $F_1 = (F_0 \neq V_1) = (\text{False} \neq V_1) = V_1$. If $V_1$ is false, then an even number of values $(V_1)$ are true, if $V_1$ is true, then an odd n... |
206,227 | <p>I was given the following problem:</p>
<p>Let $V_1, V_2, \dots$ be an infinite sequence of Boolean variables. For each natural number $n$, define a proposition $F_n$ according to the following rules: </p>
<p>$$\begin{align*}
F_0 &= \text{False}\\
F_n &= (F_{n-1} \ne V_n)\;.
\end{align*}$$</p>
<p>Use induc... | Brian M. Scott | 12,042 | <p>$\newcommand{\T}{\text{True}}\newcommand{\F}{\text{False}}$The statement that you’re trying to prove for each $n\ge 0$ is:</p>
<blockquote>
<p>$P(n):$ $F_n$ is $\T$ if and only if an odd number of the variables $V_k$ with $k\le n$ are $\T$.</p>
</blockquote>
<p>To check the base case of your induction, observe t... |
2,055,803 | <p>Simplify $\left(\dfrac{4}{5} - \dfrac{3}{5}i\right)^{\!75}$</p>
<p>I've searched around on the internet and haven't found a very straightforward answer for this particular problem. I believe this problem has something to do with Euler's Formula, but I'm not sure how to use it in this case.</p>
<p>EDIT: We are not ... | Bill Dubuque | 242 | <p>Or, multiply by $\, y = x^{-1}$ to get $\ y^2-3y +2 = (y\!-\!2)(y\!-\!1)= 0\ $ so $\, x^{-1} = y = 2,1$</p>
|
3,891,124 | <p>When <span class="math-container">$ (1+cx)^n$</span> is expanded a series in ascending powers of <span class="math-container">$x$</span>, the first three terms are given by <span class="math-container">$1+20x+150x^2$</span>. Calculate the value of the constants <span class="math-container">$c$</span> and <span class... | peter.petrov | 116,591 | <p>You need to use the <a href="https://en.wikipedia.org/wiki/Binomial_theorem#Statement" rel="nofollow noreferrer">Binomial formula</a></p>
<p>Applying it you get:</p>
<p><span class="math-container">$$(1+cx)^n = {n \choose 0} \cdot 1^n \cdot (cx)^{0} + {n \choose 1} \cdot 1^{n-1} \cdot
(cx)^{1} + {n \choose 2} \cdot... |
3,891,124 | <p>When <span class="math-container">$ (1+cx)^n$</span> is expanded a series in ascending powers of <span class="math-container">$x$</span>, the first three terms are given by <span class="math-container">$1+20x+150x^2$</span>. Calculate the value of the constants <span class="math-container">$c$</span> and <span class... | Z Ahmed | 671,540 | <p><span class="math-container">$$(1+cx)^n=1+ncx+\frac{1}{2}n(n-1)(cx)^2+...$$</span>
So we have <span class="math-container">$nc=20~~~(1)$</span> and <span class="math-container">$\frac{1}{2}n(n-1)c^2 =150~~~(2)$</span>
<span class="math-container">$$\implies n(n-1)c^2=300 \implies 20 (n-1)c=300 \implies (n-1)C=15 \im... |
3,805,745 | <p>I am working my way through a linear algebra book and would appreciate some help verifying my proof.</p>
<p><strong>Prove that <span class="math-container">$|u \cdot v| = |u | |v |$</span> if and only if one vector is a scalar multiple of the
other.</strong></p>
<p><strong>PROOF:</strong></p>
<p>Let <span class="mat... | C Squared | 803,927 | <p>In your proof, you make the assumption that <span class="math-container">$\vec{u}=k\vec{v}$</span> and <span class="math-container">$|\vec{u}\cdot\vec{v}|=||\vec{u}||\,||\vec{v}||$</span> and then claim that this holds when <span class="math-container">$$|\vec{u}\cdot\vec{v}|=||\vec{u}||\,||\vec{v}|| \Longleftrighta... |
209,420 | <p>For instance, when trying to compute <span class="math-container">$\mathbb{E}[\sum_{i=1}^{10}X_i]$</span> where <span class="math-container">$X_i \sim N(0,1)$</span>, I input into Mathematica:</p>
<pre><code>Expectation[Sum[x[i],{i, 1, 10}], x[i] \[Distributed] NormalDistribution[]]
</code></pre>
<p>but, instead o... | SHuisman | 66,987 | <p>You would need all the assumptions:</p>
<pre><code>Expectation[Sum[x[i], {i, 1, 10}], Table[x[i] \[Distributed] NormalDistribution[], {i, 1, 10}]]
</code></pre>
<p>returning:</p>
<pre><code>0
</code></pre>
|
563,161 | <p>For something I'm working on, I have a matrix $A$ with another matrix $U$ which is unitary ($U^*U = I$), and I'm trying to show that, for the Frobenius norm, $\|A\| =\|UA\|$. Now, I can do this pretty easily if an inner product space exists. For example, $\|A\| = \sqrt{\langle A,A\rangle}$ and $\|UA\| = \sqrt{\lan... | Robert Israel | 8,508 | <p>I don't know what you mean by "if an inner product space exists". The Frobenius norm does come from an inner product, namely the Frobenius inner product
$(A, B) = {\rm trace}(A^* B)$. </p>
|
196,303 | <p>All models of space that I know from physics use real or complex manifolds. I was just wondering if it is still the case at the level of Planck scale. In string theory, physicists still use strings (circles) in a 11 dimensional manifold in order to model particles. Do they do this because there is no mathematical a... | John Baez | 2,893 | <p>There are approaches to quantum gravity where spacetime is described as a quantum superposition of labelled piecewise-linear CW complexes or other related combinatorial/algebraic entities. See for example:</p>
<ul>
<li><p>John Baez, <a href="http://arxiv.org/abs/gr-qc/9905087" rel="nofollow noreferrer">An introduc... |
54,486 | <p>Many colour schemes and colour functions can be accessed using <a href="http://reference.wolfram.com/mathematica/ref/ColorData.html"><code>ColorData</code></a>.</p>
<p>Version 10 introduced new default colour schemes, and a new customization option using <a href="http://reference.wolfram.com/mathematica/ref/PlotThe... | kglr | 125 | <p><strong>Update 2:</strong> The content and organization of <code>$PlotThemes</code> in versions 10 and 9 are very different. In Version 10</p>
<pre><code> Charting`$PlotThemes
</code></pre>
<p>gives</p>
<p><img src="https://i.stack.imgur.com/3MlWt.png" alt="enter image description here"></p>
<p>whereas in Versio... |
3,873,138 | <p>Since we have variable coefficients we will use the cauchy-euler method to solve this DE. First we substitute <span class="math-container">$y=x^m$</span> into our given DE. This then gives
"</p>
<p><span class="math-container">$9x(m(m-1)x^{m-2}) + 9mx^{m-1} = 0$</span></p>
<p>Note that:</p>
<p><span class="math... | Michael Rozenberg | 190,319 | <p>It's <span class="math-container">$$(xy')'=0$$</span> or
<span class="math-container">$$xy'=C$$</span> or
<span class="math-container">$$y=C\ln|x|+C_1.$$</span></p>
|
2,478,229 | <p>The matter of interest is</p>
<p>$$\int_{0}^{1/2} \frac{1}{|\sqrt{x}\ln(x)|^p}\, dx$$</p>
<p>I am aware that this integral converges for $p=2$ (that's not too hard to show). I also believe that this integral diverges for $p>2$...but how can I show that using elementary calculus and related techniques (compariso... | Jack D'Aurizio | 44,121 | <p>By enforcing the substitution $x=e^{-z}$ we get
$$ \int_{0}^{1/2}\frac{dx}{\left(-\sqrt{x}\log x\right)^p} = \int_{\log 2}^{+\infty}\exp\left[\left(\frac{p}{2}-1\right)z\right]\frac{dz}{z^p} $$
and we clearly need $p\leq 2$ to ensure the (improperly-Riemann or Lebesgue)-integrability of $\exp\left[\left(\frac{p}{2}-... |
2,478,229 | <p>The matter of interest is</p>
<p>$$\int_{0}^{1/2} \frac{1}{|\sqrt{x}\ln(x)|^p}\, dx$$</p>
<p>I am aware that this integral converges for $p=2$ (that's not too hard to show). I also believe that this integral diverges for $p>2$...but how can I show that using elementary calculus and related techniques (compariso... | Jürg W. Spaak | 475,371 | <p>A simple change of variables does the trick: $y=1/x$</p>
<p>$$\int_0^\frac12\frac1{(-\sqrt x\log(x))^p}dx = \int_2^\infty\frac1{-y^{-p/2}\log(\frac1y)^p}\frac1{y^2}=\\
=\int_2^\infty \frac{y^{\frac p2-2}}{\log(y)^p}dy$$</p>
<p>The last integral converges iff $\frac p2-2\leq-1$, which is equivalent to $p\leq2$.</p>... |
145,612 | <p>Why are isosceles triangles called that — or called anything? Why is their class given a name? Why did they find their way into the <em>Elements</em> and every single elementary geometry text and course ever since? Did no one ever ask himself, "What use is this, or why is it interesting?"?</p>
<p>Here are som... | Mark Bennet | 2,906 | <p>Here are a couple of very practical reasons:</p>
<p>If you are an engineer and you have two long pieces of wood or metal which you want to secure a fixed distance from each other, you might just have to hand a number of standard pieces of the same length (I think about making a crane from a Meccano set). Then you a... |
1,474,067 | <p>A silly example:</p>
<p>$\exists x (P (x, x)) \leftrightarrow \exists x\forall x (P (x, x))$</p>
<p>Intuition tells me that, because we're dealing with the same variable, the Exists on the right side is of no importance, so that side of the equation would be equivalent to $\forall x (P (x, x))$.</p>
<p>Now, conce... | Noah Schweber | 28,111 | <p>"$\exists x \forall x(P(x, x))$" is not a well-formed formula - you're not allowed to overload variables like this, precisely because it leads to ambiguity.</p>
|
3,625,627 | <p>If <span class="math-container">$S = \sum_{n=1}^{243} \frac{1}{n^{4/5}} $</span>. </p>
<p>Find the value of <span class="math-container">$\lfloor S \rfloor$</span> where <span class="math-container">$\lfloor \cdot \rfloor$</span> represents the greatest integer function.</p>
<p>By approximation using definite inte... | Misha Lavrov | 383,078 | <p>You're getting a very loose approximation with the integral if you start at <span class="math-container">$0$</span>, because <span class="math-container">$\int_0^1 x^{-4/5}\,dx = 5$</span>. That's the majority of the error.</p>
<p>To avoid this, write <span class="math-container">$$S = 1 + \sum_{n=2}^{243} \frac1{n... |
3,625,627 | <p>If <span class="math-container">$S = \sum_{n=1}^{243} \frac{1}{n^{4/5}} $</span>. </p>
<p>Find the value of <span class="math-container">$\lfloor S \rfloor$</span> where <span class="math-container">$\lfloor \cdot \rfloor$</span> represents the greatest integer function.</p>
<p>By approximation using definite inte... | sammy gerbil | 203,175 | <p>You will not find an exact value. You are probably expected to find an approximation. </p>
<p>You could use the <strong>trapezium rule</strong> to approximate the integral </p>
<p><span class="math-container">$$\int_1^{243}f(x)dx \approx \frac12(f_1+f_2)+\frac12(f_2+f_3)+...\frac12(f_{241}+f_{242})+\frac12(f_{242... |
3,625,627 | <p>If <span class="math-container">$S = \sum_{n=1}^{243} \frac{1}{n^{4/5}} $</span>. </p>
<p>Find the value of <span class="math-container">$\lfloor S \rfloor$</span> where <span class="math-container">$\lfloor \cdot \rfloor$</span> represents the greatest integer function.</p>
<p>By approximation using definite inte... | Claude Leibovici | 82,404 | <p>If you know generalized harmonic numbers
<span class="math-container">$$S_p = \sum_{n=1}^{p} \frac{1}{n^{4/5}}=H_p^{\left(\frac{4}{5}\right)}$$</span> Using asymptotics
<span class="math-container">$$S_p=5 p^{1/5}+\zeta \left(\frac{4}{5}\right)+\frac{1}{2}
\frac{1}{p^{4/5}}+\cdots$$</span></p>
<p>For <span class... |
390,532 | <p>I'm trying to solve (for $x$) some problems such as $\arctan(0)=x$, $\arcsin(-\frac{\sqrt{3}}{{2}})=x$, etc.</p>
<p>What is the best way to go about this? So far, I have been trying to solve the problems intuitively (e.g. I ask myself <em>what value of sine will give me $-\frac{\sqrt{3}}{{2}}$?</em>), maybe drawing... | DonAntonio | 31,254 | <p>You need to know the basic values of the trigonometric functions, thus for example:</p>
<p>$$\sin\left(-\frac{\pi}3\right)=\sin\left(-\frac{2\pi}3\right)=\frac{\sqrt3}2\implies\arcsin\left(-\frac{\sqrt3}2\right)\in\left\{\;-\frac{\pi}3\;,\;-\frac{2\pi}3\;\right\}$$</p>
<p>so you must know where your values' range ... |
2,574,768 | <p>There are three planes, <strong>A</strong>, <strong>B</strong>, and <strong>C</strong>, all of which intersect at a single point, <strong>P</strong>. The angles between the planes are given: $$\angle\mathbf{AB}=\alpha$$ $$\angle\mathbf{BC}=\beta$$ $$\angle\mathbf{CA}=\gamma$$ $$0\lt\alpha,\beta,\gamma\le\frac{\pi}{2... | Narasimham | 95,860 | <p>If you call the last mentioned angles as $a,b,c$ and the spherical triangle $ABC$ then from Spherical Trigonometry (sphere has center at $P$,) the Law of Sines is valid:</p>
<p>$$ \dfrac{\sin \alpha}{\sin a} = \dfrac{\sin \beta}{\sin b} =\dfrac{\sin \gamma}{\sin c}= \dfrac{6\, Volume\, PABC}{\sin a\sin b\sin c } $... |
64,613 | <p>EDIT: I meant to have the coefficients reversed, showing:
$$\frac{n}{n-1}(1-(1-x)^n)^n + (1-x)^{n-1} \leq 1$$
This version should be true.. but still trying to prove it...</p>
<p>ORIGINAL:
Is it possible to show:
$$(1-(1-x)^n)^n + \frac{n}{n-1}(1-x)^{n-1} \leq 1$$ for $0<x<1$ and $n\geq 2$ (and $n$ is an ... | Brian M. Scott | 12,042 | <p>It fails already for $n=2$. Make the substitution suggested by Thijs, and the left-hand side becomes $(1-y^2)^2+2y = 1+y^4 +2y(1-y) > 1$ for $0<y<1$.</p>
|
4,390,855 | <p>For reference: In triangle ABC, <span class="math-container">$S_1$</span> and <span class="math-container">$S_2$</span> are areas of the shaded regions. If <span class="math-container">$S_1 \cdot{S}_2=16 cm^4$</span>, calculate <span class="math-container">$MN$</span>.</p>
<p><a href="https://i.stack.imgur.com/KRiJ1... | Chris Sanders | 309,566 | <p>To avoid ambiguity, I will call the p-adic integers <span class="math-container">$\mathbb{Z}_{pa}$</span>.</p>
<p>Here's a simple question. What is the ideal <span class="math-container">$I\subset\mathbb{Z}_{pa}$</span> of scalars <span class="math-container">$x$</span> such that <span class="math-container">$x\cdot... |
4,390,855 | <p>For reference: In triangle ABC, <span class="math-container">$S_1$</span> and <span class="math-container">$S_2$</span> are areas of the shaded regions. If <span class="math-container">$S_1 \cdot{S}_2=16 cm^4$</span>, calculate <span class="math-container">$MN$</span>.</p>
<p><a href="https://i.stack.imgur.com/KRiJ1... | Captain Lama | 318,467 | <p>No, it's not possible. If <span class="math-container">$q\in \mathbb{Z}$</span> is not divisible by <span class="math-container">$p$</span>, then <span class="math-container">$1/q\in \mathbb{Z}_p$</span>, so for any <span class="math-container">$\mathbb{Z}_p$</span>-module <span class="math-container">$M$</span> and... |
2,333,702 | <p>Firstly, I have opened the brackets and solved using both compositions of trig and inverse trig functions and using right triangle (the results were the same):$$\arcsin(\frac{40}{41})=\gamma$$$$\frac{40}{41}=\sin\gamma.$$ Coming from the Pythagoras theorem, the adjacent side to $\gamma^\circ$ is 9, so $\cos(\gamma)=... | Angina Seng | 436,618 | <p>You could apply Chebotarev to $\Bbb Q(\sqrt{m},\sqrt{-3},2^{1/3})$
which is Galois with Galois group $S_2\times S_3$ of order $12$.
I reckon the Frobeniuses of the $p$ you seek are of the form $(e,\sigma)$
where $e$ is the identity of $S_2$ and $\sigma$ is a $3$-cycle. There
are two of these, so the Dirichlet densit... |
1,397,576 | <p>To me there is a hierarchy where vectors $\subset$ sequences $\subset$ functions $\subset$ operators</p>
<ul>
<li><p>All vectors are sequences, but not all sequences are vectors because
sequences are infinite dimensional</p></li>
<li><p>All sequences are functions, but not all functions are sequences
because functi... | Daniel Hast | 41,415 | <ol>
<li>Vectors are not sequences. A vector is an element of a vector space; identifying vectors with tuples of numbers requires a choice of basis of the vector space. (For example, when we write elements of $\mathbb{R}^n$ as $n$-tuples of real numbers, we're implicitly using the standard basis $(1, 0, \dots, 0), \dot... |
450,410 | <p>I'm trying to teach myself how to do $\epsilon$-$\delta$ proofs and would like to know if I solved this proof correctly. The answer given (Spivak, but in the solutions book) was very different.</p>
<hr>
<p><strong>Exercise:</strong> Prove $\lim_{x \to 1} \sqrt{x} = 1$ using $\epsilon$-$\delta$.</p>
<p><strong>My ... | math4fun | 109,143 | <p>Using your work here is another flavor of this proof:</p>
<p>Let $\epsilon >0$, and put
$\delta= \epsilon(\sqrt{x}+1)$.</p>
<p>Assume $0<|x−1|<\delta$.</p>
<p>Then
$$|F(x)−L|=|x−1|
=∣(\sqrt{x}−1)(\sqrt{x}+1)∣.$$
By our assumption that $0<|x−1|<\delta$, we have
$$|F(x) - L| ... |
920,050 | <p>The answer is $\frac1{500}$ but I don't understand why that is so. </p>
<p>I am given the fact that the summation of $x^{n}$ from $n=0$ to infinity is $\frac1{1-x}$. So if that's the case then I have that $x=\frac15$ and plugging in the values I have $\frac1{1-(\frac15)}= \frac54$.</p>
| Kim Jong Un | 136,641 | <p>$$
\sum_{n=4}^\infty\frac{1}{5^n}=\frac{1}{5^4}\sum_{n=4}^\infty\frac{1}{5^{n-4}}=\frac{1}{5^4}\sum_{m=0}^\infty\frac{1}{5^m}=\frac{1}{5^4}\frac{1}{1-1/5}=\frac{1}{500}.
$$</p>
|
708,596 | <p>Suppose that $U$ and $V$ are vector spaces, and that $f:V \to W$ is a linear map. Suppose also that $u$ and $v$ are vectors in $V$ such that $f(u)=f(v)$. Show that there is a vector $w \in \ker f $ such that $v=u+w$.</p>
<p>I roughly understand what is kernel and its definition but I have no idea how to apply it to... | user133458 | 133,458 | <p>So you have: $$f(u) = f(v)$$
$$f(v) - f(u) = 0$$
$$f(v-u) = 0$$
$$f(w) = 0$$</p>
<p>And if you know the definition of kernel, it just follows from there.</p>
|
107,915 | <p>I randomly place $k$ rooks on an (arbitrarily sized) $N$ by $M$ chessboard. Until only one rook remains, for each of $P$ time intervals we move the pieces as follows:</p>
<p>(1) We choose one of the $k$ rooks on the board with uniform probability. </p>
<p>(2) We choose a direction for the rook, $(N, W, E, S)$, w... | Omer | 9,422 | <p>Depending on the precise model (see NOTE 2), each rook has probability of order $1/n$ of capturing another in a given step (they need to be on the same row, and then the probability is $O(1)$. Note also that the positions of rooks are mixed very quickly (this random walk mixes in $O(1)$ steps.</p>
<p>This brings th... |
2,318,669 | <p>So I have this differential equation</p>
<p>$-0.4 \cdot 9.81+\frac{1}{100}v^2=0.4 v'$</p>
<p>I was able to solve it which gives me </p>
<p>$\ln(\frac{v+20}{v-20})=t+c$</p>
<p>My problem is I can't isolate $v$ after that i get it in this form and also when I try to find the constant $c$ knowing that $v(0) = 0$ I ... | Lutz Lehmann | 115,115 | <p>The full equation for free fall under air friction reads
$$
m\dot v=-c|v|v-mg
$$
which has only one stationary point $v_\infty=-\sqrt{\frac{mg}c}$. Then the equation can be reformulated in new constants as
$$
\dot v = -b(|v|v+v_\infty^2)
$$
For $v\le 0$ partial fraction decomposition and integration leads to an expr... |
3,239,185 | <p>Let <span class="math-container">$f,g$</span> be two analytic functions on the domain <span class="math-container">$\Omega$</span> such that <span class="math-container">$|f(z)|=|g(z)|$</span> throughout <span class="math-container">$\Omega$</span>.</p>
<p>I believe <span class="math-container">$h(z)=f/g$</span> on... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$f$</span> has a zero of order <span class="math-container">$n$</span> at <span class="math-container">$z_0$</span> then <span class="math-container">$f(z)=(z-z_0)^{n}h(z)$</span> with <span class="math-container">$h$</span> analytic and non-zero in a neighborhood of <span class="mat... |
375,549 | <p>I need to solve this recurrence equation with the help of Generating Functions in Combinatorics.</p>
<p>Given:
$$f(0) = 0 , f(1) = 1, f(n) = 10f(n-1) - 25f(n-2) \forall n \geq 2$$</p>
<p>So I said the following:</p>
<p>$$f(n) = \sum_{n=2}^{\infty} {10(n-1)x^n} - \sum_{n=2}^{\infty} {25(n-2)x^n}$$</p>
<p>Is that ... | Community | -1 | <p>The generating function is
$$g(x) = \sum_{k=0}^{\infty} f(k) x^k = f(0) + f(1) x + \sum_{k=2}^{\infty} f(k) x^k = x + \sum_{k=2}^{\infty}(10f(k-1) - 25f(k-2))x^k$$
Hence,
$$g(x) = x + 10 x \sum_{k=1}^{\infty} f(k) x^k -25x^2 \sum_{k=0}^{\infty} f(k) x^k = x+10x g(x) - 25x^2 g(x)$$
This gives us
$$g(x) = \dfrac{x}{(5... |
538,811 | <p>Suppose A(.) is a subroutine that takes as input a number in binary, and takes linear time (that is, O(n), where n is the length (in bits) of the number).
Consider the following piece of code, which starts with an n-bit number x.</p>
<p>while x>1:</p>
<p>call A(x)</p>
<p>x=x-1</p>
<p>Assume that the subtraction ... | Nate Neuhaus | 185,039 | <p>From the looks of it, this array will run infinitely due to the fact that the recursive element is called before x is decremented, thus the function will never terminate. </p>
<p>If you called x=x-1 before you recursively called A, then that would allow the function to effectively decrement at each level of recursi... |
2,163,948 | <p><strong>Question:</strong></p>
<blockquote>
<p>Does there exist a Riemannian manifold, with a point $p \in M$, and <strong>infinitely many</strong> points $q \in M$ such that there is <strong>more than one</strong> minimizing geodesic from $p$ to $q$?</p>
</blockquote>
<p><strong>Edit:</strong></p>
<p>As demons... | Narasimham | 95,860 | <p>Between two points $(p,q)$ on any surface of revolution there are indefinitely many geodesic trajectories possible.</p>
<p>Just as you can throw a stone between two points $(p,q)$ situated at different heights choosing a different parabola at different angle of slope or angle of attack.</p>
<p>In a boundary val... |
633,799 | <p>I am a little confused about the basic definition of inclusion.</p>
<p>I understand that, for example, $\{4\}\subset\{4\}$.</p>
<p>I also understand that $4\in\{4\}$, and that it is false to say that $\{4\}\in\{4\}$.</p>
<p>However, is it possible to say that $4\subset\{4\}$?</p>
| DonAntonio | 31,254 | <p>$\;4\;$ is an element of the set $\;\{4\}\;$, so the symbol $\;\subset\;$ doesn't fit it. You need the internationally accepted symbol of curly parentheses {} or any other accepted notation in order to make clear it is a set.</p>
<p>For example, if $\;X = \{ 1,\{1\}\}\;$ , then we both have $\;1\in X\,,\,\{1\}\in X... |
633,799 | <p>I am a little confused about the basic definition of inclusion.</p>
<p>I understand that, for example, $\{4\}\subset\{4\}$.</p>
<p>I also understand that $4\in\{4\}$, and that it is false to say that $\{4\}\in\{4\}$.</p>
<p>However, is it possible to say that $4\subset\{4\}$?</p>
| apnorton | 23,353 | <p>There are two symbols, here, and I think you may be getting them confused. I would suggest to <em>not</em> use the word "inclusion" (at least, not all the time) because that a different meaning in English than in math.</p>
<p>The $\in$ symbol is used to designate if something is <em>inside of</em> a set. That is,... |
1,718,380 | <p>Simply: How do I solve this equation for a given $n \in \mathbb Z$?</p>
<p>$x^x = n$</p>
<p>I mean, of course $2^2=4$ and $3^3=27$ and so on. But I don't understand how to calculate the reverse of this, to get from a given $n$ to $x$. </p>
| Bumblebee | 156,886 | <p>See this wikipedia article: <a href="https://en.wikipedia.org/wiki/Lambert_W_function" rel="nofollow">Lambert W function</a> </p>
<p>If $x^x=n,$ then $$x=\dfrac{\ln n}{W(\ln n)},$$ Where $W$ is the Lambert W function.</p>
|
1,698,376 | <p>I need some guidance with the following proof:</p>
<p>Let V be a finite dimensional vector space, and V* its dual.<br>
Let C = $(f1, ... , fn)\subset{V*}$ be a basis for V*.<br>
Let $w\in{V*}$.<br>
Prove that there exists $B\subset{V}$ such that C is dual for B.</p>
<p>Here's what I have so far:<br>
Since C is a b... | MooS | 211,913 | <p>The dual basis $f_1^*, \dotsc, f_n^*$ in $V^{**}$ of $C=\{f_1, \dotsc, f_n \}$ gives rise to a basis $B$ of $V$ via the canonical isomorphism $j:V \to V^{**}$. The dual of $B$ is $C$.</p>
<p><em>Proof:</em> We compute:</p>
<p>$$\delta_{ab}=f_a^*(f_b)=j(j^{-1}(f_a^*))(f_b)=f_b(j^{-1}(f_a^*)),$$</p>
<p>hence the el... |
1,698,376 | <p>I need some guidance with the following proof:</p>
<p>Let V be a finite dimensional vector space, and V* its dual.<br>
Let C = $(f1, ... , fn)\subset{V*}$ be a basis for V*.<br>
Let $w\in{V*}$.<br>
Prove that there exists $B\subset{V}$ such that C is dual for B.</p>
<p>Here's what I have so far:<br>
Since C is a b... | martini | 15,379 | <p>Define $i_V \colon V \to V^{**}$ as follows: For $w \in V^*$, $v \in V$ let
$$ i_V(v)(w) = w(v) $$
that is $i_V(v)$ is "<em>evaluation at $v$</em>". Then $i_V$ is linear and one-to-one: If $v \ne 0$, extend $v$ to a basis $B$ of $V$ and define $w \in V^*$ by $w(v) = 1$, $w(b) = 0$, $b \in B \setminus \{v\}$. Then
... |
2,265,203 | <p>I was reading the paper</p>
<p><a href="https://aimsciences.org/journals/pdfs.jsp?paperID=1058&mode=full" rel="nofollow noreferrer">Dynamical models of tuberculosis and their applications</a> </p>
<p>by Castillo-Chavez, Song B. and it says </p>
<blockquote>
<p>" it is clear that the matrix
$$
D_xf=
\begin... | Dmitry | 310,971 | <p>This matrix has a simple zero eigenvalue because of the parameter $\phi$, which is chosen in a particular way, $\phi=\frac{(k+\mu)(\mu+r+d)}{k}$. For any other choice of $\phi$ there won't be a zero eigenvalue.</p>
|
372,211 | <p>I'm trying to write an <a href="http://developer.android.com/reference/android/view/animation/Interpolator.html" rel="nofollow noreferrer">interpolator</a> for a translate animation, and I'm stuck. The animation passes a single value to the function. This value maps a value representing the elapsed fraction of an an... | André Nicolas | 6,312 | <p>This answers a problem motivated by your description, but it may not be the problem you want to solve. We assume constant velocity $k$ for $0\le t\le 0.5$. Then we have constant deceleration, so that at time $t=1$ velocity reaches $0$.</p>
<p>Under these conditions, you may want the <em>net displacement</em> at tim... |
911,075 | <p>This is one of my first proofs about fields.
Please feed back and criticise in every way (including style and details).</p>
<p>Let $(F, +, \cdot)$ be a field.
Non-trivially, $\textit{associativity}$ implies that any parentheses are meaningless.
Therefore, we will not use parentheses.
Therefore, we will not use $\te... | Community | -1 | <p>A few pointers:</p>
<ul>
<li><p>You don't have to use "Now". You could just say "Let $a\in F$."</p></li>
<li><p>Don't say "meaningless". Rather, phrase it like so:</p>
<blockquote>
<p>Non-trivially, associativity implies that any parentheses are redundant. Hence, parenthesis will be suppressed and we will thus n... |
1,425,519 | <p>I'm trying to solve <a href="http://poj.org/problem?id=2140" rel="nofollow">this problem</a> on POJ and I thought that I had it. Since I can't figure out what's wrong with my code, I'd like to test it against a huge list of correct answers. This will make my code much easier to debug.</p>
<p>If you don't want to go... | Caleb Stanford | 68,107 | <p><strong>Why your algorithm doesn't work:</strong>
You need to allow the set of consecutive integers to be negative.
For instance, you say in your code that $1$ and $2$ have only one solution.
But each of them have two solutions:
$$
1 = 1 \;;\; 1 = 0 + 1 \\
2 = 2 \;;\; 2 = -1 + 0 + 1 + 2
$$
In fact, the answer will a... |
1,425,519 | <p>I'm trying to solve <a href="http://poj.org/problem?id=2140" rel="nofollow">this problem</a> on POJ and I thought that I had it. Since I can't figure out what's wrong with my code, I'd like to test it against a huge list of correct answers. This will make my code much easier to debug.</p>
<p>If you don't want to go... | Mark Bennet | 2,906 | <p>Either the sum of consecutive integers will contain an odd number of integers or an even number. Let's deal with the odd case first - and let the middle number be $n$ with the total being $N$ and $2r+1$ consecutive integers involved. Then the sum is $$N=(n-r)+(n-r+1)+\dots +(n-1)+n+(n+1)+\dots +(n+r)=(2r+1)n$$</p>
... |
70,582 | <p>For which n can $a^{2}+(a+n)^{2}=c^{2}$ be solved, where $a,b,c,n$ are positive integers?
I have found solutions for $n=1,7,17,23,31,41,47,79,89$ and for multiples of $7,17,23$...
Are there infinitely many prime $n$ for which it is solvable? </p>
| Peđa | 15,660 | <p>If you solve expression for $n$ you get </p>
<p>$n=\sqrt{c^2-a^2}-a$, let's denote $b=\sqrt{c^2-a^2}$,so we have that $n=b-a$</p>
<p>Now,take look at picture bellow.Note that $AD=a$,and $BD=b-a=n$</p>
<p>If you change value of $b$ and keep $a$ to be constant you will get a infinite number of right triangles,and t... |
1,990,670 | <blockquote>
<p>Assume that $0 < \theta < \pi$. Solve the following equation for $\theta$. $$\frac{1}{(\cos \theta)^2} = 2\sqrt{3}\tan\theta - 2$$ </p>
</blockquote>
<p><a href="https://i.stack.imgur.com/SoU8A.png" rel="nofollow noreferrer">Question and Answer</a></p>
<p>Regarding to the attached image, that... | hamam_Abdallah | 369,188 | <p>Your equation can be written as</p>
<p>$$\frac{1}{\cos^2(x)}=$$</p>
<p>$$1+\tan^2(x)=2\sqrt{3}\tan(x)-2$$
or</p>
<p>$$\tan^2(x)-2\sqrt{3}\tan(x)+3=0$$</p>
<p>the reduced discriminant is</p>
<p>$$\delta=3-3=0$$</p>
<p>thus, there is one solution given by</p>
<p>$\tan(x)=\sqrt{3}$ which gives</p>
<p>$$x=\frac{... |
1,774,294 | <p>If you have $$y^2=2x^2+C$$</p>
<p>why is this not equivalent to</p>
<p>$$y=\sqrt{2x^2}+C$$</p>
| Kenny Lau | 328,173 | <p>When you square the second equation, you would get:</p>
<p>$$y^2=2x^2+2C\sqrt{2x^2}+C^2$$</p>
<p>The rest of the proof is left to the reader as an exercise.</p>
<hr>
<p>Extra:</p>
<hr>
<p>$$y^2=2x^2+C$$</p>
<p>$$2y\mathrm dy=2x\mathrm dx$$</p>
<p>$$\frac{\mathrm dy}{\mathrm dx}=\frac xy=\pm\frac{x}{\sqrt{2x^... |
1,774,294 | <p>If you have $$y^2=2x^2+C$$</p>
<p>why is this not equivalent to</p>
<p>$$y=\sqrt{2x^2}+C$$</p>
| Soham | 242,402 | <p>Hint:-</p>
<p>The first $C$ and the second $C$ are different.</p>
<p>Its $y=\sqrt{2x^2+C}$ and $\sqrt{a+b}\neq \sqrt a+\sqrt b$</p>
|
1,774,294 | <p>If you have $$y^2=2x^2+C$$</p>
<p>why is this not equivalent to</p>
<p>$$y=\sqrt{2x^2}+C$$</p>
| Rebellos | 335,894 | <p>Because by square rooting you get : $\sqrt{2x^2 + C}$. Then constant C is "inside" the square root.</p>
|
3,270,944 | <p>Let <span class="math-container">$A$</span> be a bounded linear operator on a separable Hilbert space <span class="math-container">${\cal H}$</span>, and suppose that <span class="math-container">$A$</span> is distinct from its adjoint <span class="math-container">$A^*$</span>. </p>
<p><strong>Question:</strong> Ca... | Dr. Sonnhard Graubner | 175,066 | <p>It is <span class="math-container">$$\frac{2(x+h)^2+1-2x^2-1}{h}=\frac{2x^2+4xh+2h^2-2x^2}{h}$$</span></p>
|
70,176 | <p>So I can do something like this which I like:</p>
<pre><code>Manipulate[i, {i, {1,2,3,4}}]
</code></pre>
<p>It lets me pick which specific values I want to allow to be chosen for my function. But that list appears to be very limiting.</p>
<p>Lets say I have a list and each element contains a list of two elements ... | Rom38 | 10,455 | <p>I suspect that you can obtain what you need by following way:</p>
<pre><code>list = {{1, 2}, {3, 4}, {5, 6}};
Manipulate[list[[i]], {i, 1, Length@list, 1}]
</code></pre>
<p>This code always gives you the element (sublist) of initial list. </p>
|
1,177,349 | <p>Let $\gamma = e^{2 \pi i/5} + (e^{2 \pi i/5})^4
%γ = e<sup>2πi / 5</sup> + (e<sup>2πi / 5</sup>) <sup>4</sup>
$.</p>
<p>I am looking for the basis for $[\mathbb{Q}(\gamma):\mathbb{Q}] = 2$, and then looking for a dependence between $\gamma^2,\gamma$, and $1$. </p>
<p>I've worke... | user26486 | 107,671 | <p>$$x^2-y^2=(x-y)(x+y)$$</p>
<p>This is less than $0$, since it is given that $x<y\iff x-y<0$ and $x,y>0\implies x+y>0$.</p>
|
4,285,426 | <p>Intuitively it is quite easy to see why <span class="math-container">$$a \equiv (a \bmod m) \pmod m.$$</span></p>
<p>When you divide a by m you get a remainder in the range <span class="math-container">$0, \dots, m-1.$</span> When you divide the remainder by m again, you get the same number again as the remainder, ... | David | 651,991 | <p>Let <span class="math-container">$q$</span> and <span class="math-container">$r$</span> be integers such that <span class="math-container">$a=qm+r$</span> (with <span class="math-container">$0 \leq r < m$</span>). It's easy to see that <span class="math-container">$a \mod m$</span> is precisely <span class="math-... |
1,271,935 | <p>Let $(e_n)$ (where $ e_n $ has a 1 in the $n$-th place and zeros otherwise) be unit standard vectors of $\ell_\infty$. </p>
<p>Why is $(e_n)$ not a basis for $\ell_\infty$?</p>
<p>Thanks.</p>
| Ian | 83,396 | <p>There are basically two things to note here. First you need to understand what it means for a set to be a basis of an infinite dimensional normed space. In your context, I am all but certain that this means that it is a Schauder basis, which is a linearly independent set such that the set of all finite linear combin... |
28,195 | <p>So yesterday I came across a question. Something seemed suspicious (a badly worded question and an incorrect answer accepted), so I did some snooping. It appears that every question from the OP has been answered by the same user within minutes of posting, and subsequently upvoted and accepted. </p>
<p>I suspect ... | Community | -1 | <p>Yes this is bad. Yes this is against the rules. You should flag for moderator intervention (I did). Thanks for bringing this up.</p>
|
2,392,114 | <p>It is possible to rewrite the equation $x^3+ax^2+bx+c=0$ as $y^3+3hy+k=0$ by setting $y=x+a/3$</p>
<p>How do you find the coefficient h in the equation $y^3+3hy+k=0$?</p>
| Khosrotash | 104,171 | <p>If you change the (0,o) origin to the Inflection point of cubic equation , you will have that form
$$y=x^3+ax^2+bx+c \\y'=3x^2+2ax+b\\y''=6x+2a=0 \to x=-\frac{a}{3}$$ you must change $$x \mapsto x-\frac{a}{3}$$
$$y=f(x)=y=x^3+ax^2+bx+c \to \\(x-\frac{a}{3})^3+a(x-\frac{a}{3})^2+b(x-\frac{a}{3})+c\\=x^3-3x^2\frac{a}... |
311,849 | <p>How to evaluate:
$$ \int_0^\infty e^{-x^2} \cos^n(x) dx$$</p>
<p>Someone has posted this question on fb. I hope it's not duplicate.</p>
| Shobhit Bhatnagar | 59,380 | <p>I found a way to do it for $n \in \mathbb{N}$. We begin with</p>
<p>$$\cos^n(x)=\left(\frac{e^{ix}+e^{-ix}}{2}\right)^n = \frac{1}{2^n e^{inx}}(1+e^{2ix})^n = \frac{1}{2^n e^{inx}}\sum_{r=0}^n \binom{n}{r}e^{2irx}$$</p>
<p>Therefore</p>
<p>$$\begin{aligned}\int_{-\infty}^\infty e^{-x^2}\cos^n(x)dx &=\int_{-\i... |
266,124 | <p>A palindrome is a number or word that is the same when read forward
and backward, for example, “176671” and “civic.” Can the number obtained by writing
the numbers from 1 to n in order (n > 1) be a palindrome?</p>
| Mark Bennet | 2,906 | <p>Here's a beginning of the task where the numbers do not have to be in order. Note that a palindrome can have at most one digit which occurs an odd number of times - the centre digit if the number of digits is odd.</p>
<p>Now after 1, you have to have all digits 1-9 - nine digits. If you stop below 100 you will alwa... |
3,475,893 | <p>Is <span class="math-container">$ \mathbb{Q} \times \mathbb{Q[i]}$</span> an integral domain ?</p>
<p>My attempt : I know that <span class="math-container">$ \mathbb{Q} \times \mathbb{Q}$</span> is not integral domain take <span class="math-container">$(0,1) \times (1,0) =( 0,0)$</span></p>
<p>But im confuse... | MANI | 464,799 | <p>You may use same argument to show <span class="math-container">$\mathbb{Q}\times \mathbb{Q}[i]$</span> is not an integral domain as <span class="math-container">$(q,0)\times (0,q')=(0,0),$</span> for any two non zero rational number <span class="math-container">$q,q'.$</span></p>
<p>Infact if <span class="math-cont... |
1,044,910 | <blockquote>
<p>Prove that $$\sum_{i = 2^{n-1} + 1}^{2^n}\frac{1}{a + ib} \ge \frac{1}{a + 2b}$$</p>
</blockquote>
<p>I tried to to prove the above statement using the AM-HM inequality:</p>
<p>$$\begin{align}\frac{1}{2^n - 2^{n-1}}\sum_{i = 2^{n-1} + 1}^{2^n}\frac{1}{a + ib} &\ge \frac{2^n - 2^{n-1}}{\sum_{i = ... | robjohn | 13,854 | <p>Assuming $a,b\gt0$, we get
$$
\begin{align}
\sum_{i=2^{n-1}+1}^{2^n}\left(\frac1{a+ib}-\frac{2^{-n+1}}{a+2b}\right)
&\ge\sum_{i=2^{n-1}+1}^{2^n}\left(\frac1{a+2^nb}-\frac{2^{-n+1}}{a+2b}\right)\\
&=\sum_{i=2^{n-1}+1}^{2^n}\frac{a(1-2^{-n+1})}{(a+2^nb)(a+2b)}\\
&=\frac{a(2^{n-1}-1)}{(a+2^nb)(a+2b)}\\[12pt... |
1,044,910 | <blockquote>
<p>Prove that $$\sum_{i = 2^{n-1} + 1}^{2^n}\frac{1}{a + ib} \ge \frac{1}{a + 2b}$$</p>
</blockquote>
<p>I tried to to prove the above statement using the AM-HM inequality:</p>
<p>$$\begin{align}\frac{1}{2^n - 2^{n-1}}\sum_{i = 2^{n-1} + 1}^{2^n}\frac{1}{a + ib} &\ge \frac{2^n - 2^{n-1}}{\sum_{i = ... | Did | 6,179 | <p>A generalized version of the result might be easier to prove:</p>
<blockquote>
<p>For every positive integers $k$ and $m$, $$\sum_{i=k+1}^{k(m+1)}\frac1{a+ib}\geqslant\frac{m}{a+(m+1)b}.$$</p>
</blockquote>
<p>The question asks about the case $k=2^{n-1}$ and $m=1$.</p>
<p>To prove the claim, note that $a+ib\leq... |
162,836 | <p>I would like to find the surface normal for a point on a 3D filled shape in Mathematica. </p>
<p>I know how to calculate the normal of a parametric surface using the cross product but this method will not work for a shape like <code>Cone[]</code> or <code>Ball[]</code>.</p>
<ol>
<li>Is there some sort of <code>Reg... | Tomi | 36,939 | <p>Wether this is enough to warrant an "answer" is debatable and it relies on MichaelE2's work, but I felt it was helpful to share. </p>
<p>Using Michael E2's solution, we can plot and clearly see the normals for 3D shapes. </p>
<pre><code>numberofpoints = 50;
pts = RandomPoint[RegionBoundary[shape], numberofpoints]... |
499,044 | <p>I "know" that $\mathbb{C} \otimes_\mathbb{R} \mathbb{C} \cong \mathbb{C} \oplus \mathbb{C}$ as rings, but I don't really know it, what I mean with this is that I don't know any explicit isomorphism $f: \mathbb{C} \otimes_\mathbb{R} \mathbb{C} \rightarrow \mathbb{C} \oplus \mathbb{C}$. I suspect that such an isomorph... | Henry T. Horton | 24,934 | <p>One example of an isomorphism $\varphi: \Bbb C \oplus \Bbb C \longrightarrow \Bbb C \otimes_{\Bbb R} \Bbb C$ is given on generators by
$$\varphi(1, 0) = \tfrac{1}{2}(1 \otimes 1 + i \otimes i),$$
$$\varphi(0, 1) = \tfrac{1}{2}(1 \otimes 1 - i \otimes i).$$</p>
|
4,566,254 | <p>Let <span class="math-container">$F$</span> be a functor <span class="math-container">$\mathscr{C}^\text{op}\times\mathscr{C}\to\mathbf{Set}$</span>, and let <span class="math-container">$S$</span> be an arbitrary set. Can we write the following?
<span class="math-container">$$
\int^{C:\mathscr{C}} S\times F(C,C) \c... | fosco | 685 | <p>The coend <span class="math-container">$\int^c F(c,c)$</span> is a colimit in a category <span class="math-container">$D$</span>, so every left adjoint functor <span class="math-container">$L : D\to E$</span> (a particular example of which is <span class="math-container">$S\times-$</span> in a cartesian closed categ... |
3,018,388 | <p>It is given that the series <span class="math-container">$ \sum_{n=1}^{\infty} a_n$</span> is convergent but not absolutely convergent and <span class="math-container">$ \sum_{n=1}^{\infty} a_n=0$</span>. Denote by <span class="math-container">$s_k$</span> the partial sum <span class="math-container">$ \sum_{n=1}^{k... | Kavi Rama Murthy | 142,385 | <p>The sum is nothing but a Riemann sum for <span class="math-container">$\int_0^{1}\sqrt{1-t^{2}}\, dt$</span>. You can evaluate this by making the substitution <span class="math-container">$t=\sin\, \theta$</span> and using the formula <span class="math-container">$2 \cos ^{2}\, \theta =1+\cos\, (2\theta)$</span> and... |
3,018,388 | <p>It is given that the series <span class="math-container">$ \sum_{n=1}^{\infty} a_n$</span> is convergent but not absolutely convergent and <span class="math-container">$ \sum_{n=1}^{\infty} a_n=0$</span>. Denote by <span class="math-container">$s_k$</span> the partial sum <span class="math-container">$ \sum_{n=1}^{k... | Mostafa Ayaz | 518,023 | <p>This directly leads<br> from fundamental theorem of calculus <br>(<a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus</a>) <br>and<br> the definition of Reimannian sum<br> (<a href="https://en.wikipedia.org/wi... |
2,698,098 | <p>The question:</p>
<blockquote>
<p>Suppose <span class="math-container">$0< \delta < \pi$</span>, <span class="math-container">$f(x) = 1$</span> if <span class="math-container">$|x| \leq \delta$</span>, <span class="math-container">$f(x) = 0$</span> if <span class="math-container">$\delta < |x| \leq \pi$</sp... | mathematics2x2life | 79,043 | <p>For part (a), it might be a bit more helpful to notice that $f$ is even and real-valued so that you may use the real version of the Fourier series. Also because $f$ is even, $b_n=0$ for all $n$. You should then be able to calculate $a_n$ for $n \geq 0$ (the answer, no work, below)</p>
<blockquote class="spoiler">
... |
3,368,655 | <p>I came across a problem that asked if it is posible for a function to be Riemann integrable function in <span class="math-container">$[0,+\infty)$</span> but also <span class="math-container">$|f(x)|\geq 1$</span> for all <span class="math-container">$x\geq 0$</span>. </p>
<p>At first I thought it was imposible, bu... | copper.hat | 27,978 | <p>Here is a more elementary example:</p>
<p>Let <span class="math-container">$\phi$</span> be a <span class="math-container">$1$</span>-periodic function such that <span class="math-container">$\phi(x)=-1$</span> for <span class="math-container">$x \in [0,{1 \over 2})$</span> and <span class="math-container">$\phi(x)... |
2,168,906 | <blockquote>
<p>The task is to find necessary and sufficient condition on <span class="math-container">$b$</span> and <span class="math-container">$c$</span> for the equation <span class="math-container">$x^3-3b^2x+c=0$</span> to have three distinct real roots.</p>
</blockquote>
<p>Are there any formulas (such as <spa... | Tsemo Aristide | 280,301 | <p>$f(x)=e^{g(x)}$ where $g(x)=x^x=e^{xln(x)}$, $f'(x)=e^{g(x)}g'(x)$, $g'(x)=e^{xln(x)}(ln(x)+1)$.</p>
<p>Your mistake is when you compute $ln(y)$.</p>
|
315,457 | <p>I am trying to evaluate $\cos(x)$ at the point $x=3$ with $7$ decimal places to be correct. There is no requirement to be the most efficient but only evaluate at this point.</p>
<p>Currently, I am thinking first write $x=\pi+x'$ where $x'=-0.14159265358979312$ and then use Taylor series $\cos(x)=\sum_{i=1}^n(-1)^n\... | Renko Usami | 538,693 | <p>A theorem may help you: </p>
<p>Let $A ∈ M_n$. The following are equivalent:<br>
(a) A is irreducible.<br>
(b) $(I + |A|)^{n-1} > 0$.<br>
(c) $(I + M(A))^{n−1} > 0$.<br>
(d) $\Gamma(A)$ is strongly connected. </p>
<p>It is Theorem 6.2.24 in <em>Matrix Analysis, 2nd edition</em>. Go check it if you need a c... |
315,457 | <p>I am trying to evaluate $\cos(x)$ at the point $x=3$ with $7$ decimal places to be correct. There is no requirement to be the most efficient but only evaluate at this point.</p>
<p>Currently, I am thinking first write $x=\pi+x'$ where $x'=-0.14159265358979312$ and then use Taylor series $\cos(x)=\sum_{i=1}^n(-1)^n\... | Community | -1 | <p>Let $A=[a_{i,j}]\in M_n(\mathbb{R})$ and $|A|=[|a_{i,j}|]$. $A$ is irreducible IFF $|A|$ is too. Then we may assume that the $a_{i,j}$ are $\geq 0$. We have a look at the complexity of the problem: "decide whether $A$ is irreducible or not".</p>
<p>Of course, we do not look for a permutation of the basis vectors th... |
1,453,010 | <p>A certain biased coin is flipped until it shows heads for the first time. If the probability of getting heads on a given flip is $5/11$ and $X$ is a random variable corresponding to the number of flips it will take to get heads for the first time, the expected value of $X$ is:
$$E[x] = \sum_{x=1}^\infty{x\frac{5}{1... | David K | 139,123 | <p>The expectation is not a geometric series (at least not when you write
it directly), but its resemblance to a geometric series is a good observation.</p>
<p>First let's get that factor of $\frac{5}{11}$ out of the way,
because it will become annoying at some point if we keep it inside the
summation.
$$E[x] = \sum_{... |
184,601 | <p>A user on the chat asked how could he make something that would cap when it gets a specific value like 20. Then the behavior would be as follows:</p>
<p>$f(...)=...$</p>
<p>$f(18)=18$</p>
<p>$f(19)=19$</p>
<p>$f(20)=20$</p>
<p>$f(21)=20$</p>
<p>$f(22)=20$</p>
<p>$f(...)=20$</p>
<p>He said he would like to pe... | Marc van Leeuwen | 18,880 | <p>$ x \mapsto \min ( x , 20 ) $ </p>
|
245,464 | <p>I only have one region plot and still want to get the legend (both marker and label). I tried the following, but why the legend market does not show up?</p>
<p><code>RegionPlot[x^2 < y^3 + 1 && y^2 < x^3 + 1, {x, -2, 5}, {y, -2, 5}, PlotLegends -> Placed["MyLegend", {0.15, 0.08}]]</code>... | Bob Hanlon | 9,362 | <pre><code>RegionPlot[x^2 < y^3 + 1 && y^2 < x^3 + 1,
{x, -2, 5}, {y, -2, 5},
PlotLegends ->
Placed[SwatchLegend[{x^2 < y^3 + 1 && y^2 < x^3 + 1}], {0.3, .07}]]
</code></pre>
<p><a href="https://i.stack.imgur.com/mZ5Q7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.co... |
962,287 | <p>I am trying to isolate x in the equation $$(x-20)^{2} = -(y-40)^{2} - 525.$$ How can I do it?</p>
| please delete me | 168,166 | <p>If $x$ and $y$ are real, the right side is negative while the left side is non-negative, so the equation never holds.</p>
|
19,586 | <p>I am looking for resources for teaching math modeling to high school teachers with rusty math background. It will be a 6-week course. Some tips/directions on simple projects would be helpful. I would want to introduce coding for numerically solving systems of ODEs.</p>
| mweiss | 29 | <p>I would recommend checking in with the <a href="https://www.mtsu.edu/jstrayer/modules/modules2.php" rel="nofollow noreferrer"><em>MODULE</em><span class="math-container">$(S^2)$</span></a> project (Mathematics Of Doing, Understanding, Learning and Educating for Secondary Schools). A few notes:</p>
<ul>
<li>This pro... |
87,583 | <p>Next task to complete:</p>
<ul>
<li><p>Count <code>*</code>-symbol in such expression as <code>a + s^2*b - c/y + o^3 + n*m*u</code> (in this case count of <code>*</code> should be 6)</p></li>
<li><p>Powers such $o^3$ should be expand to $o*o*o$</p></li>
</ul>
<p>I try, but my code is pretty ugly.</p>
<p><img src=... | Andy Ross | 43 | <p>This doesn't give the result you are looking for exactly because it uses the full form of the expression you give it.</p>
<pre><code>SetAttributes[countTimes, HoldAll];
countTimes[expr_] := Block[{Times, Power, power, times},
power[a_, b_ /; b > 0] := Nest[times[a, #] &, a, b - 1];
power[a_, b_ /; b <... |
87,583 | <p>Next task to complete:</p>
<ul>
<li><p>Count <code>*</code>-symbol in such expression as <code>a + s^2*b - c/y + o^3 + n*m*u</code> (in this case count of <code>*</code> should be 6)</p></li>
<li><p>Powers such $o^3$ should be expand to $o*o*o$</p></li>
</ul>
<p>I try, but my code is pretty ugly.</p>
<p><img src=... | Mr.Wizard | 121 | <p>Accepting that in <em>Mathematica</em> <code>-c/y</code> is automatically converted to <code>-1*c*y^-1</code> and permitting the result shown in Andy's answer I believe we can use a simpler approach, at least for the kind of expression given in example.</p>
<p>Define <code>rules</code> that determine how a <code>Ti... |
2,893,568 | <p>I need some help finding the standard deviation using Chebyshev's theorem. Here's the problem:</p>
<blockquote>
<p>You have concluded that at least $77.66\%$ of the $3,075$ runners took between $60.5$ and $87.5$ minutes to complete the $10$ km race. What was the standard deviation of these $3,075$ runners?</p>
... | Arnaud Mortier | 480,423 | <p>Given a random variable $X$ of finite expectation $\mu$ and standard deviation $\sigma$, Chebyshev's theorem states that $$P(X\not\in (\mu-k\sigma,\mu+k\sigma))\leq \frac{1}{k^2}$$</p>
<blockquote>
<p>The probability of $X$ lying at least $k$ standard deviations away
from the mean is less than or equal to $\fra... |
3,773,695 | <p>I have been trying to get some upper bound on the coefficient of <span class="math-container">$x^k$</span> in the polynomial
<span class="math-container">$$(1-x^2)^n (1-x)^{-m}, \text{ $m \le n$}.$$</span></p>
<p>A straightforward calculation shows that for even <span class="math-container">$k$</span>, the coefficie... | Ned | 67,710 | <p>Consider <span class="math-container">$y=x^3$</span> and <span class="math-container">$y=x^3-x$</span>. For each one:</p>
<p>Let <span class="math-container">$T$</span> be the tangent line through the inflection point at the origin.</p>
<p>Let <span class="math-container">$L$</span> be the line through the origin ro... |
34,724 | <h3>Overview</h3>
<p>For integers n ≥ 1, let T(n) = {0,1,...,n}<sup>n</sup> and B(n)= {0,1}<sup>n</sup>. Note that |T(n)|=(n+1)<sup>n</sup> and |B(n)| = 2<sup>n</sup>.
A certain set S(n) ⊂ T(n), defined below, contains B(n). The question is about the growth rate of |S(n)|. Does it grow exponentially, like |B(n)... | Louigi Addario-Berry | 3,401 | <p><b>Edit</b>: I worked out the details of this exponential upper bound a bit more precisely. It is the case that $S(n) \leq 11*10^n$. </p>
<p>I can only prove an exponential upper bound (rather than an exponential asymptotic), but it can be obtained by weakening your restriction on the vectors $S(n)$ as follows: th... |
3,702,649 | <p>It's obvious that it's symmetric because <span class="math-container">$a_{\left(i+1\right)j}=\left(m+1\right)\left(i+1+j\right) = a_{i\left(j+1\right)}=\left(m+1\right)\left(i+j+1\right)$</span>, but how can I prove that it's a Latin square and that it's diagonal consists of different elements?</p>
<p>I thought abo... | Jeane Z | 818,754 | <p><span class="math-container">$(m+1)$</span> is any number <span class="math-container">$\in$</span> Z. I think , there is relation between <span class="math-container">$m$</span> and <span class="math-container">$n$</span> to get the require . Also, you mean the operation * is usual multiplication.</p>
|
438,070 | <p>I stumbled across this question and I cannot figure out how to use the value of $\cos(\sin 60^\circ)$ which would be $\sin 0.5$ and $\cos 0.5$ seems to be a value that you can only calculate using a calculator or estimate at the very best.</p>
| Emanuele Paolini | 59,304 | <p>Is any linear combination of bounded sequences, a bounded sequence? If yes, $F_1$ is a linear subspace.</p>
<p>Notice that $F_6$ is not a linear subspace...</p>
|
564,195 | <p>Using continuity I was able to show the sequence $x_0 = 1$, $x_{n+1} = sin(x_n)$ converges to 0, but I was wondering if there was a way to prove it using only properties and theorems related to sequences and series, without using continuity.</p>
<p>So far, I know the sequence is monotonically decreasing and bounded... | Ben Grossmann | 81,360 | <p>We know that for all $x \in (0,1]$, we have
$$
0 < \sin x < x
$$
From there, you can show that $x_0,x_1,\dots$ is a <strong>strictly</strong> monotonically decreasing sequence. Is this enough? That is, can we forgo continuity? No. As a counterexample, consider the seuqence $x_k = f(x_{k-1}); x_0 =1$ with
$$
... |
148,032 | <p>What is the larger of the two numbers?</p>
<p>$$\sqrt{2}^{\sqrt{3}} \mbox{ or } \sqrt{3}^{\sqrt{2}}\, \, \; ?$$
I solved this, and I think that is an interesting elementary problem. I want different points of view and solutions. Thanks!</p>
| PolyaPal | 22,004 | <p>In general, we can state two pertinent results: (1) If $a$ and $b$ are positive real numbers such that $b > a \ge e,$, then $a ^ {b} > b ^ {a}$; (2) If a and b satisfy $e \ge b > a > 0$, then $b ^ {a} > a ^ {b}.$</p>
|
88,469 | <p>These vectors form a basis on $\mathbb R^3$: $$\begin{bmatrix}1\\0\\-1\\\end{bmatrix},\begin{bmatrix}2\\-1\\0\\\end{bmatrix} ,\begin{bmatrix}1\\2\\1\\\end{bmatrix}$$</p>
<p>Can someone show how to use the Gram-Schmidt process to generate an orthonormal basis of $\mathbb R^3$?</p>
| Community | -1 | <p>Let's look at this in two dimensions first. After this you should know how to do it in three!</p>
<p>Suppose that you are working in the plane and have two linearly independent vectors $v$ and $w$. You want to make $v$ and $w$ orthogonal to each other in terms of the standard euclidean inner product. How can you do... |
1,243,661 | <p>Let $\Theta$ be an unknown random variable with mean $1$ and variance $2$. Let $W$ be another unknown random variable with mean $3$ and variance $5$. $\Theta$ and $W$ are independent.</p>
<p>Let: $X_1=\Theta+W$ and $X_2=2\Theta+3W$. We pick measurement $X$ at random, each having probability $\frac{1}{2}$ of being c... | drhab | 75,923 | <p><strong>Hint</strong>:</p>
<p>Denoting the random index by $I$ we have:</p>
<p>$$\mathbb EX=\mathbb E(X\mid I=1)P(I=1)+\mathbb E(X\mid I=1)P(I=1)=\mathbb EX_1.\frac12+\mathbb EX_2.\frac12$$and:</p>
<p>$$\mathbb EX^2=\mathbb E(X^2\mid I=1)P(I=1)+\mathbb E(X^2\mid I=1)P(I=1)=\mathbb EX_1^2.\frac12+\mathbb EX^2_2.\f... |
3,963,479 | <p>In a quadrilateral <span class="math-container">$ABCD$</span>, there is an inscribed circle centered at <span class="math-container">$O$</span>. Let <span class="math-container">$F,N,E,M$</span> be the points on the circle that touch the quadrilateral, such that <span class="math-container">$F$</span> is on <span cl... | sirous | 346,566 | <p><a href="https://i.stack.imgur.com/exKdU.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/exKdU.jpg" alt="enter image description here" /></a></p>
<p>COMMENT:</p>
<p>As can be seen ratio <span class="math-container">$\frac{AP}{PC}$</span> depends on the positions of C related to A. So more constrai... |
1,567,229 | <p>Let Y1 and Y2 have the joint probability density function given by:</p>
<p>$ f (y_1, y_2) = 6(1−y_2), \text{for } 0≤y_1 ≤y_2 ≤1$</p>
<p>Find $P(Y_1≤3/4,Y_2≥1/2).$</p>
<p>Answer:</p>
<p>$$\int_{1/2}^{3/4}\int_{y_1}^{1}6(1− y_2 )dy_2dy_1 + \int_{1/2}^{1}\int_{1/2}^{1}6(1− y_2 )dy_1dy_2 = 7/64 + 24/64 = 31/64 $$... | Pieter21 | 170,149 | <p>(1) Permute the non-zeroes (5*4), pick $7 \choose 3$ places to insert zeroes: 5*4*35 = 700. </p>
<p>(2) Out of $7 \choose 3$ selections for the zeroes, there is only 1 correct, so $1/35$.</p>
|
1,697,206 | <p>In the figure, $BG=10$, $AG=13$, $DC=12$, and $m\angle DBC=39^\circ$.</p>
<p>Given that $AB=BC$, find $AD$ and $m\angle ABC$.</p>
<p>Here is the figure:</p>
<p><a href="https://i.stack.imgur.com/u05wa.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/u05wa.jpg" alt="enter image description here">... | Intelligenti pauca | 255,730 | <p>You can use the RHS condition for the congruence of two right-angled triangles:</p>
<p><a href="https://en.wikipedia.org/wiki/Congruence_(geometry)#Congruence_of_triangles" rel="nofollow">https://en.wikipedia.org/wiki/Congruence_(geometry)#Congruence_of_triangles</a></p>
<p>It's also called "hypotenuse-leg test (H... |
152,336 | <p>Let $G$ be an algebraic group. Choose a Borel subgroup $B$ and
a maximal Torus $T \subset B$. Let $\Lambda$ be the set of weights wrt $T$ and let $\mathfrak{g}$ be the lie algebra of $G$.
Now, consider the following two sets,</p>
<p>1) $\Lambda^+$, the set of dominant weights wrt $B$,</p>
<p>2) The set $N_{o,r}$ ... | Jim Humphreys | 4,231 | <p>Like Jay, I don't see any reasonable way to address all parts of your wide-ranging question. You are looking at the intersection of numerous lines of research, motivated in different ways for different people. For myself, the primary motivation comes indirectly from modular representations of Lie algebras attach... |
152,336 | <p>Let $G$ be an algebraic group. Choose a Borel subgroup $B$ and
a maximal Torus $T \subset B$. Let $\Lambda$ be the set of weights wrt $T$ and let $\mathfrak{g}$ be the lie algebra of $G$.
Now, consider the following two sets,</p>
<p>1) $\Lambda^+$, the set of dominant weights wrt $B$,</p>
<p>2) The set $N_{o,r}$ ... | wky | 14,226 | <p>There are some conversations on the affine Weyl group cells perspective of the bijection. I guess I can contribute a very little bit on the primitive ideals side of the story, if it is not too late to do so.</p>
<p>My first encounter of the Lusztig's quotient comes from the paper of Barbasch and Vogan in 1985 <a hr... |
3,112,682 | <p>I was looking at</p>
<blockquote>
<p><em>Izzo, Alexander J.</em>, <a href="http://dx.doi.org/10.2307/2159282" rel="nofollow noreferrer"><strong>A functional analysis proof of the existence of Haar measure on locally compact Abelian groups</strong></a>, Proc. Am. Math. Soc. 115, No. 2, 581-583 (1992). <a href="htt... | Paras Khosla | 478,779 | <p><span class="math-container">$$z+\frac{1}{z}=2\cos\theta\iff z^2+1=2z\cos\theta \\ z_{1,2}=\cos\theta\pm i\sin\theta=e^{\pm i\theta}\implies \frac{1}{z_{1,2}}=\cos\theta\mp i\sin\theta=e^{\mp i\theta}$$</span>Using De Moivre's formula, we get the following <span class="math-container">$$ z_{1,2}^n=\cos n\theta\pm i\... |
978,114 | <p>From $ax\geq 0$ for $a>0$, we have $x\geq 0$. So I suggest that if $Ax\geq 0$ for $A$ positive definite matrix, $x$ a column vector, $0$ is the column vector with $0$ as elements, then $x\geq 0$, that is, the coordinate of $x$ is greater than $0$.</p>
<p>However, I could not prove it...</p>
| Algebraic Pavel | 90,996 | <p>Positive definiteness is rather a spectral property than a "component-wise" one.
A randomly generated example shows that the statement is not true:
$$
A=\begin{bmatrix}2 & 3 \\ 3 & 5\end{bmatrix},
x=\begin{bmatrix}-1\\3\end{bmatrix},
Ax=\begin{bmatrix}7\\12\end{bmatrix}.
$$</p>
<p>This is true for <a href="... |
1,033,208 | <p>What do square brackets mean next to sets? Like $\mathbb{Z}[\sqrt{-5}]$, for instance. I'm starting to assume it depends on context because google is of no use.</p>
| Henno Brandsma | 4,280 | <p>There is no general notation like that in set theory, that I'm aware of.</p>
<p>In your case, it's a notation from algebra, and it means: $\{m + n\sqrt{5}: m,n \in \mathbb{Z}\}$. We add a new number (here $\sqrt{5}$) to the integers and generate the minimal ring that contains them both.</p>
|
682,741 | <p>Use the Mean Value Theorem to prove that if $p>1$ then $(1+x)^p>1+px$ for $x \in (-1,0)\cup (0,\infty)$</p>
<p>How do I go about doing this?</p>
| sirfoga | 83,083 | <p>Sorry, actually I wrote the answer quickly and a bit carelessly ... but let's take a different path.</p>
<p>Let $f(x) = (1+x)^p$, and $g(x) = 1+px$; clearly $f'(x) = p(1+x)^{p-1}$ and $g'(x) = p$. </p>
<p>It means that $f'(x) > g'(x)$ for $x > 0$ ( because $1 + x > 1$), and $f(0) = g(0) = 1$, so $f(x) >... |
129,530 | <p>This book, which needs to be returned quite soon, has a problem I don't know where to start. How do I find a 4 parameter solution to the equation</p>
<p>$x^2+axy+by^2=u^2+auv+bv^2$</p>
<p>The title of the section this problem comes from is entitled (as this question is titled) "Numbers of the Form $x^2+axy+by^2$"... | Mike | 17,976 | <p>Okay, let's see if I can fix my previous answer. Again if i let $z=u_1^2+au_1v_1+bv_1^2$, we get</p>
<p>$z^2=(u_1^2+au_1v_1+bv_1^2)(u_1^2+au_1v_1+bv_1^2)=r^2+ars+bs^2$</p>
<p>where $r=u_1^2-bv_1^2,s=2u_1v_1+av_1^2$.</p>
<p>I will now multiply this by another number of the form $m^2+amn+bn^2$ to get yet another n... |
358,786 | <p>Why does Egorov's theorem not hold in the case of infinite measure? It turns out that, for example, $f_n = \chi_{[n,n+1]}x$ does not converge nearly uniformly, that is, it does not converge on E such that for a set F m(E\F) < $\epsilon$. Is this simply true because it takes on the value 1 for each n but suddenl... | chango | 20,376 | <p>$f_n$ converges pointwise to the zero function on $\mathbb{R}$ (here $E = \mathbb{R})$. However there doesn't exist a set of finite measure $F$ such that $f_n$ converges uniformly on $\mathbb{R} \setminus F$. To see this note that for large enough $n$, $f_n$ will take both the values $0$ and $1$ on $\mathbb{R} \setm... |
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