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 |
|---|---|---|---|---|
7,223 | <p>I want to produce a <em>Mathematica</em> Computable Document in which <code>N</code> appears as a variable in my formulae. But <code>N</code> is a reserved word in the <em>Mathematica</em> language. Is there a way round this other than using a different symbol? It seems a severe limitation if you cannot use <em>Math... | István Zachar | 89 | <p>As it is often voiced here, modifying built-in variables is not a good idea most of the times, especially in case of such fundamental symbols as <code>N</code>. It is used heavily through millions of underlying code lines, and you will never know where your change can (and will) cause any mischief (or catastrophe).<... |
3,952,702 | <p>The problem is to make the following integral stationary:
<span class="math-container">$$ \int_{x_1}^{x_2} \frac{\sqrt{1+y'^2}}{y^2}dx $$</span>
to simplify the Euler equation, I tried to change the independent variable:
<span class="math-container">$$ \int_{y_1}^{y_2} \frac{\sqrt{1+x'^2}}{y^2}dy, \: \: \: \: \: \: ... | Claude Leibovici | 82,404 | <p>As said in comments and answer, using <span class="math-container">$\frac{u}{\sqrt{k}}$</span> you will end with
<span class="math-container">$$\int \frac{ky^2}{\sqrt{1-k^2y^4}}\, dy=\frac{E\left(\left.\sin ^{-1}(u)\right|-1\right)-F\left(\left.\sin
^{-1}(u)\right|-1\right)}{\sqrt{k}}$$</span></p>
<p>If you need ... |
4,181,442 | <p>I know this is a dumb question but...</p>
<p>Is <span class="math-container">$x ≠ 1,3,5$</span> the same as <span class="math-container">$x$</span> does not belong to {<span class="math-container">$1,3,5$</span>}, for example ?</p>
<hr />
<p>Sorry for the formatting.</p>
<p>Btw, anyone has a link with mathjax's comm... | Community | -1 | <p><span class="math-container">$x \neq 1, 3, 5$</span> isn't super standard notation, but if someone wrote it out at random I'd assume they meant <span class="math-container">$x \neq 1$</span> and <span class="math-container">$x \neq 3$</span> and <span class="math-container">$x \neq 5,$</span> or <span class="math-co... |
631,053 | <p>I have a container of 100 yellow items.</p>
<p>I choose 2 at random and paint each of them blue.</p>
<p>I return the items to the container.</p>
<p>If I repeat this process, on average how many cycles will I make before all 100 items are painted?</p>
<p>It is obviously 50 (100/2) if there is no replacement. But ... | Zur Luria | 117,481 | <p>If you choose 1 item each time, then the expected time until the items are all painted is $n \ln(n)$; This is the coupon collector problem.</p>
<p><a href="http://en.wikipedia.org/wiki/Coupon_collector%27s_problem" rel="nofollow">http://en.wikipedia.org/wiki/Coupon_collector%27s_problem</a></p>
<p>If you take 2 at... |
1,009,082 | <p>Given is a linear map f from V to W, whereby V has dimension n and W has dimension m.</p>
<p>Now given n > m, can the map be injective,surjective or invertible?
And what about the same questions, given that m > n?</p>
<p>My thoughts so far: </p>
<ul>
<li><p>Invertibility should be possible in the second case, if ... | anomaly | 156,999 | <p>Extending $f$ to a continuous, even function on $[-1, 1]$, we then have $\int_{-1}^1 x^n f = 0$ for all $n$, as $\int_{-1}^1 x^{2n} f = 2\int_0^1 x^{2n}f = 0$. Thus $f = 0$. For the case of odd $n$, consider $g(x) = x f(x)$.</p>
|
2,291,852 | <p>$$\int_\pi^\infty{\frac{x \cos x}{x^2-1}dx}$$</p>
<p>So the only think I came up with was to take an absolute value of ${\frac{x \cos x}{x^2-1}}$ and by comparison test the integral does not converge. </p>
<p>But I see it's not very close to the solution, so what should I do?</p>
| Hagen von Eitzen | 39,174 | <p>The function $x\mapsto x^2+e^x$ is continuous, hence your set (as inverse image of the closed set $\{1\}$) is closed. It is also bounded because $e^x>0$ and so $x^2<1$ (i.e., $-1<x<1$) for points in the set.</p>
|
1,424,198 | <p>My mathematical logic textbook defines $\{x \ | \ \text {_} x \text {_} \ \}$, but I'm not sure what the $\text {_} x \text {_}$ means. </p>
<p>Do the _ just mean 'for any expression involving $x$', or is there something I'm missing?</p>
| Mauro ALLEGRANZA | 108,274 | <p>You can supplement Enderton's explanation with some examples from :</p>
<ul>
<li>Herbert Enderton, <a href="https://books.google.it/books?id=LXA_avkJAv8C&pg=PA4" rel="nofollow">Elements of set theory</a> (1977), page 4 :</li>
</ul>
<blockquote>
<p>The notation used for the set of all objects $x$ such that th... |
1,484,736 | <p>[<img src="https://i.stack.imgur.com/i5iPj.jpg" alt="enter image description here">]</p>
<p>$$\lim_{x\to 0}f\big(f(x)\big)$$</p>
<p>(Original image <a href="https://imgur.com/a/Uogqk" rel="nofollow noreferrer">here</a>.)</p>
<p>I don't need an answer, I just want to know how I can calculate the limit based on the... | Faraz Masroor | 163,745 | <p>I want to disagree with Stefan. You got it right that $\lim_{x\to 0}f(x)$ is 2, but notice that it approaches 2 from the negative direction, going from below 2 towards 2. Therefore you need to calculate $\lim_{x\to 2}f(x)$ as it approaches 2 from the negative direction, which is -2. </p>
|
2,163,494 | <p>Let $f: A\to B; \ g,h:B\to A$ and $f\circ g = I_B$ and $h \circ f = I_A$</p>
<p>I want to simply state that for any function $f$ if $f \circ h = I_A$ then it must be that $h = f^{-1}$ but that seems incomplete to me. What can I do for fixing this?</p>
| Yiorgos S. Smyrlis | 57,021 | <p>If $f(A)$ is not bounded, then there exists a sequence $\{x_n\}\subset A$, such that $|f(x_n)|\to\infty$. But, as $\{x_n\}$ is bounded, it possesses a convergent subsequence $x_{k_n}\to x$. Since $f$ is continuous, $f(x_{k_n})\to f(x)$, which contradicts the fact that $|f(x_n)|\to\infty$.</p>
|
4,292,427 | <p><span class="math-container">$$ \frac{d^{2}y}{dt^2}+ 2t \frac{dy}{dt}+ t y=0 ~~ \tag{1} $$</span></p>
<p>At least I know that in this case of ODE can be solved by finding out 2 particular solutions.</p>
<p>As those 2 particular solutions are known, the general solution for this ODE can be written as below form.... | Cesareo | 397,348 | <p>You can ask for series solutions. In this case proposing <span class="math-container">$y_n = \sum_{k=0}^n a_k t^k$</span> and substituting into the ODE we have</p>
<p><span class="math-container">$$
\left(\sum_{k=0}^n a_k t^k\right)''+2t\left(\sum_{k=0}^n a_k t^k\right)'+t\sum_{k=0}^n a_k t^k=0
$$</span></p>
<p>grou... |
76,420 | <p>The following code segment shows what I'd like to do. I'm a procedural programmer trying to learn the Mathematica functional style. Any help on this would be appreciated.</p>
<pre><code>B = {{1, 3}, {1, 5}, {4, 2}, {5, 2}, {5, 5}}
u = SparseArray[{{1, 2} -> 5, {1, 3} -> 9, {1, 4} -> 6,
{1, 5} ->... | Dr. belisarius | 193 | <p>What Bill posted as a comment, or for example:</p>
<pre><code>{r[[#1]] -= #3, r[[#2]] += #3} & @@@ (Transpose@Join[Transpose@B, {Extract[u, B]}]);
r
(* {-∞, ∞, 7, 1, Indeterminate} *)
</code></pre>
<p>The <strong>Indeterminate</strong> thingy comes from <code>∞ - ∞</code></p>
|
76,420 | <p>The following code segment shows what I'd like to do. I'm a procedural programmer trying to learn the Mathematica functional style. Any help on this would be appreciated.</p>
<pre><code>B = {{1, 3}, {1, 5}, {4, 2}, {5, 2}, {5, 5}}
u = SparseArray[{{1, 2} -> 5, {1, 3} -> 9, {1, 4} -> 6,
{1, 5} ->... | Karsten 7. | 18,476 | <p>Purely functional</p>
<pre><code>func[lastr_, {i_, j_}] := MapAt[# + u[[i, j]] &, MapAt[# - u[[i, j]] &, lastr, i], j]
Fold[func, r, B]
</code></pre>
<blockquote>
<p><code>{-∞, ∞, 7, 1, Indeterminate}</code></p>
</blockquote>
<p>or</p>
<pre><code>Fold[Function[{lastr, ind}, MapAt[# + u[[Sequence @@ in... |
2,466,527 | <p>Let $A$ be the matrix of $T:P_2\to P_2$ with respect to basis $B=\{v_1,v_2,v_3\}$. Find $T(v_1)$</p>
<p>$$A=\begin{bmatrix}
1 & 3 & -1 \\
2 & 0 & 5 \\
6 & -2 & 4
\end{bmatrix}$$</p>
<p>$v_1=3x+3x^2$</p>
<p>$v_2=-1+3x+2x^2$</p>
<p>$v_3=3+7x+2x^2$</p>
<hr>
<p>First part of question asks t... | marty cohen | 13,079 | <p>Suppose
$x_n = ux_{n-1}+v
$.</p>
<p>If $u=1$,
this is
$x_n = x_{n-1}+v
$,
so
$x_n = x_0+nv$.</p>
<p>From now on,
I will assume that
$u \ne 1$.</p>
<p>I now apply a standard trick
and divide by
$u^{n}$.
This becomes
$\dfrac{x_n}{u^n}
= \dfrac{x_{n-1}}{u^{n-1}}+\dfrac{v}{u^n}
$.</p>
<p>Now, let
$y_n
= \dfrac{x_n}{... |
817,934 | <p>How to prove</p>
<p>$$\int\frac{12x\sin^{-1}x}{9x^4+6x^2+1}dx=-\frac{2\sin^{-1}x}{3x^2+1}+\tan^{-1}\left(\frac{2x}{\sqrt{1-x^2}}\right)+C$$</p>
<p>where $\sin^{-1}x$ and $\tan^{-1}x$ are inverse of trig functions. I don't know how to find the integral because of inverse of trig functions. I missed calc class twice... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>As said in comments, integration by parts is the only way to get rid of the inverse trigonometric functions.</p>
<p>So, let $$u=\sin ^{-1}(x)$$ $$v'=\frac{12x}{9x^4+6x^2+1}=\frac{12x}{(3x^2+1)^2}$$ Then $$u'=\frac{1}{\sqrt{1-x^2}}$$ $$v=-\frac{2}{3 x^2+1}$$ Now, the answer which is giv... |
619,370 | <p>Let $C,Q$ is complex numbers Field and Rational number
Field,respectively,if $f(x),g(x)\in Q[x]$,</p>
<p>if $g(x)|f(x)$ on $C[x]$,show that
$$g(x)|f(x)$$ on $Q[x]$</p>
<p>My try: since
$g(x)|f(x)$,then we have
$$f(x)=g(x)h(x)$$
where $h(x)\in C[x]$.
Then I can't prove also have
$$g(x)|f(x)$$ on $Q[x]$.</p>
<p>... | Prahlad Vaidyanathan | 89,789 | <p>Induct on the degree of $h$ : if $deg(h) = 0$, then $h = a_0$, a constant, which must be rational.</p>
<p>If the result is true for $deg(h) \leq n-1$, assume $deg(h) = n$, then
$$
h(x) = a_0 + a_1x + \ldots + a_nx^n
$$
Then, compare the leading coefficients on both sides to conclude that $a_n \in \mathbb{Q}$. Now c... |
2,485,997 | <p><a href="https://i.stack.imgur.com/OkhoU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OkhoU.png" alt="enter image description here"></a></p>
<p>Question: What if we consider (1,4,5) and (1,5,4) as non-distinct possibilities, then what should we do?</p>
<p>$${{9}\choose{2}}-2\cdot\frac{{9}\cho... | Christian Blatter | 1,303 | <p>Since solutions obtained by a permutation are considered the same we may assume $1\leq a\leq b\leq c$ to begin with. We therefore put
$$a=1+x_1,\quad b=a+x_2=1+x_1+x_2,\quad c=b+x_3=1+x_1+x_2+x_3\ ,$$
whereby $x_i\geq0$ $\>(1\leq i\leq3)$ and
$a+b+c=3+3x_1+2x_2+x_3=10$,
or $$3x_1+2x_2+x_3=7\ .$$
If $x_1=0$ then... |
300,531 | <p>Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$</p>
<p>where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$).</p>
<hr>
<p>This integral was mentioned in <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant">Wikipedia</a> as in <a href="http://mathw... | Argon | 27,624 | <p>$$I = \int_0^1 \log (-\log x)\,dx = \int_0^\infty e^{-x} \log(x)\,dx$$</p>
<p>Noting that</p>
<p>$$\Gamma(s) = \int_0^\infty e^{-x} x^{s-1}\, dx$$</p>
<p>we find that </p>
<p>$$\Gamma'(1) = I = -\gamma$$</p>
|
300,531 | <p>Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$</p>
<p>where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$).</p>
<hr>
<p>This integral was mentioned in <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant">Wikipedia</a> as in <a href="http://mathw... | Pedro | 23,350 | <p>You can see a proof <a href="https://math.stackexchange.com/questions/126713/proof-that-1/127717#127717">here</a> where we use that $$\Gamma(z) = \frac{\exp{(-\gamma z)}}{z}\prod\limits_{n=1}^\infty\frac{\exp \left({\frac z n}\right)}{1+\dfrac z n }$$</p>
<p>There is another proof <a href="https://math.stackexchang... |
474,807 | <p>I am studying a text on permutation groups, which has the following example in a section on regular normal subgroups: </p>
<blockquote>
<p>If $Z(N)=1$, then $N \cong \mathrm{Inn}(N)$, the group of inner automorphisms of $G$, and the semidirect product $N \rtimes \mathrm{Inn}(N)$ is the diagonal group $N^*=N \tim... | Derek Holt | 2,820 | <p>In the semidirect product $N \rtimes \operatorname{Inn} N$, the normal subgroup $N \times 1$ is centralized by the subgroup $\{ (h^{-1},h) \mid h \in N \}$, which is isomorphic to $N$. So this subgroup is equal to the second factor $1 \times N$ in the direct product $1 \times N$.</p>
<p>The isomorphism $N \times N ... |
103,970 | <p>Here's a very bizarre inconsistency I've just struggled with and I'm wondering why it exists or if I'm missing something.</p>
<p>I have some noisy data and I wish to make a framed plot of the data but allow the data to extend outside the vertical limits of the frame (for stylistic reasons). Like so:</p>
<pre><code... | bbgodfrey | 1,063 | <p>I, too, could not find a simple, transparent solution, so here is a simple but not transparent solution.</p>
<pre><code>ListLinePlot[Reverse[data, 2], PlotRange -> {All, {0, 0.5}}];
Reverse[Cases[%, Line[{z__}] -> z, Infinity], 2];
ListLinePlot[{}, opts, Prolog -> First@ListLinePlot[%, PlotRange -> All]... |
220,139 | <p>Is there a way to either make <code>FindMinimum</code> do an exact computation or <code>Minimize</code> find also the local minima? Or other ideas to find local minima exactly?</p>
<p><strong>Example:</strong> find all local minima (exact values, not approximations) of <span class="math-container">$f(x,y)=x^2 − x +... | rmw | 57,128 | <p><strong>Extrema on the edge.</strong></p>
<pre><code>f = x^2 - x + 2 y^2;
g = x^2 + y^2 - 1;
L = f + \[Lambda] g;
pts = Solve[{Grad[L, {x, y}] == 0, g == 0}, {x, y, \[Lambda]}];
points = pts[[All, 1 ;; 2]];
critpts = Thread@{f /. points, points};
</code></pre>
<p>bordered Hessian:</p>
<pre><code>hesseMatrix = {{0, ... |
1,812,468 | <p>Let $x=\{a,b\}$ be a set. Then, $x\in\{a,b\}$?</p>
<p>I think: Yes. So, why?</p>
| Fnacool | 318,321 | <p>Begin with </p>
<p>$$E [e^{a W_s} ] = e^{\frac{a^2s}{2}}.$$ </p>
<p>This is true for all complex-valued $a$. Now assume $a$ is real. </p>
<p>Set $X_s^a = e^{a W_s}$. Note that $(X_s^a)^2 =X_s^{2a}$. </p>
<p>Therefore for $T<\infty$</p>
<p>$$ E [\int_0^T (X_s^a)^2 ds ] =\int_0^T E [ e^{2aW_s} ]ds = \int_0^T ... |
239,863 | <p>I've to study this series:</p>
<p>$$\sum_{n=1}^\infty e^{\sqrt n\,x}$$ </p>
<p>My teacher wrote that with the asymptotic comparison with this series:</p>
<p>$$\sum_{n=1}^\infty\frac{1}{n^2}$$<br>
My series converges for every </p>
<p>$$x<0$$</p>
<p>I don't understand the motivation, hoping for someone to... | Hui Yu | 19,811 | <p>Let $X$ be a general Banach space and $X^*$ its dual. Then the weak topology on a bounded subset of $X$ is determined by a dense subset of $X^*$ in the sense:</p>
<blockquote>
<p>If $D\subset X^*$ is dense and $(x_\alpha)\subset X$ is uniformly bounded in norm, then $x_\alpha$ converges to $x$ weakly if and only ... |
239,863 | <p>I've to study this series:</p>
<p>$$\sum_{n=1}^\infty e^{\sqrt n\,x}$$ </p>
<p>My teacher wrote that with the asymptotic comparison with this series:</p>
<p>$$\sum_{n=1}^\infty\frac{1}{n^2}$$<br>
My series converges for every </p>
<p>$$x<0$$</p>
<p>I don't understand the motivation, hoping for someone to... | Focus | 254,076 | <p>We claim that <span class="math-container">$\textrm{Ball}(\ell^p)$</span> is weakly compact and weakly metrizable.</p>
<p>Since <span class="math-container">$1<p<\infty$</span>, we know <span class="math-container">$\ell^p$</span> is reflexive, so by Alaoglu's theorem, <span class="math-container">$\textrm{Bal... |
347,315 | <p>How can I find </p>
<ol>
<li>The image of the upper half plane $\mathrm{Im}(z)>0$ under the linear fractional transformation $w=\dfrac{3z + i}{-iz + 1}$.</li>
<li>The image of the set {${z∈C-0:\{\mathrm{Im}(z)} = \mathrm{Re}(z)\}$} under $w=z + \dfrac{1}{z}$.</li>
</ol>
<p>For 1.,
I consider $y>0, x \in R$ a... | Andreas Caranti | 58,401 | <p>$\operatorname{PGL}_{2}(q)$ is centreless, while $\operatorname{SL}_{2}(q)$ has a centre $D_{2}$ of order $2$ for $q$ odd.</p>
<p>As a reference, see the <a href="http://en.wikipedia.org/wiki/Projective_special_linear_group#Finite_fields" rel="nofollow">Wikipedia article about the projective special linear group</a... |
347,315 | <p>How can I find </p>
<ol>
<li>The image of the upper half plane $\mathrm{Im}(z)>0$ under the linear fractional transformation $w=\dfrac{3z + i}{-iz + 1}$.</li>
<li>The image of the set {${z∈C-0:\{\mathrm{Im}(z)} = \mathrm{Re}(z)\}$} under $w=z + \dfrac{1}{z}$.</li>
</ol>
<p>For 1.,
I consider $y>0, x \in R$ a... | Stephen | 146,439 | <p>There are natural maps $\mathrm{SL}_2(q) \hookrightarrow \mathrm{GL}_2(q) \twoheadrightarrow \mathrm{PGL}_2(q)$; composing these gives a map from $\mathrm{SL}_2(q)$ to $\mathrm{PGL}_2(q)$. For $q$ not a power of $2$ this map definitely has a kernel, the subgroup of order $2$ generated by the diagonal matrix $-1$. Fo... |
69,961 | <p>I want to determine the set of natural numbers that can be expressed as the sum of some non-negative number of 3s and 5s.</p>
<p>$$S=\{3k+5j∣k,j∈\mathbb{N}∪\{0\}\}$$</p>
<p>I want to check whether that would be:
0,3, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and so on.</p>
<p>Meaning that it would include 0, 3, 5,... | Austin Mohr | 11,245 | <p>By inspection, you can see how to represent 0, 3, 5, and 8. Now, given any integer $n \geq 9$, classify $n$ into one of the following cases.</p>
<p>Case $n \equiv 0$ (mod 3): In this case, $n = 3k$ for some $k$, so there is nothing to show (it is already a certain multiple of 3).</p>
<p>Case $n \equiv 1$ (mod 3): ... |
69,961 | <p>I want to determine the set of natural numbers that can be expressed as the sum of some non-negative number of 3s and 5s.</p>
<p>$$S=\{3k+5j∣k,j∈\mathbb{N}∪\{0\}\}$$</p>
<p>I want to check whether that would be:
0,3, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and so on.</p>
<p>Meaning that it would include 0, 3, 5,... | Jyrki Lahtonen | 11,619 | <p>This type of a question falls under the umbrella of <a href="http://en.wikipedia.org/wiki/Numerical_semigroup" rel="nofollow">numerical semigroups.</a> The numerical semigroup generated by a set of positive integers consists of their linear combinations with non-negative integer coefficients. So your set is the nume... |
2,864,585 | <p>I tried to calculate the Hessian matrix of linear least squares problem (L-2 norm), in particular:</p>
<p>$$f(x) = \|AX - B \|_2$$
where $f:{\rm I\!R}^{11\times 2}\rightarrow {\rm I\!R}$</p>
<p>Can someone help me?<br>
Thanks a lot.</p>
| littleO | 40,119 | <p>Let $f:\mathbb R^n \to \mathbb R$ be defined by
$$
f(x)=\frac12 \|Ax-b\|^2.
$$
Notice that $f(x)=g(h(x))$, where $h(x)=Ax-b$ and $g(y) = \frac12 \|y\|^2$. The derivatives of $g$ and $h$ are given by
$$
g'(y)=y^T, \quad h'(x)=A.
$$
The chain rule tells us that
$$
f'(x)=g'(h(x))h'(x) = (Ax-b)^T A.
$$
If we use the con... |
4,108,996 | <p>I am a little confused about how the author came to this discriminant of this quadratic polynomial. I understand that the discriminant comes from solving for a variable, and the discriminant is whatever is under the square root. Below is information directly from the PDF:</p>
<p>(1) <span class="math-container">$f(y... | Amirali | 794,843 | <p>We have <span class="math-container">$f'(y)=3y^2-2uy-v^2$</span>. it is a quadratic equation in <span class="math-container">$y$</span>. so discriminant is <span class="math-container">$$\Delta=(-2u)^2-4(3)(-v^2)=4u^2+4(3v^2)=4(u^2+3v^2)$$</span></p>
|
4,108,996 | <p>I am a little confused about how the author came to this discriminant of this quadratic polynomial. I understand that the discriminant comes from solving for a variable, and the discriminant is whatever is under the square root. Below is information directly from the PDF:</p>
<p>(1) <span class="math-container">$f(y... | Octonions | 914,583 | <p>There are many ways to find the roots of a quadratic polynomial. First thing you want to figure out is how many real roots it has. How to do that? The discriminant tells you this exact type of information.</p>
<p>First, I would like to figure out the roots of the polynomial <span class="math-container">$f(x) = ax^2 ... |
814,020 | <p><strong>Preamble</strong></p>
<p>The <a href="http://mathworld.wolfram.com/CassiniOvals.html" rel="nofollow">Cassinian curves</a> are the pre-images of concentric circles (centered at $1+0\,i$) under the map $z\mapsto z^2$. Using this fact and the fact that complex polynomials are conformal we can deduce that the o... | Daniel Fischer | 83,702 | <p>The preimage of the real line under $f(z) = z^2$ is not a hyperbola, it's the limiting case, the two coordinate axes.</p>
<p>For all other straight lines passing through $1$, let us describe them by an equation: $L_c = \{ u+i v : u-1 = cv\}$, and let $H_c = f^{-1}(L_c)$. For $H_c$, we then obtain the describing equ... |
4,627,334 | <p>To my understanding that a primitive triple <span class="math-container">$x$</span> and <span class="math-container">$y$</span> can be written as <span class="math-container">$x = q^2 - p^2$</span> while <span class="math-container">$y=2pq$</span> for relatively prime opposite parity <span class="math-container">$q ... | Hagen von Eitzen | 39,174 | <p>If <span class="math-container">$a^2+b^2=c^2$</span> with integers <span class="math-container">$a,b,c$</span>, recall that the square of an even number is <span class="math-container">$\equiv 0\pmod4$</span> and the square of an odd number is <span class="math-container">$\equiv1\pmod8$</span>. Therefore, either <s... |
186,240 | <p>I need some notion about topology(I'm very interested in boundary points, open sets) and few examples of solved exercises about limits of functions($f:\mathbb{R}^{n}\rightarrow \mathbb{R}^m$) using $\epsilon, \delta$ and also some theory for continous functions.
Please give me some links or name of the books which ... | user642796 | 8,348 | <p>If you don't want to go straight to general topology, you could look at a book more specifically about metric spaces, like Mícheál O'Searcoid's <a href="http://books.google.com/books?id=aP37I4QWFRcC&printsec=frontcover&hl=de#v=onepage&q&f=false" rel="nofollow"><em>Metric Spaces</em></a>.</p>
|
1,278,860 | <p>Use the process of implicit differentiation to find $dy/dx$ given that:</p>
<p>$$x^2e^y − y^2e^x=0 $$</p>
<p>I am trying first to find $y$, </p>
<p>$$y^2e^x = x^2e^y$$</p>
<p>$$y^2 = (x^2e^y)/e^x$$</p>
<p>$$y = \sqrt{(x^2e^y)/e^x}$$</p>
<p>Is this correct? I have the feeling it is not.</p>
| Autolatry | 25,097 | <p>So, implicitly differentiating;
$$x^{2}\frac{dy}{dx}e^{y}+2x e^{y}-\left(y^{2}e^{x}-2y\frac{dy}{dx}e^{x}\right)=0$$
Collecting like terms
$$\frac{dy}{dx}\left(x^{2}e^{y}-2ye^{x} \right)=y^{2}e^{x}-2xe^{y}$$
From which it is readily seen that
$$\frac{dy}{dx}=\frac{y^{2}e^{x}-2xe^{y}}{\left(x^{2}e^{y}-2ye^{x} \right)}... |
1,302,990 | <p>I want to ask basic question. In our mathematics classes ,while teaching the Fourier series and transform topic,the professor says that when the signal is periodic ,we should use Fourier series and Fourier transform for aperiodic signals.</p>
<p>My question is can't we use Fourier transform formula in case of per... | Gyu Eun Lee | 52,450 | <p>A Fourier series is only defined for functions defined on an interval of finite length, including periodic signals, as you can see from the definition of the Fourier coefficients (in the basis $\{e^{inx}\}_{n\in\mathbb{Z}}$)
$$
a_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}~dx.
$$</p>
<p>You can't define an aperi... |
306,011 | <p>Does anyone have a proof for $$\int_0^{\infty}\frac{\sin(x^2)}{x^2}\,dx=\sqrt{\frac{\pi}{2}}.$$
I tried to get it from contour integrating $$\frac{e^{iz^2}-1}{z^2},$$ but failed.
Thanks.</p>
| Jack D'Aurizio | 44,121 | <p>Since $\mathcal{L}(\sin x)(s)=\frac{1}{1+s^2}$ and $\mathcal{L}^{-1}\left(\frac{1}{x\sqrt{x}}\right)=\frac{2}{\sqrt{\pi}}\sqrt{s}$ we have
$$ \int_{0}^{+\infty}\frac{\sin(x^2)}{x^2}\,dx = \frac{1}{2}\int_{0}^{+\infty}\frac{\sin x}{x\sqrt{x}}\,dx = \frac{1}{\sqrt{\pi}}\int_{0}^{+\infty}\frac{\sqrt{s}}{1+s^2}\,ds=\fra... |
2,161,294 | <p>I was wondering... $1$, $\phi$ and $\frac{1}{\phi}$, they have something in common: they share the same decimal part with their inverse. And here it comes the question:</p>
<p>Are these numbers unique? How many other members are in the set if they exist? If there are more than three elements: is it finite or infin... | Donald Splutterwit | 404,247 | <p>There is a pair for every $n \in N$
\begin{eqnarray*}
x_{\pm} = \frac{n \pm \sqrt{n^2+4}}{2}
\end{eqnarray*}</p>
<p>whence
\begin{eqnarray*}
1/(x_{\pm}) = \frac{-n \pm \sqrt{n^2+4}}{2} = x_{\pm}-n
\end{eqnarray*}</p>
<p>Eg $n=2$ ... $x_+=2.414 \cdots$ & $x_-=0.414 \cdots$.</p>
|
2,455,408 | <p>I am encountering questions like this below.</p>
<p>$$\frac{dP}{dt}=(a-b\cos t ) \left(P+ \frac{P^2}{M}\right)$$</p>
<p>Then there is information stating $M$ is a positive integer and $a$ and $b$ are positive. That when $t=0$, $P=P_0$.</p>
<p>It wants me to solve the differential equation and show the process.</p... | Alex S | 484,637 | <p>The simplest way to solve this problem is to first draw a picture. Here's a photograph of the curves described in your problem:</p>
<p><a href="https://i.stack.imgur.com/wndiNm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/wndiNm.png" alt="enter image description here"></a></p>
<p>The region b... |
3,754,548 | <p>Suppose there is a strictly convex continuous function <span class="math-container">$f$</span>: <span class="math-container">$R^n$</span> <span class="math-container">$\rightarrow$</span> <span class="math-container">$R$</span>.</p>
<p>Is the supremum of <span class="math-container">$f$</span> always infinity? How c... | Z Ahmed | 671,540 | <p>You can do it by the tabulated integral <span class="math-container">$\int e^{kx} dx=\frac{e^{kx}}{k}$</span> as below
<span class="math-container">$$I=\int e^{ax} \sin bx dx= \Im \left( \int e^{(a+ib)x}~dx\right)= \Im \left( \frac{e^{(a+ib)x}}{a+ib} \right).$$</span>
<span class="math-container">$$\implies I=\frac{... |
3,754,548 | <p>Suppose there is a strictly convex continuous function <span class="math-container">$f$</span>: <span class="math-container">$R^n$</span> <span class="math-container">$\rightarrow$</span> <span class="math-container">$R$</span>.</p>
<p>Is the supremum of <span class="math-container">$f$</span> always infinity? How c... | Koro | 266,435 | <p><span class="math-container">$I=\int f(x)g(x) dx$</span>. Suppose that <span class="math-container">$f$</span> is differentiable and <span class="math-container">$g$</span> is integrable. <br/>
Applying integration by parts: <br/></p>
<blockquote>
<p><span class="math-container">$I=f(x) \int g(x)dx-\int f'(x)(\int ... |
315,844 | <p>What is the probability P(X>Y) given that X,Y are Uniformly distributed between [0,1]?</p>
| gt6989b | 16,192 | <p>A more general approach is based on symmetry -- since $X,Y$ have the same distribution and $\mathbb{P}[X=Y] = 0$ (here, because both are continuous random variables), any outcome $(x,y)$ where $x>y$ is just as likely as the outcome $(y,x)$, so $X>Y$ exactly half the time.</p>
<p>Note that this is independent ... |
2,814,703 | <p>I am reading <a href="https://en.wikipedia.org/wiki/Lower_limit_topology" rel="nofollow noreferrer">lower limit topology</a> on Wikipedia, which states that the lower limit topology </p>
<blockquote>
<p>[...] is the topology generated by the basis of all half-open intervals $[a,b)$, where a and b are real numbers... | CiaPan | 152,299 | <p>Here's an example of a union of disjoint intervals:</p>
<p>$$ (a,b) = \bigcup_{n=1}^\infty\left[a+\frac{b-a}{n+1}, a+\frac{b-a}n\right) $$</p>
<p>I wonder, why nobody presented it yet?</p>
|
866,654 | <p>Reading various betting forum I came across different threads claiming <strong><em>betting multiple is worse than betting on single events</em></strong>.</p>
<p>Could you explain why?</p>
<p>[Clairification for the ones not familiar with betting:
Betting on a single event: predict the outcome of a single match.
Be... | Graham Kemp | 135,106 | <p>Assuming the outcomes of all the matches are independent, with $p_k$ being the probability of a favorable outcome of the $k^{th}$ match, for matches numbered $1$ to $n$. Then probability of favourable outcomes occurring in all matched will be:$$P_{\text{all}}=\prod\limits_{k=1}^n p_k$$</p>
<p>Since $0\leq p_j\leq ... |
2,489,498 | <p>A={a,b,c,d}</p>
<p>R={(a,b),(a,c),(c,b)}</p>
<p>According to the definition for transitive relation, if there is (a,b) and (b,c) there should be (a,c)</p>
<p>In the above relation there is (a,c),(c,b) as well as (a,b). Shouldn't it be transitive?</p>
| Mark Viola | 218,419 | <blockquote>
<p><strong>I thought it would be instructive to present a way forward that relies on elementary, pre-calculus tools only. To that end we proceed.</strong></p>
</blockquote>
<hr>
<p>To show that $\sqrt x\log(x) \to 0$ as $x\to 0$, we simply exploit the inequality</p>
<p>$$\frac{x-1}{x}\le\log(x)\le x-... |
2,505,863 | <p>I have to find one affine transformation that maps the point P=(1,1,1) to P'=(-1,-1,-1), the point P=(-1,-1,-1)' to P=(1,1,1) and the point Q=(0,0,0) to Q'=(2,2,2).
I started with a sketch and think that it is not possible to map both points with one affine transformation, but I must somehow prove that.
So I take th... | lab bhattacharjee | 33,337 | <p>Let $\sqrt{2x+1}-3=u\implies2x=u^2+6u+8$</p>
<p>and $\sqrt{x-2}-\sqrt2=v\implies x= v^2+2\sqrt2v+4$</p>
<p>and as $x\to0; u,v\to0$</p>
<p>On division, $2v^2+4\sqrt2v=u^2+6u\iff v(2v+4\sqrt2)=u(u+6)$</p>
<p>$$\lim_{x\to 4} \frac{\sqrt{2x+1}-3}{\sqrt{x-2}-\sqrt{2}}=\lim_{u,v\to0}\dfrac uv= \lim_{u,v\to0}\dfrac{2v+... |
3,778,024 | <p>Let <span class="math-container">$(\Omega, \mathcal{F}, P)$</span> be a probability space, <span class="math-container">$X$</span> a random variable and <span class="math-container">$F(x) = P(X^{-1}(]-\infty, x])$</span>. The statement I am trying to prove is</p>
<blockquote>
<p>The distribution function <span class... | Masacroso | 173,262 | <p>Let <span class="math-container">$P_X:=P\circ X^{-1}$</span>, then <span class="math-container">$(\mathbb{R},\mathcal{B}(\mathbb{R}),P_X)$</span> is a probability space (that is, <span class="math-container">$P_X$</span> is a probability measure in the Borel <span class="math-container">$\sigma $</span>-algebra of t... |
3,778,024 | <p>Let <span class="math-container">$(\Omega, \mathcal{F}, P)$</span> be a probability space, <span class="math-container">$X$</span> a random variable and <span class="math-container">$F(x) = P(X^{-1}(]-\infty, x])$</span>. The statement I am trying to prove is</p>
<blockquote>
<p>The distribution function <span class... | Kavi Rama Murthy | 142,385 | <p>It is a basic fact that for any finite measure <span class="math-container">$\mu$</span> the condition <span class="math-container">$A_n$</span> decreasing to <span class="math-container">$A$</span> implies that <span class="math-container">$\mu (A_n) \to \mu (A)$</span>. [Lebesgue measure is an infinite measure an... |
2,243,083 | <p>I'm writing an advanced interface, but I don't yet have a concept of derivatives or integrals, and I don't have an easy way to construct infinite many functions (which could effectively delay or tween their frame's contributing distance [difference between B and A] over the next few frames).</p>
<p>I can store valu... | Community | -1 | <p>You can truly simulate the physics of a dampened spring, which leads to a differential equation.</p>
<p>$$m\ddot x_A+d\dot x_A+k(x_A-x_B)=0$$</p>
<p>where $x$ is the position, $m$ the mass, $d$ a damping coefficient and $k$ the stiffness constant of the spring.</p>
<p>As you probably have enough with a qualitativ... |
1,362,860 | <p>$$\frac{1}{3}+\frac{1}{13}+\frac{1}{23}+\frac{1}{31}+\frac{1}{37}+\frac{1}{43}\cdots$$
Intuitively, I feel that this sum converges, but I really don't know why, (or if I am correct). Can I have a somewhat rigorous proof of whether or not this sum converges or diverges? Thank you lots for any help.</p>
| paw88789 | 147,810 | <p>The sum of reciprocals of all positive integers without the digit $3$ converges. (See for instance <a href="https://math.stackexchange.com/questions/387/sum-of-reciprocals-of-numbers-with-certain-terms-omitted">Sum of reciprocals of numbers with certain terms omitted</a>)</p>
<p>Hence the sum of reciprocals of all... |
3,808,575 | <p>Assuming I have the statement ∀x(∀y¬Q(x,y)∨P(x)), can I pull the universal quantifier ∀y out of the parenthesis? Meaning, is this statement equivalent to ∀x∀y(¬Q(x,y)∨P(x)) ?</p>
<p>An approach I tried so far:</p>
<ol>
<li>∀x((∃y Q(x,y) ) => P(x)). (original eq.)</li>
<li>∀x((∀y¬Q(x,y))∨P(x)) (... | Shaun | 104,041 | <p>They're equivalent.</p>
<p>Here is a proof:</p>
<p><img src="https://i.stack.imgur.com/dIKSM.jpg" alt="enter image description here" /></p>
<p>This tree was generated <a href="https://www.umsu.de/trees/#(%E2%88%80x(%E2%88%80y%C2%ACQ(x,y)%E2%88%A8P(x)))%E2%86%94(%E2%88%80x%E2%88%80y(%C2%ACQ(x,y)%E2%88%A8P(x)))" rel="... |
3,684,917 | <p>Let <span class="math-container">$C_{1}$</span> and <span class="math-container">$C_{2}$</span> be polytopes in <span class="math-container">$\mathbb{R}^{n}$</span> such that
<span class="math-container">$C_{1}=conv\left( V\right) $</span> with <span class="math-container">$V$</span> being a set of vertices. If
<s... | JMP | 210,189 | <p>Given the range <span class="math-container">$[a,b]$</span> and the congruence <span class="math-container">$k \mod n$</span>, then first, subtract <span class="math-container">$k$</span> from each of <span class="math-container">$a$</span> and <span class="math-container">$b$</span> to create a new range <span clas... |
997,602 | <blockquote>
<p>Prove that the function <span class="math-container">$x \mapsto \dfrac 1{1+ x^2}$</span> is uniformly continuous on <span class="math-container">$\mathbb{R}$</span>.</p>
</blockquote>
<p>Attempt: By definition a function <span class="math-container">$f: E →\Bbb R$</span> is uniformly continuous iff for ... | Community | -1 | <p>First:$f(x)=\frac{1}{1+x^2}\implies f'(x)=-\frac{2x}{(1+x^2)^2} $</p>
<p>Then observes that; For $|x|\le1$
$$\frac{|x|}{(1+x^2)^2} \le\frac{1}{(1+x^2)^2}\le 1$$</p>
<p>and for $|x|\ge1$
$$|x|\le x^2 \le (1+x^2)^2\implies \frac{|x|}{(1+x^2)^2} \le 1$$</p>
<p>Hence,</p>
<p>$$|f'(x)|=\frac{2|x|}{(1+x^2)^2} \le 2... |
2,745,570 | <p>Use the mathematical Induction show that $H_{2^n}\le n+1$</p>
<p>here $H$ is harmonic numbers ie. $H_n=1+\frac{1}{2}+\frac{1}{3}+.....\frac{1}{2^n}$</p>
<p><strong>my idea</strong></p>
<p>so for $n=0$ L.H.S=R.H.S</p>
<p>Suppose this is true for $n$</p>
<p>we prove for $n+1$</p>
<p>So $H_{2^{n+1}}=1+\frac{1... | Peter Szilas | 408,605 | <p>Rephrasing a bit:</p>
<p>1) $n=0$, ok.</p>
<p>2) Hypothesis: $H_{2^n} \le n+1$.</p>
<p>3)Step: $n+1$:</p>
<p>$H_{2^{n+1}} =$</p>
<p>$H_{2^n} + \dfrac{1}{2^n +1}+.....\dfrac{1}{2^{n+1}} =$ </p>
<p>$H_{2^n} + \dfrac{1}{2^n+1}+...\dfrac{1}{2^n+ 2^n} \lt$</p>
<p>$H_{2^n} + 2^n \dfrac{1}{2^n+1} \lt $</p>
<p>$H_{2... |
2,745,570 | <p>Use the mathematical Induction show that $H_{2^n}\le n+1$</p>
<p>here $H$ is harmonic numbers ie. $H_n=1+\frac{1}{2}+\frac{1}{3}+.....\frac{1}{2^n}$</p>
<p><strong>my idea</strong></p>
<p>so for $n=0$ L.H.S=R.H.S</p>
<p>Suppose this is true for $n$</p>
<p>we prove for $n+1$</p>
<p>So $H_{2^{n+1}}=1+\frac{1... | heropup | 118,193 | <p>$$H_n = \sum_{k=1}^n \frac{1}{k}.$$ Then
$$\begin{align*}
H_{2^{n+1}} &= \sum_{k=1}^{2^{n+1}} \frac{1}{k} \\
&= \sum_{k=1}^{2^n} \frac{1}{k} + \sum_{k=2^n + 1}^{2^{n+1}} \frac{1}{k} \\
&= H_{2^n} + \sum_{j=1}^{2^{n+1} - 2^n} \frac{1}{2^n + j} \\
&\overset{\text{i.h.}}{\le} (n+1) + \sum_{j=1}^{2^n} ... |
2,509,308 | <p>My friend asked me a so called modified version of <a href="https://math.stackexchange.com/questions/96826/the-monty-hall-problem#">Monty Hall problem</a> in his opinion.
But I find the description a bit spooky and maybe someone here can enlighten
us with what is the problem with the description of the problem, or m... | Dean | 393,411 | <p>In the modified game, identify the 3 cards as (1) the card you choose at random (2) the card the host chooses at random from the remaining two cards (3) the other card. It is assumed that the two random selections are done without any information about which card is the winning card, unlike in the actual Monte Hall ... |
2,509,308 | <p>My friend asked me a so called modified version of <a href="https://math.stackexchange.com/questions/96826/the-monty-hall-problem#">Monty Hall problem</a> in his opinion.
But I find the description a bit spooky and maybe someone here can enlighten
us with what is the problem with the description of the problem, or m... | fleablood | 280,126 | <p>The "quit" is simply throwing away one third of the games and not considering them. This is because you are "viewing" the problem at a specific point of time; after the host says his card and before you the player decide to switch.</p>
<p>Imagine instead of quitting the host takes out an anti-matter vial and destr... |
1,123,050 | <p>This is the same problem asked here. - <a href="https://math.stackexchange.com/questions/1105927/next-step-to-take-to-reach-the-contradiction">Next step to take to reach the contradiction?</a>
Here is it again.</p>
<p><img src="https://i.stack.imgur.com/onqzq.png" alt="enter image description here"></p>
<p>I under... | jdods | 212,426 | <p>I actually struggled with this concept in grad school since I was studying applied math and was sort of thrust into higher level theory without building it up rigorously like I assume would be done in a pure math program.</p>
<p>If we are integrating over a space <span class="math-container">$X$</span>, I sometimes ... |
1,351,350 | <p>Assume that probability of $A$ is $0.6$ and probability of $B$ is at least $0.75$. Then how do I calculate the probability of both $A$ and $B$ happening together?</p>
| JP McCarthy | 19,352 | <p>If $A$ and $B$ are two events then</p>
<p>$$\mathbb{P}[A\cap B]=\mathbb{P}[A]\cdot \mathbb{P}[B\,|\, A]=\mathbb{P}[B]\cdot \mathbb{P}[A\,|\,B],$$</p>
<p>where $A\cap B$ is the event $A$ AND $B$ and $\mathbb{P}[A\,|\,B]$ is the probability of $A$ <em>given that $B$ is true</em>.</p>
<p>When $A$ and $B$ are <em>ind... |
1,618,753 | <p>Trying to expand $f(x)=\cot(x)$ to Taylor series (Maclaurin, actually).
But I keep "adding up" infinities when using the formula. (Because of $\cot(0)=\infty$) Could you perhaps give me a hint on how to proceed?</p>
| GaussTheBauss | 104,620 | <p>The function $\cot x$ is not continuous at zero, and therefore has no power series around zero.</p>
<p>If you know complex analysis, you should look for the Laurent series of $\cot z$ at $z=0$ instead. </p>
|
1,500,827 | <p>A composite number $n$ is a Fermat-pseudoprime to base $a$, if</p>
<p>$$a^{n-1}\equiv\ 1\ (\ mod\ n)$$</p>
<p>If $n-1=2^s\times t$ , $t$ odd , $n$ is a strong a-PRP, if either
$2^t\equiv 1\ (\ mod\ n)$ or there is a number $u$ with $0\le u<s$ and
$\large 2^{2^u\times t}\equiv -1\ (\ mod\ n\ )$</p>
<p>I want t... | mrprottolo | 84,266 | <p>From the fact that $a_n=1+\frac{1}{2^{n-1}}$ we have that $a_{n+1}=1+\frac{1}{2^{n}}$. Now notice that
$$\frac{a_n}{2}=\frac{1}{2}+\frac{1}{2^{n}}=a_{n+1}-\frac{1}{2}.$$
Therefore we get the following recurrence relation
$$a_{n+1}=a_n/2+1/2$$</p>
|
3,484,052 | <p>Let's say you have a series that looks like <span class="math-container">$\sum^\infty_{n=N}f(n)$</span>, where <span class="math-container">$f(n)$</span> is some <span class="math-container">$n$</span>-dependent thing. If you take the limit of this series as <span class="math-container">$N$</span> approaches infinit... | Peter Szilas | 408,605 | <p>Hopefully not too trivial:</p>
<p>Assume <span class="math-container">$\lim_{N \rightarrow \infty} \sum_{k=N}^{\infty}f(k)=0$</span>:</p>
<p><span class="math-container">$\epsilon/2$</span> given.</p>
<p>There is a <span class="math-container">$N_0$</span> s.t. for <span class="math-container">$N>N_0$</span></... |
1,722,964 | <p>Expression :$$(p\rightarrow q)\leftrightarrow(\neg q\rightarrow \neg p)$$
What does the symbol $\leftrightarrow$ mean ? Please explain by drawing the truth table for this expression and also with other examples if possible. <strong>I'm in a desperate situation so I'd really appreciate a quick response !</strong></p>... | Rubicon | 318,714 | <p>All the answers above are great, and should help you. I will show you how I would do the full truth table for the logical statement: $$(p \Rightarrow q)\Longleftrightarrow(\neg q \Rightarrow \neg p)$$</p>
<p>\begin{array}{cc|c|c|c|c}
p & q & \neg q & \neg p & \neg q \Rightarrow \neg p & p \Right... |
3,347,342 | <blockquote>
<p><span class="math-container">$$\frac{2}{5}^{\frac{6-5x}{2+5x}}<\frac{25}{4}$$</span></p>
</blockquote>
<p>I can write this as
<span class="math-container">$$\frac25 ^{\frac{6-5x}{2+5x}} <\frac25 ^{-2}$$</span>
Therefore <span class="math-container">$$\frac{6-5x}{2+5x}<-2$$</span>
Solving it... | Community | -1 | <p>Because <span class="math-container">$\frac25<1$</span>, higher exponents will result in smaller numbers. That is why the inequality needs to be reversed when you switch to the inequality with just the exponents.</p>
|
2,811,870 | <p>This is a question from Brilliant.org</p>
<blockquote>
<p>The triangle $ABC$ has $AB = 9$ and $AC:BC = 40:41$. What is the maximum possible area of $ABC$?</p>
</blockquote>
<p>For this question, I considered the equation $A=\frac 12ab\sin\theta$.</p>
<p>Since $\sin\theta\le 1$, then $A$ is maximised when $\sin\... | Alex R. | 22,064 | <p>Your confusion stems from the fact that the area is maximized at $\theta=90$, <em>if you keep the side lengths $BC,AC$ fixed</em>. However in this problem $BC,AC$ can vary in length. </p>
|
2,811,870 | <p>This is a question from Brilliant.org</p>
<blockquote>
<p>The triangle $ABC$ has $AB = 9$ and $AC:BC = 40:41$. What is the maximum possible area of $ABC$?</p>
</blockquote>
<p>For this question, I considered the equation $A=\frac 12ab\sin\theta$.</p>
<p>Since $\sin\theta\le 1$, then $A$ is maximised when $\sin\... | g.kov | 122,782 | <p>Since we have expressions for the side lengths as
$a,b=px,c=qx$ ($a=9,p=40,q=41$),
we can ignore all the angles and
use Heron’s formula for the area
\begin{align}
S&=\tfrac14\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}
\tag{1}\label{1}
,\\
S(x)&=\tfrac14\sqrt{2x^2a^2(q^2+p^2)-x^4(p-q)^2(q+p)^2-a^4}
\tag{2}\label{2}
.
\en... |
2,811,870 | <p>This is a question from Brilliant.org</p>
<blockquote>
<p>The triangle $ABC$ has $AB = 9$ and $AC:BC = 40:41$. What is the maximum possible area of $ABC$?</p>
</blockquote>
<p>For this question, I considered the equation $A=\frac 12ab\sin\theta$.</p>
<p>Since $\sin\theta\le 1$, then $A$ is maximised when $\sin\... | farruhota | 425,072 | <p>Let the sides $AC=40x, BC=41x$.
Using the Heron's formula:
$$S=\sqrt{\frac{81x+9}{2}\cdot \frac{81x-9}{2}\cdot \frac{9+x}{2}\cdot \frac{9-x}{2}}=\frac{81}{4}\sqrt{\left(x^2-\frac 1{81}\right)\left(81-x^2\right)}\overbrace{\le}^{GM-AM} \\
\frac{81}{4}\cdot \frac{\left(x^2-\frac 1{81}\right)+\left(81-x^2\right)}{2}=82... |
672,744 | <p>Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis.</p>
<p>Having trouble evaluating the integral: </p>
<p>Solved for $x$:</p>
<ul>
<li>$x=0, y=1$</li>
<li>$x=2, y=13$</li>
</ul>
<p>$$\int_1^{13} 2\pi\sqrt\frac{y-1}3 \cdot \sqrt{1+\sqrt\frac{y-1}3'}^2\,dy$$</p>
<p>I... | robjohn | 13,854 | <p>This can be done using <a href="http://en.wikipedia.org/wiki/Pappus%27s_centroid_theorem" rel="nofollow">Pappus's Theorem</a> and integrating in $x$:
$$
\begin{align}
\int_0^22\pi x\,\overbrace{\sqrt{y'^2+1}\,\mathrm{d}x}^{\mathrm{d}s}
&=\int_0^22\pi x\sqrt{36x^2+1}\,\mathrm{d}x\\
&=\frac\pi{36}\int_0^2\sqrt... |
2,710,681 | <p>If I have a function of three variables and I want to create a new function in which it equals the other function squared, could I literally just square the other function or does this violate any rules? Would this also mean its gradient vector is just squared at a certain point?</p>
| Emilio Novati | 187,568 | <p>A function is not characterized only by the number of independent variables in the domain, but also by the number of dependent variables in the codomain and this make some difference in defining the ''square'' of a function.</p>
<p>If we have a function $f:\mathbb{R}^3 \to \mathbb{R}$ than its value is areal number... |
3,670,240 | <p>It' not a physics question, just ..coincidence ;) (i'm concerned about mathematical rightness of it)</p>
<p>Let's consider <span class="math-container">$U,T,S,P,V\in\mathbb{R_{>0}}$</span> such that
<span class="math-container">$$dU=TdS-PdV$$</span></p>
<ul>
<li>Based on this, how we can rigorously proof that <... | MasterYoda | 418,429 | <p>The OP's solution is rigorous and sound. Here I would like to point out that <span class="math-container">$U=U(S,V)$</span> <em>by construction</em>.</p>
<p>The most complete equation for the internal energy would read</p>
<p><span class="math-container">$$dU = d(TS) - d(PV)$$</span></p>
<p>In this case, the inte... |
476,899 | <p>Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$?</p>
<p>The proof should not use that $e$ is transcendental.</p>
<p>$e:$ Euler's number.</p>
<p><a href="http://paramanands.blogspot.com/2013/03/proof-that-e-is-not-a-quadratic-irrationality.html#.Uhv87tJFUnl">$\{1,e,e^2\}... | abnry | 34,692 | <p>Since I've spent enough time thinking about this, yet not getting a proof, I might as well show what I've got. Others can comment on whether or not more can be done.</p>
<p>Your problem is solved if you can show that for any integers $a, b, c$, we have $$\sum^\infty_{n=0} \frac{1}{n!} (a + b 2^n + c 3^n)$$ irration... |
476,899 | <p>Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$?</p>
<p>The proof should not use that $e$ is transcendental.</p>
<p>$e:$ Euler's number.</p>
<p><a href="http://paramanands.blogspot.com/2013/03/proof-that-e-is-not-a-quadratic-irrationality.html#.Uhv87tJFUnl">$\{1,e,e^2\}... | ParaH2 | 164,924 | <p>Using algebra, let $D$ be the differentiate operator for $C^{\infty}$ functions. </p>
<p>So with $$f_n(x)=e^{\lambda_n x}$$</p>
<p>Then $$\forall i \in \{1,...,n\}, \forall x \in \mathbb{R}, D(f_n(x))=\lambda_n \cdot f_n(x) $$</p>
<p>If all $\lambda_i$ are differentant $n$ is the space's dimension then the famill... |
489,907 | <p>I've only got the following parts of a triangle:</p>
<ul>
<li>Line A to B </li>
<li>Line B to C</li>
</ul>
<p>And optionally the Line from A to C if needed?</p>
<p>I'm trying to get the point X</p>
<p><a href="https://i.stack.imgur.com/f6leO.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/f6le... | Pocho la pantera | 92,051 | <p>$X=B+\frac{(A-B)\cdot (C-B)}{||C-B||^2} (C-B)$</p>
|
4,051,403 | <p>I'm not a math major, but a philosophy major that likes to know that he knows what he's talking about. This may seem like a super stupid question, but here I go.</p>
<p>So Euclid made a lot of sense when he gave the example of the nature of multiplication. For example.
"2 x 3" is really 2 added to itself 3... | Lutz Lehmann | 115,115 | <p>If you have a separable situation like the present, then the connection between first integral/Hamiltonian/energy and the potential is
<span class="math-container">$$
H(x,y)=\frac12y^2+P(x)
$$</span>
So in this case you can chose <span class="math-container">$P(x)=\frac14(x^2-λ)^2$</span>.</p>
|
2,066,455 | <p>I ask this question mainly to resolve (hopefully) and error with the following problem. </p>
<p>The United States Court consists of $3$ women and $6$ men. In how many ways can a $3$-member committee be formed if each committee must have at least one woman?</p>
<p>My approach:
Since each group needs at least one wo... | Kiran | 82,744 | <p>In how many ways can ten people be arranged in a line if neither of two particular people can sit on either end of the row?</p>
<p>Without any restrictions, $10!$ ways you can arrange them</p>
<p>$9!\times 2 $ ways where first person is at any end
$9!\times 2 $ ways where second person is at any end</p>
<p>$8!\ti... |
88,565 | <p>Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:</p>
<p><span class="math-container">$$x^2\ne x\implies x\ne 1$$</span></p>
<p>I immediately answered true, but for some reason, everyone (including my classmates and ... | The Chaz 2.0 | 7,850 | <p>First, some general remarks about logical implications/conditional statements. </p>
<ol>
<li><p>As you know, $P \rightarrow Q$ is true when $P$ is false, or when $Q$ is true. </p></li>
<li><p>As mentioned in the comments, the <em>contrapositive</em> of the implication $P \rightarrow Q$, written $\lnot Q \righta... |
88,565 | <p>Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:</p>
<p><span class="math-container">$$x^2\ne x\implies x\ne 1$$</span></p>
<p>I immediately answered true, but for some reason, everyone (including my classmates and ... | Phira | 9,325 | <p>The short answer is: Yes, it is true, because the contrapositive just expresses the fact that $1^2=1$.</p>
<p>But in controversial discussions of these issues, it is often (but not always) a good idea to try out non-mathematical examples:</p>
<hr>
<p>"If a nuclear bomb drops on the school building, you die."</p>
... |
3,051,480 | <p>Now we have the equation
<span class="math-container">$$\sum_{i}(x_i-\hat x_i)^2,$$</span>
where <span class="math-container">$x_i$</span> is the observed value of a data sample <span class="math-container">$S$</span>. Here is the question:</p>
<blockquote>
<p>Why does this expression get its minimum value when <... | marty cohen | 13,079 | <p>This can be solved
without calculus.</p>
<p>Let
<span class="math-container">$f(z)
=\sum_{i}(x_i-z)^2
$</span>.</p>
<p>Then,
since
<span class="math-container">$\sum_{i}x_i
=n\hat x$</span>,</p>
<p><span class="math-container">$\begin{array}\\
f(z)-f(\hat x)
&=\sum_{i}(x_i-z)^2-\sum_{i}(x_i-\hat x)^2\\
&=... |
2,960,501 | <p><span class="math-container">$(0^n 1)^* \ \ , n\geq 0 $</span></p>
<p>According to wiki</p>
<blockquote>
<p>If V is a set of strings, then V* is defined as the smallest superset of V that contains the empty string ε and is closed under the string concatenation operation</p>
<p>If V is a set of symbols or characters,... | vonbrand | 43,946 | <p>I take it that you mean <span class="math-container">$L = \bigcup_{n \ge 0} \mathcal{L}((1^n 0)^*)$</span>, i.e., arbitary repeats of <span class="math-container">$1^n 0$</span> for each <span class="math-container">$n$</span>. If you try to dream up an DFA to recognize this, you'll see it would need to record <span... |
1,931,754 | <p>I am trying to show that the interval $[0,1)$ is a closed subset of $(-1,1)$ by using the definition that a closed subset contains all of its limit points.
So for a convergent sequence $\{x_n\}$ in $[0,1)$ we have that $0 \leq x_{n} < 1$ for all $n \in \mathbf{N}$. How can I show that $\lim_{n \rightarrow \infty... | Amarildo | 307,377 | <p>$$\lim _{n\to \infty }\:\left(1+\log \left(\frac{n}{n-1}\right)\right)^n\: = \lim _{n\to \infty }\:\left(e^{n\cdot \:ln\left(1+\log \left(\frac{n}{n-1}\right)\right)}\right)\: \approx$$</p>
<p>$$\lim _{n\to \infty }\:\left(e^{n\cdot \:ln\left(\frac{n}{n-1}\right)}\right)\: \approx \lim _{n\to \infty }\:\left(e^{n\c... |
1,931,754 | <p>I am trying to show that the interval $[0,1)$ is a closed subset of $(-1,1)$ by using the definition that a closed subset contains all of its limit points.
So for a convergent sequence $\{x_n\}$ in $[0,1)$ we have that $0 \leq x_{n} < 1$ for all $n \in \mathbf{N}$. How can I show that $\lim_{n \rightarrow \infty... | Student | 255,452 | <p>Let the Limit be equal to $L$. Then $\lim n(1+ \log(n/(n-1))=\ln L$. This implies: </p>
<p>$$\lim\frac{1+\log(1/(1-1/n))}{1/n}$$ Let $y=1/n$.$$\lim\frac{1-\log(1-y)}{y}=\lim \frac{1/(1-y)}{1}=1$$ But $1=\ln L$ which implies $L=e$.</p>
|
1,596,264 | <p>Let $p$ be prime and $d \ge 2$. I want to show that
$$
\frac{(p^d - 1)(p^{d-1} - 1)}{(p-1)(p^2 - 1)} \equiv 1 \pmod{p}.
$$
I have a proof, but I think it is complicated, and the statement appears in a book as if it is very easy to see. So is there any easy argument to see it?</p>
<p>My proof uses
$$
\frac{p^n - 1... | Redundant Aunt | 109,899 | <p>You need to prove that $p\mid\frac{(p^{d-1}-1)(p^d-1)}{(p-1)(p^2-1)}-1$</p>
<p>It is not difficult to see that $p\mid (p^{d-1}-1)(p^d-1)-(p-1)(p^2-1)$. Furthermore, we have that $(p-1)(p^2-1)\mid (p^{d-1}-1)(p^d-1)$ because either $d$ or $d-1$ is even and so either $p^{d}-1$ or $p^{d-1}-1$ is divisible by $p^2-1$ a... |
1,596,264 | <p>Let $p$ be prime and $d \ge 2$. I want to show that
$$
\frac{(p^d - 1)(p^{d-1} - 1)}{(p-1)(p^2 - 1)} \equiv 1 \pmod{p}.
$$
I have a proof, but I think it is complicated, and the statement appears in a book as if it is very easy to see. So is there any easy argument to see it?</p>
<p>My proof uses
$$
\frac{p^n - 1... | Stella Biderman | 123,230 | <p>There are even easier proofs than the answers others have supplied. If you look at the numerator, one of the terms is guaranteed to be divisible by $p^2-1$, whichever has an even exponent. This is because $(x^2-1)|(x^{2n}-1)$. The division clearly results in a polynomial with constant term $1$, as does the division ... |
70,603 | <p>We were shown in class this next calculation: (Here, $V_n(RB^n)$ is the volume of an $n$ dimensional ball of radius $R$, likewise $S_{n-1}$ is the surface area of the $n$ dimensional sphere in $\mathbb{R}^n$. $rS^{n-1}$ denotes the $n$ dimensional sphere of radius $r$ and integrating $d\textbf{S}$ means a surface in... | Christian Blatter | 1,303 | <p>The third equality comes from the fact that the map $$f:\quad{\mathbb R}^n\to{\mathbb R}^n, \quad u\mapsto x:=r\>u$$
($r$ is constant here) multiplies volume elements by its Jacobian $r^n$ but multiplies $(n-1)$-dimensional surface elements by $r^{n-1}$.</p>
<p>While we are at it: The set $B^n:=\{x\in{\mathbb R}... |
1,406,878 | <p>Given is following sequence:</p>
<p>$a_{n+1} = a_n - \frac{a_n - v}{s}$</p>
<p>I found out that</p>
<p>$\forall a_0, v, s \in \mathbb{R}, s>0: \lim\limits_{n \to \infty}a_n=v$</p>
<p>But I do not know why. I tried to write down $a_2$ , $a_3$, but the term becomes very long and complex, and it doesn't help me ... | user1337 | 62,839 | <p>This is a first order linear recurrence relation, and the solution can be found explicitly:</p>
<p>$$a_n=a_0 \left(\frac{s-1}{s}\right)^n-v \left(\frac{s-1}{s}\right)^n+v.$$</p>
<p>Can you see that the limit doesn't necessarily exist?</p>
|
592,912 | <p>I need to describe the minimal field extension $\mathbb Q(\sqrt[3] {2})$ of the rational numbers $\mathbb Q$ that contain $\sqrt[3] {2}$.</p>
<p>$\mathbb Q(\sqrt[3] {2}) =\{a+b\sqrt[3] {2}+c(\sqrt[3] {2})^2|a,b,c \in \mathbb{Q}\}$.</p>
<p>I tried to use the rationalization of $x^3 + y^3 + z^3 - 3xyz$ ?</p>
| C-star-W-star | 79,762 | <p>Systematic Approach:</p>
<p><a href="https://i.stack.imgur.com/K9ANo.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/K9ANo.jpg" alt="Extensions"></a></p>
<p><em>(Comment for questions!)</em></p>
|
459,374 | <p>Let $X$ be the random variable which denotes the number of times a die has been rolled till each side has appeared. The order does not matter.
We are trying to find $E[X]$.</p>
<p>Let $X_i$ be a random variable which denotes how many times a die has to be rolled till side i has appeared.</p>
<p>So,</p>
<p>$$E[X]=... | HK Lee | 37,116 | <p>[intuitive answer]</p>
<p>If $y$ is fixed and if $x$ is a point in interior, we draw a small ball $B(x,r)$. </p>
<p>Then there exists a point $x'\in \partial B(x,r)$ such that </p>
<p>$$ d(x',y) =d(x,y)+r $$</p>
|
1,860,459 | <blockquote>
<p>Prove that $4k < 2^k$ by induction.</p>
</blockquote>
<p>It holds for $k = 5$. Assume $ k = n + 1 $. Then</p>
<p>$4(n+1) < 2^{(n+1)}$</p>
<p>$4n + 4 < 2^n * 2$</p>
<p>$2n + 2 \leq 2^n$</p>
<p>Now I just need to show that</p>
<p>$2n + 2 \leq 4n$</p>
<p>$n + 1 \leq 2n$</p>
<p>$1 \leq n$... | Henrik supports the community | 193,386 | <p>Your choice of $n$ should only be for the basis of the induction, not for the inductive step.</p>
<p>What you need to do is to show that if $4n<2^n$ then $4(n+1)<2^{n+1}$. One way of doing that is to say that $4(n+1)=4n+4<2^n+4$ and then argue that $x+4<2x$ for the relevant values.</p>
|
4,071,619 | <blockquote>
<p>There are two German couples, two Japanese couples and one unmarried person. If all 9 persons are two be interviewed one by one then the total number of ways of arranging their interviews such that no wife gives an interview before her husband is?</p>
</blockquote>
<p>I tried using the string method, bu... | Bill Dubuque | 242 | <p><a href="https://math.stackexchange.com/q/87383/242">Recall</a> a simple form (with simple proof) of LTE = Lifting The Exponent is</p>
<p><span class="math-container">$$a\equiv b\!\!\! \pmod{kn}\,\Rightarrow\, a^k\equiv b^k\!\!\!\! \pmod{k^2n}$$</span></p>
<p>Applied inductively for <span class="math-container">$\... |
4,071,619 | <blockquote>
<p>There are two German couples, two Japanese couples and one unmarried person. If all 9 persons are two be interviewed one by one then the total number of ways of arranging their interviews such that no wife gives an interview before her husband is?</p>
</blockquote>
<p>I tried using the string method, bu... | cansomeonehelpmeout | 413,677 | <p>The statement is equivalent to proving <span class="math-container">$$10^{3^k}-1\equiv_{3^{k+2}}0$$</span>
Notice that <span class="math-container">$\phi(3^{k+2})=3^{k+2}-3^{k+1}=2\cdot 3^{k+1}$</span>, so that for a unit <span class="math-container">$a$</span> we have <span class="math-container">$$a^{2\cdot 3^{k+1... |
2,558,870 | <p>Suppose $f:[0,1]\to \mathbb{R}$ is uniformly continuous, and $(p_n)_{n\in\mathbb{N}}$ is a sequence of polynomial functions converging uniformly to $f$.</p>
<p>Does it follow that $\mathcal{F}=\{p_n\mid n\in\mathbb{N}\}\cup \{f\}$ is equicontinuous?</p>
<p>Also, if $C_n$ are the Lipschitz constants of the polynomi... | Karn Watcharasupat | 501,685 | <p>Actually this is a combination of a few different results that are more easily proven separately but just for this I will prove everything in one go.</p>
<p>Suppose $e_i$ is an eigenvector of $(A - pI)^{-1}$ then
\begin{align}
(A - pI)^{-1}e_i&=(q_i - p)^{-1}e_i\\
(A - pI)(A - pI)^{-1}e_i&=(A - pI)(q_i - p)... |
2,970,370 | <p>For <span class="math-container">$f \in C^0([0,1])$</span>, I have the following partial differential equation:</p>
<p><span class="math-container">$$u''(x) = f(x)$$</span> in <span class="math-container">$\Omega = (0,1)$</span>
<span class="math-container">$$u'(0) = u'(1) = 0$$</span></p>
<p>Why is this equation ... | Giuseppe Negro | 8,157 | <p>That problem can be interpreted in a <em>weak formulation</em>, that is, a <em>solution</em> to that equation is a <span class="math-container">$u\in H^1(\Omega)$</span> such that
<span class="math-container">$$
\int_\Omega - u'\phi' =\int_{\Omega} f\phi,\qquad \forall \phi \in H^1(\Omega).$$</span>
Thus, there are... |
1,116,496 | <blockquote>
<p>Let $H$ be a Hilbert space with a countable basis $B$. Does it mean that any vector $x\in H$ can be expressed as a <strong>finite</strong> linear combination of elements from $x$, or as an <strong>infinite</strong> linear combination?</p>
</blockquote>
<p>Thanks in advance</p>
| Math1000 | 38,584 | <p>Take $\ell^2$ for example, i.e. the square-summable sequences of complex numbers with inner product
$$\langle x,y\rangle = \sum_{n=1}^\infty x_n\overline{y_n}. $$
This has the countable orthonormal basis
$$\{(1,0,0,\ldots), (0,1,0,\ldots), (0,0,1,0,\ldots),\ldots\}. $$
As
$$\sum_{n=1}^\infty 2^{-n} = 1<\infty, $$... |
982,780 | <p>I have the following system of <span class="math-container">$M$</span> linear equations in <span class="math-container">$N$</span> unknowns.</p>
<p><span class="math-container">$$
\begin{bmatrix}
3 & 0 & 1 & 0 & -1 & -3 & 2\\
1 & 2 & 0 & 4 & 0 & 0 & -1\\
1 & 1 &a... | Tomasz Kania | 17,929 | <p>I'd use the fact that $F\subset X$ is closed if and only if for each strictly increasing (possibly transfinite) sequence $(x_\alpha)_{\alpha<\lambda}$ with entries in $F$ we have</p>
<p>$$\big( \lim_{\alpha<\lambda} x_\alpha = \big) \sup_{\alpha<\lambda}x_\alpha\in F.$$</p>
<p>Now, if $F$ is closed in $X$... |
2,569,557 | <p>I'm still confused by the use of $\Rightarrow$ in (ε,δ)-definition of limit. <br/>
Take for example the definition of $\underset{x\rightarrow x_{0}}{\lim}f\left(x\right)=l$ :<br/></p>
<blockquote>
<p>$$\forall\varepsilon>0,\;\exists\delta>0\quad\mathrm{such\:that\quad}\forall x\in\mathrm{dom}\,... | hmakholm left over Monica | 14,366 | <p>Changing the definition in that way would mean that a <em>constant function</em> cannot have a limit, for example.</p>
<p>Or as a less trivial example, consider for example $\lim\limits_{x\to 1}\frac1x$. Intuitively this ought to be $1$, but with your addition to the definition the limit would not exist. Namely, if... |
2,569,557 | <p>I'm still confused by the use of $\Rightarrow$ in (ε,δ)-definition of limit. <br/>
Take for example the definition of $\underset{x\rightarrow x_{0}}{\lim}f\left(x\right)=l$ :<br/></p>
<blockquote>
<p>$$\forall\varepsilon>0,\;\exists\delta>0\quad\mathrm{such\:that\quad}\forall x\in\mathrm{dom}\,... | Ennar | 122,131 | <p>Already given example of constant function should (in my opinion) be enough to shoot the whole idea down in a blazing glory, but parabola might be more convincing visually:</p>
<p><a href="https://i.stack.imgur.com/clerf.png" rel="noreferrer"><img src="https://i.stack.imgur.com/clerf.png" alt="enter image descripti... |
3,995,492 | <p>I have no clue how to do this, I manage to get I get that <span class="math-container">$11^{36} \equiv 1 \hspace{0.1cm} \text{mod} (13)$</span> but I can't get anywhere from there.</p>
| Joffan | 206,402 | <p>The exercise here is to calculate the <a href="https://en.wikipedia.org/wiki/Multiplicative_inverse" rel="nofollow noreferrer">multiplicative inverse</a> of <span class="math-container">$11$</span>, written <span class="math-container">$11^{-1}$</span>, in <span class="math-container">$\bmod 13$</span> arithmetic. T... |
214,766 | <p>Is there an efficient way to check a number x and remove all prime factors in the number which are less than some n? For example for n = 200:</p>
<pre><code>x=88984589931961415442566827779929187431222364934742868664124547963532933
FactorInteger[x]
{{29, 2}, {31, 1}, {37, 2}, {269, 1}, {271,
1}, {3420047160553... | Fraccalo | 40,354 | <p>It's still not fully clear what are you exactly looking for, but this is a piece of code that might help you (it gives you the first set of lists you give in your question). It can easily be readapted for the other cases:</p>
<pre><code>list = {{1, {0}}, {2, {0}}, {3, {-2, 0, 2}}, {4, {-2, 0, 2}}, {5, {-2,0, 2}}};
... |
214,766 | <p>Is there an efficient way to check a number x and remove all prime factors in the number which are less than some n? For example for n = 200:</p>
<pre><code>x=88984589931961415442566827779929187431222364934742868664124547963532933
FactorInteger[x]
{{29, 2}, {31, 1}, {37, 2}, {269, 1}, {271,
1}, {3420047160553... | Alucard | 18,859 | <pre><code>List@@Flatten/@ (list /. {a_, {b_, 0, d_}} -> { a, 0} )
List@@Flatten/@ (list /. {a_, {b_, 0, d_}} -> { a, b} )
List@@Flatten/@ (list /. {a_, {b_, 0, d_}} -> { a, d} )
</code></pre>
|
3,738,579 | <blockquote>
<p>What is the cardinality of set <span class="math-container">$\big\{(x,y,z)\mid x^2+y^2+z^2= 2^{2018}, xyz\in\mathbb{Z} \big\}$</span>?</p>
</blockquote>
<p>Since I have very limited knowledge in number theory, I tried using logarithms and then manipulating the equation so that we get <span class="mat... | Alan | 696,271 | <p>Note that <span class="math-container">$2^{2n+1}=2\cdot 2^{2n}=2\cdot 4^n$</span>. Now,
<span class="math-container">$$\frac{2}{3}(4^n-1)+2^{2n+1}=\frac{2}{3}(4^n-1)+2\cdot 4^n =\frac{8}{3}\cdot 4^n -\frac{2}{3} = \frac{2}{3}(4^{n+1}-1)$$</span></p>
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