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 |
|---|---|---|---|---|
18,048 | <p>When taking a MOOC in calculus the exercises contain 5 options to select from. I then solve the question and select the option that matches my answer. Obviously only one of the options is correct. But there are (quite a few) times where my solution is wrong even though it is one of the available options. </p>
<p>My... | ytrkptl | 13,678 | <p>One thing common, especially at middle school/high school levels is to list the unknown (for example, x = 3) as one of the options, especially when the question actually asks you to solve for something else that requires knowing x. At the middle/high school levels, several students have a habit of thinking that "x i... |
2,919,266 | <p>Let $(x_n)$ be a sequence in $(-\infty, \infty]$. </p>
<p>Could we define the sequence $(x_n)$ so that limsup$(x_n) = -\infty$? </p>
<p>My intuitive thought is no, but I’m not 100% sure. </p>
| Donald Splutterwit | 404,247 | <p>Rewrite the sum
\begin{eqnarray*}
\sum_{i=1}^{n} \frac{i}{(i+1)(i+2)} 2^{i}
\end{eqnarray*}
Now do partial fractions
\begin{eqnarray*}
\frac{i}{(i+1)(i+2)} =\frac{2}{n+2} - \frac{1}{n+1}.
\end{eqnarray*}
Now not that the powers of $2$ cause this sum to be telescoping, so
Rewrite the sum
\begin{eqnarray*}
\sum_{i=1}^... |
3,745,159 | <p>Let <span class="math-container">$A,B,C$</span> be <span class="math-container">$n\times n$</span> matrices with real entries such that their product is pairwise commutative. Also <span class="math-container">$ABC=O_{n}$</span>. If
<span class="math-container">$$k=\det\left(A^3+B^3+C^3\right).\det\left(A+B+C\right)$... | Yes it's me | 767,521 | <p><span class="math-container">$\textbf{Hint:}$</span>Use euclidean algorithm.Denote by <span class="math-container">$(a,b)$</span> as the gcd of <span class="math-container">$a,b$</span></p>
<p>Observe::<span class="math-container">$\quad\begin{align}(2^m-1,2^n-1)&=(2^m-1,2^n-1-2^m+1)\\&=(2^m-1,2^m*(2^{n-m}-1... |
1,926,423 | <p>My problem with this is the final step. Through iterative substitution, I come up with the following:
$$T(n) = T(n-4) + (n-3) + (n-2) + (n-1) + n$$</p>
<p>which leads to the general form:
$$T(n) = T(n-k) + kn - \frac{[k(k-1)]}{2}$$</p>
<p>The restrictions are $T(1)=1$ and $n=2^k-1$. What I am doing at this point i... | GoodDeeds | 307,825 | <p>$$\begin{align}T(n)&=T(n-1)+n\\T(n)&=T(n-2)+(n-1)+n\\T(n)&=T(n-3)+(n-2)+(n-1)+n\\ &\vdots\\T(n)&=T(n-k)+(n-k+1)+(n-k+2)+\cdots+n\\ &\vdots\\T(n)&=T(1)+2+3+\cdots+n\\T(n)&=1+2+3+\cdots+n\\T(n)&=\frac{n(n+1)}{2}\end{align}$$</p>
|
1,134,177 | <p>Consider this primality test: Fix an initial segment of primes (e.g. 2,3,5,7), and combine a $b$-pseudoprime test for each b in that list. For several such initial segments, find the first $n$ for which the test gives an incorrect answer.</p>
<p>Hey all! I'm not quite understanding what the aforementioned question ... | Asinomás | 33,907 | <p>I'm assuming what it means is given an initial segment of primes (the first $n$ primes). for example the numbers $2,3,5$ Use this number to test if a number is prime or not. That is: check wheter a number is divisible by $2,3$ and $5$. Clearly if it is the number is not prime.</p>
<p>If the number is not divisible ... |
2,369,081 | <blockquote>
<p>Evaluate the integral $$\int_0^1\frac{x^7-1}{\log (x)}\,dx $$</p>
</blockquote>
<p>[1]: <a href="https://i.stack.imgur.com/lcK2p.jpgplz" rel="nofollow noreferrer">https://i.stack.imgur.com/lcK2p.jpgplz</a> I'm trying to solve this definite integral since 2 hours. Please, I need help on this.</p>
| Mark Viola | 218,419 | <p>Note that using $\int_0^1 x^t \,dt=\frac{x-1}{\log(x)}$ we can write</p>
<p>$$\begin{align}
\int_0^1 \frac{x^7-1}{\log(x)}\,dx&=\int_0^1 (x^6+x^5+x^4+x^3+x^2+x+1)\left(\int_0^1 x^t\,dt\right)\,dx\tag 1\\\\
&=\int_0^1\int_0^1 (x^{t+6}+x^{t+5}+x^{t+4}+x^{t+3}+x^{t+2}+x^{t+1}+x^t)\,dx\,dt\tag2 \\\\
&=\int_... |
1,849,608 | <p>Given the irreducible fraction $\frac a b$, with $a, b \in \mathbb N$, what is the expression that enumerates all the irreducible fractions of integers that add up to $\frac a b$? Namely, an expression (in terms of $a$ and $b$) for all the $\frac c d$ and $\frac e f$, with $c,d,e,f \in \mathbb N$, such that $\frac c... | naslundx | 130,817 | <p>The best you can do is simplify your expression:</p>
<p>$$\frac{c}{d} + \frac{e}{f} = \frac{cf + ed}{df} = \frac{a}{b}$$</p>
<p>The numerator and denominator will give you a system of two linear equations. With $a$ and $b$ fixed there are four unknowns $c,d,e,f$. As $4>2$ this means we have infinitely many solu... |
277,135 | <p><code>Graphics[{{Blue, Line[{{-5, 3}, {5, 8}}]}, {Dashed, Arrow[{{0, 0}, P}]}, Red, Arrow[{{0, 0}, v}]}, Axes -> True, AxesLabel -> {x, y}]</code>
I've tried combining Graphics and Plot to manipulate my graphic but it says that I can't.</p>
<p><a href="https://i.stack.imgur.com/8slBl.png" rel="nofollow noref... | user293787 | 85,954 | <p>Code:</p>
<pre><code>(* General *)
alldice[{nfaces_,min_,max_}]:=Select[Tuples[Range[min,max],nfaces],OrderedQ];
findBruteForce[admissibleDice_,ndice_,diceinfo:{nfaces_,min_,max_}]:=Catch[
Outer[If[admissibleDice[##],Throw[{##}]]&,
Sequence@@ConstantArray[alldice[diceinfo],ndice],1];{}];
(* Example ... |
3,336,311 | <p><strong>Help me factor these polynomials</strong> </p>
<ul>
<li><span class="math-container">$(x+\sqrt{2})^2$</span> - 8</li>
<li>14a - 49<span class="math-container">$a^2$</span> + 100<span class="math-container">$b^2$</span> - 1</li>
</ul>
| Matteo | 686,644 | <p>For the first, use <span class="math-container">$a^2-b^2=(a+b)(a-b)$</span>, so: <span class="math-container">$$(x+\sqrt{2})^2-8=(x+\sqrt2+2\sqrt2)(x+\sqrt2-2\sqrt2)=(x+3\sqrt2)(x-\sqrt2)$$</span>
For the second case pick a <span class="math-container">$a$</span> and use the same rule as the previous answer, so: <sp... |
2,876,050 | <p>I have There was a cowboy and 3 donkeys (Donkey A, Donkey B, Donkey C). The cowboy wears an eye cover and shoots randomly. What is the probability of donkey A to still be there after the cowboy shot 2 bullets? There is equal chances that the cowboy hits Donkey A, Donkey B, Donkey C or a Miss.</p>
<p>I am helpless r... | Christian Blatter | 1,303 | <p>As remarked by hardmath the protocol is somewhat ambiguously defined. Since the cowboy is blind he may shoot at a dead donkey the second time. I add a dummy donkey D to account for the misses. Then donkey A survives the first as well as the second shot with probability ${3\over4}$ each. Assuming these shots to be in... |
1,823,736 | <p><a href="http://www.math.drexel.edu/~dmitryk/Teaching/MATH221-SPRING'12/Sample_Exam_solutions.pdf" rel="nofollow">Problem 10c from here</a>.</p>
<blockquote>
<p>Thirteen people on a softball team show up for a game. Of the $13$ people who show up, $3$ are women. How many ways are there to choose $10$ players ... | Noble Mushtak | 307,483 | <p>As you said, we have $(0, 25)$, $(3, 0)$, and $(4, 9)$ as our intersection points. Therefore, we need to look at the regions of area in between those intersections points.</p>
<p>Between $x=0$ and $x=3$, the area is between the blue curve, $y=25-x^2$, and the purple curve, $y=25-\frac{25x}{3}$. Thus, we have the fo... |
71,166 | <p>This question have been driving me crazy for months now. This comes from work on multiple integrals and convolutions but is phrased in terms of formal power series.</p>
<p>We start with a formal power series</p>
<p>$P(C) = \sum_{n=0}^\infty a_n C^{n+1}$</p>
<p>where $a_n = (-1)^n n!$</p>
<p>With these coefficien... | Gottfried Helms | 7,710 | <p>[update] <em>remark: I shifted the original answer to the bottom although it was missing the question to keep it as reference.</em><br>
I've done a nice table, which shows the coefficients of a powerseries with constant term <em>=1</em>; for any <em>p</em> 'th integer power of the function you insert p into the for... |
3,109,482 | <p><a href="https://i.stack.imgur.com/awp2x.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/awp2x.png" alt="enter image description here"></a></p>
<p>I'm stumped on determining P(ABC) of Part A. My understanding is:</p>
<ul>
<li><p>Calculate the total number of patients (100)</p></li>
<li><p>Calcul... | Robert Lewis | 67,071 | <p>Since</p>
<p><span class="math-container">$A \ne B, \tag 1$</span></p>
<p>the only way <span class="math-container">$A$</span> and <span class="math-container">$B$</span> can be collinear is if</p>
<p><span class="math-container">$A = -B; \tag 2$</span></p>
<p>in this case <span class="math-container">$A$</span>... |
491,926 | <p>Let I have $(S,\Sigma,\mu)$ be a probability space, then $X,Y \in \Sigma$. Define $\rho (X,Y)$ by $\rho (X,Y)$ = correlation between random variable $I_X$ and $I_Y$, where $I_X$ and $I_Y$ are the indicator function of $X$ and $Y$. Express $\rho (X,Y)$ in term of $\mu (X)$, $\mu (Y)$, $\mu(XY)$. Conclude that $\rho(X... | Jonathan Y. | 89,121 | <p>This is not an answer for the quote, but definition we have
$$\rho(X,Y) = \frac{\operatorname{var}(I_X,I_Y)}{\sigma_{I_X}\sigma_{I_Y}} = \frac{\mathbb{E}(I_XI_Y)-\mathbb{E}(I_X)\mathbb{E}(I_Y)}{\sqrt{\left(\mathbb{E}(I_X^2)-\mathbb{E}(I_X)^2\right)\left(\mathbb{E}(I_Y^2)-\mathbb{E}(I_Y)^2\right)}}=\\
= \frac{\mathbb... |
3,553,681 | <p>consider the identity
<span class="math-container">$$\frac{e^{-x}}{1-x}=\sum_{n=0}^{\infty}c_nx^n$$</span></p>
<p>Show that for each <span class="math-container">$n\ge0$</span>
<span class="math-container">$$\sum_{k=0}^{n}\frac{c_k}{(n-k)!}=1$$</span></p>
<p>My trial :
By cauchy product, </p>
<p><span class="mat... | Ninad Munshi | 698,724 | <p>Just apply the integral twice and show the double integral reduces back down to the single integral via Fubini's theorem like so:</p>
<p><span class="math-container">$$(P^2f)(x) = \frac{1}{4\pi^2}\int_{-\pi}^\pi \int_{-\pi}^\pi [4\cos^2(x-y)-1][4\cos^2(y-z)-1]f(z)\:dz\:dy$$</span></p>
<p><span class="math-containe... |
3,553,681 | <p>consider the identity
<span class="math-container">$$\frac{e^{-x}}{1-x}=\sum_{n=0}^{\infty}c_nx^n$$</span></p>
<p>Show that for each <span class="math-container">$n\ge0$</span>
<span class="math-container">$$\sum_{k=0}^{n}\frac{c_k}{(n-k)!}=1$$</span></p>
<p>My trial :
By cauchy product, </p>
<p><span class="mat... | Jean-Claude Colette | 742,526 | <p>Before you show that P is a projector, you may notice that</p>
<p><span class="math-container">$2\cos^2(2x-2y)-1=2\cos(2x-2y)+1$</span></p>
<p><span class="math-container">$=2\cos(2x)\cos(2y)+2\sin(2x)\sin(2y)+1$</span></p>
<p>Hence</p>
<p><span class="math-container">$P(f)(x)=\frac{1}{\pi}\cos(2x)\int_{-\pi}^{\... |
3,353,826 | <p>All the vertices of quadrilateral <span class="math-container">$ABCD$</span> are at the circumference of a circle and its diagonals intersect at point <span class="math-container">$O$</span>. If <span class="math-container">$∠CAB = 40°$</span> and <span class="math-container">$∠DBC = 70°$</span>, <span class="math-c... | B. Goddard | 362,009 | <p>I have seen <span class="math-container">$(\!(x)\!)$</span> for "nearest integer." My memory is dim, but maybe it was Emil Grosswald's elementary number theory text. I like it because it's easy to type and it's not likely to be confused with another function.</p>
|
119,904 | <p>It is possible to do simple math between TemporalSeries objects. For example</p>
<pre><code>es=EventSeries[{{{2016, 1, 1}, 2}, {{2016, 1, 3}, 2.1}}];
td=TemporalData[{{{2016, 1, 1}, 2}, {{2016, 1, 3}, 3.1}}];
es*td (* works fine *)
</code></pre>
<p>This returns a TemporalData object with the path <code>{{366059520... | Gosia | 5,372 | <p>If you want to interpolate then use TimeSeriesThread to combine TimeSeries (by default TimeSeries uses linear interpolation):</p>
<pre><code>In[96]:= ts1 = TimeSeries[{{{2016, 1, 1}, 2}, {{2012, 1, 3}, 1.9}, {{2016, 1, 3},
2.1}}];
ts2 = TimeSeries[{{{2016, 1, 1}, 2}, {{2016, 1, 3}, 3.1}}];
In[100]:= res = Ti... |
2,756,139 | <p>Let $I=[0,1]$ and $$X = \prod_{i \in I}^{} \mathbb{R}$$
That is, an element of $X$ is a function $f:I→\mathbb{R}$.</p>
<p>Prove that a sequence $\{f_n\}_n ⊆ X$ of real functions converges to some $f ∈ X$ in the product topology on $X$, if and only if it converges pointwise, i.e. for every $x ∈ I$, $f_n(x) → f(x)$ i... | Community | -1 | <p>To make things easier to understand, let $I = {0, 1, 2}$. To say
$$X = \prod_{i \in I} \mathbb{R}$$
means that an element of $X$ is an technically an element of $\mathbb{R^3}$. But this can be interpreted as a function from $I \to \mathbb{R}$, in the sense that</p>
<p>$$(a, b, c) \cong f \text{ such that} f(0) = a... |
1,480,179 | <p>Let $n ≥ 2$, and $A ∈ M_n$ be Hermitian, and let ${\rm{B }} \in {\rm{ }}{{\rm{M}}_{n - 1}}$ be a leading principal
submatrix of A.</p>
<p>If $B$ is positive semidefinite and $rank B = rank A$, why does $A$ is positive
semidefinite?</p>
| Bento | 46,364 | <p>The hermitian form $a$ associated with $A$ has a diagonalizing basis $\mathbf{b}$, so the matrix which rapresents the form can be considered diagonal. The rank is $n-1$ so we can assume (if needed, we can interchange the basis vectors order $\mathbf{b}_{i}$ to achieve this) the last row of the matrix as zero vector ... |
3,973,611 | <p>Let <span class="math-container">$$F(x)=\int_{-\infty}^x f(t)dt,$$</span>
where <span class="math-container">$x\in\mathcal{R}$</span>, <span class="math-container">$f\geq 0$</span> is complicated (it cannot be integrated analytically).</p>
<p>Can I used the Simpson's rule to approximate this integral, knowing that <... | Robert Z | 299,698 | <p>There is no contradiction, just note that <span class="math-container">$C\cap D=\emptyset$</span> and therefore all intersection sets including <span class="math-container">$C\cap D$</span> have zero cardinality. Moreover, by a similar reason, also <span class="math-container">$A\cap B\cap C=\emptyset$</span> and <s... |
1,493,965 | <p>We're given the power series $$ \sum_1^\inf \frac{j!}{j^j}z^j$$</p>
<p>and are asked to find radius of convergence R. I know the formula $R=1/\limsup(a_n ^{1/n})$, which leads me to compute $\lim \frac{j!^{1/j}}{j}$, and then I'm stuck.</p>
<p>The solution manual calculates R by $1/\lim|\frac{a_{j+1}}{a_j}|$, but ... | Tsemo Aristide | 280,301 | <p>We suppose $t_0\in [a,b]$ and $f,f'_x$ continuous on $[a,b]$</p>
<p>$\|T(x(s)-T(y(s))\| =\|\int_{t_0-\delta}^{t_0+\delta}(f(x(s),s)-f((y(s),s))ds \|$.</p>
<p>Since $f'_x$ is continuous, $\| f(x(s),s)-f((y(s),s)\|\leq$ $Sup \|f'_x\|_{x\in [a,b]}\|x(s)-y(s)\|$.</p>
<p>$|T(x(s)-T(y(s))\|\leq 2\delta Sup \|f'_x\|_{x\... |
1,224,180 | <p>Q: evaluate $\lim_{x \to \infty}$ $ (x-1)\over \sqrt {2x^2-1}$</p>
<p>What I did:</p>
<p>when $\lim_ {x \to \infty}$ you must put the argument in the form of $1/x$ so in that way you know that is equal to $0$</p>
<p>but in this ex. the farest that I went was</p>
<p>$\lim_{x \to \infty}$ $x \over x \sqrt{2}$ $1-(... | Clement C. | 75,808 | <p><strong>Hint:</strong> write $$
\frac{(x-1)}{\sqrt {2x^2-1}} = \frac{x(1-\frac{1}{x})}{x\sqrt{2-\frac{1}{x^2}}}= \frac{1-\frac{1}{x}}{\sqrt{2-\frac{1}{x^2}}}
$$
for $x > 0$. What happens to numerator and denominator when $x\to\infty$?</p>
|
1,685,423 | <p>$$
x_n=\begin{cases}\frac{1}{n^2} & \text{if n is even} \\ \frac{1}{n} & \text{if n is odd}\end{cases}$$.</p>
<p>How can I show that $$ \sum x_n$$ is convergent?</p>
| DonAntonio | 31,254 | <p>I think you have</p>
<p>$$\sum_{n=1}^\infty x_n\;\;,\;\;\;x_n=\begin{cases}\frac1{n^2},\,&n\text{ is even}\\{}\\\frac1n,\,&n\text{ is odd}\end{cases}$$</p>
<p>If the series was convergent, supose to $\;S\;$ , then as it is a positive one, you can "split it":</p>
<p>$$S=\sum_{n=1}^\infty x_n=\frac14\sum_{n... |
480,504 | <p>If <span class="math-container">$M$</span> is a symmetric positive-definite matrix, is it possible to get a <strong>positive</strong> lower bound on the smallest eigenvalue of <span class="math-container">$M$</span> in terms of a matrix norm of <span class="math-container">$M$</span> or elements of <span class="math... | Saeed Manaffam | 124,075 | <p>There is one lower bound on minimum eigenvalue of symmetric p.d. matrix given in [Applied Math. Sc., vol. 4, no. 64] which is based on Frobenius norm (F) and Euclidean norm (E)</p>
<p>$$ \lambda_{min} \gt \sqrt{\frac{||A||_F^2-n||A||_E^2}{n(1-||A||_E^2/|det(A)|^{2/n})}} $$</p>
<p>if it helps.</p>
<p>[reference]: ... |
1,074,360 | <p>I know that $\phi(n)$, Euler's totient function, defines the number of all integers less than or equal to $n$ that are relatively prime to $n$. </p>
<p>I know that there is a trick to finding this with the larger non-prime numbers, but now I cannot find it anywhere. Could someone please explain how to find $\phi(n)... | Casteels | 92,730 | <p>There are two basic facts that you should know that will help you remember the formula. </p>
<p>First, if $n$ is a power of a prime, say $n=p^k$, then $\phi(p^k)=p^k-p^{k-1}$. This is because the only numbers less than or equal to $p^k$ that are <em>not</em> relatively prime to $p$ are $p,p^2,\ldots,p^{k-1}$. Note ... |
1,074,360 | <p>I know that $\phi(n)$, Euler's totient function, defines the number of all integers less than or equal to $n$ that are relatively prime to $n$. </p>
<p>I know that there is a trick to finding this with the larger non-prime numbers, but now I cannot find it anywhere. Could someone please explain how to find $\phi(n)... | lhf | 589 | <p>The main tool is multiplicativity: $\phi(ab)=\phi(a)\phi(b)$ if $\gcd(a,b)=1$.</p>
<p>Also, if $p$ is a prime, then $\phi(p^k)=p^{k-1}(p-1)$.</p>
<p>In particular, if $n=2^k m$, with $m$ odd, then $\phi(n)=2^{k-1} \phi(m)$.</p>
<p>For $n=352=2^5\cdot 11$, we get $\phi(352)=2^4 \phi(11)=2^4 \cdot 10 = 160$.</p>
|
4,558,387 | <p>So I need to determine the scalarform of a plane that contain <span class="math-container">$L:\begin{cases} 3x + 4y + z = 5 \\
x − y = −6 \end{cases}$</span> and goes through the midpoint of <span class="math-container">$(1, 1, 2)$</span> and <span class="math-container">$(3, 1, 4)$</span>.</p>
<p>My try solving thi... | Robert Z | 299,698 | <p>Your approach is not correct. <span class="math-container">$x+y−7z=-18$</span> is the plane which is <em>orthogonal</em> to the line <span class="math-container">$L$</span> and passes through <span class="math-container">$(2,1,3)$</span>.</p>
<p>Notice that all the planes which <em>contain</em> the line <span class=... |
4,558,387 | <p>So I need to determine the scalarform of a plane that contain <span class="math-container">$L:\begin{cases} 3x + 4y + z = 5 \\
x − y = −6 \end{cases}$</span> and goes through the midpoint of <span class="math-container">$(1, 1, 2)$</span> and <span class="math-container">$(3, 1, 4)$</span>.</p>
<p>My try solving thi... | Glorious Nathalie | 948,761 | <p><span class="math-container">$(1, 1, -7)$</span> is the direction vector of the line of intersection of the two given planes (line <span class="math-container">$L$</span>). It is not the normal vector of the required plane.</p>
<p>In addition to the direction vector of line <span class="math-container">$L$</span>, ... |
2,950,813 | <blockquote>
<p>Take <span class="math-container">$G$</span> to be a group of order <span class="math-container">$600$</span>. Prove that for any element <span class="math-container">$a$</span> <span class="math-container">$\in$</span> G there exist an element <span class="math-container">$b$</span> <span class="math... | lhf | 589 | <p><em>Hint:</em> <span class="math-container">$\gcd(600,7)=1 \implies 1=600m+7n$</span>. Indeed, <span class="math-container">$ 1 = 3 \cdot 600 - 257 \cdot 7$</span>.</p>
|
1,817,367 | <blockquote>
<p>Prove that $\forall k = m^2 + 1. \space m \in \mathbb{Z}^+$, if $k$ is divisible by any prime then that prime is congruent to $1, 2 \pmod 4$.</p>
</blockquote>
<p>I am unable to realize why it can't have $2$ prime factors congruent to $3 \pmod 4$. Can anyone please help me proceed?</p>
<p>Thanks.</p... | JMoravitz | 179,297 | <p><em>Hiding the major revelations in spoiler quotes.</em></p>
<hr>
<p>Suppose that $p\mid m^2+1$</p>
<p>Suppose for contradictory purposes that $p=4z+3$ for some $z\in\Bbb Z$.</p>
<p>Note that $p\not\mid m$ and $2\neq p$</p>
<blockquote class="spoiler">
<p> This is because $4z+3$ is always odd. Further $k=m^2... |
1,817,367 | <blockquote>
<p>Prove that $\forall k = m^2 + 1. \space m \in \mathbb{Z}^+$, if $k$ is divisible by any prime then that prime is congruent to $1, 2 \pmod 4$.</p>
</blockquote>
<p>I am unable to realize why it can't have $2$ prime factors congruent to $3 \pmod 4$. Can anyone please help me proceed?</p>
<p>Thanks.</p... | lhf | 589 | <p>If $p$ is an odd prime that divides $m^2+1$, then $m^2 \equiv -1 \bmod p$ and $m^4 \equiv 1 \bmod p$. Therefore, $m$ has order $4$ mod $p$. By Lagrange's theorem, we must have that $4$ divides $p-1$. In other words, $p \equiv 1 \bmod 4$.</p>
|
1,305,935 | <p>Let $f(n)$ be non-negative real valued function defined for each natural number $n$.</p>
<p>If $f$ is convex and $lim_{n\to\infty}f(n)$ exists as a finite number, then can we conclude that $f$ is non-increasing?</p>
| Jacky1205 | 305,093 | <p><a href="https://i.stack.imgur.com/oDLS5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/oDLS5.png" alt="SVM" /></a></p>
<p>Let <span class="math-container">$\textbf{x}_+$</span> be a positive example on one gutter, such that
<span class="math-container">$$\textbf{w} \cdot \textbf{x}_+ - b = 1$$</... |
2,260,051 | <p>Let $M$ and $N$ be two square matrices of same order, and $M^2 = N^4$.</p>
<p>Can any such $M,N$ exist when the following relations do not hold?</p>
<ol>
<li><p>$M = N^2$, and </p></li>
<li><p>$M = -N^2$ ?</p></li>
</ol>
| Gerry Myerson | 8,269 | <p>$$M=N=\pmatrix{0&1\cr0&0\cr}$$</p>
|
198,116 | <p>A finite <a href="http://en.wikipedia.org/wiki/Lattice_(order)" rel="nofollow noreferrer">lattice</a> is planar if it admits a <a href="http://en.wikipedia.org/wiki/Hasse_diagram" rel="nofollow noreferrer">Hasse diagram</a> which is a <a href="http://en.wikipedia.org/wiki/Planar_graph" rel="nofollow noreferrer">pla... | Joseph O'Rourke | 6,094 | <p>If I may be permitted to extract Wille's theorem
("On modular lattices of order dimension two")
to which Professor Stanley refers
(remember, this is 1973!):</p>
<hr>
<blockquote>
<p><strong>Theorem:</strong> A modular lattice $\mathfrak{M}$ is planar iff $\mathfrak{M} - \{ d \in \mathfrak{M} \mid d \text{ is do... |
1,055,832 | <p>How can one prove distributivity of a <a href="http://ncatlab.org/nlab/show/Heyting+algebra" rel="nofollow"> Heyting Algebra</a> via the <a href="http://ncatlab.org/nlab/show/Yoneda+lemma" rel="nofollow"> Yoneda lemma</a>?</p>
<p>I'm able to prove it using the Heyting algebra property $(x \wedge a) \leq b$ if and o... | Hanno | 81,567 | <p>For a Heyting algebra $H$, the category ${\mathcal H}$ associated to the poset underlying $H$ is a small, skeletal, Cartesian closed category with finite coproducts in which any two parallel morphisms are equal - and in fact this assignment is a bijection. Now if $x\in H$ then $x\wedge -: {\mathcal H}\to{\mathcal H}... |
4,092,473 | <p>I am trying to prove if the following set is bounded <span class="math-container">$$S=\bigcup_{a\in(0,1)} M_{a},$$</span> where
<span class="math-container">$$M_{a}=\{(x,y) \in \mathbb{R}^2: ax+(1-a)y=b, x>0, y>0, b \text{ is a fixed positive real number}\}.$$</span>
I think this set is bounded since all lines... | 5xum | 112,884 | <p>According to your definition of <span class="math-container">$S$</span>, which is <span class="math-container">$$S=\cup_{a\in(0,1)} \{(x,y)\in \mathbb R_+^2| ax+(1-a)y = b\},$$</span></p>
<p>the set <span class="math-container">$S$</span> is <strong>not</strong> bounded.</p>
<p>For an intuitive idea, think of it thi... |
3,540,613 | <p>The integral to solve:</p>
<p><span class="math-container">$$
\int{5^{sin(x)}cos(x)dx}
$$</span></p>
<p>I used long computations using integration by parts, but I don't could finalize:</p>
<p><span class="math-container">$$
\int{5^{sin(x)}cos(x)dx} = cos(x)\frac{5^{sin(x)}}{ln(5)}+\frac{1}{ln(5)}\Bigg[ \frac{5^{s... | lab bhattacharjee | 33,337 | <p>Let <span class="math-container">$5^{\sin x}=y,$</span></p>
<p>As <span class="math-container">$5=e^{\ln5},$</span></p>
<p><span class="math-container">$dy=5^{\sin x}\cos x\ln 5\ dx$</span> right?</p>
|
1,406,280 | <p>Why does taylor series have ample amount of importance in calculus? </p>
<p>I like to know some insights behind taylor series. </p>
| N. F. Taussig | 173,070 | <p>I will assume that by the natural numbers that you mean the positive integers (as opposed to the nonnegative integers).</p>
<p>Observe that $a + c = 100 - 2b$ is an even number. Thus, $a$ and $c$ must have the same parity.</p>
<p><strong>Case 1:</strong> The numbers $a$ and $c$ are both even. Let $a = 2u$; let ... |
317,601 | <blockquote>
<p>Let <span class="math-container">$F$</span> be a ring, let <span class="math-container">$f(x)=a_0+a_1x+\cdots+a_nx^n$</span> be in <span class="math-container">$F[x]$</span>, and <span class="math-container">$f'(x)$</span> be the regular derivative of <span class="math-container">$f(x)$</span>.</p>
<p>P... | Jesko Hüttenhain | 11,653 | <p>The condition that $f'$ does not vanish <a href="http://en.wikipedia.org/wiki/Inseparable_extension#Separable_and_inseparable_polynomials" rel="nofollow">is equivalent to $f$ being separable</a>.</p>
|
4,550,991 | <p>This is question is taken from an early round of a Norwegian national math competition where you have on average 5 minutes to solve each question.</p>
<p>I tried to solve the question by writing every number with four digits and with introductory zeros where it was needed. For example 0001 and 0101 would be the numb... | Henry | 6,460 | <p>One handwaving quick approach</p>
<ul>
<li>How many digits in total do you write for <span class="math-container">$0000$</span> to <span class="math-container">$9999$</span>?
<ul>
<li><span class="math-container">$4 \times 10^4$</span></li>
</ul>
</li>
<li>How many of those are <span class="math-container">$1$</span... |
302 | <p>I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first found? And what is the...significance?</p>
| Tracy Hall | 7,936 | <p>What is the significance? Most of the nice properties of the golden mean can be attributed to the fact that its continued fraction coefficients are uniformly bounded, as will be true in particular for any periodic continued fraction, which is to say any quadratic irrational, such as arises as the spectral radius of... |
255,773 | <p>As it is known that
Integrate[A+B]= Integrate[A] + Integrate[B]</p>
<p>I am facing problem with the following integral, when I integrate</p>
<pre><code>Integrate[(-(1/2) b^2 x^2 (-1 + EulerGamma + Log[(b x)/2]) -
2 (EulerGamma + Log[(b x)/2])) 1/
x ((-1 + x) Log[-1 + x]^2 - 2 (-1 + x) Log[-1 + x] (1 + Log[x])... | Akku14 | 34,287 | <p>Don't know why the definite integral of the part integral does not work, but with indefinite integration you get the right result (in MMA version 8.0)</p>
<pre><code>f[x_, b_] =
(-(1/2) b^2 x^2 (-1 + EulerGamma + Log[(b x)/2]) -
2 (EulerGamma + Log[(b x)/2])) 1/
x ((-1 + x) Log[-1 + x]^2 - 2 (-1 + x) Log[-1 + x] ... |
1,853,846 | <p>Prove that the equation <span class="math-container">$$x^2 - x + 1 = p(x+y)$$</span> has integral solutions for infinitely many primes <span class="math-container">$p$</span>.</p>
<p>First, we prove that there is a solution for at least one prime, <span class="math-container">$p$</span>. Now, <span class="math-cont... | Majid | 254,604 | <p>Replacing $cos\theta=\pm\sqrt{1-sin^2\theta}$ in your equation, $8\sin\theta = 4 + \cos\theta$, and considering $u=sin\theta$, we have</p>
<p>$8u=4\pm\sqrt{1-u^2}$ which leads to $64u^2 +16 - 64u=1-u^2\to 65u^2-64u+15=0$.</p>
<p>Now, you only need to solve this equation to find $u$ that is $sin\theta$.</p>
|
1,853,846 | <p>Prove that the equation <span class="math-container">$$x^2 - x + 1 = p(x+y)$$</span> has integral solutions for infinitely many primes <span class="math-container">$p$</span>.</p>
<p>First, we prove that there is a solution for at least one prime, <span class="math-container">$p$</span>. Now, <span class="math-cont... | Burrrrb | 322,248 | <p>$$8\sin(\theta)- \cos(\theta) =4$$
Dividing by $\sqrt{1^2 +8^2}$
$$\frac{8\sin(\theta)}{\sqrt{65}}- \frac{\cos(\theta)}{\sqrt{65}} =\frac{4}{\sqrt{65}}$$
Let $\sin(\alpha) = \frac{1}{\sqrt{65}}$, we $\cos(\alpha) = \frac{8}{\sqrt{65}}$.</p>
<p>Thus
$$\sin(\theta)\cos(\alpha)-\cos(\theta)\sin(\alpha) = \frac{4}{\s... |
542,148 | <p>Calculate the determinant of the following matrix as an explicit
function of $x$. (It is a polynomial in $x$. You are asked to find
all the coefficients.)</p>
<p>\begin{bmatrix}1 & x & x^{2} & x^{3} & x^{4}\\
x^{5} & x^{6} & x^{7} & x^{8} & x^{9}\\
0 & 0 & 0 & x^{10} &... | nbubis | 28,743 | <p><strong>Hint</strong>:
$$\det\begin{pmatrix}A & 0\\ C & D\end{pmatrix} = \det\begin{pmatrix}A & B\\ 0 & D\end{pmatrix} = \det(A) \det(D)$$
Where $A,B,C,D$ are block matrices. </p>
|
4,122,732 | <p>I was solving exercise 3.125 of Wackerly's Probability book and i did not understand the solution given in the solutions manual.</p>
<p>The problem says:</p>
<p>Customers arrive at a shop following a poisson distribution for an average of 7 customers per hour. What is the probability that exactly two clients arrive ... | angryavian | 43,949 | <p>The question is asking that the probability that exactly two people arrive in the combined two-hour period. In particular, it is ok for one person to arrive at 1:30 and one other to arrive at 2:30.</p>
<p>You computed the probability that exactly two people arrived between 1:00 and 2:00 and added it to the probabili... |
8,567 | <p>When highlighting text using <code>Style</code> and <code>Background</code>, as in <code>Style["Test ", White, Background -> Lighter@Blue]</code> is there a way to pad (ie, enlarge) the bounding box? </p>
<p>The bottom of the background seems coincident with the base of the text: <img src="https://i.stack.imgu... | Rojo | 109 | <p>Does this work as you want to?</p>
<pre><code>SetAttributes[f, HoldAllComplete];
{first, rest___} ^:= HoldComplete[rest]
f[args___] := {first, args}
f[own, down[1], sub[1][2], N[n], up]
</code></pre>
<blockquote>
<p>HoldComplete["OwnValue", "DownValue", "SubValue", 3.14, up]</p>
</blockquote>
|
161,678 | <p>Assume a process with Itô dynamics of the generic form
$$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$$</p>
<p>and let $f:\mathbb{R}\to\mathbb{R}$ be borel-measurable. Is the following function smooth ?
$$g(t,x)=\mathbb{E}[f(X_T)|\mathcal{F}_t]$$</p>
<p>I remember comming upon the proof of above once but I cannot find it ... | Dan | 44,169 | <p>This depends on what you assume of $\mu$, $\sigma$, and $f$. I'll provide a few examples. Let's assume $f$ is bounded throughout, since what I'll say below is still true in the unbounded case under suitable growth assumptions.</p>
<p>First, as long the SDE is well-posed (in the sense of weak existence and uniquenes... |
3,009,112 | <p>I am a geographer/ecologist and I want to know how to accurately calculate volume of a lake or a reservoir? I am not looking for a vague estimate which is generally calculated using surface area and mean height parameters assuming the body is of a certain shape (truncated cone/triangle or circular). Since reservoirs... | Derek Holt | 2,820 | <p>I think the following is essentially the same argument as in the proof of the Brauer-Fowler result cited by the_fox.</p>
<p>Let <span class="math-container">$I$</span> be the set of involutions in <span class="math-container">$G$</span> and <span class="math-container">$m=|I|$</span>, <span class="math-container">$... |
3,335,060 | <blockquote>
<p>The numbers of possible continuous <span class="math-container">$f(x)$</span> defiend on <span class="math-container">$[0,1]$</span> for which <span class="math-container">$I_1=\int_0^1 f(x)dx = 1,~I_2=\int_0^1 xf(x)dx = a,~I_3=\int_0^1 x^2f(x)dx = a^2 $</span> is/are</p>
<p><span class="math-container"... | rrainbow | 642,066 | <p>Similar to your last proof, I found a positive function whose integral is <span class="math-container">$e^3-20$</span>.</p>
<p>For <span class="math-container">$$f(x)=\frac{1}{186}(x-1)^2(x-2)^4e^x\ge0$$</span></p>
<p>we have <span class="math-container">$$\int_{0}^{3}f(x)dx=e^3-20$$</span></p>
|
2,777,555 | <p>Prove that if $f(0)=0$ and $f'(0)=0$, then $f(x)=0$ for all $x$. </p>
<p>Hint: The idea
is to multiply both sides of the equation $f''(x)+ f(x) = 0$ by something that
makes the left-hand side of the equation into the derivative of something.</p>
<p>I'm not sure how to proceed and don't really understand the hint.<... | marty cohen | 13,079 | <p>Multiply by f'.
It becomes
$((f^2)'+(f'^2)')/2$.</p>
|
264,587 | <p><strong>NOTE</strong></p>
<p>I'm sorry, my question was not clear. I want to know all the ways to split a list with a given length simply, <strong>rather than split a cyclic substitution</strong>.
If a given list has length <span class="math-container">$N$</span> and the rule is <span class="math-container">${m, n, ... | Simon Woods | 862 | <p>This isn't any faster than the original code, just a different approach.</p>
<pre><code>partitionList2[list_, pat_] := Module[{rn, p},
rn = Range@Length@pat;
p = Flatten[MapThread[ConstantArray, {rn, pat}]];
DeleteDuplicates@
Table[Flatten /@ Reap[MapThread[Sow, {list, q}], rn][[2]], {q, Permutations[p]}]... |
838,400 | <p>One question asking if $\mathbb{Z}^*_{21}$ is cyclic.</p>
<p>I know that the cyclic group must have a generator which can generate all of the elements within the group.</p>
<p>But does this kind of question requires me to exhaustively find out a generator? Or is there any more efficient method to quickly determine... | Ittay Weiss | 30,953 | <p>A finite group is cyclic if, and only if, it has precisely one subgroup of each divisor of its order. So if you find two subgroups of the same order, then the group is not cyclic, and that can help sometimes. </p>
<p>However, $Z^*_{21}$ is a rather small group, so you can easily check all elements for generators.</... |
2,269,042 | <p>Consider $f(x) = \lfloor x \rfloor + \lfloor -x \rfloor $ . Now find value of $\lim_{x \to \infty} f(x) $ . I know that if $x_0 \in \mathbb{R}$ then $\lim_{x \to x_0} f(x) = -1$ but I don't know whether it is true or not in the infinity . </p>
| G Cab | 317,234 | <p>Defining with $ \left\{ x \right\}$ the fractional part of $x$, i.e.
$$
x = \left\lfloor x \right\rfloor + \left\{ x \right\}
$$
and denoting by $[P]$ the <a href="https://en.wikipedia.org/wiki/Iverson_bracket" rel="nofollow noreferrer"><em>Iverson bracket</em></a>
$$
\left[ P \right] = \left\{ {\begin{array}{*{20}... |
404,574 | <p>Suppose that:</p>
<p>$Y \pmod B = 0$</p>
<p>$Y \pmod C = X$</p>
<p>I know $B$ and $C$. $Y$ is unknown, it might be an extremely large number, and it does not interest me. </p>
<p>The question is: Is it possible to find $X$, and if so, how?</p>
| Key Ideas | 78,535 | <p>$\bigg\lbrace\begin{eqnarray}y\equiv 0\pmod b\\ y\equiv x\pmod c\end{eqnarray}\bigg\rbrace $ $\!\iff\! \bigg\lbrace\begin{eqnarray} y &=& j b,\ \ &{\rm some}\ \ j\in \Bbb Z\\ y &=& x\! +\! k c,\ \ &{\rm some}\ \ k\in \Bbb Z\end{eqnarray}\bigg\rbrace\!\iff\! x = jb\!-\!kc,\ \ {\rm some}\ \ j,... |
363,767 | <p>An ellipse is specified $ x^2 + 4y^2 = 4$, and a line is specified $x + y = 4$. I need to find the max/min distances from the ellipse to the line.</p>
<p>My idea is to find two points $(x_1, y_1)$ and $(x_2,y_2)$ such that the first point is on the ellipse and the second point is on the line. Furthermore, the line ... | Will Jagy | 10,400 | <p>much easier. if your line and ellipse intersect, the minimum distance is zero. if not, it occurs on a line segment that is orthogonal to both your line and to the ellipse. which is not hard to find. there should be two such line segments, one gives the actual min, one gives a local max...draw some graphs! also, no m... |
1,278,329 | <p>Solve the recurrence $a_n = 4a_{n−1} − 2 a_{n−2}$</p>
<p>Not sure how to solve this recurrence as I don't know which numbers to input to recursively solve?</p>
| Scott | 89,128 | <p>We can represent this recursion with matrices:</p>
<p>$$
\begin{bmatrix}a_n \\ a_{n-1} \end{bmatrix}
=
\begin{bmatrix}4 & -2 \\ 1 & 0 \end{bmatrix}
\begin{bmatrix}a_{n-1} \\ a_{n-2} \end{bmatrix}
$$</p>
<p>Or $x_n = A\cdot x_{n-1}$ where $x_n$ is the vector specified above. This is a simple difference equ... |
4,121,607 | <p>I want to find a function which satisfies certain following limits.</p>
<p>The question is:
Find a function which satisfies</p>
<p><span class="math-container">$$
\lim_{x\to5} f(x)=3, \text{ and } f(5) \text{ does not exist}
$$</span></p>
<p>I would think that because it says <span class="math-container">$f(5)$</spa... | hamam_Abdallah | 369,188 | <p><span class="math-container">$$f(x)=\frac{\sin(3(x-5))}{x-5}$$</span>
or
<span class="math-container">$$f(x)=\frac{e^{3(x-5)}-1}{x-5}$$</span>
or
<span class="math-container">$$f(x)=\frac{\ln(3x-14)}{x-5}$$</span>
satifies the wanted conditions.</p>
|
4,408,507 | <p>We study the definition of Lebesgue measurable set to be the following:</p>
<p>Let <span class="math-container">$A\subset \mathbb R$</span> be called Lebesgue measurable if <span class="math-container">$\exists$</span> a Borel set <span class="math-container">$B\subset A$</span> such that <span class="math-container... | L. F. | 620,160 | <p>Here is an alternative trigonometric proof of the inequality for fun. Denote <span class="math-container">$$
x = \frac{A}{2}, \qquad y = \frac{B}{2}, \qquad \{x, y, x + y\} \subset [0, 90^\circ].
$$</span>
Then, using the <a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum... |
4,526,852 | <p>I need some help understanding the fallacy in the following reasoning (thank you in advance!). It is essentially implying that for a single sample from a population, you can know a population parameter (like the EV) precisely with a few seemingly reasonable assumptions. I’ll frame it as an example.</p>
<p>Let’s say ... | Lelouch | 991,491 | <p><span class="math-container">$\frac{1}{\sin(x)}$</span> is surely not bounded between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>.</p>
<p>You are falling into a pitfall since the inverse function is undefined in <span class="math-container">$0$</span>, you actually have to app... |
4,526,852 | <p>I need some help understanding the fallacy in the following reasoning (thank you in advance!). It is essentially implying that for a single sample from a population, you can know a population parameter (like the EV) precisely with a few seemingly reasonable assumptions. I’ll frame it as an example.</p>
<p>Let’s say ... | Bernkastel | 551,169 | <p>The reciprocal function <span class="math-container">$t \mapsto 1/t$</span> is a decreasing functions on the two separate intervals <span class="math-container">$(-\infty,0)$</span> and <span class="math-container">$(0,\infty$</span>). So, you cannot take reciprocals and reverse inequality sign if your functions not... |
2,352,313 | <p>If $f_n$ is the number of permutations of numbers $1$ to $n$ that no number is in it's place(I think same as $D_n$)and $g_n$ is the number of the same permutations with exactly one number in it's place Prove that $\mid f_n-g_n \mid =1$.</p>
<p>I need a proof using mosly combinatorics not mostly algebra.I think we s... | drhab | 75,923 | <p>Concerning <a href="https://en.wikipedia.org/wiki/Derangement#Counting_derangements" rel="nofollow noreferrer">derangements</a> we can prove with inclusion/exclusion that:
$$!n=n!\sum_{i=0}^n\frac{(-1)^i}{i!}$$</p>
<p>The RHS can also be written as:$$n\left[(n-1)!\sum_{i=0}^{n-1}\frac{(-1)^i}{i!}\right]+(-1)^n=n\le... |
1,928,439 | <p>Is there a space whose dual is $F^m$? ($F$ is the field w.r.t. which the original set is a vector space)</p>
<p>I'm trying to do the following exercise:</p>
<p><a href="https://i.stack.imgur.com/j9WMP.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/j9WMP.png" alt="enter image description here"><... | quid | 85,306 | <p>Let $e_1, \dots, e_m$ a basis of $F^m$. </p>
<p>Let $f:F^m \to F$ be a linear form, that is an element of the dual space. </p>
<p>Then for any $v= a_1e_1 + \dots + a_me_m$ we have $f(v)= f(a_1e_1 +\dots + a_me_m) = a_1f(e_1) +\dots + a_mf(e_m)$. Thus $f$ is uniquely determined by
$f(e_1), \dots, f(e_m)$ that is a... |
2,386,602 | <p>This is a question from an exam I recently failed. </p>
<p>What is the radius of convergence of the following power series? $$(a) \sum_{n=1}^\infty(n!)^2x^{n^2}$$ and $$(b) \sum_{n=1}^\infty \frac {x^{n^2}}{n!}$$</p>
<p>Edit: Here's my attempt at the first one, if someone could tell me if it's any good...</p>
<p>... | J.G. | 56,861 | <p>Use the ratio test. The ratio of consecutive terms is of the form $n^p x^{2n-1}$. For $|x|<1$, the $n\to\infty$ limit is $0$; for $|x|>1$, it is $\infty$. Thus the radius of convergence is $1$.</p>
|
883,620 | <p>$a,b,c \geq 0$ and $a^2+b^2+c^2+abc=4$ prove that $ab+bc+ac-abc \leq 2$
can any one help me with this problem,I believe Dirichlet's theorem is the key for this
sorry for making mistake over and over again,but i'm certain that the inequality is true now.</p>
| Macavity | 58,320 | <p>Note $a^2+b^2+c^2+abc=1 \implies 0\le a,b,c \le 1$. Hence
$$(1-a)(1-b)c\ge 0 \implies c\ge bc+ca-abc$$ </p>
<p>So now it is enough to prove $a+b+c \le 2$, which is obvious from AM-QM
$$\frac{a+b+c}3 \le \sqrt{\frac{a^2+b^2+c^2}3}\le \sqrt{\frac13}$$ </p>
|
883,620 | <p>$a,b,c \geq 0$ and $a^2+b^2+c^2+abc=4$ prove that $ab+bc+ac-abc \leq 2$
can any one help me with this problem,I believe Dirichlet's theorem is the key for this
sorry for making mistake over and over again,but i'm certain that the inequality is true now.</p>
| Macavity | 58,320 | <p>For your last revision of the question (you'll need to change the title too)...</p>
<p>Note among any three numbers $a,b,c$, some two will be either $\le 1$ or $\ge 1$. WLOG let these be $a,b$. So $(1-a)(1-b)c \ge 0 \implies bc + ca \le c+abc \implies ab+bc+ca -abc\le ab+c$. Now you need to show $ab \le 2-c$. For ... |
1,647,327 | <p>suppose S is a metric space and $B(S)$ is the set of bounded functions and $C_b(S)$ is the set consisting of bounded continuous functions.</p>
<p>Prove that $C_b(S)$ is a closed subspace of $B(S)$.</p>
<p>I thought of looking at the complement $B(S) \backslash C_b(S) = \{f| \text{ f is bounded and not continuous}\... | noctusraid | 185,359 | <p>An alternative answer (based more on OP's knowledge):</p>
<p>In a metric space $(X, \Vert \cdot \Vert_X)$ and $A \subset X$ we have the following assertion:</p>
<p>$$ x \in \bar{A} \iff \exists (x_n)_{n \in \mathbb N} \subset A: x_n\to x \in X $$</p>
<p>This means in words: an element $x$ of a subspace $A \subset... |
1,386,367 | <p>I'm interested in the definite integral</p>
<p>\begin{align}
I\equiv\int_{-\infty}^{\infty} \frac{1}{x^2-b^2}=\int_{-\infty}^{\infty} \frac{1}{(x+b) (x-b)}.\tag{1}
\end{align}</p>
<p>Obviously, it has two poles ($x=b, x=-b$) on the real axes and is thus singular. I tried to apply the contour integration methods me... | sebigu | 32,185 | <p>If $0 \neq 2$ in the field and $P^2=P$, then the minimal Polynomial of $P$ divides $f := x^2-x$, which means it is $f$, $x$, or $x-1$. If it is $x$, $P=0$, and if it is $x-1$, $P=1$. Those cases are clear.</p>
<p>So suppose it is $x^2-x$. Then $I+P$ has minimal polynomial $(x-1)(x-2)=x^2-3x+2$. This means that $I$ ... |
3,944,628 | <p>I'm reading a book and, in its section on the definition of a stopping time(continuous), the author declares at the start that for the whole section every filtration will be complete and right-continuous.</p>
<p>So, in the definition of a Stopping Time, how important are these conditions? Why would they matter?</p>
| J.G. | 56,861 | <p>Since <span class="math-container">$\sin x$</span> is <a href="https://en.wikipedia.org/wiki/Concave_function" rel="nofollow noreferrer">concave</a> on acute <span class="math-container">$x$</span>, by <a href="https://en.wikipedia.org/wiki/Jensen%27s_inequality" rel="nofollow noreferrer">Jensen's inequality</a> the... |
4,090,408 | <p>Show that <span class="math-container">$A$</span> is a whole number: <span class="math-container">$$A=\sqrt{\left|40\sqrt2-57\right|}-\sqrt{\left|40\sqrt2+57\right|}.$$</span>
I don't know if this is necessary, but we can compare <span class="math-container">$40\sqrt{2}$</span> and <span class="math-container">$57$<... | fleablood | 280,126 | <p>First of all is <span class="math-container">$40\sqrt 2$</span> greater or less than <span class="math-container">$57$</span>? One assumes it's a little less.</p>
<p><span class="math-container">$(\frac {57}{40})^2 = \frac {(60-3)^2}{1600} = \frac {3600 +9 -2\cdot 3\cdot 60}{1600}= \frac {3600-360+ 9}{1600}= \frac {... |
165,900 | <p>Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) \in k p$. My question: is $G$ some well known algebraic group? </p>
| Steven Sam | 321 | <p>To elaborate on abx's comment: modding out by scalars, i.e., working in $PGL_3$ instead of $GL_3$, by definition the stabilizer of $p$ is the group of projective automorphisms of the curve $p=0$ which preserves the embedding of the curve.</p>
<p>If we assume that the curve is smooth and that there is a rational poi... |
4,436,210 | <p>I have been given this exercise: Calculate the double integral:</p>
<blockquote>
<p><span class="math-container">$$\iint_D\frac{\sin(y)}{y}dxdy$$</span>
Where <span class="math-container">$D$</span> is the area enclosed by the lines: <span class="math-container">$y=2$</span>, <span class="math-container">$y=1$</span... | Doug M | 317,176 | <p>Rather than proceeding with Gauss-Jordan elimination....</p>
<p>For what values of <span class="math-container">$a$</span> are the vectors <span class="math-container">$(1,2,-3),(3,-1,5),(4,1,a^2-14)$</span> linearly independent or dependent?</p>
<p>If they are independent we have a unique solution.</p>
<p>If they a... |
4,436,210 | <p>I have been given this exercise: Calculate the double integral:</p>
<blockquote>
<p><span class="math-container">$$\iint_D\frac{\sin(y)}{y}dxdy$$</span>
Where <span class="math-container">$D$</span> is the area enclosed by the lines: <span class="math-container">$y=2$</span>, <span class="math-container">$y=1$</span... | user2661923 | 464,411 | <p>Alternative approach:</p>
<p>Compute the determinant of the matrix.</p>
<p>If the determinant is non-zero, then there will automatically be exactly <span class="math-container">$1$</span> solution.</p>
<p>If the determinant is <span class="math-container">$0$</span>, then there will either be <span class="math-conta... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Vincent | 41,139 | <p>I'm a bit surprised not to see the <a href="http://en.wikipedia.org/wiki/Weil_conjectures">Weil conjectures</a> here since their proof by Deligne is so often mentioned as a primary example of something Very Hard. Is there a more simple recent proof that I haven't heared of?</p>
|
27,490 | <h2>Motivation</h2>
<p>The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy groups are how higher dimensional spheres behave (up to homotopy in both cases, of course). Even better, for nic... | Donu Arapura | 4,144 | <p>I'm not sure how to answer this, because it already seems pretty geometric to me. So let me
answer a slightly different question: what is the fundamental groupoid good for?
Since one knows that the fundamental group and groupoid are equivalent as categories for
path connected spaces, it's tempting to view the groupo... |
27,490 | <h2>Motivation</h2>
<p>The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy groups are how higher dimensional spheres behave (up to homotopy in both cases, of course). Even better, for nic... | The Mathemagician | 3,546 | <p>A very geometric approach to the fundamental groupoid can be found in Ronald Brown's <em>Topology and Groupoids</em>. Since EVERYTHING is expressed from the beginning in terms of the category of equivelence classes of paths,the formulation is very straightforward and simple. I highly recommend the book to all mathem... |
28,347 | <p>I often end up with a function that contains the term $1/(1 + x^2/y^2)$, and I need to evaluate this in the limit $y\rightarrow 0$. By hand, I can rewrite this as $y^2/(y^2 + x^2)$, but how can I tell <em>Mathematica</em> to make such a simplification?</p>
<p>I have tried using <code>1/(1 + x^2/y^2) // Simplify</co... | bill s | 1,783 | <p>Like many limits involving two variables, this one is not as straightforward as it might seem. Consider</p>
<pre><code>Limit[1/(1 + x^2/y^2), x -> 0]
1
</code></pre>
<p>while</p>
<pre><code>Limit[1/(1 + x^2/y^2), y -> 0]
0
</code></pre>
<p>Coupling the two:</p>
<pre><code>Limit[1/(1 + x^2/y^2) /. y ->... |
2,069,507 | <p><a href="https://i.stack.imgur.com/B4b88.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/B4b88.png" alt="The image of parallelogram for help"></a></p>
<p>Let's say we have a parallelogram $\text{ABCD}$.</p>
<p>$\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two... | Narasimham | 95,860 | <p>Area of a triangle is</p>
<p>$$ Area = \frac12 b\, c \sin A $$</p>
<p>Among the three, two parameter values excepting $A$ are given, then the remaining angle $A$ and <em>its supplement</em> can satisfy this relation.</p>
<p>If you take base $b$ and $ h= c \sin A $ as height all the following shaded areas hav... |
11,698 | <p><strong>Bug introduced in 8.0 or earlier and fixed in 13.2.0 or earlier</strong></p>
<hr />
<p>So I have been fighting with this for a while. I'm trying to get custom frame ticks on both the left and right side of a <code>DistributionChart</code>. It's not going very well. It just keeps throwing errors saying tick p... | Sjoerd C. de Vries | 57 | <p>There seems to be a bug here as even </p>
<pre><code>DistributionChart[{{1, 2, 3, 4, 5}, {1, 1, 2, 3}}, FrameTicks -> {{None, None}, {None, None}}]
</code></pre>
<p>yields an error, but you can work around it as follows:</p>
<pre><code>With[{fps = {120, 60, 50, 40, 30, 25, 20, 15, 10}},
DistributionChart[
... |
1,674,676 | <p>Let $p$ be an odd prime number. I want to show that $\mathbb{F}_{p^2}$ has a primitive 8th root of unity $\zeta$. </p>
<ul>
<li>I know that $\zeta^8 = 1$. So my idea is to define $f = X^8 - 1$ such that $\zeta$ is a root of $f$. But this is for a field extension of degree 8 and $p^2$ is at least 9.</li>
</ul>
<p>a... | DonAntonio | 31,254 | <p>We have the cyclic group $\;\Bbb F_{p^2}^*\;$ of order $\;p^2-1=0\pmod8\;$ and thus there exists a (unique) subgroup of order eight there. Any generator of this subgroup will do it.</p>
|
4,440,233 | <blockquote>
<p>Find all the functions <span class="math-container">$f:\mathbb{Z}^+ \to \mathbb{Z}^+$</span> such that <span class="math-container">$f(f(x)) = 15x-2f(x)+48$</span>.</p>
</blockquote>
<p>If <span class="math-container">$f$</span> is a polynomial of degree <span class="math-container">$n$</span>, we have ... | Jyrki Lahtonen | 11,619 | <p>Let us study the two variable affine transformation <span class="math-container">$g:\Bbb{R}^2\to\Bbb{R}^2$</span>
<span class="math-container">$$
g(x,y)=(y,15x-2y+48).
$$</span>
The connection to the problem is, of course, that <span class="math-container">$g(x,f(x))=(f(x),f(f(x)))$</span>. I am using the idea from ... |
3,618,791 | <p>Given <span class="math-container">$p\in[0,1]$</span>, prove or disprove that the sum
<span class="math-container">$$\sum_{n=k}^\infty\sum_{j=0}^k\left(\matrix{n\\j}\right)p^j(1-p)^{n-j}$$</span>
is bounded by a constant that does not depend on <span class="math-container">$k$</span>.</p>
<p>The terms <span class="... | lonza leggiera | 632,373 | <p><strong>Hint:</strong> The sum in part (b) is not a geometric series. But we do know that <span class="math-container">$\ \ln\left(a^n\right)=n\ln(a)\ $</span> for any positive <span class="math-container">$\ a\ $</span>, so <span class="math-container">$\ \sum_\limits{n=0}^{10}\ln\left(0.5^n\right)= \sum_\limits{n... |
192,072 | <p>Bonjour!<br>
I'm trying this number-theory problem, but i don't have any idea how to solve it.<br>
Can you give me some hints ?</p>
<p>We have got any $\mathbb{Z_+}$ number. Let it be $n$.<br>
Then we must proof that $2 \nmid \sigma(n) \implies n = k^2 \vee n = 2k^2$.<br>
Thanks for any help </p>
| André Nicolas | 6,312 | <p>Let $n=2^e m$ where $m$ is odd. Note that $\sigma(2^e)$ is odd. So by the multiplicativity of $\sigma$, $\sigma(n)$ is odd iff $\sigma(m)$ is odd. Any power of $2$ is a square or twice a square. So we need only show that if $\sigma(m)$ is odd for the odd number $m$, then $m$ is a perfect square. </p>
<p>If $m$ is ... |
1,079,995 | <p>I can't understand how: $$ \frac {2\times{^nC_2}}{5} $$</p>
<p>Equals:</p>
<p>$$ 2\times \frac {^nC_2}{5} $$</p>
<p>If we forget the combination and replace it with a $10$, the result is clearly different. $1$ in the first example and and $0.5$ in the second.</p>
| DeepSea | 101,504 | <p>Remember some fact about fractions: $\dfrac{a}{b}\times \dfrac{c}{d} = \dfrac{ac}{bd} \Rightarrow \dfrac{2\times \binom{n}{2}}{5} = \dfrac{2\times \binom{n}{2}}{1\times 5} = \dfrac{2}{1}\times \dfrac{\binom{n}{2}}{5} = 2\times \dfrac{\binom{n}{2}}{5}$</p>
|
1,133,817 | <p>Using Legendre polynomial generating function
\begin{equation}
\sum_{n=0}^\infty P_n (x) t^n = \frac{1}{\sqrt{(1-2xt+t^2)}}
\end{equation}
Or $$ P_n(x)=\frac{1}{2^n n!} \frac{d^n}{dx^n} [(x^2-1)^n] $$ </p>
<p>Show$$ P_{2n}(0)=\frac{(-1)^n (2n)!}{(4)^n (n!)^2} $$
And $$ P_{2n+1}(0)=0$$</p>
<p>I expressed ... | Ivo Terek | 118,056 | <p>If $x \to 0$, then $x$ is close to zero, but it is not zero. So we can do: $$\lim_{x \to x_0} f(x) = \lim_{x \to x_0}\frac{f(x)}{x}x = \lim_{x \to x_0}\frac{f(x)}{x}\cdot \lim_{x \to x_0}x = 0 \cdot x_0 = 0.$$</p>
|
2,296,256 | <p>I need help how to mathematically interpret an ODE (Newton's second law). I used to the ODE in this form:
$$
m\ddot x(t)=F(t)\tag{1}
$$</p>
<p>However, in another book they wrote:
$$
m\ddot x=F(x,\dot x) \tag{2}
$$
where $F: \mathbb{R}^n \times \mathbb{R}^n\rightarrow \mathbb{R}^n$.</p>
<p><strong>Questions:</stro... | 5xum | 112,884 | <p>First of all, your two conditions are redundant, because the second follows from the first:</p>
<ul>
<li>Just to make it more readable, the first condition is equivalent to $$\forall x,y: x>y \implies f(x)>f(y)$$</li>
<li>If $i < j$, then $j>i$.</li>
<li>Set $x=j$ and $y=i$.</li>
<li>Therefore, $x>y$... |
2,510,723 | <p>Does the series $\displaystyle\sum_{n=1}^{\infty} \dfrac{19+(n+5)!}{(n+7)!}$ Converge or diverge</p>
<blockquote>
<p>I tried using the ratio test, but it gave me this which I don't see how you'd be able to solve.
$$\lim_{n\to\infty} |\frac{a_{n+1}}{a_{n}}| = \lim_{n\to\infty}\frac{19+(n+6)!}{19+(n+5)!*(n+8)}$$... | ajotatxe | 132,456 | <p>$$\sum_{n=1}^\infty\frac{19+(n+5)!}{(n+7)!}=\sum_{n=1}^\infty\frac{1+\frac{19}{(n+5)!}}{(n+6)(n+7)}<\sum_{n=1}^\infty\frac2{n^2}$$</p>
|
2,125,018 | <blockquote>
<p>You toss a fair coin 3x, events:</p>
<p>A = "first flip H"</p>
<p>B = "second flip T"</p>
<p>C = "all flips H"</p>
<p>D = "at least 2 flips T"</p>
<p><strong>Q:</strong> Which events are independent?</p>
</blockquote>
<p>From the informal def. it is where one doesnt affect th... | Mick | 42,351 | <p>The picture clearly show $\dfrac {\beta}{\alpha}$ is not constant.</p>
<p><a href="https://i.stack.imgur.com/Y0ikm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Y0ikm.png" alt="enter image description here"></a></p>
|
2,125,018 | <blockquote>
<p>You toss a fair coin 3x, events:</p>
<p>A = "first flip H"</p>
<p>B = "second flip T"</p>
<p>C = "all flips H"</p>
<p>D = "at least 2 flips T"</p>
<p><strong>Q:</strong> Which events are independent?</p>
</blockquote>
<p>From the informal def. it is where one doesnt affect th... | Mick | 42,351 | <p>Since A is fixed, I assume that it is located at (h, k) where h and k are known quantities.</p>
<p><a href="https://i.stack.imgur.com/CiY2R.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CiY2R.png" alt="enter image description here"></a></p>
<p>Further assumptions:- OY = m, OA = n, $\angle XOY ... |
154,893 | <p>I am having trouble figuring this out.</p>
<p>$$\sqrt {1+\left(\frac{x}{2}- \frac{1}{2x}\right)^2}$$</p>
<p>I know that $$\left(\frac{x}{2} - \frac{1}{2x}\right)^2=\frac{x^2}{4} - \frac{1}{2} + \frac{1}{4x^2}$$ but I have no idea how to factor this since I have two x terms with vastly different degrees, 2 and -2.<... | DonAntonio | 31,254 | <p>Hint: for any (real, complex) numbers $\,a,b,\,$: $$4ab+(a-b)^2=(a+b)^2$$</p>
|
42,881 | <p>In short, my question is the same as my previous one except that everything is now wrapped up in a module.</p>
<p>The relevant code I'm working with is:</p>
<pre><code>getinter[a_, b_, u0_, k_, m_, hbar_, Nu_, Np_, up_] :=
Module[{ekp, ms, LUs, env, eenv, envpart, f, kppart, g, approx,
approx1, papprox, appro... | Dr. belisarius | 193 | <p>Use pure functions as a return value :</p>
<pre><code>fOne[x_] :=(*just return a piecewise*)
Piecewise[{{#^2, # < x}, {#^2 + (x - #)^3 Sin[3 #], # > x}}] &
fTwo[y_, z_, w_] := Module[{vars},
Plot[fOne[y][x], {x, z, w}]]
fTwo[3, -3, 8]
</code></pre>
<p><img src="https://i... |
2,268,299 | <p>Let's say I have a series of real values $y_0,y_1,y_2\cdots$.
My question is if it's always possible to find (at least one) $C^\infty$ real function such that
\begin{equation}
f^{(n)}(0)=y_n
\end{equation}
and in the affirmative case, how.
It is a kind of "reverse taylor" problem... any hints?</p>
| Jean Marie | 305,862 | <p>I am going to address a complementary issue: the fact that if a solution to this problem exists, it is not unique.</p>
<p>This is due to the existence of (Laurent) Schwartz function defined by </p>
<p>$\varphi(x)=\exp(-\tfrac{1}{x^2})$ for $x \neq 0$ and $\varphi(0) = 0$, which is such that $\forall n\in\mathbb{N}... |
474,048 | <p>I am stuck with the following problem from a book.</p>
<p>It asks whether or not $f_n \rightarrow f$ converges uniformly on $A$ if for every $[a,b], f_n\rightarrow f$ uniformly on $A\cap [a,b]$.</p>
<p>The statement seems false to me (i.e. not necessarily true) because of this intuition I had:</p>
<p>If $A$ is no... | Colliot | 80,507 | <p>Write $A$ (say non-negative real numbers) as a infinite union of closed sets (say intervals) $A_j$, denote the (maximal) distance from limit on $A_j$ down to $f_i$ as $D(i,j)$. We know already $\forall\epsilon>0,\forall j, \exists N(j,\epsilon):i>N(j,\epsilon)\Rightarrow D(i,j)<\epsilon$. We don't whant $\f... |
208,744 | <p>I was asked to show that $\frac{d}{dx}\arccos(\cos{x}), x \in R$ is equal to $\frac{\sin{x}}{|\sin{x}|}$. </p>
<p>What I was able to show is the following:</p>
<p>$\frac{d}{dx}\arccos(\cos(x)) = \frac{\sin(x)}{\sqrt{1 - \cos^2{x}}}$</p>
<p>What justifies equating $\sqrt{1 - \cos^2{x}}$ to $|\sin{x}|$?</p>
<p>I ... | bradhd | 5,116 | <p>In general, $\sqrt{a^2} = |a|$, as $\sqrt y$ refers to the <em>non-negative</em> number whose square is $y$. Then since $1-\cos^2x = \sin^2x$, we have $\sqrt{1-\cos^2 x} = |\sin x|$.</p>
|
10,600 | <p>As mentioned in <a href="https://matheducators.stackexchange.com/questions/1538/counterintuitive-consequences-of-standard-definitions">this question</a> students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \t... | Amir Asghari | 1,217 | <p>Have a look at the paper written by Nunez et all: </p>
<p><a href="http://link.springer.com/article/10.1023%2FA%3A1003759711966">EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS AND CONTEXT IN MATHEMATICS EDUCATION</a>. </p>
<p>In essence, they argue that it is better to be causious if you want to "motivate the fo... |
1,354,953 | <p>Solve for the function f(x):</p>
<p>$$f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$$
I'm not able to solve this. </p>
<p>[For instance, I tried solving for $f(\frac{x}{x+f(x)})$, but this doesn't lead me anywhere as the value obtained, when substituted into the original equation, just yields $f(x)=f(x)$]</p>
<... | Servaes | 30,382 | <p>If such an $f$ exists, plugging in $x=0$ we find that
$$f(0)=\frac{0}{0+f\left(\tfrac{0}{0+f(0)}\right)}=0,$$
and to avoid having to divide by $0$ this requires that
$$0+f(0)\neq0\qquad\text{ and }\qquad0+f\left(\frac{0}{0+f(0)}\right)\neq0.$$
But then $f(0)=0$ so $0+f(0)=0$, a contradiction. Hence no such $f$ exist... |
3,671,223 | <p>First and foremost, I have already gone through the following posts:</p>
<p><a href="https://math.stackexchange.com/questions/2463561/prove-that-for-all-positive-integers-x-and-y-sqrt-xy-leq-fracx-y">Prove that, for all positive integers $x$ and $y$, $\sqrt{ xy} \leq \frac{x + y}{2}$</a></p>
<p><a href="https://ma... | Narasimham | 95,860 | <p>Let <span class="math-container">$ x=u^2, y=v^2$</span> since <span class="math-container">$ (x,y)$</span> are given positive.</p>
<p>You have to prove that</p>
<p><span class="math-container">$$ u^2+v^2-2 u v \ge 0$$</span></p>
<p>or that</p>
<p><span class="math-container">$$(u-v)^2\ge 0 $$</span></p>
<p>whic... |
3,671,223 | <p>First and foremost, I have already gone through the following posts:</p>
<p><a href="https://math.stackexchange.com/questions/2463561/prove-that-for-all-positive-integers-x-and-y-sqrt-xy-leq-fracx-y">Prove that, for all positive integers $x$ and $y$, $\sqrt{ xy} \leq \frac{x + y}{2}$</a></p>
<p><a href="https://ma... | Aderinsola Joshua | 395,530 | <p><span class="math-container">$$\sqrt(xy) \leq \frac{x+y}{2}$$</span>
<span class="math-container">$$xy \leq \frac{(x+y)^2}{4}$$</span>
<span class="math-container">$$4xy \leq x^2+2xy+y^2$$</span>
<span class="math-container">$$2xy \leq x^2+y^2$$</span>
<span class="math-container">$$2 \leq \frac{x^2}{xy}+\frac{y^2}{... |
2,885,754 | <p>I just sat a real analysis exam and this was a question in it that I couldn't answer...</p>
<p>Prove that $\left|e^\frac{-x^2}{2t}-e^\frac{-y^2}{2t}\right| \leq \frac{|x-y|}{t}$ for $x,y \in [-1,1] ,t>0$</p>
<p>I ended up trying to set $f(x,y)=e^\frac{-x^2}{2t}-e^\frac{-y^2}{2t}$, then attempted trying $f(-1,-1... | xbh | 514,490 | <p>Hint. consider $F(z)= \exp(-z^2/2t)\;[|z|\leqslant 1]$, $t$ is fixed. Then MVT. </p>
|
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