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 |
|---|---|---|---|---|
2,223,163 | <p>I don't have any idea on how to prove it, and I need it for one of my questions which is still unanswered: <a href="https://math.stackexchange.com/questions/2192947/what-is-the-largest-number-smaller-than-100-such-that-the-sum-of-its-divisors-is?noredirect=1#comment4521040_2192947">What is the largest number smaller... | Mark Pineau | 432,691 | <p>Here is the standard proof for the above claim:</p>
<p>Prove $10^n-4$ is always a multiple of 6, for $n\in \mathbb{N}$.</p>
<p>That is: $10^n-4=6m$, for $m\in \mathbb{N}$.</p>
<p>We prove the above via induction:</p>
<p>Consider the base case, that is, when n = 1, we thus have:</p>
<p>$$10^{1}-4=10-4=6=6m \ for... |
2,868,172 | <p>The power rule states that for any real number $r$, </p>
<p>$$\frac{d}{dx}x^r=rx^{r-1}$$</p>
<p>Now one common way to prove this is to use the definition $x^r=e^{r\ln x}$, where $e^x$ is defined as the inverse function of $\ln x$, which is in turn defined as $\int_1^x\frac{dt}{t}$.</p>
<p>But this puts the cart b... | Eric Wofsey | 86,856 | <p>Yes. In general, if you have a sequence of $C^1$ functions $f_n$ on some interval and functions $f$ and $g$ such that $f_n\to f$ pointwise and $f_n'\to g$ uniformly, then $f'=g$. (Quick sketch of how to prove this without using integration: fix $x$ and $h$ and pick $n$ so that $f_n$ is sufficiently close to $f$ at... |
2,113,062 | <blockquote>
<p>Find $x^2+y^2$ if $x^2-\frac{2}{x}=3y^2$ and $y^2-\frac{11}{y}=3x^2$.</p>
</blockquote>
<p>My try:</p>
<p>$$\frac{y^2-\frac{11}{y}}{3}-\frac{2}{x}=3y^2$$</p>
<p>then?</p>
| Batominovski | 72,152 | <p>Assume that $x,y\in\mathbb{R}$ and $a,b\in\mathbb{R}$ satisfy $$x^2-\frac{a}{x}=3y^2\text{ and }y^2-\frac{b}{y}=3x^2\,.$$
Then, taking $z:=x+\text{i}y$, we have
$$z^3=\left(x^3-3xy^2\right)-\text{i}\left(y^3-3x^2y\right)=a-b\text{i}\,.$$
Thus,
$$x^2+y^2=|z|^2=\sqrt[3]{\left|z^3\right|^2}=\sqrt[3]{|a-b\text{i}|^2}=\s... |
2,312,913 | <p>In the triangle $ABC$, let $E$ be a point on $BC$ such that $BE : EC =
3: 2$. Pick points $D$ and $F$ on the sides $AB$ and $AC$ , correspondingly, so that
$3AD = 2AF$ . Let $G$ be the point of intersection of $AE$ and $DF$ . Given that $AB = 7$
and $AC = 9$, find the ratio $DG: GF$.</p>
<p>I have been working on t... | StatsSorceress | 259,187 | <p>You need "at least 1 girl". From a probability standpoint, P(at least one girl) = 1 - P(no girls)</p>
<p>For combinatorics, that would mean there can be 19 boys and 1 girl, or 18 boys and 2 girls, or ... up to 0 boys and 20 girls.</p>
<p>${25 \choose 19}{25 \choose 1}$ is the number of ways to choose 19 boys from ... |
2,374,282 | <p>I am trying to find all connected sets containing $z=i$ on which $f(z)=e^{2z}$ is one to one.
I have no idea how to approach.
Can someone give me some hints?
Thank you</p>
| gary | 6,595 | <p>I don't know if this is an acceptable answer, but you can do
1) An SQL Server query using the DateDiff function, a "" function with inputs ( DatePart, Date 1, Date 2) where DatePart is the "Gradation" : Years, Months, Days, etc. </p>
<p>Select DateDiff( Days, Date1, Date2) , where Date1 is your DOB, Date2 is the da... |
85,126 | <p>Does anyone have an implementation for <code>AnglePath</code> (see <a href="http://reference.wolfram.com/language/ref/AnglePath.html" rel="nofollow"><code>AnglePath</code> Documentation</a> and <a href="http://blog.wolfram.com/2015/05/21/new-in-the-wolfram-language-anglepath/" rel="nofollow">example usage</a>) in <e... | J. M.'s persistent exhaustion | 50 | <p>I managed to figure out how to re-implement <code>AnglePath[]</code>, since I needed to do something turtle-related for a friend, and I still do not have access to a computer with version 10.</p>
<p>First, the special cases:</p>
<pre><code>anglePath[steps_] := anglePath[{{0, 0}, 0}, steps]
anglePath[p_?VectorQ, st... |
2,585,466 | <p>I have two growth curve data sets, A (Martians) and B (Venusians). Data point sets of age (0 (birth) - 250 months, X axis) against height (0 - 200 centimeters, Y axis). The first set (A) contains 67 X Y point pairs, the second set (B) contains 27 point pairs. I have fit both data sets to my favorite version of the L... | jonsno | 310,635 | <p>Let point of intersection be $(x_o, y_o)$. The slope of parabola is $y' = \frac{2a}{y_o}$ and of hyperbola is $\frac{-c^2}{x_o^2}$.
Product of slope is $-1$ for perpendicular intersection:</p>
<p>$$\frac{2a}{y_o}=\frac{x_o^2}{c^2}\\
x^2_o y_o = 2ac^2 $$
The point $(x_o, y_o)$ satisfies both the parabola and hyperb... |
2,585,466 | <p>I have two growth curve data sets, A (Martians) and B (Venusians). Data point sets of age (0 (birth) - 250 months, X axis) against height (0 - 200 centimeters, Y axis). The first set (A) contains 67 X Y point pairs, the second set (B) contains 27 point pairs. I have fit both data sets to my favorite version of the L... | lab bhattacharjee | 33,337 | <p>WLOG any point on $y^2=4ax$ is $(at^2,2at)$</p>
<p>At the point$(P)$ of intersection $$c^2=at^2\cdot2at\iff t^3=\dfrac{c^2}{2a^2}\ \
\ \ \ (1)$$</p>
<p>Now the gradient of $y^2=4ax$ at $P$ is $$\dfrac{4a}{2(2at)}=\dfrac1t$$</p>
<p>and that of $xy=c^2$ $$-\dfrac{2at}{at^2}=-\dfrac2t$$</p>
<p>We need $-\dfrac2t... |
815,661 | <p>Let $m$ be the product of first n primes (n > 1) , in the following expression :</p>
<p>$$m=2⋅3…p_n$$</p>
<p>I want to prove that $(m-1)$ is not a complete square.</p>
<p>I found two ways that might prove this . My problem is with the SECOND way . </p>
<p><strong>First solution (seems to be working) :</strong> <... | Mathmo123 | 154,802 | <p>If there is a solution to $x^2 \equiv 2$ (mod $3$), then $x$ must be $0$, $1$ or $2$ (mod $3$). Now you can try each case and see that none of these will give you a solution:</p>
<ul>
<li>If $x \equiv 1,2$ (mod $3$) then $x^2 \equiv$ $1$ (mod $3$)</li>
<li>If $x \equiv 0$ (mod $3$) then $x^2 \equiv$ $0$ (mod $3$)<... |
813,715 | <p>Say I am asked to find, in expanded form without brackets, the equation of a circle with radius 6 and centre 2,3 - how would I go on about doing this?</p>
<p>I know the equation of a circle is $x^2 + y^2 = r^2$, but what do i do with this information?</p>
| 5xum | 112,884 | <p>The equation of a circle with the centre at $(0,0)$ is $x^2+y^2=r^2$. This is because the circle with the radius $r$ is composed of all points which are $r$ away from $(0,0)$, and since the distance of a point $(x,y)$ from $(0,0)$ is $\sqrt{x^2+y^2}$, this means that the equation will be $$\sqrt{x^2+y^2}=r,$$
or, sq... |
4,077,917 | <p>If you have
<span class="math-container">$$
\int_0^2 \int_0^{\sqrt{4 - x^2}} e^{-(x^2 + y^2)} dy \, dx
$$</span>
and you convert to polar coordinates, you integrate from <span class="math-container">$0$</span> to <span class="math-container">$\pi/2$</span>) with respect to theta.</p>
<p>But, if you have
<span class=... | David G. Stork | 210,401 | <p>Sometimes a picture is worth 1000 words:</p>
<p><a href="https://i.stack.imgur.com/6m1OZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6m1OZ.png" alt="enter image description here" /></a></p>
<hr />
<pre><code>Graphics[
{Red, Disk[{0, 0}, 2, {0, \[Pi]/2}],
Opacity[0.1], Green, Disk[{0, 0}, 6,... |
1,987,358 | <p>We know that Riemann sum gives us the following formula for a function <span class="math-container">$f\in C^1$</span>:</p>
<blockquote>
<p><span class="math-container">$$\lim_{n\to \infty}\frac 1n\sum_{k=0}^n f\left(\frac kn\right)=\int_0^1f(x) dx.$$</span></p>
</blockquote>
<p>I am looking for an example where ... | RRL | 148,510 | <p>When definite integrals are amenable to exact valuation, it is typically the case that the more expedient approach involves an anti-derivative rather than the limit of a Riemann sum. Often computation of the limit may be straightforward or even trivial, but somewhat tedious, as is the case for integrals of $f: x \ma... |
871,744 | <p>When A and B are square matrices of the same order, and O is the zero square matrix of the same order, prove or disprove:-
$$AB=0 \implies A=0 \text{ or } \ B=0$$</p>
<p>I proved it as follows:-</p>
<p>Assume $A \neq O$ and $ B \neq O$:
then, $$ |A||B| \neq 0 $$
$$ |AB| \neq 0 $$
$$ AB \neq O $$
$$ \therefore A \... | Marco Flores | 164,575 | <p>The problem with your first approach is that there exist non-zero matrices with zero determinant. For instance, $A$ and $B$ in your example.</p>
|
1,111,168 | <p>Do we say that $0$ has $n$ $n$th roots, all nondistinct, or only one?</p>
<p>I don't think it makes any difference, but I'm curious what the convention is.</p>
| Mariano Suárez-Álvarez | 274 | <p>As many as roots of the polynomial equation $X^n=0$, that is, $n$.</p>
|
1,111,168 | <p>Do we say that $0$ has $n$ $n$th roots, all nondistinct, or only one?</p>
<p>I don't think it makes any difference, but I'm curious what the convention is.</p>
| hmakholm left over Monica | 14,366 | <p>The terminology varies a bit between people and fields, but what I would say is that $x^n=0$ has one root (namely $0$) of <em>multiplicity</em> $n$.</p>
<p>If we explicitly say that we count that root "with multiplicity", of course, there are $n$ of it.</p>
|
139,105 | <p>Can a (finite) collection of disjoint circle arcs in $\mathbb{R}^3$ be interlocked in the sense in that they cannot be separated, i.e. each moved arbitrarily far from one another while remaining disjoint (or at least never crossing) throughout?
(Imagine the arcs are made of rigid steel; but infinitely thin.)
The arc... | Joseph O'Rourke | 6,094 | <p>@Cristi: Perhaps your first example needs an argument to exclude the following motion,
which might only be possible with my generous arc-gap?
<br /><img src="https://i.stack.imgur.com/LFbnJ.jpg" alt="TwoArcsMoving"></p>
|
3,712,256 | <p>I am trying to prove that: </p>
<blockquote>
<p>For nonempty subsets of the positive reals <span class="math-container">$A,B$</span>, both of which are bounded above, define
<span class="math-container">$$A \cdot B = \{ab \mid a \in A, \; b \in B\}.$$</span>
Prove that <span class="math-container">$\sup(A \cd... | alepopoulo110 | 351,240 | <p>If <span class="math-container">$\varepsilon>0$</span>, take <span class="math-container">$a\in A,b\in B$</span> such that <span class="math-container">$\sup A-\varepsilon<a$</span> and <span class="math-container">$\sup B-\varepsilon<b$</span>. Then it is</p>
<p><span class="math-container">$$(\sup A-\var... |
25,707 | <p>[Edit: to explain some things that were not so clear in the original post]</p>
<p>I believe in being straightforward. Without linking to the specific question, people will just treat this as another vague gripe with nothing to discuss. But to talk about the specific issues with a specific post is equivalent to poin... | Brick | 263,389 | <p>On one hand, I tend to agree with your concern generally across all SE sites. There are cases where questions are asked that clearly have fact-based answers, and the voting system is not always good about ensuring that the factually correct answers move to the top. I think that Math is better than some of the othe... |
2,117,054 | <p>Find all prime solutions of the equation $5x^2-7x+1=y^2.$</p>
<p>It is easy to see that
$y^2+2x^2=1 \mod 7.$ Since $\mod 7$-residues are $1,2,4$ it follows that $y^2=4 \mod 7$, $x^2=2 \mod 7$ or $y=2,5 \mod 7$ and $x=3,4 \mod 7.$ </p>
<p>In the same way from $y^2+2x=1 \mod 5$ we have that $y^2=1 \mod 5$ and $x=0... | Kelly Lowder | 406,721 | <p>If there is a solution it has more than 4000 digits, which makes me think there is no solution beyond the one two already mentioned.</p>
<p>In Mathematica,</p>
<pre><code>i=1;
ans=Solve[5 x ^2-7x+1==y^2&&x>0&&y>0,{x,y},Integers];
ans2=ans/.C[1]->i//RootReduce
Dynamic@i
Dynamic[IntegerLengt... |
2,117,054 | <p>Find all prime solutions of the equation $5x^2-7x+1=y^2.$</p>
<p>It is easy to see that
$y^2+2x^2=1 \mod 7.$ Since $\mod 7$-residues are $1,2,4$ it follows that $y^2=4 \mod 7$, $x^2=2 \mod 7$ or $y=2,5 \mod 7$ and $x=3,4 \mod 7.$ </p>
<p>In the same way from $y^2+2x=1 \mod 5$ we have that $y^2=1 \mod 5$ and $x=0... | math | 382,933 | <p>Since <span class="math-container">$(0,1)$</span> is a solution to <span class="math-container">$5x^2-7x+1=y^2$</span>, it can be used to parametrize all rational solutions to <span class="math-container">$5x^2-7x+1=y^2$</span>. That will give us:</p>
<p><span class="math-container">$$x:=\frac{-2ab - 7b^2}{a^2 - ... |
2,117,054 | <p>Find all prime solutions of the equation $5x^2-7x+1=y^2.$</p>
<p>It is easy to see that
$y^2+2x^2=1 \mod 7.$ Since $\mod 7$-residues are $1,2,4$ it follows that $y^2=4 \mod 7$, $x^2=2 \mod 7$ or $y=2,5 \mod 7$ and $x=3,4 \mod 7.$ </p>
<p>In the same way from $y^2+2x=1 \mod 5$ we have that $y^2=1 \mod 5$ and $x=0... | Joffan | 206,402 | <p>Can we just charge straight at this?</p>
<p>$y$ is odd. $x=2 \ (\Rightarrow y^2=7)$ is not a solution, so $x$ is an odd prime.</p>
<p>$x(5x-7) = (y-1)(y+1)$, so $x \mid (y-1) $ or $x \mid (y+1)$ ($x$ is prime)
so $kx=y\pm1$, $k$ even</p>
<p>$k\ge4$ is too large: $(kx\pm1)^2\ge (4x-1)^2 $ $= 16x^2-8x+1$ $>5x^... |
939,747 | <p>Denote by $a_n$ the sum of the first $n$ primes. Prove that there is a perfect square between $a_n$ and $a_{n+1}$, inclusive, for all $n$.</p>
<p>The first few sums of primes are $2$, $5$, $10$, $17$, $28$, $41$, $58$, $75$. It seems there is a perfect square between each pair of successive sums. In addition, we ca... | Travis Willse | 155,629 | <p>Not all matrices $A$ arise way. As tetori points out, all matrices of the form $XX^T$ for some vector $X \in \mathbb{R}$ are symmetric, they all have rank (at most) one, and the diagonal entries are all nonnegative.</p>
<p>These properties actually characterize such matrices, however, and we can show this by descri... |
3,577,249 | <p>Let be <span class="math-container">$X$</span> a topological space and let be <span class="math-container">$Y\subseteq X$</span> such that <span class="math-container">$\mathscr{der}(Y)=\varnothing$</span>: so for any neighborhood <span class="math-container">$I_y$</span> of <span class="math-container">$y\in Y $</s... | DanielWainfleet | 254,665 | <p>der(<span class="math-container">$Y$</span>) is empty iff <span class="math-container">$Y$</span> is a closed discrete subspace of <span class="math-container">$X$</span>.</p>
<p>(1). If der(<span class="math-container">$Y$</span>) is empty: (i). Any <span class="math-container">$p\in \overline Y$</span> \ <span ... |
1,165,147 | <p>I would like to find the area under the curve of $\frac{\sin(ax/2)}{\sin(x/2)}$, namely between the first zero crossing on the left and right:</p>
<p>$$
\int_{-\frac{2\pi}{a}}^{\frac{2\pi}{a}} \frac{\sin(\frac{ax}{2})}{\sin(\frac{x}{2})}
$$</p>
<p>I realized from Wolfram Alpha that this was not a simple solution,... | Julián Aguirre | 4,791 | <p>Using symmetry and the change of variable $x=2\,t$ we have
$$
\int_{-\frac{2\pi}{a}}^{\frac{2\pi}{a}} \frac{\sin(\frac{ax}{2})}{\sin(\frac{x}{2})}\,dx=
4\int_0^{\pi/a} \frac{\sin(a\,x)}{\sin x}\,dx.
$$
If $a>2$ then
$$
\frac{a}{\pi}\,\Bigl(\sin\frac{\pi}{a}\Bigr)\,x\le\sin x\le x\quad 0\le x\le\frac{\pi}{a}
$$
an... |
870,583 | <p><strong>Question:</strong>
Each user on a computer system has a password, which is six to eight characters long, where each character is an upper-case letter or a digit. Each password must contain at least one digit. How many possible passwords are there?</p>
<blockquote>
<p>I'm in the <strong>Basic of Counting</... | amanpandeyap | 193,162 | <p>$36^6$ gives you the number of passwords $6$ characters long, including passwords which are alphanumeric. But it also includes password which contains only alphabet, therefore subtract subtract $26^6$. </p>
<p>Similarly for passwords of length $7$ and $8$.</p>
|
1,255,368 | <p>How do I solve this? What steps? I have been beating my head into the wall all evening. </p>
<p>$$ x^2 + y^2 = \frac{x}{y} + 4 $$</p>
| CivilSigma | 229,877 | <p>We have:</p>
<p>$$ \frac{d(x^2)}{dx} + \frac{d(y^2)}{dy}\cdot \frac{dy}{dx} = \frac{d( \frac{x}{y})}{dx} + \frac{d(4)}{dx}$$</p>
<p>$$2x+2y\frac{dy}{dx} = \frac{y-x \frac{dy}{dx} }{ y^2} +0$$</p>
<p>Can you finish?</p>
<hr>
<p>Okay, to finish it up:</p>
<p>$$2y\frac{dy}{dx} - \frac{y-x \frac{dy}{dx} }{ y^2} = ... |
2,378,508 | <p>I am reading about Arithmetic mean and Harmonic mean. From <a href="https://en.wikipedia.org/wiki/Harmonic_mean#In_physics" rel="nofollow noreferrer">wikipedia</a>
I got this comparision about them:</p>
<blockquote>
<p>In certain situations, especially many situations involving rates and ratios, the harmonic mea... | Bloch | 462,921 | <p>Your equation is not even a differential equation, as only the second derivative but no other derivatives (or the function itself) appears. You can simply integrate twice.</p>
<p>However, in my opinion:</p>
<p>$(y'')^{2/3}=-(y'')^{3/2}$</p>
<p>$(y'')^{3/2}\ (y'')^{-2/3} = (y'')^{5/6}=-1$</p>
<p>so $(y'')^{5/6}+1... |
2,378,508 | <p>I am reading about Arithmetic mean and Harmonic mean. From <a href="https://en.wikipedia.org/wiki/Harmonic_mean#In_physics" rel="nofollow noreferrer">wikipedia</a>
I got this comparision about them:</p>
<blockquote>
<p>In certain situations, especially many situations involving rates and ratios, the harmonic mea... | Raghav Madan | 932,208 | <p>I dont think any existing answers answer your question .
The degree of the differential equation you asked is 9 because when you try to convert it to a polynomial form , you get the form as <span class="math-container">$(y")^9-(y")^4=0$</span> which is similar to finding degree of polynomial <span class="math-contai... |
2,735,001 | <p>I was asked to find the corresponding series for the function $\ln(x^2+4)$</p>
<p>The obvious solution to me was to use the well known fact $$\ln(1+x)=\sum_{n=1}^\infty (-1)^{n-1}\frac{x^n}{n}$$
And substituting $x^2+3$ for $x$
$$\ln(1+(x^2+3))=\sum_{n=1}^\infty (-1)^{n-1}\frac{(x^2+3)^n}{n}$$
Using binomial theore... | Przemysław Scherwentke | 72,361 | <p>HINT:
$$
4+x^2=4\left(1+\frac{x^2}{4}\right)
$$
and $\ln ab=\ln a+\ln b$.</p>
|
2,568,157 | <p>Consider the following:</p>
<p>$$(1^5+2^5)+(1^7+2^7)=2(1+2)^4$$</p>
<p>$$(1^5+2^5+3^5)+(1^7+2^7+3^7)=2(1+2+3)^4$$</p>
<p>$$(1^5+2^5+3^5+4^5)+(1^7+2^7+3^7+4^7)=2(1+2+3+4)^4$$</p>
<p>In General is it true for further increase i.e.,</p>
<p>Is</p>
<p>$$\sum_{i=1}^n i^5+i^7=2\left( \sum_{i=1}^ni\right)^4$$ true $\f... | Guy Fsone | 385,707 | <p>The formula is already true for $n=1,2,....,5$ and we know that
$$ \sum_{i=1}^ni= \frac{n\left(n+1\right)}{2}$$
Assume
$$\sum_{i=1}^n i^5+i^7=2\left( \sum_{i=1}^ni\right)^4 =\frac{n^4\left(n+1\right)^4}{8}$$</p>
<p>then,
$$\begin{align}\sum_{i=1}^{n+1} i^5+i^7&=\sum_{i=1}^{n} i^5+i^7 +(n+1)^5 +(n+1)^7\\&... |
438,490 | <p>I'm trying to find the eigenvalues and eigenvectors of the following <span class="math-container">$n\times n$</span> matrix, with <span class="math-container">$k$</span> blocks.
<span class="math-container">\begin{gather*}
X = \left( \begin{array}{cc}
A & B & \cdots & \\
B & A & B & \cdots \... | Joseph Van Name | 22,277 | <p><span class="math-container">$X$</span> is simply a tensor product <span class="math-container">$C\otimes D$</span> where <span class="math-container">$C$</span> is the matrix with all diagonal entries <span class="math-container">$a$</span> and non-diagonal entries <span class="math-container">$b$</span> and where ... |
2,365,666 | <blockquote>
<p>Question: Let $\phi: \mathbb{Z}\rightarrow \mathbb{Z}$ be a ring homomorphism.</p>
</blockquote>
<p>Show that $\forall x \in \mathbb{Z}:$ either $\phi\left ( x \right )=0$ or $\phi\left ( x \right )=x.$</p>
<p>I've shown that $\phi\left ( x \right )=x$.</p>
<p>Now I want to show that $\phi\left ( ... | Chinnapparaj R | 378,881 | <p>Any homomorphism $\phi:\mathbb{Z} \rightarrow \mathbb{Z}$ is completely determined by the image of $1$</p>
<p>Note that $\phi(1)=\phi(1^2)=\phi(1)\phi(1)=(\phi(1))^2$ and so $\phi(1)=0$ or $1$</p>
<p>If $\phi(1)=0$, then $\phi(x)=\phi(x.1)=\phi(x).\phi(1)=0$</p>
<p>If $\phi(1)=1$, then $\phi(x)=x$ </p>
|
2,365,666 | <blockquote>
<p>Question: Let $\phi: \mathbb{Z}\rightarrow \mathbb{Z}$ be a ring homomorphism.</p>
</blockquote>
<p>Show that $\forall x \in \mathbb{Z}:$ either $\phi\left ( x \right )=0$ or $\phi\left ( x \right )=x.$</p>
<p>I've shown that $\phi\left ( x \right )=x$.</p>
<p>Now I want to show that $\phi\left ( ... | Robert Lewis | 67,071 | <p>Look at this: </p>
<p>$\phi(1) = \phi(1^2) = \phi(1) \cdot \phi(1) = (\phi(1))^2; \tag{1}$</p>
<p>(1) shows that $\phi(1) = 0$ or $\phi(1) = 1$; if $\phi(1) = 0$ then</p>
<p>$\phi(x) = \phi(1x) = \phi(1)\phi(x) = 0(\phi(x)) = 0 \tag{2}$</p>
<p>for all $x \in \Bbb Z$; if $\phi(1) \ne 0$, then $\phi(1) = 1$; if ... |
2,881,914 | <p>Using a computer I found the double sum</p>
<p>$$S(n)= \sum_{j=1}^n\sum_{k=1}^n \frac{j^2 + jk + k^2}{j^2(j+k)^2k^2}$$
has values</p>
<p>$$S(10) \quad\quad= 1.881427206538142 \\ S(1000) \quad= 2.161366028875634 \\S(100000) = 2.164613524212465\\$$</p>
<p>As a guess I compared with fractions $\pi^p/q$ where $p,q$ ... | Tom Himler | 457,359 | <p>So before I start, I've never even attempted to evaluate a double sum before, so there could very well have been an easier way.</p>
<p>$$\sum_{j=1}^\infty\sum_{k=1}^\infty \frac{j^2+jk+k^2}{j^2k^2(j+k)^2} = \sum_{j=1}^\infty\sum_{k=1}^\infty \frac{1}{k^2(j+k)^2} +\sum_{j=1}^\infty\sum_{k=1}^\infty \frac{1}{jk(j+k)... |
2,881,914 | <p>Using a computer I found the double sum</p>
<p>$$S(n)= \sum_{j=1}^n\sum_{k=1}^n \frac{j^2 + jk + k^2}{j^2(j+k)^2k^2}$$
has values</p>
<p>$$S(10) \quad\quad= 1.881427206538142 \\ S(1000) \quad= 2.161366028875634 \\S(100000) = 2.164613524212465\\$$</p>
<p>As a guess I compared with fractions $\pi^p/q$ where $p,q$ ... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
3,881,029 | <p><span class="math-container">$R(A)$</span> is range of <span class="math-container">$A$</span> <br>
<span class="math-container">$N(A)$</span> is nullspace of <span class="math-container">$A$</span> <br>
<span class="math-container">$R(A^T)$</span> is range of <span class="math-container">$A^T$</span> <br>
<span cla... | Community | -1 | <p><strong>Hint:</strong> The range is the span of the columns.</p>
|
2,137,530 | <blockquote>
<p>Given $N$ trials of a die roll, where we have defined $D$ as the number of distinct outcomes, what would be the mean and standard deviation of $D$?</p>
</blockquote>
<p>If we have defined $I(k)$ as an indicator random variable which equals 1 if outcome $k$ (such as 6) appears at least once, and 0 oth... | n88k | 1,039,908 | <p>I just wanted to give another way to answer the question. Instead of partitioning by whether the number <span class="math-container">$i$</span>, <span class="math-container">$1 \leq i \leq 6$</span>, has appeared, one can consider <span class="math-container">$X_i$</span> to be the indicator random variable correspo... |
88,511 | <p>In version <a href="http://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn102.html">10.2</a> there is a new experimental function: <a href="http://reference.wolfram.com/language/ref/FindFormula.html"><code>FindFormula[]</code></a>.</p>
<p>I suspect that a <a href="https://en.wikipedia.org/wiki/Symbolic_... | Jacob Akkerboom | 4,330 | <p>The following reveals definitions</p>
<pre><code><< GeneralUtilities`
PrintDefinitions@FindFormula
</code></pre>
<p>As usual one can click the symbols to find definitions of functions "further down". It should also be noted that <code>FindFormula</code> is listed in the <a href="https://reference.wolfram.com... |
110,896 | <p>Now I have a function, say $f(k,z)=e^{-kz}(1+kz)$</p>
<p>I want to find the $n$th $\log$ derivative with respect to z.
like $(z\partial_z)^{(n)}f(k,z)$ (or $(\partial_{\ln z})^{(n)}f(k,z)$
if you like), where the $(n)$ denotes that we take the derivative $n$
times. </p>
<p>I found the answer in <a href="https://... | bbgodfrey | 1,063 | <p>Consider</p>
<pre><code>logDiv[f_, n_] := Nest[z D[#, z] &, f, n]
</code></pre>
<p>which applies <code>z D[#, z] &</code> <code>n</code> times to <code>f</code>. Then, for instance,</p>
<pre><code>logDiv[k z, 2]
(* k z *)
</code></pre>
<p>or</p>
<pre><code>logDiv[Exp[-kz] (1 + k z), 2]
(* E^-kz k z *)
... |
3,409,068 | <p>I have to find (and prove) the infimum and supremum of the following set:</p>
<p><span class="math-container">$M_1:=\{x\in\mathbb{Q} \mid x^2 < 9\}$</span></p>
<p>On first glance, I would say:</p>
<p><span class="math-container">$\inf M_1=-3 $</span><br>
<span class="math-container">$\sup M_1=3$</span></p>
<p... | Eric Towers | 123,905 | <p>If <span class="math-container">$f(x) = x\ln|x|$</span> is analytic at <span class="math-container">$x = 0$</span>, you can differentiate it there. What's <span class="math-container">$f'(0)$</span>?</p>
<blockquote class="spoiler">
<p> <span class="math-container">$f'(0)$</span> is undefined. <span class="math... |
715,809 | <p>I have occasionally come across Leibniz's Law (left to right, ie the indiscernibility of identicals) written with schematic letters in the consequent, and occasionally with a bound predicate-variable taking the place of the schematic letters. What is the relevant difference between these formulations? If there is ... | Jon | 124,068 | <p>I take it that by the schematic formulation you mean a meta-language statement such as the following:</p>
<ul>
<li>For any $r+1$-place relation symbol $R$ of the object language the sentence $\forall x_1 \ldots x_r \forall xy(x =y \rightarrow (Rx x_1 \ldots x_r \leftrightarrow Ry x_1 \ldots x_r))$ is true in every ... |
174,655 | <p>So I have 2 lists of 10000+ lists of 3 numbers, e.g.</p>
<pre><code>{{1,2,3},{4,5,6},{7,8,9},...}
{{2,1,3},{4,5,6},{41,2,0},...}
</code></pre>
<p>Wanting a result like </p>
<pre><code>{2,...}
</code></pre>
<p>Getting some sort of list of <code>True</code>/<code>False</code> is also probably enough, like this:</p... | Fraccalo | 40,354 | <p>One possible way of doing it:</p>
<pre><code>a = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {1, 2, 3}};
b = {{2, 1, 3}, {4, 5, 6}, {41, 2, 0}, {1, 2, 3}};
Position[MapThread[Equal, {a, b}], True]
</code></pre>
<p>The output reads: <code>{{2}, {4}}</code></p>
<p>In my code, it compares the elements for each position, but ... |
73,375 | <p>The example cells in the documentation each have a count of the cells inside their section:</p>
<pre><code> Cell[TextData[{"Basic Examples", " ", Cell["(4)", "ExampleCount"]}],
"ExampleSection", "ExampleSection"]
</code></pre>
<p>But this is static content, how exactly would this work dynamically? I'd like to ... | Carl Woll | 45,431 | <p>Here is a stylesheet solution that gives "Section" cells a dingbat that counts the number of "Input" cells in the section:</p>
<pre><code>SetOptions[
EvaluationNotebook[],
StyleDefinitions -> Notebook[
{
Cell[StyleData[StyleDefinitions -> "Default.nb"]],
Cell[StyleData["Input"... |
3,657,026 | <p>I need to prove expression using mathematical induction <span class="math-container">$P(1)$</span> and <span class="math-container">$P(k+1)$</span>, that:</p>
<p><span class="math-container">$$
1^2 + 2^2 + \dots + n^2 = \frac{1}{6}n(n + 1)(2n + 1)
$$</span></p>
<p>Proving <span class="math-container">$P(1)$</span>... | Isaac Ren | 415,180 | <p>If you look carefully at <span class="math-container">$\frac16k(k+1)(2k+1)+(k+1)^2$</span>, you'll see that it's a sum of two terms, <span class="math-container">$\frac16k(k+1)(2k+1)$</span> and <span class="math-container">$(k+1)^2$</span>. Both of these terms have a factor <span class="math-container">$(k+1)$</spa... |
2,758,965 | <p>Show $\log(1-x)=-\sum_{n=1}^{\infty}\frac{x^n}{n}\,\forall x\in(-1,1)$. Which value does $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}$ take?</p>
<hr>
<p>Now because I skipped forward in my (personal) textbook I know that I could tackle this using knowledge of the Maclaurin/Taylor series. However, it was not covered in the... | dxiv | 291,201 | <p>Alt. hint: by AM-GM:</p>
<p>$$
\frac{1}{a}+\frac{1}{bc}=\frac{1}{a}+\frac{a+b+c}{bc} = \frac{1}{a} + \frac{1}{b}+\frac{1}{c}+\frac{a}{bc}\ge 4 \sqrt[4]{\frac{1}{b^2c^2}} = 4 \sqrt{\frac{1}{bc}}
$$</p>
|
2,758,965 | <p>Show $\log(1-x)=-\sum_{n=1}^{\infty}\frac{x^n}{n}\,\forall x\in(-1,1)$. Which value does $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}$ take?</p>
<hr>
<p>Now because I skipped forward in my (personal) textbook I know that I could tackle this using knowledge of the Maclaurin/Taylor series. However, it was not covered in the... | Macavity | 58,320 | <p>Another way, is to use a more general Holder’s inequality:
$$LHS \geqslant \left( \frac1{\sqrt[3]{abc}}+\frac1{\sqrt[3]{a^2b^2c^2}}\right)^3\geqslant (3+9)^3$$</p>
|
411,868 | <p>Laver showed in 1995 that the period of the first row of certain <a href="https://en.wikipedia.org/wiki/Laver_table" rel="noreferrer">Laver tables</a> is unbounded, assuming that a rank-into-rank cardinal exists.</p>
<p>The most accessible proof of his result that I was able to find is in chapter 12 of Patrick Dehor... | Joseph Van Name | 22,277 | <p>Richard Laver was a set theorist throughout his entire career, so he was originally motivated to investigate very large cardinals and the resulting finite algebras from a set theoretic perspective, but today it is probably best to think of the Laver tables <span class="math-container">$A_{n}$</span> as algebraic obj... |
1,743,542 | <p>I'm looking for a continuous random variable with the following properties</p>
<ul>
<li>It is not bounded towards $+\infty$.</li>
<li>The expected value of the <em>maximum</em> of x-many draws out of that random variable has a closed-form solution.</li>
</ul>
<p>The more standard and well-known it is, the better. ... | user21820 | 21,820 | <p>For a distribution $X$ with cumulative density function $F$, the probability that $n$ variables independently drawn from $X$ are all at most $r$ would be $F(r)^n$. That gives you the cumulative density function for the maximum of those $n$ variables, and from there you can get everything you want.</p>
<p>Now the ex... |
211,623 | <p>Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by
\begin{eqnarray*}
s_0 &=& a, \\
s_1 &=& b, \text{and} \\
s_{n+2} &=& s_{n+1} s_{n} \text{ for $n \geq 0$}.
\end{eqnarray*}
This sequence is what we get when replace the letters in the algae L-s... | Joonas Ilmavirta | 55,893 | <p>Here are some observations about the sequence when $a$ and $b$ commute.</p>
<p>If $a=e$, then $s_n=b^{F_n}$, where $F_n$ is the $n$th Fibonacci number.
Let $|b|$ denote the order of $b$; we are interested in Fibonacci numbers modulo $|b|$.
The Fibonacci numbers are periodic modulo any finite number, so the sequence... |
1,977,031 | <p>How would I parameterise this curve in 3D?
I am confused since the diagrams deal with three variables in total – should I use complex numbers? I'm only used to two diagrams and haven't encountered a problem with three like this.</p>
| Jyrki Lahtonen | 11,619 | <p>The projection in the $xy$-plane looks like an <a href="https://en.wikipedia.org/wiki/Archimedean_spiral" rel="nofollow noreferrer">Archimedean spiral</a>. Except that everything in the $y$-direction is doubled. The formulas
$$
x=t \cos t,\qquad y=2t\sin t
$$
match with the first figure perfectly. The given points ... |
4,028,534 | <p>I have three points with coordinates: <span class="math-container">$A (5,-1,0),B(2,4,10)$</span>, and <span class="math-container">$C(6,-1,4)$</span>.</p>
<p>I have the following vectors <span class="math-container">$\overrightarrow {CA} = (-1, 0, -4)$</span> and <span class="math-container">$\overrightarrow{CB} = (... | ultralegend5385 | 818,304 | <p>Your book got a typo: it's <span class="math-container">$101$</span> and not <span class="math-container">$102$</span>. And that's it, that is approximately your answer.</p>
<p>Hope this helps. Ask anything if not clear :)</p>
|
756,662 | <p>I am trying to figure out what random variables are measurable with respect to sigma algebra given by $[1-4^{-n}, 1]$ where $n= 0, 1, 2, ....$ if $[0,1]$ is the sample space. I believe I can do with with indicator functions but I'm not sure how to write this.
Thanks!</p>
| drhab | 75,923 | <p>If $\mathcal{P}$ is a countable partition of the space then $\mathcal{F}=\left\{ \cup\mathcal{P}'\mid\mathcal{P}'\subset\mathcal{P}\right\} $
is the $\sigma$-algebra generated by the elements of $\mathcal{P}$.</p>
<p>A function $f$ is $\mathcal{F}$-measurable iff it is constant on
elements of $\mathcal{P}$. </p>
<... |
1,642,029 | <p>I'm looking at my textbooks steps for calculating the complexity of bubble sort...and it jumps a step where I don't know what exactly they did. </p>
<p><a href="https://i.stack.imgur.com/XaztP.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XaztP.png" alt="enter image description here"></a></p>
... | MPW | 113,214 | <p>If $(b_n)$ converges and you have only omitted finitely many terms from $(a_n)$ to get $(b_n)$, then the original sequence $(a_n)$ must have been convergent to begin with. Convergence only depends on the tail.</p>
<p>So if $(a_n)$ is not convergent, you can repeat the process infinitely many times on the "leftovers... |
2,709,832 | <p><a href="https://en.wikipedia.org/wiki/Cantor_set#Construction_and_formula_of_the_ternary_set" rel="noreferrer">Cantor set</a>
See the link, I am referring to cantor set on the real line. I wish to show that it is compact. I am doing this by pointing following arguments. I am not sure if this is enough.</p>
<ol>
<li... | Cyriac Antony | 120,721 | <p>Cantor set is defined as $C=\cap_n C_n$ where $C_{n+1}$ is obtained from $C_n$ by dropping 'middle third' of each closed interval in $C_n$</p>
<p>As you have noted, Cantor set is bounded.</p>
<p>Since each $C_n$ is closed and $C$ is an intersection of such sets, $C$ is closed (arbitrary intersection of closed sets... |
456,581 | <p>If one requires simply the existence of partial derivatives of first order rather than all orders, then a standard example is the function</p>
<p>$$ f(x,y) = \left\{\begin{array}{l l}
\frac{2xy}{x^2+y^2} & \quad \text{if $(x,y)\neq(0,0),$}\\
0 & \quad \text{if $(x,y)=(0,0).$}
\end{array} \... | paul garrett | 12,291 | <p>@DavidL.Renfro's comment's link gives a very nice example... But/and there may be some reason to give a complementary sort of answer. Namely, if we look at (let's say <em>tempered</em> for simplicity, to avoid worrying about smooth truncations and such) <em>distributions</em>, first (as warm-up to the point I want t... |
1,253,778 | <p>When we say that an inequality is sharp, does it mean that it is "the best" inequality we can get between the two quantities involved?</p>
<p>For example, I read that we would say that the inequality
$$
\frac{a^2+b^2}{2}\geq ab
$$
is sharp, but wouldn't $\frac{a^2+b^2}{2}$ on the RHS be sharper than $ab$?</p>
<p>D... | Pietro Paparella | 414,530 | <p>Here is another definition of sharpness that demonstrates that the inequality is <em>optimal</em> (i.e., best-possible).</p>
<p>If <span class="math-container">$f,g:\mathbb{R}^n \longrightarrow \mathbb{R}$</span> and
<span class="math-container">\begin{equation}
f(x) \geq g(x),~\forall x \in S \subseteq\mathbb{R}^n... |
657,047 | <p>So I have $a^n = b$. When I know $a$ and $b$, how can I find $n$?</p>
<p>Thanks in advance!</p>
| bsdshell | 54,105 | <p>$$
\begin{align}a^n &= b\\
\Rightarrow \log_{a}{a^n} &= \log_{a}{b}\quad(\because \log_{a}a^n = n)\\
\Rightarrow n &= \log_{a}{b} \end{align}
$$</p>
|
271,259 | <p>My goal is to have NumericQ[h[j]]=True for any j regardless of whether j may be symbolic with no defined value.</p>
<p>Setting NumericQ[h[j_]]=True does not work and as I understand the NumericFunction attribute only works when the input is also numeric.</p>
<p>One solution might be to unprotect NumericQ and then se... | user293787 | 85,954 | <p><em>This is an extended comment and not strictly an answer. But I thought I put this here, since people that end up here for whatever reason might find it useful. For reference, I am currently using Version 12.3.</em></p>
<p>While writing the other answer, I was surprised to find that the assignment</p>
<pre><code>N... |
163,816 | <p>I'm looking for a good exercise book for probability theory, preferably at least partially with solutions to it. I want it to be detailed, not trivial, providing me solid fundamentals in the topic to be developed in the future. I'd wish it to be more of a "applied" technical university approach than the highly "abst... | nanme | 33,515 | <p>"A Collection of Exercises in Advanced Probability Theory" - the solutions manual of all even-numbered exercises from the book “A First Look at Rigorous Probability Theory” (second edition, 2006) by Jeffrey S. Rosenthal :</p>
<p><a href="http://www.worldscibooks.com/mathematics/6300_solutionsmanual.pdf" rel="nofoll... |
482,003 | <p>I need help with the following limit $$\lim_{n\to\infty}\sum_{k=1}^n \frac{1}{\sqrt{kn}}$$</p>
<p>Thanks.</p>
| GEdgar | 442 | <p>mrf has the main idea. But since the integral is improper (as mrf notes) some care is required. Here is one way, using the Lebesgue theory...</p>
<p>Let <span class="math-container">$f(x) = 1/\sqrt{x}$</span>. For fixed <span class="math-container">$n$</span>, let <span class="math-container">$g_n$</span> be def... |
26,062 | <p>Given a <code>TabView</code> panel like this one</p>
<pre><code>TabView[{
DynamicModule[{x = False}, {Checkbox[Dynamic[x]], Dynamic[x]}],
"foo"}]
</code></pre>
<p>I would like to reset the value of <code>x</code> to its initial value (<code>False</code>) every time the first tab gets selected - that is, I basi... | Kuba | 5,478 | <h3>edit</h3>
<p>This is an old question but it came to my mind I know the solution now.</p>
<pre><code>DynamicModule[{tab = 1},
TabView[{
DynamicModule[{x = False},
Row[{ Dynamic[tab; x = False; "", TrackedSymbols :> {tab}],
{Checkbox[Dynamic[x]], Dynamic[x]}}]]
,
"foo"... |
3,048,702 | <p>Let <span class="math-container">$S$</span> be a orientable closed surface with genus <span class="math-container">$g \geq 1$</span> and let <span class="math-container">$\gamma \subset S$</span> be an immersed curve. Does there exist a finite cover of <span class="math-container">$S$</span> where <span class="math... | Lee Mosher | 26,501 | <p>The existence of the finite cover you are asking for is a well known consequence of the residual finiteness of surface groups, a theorem with a long history. </p>
<p>For a discussion of that theorem, which starts out with the proof that answers your first question, see <a href="https://260p.wordpress.com/2014/04/02... |
223,955 | <p>How can we convert a list to an integer correctly? </p>
<p><strong>{5, 22, 4, 5} -> 52245?</strong></p>
<p>When I use the command <code>FromDigits</code> in Mathematica </p>
<pre><code>FromDigits[{5, 22, 4, 5}]
</code></pre>
<p>The result is incorrect, namely <strong>7245</strong></p>
| kglr | 125 | <pre><code>{5, 22, 4, 5} ~ StringRiffle ~ "" // FromDigits
</code></pre>
<blockquote>
<pre><code>52245
</code></pre>
</blockquote>
|
1,043,090 | <p>A rectangle $ABCD$, which measure $9 ft$ by $12 ft$, is folded once perpendicular to diagonal AC so that the opposite vertices A and C coincide. Find the length of the fold. So I tried to fold a rectangular paper but there are spare edges. So the gray is my fold and I'm not sure if its in the middle of my diagonal ... | MvG | 35,416 | <p>The length is</p>
<p>$$\frac{45}4\,\text{ft}=11.25\,\text{ft}$$</p>
<p><img src="https://i.stack.imgur.com/dVhsr.png" alt="Figure"></p>
<p>One way to obtain this result is using coordinates. Use $E$ as the center of the coordinate system, then the corners are $A,B,C,D=\left(\pm\frac92,\pm\frac{12}2\right)$. The d... |
2,024,097 | <blockquote>
<p>Which prime numbers $p \in \mathbb{Z}$ are reducible in the unique factorization domain $\mathbb{Z}\left[\frac{1 + \sqrt{-3}}{2}\right]$ ?</p>
</blockquote>
<p>Suppose $p$ is a prime integer and $p = \alpha \beta$ in $\mathbb{Z}\left[\frac{1 + \sqrt{-3}}{2}\right] = \mathbb{Z}[\omega]$. Then $f(p) = ... | Mr. Brooks | 162,538 | <p>Very minor detail, the function $f$ is more commonly notated $N$, for "norm."</p>
<p>More importantly, however, I think the norm function reveals itself more clearly if you regard it in this manner: given $a + b \sqrt{-3}$, for $\{a, b\} \in \mathbb{Z}$, then $N(a + b \sqrt{-3}) = a^2 + 3b^2$; or given $$\frac{a + ... |
1,215,273 | <blockquote>
<p>Describe the cosets of the subgroup $\langle 3\rangle$ of $\mathbb{Z}$</p>
</blockquote>
<p>The problem I have is $\mathbb{Z}$ is infinite.</p>
<p>So we know that $\langle 3\rangle=\{0,3,6,9,12,\ldots\}$ and I know the definition of cosets (in this case right cosets) is the set of all products of ha... | John Hughes | 114,036 | <p>The operation on $Z$ in this case is ADDITION. So
$$
<3> = \{ \ldots, -6, -3, 0, 3, 6, \ldots \}
$$
and a typical coset is
$$
<3> + 0 = <3>
$$
while another is
$$
<3> + 1 = \{ \ldots, -6+1, -3+1, 0+1, 3+1, 6+1, \ldots\}.
$$
With that, can you write down all the other cosets? How many are t... |
4,042,250 | <p>My idea is to use disjoint events and calculating the probability of getting at least two heads for each number rolled. For example, if I roll a 3, I would calculate the probability with the expression <span class="math-container">$(\frac{1}{6}) (\frac{1}{2})^3 \binom{3}{2} + (\frac{1}{6}) (\frac{1}{2})^3\binom{3}{3... | Vons | 274,987 | <p>It is valid. You find that P(at least 2 heads|die=1) = 0, P(at least 2 heads|die=2)=1/4, P(at least 2 heads|die=3)=1/2, P(at least 2 heads|die=4)=11/16, P(at least 2 heads|die=5)=13/16, and P(at least 2 heads|die=6)=57/64. Then 1/6*(0+1/4+1/2+11/16+13/16+57/64)=67/128.</p>
<p>There is a way you can numerically appro... |
4,042,250 | <p>My idea is to use disjoint events and calculating the probability of getting at least two heads for each number rolled. For example, if I roll a 3, I would calculate the probability with the expression <span class="math-container">$(\frac{1}{6}) (\frac{1}{2})^3 \binom{3}{2} + (\frac{1}{6}) (\frac{1}{2})^3\binom{3}{3... | user247327 | 247,327 | <p>There is a 1/6 chance you will roll a 1. Then you CAN'T flip two heads. The probability is 0.</p>
<p>There is a 1/6 chance you will roll a 2. The probability of flipping two head in two flips is (1/2)(1/2)= 1/4. The probability is (1/6)(1/4)= 1/24.</p>
<p>There is a 1/6 chance you will roll a 3. The probability of... |
3,853,980 | <p>Let <span class="math-container">$A$</span> be an <span class="math-container">$n×n $</span> complex matrix such that the three matrices <span class="math-container">$A+I$</span> , <span class="math-container">$A^2+I $</span> , <span class="math-container">$ A^3+I$</span> are all unitary .Prove that<span class="mat... | Matthew H. | 801,306 | <p>Set <span class="math-container">$X:=\max(X_1,X_2)$</span>. Notice from symmetry <span class="math-container">$$\operatorname{cov}(X_1+X_2,X)=\operatorname{cov}(X_1,X)+\operatorname{cov}(X_2,X)=2\operatorname{cov}(X_1,X)$$</span> Let's take a closer look at <span class="math-container">$\operatorname{cov}(X_1,X)$</s... |
56,103 | <p>A referee asked me to include a reference or proof for the following classical fact. It's not hard to prove, but I'd prefer to just give a reference -- does anyone know one?</p>
<p>Let $X$ be a nice space (eg a smooth manifold, or more generally a CW complex). The topological Picard group $Pic(X)$ is the set of i... | Dan Ramras | 4,042 | <p>It's probably not exactly what you want (in particular, they're dealing with real bundles and the Stiefel Whitney classes), but something sort of close is discussed in the appendix to </p>
<p>MR2003827 (2004h:53116)
Ho, Nan-Kuo(3-TRNT); Liu, Chiu-Chu Melissa(1-HRV)
Connected components of the space of surface group... |
667,903 | <p>$$D=\left\{z\ \left|\ \right. 1<|z|<2, \ \Re(z)>-\frac{1}{2}\right\}$$</p>
<p>I can try to prove that each disc with radius 1, and 2 is open (?), so I need to show that the ball centered at $w$ is contained in the disc with radius 2 </p>
<p>If $w\in \{z: |z|<2\}$, then $|z_0-w|<2$, let $\epsilon = 2... | Martín-Blas Pérez Pinilla | 98,199 | <p>Is the intersection of three open sets. Each one is the inverse image of a open set by a continuous function.</p>
<p>EDIT: intersection of open sets, the general case.</p>
<p>Let be $O_1,O_2$ two open sets.</p>
<p>If $z_0\in O_1\cap O_2$, then $z_0\in O_1$ and $z_0\in O_2$ by definition, so
$$\exists r_1>0: \ ... |
2,856,180 | <p>I've been learning some about counting and basic combinatorics. But some scenarios were not explained in my class...</p>
<p><strong>Example problem:</strong> You are given 6 tiles. 1 is labeled "1", 2 are labeled "2", and 3 are labeled "3".</p>
<p><strong>Problem 1:</strong> How many different ways can you arrange... | Tengu | 58,951 | <p>Don't try to find a new formula every time you encounter a new counting problem! Counting problems have many variations so it's better if you know <strong>how</strong> to count rather than knowing the formula. For example, formulas for combinations and permutations are essentially coming from <a href="https://en.wik... |
3,852,362 | <p>Let <span class="math-container">$~X = \{(x,y)∈ℝ^2∶|x| ≤1,~|y|≤1\}$</span> and function <span class="math-container">$f : X →ℝ$</span> defined by <span class="math-container">$$f(x,y)=\dfrac{x\cos x + y \sin y}{x^2+y^2+\alpha}$$</span>where <span class="math-container">$\alpha\gt0$</span>, then the range of <span cl... | Lutz Lehmann | 115,115 | <p>In <span class="math-container">$Y(s)=G(s)X(s)$</span> transfer the denominator to the left side and apply the inverse Laplace transform to get
<span class="math-container">$$
y'(t)+5y(t)=5x(t-5.8)
$$</span>
for the input function <span class="math-container">$x$</span> and the output function <span class="math-cont... |
3,852,362 | <p>Let <span class="math-container">$~X = \{(x,y)∈ℝ^2∶|x| ≤1,~|y|≤1\}$</span> and function <span class="math-container">$f : X →ℝ$</span> defined by <span class="math-container">$$f(x,y)=\dfrac{x\cos x + y \sin y}{x^2+y^2+\alpha}$$</span>where <span class="math-container">$\alpha\gt0$</span>, then the range of <span cl... | user577215664 | 475,762 | <p><span class="math-container">$$G(s) = e^{-5.8s} \frac{5}{s+5}$$</span>
<span class="math-container">$$\implies \dfrac {X(s)}{U(s)}= e^{-5.8s} \frac{5}{s+5}$$</span>
<span class="math-container">$$sX(s)+5X(s)= 5e^{-5.8s}U(s)$$</span>
Since <span class="math-container">$x(0)=0$</span>:
<span class="math-container">$$(... |
105,071 | <p>As one may know, a <b>dynamical system</b> can be defined with a monoid or a group action on a set, usually a manifold or similar kind of space with extra structure, which is called the <i>phase space</i> or <i>state space</i> of the dynamical system. The monoid or group doing the acting is what I call the <i>time s... | John B | 74,138 | <p>This question is from long ago, but I will give it a try. I want to note that multi-dimensional times do occur "naturally", in some reasonable manner. As an example, take coupled map lattices, with two times involved: the actual time and the space translation, along which different systems may interact. In this sett... |
271,343 | <p>I need help finding the integral of $\sin(\sqrt{x})dx$. I have the answer here but would like to know how to get there. </p>
| cderwin | 50,816 | <p>Use the substitution $t=\sqrt{x}$, and then use integration by parts. From the substitution, you should get:</p>
<p>$$\int\sin\sqrt{x}dx=2\int t\sin (t) dt$$</p>
<p>Then, use integration by parts. Let $u=t$ and $dv=\sin t$. Then $du=dt$ and $v=-\cos t$, and we have:</p>
<p>$$\int t\sin (t) dt=-t\cos t +\int \c... |
35,585 | <p>Let H be an infinite dimensional and separable Hilbert space. Let C be a closed and
connected subset of H containing more than one point. Can C ever be the countable union
of closed and totally disconnected subsets of H?</p>
| Gerald Edgar | 454 | <p>$C$ is simply a connected complete separable metric space with more than one point.
Now the union of countably many closed sets of topological dimension zero must again have dimension zero. But I suppose "totally disconnected" is not quite the same as "zero dimensional" so this is not yet a complete answer.</p>
|
35,585 | <p>Let H be an infinite dimensional and separable Hilbert space. Let C be a closed and
connected subset of H containing more than one point. Can C ever be the countable union
of closed and totally disconnected subsets of H?</p>
| Bill Johnson | 2,554 | <p>Let $C$ be an explosion point space such as the Knaster–Kuratowski fan (<a href="http://en.wikipedia.org/wiki/Knaster" rel="nofollow">http://en.wikipedia.org/wiki/Knaster</a>–Kuratowski_fan) and $p$ the explosion point in $C$.<br>
Set $A_n = \{p\}\cup (C\sim B(p; 1/n))$. </p>
|
1,003,096 | <p>Let $G=(\mathbb{Q}-\{0\},*)$ and $H=\{\frac{a}{b}\mid a,b\text{ are odd integers}\}$.</p>
<ol>
<li>Show $H$ is a normal subgroup of $G$.</li>
<li>Show that $G/H \cong (\mathbb{Z},+)$</li>
</ol>
<p>I know that there are multiple definitions for normal subgroup and I am having a hard time to develop the proof for th... | anomaly | 156,999 | <p>1) $G$ is abelian, so any subgroup is normal.</p>
<p>2) Show that every element of $G/H$ is of the form $2^kH$ with $k\in \mathbb{Z}$; the map $2^k H \to k$ then gives you the required isomorphism.</p>
|
57,131 | <p>Evaluating the following lines in my computer takes near six seconds in Mathematica 10 and near 5 in Mathematica 9. I consider this very slow as only quantiles are being calculated from data to create this simple set of charts. I think that Mathematica should do much better than this. Do you agree? Thoughts?</p>
... | paw | 4,539 | <p>You could build your own plot function and calculate the <code>Quantiles</code> seperately as @Verbeia suggested.</p>
<p>Here is a little example you could expand on:</p>
<pre><code>WhiskerChart[data_List] := Module[{n = Length[data], max, min, mean, min25, max75},
max = Max[data];
min = Min[data];
mean = Quant... |
65,582 | <p>Let $G$ be a locally compact group on which there exists a Haar measure, etc..</p>
<p>Now I am supposed to take such a metrisable $G$, and given the existence of some metric on $G$, prove that there exists a translation-invariant metric, i.e., a metric $d$ such that $d(x,y) = d(gx,gy)$ for all $x,y,g \in G$. How to... | Makoto Kato | 28,422 | <p>Though the answer was essentially given in the comment area, I think it's nice to have the full proof here.
We follow largely Bourbaki.</p>
<p><strong>Notation</strong>
Let $X$ be a set.
Let $U$ and $V$ be subsets of $X \times X$.
We define $UV$ = {$(x, y) \in X \times X;$ there exists $z \in X$ such that $(x, z) \... |
3,106,574 | <p>Let <span class="math-container">$(a_n) _{n\ge 0}$</span> <span class="math-container">$a_{n+2}^3+a_{n+2}=a_{n+1}+a_n$</span>,<span class="math-container">$\forall n\ge 1$</span>, <span class="math-container">$a_0,a_1 \ge 1$</span>. Prove that <span class="math-container">$(a_n) _{n\ge 0}$</span> is convergent.<br>
... | Dr. Wolfgang Hintze | 198,592 | <p>This is the first time I see this type of recusrion relation where the next term in the sequence is not given directly but instead a function of it is given. This makes the problem particularly interesting (for me, at least).</p>
<p>Hence an uncommon approach seems to be justified. I am going to relate convergence ... |
3,514,494 | <p>Expand into Laurent's series after the power of z-a for the following function</p>
<p><span class="math-container">$$ f(z)=\frac{z+z^2}{(1-z)^3} $$</span>for a = 0 .</p>
<p>I know the formula for the Laurent's series but I don't know how to apply it. </p>
| kludg | 42,926 | <p>If you need expected value of won prizes, it is
<span class="math-container">$$\frac{17}{36}\cdot 1 + \frac{8}{36}\cdot 2 + \frac{1}{36}\cdot 3$$</span></p>
|
3,514,494 | <p>Expand into Laurent's series after the power of z-a for the following function</p>
<p><span class="math-container">$$ f(z)=\frac{z+z^2}{(1-z)^3} $$</span>for a = 0 .</p>
<p>I know the formula for the Laurent's series but I don't know how to apply it. </p>
| Math1000 | 38,584 | <p><span class="math-container">$X$</span> does not have a Poisson distribution because it only takes finitely many values. Observe that <span class="math-container">$X=X_1+X_2+X_3$</span> where <span class="math-container">$X_1\sim\mathrm{Ber}\left(\frac16\right)$</span>, <span class="math-container">$X_2\sim\mathrm{B... |
3,910,345 | <p>Recently a lecturer used this notation, which I assume is a sort of twisted form of Leibniz notation:</p>
<p><span class="math-container">$$y\,\mathrm{d}x - x\,\mathrm{d}y \equiv -x^2\,\mathrm{d}\left(\frac{y}{x}\right)$$</span></p>
<p>The logic here was that this could be used as:</p>
<p><span class="math-container... | Ethan Bolker | 72,858 | <p>In the context for this discussion you have two variables <span class="math-container">$x$</span> and <span class="math-container">$y$</span> that are somehow related. Perhaps you have <span class="math-container">$y$</span> as a function of <span class="math-container">$x$</span>, or perhaps both depend on some oth... |
3,995,986 | <p>Need help integrating:
<span class="math-container">$$\int _0^{\infty }\:\:\frac{6}{\theta}xe^{-\frac{2x}{\theta }}\left(1-e^{-\frac{x}{\theta }}\right)dx$$</span></p>
<p>I think I should multiply the <span class="math-container">$$xe^{-\frac{2x}{\theta }}$$</span> out and then use integration by parts but it is not... | Quanto | 686,284 | <p>Integrate by parts as follows</p>
<p><span class="math-container">\begin{align}
& \int _0^{\infty }\:\:\frac{6}{\theta}xe^{-\frac{2x}{\theta }}\left(1-e^{-\frac{x}{\theta }}\right)dx\\
=& \frac6{\theta} \int _0^{\infty }
x\> d \left( -\frac {\theta}2 e^{-\frac{2x}{\theta } }
+ \frac {\theta}3e^{-\frac{3x... |
642,324 | <p>According to <a href="http://en.wikipedia.org/wiki/Regression_analysis" rel="nofollow">wikipedia</a>, regression analysis is a statistical process for estimating the relationships among variables. Regression analysis is widely used for prediction and forecasting. So why is regression analysis also used as statistica... | JJacquelin | 108,514 | <p>On my viewpoint, regression analysis is a mathematical tool which can be used in various circonstances and for various kind of problems. Of course, in many statistical problems a regression technic is currently used. But they are also problems involving regressions where nothing is randomly specified and where no st... |
29,255 | <p>sorry! am not clear with these questions</p>
<ol>
<li><p>why an empty set is open as well as closed?</p></li>
<li><p>why the set of all real numbers is open as well as closed?</p></li>
</ol>
| sihong xie | 44,881 | <p>$\emptyset$ is open. By definition, $X$ is open if $\forall x\in X$, there is a open set $U\subset X$ such that $x\in U$. So there is not any point in $\emptyset$, the condition of the definition is automatically satisfied (a logical convention).</p>
<p>$\mathbb{R}$ is open (check Andrea's answer), so its complemen... |
2,128,588 | <p>The question;</p>
<p>$U = \{x |Ax = 0\}$ If $ A = \begin{bmatrix}1 & 2 & 1 & 0 & -2\\ 2 & 1 & 2 & 1 & 2\\1 & 1 & 0 & -1 & -2\\ 0 & 0 & 2 & 0 & 4\end{bmatrix}$</p>
<p>Find a basis for $U$.</p>
<p><hr>
To make it linearly independent, I reduce the rows... | Alvin Jin | 276,134 | <p>Hint: $n=1$ gives us two "successes" (3 and 6) and thus a probability of 1/3.</p>
<p>$n=2$ gives us $2 + 5 + 4 + 1 = 12$ successes (corresponding to 3,6,9,12). There are a total of 36 outcomes. This gives a probability of $1/3$.</p>
<p>$n=3$ gives us $1 + 10 + 25 +25 +10 + 1 = 72$ successes (corresponding to 3,... |
480,727 | <p>If $$2^x=3^y=6^{-z}$$ and $x,y,z \neq 0 $ then prove that:$$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$$</p>
<p>I have tried starting with taking logartithms, but that gives just some more equations.</p>
<p>Any specific way to solve these type of problems?</p>
<p>Any help will be appreciated.</p>
| lab bhattacharjee | 33,337 | <p>As $\log_aa=1$ and $\log_a(bc)=\log_ab+\log_ac,$</p>
<p>applying logarithm wrt $2,$ </p>
<p>$x=y\log_23=-z\log_26=-z(1+\log_23)$</p>
<p>$x=y\log_23\implies \log_23=\frac xy$</p>
<p>and put this value of $\log_23$ in $\displaystyle x=-z(1+\log_23)$ </p>
<p>to eliminate $\log_23$ and simplify.</p>
|
1,386,261 | <p>They start by choosing $m$ s.t. $|s_n - s| \lt \frac{1}{2} |s|$ if $n > m$.
From here, it looks like they use the triangular inequality $|s_n - s| + |s_n| < |s|$ to come up with this statement but I'm not sure as they introduce some variable m.</p>
<p>Next, given $\epsilon > 0$, there is a $N > m$ s.t $... | Travis Willse | 155,629 | <p>The Zariski topology is a topology well-suited to the study of algebraic varieties, the principal objects of study in algebraic geometry:</p>
<blockquote>
<p>For any field $\Bbb F$, a set $X \subseteq \Bbb F^n$ is defined to be closed in the Zariski topology on iff it has the form $$V(S) := \{(x_1, \ldots, x_n) \... |
1,029,650 | <p>In Four-dimensional space, the Levi-Civita symbol is defined as:</p>
<p>$$ \varepsilon_{ijkl } =$$
\begin{cases}
+1 & \text{if }(i,j,k,l) \text{ is an even permutation of } (1,2,3,4) \\
-1 & \text{if }(i,j,k,l) \text{ is an odd permutation of } (1,2,3,4) \\
0 & \text{otherwise}
\end{cases}
</p>
<p>Let'... | Baalateja Kataru | 620,119 | <p>There is a method by which we can formally verify your guess that,</p>
<p><span class="math-container">$$
\epsilon^{ijk4} = \epsilon^{ijk}
$$</span></p>
<p>And determine whether,</p>
<p><span class="math-container">$$ \epsilon^{ijk4} = \pm \epsilon^{ij4k} $$</span></p>
<p>The <span class="math-container">$n$</span>-... |
1,032,535 | <p>I know $n \in \mathbb{N}$ and...</p>
<p>$$
a_n = \begin{cases}
0 & \text{ if } n = 0 \\
a_{n-1}^{2} + \frac{1}{4} & \text{ if } n > 0
\end{cases}
$$</p>
<ol>
<li><strong>Base Case:</strong></li>
</ol>
<p>$$a_1 = a^2_0 + \frac{1}{4}$$</p>
<p>$$a_1 = 0^2 + \frac{1}{4} = \frac{1}{... | mvggz | 167,171 | <p>Ok I'll right it down so that it's clear to you :) </p>
<p>I want to prove the property P: " 0 < $a_n$ < 1 "</p>
<p>I look at the property A: $0 < a_n < \frac{1}{2}$</p>
<p>A => P, that is : If A is true then P is true </p>
<p>I'll prove A using induction (so technically I don't prove P by induction,... |
717,664 | <p>I need a step by step answer on how to do this. What I've been doing is converting the top to $2e^{i(\pi/4)}$ and the bottom to $\sqrt2e^{i(-\pi/4)}$. I know the answer is $2e^{i(\pi/2)}$ and the angle makes sense but obviously I'm doing something wrong with the coefficients. I suspect maybe only the real part goes ... | Michael Hoppe | 93,935 | <p>What about
$$\frac{1+i}{1-i}=\frac{2\sqrt{2}(\sqrt{2}/2+i\sqrt{2}/2)}{\sqrt{2}(\sqrt{2}/2-i\sqrt{2}/2)}
=2\frac{e^{i\frac{\pi}{4}}}{e^{-i\frac{\pi}{4}}}=2e^{i\frac{\pi}{2}}=2i?$$</p>
|
1,109,853 | <blockquote>
<p><em>The below proof is incorrect. See the answers for more information.</em></p>
</blockquote>
<p>This question is in the context of exploring how to explain the process of developing a proof.</p>
<p>When reading a proof on the irrationality of $ \sqrt{3} $, I came across the following statement, wh... | Glare | 208,225 | <p>As @mapierce271 pointed out already, your proof doesn't actually accomplish what you've set out to show. I would also like to add that your "theorem" (if $n$ divides $a^2$, then $n$ divides $a$) is not true for just <em>any</em> integer $n$. For example, $8^2=64$ is divisible by $32$, but $8$ is definitely not divi... |
199,148 | <p><a href="https://i.stack.imgur.com/9BuHp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9BuHp.png" alt="enter image description here"></a> </p>
<pre><code>pbdomains = <|
"Overall " -> Around[2.6, 0.04],
"PB" -> Around[4.25, 0.06]
|>;
BarChart[pbdomains, ChartStyle ... | kglr | 125 | <p><strong>1.</strong> Use the (undocumented) option <code>"FixedBarSpacing"</code> as <code>"FixedBarSpacing" -> True</code> or as <code>Method -> {"FixedBarSpacing" -> True}</code></p>
<pre><code>BarChart[pbdomains, ChartStyle -> "BrightBands",
LabelStyle ->... |
4,107,396 | <p>I'm stuck at solving the following problem: launch 3 fair coins independently. Let A the event: "you get at least a head" and B "you get exactly one tail".
Then what is the probability of the event <span class="math-container">$A \cup B$</span>?</p>
| I am a person | 806,777 | <p>The probability that you want is everything except if you get all <span class="math-container">$3$</span> tails, since everything else is in the union of <span class="math-container">$A$</span> and <span class="math-container">$B$</span>. So the answer = <span class="math-container">$1 - \frac{1}{8} = \boxed{\frac{7... |
442,759 | <p>I was reading a book on groups, it points out about the uniqueness of the neutral element and the inverse element. I got curious, are there algebraic structures with more than one neutral element and/or more than one inverse element?</p>
| Brian Rushton | 51,970 | <p>In an interesting way, yes; in matrix theory, you can have a ring of matrices that has a single identity, but has subspaces which have distinct right identities. For instance, in 2X2 matrices, the subset of all matrices which have 0's in the left column has a right identity given by $\begin{bmatrix}0 & 0\\0 &am... |
1,576,713 | <p>$X$ and $Y$ are two sets and $f:X\to Y$. If $f(C)=\{f(x):x\in C\}$ for $C\subseteq X$ and $f^{-1}(D)=\{x:f(x)\in D\}$ for $D\subseteq Y$, then the true statement is </p>
<p>(A) $f(f^{-1}(B))=B$</p>
<p>(B) $f^{-1}(f(A))=A$</p>
<p>(C) $f(f^{-1}(B))=B$ only if $B\subseteq f(X)$</p>
<p>(D) $f^{-1}(f(A))=A$ only if $... | Alex M. | 164,025 | <p>Let $X = \Bbb R, \ Y = [0, \infty)$ and $f(x) = x^2$. If $A = (-\infty, 0]$, then $f(A) = [0, \infty)$ and $f^{-1} (f(A)) = f^{-1} ([0, \infty)) = \Bbb R \ne A$, therefore $f^{-1} (f(A)) \ne A$, so (B) is false.</p>
<p>Let $X = [0, \infty), \ Y = \Bbb R$ and $f(x) = x$. If $A \subseteq X$, then $f(A) = A \subseteq ... |
156,285 | <p>I have been working on this exercise for a while now. It's in B.L. van der Waerden's <em>Algebra (Volume I)</em>, page $19$. The exercise is as follows:</p>
<blockquote>
<p>The order of the symmetric group $S_n$ is $n!=\prod_{1}^{n}\nu$. (Mathematical induction on $n$.)</p>
</blockquote>
<p>I don't comprehend ho... | Eugene | 31,288 | <p>You can actually just use a combinatorial argument for this. The permutation group is a bijection from a set of $n$ elements to itself. So look at the first element in the permutation. There are $n$ choices to send that element to. Now for the second element, there are only $n-1$ choices left (because it is a biject... |
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