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 |
|---|---|---|---|---|
3,627,704 | <p>The pmf of a negative binomial distribution is</p>
<p><span class="math-container">$$p_X(x)= {x-1 \choose r-1}~ p^r~ (1-p)^{x-r}\quad x=r,r+1,\cdots$$</span></p>
<p>I want to verify that </p>
<p><span class="math-container">$$\sum \limits_{x=r}^{\infty} p_X(x)= 1$$</span></p>
<p>I start with</p>
<p><span class=... | pico | 666,807 | <p><span class="math-container">$$(a+1)^{-n} =\sum \limits_{k=0}^{\infty} (-1)^k~{n+k-1\choose k}~a^k$$</span></p>
<p>Now let a=-b where b >0</p>
<p><span class="math-container">$$(-b +1)^{-n} =\sum \limits_{k=0}^{\infty} (-1)^k~{n+k-1\choose k}~(-b)^k$$</span></p>
<p><span class="math-container">$$(1-b)^{-n} = \sum... |
196,024 | <p>I am an undergrad and I know that the conjecture may have been proven recently. But in reading about it, I am entirely confused as to what it means and why it is important. I was hoping some of you kind people could help me.</p>
<p>I know there are several formulations of the conjecture.</p>
<p>Wolfram says:</p>
... | Charles | 1,778 | <p>Suppose a, b, and c are coprime positive integers: $0<a<b<c=a+b$ with $\gcd(a,b)=\gcd(a,c)=\gcd(b,c)=1.$ Then (under the abc conjecture) there are only finitely many such a, b, and c such that $c>\operatorname{rad}(abc)^{1.1}$, only finitely many such that $c>\operatorname{rad}(abc)^{1.01}$, only fini... |
3,500,889 | <p>Given natural numbers <span class="math-container">$a,b,c,d$</span>, let <span class="math-container">$a,b$</span> be coprime with <span class="math-container">$b>a$</span> and let <span class="math-container">$c,d$</span> be coprime with <span class="math-container">$d>c$</span>. Define a function <span class... | Patrick Stevens | 259,262 | <p>An "and" here would mean "for every <span class="math-container">$x$</span>, if <span class="math-container">$x$</span> is a teacher and <span class="math-container">$x$</span> is a student, then <span class="math-container">$x$</span> is here". In English the "and" is intended as a union, which is expressed in logi... |
1,502,610 | <p>My exercise book and Wolfram Alpha give:</p>
<p>$$\lim\limits_{x\to\infty}(\sqrt{9x^2+x}-3x)=\frac{1}{6}$$</p>
<p>When I work it out I get 0:</p>
<p>$$(\lim\limits_{x\to\infty}x\sqrt{9\frac{x^2}{x^2}+\frac{x}{x^2}}-\lim\limits_{x\to\infty}3x)$$</p>
<p>$$(\lim\limits_{x\to\infty}x*\sqrt{\lim\limits_{x\to\infty}9+... | Mark Viola | 218,419 | <p>Although this doesn't address the specific question on why the procedure in the OP is flawed, I thought it might be instructive to present an approach using a powerful general method. To that end, we proceed.</p>
<p>One approach is to use the <a href="https://en.wikipedia.org/wiki/Binomial_theorem#Newton.27s_gener... |
3,421,503 | <blockquote>
<p>Let <span class="math-container">$\lim\limits_{n \to \infty}(a_n+b_n+c_n)=\sqrt{3}$</span> and
<span class="math-container">$\lim\limits_{n \to \infty} (a_n^2+b_n^2+c_n^2)=1$</span>. Prove <span class="math-container">$\{a_n\}$</span> is
convergent.</p>
</blockquote>
<p>I think this should apply ... | Conrad | 298,272 | <p>Consider <span class="math-container">$f_n=3(a_n^2+b_n^2+c_n^2), g_n=(a_n+b_n+c_n)^2, h_n=(a_n-b_n)^2+(a_n-c_n)^2+(c_n-b_n)^2$</span></p>
<p>We notice that <span class="math-container">$f_n-g_n=h_n \ge 0, f_n, g_n \to 3$</span>, hence <span class="math-container">$h_n \to 0$</span>, hence for every <span class="mat... |
268,544 | <p>Here's an interesting inequality involving binomial coefficient and Stirling numbers of the second kind that I believe holds for all $n,k$:
$$ k^n {n \choose k} \leq n^k {n \brace k} $$
On the left-hand side we choose a $k$-element subset of the set $[n]=\{1,\ldots,n\}$, and then choose a function from $[n]$ into th... | Nicholas Kuhn | 102,519 | <p>Doesn't this admit the following easy argument: given an injection $f:[k] \rightarrow [n]$, let $\sigma(f): [n] \rightarrow [k]$ send $i \in [n]$ to $j \in [k]$ if $f(j)=i$ and to 1 otherwise. $\sigma(f)$ is obviously surjective, and it is also clear that distinct injections go to distinct surjections.</p>
|
9,648 | <p>To motivate my question, recall the following well-known fact: Suppose that $p\equiv 1\pmod 4$ is a prime number. Then the equation $x^2\equiv -1\pmod p$ has a solution.</p>
<p>One can show this as follows: Consider the following polynomial in ${\mathbb Z}_p[x]$: $x^{4k}-1$, where $p=4k+1$. The roots of this polyno... | Robin Chapman | 226 | <p>If $p=8k+1$ then consider $a$ with $a^{4k}\equiv-1$ (mod $p$).
Let $b=a^k-a^{7k}$. Then
$$b^2=a^{2k}-2a^{8k}+a^{14k}\equiv a^{2k}(1+a^{4k})-2
\equiv-2\pmod{p}.$$
For $p\equiv3$ (mod $8$) this argument will also work but you need to use the
Galois field $GF(p^2)$ rather than the integers modulo $p$.</p>
|
4,086,894 | <p>Let <span class="math-container">$G$</span> be a finite group and <span class="math-container">$V,W$</span> be vector spaces. Let <span class="math-container">$\rho: G \to GL(V)$</span> and <span class="math-container">$\tilde{\rho}: G \to GL(W)$</span> be two representations of <span class="math-container">$G$</spa... | azif00 | 680,927 | <p>Given a group <span class="math-container">$G$</span> we can form two new structures:</p>
<ul>
<li>for a given field <span class="math-container">$k$</span>, a <strong><span class="math-container">$k$</span>-representation of <span class="math-container">$G$</span></strong>, which is a pair <span class="math-contain... |
2,408,703 | <p>Is it possible to solve for <code>b</code> in the following?</p>
<pre><code>a = log2(b + c)
</code></pre>
<p>For example:</p>
<pre><code>3.907 = log2(b + 10)
b = 5
</code></pre>
| C. Falcon | 285,416 | <p>Recall that one has the following:</p>
<blockquote>
<p><strong>Proposition.</strong> Let $(X,d)$ be a topological metric space and $A$ be a subset of $X$, then one has:
$$\forall x\in X,d(x,A)=0\iff x\in\overline{A}.$$</p>
</blockquote>
<p><strong>Proof.</strong> Let us proceed by double implication.</p>
<ul>... |
1,811,907 | <p>I tried finding the maxima of $f(x)=\frac{3}{4}-x-x^2$ by taking the derivative and so on and use the fact that $\displaystyle\int_{a}^{b}f(x)\,dx \leq M(b-a)$ where $M$ is the global maximum, but then the maximum value depends on the values of $a$ and $b$.</p>
| marwalix | 441 | <p>Let's look at a brute force approach.</p>
<p>$$f(a,b)=\int_a^b\left({3\over 4}-x-x^2\right)dx={3\over 4}(b-a)-{b^2-a^2\over 2}-{b^3-a^3\over 3}$$</p>
<p>Now this is maximum when</p>
<p>$$\begin{align} {{\partial{f(a,b)}\over\partial{a}}}&=0\\{{\partial{f(a,b)}\over\partial{b}}}&=0\end{align}$$</p>
<p>Whi... |
2,345,502 | <p>I am going crazy trying to figure out what I am doing wrong on this basic problem. I need to find the $y$ coordinate of the center of mass of a pan of water that is sloshing back and forth. Let the equilibrium height be $h$, the length of the pan be $L$, and the height at $x=L$ be $y=h+b$. The coordinate of the cent... | Ronald Blaak | 458,842 | <p>There is the factor $\frac{1}{2}$ missing due to not doing the integration over $y$. Due to insight in physics you already noticed that there was a mistake, which is always good. (Correct result should be $y_c=\frac{b}{6 h} + \frac{h^2}{2}$.)</p>
<p>You also came up with a second option to get the result, via the a... |
65,944 | <p>How can I prove the identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$?</p>
<p>I have to prove it using lattice paths, it should be related to Catalan numbers</p>
<p>The $n$th Catalan number $C_n$ counts the number of monotonic paths along the edges of a grid with $n\times n$ square... | Brian M. Scott | 12,042 | <p>The right-hand side counts the number of monotonic paths from $(0,0)$ to $(n-1,n+1)$. Since $(n-1,n+1)$ is above the diagonal, every one of these paths must cross the diagonal at some point. Suppose that the first ‘bad’ step is from $(k-1,k-1)$ to $(k-1,k)$.</p>
<ul>
<li><p>How many ways are there to get from $(0,0... |
2,552,005 | <p>As long as I've been working with mathematical notation, it seems like I should be embarrassed about being confused about this, but...</p>
<p>Does $\sqrt[n]{a}$ with $n \in \mathbb{N}$ in general stand for the set of all $n$ roots, or just one of them? When $a \in \mathbb{Z}$ and it's clear that the surd is also me... | lhf | 589 | <p>If $a \in \mathbb R$, then $\sqrt[n]{a}$ always stands for a single value:</p>
<ul>
<li><p>When $n$ is odd, there is no ambiguity because $x^n=a$ has only one real solution.</p></li>
<li><p>When $n$ is even, $\sqrt[n]{a}$ is the positive solution of $x^n=a$. This is only defined when $a\ge 0$.</p></li>
</ul>
|
2,552,005 | <p>As long as I've been working with mathematical notation, it seems like I should be embarrassed about being confused about this, but...</p>
<p>Does $\sqrt[n]{a}$ with $n \in \mathbb{N}$ in general stand for the set of all $n$ roots, or just one of them? When $a \in \mathbb{Z}$ and it's clear that the surd is also me... | José Carlos Santos | 446,262 | <p>The standard meaning of $\sqrt[n]a$, when $a\in[0,+\infty)$, is the only real number greater than or equal to $0$ such that its $n$<sup>th</sup> power is equal to $a$, nor the set of all $n$<sup>th</sup> roots of $a$. With this convention, both$$\overbrace{\sqrt[n]a\times\cdots\times\sqrt[n]a}^{n\text{ times}}\text{... |
283,616 | <p>I meant to assign to my class the following homework problem:</p>
<blockquote>
<p>If $u\in C^2((0,T)\times \Omega) \cap C^0([0,T]\times\bar{\Omega})$ where $\Omega$ is an open, bounded domain, is such that $\partial_t u - \triangle u \leq - \epsilon < 0$ for some constant $\epsilon > 0$, then $u$ cannot hav... | Mateusz Kwaśnicki | 108,637 | <p>A potential-theoretic argument seems to be applicable. I come from probability, so I use the language of probabilistic potential theory below. However, everything can be translated into the language of harmonic (or ``caloric'') measures.</p>
<p>Let $X_t$ be the standard Brownian motion, generated by the Laplacian $... |
2,204,221 | <p>This random thought just struck me: how can one calculate $(-1)^{\sqrt{2}}$? </p>
<p>Different tools gives different results: according to google, the result is <code>undefined</code> (as it does not support complex numbers); python gives me a complex number <code>(-0.2662553420414156-0.9639025328498773j)</code> w... | mrnovice | 416,020 | <p>Let $c = (-1)^{\sqrt{2}}$</p>
<p>Note that $e^{i(\pi+2k\pi)} = -1 , k\in\mathbb{Z}$ (this is where $e$ comes into it)</p>
<p>Then:</p>
<p>$$c = [e^{i(\pi+2k\pi)}]^{\sqrt{2}}$$</p>
<p>$$c = e^{\pi\sqrt{2}i(1+2k)} $$</p>
<p>$$c = \cos(\pi\sqrt{2}(1+2k))+i \sin(\pi\sqrt{2}(1+2k))$$</p>
|
2,204,221 | <p>This random thought just struck me: how can one calculate $(-1)^{\sqrt{2}}$? </p>
<p>Different tools gives different results: according to google, the result is <code>undefined</code> (as it does not support complex numbers); python gives me a complex number <code>(-0.2662553420414156-0.9639025328498773j)</code> w... | Thomas Andrews | 7,933 | <p>Basically, it depends on how you define $x^y$ generally. In some cases, it is undefined, because we choose the negative numbers as a "branch cut" for the complex logarithm, so we don't define $x^y$ when $x$ is a negative number.</p>
<p>In other definitions, we define it as a multivalued function, so that $x^y$ migh... |
1,179,036 | <p>I'm researching the mathematics behind GPS, and at the moment I'm trying to get my head around how to solve the following system of equations:</p>
<p>$\sqrt{(x-x_1)^2+(y-y_1)^2+(z-z_1)^2}=r_1$</p>
<p>$\sqrt{(x-x_2)^2+(y-y_2)^2+(z-z_2)^2}=r_2$</p>
<p>$\sqrt{(x-x_3)^2+(y-y_3)^2+(z-z_3)^2}=r_3$</p>
<p>$(x,y,z)$ is ... | Claude Leibovici | 82,404 | <p>I think that you can make the problem simpler the solution of which not requiring Newton-Raphson or any root finder method; the solution is explicit and direct (it does not require any iteration). </p>
<p>Start squaring the expressions to define $$f_i={(x-x_i)^2+(y-y_i)^2+(z-z_i)^2}-r_i^2=0$$ Now, develop $(f_2-f_1... |
1,417,155 | <p>Let $X_0=1$, define $X_n$ inductively by declaring that $X_{n+1}$ is uniformly distributed over $(0,X_n)$.
Now I can't understand how does $X_{n}$ gets defined. If someone would just derive the distribution of $X_2$ that would be helpful. I saw this in a problem and I can't really start trying it. Thanks for any he... | Dominik | 259,493 | <p>Here's a way to define those random variables rigorously:</p>
<p>Let $\theta_1, \theta_2, \ldots$ be an i.i.d. sequence of random variables that are uniformly distributed on $[0, 1]$. Now define $X_0 = 1$ and $X_n = X_{n - 1} \theta_n = \prod \limits_{i = 1}^n \theta_i$ for $n > 0$.</p>
|
1,004,801 | <p>I have a question which I'm deeply confused about. I was trying to do some problems my professor gave us so we could practice for exam, one of them says:</p>
<p>Give a partition of ω in ω parts, everyone of them of cardinal ω.</p>
<p>I know that $ ω=\left \{ 1,2,3,...,n,n+1,.... \right \}$ , but I thought that... | Brian M. Scott | 12,042 | <p>There are many ways to do it, since $|\omega\times\omega|=\omega$. Here’s one. Every $n\in\Bbb Z^+$ can be written uniquely in the form $n=2^km$, where $m$ is odd, and you can start by letting $S_k=\{2^km:m\in\omega\text{ is odd}\}$: $S_0$ is the set of odd positive integers, $S_1$ the set of even positive integers ... |
649,169 | <p>I know this has to do with Euclidean division, I just can't prove the => direction.
Any tips would be appreciated!</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $ $ If $\, d\mid n\ $ then $\ d\mid \color{#0a0}{n-a}\iff d\mid \color{#c00}a.\ $ Thus $\ \color{#0a0}{n-a},n\ $ and $\ \color{#c00}a,n\ $ have the same set $S$ of common divisors $\,d,\,$ so they have the same <em>greatest</em> common divisor $\,(= \max S).$</p>
<p><strong>Alternatively</stro... |
3,156,557 | <p>I've having trouble understanding what the question is trying to ask. And I am not sure how to start to answer the question.</p>
<hr>
<p>The diagram below shows what happens for waves on the surface of a pond. If you
drop a stone in the point at the point <span class="math-container">$F_1$</span> at time <span cla... | Bib-lost | 294,785 | <p>It is true that an algebraically closed field of characteristic 0 with the same cardinality as <span class="math-container">$\mathbb{C}$</span> is isomorphic to <span class="math-container">$\mathbb{C}$</span>, so your reasoning works. This follows from the following more general fact, which can be found in many alg... |
482,061 | <p>Given the continuous Probability density function $f(x)=\begin{cases} 2x-4, & 2\le x\le3 \\ 0 ,& \text{else}\end{cases}$</p>
<p>Find the cumulative distribution function $F(x)$.</p>
<p>The formula is $F(x)=\int _{ -\infty }^{ x }{ f(x) } $</p>
<hr>
<p><h3>My Solution</h3> <br>
The first case is when $2... | Stefan Hansen | 25,632 | <p>A general technique to deal with such type of questions is to split up the integral at the points where the density $f$ has different specifications, i.e. at $y=2$ and $y=3$. If $y<2$, then
$$
F(y)=\int_{-\infty}^yf(x)\,\mathrm dx=\int_{-\infty}^y0\,\mathrm dx=0,
$$
and if $2\leq y\leq 3$, then
$$
\begin{align}
F... |
2,368,300 | <p>If I have the equation $A=PDP^{-1}$ for $3 \times 3$ matrices, where $P$ and $D$ are known and $D$ is a diagonal matrix, how can I use this information to find the eigenvalues of $A$?</p>
| Robert Lewis | 67,071 | <p>For any two square $n \times n$ matrices $M$ and $N$ such that</p>
<p>$M = PNP^{-1}, \tag{1}$</p>
<p>the eigenvalues are the same. For if</p>
<p>$N \vec v =\mu \vec v, \tag{2}$</p>
<p>with $\vec v \ne 0$, then</p>
<p>$M(P\vec v) = PNP^{-1}(P\vec v) = PN\vec v = P(\mu \vec v) = \mu P \vec v, \tag{3}$</p>
<p>sh... |
2,368,300 | <p>If I have the equation $A=PDP^{-1}$ for $3 \times 3$ matrices, where $P$ and $D$ are known and $D$ is a diagonal matrix, how can I use this information to find the eigenvalues of $A$?</p>
| Brian Fitzpatrick | 56,960 | <p>Recall that the <em>characteristic polynomial</em> of a square matrix $M$ is the polynomial $\chi_M(t)=\det(t\cdot I-M)$. The eigenvalues of $M$ are the roots of $\chi_M(t)$.</p>
<p>Now, suppose that $A=PBP^{-1}$. Then
\begin{align*}
\chi_A(t)
&= \det(t\cdot I-A) \\
&= \det\left(t\cdot I-PBP^{-1}\right) \\
... |
41,795 | <p>Every matrix $A\in SL_2(\mathbb{Z})$ induces a self homeomorphism of $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$. For different matrices these homeomorphisms are not homotopic, as the induced map on $\pi_1(S_1\times S_1)$ is given by $A$ (w.r.t the induced basis).</p>
<p>So I am wondering, whether a similar construct... | Greg Kuperberg | 1,450 | <p>The matrix $\text{SL}(2,\mathbb{Z})$ acts on $H^n(S^n \times S^n)$; one interpretation of your question is whether this action lifts to $\text{Diff}(S^n \times S^n)$. There is a simple reason that it doesn't when $n$ is even: The intersection form on $H^n(S^n \times S^n)$ is symmetric rather than anti-symmetric, a... |
731,421 | <p>I'm having some trouble finding $\tan^{-1}(2i)$. The formula the book has is $\tan^{-1}z=\dfrac{i}{2} \log\dfrac{i+z}{i-z}$. But when I use this I get a different answer than what the book has. This is what I have so far</p>
<p>$\log\dfrac{i+2i}{i-2i}=\log(-3)=\ln3+i(\pi+2\pi n)$ where $n\in\mathbb{Z}$. Thus $\dfr... | lab bhattacharjee | 33,337 | <p>I will follow another method:</p>
<p>Let $\displaystyle\tan^{-1}(2i)=a+ib$ where $a,b$ are real</p>
<p>$\displaystyle\implies \tan(a+ib)=2i\iff\tan(a-ib)=-2i$</p>
<p>So, $\displaystyle\tan2a=\tan(a+ib+a-ib)=\frac{\tan(a+ib)+\tan(a-ib)}{1-\tan(a-ib)\tan(a+ib)}=0$</p>
<p>So, $2a=n\pi$ where $n$ is any integer</p>
... |
2,285,303 | <p>i am looking for a easy way to read if a transformation is bijective, surjective or injective based solely on the reduced echelon form? someone told me that this was possible, but i cannot figure it out.</p>
<p>I have the following transformation:
$$
T(X) = \left(
\begin{array}{c}
x_{1} \\
x_{2} \\
... | Sri-Amirthan Theivendran | 302,692 | <p>Is the null space of the matrix trivial(consisting only of the zero vector)? If yes, the linear transformation is injective. Otherwise it is not injective.</p>
<p>Next for a linear transformation $T:\mathbb{R}^n\to \mathbb{R}^m$, we have that
$$
n=\dim(\ker T)+\dim \,(\text{im}\, T)\tag{1}
$$
Based on the null spac... |
2,285,303 | <p>i am looking for a easy way to read if a transformation is bijective, surjective or injective based solely on the reduced echelon form? someone told me that this was possible, but i cannot figure it out.</p>
<p>I have the following transformation:
$$
T(X) = \left(
\begin{array}{c}
x_{1} \\
x_{2} \\
... | Alex Mathers | 227,652 | <p>Recall that by definition, the transformation is injective if $T(X)=T(X')$ implies $X=X'$. Now, check that this is <strong>not</strong> true in the case $X=(2,-1,0)$ and $X'=(0,0,1)$. Looking at the reduced echelon form, can you deduce how I came up with those vectors?</p>
<p>For surjectivity, you want the matrix t... |
1,259,383 | <p>I have a distribution with literally an infinite number of potential data points. I need the standard deviation. I generate about a hundred points and take the standard deviation of the points. This gives a hopefully good approximation of the true standard deviation, but it won't, of course, be exact. How do I e... | Lior Avrahami | 828,954 | <p>the answer to OP's question depends on whether or not the mean of the distribution is known. if the mean is known ( for example if you know that the mean of you sampled population should eventually average out to be zero) than the problem is a little different, not by much but I did not do the research to find out t... |
1,692,987 | <p>General solutions of trigonometric equations are given by: </p>
<blockquote>
<p>If $\sin(x) = \sin(y)$, then $x = n \pi + (-1)^ny$<br>
If $\cos(x) = \cos(y)$, then $x = 2n \pi \pm y$ </p>
</blockquote>
<p>If we consider an example, </p>
<p>$$
\sin(x) = \sin(30^\circ) \\
\implies x = n \pi + (-1)^n30^\cir... | lab bhattacharjee | 33,337 | <p>$\sin x=\sin y,x=n\pi+(-1)^ny$ where $n$ is any integer</p>
<p>For even $n=2m$(say) $x=2m\pi+y\ \ \ \ (1)$</p>
<p>For odd $n=2m+1$(say) $x=(2m+1)\pi-y$</p>
<p>Now let $X=\dfrac\pi2-x, Y=\dfrac\pi2-y\implies\cos x=\cos y$</p>
<p>$(1)\implies\dfrac\pi2-X=2m\pi+\dfrac\pi2-Y\iff X=-2m\pi+Y$</p>
<p>$(2)\implies\dfra... |
1,718,623 | <p>I am currently writing a NURBS ray tracer. What I do is convert the NURBS into rational Bézier patches and then perform the intersection test using Newton's method. To do this fast (the ray tracer should be later implemented using GPGPU programming and should run in real time) a point on the surface and first deriva... | bubba | 31,744 | <p>NURBs are vastly over-rated, in my view. Since they're rational, computing derivatives is a mess. Computing integrals is even worse. If you have the option, I'd strongly recommend sticking with polynomials. People will tell you that polynomials can't represent conics and quadrics exactly. This is true, but we're usi... |
172,924 | <p>Are there any known or conjectured bounds on the exponent $d(r)$ such that $x^{d(r)} = 0$ for all $x \in \pi_r^S(S^0)$?</p>
| Drew Heard | 16,785 | <p>If one uses the Adams-Spectral sequence based on cohmology theories other than $BP$ it is possible to say a little more. In <a href="http://ac.els-cdn.com/S0040938399000026/1-s2.0-S0040938399000026-main.pdf?_tid=925ef31a-273a-11e4-be46-00000aacb362&acdnat=1408409504_3ee3458e72f9e3556e672c0e21451eea" rel="nofollo... |
54,634 | <p>As you know, <em>Mathematica</em> V10 has just been released and I think a lot of updates will be released. However, I don't see any command in the help menu to update (which is present in most other software).</p>
<p>How would we know if an update is available or not?</p>
<p>I am using trial version of V10.</p>
... | Eight Hour Lunch | 20,167 | <p>Update: The point releases are <em>supposed</em> to be automatically available through your Wolfram User Portal under My Products and Services. In my case, it hadn't been updated. I contacted support via their chat, and they changed the link for me manually. If you haven't gotten the email yet, you might want to pur... |
1,339,332 | <p>I would like to integrate this in my research:</p>
<p>$\int_0^\infty s e^{i bs^2}J_0(a s)$, where a and b are both real and greater than zero. Integration by parts seems like the obvious first step, but that leaves a term like $\int_0^\infty e^{i bs^2}J_1(a s)$ , which seems even more complicated.</p>
<p>The topic... | Jack D'Aurizio | 44,121 | <p><strong>Hint:</strong>
$$\mathcal{L}\left(J_0(a\sqrt{x})\right) = \frac{1}{t} e^{-\frac{a^2}{4t}},\tag{1}$$</p>
<p>$$\mathcal{L}\left(J_1(a\sqrt{x})\right) = \frac{|a|\sqrt{\pi}}{t^{3/2}}\left(I_0\left(e^{-\frac{a^2}{8t}}\right)-I_1\left(e^{-\frac{a^2}{8t}}\right)\right) e^{-\frac{a^2}{8t}}\tag{2}$$</p>
<p>$(1)$ ... |
4,351,610 | <p>I'm trying to solve this problem, I took the basis <span class="math-container">$B=\langle(2,7)^T, (1,3)^T\rangle$</span> and then build the matrix of <span class="math-container">$f$</span> application in <span class="math-container">$B$</span>:
<span class="math-container">$$
M_b(f)=\left(\begin{array}{cc}
7&4... | Sourav Ghosh | 977,780 | <p>I assume <span class="math-container">$f:\Bbb{R^2}\to \Bbb{R^2}$</span> be a linear map.</p>
<p>Given <span class="math-container">$f(2, 7) =(7, 5) $</span> and <span class="math-container">$f(1, 3) =(4, 1) $</span></p>
<p>To find <span class="math-container">$f(3, 5) $</span> , first we have to write <span class="m... |
2,669,617 | <p>The bilinear axiom is:</p>
<pre><code> <cu + dv,w> = c<u,w> + d<v,w>
<u,cv + dw> = c<u,v> + d<u,w>
</code></pre>
<p>Where c and d are scalars and u, v, and w are vectors.</p>
<p>Can this be extended to something like</p>
<pre><code> <cu + dv, ew + fx> = ?
</code></pre>
| Hagen von Eitzen | 39,174 | <p><em>Hint:</em> ${n\choose k}={n-1\choose k-1}+{n-1\choose k}$</p>
|
4,476,229 | <p>In Theorem 5.5, Rudin proves the chain rule, but does so in a somewhat different fashion than expected. It seems we can prove the chain rule more easily.</p>
<p>Theorem:
Suppose <span class="math-container">$f:[a,b]\to\mathbb{R}$</span> is continuous on <span class="math-container">$[a,b]$</span> and <span class="m... | egglog | 626,167 | <p>Here is a method.</p>
<p><span class="math-container">$x^2 + y^2 = $</span></p>
<p><span class="math-container">$d^2(\cos^2\theta + \sin^2\theta) + b^2(\cos^2\phi + \sin^2\phi) - 2db (\cos\theta\cos\phi - \sin\theta\sin\phi) = $</span></p>
<p><span class="math-container">$ d^2 + b^2 - 2db\cos(\theta + \phi)$</span><... |
874,859 | <p>let $R$ be a finite boolean ring.<br>
prove that $|R|=2^n$ for some $n\in\mathbb N$.
<br><br>
I know that $R$ is commutative and for every element $a\in R\space a+a=0$ and $a^2=a$</p>
| egreg | 62,967 | <p>A boolean ring is an algebra, in particular a vector space, over the two element field $\{0,1\}$.</p>
<p>Alternatively, the additive group of $R$ is a $2$-group: the order of every element is a power of $2$. A finite $p$-group ($p$ a prime) has order $p^n$ for some integer $n$.</p>
|
3,422,670 | <p>I was factorizing <span class="math-container">$$f(x) = \frac{4x^2+7x+2}{4x^3+4x^2+2x}$$</span><br>
I approached this like below<br>
<span class="math-container">$$f(x) = \frac{a}{2x} + \frac{bx+c}{2x^2+2x+1}$$</span>
and got a = 2, b = 0, c = <span class="math-container">$\frac{3}{2}$</span>. So, f(x) becomes </p>... | merelymyself | 722,119 | <p>Isolating the multiples of 3 is the way to go. In 30!, </p>
<p>3*6*9*12*15*18*21*24*27*30 </p>
<p>includes</p>
<p>1+1+2+1+1+2+1+1+3+1 '3's in its index notation, which is equal to 14.</p>
<p>Thus, the answer is 14.</p>
|
3,406,164 | <p><span class="math-container">$$a_n=\frac{sin1!}{1*2}+\frac{sin2!}{2*3}+\frac{sin3!}{3*4}+...+\frac{sinn!}{n(n+1)}$$</span>
I have this series and I don't understand how to apply the cauchy criterion
<span class="math-container">$$\lvert a_{n+p}-a_n \rvert \lt ε$$</span>
The only result I get is this one:
<span class... | Robert Israel | 8,508 | <p>Hint: <span class="math-container">$|\sin(x)|\le 1$</span>. Estimate <span class="math-container">$\left|\sum_{k=n+1}^{n+p} \frac{1}{k(k+1)}\right|$</span>.</p>
|
4,575,800 | <p>Could maybe someone help me here?</p>
<blockquote>
<p>For <span class="math-container">$p\in [1,2)$</span> I need to show that <span class="math-container">$l^p$</span> has empty interior in <span class="math-container">$l^2$</span>.</p>
</blockquote>
<p>I know that I need to show that there is no open ball <span cl... | Anne Bauval | 386,889 | <p>By contradiction, if this interior was not empty, it would contain some point <span class="math-container">$y\in\ell^p$</span> and some <span class="math-container">$\ell^2$</span>-ball <span class="math-container">$B(y,r)$</span> (<span class="math-container">$r>0$</span>) around it. Let <span class="math-contai... |
107,443 | <p>Let $T \colon \mathbb{R}^n \to \mathbb{R}^n$ be a linear map, $H^{m}$ be a Hausdorff measure.
Is it true that
$$
\int\limits_{T(M)} f(x) H^{m}(dx) = |\det{T}| \int\limits_{M} f(T(x)) H^{m}(dx)
$$
where $f(x)$ is some continuous function?</p>
| Robert Haraway | 9,502 | <p>No; it must be slightly more interesting than that. Consider the Cantor set, which has $\frac{\log 2}{\log 3}$-dimensional measure 1. Expanding the set by a factor of 3 produces two identical copies of the original set, presumably doubling the $\frac{\log 2}{\log 3}$-dimensional measure. </p>
<p>One might <em>suspe... |
666,229 | <p>What is ML inequality property in complex integral which says $|\int_{c}f(z)dz| \leq ML$. I can't understand a thing from this expression.
I want to understand it conceptually(if that helps).<br> How can we find the upper bound of a complex integral</p>
| Zaid Alyafeai | 87,813 | <p>$$\left|\int_c f(z) \, dz \right| \leq \int_c |f(z)| \cdot |dz|$$</p>
<p>Now assume that $|f(z)| \leq M$ that means the function is bounded on the curve </p>
<p>$$\int_c |f(z)| \cdot |dz| \leq M \int_c |dz|$$</p>
<p>Now assume that the $c=\gamma(t)$ is a parametrization of the curve then </p>
<p>$$\int_c |dz| =... |
3,887,111 | <p>If <span class="math-container">$$n \in \mathbb{Z}, \geq3$$</span>
<span class="math-container">$${A_{n}}=\left \{ {an+1: a \in \mathbb{Z}} \right \}$$</span></p>
<p><span class="math-container">$${B_{n}}=\left \{ n|(b+1): b \in \mathbb{Z} \right \}$$</span></p>
<p>How do we prove that the set is disjoint? So This i... | DeepSea | 101,504 | <p>Hint: <span class="math-container">$\dfrac{a}{a+b} > \dfrac{a}{a+b+c}$</span></p>
|
3,887,111 | <p>If <span class="math-container">$$n \in \mathbb{Z}, \geq3$$</span>
<span class="math-container">$${A_{n}}=\left \{ {an+1: a \in \mathbb{Z}} \right \}$$</span></p>
<p><span class="math-container">$${B_{n}}=\left \{ n|(b+1): b \in \mathbb{Z} \right \}$$</span></p>
<p>How do we prove that the set is disjoint? So This i... | Michael Rozenberg | 190,319 | <p>We can use also your idea, but in another writing:
<span class="math-container">$$\sum_{cyc}\sqrt{\frac{a}{a+b}}=\sum_{cyc}\frac{\sqrt{a(a+b)}}{a+b}>\sum_{cyc}\frac{\sqrt{a\cdot a}}{a+b}>\sum_{cyc}\frac{a}{a+b+c}=1.$$</span></p>
|
3,339,861 | <p><span class="math-container">$\mathbf{Question}:$</span> Let <span class="math-container">$f$</span> be a continuous function on <span class="math-container">$[0,1]$</span>. Then prove that the limit <span class="math-container">$\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$</span> is equal to <span class="math-containe... | Stefan Lafon | 582,769 | <p>Another way is to change the variable <span class="math-container">$u=x^n$</span>:
<span class="math-container">$$\int_0^1nx^nf(x)dx=\int_0^1 u^{\frac 1 n}f(u^{\frac 1 n})du$$</span>
The integrand converges pointwise to <span class="math-container">$f(1)$</span> for <span class="math-container">$u\in (0,1]$</span> a... |
3,774,507 | <p>How to prove what follows?</p>
<blockquote>
<p><span class="math-container">$$\sum_{n=0}^{\infty}\frac{1}{2^{n}(3n+1)}=\frac{2^{\frac{1}{3}}}{3}\ln\left(\frac{\sqrt{2^{\frac{2}{3}}+2^{\frac{1}{3}}+1}}{2^{\frac{1}{3}}-1}\right)+\frac{\sqrt[3]{2}}{3}\arctan\left(\frac{2^{\frac{2}{3}}+1}{\sqrt{3}}\right)-\frac{2^{\frac... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One may prove that
<span class="math-container">$$
\sum_{n=0}^{\infty}\frac{1}{2^{n}(3n+1)}=\sum_{n=0}^{\infty}\int_0^1\frac{x^{3n}}{2^{n}}dx=\int_0^1\sum_{n=0}^{\infty}\left(\frac{x^3}{2}\right)^{n}dx=\int_0^1\frac{2}{2-x^3}\:dx
$$</span>
Hope you can take it from here.</p>
|
3,774,507 | <p>How to prove what follows?</p>
<blockquote>
<p><span class="math-container">$$\sum_{n=0}^{\infty}\frac{1}{2^{n}(3n+1)}=\frac{2^{\frac{1}{3}}}{3}\ln\left(\frac{\sqrt{2^{\frac{2}{3}}+2^{\frac{1}{3}}+1}}{2^{\frac{1}{3}}-1}\right)+\frac{\sqrt[3]{2}}{3}\arctan\left(\frac{2^{\frac{2}{3}}+1}{\sqrt{3}}\right)-\frac{2^{\frac... | Community | -1 | <p>Observe that</p>
<p><span class="math-container">$$f(x):=\sum_{n=0}^\infty\frac{x^n}{3n+1}\implies f'(x)=\sum_{n=0}^\infty\frac{nx^{n-1}}{3n+1}$$</span> and
<span class="math-container">$$3xf'(x)+f(x)=\sum_{n=0}^\infty\frac{3n+1}{3n+1}x^n=\frac1{1-x}.$$</span></p>
<p>The solution of the homogeneous part of this line... |
3,774,507 | <p>How to prove what follows?</p>
<blockquote>
<p><span class="math-container">$$\sum_{n=0}^{\infty}\frac{1}{2^{n}(3n+1)}=\frac{2^{\frac{1}{3}}}{3}\ln\left(\frac{\sqrt{2^{\frac{2}{3}}+2^{\frac{1}{3}}+1}}{2^{\frac{1}{3}}-1}\right)+\frac{\sqrt[3]{2}}{3}\arctan\left(\frac{2^{\frac{2}{3}}+1}{\sqrt{3}}\right)-\frac{2^{\frac... | Đào Minh Dũng | 516,480 | <p>You can modify your attempt a little bit to get an easier way of solving this problem:
<span class="math-container">$$\sum_{n=0}^{\infty} \dfrac{1}{2^n(3n+1)} = \sum_{n=0}^{\infty} \dfrac{1}{(\sqrt[3]{2})^{3n}(3n+1)} = \sqrt[3]{2} \sum_{n=0}^{\infty} \dfrac{1}{(\sqrt[3]{2})^{3n+1}(3n+1)}$$</span>
And now you can fin... |
3,725,571 | <p>With the rise of factory games like <a href="https://www.satisfactorygame.com/" rel="nofollow noreferrer">Satisfactory</a> and <a href="https://factorio.com/" rel="nofollow noreferrer">Factorio</a>, many people have started wondering about problems like these.</p>
<p>Factorio already has a very interesting analysis ... | vepsankel | 837,999 | <p>If we take our initial flow rate and estimate it in N steps with "splitters", each having K>0 multiplier rate, then on n-th step maximum error can be estimated with (K^n)/2. Error converges to 0 for K<1 => numbers can be achieved as precisely as we want as long as they're smaller than input.</p>
<... |
1,382,366 | <p>Can anybody pass me on a good source to see the steps in proving,
<span class="math-container">\begin{equation}
\zeta(2n) = \frac{(-1)^{k-1}B_{2k}(2\pi)^{2k}}{2(2k)!}
\end{equation}</span></p>
<p>I know how we start by looking at the product of sine and use the generatinf function for the Bernoulli numbers to connec... | user2520938 | 95,162 | <p>This happens to have been the topic of an article in the May 2015 American Mathematical Monthly. You can get access <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.444" rel="nofollow">here</a>. </p>
<p>Edit: instead of me sumerizing the article, you can find it <a href="http://arxiv.org/abs/120... |
3,132,736 | <p>Good evening,</p>
<p>I'm struggling with understanding a proof:</p>
<p>I know, that a solution of <span class="math-container">$y'=c \cdot y$</span> is <span class="math-container">$y=a \cdot e^{ct}$</span> and it's clear how to calculate this.</p>
<p>I want to proof, that all solutions of a function describing a... | Botond | 281,471 | <p>Let's look for a solution in the form of
<span class="math-container">$$y(t)=x(t)\exp(ct)$$</span>
Let's assume that it satisfies the differential equation. But it means that:
<span class="math-container">$$x'(t)\exp(ct)+x(t)c\exp(ct)=cx(t)\exp(ct)$$</span>
<span class="math-container">$$x'(t)\exp(ct)=0$$</span>
But... |
3,132,736 | <p>Good evening,</p>
<p>I'm struggling with understanding a proof:</p>
<p>I know, that a solution of <span class="math-container">$y'=c \cdot y$</span> is <span class="math-container">$y=a \cdot e^{ct}$</span> and it's clear how to calculate this.</p>
<p>I want to proof, that all solutions of a function describing a... | D.R. | 405,572 | <p>The motivation behind doing <span class="math-container">$$\frac{g(t)}{e^x}$$</span>
is because we <em>think</em> that for <strong>any <span class="math-container">$g(t)$</span> that satisfies <span class="math-container">$g(t)=g'(t)$</span></strong>, <span class="math-container">$g(t)$</span> must be of the form <s... |
351,170 | <p>The <em>genus <span class="math-container">$g$</span> handlebodies</em> are building blocks of <span class="math-container">$3$</span>-manifolds. They are constructed from <span class="math-container">$3$</span>-ball <span class="math-container">$B^3$</span> by adding <span class="math-container">$g$</span>-copies o... | Community | -1 | <p>This is not an answer, just a comment. It is from Manolescu's <a href="https://www.math.ucla.edu/~cm/matlab/234.jpeg" rel="nofollow noreferrer">website</a>. It seems to be related to your way of thinking Brieskorn spheres, but <span class="math-container">$2,3$</span> and <span class="math-container">$4$</span> are ... |
4,601,270 | <p>I need to evaluate the following integral, using Gamma and Beta functions:
<span class="math-container">$$\int_0^\infty\frac{\sqrt[5]{x}}{(x^2+1)(x+1)^2}dx$$</span>
I tried to use partial fractions, but I am not sure if that is correct, since doing that I get that the integral is divergent.</p>
<p>Since
<span class=... | dxdydz | 239,024 | <p>We can make your approach work by modifying it slightly. Consider the integral</p>
<p><span class="math-container">$$I(s)=\int_0^\infty\frac{x^{s-1}}{(1+x^2)(1+x)^2}\,\mathrm dx$$</span></p>
<p>instead. Applying your partial fraction expansion, an integral representation of the beta function <span class="math-contai... |
2,388,492 | <p>If we had a function f(x) and its indefinite integral is F(x). What is the physical interpretation of F(N), where N is just a value in f(x)'s domain?</p>
<p>Is it an area under f(x)? If so, with respect to which bounding points?(i.e. negative infinity?, zero?). Or can integrals only be used to find area if it's use... | Community | -1 | <p>$\int_a^b f (x)dx $ is the area under $f (x) $ from $x=a $ to $x=b $, assuming $f $ is integrable and everything. .. $F (x)=\int_a^x f (t)dt $ is a function whose derivative $F'(x)=f (x) $ by the <em>fundamental theorem of calculus</em>... Of course, you could say that $F (x) $ is the area between $a $ and $x... |
1,507,187 | <p>I am reading "Topology without tears " book, and got confused about proposition 2.3.4 page 60:</p>
<p>$\tau_1 =\tau_2$ iif</p>
<p>1) for each $B \in \mathscr B_1 $ and $ \forall x \in B , \exists B' \in \mathscr B_2 $ such that $ x\in B' \subseteq B $</p>
<p>2) for each $B \in \mathscr B_2 $ and $ \forall x \in... | Jack D'Aurizio | 44,121 | <p>Since
$$\int \arctan(x)\,dx = x\arctan x-\int \frac{x}{1+x^2}\,dx = x \arctan(x)-\frac{1}{2}\log(1+x^2) $$
we clearly have:
$$\int x\arctan(x^2)\,dx = \frac{x^2}{2}\arctan(x^2)-\frac{1}{4}\log(1+x^4).$$</p>
|
3,479,232 | <p>Here's a problem from my textbook:</p>
<blockquote>
<p>On the island of Mumble, the Mumblian alphabet has only 5 letters, and every word in the Mumblian language has no more than three letters in it. How many words are possible if letters can be repeated?</p>
</blockquote>
<p>I know we can break this problem int... | gdepaul | 623,989 | <p>number of single letter words: 5</p>
<p>number of two letter words without repetition: 5*5</p>
<p>number of three letter words without repetition: 5*5*5</p>
<p>Taking the sum of these combinations = 155.</p>
<p>We can also instead expand our alphabet to include a null or space character. But this requires a smal... |
2,436,634 | <p>The limit is $$ \lim_{(x,y)\to(0,0)} \frac{x\sin(y)-y\sin(x)}{x^2 + y^2}$$</p>
<p>My calculations: I substitute $y=mx$</p>
<p>\begin{align}\lim_{x\to 0} \frac{x\sin(mx)-mx\sin(x)}{x^2 + (mx)^2} &= \lim_{x\to 0} \frac{x(\sin(mx)-m\sin(x)}{x^2(1 + m^2)}\\ &= \lim_{x\to 0} \frac{1}{1+m^2}\bigg[\frac{\sin(mx)}... | marty cohen | 13,079 | <p>I would just
throw in the
first terms of
the power series
and see what happens.</p>
<p>Since
$\sin(x)
= x-x^3/6+O(x^5)
$,</p>
<p>\begin{align}
\frac{x\sin(y)-y\sin(x)}{x^2 + y^2}
&=\frac{x(y-y^3/6+O(y^5))-y(x-x^3/6+O(x^5))}{x^2 + y^2}\\
&=\frac{xy-xy^3/6+O(xy^5)-yx+yx^3/6+O(x^5y))}{x^2 + y^2}\\
&=\frac... |
1,768,707 | <p>The question is:</p>
<blockquote>
<p>Are there square matrices $A,B$ over $\mathbb{C}$ s.t. $A^2+B^2=AB$ and $BA-AB$ is non-singular?</p>
</blockquote>
<p>From $A^2+B^2=AB$ one could obtain $A^3+B^3=0$. Can we get something from this?</p>
<p><strong>Edit:</strong>
$$A^2+B^2=AB\implies\\
A(A^2+B^2)=A^2B \implie... | Community | -1 | <p>We consider the equations<br />
(1) <span class="math-container">$A^2+B^2=AB$</span>,<br />
(2) <span class="math-container">$A^2+B^2=2AB$</span>.<br />
A couple <span class="math-container">$(A,B)$</span>, solution of (2), is simultaneously triangularizable, and, moreover, when <span class="math-container">$n=2$</s... |
3,324,375 | <p>What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?</p>
<p>In the beginning, when there were just reals, these operations were defined for them. Then, <em>i</em> was created, literally a number whose value is undefined, like e.g. one divided by zero is undefi... | Martín-Blas Pérez Pinilla | 98,199 | <p>Why "adding a new element" <span class="math-container">$i$</span> with <span class="math-container">$i^2 = -1$</span> to <span class="math-container">$\Bbb R$</span> "works", the technical cause.</p>
<p>What means "works": we want a <a href="https://en.wikipedia.org/wiki/Field_extension" rel="noreferrer">field ext... |
3,324,375 | <p>What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?</p>
<p>In the beginning, when there were just reals, these operations were defined for them. Then, <em>i</em> was created, literally a number whose value is undefined, like e.g. one divided by zero is undefi... | Wuestenfux | 417,848 | <p>Well, coming from the algebraic side, consider field extensions of the form
<span class="math-container">$${\Bbb Q}(\sqrt n)=\{a+b\sqrt n\mid a,b\in{\Bbb Q}\},$$</span>
where <span class="math-container">$n\ne0,1$</span> is a square-free integer.
The addition is ''componentwise'',
<span class="math-container">$$(a... |
3,324,375 | <p>What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?</p>
<p>In the beginning, when there were just reals, these operations were defined for them. Then, <em>i</em> was created, literally a number whose value is undefined, like e.g. one divided by zero is undefi... | Draconis | 277,258 | <blockquote>
<p>In the beginning, when there were just reals, these operations were defined for them. Then, i was created, literally a number whose value is undefined, like e.g. one divided by zero is undefined.</p>
</blockquote>
<p>Not quite: in the beginning were the natural numbers, and addition over the natural ... |
1,129,070 | <p>In a recent examination this question has been asked, which says:</p>
<p>$a^2+b^2+c^2 = 1$ , then $ab + bc + ca$ gives = ?</p>
<p>What should be the answer? I have tried the formula for $(a+b+c)^2$, but gets varying answer like $0$ or $0.25$, on assigning different values to variables.</p>
<p><em>How to approach... | Yogeshwar Bala | 231,659 | <p>$$
(a+b+c)^2 = a^2 + b^2 +c^2 + 2(ab + bc + ca)
$$</p>
<p>The LHS of the above identity is a perfect square, hence it is always positive or 0.</p>
<p>Thus,
$$
a^2 + b^2 +c^2 + 2(ab + bc + ca) ≥ 0
$$</p>
<p>It is given that $a^2 + b^2 +c^2 =1 $ .
$$
1 + 2(ab + bc + ca) ≥ 0
$$
Therefore,
$$
ab + bc + ca ≥ -1/2
$$
... |
1,714,965 | <p>I'm wondering, is the function $f=(\sin{x})(\sin{\pi x})$ is periodic?</p>
<p>My first inclination would be two assume that if the periods of the individual sine expressions, $p_1 \text{and}\space p_2$ have the quality that $p_1 \times a = p_2 \times b$ where $a \space\text{and}\space b$ are integers, then the enti... | Henricus V. | 239,207 | <p>Using the product-to-sum formula,
$$ \sin x \sin\pi x = \frac{1}{2} \left( \cos ((1-\pi)x) - \cos((1 + \pi)x) \right)
$$
but $\frac{1-\pi}{1+\pi} \not\in \mathbb{Q}$ and $\sin x \sin \pi$ is continuous, so this function is not periodic.</p>
|
3,086,741 | <p>By transforming to polar coordinates, show that</p>
<p><span class="math-container">$$\int_{0}^{1} \int_{0}^{x}\frac{1}{(1+x^2)(1+y^2)} \,dy\,dx$$</span></p>
<p>Is equal to</p>
<p><span class="math-container">$$ \int_{0}^{\pi/4}\frac{\log(\sqrt{2}\cos(\theta))}{\cos(2\theta)} d\theta$$</span></p>
<p>I have tried... | J.G. | 56,861 | <p>Or even easier than omegadot's argument, note this is the <span class="math-container">$x\ge y$</span> half of <span class="math-container">$\left(\int_0^1\frac{dx}{1+x^2}\right)^2$</span>, i.e. <span class="math-container">$\frac{(\pi/4)^2}{2}=\frac{\pi^2}{32}$</span>.</p>
|
19,797 | <p>I'm trying to compute a multidimensional integral with a variable number of dimensions.</p>
<p>The integral is as follows:
$$
\int d^{3N}\!p~e^{-\frac{\beta}{2m}\vec p^2}.
$$</p>
<p>I have tried this</p>
<pre><code>Integrate[e^(-a*{p1,p2,p3}^2),{{p1,p2,p3}^N,-Infinity,Infinity}]
</code></pre>
<p>but it's not wor... | Jens | 245 | <p>Although the original question is a little too narrow, I'm going to interpret it as asking about general multidimensional integrals in which Gaussians appear together with arbitrary factors in the integrand:</p>
<p>$$\iiint f(\,\vec{r}\,)\,\exp(\,-\frac{1}{2}\vec{r}^T\Sigma^{-1}\vec{r}\,)\,dx\,dy\,dz$$</p>
<p>Here... |
19,797 | <p>I'm trying to compute a multidimensional integral with a variable number of dimensions.</p>
<p>The integral is as follows:
$$
\int d^{3N}\!p~e^{-\frac{\beta}{2m}\vec p^2}.
$$</p>
<p>I have tried this</p>
<pre><code>Integrate[e^(-a*{p1,p2,p3}^2),{{p1,p2,p3}^N,-Infinity,Infinity}]
</code></pre>
<p>but it's not wor... | chris | 1,089 | <p>Stealing from @carl's answer, we can do it for a few dimensions and 'guess' the trend.</p>
<pre><code>tt = Table[
Integrate[Exp[-(\[Beta]/(2 m)) p.p], p \[Element] FullRegion[i],
Assumptions -> \[Beta] > 0 && m > 0], {i, 2, 10}]
</code></pre>
<p><a href="https://i.stack.imgur.com/NMJSM.png" ... |
4,610,892 | <p>Find all <strong>positive integers</strong> <span class="math-container">$x, y, z$</span> such that <span class="math-container">$x^2y+y^2z+z^2x = 3xyz$</span>.</p>
<blockquote>
<p>I first tried to solve it by spilting <span class="math-container">$xyz$</span> to every expression and factor it. But it fails.</p>
</b... | lone student | 460,967 | <p>By <a href="https://en.m.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means" rel="noreferrer">AM-GM</a> inequality, you have:</p>
<p><span class="math-container">$$x^2y+y^2z+z^2x=3xyz≥3xyz$$</span></p>
<p>Equality holds iff, when <span class="math-container">$x^2y=y^2z=z^2x\thinspace.$</span></p>
<p>Thu... |
100,910 | <p>A central advantage of cohomology theory over homology -- at least in terms of richness of structure and strength as an invariant -- is the additional ring structure from the cup product. Recall that this arises from applying the cohomology functor to the following inclusion map of topological spaces $$X \hookrighta... | Peter May | 14,447 | <p>I am disturbed by this question. The literature of algebraic topology has abounded in use of the product in homology for more than the half century that I've been working in it. Even a cursory knowledge of the subject makes that clear. Shouldn't some effort be made to know just a tiny bit about a subject before as... |
2,177,581 | <p>Let $S$ be that part of the surface of the paraboloid $z=x^2+y^2$ between the planes $z=1$ and $z=4$.</p>
<p>Now given $\vec{V}=x^3\hat j+z^3\hat k$ and evaluate the line integrals $\int_{C}{_1}\vec{V}.dr+\int_{C}{_2}\vec{V}.dr$ where $C_1$ and $C_2$ are the curves bounding $S$</p>
<p>I know the answer should be $... | Filipe Nóbrega | 416,362 | <p>One can prove that if $T: V \to W$ is not surjective then $T^*: W^* \to V^*$ is not one-to-one.</p>
<p>Consider such a $T$. We know that $imT \neq W$, so there is a non-trivial linear subspace $U \subset W$ that is not reached by T. Consider $f \in W^*, f \neq 0$ such that $f|_{imT}: imT \to \mathbb{R}$ satisfies $... |
2,574,922 | <p>I'm sorry this is really basic. I'm terrible with math and I'm struggling to help my son with homework:</p>
<p>Find an expression in terms of n for the nth term in this sequence:
$0 , 9 , 22 , 39, 60, \ldots$</p>
<p>We can get the $2n^{2}$ part of the answer but just can't get the rest. Any help gratefully appreci... | heropup | 118,193 | <p><strong>Every quadratic sequence has successive differences in arithmetic progression.</strong> That is to say, if the original terms are $$a_0, a_1, a_2, a_3, \ldots,$$ then the sequence defined by $$b_k = a_{k+1} - a_k, \quad k = 0, 1, 2, \ldots$$ is an arithmetic progression, which in turn implies $$c_k = b_{k+1... |
2,574,922 | <p>I'm sorry this is really basic. I'm terrible with math and I'm struggling to help my son with homework:</p>
<p>Find an expression in terms of n for the nth term in this sequence:
$0 , 9 , 22 , 39, 60, \ldots$</p>
<p>We can get the $2n^{2}$ part of the answer but just can't get the rest. Any help gratefully appreci... | John | 7,163 | <p>Another way to get the answer is to see what you have left.</p>
<p>If you suspect that you have the $2n^2$ part, then write that out:</p>
<p>$$0, 2, 8, 18, 32$$</p>
<p>Subtract this sequence from what you have to get</p>
<p>$$(0-0), (9-2), (22-8), (39-18), (60-32) \\ \to 0, 7, 14, 21, 28$$</p>
<p>Now it's fairl... |
506,499 | <p>If $x$ is fixed within $[0,1]$, the limit of $n^3x^n(1-x)$ as n tends to infinity is $0$.</p>
<p>But how do I show this? $n^3$ tends to inifinity, but the other term tends to zero, and surely I cannot simply assume that their product tends to $0$?</p>
<p>And if I reason that because $x^n$ dominates $n^3$ therefore... | Dan | 79,007 | <p>A zero divisor of a ring $R$ is an element $r \neq 0$ of $R$ which satisfies $rs = 0$ for some nonzero $s$ in $R$. So you haven't answered question (1). Yes, there is only one $0$ in $R$ (this is part of the definition of a ring), but some of the $r \neq 0$ in $R$ might still be zero divisors.</p>
<p>Here's a hint:... |
506,499 | <p>If $x$ is fixed within $[0,1]$, the limit of $n^3x^n(1-x)$ as n tends to infinity is $0$.</p>
<p>But how do I show this? $n^3$ tends to inifinity, but the other term tends to zero, and surely I cannot simply assume that their product tends to $0$?</p>
<p>And if I reason that because $x^n$ dominates $n^3$ therefore... | D Left Adjoint to U | 26,327 | <p>If $a$ is not a zero divisor then $a^k = 1$ has a solution. Conversely if $a$ is a unit it is not a zero divisor. So $a$ is a zero divisor iff it's not a unit. So take $|R| - |$ <em>units of R</em> $|. \ $ $a$ is a unit iff $\gcd(a, |R|) = 1$.</p>
<p>Since $S$ contains $1$, $RS = R$. So the problem becomes cou... |
2,568,860 | <p>How to determine whether the given real number $\alpha =3-\sqrt[5]{5}-\sqrt[5]{25}$ is algebraic or not. And, $[\mathbb{Q}(\alpha):\mathbb{Q}]=?$</p>
<p>Let $x=3-\sqrt[5]{5}-\sqrt[5]{25}$,
\begin{align*}
& x = 3-\sqrt[5]{5}-\sqrt[5]{25}\\
\implies & (x-3) = -\sqrt[5]{5}-\sqrt[5]{25}\\
\implies & (x-3)... | Angina Seng | 436,618 | <p>$$\alpha\pmatrix{1\\5^{1/5}\\5^{2/5}\\5^{3/5}\\5^{4/5}}
=\pmatrix{3&-1&-1&0&0\\0&3&-1&-1&0\\0&0&3&-1&-1\\-5&0&0&3&-1\\-5&-5&0&0&3}
\pmatrix{1\\5^{1/5}\\5^{2/5}\\5^{3/5}\\5^{4/5}}
$$
so that $\alpha$ is an eigenvalue of
$$\pmatrix{3&-... |
2,452,783 | <p>How to prove that $(a^2 + 1)(b^2 + 1)(c^2 + 1) \ge (a + b)(b + c)(c + a)$ for $a, b, c \in \mathbb{R}$ ? I have tried AM-GM but with no effect.</p>
| Donald Splutterwit | 404,247 | <p>\begin{eqnarray*}
(abc-1)^2 + \frac{1}{2}\sum_{perms} a^2(b-1)^2 \geq 0.
\end{eqnarray*}</p>
|
2,794,945 | <p>Let's say there's a 30% chance of some event happening, and if it happens then there's a 30% chance of it happening again (but it can only occur twice). I want to calculate the expected value for the number of times it happens. I think I can do this:</p>
<p>Chance of 0 occurences: 0.7</p>
<p>Chance of only 1 occur... | Arnaud Mortier | 480,423 | <p>Call $X$ the number of times your event occurs. Assume first that the number of occurrences is theoretically unlimited. Then $X+1$ follows a <a href="https://en.wikipedia.org/wiki/Geometric_distribution" rel="nofollow noreferrer">geometric distribution</a> of parameter $1-p$: $$\Bbb P(X+1=n)=p^{n-1}(1-p)\qquad \text... |
3,100,738 | <p>Let (A,☆) be a semi-group such that the following 2 conditions are true:</p>
<ol>
<li><p>For any <span class="math-container">$a,b\in A$</span>, there exists a <span class="math-container">$x\in A$</span> such that <span class="math-container">$a☆x=b$</span></p></li>
<li><p>For any <span class="math-container">$a,b... | P Vanchinathan | 28,915 | <p>You can first use intuition from polynomials where coefficients are from the <em>real numbers</em>. Being every where continuous the image of <span class="math-container">$[-1,1]$</span> is a compact set, which is bounded for a polynomial. But the image of the same set under <span class="math-container">$f(x)= \f... |
3,100,738 | <p>Let (A,☆) be a semi-group such that the following 2 conditions are true:</p>
<ol>
<li><p>For any <span class="math-container">$a,b\in A$</span>, there exists a <span class="math-container">$x\in A$</span> such that <span class="math-container">$a☆x=b$</span></p></li>
<li><p>For any <span class="math-container">$a,b... | Robert Lewis | 67,071 | <p>The short answer, as pointed out by Randall in his comment, is that polynomials are by definition sums of terms of the form <span class="math-container">$ax^k$</span> where <span class="math-container">$k \ge 0$</span>; since <span class="math-container">$x^{-1}$</span> is not of this type, it is not polynomial. Th... |
2,771,081 | <p>In a previous question I described <a href="https://math.stackexchange.com/q/2767118/121988"><span class="math-container">$n$</span>-robot walks and <span class="math-container">$(i,j)$</span>-paths</a>:</p>
<blockquote>
<blockquote>
<p>A [<span class="math-container">$5$</span>-]robot moves in a series of one-fifth... | Asinomás | 33,907 | <p>Let me provide a method for calculating the number of permutations in <span class="math-container">$S_{n}$</span> that contain a <span class="math-container">$k$</span>-cycle when <span class="math-container">$2k>n$</span>.</p>
<p>Clearly at most one of these cycles can exist, we can select it in <span class="ma... |
2,771,081 | <p>In a previous question I described <a href="https://math.stackexchange.com/q/2767118/121988"><span class="math-container">$n$</span>-robot walks and <span class="math-container">$(i,j)$</span>-paths</a>:</p>
<blockquote>
<blockquote>
<p>A [<span class="math-container">$5$</span>-]robot moves in a series of one-fifth... | dtldarek | 26,306 | <p>The answer of @Acccumulation already mentions it, but I think it needs more emphasis: <strong>dependence</strong> is the key. Assuming the permutation of numbers is completely random, it does not matter which half of the drawers the first prisoner opens – as long as these are 50 different drawers (opening the same d... |
5,628 | <p>In <a href="http://www.ams.org/notices/200501/fea-grossman.pdf" rel="noreferrer">this 2005 Notices article</a>, Jerold Grossman tracks the proportion of papers in Math Reviews with 1, 2, 3, and >3 authors over time. His data set ends in 1999. I seem to recall reading that in 200k, for some value of k, the number o... | Andrew Stacey | 45 | <p>According to the article, the original data was provided by the AMS. I don't think that the AMS leaves this sort of data lying around on laptops on trains, so do to it again you'd have to go and ask them. I suspect that, quite reasonably, the AMS likes to know what uses their data is put to.</p>
<p>On the other h... |
3,099,652 | <p>Can anyone help me with this!?
If <span class="math-container">$n=p_1^{k_1},p_2^{k_2},\ldots $</span>
Then I applied the given condition of divisibility of <span class="math-container">$\varphi(n)$</span> but can't reach to a conclusion.</p>
| Akash Patel | 638,127 | <p>As you're in high school, you have probably not covered the topic of differential equations but you can use one to find the analytical solution to your question. <span class="math-container">$dy/dx + y = 0$</span> and you will find that the solution is <span class="math-container">$y=ce^{-x}$</span> (where c is any ... |
2,146,252 | <p>How can one prove the following equality?</p>
<p>$$\int_{0}^{1} x^{\alpha} (1-x)^{\beta-1} \,dx = \frac{\Gamma(\alpha+1) \Gamma(\beta)}{\Gamma(\alpha + \beta + 1)}$$</p>
| Zaid Alyafeai | 87,813 | <p>We show that using Convolution </p>
<p>$$\beta(x+1,y+1)=\int^{1}_{0}t^{x}\, (1-t)^{y}\,dt= \frac{\Gamma(x+1)\Gamma
{(y+1)}}{\Gamma{(x+y+2)}}$$ </p>
<p>$$proof$$</p>
<p>Let us choose some functions $f(t) = t^{x} \,\, , \, g(t) = t^y$</p>
<p>Hence we get </p>
<p>$$(t^x*t^y)= \int^{t}_0 s^{x}(t-s)^{y}\,ds $$ </p>
... |
52,145 | <p>I've decided it's time to start learning how to use a computer to do calculations... I've used Singular to some small extent so far, but I want to start relying on computer algebra systems more.</p>
<h3>Question</h3>
<p>Which computer algebra system is best for what, and what is the easiest/most fun(?) way to lear... | J.C. Ottem | 3,996 | <p>I use <a href="http://www.math.uiuc.edu/Macaulay2/" rel="noreferrer">Macaulay2</a> for standard computations in commutative algebra/algebraic geometry, like Gröbner bases, graded free resolutions, Tor/Ext groups etc. There are also a lot of add-on packages you can import for working with say, intersection theory, to... |
1,508,340 | <p>Prove that if $G$ is a group and $a\in G$, then we have $\forall m,n\in\mathbb{Z} $ that
$$a^{m} a^{n} = a^{m+n}.$$</p>
<p>I've proved the case when $m,n>0$ but I'm stuck on how to prove the case when one or both are negative without assuming that $(a^{-1})^n = a^{-n} $. </p>
| A.P. | 65,389 | <p>I assume that you would like to use $(a^{-1})^n = a^{-n}$ by saying that if $n > 0$ then
$$
a^m a^{-n} = a^m (a^{-1})^n = a^{m-n}
$$
by leveraging the case for $m,n > 0$. The problem is that you can't really do this, because in that case you had two powers with the same basis, but usually $a \neq a^{-1}$.</p>
... |
57,498 | <p>I have a list with points coordinates. And I'm trying to traverse it and perform some matrix operations on each point. But I have a problem with storing modified points in the initial list instead of the original points.</p>
<p>Here is the complete entry point example:</p>
<pre><code>(* the matrix of the linear op... | Mr.Wizard | 121 | <p>From your updated example this does what you desire:</p>
<pre><code>N[f.t2.rx.t1.#] & /@ points
</code></pre>
<blockquote>
<pre><code>{
{{100.}, {29.2893}, {0.}, {1.}},
{{155.}, {29.2893}, {0.}, {1.}}
}
</code></pre>
</blockquote>
<p>You can eliminate some redundancy by precomputing the fixed part of that o... |
906,103 | <h1>Context:</h1>
<p>I'm trying to <strong>algebraically</strong> prove that an <strong>open interval</strong> is an <strong>open set</strong>. If I sketch it, as suggested by @rschwieb in this <a href="https://math.stackexchange.com/a/301381/688539">answer</a>, then it seems quite obvious that this is indeed true. But... | vociferous_rutabaga | 164,345 | <p>Here's an alternate way to go about proving that $V_\epsilon(a) \subset (c,d)$ for some $\epsilon >0$: instead of "measuring distance to the edges," you can "measure distance to the center." Let $p$ be the midpoint of $(c,d)$. Then</p>
<p>$$(c,d) = V_\delta(p),$$</p>
<p>where $\delta:= |p-c|=|p-d|.$ Given $a \i... |
906,103 | <h1>Context:</h1>
<p>I'm trying to <strong>algebraically</strong> prove that an <strong>open interval</strong> is an <strong>open set</strong>. If I sketch it, as suggested by @rschwieb in this <a href="https://math.stackexchange.com/a/301381/688539">answer</a>, then it seems quite obvious that this is indeed true. But... | Ishfaaq | 109,161 | <p>Choose $\epsilon \lt \mathrm{min} \{ a-c,d-a \}$.</p>
<p>Picture it geometrically by drawing a real line. $ |x - a| \lt \epsilon $ represents all points on the line that are $\epsilon$-distant from the point $a$. By picking $\epsilon \lt \mathrm{min} \{ a-c,d-a \}$ what you do is to pick the smallest distance from ... |
4,476,130 | <p>Is C following an identity matrix?</p>
<p><span class="math-container">$$A = \begin{bmatrix}
2& -1 & 12 \\
3 & 6 & -9\\
1& 1& 3
\end{bmatrix}$$</span></p>
<p><span class="math-container">$$B = \begin{bmatrix}
9 & 5 & -21 \\
-6 & -2 & 18\\
-1& -1& 5
\end{b... | Greg Martin | 16,078 | <p>The fatal flaw is in this line:</p>
<blockquote>
<p>the smallest closed set (interval) <span class="math-container">$A$</span> s.t <span class="math-container">$D \subset A $</span> is <span class="math-container">$\mathbb{R}$</span>, therefore <span class="math-container">$\overline D = \mathbb{R}$</span></p>
</blo... |
3,624,267 | <p>Let <span class="math-container">$f$</span> be an entire function which is not a polynomial, and <span class="math-container">$a \neq b$</span>. Can <span class="math-container">$f^{-1}(\{a,b\})$</span> be a finite set? </p>
| Robert Israel | 8,508 | <p>Hint: the
Great Picard Theorem.</p>
|
3,995,483 | <p>I am trying to understand what is the best approach to solve this binary programming problem</p>
<p><span class="math-container">$$ \max_{X\in \left\{0,1 \right\}^{N}} \left(\sum_{i=1}^{N} a_{i}X_{i}\right)^{\beta} - \sum_{i=1}^{N}c_{i}X_{i} $$</span></p>
<p>with <span class="math-container">$X\in \left\{0,1 \right\... | prubin | 458,896 | <p>You could perform no-frills branch-and-bound on the problem. Let level 0 contain just the root node. At each node in level <span class="math-container">$i$</span>, you create one child node with <span class="math-container">$X_{i+1} = 1$</span> and one with <span class="math-container">$X_{i+1} = 0$</span>. Suppose ... |
2,644,104 | <p>If $\tan^{-1} \left(\dfrac {\sqrt {1+x^2} - \sqrt {1-x^2}}{\sqrt {1+x^2} + \sqrt {1-x^2}}\right) = \alpha$ then prove that: $x^2= \sin (2\alpha) $</p>
<p>My Attempt:
$$\tan^{-1} \left(\dfrac {\sqrt {1+x^2}-\sqrt {1-x^2}}{\sqrt {1+x^2} + \sqrt {1-x^2}}\right) =\alpha$$
$$\dfrac {\sqrt {1+x^2}-\sqrt {1-x^2}}{\sqrt {... | adfriedman | 153,126 | <p>Let $A=\sqrt{1+x^2}$ and $B=\sqrt{1-x^2}$, then
$$\tan(\alpha) = \frac{A-B}{A+B}, \quad A^2+B^2=2,\quad A^2-B^2 = 2x^2$$
and also note
$$\tan(a) = \underbrace{\frac{\sin(2a)}{2\sin(a)\cos(a)}}_{=\,1} \cdot \frac{\sin(a)}{\cos(a)}= \sin(2a)\cdot\frac{1}{2}\sec^2(a) = \sin(2a) \frac{1}{2} \left(1+\tan^2(a)\frac{}{}\ri... |
325,192 | <p>I was stunned when I first saw the article <a href="http://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/home.html" rel="noreferrer">Counterexample to Euler's conjecture on sums of like powers</a> by L. J. Lander and T. R. Parkin:.</p>
<p><a href="https://i.stack.imgur.com/VQL6D.png" rel="noreferrer"... | Reid Barton | 126,667 | <p>Even simply generating all quadruples <span class="math-container">$(a, b, c, d)$</span> with <span class="math-container">$1 \le a \le b \le c \le d \le 133$</span> should work fine. There are only about 13 million such quadruples. For each, we need to add together the fifth powers (which can be looked up in a smal... |
325,192 | <p>I was stunned when I first saw the article <a href="http://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/home.html" rel="noreferrer">Counterexample to Euler's conjecture on sums of like powers</a> by L. J. Lander and T. R. Parkin:.</p>
<p><a href="https://i.stack.imgur.com/VQL6D.png" rel="noreferrer"... | DrBRE | 137,565 | <p>The longer and more detailed paper reveals that where possible, various modulo constraints were used to shrink the number of tuples to search through further:</p>
<p>L. J. Lander; T. R. Parkin; J. L. Selfridge (1967). "A Survey of Equal Sums of Like Powers". Mathematics of Computation. 21 (99): 446–459. doi:10.1090... |
1,170,477 | <p>Show that if $A$ is an $m \times n$ matrix and $A(BA)$ is defined, then $B$ is an $n \times m$ matrix.</p>
<p>I know that $A$ is a $m \times n$ matrix and to be able to multiply $B$ with $A$, $B$ must be a $n \times m$ matrix. I am confused though because I can't just assume that. </p>
| sardoj | 101,327 | <blockquote>
<p>I am confused though because I can't just assume that. </p>
</blockquote>
<p>Of course you can assume that! That is how matrix multiplication is <em>defined</em>. That reasoning is sufficient to deduce to that $B$ must have $m$ columns, but you need to take into account the multiplication of $BA$ by ... |
3,386,218 | <p>I was trying to get a better understanding for e and pi, and came across Alon Amit's explanation here: <a href="https://www.quora.com/q/bzxvjykyriufyfio/What-is-math-pi-math-and-while-were-at-it-whats-math-e-math" rel="nofollow noreferrer">https://www.quora.com/q/bzxvjykyriufyfio/What-is-math-pi-math-and-while-were-... | Oscar Lanzi | 248,217 | <p>You don't really have to set <span class="math-container">$f(0)=1$</span> to retrieve <span class="math-container">$e$</span>. Any nonzero value of <span class="math-container">$f(0)$</span> works quite well if you then render <span class="math-container">$e$</span> as the ratio <span class="math-container">$f(1)/f... |
3,981,000 | <p>I have a set of points forming a polygon. However, any 3 points in this polygon can also be represented as an arc (starting at point 1, through point 2, to point 3).</p>
<p><a href="https://i.stack.imgur.com/Tz9L6.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Tz9L6.png" alt="Polygon with Arcs" /... | lhf | 589 | <p><em>Hint:</em> Find a parametrization of the boundary and use Green's theorem.</p>
|
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