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 |
|---|---|---|---|---|
255,660 | <p>I want to get a solution of a equation using NSolve.</p>
<p><span class="math-container">$BesselI[1,x]/(x*BesselI[0,x])=0.2$</span></p>
<p>So I plugged this equation to NSolve:</p>
<pre><code>NSolve[BesselI[1,x]/(x*BesselI[0,x])==0.2, x]
</code></pre>
<p>But when I use this, the Mathematica gives the same expression... | Ulrich Neumann | 53,677 | <p>Often the solver of <code>NMinimize</code>is more robust and calculates one solution</p>
<pre><code>NMinimize[{1, BesselI[1, x]/(x*BesselI[0, x]) == 0.2}, x]
(*{1., {x -> -4.38412}}*)
</code></pre>
<p>without restriction.</p>
<p>Second solution follows to</p>
<pre><code>NMinimize[{1, BesselI[1, x]/(x*BesselI[0, x... |
1,706,939 | <p>Can anyone share an easy way to approximate $\log_2(x)$, given $x$ is between $0$ and 1?</p>
<p>I'm trying to solve this using an old fashioned calculator (i.e. no logs)</p>
<p>Thanks!</p>
<p>EDIT: I realize that I stepped a bit ahead. The x comes in the form of a fraction, e.g. 3/8, which is indeed between 0 and... | ax_9 | 564,701 | <p>$$\log(2)=\sum_{n=1}^{\infty}\frac{1}{n2^n}$$</p>
<p>$$(1/2)+(1/(2\cdot2^2))+(1/(3\cdot2^3))+(1/(4\cdot2^4))+(1/(5\cdot2^5))+(1/(6\cdot2^6))+(1/(7\cdot2^7))+(1/(8\cdot2^8))+(1/(9\cdot2^9))+(1/(10\cdot2^{10}))=0.6930...$$</p>
<p>Even by hand this is giving a simple procedure of finding $\log(2)$ in base $2$</p>
|
2,602,799 | <p>This is exactly what is written in Walter Rudin chapter 2, Theorem 2.41:</p>
<p>If $E$ is not closed, then there is a point $\mathbf{x}_o \in \mathbb{R}^k$ which is a limit point of $E$ but not a point of $E$. For $n = 1,2,3, \dots $ there are points $\mathbf{x}_n \in E$ such that $|\mathbf{x}_n-\mathbf{x}_o| < ... | M A Pelto | 171,159 | <p>We say that $\mathbf{x}_o$ is a <strong>limit point of</strong> $E \subseteq \mathbb{R}^k$ if and only if for every $\varepsilon>0$ there exists $\mathbf{x}_{\varepsilon} \in E$ such that $\mathbf{x}_{\varepsilon} \in B(\mathbf{x}_o,\varepsilon) \setminus \{\mathbf{x}_o\}$.
A set $E \subseteq \mathbb{R}^k$ is clo... |
1,079,262 | <p>How many permutations exist in the string $ABCDEFG$, starting from the smallest possible combination if the only direction allowed is forward? For example, B is the smallest possible combination in the string $BDEF$. The only direction being forward, $BD, BE, BF$ are larger permutations, etc.</p>
<p>NOTE: You can't... | Jack D'Aurizio | 44,121 | <p>You are making everything a bit too lenghty. Since $e^z$ is an entire function and $\sin^2(z)$ is a periodic function with period $\pi$, the residue is just $e^{k\pi}$ times the residue of $\frac{e^z}{\sin^2 z}$ at $z=0$, that is:
$$\frac{d}{dz}\left.\frac{z^2 e^z}{\sin^2 z}\right|_{z=0}=[z](1+z)=1$$
since both $\fr... |
1,180,854 | <p>I was solving some probability and combination problems and I came across this one,
that I couldn't solve. Any tips appreciated.</p>
<blockquote>
<p>Chance to win with one lottery ticket is <span class="math-container">$\frac13$</span>. How many tickets must be purchased for the probability of winning at least with ... | DeepSea | 101,504 | <p><strong>hint</strong>: $z(1-i) = -1 \to z = \dfrac{-1}{1-i}$. Can you take it further from here?</p>
|
22,340 | <p>Prove that for all natural numbers statement n, statement is dividable by 7 </p>
<p>$$15^n+6$$</p>
<p><strong>Base.</strong> We prove the statement for $n = 1$</p>
<p>15 + 6 = 21 it is true</p>
<p><strong>Inductive step.</strong></p>
<p><em>Induction Hypothesis.</em> We assume the result holds for $k$. That is,... | Apostolos | 1,772 | <p>Observe that $14$ is divisible by 7. Then let $15^k\cdot 15+6=15^k\cdot 14+ 15^k+6$. </p>
|
4,263,629 | <blockquote>
<p>Let <span class="math-container">$A=\{(x,y) \in \Bbb R^2 \mid x \ge 1, 0<y<\frac{1}{x^2}\}$</span>. Show that <span class="math-container">$m_2(A) < \infty$</span> where <span class="math-container">$m$</span> is the Lebesgue measure.</p>
</blockquote>
<p>I now that the integral <span class="ma... | Lázaro Albuquerque | 85,896 | <p>Consider the cover <span class="math-container">$\cup [k, k+1] \times [0, \frac{1}{k^2}]$</span>. The sum of these rectangle's measures is <span class="math-container">$\sum \frac{1}{k^2} < \infty$</span>.</p>
|
194,134 | <p>For some FittedModel, the "BestFitParameters" are given in terms of the symbols used to define the model. </p>
<pre><code>fit = NonlinearModelFit[{10,11,12},a*x+c,{a,c},x];
fit["BestFitParameters"]
</code></pre>
<p>returns <code>{a->1.,c->9.}</code></p>
<p>This can be problematic if I define <code>a</code> ... | Carl Woll | 45,431 | <p>Why not use formal symbols, e.g., <a href="http://reference.wolfram.com/language/ref/character/FormalA" rel="nofollow noreferrer"><code>\[FormalA]</code></a> or <a href="http://reference.wolfram.com/language/ref/character/FormalC" rel="nofollow noreferrer"><code>\[FormalC]</code></a>? These symbols are protected, so... |
3,637,085 | <p>I have found this limit in <a href="https://oeis.org/A019609" rel="nofollow noreferrer">https://oeis.org/A019609</a> and I was wondering how to prove it (if it is actually correct):
<span class="math-container">$$\lim_{n\to\infty} \frac{4n}{a^2_n}=\pi e$$</span>
where
<span class="math-container">$$a_1=0,a_2=1, a_... | Claude Leibovici | 82,404 | <p><span class="math-container">$$a_n=\sum_{i=0}^{n-2}\frac{(-1)^n}{2^i i!}\binom{-3/2}{n-i-2}=(-1)^n \binom{-\frac{3}{2}}{n-2} \, _1F_1\left(2-n;\frac{3}{2}-n;-\frac{1}{2}\right)$$</span>
<span class="math-container">$$\frac {4n}{a_n^2}=\frac{4 n}{\binom{-\frac{3}{2}}{n-2}^2 \,\,\Big[\,
_1F_1\left(2-n;\frac{3}{2}-n... |
2,621,901 | <blockquote>
<p>Let $f:\mathbb{R}\to \mathbb{R}$ be a function
$$f(x)=
\begin{cases}
(x-1)\min(x,x^2)&\text{if $x\geq 0$,}\\
x\min\left(x,\dfrac{1}{x}\right)&\text{if $x< 0$.}
\end{cases}$$
Choose the correct option:</p>
<p>a) $f$ is differentiable everywhere;</p>
<p>b) $f$ is not differentia... | Darío A. Gutiérrez | 353,218 | <p>By definition </p>
<blockquote>
<p>$$f(x) = \mathcal{O}(g(x)) \iff f(x) \leq cg(x)$$</p>
</blockquote>
<p>In your case let $f(N)=\dfrac{1}{N^a}$ with $a \in \mathbb{N} $ and $\,g(N)=\dfrac{1}{\sqrt{N}}$, then
$$\lim_{N\to\infty}\frac{f(N)}{cg(N)} = \frac{\frac{1}{N^a}}{\frac{c}{\sqrt{N}}} = \frac{\sqrt{N}}{cN^... |
2,621,901 | <blockquote>
<p>Let $f:\mathbb{R}\to \mathbb{R}$ be a function
$$f(x)=
\begin{cases}
(x-1)\min(x,x^2)&\text{if $x\geq 0$,}\\
x\min\left(x,\dfrac{1}{x}\right)&\text{if $x< 0$.}
\end{cases}$$
Choose the correct option:</p>
<p>a) $f$ is differentiable everywhere;</p>
<p>b) $f$ is not differentia... | user8469759 | 197,817 | <p>When you say $f(N) = \mathcal{O} \left( g(N) \right)$ this means that $f(N)$ is equal to some function that has the property</p>
<p>$$
f(N) \leq c g(N)
$$</p>
<p>for all $N \geq N_0$, and $c$ some constant. In your case this means that there's an $N_0$ and constant $c$ for which</p>
<p>$$f(N) \leq \frac{c}{\sqrt{... |
3,725,385 | <p><span class="math-container">$\{(x, y, z)\} \space$</span> with <span class="math-container">$\space x + y + z = 0$</span></p>
<p>Working through some problems in a textbook and I'm not very confident about checking if subsets are subspaces. I know that for a subset to be a subspace of <span class="math-container">$... | copper.hat | 27,978 | <p>Well, if <span class="math-container">$x+y+z = 0$</span> then <span class="math-container">$t x+t y +t z = 0$</span>.</p>
<p>If <span class="math-container">$x_k+y_k+z_k = 0$</span> for <span class="math-container">$k=1,2$</span> then
<span class="math-container">$(x_1+x_2)+(y_1+y_2)+(z_1+z_2) = 0$</span>.</p>
|
3,725,385 | <p><span class="math-container">$\{(x, y, z)\} \space$</span> with <span class="math-container">$\space x + y + z = 0$</span></p>
<p>Working through some problems in a textbook and I'm not very confident about checking if subsets are subspaces. I know that for a subset to be a subspace of <span class="math-container">$... | ClassicBeavs | 801,243 | <p>With the help of these comments, I now have the answer! The subset IS a subspace of R3.
To check if it is closed under scalar multiplication: If <span class="math-container">$x + y + z = 0$</span>, then the following is true for any scalar multiple: <span class="math-container">$ax + ay + az = 0$</span>
To check for... |
3,733,229 | <p>In a probability space, it is said that a set of events should be <span class="math-container">$\sigma$</span>-algebra, meaning:</p>
<p><a href="https://i.stack.imgur.com/BgP8O.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BgP8O.png" alt="definition of sigma-algebra" /></a>
<em>This is from <a h... | nicomezi | 316,579 | <p>You are right, a <span class="math-container">$\sigma$</span>-algebra does not contain necessarily all the possible outcomes as events but all outcomes are necessarily contained in <span class="math-container">$\Omega$</span>. Ideally, a <span class="math-container">$\sigma$</span>-algebra contains all the things th... |
3,245,796 | <p>Let <span class="math-container">$f$</span> have a continuous second derivative. Prove that</p>
<p><span class="math-container">$$f(x) = f(a) + (x - a)f'(a) + \int_a^x(x - t)f''(t) dt.$$</span></p>
<p>This is a modification of exercise 6.6.4 from Advanced Calculus by Fitzpatrick. I have seen that this question has... | Olba12 | 229,160 | <p>You did a mistake in the integration by parts</p>
<p><span class="math-container">$$
\int_a^x f'(t) dt = \left[t f'(t) \right]_a^x - \int_a^x tf''(t) dt = \\
xf'(x) - af'(a) - \int_a^x tf''(t) dt = \\
(x-a)f'(a) + (f'(x) - f'(a))x - \int_a^x tf''(t) dt = \\
(x-a)f'(a) + x \int_a^x f''(t) dt - \int_a^x tf''(t) dt = ... |
3,245,796 | <p>Let <span class="math-container">$f$</span> have a continuous second derivative. Prove that</p>
<p><span class="math-container">$$f(x) = f(a) + (x - a)f'(a) + \int_a^x(x - t)f''(t) dt.$$</span></p>
<p>This is a modification of exercise 6.6.4 from Advanced Calculus by Fitzpatrick. I have seen that this question has... | trancelocation | 467,003 | <p>Just note that applying partial integration to the integral gives</p>
<p><span class="math-container">\begin{eqnarray*} \int_a^x(x - t)f''(t) dt
& = & \left[(x-t)f'(t) \right]_a^x +\int_a^xf'(t) dt\\
& = & -(x-a)f'(a) + f(x) - f(a)
\end{eqnarray*}</span></p>
<p>Now, solve for <span class="math-cont... |
1,595,118 | <p>What is value of $a+b+c+d+e$? If given :</p>
<p>$$abcde=45$$</p>
<p>And $a,b, c, d, e$ all are distinct integer.</p>
<hr>
<p>My attempt :</p>
<p>I calculated, $45 = 3^2 \times 5$.</p>
<blockquote>
<p>Can you explain, how do I find the distinct values of $a,b, c, d, e$ ?</p>
</blockquote>
| Darth Geek | 163,930 | <p>The divisors of $45$ are:</p>
<p>$$\pm 1 \qquad \pm 3 \qquad \pm 5 \qquad \pm 9 \qquad \pm 15 \qquad \pm 45 $$</p>
<p>The only way to multiply $5$ distict ones and make $45$ is</p>
<p>$$(-1)\cdot 1\cdot(-3)\cdot 3\cdot 5$$
So the sum is $5$.</p>
|
2,749,539 | <p>I have tried for some time to solve this problem and I'm stuck, so any help would be greatly appreciated. I'm not a math guy, so I apologize if I am missing something basic.</p>
<p>I have a function $f(x)$ $$f(x) = \sum_{i=1}^x dr^{i-1}$$ where<br>
$x$ is a positive integer<br>
$d$ is an initial delta and<br>
$r$ ... | Siong Thye Goh | 306,553 | <p>$$f(n)=\sum_{i=1}^n dr^{i-1}=\frac{d(1-r^{n})}{1-r}$$</p>
<p>$$f(n)(1-r)=d(1-r^n)$$</p>
<p>which is an $n$-th degree polynomial.</p>
<p>Hence your problem is equivalent to solving the roots, $r$, of the following:</p>
<p>$$dr^n-f(n)r+f(n)-d=0$$</p>
<p>Note that for positive $r$, $\frac{1-r^n}{1-r}$ is an incr... |
3,043,296 | <p>Prop: For sets A and B, say A ~ B iff there exists a bijection from A to B. Then ~ is an equivalence relation on sets.</p>
<p>I understand that an equivalence relation holds the properties of reflexive, symmetric, and transitive. I am also aware of their definitions, however, I am struggling to write a proof for th... | J.G. | 56,861 | <p>Since <span class="math-container">$f(x)=x$</span> bijects <span class="math-container">$A$</span> to <span class="math-container">$A$</span>, <span class="math-container">$\sim$</span> is reflexive. Write a similar proof <span class="math-container">$\sim$</span> is symmetric, using the fact bijections have inverse... |
4,066,265 | <p>Consider the perpetuity that pays <span class="math-container">$3$</span> by the end of the 2nd year and then every <span class="math-container">$4$</span> years. I need to calculate the present value of it when <span class="math-container">$i=0.05$</span>. So the second payment will be at 6th year. I thought about ... | Billy J. | 901,888 | <p>Your answer is correct. However, the perpetuity formula works for the second term.</p>
<p><span class="math-container">$$PV = 3v^2 + v^2\cdot \frac{3}{1.05^4-1} = 15.348$$</span></p>
<p>The second term is the perpetuity formula discounted the first two years. Saves from using the infinite sum.</p>
|
4,066,265 | <p>Consider the perpetuity that pays <span class="math-container">$3$</span> by the end of the 2nd year and then every <span class="math-container">$4$</span> years. I need to calculate the present value of it when <span class="math-container">$i=0.05$</span>. So the second payment will be at 6th year. I thought about ... | heropup | 118,193 | <p>This is just a deferred perpetuity-due with deferral period of <span class="math-container">$2$</span> years, and periodic payments of <span class="math-container">$4$</span> years: <span class="math-container">$$\require{enclose} PV = 3v^2 \ddot a_{\enclose{actuarial}{\infty} j} = 3\frac{1+j}{(1+i)^2 j}$$</span> w... |
1,528,235 | <p>Recall that <a href="http://en.wikipedia.org/wiki/Tetration" rel="noreferrer">tetration</a> ${^n}x$ for $n\in\mathbb N$ is defined recursively: ${^1}x=x,\,{^{n+1}}x=x^{({^n}x)}$. </p>
<p>Its inverse function with respect to $x$ is called <a href="http://en.wikipedia.org/wiki/Tetration#Super-root" rel="noreferrer">s... | Gottfried Helms | 1,714 | <p><em>I have some observations, from where someone more experienced might be able to derive the proof - maybe this is helpful)</em> </p>
<p>Let the iterated functional root ("superroot of order") $B(z,n)$ (which finds the " <strong><em>B</em></strong> "ase of the powertower) be defined as
$$ \;^n b = z ... |
2,097,584 | <blockquote>
<p>How many ways can the number $2160$ be written as a product of factors which are relatively prime to each other?</p>
</blockquote>
<p>I was confused by this question because couldn't we just add $1$ into the factorization every time? For example, $2160$ and $2160 \cdot 1$ would count as distinct fact... | Med | 261,160 | <p>If the number is $n=p_1^{\alpha_1}p_2^{\alpha_2}...p_n^{\alpha_n}$, you can forget about the powers of the primes and find the number of partitions of the set $\{p_1,p_1,...,p_n\}$</p>
|
64,406 | <p>It's often said that there are only two nonabelian groups of order 8 up to isomorphism, one is the quaternion group, the other given by the relations $a^4=1$, $b^2=1$ and $bab^{-1}=a^3$. </p>
<p>I've never understood why these are the only two. Is there a reference or proof walkthrough on how to show any nonabelian... | Jyrki Lahtonen | 11,619 | <p>A very down-to-Earth approach might be:</p>
<p>Let $G$ be a group of order 8.</p>
<p>Exercise 1: Show that the maximal order $m$ of an element $x$ of $G$ is either 2, 4, or 8.</p>
<p>Exercise 2: Show that if $m=2$, then the group $G$ is abelian.</p>
<p>Exercise 3: Show that if $m=8$, then the group $G$ is abelia... |
39,476 | <p>Fold is an extension of Nest for 2 arguments. How does one extend this concept to multiple arguments. Here is a trivial example:</p>
<pre><code>FoldList[#1 (1 + #2) &, 1000, {.01, .02, .03}]
</code></pre>
<p>Say I want do something like:</p>
<pre><code>FoldList[#1(1+#2)-#3&,1000,{.01,.02,.03},{100,200,3... | Mr.Wizard | 121 | <p>To achieve the specific syntax you requested we can use something like this:</p>
<pre><code>multiFoldList[f_, start_, args__List] :=
FoldList[f @@ Prepend[#2, #] &, start, {args}\[Transpose]]
</code></pre>
<p>Example:</p>
<pre><code>multiFoldList[#1 (1 + #2) - #3 &, 1000, {.01, .02, .03}, {100, 200, 300... |
21,752 | <blockquote>
<p>"Let $P$ be the change-of-basis matrix
from a basis $S$ to a basis $S'$ in a
vector space $V$. Then, for any vector
$v \in V$, we have $$P[v]_{S'}=[v]_{S}
\text{ and hence, } P^{-1}[v]_{S} =
[v]_{S'}$$</p>
<p>Namely, if we multiply the coordinates
of $v$ in the original b... | Ztysjdjdksksmdm | 572,417 | <p>My belief is that this is merely convention. It is indeed natural to call matrix <span class="math-container">$P$</span> in <span class="math-container">${P[v]}_{s'}=[v]_s$</span> the change-of-basis matrix from a basis <span class="math-container">$S'$</span> to a basis <span class="math-container">$S$</span>, rath... |
4,398,864 | <p>How many ways are there to select a three digit number <span class="math-container">$\underline{A}\ \underline{B}\ \underline{C}$</span> so that <span class="math-container">$A \neq B$</span>, <span class="math-container">$B \leq C$</span>, and <span class="math-container">$A < C$</span>?</p>
<hr />
<p>I found th... | Mor A. | 475,545 | <p>Let us split this into two cases: <span class="math-container">$A<B$</span> and <span class="math-container">$B<A$</span>.</p>
<p>To count the number of solutions with <span class="math-container">$A<B$</span> we have <span class="math-container">$0<A<B\leq C\leq 9$</span>, so we instead pick <span cl... |
49,339 | <p>Are there any known non-slow methods for solving diophantine systems?</p>
<p>I can't find books of mathematics that appear methods explaining how to solve diophantine systems in a manner "not slow", e.g. not force brute enumeration.</p>
| h10 | 11,556 | <p><a href="http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem" rel="nofollow">http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem</a></p>
|
1,731,978 | <p>For two complex numbers $z_1$ and $z_2$, it is given that: </p>
<blockquote>
<p>$$|z_1+z_2|>|z_1-z_2|$$</p>
</blockquote>
<p>How could we prove that $-\frac{\pi}{2}<arg\big(\frac{z_1}{z_2}\big)<\frac{\pi}{2}$</p>
<p>If I take $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ I get $x_1x_2+i y_1y_2=0$ but it does help... | Archis Welankar | 275,884 | <p>Hint we can write $arg(z_1/z_2)=arg(z_1)-arg(z_2)=\tan^{-1}(y_1/x_1)-\tan^{-1}(y_2/x_2)$ so now by inverse trigo we have $arg(z_1)-arg(z_2)=\tan^{-1}(\frac{x-y}{1+xy})$ so now we know that range of $\tan^{-1}(l)$ is $(\frac{-\pi}{2},\frac{\pi}{2})$ also note that here $xy>-1$ you can now prove it for other cases ... |
1,137,336 | <p>This is from Discrete Mathematics and its applications
<img src="https://i.stack.imgur.com/jjJiF.png" alt="enter image description here"></p>
<p>I was able to get sum pretty easy. </p>
<p>I am trying to follow this example in the book to get the product of the two binary numbers
<img src="https://i.stack.imgur.c... | Daniel W. Farlow | 191,378 | <p>One very easy way to get your desired product is to convert both numbers to base $10$ and <em>then</em> to multiply them and <em>then</em> to express that product in base $2$. It's not the cleanest thing to do in the world, but neither is your approach. Consider that
$$
(1000111)_2 = 1(2^6)+1(2^2)+1(2)+1 = 71,
$$
an... |
35,321 | <p>I need to do some simplification of an expression involving averages over a stochastic variable (in order to verify a long analytical calculation).
The easiest way to do that, I figured, were if I could implement an operator which would basically be short-hand for the averaging procedure, with all the appropriate p... | ybeltukov | 4,678 | <p>Just define exactly what you want:</p>
<pre><code>av[expr_] := Integrate[f[x] expr, {x, -∞, ∞}];
</code></pre>
<p>Here <code>f[x]</code> is undefined function.</p>
<pre><code>D[av[Exp[-x y]], y]
</code></pre>
<p><img src="https://i.stack.imgur.com/KzuIb.png" alt="enter image description here"></p>
<pre><code>D[... |
3,446,084 | <p>Suppose we have coprime integers <span class="math-container">$(a,b)$</span> and let <span class="math-container">$\ell \in \mathbb{Z}$</span> be arbitrary. The general solution to the linear Diophantine equation <span class="math-container">$ax+by=\ell$</span> is given by <span class="math-container">$x=\ell x' + b... | RobPratt | 683,666 | <p>You can solve this problem via integer linear programming as follows. Let decision variable <span class="math-container">$z$</span> represent <span class="math-container">$\max(|x|,|y|)$</span>, to be linearized. The problem is to minimize <span class="math-container">$z$</span> subject to:
<span class="math-contain... |
2,183,321 | <p>I have a proof of the fact that an abelian group $I$ is injective $\iff$ it is divisible.</p>
<p>If $I$ is injective, then applying the definition of injective to the inclusion $n\mathbb{Z} \hookrightarrow \mathbb{Z}$ and the homomorphism $n\mathbb{Z}\rightarrow \mathbb{Z}$ taking $nk\mapsto kd$ shows that $I$ is d... | rschwieb | 29,335 | <p>Take the homomorphism $f:n\mathbb Z\to I$ given by $n\mapsto d$.</p>
<p>It extends to $\hat f:\mathbb Z\to I$ such that $\hat f(nk)=f(nk)$.</p>
<p>As $\hat f$ is $\mathbb Z$-linear, you have that $\hat f(nk)=n\hat f(k)$.</p>
<p>When $k=1$, you have $d=f(n)=\hat f(n\cdot 1)=n\hat f(1)$.</p>
<p>Thus the element $\... |
2,183,321 | <p>I have a proof of the fact that an abelian group $I$ is injective $\iff$ it is divisible.</p>
<p>If $I$ is injective, then applying the definition of injective to the inclusion $n\mathbb{Z} \hookrightarrow \mathbb{Z}$ and the homomorphism $n\mathbb{Z}\rightarrow \mathbb{Z}$ taking $nk\mapsto kd$ shows that $I$ is d... | egreg | 62,967 | <p>Consider an injective module $I$ and $x\in I$. Suppose $n>0$ and consider</p>
<ol>
<li><p>the homomorphism $f\colon\mathbb{Z}\to I$, $f(z)=zx$</p></li>
<li><p>the monomorphism $g\colon\mathbb{Z}\to\mathbb{Z}$, $g(z)=nz$</p></li>
</ol>
<p>By injectivity, there exists $h\colon\mathbb{Z}\to I$ such that $hg=f$.</p... |
3,460,843 | <p>I understand that the way to calculate the cube root of <span class="math-container">$i$</span> is to use Euler's formula and divide <span class="math-container">$\frac{\pi}{2}$</span> by <span class="math-container">$3$</span> and find <span class="math-container">$\frac{\pi}{6}$</span> on the complex plane; howeve... | Community | -1 | <p>-3/2arctanx/2 is a primitive function of -3/（x^2+4）.
It is easy to check this because （-3/2arctanx/2）’＝-3/（x^2+4）</p>
|
1,705,159 | <blockquote>
<p>Find necessary and sufficient conditions for a Mobius transformation <span class="math-container">$T(z)=\frac{az+b}{cz+d}$</span> to map the unit circle to itself. So if <span class="math-container">$\gamma$</span> is a circle, <span class="math-container">$T(\gamma)=\gamma$</span>.</p>
<p>I've worked o... | Paul palmer | 613,650 | <p>here we have a unit circle <span class="math-container">$\gamma$</span>. we want to find the sufficient and necessary condition such that <span class="math-container">$T(\gamma)=\gamma$</span>.</p>
<p>we know that for every point <span class="math-container">$z$</span> in the unit circle, <span class="math-containe... |
95,741 | <p>I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to $\mathbb R$. But I am not ok with this answer. I need a simple way to explain the differences between mapping and func... | Nii | 44,288 | <p>By Nii: To my best understanding, mapping is just a process of matching elements of one set to elements of another set. Mapping is not a function unless some conditions are defined. Thus every mapping is a retation but not necessary a function.</p>
|
4,251,332 | <p>Since <span class="math-container">$2^4 \equiv 16 \equiv 1 \pmod 5$</span> then <span class="math-container">$2^{4n} \equiv (2^4)^n \equiv 1^n \equiv 1 \pmod 5$</span> so <span class="math-container">$\frac{2^{4n} - 1}{5} \in \mathbb{Z}$</span></p>
<blockquote>
<p>Let <span class="math-container">$m \in \mathbb{N}$<... | Merosity | 741,168 | <p>I will assume that it doesn't matter if we write our answer in any particular base, since base 2 is much more convenient than base 10 to answer this question. Otherwise, you'll need to look for a different answer.</p>
<p>It helps whenever working modulo a power of a prime, to think in terms of writing your numbers i... |
4,251,332 | <p>Since <span class="math-container">$2^4 \equiv 16 \equiv 1 \pmod 5$</span> then <span class="math-container">$2^{4n} \equiv (2^4)^n \equiv 1^n \equiv 1 \pmod 5$</span> so <span class="math-container">$\frac{2^{4n} - 1}{5} \in \mathbb{Z}$</span></p>
<blockquote>
<p>Let <span class="math-container">$m \in \mathbb{N}$<... | Mastrem | 253,433 | <p>Here's what I think is going on. <span class="math-container">$(2^{4n}-1)/5$</span> is an integer, because
<span class="math-container">$$\frac{2^{4n}-1}{5}=3\cdot \frac{16^n-1}{16-1}=3\sum_{i=0}^{n-1}16^i.$$</span>
However, this integer may not be an element of <span class="math-container">$\{0,1,\ldots,2^m-1\}$</s... |
2,038,323 | <p>I am a first year college student studying linear algebra. </p>
<p>I understand that all linear transformations can be represented by a matrix mapping, and more specifically, the matrix mapping can be constructed by taking the column vectors of the images of the standard basis vectors. However, if the transformatio... | angryavian | 43,949 | <ol>
<li><p>All linear transformations (from $\mathbb{R}^n$ to $\mathbb{R}^m$) can be represented as a ($m \times n$) matrix (as you described).</p></li>
<li><p>Conversely, any matrix represent a linear transformation. (If $A$ is a $m \times n$ matrix, then the transformation sends a vector $v \in \mathbb{R}^n$ to $Av$... |
3,640,307 | <p>The extended reals are taken to be the set <span class="math-container">$\mathbb{R}\cup\{+\infty,-\infty\}$</span>. But is there a <em>natural</em> way to define <span class="math-container">$+\infty$</span> and <span class="math-container">$-\infty$</span> as sets, in a pure set-theoretic theme, following the const... | Noah Schweber | 28,111 | <p>Sure - you can interpret <span class="math-container">$+\infty$</span> as the set <span class="math-container">$S_{+\infty}$</span> of all sequences of rationals whose limit is <span class="math-container">$+\infty$</span> in the usual sense. That is, <span class="math-container">$$S_{+\infty}=\{(a_i)_{i\in\mathbb{... |
3,643 | <p>Is there a quick method to transpose uneven lists without conditionals?</p>
<p>With:</p>
<pre><code>Drop[Table[q, {10}], #] & /@ Range[10]
</code></pre>
<p>Thus the first list would have the first element of all the lists, the 2nd list would have all the 2nd elements of all the lists, etc. If there are no ele... | rm -rf | 5 | <p>Yes, but it is not trivial to comprehend. You would have to use the second argument of <code>Flatten</code> to implement a generalized transpose of uneven lists. For example:</p>
<pre><code>(* Uneven list *)
list = Range ~Array~ 5
Out[1]= {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}}
(* Transpose the lis... |
654,968 | <p>Let $g(x)=x+6$ and $h(x)=\frac{4}{x}$. Compute $\displaystyle\left(\frac{h}{g}\right)(5)$.</p>
<p>I've plugged $5$ in for $x$ but I keep coming up with $.07$ and thanks to webassign I know that is wrong. I'm sure I'm missing something basic but what is it?</p>
| Davide Giraudo | 9,849 | <ul>
<li>A countable union of separable sets is separable (if <span class="math-container">$S_j$</span> is separable, let <span class="math-container">$\left(x^{j}_n\right)_{n\geqslant 1}$</span> be a dense sequence in <span class="math-container">$S_j$</span>; then <span class="math-container">$\left\{x_n^{j},n,j\geqs... |
203,378 | <p>I am trying to solve this equation where I need the solution of K in term of v</p>
<pre><code> Solve[1 -
K - (54 (20 -
K) v (2 (-10 +
K) (5 (1300 - 10 K + 3 K^2 - 100 (2 + K)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
3 + (Sqrt[3] (-10 + K))/Sqrt[... | Alex Trounev | 58,388 | <p>Addendum to the answer @MariuszIwaniuk. It is possible to construct a function for calculating <span class="math-container">$k(v)$</span> with a given <code>WorkingPrecision</code></p>
<pre><code>eq = 1 - k - (108 (20 - k) (-20 + k)^3 v Hypergeometric2F1[-1 -
Sqrt[3], -1 + Sqrt[3],
1, -(10/(-20 + k... |
1,138,212 | <p>I am given $f(x) = 1 + x - \frac{sin(x)}{(x e^x)} $ and am asked to solve this for when x ≃ 0.</p>
<p>I'm doing the following steps but am getting stuck halfway through:</p>
<p>$$f(x) = 1 + x - \frac {x - \frac{x^3}{6} + \frac{x^5}{120}}{xe^x} $$</p>
<p>$$= 1 + x - \frac{e^{-x} (x - \frac{x^3}{6} + \frac{x^5}{120... | Michael Jørgensen | 188,373 | <p>Well, you DO need to expand the numerator. But you only need terms up to x^3 in both factors.</p>
|
194,671 | <p>I'm searching for two symbols - considering they exist - (1) unknown value; (2) unknown probability.</p>
<p><strong>Note</strong>: I thought that $x$ was used in a temporary context, whenever I see it, it remains unknown until an evaluation is made. I was thinking in a "unknown and impossible to be known" context. ... | d4v3y0rk | 40,134 | <p>X,Y,Z unknown or variable quantities R. Descartes 1637</p>
<p>taken from: <a href="https://www.encyclopediaofmath.org/index.php/Mathematical_symbols" rel="nofollow">Encyclopedia of Math</a></p>
|
1,652,929 | <p>The question I'm trying to solve is $$\left(y-4y^6\right)=\left(y^4+5x\right)y'$$ where $y(0)=1$ </p>
<p>I want to find the solution explicitly for $x$. I found the integration factor to be $u=y^-6$. Multiplying the equation by the integrating factor, I get $(y^{-5}-4)+(-y^{-2}-5xy^{-6})y'=0$ and then I solved $\in... | JJacquelin | 108,514 | <p>The mistake is marked on the screen copy below :</p>
<p><a href="https://i.stack.imgur.com/fphPg.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fphPg.jpg" alt="enter image description here"></a></p>
|
2,878,777 | <p>Usually mathematicians consider isomorphic fields as equal fields. That is, if the $(A,+,\cdot)$ is isomorphic to $(B,\oplus,\odot)$, then I can consider those fields as equals. Thinking about it, I thought about the following interpretation:</p>
<p>Let $A$ and $B$ be two sets. I think we can interpret that $A$ and... | Paul Frost | 349,785 | <p>I suppose you know the concept of a <em>category</em>. If not, I strongly recommend to consult any book on this thematic area. Your question suggests that you regard objects of a category $\mathcal{C}$ as <em>essentially equal</em> if they are isomorphic in $\mathcal{C}$.</p>
<p>The category $Set$ has as objects al... |
14,583 | <p>Given a <code>Graph</code> with an automatically computed layout (i.e. not explicitly given <code>VertexCoordinates</code>, but using a <code>GraphLayout</code> method), how can we extract the coordinates of the vertices?</p>
<pre><code>In[]:= g = RandomGraph[{10, 20}, GraphLayout -> "SpringEmbedding"]
Out[]= &l... | cormullion | 61 | <p>Perhaps:</p>
<pre><code>Table[PropertyValue[{g, n}, VertexCoordinates],
{n, 1, VertexCount[g]}]
(*
{{1.93552, 0.76408}, {2.51085, 1.17051}, {1.48194, 1.6304}, {1.90242,
1.64263}, {0.92823, 1.47388}, {2.31252, 0.126716}, {0.,
1.08036}, {1.42818, 0.}, {0.554302, 0.210118}, {1.37128, 0.758598}}
*)
</code></pr... |
1,631,505 | <p>Give the precise meaning of the limit- $$\lim_{x\to-\infty} f(x) = +\infty$$ (x is going to negative infinity, the symbol is hard to see)</p>
<p>I know that as $x$ gets smaller and smaller, $f(x)$ gets larger and larger, but how do I put that in terms of a precise definition?</p>
| Community | -1 | <p>It means that, for all $M\in\Bbb R$ there exists $H\in\Bbb R$ such that for all $x<H$ it holds $f(x)>M$. In symbols:
$$\forall M\in\Bbb R,\exists H\in\Bbb R\ \ \forall x<H,\ f(x)>M$$</p>
|
2,573,492 | <p>In my course "Introduction To Algebraic Topology" I had following test problem:</p>
<blockquote>
<p>Exemplify a topological space with fundamental group $\mathbb{Z}/3\mathbb{Z}$.</p>
</blockquote>
<p>I was supposed to use this theorem:</p>
<blockquote>
<p>Let $Y$ be a simply connected topological space. If a ... | Angina Seng | 436,618 | <p>How about the unit sphere $S^3$? If $A$ is a $2$-by-$2$ rotation matrix
for angle $2\pi/n$, then the block matrix $\pmatrix{A&0\\0&A}$ acts on $S^3$ without fixed points.</p>
|
912,426 | <p>A bag contains six chips, numbered 1 through 6. If two chips are chosen at random without replacement and the values on those two chips are multiplied, what is the probability that this product will be greater than 20?</p>
<p>I tried to solve by counting the total possibilities (36) and solving for 6 choices that w... | Alex | 38,873 | <p>Multiply both numerator and denominator by $(2n)!!$, rewrite the denominator as $n! 2^n$ then use Stirling's formula for factorial and the upper bound. What do you get?</p>
|
478,566 | <p>I'm reading a book about combinatorics. Even though the book is about combinatorics there is a problem in the book that I can think of no solutions to it except by using number theory.</p>
<p>Problem: Is it possible to put $+$ or $-$ signs in such a way that $\pm 1 \pm 2 \pm \cdots \pm 100 = 101$?</p>
<p>My proof... | coffeemath | 30,316 | <p>If $T_n=n(n+1)/2$ is the $n^{th}$ triangular number, an inductive proof (using $T_n+(n+1)=T_{n+1}$) shows the attainable numbers at step $n$ are
$$-T_n,\ -T_n+2,\ \cdots , T_n-2, \ T_n,$$
in particular they all have the same parity as $T_n$. Since $T_{100}=5050$ is even, we see that $101$ cannot be attained in any ... |
478,566 | <p>I'm reading a book about combinatorics. Even though the book is about combinatorics there is a problem in the book that I can think of no solutions to it except by using number theory.</p>
<p>Problem: Is it possible to put $+$ or $-$ signs in such a way that $\pm 1 \pm 2 \pm \cdots \pm 100 = 101$?</p>
<p>My proof... | pre-kidney | 34,662 | <p>Consider both sides modulo $2$. Then the right side is $1$, whereas the left side is $0$ (since it consists of $50$ ones and $50$ zeros).</p>
|
4,253,640 | <p>For example:
suppose we need to find <strong>x</strong> given that <strong>x mod 7 = 5</strong> and <strong>x mod 13 = 8</strong>.</p>
<p><strong>x = 47</strong> is a solution but needs hit and trial.</p>
<p>Is there any shortcut to calculate such number?</p>
| Alessio K | 702,692 | <p>In general, if we are taking the moduli of pairwise coprime integers greater than <span class="math-container">$1$</span>, then you can use the <a href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem" rel="nofollow noreferrer">Chinese remainder theorem</a>.</p>
<p>However for the case above, I will consider... |
233,367 | <p>A set of $m$ non-zero <strong><em>rationals</em></strong> {$a_1, a_2, ... , a_m$} is called a <em><a href="https://web.math.pmf.unizg.hr/~duje/intro.html" rel="nofollow">rational Diophantine $m$-tuple</a></em> if $a_i a_j+1$ is a square. It turns out an $m$-tuple can be extended to $m+2$ if it has certain properties... | Philip Gibbs | 99,093 | <p>There is also an equation for extending quintuples to sextuples. </p>
<p>$(abcde+abcdf+abcef-abdef-acdef-bcdef+2abc-2def+a+b+c-d-e-f)^2 =
4(ab+1)(ac+1)(bc+1)(de+1)(df+1)(ef+1)$</p>
<p>This can be solved for $f$ with two rational roots when $\{a,b,c,d,e\}$ is a rational Diophantine quintuple (except in a few except... |
1,640,678 | <p>How do you find the maximum of a quadratic function? Specifically,
$R(x) = -4x^2 + 4000x$</p>
| Mark Fischler | 150,362 | <p>I assume from your question that you have not had differential calculus, or that this question is posed in a calculus course before coming to the concept of derivatives, which would make it easier.</p>
<p>You find the maximum by the trick of completing the square:</p>
<p>$$R(x) = -4x^2 + 4000x = -4 (x^2 - 1000x) =... |
715,706 | <p>I try to find a partial fraction expansion of $\dfrac{1}{\prod_{k=0}^n (x+k)}$ (to calculate its integral).
After checking some values of $n$, I noticed that it seems to be true that $\dfrac{n!}{\prod_{k=0}^n (x+k)}=\sum_{k=0}^n\dfrac{(-1)^k{n \choose k}}{x+k}$. However, I can't think of a way to prove it. Can someb... | xyzzyz | 23,439 | <p>Notice that:
$$
\frac{1}{x(x+1)\cdots(x+n)} = \frac{1}{n} \frac{(x+n) - x}{x(x+1)\cdots(x+n)} = \frac{1}{n} \left(\frac{1}{x(x+1)\cdots(x+n-1)} - \frac{1}{(x+1)\cdots(x+n)}\right)
$$</p>
<p>This gives you a recursion formula for partial fraction expansion. You can use it to check your conjecture, and prove it by in... |
1,600,428 | <p>Find the equation of a line that passes through the origin, with positive slope, and its tangent to the parabola given by :$ y = x^2 - 2x + 2$</p>
<p>My approach to this problem was to differentiate the equation of the parabola, so I can et an expression, that determines the tangent line anywhere on the parabola.... | Maffred | 279,068 | <p>A generic line is given by $y = ax + b$. </p>
<p>If it passes in $(x=0,y=0)$ you can put this value in your equation and find $0 = a0 + b$, thus $b = 0$, and your line is $y =ax$. </p>
<p>The common points of the line and the parabola are given by $ax = y = x^2 -2x +2$ thus $x$ must solve $ax = x^2 -2x +2$. Rewrit... |
3,915,141 | <p>Let <span class="math-container">$(G,.)$</span> be a group with <span class="math-container">$e$</span> as an identity element. If <span class="math-container">$x\in G$</span> and <span class="math-container">$C_G(x)=\left\{g \in G: gx=xg\right\}$</span> and <span class="math-container">$|x|=5$</span>. In the proces... | hunter | 108,129 | <p>Yes. Here is a lemma. Let <span class="math-container">$G$</span> be a group, and <span class="math-container">$g \in G$</span>. Then for any <span class="math-container">$m$</span>,
<span class="math-container">$$
C(g) \subset C(g^m).
$$</span>
Proof: If <span class="math-container">$y \in C(g)$</span>, then <span... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Paul Siegel | 4,362 | <p>My favorite theorem, the Atiyah-Singer index theorem, seems to have the desired property. The theorem states that the Fredholm index of the Dirac operator on a compact spin manifold $M$ is equal to the $\hat{A}$ genus. There are two essentially different types of proofs: a global, conceptual argument based on litt... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Anweshi | 2,938 | <p>I mentioned this <a href="https://mathoverflow.net/questions/32968/slick-ways-to-make-annoying-verifications/34690#34690">somewhere else</a> too.</p>
<p>Many general statements in algebraic geometry can be proved via direct tedious verification or by abstract thought. In fact, the notions of abstract algebraic varie... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Nick S | 9,313 | <p>I think that Gauss Theorem on constructible polygons fit this category.</p>
<p>For more than 2000 years the actual construction only lead to 4 classes: $2^n; 2^n\cdot 3; 2^n \cdot 5; 2^n \cdot 15$.</p>
<p>Gauss' abstract approach solved the problem. The interesting case $n=17$ becomes easy to understand, and easy ... |
432,208 | <p>I want to grasp the moving frames method but I find some obstacles. I don't know if this question is suitable for MO, if it is not the case please let me know and I will move it.<br />
I am aware there are other related questions here like <a href="https://mathoverflow.net/questions/337294/moving-frames-method-for-n... | A. J. Pan-Collantes | 129,995 | <p>I am going to answer my own question with my conclusion after some reading of Bryant references and other papers I have found.</p>
<p><strong>Question 1</strong></p>
<p>The answer to question 1 is no, there is not an standard way to assign a lift which is equivariant.</p>
<p>Indeed, this lack has given rise to lots ... |
238,128 | <p>Let $G$ be an abelian group. <br/>
Show that $\{x\in{G} | |x| < \infty\}$ is a subgroup of $G$. Give an example of a non-abelian group where this fails to be a subgroup.</p>
| Mikasa | 8,581 | <p>There is another counter example which I think you may find it interesting. Please see Exercise 2.17 of J.J.Rotman's well-known book. There; he gave us $G=GL(2,\mathbb Q)$ such that $tG$ is not a subgroup.. </p>
|
78,569 | <p><img src="https://i.stack.imgur.com/FhX2B.png" alt="Limit of both sides of function"></p>
<p>I need to solve for <code>c</code> such that the function is continuous at <code>x=2</code>. How do I do this automatically?</p>
<p>I have expressions for the limit of both sides of the function as x->2, but how would i u... | djp | 25,325 | <p>Seeing you got your two limits, you could just solve them like this:</p>
<pre><code>sol = First@Solve[8 - 2 c == 4 (1 + c)]
(* c -> 2/3 *)
</code></pre>
<p>and then</p>
<pre><code>myfuncC[x_] := myfunc[x] /. sol
</code></pre>
|
78,569 | <p><img src="https://i.stack.imgur.com/FhX2B.png" alt="Limit of both sides of function"></p>
<p>I need to solve for <code>c</code> such that the function is continuous at <code>x=2</code>. How do I do this automatically?</p>
<p>I have expressions for the limit of both sides of the function as x->2, but how would i u... | Jinxed | 24,763 | <p>Just solve for <code>c</code>, given your <code>Limit</code>-conditions:</p>
<pre><code>cc=Solve[
Limit[myfunc[x], x->2, Direction->-1]
==Limit[myfunc[x], x->2, Direction->1], c]
(* {{c -> 2/3}} *)
</code></pre>
<p>and apply it (<code>myfunc[x]/.First@cc</code>) with the result</p>
<pre... |
2,910,101 | <p>A positive integer X is said to be a cube-loving number if it can be written as $(a^3) \cdot b$, for some positive integers $a$ and $b$ ($a>1$,$b \ge 1$). Given a positive integer $n$, determine the number of Cube-loving numbers less than or equal to $n$.</p>
| Misha Lavrov | 383,078 | <p>The degree distribution of a node is <span class="math-container">$\text{Binomial}(n-1, p)$</span>. Here, the mean is constant, so it is a very skewed binomial distribution (approximately Poisson), and the usual Chernoff bound does not do well in such cases. You can try it, but it will not work.</p>
<p>Instead, we c... |
189,744 | <p>The following procedure can be easily aborted using Evaluation > Abort Evaluation menu item:</p>
<pre><code>Do[j + k, {j, 1, 10000}, {k, 1, 10000}]
</code></pre>
<p>But it is not possible if <code>NotebookEvaluate</code> was used:</p>
<pre><code>nb = CreateDocument[ ExpressionCell[Defer[Plot[Sin[x], {x, 0, 2 Pi}]... | MassDefect | 42,264 | <p>Here is my answer:</p>
<pre><code>myshallow[mat_List, dims_: {20, 20}] :=
Module[{matrix, rows, cols, matrows, matcols, splitrow, splitcol},
If[! And @@ IntegerQ /@ dims,
Return[HoldForm[myshallow[mat, dims]]]];
If[Length[Dimensions[mat]] == 1, matrix = {mat}, matrix = mat];
Switch[
Length[dims],
0... |
2,990,947 | <p>If r stands for counter-clockwise 90 degree rotation, s stands for horizontal flip.
<span class="math-container">$D_4= \{1, r, r^2, r^3, s, rs, r^2s, r^3s\}$</span>. What rule should I apply to find the subgroups of <span class="math-container">$D_4$</span>? Should I just put elements with same order in the same sub... | Travis Willse | 155,629 | <p>The group <span class="math-container">$D_8$</span> is small enough that we can proceed naively, splitting cases according to whether each element occurs within a given subgroup.</p>
<p><strong>Hint</strong> Let <span class="math-container">$H$</span> be a subgroup of <span class="math-container">$D_8$</span>.</p>
... |
3,439,298 | <blockquote>
<p>Let <span class="math-container">$G = (V, E)$</span> be a directed graph. For any subsets <span class="math-container">$X, Y \subseteq V$</span> of vertices, let
<span class="math-container">\begin{align}
E(X, Y ) := \left\{(u, v) \in E \mid u \in X \text{ and } v \in Y \right\}
\end{align}</span>
... | Calvin Lin | 54,563 | <p>Consider the disjoint subsets <span class="math-container">$A\cap B, A-B, B-A, S-A - B$</span>.<br />
Consider an edge that goes from one subset to another distinct subset.<br />
How many times is this edge counted on the LHS?<br />
How many times is this edge counted on the RHS?</p>
<p>Is the first number always la... |
3,439,298 | <blockquote>
<p>Let <span class="math-container">$G = (V, E)$</span> be a directed graph. For any subsets <span class="math-container">$X, Y \subseteq V$</span> of vertices, let
<span class="math-container">\begin{align}
E(X, Y ) := \left\{(u, v) \in E \mid u \in X \text{ and } v \in Y \right\}
\end{align}</span>
... | AlDante | 255,873 | <p>Let <span class="math-container">$C=V-A-B$</span>, i.e. the set of all vertices not in <span class="math-container">$A$</span> or <span class="math-container">$B$</span> and <span class="math-container">$D=A \cap B$</span>, i.e. the set of all vertices in both <span class="math-container">$A$</span> or <span class="... |
3,527,350 | <p>I need to calculate the spectrum of the operator <span class="math-container">$T$</span> for <span class="math-container">$f \in L^2([0,1])$</span> defined by: </p>
<p><span class="math-container">\begin{equation}
(Tf)(x) = \int_0^1 (x+y)f(y)dy.
\end{equation}</span></p>
<p>I know that <span class="math-container"... | Oliver Díaz | 121,671 | <p>Notice that the range of <span class="math-container">$T$</span> is the span of <span class="math-container">$\{1,x\}$</span>, that is is the space of linear functions on <span class="math-container">$[0,1]$</span>. So if <span class="math-container">$f(y)=ay+b$</span> is an eigenvector of <span class="math-containe... |
2,662,554 | <p>I have to use Proof by contradiction to show what if $n^2 - 2n + 7$ is even then $n + 1$ is even. </p>
<p>Assume $n^2 - 2n + 7$ is even then $n + 1$ is odd. By definition of odd integers, we have $n = 2k+1$. </p>
<p>What I have done so far:</p>
<p>\begin{align}
& n + 1 = (2k+1)^2 - 2(2k+1) + 7 \\
\implies &am... | Roby5 | 243,045 | <p>As pointed out bt @Bram28 and others you use contradiction by making the appropriate changes. </p>
<p>However, I suggest that you try to use contrapositive instead of contradiction, whenever possible as it makes fewer assumptions. </p>
<p>The easiest (and cleanest) way to solve this is proving the contrapositive ... |
1,448,213 | <p>In other words, consider $A_n$, the alternating group of the $n$-th symmetrical group $S_n$, is it true that
$$A_n=\{a^2\mid a\in A_n\}$$?
I tested for $S_3$ and it seemed to hold. If it is true, it will be very helpful to me for solving another problem. </p>
| Ian | 83,396 | <p>Let $X$ be a normed vector space and $Y$ be a normed vector space, $Y \neq \{ 0 \}$. If $X$ is infinite dimensional and incomplete, then there exist discontinuous linear maps $L : X \to Y$. Often, these can be explicitly constructed. For instance, if $X=C^1$ with the sup norm and $Y=C^0$ with the sup norm, then $L(f... |
3,806,122 | <p>I tried using Chinese remainder theorem but I kept getting 19 instead of 9.</p>
<p>Here are my steps</p>
<p><span class="math-container">$$
\begin{split}
M &= 88 = 8 \times 11 \\
x_1 &= 123^{456}\equiv 2^{456} \equiv 2^{6} \equiv 64 \equiv 9 \pmod{11} \\
y_1 &= 9^{-1} \equiv 9^9 \equiv (-2)^9 \equiv -512... | Evariste | 239,682 | <p>Your calculations look correct except for the last line which I don't understand.</p>
<p>One you get <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>, you could simply write</p>
<p><span class="math-container">$x=123^{456}=9+11k$</span> (from <span class="math-container">$x_1$<... |
3,806,122 | <p>I tried using Chinese remainder theorem but I kept getting 19 instead of 9.</p>
<p>Here are my steps</p>
<p><span class="math-container">$$
\begin{split}
M &= 88 = 8 \times 11 \\
x_1 &= 123^{456}\equiv 2^{456} \equiv 2^{6} \equiv 64 \equiv 9 \pmod{11} \\
y_1 &= 9^{-1} \equiv 9^9 \equiv (-2)^9 \equiv -512... | J. W. Tanner | 615,567 | <p><span class="math-container">$123^{456}\equiv 2^6=64\equiv9\bmod 11$</span>.</p>
<p><span class="math-container">$123^{456}\equiv 3^0=1\equiv9\bmod 8$</span>.</p>
<p>Therefore, by the constant case of the Chinese Remainder Theorem, <span class="math-container">$123^{456}\equiv9\bmod88$</span>.</p>
|
3,525,621 | <p>Find all integral solutions to the equation <span class="math-container">$x^2 + 4xy - y^2 = m$</span> with <span class="math-container">$-5 \leq m \leq 10$</span>.</p>
<p>I know that I can set <span class="math-container">$m = -5$</span> to <span class="math-container">$m = 10$</span> and solve all of the equations... | Donald Splutterwit | 404,247 | <p><span class="math-container">\begin{eqnarray*}
\ln \left( \frac{1+x}{1-x} \right) = 2 \sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1}.
\end{eqnarray*}</span>
Now invert the order of the sum & integration ( and perform the integrations)
<span class="math-container">\begin{eqnarray*}
\int_0^{1} \frac{1}{x} \ln \left( ... |
2,589,938 | <p>Suddenly I am confused by a very elementary question:</p>
<blockquote>
<p>Let $a, b, c$ be the sides of a triangle. How about this inequality:
$$ a+b > c. $$</p>
</blockquote>
<p>Is it the definition of a triangle or is it a theorem?</p>
| Arthur | 15,500 | <p>Whether it's a theorem or an axiom depends very much on how you approach your geometry. If you approach it from analysis, basing everything on a defined notion of distance, then it's an axiom. (It's one of the relatively few properties we generally require that this "distance", called a <em>metric</em>, fulfills. Al... |
348,532 | <p>Consider the following integral
<span class="math-container">$$
I_\delta(\lambda)=\int_0^\delta e^{i\lambda \exp(-x^{-2})}dx.
$$</span>
Here, <span class="math-container">$\phi(x)=\exp(-x^{-2})$</span> is the phase function. I would like to study the rate of decay of <span class="math-container">$I(\lambda)$</span> ... | Zarrax | 149,955 | <p>Change variables from <span class="math-container">$x$</span> to <span class="math-container">$y = e^{-{1 \over x^2}}$</span>, so that <span class="math-container">$x = (-\ln y)^{-{1 \over 2}}$</span> and <span class="math-container">$dx = {1 \over 2y (-\ln y)^{3 \over 2}}dy$</span>. So the integral becomes
<span cl... |
135,252 | <p>Evaluate $\displaystyle \lim_{n \to +\infty}\sum_{k=n+1}^{2n}\frac{1}{k}$. What are the ways of counting such things? My last topic in school was Riemann integral, can I use it here?</p>
| Wayson Kong | 29,081 | <p>Yes, you can use it.</p>
<p>The technique is
$$\int_{k+1}^{k+2}\frac{1}{x}d x \leq \frac{1}{k+1}\leq \int_{k}^{k+1}\frac{1}{x}d x.$$</p>
|
2,258,139 | <p>A natural number $n>1$ is called <em>good</em> if$$n \mid 2^n+1.$$ For example, $n=3$ is good, as $3 \mid 2^3+1=9$. Prove that if $N_1$ and $N_2$ are good, then:</p>
<ul>
<li>$\mathrm{lcm}(N_1,N_2)$ and $\gcd(N_1,N_2)$ are good,</li>
<li>$N_1\cdot N_2$ is good. </li>
</ul>
<p>This seems pretty difficult for me.... | sanketalekar | 327,455 | <p>You can prove "good" ness pretty easily without knowing much more about <span class="math-container">$N_1 $</span> and <span class="math-container">$N_2$</span>, but I thought I'd share the properties that make numbers "good", and thus you can see that your problem becomes trivial.</p>
<p>We are ... |
378,735 | <p>Let $A \in M_n(\mathbb C)$. Let $\langle \; \cdot\; , \; \cdot\; \rangle$ be the standard inner product in $ \mathbb C^n$, viewed either as row vectors or as column vectors.</p>
<p>Let $r_j$ be the $j$-th row of A, and let $c_j$ be the $j$-th column of $A$.</p>
<p>Show that A is normal, if and only if $\langle r_... | copper.hat | 27,978 | <p>The $i$th row of the matrix $B$ is given by $B^* e_i$, similarly, the $i$th column $B$ is given by $B e_i$.</p>
<p>$\langle e_i, A^* A e_j \rangle = \langle A e_i, A e_j \rangle$</p>
<p>$\langle e_i, A A^* e_j \rangle = \langle A^* e_i, A^* e_j \rangle$</p>
|
2,213,626 | <p>How can you prove that if the gcd(a,b) = 1 then gcd(a,bi) = 1 in the Gaussian integers? I know that $i$ is a unit in the ring, but how can you rigorously prove this?</p>
| szw1710 | 130,298 | <p>Substitute $y'=p(y)$ with the new unknown function $p$ of a variable $y$. Then separate variables.</p>
|
1,458,144 | <p><a href="https://i.stack.imgur.com/pJcCp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pJcCp.png" alt="enter image description here"></a></p>
<p>I have this function and I'm trying to write a program to compute it as n approaches 100. The problem is it overflows once it reaches around 50. The h... | Mesmerized student | 221,927 | <p>Try something like this, assume $x>0$:</p>
<p>$$\sqrt x-1= (x^{1/4})^2-1=(x^{1/4} -1)(x^{1/4}+1). $$</p>
<p>But I am not sure that it will be easier now. </p>
|
4,286,296 | <p>I am trying to prove the following claim:</p>
<blockquote>
<p>Let <span class="math-container">$ 0\leq n \in \Bbb Z$</span> and suppose that there exists a <span class="math-container">$k \in \Bbb Z$</span> such that <span class="math-container">$n=4k+3$</span>.
Prove or disprove: <span class="math-container">$\sqr... | gnasher729 | 137,175 | <p>The square of an even integer is <span class="math-container">$4k$</span>, the square of an odd integer is <span class="math-container">$8k+1$</span>. <span class="math-container">$4k+3$</span> is never the square of an integer, neither is <span class="math-container">$4k+2$</span> nor <span class="math-container">$... |
1,353,432 | <p>I know two proofs about the approximation of Euler-Mascheroni constant $\gamma$ that are very technical. So I would like to know if someone has a strategic proof to show that $0.5<\gamma< 0.6$.</p>
<blockquote>
<p>Let be $\gamma\in \mathbb{R}$ such that</p>
<p>$$\large\gamma= \lim_{n\to +\infty}\left[... | Euler88 ... | 252,332 | <p>Let $f(x)=\frac{1}{x}$ and $H_n=\sum_{k=1}^nf(k)$. For every $k$ take the segments $\overline{P_kP_{k+1/2}}$ and $\overline{P_{k+1/2}P_{k+1}}$, where $P_k=(k,f(k))$. Note that for every $k$ the sum of area of trapezes $Q_kP_kP_{k+1/2}Q_{k+1/2}$, and $Q_{k+1/2}P_{k+1/2}P_{k+1}Q_{k+1}$, where $Q_k=(0,k)$, is greater ... |
3,767,452 | <blockquote>
<p>Suppose we have
<span class="math-container">$$\begin{align}
\cos x + \cos y + \cos z &= \frac{3}{2}\sqrt{3} \\[4pt]
\sin x + \sin y + \sin z &= \frac{3}{2}
\end{align}$$</span></p>
<p>How can we solve for <span class="math-container">$x$</span>, <span class="math-container">$y$</span> and <sp... | Angina Seng | 436,618 | <p>We can combine these equations to state
<span class="math-container">$$e^{ix}+e^{iy}+e^{iz}=3\frac{\sqrt3+i}{2}.$$</span>
But
<span class="math-container">$$|e^{ix}|+|e^{iy}|+|e^{iz}|=3=\left|3\frac{\sqrt3+i}{2}\right|
=|e^{ix}+e^{iy}+e^{iz}|.$$</span>
So equality holds in the triangle inequality; if
<span class="ma... |
136,264 | <p>I have a question concerning the stability analysis for a kind of differential equation taking the form
$$\dot x=Ax+Bw,$$
where $A\in \mathbb{R}^{n \times n}$, $B\in \mathbb{R}^{n \times m}$ are constant matrices and
$w \in \mathbb{R}^m$ is a normal random variable, i.e., $w\sim \mathcal{N}(0,W)$ with $W$ ... | Peter Michor | 26,935 | <p>There is the <a href="http://www.mathunion.org/fileadmin/IMU/Report/WG_JRP_Report_01.pdf">Report of the IMU/ICIAM Working Group on Journal Ranking (June 2011)</a>,
and the <a href="http://blog.mathunion.org/rating/">IMU blog on mathematical journals</a>, discussing exactly these questions, and giving lists of such ... |
3,555,084 | <blockquote>
<p>Let
<span class="math-container">$$f(z) = e^z (1+\cos\sqrt{z} ) $$</span>
<span class="math-container">$\Omega=\{z\in\Bbb C: |z|\gt r\}$</span>, <span class="math-container">$r\gt 0$</span>. What is <span class="math-container">$f(\Omega)$</span>?</p>
<p>where <span class="math-container">$... | lab bhattacharjee | 33,337 | <p>Set <span class="math-container">$1/x=h$</span> to find</p>
<p><span class="math-container">$$\lim_{h\to0^+}\dfrac{1-\sqrt{1+6h^2}+3h^2}{h^4}$$</span></p>
<p><span class="math-container">$$=\lim\dfrac{(1+3h^2)^2-(1+6h^2)}{h^4}\cdot\lim\dfrac1{1+3h^2+\sqrt{1+6h^2}}$$</span> rationalizing the numerator</p>
<p><span... |
2,614,969 | <p>I wonder whether there is a general method for accurately estimating the limit of the sequence:</p>
<p>\begin{equation}
x_{n+1} = x_n - x_{n}^{n+1}, \forall x_1 \in (0,1)
\end{equation}</p>
<p>After showing that the limit exists, since $ x_n $ is decreasing and bounded, I managed to derive a lower-bound. In partic... | hamam_Abdallah | 369,188 | <p>As you noted, $(x_n) $ is convergent as a decreasing positive sequence.</p>
<p>We should have
$$\lim_{n\to\infty}x_n^{n+1}=$$
$$\lim_{n\to\infty}e^{(n+1)\ln (x_n)}=0$$</p>
<p>for $A <0$ ans great enough $n,$</p>
<p>$$\ln (x_n)<\frac {A}{n+1}$$</p>
|
408,128 | <p>Suppose a vector $y$ and a <em>symmetric</em> matrix $M$ are given.</p>
<p>\begin{equation}
\forall x; \quad x^Ty=0 \implies x^TMx \ge 0
\end{equation}</p>
<p>Prove that $M$ has at most one negative eigenvalue.</p>
| Ma Ming | 16,340 | <p>First $y\neq 0$, otherwise this is trivially true.</p>
<p>WOLOG, suppose $y=(0,\dots,0,1)$, then $x^T y=0$ means $x_n=0$, say $x=(x_1,\dots, x_n)$. Let $x'=(x_1,\dots,x_{n-1})$ and $M'$ be the matrix deleting $n$ row and $n$ column of $M$. Note that
$$
x^TMx=x'^{T}M'x' \ge 0
$$
the eigenvalue of $M'$ is non-negati... |
4,466,733 | <p><span class="math-container">$$\frac{df(x)}{dx}=f(x+5)$$</span>
I am unable to solve this kind of integration using high school mathematics.
Please help.</p>
| eyeballfrog | 395,748 | <p>Since
<span class="math-container">$$ \frac {d e^{kx}}{dx} = k e^{kx} , \space e^ {k(x + 5)} = e^ {5k} e^ {kx},
$$</span>
it seems like an exponential is the way to go. This leads to the equation
<span class="math-container">$$
k = e^{5k}.
$$</span>
There's no elementary way to express the solution to this equation.... |
2,548,177 | <p>I'd like to define <code>sumdiv</code> in Maple such that this:</p>
<pre><code>with(numtheory);
f:=x->x^2;
sumdiv(f(d)*mobius(100/d), d=1..100);
</code></pre>
<p>would do a sum on all divisors <code>d</code> of $100$.</p>
<p><strong>How to do such a sum over divisors in Maple?</strong></p>
<p>Here's what I've... | Sil | 290,240 | <p>Just use <code>divisors</code> function:</p>
<pre><code>> divisors(100);
{1, 2, 4, 5, 10, 20, 25, 50, 100}
</code></pre>
<p>So for example:</p>
<pre><code>> add(f(d)*mobius(100/d), d in divisors(100));
7200
</code></pre>
<p>The <code>add</code> allows to iterate over set as oposed to <code>sum</code> which... |
2,573,487 | <p>I have given this set</p>
<blockquote>
<p>$$ M = \{ x \in [1,2]\times [3,4] ~|~ x\in\mathbb{Q}^2 \} \subset \mathbb{R}^2 $$</p>
</blockquote>
<p>First I have to identify the boundary $\partial M$ and then tell if it is open or closed.</p>
<p>I think that $$ \partial M = \{ (x,y) ~|~ x\not\in\mathbb{Q}^2, 1\leq ... | lhf | 589 | <p><em>Hint:</em> Can you find an open ball containing $(1.5, 3.5)$ and totally contained in $M$?</p>
|
1,244,564 | <p>For n > 1 Let $F_n = 2^{2^n} + 1$ be a fermat number and b = $2^{2^{n - 2}}$ * ($2^{2^{n - 1}}$ - 1 ).</p>
<p>Then $b^2$ $\equiv$ 2 (mod $F_n$)</p>
<p>I tried to square the original expression I got something ugly that I couldn't simplify further.</p>
<p>I got $b^2$ = $2^{2^{n - 1}}$ * ($2^{2^n}$ - $2 * 2^{2^{n -... | Ragnar | 232,420 | <p>Your simplification isn't working out. Let's go with the first expression for $b^2$</p>
<p>$2^{2^n}$ mod $F_n$ = -1 </p>
<p>so, (all mod $F_n$) $b^2$ = -$2^{2^{n-1}}$*$2^{2^{n-1}+1}$ = -$2^{2^n +1}$ = -$2$ ($2^{2^n}$) = -($2$)($-1$) mod $F_n$ = $2$ mod $F_n$</p>
|
1,878,975 | <p>X is for continuous random variable and it's nonnegative.
Then this is the formula.</p>
<p>$$E(X)=\int_0^\infty(1-F(x))dx$$</p>
<p>Does anyone know the proof?
I appreciate any help.</p>
| Cato | 357,838 | <p>I got the answer </p>
<p>$$\text{arctanh} = \frac12 \ln \frac{1+x}{1-x}$$</p>
<p>consider $\ln(x)$ where $x < 0 $</p>
<p>$$\ln(x) = \ln(|x| (-1))=$$
$$ = \ln(x) + \ln(-1)= \qquad \text{(properties of logs)} $$
$$= \ln(x) + \ln(e^{i \pi}) $$
$$= \ln(x) + i \pi$$</p>
<p>so in your integration, the log of a neg... |
735,101 | <p>Let $X_1$ and $X_2$ be independent random variables having the uniform density with $\alpha = 0$ and $\beta = 1$. Find expressions for the function $Y =X_1 + X_2$.</p>
<p>(a)$y \le 0$</p>
<p>(b)$0<y<1$</p>
<p>(c)$1<y<2$</p>
<p>(d)$y\ge2$</p>
<p>I'm thinking $f(x_1)=f(x_2) = 1$ for $0 \le x_1\le 1$ a... | 51st | 139,495 | <p>This problem can be generalized to finding $x$, $y$, and $z$ in the equation $ax + by + cz = n$ where $a$, $b$, $c$, and $n$ are given and $\gcd(a, b, c) = 1$. If $\gcd(a, b, c) \nmid n$, then there is no solution, otherwise, we can divide both sides by $\gcd(a, b, c)$. </p>
<p>The method to solve for $x$, $y$, and... |
3,247,563 | <p>Given a segment AB on the plane let <span class="math-container">$r$</span> and <span class="math-container">$s$</span> parallel lines with <span class="math-container">$A \in r$</span> and <span class="math-container">$B \in s $</span>. What is the locus of the circles tangent to <span class="math-container">$AB$</... | zhw. | 228,045 | <p>Let <span class="math-container">$x\in [-1/2,1/2].$</span> By the MVT we have</p>
<p><span class="math-container">$$\ln (1-x) = \ln (1) -\frac{1}{1-c_x}x$$</span></p>
<p>for some <span class="math-container">$c_x$</span> between <span class="math-container">$0$</span> and <span class="math-container">$x.$</span> T... |
3,413,837 | <p>Jerry the mouse is hungry and according to some confidential information, there is a tempting piece of cheese at the end of one of the three paths after the junction he just found himself!</p>
<p>Fortunately, Tom is standing right there and Jerry hopes he can get some useful information as to which path he must get... | michalis vazaios | 540,891 | <p>Let's say the doors are numbered <span class="math-container">$1,2,3$</span>.</p>
<p>First question: "Is door 1 the correct door XOR are you telling the truth?"</p>
<p><a href="https://i.stack.imgur.com/nu42l.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nu42l.png" alt="enter image description... |
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