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 |
|---|---|---|---|---|
40,532 | <p>I'm having some problems understanding the following paragraph, which I read in a analysis script (hopefully I haven't made any translation errors):</p>
<blockquote>
<p>"A map $f:U \rightarrow Y$, where $U$ is open and $X,Y$ are Banach spaces, is continuous at $x' \in U$ if $$f(x')=\lim_{x\rightarrow x... | mac | 9,390 | <p>I guess in (1) it should be $t=\|h\|$, so the analogy would be with the polar decomposition of a complex number $z=ts$ where $t=|z|$ and $s=z/|z|$.</p>
<p>In (2), the uniform convergence probably means that $\sup\limits_{\|s\|=1} \|f(x'+ts)-f(x')\|\to 0$ as $t\to 0_+$.</p>
<p>I suppose the minimal speed th... |
65,083 | <p>Rotman's book <em>An Introduction to the Theory of Groups</em> (Fourth Edition) asks, on page 22, Exercise 2.8, to show that <span class="math-container">$S(n)$</span> cannot be embedded in <span class="math-container">$A(n+1)$</span>, where <span class="math-container">$S(n)$</span> = the symmetric group on <span c... | Aaron Meyerowitz | 8,008 | <p>One could ask Rotman. It may be that in a reorganization of the material in the book that problem ended up earlier than the material needed for the (intended) answer. On the other hand it is not a bad experience for students to see problems where the complete solution seems slightly out of reach. Here, one can prove... |
1,712,481 | <p><span class="math-container">$$I=\displaystyle\int\frac{dx}{(3 + 2\sin x - \cos x)}$$</span></p>
<p>If <span class="math-container">$$\tan\left(\frac{x}{2}\right)=u$$</span></p>
<p>or <span class="math-container">$$x=2\cdot\tan^{-1}(u)$$</span></p>
<p>Then,</p>
<p><span class="math-container">$$\sin{x}=\dfrac{2u}{1+... | jim | 289,829 | <p>You could write $2\sin(x)-\cos(x) = \sqrt{5} (\frac{2}{\sqrt{5}} \sin x - \frac{1}{\sqrt{5}} \cos x) = \sqrt{5} \sin (x - \alpha)$, with $\tan \alpha = \frac{1}{2}$, so that the required integral becomes $\int \dfrac{dx}{3 + \sqrt{5} \sin(x-\alpha)}$ and then change the integration variable from $x$ to $y = x - \alp... |
131,482 | <p>Noticed strange behaviour of <code>Export[]</code> when saving graphics to EPS. Here is an example figure I am exporting:</p>
<pre><code>fig =
Show[
ListPolarPlot[
Table[{a, .9}, {a, 0, 2 Pi, .001}],
PlotRange -> {{0, 1.1}, {0, 1.1}}, GridLines -> Automatic,
PlotStyle -> PointSize[.005]],
... | Jose Enrique Calderon | 11,974 | <p>First do correct exporting procedure</p>
<pre><code>Export["/Users/jecalderon1/Documents/testFigreu.eps", fig, "EPS"]
</code></pre>
<p>Off course, you have to change the direction of the directory path../user/ to your local directory.</p>
<p>Then open using Adobe Illustrator . Will render the correct way.</p>
|
3,272,030 | <p>Let's X, Y are random variables (i.e. each one maps elements of a sample space to real numbers). In particular, let's X is such that <span class="math-container">$X:\Omega \rightarrow \mathcal{R}$</span>, where <span class="math-container">$\Omega=\{\omega_1, ..., \omega_n \}$</span> and <span class="math-contain... | pre-kidney | 34,662 | <p>By definition,
<span class="math-container">$$
\textrm{Var}(Y\mid X):=\mathbb E\bigl[\bigl(Y-\mathbb E(Y\mid X)\bigr)^2\mid X\Bigr]=\mathbb E(Y^2\mid X)-\mathbb E(Y\mid X)^2.
$$</span>
Thus, the conditional variance is a random variable, in the same way that the conditional expectation <span class="math-container">$... |
3,551,976 | <p>Let <span class="math-container">$E$</span> be a uniformly convex banach space, <span class="math-container">$K$</span> convex and closed in <span class="math-container">$E$</span>, and <span class="math-container">$x\in B\setminus K$</span>.</p>
<p>Can someone give a concise proof that there is a unique <span clas... | orangeskid | 168,051 | <p>HINT:</p>
<p>We may assume that <span class="math-container">$x=0$</span> and <span class="math-container">$d(0, K) = 1$</span>. </p>
<p>Consider <span class="math-container">$y_n \in K$</span> so that <span class="math-container">$\|y_n \| < 1 + \frac{1}{n}$</span>.
Since <span class="math-container">$K$</spa... |
1,988,021 | <p>Consider $n$ objects each with an associated probability $p_i$, $i\in\{1,\dots,n\}$. If I sample objects $k$ times independently with replacement according to the probability distribution defined by the $p_i$, how does one compute the expected number of times you sample an object you have sampled before?</p>
<p>W... | Jan Eerland | 226,665 | <p>HINT, substitute $u=x^2$ and $\text{d}u=2x\space\text{d}x$:</p>
<p>$$\mathcal{I}(x)=\int\sqrt{16x^2+8+\frac{1}{x^2}}\space\space\text{d}x=\int\frac{\sqrt{16x^4+8x^2+1}}{x}\space\text{d}x=$$
$$\frac{1}{2}\int\frac{\sqrt{16u^2+8u+1}}{u}\space\text{d}u=\frac{1}{2}\int\frac{\sqrt{(1+4u)^2}}{u}\space\text{d}u$$</p>
|
184,142 | <p>I am currently an undergraduate math student. (In fact, freshmen.)</p>
<p>I know that usually abstract algebra is taught somehow late in the undergraduate course, and curious how studies of abstract algebra at graduate level differ from studies at undergraduate level.</p>
<p>So, things like what gets new treatment... | rschwieb | 29,335 | <p>It differs in all the same ways that all graduate courses differ from undergraduate courses. They usually cover more material more quickly, and there is an expectation of better performance and ability from the students. Refining ability to write proofs well is the best skill develop. </p>
<p>If you're asking what ... |
279,238 | <p>I want to create a regular polygon from the initial two points <span class="math-container">$A$</span>, <span class="math-container">$B$</span> and number of vertices <span class="math-container">$n$</span>,<br />
<a href="https://i.stack.imgur.com/dKr9A.png" rel="noreferrer"><img src="https://i.stack.imgur.com/dKr9... | kglr | 125 | <p>For variety's sake, we can also use the given inputs to construct the circumradius, circumcenter and starting angle and use these with the three-argument form of <code>RegularPolygon</code>:</p>
<pre><code>ClearAll[regularPoly]
regularPoly[p1_, p2_, n_] :=
Module[{startingangle = Pi/n - Pi/2 + ArcTan @@ (p2 - p1)... |
2,735,179 | <p>I am trying to sketch the curve given by the following two parametric equations.</p>
<p>$x=cos^3\theta$</p>
<p>$y=sin^3\theta$</p>
<p>Or the single cartesian equation:</p>
<p>$x^{\frac{2}{3}}+y^{\frac{2}{3}}=1$</p>
<p>So,</p>
<p>I put the graph in Desmos and got:</p>
<p><a href="https://i.stack.imgur.com/HtFN... | David Quinn | 187,299 | <p>The more times you repeat the experiment the better the estimate of the true probability. This is the Strong Law of Large Numbers <a href="https://en.m.wikipedia.org/wiki/Law_of_large_numbers" rel="nofollow noreferrer">https://en.m.wikipedia.org/wiki/Law_of_large_numbers</a></p>
|
3,243,655 | <p>Question:</p>
<blockquote>
<p>Prove the equation <span class="math-container">$2x - 6y = 3$</span> has no integer solution to <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
</blockquote>
<p>I need to verify my proof I think I did it correctly, but am not fully sure since... | heropup | 118,193 | <p>Alternatively, you may write <span class="math-container">$$2x - 6y = 2(x-3y).$$</span> Since <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are integers, so must be <span class="math-container">$x-3y$</span>. So <span class="math-container">$2(x-3y)$</span> must be an even int... |
3,337,147 | <p>Let <span class="math-container">$R$</span> be a unique factorization domain (UFD). Given <span class="math-container">$a,b \in R$</span> not simultaneously equal to zero, an element <span class="math-container">$d \in R$</span> is by definition a greatest common divisor (GCD) of <span class="math-container">$a$</sp... | 0CT0 | 699,175 | <p>No it is not correct. For the third, what are the chances that there are 2 balls left of the same color and what are the chances that there are two left that are not the same color? They are not both equally likely, so you do not have <span class="math-container">$1/2*(1+1/2))$</span>.</p>
|
469,521 | <p>$\theta$, $\phi$ are integrable random variables on a probability space $(\Omega,\mathcal{F},P)$ and $\mathcal{G}$ is $\sigma$-field on $\Omega$ contained in $\mathcal{F}$.
Now we want to prove $E(\theta\mid\mathcal{G})=E(\theta)$ if $\theta$ is independent of $\mathcal{G}$. The proof is, for any $B\in \mathcal{G}$,... | Did | 6,179 | <p>For every real number $a$, one has: $\int\limits_Ba\mathrm dP=a\int\limits_B\mathrm dP=aP[B]$. Furthermore, $P[B]=E[I_B]$. Apply these to $a=E[\theta]$.</p>
|
3,253,145 | <p>I am doing this question
<a href="https://i.stack.imgur.com/AEaIn.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/AEaIn.jpg" alt="enter image description here" /></a>
As it can't be solved using separation of variables (my assumption according to what i did, after checking by substituting <span c... | StephenG - Help Ukraine | 298,172 | <p>Any odd integer.</p>
<p><span class="math-container">$$\sum_{i=0}^n 2^i = 2^{n+1}-1$$</span></p>
<p>That's an odd number.</p>
<p>Now you're subtracting :</p>
<p><span class="math-container">$$2\sum_{i=0}^n 2^i a_i$$</span></p>
<p>where <span class="math-container">$a_i$</span> is either <span class="math-contai... |
301,038 | <p>If you want to show that a sequence $(a_{n})$ in $\mathbb{R}$ is convergent, when is it sufficient to show that there is a number $b\in\mathbb{R}$ such that
$$ \liminf a_{n} \geq b \geq \limsup a_{n}$$</p>
<p>In particular, I have a situation where my sequence is bounded and I wanted to use this approach, but I'm n... | Aeolian | 58,941 | <p>One way to think about this is that the $\liminf a_n$ is the smallest subsequential limit (the smallest limit of any subsequence of $a_n$), and the $\limsup a_n$ is the largest subsequential limit. So if you can show that $\liminf a_n \geq b \limsup a_n$, then you will have in fact shown $\liminf a_n = b = \limsup a... |
3,232,766 | <blockquote>
<p>Is <span class="math-container">$10^{100}$</span> (Googol) bigger than <span class="math-container">$100!$</span>?</p>
<p>If <span class="math-container">$10^{100}$</span> is called as Googol, does <span class="math-container">$100!$</span> have any special name to be called, apart from being called as ... | Thomas | 89,516 | <p>Before there was an error on the algebra, as pointed out in the comments. I try to fix the error following the same approach:</p>
<p><span class="math-container">$100!=(1\times..\times 10)\times(11\times..\times 20)\times...\times(91\times..\times 100)=A_1...A_{10}$</span></p>
<p>so we estimate <span class="math-c... |
3,232,766 | <blockquote>
<p>Is <span class="math-container">$10^{100}$</span> (Googol) bigger than <span class="math-container">$100!$</span>?</p>
<p>If <span class="math-container">$10^{100}$</span> is called as Googol, does <span class="math-container">$100!$</span> have any special name to be called, apart from being called as ... | jacky | 14,096 | <p>Using <span class="math-container">$$n!>\bigg(\frac{n}{3}\bigg)^{n}, n>8$$</span></p>
<p><span class="math-container">$$100!>\bigg(\frac{100}{3}\bigg)^{100}>10^{100}$$</span></p>
|
794,032 | <p>Prove by induction that the complement of $ A1 \cup A2...An = A1^c \cap A2^c ...\cap An^c$</p>
<p>My approach: basic step is true, $\overline A1 = A1^c$,</p>
<p>then assume $ A1 \cup A2...Ak = A1^c \cap A2^c ...\cap Ak^c$, prove the case of $k+1$ is true. How should I do that?</p>
| Valent | 92,608 | <blockquote>
<p><strong>Hint:</strong> Your basic step must be $n=2$. Then since $$\overline{\bigcup_{i=1}^{n}A_i}=\overline{\left(\bigcup_{i=1}^{n-1}A_{i}\right)\cup A_n}$$ you can use the case $n=2$ and the induction hypothesis.</p>
</blockquote>
|
3,268,398 | <blockquote>
<p>General solution of the equation
<span class="math-container">$$x\left(\frac{dy}{dx}\right)^2+\left(y-x\right)\frac{dy}{dx}\:-y=0
$$</span>is</p>
</blockquote>
<p>Option are as follows:</p>
<p><span class="math-container">$a)\qquad (x-y+c)(xy-c)=0$</span></p>
<p><span class="math-container">$b)... | Archis Welankar | 275,884 | <p>The equations of such forms are Clairauts equations.Hint let <span class="math-container">$\frac{dy}{dx}=p$</span> we have <span class="math-container">$xp^2+(y-x)p-y=0$</span> thus <span class="math-container">$$p=\frac{(x-y)\pm\sqrt{y^2-2xy+x^2-4(x)(-y)}}{2x}=\frac{(x-y)\pm(y+x)}{2x}$$</span> now continue from her... |
500,446 | <p>Let $p$ be a prime and $p \geq 5$, Consider the congruence $x^3 \equiv a$ (mod p) with $\gcd(a,p)=1$. Show that The congruence has either no solution or three incongruent solutions modulo $p$ if $p \equiv 1$ (mod 6) and has unique solution modulo $p$ if $p \equiv 5$ (mod 6).</p>
<p>My attempt: By Lagrange theorem... | André Nicolas | 6,312 | <p>We give an extremely detailed argument. Sorry about the length!</p>
<hr>
<p>Suppose first that $p\equiv 1\pmod{6}$. So $p=6k+1$ for some $k$.</p>
<p>We will use a <em>primitive root</em> argument. Let $g$ be a primitive root of $p$. In group-theoretic terms, let $g$ be a generator of the multiplicative group modu... |
1,863,868 | <p>Coordinates: <span class="math-container">$(0,0), (3,3), (6,4.5), (9, 5.25)$</span></p>
<p>If this is a curve is there a formula for determining the <span class="math-container">$y$</span> value for any given <span class="math-container">$x$</span> within the range <span class="math-container">$0$</span> to <span c... | Christian | 332,403 | <p>I recommend looking at <a href="http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html" rel="nofollow">Lagrange's Interpolation Formula</a>. Given any $n$ points, they can always be interpolated by a polynomial of degree $n-1$ or less. This means that those points will always lie on the curve of that poly... |
275,526 | <p>What would be an easy example of a sequence of functions defined on a compact interval so that $f_n$ goes to $f$ pointwise but $\sup f_n$ does not go to $sup f$.</p>
<p>I thought of the usual example we take to show that the limits in integration can't be interchanged when we only have pointwise convergence. Is th... | Elias Costa | 19,266 | <p>Let's $f:[0,1]\to \mathbb{R}$. Set $f_n(x)=x^n$ if $x<1-\frac{1}{n}$ and $f_n(x)=0$ if $x\geq 1-\frac{1}{n}$. Note that $\lim_{n\to\infty}f_n=f\equiv 0$ Then
$$
\sup_{x}f_n(x)=1-\frac{1}{n} \mbox{ and } \sup_{x}f(x)=0
$$</p>
|
1,694,991 | <p>I tried doing $u$-substitution and got $-20e$ as my final answer, but I think the correct answer is just $20$. I'm not sure what I did wrong, but probably had to do with plugging in infinity... could someone explain the process of solving this integral?</p>
| Enrico M. | 266,764 | <p><strong>Hint</strong></p>
<p>$$\frac{x}{20} = y$$</p>
<p>$$20\int_0^{+\infty}ye^{-y}\ \text{d}y$$</p>
<p>Many ways to evaluate it. By parts once, or just knowing it's the gamma function:</p>
<p>$$20\int_0^{+\infty}ye^{-y}\ \text{d}y = 20\cdot\Gamma(2) = 20\cdot 1 = 20$$</p>
<p><strong>What Gamma Function is</st... |
2,253,501 | <p>The question is to find out the sum of the series $$\sum_{n=1}^\infty n^2 e^{-n}$$</p>
<p>I tried to bring the summation in some form of telescoping series but failed. I then tried approximating the sum by the corresponding integral(which I am not sure about) to get the value as $2/e$ indicating that the sum conver... | Chinny84 | 92,628 | <p>$$
\frac{d}{da}\mathrm{e}^{an} = n\mathrm{e}^{an}
$$
and
$$
\frac{d^2}{da^2}\mathrm{e}^{an} = n^2\mathrm{e}^{an}
$$
so I posit that we can use
$$
\sum_{n=1}^\infty\frac{d^2}{da^2}\mathrm{e}^{an} = \sum_{n=1}^\infty n^2\mathrm{e}^{an}
$$
we can pull the derivative out of the sum to find
$$
\frac{d^2}{da^2}\sum_{n=1}^... |
2,253,501 | <p>The question is to find out the sum of the series $$\sum_{n=1}^\infty n^2 e^{-n}$$</p>
<p>I tried to bring the summation in some form of telescoping series but failed. I then tried approximating the sum by the corresponding integral(which I am not sure about) to get the value as $2/e$ indicating that the sum conver... | Isaac Browne | 429,987 | <p>Just for kicks, here's the way I like to solve these without calculus:
As long as $|x|<1$, we have
$$S=\sum_{n=1}^{\infty}n^2x^n$$
$$S(1-x) = \sum_{n=1}^{\infty}n^2x^n - \sum_{n=2}^{\infty}(n-1)^2x^n = \sum_{n=1}^{\infty}(2n-1)x^n =\sum_{n=1}^{\infty}2nx^n-\sum_{n=1}^{\infty}x^n$$
$$S(1-x) + \frac{x}{1-x} = 2\sum... |
2,253,501 | <p>The question is to find out the sum of the series $$\sum_{n=1}^\infty n^2 e^{-n}$$</p>
<p>I tried to bring the summation in some form of telescoping series but failed. I then tried approximating the sum by the corresponding integral(which I am not sure about) to get the value as $2/e$ indicating that the sum conver... | Jack D'Aurizio | 44,121 | <p>We have
$$ \sum_{n\geq 0} e^{-nx} = \frac{1}{1-e^{-x}}\tag{1} $$
hence by applying $\frac{d^2}{dx^2}$ to both sides
$$ \sum_{n\geq 0} n^2 e^{-nx} = \frac{e^x(e^x+1)}{(e^x-1)^3}\tag{2} $$
and by evaluating at $x=1$
$$ \sum_{n\geq 1} n^2 e^{-n} = \color{red}{\frac{e(e+1)}{(e-1)^3}}.\tag{3}$$</p>
|
1,318,880 | <p>I'm trying to prove that $\operatorname{lcm}(n,m) = nm/\gcd(n,m)$
I showed that both $n,m$ divides $nm/\gcd(n,m)$
but I can't prove that it is the smallest number.
Any help will be appreciated.</p>
| Steven Alexis Gregory | 75,410 | <p>Let's just do this directly. Let $g = \gcd(m,n)$. We need to prove that
$\operatorname{lcm}(m,n) = \dfrac{mn}{g}$.</p>
<hr>
<p>STEP $0$. (Preliminary stuff.)</p>
<p>DEFINITION $1$. $L = \operatorname{lcm}(m,n)$ if and only if</p>
<pre><code> 1. L is a multiple of m and of n.
2. If C is a multiple of m and of n,... |
1,318,880 | <p>I'm trying to prove that $\operatorname{lcm}(n,m) = nm/\gcd(n,m)$
I showed that both $n,m$ divides $nm/\gcd(n,m)$
but I can't prove that it is the smallest number.
Any help will be appreciated.</p>
| Community | -1 | <p>Here is one way without using the Fundmental theorem of arithmetic just using the definitions </p>
<p>The definition of lcm(a,b) is as follows:</p>
<p>t is the lowest common multiple of a and b if it satisfies the following:</p>
<p>i)a | t and b | t </p>
<p>ii)If a | c and b | c, then t | c.</p>
<p>Similiarly f... |
1,549,506 | <p>To Prove $$\lim_{x \to 0}\frac{x^2+2\cos x-2}{x\sin^3 x}=\frac{1}{12}$$
I tried with L'Hospital rule but in vain.</p>
| Paramanand Singh | 72,031 | <p>Here is how you can do it via a single application of L'Hospital's Rule. The denominator in the expression can be replaced by $x^4$ via the standard limit $\lim_{x\to 0}(\sin x) /x=1$. And the resulting expression can be rewritten as $$\frac{x^2-4\sin^2(x/2)}{x^4}=\frac{t^2-\sin^2t}{4t^4} =\frac{t-\sin t} {4t^3}\cdo... |
2,607,333 | <p>Let $L$ be the splitting field of $x^3+2x+1$. I want to know how 59 splits in $L$. I calculated the discriminant of $\mathbb{Z}[\alpha]$ to be $-59$, (where $\alpha$ is a root of the polynomial), which is squarefree therefore $\mathcal{O}_K = \mathbb{Z}$ (because $d(\mathbb{Z}[\alpha]) = (\mathcal{O}_K:\mathbb{Z}[\a... | bof | 111,012 | <p>Let $M$ be the set of all numbers $m\in\mathbb N$ such that $A$ can be partitioned into $m$ disjoint nonempty sets, each clopen in $A.$</p>
<p>It is clear that $1\in M,$ and that $m\in M\implies\{1,2,3,\dots,m\}\subseteq M.$</p>
<p>If $M$ has no greatest element, then $M=\mathbb N$ and we're done. Otherwise, let $... |
299,140 | <p>Is there a closed form sum of </p>
<p>$\sum_{k=0}^{\infty} \frac{x^k}{(k!)^2}$</p>
<p>It is trivial to show that it is less than $e^x$ but is there a tighter bound?</p>
<p>Thanks</p>
| Iosif Pinelis | 36,721 | <p>Here there are many possibilities. One of them is as follows. Note that for $k=0,1,\dots$
\begin{equation}
\frac1{(k!)^2}=\binom{2k}k\,\frac1{(2k)!}\le\frac{2^{2k}}{(2k)!},
\end{equation}
whence for $x\ge0$ the sum of your series is no greater than
\begin{equation}
B(x):=\sum_{k=0}^{\infty} \frac{(4x)^k}{(2k)!}=... |
660,034 | <p>I wondered if all decimal expansions of $\frac{1}{n}$ could be thought of in such a way, but clearly for $n=6$,</p>
<p>$$.12+.0024+.000048+.00000096+.0000000192+...\neq.1\bar{6}$$</p>
<p>Why does it work for 7 but not 6? Is there only one such number per base, <em>i.e.</em> 7 in base 10? If so what is the general ... | ShreevatsaR | 205 | <p>You're trying to extend your observation that
$$.14+.0028+.000056+.00000112+\dots = \frac17 = \frac7{49}$$
in two directions, that don't match. The right identities are:
$$.12 + 0.0024 + 0.000048 + \dots = \frac{6}{49}$$
and
$$.16 + 0.0064 + 0.000256 + \dots = \frac{1}{6}.$$</p>
<p>These three numbers are solution... |
199,842 | <p>I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below:</p>
<p>The area of a square is like a line, the height (one dimension, length) placed several times next to each other up to the square all the way until the square length thus we have heig... | Timothy | 137,739 | <p>For each shape like a square or a circle that for sufficiently large <span class="math-container">$n$</span> can be split into lines of thickness of approximately width <span class="math-container">$\frac{1}{n}$</span>, you can find a function that assigns to each sufficiently large positive integer <span class="mat... |
134,815 | <p>Assume I have a shuffled deck of cards (52 cards, all normal, no jokers) I'd like to record the order in my computer in such a way that the <em>ordering</em> requires the least bits (I'm not counting look up tables ect as part of the deal, just the ordering itself. </p>
<p>For example, I could record a set of strin... | Ross Millikan | 1,827 | <p>As carlop says, you can store the order number in 226 bits. Then to find the order of the cards, divide the order number by 51!. The quotient is the card number of the first card and the remainder is the order number of the remaining 51 cards.</p>
<p>Added: If you just store a card number (0-51) for each positio... |
2,217,295 | <p>I thought about defining a function $ g(x) = \dfrac{1}{\epsilon} x^{\epsilon} - \ln(x) $ and show that it's larger than $0$. I saw that if I substitute
$1$, I get $\dfrac{1}{\epsilon}$ which is positive and so on. I also tried to derive the function and I got $g'(x) = x^{\epsilon -1} - \dfrac{1}{x} $</p>
<p>I'... | Adren | 405,819 | <p>If you are allowed to use :</p>
<p>$$\lim_{x\to+\infty}\frac{\ln(x)}x=0$$</p>
<p>then, given $\epsilon>0$, you can see that :</p>
<p>$$\forall x>0,\,\frac{\ln(x)}{x^\epsilon}=\frac 1\epsilon\frac{\ln(x^\epsilon)}{x^\epsilon}$$and since $\lim_{x\to+\infty}x^\epsilon=+\infty$, composition of limits shows that... |
3,725,638 | <p>The exterior derivative of a scalar function is</p>
<p><span class="math-container">$d f(x,y,z) = (
\frac{\partial f}{\partial x} dx
+ \frac{\partial f}{\partial y} dy
+ \frac{\partial f}{\partial z} dz
)$</span></p>
<p>Am I correct in assuming then that</p>
<p><span class="math-container">$d\left( F_x(x,y,z) e_x ... | peek-a-boo | 568,204 | <p>We only talk about exterior derivatives of differential <span class="math-container">$k$</span>-forms, not vector fields. However, what we can do is the following: given a vector field <span class="math-container">$F: \Bbb{R}^3 \to \Bbb{R}^3$</span>, <span class="math-container">$F = (F_x, F_y, F_z)$</span>, we can ... |
3,836,662 | <p>I understand basic group theory. I would say that I've seen most of the standard stuff up to, say, the quotient group.</p>
<p>I feel like I've seen in more than one place the suggestion that group theory is the study of symmetries, or actions that leave something (approximately) unchanged. Unfortunately I can only f... | AlexanderGrey | 675,096 | <p>There is a quite easy way to prove this problem. If you understand chinese, there is a classic proof on Yang Zixu's 'Abstract Algebra'. Here is the proof:</p>
<p>Since <span class="math-container">$H\cap K\le H$</span>, let <span class="math-container">$|H|/|H\cap K|=m$</span> and
<span class="math-container">$H = h... |
902,407 | <p>Exercise</p>
<p>Let $f:G \to G'$ be an isomorphism and let $H\unlhd G$. If $H'=f(H)$, prove that $G/H \cong G'/H'$.</p>
<p>As I've shown that $H'\unlhd G'$, I thought of defining $$\phi(Ha)=H'f(a)$$I was trying to show that this function is well defined and that it is an isomorphism.</p>
<p>I'll work with right c... | Asaf Karagila | 622 | <p>It's easy to get bogged down with the details here. But the simplest way would be to find a bijection between $A$ and $B$.</p>
<p>We are given that $|A-B|=|B-A|$, therefore there is a function $f\colon (A-B)\to (B-A)$ which is a bijection. Can you think of a way to extend $f$ to be a bijection between $A$ and $B$?<... |
4,024,871 | <p>I'm looking to build a function <span class="math-container">$f:S^2 \to \mathbb R^2$</span> such that <span class="math-container">$f(x)\neq f(−x)$</span> for all <span class="math-container">$x\in S^2$</span>.</p>
<p>By Borsuk-Ulam Theorem, this function must be discontinuous. I was trying to build a not too compli... | Aryaman Maithani | 427,810 | <p>Fix two distinct elements <span class="math-container">$a$</span> and <span class="math-container">$b$</span> in <span class="math-container">$\Bbb R^2$</span>. Consider an arbitrary <span class="math-container">$(x, y, z) \in S^2.$</span></p>
<ul>
<li>If <span class="math-container">$x > 0$</span>, map it to <sp... |
2,940,306 | <p>I understand that it would be n! permutations for the given amount of elements, but I am not sure calculate it with these parameters.</p>
| maveric | 590,250 | <p>3 options for last place
simultaneously 3 for first place,
now we have 7 objects left. so 7 objects, 7 places
which is 7!
so answer is 3 <em>3</em> 7!</p>
|
3,126,080 | <p>Is there any particular equation which doesn't work on the real plane of numbers but works on other planes?</p>
| Theo Bendit | 248,286 | <p>Freshman's Dream, <span class="math-container">$(x + y)^n = x^n + y^n$</span>, holds in fields of characteristic <span class="math-container">$n$</span>, when <span class="math-container">$n$</span> is prime.</p>
|
1,745,180 | <p>I want to prove that the polynomial </p>
<p>$$
f_p(x) = x^{2p+2} - cx^{2p} - dx^p - 1
$$</p>
<p>,where $c>0$ and $d>0$ are real numbers, has distinct roots. Also $p>0$ is an even integer. How can I prove that the polynomial $f_p(x)$ has distinct roots for any $c$,$d$ and $p$.</p>
<p>PS: There is a simila... | Bernard | 202,857 | <p>‘All’ you have to do is computing the g.c.d. of $f_p(x)$ and $f'_p(x)$ via the <em>Euclidean algorithm</em>. A polynomial has only simple roots (in $\mathbf C$) if and only if this polynomial and its derivative are coprime.</p>
|
372,401 | <p>Let us assume that the boundary of the domain in the definition of the Sobolev spaces $L^2$ and $H_0^1$ is sufficiently smooth.</p>
<p>Let $|\cdot |$ denote the norm in $L^2$. Then for a function $v$ in $H_0^1$, the norm is given via $\|v\|^2=|v|^2+|\nabla v|^2$. </p>
<p>In general, one cannot bound the $H_0^1$-no... | Wintermute | 67,388 | <p>Suppose this set is path connected. Consider the point $(0,0)$ and the point $(1,\sin(1))$. The only way to link these two points is along the path $(x,\sin(\frac{1}{x}))$. However $\lim_{x \rightarrow 0}\sin(\frac{1}{x})=$DNE. Hence there is no path connecting $(0,0)$ to $(1,\sin(1))$, a contradiction. Therefore th... |
3,443,672 | <p>The equation is
<span class="math-container">$$\tan\frac{5\pi}{6} \cos x=1-\sin x$$</span>
<span class="math-container">$$\sin\frac{5\pi}{6} \cos x=\cos\frac{5\pi}{6}-\cos\frac{5\pi}{6} \sin x$$</span>
<span class="math-container">$$\sin\left(\frac{5\pi}{6}+x\right)=\cos \frac{5\pi}{6}$$</span> which looks weird to... | johnny09 | 285,238 | <p>Let <span class="math-container">$\theta = \frac{5 \pi}{6}$</span>. Then continuing from where you stopped and noting that <span class="math-container">$\cos(\theta) = -\frac{\sqrt(3)}{2}$</span>, we get
<span class="math-container">$$\sin(\theta + x) = \cos(\theta)
= -\frac{\sqrt{3}}{2}$$</span>
and so,
<span class... |
1,307,460 | <p>Need to integrate this function. Need help with my assignment. Thanks</p>
| DeepSea | 101,504 | <p>Let $I = \displaystyle \int_{0}^{\frac{\pi}{2}} \dfrac{\sin^2x}{\cos x+\sin x}dx$ and $J = \displaystyle \int_{0}^{\frac{\pi}{2}} \dfrac{\cos^2x}{\cos x+ \sin x}dx$. Then you can evaluate: $I+J, J-I$ quicky and then <strong>solve</strong> for $I$</p>
|
2,013,650 | <p>I am trying to prove that
$$n^{n/2}<n!,\text{ for } n\ge2.$$
I can't really figure it out. </p>
| Chill2Macht | 327,486 | <p>$\newcommand{\Vec}{\mathsf{Vec}}$$\newcommand{\Hom}{\operatorname{Hom}}$$\newcommand{\id}{\operatorname{id}}$My answer exceeded the character limit, so here's the last part:</p>
<p>Finally, to get the general case, consider the following functor: $$\bigotimes: \Vec^{\times k} \to \Vec, $$ which acts by: $$\bigotime... |
3,648,315 | <p><a href="https://i.stack.imgur.com/IKnlD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IKnlD.png" alt="enter image description here"></a></p>
<p>I have an item, let's say sword. </p>
<p>As described above, <br/>
If my sword is at state 1, I can try upgrading it. There are 3 possibilities. <br/... | fleablood | 280,126 | <p>Wolog assume <span class="math-container">$x \le y$</span> and <span class="math-container">$y-x = m\ge 0$</span> then</p>
<p><span class="math-container">$2x + m + 2x(x+m) = 83$</span> and </p>
<p><span class="math-container">$2x^2 + (2+2m)x + (m-83) = 0$</span></p>
<p><span class="math-container">$x = \frac {-(... |
471,710 | <p>Why do small angle approximations only hold in radians? All the books I have say this is so but don't explain why.</p>
| Community | -1 | <p>You need to clarify what you are really asking: in physics the small angle approx. is typically used to approximate a non-linear differential equation by a linear one - which is much easier to solve - by allowing us to approximate the sin(x) function by x. The resulting equation is only a good one for small angles... |
4,524,393 | <p>I don't know if this question is best suited to this stack exchange. If it isn't, feel free to migrate it or close it. This question was inspired by a mistake I saw in a math class. I corrected the professor, and he acknowledged it. I then said, "Some students think professors never make mistakes". And he ... | Jean-Armand Moroni | 1,064,750 | <p>Grothendieck is famous for having chosen <span class="math-container">$57$</span> as a prime number example, when answering a question from a student.</p>
<p>And actually the same mistake had been made before, by Weyl, in a paper, as can be read here.
<a href="https://hsm.stackexchange.com/questions/6358/story-of-gr... |
440,528 | <p>My question is about group theory:</p>
<blockquote>
<p>How many subgroups does a non-cyclic group contain whose order is 25?</p>
</blockquote>
<p>How can i answer that question?</p>
<p>Can you generalize the answer?</p>
<p>Thanks for your help.</p>
| N. S. | 9,176 | <p>If $H$ is a non-trivial subgroup, then $H$ has $5$ elements (why?).</p>
<p>As the subgroup is not cyclic, and $25=5^2$, the order of every element $x \neq e$ is .....</p>
<p>Last but not least, if the order of $x$ is 5, there are 4 other elements of order 5 which generate the same group.</p>
|
1,955,505 | <blockquote>
<p>$\sum_{n=0}^\infty \frac{n^2+3n+2}{4^n} = \frac{128}{27}$ Given hint: $(n^2+3n+2) = (n+2)(n+1)$</p>
</blockquote>
<p><strong>I've tried</strong> converting the series to a geometric one but failed with that approach and don't know other methods for normal series that help determine the actual conver... | Hazem Orabi | 367,051 | <p>$$
\begin{aligned}
& S_{0} = \sum_{n=0}^{\infty} \frac{n^{0}}{4^{n}} = \sum_{n=0}^{\infty} \frac{1}{4^{n}} = \sum_{n=0}^{\infty} (1/4)^{n} = \frac{1}{1 - (1/4)} \Rightarrow \color{red}{S_{0} = \frac{4}{3}} \\ \\
& S_{1} = \sum_{n=0}^{\infty} \frac{n^{1}}{4^{n}} = \sum_{n=0}^{\infty} \frac{n + 1 - 1}{4^{n}} =... |
1,955,505 | <blockquote>
<p>$\sum_{n=0}^\infty \frac{n^2+3n+2}{4^n} = \frac{128}{27}$ Given hint: $(n^2+3n+2) = (n+2)(n+1)$</p>
</blockquote>
<p><strong>I've tried</strong> converting the series to a geometric one but failed with that approach and don't know other methods for normal series that help determine the actual conver... | robjohn | 13,854 | <p>Using the formula for the sum of a geometric series:
$$
\sum_{k=0}^\infty x^k=\frac1{1-x}\tag{1}
$$
Taking the derivative of $(1)$ and tossing the terms which are $0$:
$$
\sum_{k=1}^\infty kx^{k-1}=\frac1{(1-x)^2}\tag{2}
$$
Taking the derivative of $(2)$ and tossing the terms which are $0$:
$$
\sum_{k=2}^\infty k(k-... |
3,080,566 | <p>please tell me how I can solve the following equation. </p>
<p><span class="math-container">$$z^3+\frac{(\sqrt2+\sqrt2i)^7}{i^{11}(-6+2\sqrt3i)^{13}}=0$$</span></p>
<p>What formula should I use? If possible, tell me how to solve this equation or write where I can find a formula for solving such an equation. I sear... | Community | -1 | <p><span class="math-container">$\sqrt2+\sqrt2i=2e^{\frac{\pi i}4}$</span>.</p>
<p>And <span class="math-container">$-6+2\sqrt3i=4\sqrt3e^{\frac{\pi i}6}$</span>.</p>
<p>And <span class="math-container">$i^{11}=-i$</span>.</p>
<p>So we have <span class="math-container">$z^3-\frac{2^7e^{\frac{7\pi i}4}}{i\cdot (4\sqr... |
2,729,364 | <p>What are direct methods for proving that a ring is a UFD in general without proving that it's a PID/Euclidean domain/field and using the fact that all those things are UFDs?</p>
<p>As an example, we can take <span class="math-container">$\mathbb{Z}[i]$</span> or <span class="math-container">$\mathbb{Z}[\sqrt{-2}]$<... | Jordan Hardy | 202,814 | <p>This would be extremely wasteful, and nobody would do it before showing they were Euclidean domains or PIDs, but you could show that they have class number 1 through some indirect means. $\mathbb Z[i]$ is a Dedekind domain so it is a UFD if and only if its class number is 1.</p>
|
3,113,850 | <blockquote>
<p><span class="math-container">$$f(x)=\sum_{n=1}^{\infty}{\frac{x^{n-1}}{n}},$$</span></p>
<p>Prove that <span class="math-container">$f(x)+f(1-x)+\log(x)\log(1-x)=\frac{{\pi}^2}{6}$</span></p>
</blockquote>
<p>In my mind though,I think that this is related to Basel problem<span class="math-container">$\l... | user1551 | 1,551 | <p>Let <span class="math-container">$A$</span> be <span class="math-container">$n\times n$</span>. Since each entry of <span class="math-container">$\operatorname{adj}(A)$</span> is a signed multiple of a <span class="math-container">$(n-1)$</span>-rowed minor, <span class="math-container">$\operatorname{adj}(A)=0$</sp... |
593,746 | <p>In general, if a random process is ergodic, does it imply that it is also stationary in any sense?</p>
| Val | 39,930 | <p>This man says that it is <a href="https://www.youtube.com/watch?v=k6y2kzayV6A&#t=1433" rel="nofollow">https://www.youtube.com/watch?v=k6y2kzayV6A&#t=1433</a>. So, ergodicity implies stationarity.</p>
|
3,161,874 | <p>Find the vector equation of the plane which contains points <span class="math-container">$A(0,1,1)$</span>, <span class="math-container">$B(-1,2,1)$</span> and <span class="math-container">$C(2,0,2)$</span>. </p>
<p>I did this by first finding <span class="math-container">$AB$</span> and <span class="math-container... | fleablood | 280,126 | <p>Suppose the fly is at the wall. And suppose this will be the last trip or partial trip of the fly. Suppose the car is <span class="math-container">$h$</span> miles away.</p>
<p>The fly and and car have a combined speed of <span class="math-container">$140 \frac {km}{hr}$</span> so the fly reaches the car in <span... |
3,161,874 | <p>Find the vector equation of the plane which contains points <span class="math-container">$A(0,1,1)$</span>, <span class="math-container">$B(-1,2,1)$</span> and <span class="math-container">$C(2,0,2)$</span>. </p>
<p>I did this by first finding <span class="math-container">$AB$</span> and <span class="math-container... | Guest3711 | 657,760 | <p>Simple way to calculate it: It takes the car 30 minutes to travel the 20km to the wall at 40km/hr. The fly, traveling at 100km/hr will travel 50km in those same 30 minutes.</p>
<p>Edit: There will be an infinite number of trips. It's similar to how Zeno's paradox works where the trips get shorter and shorter and ev... |
87,319 | <p>How might I show that there's no metric on the space of measurable functions on $([0,1],\mathrm{Lebesgue})$ such that a sequence of functions converges a.e. iff the sequence converges in the metric?</p>
| pgassiat | 7,406 | <p>In a metric space, a sequence such that any subsequence has a converging subsequence must be convergent.</p>
<p>Considering a sequence converging in measure but not a.e. shows that this property is not true for a.e. convergence.</p>
|
3,908,795 | <p>This was left as an exercise in Apostol's Calculus Vol II and I'm not very sure how to proceed and have my doubts on whether the following would be sufficient due to step <span class="math-container">$(1)$</span>:
<span class="math-container">$$-(x+y) \stackrel{(1)}{=} -1(x+y) \stackrel{(2)}{=} -1x + -1y \stackrel{(... | blackmirror7 | 849,727 | <p>You can simply say</p>
<p>-(x+y)=-(x+y)+x-x+y-y=-(x+y)+(x+y)-x-y (since we add x-x and y-y there is no problems)</p>
<p>which then would imply -(x+y)=-x-y</p>
|
3,908,795 | <p>This was left as an exercise in Apostol's Calculus Vol II and I'm not very sure how to proceed and have my doubts on whether the following would be sufficient due to step <span class="math-container">$(1)$</span>:
<span class="math-container">$$-(x+y) \stackrel{(1)}{=} -1(x+y) \stackrel{(2)}{=} -1x + -1y \stackrel{(... | Mirzathecutiepie | 791,050 | <p>Notice that <span class="math-container">$(x + y) - (x + y) = 0$</span> but we also have <span class="math-container">$x + y - x - y = 0$</span>. Hence we have <span class="math-container">$(x + y) - (x + y) = x + y - x - y = (x + y) - x - y$</span>. This can be reduced to <span class="math-container">$(x + y) - (x+... |
4,065,868 | <p>Can anyone here recommend a low-dimensional topology textbook that contains knot theory and 3,4-manifolds?Or should I look for these subjects in separate textbooks?</p>
| user901807 | 901,807 | <p>These are diverse subjects studied with different tools. You will need a <em>strong</em> background in algebraic topology, as well as manifold topology (PL or differentiable, most likely the latter), to do well with the material.</p>
<p>Here are some very good introductory texts in each.</p>
<p>Knot theory: Rolfsen'... |
1,105,056 | <p>There's something I've never understood about polynomials.</p>
<p>Suppose $p(x) \in \mathbb{R}[x]$ is a real polynomial. Then obviously,</p>
<p>$$(x-a) \mid p(x)\, \longrightarrow\, p(a) = 0.$$</p>
<p>The converse of this statement was used throughout high school, but I never really understood why it was true. I ... | idmercer | 7,635 | <p>Here's what helped make this intuitive for me when I took undergraduate abstract algebra:</p>
<p>You can always <em>try</em> to divide a polynomial by another polynomial. There is a division algorithm.</p>
<p>Of course, the division might not come out evenly, and there might be a remainder. But what would that rem... |
4,431,764 | <p>I was solving one problem in which I had to find the value of <span class="math-container">$x$</span> and at the last step my result came out to be <span class="math-container">$x^{\log_b{a}}=2$</span> but I was not able to get to the correct answer after this. Here's what I did:</p>
<p><span class="math-container">... | 5xum | 112,884 | <p>In the reals, if <span class="math-container">$\beta > 0$</span>, the solution to the equation <span class="math-container">$$x^{\alpha} = \beta$$</span> is <span class="math-container">$x=\beta^\frac{1}{\alpha}$</span> which is also sometimes written as <span class="math-container">$x=\sqrt[\alpha]{\beta}$</span... |
1,047,292 | <p><img src="https://i.imgur.com/2xwGbMS.png" alt="Sum from n = 0 to infinity of Cn*(4^n)">
<img src="https://i.imgur.com/z9qZ5Wh.png" alt="Sum from n = 0 to infinity of Cn*(-4^n) and sum from n = 0 to infinity of Cn*(-2^n)"></p>
<p>I feel the coefficient Cn has to be zero in order for the original series to converge,... | Eoin | 163,691 | <p>It is not true that the values of $C_n$ must be $0$. For example, if $C_n=(\frac{1}{8})^n$ then our series would be $\sum_{n=0}^\infty(\frac{1}{8})^n4^n=\sum_{n=0}^\infty(\frac{1}{2})^n$ which does converge.</p>
<p>It is not true that $\sum_{n=0}^\infty C_n (-4)^n$ must converge. For an example, suppose $C_n=(-\fra... |
1,047,292 | <p><img src="https://i.imgur.com/2xwGbMS.png" alt="Sum from n = 0 to infinity of Cn*(4^n)">
<img src="https://i.imgur.com/z9qZ5Wh.png" alt="Sum from n = 0 to infinity of Cn*(-4^n) and sum from n = 0 to infinity of Cn*(-2^n)"></p>
<p>I feel the coefficient Cn has to be zero in order for the original series to converge,... | marty cohen | 13,079 | <p>If
$c_n
=\frac{(-1)^n}{n4^n}
$,
then
$\sum_{=1}^{\infty} c_n 4^n$
converges
(it's a alternating sum
with the terms decreasing to zero),
and
$\sum_{=1}^{\infty} (-1)^n c_n 4^n$
does not converge
(it's the well-known
harmonic sum).</p>
<p>If
$\sum_{=1}^{\infty} c_n 4^n$
converges,
then
$c_n 4^n \to 0$.
This is <em>n... |
936,830 | <p>I made the differential equation : $$dQ = (-1/100)2Q dt$$ </p>
<p>I separate it and get: $\int_a^b x (dQ/Q) = \int_a^b x (-2/100)dt$</p>
<p>this leads me to: $\log(|Q|) = (-t/50) + C$</p>
<p>I simplify that to $Q = e^{-t/50}$</p>
<p>My TI-Nspire differential equation solver, however, gives me: $Q = Ce^{-t/50}$<... | Did | 6,179 | <p>Forgetting some irrelevant factorial and exponential terms, you seem to be trying to factor the ratio $$R=\frac{\theta^{u-v}\lambda^v}{(\theta+\lambda)^u}.$$
You are not very specific about the <em>complete</em> expression you get for $R$, but indeed, $$R=\left(\frac{\lambda}{\theta}\right)^v\left(\frac{\theta}{\the... |
2,053,255 | <p>Let $f_{X,Y}(x,y)=\frac{1}{8}$ for $-2<x<2$, $0<y<2$. Find $f(z)$ where $Z=X+Y$. </p>
<p>I am having difficulty solving the above problem. My attempt at a solution relies on a convolution formula, which says that the $pdf$ of $Z=X+Y$, given the joint $pdf$ $f(x,y)$ is given by $$f_Z(z)= \int_{s+t=z} f(s... | Graham Kemp | 135,106 | <p>First note that when <span class="math-container">$X\in[-2..2]$</span> and <span class="math-container">$Y\in[0..2]$</span>, then <span class="math-container">$X+Y\in[-2..4]$</span>, thus this is the support for the pdf of <span class="math-container">$Z$</span>.</p>
<p>You have obtained that: <span class="math-cont... |
3,672,948 | <p>Consider the series
<span class="math-container">$$
S = \sum_{n=1}^\infty e^{-n^2x}
$$</span>
then I have to argue for that <span class="math-container">$S$</span> is convergent if and only if <span class="math-container">$x>0$</span>.</p>
<p>As this is if and only if I think I have to assume first that S is con... | Alex | 38,873 | <p>Another way to see it is to compare the sum in question to the integral</p>
<p><span class="math-container">$$
\int_{1}^{\infty} e^{-(t\sqrt{x})^2} dx =\frac{1}{\sqrt{x}}\int_{\sqrt{x}}^{\infty} e^{-s^2} ds
$$</span>
Which exists only for <span class="math-container">$x>0$</span>.</p>
|
1,618,952 | <p>How to prove this exercise?</p>
<p>Let $A$ be a $n \times n$ diagonal matrix with characterist polynominal</p>
<p>$$(x-c_1)^{d_1}...(x-c_k)^{d_k}$$</p>
<p>where $c_i$ are distinct. Let $V$ be the space of $n \times n$ matrixes $B$ such that $AB=BA$. Prove that dimension of $V$ is $d_{1}^{2}+...+d_{k}^{2}$.</p>
| Daniel Buck | 293,319 | <p>Consider the convergent series
$$\sum_{s\in\mathbb{N}} f(s)$$
with sum $a$, and label this series $A$.
Now let us order the terms in series $A$ as $f(n)$, for $n=1,2,\dotsc,$ to give the series as follows:
$$\sum_{n=1}^{\infty} f(n),$$
and label this series $B$. For every partial sum $B_M=\sum_{n=1}^{M} f(n)$ of se... |
4,320,437 | <p>I was thinking about linear tramsformations and i came up with this example:
<span class="math-container">$$f:\mathbb{R}^n \to \mathbb{C}^n\\
f(x)=ix$$</span>
for this example, domain and co-domain are not defined over the same field and all linear transformations that i encountered by now had domain and co-domain ... | Wuestenfux | 417,848 | <p>Hint: <span class="math-container">$\mathbb{R}^n$</span> is an <span class="math-container">$n$</span>-dim. <span class="math-container">$\mathbb{R}$</span>-vector space and <span class="math-container">$\mathbb{C}^n$</span> is a <span class="math-container">$2n$</span>-dim, <span class="math-container">$\mathbb{R}$... |
1,330,078 | <p>Given a quadrilateral $MNPQ$ for which $MN=26$, $NP=30$, $PQ=17$, $QM=25$ and $MP=28$ how do I find the length of $NQ$?</p>
| Rory Daulton | 161,807 | <p>For the answer to be unique we must assume the quadrilateral is convex and/or simple, which you said in a comment that we can assume.</p>
<p>One way to solve the problem is to place your quadrilateral in a Cartesian coordinate system, perhaps placing $M$ at the origin and $P$ at $(28,0)$, and assume that point $N$ ... |
2,758,338 | <p>I need to find the laurent series and the residue of the following complex function
$$f(z)=(z+1)^2e^{3/z^2}$$
at $z=0$.</p>
<p>Since $e^z=\sum z^n/n!$, then
$$e^{3/z^2}=\sum_{n=0}^\infty \frac{3^n/n!}{z^{2n}}$$
thus
$$f(z)=(z^2+2z+1)\sum_{n=0}^\infty \frac{3^n/n!}{z^{2n}}=\sum_{n=0}^\infty \frac{3^n/n!}{z^{2(n-1)}... | B. Mehta | 418,148 | <p>In $S_6$, the group generated by the cycle $(123456)$ is cyclic of order 6, and the group generated by $(12)(345)$ is cyclic of order 6. But, they are not the same. This example shows also that they may not even be conjugate subgroups. </p>
|
42,854 | <p>How much money would you have if the amount of money you started with was 5 and it increased by 5 a day for 365 days.
So January 1st you receive 5, Jan 2nd you receive 10, the third 15.. etc.
I'm wondering what the formula is</p>
| Arturo Magidin | 742 | <p>$$5 + 2\times 5 + 3\times 5 + \cdots + 365\times 5 = 5\times\Bigl( 1+2+\cdots + 365\Bigr)$$
at which point it comes down to figuring out how much is the sum of $n$ consecutive integers, starting with $1$.</p>
|
2,028,646 | <p>Let two random variables: $$x_1 \sim Bin(100, 0.5) \\ x_2 \sim Bin(100, 0.6)$$</p>
<p>Now, we define a third random variable, $x_{12}$ which it's distribution is the aggregated distributions of $x_1$ and $x_2$, so it's <strong>not</strong> quite like $x_1 + x_2$ even though empirically the variance seems like the s... | Henry | 6,460 | <p>It seems you may be describing a mixture of distributions rather than a sum</p>
<p>For the individual distributions you have </p>
<ul>
<li>$E[X_1] =50$, $\text{Var}[X_1]=25$, $E[X_1^2] = 2525$ </li>
<li>$E[X_2] = 60$, $\text{Var}[X_2]=24$, $E[X_2^2] = 3624$</li>
</ul>
<p>For the sum you would have</p>
<ul>
<li... |
743,465 | <p>Suppose $l,t\in[0,1]$ and $l+t\leq1$ I want to prove $1+l+t>6lt$. When $t=0$ or $l=0$, it is trivial, so I started with $l,t\neq0$ but I couldn't reach anywhere. I don't have time to write in detail what I have already tried, but I tried to manipulate $(l-t)^2$ mostly. Anyway, if anyone help me with the proof tha... | rlartiga | 93,314 | <p>I think your problem is possible to state it as a optimization problem:</p>
<p>$$\min 1+l+t-6lt$$
$$s.t.$$
$$ l+t\leq 1$$
$$0\leq l\leq1$$
$$0\leq t\leq1$$</p>
<p>Which have a global minimum at $(l,t)=\left(\frac{1}{2},\frac{1}{2}\right)$ then $$1+l+t-6lt\geq1+\frac{1}{2}+\frac{1}{2}-2\frac{1}{2}\frac{1}{2}=\frac{... |
4,337,887 | <p>I am solving an exercise:</p>
<p>Let <span class="math-container">$T: V \rightarrow W$</span> be a linear transformation.
<span class="math-container">$V$</span> and <span class="math-container">$W$</span> are finite-dimensional inner product spaces. Prove <span class="math-container">$T^*T$</span> and <span class="... | I am a person | 806,777 | <p>This "it" is called the <strong>number</strong> of <strong>factors</strong> or <strong>divisors</strong> of a number.</p>
<p>As said in the comments however, I don't think there is a singular word that describes this.</p>
|
1,344,892 | <p>Let $K$ be a field, and let $A$ be a $K$-algebra such that $\alpha \in A$. Then the natural homomorphism $$ \phi: K[x] \to K[\alpha], \hspace{3mm} (x \mapsto \alpha )$$ has a kernel which is a principal ideal $ \langle f \rangle$ and so $$ K[x] / \langle f \rangle \cong K[\alpha]$$</p>
<p>Notice that $K[\alpha]$ is... | lhf | 589 | <p>Every $g \in K[x]$ can be written as $g=fq+r$, with $r=0$ or $\deg r<n$.
Since $fq\in \ker\phi$, we have $\phi(g)=\phi(r)$.</p>
<p>In other words, the image of $\phi$ is generated by the image of $\{1, x, x^2, \dots, x^{n-1} \}$, with of course is $\{1, \alpha, \alpha^2, \dots, \alpha^{n-1} \}$.</p>
<p>That $\{... |
3,800,833 | <p><strong>Question: Bob invests a certain sum of money in a scheme with a return of 22% p.a . After one year, he withdraws the entire amount (including the interest earned) and invests it in a new scheme with returns of 50% p.a (compounded annually) for the next two years. What is the compounded annual return on his ... | Z Ahmed | 671,540 | <p>Use <span class="math-container">$\beta$</span> integral
<span class="math-container">$$\int_{0}^{\pi/2} \sin^{n-1} x \cos^{1-n} d x=\frac{1}{2} \frac{\Gamma(n/2) \Gamma(1-n/2)}{\Gamma(1)}= \frac{1}{2}\frac{\pi}{\sin n \pi/2}.$$</span> if <span class="math-container">$1-n/2>0$$ \implies 0<n<2.$</span></p>
<... |
241,612 | <p>I have the double inequality
<span class="math-container">$$G(n) = \left( \frac{1}{2} \right)^{\omega(n)} \tau(n)^2 <
F(n) \leq
\left( \frac{3}{4} \right)^{\omega(n)} \tau(n)^2 = H(n),$$</span>
where <span class="math-container">$F(n) = \sum_{d \mid n} \tau(d)$</span>. (What <span class="math-container">$\omega... | thorimur | 63,584 | <p>Building off of what you have, I think it's nice to put <span class="math-container">$F(n)$</span> in the denominator each time, and shift the origin of the axes to <code>{0,1}</code>. Then the one bigger than <span class="math-container">$F(n)$</span> is above the axis, and the one smaller than <span class="math-co... |
241,612 | <p>I have the double inequality
<span class="math-container">$$G(n) = \left( \frac{1}{2} \right)^{\omega(n)} \tau(n)^2 <
F(n) \leq
\left( \frac{3}{4} \right)^{\omega(n)} \tau(n)^2 = H(n),$$</span>
where <span class="math-container">$F(n) = \sum_{d \mid n} \tau(d)$</span>. (What <span class="math-container">$\omega... | kglr | 125 | <p>A variation on thorimur's second method: We can use custom ticks for the vertical axis and put a gap between the two curves to have the horizontal axis visible:</p>
<pre><code>ClearAll[g, h, F]
g[n_] := (1/2)^PrimeNu[n] DivisorSigma[0, n]^2
h[n_] := (3/4)^PrimeNu[n]*DivisorSigma[0, n]^2
F[n_] := DivisorSum[n, Diviso... |
98,317 | <p>This comes from Artin Second Edition, page 219. Artin defined $G = \langle x,y\mid x^3, y^3, yxyxy\rangle$, and uses the Todd-Coxeter Algorithm to show that the subgroup $H = \langle y\rangle$ has index 1, and therefore $G = H$ is the cyclic group of order 3.</p>
<p>That being the case, $x$ cannot be either $y$ or ... | Mikasa | 8,581 | <p>Applying the way Artin used, you see from the following TC algorithm that $[G:H]=1$.</p>
<p><img src="https://i.stack.imgur.com/otRbr.jpg" alt="enter image description here"></p>
|
98,317 | <p>This comes from Artin Second Edition, page 219. Artin defined $G = \langle x,y\mid x^3, y^3, yxyxy\rangle$, and uses the Todd-Coxeter Algorithm to show that the subgroup $H = \langle y\rangle$ has index 1, and therefore $G = H$ is the cyclic group of order 3.</p>
<p>That being the case, $x$ cannot be either $y$ or ... | i. m. soloveichik | 32,940 | <p>$e=y^{-1}(yxyxy)y=xyxy^{-1}$ so $y=xyx$ and $xy=yx^{-1}$; therefore $e=y^3=(xyx)^3=xyx^{-1}yx^{-1}yx=xxyxyyx=x^{-1}yxy^{-1}x$. Cancelling on the left and right yields $x=e$ and the result follows.</p>
|
586,724 | <p>Please how to integrate this $$\frac{1}{(1-e^{2x})^{1/2}}$$
I have tried $u= e^x$
But I think that is wrong
So can anyone help me ?</p>
| Stefan Smith | 55,689 | <p>Hint: $u = e^x$ should work. You get</p>
<p>$$\int \frac{e^x}{e^x\sqrt{1-e^{2x}}}\,dx=\int\frac{1}{u\sqrt{1-u^2}} \,du.$$</p>
<p>Then try trig substitution with $u = \sin\theta$.</p>
|
1,115,793 | <p>Let's consider a number of linear operators, defined on a finite dimensional complex vector space, which two by two commutes with each other. (the amount of them can be infinite). How to prove that that will have a common eigenvector?</p>
<p>The finite case can be done by induction:
1) $n=2$, $AB=BA$, then let $x$ ... | copper.hat | 27,978 | <p>Here is an inductive proof:</p>
<p>Let ${\cal C}$ be the commuting family of operators.</p>
<p>Let me call an operator $A$ on a subspace $S$ a multiplier operator on $S$ <strong>iff</strong> there exists some $\lambda$ such that $Ax=\lambda x$ for all $x \in S$.</p>
<p>Let me call a subspace $S$ invariant <strong... |
176,613 | <p>Working on Harmonic numbers, I found this very interesting recurrence relation :
$$
H_n = \frac{n+1}{n-1} \sum_{k=1}^{n-1}\left(\frac{2}{k+1}-\frac{1}{1+n-k}\right)H_k
,\quad \forall\ n\in\mathbb{N},n>1$$
My proof of this is quite long and complicated, so I was wondering if someone knows an elegant or concise one... | J. M. ain't a mathematician | 498 | <p>Here's a generating function route: as I already mentioned in the comments,</p>
<p>$$\begin{align*}
H_n&=\frac{n+1}{n-1}\sum_{k=1}^{n-1}\left(\frac{2}{k+1}-\frac{1}{n-k+1}\right)H_k\\
H_n&=\frac{n+1}{n-1}\left(2\sum_{k=1}^{n-1}\frac{H_k}{k+1}-\sum_{k=1}^{n-1}\frac{H_k}{n-k+1}\right)\\
H_n&=\frac{n+1}{n-... |
176,613 | <p>Working on Harmonic numbers, I found this very interesting recurrence relation :
$$
H_n = \frac{n+1}{n-1} \sum_{k=1}^{n-1}\left(\frac{2}{k+1}-\frac{1}{1+n-k}\right)H_k
,\quad \forall\ n\in\mathbb{N},n>1$$
My proof of this is quite long and complicated, so I was wondering if someone knows an elegant or concise one... | M. M. | 36,778 | <p>Thanks for your help. Here is another proof that I found today :</p>
<p>From the well known $\zeta(3)=\frac{1}{2}\sum_{k=1}^{\infty}\frac{H_k}{k^2}$, we find
$$
\zeta(3)=\sum_{k=1}^{\infty}\frac{H_k}{(k+1)^2} \quad\quad(1)
$$
Were $\zeta(z)$ is the Riemann zeta function. But also,
$$
\zeta(3)=\frac{1}{2}\int_{0}^{\... |
36,272 | <p>Is there a characterisation for which $x\in\mathbb{R}$ the value $\arctan(x)$ is a rational multiple of $\pi$? </p>
<p>Or reformulated: What is the "structure" of the subset $A\subseteq\mathbb{R}$ which fulfils
$$ \arctan(x) \in \pi\mathbb{Q} \Leftrightarrow x\in A$$
for all $x\in\mathbb{R}$?</p>
| Joseph O'Rourke | 6,094 | <p>A partial answer was provided in response to my MSE question,
"<a href="https://math.stackexchange.com/questions/79861/">ArcTan(2) a rational multiple of $\pi$</a>?"</p>
<p>There Thomas Andrews showed that $\arctan(x)$ is not a rational multiple of $\pi$ for any
$x$ rational, except for $-1,0,1$.
More specifically:... |
4,494,199 | <p>This is a problem from a past qualifying exam in complex analysis. I'm working through these to study for my own upcoming qual. For this question, I think my proof is fairly straightforward, but I'd like to know whether or not it is correct and complete. I'm also interested in other ways of answer the question. Than... | Adam Rubinson | 29,156 | <p>An easier proof for me is the following:</p>
<p>Firstly, <span class="math-container">$\frac a{1+b}+\frac b{1+a}\geq \frac{b}{1+a}>0,\ $</span> as numerator and denominator of <span class="math-container">$\frac{b}{1+a}$</span> are both positive.</p>
<p>Secondly,</p>
<p><span class="math-container">$$\frac a{1+b}... |
514,352 | <p>Can anybody give me an example of a multiplicative function $f$ such that
$$\prod_p \sum_{k=0}^\infty f(p^k)$$
converges absolutely and such that
$$\sum_{n=1}^\infty f(n)$$
diverges?</p>
| PITTALUGA | 94,471 | <p>Wherever $\sum_n f(n)$ converges also $\prod_p\sum_k f(p^k)$ does and vice versa.
I emphasize that this is true <em>only inside the convergence region.</em>
This comes from the fact that $f$ is multiplicative and from the fundamental theorem of arithmetic.
So the answer is: no.</p>
|
889,628 | <p>The question and answer is shown but I don't fully understand the answer for part a. Could someone please explain to me why the integral setup for the marginal density function of y1 is from y1 to 1, and not 0 to 1? And also the same thing for y2. Thank you
<img src="https://i.stack.imgur.com/ECQQ7.png" alt="Questio... | Sara | 220,380 | <p>To tell someone what infinity ... Take a sheet of $A4$ paper and divide it into two halves. Now take one of the halves and divide it again. Repeat this step indefinitely. Here ask the question 'Will this process finish?'.</p>
|
874,607 | <blockquote>
<p>There were 10 questions on a test. A student gets 5 points for every correct answer and 3 points for every partially correct answer. If the student got 19 points, how many correct and partial answers did they have?</p>
</blockquote>
<p>To solve the problem I express the total number of points as the ... | Rebecca J. Stones | 91,818 | <p>We see that $19 \equiv 4 \pmod 5$, so we must also have $3y \equiv 4 \pmod 5$. We check that the multiplicative inverse of $3$ modulo $5$ is $2$ (since $2 \times 3=6 \equiv 1 \pmod 5$), so we have $2 \times 3y \equiv 2 \times 4 \pmod 5$, i.e., $y \equiv 3 \pmod 5$.</p>
<p>Since $0 \leq y \leq \lfloor 19/3 \rfloor=... |
183,768 | <p>Prove convergence\divergence of the series:
$$\sum_{n=1}^{\infty}\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$$</p>
<p>Here is what I have at the moment:</p>
<p><strong>Method I</strong></p>
<p>My first way uses a result that is related to <strong><a href="http://en.wikipedia.org/wiki/Wallis_product" rel=... | Morgan Sherman | 21,512 | <p>Since
$$ \frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)}{2 \cdot 4 \cdot \ldots \cdot (2n)} \ge \frac{1 \cdot 2 \cdot 4 \cdot \ldots \cdot (2n-2)}{2 \cdot 4 \cdot \ldots \cdot (2n)} = \frac1{2n} $$
the series diverges by comparison to the Harmonic series.</p>
|
2,259,243 | <p>How to solve T(n) = T(n-2) + n using iterative substitution</p>
<pre><code>Base case:
T(0) = 1
T(1) = 1
Solve:
T(n) = T(n-2) + n
</code></pre>
<p>Currently I have:</p>
<pre><code>T(n) = T(n-2) + n
= T(n-4) + n - 2 + n = T(n-4) + 2n - 2
= T(n-6) + n - 4 + n - 2 + n = T(n-6) + 3n - 6
= T(n-8) + n - ... | user1015113 | 1,015,113 | <p>Let P be a permutation... P is either odd or even so if P is odd then it's inverse also odd.. We know that identity permutation is product of a permutation and it's inverse .. So identity permutation will be product of P and it's inverse.. Since both permutation is odd so their product is even... Similarly we can ... |
29,926 | <p>Is there an algorithm which will allow me to find an isomorphism between two graphs if I have their adjacency lists?</p>
| Reed Richards | 7,041 | <p>It seems like it can't be done. If the $K^{-1}DK^{-1}$ is a m-by-n matrix then the map can be seen as just $m$ scalar products. However the <em>log</em> of a scalar product can't be tampered with since $\log(a \cdot b) = \log(a_{1} b_{1} + a_{2} b_{2} +...)$. Further on if we take the <em>log</em> of the individual ... |
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | hce | 2,781 | <p>Milnor - Lectures on the h-cobordism theorem</p>
|
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | hce | 2,781 | <p>Atiyah - K-Theory</p>
|
79,658 | <p>My knowledge is very limited for complex geometry. I have the following question:</p>
<p>If we have two complex vector bundles $E\to X$ and $F\to X$ such that we have an isomorphism $\mathcal O\left(E\right) \cong \mathcal O\left(F\right)$ between the sheaf of holomorphic sections, do we have an isomorphism $E \con... | Georges Elencwajg | 450 | <p>No. On $\mathbb P^1=\mathbb P^1(\mathbb C)$ we have $\Gamma(\mathbb P^1,\mathcal O_{\mathbb P^1}(-1))=\Gamma(\mathbb P^1,\mathcal O_{\mathbb P^1}(-2)=0$, but $O_{\mathbb P^1}(-1)$ and $O_{\mathbb P^1}(-2)$ are not isomorphic. </p>
<p>However on an <em>affine algebraic</em> variety $X$, the answer is "yes". There ... |
79,658 | <p>My knowledge is very limited for complex geometry. I have the following question:</p>
<p>If we have two complex vector bundles $E\to X$ and $F\to X$ such that we have an isomorphism $\mathcal O\left(E\right) \cong \mathcal O\left(F\right)$ between the sheaf of holomorphic sections, do we have an isomorphism $E \con... | user2035 | 2,035 | <p>Another way of looking at the problem is to consider the (set-valued) sheaf of isomorphisms $E\to F$ and the sheaf of isomorphisms $\mathcal O(E)\to\mathcal O(F)$. There is clearly a map from the former to the latter, so we only need to show that it is an isomorphism locally, i.e., we may assume that both $E$ and $F... |
2,962,377 | <p>Rewrite <span class="math-container">$f(x,y) = 1-x^2y^2$</span> as a product <span class="math-container">$g(x) \cdot h(y)$</span> (both arbitrary functions)</p>
<p>To make more clear what I'm talking about I will give a example.</p>
<p>Rewrite <span class="math-container">$f(x,y) = 1+x-y-xy$</span> as <span class... | Daniel Schepler | 337,888 | <p>Suppose we had functions <span class="math-container">$g,h$</span> with <span class="math-container">$g(x) h(y) = 1 - x^2 y^2$</span>. Then substituting in <span class="math-container">$x:=1, y:=1$</span>, we would have <span class="math-container">$g(1) h(1) = 0$</span>, so either <span class="math-container">$g(1... |
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