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 |
|---|---|---|---|---|
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | Joe Silverman | 11,926 | <p>It's 12:40am New Year's Day, so maybe not the best time to be writing MO responses, but I loved reading everyone's encouraging answers to this question. (BTW: @Lubin: Taking algebra, with a dollop of algebraic number theory, from you was inspiring.) Anyway, my answer to the OP would be to learn lots and to work hard... |
328,868 | <p>It is easily shown that the function $$\begin{cases} \exp \left(\frac{1}{x^2-1} \right) & |x| < 1 \\ 0 & \text{otherwise} \\ \end{cases}$$
is smooth and has compact support in $\mathbb R$. I tried playing with it to find a function with the following properties:</p>
<p>a. $f(x)=0$ for $x \le 0$</p>
<p>b... | user1337 | 62,839 | <p>I've found this other explicit construction of a transition function (I believe it's from Spivak's Calculus on Manifolds).</p>
<ol>
<li><p>Let $$g(x)=\begin{cases} \exp \left( \frac{1}{x^2-1} \right) & |x|<1 \\ 0 & |x| \geq 1 \end{cases}$$
and let $A:=\int_{-\infty}^\infty g(x) \mathrm{d}x$.</p></li>
<li... |
57,223 | <p><strong>No error in 10.0.2</strong></p>
<hr>
<p>When using <code>Correlation[]</code> function I sometimes get the strange warning:</p>
<pre><code>CorrelationTest::nortst: "At least one of the p-values in {0.0527317}, resulting from a test for normality, is below 0.05`. The tests in \!\({\"PearsonCorrelation\"}\)... | Andy Ross | 43 | <p>It appears to be a bug in the reporting mechanism. The individual components <code>x</code> and <code>y</code> are being tested for normality but the reported value is that of a joint test for multivariate normality. The conclusion is correct, the message is wrong.</p>
<pre><code>SeedRandom[2154];
x = RandomReal[... |
2,948,557 | <p>So I've been trying to prove that
<span class="math-container">$$\sum^{\infty}_{n=1}\frac{(-1)^{n}\sin(n)}{n}=-\frac{1}{2}$$</span></p>
<p>I've tried putting various bounds on it to see if I can "squeeze" out the result. Say something like (one of many tried examples):
<span class="math-container">$$ -\frac{1}{n}-\... | Siong Thye Goh | 306,553 | <p>Guide:</p>
<p>Verify by computing the fourier series of <span class="math-container">$f(x) = \frac{x}{\pi}, -\pi < x<\pi$</span></p>
<p><span class="math-container">$$\frac{x}{\pi} = \frac{2}{\pi}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\sin(nx)$$</span></p>
<p>and choose an appropriate value of <span class="m... |
2,948,557 | <p>So I've been trying to prove that
<span class="math-container">$$\sum^{\infty}_{n=1}\frac{(-1)^{n}\sin(n)}{n}=-\frac{1}{2}$$</span></p>
<p>I've tried putting various bounds on it to see if I can "squeeze" out the result. Say something like (one of many tried examples):
<span class="math-container">$$ -\frac{1}{n}-\... | Parcly Taxel | 357,390 | <p>A direct approach:
<span class="math-container">$$\sum_{n=1}^\infty\frac{(-1)^n\sin n}n=\sum_{n=1}^\infty\frac{(-1)^n\Im (e^{in})}n=-\Im\sum_{n=1}^\infty\frac{(-1)^{n+1}(e^i)^n}n=-\Im\ln(1+e^i)$$</span>
where the Maclaurin series of <span class="math-container">$\ln(1+z)$</span> was used. As can be seen by drawing a... |
3,380,479 | <p>i'm doing a seminar about Galois Theory but I have a problem with the definition of purely inseparable element and the one of purely inseparable extension, I read the definitions but i would like to see an example for each one of them. If somebody can help me please. </p>
| lhf | 589 | <p>Instead of the extended euclidean algorithm, you can also use Fermat's theorem.</p>
<p>For instance, <span class="math-container">$1 \equiv 7^{12} = 7 \cdot 7^{11}$</span>. Therefore, the inverse of <span class="math-container">$7$</span> mod <span class="math-container">$13$</span> is <span class="math-container">... |
2,090,452 | <p>I have a signature given with the real numbers as its universe and addition and multiplication as functions. I need to write the following expression in First Order Logic.</p>
<ul>
<li>$x$ is a rational number</li>
</ul>
<p>My idea: $\varphi(x) = \exists a \, \exists b \, x = \frac{a}{b}$.</p>
<p>The problem is ... | Noah Schweber | 28,111 | <p>Let me strengthen what others have already said:</p>
<p>The structure $\mathcal{R}=(\mathbb{R}; +, \times)$ is <em>decidable</em> (this is <a href="https://en.wikipedia.org/wiki/Real_closed_field#Model_theory:_decidability_and_quantifier_elimination" rel="nofollow noreferrer">due to Tarski</a>). This immediately ru... |
969,871 | <p>First, by definition of odd functions I have $-f(x) = f (-x)$. It would follow that $|–f(x)|= |f(-x)|$. Then given $\epsilon > 0$ there exists $\delta > 0$ s.t. when $0 < |x| < \delta$, it follows that $|f(x)-M| < \epsilon$. With $M = 0$, then $|f(x)| < \epsilon$. (approaching $0$ from the left... | Some Math Student | 181,130 | <p>I have the slight impression you're making your job a bit harder than it seems to be.</p>
<p>First, because our function is continuous, we know that our limit does exist.</p>
<p>Second, I suggest you assume your limit is anything other than zero. Without loss of generality, we can say it could be a $\varepsilon &g... |
1,072,750 | <p>I'm looking at this solution to this problem:</p>
<p><img src="https://i.stack.imgur.com/qQTTU.png" alt="enter image description here"></p>
<p>I'm getting thrown off by the special case where $n = 2$. If $n = 2$, why must it be that $x = 1$? All that we then know is that $x^2 = 1$ or that $x = x^{-1}$. However, I ... | rschwieb | 29,335 | <p>Probably what is throwing you off is that $x\in R/P$ and <em>not</em> $x\in R$.</p>
<p>Of course, $x^2=x$ in $R$ without $x$ being $1$. But in the quotient, there are no zero divisors (because you're quotienting by a prime ideal.) So you must examine $x^2-x=0$.</p>
|
1,201,717 | <p>Inn$(G)=\{\varphi_g \in \text{Aut}(G) \mid g \in G\}$</p>
<p>If $\varphi_g, \varphi_h \in \text{Inn}(G)$, then
$$\varphi_g \varphi_h (x) =\varphi_g(hxh^{-1})=ghxh^{-1}g^{-1}=ghx(gh)^{-1}=\varphi_{gh} \in \text{Inn}(G)$$
Also, since $\varphi_g\varphi_g^{-1}=x$, and $\varphi_g\varphi_{g^{-1}}=x$,
$$\varphi_g^{-1}=\va... | Hagen von Eitzen | 39,174 | <p>Well-definedness is not needed to be shown as there is no choice involved: Given $x\in G$ you are to exhibit some $\psi(x)\in\operatorname{Inn}(G)$. Since you are given $x\in G$, no-one can prevent you from considering the inner automorphism $\phi_x$.</p>
<hr>
<p>Just to make clear where we do need to show well-d... |
1,316,661 | <p>Let $\{ h_n :X \to Y\}_{n \in \mathbb{Z^+}} $ be a sequence of continuous functions from a topological space $X$ to another topological space $Y$, and for each $n$ let $U_n$ be an open subset of $Y$. </p>
<p>Does
$$
\bigcup_{n=1}^{\infty} {h_n}^{-1} (\overline{U_n}) \subset \overline{ \bigcup_{n=1}^{\infty} {h_n}... | Rory Grice | 242,544 | <p>If a is a real number then you can make $x^a$ as small or as large as you like my making a more and more negative or more and more positive, regardless of the value of x. consider:</p>
<p>1)If x is between zero and one, then you can make your number a largely positive to get a number that is very close to zero and ... |
3,174,153 | <p>Let <span class="math-container">$X_1,X_2,...X_n$</span> be a random sample from <span class="math-container">$f(x,\theta)=\frac{1}{2 \theta}e^{\frac{-|x|}{\theta}}$</span>.We know by Factorisation theorem that <span class="math-container">$\frac{\sum |X_i|}{n}$</span> is sufficient for <span class="math-container"... | Masoud | 653,056 | <p>let <span class="math-container">$S$</span> is minimal sufficient. <span class="math-container">$T$</span> is not sufficient if exist <span class="math-container">$x,y\in support$</span> </p>
<p><span class="math-container">$T(x)=T(y)$</span> but <span class="math-container">$S(x)\neq S(y)$</span> </p>
<p>let n=2,... |
653,451 | <p>For $X$ a Banach space, let me define the space $C^0([0,T];X)$ to consist of elements $u:[0,T] \to X$ such that
$$\lVert u \rVert_{C^0} := \max_{t \in [0,T]}\lVert u(t) \rVert_X < \infty.$$
So the difference is that I don't care about continuity of $u$ in $t$. This defines a norm. </p>
<p>For completeness, let $... | Giorgio Mossa | 11,888 | <p>It not so clear what is they mean, but I guess what they mean is that if you consider the graph above (in which edges with different labels are different) then you cannot put on that graph a structure of a category.</p>
<p>To prove that you have to use reductio ab absurdum:
if there where any category structure on ... |
2,029,279 | <blockquote>
<p>Let <span class="math-container">$T:\Bbb R^2\to \Bbb R^2$</span> be a linear transformation such that <span class="math-container">$T (1,1)=(9,2)$</span> and <span class="math-container">$T(2,-3)=(4,-1)$</span>.</p>
<p>A) Determine if the vectors <span class="math-container">$(1,1)$</span> and <span cla... | user5216 | 250,930 | <p>a) hint: Check linear independence.
b) Write any vector (x,y) as linear combination of basis you have and use the property of linear operator.</p>
|
4,117,954 | <p>Let <span class="math-container">$l^\infty$</span> be the space of all real bounded sequences equipped with supremum norm. Let <span class="math-container">$S$</span> be the shift operator defined on <span class="math-container">$l^\infty$</span> by <span class="math-container">$(Sx)_n=x_{n+1}$</span>, <span class="... | Theo Bendit | 248,286 | <p>As I said in the comments, you need to show that no <span class="math-container">$(y_n)_n \in \ell^1$</span> exists so that
<span class="math-container">$$L(x_n) = \sum_{n=1}^\infty x_n y_n$$</span>
holds for all <span class="math-container">$(x_n)_n \in \ell^\infty$</span>. Suppose this were the case, and consider ... |
2,260,172 | <p>I know that if two topological spaces $X$ and $Y$ are homeomorphic then so are their one point compactifications $X^*$ and $Y^*$. If $X$ and $Y$ (say both are smooth manifolds) are diffeomorphic what do we know about $X^*$ and $Y^*$? Are they diffeomorphic too or only homeomorphic? If the latter is true, are there ... | Community | -1 | <p>You can't put a differentiable structure on the one-point compatification, so your question is not well defined. For example, consider the union of several open segments $(0,1)$, the one-point compactification will be a wedge of circle which is not a smooth manifold. I also think that one point compactification of $... |
280,292 | <p>Find $a,c$ such that:</p>
<p>$$f(x)= \begin{cases} a\frac{\exp(tgx)}{(1+\exp(tgx))} &\text{for }|x|<\pi/2 \\[2ex] \exp(cx)-2 &\text{for } |x|\ge\pi/2 \end{cases}$$</p>
<p>is continuous. </p>
<p>How do I evaluate the left- and right-hand limits to see if they are equal?</p>
| Mhenni Benghorbal | 35,472 | <p>To prove the function is continuous, you need to fullfill the condition </p>
<p>$$ \lim_{x\to x_0}f(x)=f(x_0). $$</p>
<p>So, as a first step, you need to prove that the limit exists at the point $x_0$, that is why you need to prove
$$\lim_{x\to \pi/2^{-}}f(x)= \lim_{x\to \pi/2^{+}}f(x).$$ </p>
|
3,305,861 | <p>This is an answer from Slader for <em>Understanding Analysis</em> by Abbott. The book suggests to use <span class="math-container">$a = a-b+b$</span> but the steps for this answer doesn't make any sense to me. For one, why does s/he write use <span class="math-container">$a=a-b+b$</span> when it looks like they used... | Quanto | 686,284 | <p>WolframAlpha, though handy, is not yet advanced to perform certain integrals satisfactorily. For less familiar integrands, it tends to fall back to complex variables, leading often to uninteresting, perhaps only symbolically useful, results.</p>
<p>The integral in question can be integrated in real variables explici... |
2,115,451 | <p>$$\lim_{n\to \infty}\frac{\frac{1}{2\ln2}+\frac{1}{3\ln3}+\ldots+\frac{1}{n\,\ln\,n}}{\ln(\ln\,n)}$$</p>
<p>The result is $1$ (according to the book, though it does not show the steps, which I'm interested in).</p>
<p>I've applied the theorem and it led me to an equally unhelpful limit.</p>
| JoeyJoey | 410,414 | <p>We apply the Stolz-Cesaro theorem: $$lim_{n\to \infty}\frac{(\frac{1}{2ln2}+\frac{1}{3ln3}+...+\frac{1}{n\,ln\,n}+\frac{1}{(n+1)\,ln(n+1)})-(\frac{1}{2ln2}+\frac{1}{3ln3}+...+\frac{1}{n\,ln\,n})}{ln(ln(n+1))-ln(ln\,n)}=lim_{n\to \infty}\frac{\,\frac{1}{(n+1)\,ln(n+1)}\,}{\,ln(\frac{ln(n+1)}{ln\,n})\,}$$</p>
<p>We k... |
1,139,885 | <p>If $a$ and $b$ are in a group $G$ and $ab=ba$, show that $xax^{-1}$ commutes with $xbx^{-1}$ for any $x \in G$.</p>
<p>So I wrote: </p>
<p>WWTS: $\bf{xax^{-1} \times xbx^{-1}=xbx^{-1}\times xax^{-1} }$</p>
<p>Now, the problem I have is I don't know where to start. Let's say if I start with what is given:</p>
<p>... | Milo Brandt | 174,927 | <p>A very useful idea to such problems is that the map
$$f(a)=xax^{-1}$$
is called "conjugation" and is an homomorphism from the group to itself. That is to say, we can prove, for all $a,b$ that:
$$f(a)f(b)=f(ab).$$
This is very clear if you expand both expressions. Now, if we start with
$$ab=ba$$
We can apply $f$ to b... |
134,424 | <p>I would like to create a function where I can define which case I want to use to create a path.</p>
<pre><code>p1 = {40, 48}; p2 = {50, 116}; p3 = {63, 160};
listPurple = Symbol["p" <> ToString[#]] & /@ Range[3];
disksPurple = {Purple, Disk[#, 2] & /@ listPurple};
Graphics[{disksPurple}, ImageSize -&g... | corey979 | 22,013 | <p><strong>Function:</strong></p>
<pre><code>path[points_List, directions_List] :=
Block[{disksPurple, pseudoCross, v, h, ints, dirs},
If[Length@directions != Length@points - 1, Print["incorrect input"]; Abort[]];
disksPurple = {Purple, Disk[#, 2] & /@ points};
pseudoCross[p1_, p2_] := {{p1[[1]], p2[[2]]},... |
877,850 | <p>I'm trying to calculate the expected area of a random triangle with a fixed perimeter of 1. </p>
<p>My initial plan was to create an ellipse where one point on the ellipse is moved around and the triangle that is formed with the foci as the two other vertices (which would have a fixed perimeter) would have all the ... | Taladris | 70,123 | <p>A calculus proof:</p>
<p>Let $f(x)=\arctan(x)+\textrm{arcot}(x)$ for all $x\in {\mathbb R}$. The function $f$ is differentiable and $f'(x)=\frac{1}{1+x^2}+\frac{-1}{1+x^2}=0$, so $f$ is constant. The value of the constant is $f(0)=\arctan(0)+\textrm{arccot}(0)=0+\frac{\pi}{2}$ (remember that $\tan(0)=0$ and $\cot(\... |
200,063 | <p>I am looking to evaluate</p>
<p>$$\int_0^1 x \sinh (x) \ \mathrm{dx}$$</p>
| Aang | 33,989 | <p>$2\sinh(x)=e^x-e^{-x}$ and $\int e^x(f(x)+f'(x))dx=e^xf(x)$</p>
<p>Thus, $$2\int_0^1 x \sinh (x)dx=\int_0^1 xe^xdx-\int_0^1 xe^{-x}dx$$ $$=\int_0^1 e^x(x+1)dx-\int_0^1 e^{-x}(-x+1)d(-x)-\int_0^1 e^{x}dx+\int_0^1 e^{-x}d(-x)$$ $$=xe^x|_0^1-(-xe^{-x})|_0^1-e^x|_0^1+e^{-x}|_0^1$$ $$=e+\frac{1}{e}-(e-1)+(\frac{1}{e}-1)... |
3,949,525 | <p>This question is from Ponnusamy and silvermann complex analysis and I am making some mistake in this question and unable to find it.</p>
<blockquote>
<p>Show that <span class="math-container">$\prod_{n=1}^{\infty} (1-z/n) e^{z/n +5z^2/n^2}$</span> is entire.</p>
</blockquote>
<p><span class="math-container">$1+ f_n ... | reuns | 276,986 | <p>Because <span class="math-container">$f(z)=\frac{\log(1-z)+z}{z^2}$</span> is analytic thus continuous for <span class="math-container">$|z|< 1$</span> then <span class="math-container">$$C= \sup_{|z|\le 1/2}|f(z)|<\infty$$</span>
gives <span class="math-container">$|\log(1-z/n)+z/n|\le Cz^2/n^2$</span> for <s... |
3,213,253 | <p>We're given dynamical system:</p>
<p><span class="math-container">$$
\dot x = -x + y + x (x^2 + y^2)\\
\dot y = -y -2x + y (x^2 + y^2)
$$</span></p>
<p>Question is what's the largest constant <span class="math-container">$r_0$</span> s.t. circle <span class="math-container">$x^2+y^2 < r_0^2$</span> lies in the ... | Maxim | 491,644 | <p>Since the limit cycle looks like an ellipse centered at the origin, we'll try to find a Lyapunov function <span class="math-container">$V = a x^2 + b x y + c y^2$</span> s.t. <span class="math-container">$\dot V = (A x^2 + B x y + C y^2 + D) (V - V_0)$</span>. This only requires equating the coefficients of two poly... |
1,498,952 | <p>How does one compute the bias for the estimator given by the sample geometric mean for a gamma distribution with parameters ($\theta$,1)?</p>
<p>i.e:
Given $X_1,...,X_n$ are iid with distribution Gamma($\theta$,1), what is the bias of the estimator given by: $$\hat\theta = \left(\prod_{i=1}^n X_i\right)^{1/n}$$</p>... | Narek Margaryan | 203,307 | <p>For a $\Gamma(k,\lambda)$ you have to count the integral:</p>
<p>$$\int_0^\infty \frac {X^{k-1+1/n}e^{-\lambda X}\lambda^k}{\Gamma(k)}dX$$ </p>
<p>All we need is to multiply and divide the fraction by $\lambda^{1/n-1}$</p>
<p>$$\frac{1}{\lambda^{1/n}}\int_0^\infty \frac {(X\lambda)^{k+1/n-1}e^{-\lambda X}}{\Gamma... |
2,484,027 | <p>I've been trying to solve this permutation problem. I know that it's been posted on this site before, but my question is about the specific approach I take to solving it.</p>
<p>Here's what I thought I'd do : I could first figure out the total number of permutation where AT LEAST $2$ girls are together, and then su... | Vysakh Mohan | 707,497 | <p>You're overcounting the number of ways "at least two" girls can sit together, because for example</p>
<p>Boy1 Boy2 <em>Girl1 Girl2 Girl3</em> Boy3 Boy4 Boy5
is produced twice: Once considering <strong>Girl1 Girl2</strong> as the unit, and once considering <strong>Girl2 Girl3</strong> as the unit.
so Girl1 Girl2 an... |
3,244,365 | <blockquote>
<p>Prove that the product of any four consecutive integers is one less than a perfect square.</p>
</blockquote>
<p>My first idea is the let k be a member of the integers. Let <span class="math-container">$m$</span> which also belongs to the integers be equal to <span class="math-container">$k(k+1)(k+2)(... | Dan Uznanski | 167,895 | <p>We know that a number that's one less than a perfect square has two factors that are two apart: <span class="math-container">$y^2-1 = (y-1)(y+1)$</span>. If we can find a way to make the factors of <span class="math-container">$x(x+1)(x+2)(x+3)$</span> combine into two factors that are two apart, we win.</p>
<p><s... |
356,583 | <p>In the paper <a href="https://arxiv.org/abs/1811.02002" rel="nofollow noreferrer">"Finding Mixed Nash Equilibria of Generative Adversarial Networks"</a> the authors write in equation (1) on page 2:</p>
<blockquote>
<p>Consider the classical formulation of a two-player game with
<em>finitely</em> many strategies... | Eilon | 64,609 | <p>The classical formulation of a two player zero sum game with finitely many strategies is <span class="math-container">$\langle q, Ap \rangle$</span>. There is no need to introduce the vector <span class="math-container">$a$</span>, as it can be incorporated in the matrix <span class="math-container">$A$</span>. In t... |
2,013,459 | <p>The period of $\sin(x)$ is $2\pi$. Thus the period of $\sin(\pi x)$ will become $T_1=2$, similarly the period of $\sin(2\pi x)$ is $T_2=1$ and for $\sin(5\pi x)$, the period is $T_3=\frac{2}{5}$.</p>
<p>To find the period of $\sin(\pi x)-\sin(2\pi x)+\sin(5\pi x)$, we can have
$$\frac{T_1}{T_{2}}=\frac{2}{1}\Righta... | String | 94,971 | <p>You have
$$
(T_1,T_2,T_3)=\left(\tfrac{10}5,\tfrac55,\tfrac25\right)
$$
where $\operatorname{LCM}(10,5,2)=10$ (least common multiple). Thus the minimal common integer multiple where those fractions coincide must be
$$
\frac{10}5=2\cdot \frac 55=5\cdot\frac25=2
$$
So your suggestion is correct.</p>
|
2,648,526 | <p>I really cannot figure this question out. Can anyone help me please!?</p>
<blockquote>
<p>Prove that the length of the median $m_a$ of obtuse triangle $△ABC$ with the obtuse $∠CAB$ is smaller than $\dfrac{1}{2}|BC|$.</p>
</blockquote>
<p>Thank you very much!</p>
| Dr. Sonnhard Graubner | 175,066 | <p>by the theorem of cosines we have
$$,m_a^2=c^2+\frac{a^2}{4}-2\frac{a}{2}c\cos(\beta)$$
with $$\cos(\beta)=\frac{a^2+c^2-b^2}{2ac}$$
we get
$$m_a^2=c^2+\frac{a^2}{4}-ac\left(\frac{a^2+c^2-b^2}{2ac}\right)$$
or $$m_a^2=\frac{2(c^2+b^2)-a^2}{4}$$
and we get
$$m_a^2>\frac{a^2}{4}$$ and this is equivalent to
$$b^2+c^... |
1,540,756 | <p>I need help trying to solve this question, been cracking my head for the whole week and my professor said he used an online solver but in exams we have to solve by hand!</p>
<p>Given these 8 equations, we are supposed to solve for $i_0, i_1, \dots, i_7$:
$$\begin{array}{rl}
i_1+i_2 &= 12 \\
i_2+i_5+i_6 &= 0... | amd | 265,466 | <p>The above equations can be represented by the following augmented matrix:
$$\left(\begin{array}{rrrrrrrr|r}
0&1&1&0&0&0&0&0 & 12 \\
0&0&1&0&0&1&1&0 & 0 \\
0&0&0&1&0&1&0&1 & 0 \\
0&0&1&0&-1&1&0&... |
3,694,719 | <p>source: BMO2 2004 q4</p>
<p>The real number <span class="math-container">$x$</span> between <span class="math-container">$0$</span> and <span class="math-container">$1$</span> has decimal representation <span class="math-container">$0.a_1a_2a_3a_4\dots$</span>
And enjoys the following property: the number of distinc... | Balaji sb | 213,498 | <p>A reduction of problem (not a full answer):
Let us retain only the sequences of length 2004 that repeat infinitely often. Hence w.l.o.g there exists a smallest <span class="math-container">$\ell$</span> such that <span class="math-container">$a_k,a_{k+1},...,a_{k+\ell}$</span> is one of <span class="math-container">... |
79,317 | <p>This concerns one of those "well known" facts, referred to in a recent preprint I've been looking at. In principle it's elementary, but I can't pin down an explicit textbook reference for it. Start with two finite groups $A,B$ and their product $G:=A \times B$, working over a splitting field $K$ for the groups i... | Benjamin Steinberg | 15,934 | <p>The paper Representations of direct products of finite groups.
Burton Fein
Source: Pacific J. Math. Volume 20, Number 1 (1967), 45-58.</p>
<p><a href="https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-20/issue-1/Representations-of-direct-products-of-finite-groups/pjm/1102992967.full" rel="nofo... |
9,302 | <p>Say I have a symmetric matrix. I have the concept of 2-norm as defined on wikipedia. Now I want to prove (disprove?) that the norm of a symmetric matrix is maximum absolute value of its eigenvalue. I would really appreciate if this can be done only using simple concepts of linear algebra.</p>
<p>I am quite new to m... | mpiktas | 4,742 | <p>Here is a simple explanation not necessarily from linear algebra. We have</p>
<p>$$\|A\|_2=\max_{\|x\|=1}\|Ax\|$$</p>
<p>where $\|\cdot\|$ is simple euclidean norm. This is a constrained optimisation problem with Lagrange function:</p>
<p>$$L(x,\lambda)=\|Ax\|^2-\lambda(\|x\|^2-1)=x'A^2x-\lambda(x'x-1)$$<... |
3,435,511 | <p>I'm looking to write down formally that a multiset of elements contains at least two elements that differs in value.
e.g.,
S1 = {1,1,1,1,1,1} and S2={1,1,1,1,0,1}
S1 has all identical elements, S2 has at least two elements that differs in value.</p>
| Robo300 | 718,762 | <p><img src="https://i.stack.imgur.com/nBazV.jpg" alt="the work">
I believe that the first derivative at zero is zero, my justification is in the attached image. It basically boils down to defining a multivariate function <span class="math-container">$F(x,y)$</span> such that <span class="math-container">$f(x) = F(x,x)... |
3,435,511 | <p>I'm looking to write down formally that a multiset of elements contains at least two elements that differs in value.
e.g.,
S1 = {1,1,1,1,1,1} and S2={1,1,1,1,0,1}
S1 has all identical elements, S2 has at least two elements that differs in value.</p>
| ComplexYetTrivial | 570,419 | <p><strong>Some integral representations</strong></p>
<p>For <span class="math-container">$x > 0$</span> we have
<span class="math-container">\begin{align}
f(x) &= \int \limits_x^\infty - \log \left(1 - \mathrm{e}^{-k}\right) \sqrt{1 - \frac{x^2}{k^2}} \, \mathrm{d}k \tag{1} \\
&\!\!\!\!~\stackrel{k = x u}{... |
1,077,119 | <p>I am starting to learn about tensor products of abelian groups.</p>
<p>Why is the tensor product defined for <strong>abelian</strong> groups? In which part of the construction the commutativity of the groups is needed?</p>
| Mister Benjamin Dover | 196,215 | <p>There is a version of the tensor product for nonabelian groups, but this notion is much more specialized. See <a href="http://www-irma.u-strasbg.fr/~loday/PAPERS/87BrownLoday%28vanKampen%29.pdf" rel="nofollow">http://www-irma.u-strasbg.fr/~loday/PAPERS/87BrownLoday%28vanKampen%29.pdf</a>, section 2. In the construct... |
2,049,714 | <p>Does it make sense when people say "statistically impossible"?</p>
| Peter Darmis | 416,428 | <p>In <code>statistics</code> there is nothing <code>impossible</code>, you may want to change this to <code>improbable</code> which is something completely different. Theoretically speaking everything could have <strong>1÷∞ = ~0</strong> probability but that just not mean it could actually happen... |
4,030,877 | <p><strong>Question:</strong> Ten different candies are given to three children <code>A</code>, <code>B</code> and <code>C</code>. each child has at least one. How many different ways are there?</p>
<p>I use two different methods to solve this problem:</p>
<pre><code> f[1, n_] := n;
f[n_, 1] := 1;
f[n_, m_] ... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p><span class="math-container">$$x>0\implies x^n=e^{n\ln(x)}$$</span>
So,
<span class="math-container">$$\lim_{n\to+\infty}x^n=0\iff \ln(x)<0$$</span>
<span class="math-container">$$\iff x<1$$</span></p>
<p>and
<span class="math-container">$$x<0\implies x^n=(-1)^ne^{n\ln(-x)}$$... |
2,706,504 | <p>In Lang's Algebra, I had hard times with this question:</p>
<blockquote>
<p>Let $K$ be field with characteristic $p$ (a prime). Let $L|K$ be a finite extension of $K$, and suppose $gcd([L:K],p)=1$. Show $L$ is separable over $K$.</p>
</blockquote>
<p>In order to have $L$ separable over $K$, one needs to have $K(... | Zev Chonoles | 264 | <p>The extension $K(\alpha)/K$ is separable when the minimal polynomial of $\alpha$ over $K$ has no repeated roots.</p>
<p>An <em>irreducible</em> polynomial $f\in K[x]$ has repeated roots if and only if $f(x)=g(x^p)$ for some $g\in K[x]$.</p>
<p>Do you see how to proceed from here?</p>
|
2,970,773 | <p>Let's assume I am given a positive integer <span class="math-container">$n$</span>, as well as an upper limit <span class="math-container">$L$</span>.</p>
<p>How could one find all, or at least one, possible solutions for <span class="math-container">$a$</span> and <span class="math-container">$b$</span> such that ... | Henno Brandsma | 4,280 | <p>He doesn't. You could say he defines the set <span class="math-container">$G=\{(n,m) \in \mathbb{N}: \overline{B_n} \subseteq B_m\}$</span>, the set of "good" pairs. One you've fixed your enumeration of a countable base, there will be some pairs like this, a randomly chosen one won't satisfy it, but there are many p... |
3,412,217 | <p>The category of abelian groups <span class="math-container">$\mathbb{Ab}$</span> has a monoidal (closed) structure <span class="math-container">$(\otimes, \mathbb{Z})$</span>. Moreover, it is monadic over the category of sets via the free abelian group monad <span class="math-container">$$\mathbb{Z}[\_]: \text{Set} ... | Simon | 716,796 | <p>Let <span class="math-container">$ (x_0,y_0) $</span> be a solution of <span class="math-container">$ ax+by = n $</span>, then by some elementary number theory, all the integer solution is given by
<span class="math-container">$$ x=x_0+\frac{b}{(a,b)}t,y = y_0-\frac{a}{(a,b)}t $$</span>
where <span class="math-cont... |
2,001,441 | <p>I am having the hardest time solving the following trigonometric equation. Can anyone help please? Thank you.</p>
<p>Solve for x. [Hint: Let $\\u = \tan^{-1}(x)$ and $\\v = tan^{-1}(2x)$. Solve the equation $\\u+v = \frac{π}{4}$ by taking the tangent of each side.]</p>
<p>$\tan^{-1}(x) +\tan^{-1}(2x)= \frac{π}{... | Tyler | 383,143 | <p><strong>Hint:</strong> Use their hint and take the tangent of each side - you get $\tan(u+v)=\tan(\frac{\pi}{4})$. The following identity may be useful: $$\tan(a+b)=\frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}$$</p>
|
655,064 | <p>Let $G$ be a group and $a\in G$. Define the centralizer of $a$ to be </p>
<p>$\hspace{150pt} C(a)=\{g\in G : ga=ag\}$.</p>
<p>That is, $C(a)$ consists of all the elements that commute with $a$. Show that $C(a)$ is a subgroup of $G$. </p>
<hr>
<p>Clearly $C(a)$ is nonempty. Since $G$ is a group, $\exists e\in G$ ... | Robert Lewis | 67,071 | <p>I had a hard time following the OP's extended calculation; I think he/she was trying to show that for $a, g \in C(a)$, $ag^{-1} \in C(a)$. It is certainy true that for $S \subset G$, $S$ is a subgroup if and only if $s_1, s_2 \in S$ implies $s_1s_2^{-1} \in S$, but I think in the present case it is easier to procee... |
335,611 | <p>find this limit:
$$\displaystyle\lim_{n\to+\infty}\left[\sum_{k=1}^{n}\left(\dfrac{1}{\sqrt{k}}- \int_{0}^{\large {1/\sqrt k}}\dfrac{t^2}{1+t^2}dt\right)-2\sqrt{n}\right]$$</p>
| Ian Coley | 60,524 | <p>First, you can check that
$$
\int_0^{1/\sqrt k} \frac{t^2}{1+t^2}\,dt=\int_0^{1/\sqrt k} 1-\frac{1}{1+t^2}\,dt=\frac{1}{\sqrt k}-\arctan\frac{1}{\sqrt k},
$$
so you may rewrite your limit as
$$
\lim_{n\to\infty}\left[\sum_{k=1}^n\left(\arctan\frac{1}{\sqrt k}\right)-2\sqrt n\right].
$$</p>
<p>This is at least somew... |
3,992,972 | <p>I want to draw the set <span class="math-container">$M_{4}$</span>={(x, y) ; |xy|< 1/4}</p>
<p>My attempt is that I evaluate the expression of |xy|<1/4 depending on the values of x an y, i.e depending on the quadrant.</p>
<p>For the second quadrant I get the following:</p>
<p>x < 0 <span class="math-contain... | Community | -1 | <p>You are right to consider the set in four different quadrants.</p>
<p>For <span class="math-container">$xy<0$</span>, you have <span class="math-container">$-xy<\frac14$</span>.</p>
<p>For <span class="math-container">$x>0$</span> and <span class="math-container">$y<0$</span>,
<span class="math-container... |
4,343,249 | <p>In a variant of Russian Roulette, where you put 2 bullets in 2 adjacent chambers, like that:<a href="https://i.stack.imgur.com/2pVxr.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2pVxr.png" alt="This is how it looks might look like, red circles represent bullet" /></a></p>
<p>Image credit: Brill... | Vincent | 101,420 | <blockquote>
<p>P(dying in second scenario) = P(1st surviving)*P(you being hit)</p>
</blockquote>
<p>(quote from your question)</p>
<p>holds if you compute the probability of you dying <em>on forehand</em>, before the first person shoots. However, from the wording of the question I think they want you to compute the pr... |
1,958,152 | <blockquote>
<p>I want to know why $\frac{\log4}{\log b}$ can't be simplified to $\frac4b$. </p>
</blockquote>
<p>I am a high school student. Please do not quote some theories that are too advanced for me. Thank you!</p>
| OFRBG | 42,793 | <p>Assuming you don't have a good professor and something of the sort, remember that $\log$ is the name of an operation. It's like $\cos(x)$. You can't, in general, simplify quotients of functions. So $\frac{\cos30º}{\cos45º}$ is not $\frac{30º}{45º}$. Same thing happens with logarithm.</p>
<p>There are other relation... |
1,958,152 | <blockquote>
<p>I want to know why $\frac{\log4}{\log b}$ can't be simplified to $\frac4b$. </p>
</blockquote>
<p>I am a high school student. Please do not quote some theories that are too advanced for me. Thank you!</p>
| Thomas Andrews | 7,933 | <p>The cancellation rule you know, $$\frac{ab}{ac}=\frac{b}{c}$$ is only true if $ab$ means "$a$ times $b$."</p>
<p>But $\log 4$ doesn't mean multiplying something called "log" by the number $4$. It means we are applying an operation, "logarithm," to the value $4$. As notation, it looks similar, but it does not <stro... |
1,958,152 | <blockquote>
<p>I want to know why $\frac{\log4}{\log b}$ can't be simplified to $\frac4b$. </p>
</blockquote>
<p>I am a high school student. Please do not quote some theories that are too advanced for me. Thank you!</p>
| Community | -1 | <p>Well, suppose you <em>could</em> do such simplification:
$$
\frac{\log 4}{\log b}=\frac{4}{b}\tag{1}
$$
You would end up with (do you know why?)
$$
b\cdot \log 4=4\cdot\log b,
$$
which implies (do you know why?) that
$$
\log 4^b=\log b^4\tag{2}.
$$
If (1) were true for every $b>0$, then (2) must also be true for ... |
3,755,638 | <p>Given a point <span class="math-container">$A$</span>, a circle <span class="math-container">$O$</span> and conic section <span class="math-container">$e$</span>, if <span class="math-container">$BC$</span> is a moving chord of the circle <span class="math-container">$O$</span> tangent to <span class="math-container... | brainjam | 1,257 | <p>This an observation, not an answer, that is too long to put in a comment. It introduces a new conic, and runs the OP's construction in reverse. The diagram below is based on the the original diagram, and adds a given conic <span class="math-container">$e$</span> (in black) with foci <span class="math-container">$O$... |
804,414 | <p>A Fair Dice is Thrown Repeatedly. Let $X$ be number of Throws required to get a '$6$' and $Y$ be number of throws required to get a '$5$'. Find $$E(X|Y=5)$$</p>
| Graham Kemp | 135,106 | <p>We cannot get both 5 and 6 on the fifth trial, so $P(X=5 \mid Y=5)=0$</p>
<p>$\mathrm{\large E}(X\mid Y=5) = \sum\limits_{x=1}^{4} x\cdot\mathrm{\large P}(X=x\mid Y=5) + \sum\limits_{x=6}^{\infty} x\cdot\mathrm{\large P}(X=x\mid Y=5)$</p>
<p>To unconditionally get a 5 on the fifth trial, we need four not 5 before ... |
223,087 | <p>Given a list of numbers in decimal form, what is the most efficient way to determine if there are any consecutive 1s in the binary forms of those numbers? My solution so far:</p>
<pre><code>dim = 3;
declist = Range[0, 2^dim - 1];
consecutiveOnes[binary_] := AnyTrue[Total /@ Split[binary], # > 1 &];
consecuti... | Nasser | 70 | <p>You can use <code>SequenceCases</code> to check if there is <code>1,1</code> anywhere. For example</p>
<pre><code> SequenceCases[{0, 1, 1, 1, 0, 0, 1, 0, 1}, {___, 1, 1, ___}]
</code></pre>
<p>And check if the result is <code>{}</code> or not since you only care if there is at least one such case within.</p>
<p>... |
223,087 | <p>Given a list of numbers in decimal form, what is the most efficient way to determine if there are any consecutive 1s in the binary forms of those numbers? My solution so far:</p>
<pre><code>dim = 3;
declist = Range[0, 2^dim - 1];
consecutiveOnes[binary_] := AnyTrue[Total /@ Split[binary], # > 1 &];
consecuti... | flinty | 72,682 | <p><strong>Update</strong>: just use <code>BitAnd[x, BitShiftRight[x, 1]] > 0</code>. It's 10x faster than below. Bit-level parallelism beats multiple shifts every time.</p>
<p>This method is super fast and uses little memory all the way out to truly astronomical numbers like <span class="math-container">$2^{8192} ... |
386,655 | <p>Recently Ernest Davis asked me about the following computational problem: we're given as input a composite integer <span class="math-container">$n$</span>, a divisor <span class="math-container">$k$</span> of <span class="math-container">$n$</span>, and a subset <span class="math-container">$S \subset \mathbb{Z}_n$<... | Jan Kyncl | 24,076 | <p>Here are a few references; the right keywords seem to be "tiling by translation" or a "factorization of a group":</p>
<p>Mihail N. Kolountzakis and Máté Matolcsi, Algorithms for translational tiling, Journal of Mathematics and Music, 3:2 (2009), 85-97, <a href="https://doi.org/10.1080/17459730903... |
1,989,229 | <p>My approach : I tried using integral calculus and using infinite geometric series..however it didn't match..any trick?</p>
| MrYouMath | 262,304 | <p>Hint: $$\frac{1}{1-x}=1+x+x^2+x^3+\dots$$ Differentiating with respect to $x$ and assuming $|x|<1$, which guarantees uniform convergence:</p>
<p>$$\dfrac{d}{dx}\left(\frac{1}{1-x}\right)=1+2x+3x^2+\dots.$$ (you should also check the radius of convergence of the resulting expression.)</p>
<hr>
<p>An Alternative... |
272,752 | <p><strong>Bug introduced in 12.2.0 and persisting through 13.1</strong></p>
<hr />
<p>I have selected the following minimal example from the documentation. I am running v12.2.0 Win7-x64.</p>
<pre><code>GeoRegionValuePlot[
EntityClass["Country", "SouthAmerica"] -> "MerchantShips"]
</co... | MarcoB | 27,951 | <p>I am not sure why it doesn't work in the "obvious" way, i.e. by just adding the <code>GeoScaleBar</code> option to <code>GeoRegionValuePlot</code>, but it can be accomplished using <code>Show</code> after the fact:</p>
<pre><code>Show[
GeoRegionValuePlot[EntityClass["Country", "SouthAmeric... |
96,198 | <p>I'm trying to prove that the variance of a RV whose values are discrete 1's or 0's is greater than the variance of a RV who's values are 0's or continuous on the domain (0,1], where any "1" in the Bernoulli RV corresponds to a value on (0,1] in the other RV. Intuitively, I think this is the case, but I'm trying to d... | Michael Hardy | 11,667 | <p>$$
\begin{align}
1 & = E(X+Y+Z\mid X+Y+Z=1) \\ \\
& = E(X\mid X+Y+Z=1) + E(Y\mid X+Y+Z=1) + E(Z\mid X+Y+Z=1).
\end{align}
$$
If they're <b>independent</b>, you've got symmetry that justifies the conclusion that the three terms are equal. You can get by with much weaker hypotheses than independence if... |
4,630,145 | <p>We need to determine all <span class="math-container">$\mathbb{Q}$</span>-homomorphisms from <span class="math-container">$\mathbb{Q}(\sqrt[5]2)$</span> to <span class="math-container">$\mathbb{Q}(\sqrt[5]2, \omega)$</span>, where <span class="math-container">$\omega=e^{2\pi i/5}$</span>.</p>
<p>I can figure it out ... | cigar | 1,070,376 | <p>We need <span class="math-container">$\varphi (\sqrt[5]2)^5=\varphi (\sqrt[5]2^5)=\varphi (2)=2,$</span> by homomorphismness.</p>
<p>Thus there are <span class="math-container">$5:$</span></p>
<p><span class="math-container">$$\sqrt [5]2\to\sqrt [5]2\omega ^i,i=1,\dots, 5.$$</span></p>
|
1,864,939 | <p>I am surprised that this question hasn't been asked on here</p>
<p><strong>I need to show that</strong> </p>
<blockquote>
<p>$$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$</p>
</blockquote>
<p>is a metric on $C[0,1]$</p>
<hr>
<p><strong>Proof:</strong></p>
<ul>
<li><p>As usual, positive semidefinit... | Jack D'Aurizio | 44,121 | <p>It is an overkill, but I think it is an interesting one. $C^0(0,1)\subset L^2(0,1)$, hence we may assume that the functions we are dealing with have a Fourier-Legendre expansion in terms of <a href="https://en.wikipedia.org/wiki/Legendre_polynomials" rel="nofollow">shifted Legendre polynomials</a>: they give an orth... |
1,864,939 | <p>I am surprised that this question hasn't been asked on here</p>
<p><strong>I need to show that</strong> </p>
<blockquote>
<p>$$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$</p>
</blockquote>
<p>is a metric on $C[0,1]$</p>
<hr>
<p><strong>Proof:</strong></p>
<ul>
<li><p>As usual, positive semidefinit... | avs | 353,141 | <p>First, let's show that $||f||_2 = \sqrt{\int_{[0,1]} f(x)^2 dx}$ is a norm. I will only prove that it satisfies the triangle inequality for norms:
$$
||f + g||_2 \leq ||f||_2 + ||g||_2.
$$
This follows from noting that the scalar product
$$
(f, g) = \int_{[0,1]} f(x) g(x) dx
$$
satisfies the Cauchy-Schwarz inequal... |
2,270,019 | <p>Is $\displaystyle \int _ a^ b \left(\int_c^d f_{X,Y}(x,y)\, dx\right)\, dy$</p>
<p>a) $\displaystyle \int _ a^ b dy \int_ c^ d f(x,y)\,dx$</p>
<p>or</p>
<p>b) $\displaystyle \int _ a^ b dx \int_ c^ d f(x,y)\,dy$</p>
<p>?</p>
| Gio67 | 355,873 | <p>You are right, it would have been better if in that reply they had separated $n$ into even and odds. If $n=2k$ you get
$$\int_{(2k-1)\pi}^{2k\pi}\frac{(\sin x)^-}{x}dx\ge \frac{1}{k\pi}$$ and so if you sum over all $k$ you get that $$\int_{0}^{\infty}\frac{(\sin x)^-}{x}dx=\infty.$$
Similarly, if you take $n=2k-1$ y... |
3,525,017 | <p>a) Show that the generating function by length for binary strings where every block of 0s has length at least 2, each block of ones has length at least 3 is:</p>
<p><span class="math-container">$$\frac{(1-x+x^3)(1-x+x^2)}{1-2x+x^2-x^5}$$</span></p>
<p>b) Give a recurrence relation and enough initial conditions to ... | Henry | 6,460 | <p>robjohn has given a generating function. Here is a recurrence based approach. </p>
<p>Suppose <span class="math-container">$b_n$</span> is the number of strings ending with <span class="math-container">$0$</span> and <span class="math-container">$c_n$</span> the number ending with <span class="math-container">$1$... |
415,150 | <blockquote>
<p>Let $x$ and $y$ be positive numbers. Let $a_0=y$, and let $$a_n=\frac{(x/a_{n-1})+a_{n-1}}{2}$$Prove that the sequence $\{a_n\}$ has limit $\sqrt{x}$. Generalize to $n$th roots.</p>
</blockquote>
<p>I already solved the first part <a href="https://math.stackexchange.com/questions/415125/limit-of-recu... | Zen | 72,576 | <p>This is Evariste's answer translated into Spanish.</p>
<p>La integral que has computado es correcta, pero tal vez te saldria mejor si incluyes los limites de integracion antes de evaluar la integral, es decir, no dejarla en forma indefinida asi: $$\frac{4}{25}\frac{x^{2+1}}{2+1}+x$$ </p>
<p>Si dudaste que todo lo... |
4,482,285 | <p>I'm currently making a game and have run into a problem I'm not quite sure how to solve, I'll try to lay it out as a maths question. None of the values are fixed, so I'm looking for an equation that solves the below question:</p>
<p>A plane lies on the position vector <code>p0</code> <code><x0, y0, z0></code> ... | tukars | 397,739 | <p>It took some thinking, but I managed to find a solution to finding <code>sy</code>.</p>
<p>Essentially, the length of the vector between <code>t</code> and <code>s</code>, which I will call vector R can be defined using:</p>
<p><code>abs|n ⋅ (s - p0)|</code>.</p>
<p>Since the length of the vector is already known, t... |
4,645,950 | <p>The question is: Peter and Mary take turns rolling a fair die. If Peter rolls 1 or 2 he wins and the game stops. If Mary rolls 3,4,5, or 6, she wins and the game stops. They keep rolling until one of them wins. Suppose Peter rolls first.</p>
<p>a) What is the probability that Peter wins and rolls at most 4 times?</p... | Pavan C. | 914,078 | <p>The probability that Peter wins in any given round is <span class="math-container">$\frac{2}{6}$</span>. The probability that Peter and Mary both don't win on any given round is</p>
<p><span class="math-container">$$\mathbb{P}(\text{Peter loses and Mary loses}) = \mathbb{P}(\text{Peter loses}) \cdot \mathbb{P}(\text... |
1,717,169 | <p>I know that if a vector field is conservative, a potential function exists, but does this relation also hold the other way around? In other words, does a potential function exist for a certain vector field if and only if the field in question is conservative?</p>
| Captain Lama | 318,467 | <p>EDIT : it seems you are looking for the "easy" implication that if a vector field is derived from a potential then it's conservative, which is always true. My answer actually says that what you are taking for granted (the reverse implication) is false in general. I guess you are working in a case where it's true, bu... |
1,717,169 | <p>I know that if a vector field is conservative, a potential function exists, but does this relation also hold the other way around? In other words, does a potential function exist for a certain vector field if and only if the field in question is conservative?</p>
| SchrodingersCat | 278,967 | <p>Say a vector field $\vec A$ has the potential $\phi$.</p>
<p>So we can write that $\vec A=\nabla \phi$</p>
<p>Now, line integral around the field from $M$ to $N$<br>
$= \int_M^N \vec A \cdot \vec{dl}$ <br>
$= \int_M^N \nabla \phi \cdot \vec{dl}$ <br>
$= \int_M^N \left(\frac{\partial \phi}{\partial x}\hat i+ \frac{... |
196,155 | <p>I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$. This may be able to see through easily, but how can we denest such a complicated one $\sqrt{61-24\sqrt{5}}(=4-3\sqr... | Bill Dubuque | 242 | <p>There do exist <a href="https://math.stackexchange.com/a/194148/242">general denesting algorithms</a> employing Galois theory, but for the simple case of quadratic algebraic numbers we can employ a simple rule that I discovered as a teenager.</p>
<hr />
<p><strong>Simple Denesting Rule</strong> <span class="math-con... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Blue | 409 | <p>"I'm also thinking of one of these numbers. Is your number, raised to my number, bigger than $2$?"</p>
<p>Let $n$ be the girl's number (unknown to me), and let $m$ be my number (unknown to her).</p>
<ul>
<li>$n = 1 \implies $ <strong>NO</strong>: $1^m = 1 \not > 2$ for all $m \in \{1,2,3\}$.</li>
<li>$n = 3 \im... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | SamYonnou | 91,283 | <p>For an unknown $x$, is your number 1 or (2 and $x$)?</p>
<p><b>YES</b> means that the number is 1<br>
<b>NO</b> means that the number is not 1 and not 2 -- so it's 3<br>
<b>I DON'T KNOW</b> means that the number is 2 (since the (2 and $x$) clause will be true if $x=2$ and false otherwise)</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Minuteman | 94,593 | <blockquote>
<p>If k is your number,
does each continuous mass distribution µ in $\mathbb{R}^{3k-2}$ admit
an equipartition by hyperplanes ?</p>
</blockquote>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Kaz | 28,530 | <p>Does the girl know computing addition to math? Probably. </p>
<p>"Here is an array of three character strings, indexed from <code>[1]</code>:</p>
<pre><code>s[] = { "Yes", "No", "I don't know" }
</code></pre>
<p>what is the value of <code>s[x]</code> where <code>x</code> is the number you are thinking of?"</p>
<... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Joe Z. | 24,644 | <p>Here's one involving probabilistic trials.</p>
<p>If I were to choose a random variable $x$ distributed at $N(2, 0.01)$ but cut off at $[1.5, 2.5]$, would the number you're thinking of be higher than $x$?</p>
<p>If you're thinking of $1$, your answer is "no".</p>
<p>If you're thinking of $2$, your answer is "I do... |
174,339 | <p>This one is somewhat hard to explain, but I'll try my best to.<br>
I'm trying to generate a list of numbers (containing only pi/10, 0, -pi/10). Numbers are randomly selected from these 3, where the probability of getting 0 is always 70%. But the probability of getting pi/10 and -pi/10 depend on the previous outcomes... | mef | 449 | <p>If I have understood correctly, then I think the following code does what you want. (To keep things simple, I have used {-1, 0, 1} instead of {-Pi/10, 0, Pi/10}. One can easily modify the output as shown at the end.) I have compiled the function for speed.</p>
<pre><code>cf = Compile[{{n, _Integer}},
Module[{
... |
3,768,836 | <blockquote>
<p><strong>Exercise 16 (Stein): The Borel-Cantelli Lemma: Suppose <span class="math-container">$\{E_k\}_{k=1}^\infty$</span> is a countable family of measurable subsets of <span class="math-container">$\mathbb{R}^d$</span> and that
<span class="math-container">\begin{equation*}
\sum_{k=1}^\infty m(E_k) &l... | QuantumSpace | 661,543 | <p>We have
<span class="math-container">$E= \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_k$</span> (why?)
Hence, <span class="math-container">$E$</span> is trivially measurable by the axioms of <span class="math-container">$\sigma$</span>-algebra. The assumption <span class="math-container">$\sum_k m(E_k) < \infty$<... |
763,381 | <p>There are 10 men and 7 women working as supervisors in a company. The company has recently decided to form a committee to represent all the employees. The committee has to consist of 3 members, all of whom must be supervisors. The members will be President, General Secretary and Coordinator respectively. Answer the ... | Thomas | 26,188 | <p>Hint: If $x$ is a root of $f$, then $f(x) = 0$. That means that $f(x^2 +1) = f(x)g(x) = 0$. So that means that $x^2 + 1$ is a solution. But $x < x^2 + 1$. You can see by noting that $x = x^2 +1$ doesn't have any roots and since $1^2 + 1 > 1$ you have $x < x^2 +1$ for all real numbers $x$.</p>
<p>So you mus... |
2,173,586 | <p>Let $R$ be an unique factorization domain. Being an integral domain, it has a field of quotients $F$. We can consider $R[x]$ to be a subring of $F[x]$. </p>
<blockquote>
<p>Given any polynomial $f(x)\in F[x]$, then $f(x)=(f_0(x)/a)$, where $f_0(x)\in R[x]$ and where $a\in R$.</p>
</blockquote>
<p>I don't underst... | Bram28 | 256,001 | <p>First of all, I am not sure why you can't sum an infinite number of terms (other than summing them one by one of course). If $a_i = 0$ for all $i$, then the sum is $0$ ... no limit needed.</p>
<p>But yes, typically we have to use a limit to evaluate the value of the sum.</p>
<p>The difference with sets is that we ... |
3,980,796 | <p>Given <span class="math-container">$f: R \to (0,\infty]$</span>, twice differentiable with <span class="math-container">$f(0) = 0, f'(0) = 1, 1 + f = 1/f''$</span></p>
<p>Prove that <span class="math-container">$f(1) < 3/2$</span></p>
<p>I found some useless (I believe) facts about <span class="math-container">$f... | Community | -1 | <p>As you've noticed, this can be written as
<span class="math-container">$$\frac12(f'^2-1)=\ln(1+f).$$</span> If we want a parametric equation, we might choose <span class="math-container">$f'=t$</span> as a parameter, and then, we have (denoting the derivative with respect to <span class="math-container">$t$</span> b... |
624,672 | <blockquote>
<p>Let $K$ be a cyclic group. Let $\phi,\psi: K\rightarrow Aut(H)$ be group homomorphisms such that there exists $\zeta\in Aut(H)$ satisfying $\phi(K)=\zeta \psi(K)\zeta^{-1}$. Then can we prove $H\rtimes_{\phi}K\simeq H\rtimes_{\psi}H$?</p>
</blockquote>
<p>My idea is to use the following</p>
<p>Theor... | Derek Holt | 2,820 | <p>The problem is that the action of $\zeta$ might not lift to the whole of $K$.</p>
<p>As an example, let $H=\langle z \mid z^7=1 \rangle = C_7$ and $K = \langle x,y \mid x^3=y^7=1,x^{-1}yx=y^2 \rangle$ be the nonabelian group of order $21$.</p>
<p>Define $\phi,\psi$ by $\phi(x)= (z \mapsto z^2), \phi(y)= 1$, and
$\... |
2,768,308 | <p>If there is an easier way to see this I would be thrilled to know that way. I think the trick is to make sure the maximum of $f_n$ occurs to the right of a fixed $x_0$ so that we can avoid the function converging to $\frac{1}{e}$.</p>
<p><strong>Attempt.</strong></p>
<p>For x=1 and x=0, $f_n(x)=0$. Fix an $x_0\in(... | XPenguen | 293,908 | <p>$nx^n(1-x)\leq nx^n$ for all $x\in (0,1)$ <br><br> $x^n=\frac{1}{(1+y)^n}$ for some $y\gt 1$</p>
<p>$(1+y)^n \ge \frac{n(n-1)}{2}y^2$ We get $nx^n\le n\frac{2}{n(n-1)y^2}=\frac{2}{(n-1)y^2}$</p>
<p>You can continue from here</p>
|
810,514 | <p>How to compute the following series:</p>
<p>$$\sum_{n=1}^{\infty}\frac{n+1}{2^nn^2}$$</p>
<p>I tried</p>
<p>$$\frac{n+1}{2^nn^2}=\frac{1}{2^nn}+\frac{1}{2^nn^2}$$</p>
<p>The idea is using Riemann zeta function</p>
<p>$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$$</p>
<p>but the term $2^n$ makes complicated. I k... | Tunk-Fey | 123,277 | <p>Consider <a href="http://en.wikipedia.org/wiki/Taylor_series#List_of_Maclaurin_series_of_some_common_functions">Maclaurin series of natural logarithm</a>
$$
\ln(1-x)=-\sum_{n=1}^\infty\frac{x^n}{n}.
$$
Taking $x=\dfrac12$ yields
\begin{align}
\ln\left(1-\frac12\right)&=-\sum_{n=1}^\infty\frac{1}{2^n\ n}\\
\ln2&a... |
3,256,879 | <p>I've only been exposed to basic abstract algebra (Like Definition of a group + Subgroup lemma etc) and some first year linear algebra. (We have not seen lagranges theorem, incase that is required for this question).</p>
<p>I was hoping if someone could show an elementary way of doing this question:</p>
<blockquote... | ArsenBerk | 505,611 | <p>I don't know you will find this one short or not but it is elementary I can say.</p>
<p>Say <span class="math-container">$A = \pmatrix{0 & 1 \\ -1 & 0}$</span> and <span class="math-container">$ B= \pmatrix{0 & 1 \\ 1 & 0}$</span>. Then, first we can multiply and element with itself until we get ide... |
2,839,388 | <p>Given that the formula for the odds of an even split for $2n$ flips is $\binom{2n}{n}=\frac{2n!}{n!^2}$, this means that as the</p>
<p>the number of flips increases, the chances of obtaining an even split become smaller. If you flip a coin<br>
$100$ times the odds of getting $50$ heads are $0.079$. So why then if ... | Joseph Eck | 387,311 | <p>From <a href="https://en.m.wikipedia.org/wiki/Law_of_large_numbers" rel="nofollow noreferrer">Wikipedia’s Law of Large Numbers Page</a></p>
<blockquote>
<p>For example, a fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to 1/2.... |
2,839,388 | <p>Given that the formula for the odds of an even split for $2n$ flips is $\binom{2n}{n}=\frac{2n!}{n!^2}$, this means that as the</p>
<p>the number of flips increases, the chances of obtaining an even split become smaller. If you flip a coin<br>
$100$ times the odds of getting $50$ heads are $0.079$. So why then if ... | user21820 | 21,820 | <blockquote>
<p>Does the basis for that intuition come from the fact that we sum the probabilities of all combinations that we consider close to an even split (such as 485 H: 515 T) and combine them? </p>
</blockquote>
<p>Short answer: More or less.</p>
<p>Long answer: The <strong>central limit theorem</strong> tel... |
41,707 | <p>Is there a slick way to define a partial computable function $f$ so that $f(e) \in W_{e}$ whenever $W_{e} \neq \emptyset$? (Here $W_{e}$ denotes the $e^{\text{th}}$ c.e. set.) My only solution is to start by defining $g(e) = \mu s [W_{e,s} \neq \emptyset]$, where $W_{e,s}$ denotes the $s^{\text{th}}$ finite approxim... | Qiaochu Yuan | 232 | <p>A fundamental example is the <a href="http://en.wikipedia.org/wiki/Chain_rule#The_chain_rule_in_higher_dimensions">multivariate chain rule</a>. A basic principle in mathematics is that if a problem is hard, you should try to linearize it so that you can reduce as much of it as possible to linear algebra. Often this ... |
41,707 | <p>Is there a slick way to define a partial computable function $f$ so that $f(e) \in W_{e}$ whenever $W_{e} \neq \emptyset$? (Here $W_{e}$ denotes the $e^{\text{th}}$ c.e. set.) My only solution is to start by defining $g(e) = \mu s [W_{e,s} \neq \emptyset]$, where $W_{e,s}$ denotes the $s^{\text{th}}$ finite approxim... | Gerben | 6,004 | <p>High-dimensional problems in statistical physics can sometimes be solved directly using matrix multiplication, see <a href="http://en.wikipedia.org/wiki/Transfer_matrix_method" rel="noreferrer">http://en.wikipedia.org/wiki/Transfer_matrix_method</a>. The best-known example of this trick is the one-dimensional Ising ... |
1,832,887 | <p>Consider the conjunction introduction and implication elimination rules of natural deduction:</p>
<p>$$\frac{\Gamma\vdash\alpha \quad \Gamma\vdash\beta}{ \Gamma\vdash \alpha \land \beta} (\land I)
\qquad
\frac{ \Gamma \vdash \alpha \to \beta \quad \Gamma \vdash \alpha} {\Gamma,\vdash\beta} (\to E) \qquad \text{(... | Vineet Mangal | 346,869 | <p>Assume $p(x)$ of the form
$$p(x)=a\prod_{r=1}^5(x-r)+b\prod_{r=1}^4(x-r)+c\prod_{r=1}^3(x-r)+d\prod_{r=1}^2(x-r)+e(x-1)+f$$ Now put the values of $x$ i.e. $x=1,2,3,4,5,6$ , then values of $a,b,c,d,e,f$ will be $[\frac{-1}{40},\frac{1}{12},\frac{-1}{6},\frac{1}{2},0,1]$ respectively. you can get these values very eas... |
3,796,937 | <p>Prove that <span class="math-container">$2^n+1$</span> is not a cube for any <span class="math-container">$n\in\mathbb{N}$</span>.</p>
<p>I managed to prove this statement but I would like to know if there any other approaches different from mine.</p>
<p>If existed <span class="math-container">$k\in\mathbb{N}$</span... | Brian M. Scott | 12,042 | <p>Here is a parity-based solution that avoids the rational root test.</p>
<p>If <span class="math-container">$2^n+1=m^3$</span>, then <span class="math-container">$2^n=m^3-1=(m-1)(m^2+m+1)$</span>, so <span class="math-container">$m-1=2^k$</span> for some <span class="math-container">$k\le n$</span>, and</p>
<p><span ... |
3,796,937 | <p>Prove that <span class="math-container">$2^n+1$</span> is not a cube for any <span class="math-container">$n\in\mathbb{N}$</span>.</p>
<p>I managed to prove this statement but I would like to know if there any other approaches different from mine.</p>
<p>If existed <span class="math-container">$k\in\mathbb{N}$</span... | J. W. Tanner | 615,567 | <p>Invoking an argument more powerful than needed for this:</p>
<p>there cannot be any solutions to <span class="math-container">$2^n+1=m^3$</span> (i.e., <span class="math-container">$m^3-2^n=1$</span>) by <a href="https://en.wikipedia.org/wiki/Catalan%27s_conjecture" rel="noreferrer">Mihăilescu's theorem</a>,</p>
<p>... |
881,013 | <p>I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. </p>
<p><strong>Question:</strong> What are some examples of mathematical logic solving open problem outside of mathematical logic?</p>
<p>Note that the <a href="//en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck... | Conifold | 152,568 | <p>I was impressed by Bernstein and Robinson's 1966 proof that if some polynomial of an operator on a Hilbert space is compact then the operator has an invariant subspace. This solved a particular instance of <a href="http://en.wikipedia.org/wiki/Invariant_subspace_problem" rel="nofollow noreferrer">invariant subspace ... |
881,013 | <p>I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. </p>
<p><strong>Question:</strong> What are some examples of mathematical logic solving open problem outside of mathematical logic?</p>
<p>Note that the <a href="//en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck... | apurv | 49,824 | <p>Not sure if this qualifies, but <a href="http://en.m.wikipedia.org/wiki/Goodstein's_theorem">Goodstein's theorem</a> which appears quite number theoretic and states that any <a href="http://mathworld.wolfram.com/GoodsteinSequence.html">Goodstein sequence</a> converges, was proved using ordinals. </p>
|
881,013 | <p>I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. </p>
<p><strong>Question:</strong> What are some examples of mathematical logic solving open problem outside of mathematical logic?</p>
<p>Note that the <a href="//en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck... | Mikhail Katz | 72,694 | <p>Terry Tao has been using Abraham Robinson's framework and more generally ultraproducts (often considered to be "logic" as you put it), as for example his solution of Hilbert's Fifth problem in <a href="http://bookstore.ams.org/gsm-153" rel="nofollow">this recent book</a>.</p>
|
847,316 | <p>Why is the solution to linear differential equations with constant coefficients sought in the form of $Ce^{kx}$ ?</p>
| Gaussler | 129,649 | <p>I'm making a long reply to one of joo's comments:</p>
<p>Well, a linear differential equation with constant coefficients has the form $y' = Ay$ for some matrix $A$. Now the general solution with the initial value $y(0) = v$ is $y(t) = \exp(tA)v$, where $\exp(X) = \sum_{k=0}^\infty \frac 1{k!} X^k$ for all square ma... |
3,118,282 | <p>I'm trying to do the following problem in my book, but I don't understand how the book got their answer.</p>
<p>The problem:
Determine whether the following relations are equivalence relations:<span class="math-container">$\newcommand{\relR}{\mathrel{R}}$</span></p>
<p>The relation <span class="math-container">$\... | Graham Kemp | 135,106 | <blockquote>
<p>Reflexive: For any x such that xRx ---> x <= 1</p>
</blockquote>
<p>No, reflexivity requires that <span class="math-container">$\def\R{\operatorname R}\forall x{\in}\Bbb R~(x\R x)$</span>, which is clearly true (given the definitions for absolute value, substraction, and the <span class="math-cont... |
1,571,478 | <p>Let $X, Y$ have the joint pdf</p>
<p>$f(x, y) = 2, \quad 0 < x < y < 1 \quad 0, \quad$ otherwise</p>
<p>Find $P(0 < X < 1/2$ | $y = 3/4)$</p>
<p>The solutions say</p>
<p>$\int_0^{1/2} f_{X|Y}(x | y = 3/4)dx$</p>
<p>I know that $f_{X|Y}(x | Y = y) = 2$, but how do I find $f_{X|Y}(x | y = 3/4)$?</p... | Graham Kemp | 135,106 | <p>$f_{X\mid Y}(x\mid y) = \dfrac{f_{X,Y}(x,y)}{\displaystyle \int_0^y f_{X,Y}(x,y)\operatorname d x} = \dfrac{1}{y}\quad\mathbf 1_{0< x< y< 1}$</p>
<p>That is $X$ (conditioned on $Y=y$) is uniformly distributed on the interval $(0;y)$</p>
|
633,522 | <ol>
<li><p>$ \lim_{n\to \infty} \sqrt[n]{3^n+4^n} $ . I think the limit is $4$. I did :
$ \sqrt[n]{3^n+4^n} = 4 \sqrt[n]{(\frac{3}{4}) ^n+1}$ .Am I right?</p></li>
<li><p>$ \lim_{n\to \infty} \frac{1}{1\cdot 4 } + \frac{1}{4\cdot 7} +...+\frac{1}{(3n-2)(3n+1)} $. I know that for each $k$ , this sequence is the sum of... | Yiorgos S. Smyrlis | 57,021 | <p>For every $n$
$$
4^n<3^n+4^n< 2\cdot 4^n,
$$
thus
$$
4<\sqrt[n]{3^n+4^n}< 2^{1/n}\cdot 4.
$$</p>
|
3,967,862 | <p>If <span class="math-container">$A,B$</span> commute, that <span class="math-container">$e^A\leq e^B$</span> follows from functional calculus.</p>
<p>Is this still true when <span class="math-container">$A,B$</span> do not commute?</p>
<p>Also I wonder if <span class="math-container">$A^{2n+1}\leq B^{2n+1}$</span> i... | Community | -1 | <p><span class="math-container">$
\dfrac{\cos x + \sin x}{\cos x - \sin x} = \dfrac{1+ \tan x}{1 - \tan x}
$</span></p>
<p>Now, <span class="math-container">$\tan (x+y) = \dfrac{\tan x+ \tan y}{1 - \tan x \tan y}$</span><br><br></p>
<p>Therefore, <span class="math-container">$\tan (x+ \frac{\pi}{4}) = \dfrac{\tan x+ 1... |
907,055 | <p>I've begun a course in "Real Analysis" recently and I have this trivial exercise. Could someone check if my proof is correct?</p>
<p>Proposition: There exists Injective function $ f: A \rightarrow B \iff $ there exists function $ g: B \rightarrow A $ is surjective</p>
<p>Proof: Firstly, we prove injective function... | forallepsilon | 170,777 | <p>There are several problems with your proof.</p>
<p>Firstly, in both directions of your proof, your statements "By hypothesis, ..." seem to assume that <em>all</em> proofs are by contradiction, and that you must refute the negation of the statement. What you are assuming is not a hypothesis.</p>
<p>Secondly, your n... |
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