qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,927,502 | <p><span class="math-container">$\lim\limits_{n\to\infty}\dfrac{n^2-n+2}{3n^2+2n-4}=\dfrac{1}{3}$</span>.</p>
<p>With epsilon definition I get my answer as <span class="math-container">$N=\left[ \dfrac{5}{9\varepsilon }\right] +1$</span>. But then I thought that how can I evaluate this sequence, in functions <span clas... | Claude Leibovici | 82,404 | <p>If you use the long division, you have
<span class="math-container">$$\frac{n^{2}-n+2}{3n^{2}+2n-4}=\frac{1}{3}-\frac{5}{9 n}+\frac{40}{27
n^2}+O\left(\frac{1}{n^3}\right)$$</span> Therefore
<span class="math-container">$$ \frac{1}{3}-\frac{5}{9 n}<\frac{n^{2}-n+2}{3n^{2}+2n-4}<\frac{1}{3}-\frac{5}{9 n}+\fr... |
3,927,502 | <p><span class="math-container">$\lim\limits_{n\to\infty}\dfrac{n^2-n+2}{3n^2+2n-4}=\dfrac{1}{3}$</span>.</p>
<p>With epsilon definition I get my answer as <span class="math-container">$N=\left[ \dfrac{5}{9\varepsilon }\right] +1$</span>. But then I thought that how can I evaluate this sequence, in functions <span clas... | Angelo | 771,461 | <p>We have to prove that</p>
<p><span class="math-container">$\forall\;\varepsilon>0\;\;\exists\;M>0\;$</span> such that <span class="math-container">$\;x>M\;$</span> implies <span class="math-container">$\;\left|f\left(x\right)-l\right|<\varepsilon\;.$</span></p>
<p>For any <span class="math-container">$\;... |
73,277 | <p>Let $\boldsymbol{\theta}=(\theta_1,\ldots,\theta_m)$ be a vector of real numbers in $[-\pi,\pi]$. For $t\ge 0$, define
$$ f(t,\boldsymbol{\theta}) = \binom{m+t-1}{t}^{-1}
\sum_{j_1+\cdots+j_m=t} \exp(ij_1\theta_1+\cdots+ij_m\theta_m),$$
where the sum is over non-negative integers $j_1,\ldots,j_m$ with sum $t$.
Note... | Noam D. Elkies | 14,830 | <p>Another approach is to write ${m+t-1 \choose t} f(t,\theta)$ as a Schur function of the $z_j := \exp i \theta_j$, and thus as a quotient $\Delta' / \Delta$ of $m\times m$ determinants with unit-norm entries. Then $|\Delta'| \leq m^{m/2}$ by Hadamard, and $\Delta$ is the Vandermonde determinant of the $z_j$ so
$$
|\... |
3,910,345 | <p>Recently a lecturer used this notation, which I assume is a sort of twisted form of Leibniz notation:</p>
<p><span class="math-container">$$y\,\mathrm{d}x - x\,\mathrm{d}y \equiv -x^2\,\mathrm{d}\left(\frac{y}{x}\right)$$</span></p>
<p>The logic here was that this could be used as:</p>
<p><span class="math-container... | Bernard | 202,857 | <p>You should know that the <em>differential</em> at a point <span class="math-container">$\mathbf x_0$</span> of a function <span class="math-container">$\;\mathbf R^m\longrightarrow \mathbf R^n$</span> is the <em>linear map</em> <span class="math-container">$\:\ell:\mathbf R^m\longrightarrow \mathbf R^n$</span>, that... |
351,030 | <p>for positive integer $n$, how can we show</p>
<p>$$ \sum_{d | n} \mu(d) d(d) = (-1)^{\omega(n)} $$</p>
<p>where $d(n)$ is number of positive divisors of $n$ and $mu(n)$ is $(-1)^{\omega(n)} $ if $n$ is square free, and $0$ otherwise. Also, what is</p>
<p>$$ \sum_{d | n} \mu(d) \sigma (d) $$ where $\sigma(n)$ is t... | Harald Hanche-Olsen | 23,290 | <p><strong>Hint:</strong> It is enough to compute $$\lim_{n\to\infty}\int_0^1 A_n(x)f(x)\,dx$$ for continuous functions $f\colon[0,1]\to\mathbb{R}$, since these are dense in $L^1(\Omega)$. You will find that the limit is $$\frac{\alpha+\beta}{2}\int_0^1 f(x)\,dx\tag{1}$$ – just take the difference between the two integ... |
3,546,773 | <p>what are the real/complex zeros for:</p>
<p><span class="math-container">$t^9 - 1$</span></p>
<p>I also need to use the exponential form of complex numbers</p>
| Andrew Chin | 693,161 | <p><span class="math-container">\begin{align}
t^9-1&=(t^3-1)(t^6+t^3+1)\\
&=(t-1)(t^2+t+1)(t^6+t^3+1)
\end{align}</span></p>
<p>The complex zeroes can be solved by means of quadratics.</p>
|
3,546,773 | <p>what are the real/complex zeros for:</p>
<p><span class="math-container">$t^9 - 1$</span></p>
<p>I also need to use the exponential form of complex numbers</p>
| fleablood | 280,126 | <p>The truly <em>WONDERFUL</em> thing about exponential form of complex numbers is that if</p>
<p><span class="math-container">$z = re^{i\theta}$</span> and <span class="math-container">$w = se^{i\phi}$</span> then <span class="math-container">$z\cdot w = (rs)e^{i(\theta + \phi \pm\text{some multiples of }2\pi\text{ t... |
519,325 | <p>Evaluate $\displaystyle\int \dfrac{1}{x^2+9} \, dx$.
I've only learned the normal way of solving integrals but it does not work.
I haven't learned how to use trigonometry to solve these problem.</p>
<p>I know you have to rearrange it into the form ${[f(x)]² + 1}$ and then integrate.</p>
<p>Can someone point me s... | Mhenni Benghorbal | 35,472 | <p>Make the substitution</p>
<p>$$ x=3\tan t \implies dx = 3\sec^2 t \,dt .$$</p>
<p>Subs back in the integral and you need to use the identity</p>
<p>$$ 1+\tan^2 t = \sec^2 t. $$</p>
|
3,695,439 | <p>So I know that we can find <span class="math-container">$dy/dx$</span> of a curve in polar coordinates by leveraging the fact that <span class="math-container">$x=rcos\theta$</span> and <span class="math-container">$y=rsin\theta$</span>, and since <span class="math-container">$r$</span> is a function of <span class=... | SarGe | 782,505 | <p><span class="math-container">$\frac{dy}{dx}$</span> is, by definition, the limit of a secant line as the distance between two points approaches zero - it simply is the slope, nothing more to prove really (other than that the derivative actually exists, which is beyond the scope of this question). </p>
<p>Also, <spa... |
29,255 | <p>sorry! am not clear with these questions</p>
<ol>
<li><p>why an empty set is open as well as closed?</p></li>
<li><p>why the set of all real numbers is open as well as closed?</p></li>
</ol>
| Andrea Mori | 688 | <p>By definition, a set $A$ of real numbers is <em>open</em> when the following condition is met:
$$
\hbox{$\forall x\in A, \exists\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\subset A$,}
$$
where $(a,b)$ denotes the open interval $\{x\in{\Bbb R}\,|\,a<x<b\}$. It should be not hard to convince yourself that ... |
29,255 | <p>sorry! am not clear with these questions</p>
<ol>
<li><p>why an empty set is open as well as closed?</p></li>
<li><p>why the set of all real numbers is open as well as closed?</p></li>
</ol>
| Ari Royce Hidayat | 435,467 | <p>By definition, a set <span class="math-container">$A$</span> of real numbers is open when the following condition is met:</p>
<p>(<em>Note that this applies equally well to the set of real numbers, just substitute <span class="math-container">$A = R$</span>.</em>)</p>
<p><span class="math-container">$$
\hbox{$\foral... |
29,255 | <p>sorry! am not clear with these questions</p>
<ol>
<li><p>why an empty set is open as well as closed?</p></li>
<li><p>why the set of all real numbers is open as well as closed?</p></li>
</ol>
| Joe | 623,665 | <p>Of course, the answer depends on how you define "open" and "closed" sets of <span class="math-container">$\mathbb R$</span>. There are many equivalent definitions.</p>
<p>Here is a common one when we are considering <span class="math-container">$\mathbb R$</span> as a metric space with the absolu... |
2,310,441 | <p>I consider the sequence of composite odd integers: 9, 15, 21, 25, 27, 33, 35, 41, ...</p>
<p>I observe that there are certain large gaps between the composite odd integers and this may contribute towards the solution.</p>
<p>So I start by considering some sums first:</p>
<p>9 + 9 + 9 = 27, 9 + 9 + 15 = 33. So thi... | lokodiz | 70,984 | <p>The largest such number is $47$.</p>
<p>Let $C$ be the set of positive odd composite numbers, so $C = \{9,15,21,25, \dots \}$. First check that $47$ can't be written as a sum of three elements of $C$. Now observe that $C$ contains $\{ 6k+3 \mid k \geqslant 1 \}$, so if we can write a prime $p$ as a sum of three ele... |
480,727 | <p>If $$2^x=3^y=6^{-z}$$ and $x,y,z \neq 0 $ then prove that:$$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$$</p>
<p>I have tried starting with taking logartithms, but that gives just some more equations.</p>
<p>Any specific way to solve these type of problems?</p>
<p>Any help will be appreciated.</p>
| Balbichi | 24,690 | <p>$2^x=3^y=6^{-z}=k $ say, then $2= k^{1\over x},3=k^{1\over y},6=k^{-1\over z}$ now can you go on?</p>
<p>then $k^{-1\over z}=6=2\times 3 = k^{1\over x}\times k^{1\over y}=k^{{1\over x}+{1\over y}}$</p>
|
480,727 | <p>If $$2^x=3^y=6^{-z}$$ and $x,y,z \neq 0 $ then prove that:$$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$$</p>
<p>I have tried starting with taking logartithms, but that gives just some more equations.</p>
<p>Any specific way to solve these type of problems?</p>
<p>Any help will be appreciated.</p>
| Harish Kayarohanam | 30,423 | <p>$$2^x = 3^y = 6^{-z} = k $$</p>
<p>so$$x = \log_2k$$
$$ y = \log_3k$$
$$z= -\log_6k$$</p>
<p>so $$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \log_k2 + log_k3 -\log_k6$$
$$=\log_k{\frac{2\times3}{6}}$$
$$=0$$</p>
|
3,057,517 | <p>Hy everybody ! </p>
<p>I'm studying population dynamics for my calculus exam, and I don't understand something that seems really easy, so I thought you might be able to help me out ;)</p>
<p>Here's the thing. I have this differential equation <span class="math-container">$\frac{dN}{dt} = \sqrt{N}$</span>.</p>
<p>... | user619894 | 617,446 | <p>You are correct. There are indeed two solutions. If the population is zero it will stay zero. The only way it will grow is if the initial conditions are positive. However, the population zero solution is "unstable" in the sense that even if the initial condition is slightly positive, the <span class="math-container"... |
1,386,261 | <p>They start by choosing $m$ s.t. $|s_n - s| \lt \frac{1}{2} |s|$ if $n > m$.
From here, it looks like they use the triangular inequality $|s_n - s| + |s_n| < |s|$ to come up with this statement but I'm not sure as they introduce some variable m.</p>
<p>Next, given $\epsilon > 0$, there is a $N > m$ s.t $... | Sinister Cutlass | 235,860 | <p>Yes, the space of all n-by-n matrices with entries in field F itself admits the structure of an affine space and hence also admits the Zariski topology. The general linear group is, as the previous poster said, a Zariski-open subset of the affine space. The general linear group may <em>itself</em> be given the sub... |
3,350,251 | <p>The integral of velocity plots position and not change in position. But the definition of the integral is the area under the velocity curve and the area under the velocity curve is change in position. So why doesn't the integral of velocity plot change in position?</p>
| Saketh Malyala | 250,220 | <p>The information presented in the question seems a bit misguided. The plot of the integral of velocity, that is <span class="math-container">$\displaystyle f_1(t) = \int_{a}^{t}v(x)\,dx$</span> does show the change in position. In this case, we will have <span class="math-container">$f(a)=0$</span>. In fact, if we ch... |
565,046 | <blockquote>
<p>The center of $D_6$ is isomorphic to $\mathbb{Z}_2$.</p>
</blockquote>
<p>I have that
$$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$
$$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$
My method for trying to do this has been just checking elements that could be candid... | Jay | 608,546 | <p>Check that <span class="math-container">$a^{k}b=(a^{k}b)^{-1},k=1,2,3,4,5$</span>. Then since <span class="math-container">$ba^k=(a^{6-k}b)^{-1}$</span> (multiply them), then since <span class="math-container">$(a^{6-k}b)^{-1}=a^{6-k}b, a^{6-k}b=ba^{k}\Rightarrow $</span>
every element of <span class="math-container... |
2,840,333 | <p>I know that the easy way to evaluate the mean and variance of the Binomial distribution is by considering it as a sum of Bernoulli distributions.</p>
<p>However, I was wondering just for fun if there is a way to evaluate them directly. I got the mean easily: it only involves some fiddling around with the binomial c... | Robert Israel | 8,508 | <p>More generally, $$\sum_{k=0}^n k a_k r^k= r \dfrac{d}{dr} \sum_{k=0}^n a_k r^k$$
and so $$\sum_{k=0}^n k^2 a_k r^k = r \dfrac{d}{dr} \left( r \dfrac{d}{dr} \sum_{k=0}^n a_k r^k \right)$$
Here
$$ \sum_{k=0}^n {n \choose k} r^k = (1+r)^n $$
so
$$ \sum_{k=0}^n k^2 {n \choose k} r^k = r \dfrac{d}{dr} \left( r \dfrac{d... |
1,571,099 | <blockquote>
<p>Consider the rectangle formed by the points $(2,7),(2,6),(4,7)$ and
$(4,6)$. Is it still a rectangle after transformation by $\underline
A$= $ \left( \begin{matrix} 3&1 \\ 2&\frac {1}{2} \\ \end{matrix}
\right) $ ?By what factor has its area changed ?</p>
</blockquote>
<p>I've defined th... | Charles Bronson | 161,483 | <p>Just to expand what Rob Arthan said above.</p>
<p>So, you have got the following Kripke model $M= (S, R, V)$:</p>
<h2><ul></h2>
<ul>
<li>$S = \{u_1, u_2, u_3, u_4\}$, </li>
<li>$R (a) = \{u_1 \xrightarrow {a} u_1, u_2 \xrightarrow {a} u_1, u_4 \xrightarrow {a} u_1\}$,
<li>$V(p) = \{u_1, u_2\}, V(q) = \{u_3, u_4... |
11,457 | <p>In their paper <em><a href="http://arxiv.org/abs/0904.3908">Computing Systems of Hecke Eigenvalues Associated to Hilbert Modular Forms</a></em>, Greenberg and Voight remark that</p>
<p>...it is a folklore conjecture that if one orders totally real fields by their discriminant, then a (substantial) positive proporti... | Jonah Sinick | 683 | <p>For ordinary class number 1 in the real quadratic case, see Cohen and Lenstra's <em>Heuristics on Class Groups of Number Fields</em>
<a href="https://openaccess.leidenuniv.nl/retrieve/2845/346_069.pdf" rel="nofollow">https://openaccess.leidenuniv.nl/retrieve/2845/346_069.pdf</a></p>
<p>Maybe it's not so much of a j... |
3,436,515 | <p>Please help!</p>
<p>How to show that <span class="math-container">$ \lim _{n→∞} \frac{x_{(n+1)}}{x_n} =\frac{1+\sqrt 5}{2}$</span> for a dynamical system
<span class="math-container">$$x_{(n+1)}=x_n + y_n\\
y_{(n+1)}=x_n$$</span></p>
<p>Thank you!</p>
| Brian S. | 1,065,386 | <p>Let <span class="math-container">$ \lim _{n\to\infty} x(n+1)/x(n) = a$</span></p>
<p><span class="math-container">$\lim_{n\to\infty} \ { x(n+1)/x(n)= \lim_{n\to\infty} \ x(n)/x(n-1) = a} $</span></p>
<p>Given <span class="math-container">$y(n)=x(n-1)$</span></p>
<p><span class="math-container">$x(n+1)=x(n)+x(n-... |
442,759 | <p>I was reading a book on groups, it points out about the uniqueness of the neutral element and the inverse element. I got curious, are there algebraic structures with more than one neutral element and/or more than one inverse element?</p>
| Hagen von Eitzen | 39,174 | <p>A structure can have more than one left neutral element ($e$ with $e\circ x=x$ for all $x$) or more than one right neutral element ($e$ with $x\circ e=x$ for all $x$).
For example consider the set of functions $f\colon \mathbb N_0\to\mathbb N$ under composition (using $\mathbb N\subset \mathbb N_0$ of course).
Then ... |
1,798,855 | <p>I'm trying to understand this proof that:</p>
<p>$M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time</p>
<p>Which is:</p>
<p>If $M=A\cup B$ is a separation, then $A$ and $B$ are open and closed. Recriprocally, if $A\subset M$ is open and closed, then $M = A\cup(M-... | user332239 | 332,239 | <p>For the first proof, if you have a separation, $M = A \cup B$, then $A$ and $B$ are both open, and $A \cap B = \emptyset$. But, $A$ is also closed since $B$ is open, and $A = M \setminus B$. Same goes for $B$. So this is the contrapositive of the reverse direction of the statement. </p>
<p>When they say reciprocall... |
2,835,175 | <p>I have to find the limit of $\lim\limits_{x \to \pi/4}\frac{8-\sqrt{2}(\sin x+\cos x)^5}{1-\sin 2x}$</p>
<p>here is my try $\lim\limits_{x \to \pi/4}\frac{8-\sqrt{2}(\sin x+\cos x)^5}{1-\sin 2x}=\lim\limits_{x \to \pi/4}\frac{4(1-\cos^5(\frac{\pi}{4}- x))}{\sin^2 (\frac{\pi}{4}- x)}$</p>
<p>now observe that for $\... | BruceET | 221,800 | <p><strong>Extended Comment:</strong> As indicated in the Comment by @HagenvonEitzen, one way to work the initial problem (on the probability D6 shows a larger value than D10) is to enumerate
cases. In particular, you might make a $10 \times 6$ array of possible pairs
of outcomes and highlight the pairs that satisfy yo... |
2,835,175 | <p>I have to find the limit of $\lim\limits_{x \to \pi/4}\frac{8-\sqrt{2}(\sin x+\cos x)^5}{1-\sin 2x}$</p>
<p>here is my try $\lim\limits_{x \to \pi/4}\frac{8-\sqrt{2}(\sin x+\cos x)^5}{1-\sin 2x}=\lim\limits_{x \to \pi/4}\frac{4(1-\cos^5(\frac{\pi}{4}- x))}{\sin^2 (\frac{\pi}{4}- x)}$</p>
<p>now observe that for $\... | Graham Kemp | 135,106 | <p>Use the Law of Total Probability: for example $d6, d10$ the results of independen six and ten sided dice.</p>
<p>$$\begin{align}\mathsf P(d6>d10) &= \mathsf P(d10>6)\mathsf P(d6>d10\mid d10>6)+\mathsf P(d10\leq 6)\mathsf P(d6>d10\mid d10\leq 6) \\ &=\tfrac 4{10}\cdot 1+\tfrac 6{10}\cdot\maths... |
694,279 | <p>I am learning convex analysis by myself and I need help.</p>
<p>How to show that if $X=U=\mathbb{R}$
and $f\left(x\right)=\frac{|x|^{p}}{p}$
then the convex conjugate $f^{*}\left(u\right)=\frac{|u|^{q}}{q}$
when $\frac{1}{p}+\frac{1}{q}=1$?
There exists a particular technique that I have to apply in order to ... | Falcon | 766,785 | <p>Another nice way to show it is by using Young's inequality: For <span class="math-container">$x, \xi \in \mathbb R^n$</span>, we have
<span class="math-container">$$\langle\xi, x \rangle \le \frac{|\xi|^p}{p} + \frac{|x|^q}{q},$$</span>
with equality if <span class="math-container">$|\xi|^p = |x|^q$</span>. Therefor... |
7,761 | <p>Our undergraduate university department is looking to spruce up our rooms and hallways a bit and has been thinking about finding mathematical posters to put in various spots; hoping possibly to entice students to take more math classes. We've had decent success in finding "How is Math Used in the Real World"-type po... | Gerhard Paseman | 3,468 | <p>Various conference attendees sometimes have informative posters as part of their advertising campaigns. I have an old poster (somewhere!) of the graph of the real and imaginary parts of the zeta function on the critical line, produced I think by Wolfram Research. You might ask colleagues about promotional material... |
2,629,744 | <p>I have done the sum by first plotting the graph of the function in the Left Hand Side of the equation and then plotted the line $y=k$. For the equation to have $4$ solutions, both these two curves must intersect at $4$ different points, and from the two graphs, I could see that for the above to occur, the value of $... | Renji Rodrigo | 522,531 | <p>I will show you a more general way to find solutions, you can apply that method later to your problem ( and others of the same type)</p>
<blockquote>
<p><strong>Theorem(Solution of the recurrence)</strong>
Given sequences $g(n) \neq 0$ and $b(n)$, we have that $f(n)$
the solution of the recurrence
$$f(n+1)=... |
246,862 | <p>I have stumbled upon this problem which keeps me from finishing a proof:</p>
<p>$(\sum_{n} {|X_n|})^a \leq \sum_{n} {|X_n|}^a$,
where $n \in \mathbb{N}$ and $ 0 \leq a \leq 1 $</p>
<p>I have no idea how to prove this. It is something like the Cauchy-Schwarz inequality which applies in case $0 \leq a \leq 1$?</p>
... | Robert Israel | 8,508 | <p>I'll leave the case $a=0$ to you. Otherwise, let $a = 1/b$, $b \ge 1$. If $y_n = |X_n|^a$, we have
$|X_n| = y_n^b$, and your inequality says
$$ \left(\sum_n y_n^b\right)^{1/b} \le \sum_n y_n$$
which is essentially Minkowski's inequality for counting measure: if
$v(n)$ is the vector with $v(n)_n = y_n$, $v(n)_j = ... |
2,185,585 | <p>Triangular numbers (See <a href="https://en.wikipedia.org/wiki/Triangular_number" rel="noreferrer">https://en.wikipedia.org/wiki/Triangular_number</a> )</p>
<p>are numbers of the form $$\frac{n(n+1)}{2}$$</p>
<p>In ProofWiki I found three claims about triangular numbers. The three claims are that a triangular num... | Pete Caradonna | 164,325 | <p>Here is a somewhat lower-brow 'proof by picture' of how such a process of gluing might go. To save you from having to read my poor handwriting, the steps are:</p>
<p>1) Embed $S^1$ into the $(x,y)$-plane in $\mathbb{R}^3$.</p>
<p>2) Pick an antipodal pair and glue (together) to the $z$-axis. </p>
<p>3) Pick anoth... |
255,252 | <p>Let $\mathfrak{S}_n$ be the permutation group on an $n$-element set. For each fixed $k\in\mathbb{N}$, consider the two sets
$$A_n(k)=\{\sigma\in\mathfrak{S}_n\vert\,\, \text{$\exists i,\,\, 1\leq i\leq n\,$ such that $\,\sigma(i)-i=k$}\}$$
and
$$B_n(k)=\{\sigma\in\mathfrak{S}_n\vert\,\, \text{$\exists i,\,\, 1\leq ... | Richard Stanley | 2,807 | <p>A simple variant of the "transformation fondamentale" of Rényi and
of Foata-Schützenberger does the trick. Write a permutation $\sigma$ in
disjoint cycle form, with the smallest element of each cycle first,
and the cycles arranged in decreasing order of the smallest element,
e.g., $(7,8)(5,6,9)(3)(1,4,2)$. Erase the... |
287,947 | <p>For example, $\sqrt 2 = 2 \cos (\pi/4)$, $\sqrt 3 = 2 \cos(\pi/6)$, and $\sqrt 5 = 4 \cos(\pi/5) + 1$. Is it true that any integer's square root can be expressed as a (rational) linear combinations of the cosines of rational multiples of $\pi$?</p>
<p>Products of linear combinations of cosines of rational multiples... | David E Speyer | 297 | <p>Someone should actually record the formula. If $p$ is a prime $\equiv 1 \bmod 4$, then
$$\sqrt{p} = \sum_{k=1}^{p-1} \left( \frac{k}{p} \right) \cos \frac{2 k \pi}{p}$$
where $\left( \tfrac{k}{p} \right)$ is the quadratic residue symbol. Note that $\left( \tfrac{k}{p} \right) = \left( \tfrac{p-k}{p} \right)$, so ev... |
50,521 | <p>I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics. </p>
<p>Is there something that still needs to be justified mathematically in order to have solid foundations? </p>
| timur | 2,473 | <p>There are some problems related to initial data, such as if gravity can be "insulated", or if there exist generic set of spacetimes which do not admit CMC spacelike hypersurface.</p>
|
158,810 | <p>Let $ (\hat i, \hat j, \hat k) $ be unit vectors in Cartesian coordinate and $ (\hat e_\rho, \hat e_\theta, \hat e_z)$ be on spherical coordinate.
Using the relation, $$ \hat e_\rho = \frac{\frac{\partial \vec r}{\partial \rho}}{ \left | \frac{\partial \vec r}{\partial \rho} \right |}, \hat e_\theta = \frac{\frac{... | Valentin | 31,877 | <p>In fact unit vectors are components of the <em>determinant</em>, not the <em>matrix</em> $A$. There is nothing wrong with it. Determinant is really an antisymmetric linear form, so you still have vector quantities on both sides of the relation.
<strong>EDIT</strong>
After a closer look the formulae do not seem entir... |
66,951 | <p>I am asked to find all rows in a matrix in reduced row echelon form which contain nothing but pivots (pivot is $1$, all other entries are $0$).</p>
<p>For example, in this matrix:</p>
<p>$$
\begin{bmatrix}
1 & 1 & 1 & 1 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\sim
\begin... | Silvia | 17 | <h1>Update</h1>
<p>According to <a href="https://mathematica.stackexchange.com/users/3246/kennycolnago">KennyColnago</a>'s advice, post-processing is not needed, as <code>StreamColorFunction</code> can handle it essentially by using <code>VertexColors</code> on <code>Line</code>-s:</p>
<pre><code>ListStreamPlot[
... |
46,631 | <p>I'm writing a program to play a game of <a href="http://en.wikipedia.org/wiki/Pente" rel="noreferrer">Pente</a>, and I'm struggling with the following question:</p>
<blockquote>
<p>What's the best way to detect patterns on a two-dimensional board?</p>
</blockquote>
<p>For example, in Pente a pair of neighboring ... | george2079 | 2,079 | <p>This may be a bit un-mathematicaesque, but it turns out to be convenient to store the board as a flat vector:</p>
<p>(larger board for illustration)</p>
<pre><code> n = 12;
board0 = Flatten[ Table[0, {n^2}], 1];
v[icol_, jrow_] = icol + n (jrow - 1);
</code></pre>
<p>Now we can create lists of indices represent... |
394,085 | <p>How is it possible to establish proof for the following statement?</p>
<p>$$n = \frac{1}{2}(5x+4),\;2<x,\;\text{isPrime}(n)\;\Rightarrow\;n=10k+7$$</p>
<p>Where $n,x,k$ are $\text{integers}$.</p>
<hr>
<p>To be more verbose:</p>
<p>I conjecture that;</p>
<p>If $\frac{1}{2}(5x+4),\;2<x$ is a prime number, ... | Alex R. | 22,064 | <p>If $n$ is prime , clearly $x$ must be even and moreover $x$ has only one power of 2, eg $x=2y$ where $y$ is odd, $y=2z+1$. Thus we have </p>
<p>$$n\equiv 5y+2 \pmod{10}\equiv 5+2\equiv 7$$ </p>
|
425,969 | <p>It seems striking that the cardinalities of <span class="math-container">$\aleph_0$</span> and <span class="math-container">$\mathfrak c = 2^{\aleph_0}$</span> each admit what I will call a "homogeneous cyclic order", via the examples of <span class="math-container">$ℚ/ℤ$</span> and <span class="math-conta... | Alessandro Codenotti | 49,381 | <p>While Andreas's answer is a very simple construction let me point out that one can get even more.</p>
<p><strong>Theorem:</strong> for every cardinal <span class="math-container">$\lambda$</span> there exists a ultra-homogeneous circularly ordered set <span class="math-container">$(X,R)$</span> with <span class="mat... |
1,181,631 | <p>Let $f : \mathbb R \to \mathbb R$ continuous. Prove that graph $G = \{(x, f(x)) \mid x \in \mathbb R\}$ is closed.</p>
<p>I'm a little confused on how to prove $G$ is closed. I get the general strategy is to show that every arbitrary convergent sequence in $G$ converges to a point in $G$.</p>
<p>Here is what I tri... | Gregory Grant | 217,398 | <p>Almost, but better to start the argument with a general convergent sequence $(x_n,y_n)\rightarrow (x,y)$ and then write $y_n$ as $f(x_n)$ to show $y_n$ converges to $f(x)$. Then conclude $(x_n,y_n)$ converges to $(x,f(x))$ which is in $G$ and therefore an arbitrary sequence in $G$ that converges, converges in $G$. ... |
1,529,324 | <p>I've read that if $\Phi$ is a Poisson point process (on $\mathbb{R}^d$, say), then conditional on there being $k$ points in some $A \subseteq \mathbb{R}^d$, the positions $X_1,\ldots,X_k$ of these points are uniformly distributed in $A$.</p>
<p>I'm having trouble making sense of what this means. "Conditional on $\P... | MaxW | 23,782 | <blockquote>
<blockquote>
<p>I've read that if $\Phi$ is a Poisson point process (on $\mathbb{R}^d$, say), then conditional on there being $k$ points in some $A \subseteq \mathbb{R}^d$, the positions $X_1,\ldots,X_k$ of these points are uniformly distributed in $A$.</p>
</blockquote>
</blockquote>
<p>$\Phi$ is... |
76,753 | <p>I am having a hard time getting to factor this binomial: I have tried other methods but they do not seem to work... ah well.
$$4m^2-\frac{9}{25}.$$</p>
<p>Thanks.</p>
| picakhu | 4,728 | <p>I assume this is homework, so only ideas from me here. </p>
<p>Note that the formula is of the form $A^2-B^2$.</p>
<p>and we know that $A^2-B^2$ can be factored easily. </p>
|
918,788 | <p>How to do this integral</p>
<p>$$\int_{-\infty}^{\infty}{\rm e}^{-x^{2}}\cos\left(\,kx\,\right)\,{\rm d}x$$</p>
<p>for any $k > 0$ ?.</p>
<p>I tried to use gamma function, but sometimes the series doesn't converge.</p>
| gaoxinge | 94,751 | <p>We assume
$$F(k)=\int_{-\infty}^{\infty}e^{-x^2}\cos kxdx$$
Consider $F'(k)$, we have
$$F'(k)=\int_{-\infty}^{\infty}-xe^{-x^2}\sin kxdx$$
$$=\frac{1}{2}(e^{-x^2}\sin kx|_{-\infty}^{\infty}-\int_{-\infty}^{\infty}ke^{-x^2}\cos kxdx)$$
$$=-\frac{1}{2}kF(k)$$
Then we solve the ordinary differential equation with $F(0)... |
3,511,118 | <p>I can't see how <span class="math-container">$$e^\left(2i\pi\right) = 1$$</span>
will result in:
<span class="math-container">$$e^\left(i\pi\right) +1 = 0$$</span>
thanks</p>
| Fred | 380,717 | <p>If <span class="math-container">$f$</span> is continuous, then</p>
<p><span class="math-container">$$\frac{d}{d x} \int_{3}^{x} f(t)^{2} d x= f(x)^2,$$</span></p>
<p>by the Fundamental Theorem of Calculus. </p>
|
2,772,895 | <p>I've noticed one classical way of defining certain topologies is to define them as the "weakest" (or coarsest) topology such that a certain set of functions is continuous. For example,</p>
<blockquote>
<p>The <strong>product topology</strong> on <span class="math-container">$X=\prod X_i$</span> is the weak... | Martin Argerami | 22,857 | <blockquote>
<p>If $τ$ is the weakest topology on $X$ such that $f:X→Y$ is continuous, is it correct to imagine a base for the open sets to be the preimage of all open sets under $Y$? This follows directly from the definition of a "continuous" function. Is this always the coarsest topology?</p>
</blockquote>
<p>Yes.... |
2,111,833 | <p>Let $A=(a_{ij})\in M_n$ be an arbitrary matrix and let
$A_1=\begin{pmatrix}
a_{11}\\
a_{21}\\
\vdots\\
a_{n1}\\
\end{pmatrix}$
$A_2=\begin{pmatrix}
a_{12}\\
a_{22}\\
\vdots\\
a_{n2}\\
\end{pmatrix}\ldots$
$A_n=\begin{pmatrix}
a_{1n}\\
a_{2n}\\
\vdots\\
a_{nn}\\
\end{pmatrix}\in M_{n1}$ be columns of $A$. Prove tha... | egreg | 62,967 | <p>Since exchanging two columns only switches sign to the determinant, it is not restrictive to assume that the last column is a linear combination of the previous $n-1$ columns:
$$
A_n=\alpha_1A_1+\dots+\alpha_{n-1}A_{n-1}
$$
By multilinearity of the determinant, you have
$$
\det A=
\det\begin{bmatrix} A_1 & \dots... |
2,111,833 | <p>Let $A=(a_{ij})\in M_n$ be an arbitrary matrix and let
$A_1=\begin{pmatrix}
a_{11}\\
a_{21}\\
\vdots\\
a_{n1}\\
\end{pmatrix}$
$A_2=\begin{pmatrix}
a_{12}\\
a_{22}\\
\vdots\\
a_{n2}\\
\end{pmatrix}\ldots$
$A_n=\begin{pmatrix}
a_{1n}\\
a_{2n}\\
\vdots\\
a_{nn}\\
\end{pmatrix}\in M_{n1}$ be columns of $A$. Prove tha... | Bernard | 202,857 | <p>I'll give a variant of the proof, hoping you'll understand better.</p>
<p>Suppose there's a non-trivial linear relation between the columns:
$$\lambda_1A_1+\lambda_2A_2+\dots+\lambda_nA_n=0.$$
Say $\lambda_1\ne 0$. By linearity w.r.t. the 1st column, $\;\det(
\lambda_1A_1,A_2,\dots,A_n)=\lambda_1\det(A_1,A_2,\dots... |
2,111,833 | <p>Let $A=(a_{ij})\in M_n$ be an arbitrary matrix and let
$A_1=\begin{pmatrix}
a_{11}\\
a_{21}\\
\vdots\\
a_{n1}\\
\end{pmatrix}$
$A_2=\begin{pmatrix}
a_{12}\\
a_{22}\\
\vdots\\
a_{n2}\\
\end{pmatrix}\ldots$
$A_n=\begin{pmatrix}
a_{1n}\\
a_{2n}\\
\vdots\\
a_{nn}\\
\end{pmatrix}\in M_{n1}$ be columns of $A$. Prove tha... | user646399 | 646,399 | <p>The determinant of <span class="math-container">$A$</span> is the product of the elements of the main diagonal when <span class="math-container">$A$</span> is converted to row echelon form. For a linearly dependent set of columns, when <span class="math-container">$A$</span> is converted to row echelon form, there w... |
1,734,102 | <blockquote>
<p>Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$.</p>
</blockquote>
<p>We know that $f(0) = 0$. Now set $x = n$ and $y=n$ to get $f(2n) = 2f(n)$ where $n$ is a rat... | Derek Allums | 17,736 | <p>Notice:
$$
f\left(\frac{m}{n}\right) = m\cdot f\left(\frac{1}{n}\right) = m\cdot \left( \frac{f(1)}{n}\right) = f(1)\cdot \frac{m}{n}
$$
where $f\left(\frac{1}{n}\right) = \frac{f(1)}{n}$ because
$$
f\left(\frac{1}{n}\right) = f\left( \frac{n}{n} - \frac{n-1}{n}\right) = f(1) - (n-1)f\left(\frac{1}{n}\right)
$$
by t... |
1,734,102 | <blockquote>
<p>Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$.</p>
</blockquote>
<p>We know that $f(0) = 0$. Now set $x = n$ and $y=n$ to get $f(2n) = 2f(n)$ where $n$ is a rat... | bgins | 20,321 | <p>You can demonstrate that for $n\in\mathbb{Z}$, $f(n)=nf(1)$ and $f(1)=nf(\frac1n)$ for $n\ne0$, so that for $x=\frac{p}{q}\in\mathbb{Q}$, $f(x)=\frac{p}{q}f(1)$, so $c=f(1)$.</p>
|
68,428 | <p>I am looking at the description of LTI systems in the time domain.</p>
<p>Intuitively, I'd have guessed it would be the composition of the input function and some "system function".
$$ y(t) = f(x(t)) = (f\circ x)(t)$$
Where $x(t)$ is the input, $y(t)$ output and $f(x)$ a "system function".</p>
<p>Why is it not tha... | Mike Stay | 756 | <p>Simon Willerton explains it all very well here: <a href="http://www.simonwillerton.staff.shef.ac.uk/ftp/TwoTracesBeamerTalk.pdf" rel="nofollow">http://www.simonwillerton.staff.shef.ac.uk/ftp/TwoTracesBeamerTalk.pdf</a></p>
|
3,207,767 | <p>What is the general solution of differential equation <span class="math-container">$y\frac{d^{2}y}{dx^2} - (\frac{dy}{dx})^2 = y^2 log(y)$</span>.</p>
<p>The answer to this DE is <span class="math-container">$log(y) = c_1 e^x + c_2 e^{-x}$</span></p>
<p>I don't know the method to solve differential equation with d... | Michael Rozenberg | 190,319 | <p>We'll prove that
<span class="math-container">$$\frac{a+b+c+d}{\sqrt{(1+a^2)(1+b^2)(1+c^2)(1+d^2)}}\leq\frac{3\sqrt3}{4},$$</span> where the equality occurs for <span class="math-container">$a=b=c=d=\frac{1}{\sqrt3}$</span> only.</p>
<p>Indeed, let <span class="math-container">$a=\frac{x}{\sqrt3},$</span> <span cla... |
1,684,095 | <p>How can I evaluate the following series.</p>
<p>$$\sum_{k=1}^{\infty}\frac{1}{(k+1)(k-1)!}.$$</p>
| Stefan Mesken | 217,623 | <p>A trivial example is $\operatorname{On} \cup \{*\}$ for some $* \not \in \operatorname{On}$ that is put "on top", i.e. we extend the order on $\operatorname{On}$ by letting $\alpha < *$ for all $\alpha \in \operatorname{On}$. Clearly $<$ remains to be a strict well order and its order type is "$\operatorname{O... |
3,747,453 | <p>Isn't it wrong to write the following with only the percent sign? Instead of <span class="math-container">$100 \%$</span>?</p>
<blockquote>
<p>The change in height as a percentage is
<span class="math-container">$$
\frac{a - b}{a} \% \tag 1
$$</span>
where <span class="math-container">$a$</span> is the initial heigh... | Brian M. Scott | 12,042 | <p>Let <span class="math-container">$S=\wp(\Bbb N)$</span>, with symmetric difference as addition and intersection as multiplication. Let <span class="math-container">$R=T=\wp(2\Bbb N)$</span> with the same operations. Let <span class="math-container">$f:R\to S:x\mapsto x$</span>, and let <span class="math-container">$... |
3,188,298 | <blockquote>
<p>Let <span class="math-container">$f: X\to Y$</span> be bijective, and <span class="math-container">$f^{-1}: Y\to X$</span> be it's inverse. If
<span class="math-container">$V\subseteq Y$</span>, show that the forward image of <span class="math-container">$V$</span> under <span class="math-container"... | David | 119,775 | <p>Actually, what you have to prove is that
<span class="math-container">$$f^{-1}(V)=f^{-1}(V)\ .$$</span>
To see why this is <strong>not obvious</strong>, you have to carefully unpack the meanings of all terms, noting in particular that both <span class="math-container">$f$</span> and <span class="math-container">$f^{... |
78,443 | <p>Let $F$ be a number field and $A$ an abelian variety over $F$. It is known that if $A$ has complex multiplication, then it has potentially good reduction everywhere, namely there exists a finite extension $L$ of $F$ such that $A_L$ has good reduction over every prime of $L$.</p>
<p>And what about the inverse: if $A... | David Roberts | 4,177 | <p>In his <a href="http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf" rel="nofollow">recently posted draft of <em>Spectral Algebraic Geometry</em></a>, Lurie defines Nisnevich covers for quasi-compact, quasi-separated spectral algebraic spaces as follows. </p>
<p>Consider a family of étale morphisms $\{X_\alp... |
3,984,930 | <p>I am studying maths purely out of interest and have come across this question in my text book:</p>
<p>A rectangular piece of paper ABCD is folded about the line joining points P on AB and Q on AD so that the new position of A is on CD. If AB = a and AD = b, where <span class="math-container">$a \ge\frac{2b}{\sqrt3}$... | Will Orrick | 3,736 | <p>I think you were almost there. You don't need <span class="math-container">$y$</span> to compute the area, so you can omit that part of your calculation. So you have two variables, <span class="math-container">$x$</span> and <span class="math-container">$m$</span>, but these are related. You've solved for <span clas... |
4,246,048 | <p>As I understand it, Cantor defined two sets as having the same cardinality iff their members can be paired 1-to-1. He applied this to infinite sets, so ostensibly the integers (Z) and the even integers (E) have the same cardinality because we can pair each element of Z with exactly one element of E.</p>
<p>For infi... | Bertrand Einstein IV | 824,733 | <p>You are approaching cardinality and infinity in the wrong way. For two finite sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span> the sets are of the same size if and only if there exists a bijection between the two sets. This is somewhat obvious, and the proof is done in the fir... |
3,579,065 | <p>It is known that the quantity <span class="math-container">$\cos \frac{2π}{17}$</span> is a root of the <span class="math-container">$8$</span>'th degree equation,
<span class="math-container">$$x^8 + \frac{1}{2} x^7 - \frac{7}{4} x^6 - \frac{3}{4} x^5 + \frac{15}{16} x^4 + \frac{5}{16} x^3 - \frac{5}{32} x^2 -... | J. W. Tanner | 615,567 | <p>You know <span class="math-container">$\dfrac1{(1-x)^2}=1+2x+3x^2+4x^3+\cdots.$</span></p>
<p>Therefore, <span class="math-container">$\dfrac x{(1-x)^2}=x(1+2x+3x^2+4x^3+\cdots)=x+2x^2+3x^3+\cdots$</span></p>
<p>and <span class="math-container">$\dfrac {x^2}{(1-x)^2}=x^2(1+2x+3x^2+4x^3+\cdots)=x^2+2x^3+\cdots.$</s... |
2,962,203 | <p>I got stuck at : <span class="math-container">$a^2/b^2 = 12+2 \sqrt 35$</span></p>
<p>I understand that <span class="math-container">$12$</span> is rational and now I need to prove that <span class="math-container">$\sqrt{35}$</span> is irrational.</p>
<p>so I defined <span class="math-container">$∀c,d∈R$</span> ... | Connor Harris | 102,456 | <p>If <span class="math-container">$a = \sqrt{5} + \sqrt{7}$</span> is rational, then <span class="math-container">$b = \sqrt{5} - \sqrt{7} = a - 2 \sqrt{7}$</span> is irrational. But <span class="math-container">$ab = -2$</span>, so <span class="math-container">$a$</span> and <span class="math-container">$b$</span> mu... |
42,787 | <p>I am using <code>ListPlot</code> to display from 5 to 12 lines of busy data. The individual time series in my data are not easy to distinguish visually, as may be evident below, because the colors are not sufficiently different.</p>
<p><img src="https://i.stack.imgur.com/PiMMh.png" alt="enter image description here... | Dr. belisarius | 193 | <p>In case all else fails :) ...</p>
<pre><code>n = 8;
Manipulate[
ListPlot[d,
Frame -> True,
Joined -> True,
PlotRange -> All,
PlotLegends -> SwatchLegend[Characters["ABCDEFGHIGKLMNOP"]],
PlotStyle -> cs],
{{cs, Array[RGBColor[0.`, 0.`, 0.`] &, n]}, ControlType -> None},
Column[Oute... |
227,797 | <p>I have this function and I want to see where it is zero.
<span class="math-container">$$\frac{1}{16} \left(\sinh (\pi x) \left(64 \left(x^2-4\right) \cosh \left(\frac{2 \pi x}{3}\right) \cos (y)+\left(x^2+4\right)^2+256 x \sinh \left(\frac{2 \pi x}{3}\right) \sin (y)\right)+\left(x^2-12\right)^2 \sinh \left(\frac... | Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
f[x_, y_] :=
2 (-4 + x^2) Sinh[(π x)/3] +
1/16 (((4 + x^2)^2 + 64 (-4 + x^2) Cos[y] Cosh[(2 π x)/3] +
256 x Sin[y] Sinh[(2 π x)/3]) Sinh[π x] -
2 (4 + x^2)^2 Sinh[(5 π x)/
3] + (-12 + x^2)^2 Sinh[(7 π x)/3]);
ContourPlot[f[x, y] == 0,
{x, 3464/1... |
227,797 | <p>I have this function and I want to see where it is zero.
<span class="math-container">$$\frac{1}{16} \left(\sinh (\pi x) \left(64 \left(x^2-4\right) \cosh \left(\frac{2 \pi x}{3}\right) \cos (y)+\left(x^2+4\right)^2+256 x \sinh \left(\frac{2 \pi x}{3}\right) \sin (y)\right)+\left(x^2-12\right)^2 \sinh \left(\frac... | Akku14 | 34,287 | <p><code>f[x, y] == 0</code> can be separated in lhs[y] == rhs[x]. This shows, rhs gets immaginary in a small range of x. So curves do not intersect.</p>
<pre><code>f[x_, y_] =
2 (-4 + x^2) Sinh[(\[Pi] x)/3] +
1/16 (((4 + x^2)^2 + 64 (-4 + x^2) Cos[y] Cosh[(2 \[Pi] x)/3] +
256 x Sin[y] Sinh[(2 \[Pi] x)... |
2,642,144 | <p>How would I prove or disprove the following statement?
$ \forall a \in \mathbb{Z} \forall b \in \mathbb{N}$ , if $a < b$ then $a^2 < b^2$</p>
| AbstractNonsense | 429,931 | <p>take $a=-1$ and $b=1$, then $a^2=1 \nless1=b^2$</p>
|
1,716,656 | <p>I am having trouble solving this problem</p>
<blockquote>
<p>Julie bought a house with a 100,000 mortgage for 30 years being repaid with payments at the end of each month at an interest rate of 8% compounded monthly. If Julie pays an extra 100 each month, what is the outstanding balance at the end of 10 years im... | Em. | 290,196 | <p>To get yourself started, you could draw a table. The rows could be one roll, and the columns could be the other roll. Then the checkmark shows where the rolls are "two away" from each other.</p>
<p>\begin{array}{r|c|c|c|c|c|c}
&1&2&3&4&5&6\\\hline
1&&&\checkmark&&&\\\... |
1,716,656 | <p>I am having trouble solving this problem</p>
<blockquote>
<p>Julie bought a house with a 100,000 mortgage for 30 years being repaid with payments at the end of each month at an interest rate of 8% compounded monthly. If Julie pays an extra 100 each month, what is the outstanding balance at the end of 10 years im... | user | 79,303 | <p>If the first die is 1, the other can only be 3, probability = 1/6</p>
<p>If the first die is 2, the other can only be 4, probability = 1/6</p>
<p>If the first die is 5, the other can only be 3, probability = 1/6</p>
<p>If the first die is 6, the other can only be 4, probability = 1/6</p>
<p>If the first die is 3... |
435,298 | <p>Define
$$\langle X,Y \rangle := \operatorname{tr}XY^t,$$
where $X,Y$ are square matrices with real entries and $t$ denotes transpose.</p>
<p>I have some troubles in proving that
$$ \langle [X,Y],Z \rangle = - \langle Y,[X,Z] \rangle,$$
where square brackets denote commutator.</p>
<p>Let me update my <strong>questi... | Avitus | 80,800 | <p>The key properties to use are </p>
<p>$$\langle A,B\rangle=\langle B, A\rangle,$$</p>
<p>i.e. with $tr((AB)^t)=tr(AB),$ and</p>
<p>$$tr(ABC^t)=tr(BC^tA),$$</p>
<p>for all $A,B\in M_n(\mathbb R)$.</p>
|
2,565,204 | <p>I have following problem that I cannot solve... I have a triangle with sides <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, and <span class="math-container">$c$</span> which is split into two smaller triangles, <span class="math-container">$E$</span> and <span class="math-container... | induction601 | 444,022 | <p>For any $j$, there is $N_j$ such that for all $n > N_j$,
$$ |g_n^j(x) - g(x)| \le \epsilon/2$$
Then we have
\begin{align}
|g^i_n(x) - g(x)| &\leq |g^i_n(x) - g^j_n(x)| + |g^j_n(x) - g(x)|\\
&\leq |g^i_n(x) - g^j_n(x)| + \epsilon/2.
\end{align}
By taking sup only w.r.t. $i$, we have
\begin{align}
\sup_{i\i... |
2,565,204 | <p>I have following problem that I cannot solve... I have a triangle with sides <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, and <span class="math-container">$c$</span> which is split into two smaller triangles, <span class="math-container">$E$</span> and <span class="math-container... | grand_chat | 215,011 | <p>The claim you're trying to prove is intended to hold for each $x$, so you can suppress $x$. Pick <em>any</em> $j$, say $j=J$. Then for each $i$ and $n$ we have
$$
|g_n^i-g|\le |g_n^i-g_n^J| + |g_n^J-g|\le \sup_{i,j}|g_n^i-g_n^j|+|g_n^J-g|\tag1
$$
The RHS of (1) depends on $n$ only, which implies the sup over all $i$... |
474,587 | <p>Does $\|Tv\|\leq\|v\|$ (for all $v \in V$) leads to $T$ is normal?</p>
<p>If not, when I add the additional information that every e.e of $T$ is of the absolute value 1, can I prove $T$ is unitary? </p>
<p>Thanks!</p>
| M Turgeon | 19,379 | <p>As Daniel Fischer has noted in the comments, the answer is no. One way to see this is as follows: on the vector space of linear operators on $V$, you can define a <em>norm</em>. The norm of $T$ is the smallest non-negative real number $c$ such that
$$\|Tv\|\leq c\|v\|,$$
for all $v\in V$. Since this is a norm, for ... |
122,471 | <p>Can anyone explain how I can prove that either $\phi(t) = \left|\cos (t)\right|$ is characteristic function or not? And which random variable has this characteristic function? Thanks in advance.</p>
| GEdgar | 442 | <p>W. Feller, <em>An Introduction to Probability Theory and Applications</em>, Volume I, XIX.4, Theorem 1.<br>
A continuous function $\phi$ with period $2\pi$ is a characteristic function iff its Fourier coefficients (4.2) satisfy $\phi_k \ge 0$ and $\phi(0) = 1$. </p>
<p>$$
\phi_k = \frac{1}{2\pi}\int_{-\pi}^{\pi} ... |
12,717 | <p>In the familiar case of (smooth projective) curves over an algebraically closed fields, (closed) points correspond to DVR's.</p>
<p>What if we have a non-singular projective curve over a non-algebraically closed field? The closed points will certainly induce DVR's, but would all DVR's come from closed points? Is th... | Emerton | 2,874 | <p>Suppose that $X$ is a projective variety, and that $v$ is a discete valuation on $K(X)$
(trivial on $k$) whose corresponding valuation ring we will denote by $R$. The valuative criterion shows that
the map Spec $K(X) \rightarrow X$ extends to a map Spec $R \rightarrow X$. If I have the terminology correct, the im... |
187,959 | <p>This is the question:</p>
<p>Use the integral test to determine the convergence of $\sum_{n=1}^{\infty}\frac{1}{1+2n}$.</p>
<p>I started by writing:</p>
<p>$$\int_1^\infty\frac{1}{1+2x}dx=\lim_{a \rightarrow \infty}\left(\int_1^a\frac{1}{1+2x}dx\right)$$</p>
<p>I then decided to use u-substitution with $u=1+2n$ ... | DonAntonio | 31,254 | <p>$$u=1+2x\Longrightarrow du=2dx\Longrightarrow dx=\frac{1}{2}du$$</p>
<p>Remember, not only you substitute the variable and nothing more: you also have to change the $\,dx\,$ and the integral's limits:
$$u=1+2x\,\,,\,\text{so}\,\, x=1\Longrightarrow u=1+2\cdot 1 =3$$</p>
|
187,959 | <p>This is the question:</p>
<p>Use the integral test to determine the convergence of $\sum_{n=1}^{\infty}\frac{1}{1+2n}$.</p>
<p>I started by writing:</p>
<p>$$\int_1^\infty\frac{1}{1+2x}dx=\lim_{a \rightarrow \infty}\left(\int_1^a\frac{1}{1+2x}dx\right)$$</p>
<p>I then decided to use u-substitution with $u=1+2n$ ... | Santosh Linkha | 2,199 | <p>Let $1 + 2x = u$ then $du = 2 dx \implies du = {1 \over 2} dx$</p>
<p>$x $ goes from $1$ to $\infty$, since we are using new variable $u$ here, it would have different bound. When $x \rightarrow \infty$, $ u \rightarrow \infty$ and when $x = 1$, $ u = 1 + 2 \cdot ( 1) = 3 $, so $u$ goes from $3$ to $\infty$</p>
|
187,959 | <p>This is the question:</p>
<p>Use the integral test to determine the convergence of $\sum_{n=1}^{\infty}\frac{1}{1+2n}$.</p>
<p>I started by writing:</p>
<p>$$\int_1^\infty\frac{1}{1+2x}dx=\lim_{a \rightarrow \infty}\left(\int_1^a\frac{1}{1+2x}dx\right)$$</p>
<p>I then decided to use u-substitution with $u=1+2n$ ... | Thomas Russell | 32,374 | <p>You have $\int_{1}^{\infty}{\frac{1}{1+2n}\:dn}=\lim_{a\to\infty}\left(\int_{1}^{a}{\frac{1}{1+2n}\:dn}\right)$. Using the substitution you mentioned $u=1+2n$, we have our lower bound as $u(1)=1+2(1)=3$ and our upper bound as $u(a)=1+2(a)=1+2a$.</p>
<p>We also have $\frac{du}{dn}=2\implies dn=\frac{du}{2}$, therefo... |
202,034 | <p>Is finding the largest prime factor of a number computationally easier than factoring the number into powers of primes? </p>
| gnasher729 | 137,175 | <p>Of course it is easier, since factoring a number into prime powers gives you the largest prime factor for free. In general, it's only a <em>tiny</em> bit easier. There will be extreme examples, like $4^k-1 = (2^k-1)(2^k+1)$, where it might be much easier to prove that one of the numbers is a prime and therefore the ... |
121,245 | <p><em>What is the difference between how Matlab and Mathematica solve <strong>State-Space</strong> and <strong>Transfer Function</strong> models</em>?</p>
<p>I have a $16 \times 16$ state space system for which I am calculating transfer function. Mathematica and Matlab give me completely different answers. I can imag... | JimB | 19,758 | <p>Using <em>Mathematica</em> 10.4.1 (Windows 7) the following code finds a pretty good match with the Matlab results. (And I'd argue that any differences between the two are due to differences in how the precision of the input numbers are handled.)</p>
<pre><code>a = Rationalize[a, 0.00001];
b = Rationalize[b];
sys... |
121,245 | <p><em>What is the difference between how Matlab and Mathematica solve <strong>State-Space</strong> and <strong>Transfer Function</strong> models</em>?</p>
<p>I have a $16 \times 16$ state space system for which I am calculating transfer function. Mathematica and Matlab give me completely different answers. I can imag... | Steffen Jaeschke | 61,643 | <p>In M11.3 this is much much easier to solve. Simply: use instead of d Automatic:</p>
<pre><code>a = {{-(350103/48500), -14.9811, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0}, {1.75898, -6.08528, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0,
0, 0... |
674,259 | <p>I am trying to understand why the derivative of $f(x)=x^\frac{1}{2}$ is $\frac{1}{2\sqrt{x}}$ using the limit theorem. I know $f'(x) = \frac{1}{2\sqrt{x}}$, but what I want to understand is how to manipulate the following limit so that it gives this result as h tends to zero:</p>
<p>$$f'(x)=\lim_{h\to 0} \frac{(x+h... | Ben Grossmann | 81,360 | <p><strong>Alternative approach:</strong></p>
<p>Noting that $h$ and $x+h$ need to be positive in order for this to make sense, we have
$$
\lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt x}{h} =
\lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt x}{(\sqrt{x+h})^2 - (\sqrt{x})^2} =
\lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt x}
{(\sqrt{x+h}+... |
3,316,730 | <p>I'm taking a course in Linear Algebra right now, and am having a hard time wrapping my head around bases, especially since my prof didn't really explain them fully. I would really appreciate any insight you could give me as to what bases are! Also, can there can be multiple different bases for a single subspace?</p>... | jack.f | 269,635 | <p>While a bit late to the game, I thought another perspective might help.</p>
<hr>
<p>Consider the following physical example. Now, without being too pedantic about definition, a basis for a vector space is much like a building block of a biological system. We can <i>build</i> a human body from a set of cells. That ... |
909,741 | <blockquote>
<p><strong>ALREADY ANSWERED</strong></p>
</blockquote>
<p>I was trying to prove the result that the OP of <a href="https://math.stackexchange.com/questions/909712/evaluate-int-0-frac-pi2-ln1-cos-x-dx"><strong><em>this</em></strong></a> question is given as a hint.</p>
<p>That is to say: <em>imagine tha... | Matthias | 164,923 | <p>Hint: with $\cos x=u$
$$\int_0^{\pi/2}\log\cos x\mathrm{d}x=-\int_0^1\frac{\log u}{\sqrt{1-u^2}}\mathrm{d}u$$</p>
|
792,924 | <p>If a quantity can be either a scalar or a vector, how would one call that property? I could think of scalarity but I don't think such a term exists.</p>
| Gabriel Romon | 66,096 | <p>Suppose for contradiction that such a function exists. </p>
<p>Without resorting to Cauchy-Schwarz, since $f$ is continuous on a closed/compact interval, it is bounded by some positive $M$. </p>
<p>Now $$1=\left|\int_{0}^1f(t)t^n dt \right| \leq \frac{M} {n+1}$$</p>
<p>The RHS goes to $0$ whereas the LHS is $1$... |
140,754 | <p>Pleas tell me that what a "Kink" is and what this sentence means: </p>
<blockquote>
<p>Distance functions have a kink at the interface where $d = 0$ is a local minimum.</p>
</blockquote>
| Alex Becker | 8,173 | <p>In this case, I believe a "kink" in the function refers to a point at which the function fails to be differentiable. For example, the function $f(x)=|x|$ (which gives the distance between $x$ and $0$) is not differentiable at $x=0$, where the function is $0$ as well.</p>
|
152,880 | <p>I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$.</p>
<blockquote>
<p>I am interested in counting how many such $p(x)$ there exist (that is, given $n\in\mathbb{N}$, $n\ge 1$, how many irreducible polynomials of... | Qiaochu Yuan | 232 | <p><strong>Theorem:</strong> Let $\mu(n)$ denote the Möbius function. The number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_q$ is the <a href="http://en.wikipedia.org/wiki/Necklace_polynomial" rel="noreferrer">necklace polynomial</a>
$$M_n(q) = \frac{1}{n} \sum_{d | n} \mu(d) q^{n/d}.$$</p>
<p>(To... |
4,344,571 | <p>In a previous exam assignment, there is a problem that asks for a proof that <span class="math-container">$\mathbb{Z}_{24}$</span> and <span class="math-container">$\mathbb{Z}_{4}\times\mathbb{Z}_6$</span> are <strong>not</strong> isomorphic.</p>
<p>We have <span class="math-container">$\mathbb{Z}_{24}$</span> is is... | Shaun | 104,041 | <p>Since <span class="math-container">$(\Bbb Z_{24},+)$</span> has an element of order <span class="math-container">$24$</span> and <span class="math-container">$(\Bbb Z_4\times \Bbb Z_6, +)$</span> does not, the two groups cannot be isomorphic as groups; hence the respective rings cannot be isomorphic as rings.</p>
|
544,464 | <p>Show that any subset of $\{1, 2, 3, ..., 200\}$ having more than $100$ members must contain at least one pair of integers which add to $201$.</p>
<p>I think it is doable using the Pigeonhole Principle.</p>
| Ross Millikan | 1,827 | <p>Hint: think about the pairs that add to $201$. How many such pairs are there?</p>
|
1,436,867 | <p>I don´t know an example wich $ \rho (Ax,Ay)< \rho (x,y) $ $ \forall x\neq y $ is not sufficient for the existence of a fixed point .
can anybody help me? please</p>
| Hamou | 165,000 | <p>$f:(0,1)\to (0,1)$ with $f(x)=x/2$.</p>
|
71,636 | <p>For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each case $a_n$ represents some appropriate quantity (see, for example, <a href="https://mathoverflow.net/questions/69218/if... | Ian Morris | 1,840 | <p>For the topological entropy of a subshift on finitely many symbols, I think that this limit will typically be infinite. Here is an example where this is the case.</p>
<p>Let $\Sigma_2= \{0,1\}^{\mathbb{N}}$ with the infinite product topology and let $T \colon \Sigma_2 \to \Sigma_2$ denote the shift transformation g... |
4,157,841 | <p>Q is to prove that integer just above(<span class="math-container">$\sqrt{3} + 1)^{2n}$</span> is divisible by <span class="math-container">$2^{n+1}$</span> for all n belongs to natural numbers.</p>
<p>In Q , by integer just above means that:
For an example , which is the integer just above 7.3 . It is 8. Then , Q w... | robjohn | 13,854 | <p>Part of the problem is that <a href="https://i.stack.imgur.com/9nPlD.jpg" rel="nofollow noreferrer">the image</a> was a bit obscured and so the transcription was a bit off. Here is a more accurate transcription with tags added for reference:</p>
<blockquote>
<p><span class="math-container">$$\require{cancel}
\left(\... |
126,120 | <p>Python has generators which save memory, is there a technique for generating in memory examples for your training set "on the fly".</p>
<p>For example purposes, I constructed here a regressor for blur:</p>
<pre><code>randomMask[img_] :=
Module[{t, h, g, d = ImageDimensions[img]},
t = Table[{PointSize@RandomRe... | Taliesin Beynon | 7,140 | <p>It's not very well tested but you can supply a <code>"$AugmentationFunction" -> f</code> option to the Image <code>NetEncoder</code> in which you can put a <code>Blur</code> or whatever (anything that takes an image and produces an image). This option is not officially supported and it'll probably be replaced wit... |
3,031,460 | <blockquote>
<p>Give an example of an assertion which is not true for any positive
integer, yet for which the induction step holds.</p>
</blockquote>
<p>First of all, definition.</p>
<blockquote>
<p>In <strong>inductive step</strong>, we suppose that <span class="math-container">$P(k)$</span> is true for some p... | Bram28 | 256,001 | <p><span class="math-container">$n < n$</span> does not hold for any <span class="math-container">$n$</span> but the inductive step holds: </p>
<p>If <span class="math-container">$k <k$</span> then <span class="math-container">$k+1<k+1$</span></p>
|
22,753 | <p>I've learned the process of orthogonal diagonalisation in an algebra course I'm taking...but I just realised I have no idea what the point of it is.</p>
<p>The definition is basically this: "A matrix <span class="math-container">$A$</span> is orthogonally diagonalisable if there exists a matrix <span class="math-co... | Community | -1 | <p>Arturo has given a nice answer. Here are couple of my additions to Arturo's answer.</p>
<p>Any symmetric matrix which can be diagonalized can be re-written in an orthogonal diagonalized form. Another aspects of this orthogonal diagonalization is that this essentially means that the left and right singular vectors a... |
164,896 | <p>I want to create a list length <code>l</code> with the function $f(x_n)=x_{n-1}+r$ where $r$ is a random real number between -1 and 1 and $x_0=1$. It got it working like this:</p>
<pre><code>l = 50; a = Range[l]; a[[1]] = 0;
For[i = 2, i <= l, i++, a[[i]] = a[[i - 1]] + RandomReal[{-1, 1}]];
a
</code></pre>
<p>... | user42582 | 42,582 | <p>It's not clear if the first entry should be zero or one (the body of the question uses $x_0=1$ but in the code there is <code>a[[1]]=0</code>) and if it's <code>RandomReal</code> or <code>RandomInteger</code> the relevant function to use; here, the first entry in the list is set to <em>zero</em> (see <code>xo</code>... |
1,702,616 | <p>I was working on a programming problem to find all 10-digit perfect squares when I started wondering if I could figure out how many perfects squares have exactly N-digits. I believe that I am close to finding a formula, but I am still off by one in some cases.</p>
<p>Current formula where $n$ is the number of digit... | ALam | 685,033 | <p>I'm interested in the question that you're asking. I've been thinking about this for the last few days and couldn't find anything online about this subject. If you have anything references, I would appreciate it. Here is what I have so far and sorry if the formatting is off, this is my first post.</p>
<p><span clas... |
1,491,484 | <p>Let $a,b,x \in Z^+$. Prove that $\operatorname{lcm}(ax,bx) = \operatorname{lcm}(a,b)\cdot x$.</p>
<p>Here are my thoughts: </p>
<p>Let $d = \operatorname{lcm}(ax, bx)$. By definition $ax|d$ and $bx|d$. Now it can be seen that $a|d$ and $b|d$. So, let e = lcm(a,b). e is merely the lcm(ax, bx) (which equals d) multi... | 5xum | 112,884 | <p>Your proof, as written now, is incorrect:</p>
<blockquote>
<p>Now it can be seen that $d|a$ and $d|b$.</p>
</blockquote>
<p>This is false, since if $a=6$, $b=4$, then $d=12$, and it is not true that $d|a$.</p>
|
1,942,364 | <p>How many squares exist in an $n \times n$ grid? There are obviously $n^2$ small squares, and $4$ squares of size $(n-1) \times (n-1)$.</p>
<p>How can I go about counting the number of squares of each size?</p>
| Jean-Claude Arbaut | 43,608 | <p>Another way to view this: for each size, there is one square that you move inside the larger one.</p>
<p>How much can it move? A square of size $k\times k$ inside a square of size $n\times n$ has $n-k+1$ possible positions in each direction (up-down and left-right): consider the position of the top left corner for ... |
1,942,364 | <p>How many squares exist in an $n \times n$ grid? There are obviously $n^2$ small squares, and $4$ squares of size $(n-1) \times (n-1)$.</p>
<p>How can I go about counting the number of squares of each size?</p>
| Sahil Kumar | 363,778 | <p><strong>NOTE:</strong></p>
<p>For $n\times n$ grid the answer is $n^2+(n-1)^2+(n-2)^2+.....+1^2$</p>
<p>So, the answer is
$$
\frac{n(n+1)(2n+1)}{6}
$$</p>
|
210,655 | <p>The lower density of $A\subseteq\mathbb{N}$ is defined to be $\lambda(A)=\lim\text{inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$. We set $${\cal C} = \{A\subseteq \mathbb{N}: \lambda(\mathbb{N}\setminus A) = 1 - \lambda(A)\}.$$</p>
<p>Do both ${\cal C}$ and ${\cal P}(\mathbb{N})\setminus {\cal C}$ have cardinal... | Stefan Kohl | 28,104 | <p>The answer is <em>yes</em>: Given a set $A$ in $\mathcal{C}$ or in
$\mathcal{P}(\mathbb{N}) \setminus \mathcal{C}$, you can take
any subset $B \subset \mathbb{N} \setminus A$ of lower density $0$,
and $A \cup B$ is still in $\mathcal{C}$ or in
$\mathcal{P}(\mathbb{N}) \setminus \mathcal{C}$, depending on
which of th... |
3,706,332 | <p>Let <span class="math-container">$f: \mathbb{R} \to \mathbb{R} $</span> be a differentiable function. Is it true that <span class="math-container">$f$</span> is strictly increasing on <span class="math-container">$\mathbb R$</span> if and only if <span class="math-container">$f'(x) \geq 0$</span> on <span class="mat... | jacob bradley | 658,280 | <p>Edit: the op has edited the question to strictly increasing from simply increasing thus my answer is no longer valid. I think this isn't true and here is why, Take <span class="math-container">$f(x) = -x^2$</span> when <span class="math-container">$x<0$</span>, <span class="math-container">$f(x) = 0$</span> when ... |
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