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 |
|---|---|---|---|---|
221,026 | <p>I need to find the value of <span class="math-container">$z$</span> for a particular value of <span class="math-container">$D_c$</span> (eg. <span class="math-container">$500$</span>), but <span class="math-container">$z$</span> is inside an integral, and I'm not able to use <code>Solve</code> since the integral is ... | Artes | 184 | <p>We will demonstrate that the exact formula for <span class="math-container">$z$</span> reads:
<span class="math-container">$$z=\wp\bigg(\frac{\sqrt{\Omega_M}}{2}D_c+\wp^{-1}\big(1;0,-\frac{4\Omega_\Lambda}{\Omega_M}\big);0,-\frac{4\Omega_\Lambda}{\Omega_M}\bigg)-1$$</span>
where <span class="math-container">$\wp(x;g... |
2,086,770 | <p>$[c^2, c^3, c^4] \text { parallel to/same direction as } [1,-2,4]$</p>
<p>Find $c$ if it exists.</p>
<p>How do I see if they are parallel and find a c?</p>
<p>Generally, we can say that $r[1,-2,4] = [c^2, c^3, c^4]$</p>
<p>But then I have two unknowns and I don't know how to solve like this?</p>
| Mark Bennet | 2,906 | <p>Consider $$c=\frac {c^3}{c^2}=\frac {c^4}{c^3}=?$$</p>
|
4,292,091 | <p>I'm reading the definition of <span class="math-container">$inf\emptyset$</span> and <span class="math-container">$sup\emptyset$</span>.</p>
<p>a) I'm wondering why <span class="math-container">$inf\emptyset = \infty$</span> and <span class="math-container">$sup\emptyset = -\infty$</span>. I would have expected both... | 311411 | 688,046 | <p>You should assume the presence of the "extended real number system" when thinking about the infimum (about the supremum) of a set <span class="math-container">$M \subset \mathbb{R}$</span>.</p>
<p>You should do this because it is helpful. I visualize it dynamically and non-rigorously, as follows. The exten... |
1,407,944 | <p>Can this be proved that $\log(n)$ is irrational for every $n=1,2,3,\dots$ ?</p>
<p>I find that question in my mind in searching for if $\log(x)$ is irrational for every rational number $x\gt0$. </p>
| Caleb Stanford | 68,107 | <p>There is a theorem (mentioned <a href="https://math.stackexchange.com/a/809913/68107">here</a>) that for any nonzero $\alpha$, at least one of $\alpha$ and $e^{\alpha}$ is transcendental.</p>
<p>In this case, take $\alpha = \log x$, where $x$ is rational.
Then either $\log x$ is transcendental, or $x$ is.
But $x$ i... |
1,999,643 | <p>The title states my question: what aspect of closed makes it attractive for optimization?</p>
| Jonasson | 378,136 | <p>Having a restricted and <strong>closed</strong> (i.e. compact) feasible set and if the (real-valued) objective <em>f</em> is furthermore continous, you are able to apply the <strong>Extreme value theorem</strong> in order to <strong>prove feasibility</strong> of your constrained optimization problem.</p>
|
478,516 | <p>$\lim_{d \to \infty} (1+\frac{w}{d})^{\frac{d}{w} } = e$. But, what if the number of bits used to encode $d$ is polynomial in length. In this model, infinity can't be encoded. However, $d$ is polynomialy much larger than $w$.
Is there any tight lower bound, a closed form function $f(d)$ such that
$$ f(d) \le (1+\fr... | Yuval Filmus | 1,277 | <p>Let $x = d/w$. Since $d \ll w$, we have $x \ll 1$. Therefore the Taylor expansion of $(1+x)^{1/x}$ should offer a good approximation to the function:
$$ (1+x)^{1/x} = e \left[1 - \frac{1}{2} x + \frac{11}{24} x^2 - \frac{7}{16} x^3 + \frac{2447}{5760} x^4 - \frac{959}{2304} x^5 + O(x^6) \right]. $$
In particular, we... |
6,990 | <p>The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.</p>
<ul>
<li><p>In how far can this be generalized to other countable sets of functions? For example, if we keep our inner product, can ... | Julián Aguirre | 1,168 | <p>There are certainly many other basis for spaces of functions on an interval, if we eliminate the periodicity condition. The more widely used are orthogonal polynomials. Given an interval $I\subset\mathbb{R}$ and a weight $w\colon I\to (0,\infty)$, there is a sequence of polynomials $\{P_n\}$ orthogonal with respect ... |
62,981 | <p><strong>Introduction.</strong> I recently revisited Shelah's model without P-points and I was wondering how "badly" Grigorieff forcing destroys ultrafilters, i.e., what kind of properties can survive the destruction of the "ultra"ness.</p>
<p><strong>An example.</strong> Given a free (ultra)filter $F$ on $\omega$, ... | Peter Krautzberger | 7,281 | <p>I wanted to add two comments that I received in 'meatspace'. I hope this isn't too inappropriate.</p>
<ul>
<li>If <span class="math-container">$F$</span> is a P-filter, and the forcing is proper, than <span class="math-container">$F$</span> generates a P-filter in the extension.</li>
<li>If <span class="math-contain... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Neves | 1,747 | <p>$\displaystyle\big(a^2+b^2\big)\cdot\big(c^2+d^2\big)=\big(ac \mp bd\big)^2+\big(ad \pm bc\big)^2$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Integral | 33,688 | <p>Facts about $\pi$ are always fun!</p>
<p>\begin{equation}
\frac{\pi}{2} = \frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\ldots\\
\end{equation}
\begin{equation}
\frac{\pi}{4} = 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\ldots\\
\end{equation}... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Austin Mohr | 11,245 | <p>The product of any four consecutive integers is one less than a perfect square.</p>
<p>To phrase it more like an identity:</p>
<p>For every integer $n$, there exists an integer $k$ such that
$$n(n+1)(n+2)(n+3) = k^2 - 1.$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| preferred_anon | 27,150 | <p>$$\frac{1}{998901}=0.000001002003004005006...997999000001...$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Felix Marin | 85,343 | <p>$$
\int_{-\infty}^{\infty}{\sin\left(x\right) \over x}\,{\rm d}x
=
\pi\int_{-1}^{1}\delta\left(k\right)\,{\rm d}k
$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| triple_sec | 87,778 | <p>$\textbf{Claim:}\quad$$$\frac{\sin x}{n}=6$$ for all $n,x$ ($n\neq 0$).</p>
<p>$\textit{Proof:}\quad$$$\frac{\sin x}{n}=\frac{\dfrac{1}{n}\cdot\sin x}{\dfrac{1}{n}\cdot n}=\frac{\operatorname{si}x}{1}=\text{six}.\quad\blacksquare$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Theemathas Chirananthavat | 66,404 | <p>For all $n\in\mathbb{N}$ and $n\neq1$
$$\prod_{k=1}^{n-1}2\sin\frac{k \pi}{n} = n$$</p>
<p>For some reason, the proof involves complex numbers and polynomials.</p>
<p>Link to proof: <a href="https://math.stackexchange.com/questions/8385/prove-that-prod-k-1n-1-sin-frack-pin-fracn2n-1">Prove that $\prod_{k=1}^{n-1}\... |
1,310,530 | <p>From: $2015$ Singapore Mathematical Olympiad Secondary 2 (Grade 8) Question 21 Round 1 on 3rd June.</p>
<blockquote>
<p>Find the value of $\sqrt {(98 \times 100+2)(100\times102+2)+(100\times2)^2}$ (No use of calculators)</p>
</blockquote>
<p>My attempt:Special cases formula -> $x^2=(x+1)(x-1)+1$</p>
<p>Therefor... | chenbai | 59,487 | <p>$((100-2)100+2)((100+2)100+2)=(100^2-2\times100+2)(100^2+2\times 100+2)=(100^2+2)^2-(2\times 100)^2$</p>
|
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Tim Porter | 3,502 | <p>This is really an add-on to David Corfield's answer.</p>
<p>Since David mentions groups and groupoids, I will mention that Ronnie Brown (<a href="https://groupoids.org.uk/hdaweb2.html" rel="nofollow noreferrer">https://groupoids.org.uk/hdaweb2.html</a>) considers some of the possible criteria as follows:</p>
<p>Tes... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | sleepless in beantown | 8,676 | <p>Alex, don't feel as if the weight of the burden of proof (of concept generalization) has to rest completely on your shoulders. I realize you already agree that curiousity and your own interest can be enough reason to pursue a topic or generalization, but...</p>
<p>Isn't it the same as asking a question on mathover... |
2,454,455 | <p>I know this is a soft and opinion based question and I risk that this question get's closed/downvoted but I still wanted to know what other persons, who are interested in mathematics, think about my question.</p>
<p>Whenever people are talking about the most beautiful equation/identity Euler's identity is cited in... | Vidyanshu Mishra | 363,566 | <p>Well said by @copperhat: Beauty lies in the eye of beholder.</p>
<p>I like the form $e^{9i\pi}+1=0$ as $9$ is the first odd composite number. People have different tastes and you cannot force someone to like apples if you do like them. </p>
<p>One problem I can think of with $e^{2i\pi}-1 = 0$ is that you can sqare... |
239,900 | <p>Hatcher states the following theorem on page 114 of his Algebraic Topology:</p>
<blockquote>
<p>If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence
$$...\longrightarrow\widetilde{H}_n(A)\overset{i_*}\longrightarrow \w... | Miha Habič | 9,440 | <p>Remember that to be a deformation retract of a neighbourhood, you have to be locally path connected. If you think about this, you'll see that the comb space gives you an example. To be precise, if $X$ is the space $[0,1]\times\{0\}\cup(\{\frac{1}{n};n\in\mathbb{N}\}\cup\{0\})\times [0,1]$ and $A$ is $\{0\}\times[a,1... |
3,151,143 | <p>I want to find a symmetric matrix <span class="math-container">$A$</span>, whose eigenvalues are <span class="math-container">$4$</span> and <span class="math-container">$-1$</span>. One of the eigenvectors corresponding to the eigenvalue <span class="math-container">$4$</span> is <span class="math-container">$(2,3)... | Widawensen | 334,463 | <p>Consider general form </p>
<p><span class="math-container">$$\begin{bmatrix} x & d \\ d & y \end{bmatrix}=
\begin{bmatrix} 2/\sqrt{13} & a \\ 3/\sqrt{13} & b \end{bmatrix}
\begin{bmatrix} 4 & 0 \\ 0 & -1 \end{bmatrix}
\begin{bmatrix} 2/\sqrt{13} & a \\ 3/\sqrt{13} & b \end{bm... |
3,151,143 | <p>I want to find a symmetric matrix <span class="math-container">$A$</span>, whose eigenvalues are <span class="math-container">$4$</span> and <span class="math-container">$-1$</span>. One of the eigenvectors corresponding to the eigenvalue <span class="math-container">$4$</span> is <span class="math-container">$(2,3)... | amd | 265,466 | <p>Recall that the eigenspaces of a symmetric matrix are mutually orthogonal. Thus, all eigenvectors with eigenvalue <span class="math-container">$-1$</span> are orthogonal to all eigenvectors with eigenvalue <span class="math-container">$4$</span>. Can you come up with a nonzero vector that’s orthogonal to <span class... |
2,125,297 | <p><a href="https://i.stack.imgur.com/cBCJe.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cBCJe.png" alt="enter image description here"></a></p>
<p>I would like to know how exactly this works. I watched a khan academy video: "Multiplying matrices" but in this case he would've done B*A and had 2 co... | Learnmore | 294,365 | <p>Note that distance of $x=(1,1,1,1)$ to $W=d(x,W)=\inf\{(d(x,w):w\in W\}$ </p>
<p>Now $(1,-1,1,1)=(0,0,1,1)+(1,-1,0,0)\in W$ and hence </p>
<p>$d((1,-1,1,1),(1,1,1,1))=\sqrt 4=2$</p>
<p><strong>ELSE</strong>:</p>
<p>Any element of $W$ is $a(0,0,1,1)+b(1,-1,0,0)=(b,-b,a,a)$</p>
<p>Now $d((b,-b,a,a),(1,1,1,1))=\sq... |
840,211 | <p>I have line integral of a vector function: $\vec{F}=-e^{-x}\sin y\,\,\vec{i}+e^{-x}\cos y\,\,\vec{j}$
The path is a square on the $xy$ plane with vertices at $(0,0),(1,0),(1,1),(0,1)$</p>
<p>Of course it is a closed line integral, and I know the result should be zero. </p>
<p>I am baffled how can you calculate $\s... | David | 119,775 | <p>For the first one: if $t$ is odd, then the polynomial $x^t+1$ can be factorised into a product of two polynomials with integer coefficients:
$$x^t+1=(x+1)(x^{t-1}-x^{t-2}+\cdots-x+1)\ .$$
Now suppose that $2^m+1$ is prime, and let $t$ be an odd factor of $m$. Writing $m=st$, we have from above that
$$2^m+1=(2^s)^t+... |
698,474 | <p>I am trying to find the phi(18). Using an online calculator, it says it is 6 but im getting four.
<br/>
The method I am using is by breaking 18 down into primes and then multiplying the phi(primes)</p>
<p>$$=\varphi (18)$$
$$=\varphi (3) \cdot \varphi(3) \cdot \varphi(2)$$
$$= 2 \cdot 2 \cdot 1$$
$$= 4$$</p>
| kmbrgandhi | 132,855 | <p>Your multiplicative property is not necessarily true when the two numbers you're multiplying share a common factor. Here's the general formula: given $N = p_1^{q_1}p_2^{q_2}\cdots p_n^{q_n}$ (where $p_1, \ldots, p_n$ are distinct primes), we can find that
$$
\phi(N) = N\left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p... |
147,338 | <p>I'm helping a student through a course in mathematics. In the course text, we came across the following problem concerning the Lagrange multiplier technique.
Given a differentiable function with continuous partial derivatives $f:\mathbb{R}\to\mathbb{R}:(x,y)\mapsto f(x,y)$ that has to be extremized and a constraint... | Christian Blatter | 1,303 | <p>Here is an example, albeit with two constraints: We want to find the minimum of the function
$$f(x,y,z):=y\ ,$$
given the constraints
$$F(x,y,z):=x^6-z=0\ ,\qquad G(x,y,z):=y^3-z=0\ .$$
The constraints define the curve
$$\gamma:\quad x\mapsto (x,x^2, x^6)\qquad(-\infty<x<\infty)\ ,$$
and it is easy to see that... |
1,888,732 | <p>When I had the calculus class about the limit, one of my classmate felt confused about this limit:</p>
<blockquote>
<p><span class="math-container">$$\lim_{x\to \infty} \sqrt{x^2+x+1}-\sqrt{x^2+3x+1} = -1$$</span></p>
</blockquote>
<p>What he thought that since <span class="math-container">$x^2 > x$</span> and <s... | imranfat | 64,546 | <p>@ZackNi As David indicated, you are assuming "infinity minus infinity = 0,which works out well because both radicals are squareroots and the headcoefficients of the polynomials are both 1.You can also get"infinity minus infinity = 0 from a squareroot and cuberoot, but using your approach will then yield to a wrong a... |
61,659 | <p>How to find: $$\lim_{x \to \frac{\pi}{2}} \frac{\tan{2x}}{x - \frac{\pi}{2}}$$ I know that $\tan(2\theta)=\frac{2\tan\theta}{1-\tan^{2}\theta}$ but don't know how to apply it here.</p>
| Community | -1 | <p>Put $x = \frac{\pi}{2} + h$. As $x \to \frac{\pi}{2}$, you have $h \to 0$.</p>
<p>Then you have
\begin{align*}
\lim_{x \to \frac{\pi}{2}} \frac{\tan{2x}}{x-\frac{\pi}{2}} &= \lim_{h \to 0} \: \frac{\tan{2\bigl(\frac{\pi}{2}+h\bigr)}}{h} \\ &=\lim_{h \to 0} \: \frac{\tan(\pi + 2h)}{h} \\ &= \lim_{h \to ... |
1,843,724 | <p>I know that the Maclaurin expansion of $e^x$ is $$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$$
But i'm not sure how to find the Maclaurin series here
I tried this</p>
<p>$$
f'_{(0)}=-2xe^{-x^2}=0
$$
And that follows to every derivative that follows, so how can I get a power series out of it?</p>
| MPW | 113,214 | <p>You're mistaken in saying "And that follows to every derivative that follows".</p>
<p>The product rule will come into play as you keep differentiating, giving terms that don't vanish. Here are a few steps:</p>
<p>$f(x)=e^{-x^2}$, so $f(0)=1$.</p>
<p>$f'(x) = -2xe^{-x^2}$, so $f'(0) = 0$.</p>
<p>$f''(x) = -2x[-2x... |
1,241,639 | <p>How do I prove that $\lim_{(x,y)\to (0,0)} \frac{x^2 y }{x^2 + y^2} = 0$?</p>
<p>I can prove this by notifying $x=rcos\theta$ and $y=rsin\theta$, but I remember that it could also be proven by squeeze theorem.</p>
<p>How do I prove this using squeeze theorem?</p>
| Timbuc | 118,527 | <p>$$0\le\left|\frac{x^2y}{x^2+y^2}\right|\le \left|\frac{x^2y}{x^2}\right|=|y|\xrightarrow[(x,y)\to (0,0)]{}0$$</p>
|
1,241,639 | <p>How do I prove that $\lim_{(x,y)\to (0,0)} \frac{x^2 y }{x^2 + y^2} = 0$?</p>
<p>I can prove this by notifying $x=rcos\theta$ and $y=rsin\theta$, but I remember that it could also be proven by squeeze theorem.</p>
<p>How do I prove this using squeeze theorem?</p>
| Beni Bogosel | 7,327 | <p>Note that $\frac{x^2}{x^2+y^2} \leq 1$ so your function is bounded by $\pm y$.</p>
|
1,090,974 | <p>We have two functions of time $f(t)$ and $g(t)$, for which convolution and correlation are defined as following:</p>
<p>Convolution: $(f(t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f(t)g(\tau-t)dt}$</p>
<p>Correlation: $(f(t)\star g(t))(\tau) = \int_{-\infty}^\infty{f^\ast(t)g(\tau+t)dt}$</p>
<p>In the english wik... | Swapnil Agarwal | 689,634 | <p>We have -<br>
Convolution - <span class="math-container">$ f(t) * g(t) = \int_{-\infty}^{\infty}f(\tau) . g(t-\tau) d\tau$</span><br>
Correlation - <span class="math-container">$ R_{fg}(\tau) = \int_{-\infty}^{\infty}f(t) . g(t-\tau) dt$</span><br></p>
<p>Now,
<span class="math-container">$$ f(t) * g(t) = \int_{-\... |
2,148,354 | <p>I've been reading <em>'How to Prove It'</em> by Velleman, in the book it states that:</p>
<blockquote>
<p>$\{x\,\in\,U\,|\,P(x)\}$; this is read 'the set of all $x$ in $U$ such that $P(x)$'.</p>
</blockquote>
<p>It then goes on to give $\{x\,\in\,\mathbb R\,|\,x^2<9\}$ as an example of such a set which would ... | Alp Uzman | 169,085 | <p><strong>TL;DR:</strong> Yes.</p>
<hr>
<p>Surely for this sweeping answer to make sense we should justify that there are (or could be) universal conventions on mathematical notation (since you don't specify by whom you mean the notation to be acceptable, I interpret your question as something along the lines of: </... |
2,148,354 | <p>I've been reading <em>'How to Prove It'</em> by Velleman, in the book it states that:</p>
<blockquote>
<p>$\{x\,\in\,U\,|\,P(x)\}$; this is read 'the set of all $x$ in $U$ such that $P(x)$'.</p>
</blockquote>
<p>It then goes on to give $\{x\,\in\,\mathbb R\,|\,x^2<9\}$ as an example of such a set which would ... | Asaf Karagila | 622 | <p><strong>TL;DR:</strong> NO!</p>
<hr>
<p>The general structure of a set-builder notation is $\{\text{term}(x)\mid\text{condition on }x\}$ or $\{\text{term}(x)\in X\mid\text{condition on }x\}$ when we want to bound the terms we allow.</p>
<p>The left side is some term defined from $x$. This can be $n^2$, or it can ... |
3,244,361 | <p>I need some help on two exercises from Kiselev's geometry, about straight lines.</p>
<blockquote>
<p>Ex 7: Use a straightedge to draw a line passing through two points given on a sheet of paper. Figure out how to check that the line is really straight. Hint: Flip the straightedge upside down.</p>
</blockquote>
<... | user21820 | 21,820 | <p>While these questions are handwavy and cannot be precisely answered, I think that the author is looking for something like the following for exercise 8:</p>
<blockquote>
<p>When you fold the paper, you can do it in two ways, corresponding to which side you fold across the line. Notice that the results of these tw... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | CaptainCodeman | 65,852 | <p>Easy, just define </p>
<p>$$\begin{array}{rcl}a \times b &=&
\hspace{10.5pt}(a-4)(b-5)(a-5)(b-6)(a-6)(b-7)(a-7)(b-8)/72 + \\&& 25(a-3)(b-4)(a-5)(b-6)(a-6)(b-7)(a-7)(b-8)/18 + \\&& 15(a-3)(b-4)(a-4)(b-5)(a-6)(b-7)(a-7)(b-8)/8 \hspace{5.25pt}+ \\&& 49(a-3)(b-4)(a-4)(b-5)(a-5)(b-6)(a-7... |
2,125,136 | <p>I understand the usual motivation behind the truth table for the logical connective <span class="math-container">$\to$</span>.</p>
<p>However, I would like to know if there is a more fundamental reason for that truth table. Something that would have to do with arguments and validity.</p>
<p>A.G.Hamilton writes in <e... | Tamar | 467,879 | <p>The total number of passwords should be the number of 4-6 digit passwords using the characters 2, 3, 5, and 7 minus the number of passwords using just 3, 5, and 7. For this you compute</p>
<p>$$(4^4 + 4^5 + 4^6) - (3^4 + 3^5 + 3^6) = 5376 - 1053 = 4323.$$</p>
<p>The probability that your friend guesses on the firs... |
1,401,760 | <p>First I tried to use integration:
$$y=\lim_{n\to\infty}\frac{a^n}{n!}=\lim_{n\to\infty}\frac{a}{1}\cdot\frac{a}{2}\cdot\frac{a}{3}\cdots\frac{a}{n}$$
$$\log y=\lim_{n\to\infty}\sum_{r=1}^n\log\frac{a}{r}$$
But I could not express it as a <em>riemann integral</em>. Now I am thinking about sandwich theorem.</p>
<p>$$... | Christian Prince | 249,512 | <p>This is the shortest proof</p>
<p>$\displaystyle \sum _{n=1} ^{\infty} \frac{a^{n}}{n!}$ converges by ratio test.</p>
<p>Let $x_{n}=\frac{a^{n}}{n!} $</p>
<p>Then the convergence of $\displaystyle \sum _{n=1} ^{\infty }x_{n} $ implies $\lbrace x_{n} \rbrace $ converges to zero</p>
|
859,493 | <p>Given that any fixed integer $n>0$, let $S=\{1,2,3,4,...,n\}$. Now a Red-Blue subset of $S$ is called $T$. Every element of $T$ is given a colour (either red or blue). For instance $\{17 (\text{red})\}, \{1 (\text{red}), 5 (\text{red})\}$ and $\{1(\text{red}),5(\text{blue})\}$ are $3$ different subsets of $S$.</p... | Ethan Splaver | 50,290 | <p>$$(2x^2+(x-3))^8=a_0+a_1x+a_2x^2+a_3x^3\cdots$$</p>
<p>$$\frac{d}{dx}(2x^2+(x-3))^8=0+a_1+a_22x+a_33x^2\cdots$$</p>
<p>$$\lim_{x\to 0}\frac{d}{dx}(2x^2+(x-3))^8=a_1$$</p>
<p>$$-17496=a_1$$</p>
|
3,858,966 | <blockquote>
<p>Let <span class="math-container">$A= \{(x,y,z) \in \Bbb R^3 \vert x+y<z < x^2+y^2 \}$</span>. Show that <span class="math-container">$A$</span> is an open set in <span class="math-container">$\Bbb R^3$</span> defined by the Euclidean metric.</p>
</blockquote>
<p>So <span class="math-container">$A$... | José Carlos Santos | 446,262 | <p>Take <span class="math-container">$(a,b,c)\in A$</span>. Then <span class="math-container">$a^2+b^2>c$</span>. If <span class="math-container">$\delta>0$</span> and if <span class="math-container">$\|(x,y,z)-(a,b,c)\|<\delta$</span>, then<span class="math-container">\begin{align}x^2+y^2-z&=\bigl((x-a)+a... |
65,892 | <p>Hello!</p>
<p>Let $M$ be an almost complex manifold. Let $TM$ denote its tangent bundle. Then we have the decomposition $TM\otimes\mathbb{C}=T^{1,0}M\oplus T^{0,1}M$ corresponding to the eigenvalues of the almost complex structure. This decomposition yields the decomposition:
$$
\Lambda^r(T^\star M\otimes\mathbb{C}... | Ben McKay | 13,268 | <p>Try Chern, <a href="http://books.google.ie/books?hl=en&lr=&id=87O81_uJ10kC&oi=fnd&pg=PP8&dq=complex+manifolds+without+potential+theory&ots=R8p5dlhq6Q&sig=SnJftV-DaK3hYnN3mwZfcw9mXlM&redir_esc=y#v=onepage&q=complex%2520manifolds%2520without%2520potential%2520theory&f=false" rel... |
264,595 | <p>I've been trying to find an asymptotic expansion of the following series</p>
<p>$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$</p>
<p>and</p>
<p>$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n(n!{\sqrt{n}}) }$$</p>
<p>around $+\infty$, in the from</p>
<p>$$\exp(x^2)\Big(1+\frac{a_1}{... | Johannes Trost | 37,436 | <p><strong>Warning</strong>: <strike>Nearly</strike> every number in this answer is wrong ! Please, read the answer posted by esg, which gives the right asymptotic expansion !</p>
<p>Such problems can be solved by Laplace's method. The starting point is the observation, that the values of the terms of the sums are uni... |
3,169,258 | <p>I need to evaluate on the circle <span class="math-container">$\left|z\right|=\pi$</span> the integral
<span class="math-container">$$\int_{\left|z\right|=\pi}\frac{\left|z\right|e^{-\left|z\right|}}{z}dz.$$</span>
The function is not holomorphic there. Anyway, I tried to integrate it using polar coordinates and sim... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$\gamma(t)=\pi e^{it}$</span> (<span class="math-container">$t\in[0,2\pi]$</span>), then<span class="math-container">\begin{align}\int_{\lvert z\rvert=\pi}\frac{\lvert z\rvert e^{-\lvert z\rvert}}z\,\mathrm dz&=\int_0^{2\pi}\frac{\bigl\lvert\gamma(t)\bigr\rvert e^{-\lvert\gamma(t)... |
2,375,714 | <p>I know this is a really easy question but for some reason I'm having trouble with it.</p>
<p>If $M$ is an object in an additive category $\mathcal C$, and $\text{Hom}_{\mathcal C}(M,M) = 0$, then $M = 0$.</p>
<p>I know that this implies $\text{id}_M =0$ but I'm having trouble showing that $M$ satisfies the conditi... | k.stm | 42,242 | <p>More generally, let $\mathcal C$ be a category. Let $x, y ∈ \operatorname{Ob} \mathcal C$ be objects of $\mathcal C$ whose only endomorphisms are their respective identities $\mathrm{id}_x$, $\mathrm{id}_y$. Then
$$x \cong y \Longleftrightarrow \text{there are arrows $x → y$ and $y → x$}.$$
This is because whenever ... |
4,331,258 | <p>I'm trying to find the median of <span class="math-container">$f(x) = 4xe^{-2x}$</span>.</p>
<p>So far, I've tried solving for <span class="math-container">$q_{50}$</span> by plugging it into an integral and setting it equal to 0.5 like so: <span class="math-container">$\int_{0}^{q_{50}} 4xe^{-2x} dx = 0.5$</span>. ... | Claude Leibovici | 82,404 | <p>Letting <span class="math-container">$x=q_{50}$</span>, as @Vítězslav Štembera answered, you want to solve for <span class="math-container">$x$</span> the equation
<span class="math-container">$$ (2 x+1)\,e^{-2 x}=k \quad\implies\quad(2x+1)\,e^{-(2 x+1)}=\frac k e$$</span> The only explicit solution of it is given b... |
457,427 | <p>Find the derivative of the functions:<br>
$$\int_{x^2}^{\sin(x)}\sqrt{1+t^4}dt$$<br></p>
<p>In class we had the following solution:<br>
By the fundamental theorem of calculus we know that <br>
$$\left(\int_a^xf(t)dt\right)'=f(x)$$ So<br>
$$\int_{x^2}^0\sqrt{1+t^4}dt+\int_0^{\sin(x)}\sqrt{1+t^4}dt=$$<br>
$$\int_0^{\... | DonAntonio | 31,254 | <p>The function $\,\sqrt{1+t^4}\;$ is defined and continuous everywhere, from where we can thus apply the Fundamental Theorem of Integral Calculus and get</p>
<p>$$\int\limits_{x^2}^{\sin x}\sqrt{1+t^4}dt=F(\sin x)-F(x^2)$$</p>
<p>with $\;F\;$ a function s.t. $\,F'(x)=\sqrt{1+x^4}\;$ (the primitive function of the in... |
1,283,037 | <blockquote>
<p>Consider the series: $$ \sum_{i=1}^\infty \frac{i}{(i+1)!} $$ Make a
guess for the value of the $n$-th partial sum and use induction to prove
that your guess is correct.</p>
</blockquote>
<p>I understand the basic principles of induction I think I would have to assume the n-1 sum to be true and... | DeepSea | 101,504 | <p>We have:$$\dfrac{i}{(i+1)!} = \dfrac{(i+1)-1}{(i+1)!} = \dfrac{i+1}{(i+1)!} - \dfrac{1}{(i+1)!} = \dfrac{1}{i!}-\dfrac{1}{(i+1)!} \Rightarrow S_n = 1-\dfrac{1}{(n+1)!}\Rightarrow S=\displaystyle \lim_{n\to \infty} S_n = 1$$</p>
|
3,563,419 | <p>What I did was writing <span class="math-container">$(1\, 2)(3\, 4)(1\, 3)(2\, 4)$</span> as <span class="math-container">$\bigl(\begin{smallmatrix}
1& 2 &3 & 4\\
2& 1 &4 & 3
\end{smallmatrix}\bigr)\bigl(\begin{smallmatrix}
1& 2 &3 &4 \\
3& 4& 1 & 2
\end{sma... | Community | -1 | <p>well it's as simple as using a theorem.
This is an alternating series which monotonically converges to 0.
So according to Leibniz it converges.
thanks to @lulu</p>
|
4,552,098 | <p>The rules: Two people rolls a single dice. If the dice rolls 1,2,3 or 4, person A gets a point. For the rolls 5 and 6, person B gets a point. One person needs a 2 point lead to win the game.</p>
<p>This is a question taken from my math book. The answer says the probability is <span class="math-container">$\frac{4}{5... | Lourrran | 1,104,122 | <p>In probability, you need to always cross-check your results.
In this game, A will win with proba PA, and B will win with proba PB.</p>
<p>And <strong>PA+PB should be equal to 1.</strong></p>
<p>If you say PA=4/9, you say, with same logic, PB=1/9</p>
<p>Do you have PA+PB=1 ? No. So there is something wrong.</p>
<p>4/... |
2,370,378 | <p>How many distinct pairs of disjoint hyperplanes of size $q^{n-1}$ exist in $\mathbb{F}_q^n$?</p>
<p>Initially I had just thought to pick n points to define a hyperplane, and divide by the number of ways to pick such points, that is:</p>
<p>$\frac{q^n \choose n}{q^{n-1} \choose n}$</p>
<p>From there each hyperplan... | Alex Meiburg | 127,777 | <p>For selecting the first and second points, you have $q^n$ and $q^n-1$ options: anything except for the same thing. For selecting the third point, you can again pick any of the $q^n$, except for the $q$ points on the line defined -- so that you have $q^n-q$ options. For the fourth point, you have $q^n-n^2$, because y... |
870,367 | <p>Suppose $A$ is a $2$-by-$2$ matrix, and $\mathcal{l}$ is an invariant line under $A$, so $(x,mx+c)$ is mapped to $(X,mX+c)$ for some variable $X$ linear in $x$. Then is there a point on the line $\mathcal{l}$ which is a fixed point of $A$, i.e. there is some $x' \in \mathcal{l}$ such that $Ax'=x'$?</p>
<p>The reaso... | Graham Kemp | 135,106 | <p>Let $S$ be the count of the 12 students selected who eat sushi.</p>
<p>$\begin{align}\Pr(S\geq 11) & = \Pr(S=11 \cup S=12) \\ & = \Pr(S=11)+\Pr(S=12) \\ & = \frac{{^{18}C_{11}}{^{10}C_1}+{^{18}C_{12}}}{^{28}C_{12}}\end{align}$ </p>
<p>This is the probability of selecting either 11 of 18 sushi( and 1 of... |
14,765 | <p>I like to make the "dominoes" analogy when I teach my students induction.</p>
<p>I recently came across the following video:</p>
<p><a href="https://www.youtube.com/watch?v=-BTWiZ7CYoI" rel="noreferrer">https://www.youtube.com/watch?v=-BTWiZ7CYoI</a></p>
<p>In this video, a sequence of concrete block wall caps ar... | Community | -1 | <p>Maybe the first question to ask is : WHY are they doing that mistake. </p>
<p>As says a famous 17th century philosopher, falsehood is nothing in itself, but just incomplete truth. So to understand the mistake, it can be usefull to look for the incomplete truth that lies in that mistake. </p>
<p>I think the " incom... |
2,060,633 | <p>Let $R=(\Bbb Z/5\Bbb Z$)[X]/ $\langle X^n \rangle $ with $n\in\Bbb N_+$. Show that $f\in R$ is a zero divisor if and only if $f_0=0$. And how many zero dividers exist?</p>
<p>I know that zero divispr means there exists non-zero $m$ such that $fm=0$ in $R$ and that $\Bbb Z/5\Bbb Z$ has no zero divisors, meaning $f_... | Michael R | 399,606 | <p>Imagine that you are randomly placing students into 4 different groups. You "draw" (or randomly place) graduate student 1 first, and you put him or her in group x. You then have to place the remaining 15 students into the 4 groups (where group x has 3 remaining spots, and the other groups have 4 remaining spots). </... |
4,224,713 | <p>Let <span class="math-container">$f : \Bbb{Z} \to \Bbb{Z}_2$</span> be any map such that <span class="math-container">$f(xy) = f(x) + f(y) + f(x)f(y)$</span> and <strong><span class="math-container">$f$</span> is non-constant.</strong></p>
<p>Then can we always write <span class="math-container">$f(x) = (q \mid x)$<... | Alan | 175,602 | <p>The map <span class="math-container">$f(x)=1$</span> satisfies your equation, but there certainly is no prime that divides every <span class="math-container">$x$</span>! So no.</p>
|
4,224,713 | <p>Let <span class="math-container">$f : \Bbb{Z} \to \Bbb{Z}_2$</span> be any map such that <span class="math-container">$f(xy) = f(x) + f(y) + f(x)f(y)$</span> and <strong><span class="math-container">$f$</span> is non-constant.</strong></p>
<p>Then can we always write <span class="math-container">$f(x) = (q \mid x)$<... | Servaes | 30,382 | <p>A particular counterexample would be
<span class="math-container">$$f(x):=\begin{cases}1&\text{ if }2\mid x\text{ or }3\mid x\\
0&\text{ otherwise }\end{cases}.$$</span>
Of course this generalizes to arbitrary primes, and arbitrarily many of them.</p>
<p>Every such function <span class="math-container">$f$</... |
2,642,547 | <p>If a subset A of $\mathbb{R}^n$ has no interior, must it be closed?</p>
<p>Can I prove this using the example of a subset A that consists of a single point, so A has no interior yet it is closed?</p>
| ajotatxe | 132,456 | <p>The set $\{\frac1n: n\in\Bbb Z_+\}$ is not closed, and its inerior is empty.</p>
|
2,008,341 | <p>Consider the standard presentation of $D_{2n}$:</p>
<p>$\langle r, s : r^n = s^2 =1, rs = sr^{-1}\rangle$.</p>
<p>I have seen the latter relation given as $sr = r^{-1}s$ a few times. Is this correct, as well?</p>
| eepperly16 | 239,046 | <p>It is correct</p>
<p>$$rs = sr^{-1} \Leftrightarrow rsr = s \Leftrightarrow sr = r^{-1}s$$</p>
|
3,624,953 | <p>Let <span class="math-container">$(X_t)_{t \in [0, T]}$</span> be a continuous stochastic process with paths which are a.s. continuous, the underlying space of which is irrelevant but is well defined. Let <span class="math-container">$a$</span> be a constant
Define two stopping times
<span class="math-container">$$\... | infinite_monkey | 776,731 | <p>In general they are not. Consider the constant process which is identically <span class="math-container">$a$</span> for all <span class="math-container">$t$</span>. Then <span class="math-container">$\tau_2 = 0$</span> a.s. but <span class="math-container">$\tau_1 = +\infty$</span>. (If you are working with the conv... |
2,699,483 | <blockquote>
<p>The probability that a fair-coin lands on either <strong>Heads</strong> or <strong>Tails</strong> on a certain day of the week is $1/14$. </p>
<p><strong>Example: (H, Monday), (H, Tuesday) $...$ (T, Monday), (T, Tuesday) $...$</strong><br>
Thus, $(1/2 \cdot 1/7) = 1/14$. There are $14$ such ou... | jgon | 90,543 | <p>Remy has already given the correct answer, but is not confident because of a missing intuition, and I already more or less answered the question in comments on NewGuy's answer, so I'll just write it up and try to give an intuition for it. </p>
<p>The sample space for a single coin flip is $\Omega=\newcommand{\set}[... |
1,123,247 | <p>let $a_n$ be a sequence of real numbers such that the series $\sum |a_n|^2$ is convergent.Find range of $p$ such that the series $\sum |a_n|^p$ is convergent.</p>
<p>My try:</p>
<p>To show the series it is necessary to show that sequence of partial sums of $\sum |a_n|^p$ is bounded.</p>
<p>Consider $S_n=|a_1|^p+|... | Umberto P. | 67,536 | <p>If you denote $M = \max\{|a_n|\}$ you have $$\sum_{n=1}^\infty |a_n|^p = \sum_{n=1}^\infty |a_n|^{p-2} |a_n|^2 \le M^{p-2} \sum_{n=1}^\infty |a_n|^2 < \infty$$whenever $p \ge 2$.</p>
|
3,029,519 | <p>I have a variable, <span class="math-container">$X$</span>, which is normally distributed</p>
<p><span class="math-container">$$X \sim \mathcal{N}(\mu, \sigma^2)$$</span></p>
<p>I also have an event, <span class="math-container">$A$</span>, which measures the probability that <span class="math-container">$X$</span... | Community | -1 | <p>I think
<span class="math-container">$$\frac{4}{n+4}-\frac{n}{(n+3)(n+4)}=\frac{4(n+3)-n}{(n+3)(n+4)}=\frac{3(n+4)}{(n+3)(n+4)}=\frac{3}{n+3}$$</span></p>
|
4,510,808 | <p>Value of p such that <span class="math-container">$\mathop {\lim }\limits_{x \to \infty } \left( {{x^p}\left( {\sqrt[3]{{x + 1}} + \sqrt[3]{{x - 1}} - 2\sqrt[3]{x}} \right)} \right)$</span> is some finite | non-zero number.</p>
<p>My approach is as follow</p>
<p><span class="math-container">$\mathop {\lim }\limits_{... | Prem | 464,087 | <p><span class="math-container">$\lim \limits_{x \to \infty } {x^{P}(\sqrt[3]{x+1} + \sqrt[3]{x-1} -2\sqrt[3]{x})} = \lim \limits_{x \to \infty } {x^{P}(\sqrt[3]{x}\sqrt[3]{1+1/x} + \sqrt[3]{x}\sqrt[3]{1-1/x} -2\sqrt[3]{x})}$</span></p>
<p><span class="math-container">$\lim \limits_{x \to \infty } {x^{P}(\sqrt[3]{x+1} ... |
411,549 | <p>Here is an modular equation</p>
<p>$$5x \equiv 6 \bmod 4$$</p>
<p>And I can solve it, $x = 2$.</p>
<p>But what if each side of the above equation times <strong>8</strong>, which looks like this</p>
<p>$$40x \equiv 48 \bmod 4$$</p>
<p>Apparently now, $x = 0$. Why is that? Am I not solving the modular equation in... | Brian M. Scott | 12,042 | <p>The congruence $40x\equiv48\pmod4$ means that $4\mid40x-48$. But $40x-48=4(10x-12)$, so this is always true: $40x\equiv48\pmod4$ for <strong>all</strong> integers $x$. Thus, $x\equiv0\pmod4$ is <strong>not</strong> the only solution.</p>
<p><strong>Added:</strong> If you have the congruence $ax\equiv b\pmod m$, and... |
1,818,557 | <p>I will be teaching some "topology" to high school students. I was wondering how to explain to such a school student that on a sphere the shortest path between 2 points is given by a great circle?</p>
<p>Also, how to explain that if they lived on a sphere they would have no notion of "above" or "below"? I cannot fin... | user7530 | 7,530 | <p>If you connect the two points by a rubber band in the shape of a meandering path on the sphere, it is intuitive the rubber band will snap into a great circle shape.</p>
<p>Alternatively you can explain the geodesic as the path a magnetic marble would take if the sphere were a steel ball, and you let the marble roll... |
198,105 | <p>I have runtimes for requests on a webserver. Sometimes events occur that cause the runtimes to skyrocket (we've all seen the occasionaly slow web page before). Sometimes, they plummet, due to terminated connections and other events. I am trying to come up with a consistent method to throw away spurious events so tha... | Michael R. Chernick | 30,995 | <p>Methods such as Chauvenet's and Peirce's are 19th century attempts at formal testing for outliers from a normal distirbution. They are not sound approaches because they do not consider the distribution of the largest and smallest values from n iid normally distributed variables. Test's such as Grubbs' and Dixon's ... |
702,506 | <p>We have an exam in $3$ hours and I need help how to solve such trigonometric equations for intervals.</p>
<p>How to solve</p>
<p>$$\sin x - \cos x = -1$$</p>
<p>for the interval $(0, 2\pi)$.</p>
| Patrick Da Silva | 10,704 | <p>Square it. You get
$$
1 - 2 \sin x \cos x = \sin^2x + \cos^2 x - 2 \sin x \cos x = (\sin x - \cos x)^2 = 1
$$
which implies $\sin x = 0$ or $\cos x = 0$. Therefore the only possible solutions are multiples of $\pi/2$. See which ones are actual solutions of your problem (we squared, so $\sin x - \cos x = \pm 1$ when ... |
702,506 | <p>We have an exam in $3$ hours and I need help how to solve such trigonometric equations for intervals.</p>
<p>How to solve</p>
<p>$$\sin x - \cos x = -1$$</p>
<p>for the interval $(0, 2\pi)$.</p>
| Barry Cipra | 86,747 | <p>Rewrite the equation as $\sin x=\cos x-1$, square both sides, and use the identity $\sin^2x=1-\cos^2x$ to get</p>
<p>$$1-\cos^2x=\cos^2x-2\cos x+1$$</p>
<p>which simplifies to</p>
<p>$$\cos x(1-\cos x)=0$$</p>
<p>so that either $\cos x=0$ or $\cos x=1$. The former corresponds to $x=\pi/2$ and $3\pi/2$ in the in... |
702,506 | <p>We have an exam in $3$ hours and I need help how to solve such trigonometric equations for intervals.</p>
<p>How to solve</p>
<p>$$\sin x - \cos x = -1$$</p>
<p>for the interval $(0, 2\pi)$.</p>
| robjohn | 13,854 | <p><strong>Hint:</strong> Use the formula for the sine of a difference to get
$$
\begin{align}
\sqrt2\sin(x-\pi/4)
&=\sqrt2\Big(\sin(x)\cos(\pi/4)-\cos(x)\sin(\pi/4)\Big)\\
&=\sin(x)-\cos(x)
\end{align}
$$</p>
|
716,767 | <p>I feel like an idiot for asking this but i can't get my formula to work with negative numbers</p>
<p>assume you want to know the percentage of an increase/decrease between numbers</p>
<pre><code>2.39 1.79 =100-(1.79/2.39*100)=> which is 25.1% decrease
</code></pre>
<p>but how would i change this fo... | Hans Knudsen | 178,134 | <p>I know this is a very old thread, but I am here for the first time so I hope it is OK to comment.</p>
<p>Let's take an example:</p>
<hr>
<p>Original value $-10$, final value $10$:</p>
<p>$\frac{Original\ value - Final\ value}{Original\ value} = 200\% \ increase $</p>
<hr>
<p>Original value $-1$, final value $1... |
18,530 | <p>Sorry about the title, I have no idea how to describe these types of problems.</p>
<p>Problem statement:</p>
<p>$A(S)$ is the set of 1-1 mappings of $S$ onto itself. Let $S \supset T$ and consider the subset $U(T) = $ { $f \in A(S)$ | $f(t) \in T$ for every $t \in T$ }. $S$ has $n$ elements and $T$ has $m$ eleme... | Neigyl Noval | 6,126 | <p>$\log$ is usually used in statistical results. Here are few examples:</p>
<ol>
<li><p>If I have obtained a data with values in orders of $10$ (like: $10, 100, 50000, 6000000000$), I can use log in my graph to shorten the length of the axis values (see: log-log graph)</p></li>
<li><p>There are results which increase... |
24,361 | <p>Let $X$ be a topological space and let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over $X$.</p>
<p>Of course, if one has a morphism $f : \mathcal{F} \to \mathcal{G}$ such that for all $x\in X$, $f_x : \mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, then it is known that $f$ itself is an isomorphism.</p>
<p... | Andrea Ferretti | 828 | <p>No, indeed there exist sheaves which are locally isomorphic to locally constant sheaves, without being locally constant. These are usually called local coefficient systems.</p>
<p>It is not hard to see that, for a nice spaces $X$, e.g. a manifold, to give a sheaf locally isomorphic to the locally constant sheaf $\u... |
24,361 | <p>Let $X$ be a topological space and let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over $X$.</p>
<p>Of course, if one has a morphism $f : \mathcal{F} \to \mathcal{G}$ such that for all $x\in X$, $f_x : \mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, then it is known that $f$ itself is an isomorphism.</p>
<p... | Qfwfq | 4,721 | <p>No: think of non-isomorphic vector bundles over $X$. They are stalkwise isomorphic even as modules (over $\mathcal{O}_{X,x}$ at various $x \in X$, where $\mathcal{O}_X$ is the sheaf of continuous functions on $X$), hence as abelian groups.</p>
|
138,866 | <p>I have data in a csv file. The first row has labels, and the first column, too.</p>
<pre><code>Datos = Import["C:\\Users\\jodom\\Desktop\\Data.csv"]
</code></pre>
<p>Tha data in the csv file is that:</p>
<pre><code>{{"No", "Vol", "Vel"}, {1, 500, 45}, {2, 700, 67}, {3, 350, 87}, {4,
123, 23}, {5, 587, 45}, {6, 4... | kglr | 125 | <pre><code>datos = {{"No", "Vol", "Vel"}, {1, 500, 45}, {2, 700, 67}, {3, 350, 87},
{4, 123, 23}, {5, 587, 45}, {6, 435, 89}, {7, 896, 65}, {8, 125, 5},
{9, 476, 27}, {10, 987, 80}};
ds = Dataset[AssociationThread[datos[[2;;,1]]->
(AssociationThread[Rest@First@datos->Rest[#]]&/@R... |
4,231,580 | <p>I have been working on th integral
<span class="math-container">$$\int_0^\infty \frac{\sin x}{1+x^2} dx$$</span>
for a short while now trying substitutions and even Laplace transforms and other stuff but gave up. I looked to Wolfram Alpha but how did it get the series expansion of the constant that it did (see <a hr... | Community | -1 | <p>In the hope that it becomes a bit more visible, what happens behind the scenes, I tried to extract all steps from the linked notebook:</p>
<p><span class="math-container">$\frac{\text{Ei}(1)-e^2 \text{Ei}(-1)}{2 e}=\sum _{k=1}^{\infty } -\frac{e^2 (-1)^k-1}{2 e k k!}-\frac{e \gamma }{2}+\frac{\gamma }{2 e}$</span></... |
3,072,142 | <p>Isn't <span class="math-container">$\mathbb Q[X]/(X^2+1)\cong \mathbb Q[i]$</span> wrong and should be <span class="math-container">$\mathbb Q[X]/(X^2+1)\cong \mathbb Q(i)$</span> ?</p>
<p>Indeed, <span class="math-container">$\mathbb Q[X]/(X^2+1)$</span> is a field whereas <span class="math-container">$\mathbb Q[i... | José Carlos Santos | 446,262 | <p>For any constant function <span class="math-container">$k$</span>, <span class="math-container">$\int_0^1k\,\mathrm d\lambda=k$</span>. So, yes, your argument is correct.</p>
|
163,153 | <p>How do i find the absolute maximum and absolute minimum values of f on this given interval.</p>
<p>$f(x) = 6x^3 − 9x^2 − 36x + 7, \
[−2, 3]$</p>
| Ilies Zidane | 38,639 | <p>For the first question: An application of the $\lambda$-lemma.</p>
<p>Theorem:
Let $c_0$ and $c_1$ be in the same component $U$ of $C \backslash \partial M$, ($M$ is the Mandelbrot set) then $J_{c_0}$ and $J_{c_1}$ (the Julia sets) are homeomorphic (and even quasiconformal-homeomorphism) and the dynamics of the two... |
2,812,122 | <blockquote>
<p>For any real numbers $x$ and $y$ satisfying $x^2y + 6y = xy^3 +5x^2 +2x$, it is known that $$(x^2 + 2xy + 3y^2) \, f(x,y) = (4x^2 + 5xy + 6y^2) \, g(x,y)$$<br>
Given that $g(0,0) = 6$, find the value of $f(0,0)$.</p>
</blockquote>
<p>I have tried expressing $f(x,y)$ in terms of $g(x,y)$. But seems ... | ned grekerzberg | 489,746 | <p>first we shall see that:</p>
<p>$\frac1n\sum_{\omega:\omega^n=1} \omega^{-\alpha}f(\omega) = \frac1n\sum_{\omega:\omega^n=1}\omega^{-\alpha }\sum_{j}a_j\omega^j = \frac1n\sum_{\omega:\omega^n=1}\sum_{j}\omega^{j-\alpha }a_j =$</p>
<p>$ = \frac1n\sum_{j}\sum_{\omega:\omega^n=1}\omega^{j-\alpha }a_j$ </p>
<p>I'm le... |
4,096,516 | <p>Calculate <span class="math-container">$\operatorname{tg}( \alpha), $</span> if <span class="math-container">$\frac{\pi}{2} < \alpha<\pi$</span> and <span class="math-container">$\sin( \alpha)= \frac{2\sqrt{29}}{29}$</span>. Please provide a hint.<br />
I know that <span class="math-container">$\operatorname{t... | PrincessEev | 597,568 | <p>José Carlos Santos provided an algebraic answer; I'll provide a more geometric one.</p>
<p>Try drawing a picture. Since <span class="math-container">$\alpha$</span> is between <span class="math-container">$\pi/2$</span> and <span class="math-container">$\pi$</span>, it lies in Quadrant II. The triangle that becomes ... |
1,990,638 | <p>This is a purely affine variety. Define $F_1=y-x^2, F_2=z-xy, F_3=xz-y^2$. </p>
<p>It is clear that $(x,x^2,x^3)$ is a solution to $F_1=F_2=F_3=0$. Since projective twisted cubic is not intersection of two quadractics which contains an extra line, I would not expect this to be the case in affine as well. However, $... | Georges Elencwajg | 3,217 | <p>1) It is perfectly correct that the affine curve $$C=\{(u:u^2:u^3)\vert u\in k\}\subset \mathbb A^3_k=\mathbb A^3_{x,y,z}=\mathbb A^3\:$$ is the complete intersection of the affine smooth quadrics $Q_1=V(F_1), Q_2=V(F_2)\subset \mathbb A^3$. </p>
<p>2) The curve $C$ has as closure in $\mathbb P^3=\mathbb P^3_{[... |
313,254 | <p>I know this question has been asked before on MO and MSE (<a href="https://mathoverflow.net/questions/59605/reference-in-riemann-surfaces">here</a>, <a href="https://math.stackexchange.com/questions/407004/good-book-for-riemann-surfaces">here</a>, <a href="https://math.stackexchange.com/questions/1839673/books-on-ri... | M.G. | 1,849 | <p>As it is evident from your bibliography list, there are two aspects of the theory: Riemann surfaces in the sense of 1-dimensional complex manifolds (which are not necessarily algebraic) and Complex Algebraic Curves (which are not necessarily smooth). It should be pointed out that some authors (old-school?) still use... |
2,460,195 | <p>I had the following question:</p>
<p>Three actors are to be chosen out of five — Jack, Steve, Elad, Suzy, and Ali. What is the probability that Jack and Steve would be chosen, but Suzy would be left out?</p>
<p>The answer given was:
Total Number of actors = $5$;
Since Jack and Steve need to be in the selection and... | iamwhoiam | 412,755 | <p>Note that $y + z$ must be even. Hence, $y$ and $z$ are either both even or both odd. </p>
<p>When they are both even (say $y = 2c_1$ and $z = 2c_2$), the equation reduces to $x + c_1 + c_2 = 8$. The number of non-negative integer solutions in this case is $\binom{10}{8}$. When they are both odd (say $y = 2c_1 + 1$ ... |
234,668 | <p>I have an equation in $x$ and I would like to determine if it has any solutions modulo a large prime $p$. Suppose $p$ is large enough that I can factor numbers up to $p$, but I cannot test all values up to $p$. (Actually, so far, I have been doing just that -- but I'd like to avoid this as it takes a long time. If y... | The Substitute | 17,955 | <p>$n=2m+1 \Rightarrow kn=2mk+k=k(2m+1)$. Note that the latter expression is the product of an integer and an odd integer, which is not always odd.</p>
<p>For example,
Even*Odd:
$(2a)\cdot (2b+1)=4ab+2a=2a(2b+1)$, which is even.</p>
|
701,176 | <p>A function $f$ is differentiable over its domain and has the following properties:</p>
<ol>
<li><p>$\displaystyle f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$</p></li>
<li><p>$\lim_{h \to 0} f(h) = 0$</p></li>
<li><p>$\lim_{h \to 0} f(h)/h = 1$</p></li>
</ol>
<p>i) Show that $f(0)=0$</p>
<p>ii) show that $f'(x)=1+[f(x)]^... | Harshal Gajjar | 132,479 | <p>$a^3−6a^2−2$ does hit the $x$-axis at $a=6.0546$, so, after placing the value of $a$ in the equation, you get the value of $a^3−6a^2−2$ to be $0.001536$ which is not equal to $0$.</p>
|
183,039 | <p>I'm pretty weak in the field of mathematics, but a strong programmer. I am looking for a mathematical solution that, given two points on a line will give me a curve between them, including those two points within the curve itself.</p>
<p>For instance, if I have a set of points { (0, 3) (1,10) } I'd like a mathemati... | MJD | 25,554 | <p>You could do that, but if a straight line is acceptable, the best-of-breed algorithm for calculating which pixels to plot is <a href="http://en.wikipedia.org/wiki/Bresenham%27s_line_algorithm" rel="nofollow">Bresenham's algorithm</a>. It is easy to program, produces good-looking output, and it is extremely efficient... |
2,965,865 | <blockquote>
<p>Finding the minimum value of <span class="math-container">$\displaystyle \frac{x^2 +y^2}{y}.$</span> where <span class="math-container">$x,y$</span> are
real
numbers satisfying <span class="math-container">$7x^2 + 3xy + 3y^2 = 1$</span></p>
</blockquote>
<p>Try: Equation <span class="math-conta... | smcc | 354,034 | <p>The problem has no solution (assuming you are looking for a (global) minimum rather than just a local minimum). As we approach the point <span class="math-container">$(1/\sqrt{7},0)$</span> along the constraint curve from below the <span class="math-container">$x$</span>-axis, the value of the function goes to <span... |
1,830,867 | <p>I'm trying to self-study some algebraic topology, reading Hatcher. His questions seem much less straightforwardly worded than Munkres - with Munkres it was always clear that you weren't expected to know much coming in, whereas Hatcher seems to assume a lot of knowledge. Often simple seeming statements of his just se... | Alex Ermolin | 292,258 | <p>What I think is meant by this question is this.
Given a topological space $X$ and a point $p \in X$ we define the fundamental group of $X$ with basepoint $p$ to be $\pi_1(X, p) = \{[\gamma] \mid \gamma \text{ is a closed loop around }p\}$. Here $[\gamma]$ denotes the homotopy class of $\gamma$.</p>
<p>Given $p, q \... |
1,460,561 | <p>Let $\lambda =1$ is the eigenvalue corresponding to the single Jordan block $J$. Prove $J^m \sim J$ with an arbitrary positive integer $m$.</p>
<p>My try: Because $\lambda = 1$ is eigenvalue, $(J-I)^m =0$. After that $(J-I)^{m-1} J = (J-I)^{m-1}$. At this point, I do not know how to continue.</p>
| Andreas Cap | 202,204 | <p>First observe that any power $J^m$ of $J$ is an upper triangular matrix with all entries on the main diagonal equal to $1$. To prove that $J^m$ is similar to $J$, you can show that $J$ is the Jordan normal form of $J^m$. To do this, it suffices to show that $(J^m-I)^k$ is non-zero provided that $(J-I)^k$ is non-zero... |
4,126,238 | <p>Prove <span class="math-container">$f$</span> is uniformly continuous <span class="math-container">$\implies$</span> there exist <span class="math-container">$C, D$</span> such that <span class="math-container">$|f(x)| < C + D|x|$</span>.</p>
<p>Proof below. Please verify or critique.</p>
<p>By definition of uni... | Siong Thye Goh | 306,553 | <p>It seems to be that you are assuming <span class="math-container">$x$</span> to be positive and you are trying to partition <span class="math-container">$[0, x]$</span> and use triangle inequalities. Be careful that <span class="math-container">$x$</span> can be negative and you are partitioning the interval correct... |
2,028,799 | <p>I've been trying to get this limit for hours. Can someone help me, please?<br>
The solution manual says it's 0 but I can't get there. I tried to use $\lim_{h\to0} {h-\cos(h)\over h} = 0$.</p>
<p>$$\lim_{x\to 0}\frac{x\cos(1/x)}{x-\sqrt{x}}.$$<br>
Thank you.</p>
| Simply Beautiful Art | 272,831 | <p>Divide numerator and denominator by $x$.</p>
<p>$$\frac{x\cos(1/x)}{x-\sqrt x}=\frac{\cos(1/x)}{1-x^{-0.5}}$$</p>
<p>Now see that</p>
<p>$$-1\le\cos(1/x)\le1$$</p>
<p>And apply squeeze theorem.</p>
<hr>
<p>Remark: Notice that $\frac{x\cos(1/x)}{x-\sqrt{x}}$ is not defined for $x<0$, so the above limit only... |
2,485,109 | <blockquote>
<p>Proof by induction that $1^2 + 2^2 + 3^2 +.......+ n^2 = \frac{1}{6}\cdot n\cdot (1+n)\cdot (1+2n)$</p>
</blockquote>
<p>I tried showing that </p>
<p>$1^2 + 2^2 + 3^2 +.......+ (k+1)^2 = \frac{1}{6}\cdot (k+1)\cdot (1+(k+1))\cdot (1+2(k+1))$</p>
<p>By using the left side:</p>
<p>$1^2 + 2^2 + 3^2 +... | Mr Pie | 477,343 | <p><strong>This is more of a comment as opposed to an answer</strong></p>
<hr>
<p>$$5x + 7y = 1234$$
What you want to do first is find integer solutions for $x$ and $y$ to satisfy $5x + 7y = 1$. A solution is found by letting $x = 3$ and $y = -2$.</p>
<p>Since $1234 = 1234\times 1$, we just multiply $3$ and $-2$ by ... |
3,709,331 | <p>How can I integrate
<span class="math-container">$$
\int \frac{x\,dx}{(a-bx^2)^2}
$$</span> I've tried to use partial fraction decomposition, but I'm getting six equations for four variables, and they don't give uniform answers.</p>
| J.G. | 56,861 | <p>Substitute <span class="math-container">$u=a-bx^2$</span> so your integral is <span class="math-container">$-\frac{1}{2b}\int\frac{du}{u^2}=\frac{1}{2bu}+C$</span>.</p>
|
806,779 | <p>Prove, without expanding, that
\begin{vmatrix}
1 &a &a^2-bc \\
1 &b &b^2-ca \\
1 &c &c^2-ab
\end{vmatrix} vanishes.</p>
<p>Any hints ?</p>
| Lutz Lehmann | 115,115 | <p>Add $(ab+bc+ca)$ times the first column to the last and find the common factor of the last column.</p>
|
2,294,991 | <p>I am trying to find the limit as $x\to 8$ of the following function. What follows is the function and then the work I've done on it. </p>
<p>$$ \lim_{x\to 8}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8}$$</p>
<hr>
<p>\begin{align}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8} &= \frac{\frac{1}{\sqrt{x +1... | StackTD | 159,845 | <blockquote>
<p>$$\frac{\frac{1}{x+1}-\frac{1}{9}}{(x-8)\left(\frac{1}{\sqrt{x +1}} + \frac{1}{3}\right)} = \frac{\color{red}{8-x}}{(x-8)\left(\frac{1}{\sqrt{x +1}} + \frac{1}{3}\right)}$$</p>
</blockquote>
<p>Careful in the numerator:
$$\frac{1}{x+1}-\frac{1}{9} \ne 8-x$$
but rather:
$$\frac{1}{x+1}-\frac{1}{9}= \f... |
156,474 | <p>Is this phrase safe to consider in general: </p>
<blockquote>
<blockquote>
<p>Equal sides of a polygon have corresponding equal angles</p>
</blockquote>
</blockquote>
<p>if not how would you refine or correct it</p>
<p>Example of a corresponding angle would be</p>
<p><img src="https://i.stack.imgur.com/o... | Robert Mastragostino | 28,869 | <p>Construct a pentagon of all equal sides. Physically, out of pieces of straws or something. Note how you can bend it in any which way so that no angles are the same. So the question of "equal sides correspond to equal angles" is true for triangles, can <em>maybe</em> be rephrased in a way that's true for quadrilatera... |
603,290 | <p>this might be a stupid question, but is any $C^2$ function $f:\mathbb{R}\to\mathbb{C}$ of period $f(x+L)=f(x)$ automatically analytic (and in particular, infinitely often differentiable)?</p>
<p>I learned that for a $L$-periodic $C^k$ function with Fourier coefficients $f_n$, they converge to zero like $|f_n| |n|^k... | Nate Eldredge | 822 | <p>Even a $C^\infty$ periodic function need not be real analytic. An example is
$$f(x) = \exp\left(\frac{-1}{\sin(x)^2}\right)$$
with $f(n \pi) = 0$. Showing $f$ is smooth is a variation on a standard exercise. But at $x=n\pi$, all derivatives of $f$ vanish, yet $f$ is not zero in any neighborhood of $x$.</p>
|
239,097 | <p>Recently, I want to buy some 30xx series GPUs which are used to do deep learning jobs, so does anyone know whether Mathematica's <code>NetTrain</code> supports 30xx series GPUs?</p>
| HyperGroups | 6,648 | <p>2021-03-12</p>
<p>Yes, I can run NetTrain with RTX3060 on Window 10</p>
<p>But, I'd waited for a long time the first time without any tips.</p>
<p><a href="https://i.stack.imgur.com/Rhguw.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Rhguw.jpg" alt="enter image description here" /></a>
<a href="... |
2,970,053 | <p>I've been trying to understand this problem for hours but not getting it. HELP!!!</p>
<p>The correct answer is <span class="math-container">$\frac{2}{3}$</span>, but I don't know why this is the correct answer.</p>
<p>Thank you in advance for your help!</p>
| M. Wind | 30,735 | <p>Please note that --ignoring the spaces in between them-- there are only six ways to order these three books: (123), (132), (213), (231), (312) and (321). Since the books are randomly arranged, each of this six orderings must have the same probability. Which is therefore 1/6. So that is your answer.</p>
|
4,605,263 | <p>If someone asks me right now, what a number is, I would say 1,2,3,-1,1+i,2.13, etc. But what I have essentially stated are just symbols. I tried to explore a bit what a number exactly is. What are these symbols pointing to?</p>
<p>As I got to know the answer is not that simple, we cannot have an all-encompassing def... | Quanto | 686,284 | <p>Let <span class="math-container">$J(a)=\int_{0}^{\infty}\frac{f(x)}x e^{-ax} dx$</span>, with periodic function
<span class="math-container">$$f(x)= f(x+2\pi k)=|\cos( x-\frac{\pi}{4} ) |- |\cos( x+\frac{\pi}{4} ) |$$</span>
Then
<span class="math-container">\begin{align}
J’(a)& =- \int_0^\infty f(x)e^{-ax}dx
= ... |
2,205,776 | <p>I am pretty sure there is an easy Counter example But i do not find One right Now.</p>
| Ethan Alwaise | 221,420 | <p>HINT: If the sequence $\{a_n\}$ is not eventually constant, then you can find arbitrarily large $n,m$ such that
$$\vert a_n - a_m \vert \geq 1.$$</p>
|
381,093 | <p>I would like to prove Chebyshev's sum inequality, which states that:</p>
<p>If <span class="math-container">$a_1\geq a_2\geq \cdots \geq a_n$</span> and <span class="math-container">$b_1\geq b_2\geq \cdots \geq b_n$</span>, then<br />
<span class="math-container">$$
\frac{1}{n}\sum_{k=1}^n a_kb_k\geq \left(\frac{1}{... | Seva | 9,924 | <p>Let <span class="math-container">$A$</span> be the random variable attaining the values <span class="math-container">$a_1,\dotsc,a_n$</span> with equal probabilities, and define <span class="math-container">$B$</span> similarly, subject to <span class="math-container">$\mathbb P(B=b_i|A=a_i)=1$</span>. Then <span cl... |
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