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 |
|---|---|---|---|---|
1,246,250 | <p>I recently learned of Cantor's diagonal argument, and was thinking about why there can't be a bijection between any infinite set of integers and any infinite set of real numbers. I understood the basic idea behind the proof, but I was thinking of a particular transformation, for which I don't see why it doesn't form... | P Vanchinathan | 28,915 | <p>For example $1/3=0.333\ldots$ is not in the image, and so the function is not onto.(This is already contained in the answer by MooS).</p>
|
2,368,179 | <p>Answer should be in radians
Like π/4 (45°) π(90°).
I used $\tan(A+B)$ formula and got $5/7$ as the answer, but that's obviously wrong.</p>
| Dando18 | 274,085 | <p>Using the $\tan(A+B)$ formula,</p>
<p>$$ \tan(A+B) = \frac{-1/2 - 1/3}{1-(1/2)(1/3)} = -1 $$</p>
<p>now use the $\arctan$,</p>
<p>$$ A+B = \arctan(-1) = n\pi + \frac{3\pi}{4}, \ \ n\in\mathbb Z $$</p>
|
3,699,439 | <p>I am interested if there is geometric meaning (using graphs) of <span class="math-container">$(1 + \frac{1}{n})^n$</span> when <span class="math-container">$n \rightarrow \infty$</span>. Also, is there visual explanation of why is <span class="math-container">$e^x = (1 + \frac{x}{n})^n$</span> when <span class="math... | Community | -1 | <p>I think of my favorite, and pretty geometric, proof of this limit, using the squeeze or sandwich theorem for limits. You can do it using an upper and lower Riemann sum with one subdivision for the integral of <span class="math-container">$1/t$</span>.</p>
<p>One has <span class="math-container">$L\le\int_1^{1+x/n}1... |
2,917,896 | <p>I think my proof is wrong but I don't know how to approach the statement differently. I hope you can help me identify where I'm mistaken/incomplete.</p>
<p>Proof:
$$\text{We need to prove: } \bigcup_{n=1}^{\infty}[3 - \frac{1}{n}, 6] = [2, 6] $$</p>
<p>$$\text{Thus, } x \in \bigcup_{n=1}^{\infty}[3 - \frac{1}{n... | eloiprime | 180,579 | <p>We have $[3-\frac{1}{n},6]\subseteq[2,6]$ for all $n\ge 1$, and thus
$$\bigcup_{n=1}^\infty\left[3-\frac{1}{n},6\right]\subseteq[2,6].$$
As for the reverse inclusion, we have $$[2,6]=\left[3-\frac{1}{1},6\right]\subseteq\bigcup_{n=1}^\infty\left[3-\frac{1}{n},6\right].$$</p>
|
3,692,083 | <p>I wish to show that the closed unit ball in <span class="math-container">$l^1$</span> is not compact, for which I believe it would be easiest to show that it is not bounded. For this I want to consider the sequence {1, 1/2, 1/3, ... , 1/n, ...}, since the harmonic series is known to be divergent. But will this seque... | obscurans | 619,038 | <p><span class="math-container">$\ell^1$</span> is the space of sequences under the norm
<span class="math-container">$$\left\|x\right\|=\sum_{n=1}^{\infty}\left|x_i\right|$$</span>
such that the norm is finite. So no, not only is the sequence <span class="math-container">$\left\{\frac{1}{n}\right\}_{n=1}^{\infty}$</sp... |
3,384,416 | <p>Consider the following algorithm: </p>
<ol>
<li><p>pick an integer <span class="math-container">$n> 0$</span>.</p></li>
<li><p>If <span class="math-container">$n$</span> is even, divide by 2. If <span class="math-container">$n$</span> is odd, find the least perfect square <span class="math-container">$m^2$</span... | Nitin Uniyal | 246,221 | <p>Geometrically the given hypothesis implies that the curvature of the surface must be zero at the intersecting curve.</p>
<p>This implies <span class="math-container">$z=f(x,y=\lambda x)=c$</span> or <span class="math-container">$dz=0$</span>.</p>
<p><span class="math-container">$\implies \frac{\partial z}{\partial... |
3,384,416 | <p>Consider the following algorithm: </p>
<ol>
<li><p>pick an integer <span class="math-container">$n> 0$</span>.</p></li>
<li><p>If <span class="math-container">$n$</span> is even, divide by 2. If <span class="math-container">$n$</span> is odd, find the least perfect square <span class="math-container">$m^2$</span... | zhw. | 228,045 | <p>WLOG, <span class="math-container">$f(0,0)=0.$</span> I'll assume the graph of <span class="math-container">$f$</span> over every line through the origin is itself a line.</p>
<p>Let <span class="math-container">$v$</span> be a nonzero vector in <span class="math-container">$\mathbb R^2.$</span> For <span class="m... |
2,107,787 | <p>I am a a student and I am having difficulty with answering this question. I keep getting the answer wrong. Please may I have a step by step solution to this question so that I won't have difficulties with answering these type of questions in the future.</p>
<p><em>n</em> is a number. 100 is the LCM of 20 and <em>n<... | Eric Wofsey | 86,856 | <p>Suppose the graph is nonempty and disconnected; say you can partition the $v_\alpha$ into two nonempty sets $A$ and $B$ with no edges between them. Let $U=\bigcup_{\alpha\in A} U_\alpha\cup X\setminus\left(\bigcup_{\alpha} U_\alpha\right)$ and $V=\bigcup_{\beta\in B}U_\beta$. Note that for any $\alpha$, either $U_... |
753,553 | <p>Let $R$ be a commutative ring and let $0 \to L \to M \to N \to0$ be an exact sequence of $R$-modules. Prove that if $L$ and $N$ are noetherian, then $M$ is noetherian. I tried considering the pre image of the map $L \to M$ and the image of the map $M \to N$ as they are submodules of $L$ and $N$ respectively, but I c... | gniourf_gniourf | 51,488 | <p>Let
$$0\longrightarrow L\overset{i}\longrightarrow M\overset{p}\longrightarrow N\longrightarrow0$$
be a short exact sequence of $R$-modules. Assume that $L$ and $N$ are Noetherian modules.</p>
<p>Let $(M_n)_{n\geq0}$ be an ascending chain of submodules of $M$.</p>
<p>Since $(p(M_n))_{n\geq0}$ is an ascending chain... |
32,088 | <h2>Motivation</h2>
<p>One of the methods for strictly extending a theory <span class="math-container">$T$</span> (which is axiomatizable and consistent, and includes enough arithmetic) is adding the sentence expressing the consistency of <span class="math-container">$T$</span> ( <span class="math-container">$Con(T)$</... | Joel David Hamkins | 1,946 | <p>Vitali famously constructed a set of reals that is not Lebesgue measurable by using the Axiom of Choice. Most people expect that it is not possible to carry out such a construction without the Axiom of Choice.</p>
<p>Solovay and Shelah, however, proved that this expectation is exactly equiconsistent with the existe... |
1,791,631 | <p>The following is stated on <a href="https://en.wikipedia.org/wiki/Constructible_universe#L_and_large_cardinals" rel="nofollow">Wikipedia</a> for <a href="https://en.wikipedia.org/wiki/Mahlo_cardinal" rel="nofollow">Mahlo cardinals</a>. Unfortunately, it's not sourced. Where can I find details? I wasn't able to googl... | Stefan Mesken | 217,623 | <p>Let $\kappa$ be a Mahlo cardinal in $V$, i.e. let $\kappa$ be inaccessible such that $S:= \{ \alpha \in \kappa \mid \alpha \text{ is regular} \}$ is stationary in $\kappa$.</p>
<p>First, note that $L \models \kappa \text{ is inaccessible}$. Indeed, if $L \models \kappa \text{ is not a cardinal}$, then there is some... |
106,000 | <p>I have the following data </p>
<pre><code> hours={38.9, 39, 38.9, 39, 39.3, 39.7, 39.2, 38.8, 39.6, 39.8, 39.9, 40.3, \
40, 40.2, 40.8, 40.7, 40.8, 41.2, 40.6, 40.7, 40.7, 40.9, 40.6, 40.8, \
40.3, 40.4, 40.7, 40.5, 40.7, 41.2, 40.3, 39.7, 40.4, 40.1, 40.3, \
40.6, 40.1, 40.5, 40.8, 40.8, 40.9, 41.7}
</code></pr... | Quantum_Oli | 6,588 | <p>It's because the different elements of your <code>PlotMarkers</code> option refer to different data sets, whereas you just have the one data set. </p>
<p>If you were only after different coloured data points then you could <code>Style</code> each element of your data set, however I don't know if its possible to do ... |
268,091 | <p><a href="https://i.stack.imgur.com/PAO6T.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PAO6T.jpg" alt="enter image description here" /></a></p>
<p>I have to solve ODE x'(t)=1/2(x(t))-t, x(0)
The existence of solutions of this IVP is equivalent to finding a fixed point of integral operator T:C[0,... | Roman | 26,598 | <p>We can define your operator <code>T</code> as a functional acting on <a href="https://reference.wolfram.com/language/howto/WorkWithPureFunctions.html" rel="nofollow noreferrer">pure functions</a>:</p>
<pre><code>T[x_] := Function[t, Evaluate[x[0] + Integrate[x[τ]/2 - τ, {τ, 0, t}]]]
</code></pre>
<p>Starting with th... |
194,096 | <p>Is it possible to find an expression for:
$$S(N)=\sum_{k=0}^{+\infty}\frac{1}{\sum_{n=0}^{N}k^n}?$$</p>
<p>For $N=1$ we have</p>
<p>$$S(1) = \displaystyle\sum_{k=0}^{+\infty}\frac{1}{1 + k} = \displaystyle\sum_{k=1}^{+\infty}\frac{1}{k}$$</p>
<p>which is the (divergent) harmonic series. Thus, $S (1) = \infty$.</p... | Sangchul Lee | 9,340 | <p>Let $T(N) = S(N-1)$. Then</p>
<p>$$ \begin{align*}T(n)
&= 1 + \frac{1}{n} + \sum_{k=2}^{\infty} \frac{1}{k^{n-1}+k^{n-2}+\cdots+k+1} \\
&= 1 + \frac{1}{n} + \sum_{k=2}^{\infty} \frac{k - 1}{k^n - 1} \\
&= 1 + \frac{1}{n} + \sum_{k=2}^{\infty} \frac{1}{n} \sum_{l=1}^{n-1} \frac{\omega_l (\omega_l - 1)}{... |
4,159,060 | <p>Let <span class="math-container">$H_1, H_2$</span> be Hilbert spaces and <span class="math-container">$T:H_1\to H_2$</span>.
We say that <span class="math-container">$T$</span> is unitary if it preserves the inner product and unto.</p>
<ol>
<li>Show that the following claims are equivalent:</li>
</ol>
<p>A. <span cl... | Marc Romaní | 179,483 | <p>I think the confusion arises from the notation itself. Note that
<span class="math-container">\begin{align}
&\,\mathbb{E}_X[\mathbb{P}(Y \neq h(X)|X=x)]\\
=&\,\mathbb{E}_X[\mathbb{P}(Y=0,h(X)=1 | X=x) + \mathbb{P}(Y=1,h(X)=0 | X=x)]\\
=&\,\mathbb{E}_X[\mathbb{P}(Y=0|X=x)\mathbb{P}(h(X)=1|X=x) + \mathbb{P... |
2,505,757 | <p>Today, I was trying to prove <a href="https://math.stackexchange.com/questions/2505714/showing-cantor-set-is-uncountable">Cantor set is uncountable</a> and I completed it just a while ago.</p>
<p>So, I know that the end-points of each $A_n$ are elements of $C$ and those end-points are rational numbers. But since $C... | Angina Seng | 436,618 | <p>It's disconnected: the equation is
$$y^2=x(x-1)(x+1).$$
In any real solution $-1\le x\le 0$ or $x\ge1$. There are real
solutions for $y$ for any $x$ in these intervals, so falls into
two components, one defined by $-1\le x\le0$ and the other by $x\ge1$.</p>
|
3,361,153 | <p>I am given two boolean expression<br>
1) <span class="math-container">$x_1 \wedge x_2 \wedge x_3$</span><br>
2) <span class="math-container">$(x_1 \wedge x_2) \vee (x_3 \wedge x_4)$</span> </p>
<p>Now I need to know which expression is trivial and which is non-trivial. I wanted to know what is the procedure of doi... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$n^2=3k$</span>, for some integer <span class="math-container">$k$</span>, then <span class="math-container">$3\mid n^2$</span>. Therefore, by Euclid's lemma, and since <span class="math-container">$3$</span> is prime, <span class="math-container">$3\mid n$</span>.</p>
|
629,347 | <p>I understand <strong>how</strong> to calculate the dot product of the vectors. But I don't actually understand <strong>what</strong> a dot product is, and <strong>why</strong> it's needed.</p>
<p>Could you answer these questions?</p>
| user127.0.0.1 | 50,800 | <p>In a general vector spaces you can define the <strong>length</strong> of a vector by the induced <strong>norm</strong> via
$$\|x\| = \sqrt{x\cdot x}$$</p>
<p>this is possible because the dot product is positive definite and thus $x\cdot x$ is not-negative. </p>
<p>It is even possible to define an <strong>angle</s... |
629,347 | <p>I understand <strong>how</strong> to calculate the dot product of the vectors. But I don't actually understand <strong>what</strong> a dot product is, and <strong>why</strong> it's needed.</p>
<p>Could you answer these questions?</p>
| Steven Alexis Gregory | 75,410 | <p>Consider two points in $\mathbb R^n$, $P=(x_1, x_2, \dots, x_m)$ and $Q=(y_1, y_2, \dots, y_n)$. Let $O=(0,0,\dots, 0)$ be the origin. How do you find $\theta = m\angle POQ$? According to the law of cosines,
$$\cos \theta = \dfrac{|P|^2 +|Q|^2-|Q-P|^2}{2|P|\cdot|Q|}$$.</p>
<p>$$|P|^2=\sum_{i=1}^n x_i^2$$
$$|Q|^2=... |
3,715,987 | <p>The domain is: <span class="math-container">$\forall x \in \mathbb{R}\smallsetminus\{-1\}$</span></p>
<p>The range is: first we find the inverse of <span class="math-container">$f$</span>:
<span class="math-container">$$x=\frac{y+2}{y^2+2y+1} $$</span>
<span class="math-container">$$x\cdot(y+1)^2-1=y+2$$</span>
<sp... | Saket Gurjar | 769,080 | <p>Alternate way to find the range : </p>
<p><span class="math-container">$$f(x) = y =\frac{x+2}{x^2+2x+1}$$</span></p>
<p><span class="math-container">$$yx^2+(2y-1)x+(y-2)=0 $$</span></p>
<p>Now this quadratic has real roots (since real points exist belonging to the function for all values of x (we can remove the ... |
2,611,855 | <p>Let $ (X,\tau) $ be a topology space, $ f : X \rightarrow Y $ a surjective function, $ A \subset Y $, and consider the topology in $ Y $: $\,$
$ \tau _{f} = \lbrace A \subset Y: f^{-1}(A) \in \tau \rbrace$. Show that:
$ A $ is closed in $ Y $ if and only if $ f^{-1}(A) $ is closed in $ X.$</p>
<p>I got stuck with t... | max8128 | 336,673 | <p>Too long for a comment (but I can delete it if you want) so :</p>
<p>In fact the RHS is a polynomial more particulary a <a href="http://mathworld.wolfram.com/JensenPolynomial.html" rel="nofollow noreferrer">Jensen polynomial</a> :</p>
<p>Why? Because the coefficients fulfill the conditions also called Turan's ineq... |
2,611,855 | <p>Let $ (X,\tau) $ be a topology space, $ f : X \rightarrow Y $ a surjective function, $ A \subset Y $, and consider the topology in $ Y $: $\,$
$ \tau _{f} = \lbrace A \subset Y: f^{-1}(A) \in \tau \rbrace$. Show that:
$ A $ is closed in $ Y $ if and only if $ f^{-1}(A) $ is closed in $ X.$</p>
<p>I got stuck with t... | Ѕᴀᴀᴅ | 302,797 | <p>$\def\e{\mathrm{e}} \def\veq{\mathrel{\phantom{=}}}$Denote $μ = λ^d$. It will be proved that\begin{align*}
(μ + (1 - μ) \e^{(d - 1)s})^{-\frac{1}{d - 1}} \leqslant \sum_{n = 0}^\infty \frac{(sμ^{\frac{1}{d}})^n}{n!} μ^{\frac{d^n - 1}{d - 1}} \e^{-sμ^{\frac{1}{d}}}
\end{align*}
holds for $0 < μ < 1$, $s > 0$... |
3,459,205 | <p>Explain why <span class="math-container">$\arccos(t)=\arcsin(\sqrt{{1}-{t^2}})$</span> when <span class="math-container">$0<t≤1$</span>. </p>
<p>I tried researching online, couldn't find anything related to this question though. Know this equation is correct and make sense, just don't know how to explain it usin... | Clement Yung | 620,517 | <p><strong>Algebraic proof</strong>:</p>
<p>Let <span class="math-container">$\theta = \arccos(t)$</span>, so <span class="math-container">$\cos(\theta) = t$</span>. Note that since <span class="math-container">$0 < t \leq 1$</span>, we have <span class="math-container">$0 \leq \theta < \frac{\pi}{2}$</span>. Re... |
3,622,508 | <p>I’m not sure exactly about the conditions needed for a subset <span class="math-container">$S$</span> to localise a ring <span class="math-container">$R$</span>. I know <span class="math-container">$S$</span> has to be multiplicative. But does <span class="math-container">$S$</span> also have to be a subset of the n... | rschwieb | 29,335 | <p>Classical localization can be extended to noncommutative rings using the <a href="https://en.wikipedia.org/wiki/Ore_condition" rel="nofollow noreferrer">Ore conditions</a>. They are a set of sufficient conditions to guarantee that the classical construction works, at least on one side.</p>
<p>That is if you have a... |
2,936,269 | <p>How do you simplify: <span class="math-container">$$\sqrt{9-6\sqrt{2}}$$</span></p>
<p>A classmate of mine changed it to <span class="math-container">$$\sqrt{9-6\sqrt{2}}=\sqrt{a^2-2ab+b^2}$$</span> but I'm not sure how that helps or why it helps.</p>
<p>This questions probably too easy to be on the Math Stack Exc... | Vladimir Vargas | 187,578 | <p>The reason for doing that is that <span class="math-container">$\sqrt{a^2-2ab+b^2} = \sqrt{(a-b)^2} = a-b$</span>. Now try to put your radical in the form your classmate suggested!</p>
|
2,936,269 | <p>How do you simplify: <span class="math-container">$$\sqrt{9-6\sqrt{2}}$$</span></p>
<p>A classmate of mine changed it to <span class="math-container">$$\sqrt{9-6\sqrt{2}}=\sqrt{a^2-2ab+b^2}$$</span> but I'm not sure how that helps or why it helps.</p>
<p>This questions probably too easy to be on the Math Stack Exc... | fleablood | 280,126 | <p>Your class mate is being.... clever.</p>
<p>If $\sqrt {9-6\sqrt 2}=a-b $ then $9-6\sqrt 2=a^2-2ab+c^3$</p>
<p>Let $2ab=6\sqrt 2$ and $a^2+b^2=9$.</p>
<p>Can we do that? </p>
<p>If we let $b^2=k $ and $a^2=9-k$ then $ab=\sqrt {k (9-k)}=3\sqrt 2=\sqrt {18} $. Solving $k (9-k)=18$ for $k $ (if it isn't visiblely o... |
1,158,970 | <p>In the lectures notes <a href="http://users.jyu.fi/~pkoskela/quasifinal.pdf" rel="nofollow">http://users.jyu.fi/~pkoskela/quasifinal.pdf</a> (Prof. Koskela has made them freely available from his webpage, so I am guessing is OK that I paste the link here) Quasiconformality is defined by saying that $\displaystyle \l... | Lee Mosher | 26,501 | <p>To answer 1), he's only using the definition of the $\mathbb{R}^2$ derivative. $Df(x) : \mathbb{R}^2 \to \mathbb{R}^2$ is a linear transformation and is characterized by the formula
$$\lim_{h \to 0} \frac{f(x+h) - f(x) - Df(x)(h)}{|h|} = 0
$$
Fix $|h|=r$ for very small $r$ and you will see from this that the ratio $... |
3,531,971 | <p>Let <span class="math-container">$T$</span> a linear operator. <span class="math-container">$T$</span> is bounded then ker(<span class="math-container">$T$</span>) is closed.</p>
<p><b>My attempt:</b></p>
<p>Let <span class="math-container">$\{x_n\}\subset \ker(T)$</span>.</p>
<p>As <span class="math-container">$... | Tsemo Aristide | 280,301 | <p>Suppose that <span class="math-container">$lim_nx_n=0$</span>, we have <span class="math-container">$\|T(x_n)\|\leq M\|x_n\|$</span> since <span class="math-container">$lim_nM\|x_n\|=0$</span>, we deduce that <span class="math-container">$lim_n\|T(x_n)\|=0$</span>.</p>
<p>In general let <span class="math-container"... |
1,397,776 | <p>Suppose $X_1,\ldots,X_n$ are iid r.v.'s, each with pdf $f_{\theta}(x)=\frac{1}{\theta}I\{\theta<x<2\theta\}$. I find the minimal sufficient statistics $(X_{(1)},X_{(n)})$. I am trying to prove it is complete. Can someone give me hint? Also are there any complete sufficient statistics in this model?</p>
| Michael Hardy | 11,667 | <p>We have $$\operatorname{E} (X_{(1)}) = \theta + \dfrac \theta {n+1} = \dfrac{n+2}{n+1} \theta$$ and $$\operatorname{E}(X_{(n)}) = 2\theta - \dfrac{\theta}{n+1} = \dfrac {2n+1} {n+1} \theta,$$ so
$$
\operatorname{E} \left( \frac{n+1}{n+2} X_{(1)} - \frac{n+1}{2n+1} X_{(n)} \right) = 0
$$
regardless of the value of $\... |
932,535 | <p>Let's denote the set of all singleton subsets of $X$(i.e. of all subsets consisting of one element) by $A$. Describe $\sigma(A)$ in the following two cases:</p>
<p>i) $X$ is countable</p>
<p>ii) $X$ is uncountable</p>
<p>I am new to this topic, so could you please help me understand the thinking/the concept behin... | Davide Giraudo | 9,849 | <p>First of all, a remark: a countable set can be written as a countable union of singletons, and $\sigma$-algebra are stable under countable unions, hence in each case, $\sigma(\mathcal A)$ contains each countable subset of $X$.</p>
<p>In the first case, we are done. </p>
<p>In the second one, we also have to consid... |
932,535 | <p>Let's denote the set of all singleton subsets of $X$(i.e. of all subsets consisting of one element) by $A$. Describe $\sigma(A)$ in the following two cases:</p>
<p>i) $X$ is countable</p>
<p>ii) $X$ is uncountable</p>
<p>I am new to this topic, so could you please help me understand the thinking/the concept behin... | Josh Keneda | 45,256 | <p>We want to find the smallest $\sigma$-algebra that contains $A$. In the first case, for any subset $Y$ of $X$, we can express $Y$ as a countable union of singletons in $X$, so $Y \in \sigma(A)$. But $Y$ was arbitrary, so every subset of $X$ is in $\sigma(A)$.</p>
<p>The second case is a bit tougher. We know that... |
18,659 | <p>I'm looking for a fast algorithm for generating all the partitions of an integer up to a certain maximum length; ideally, I don't want to have to generate <em>all</em> of them and then discard the ones that are too long, as this will take around 5 times longer in my case.</p>
<p>Specifically, given <span class="math... | Anthony Labarre | 689 | <p>If you are only interested in using an actual implementation, you could go for the <code>integer_partitions(n[, length])</code> in <a href="http://maxima.sourceforge.net/" rel="nofollow">Maxima</a>. More details can be found <a href="http://maxima.sourceforge.net/docs/manual/en/maxima_38.html" rel="nofollow">here</a... |
1,567,152 | <blockquote>
<p>Theorem: $X$ is a finite Hausdorff. Show that the topology is discrete.</p>
</blockquote>
<p>My attempt: $X$ is Hausdorff then $T_2 \implies T_1$ Thus for any $x \in X$ we have $\{x\}$ is closed. Thus $X \setminus \{x\}$ is open. Now for any $y\in X \setminus \{x\}$ and $x$ using Hausdorff property, ... | Potato | 18,240 | <p>Yes, you're completely right (although, you might want to write out your argument for $\{x\}$ being open in a little more detail.) You've shown that every point is open, so it follows form the axioms for a topology that every set, as a union of points, is open. This is precisely the definition of the discrete topolo... |
2,796,694 | <p>So for my latest physics homework question, I had to derive an equation for the terminal velocity of a ball falling in some gravitational field assuming that the air resistance force was equal to some constant <em>c</em> multiplied by $v^2.$ <br> So first I started with the differntial equation: <br>
$\frac{dv}{dt}... | Narasimham | 95,860 | <p>Taking <em>proper sign</em> of air resistance opposing gravity, we have terminal velocity when acceleration vanishes:</p>
<p>$$ \dfrac{dv}{dt}=mg-cv^2 = 0 \rightarrow v= v_{terminal}=\sqrt{\dfrac{mg}{c}}. $$</p>
<p>gets included in the coefficient of <em>tanh function</em> for velocity as an asymptotic value... |
3,235,300 | <p>I tried with , whenever <span class="math-container">$x > y$</span> implies <span class="math-container">$p(x) - p(y) =( 5/13)^x (1-(13/5)^{(x-y)}) + (12/13)^x (1- (13/12)^{(x-y)}) > 0 $</span>.
But here I don't understand why the answer is no.</p>
| Martin R | 42,969 | <blockquote>
<p>But here I don't understand why the answer is no.</p>
</blockquote>
<p>You started with
<span class="math-container">$$p(x) - p(y) =\left( \frac{5}{13}\right)^x \left( 1-\left( \frac{13}{5}\right)^{x-y}\right) + \left( \frac{12}{13}\right)^x\left(1- \left( \frac{13}{12}\right)^{x-y}\right)
$$</span>... |
4,332,812 | <p>I came across this series.</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{n!}{n^n}x^n$$</span></p>
<p>I was able to calculate its radius of convergence. If my calculations are OK, it is the number <span class="math-container">$e$</span>. Is that correct?</p>
<p>Then I started wondering if the series i... | Eric Towers | 123,905 | <p>Start with <span class="math-container">$\left(1 + \frac{1}{n} \right)^n$</span> is a strictly increasing sequence with limit <span class="math-container">$\mathrm{e}$</span>. (In some undergraduate calculus courses, this limit is the definition of <span class="math-container">$\mathrm{e}$</span>.) If we don't hav... |
4,286,136 | <p>I'm trying to find the general solution to <span class="math-container">$xy' = y^2+y$</span>, although I'm unsure as to whether I'm approaching this correctly.</p>
<p>What I have tried:</p>
<p>dividing both sides by x and substituting <span class="math-container">$u = y/x$</span> I get:</p>
<p><span class="math-cont... | Mathphys meister | 583,618 | <p>It can be done in a simpler way:</p>
<p><span class="math-container">\begin{equation}
\frac{dy}{y(y+1)} = \frac{dx}{x} \implies \mathrm{ln}\Big|\frac{y}{y+1}\Big| = \mathrm{ln}|x| + \mathrm{ln}(\tilde{c}), \\
\end{equation}</span></p>
<p>for some constant <span class="math-container">$\tilde{c}>0$</span>. Hence i... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Oscar Lanzi | 86,625 | <p>Here we have a monotonocally increasing sequence that predicts itself:</p>
<p><span class="math-container">$1,2,2,3,\color{brown}{3},4,4,4,\color{blue}{5,5,5},6,6,6,6,...$</span></p>
<p>The <span class="math-container">$n$</span>th term <span class="math-container">$a(n)$</span> predicts the number of times that ter... |
38,206 | <p>Simple question (I seem have asked a few like this...)</p>
<p>What is $\mbox{Hom}(\mathbb{Z}/2,\mathbb{Z}/n)$? (for $n \ne 2$)</p>
| Shahab | 10,575 | <p>Try to find the neccessary condition for such a homomorphism first. </p>
<p>If f is a homomorphism from Z/2 to Z/n then clearly f maps the zero element of Z/2 to the zero element of Z/n. Can it map 1 to an arbitrary element of Z/n? No, the mapping should be such that f is a homomorphism. Assuming it is, and $f(1) =... |
1,336,424 | <p>Find the minimum distance between point $M(0,-2)$ and points $(x,y)$ such that: $y=\frac{16}{\sqrt{3}\,x^{3}}-2$ for $x>0$ .</p>
<p>I used the formula for distance between two points in a plane to get: $$d=\sqrt{x^{2}+\frac{256}{3x^{6}}}$$ And this is where I cannot come up with how to proceed. I tried calculus ... | Emilio Novati | 187,568 | <p>Hint:</p>
<p>Note that the value of $x$ that minimize $d$ minimimize also $d^2=x^2+\dfrac{2^8}{3x^6}=f$ and the derivative is $f'= 2x-\dfrac{2^9}{x^7}$. </p>
|
1,336,424 | <p>Find the minimum distance between point $M(0,-2)$ and points $(x,y)$ such that: $y=\frac{16}{\sqrt{3}\,x^{3}}-2$ for $x>0$ .</p>
<p>I used the formula for distance between two points in a plane to get: $$d=\sqrt{x^{2}+\frac{256}{3x^{6}}}$$ And this is where I cannot come up with how to proceed. I tried calculus ... | André Nicolas | 6,312 | <p>We give a non-calculus approach. It is in my opinion a fair bit harder than the calculus way. </p>
<p>We want to minimize $x^2+\frac{256}{3x^6}$, or equivalently
$$\frac{x^2}{3}+\frac{x^2}{3}+\frac{x^2}{3}+\frac{256}{3x^6}.$$
By the arithmetic mean geometric mean inequality (AM/GM) we have
$$\frac{1}{4}\left( \fra... |
1,114,767 | <p>I would like a reference for the following result (you can assume more regularity and replace $C^2(\bar\Omega)$ with $C^2(\mathbb R^n)$ if needed):</p>
<blockquote>
<p>Let $\Omega\subset\mathbb R^n$ be a bounded domain with a $C^2$ boundary. Let $f\in C^2(\bar\Omega)$ and $\gamma\in C^2(\bar\Omega)$ with $\gamma&... | spatially | 124,358 | <p>I think you need to at least combine the result of existence & uniqueness with the result of regularity.</p>
<p>For existence & uniqueness w.r.t a solution $u\in H_0^1(\Omega)$, I would suggest the first existence theorem in Evans book, chapter 6.2, look for Lax-Milgram. But I think you may need to assume i... |
2,255,192 | <p>I'm going through the exercises in Georgi E Shilov's Linear Algebra book and am on chapter 1 problem 2: "Write down all the terms appearing in the determinant of order four which have a minus sign and contain $ a_{23}$"</p>
<p>the answers I have arrived at are: </p>
<p>$a_{11}$$a_{23}$$a_{32}$$a_{44}$</p>
<p>$a_{... | Onur | 414,808 | <p>$$\begin{align} S &=\sum_{n=1}^\infty a\cdot r^{n-1}\\ \\ & = \frac{1}4+\frac{3}{16}+\frac{9}{64}+\dots \\ \\
&= \frac{1}4\cdot\frac{3}{4}^{0}+\frac{1}4\cdot\frac{3}{4}^{1}+\frac{1}4\cdot\frac{3}{4}^{2}+\dots \\ \\ \end{align}$$</p>
<p>$$ a=\frac{1}4, r=\frac{3}4\implies S= 1 ... |
3,494,470 | <p>Will all units in <span class="math-container">$\mathbb{Z}_{72}$</span> be also units (modulo <span class="math-container">$8$</span> and <span class="math-container">$9$</span>) of <span class="math-container">$\mathbb{Z}_8$</span> and <span class="math-container">$\mathbb{Z}_9$</span>? </p>
<p>I think yes, becaus... | red_trumpet | 312,406 | <p>A bit more abstractly: If you have a homomorphism <span class="math-container">$\varphi: R \to S$</span> of unital commutative rings, then for any unit <span class="math-container">$r \in R$</span>, its image <span class="math-container">$\varphi(r)\in S$</span> is a unit, because <span class="math-container">$\varp... |
148,037 | <p>for example I have a data</p>
<pre><code>Clear[data];
data[n_] :=
Join[RandomInteger[{1, 10}, {n, 2}], RandomReal[1., {n, 1}], 2];
</code></pre>
<p>then <code>data[3]</code> gives</p>
<pre><code>{{4, 8, 0.264842}, {9, 5, 0.539251}, {3, 1, 0.884612}}
</code></pre>
<p>in each sublist, first two value is matrix ... | Carl Woll | 45,431 | <p>You can change the <a href="http://reference.wolfram.com/language/ref/SparseArray" rel="noreferrer"><code>SparseArray</code></a> system options to total repeated entries instead of taking the first. Here is a function that does this:</p>
<pre><code>carl[data_] := Internal`WithLocalSettings[
old=SystemOptions["S... |
148,037 | <p>for example I have a data</p>
<pre><code>Clear[data];
data[n_] :=
Join[RandomInteger[{1, 10}, {n, 2}], RandomReal[1., {n, 1}], 2];
</code></pre>
<p>then <code>data[3]</code> gives</p>
<pre><code>{{4, 8, 0.264842}, {9, 5, 0.539251}, {3, 1, 0.884612}}
</code></pre>
<p>in each sublist, first two value is matrix ... | Marius Ladegård Meyer | 22,099 | <p>I can't see any sparsity at all in this problem, given that we are adding values to <code>n</code> random elements of a 10x10 matrix, and <code>n</code> is up to $10^6$. So, given that we keep <code>data</code> as-is, the algorithm to fill the matrix is so straight-forward that it's a good candidate for compiling. I... |
2,394,815 | <p>Let $A$ be a (not necessarily finitely generated) abelian group where all elements have order 1, 2, or 4. Does it follow that $A$ can be written as a direct sum $(\bigoplus _\alpha \mathbb Z/4) \oplus (\bigoplus_\beta \mathbb Z/2)$?</p>
| hmakholm left over Monica | 14,366 | <p>Call a set $S\subseteq A$ "good" if</p>
<ul>
<li>$S$ does not contain $a^2$ for any $a\in A$.</li>
<li>Whenever $s_1^{m_1}s_2^{m_2}\cdots s_n^{m_n}=e$ where $s_1,s_2,\ldots,s_n$ are <em>different</em> elements of $S$, we have $s_1^{m_1}=e$.</li>
</ul>
<p>Apply Zorn's lemma to the family of good subsets of $A$ (ord... |
3,143,649 | <p>I am reading the book <em>Random perturbation of dynamical sustem</em> of Freidlin and Wantzell (2nd edition). On page 20, they define a Markov process as follow:</p>
<blockquote>
<p>Let <span class="math-container">$(\Omega ,\mathcal F,\mathbb P)$</span> a probability space and <span class="math-container">$(X,\... | Ѕᴀᴀᴅ | 302,797 | <p><span class="math-container">$\def\Γ{{\mit Γ}}$</span>For Q1, since<span class="math-container">$$
p(0, x, X \setminus \{x\}) = P_x(X_0 \in X \setminus \{x\}) = 1 - P_x(X_0 = x),
$$</span>
so <span class="math-container">$p(0, x, X \setminus \{x\}) = 0 \Leftrightarrow P_x(X_0 = x) = 1$</span>. The process under <spa... |
4,563,707 | <p>Sequence given : 6, 66, 666, 6666. Find <span class="math-container">$S_n$</span> in terms of n</p>
<p>The common ratio of a geometric progression can be solved is <span class="math-container">$\frac{T_n}{T_{n-1}} = r$</span>, where r is the common ratio and n is the</p>
<p>When plugging in 66 as <span class="math-c... | Arthur | 15,500 | <p>Alternately, note that if you add <span class="math-container">$\frac23=0.666\ldots$</span> to each term, then it actually becomes a geometric progression with common ratio <span class="math-container">$10$</span>. So remembering to subtract it again afterwards, we get
<span class="math-container">$$
S_n=T_1+T_2+\cd... |
3,887,156 | <p>I understand that the vertical shift is <span class="math-container">$0$</span> that is why the graph starts at <span class="math-container">$(0,0)$</span>. Also I understand that the amplitude is <span class="math-container">$3$</span> because the maximum y value is <span class="math-container">$3$</span> and the m... | Community | -1 | <p><span class="math-container">$V=\{(a,a,c)|a,c\in\Bbb R\}=\rm{span}\{(1,1,0),(0,0,1)\}\cong\Bbb R^2$</span>. And <span class="math-container">$\Bbb R^3/\Bbb R^2\cong\Bbb R$</span>.</p>
|
7,080 | <p>What is the right definition of the symmetric algebra over a graded vector space V over a field k?</p>
<p>More generally: What is the right definition of the symmetric algebra over an object in a symmetric monoidal category (which is suitably (co-)complete)?</p>
<p>Two possible definitions come to my mind:</p>
<p... | Mariano Suárez-Álvarez | 1,409 | <p>Which one is the right definition depends on what you want to do with it. </p>
<p>No matter how much technology you throw at the question, including homotopy coinvariants and quasi-triangular Hopf algebras, "right" is a relative notion :D</p>
|
3,506,982 | <p>Let <span class="math-container">${X_n, n\in N}$</span> be an iid sequence of psitive rrvs and let <span class="math-container">$K$</span> be a rrv independent of this sequence and taking its values in <span class="math-container">$N$</span> with <span class="math-container">$P(K=k)=p_k$</span>. Consider the rrv <sp... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$EZ=\sum_k E(Z|K=k)P(K=k)$</span> <span class="math-container">$=\sum_k (kEX_1) P(K=k)=E(X_1) \sum_k kP(Z=k)=EX_1EK$</span>. Note that <span class="math-container">$EX_n$</span> doe not depend on <span class="math-container">$n$</span> since <span class="math-container">$(X_I)$</span> ... |
1,037,736 | <p>$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$</p>
<p>please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums'</p>
<p>proof:</p>
<p>if we look at $\sum \limits_{v=1}^3 v=1+2+3,\sum \limits_{v=1}^4 v=1+2+3+4,\sum \limits_{v=1}^5 v=... | John Joy | 140,156 | <p>While most of the proofs that you will see are algebraic, sometimes it is useful to get a geometric view of the problem. I've always preferred getting multiple perspectives to give me deeper understanding of the problem at hand.</p>
<p>In the image, there are 5 different views of the problem. The first one has $(n+... |
3,958,133 | <p>How to simplify the following probability</p>
<p><span class="math-container">$\operatorname{P} ( { C |B,A} )P( { B |A} )P( A ) + P( { {\bar B} |A} )P( A )$</span></p>
<p><span class="math-container">$ = P\left( {A,B,C} \right) + P\left( {A,\bar B} \right)$</span></p>
<p>Can <span class="math-container">$P\left( {A,... | Graham Kemp | 135,106 | <p><span class="math-container">$$\begin{align}&~~~~~~\mathsf P(C\mid B,A)\,\mathsf P(B\mid A)\,\mathsf P(A)+\mathsf P(\overline B\mid A)\,\mathsf P(A)\\&=\big(\mathsf P(B,C\mid A)+\mathsf P(\overline B\mid A)\big)\,\mathsf P(A)&&\small\text{by definition of conditional probability}\\&=\mathsf P\big... |
2,225,150 | <p>I am seek for a rigorous proof for the following identity</p>
<p>$\sum_{i = 0}^{T} x_i \sum_{j = 0}^{i} y_j = \sum_{i = 0}^{T}y_i\sum_{j = i}^{T} x_j$. </p>
<p>By setting some small $T$ and expand the formulas, it is then clear to see the result. I am asking for help to give a formal proof of this identity, by reo... | epi163sqrt | 132,007 | <blockquote>
<p>The following representation might be helpful
\begin{align*}
\sum_{i = 0}^{T}\sum_{j = 0}^{i} x_iy_j=\sum_{0\leq j\leq i\leq T} x_iy_j=\sum_{j=0}^T\sum_{i=j}^Tx_iy_j\tag{1}
\end{align*}</p>
</blockquote>
<p>From (1) we obtain by applying the laws of associativity, distributivity and commutativity:
... |
2,455,428 | <p>While going through some exercises in my analysis textbook, I came up with an equation which looks like an identity. I strongly believe that this is the case, but I couldn't prove this.</p>
<blockquote>
<p>$$\sum_{0\leq k\leq n}(-1)^k\frac{p}{k+p}\binom{n}{k} = \binom{n+p}{p}^{-1}$$</p>
</blockquote>
<p>Can some... | Marko Riedel | 44,883 | <p>This identity has appeared on MSE on several occasions in various
forms. There is a proof using residues which goes like this (quoted
from what should be earlier posts). Start with the function</p>
<p>$$f(z) = n! (-1)^n \frac{p}{z+p} \prod_{q=0}^n \frac{1}{z-q}.$$</p>
<p>We then get</p>
<p>$$\mathrm{Res}... |
3,523,213 | <p>I've read some simple explanations of Cantor's diagonal method.</p>
<p>It seems to be:</p>
<pre><code>1) Changing the i-th value in a row.
2) Do the same to the next row with the (i+1)th element.
3) Now you get an element not in any other row. So add it to list.
4) This process never ends.
</code></pre>
<p>This l... | Frosty | 744,938 | <p>You don't need induction to prove that the new number is different than any already listed number. </p>
<p>You have a construction for the new number. For any element in the list, there is some digit that is not the same, and based on where it is in the list, you can say exactly which one. This statement does not d... |
569,012 | <p>Let $I$ be the incenter of $\triangle{ABC}$. Let $R$ be the radius of the circle that circumscribes $\triangle{IAB}$. Find a formula for $R$ in term of other elements $a, b, c, A, B, C, r, R$ of $\triangle{ABC}$. I need this formula in order to prove a geometric inequality.</p>
| Sawarnik | 93,616 | <p>Remember the extended law of sines:
<span class="math-container">$$R = \frac{a}{2\sin A} = \frac{b}{2\sin B} = \frac{c}{2\sin C}$$</span></p>
<p>The above gives the circumradius (<span class="math-container">$R$</span>) for a triangle whose one angle is <span class="math-container">$A$</span> and the opposite side i... |
216,099 | <p>$$x=\int \sqrt{\frac{y}{2a-y}}dy$$</p>
<p>According to my textbook, it says that the substitution by $y=a(1-\cos\theta)$ will easily solve the intergral. Why does this work?</p>
| Ken Dunn | 42,937 | <p>Find $\frac{dy}{d\theta}$ and then do the necessary replacement into the integral</p>
|
4,141,477 | <blockquote>
<p>Find <span class="math-container">$$\lim_{x\rightarrow0} x\tan\frac1x$$</span></p>
</blockquote>
<p>Now I tried to find the form of the limit (<span class="math-container">$0/0$</span> or <span class="math-container">$0\cdot \infty$</span> or <span class="math-container">$\infty/\infty$</span>), but as ... | cdeamaze | 429,551 | <p>The limit does not exist.
Consider the sequence <span class="math-container">$xn=\{2/n\pi\} = \{2/\pi, 1/\pi, 2/3\pi, 1/2\pi, \}$</span>
<span class="math-container">$y_n = x_n *\tan(x_n)$</span> As <span class="math-container">$x_n\to0, \tan(x_n)$</span> does NOT exist as <span class="math-container">$\tan(x_n)$</... |
29,703 | <p>For an <a href="http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf">image denoising problem</a>, the author has a functional $E$ defined </p>
<p>$$E(u) = \iint_\Omega F \;\mathrm d\Omega$$</p>
<p>which he wants to minimize. $F$ is defined as </p>
<p>$$F = \|\nabla u \|^2 = u_x^2 + u_y^2$$</p>
<p>Then, the ... | rcollyer | 2,283 | <p>You should note that a solution, $f$, to your differential equation, $\mathcal{L}[f] = 0$, is the steady state solution to the second equation, as $\partial_t f = 0$. By turning this into a parabolic equation, only the error term will depend on $t$, and it will decay with time. This can be seen by letting </p>
<p>... |
3,276,877 | <blockquote>
<p>Insert <span class="math-container">$13$</span> real numbers between the roots of the equation: <span class="math-container">$x^2 +x−12 = 0$</span> in a few ways that these <span class="math-container">$13$</span> numbers together with the roots of the equation will form the first <span class="math-co... | Amit Rajaraman | 447,210 | <p><span class="math-container">$$\angle DAE = \angle EAC + \angle DAB + \alpha$$</span>
<span class="math-container">$$= \frac{180°-\angle ACE}{2}+\frac{180°-\angle ABD}{2}+\alpha$$</span>
<span class="math-container">$$=\frac{\angle ACB+\angle ABC}{2}+\alpha$$</span>
<span class="math-container">$$=\frac{180°-\alpha}... |
3,896,345 | <p>I've been studying a paper in which the author says:</p>
<p>Fix <span class="math-container">$n$</span> such that <span class="math-container">$m^n \prod_{j=1}^n \frac{j}{j+\delta} > 1$</span>, where <span class="math-container">$1<m<\infty$</span>, and <span class="math-container">$\delta >0$</span>.</p... | Václav Mordvinov | 499,176 | <p>Find an <span class="math-container">$N$</span> such that <span class="math-container">$\frac{mN}{N+\delta}>1+\eta$</span> for some <span class="math-container">$\eta>0$</span>. Then take <span class="math-container">$M$</span> such that <span class="math-container">$(1+\eta)^M>\left(\prod_{j=1}^N\frac{j}{j... |
2,633,392 | <p>I was recently reading about sets and read that $B$ is a subset of $A$ when each member of $B$ is a member of $A$. However, I am not sure about whether this requires the members of $A$ to simply be members of $B$, or if they could be part of $A$ in some other way - i.e. embedded within a set inside $A$.</p>
<p>I tr... | Sahiba Arora | 266,110 | <blockquote>
<p>"I am not sure about whether this requires the members of $A$ to simply be members of $B$, or if they could be part of $A$ in some other way - i.e. embedded within a set inside $A$."</p>
</blockquote>
<p>Former.</p>
<blockquote>
<p>"Does this mean that $Y$ is not a subset of $X$, as $x$ is a membe... |
250,074 | <p>How can one generate a random vector <span class="math-container">$v=[v_1, v_2, v_3]^T$</span> satisfying <span class="math-container">$\sqrt{v_1v_1^* + v_2 v_2^* + v_3 v_3^*} = 1$</span>, where <span class="math-container">$T$</span> and <span class="math-container">$*$</span> denote the transpose and complex-conju... | Szabolcs | 12 | <p>Generate a single such vector:</p>
<pre><code>Complex @@@ Partition[RandomPoint@Sphere[{0, 0, 0, 0, 0, 0}], 2]
</code></pre>
<p>Generate <code>n</code> of them with good performance:</p>
<pre><code>n = 100;
#1 + I #2 & @@ Transpose[
ArrayReshape[RandomPoint[Sphere[{0, 0, 0, 0, 0, 0}], n], {n, 3, 2}],
{2, 3,... |
1,743,482 | <p>I was doing this question on convergence of improper integrals where in our book they have used the fact that $2+ \cos(t) \ge1$. Can somebody prove this?</p>
| user236182 | 236,182 | <p>If $p$ is any odd prime, then by <a href="https://en.wikipedia.org/wiki/Wilson%27s_theorem" rel="nofollow">Wilson's Theorem</a>: $$(p-1)!\equiv 1\cdot 2\cdots\left(\frac{p-1}{2}\right)\left(-\frac{p-1}{2}\right)\cdots (-2)(-1)\pmod{p}$$</p>
<p>$$\equiv (-1)^{\frac{p-1}{2}}\left(\left(\frac{p-1}{2}\right)!\right)^2\... |
2,604,178 | <p>Let $(G=(a_1,...,a_n),*)$ be a finite Group. Define for a element $a_i \in G$ a permutation $\phi = \phi(a_i)$ by left multiplication:</p>
<p>$$
\begin{bmatrix}
a_1 & a_2 & ... & a_n \\
a_i*a_1 & a_i*a_2 & ... & a_i*a_n \\
\end{bmatrix}
$$
I am struggling to understand why this is the permu... | gammatester | 61,216 | <p>You have to choose complex starting values, otherwise the method cannot converge to complex roots.</p>
<p>With the correct iteration formula
$$x_{n+1}=x_n - \frac{f(x)}{f'(x_n)} = x_n - \frac{x_n^2+1}{2x_n} = \frac{2x_n^2-x_n^2 -1}{2x_ n}=\frac{x_n^2 - 1}{2x_n}$$
and a complex starting value you get e.g.</p>
<pre... |
959,322 | <p>Solve $$ \sum_{k = 1}^{ \infty} \frac{\sin 2k}{k}$$</p>
<p>I first tried to use Eulers formula</p>
<p>$$ \frac{1}{2i} \sum_{k = 1}^{ \infty} \frac{1}{k} \left( e^{2ik} - e^{-2ik} \right)$$</p>
<p>However to use the geometric formula here, I must subtract the $k=0$ term and that term is undefinted since $1/k$. I... | lab bhattacharjee | 33,337 | <p>For finite non-zero $a,$</p>
<p>$$\sum_{k=1}^\infty\frac{e^{2iak}}k=-\ln(1-e^{2ia})$$</p>
<p>$$1-e^{2ia}=-e^{ia}(e^{ia}-e^{-ia})=-e^{ia}[2i\sin(a)]$$</p>
<p>$$\ln(1-e^{2ia})=\ln(e^{i(a+2m\pi)})+\ln2+\ln(-i)+\ln[\sin(a)]$$
$$=i(a+2m\pi)+\ln2-\frac{i\pi}2+\ln[\sin(a)]$$ </p>
<p>as $-i=\cos\left(\frac\pi2\right)+i... |
361,862 | <p>I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are extremely subtle and difficult to note at a first sight. I am looking for hypothesis or conditions that appear to be almo... | Alistair Wall | 159,000 | <p>Some of Euclid's theorems rely on axioms of betweenness that he was not aware of.</p>
<p>Hilbert's axioms: <a href="https://www.math.ust.hk/~mabfchen/Math4221/Hilbert%20Axioms.pdf" rel="noreferrer">https://www.math.ust.hk/~mabfchen/Math4221/Hilbert%20Axioms.pdf</a></p>
|
1,041,177 | <p>Proof that if $p$ is a prime odd and $k$ is a integer such that $1≤k≤p-1$ , then the binomial coefficient</p>
<p>$$\displaystyle \binom{p-1}{k}\equiv (-1)^k \mod p$$</p>
<p>This exercise was on a test and I could not do!!</p>
| Bruno Joyal | 12,507 | <p>In characteristic $p$, where $p$ is odd,</p>
<p>$$(1+X)^{p-1} = \frac{(1+X)^p}{1+X} = \frac{1+X^p}{1+X} = \frac{1-(-X)^p}{1-(-X)} = 1 -X + X^2 - \dots +X^{p-1}.$$</p>
|
1,231,365 | <p>I am from a non-English speaking country. Should we say monotonous function or monotonic function?</p>
| Warlord5 | 230,629 | <p>That would be monotonic function. Monotonic is always used in relation to the function you are talking about. </p>
<p><a href="http://mathworld.wolfram.com/MonotonicFunction.html">http://mathworld.wolfram.com/MonotonicFunction.html</a></p>
<p>Monotonic describes something this is unchanged or altered, such as the ... |
381,011 | <p>I should prove this claim:</p>
<blockquote>
<p>Every undirected graph with n vertices and $2n$ edges is connected.</p>
</blockquote>
<p>If it is false I should find a counterexample.
I was thinking to consider the complete graph with $n$ vertices. Such a graph is connected and contains $\frac{n(n-1)}{2}$ nodes. ... | Abel | 71,157 | <p>Hint: Suppose you have a non-empty graph $G$ with $n$ vertices and $2n$ edges, how many edges and vertices does $G\coprod G$ have? Is it connected?</p>
|
381,011 | <p>I should prove this claim:</p>
<blockquote>
<p>Every undirected graph with n vertices and $2n$ edges is connected.</p>
</blockquote>
<p>If it is false I should find a counterexample.
I was thinking to consider the complete graph with $n$ vertices. Such a graph is connected and contains $\frac{n(n-1)}{2}$ nodes. ... | Ma Ming | 16,340 | <p>This is false. Suppose $2n=\binom{k}{2}$ for some $k$ and $n>k$. Take a complete graph with $k$ vertex together with $n-k$ isolated vertex, then you get a graph with $n$ vertex and $2n$ wedges.</p>
|
3,628,358 | <p>As stated, I need to prove that, up to isomorphism, the only simple group of order <span class="math-container">$p^2 q r$</span>, where <span class="math-container">$p, q, r$</span> are distinct primes, is <span class="math-container">$A_5$</span> (the alternating group of degree 5).</p>
<p>Now I know the following... | Dietrich Burde | 83,966 | <p>The groups of order <span class="math-container">$p^2qr$</span> for distinct primes <span class="math-container">$p,q,r$</span> have been classified <a href="https://www.jstor.org/stable/1986340?seq=1#metadata_info_tab_contents" rel="nofollow noreferrer">here</a> by Oliver G. Glenn in <span class="math-container">$1... |
2,311,979 | <p>Let $A = (a_{i,j})_{n\times n}$ and $B = (b_{i,j})_{n\times n}$</p>
<p>$(AB) = (c_{i,j})_{n\times n}$, where $c_{i,j} = \sum_{k=1}^n a_{i,k} b_{k,j}$, so</p>
<p>$(AB)^T = (c_{j,i})$, where $c_{j,i} = \sum_{k=1}^n a_{j,k}b_{k,i} $, and
$B^T = b_{j,i}$ and $A^T = a_{j,i}$, so </p>
<p>$B^T A^T = d_{j,i}$ where $d_... | Surb | 154,545 | <p>In the world of real vector spaces, one can define $A^T$ to be the adjoint of $A$ with respect to the Euclidean inner product $\langle \cdot,\cdot \rangle$ (this adjoint is unique). More precisely, $A^T$ is the unique linear mapping so that $$\langle Ax,y\rangle = \langle x,A^Ty\rangle \qquad \forall x,y$$
Similarl... |
4,330,755 | <p>Given a convex pentagon <span class="math-container">$ABCDE$</span>, there is a unique ellipse with center <span class="math-container">$F$</span> that can be inscribed in it as shown in the image below. I've written a small program to find this ellipse, and had to numerically (i.e. by iterations) solve five quadra... | Intelligenti pauca | 255,730 | <p>There is a simple geometric construction for the ellipse inscribed in a given convex pentagon <span class="math-container">$ABCDE$</span>. One can, first of all, find tangency points <span class="math-container">$PQRST$</span>. Draw, for instance, diagonals <span class="math-container">$AC$</span> and <span class="m... |
46,462 | <p>Hi I have a simple question. How do I plot the following with Day 1 as my X axis and Day 2 as my Y axis? I need the 22 variances plotted according to the Day they were taken from (these were originally 3D measurements taken over 2 days with the same specimens each day, there were 11 specimens and 22 xyz measurements... | kglr | 125 | <pre><code>reorgdata = GatherBy[data[[1]], #[[2]] &][[2 ;;, All, 3 ;;]];
variances = Thread[Variance /@ reorgdata];
means = Thread[Mean /@ reorgdata];
Row[{ListPlot[means, PlotLabel -> "means", ImageSize -> 300],
ListPlot[variances, PlotLabel -> "variances", ImageSize -> 300]}]
</code></pre>
<p><i... |
149,049 | <p>Suppose you have a list of intervals (or tuples), such as:</p>
<pre><code>intervals = {{3,7}, {17,43}, {64,70}};
</code></pre>
<p>And you wanted to know the intervals of all numbers not included above, e.g.:</p>
<pre><code>myRange = 100;
numbersNotUed[myRange,intervales]
(*out: {{1,2},{8,16},{44,63},{71,100}}*)... | Alexey Popkov | 280 | <p>Here is another implementation. It assumes that all the intervals lie in the specified range <em>and</em> intervals are sorted <em>and</em> aren't overlapping:</p>
<pre><code>integerIntervalComplement[completeInterval : {start_, end_}, {subIntervals___}] :=
If[First[#2] - Last[#1] <= 1, Nothing, {Last[#1] + 1... |
834,949 | <p>I have this HW where I have to calculate the $74$th derivative of $f(x)=\ln(1+x)\arctan(x)$ at $x=0$.
And it made me think, maybe I can say (about $\arctan(x)$ at $x=0$) that there is no limit for the second derivative, therefore, there are no derivatives of degree grater then $2$.
Am I right?</p>
| Dario | 156,754 | <p>Use Taylor series:
$$\log(1+x)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^n}{n}$$
$$\arctan(x)=\sum_{n=1}^{\infty}(-1)^{n}\frac{x^{2n+1}}{2n+1}$$
so
$$f(x)=\log(1+x)\arctan(x)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}(-1)^{n+m+1}\frac{x^{n+2m+1}}{n(2m+1)}$$
You want to compare this with
$$f(x)=\sum_{k=0}^{\infty}\frac{f^{(k)... |
1,554 | <p>Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold structure extend across the point singularity?</p>
<p>(Penny Smith and I wrote a paper on this many years ago, but we... | Igor Belegradek | 1,573 | <p>If by "extends across the point singularity" you mean extends smoothly, then I think you may just start with the Euclidean space thought of as a warped product over (0,infinity) with sphere as a fiber and replace the warping function r by any smooth function f(r) that is near r in C^2-topology. Then the curvature wi... |
1,930,401 | <p>Are there any non-linear real polynomials $p(x)$ such that $e^{p(x)}$ has a closed form antiderivative? If not, is the value of $\int_{0}^{\infty}e^{p(x)}dx$ known for any $p$ with negative leading term other than $-x$ and $-x^2$?</p>
| Jack D'Aurizio | 44,121 | <p>In general, if $A_1 A_2\ldots A_n$ is a $n$-sided polygon and the lengths $l_1,l_2,\ldots,l_{n-1},l_n$ of $A_1 A_2,A_2 A_3,\ldots,A_{n-1} A_n, A_n A_1$ are fixed, the maximum area is achieved by the cyclic polygon, i.e. the polygon having all its vertices on a circle. You may easily prove this fact through <a href="... |
1,930,401 | <p>Are there any non-linear real polynomials $p(x)$ such that $e^{p(x)}$ has a closed form antiderivative? If not, is the value of $\int_{0}^{\infty}e^{p(x)}dx$ known for any $p$ with negative leading term other than $-x$ and $-x^2$?</p>
| robjohn | 13,854 | <p><strong>Theorem:</strong> For given sidelengths, a cyclic polygon has maximal area.</p>
<p><strong>Proof:</strong> Let <span class="math-container">$\{p_k\}_{k=1}^n$</span> be the vertices of the polygon.</p>
<p>Set <span class="math-container">$v_k=p_k-p_{k-1}$</span> and <span class="math-container">$m_k=\frac12... |
3,702,094 | <p>For a school project for chemistry I use systems of ODEs to calculate the concentrations of specific chemicals over time. Now I am wondering if </p>
<p><span class="math-container">$$ \frac{dX}{dt} =X(t) $$</span></p>
<p>the same is as </p>
<p><span class="math-container">$$ X(t)=e^t . $$</span> </p>
<p>As far a... | Fred | 380,717 | <p>The differential equation </p>
<p><span class="math-container">$$ \frac{d X}{dt}=X(t)$$</span></p>
<p>has the general solution</p>
<p><span class="math-container">$$X(t)=Ce^t$$</span></p>
<p>where <span class="math-container">$C \in \mathbb R.$</span></p>
|
2,547,488 | <p>We have always been taught that a function assigns to every element in the domain a single <em>unique</em> element in the range. If a rule of assignment assigns to one element in the domain more than one element in the range then it isn't a function.</p>
<p>Now in Munkres' <em>Topology</em>, on page 107, it says:</... | drhab | 75,923 | <p>$f^{-1}$ denoting a function needs not to be defined. </p>
<p>$f^{-1}(V)$ denoting the <em>preimage of</em> $V$ <em>w.r.t.</em> $f$ needs to be defined (and is) for every set $V\subseteq Y$.</p>
<p>It is defined as: $$f^{-1}(V):=\{x\in X\mid f(x)\in V\}$$</p>
<p>If $f$ is constant then $f^{-1}(V)\in\{\varnothing,... |
312,847 | <p>Let <span class="math-container">$k$</span> be a global field, and let <span class="math-container">$G = \mathbf G(\mathbb A_k)$</span> for a connected, reductive group <span class="math-container">$\mathbf G$</span> over <span class="math-container">$k$</span>. In <a href="https://services.math.duke.edu/~hahn/Chap... | paul garrett | 15,629 | <p>It is absolutely essential that the space of (bounded/continuous) operators be given the "strong" operator topology (strictly weaker than the norm topology), and the map <span class="math-container">$G\times V\to V$</span> to be jointly continuous.</p>
<p>This is not a pathology: even in very simple cases, such as ... |
110,078 | <p>Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then:
$ \left \| \int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha} } \right\|_{L^q(\mathbb{R}^n)}\leq$ $C\left\| f\right\| _{L^p(\mathbb{R^n})}$.</p>
| user23078 | 23,078 | <p>This is the standard Hardy-Littlewood-Sobolev inequality(or the theorem of fractional integration).A more direct approach is write
$$
\int{f(x-y)|y|^{\alpha-n}dy}=\int_{|y|<R}+\int_{|y|\ge R}
$$
For the second term on the RHS,using Holder inequality,and easy to see that it's dominated by $\|f\|_{L^p}R^{-\frac{... |
69,476 | <p>Hello everybody !</p>
<p>I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer scientist each day) that if the result is exact this may not be the computationally fastest way possible to do it... | pEquals2 | 40,735 | <p>For evaluation of a general polynomial in one variable, the provably fastest method is Horner's scheme as Emil has pointed out. It is worth mentioning that this scheme has a more popular face in the form of <a href="http://en.wikipedia.org/wiki/Polynomial_remainder_theorem" rel="nofollow">little Bézout's theorem</a>... |
823,928 | <p>Prove that all of the rings, which mediate between principal ideal ring $K$ and the field of fractions $Q$, are the principal ideal ring.</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $ $ They're localizations since $\,K[a/b] = K[1/b],\,$ by $\,(a,b) =1\,\Rightarrow\, ra+sb = 1\,\Rightarrow\, ra/b + s = 1/b$</p>
|
122,728 | <p>Suppose that I have a <a href="http://en.wikipedia.org/wiki/Symmetric_matrix" rel="nofollow noreferrer">symmetric</a> <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix" rel="nofollow noreferrer">Toeplitz</a> <span class="math-container">$n\times n$</span> matrix</p>
<p><span class="math-container">$$\mathbf{A}=\... | joriki | 6,622 | <p>Expanding $\mathbf A(\mathbf A + \mathbf B + \mathbf E)^{-1}\mathbf A$ in $\mathbf E$ yields $\mathbf A(\mathbf A + \mathbf B)^{-1}\mathbf A-\mathbf A(\mathbf A + \mathbf B)^{-1}\mathbf E(\mathbf A + \mathbf B)^{-1}\mathbf A$ up to first order. Thus</p>
<p>$$
\begin{eqnarray}
\frac{\partial\operatorname{Tr}[M]}{\... |
31,308 | <p>Apologies if my question is poorly phrased. I'm a computer scientist trying to teach myself about generalized functions. (Simple explanations are preferred. -- Thanks.)</p>
<p>One of the references I'm studying states that the space of Schwartz test functions of rapid decrease is the set of infinitely differentiabl... | Willie Wong | 3,948 | <p>From the Fourier analysis point of view, the reason is the property of the Fourier transform to interchange derivatives and multiplications, which you can read more about on Wikipedia. The crucial point is that <em>the smoothness of a function is directly related to the decay rate of its (inverse) Fourier transform<... |
31,308 | <p>Apologies if my question is poorly phrased. I'm a computer scientist trying to teach myself about generalized functions. (Simple explanations are preferred. -- Thanks.)</p>
<p>One of the references I'm studying states that the space of Schwartz test functions of rapid decrease is the set of infinitely differentiabl... | Will Jagy | 3,324 | <p>Just a quick clue. The example you want is essentially the Gaussian normal distribution from probability, $$ \frac{1}{\sqrt {2 \pi}} \; \; e^{- x^2 / 2} $$ and probably the simplest motivation is that the Fourier transform of this function is just itself (well, up to a constant multiple, depends on whose definitio... |
1,640,733 | <p>I think it is true that any power of a logarithm, no matter how big, will eventually grow slower than a linear function with positive slope.</p>
<p>Is it true that for any exponent $m>0$ (no matter how big we make $m$), the function $f(x)$ $$f(x)=(\ln x)^m$$</p>
<p>will eventually always be less than $g(x) = x$... | Clement C. | 75,808 | <p>Yes, this is true. This is equivalent to proving that, for any $a > 0$, we have
$$
\frac{\ln x}{x^a} \xrightarrow[x\to\infty]{}0
$$
(you can see it by setting $a=\frac{1}{m}$ from your question).\; which itself is equivalent to showing
$$
\frac{a\ln x}{x^a} = \frac{\ln x^a}{x^a} \xrightarrow[x\to\infty]{}0
$$
so... |
3,573,811 | <p>This is a theorem given by my professor from Artin Algebra:</p>
<p>Suppose that a finite abelian group <span class="math-container">$V$</span> is a direct sum of cyclic groups of prime orders <span class="math-container">$d_j=p_j^{r_j}$</span>. The integers <span class="math-container">$d_j$</span> are uniquely det... | Oliver Kayende | 704,766 | <p>Suppose <span class="math-container">$G$</span> is an abelian <span class="math-container">$p$</span>-group. The cases <span class="math-container">$|G|=1,p$</span> are obvious. Proceed by induction and assume every proper <span class="math-container">$G$</span> subgroup can be uniquely decomposed, up to order of s... |
88,122 | <p>For the easiest case, assume that $L/E$ is Galois and $E/K$ is Galois. Under what conditions can we conclude that $L/K$ is Galois? I guess the general case can be a bit tricky, but are there some "sufficiently general" cases that are interesting and for which the question can be answered?</p>
<p>EDIT: Since Jyrki's... | Ted | 15,012 | <p>Always. Galois = normal + separable. A tower of normal extensions is normal, and a tower of separable extensions is separable.</p>
<p>Edit: That's wrong. Separability is transitive, but not normality. See comments below.</p>
|
344,345 | <p>Are there any relationship between the scalar curvature and the simplicial volume? </p>
<p>The simplicial volume is zero (positive) on Torus (Hyperbolic manifold) and those manifolds does not admit a Riemannian metric with positive scalar curvature. What do we know about the simplicial volume of a Riemannian mani... | Grisha Papayanov | 43,309 | <p>By theorems of Wilking and Milnor (see <a href="https://www.sciencedirect.com/science/article/pii/S0926224500000309" rel="nofollow noreferrer">On fundamental groups of manifolds of nonnegative curvature</a> by Wilking), the fundamental group of a compact manifold with nonnegative <strong>sectional</strong> (not scal... |
1,651,427 | <blockquote>
<p>Let $f$ be a bounded function on $[0,1]$. Assume that for any $x\in[0,1)$, $f(x+)$ exists. Define $g(x)=f(x+)$, $x\in [0,1)$, and $g(1)=f(1)$. Is $g(x)$ right continuous? </p>
</blockquote>
<p>Prove it or give me a counterexample.</p>
<p>My ideas:</p>
<p>$(1)$If $f$ is of bounded variation, then $g... | DanielWainfleet | 254,665 | <p>For $x\in [0,1)$ let $(x_n)_n$ be a sequence in $(x,1)$ converging to $x.$</p>
<p>For each $n$ let $x_n<y_n< \min (1, x_n+2^{-n}).$</p>
<p>Let $z_n\in (x_n,y_n)$ such that $|f(z_n)-g(x_n)|<2^{-n}.$ </p>
<p>Then $(\;f(z_n)\;)_n$ converges to $g(x)$ and $(\;f(z_n)-g(x_n)\;)_n$ converges to $0,$ so $(\;g(x... |
201,163 | <p>I have a data set that contains data of the form (x0, y0, f1, f2, i1, i2, i3). The (x0, y0) are the coordinates, while the values f1 and f2 are real numbers (i1, i2, i3 correspond to some integers which are used as indices). The data can be downloaded <a href="http://www.mediafire.com/file/0xtxn8rggjorhdj/basins_%25... | Roman | 26,598 | <p>read the data:</p>
<pre><code>SetDirectory[NotebookDirectory[]];
data = Import["basins_(L4).out.txt", "Table"];
</code></pre>
<p>interpolate <span class="math-container">$f_1$</span> and <span class="math-container">$f_2$</span>: linear interpolation on irregular grid,</p>
<pre><code>F1 = Interpolation[{{#[[1]], ... |
2,354,004 | <p>I'm struggling with the following sum:</p>
<p>$$\sum_{n=0}^\infty \frac{n!}{(2n)!}$$</p>
<p>I know that the final result will use the error function, but will not use any other non-elementary functions. I'm fairly sure that it doesn't telescope, and I'm not even sure how to get $\operatorname {erf}$ out of that.</... | Simply Beautiful Art | 272,831 | <p>One might note that:</p>
<p>$$\frac{n!}{(2n)!}=\frac{\sqrt\pi}{4^n\Gamma(n+1/2)}$$</p>
<p>Indeed, this makes your series a special case of the following function:</p>
<p>$$f_\alpha(x)=\sum_{n=0}^\infty\frac{x^n}{\Gamma(n+\alpha)}$$</p>
<p>With $S=\sqrt\pi f_{1/2}(1/4)$.</p>
<p>Wonderfully, this problem serves a... |
2,354,004 | <p>I'm struggling with the following sum:</p>
<p>$$\sum_{n=0}^\infty \frac{n!}{(2n)!}$$</p>
<p>I know that the final result will use the error function, but will not use any other non-elementary functions. I'm fairly sure that it doesn't telescope, and I'm not even sure how to get $\operatorname {erf}$ out of that.</... | Marco Cantarini | 171,547 | <p>The series can be written also in terms of the Incomplete Gamma function. As noted by Simply Beautiful Art we have <span class="math-container">$$\sum_{n\geq0}\frac{n!}{\left(2n\right)!}=\sum_{n\geq0}\frac{\Gamma\left(1/2\right)}{4^{n}\Gamma\left(n+1/2\right)}$$</span> <span class="math-container">$$=1+\frac{1}{4}\s... |
866,847 | <p><strong>Question:</strong><br/>
Show that $$\sum_{\{a_1, a_2, \dots, a_k\}\subseteq\{1, 2, \dots, n\}}\frac{1}{a_1*a_2*\dots*a_k} = n$$
(Here the sum is over all non-empty subsets of the set of the $n$ smallest positive integers.)</p>
<blockquote>
<p>I made an attempt and then encountered an inconsistency with th... | rae306 | 168,956 | <p>We know that $\cos(A+B)=\cos A\cos B-\sin A\sin B$.</p>
<p>Now:</p>
<p>$\cos(B-A)=\cos(-A+B)=\\\cos(-A) \cos B-\sin(-A)\sin(B)=\cos A\cos B+\sin A \sin B$.</p>
|
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