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 |
|---|---|---|---|---|
422,941 | <p>How can we expand the following by the binomial expansion, upto the term including $x^3$? That'll be 4 terms.</p>
<p>This the expression to be expanded: $\sqrt{2+x\over1-x}$</p>
<p>I understand how to do the numerator and denominator individually. Now this is what I'm doing - having expanded the denominator (u... | marty cohen | 13,079 | <p>We will use the expansion
$\sqrt{1+x} = 1+x/2+x^2(1/2)(-1/2)/2 + x^3(1/2)(-1/2)(-3/2)/6 + ...
= 1+x/2-x^2/8+x^3/8+...
$
where "..." means "terms of higher order than $x^3$"
both in this expansion and in the math below.</p>
<p>Note: I am doing the following math
off the top of my head
as I am entering it,
so the cha... |
1,237,450 | <p>I couldn't follow a step while reading this <a href="https://math.stackexchange.com/a/1237316/135088">answer</a>. Since I do not have enough reputation to post this as a comment, I'm asking a question instead. The answer uses "partial integration" to write this $$ \int \frac{dv}{(v^2 + 1)^\alpha} = \frac{v}{2(\alpha... | Chappers | 221,811 | <p>"Partial integration" just means integration by parts. The important step here is the writing of the fraction as
$$ \frac{1}{(v^2+1)^{\alpha}} = \frac{1}{(v^2+1)^{\alpha}} - \frac{1}{(v^2+1)^{\alpha-1}} + \frac{1}{(v^2+1)^{\alpha-1}} \\
= -\frac{v^2}{(v^2+1)^{\alpha}} + \frac{1}{(v^2+1)^{\alpha-1}}. $$
Then you inte... |
122,546 | <p>There is a famous proof of the Sum of integers, supposedly put forward by Gauss.</p>
<p>$$S=\sum\limits_{i=1}^{n}i=1+2+3+\cdots+(n-2)+(n-1)+n$$</p>
<p>$$2S=(1+n)+(2+(n-2))+\cdots+(n+1)$$</p>
<p>$$S=\frac{n(1+n)}{2}$$</p>
<p>I was looking for a similar proof for when $S=\sum\limits_{i=1}^{n}i^2$</p>
<p>I've trie... | Pedro | 23,350 | <p>Since I think the solution Tyler proposes is very useful and accesible, I'll spell it out for you:</p>
<p>We know that</p>
<p>$$(k+1)^3-k^3=3k^2+3k+1$$</p>
<p>If we give the equation values from $1$ to $n$ we get the following:</p>
<p>$$(\color{red}{1}+1)^3-\color{red}{1}^3=3\cdot \color{red}{1}^2+3\cdot \color{... |
122,546 | <p>There is a famous proof of the Sum of integers, supposedly put forward by Gauss.</p>
<p>$$S=\sum\limits_{i=1}^{n}i=1+2+3+\cdots+(n-2)+(n-1)+n$$</p>
<p>$$2S=(1+n)+(2+(n-2))+\cdots+(n+1)$$</p>
<p>$$S=\frac{n(1+n)}{2}$$</p>
<p>I was looking for a similar proof for when $S=\sum\limits_{i=1}^{n}i^2$</p>
<p>I've trie... | JeremyKun | 13,528 | <p>There is a more beautiful Gauss-style proof that involves writing the numbers in triangles instead of in a line.</p>
<p><img src="https://i.stack.imgur.com/za9s2.png" alt="Gauss style proof"></p>
<p>I leave the details to you.</p>
|
280,346 | <p>I am wondering how to tell Mathematica that a function, say <code>F[x]</code>, is a real-valued function so that, e.g., the <code>Conjugate</code> command will pass through it:</p>
<pre><code>Conjugate[E^(-i k x)F[x]] = E^(i k x)F[x]
</code></pre>
<p>I tried to make a huge calculation using the <code>Conjugate</code... | Roman | 26,598 | <p>Use <a href="https://reference.wolfram.com/language/ref/Assuming.html" rel="nofollow noreferrer">assumptions</a>:</p>
<pre><code>Assuming[Element[F[_], Reals],
Conjugate[E^(-I k x) F[x]] // FullSimplify]
(* E^(I Conjugate[k] Conjugate[x]) F[x] *)
</code></pre>
<p>With several real-valued symbols:</p>
<pre>... |
1,747,696 | <p>First of all: beginner here, sorry if this is trivial.</p>
<p>We know that $ 1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2 $ .</p>
<p>My question is: what if instead of moving by 1, we moved by an arbitrary number, say 3 or 11? $ 11+22+33+44+\ldots+11n = $ ?
The way I've understood the usual formula is that the first n... | MJ73550 | 331,483 | <p>This a question of notation.</p>
<p>$1+2+3+4+\dots+n$ is a notation for $\sum_{k=1}^n k$</p>
<p>I assume that $11+22+33+44+\dots+11\times n$ is a notation for $\sum_{k=1}^n 11\times k$</p>
<p>in this case, you just get :
$$ \sum_{k=1}^n 11\times k = 11 \times \sum_{k=1}^n k = 11 \frac{n(n+1)}{2}$$</p>
<p>with an... |
2,359,292 | <p>I have been working on a problem in Quantum Mechanics and I have encountered a equation as given below.</p>
<p>$$\frac{d\hat A(t)}{dt} = \hat F(t)\hat A(t)$$</p>
<p>Where ^ denotes it is an operator </p>
<p>How will this differential equation be solved? Will the usual rules for linear homogeneous first order diff... | md2perpe | 168,433 | <p>Let us introduce an evolution operator $\hat U(t_1, t_0)$ such that
$\hat A(t_1) = \hat U(t_1, t_0) \hat A(t_0).$
It satisfies
$$\frac{\partial}{\partial t_1} \hat U(t_1, t_0) = \hat F(t_1) \, \hat U(t_1, t_0).$$</p>
<p>$\newcommand{\prodint}{{\prod}}$
We can use a <a href="https://en.wikipedia.org/wiki/Product_in... |
290,132 | <p>Let $x,a,b$ be real numbers and $f(x)$ a (nongiven) real-analytic function.</p>
<p>How to find $f(x)$ such that for all $x$ we have $f(x)+af(x+1)=b^x$ ? </p>
<p>In particular I wonder most about the case $a=1$ and $b=e$. (I already know the trivial cases $a=-1$ and $a=0$)</p>
<p>I know how to express $f(x+1)$ int... | sdcvvc | 12,523 | <p>Note that you can write $Lf = e^x$ where $(L f)(x)=f(x)+f(x+1)$ is a linear operation.</p>
<p>Therefore, it's enough to find a single function $f$ such that $Lf = e^x$ and all solutions will be of the form $f+g$ where $L g=0$. You can discover as in Haskell Curry's answer that $f$ can be taken to be $\frac{1}{1+e} ... |
3,142,417 | <p>If <span class="math-container">$a , b , c$</span> and <span class="math-container">$d$</span> are positive integers,
and <span class="math-container">$ab$</span> is greater than <span class="math-container">$cd$</span>,
then, is <span class="math-container">$a+b$</span> greater than or equal to <span class="ma... | att epl | 610,770 | <p>Nope...consider:
<span class="math-container">$$100=20 \times 5$$</span> and <span class="math-container">$$54=27 \times 2$$</span></p>
|
2,631,230 | <p>So, I'm studying mathematics on my own and I took a book about Proofs in Abstract Mathematics with the following exercise:</p>
<p>For each $k\in\Bbb{N}$ we have that $\Bbb{N}_k$ is finite</p>
<p>Just to give some context on what theorems and definitions we can use:</p>
<ol>
<li>Definition: $\Bbb{N}_k = \{1, 2, ..... | Community | -1 | <p>$$\int^{1}_{0}\int^{1}_{0}4xy\sqrt{x^2+y^2} dy \, dx=$$
$$\int^{\pi/4}_{0}\int^{\sec\theta}_{0}4r^4\sin\theta \cos \theta drd\theta+\int^{\pi/2}_{\pi/4}\int^{cosec \theta}_{0}4r^4\sin\theta \cos \theta drd\theta $$</p>
<p><strong>Explanation:-</strong></p>
<p>$$x=r\cos \theta$$
$$y=r\sin \theta$$
$$dxdy=rdrd\theta... |
1,793,231 | <p>Can you please help me on this question?
$\DeclareMathOperator{\adj}{adj}$</p>
<p>$A$ is a real $n \times n$ matrix; show that:</p>
<p>$\adj(\adj(A)) = (\det A)^{n-2}A$</p>
<p>I don't know which of the expressions below might help</p>
<p>$$
\adj(A)A = \det(A)I\\
(\adj(A))_{ij} = (-1)^{i+j}\det(A(i|j))
$$</p>
<p... | Jyothi Krishna Gudi | 616,478 | <p>We know the property adj(A).A = |A|I</p>
<p>Now consider</p>
<p><span class="math-container">$$adj(adj(A))*adj(A)=|adj A|I$$</span></p>
<p>Post multiply this with A</p>
<p><span class="math-container">$$adj(adj(A))*adj(A)*A=|A|^{n-1}.I.A$$</span></p>
<p><span class="math-container">$$adj(adj(A))*|A|=|A|^{n-1}.A$$</s... |
2,965,717 | <p>How would you prove that <span class="math-container">$$\displaystyle \prod_{k=1}^\infty \left(1+\dfrac{1}{2^k}\right) \lt e ?$$</span></p>
<p>Wolfram|Alpha shows that the product evaluates to <span class="math-container">$2.384231 \dots$</span> but is there a nice way to write this number? </p>
<p>A hint about so... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Taking <span class="math-container">$\log$</span> both sides the statement is equivalent to prove that</p>
<p><span class="math-container">$$\sum_{k=1}^\infty \log \left(1+\dfrac{1}{2^k}\right) \lt 1$$</span></p>
<p>then use <span class="math-container">$\log(1+x)<x$</span>.</p>
|
2,934,973 | <p><a href="https://i.stack.imgur.com/XQ80d.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XQ80d.png" alt="enter image description here"></a> </p>
<p>Let <span class="math-container">$\theta = \angle BAC$</span>. Then we can write <span class="math-container">$\cos \theta = \dfrac{x}{\sqrt{2}}$</sp... | nonuser | 463,553 | <p><span class="math-container">$$a = BC = \sqrt{2^2+3^2}=\sqrt{13}$$</span>
<span class="math-container">$$b = AC = \sqrt{3^2+1^2}=\sqrt{10}$$</span>
<span class="math-container">$$c = AB = \sqrt{2^2+1^2}=\sqrt{5}$$</span></p>
<p>so <span class="math-container">$$\cos \theta = {b^2+c^2-a^2\over 2bc} = {1\over 5\sqrt{... |
368,292 | <p>This question is two-fold.</p>
<p>The first question is rather specific: what are some small examples of negative surgeries on negative knots that give rise to the same 3-manifold? I know one class of examples coming from Borromean rings. By performing <span class="math-container">$-1/m$</span> and <span class="math... | Marc Kegel | 84,120 | <p><span class="math-container">$(-7)$</span>-surgery on the left-handed trefoil yields the lens space <span class="math-container">$L(7,2)$</span> which is defined to be the <span class="math-container">$(-7/2)$</span>-surgery along the unknot.</p>
<p>Similarly one can get more examples along negative torus knots prod... |
368,292 | <p>This question is two-fold.</p>
<p>The first question is rather specific: what are some small examples of negative surgeries on negative knots that give rise to the same 3-manifold? I know one class of examples coming from Borromean rings. By performing <span class="math-container">$-1/m$</span> and <span class="math... | Oğuz Şavk | 131,172 | <p>In general, to find explicit examples for the first part of your question is a hard problem, sometimes impossible. Actually, it is related to the notion of <em>cosmetic surgeries</em>, see Ni and Wu's <a href="http://www.its.caltech.edu/%7Eyini/Published/Cosmetic.pdf" rel="nofollow noreferrer">paper</a>, and further... |
75,791 | <p>When will a probabilistic process obtained by an "abstraction" from a deterministic discrete process satisfy the Markov property?</p>
<p>Example #1) Suppose we have some recurrence, e.g., $a_t=a^2_{t-1}$, $t>0$. It's a deterministic process. However, if we make an "abstraction" by just considering the one partic... | Did | 6,179 | <p>It seems both examples fit into the following setting. One starts from a (deterministic) dynamic system defined by $a_0\in A$ and $a_{t+1}=u(a_t)$ for every nonnegative integer $t$, for a given function $u:A\to A$, and one considers the $X$-valued process $(x_t)_{t\geqslant0}$ defined by $x_t=\xi(a_t)$ for every non... |
2,691,266 | <p>The quotient ring $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain by an ideal $\mathfrak{a}\ne 0$ is a principal ideal domain. </p>
<p>I am trying to show $\mathcal{O}/\mathfrak{p}^n$ is principal ring. Let $\mathfrak{p}^i/\mathfrak{p}^n$ be an ideal and choose $\pi\in\mathfrak{p}\setminus\mathfrak{p}^2$. Then how ... | Bernard | 202,857 | <p><strong>Hint</strong>:
$$\mathcal O/\mathfrak p^n\simeq\mathcal O_{\mathfrak p}/(\mathfrak pO_{\mathfrak p})^n.$$
What is the localisation of a Dedekind domain at a maximal ideal? </p>
|
1,737,674 | <p>I am trying to understand how to find all congruence classes in $\mathbb{F}_2[x]$ modulo $x^2$. How can I compute them ? Can someone get me started with this? I am having trouble understanding $\mathbb{F}_2[x] $ is it the set $\{ f(x) = a_nx^n + ...+ a_1 x + a_0 : a_i = 0,1 \} $?</p>
| imranfat | 64,546 | <p>In triangle BDC you can set up the Sine Law: $\frac{DC}{\sin15^{\circ}}=\frac{BD}{\sin45^{\circ}}$ with $DC=1$, I get $BD=2.732$ Now in triangle BDA we will have to use the Law of Cosine because you have 2 sides with an enclosed angle D so calculating AB we get $AB²=2.732²+2²-2*2.732*2*\cos60°$ which gives $AB=\sqrt... |
1,737,674 | <p>I am trying to understand how to find all congruence classes in $\mathbb{F}_2[x]$ modulo $x^2$. How can I compute them ? Can someone get me started with this? I am having trouble understanding $\mathbb{F}_2[x] $ is it the set $\{ f(x) = a_nx^n + ...+ a_1 x + a_0 : a_i = 0,1 \} $?</p>
| Senex Ægypti Parvi | 89,020 | <p>assumption $\overline{CD}=1$<br>
point C $(-1\mid 0)$<br>
point D $(0\mid 0)$<br>
point A $(2\mid 0)$<br>
point B $\left(\frac{1+\sqrt3}2\mid\frac{3+\sqrt3}2\right)\quad$ intersection of<br>
$\qquad\qquad y=x\tan {60°}$ and $y=(x+1)\tan{45°}$<br>
angle A =$\tan^{-1}{\frac{\frac{3+\sqrt3}2}{\frac{1+\sqrt3}2-2}}$
=$\t... |
3,027,286 | <p>I am a little confused as to proving that <span class="math-container">$(C^*)^{-1} = (C^{-1})^*$</span> where <span class="math-container">$C$</span> is an invertible matrix which is complex. </p>
<p>Initially, I thought that it would have something to do with the identity matrix where <span class="math-container">... | Scientifica | 164,983 | <p>It is known that <span class="math-container">$$\sum_{j=1}^\infty \dfrac{1}{j^2}=\dfrac{\pi^2}{6}.$$</span></p>
<p>So if you take <span class="math-container">$p_j=\frac{6}{(\pi j)^2}$</span>, you have <span class="math-container">$\sum_{j=1}^\infty p_j=1$</span> yet <span class="math-container">$\sum_{j=1}^\infty ... |
3,027,286 | <p>I am a little confused as to proving that <span class="math-container">$(C^*)^{-1} = (C^{-1})^*$</span> where <span class="math-container">$C$</span> is an invertible matrix which is complex. </p>
<p>Initially, I thought that it would have something to do with the identity matrix where <span class="math-container">... | jjagmath | 571,433 | <p>We have the series <span class="math-container">$\displaystyle\sum_{j=1}^\infty \frac{1}{j(j+1)} = 1$</span>, but <span class="math-container">$\displaystyle\sum_{j=1}^\infty \frac{1}{j+1}$</span> diverges, so your affirmation is false.</p>
|
3,027,286 | <p>I am a little confused as to proving that <span class="math-container">$(C^*)^{-1} = (C^{-1})^*$</span> where <span class="math-container">$C$</span> is an invertible matrix which is complex. </p>
<p>Initially, I thought that it would have something to do with the identity matrix where <span class="math-container">... | Mike Earnest | 177,399 | <p>Not all aperiodic, irreducible Markov processes have a stationary distribution. This is only true for finite state spaces. For infinite spaces, you need the process to be positive recurrent, meaning the expected time to return to a state is finite. Here, starting from <span class="math-container">$1$</span>, the exp... |
2,245,631 | <blockquote>
<p>$x+x\sqrt{(2x+2)}=3$</p>
</blockquote>
<p>I must solve this, but I always get to a point where I don't know what to do. The answer is 1.</p>
<p>Here is what I did: </p>
<p>$$\begin{align}
3&=x(1+\sqrt{2(x+1)}) \\
\frac{3}{x}&=1+\sqrt{2(x+1)} \\
\frac{3}{x}-1&=\sqrt{2(x+1)} \\
\frac{(3-x... | John Doe | 399,334 | <p>So you got to the cubic equation $f(x)=-2x^{3}-x^{2}-6x+9=0$. When you come across a cubic like this, try evaluating $f(\pm1), f(\pm2)$, etc, to try and figure out some roots so you can factor it (you know $x_0$ a root if $f(x_0)=0$). Here you can see $1$ is a root, so factoring out $(x-1)$ gives $f(x)=(x-1)\underbr... |
3,840,692 | <p>The equation is <span class="math-container">$2z^2w''+3zw'-w=0$</span></p>
<p><span class="math-container">$z_0=0$</span> is a regular singular point, so <span class="math-container">$w(z)=\sum_{n=0}^{\infty} a_nz^{n+r}$</span></p>
<p>then <span class="math-container">$w'(z)=\sum_{n=0}^{\infty} (n+r)a_nz^{n+r-1}$</s... | metamorphy | 543,769 | <p>Your last equation is all you get. There's no recurrence, and you don't need one.</p>
<p>You <em>necessarily</em> have <span class="math-container">$\color{blue}{a_n=0}$</span> for <span class="math-container">$n>0$</span>. Otherwise, if <span class="math-container">$a_n\neq 0$</span> for some <span class="math-c... |
1,685,895 | <blockquote>
<blockquote>
<p>Question: Find a value of $n$ such that the coefficients of $x^7$ and $x^8$ are in the expansion of $\displaystyle \left(2+\frac{x}{3}\right)^{n}$ are equal.</p>
</blockquote>
</blockquote>
<hr>
<p>My attempt:</p>
<p>$\displaystyle \binom{n}{7}=\binom{n}{8} $</p>
<p>$$ n(n-1)(n-... | Decaf-Math | 227,902 | <p>Reference that $$(a + b)^n = {n \choose 0}a^nb^0 + {n \choose 1}a^{n-1}b^1 + \cdots + {n \choose n-1}ab^{n-1} + {n \choose n}a^0b^n.$$</p>
<p>So we want $a = 2$ and $b = {x \over 3}$. So we are considering the terms $\displaystyle {n \choose 7}a^{n - 7}b^7$ and $\displaystyle {n\choose 8}a^{n-8}b^8.$ So, $${n \choo... |
2,554,153 | <p>I have some problem with writing character table of a group. For instance, a group $S_4$. When we write character table, we write irreducible representations of group. So, how can I quickly find them? Then how to Fill the table? Can someone explain me upon this example?</p>
| Andres Mejia | 297,998 | <p>Let's start with $S_3$.</p>
<p><strong>Step 1:</strong> Find the conjugacy classes when this is not too difficult. $1, (12), (123)$ generate the full group.</p>
<p><strong>Step 2:</strong> There are two easy representations: the trivial one, and the "alternating" representation, which is just $\mathrm{sgn}$ which ... |
3,068,934 | <blockquote>
<p>Let <span class="math-container">$A$</span> be a square matrix over <span class="math-container">$\mathbb{C}$</span>. Prove there are matrices <span class="math-container">$D$</span> and <span class="math-container">$N$</span> such that <span class="math-container">$A = D + N$</span> such that <span c... | Spitemaster | 604,925 | <p>A triangle in <span class="math-container">$n$</span> dimensions is known as an <em>n-simplex</em>.</p>
|
3,068,934 | <blockquote>
<p>Let <span class="math-container">$A$</span> be a square matrix over <span class="math-container">$\mathbb{C}$</span>. Prove there are matrices <span class="math-container">$D$</span> and <span class="math-container">$N$</span> such that <span class="math-container">$A = D + N$</span> such that <span c... | Dr. Richard Klitzing | 518,676 | <p>The <span class="math-container">$n$</span>-dimensional simplex has <span class="math-container">$n+1$</span> vertices and also <span class="math-container">$n+1$</span> facets, all of which are <span class="math-container">$n-1$</span>-dimensional simplices in turn. </p>
<p>In fact, the count of elements of an <sp... |
1,180,199 | <p>I am not quite sure how to deal with discrete IVP</p>
<p>Find self-similar solution
\begin{equation}
u_t=u u_x\qquad -\infty <x <\infty,\ t>0
\end{equation}</p>
<p>satisfying initial conditions</p>
<p>\begin{equation}
u|_{t=0}=\left \{\begin{aligned}
-1& &x\le 0,\\
1& &x> 0
\end{aligne... | doraemonpaul | 30,938 | <p>Follow the method in <a href="http://en.wikipedia.org/wiki/Method_of_characteristics#Example" rel="nofollow">http://en.wikipedia.org/wiki/Method_of_characteristics#Example</a>:</p>
<p>$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$</p>
<p>$\dfrac{du}{ds}=0$ , letting $u(0)=u_0$ , we have $u=u_0$</p>
<p>$\df... |
233,169 | <p>I had to redo the problem because there was a mistake. With the given function from a previous problem, I was solving <a href="https://mathematica.stackexchange.com/questions/231664/adding-a-point-in-a-manipulate-command">link</a>, I found that the parabola created a trajectory on the graph, ie another parabola.</p>... | N0va | 42,436 | <p>I am not sure if I understood the question correctly. Does this solve your question?</p>
<pre><code>Table[{Plot[x^2-2*(a-2)*x+a-2,{x,-20,20}],{a-2,-6+5 a-a^2}},{a,Range[-7,7,14/20]}];
Show[Flatten[{%[[All,1]],Plot[x-x^2,{x,-20,20},PlotStyle->Red],ListPlot[%[[All,2]]]}]]
-6+5 a-a^2/.a->x+2//Expand
</code></pre>... |
2,677 | <p>If <em>G</em> is a group, its <strong>abelianization</strong> is the abelian group <em>A</em> and the map <em>G</em> → <em>A</em> such that any map <em>G</em> → <em>B</em> with <em>B</em> abelian factors through <em>A</em>. Abelianization is a functor, and in general a very lossy operation. The map <em>G... | Jason DeVito | 1,708 | <p>(In some sense, this is just a restatement of what Eric said above....)</p>
<p>For compact groups, quite a lot can be said. Every compact group H' has a finite cover H which is Lie group isomorphic to $T^{k} \times G$, where $G$ is compact and simply connected.</p>
<p>Then, one can easily show that [H,H] = {$e$}$... |
108,060 | <p>Suppose:
$$\sum_{n=2}^{\infty} \left( \frac{1}{n(\ln(n))^{k}} \right) =\frac{1}{ 2(\ln(2))^{k} } +\frac{1}{ 3(\ln(3))^{k} }+...,
$$
by which $k$ does it converge?</p>
<p>When I use comparison test I get inconclusive result:</p>
<p>$\lim_{n\rightarrow\infty} \frac{u_{n+1}}{u_{n}}=\frac{n\ln(n)^{k}}{(n+1)\ln(n+1)^{... | André Nicolas | 6,312 | <p>For completeness, we sketch the <a href="http://en.wikipedia.org/wiki/Integral_test_for_convergence" rel="nofollow">Integral Test</a> approach. </p>
<p>Let $f$ be a function which is defined, non-negative, and decreasing (or at least non-increasing from some point $a$ on. Then $\sum_1^\infty f(n)$ converges if and ... |
327,860 | <p>Let <span class="math-container">$A$</span> be a symmetric <span class="math-container">$d\times d$</span> matrix with integer entries such that the quadratic form <span class="math-container">$Q(x)=\langle Ax,x\rangle, x\in \mathbb{R}^d$</span>, is non-negative definite. For which <span class="math-container">$d$</... | WKC | 29,241 | <p>This is a well-known problem, called the Waring's problem of integral quadratic forms. Every semi-positive definite quadratic form in <span class="math-container">$n \leq 5$</span> variables is a sum of <span class="math-container">$n + 3$</span> squares of linear forms. This was proved by Chao Ko, but this can be... |
580,616 | <p>The Axiom of separation states that, if A is a set then $\{a \in A ;\Phi(a)\}$ is a set.
Given a set $B \subseteq A$, Suppose I define $B=\{ a \in A ; a\notin B \}$.
This, of course leads to a contradiction. Because we define $B$ by elements not from $B$. My queation is: what part of the axioms sais that this kind o... | Peter Smith | 35,151 | <p>There is nothing at all to stop you defining a set $\Sigma$ such that $x \in \Sigma$ iff $x \in A \land x \notin B$, so $\Sigma = \{x \in A \mid x \notin B\}$. </p>
<p>But what you've shown is that $\Sigma \neq B$! </p>
<p>No problem so far.</p>
<p>What you can't do is then go on (having a knock-down argument to ... |
580,616 | <p>The Axiom of separation states that, if A is a set then $\{a \in A ;\Phi(a)\}$ is a set.
Given a set $B \subseteq A$, Suppose I define $B=\{ a \in A ; a\notin B \}$.
This, of course leads to a contradiction. Because we define $B$ by elements not from $B$. My queation is: what part of the axioms sais that this kind o... | Community | -1 | <p>You were given a set and named it $B$.</p>
<p>You defined another set, and named it $B$.</p>
<p>Just because you've given them the same names doesn't mean they are actually the same set. Your contradiction only appears because you've confused yourself and thought the two sets were the same since you gave them the ... |
638,244 | <p>In any (simple) type theory there are <strong>base types</strong> (i.e. the type of <em>individuals</em> and the type of <em>propositions</em>) and <strong>type builders</strong> (i.e. $\rightarrow$, which takes two types $t,t'$ and yields the type of <em>functions</em> $t \rightarrow t'$). </p>
<p>For each type in... | Giorgio Mossa | 11,888 | <p>Basically what you're considering seems to me as the type-operations which you can obtain from the basic <strong>type builders</strong>.</p>
<p>As you have guessed this objects should be the <a href="http://en.wikipedia.org/wiki/Type_constructor" rel="nofollow">type constructors</a>.</p>
<p>About the second part o... |
78,725 | <p>The general theorem is: for all odd, distinct primes $p, q$, the following holds:
$$\left( \frac{p}{q} \right) \left( \frac{q}{p} \right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$$</p>
<p>I've discovered the following proof for the case $q=3$:
Consider the Möbius transformation $f(x) = \frac{1}{1-x}$, defined on $F_{p}... | franz lemmermeyer | 23,365 | <p>As for your second question, a (partial) list of articles dealing with the quadratic character of small primes can be found <a href="http://www.rzuser.uni-heidelberg.de/~hb3/small.html">here</a>.</p>
|
2,481,767 | <p>Let A={$3m-1|m\in Z$} and B={$4m+2|m\in Z$} and let $f:A\rightarrow B$ is defined by </p>
<p>$f(x)=\frac{4(x+1)}{3}-2$ . Is f surjective?</p>
<p>I'm not really sure how to prove this. By trying out certain values it seems it's surjective. This is my work so far:</p>
<p>$f(x)=y \iff \frac{4(x+1)}{3}-2 = y \iff x=\... | Andres Mejia | 297,998 | <p><strong>Hint:</strong> $x=3m+2 \implies x=3(m+1)-1$.</p>
|
4,513,678 | <p>Suppose <span class="math-container">$f(x) = ax^3 + bx^2 + cx + d$</span> is a cubic equation with roots <span class="math-container">$\alpha, \beta, \gamma.$</span> Then we have:</p>
<p><span class="math-container">$\alpha + \beta + \gamma= -\frac{b}{a}\quad (1)$</span></p>
<p><span class="math-container">$\alpha\b... | Ivan Kaznacheyeu | 955,514 | <p>Quantity is not-symmetric. This results in rather complex formula:</p>
<p><span class="math-container">$$\alpha^2\beta+\beta^2\gamma+\gamma^2\alpha=t_{1}\,t_{3}^2-{{b\,t_{3}^2}\over{3\,a}}+t_{2}^2\,t_{3}-{{2\,b\,
t_{2}\,t_{3}}\over{3\,a}}-{{2\,b\,t_{1}\,t_{3}}\over{3\,a}}+\\{{b^2\,
t_{3}}\over{3\,a^2}}-{{b\,t_{2}^... |
1,936,260 | <p>We have a binary sequence of 1s and 0s, and the length is 10.
I wonder how many binary sequence of length 10 with four 1's can be created such that the 1's do not appear consecutively?</p>
| Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>$\bigl\lfloor\log_{10}x\bigr\rfloor=k\iff 10^k\le x<10^{k+1}$.</p>
|
212,240 | <p>I'm a beginner of the area of free boundary problem. Let me first give some background: </p>
<p>$\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph.
Consider the convex set $$K:＝\{v \in L^1_{loc}(\Omega): \nabla v \in L^2(\Omega) \,, v=u^0 \mbox{on $\partial \... | student | 51,546 | <p>I figured out the problem later and until today I could have time to write it down.</p>
<p>The proof of the rectifiability of free boundary $\partial \{u>0\}$ requires the Lipchitz regularity of $u$ across the free boundary and the nondegeneracy of the function $u$, see theorem 3.2-theorem 4.5 in Alt and Caffare... |
121,403 | <blockquote>
<p>A manifold $M$ of dimension n is a topological space with the following properties:<br>
a) $M$ is Hausdorff<br>
b)$M$ is locally Euclidean of dimension n<br>
c) $M$ has a countable basis of open sets. </p>
</blockquote>
<p>Why is the first property necessary? I do not have much experience with ... | davidlowryduda | 9,754 | <p>A lot of the work on smooth manifolds is to let us use Euclidean analysis to merely locally Euclidean things that come up. Things that aren't Hausdorff are terrible and scary, real intuition busters (at least in my case), so I don't mind at all that we require that. And in fact, with just these 3 requirements (and a... |
3,657,075 | <p><a href="https://i.stack.imgur.com/ytcQ3.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ytcQ3.png" alt="enter image description here"></a></p>
<blockquote>
<p>In the given figure <span class="math-container">$\angle BAE, \angle BCD$</span> and <span class="math-container">$\angle CDE$</span> a... | Quanto | 686,284 | <p>Continue with <span class="math-container">$BE = \sqrt{8^2+4^2} = 4\sqrt5$</span> and recognize that the triangle BDE is isosceles with <span class="math-container">$BD =DE=5$</span>,</p>
<p><span class="math-container">$$\cos \angle ABE =\frac{AB}{BE} = \frac1{\sqrt5}, \>\>\>\>\>
\cos \angle DBE =\f... |
66,670 | <p>I want to use:</p>
<pre><code>demand = {1.92,
2.07,
2.37,
2.72,
2.87}*10^6;
NSolve[SetV == demand[[1]]/(Cpf (1 - χ)), χ]
</code></pre>
<p>I want to make a vector of solutions for chi (χ) given each of the demand vector components.</p>
| Dr. belisarius | 193 | <p>Diophantine problems are tough and there is no silver bullet. In your example this works:</p>
<pre><code>i = IntegerPart;
sol = NMaximize[{(3 i@ n + 4)/(2 i@n + 1), n > 1}, n];
i@n /. sol[[2]]
(* 1 *)
</code></pre>
|
1,979,226 | <p>Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places.
Y1, Y2, Y3 form a partition of S.</p>
<p>P(X | Y1) = .8, P(X | Y2) = .1, P(X | Y3) = .9, P(Y1) = .1, P(Y2) = .4. </p>
<p>Find P(Y1 | X).</p>
<p>P(Y1 | X) =</p>
<p>For this one I thought that all... | hamam_Abdallah | 369,188 | <p>let</p>
<p>$$v_n=\frac{\sum_{k=0}^n u_k}{n+1}$$.</p>
<p>we have</p>
<p>$$\frac{u_{n+1}^2}{n+1}=v_n$$.</p>
<p>if $lim_{n\to +\infty}u_n=L$ then</p>
<p>$0=\lim_{n\to+\infty} v_n=L$</p>
<p>using Cesaro average.</p>
<p>Now, if $(u_n)$ is increasing and</p>
<p>$u_0=a>0$, the limit can't be $0$.</p>
<p>thus $(... |
2,740,954 | <p>Determine price elasticity of demand and marginal revenue if $q = 30-4p-p^2$, where q is quantity demanded and p is price and p=3.</p>
<p>I solved it for first part-</p>
<p>Price elasticity of demand = $-\frac{p}{q} \frac{dq}{dp}$</p>
<p>on solving above i got answer as $\frac{10}{3}$</p>
<p>But on solving for M... | Trurl | 72,915 | <p>Try this. Total Revenue TR is $pq$. Marginal revenue is the change in TR with change in $quantity$ (not price, as I incorrectly stated in my comment) so marginal revenue is $\frac{\partial TR}{\partial q}$ or
$$\frac{\partial (pq)}{\partial p}\frac{\partial p}{\partial q}$$ Revenue is $pq$, or $30p−4p^2−p^3$ so mar... |
1,805,615 | <p>I have one problem. I am sure it is not complicated, but I only need help to see am I, at least, on the right path.</p>
<p><strong>Problem: Let $S=Span\{(0,-2,3),(1,1,1),(2, -2, 8)\}\subseteq \mathbb R^3$. Find subspace $T$ of space $\mathbb R^3$ so that $\mathbb R^3=S \oplus T$.</strong></p>
<p>Here is what I hav... | M. Vinay | 152,030 | <p>Forming a matrix with the vectors in $S$ as column vectors (as you have done), we get
\begin{equation*}
A = \begin{bmatrix}
0 & 1 & 2\\
-2 & 1 & -2\\
3 & 1 & 8
\end{bmatrix}.
\end{equation*}</p>
<p>Now, the span of $S$ is the same as the column space of $A$, and we can find a basis for this ... |
48,864 | <p>I can't resist asking this companion question to the <a href="https://mathoverflow.net/questions/48771/proofs-that-require-fundamentally-new-ways-of-thinking"> one of Gowers</a>. There, Tim Dokchitser suggested the idea of Grothendieck topologies as a fundamentally new insight. But Gowers' original motivation is to ... | Allen Knutson | 391 | <p>I think the human/computer dichotomy you set up should be extended to
a human/mathematician/computer trichotomy, just because a substantial
portion of "mathematical maturity" is about learning to think like a computer,
in your sense.</p>
<p>Anyway I've just put that in place to try and shore up my example. It seems... |
48,864 | <p>I can't resist asking this companion question to the <a href="https://mathoverflow.net/questions/48771/proofs-that-require-fundamentally-new-ways-of-thinking"> one of Gowers</a>. There, Tim Dokchitser suggested the idea of Grothendieck topologies as a fundamentally new insight. But Gowers' original motivation is to ... | o a | 22,247 | <p>This paper on <a href="http://mishap.sdf.org/by:gavrilovich-and-hasson/what:a-homotopy-theory-for-set-theory/Exercises_de_style_A_homotopy_theory_for_set_theory-II.pdf" rel="nofollow"> homotopy and set theory</a> seems to take this question seriously: if you restrict yourself to posetal categories and try to do mode... |
3,538,305 | <blockquote>
<p>Given that the differential equation</p>
<p><span class="math-container">$f(x,y) \frac {dy}{dx} + x^2 +y = 0$</span> is exact and <span class="math-container">$f(0,y) =y^2$</span> , then <span class="math-container">$f(1,2)$</span> is</p>
</blockquote>
<p>choose the correct option</p>
<p><span cla... | Qurultay | 338,156 | <p>Your equation is <span class="math-container">$$(x^2+y)dx+f(x,y)dy=0$$</span>
thus from <span class="math-container">$\frac{dM}{dy}=\frac{dN}{dx}$</span> we have
<span class="math-container">$$\frac{df}{dx}=1$$</span>
or
<span class="math-container">$$f(x,y)=x+h(y)$$</span>
Now ...</p>
|
729,444 | <p>Let be two lists $l_1 = [1,\cdots,n]$ and $l_2 = [randint(1,n)_1,\cdots,randint(1,n)_m]$ where $randint(1,n)_i\neq randint(1,n)_j \,\,\, \forall i\neq j$ and $n>m$. How I will be able to found the number of elements $x\in l_1$, to select, such that the probability of $x \in l_2$ is $1/2$?. I'm trying using the bi... | Marc van Leeuwen | 18,880 | <p>If you know that change of basis is realised by conjugating by an appropriate invertible matrix, then you can reason in terms of matrices as follows. $E_{i,j}$ is the matrix with unique nonzero entry $1$ at position $i,j$.</p>
<ul>
<li><p>The (unique) matrix $M$ of $T$ can have no nonzero off-diagonal entries: if $... |
4,064,084 | <p>Does there exists a countable family of infinite sets <span class="math-container">$\{A_n:n\in\mathbb N\}\subset\mathcal P(\mathbb N)$</span> satisfying the following property:
<span class="math-container">$$\text{For every infinite set }I\in\mathcal P(\mathbb N),\text{ there is }n\in\mathbb N\text{ such that }A_n\s... | moray eel | 892,232 | <p>Using hgmath's comment, here is a way to write <span class="math-container">$n^3$</span> as a sum of five cubes of integers with absolute values <span class="math-container">$<|n|$</span> for <span class="math-container">$n=2k$</span> and <span class="math-container">$k\ge 8$</span>. We write
<span class="math-c... |
3,465,018 | <p>Compute <span class="math-container">$\pi_{2}(S^2 \vee S^2).$</span></p>
<p><strong>Hint:</strong>
Use universal covering thm. and use Van Kampen to show it is simply connected.</p>
<p>Still I am unable to solve it, could anyone give me more detailed hint and the general idea of the solution.</p>
| kamills | 497,007 | <p>Here's another quick way, using Hurewicz.</p>
<p><span class="math-container">$\pi_1(S^2 \vee S^2) \cong 0$</span> by van Kampen. Then the Hurewicz theorem asserts that <span class="math-container">$\pi_2(S^2 \vee S^2) \cong H_2(S^2 \vee S^2) \cong \mathbb{Z} \oplus \mathbb{Z}$</span>.</p>
|
2,094,596 | <p>I'm questioning myselfas to why indeterminate forms arise, and why limits that apparently give us indeterminate forms can be resolved with some arithmetic tricks. Why $$\begin{equation*}
\lim_{x \rightarrow +\infty}
\frac{x+1}{x-1}=\frac{+\infty}{+\infty}
\end{equation*} $$</p>
<p>and if I do a simple operation,</... | StackTD | 159,845 | <p>So you're looking at something of the form
$$\lim_{x \to +\infty} f(x) = \lim_{x \to +\infty}\frac{g(x)}{h(x)} $$
and if this limit exists, say the limit it $L$, then it doesn't matter how we rewrite $f(x)$. However, it's possible you can write $f(x)$ in different ways; e.g. as the quotient of different functions:
$... |
1,268,431 | <p>$$\lim_{x\to 2} \frac {\sin(x^2 -4)}{x^2 - x -2} $$</p>
<p>Attempt at solution:</p>
<p>So I know I can rewrite denominator:</p>
<p>$$\frac {\sin(x^2 -4)}{(x-1)(x+2)} $$</p>
<p>So what's next? I feel like I'm supposed to multiply by conjugate of either num or denom.... but by what value...?</p>
<p>Don't tell me ... | Jordan Glen | 225,803 | <p>$$\frac {\sin(x^2 -4)}{(x-2)(x+1)}\cdot\frac{x+2}{x+2} = \frac{(x+2)\sin(x^2 - 4)}{(x+1)(x^2 - 4)} = \dfrac{x+2}{x+1}\cdot \dfrac{\sin(x^2 - 4)}{x^2 - 4}$$</p>
|
3,115,830 | <p>So my logic to this up until now has been that for any <span class="math-container">$x$</span> the function <span class="math-container">$\left\lfloor\frac{\lceil x\rceil}{2}\right\rfloor$</span> will return an integer that is an element of <span class="math-container">$\mathbb Z$</span>. Thus since you can map any ... | Robert Lewis | 67,071 | <p>With</p>
<p><span class="math-container">$f(s) = a T + b N + c(s) B, \tag 1$</span></p>
<p>and</p>
<p><span class="math-container">$\Vert f(s) \Vert = 1, \tag 2$</span></p>
<p>it follows that <span class="math-container">$s$</span> is the arc-length along <span class="math-container">$f(s)$</span>; thus</p>
<p>... |
715,361 | <p>Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that
$$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ \|q\|_{H^1(\Omega)}\leq1,\ n\in\mathbb{N}.\quad (1) $$
Is it then possible to conclude that
$$ \sup_{n\in\mathbb{N}}\|f_n\|_{L^2(\Omega)}... | 5xum | 112,884 | <p>The trainer is right, there is no solution. Your approach of crossing the equations has an implicit demand that the denominators are nonzero, so your approach should show there are no solutions as well.</p>
|
262,173 | <p>Consider $x^2 + y^2 = r^2$. Then take the square of this to give $(x^2 + y^2)^2 = r^4$. Clearly, from this $r^4 \neq x^4 + y^4$. </p>
<p>But consider: let $x=a^2, y = b^2 $and$\,\,r = c^2$. Sub this into the first eqn to get $(a^2)^2 + (b^2)^2 = (c^2)^2$. $x = a^2 => a = |x|,$ and similarly for $b.$</p>
<p>Now ... | gt6989b | 16,192 | <p>$x = a^2$ does not imply that $a = |x|$, rather $|a| = \sqrt{x}$.</p>
|
716,036 | <blockquote>
<p>Suppose that a curve $\mathbf\gamma$ in $\mathbb R^3$ has constant strictly positive curvature function $\mathbf\kappa(s)$, and constant non-zero torsion function $\mathbf\tau(s)$. Prove that the curve is a helix.</p>
</blockquote>
<p>I think it is easier to work backward here. First I can show that ... | Yiorgos S. Smyrlis | 57,021 | <p>A curve in $\mathbb R^3$ can be uniquely (up to a rigid motion) reproduced once its curvature and torsion are known. If ${T}$, ${N}$ and ${B}$ is its moving orthogonal frame (tangent, norma and binormal), then they satisfy the system (Frenet-Serret)
$$
T'=kN,\\
N'=-kT-\tau B,\\
B'=\tau N.
$$
or
$$
H'=\left(\begin{m... |
4,521,199 | <blockquote>
<p><strong>Theorem 8.15</strong>: If <span class="math-container">$f$</span> is a continuous and <span class="math-container">$2\pi$</span>-periodic function and if <span class="math-container">$\epsilon>0$</span> is fixed, then there exists a trigonometric polynomial <span class="math-container">$P$</s... | José Carlos Santos | 446,262 | <p>Let <span class="math-container">$\log_1\colon\Bbb C\setminus[0,\infty)\longrightarrow\Bbb C$</span> be the antiderivative of <span class="math-container">$\frac1z$</span> which maps <span class="math-container">$-1$</span> into <span class="math-container">$\pi i$</span>. It is a continuous function (actually, it i... |
59,828 | <p>Is there a way to display the variable name instead of its value? for example, I need something like<code>varname = 1; function[varname];</code> and the output is <code>varname</code> instead of <code>1</code></p>
| RunnyKine | 5,709 | <p>There's also <code>Defer</code> to accomplish this:</p>
<pre><code>varname = 1;
Defer @ varname
</code></pre>
<blockquote>
<p>varname</p>
</blockquote>
|
65,658 | <p>Suppose $X_i$'s are i.i.d, with the density distribution $f(x) = e^{-x}$, $x \geq 0$. I was able to show that
$$P(\limsup X_n/\log{n} =1)=1$$ using Borel-Cantelli.</p>
<p>Define $M_n=\max \{X_1,\ldots,X_n\}$, can I claim $M_n/\log{n} \rightarrow 1$ a.s. in this case? Is it still true in general without knowing the ... | Leandro | 633 | <p>this is a small observation not an answer: </p>
<p>the distribution is in fact important, for example if the random variables are bounded almost surely the limit is zero a.s.</p>
<p>For the unbounded case, (that more likely you are thinking about), I just got the trivial lower bound
$$
1\leq \liminf_{n\to\infty}... |
3,910,013 | <p>I'm preparing for a high school math exam and I came across this question in an old exam.</p>
<p>Let <span class="math-container">$f(x) = \dfrac{1}{2(1+x^3)}$</span>.</p>
<p><span class="math-container">$\alpha \in (0, \frac{1}{2})$</span> is the only real number such that <span class="math-container">$f(\alpha) = \... | mechanodroid | 144,766 | <p><strong>Hint:</strong></p>
<p>Prove that for <span class="math-container">$x,y \in \left(0,\frac12\right)$</span> we have
<span class="math-container">$$|f(x)-f(y)| \le \frac12 |x-y|.$$</span>
and then apply this to <span class="math-container">$x = u_n$</span> and <span class="math-container">$y = \alpha$</span>.</... |
271 | <p>Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it?</p>
<p>I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers.</p>
<p>The intended u... | Michael Lugo | 173 | <p>I've long used Simon Plouffe's <a href="http://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html" rel="nofollow noreferrer">inverse symbolic calculator</a> for this purpose. It is essentially a searchable list of "interesting" numbers.</p>
<p>Edit: link updated (Mar 2022).</p>
|
8,699 | <p>I love your site.... but the your question does not meet our quality standards thing is really annoying... I have wasted lots of time trying to figure out what this message means.....maybe someone could explain it to me.....whats wrong with this question:</p>
<p>Find numbers a and b such that: </p>
<p>$ lim =((sq... | robjohn | 13,854 | <p>In addition to Asaf's suggestions, one more that deals with the readability is to use <a href="https://math.meta.stackexchange.com/questions/5020/tex-latex-mathjax-basic-tutorial-and-quick-reference">$\LaTeX$</a> with the <a href="http://www.mathjax.org/docs/2.0/tex.html" rel="nofollow noreferrer">MathJax</a> markup... |
8,699 | <p>I love your site.... but the your question does not meet our quality standards thing is really annoying... I have wasted lots of time trying to figure out what this message means.....maybe someone could explain it to me.....whats wrong with this question:</p>
<p>Find numbers a and b such that: </p>
<p>$ lim =((sq... | zyx | 14,120 | <blockquote>
<p>the your question does not meet our quality standards thing is really annoying...
I have wasted lots of time trying to figure out what this message means</p>
</blockquote>
<p>It is an automatically generated message.</p>
<p>There is an algorithmic "quality filter" for questions. StackExchange has... |
1,879,129 | <p>If $0 < y < 1$ and $-1 < x<1$, then prove that $$\left|\frac{x(1-y)}{1+yx}\right| < 1$$</p>
| ervx | 325,617 | <p>$$
\bigg|\frac{x(1-y)}{(1+yx)}\bigg|=\frac{|x|(1-y)}{1+yx}.
$$</p>
<p>We have two cases.</p>
<p>If $x\geq 0$, then the above becomes</p>
<p>$$
\frac{x(1-y)}{1+xy}.
$$</p>
<p>Note that $x(1-y)<x<1$, while $1+xy>1$. Thus, the inequality follows in this case.</p>
<p>If instead $x<0$, then the inequalit... |
1,985,905 | <p>I was wondering if the cardinality of a set is a well defined function, more specifically, does it have a well defined domain and range?</p>
<p>One would say you could assign a number to every finite set, and a cardinality for an infinite set. So the range would be clear, the set of cardinal numbers. But what about... | Asaf Karagila | 622 | <p>The cardinality function is well-defined, but it is what known as a <em>class</em> function. Since <em>every</em> set has a cardinality, the domain of the function $A\mapsto |A|$ has to be the class of all sets, so this is indeed a proper class. And since every set has a strictly large cardinal, the class of cardina... |
108,953 | <p>Given a variety $X$ over $\mathbb{Q}$ with good reduction at $p$, proper smooth base change tells us that its $l$-adic cohomology groups are unramified at $p$ (and I'd guess some $p$-adic Hodge theory tells us its p-adic cohomology is crystalline).</p>
<p>My question is to what extent it's possible to find a conver... | Joël | 9,317 | <p>(could be a comment but too long...) </p>
<p>That's quite a natural question. I am not sure it is possible to prove that the "conjecture" you state is true with the current technology (and to be sure I have no idea how to prove it),
but my intuition would differ from yours in that I believe the conjecture to be t... |
1,594,130 | <p>Does there exist a vector field $\vec F$ such that curl of $\vec F$ is $x \vec i+y\vec j+z \vec k$ ? </p>
<p>UPDATE : I did $div(curl \vec F)=0$ as the answers did ; but that assumes a lot i.e. it assumes that components of $F$ have second partial derivatives and continuous mixed partial derivatives ; whereas for c... | j.d. allen | 293,950 | <p>Using the theorem $div(curl(\vec F))= 0 $ we can show that the vector $F=x \vec i+y\vec j+z \vec k$ cannot be the curl of any field because $F=x \vec i+y\vec j+z \vec k$ has a divergence of 3. </p>
|
1,201,900 | <p>This is a rather soft question to I will tag it as such.</p>
<p>Basically what I am asking, is if anyone has a good explanation of what a homomorphism is and what an isomorphism is, and if possible specifically pertaining to beginner linear algebra.</p>
<p>This is because, in my courses we have talked about vector... | Moya | 192,336 | <p>Well the standard answer to this sort of question is that two algebraic objects (vector spaces in this case) $V$ and $W$ are isomorphic if they are basically the same, meaning that once can identify them with one another in a reasonable way. Another way to say this is that a map $f:V\to W$ is an isomorphism if it is... |
903,656 | <p>An urn has $2$ balls and each ball could be green, red or black. We draw a ball and it was green, then it was returned it to the urn. What is the probability that the next ball is red? </p>
<p>My attempt: I think it is just a probability of $1/4$ because we have 4 colors in total but on the other hand I think i ne... | David | 119,775 | <p><strong>Hint</strong>:
$$\eqalign{&P(\hbox{second ball is red})\cr
&\qquad=P(\hbox{second ball is red}\,|\,\hbox{second ball drawn is the same ball as the first})\cr
&\qquad\qquad\qquad{}\times P(\hbox{second ball drawn is the same ball as the first})\cr
&\qquad\qquad{}+P(\hbox{second ball is r... |
903,656 | <p>An urn has $2$ balls and each ball could be green, red or black. We draw a ball and it was green, then it was returned it to the urn. What is the probability that the next ball is red? </p>
<p>My attempt: I think it is just a probability of $1/4$ because we have 4 colors in total but on the other hand I think i ne... | angryavian | 43,949 | <p>I'm guessing we make the Bayesian assumption that before we draw anything, each of the two balls has an equal chance of being any of the $3$ colors.</p>
<p>Your guess of $1/3$ is incorrect; the intuition is that by drawing a red, you gain some knowledge about the urn, and sort of decreases the chance of drawing a r... |
2,809,686 | <p>Let S={1,2,3,...,20}. Find the probability of choosing a subset of three numbers from the set S so that no two consecutive numbers are selected in the set.
"I am getting problem in forming the required number of sets."</p>
| JMoravitz | 179,297 | <p>To remove from unanswered queue:</p>
<p>Consider the related problem of counting how many quadruples $(x_1,x_2,x_3,x_4)$ of non-negative integers exist such that $x_1+x_2+x_3+x_4=15$</p>
<p>Count the number of such possible quadruples using <a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)" rel... |
112,226 | <p>Prove that there are exactly</p>
<p>$$\displaystyle{\frac{(a-1)(b-1)}{2}}$$ </p>
<p>positive integers that <em>cannot</em> be expressed in the form </p>
<p>$$ax\hspace{2pt}+\hspace{2pt}by$$</p>
<p>where $x$ and $y$ are non-negative integers, and $a, b$ are positive integers such that $\gcd(a,b) =1$.</p>
| Aryabhata | 1,102 | <p>It is well known that any number $\ge (a-1)(b-1)$ is representable.</p>
<p>The number of numbers $c$ such that $0 \lt c \lt ab$ which are representable correspond exactly to the number of lattice points in the region</p>
<p>$ax + by \lt ab$, $x \ge 0$, $y \ge 0$</p>
<p>This is because if $ax + by = ax' + by&#... |
2,994,962 | <p>Recently I came through the following expansion for <span class="math-container">$\log {(x + \sqrt {x^2+1})}$</span> :
<span class="math-container">$$x - \frac {1}{2}.\frac {x^3}{3} + \frac {1}{2}.\frac {3}{4}.\frac {x^5}{5} - ……$$</span>
I think I can use the Faa di Bruno formula to get a closed form for the <span ... | lab bhattacharjee | 33,337 | <p><span class="math-container">$$\dfrac{d\ln(x+\sqrt1+x^2)}{dx}=\frac{1+\dfrac x{\sqrt{1+x^2}}}{x+\sqrt1+x^2}=(1+x^2)^{-1/2}$$</span></p>
<p>Using <a href="https://en.wikipedia.org/wiki/Binomial_series" rel="nofollow noreferrer">Binomial Series</a> for <span class="math-container">$|x^2|<1,$</span></p>
<p><span c... |
2,994,962 | <p>Recently I came through the following expansion for <span class="math-container">$\log {(x + \sqrt {x^2+1})}$</span> :
<span class="math-container">$$x - \frac {1}{2}.\frac {x^3}{3} + \frac {1}{2}.\frac {3}{4}.\frac {x^5}{5} - ……$$</span>
I think I can use the Faa di Bruno formula to get a closed form for the <span ... | Robert Z | 299,698 | <p>The function <span class="math-container">$f(x)=\log {(x + \sqrt {x^2+1})}$</span> is the <a href="http://mathworld.wolfram.com/InverseHyperbolicSine.html" rel="nofollow noreferrer">inverse hyperbolic sine</a> whose expansion is
<span class="math-container">$$\sum_{n=0}^{\infty} \frac{(-1)^n (2n-1)!!}{(2n+1)(2n)!!}\... |
760,767 | <p>I don't understand the last part of this proof:</p>
<p><a href="http://www.proofwiki.org/wiki/Intersection_of_Normal_Subgroup_with_Sylow_P-Subgroup" rel="nofollow">http://www.proofwiki.org/wiki/Intersection_of_Normal_Subgroup_with_Sylow_P-Subgroup</a></p>
<p>where they say: $p \nmid \left[{N : P \cap N}\right]$, t... | Mark Bennet | 2,906 | <p>Imagine a segment of the curve along a radius from the origin of your polar co-ordinates. That increases the arc length without changing $\theta$ at all and $rd\theta=0$ for this segment. So you need to take into account the radial component.</p>
|
1,372,376 | <p>For what values of $a$ and $b$, the two functions $f_a(x)=ax^2+3x+1$ and $g_b(x)=\frac{b}{x}$ are tangent to each other at a point where the $x\text{-coordinate}=1$.</p>
<p>The points of intersection are where:
$f_a(1)=g_b(1)$</p>
<p>which gives
$$a+4=b\text{ and } b-4=a$$</p>
<p>Now what to do with this informa... | Rory Daulton | 161,807 | <p>It looks like your approach is right, and you cannot gather any further information about $a$ and $b$.</p>
<p>There are just infinitely many pairs of $a$'s and $b$'s that satisfy the requirements of the problem, and you have shown all the requirements on the $a$'s and $b$'s.</p>
<p>So you are done! So report your ... |
2,498,628 | <p>This was a question in our exam and I did not know which change of variables or trick to apply</p>
<p><strong>How to show by inspection ( change of variables or whatever trick ) that</strong></p>
<p><span class="math-container">$$ \int_0^\infty \cos(x^2) dx = \int_0^\infty \sin(x^2) dx \tag{I} $$</span></p>
<p>Co... | Zaid Alyafeai | 87,813 | <p>Note by change of variable it suffices to show </p>
<p>$$\int^\infty_0\frac{\cos(x)}{\sqrt{x}}\,dx =\int^\infty_0\frac{\sin(x)}{\sqrt{x}}\,dx $$</p>
<p>Consider the following function</p>
<p>$$f(z)=z^{-1/2}\,e^{iz}$$</p>
<p>Where we choose the principal root for $ z^{-1/2}=e^{-1/2\log(z)}$. By integrating around... |
2,498,628 | <p>This was a question in our exam and I did not know which change of variables or trick to apply</p>
<p><strong>How to show by inspection ( change of variables or whatever trick ) that</strong></p>
<p><span class="math-container">$$ \int_0^\infty \cos(x^2) dx = \int_0^\infty \sin(x^2) dx \tag{I} $$</span></p>
<p>Co... | robjohn | 13,854 | <p>Since $e^{iz^2}$ is entire, by <a href="https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem" rel="noreferrer">Cauchy's Integral Theorem</a>, we have
$$
\int_0^R e^{iz^2}\,\mathrm{d}z
=\int_0^{(1+i)R} e^{iz^2}\,\mathrm{d}z+\int_{(1+i)R}^R e^{iz^2}\,\mathrm{d}z\tag1
$$
where, using the parameterization $z=R(1+it... |
959,201 | <p>I am confused about the following.</p>
<p>Could you explain me why if $A=\varnothing$,then $\cap A$ is the set of all sets?</p>
<p>Definition of $\cap A$:</p>
<p>For $A \neq \varnothing$:</p>
<p>$$x \in \cap A \leftrightarrow (\forall b \in A )x \in b$$</p>
<p><strong>EDIT</strong>:</p>
<p>I want to prove that... | Community | -1 | <p>Usually, $\cap A$ is defined as the class of all things that are in every element of $A$.</p>
<p>No matter what $x$ is, $\forall y \in \varnothing: x \in y$ is vacuously true, therefore, <em>all</em> sets are members of the class $\cap \varnothing$.</p>
|
959,201 | <p>I am confused about the following.</p>
<p>Could you explain me why if $A=\varnothing$,then $\cap A$ is the set of all sets?</p>
<p>Definition of $\cap A$:</p>
<p>For $A \neq \varnothing$:</p>
<p>$$x \in \cap A \leftrightarrow (\forall b \in A )x \in b$$</p>
<p><strong>EDIT</strong>:</p>
<p>I want to prove that... | Thomas Andrews | 7,933 | <p>Intuitively, if $A\subseteq B$ then $\bigcap B\subseteq \bigcap A$. Now, for any set $X$, let $B=\{\{X\}\}$. Then $\emptyset = A\subseteq B$ and $\{X\}=\bigcap B \subseteq \bigcap A$, so $X\in\bigcap A$. </p>
<p>But that definition cannot actually be done - there is no set of all sets.</p>
|
25,363 | <p>In what way and with what utility is the law of excluded middle usually disposed of in intuitionistic type theory and its descendants? I am thinking here of topos theory and its ilk, namely synthetic differential geometry and the use of topoi in algebraic geometry (this is a more palatable restructuring, perhaps), w... | Charles Matthews | 6,153 | <p>I don't know whether this will be helpful, but here goes. There used to be things called the "Laws of Thought", and they used to be equated (tendentiously) with sort-of axioms for rationality, when "axiom" still meant self-evident. After Leibniz there were four basic Laws of Thought, of which you have referenced two... |
392,835 | <p>In a concrete category (i.e., where the morphisms are functions between sets), I define a <strong>base</strong> of an object <span class="math-container">$A$</span> to be a set of elements <span class="math-container">$M$</span> of <span class="math-container">$A$</span> such that for any morphisms <span class="math... | Dominique Unruh | 101,775 | <p>At least in the context of von Neumann algebras, <em>separating</em> is used for this concept. Confer [Takesaki], Definition II.3.16 (slightly reformulated):</p>
<p><strong>Definition.</strong> Let <span class="math-container">$\mathcal M$</span> be a von Neumann algebra on <span class="math-container">$\mathfrak H$... |
1,600,597 | <p>I'm currently going through Spivak's calculus, and after a lot of effort, i still can't seem to be able to figure this one out.</p>
<p>The problem states that you need to prove that $x = y$ or $x = -y$ if $x^n = y^n$</p>
<p>I tried to use the formula derived earlier for $x^n - y^n$ but that leaves either $(x-y) = ... | Américo Tavares | 752 | <p>Let $n=2p$. For convenience let us denote $y=a$. From the algebraic identities</p>
<p>\begin{eqnarray}
x^{2p}-a^{2p} &=&(x-a)\sum_{k=0}^{2p-1}a^{k}x^{2p-1-k}, \tag{1} \\
\sum_{k=0}^{2p-1}a^{k}x^{2p-1-k} &=&(x+a)\sum_{k=0}^{p-1}a^{2k}x^{2p-2-2k},\tag{2}
\end{eqnarray}</p>
<p>we conclude that</p>
<p... |
1,600,597 | <p>I'm currently going through Spivak's calculus, and after a lot of effort, i still can't seem to be able to figure this one out.</p>
<p>The problem states that you need to prove that $x = y$ or $x = -y$ if $x^n = y^n$</p>
<p>I tried to use the formula derived earlier for $x^n - y^n$ but that leaves either $(x-y) = ... | Ennar | 122,131 | <p>We have that $x\mapsto x^n\colon \mathbb R_{\geq 0} \to \mathbb R_{\geq 0}$ is strictly increasing function and thus injective. Now,</p>
<p>$$x^n = y^n \implies |x|^n = |y|^n \implies |x| = |y| \implies x=\pm y\stackrel{\text{$n$ is even}}\implies x^n = y^n$$ therefore, $x^n = y^n\iff x =\pm y$.</p>
|
2,871,949 | <p>Let $X_1, X_2, X_3, X_4$ be independent Bernoulli random variables. Then
\begin{align}
Pr[X_i=1]=Pr[X_i=0]=1/2.
\end{align}
I want to compute the following probability
\begin{align}
Pr( X_1+X_2+X_3=2, X_2+X_4=1 ).
\end{align}
My solution: Suppose that $X_1+X_2+X_3=2$ and $X_2+X_4=1$. Then $(X_2, X_4)=(0,1)$ or $(... | tortue | 140,475 | <p>Yes, your solution is indeed correct! </p>
<hr>
<p>Minor comment: In the last sentence before the final expression "this" is probably referred to each of the outcomes $(1, 0)$ and $(0, 1)$ rather than to the event $(X_1, X_3) \in \{ (1, 0), (0, 1) \}$.</p>
|
2,871,949 | <p>Let $X_1, X_2, X_3, X_4$ be independent Bernoulli random variables. Then
\begin{align}
Pr[X_i=1]=Pr[X_i=0]=1/2.
\end{align}
I want to compute the following probability
\begin{align}
Pr( X_1+X_2+X_3=2, X_2+X_4=1 ).
\end{align}
My solution: Suppose that $X_1+X_2+X_3=2$ and $X_2+X_4=1$. Then $(X_2, X_4)=(0,1)$ or $(... | Giulio Scattolin | 580,201 | <p>Let's verify your result by simulation using a <em>Python</em> script:</p>
<pre><code>import numpy as np
N = 10**5 # number of trials
# list of N 4-ples (X1, X2, X3, X4)
XX = [np.random.randint(2, size=4) for n in np.arange(N)]
# list of trials outcomes
P = list(map(lambda X: (X[0]+X[1]+X[2]==2)&(X[2]+X[3]==... |
3,069,262 | <p>Given some quadrilateral <span class="math-container">$Q \subset \mathbb R^2$</span> defined by the vertices <span class="math-container">$P_i = (x_i,y_i), i=1,2,3,4$</span> (you can assume they are in positive orientation), is there a function <span class="math-container">$f: \mathbb R^2 \to \mathbb R^2$</span> tha... | symchdmath | 626,816 | <p>This integral is more complicated than it looks and only requires comfort with hyperbolic trigonometric definitions. My initial instinct is to look at the expression inside the <span class="math-container">$\cosh$</span> to see if I can simplify it. In fact we have by the definition of the hyperbolic trigonometric f... |
4,321,675 | <p>I'm struggling to derive the Finsler geodesic equations. The books I know either skip the computation or use the length functional directly. I want to use the energy. Let <span class="math-container">$(M,F)$</span> be a Finsler manifold and consider the energy functional <span class="math-container">$$E[\gamma] = \f... | Qmechanic | 11,127 | <p>OP's red term vanishes
<span class="math-container">$$\frac{\partial g_{ik}}{\partial v^j} \ddot{x}^j\dot{x}^i ~\stackrel{\rm EOM}{\approx}~ v^i\frac{\partial g_{ik}}{\partial v^j} \dot{v}^j ~\stackrel{(C)}{=}~v^i\frac{\partial g_{jk}}{\partial v^i} \dot{v}^j ~\stackrel{(B)}{=}~0\tag{A}$$</span> because of the metr... |
1,530,848 | <p>Let $F(\mathbb{R})$ be the set of all functions $f : \mathbb{R} → \mathbb{R}$. Define pointwise addition and multiplication as follows. For any $f$ and $g$ in $F(\mathbb{R})$ let:</p>
<p>(i) $(f + g)(s) = f(x) + g(x)$ for all $x \in \mathbb{R}$</p>
<p>(ii) $(f · g)(s) = f(x) · g(x)$ for all $x \in \mathbb{R}$</p>
... | Daniel R. Collins | 266,243 | <p>Practice. </p>
<p>I might say that the general problem-solving sequence is (per Polya): (1) Read a natural-language problem carefully, (2) Translate to math equation(s); (3) Solve the equation(s); (4) Translate back to natural language and check for reasonability. </p>
<p>Now the truth is that in step #3 you usual... |
2,214,137 | <p>How many positive integer solutions does the equation $a+b+c=100$ have if we require $a<b<c$?</p>
<p>I know how to solve the problem if it was just $a+b+c=100$ but the fact it has the restriction $a<b<c$ is throwing me off.</p>
<p>How would I solve this?</p>
| Ziad Fakhoury | 295,839 | <p>Since $a$ is the smallest, the largest number it can be is $32$, so $a$ ranges from $1$ to $32$. The remaining sum $b+c$ must be equal to $100 - a$ and since $b$ is smaller than $c$ then $b$ ranges from $a+1$ to $\lfloor \frac{100-a}{2}\rfloor $. So for a given $a$ there are $\lfloor \frac{100-a}{2}\rfloor - a -1$ d... |
2,114,276 | <p>How to show that $(x^{1/4}-y^{1/4})(x^{3/4}+x^{1/2}y^{1/4}+x^{1/4}y^{1/2}+y^{3/4})=x-y$</p>
<p>Can anyone explain how to solve this question for me? Thanks in advance. </p>
| S.C.B. | 310,930 | <p>This follows from the fact that $$x^4-y^4=(x^2-y^2)(x^2+y^2)=(x-y)(x+y)(x^2+y^2)=(x-y)(x^3+x^2y+xy^2+y^3)$$
Now just replace $x,y$ with $x^{\frac{1}{4}}$ and $y^{\frac{1}{4}}$.</p>
<p>It is known, in general that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\dots+xy^{n-2}+y^{n-1})$$
As can be seen <a href="https://en.wikipedi... |
615,275 | <p>So I'm making a star Ship bridge game where the game is rendered using a 2-D Cartesian grid for positioning logic. The player has only the attributes of position and an arbitrary look-at angle (currently degrees). A "view-port" determines if a planet is within the angular difference of $45^\circ$ so that it can rend... | Michael Albanese | 39,599 | <p>First of all, you said that $\sin x = \frac{\sqrt{2}}{2}$ when $x = \frac{\pi}{4}, \frac{3\pi}{4}$ then concluded that $\sin x > \frac{\sqrt{2}}{2}$ when $\frac{\pi}{4} < x < \frac{3\pi}{4}$. While this is true, you should give some explanation here as it could be the case that $\sin x < \frac{\sqrt{2}}{... |
615,275 | <p>So I'm making a star Ship bridge game where the game is rendered using a 2-D Cartesian grid for positioning logic. The player has only the attributes of position and an arbitrary look-at angle (currently degrees). A "view-port" determines if a planet is within the angular difference of $45^\circ$ so that it can rend... | lab bhattacharjee | 33,337 | <p>We need $$2\sin\left(x-60^\circ\right)-\sqrt2>0$$</p>
<p>But as $\sin45^\circ=\frac1{\sqrt2},$ it essentially implies and is implied by $$2\sin\left(x-60^\circ\right)-2\sin45^\circ>0$$ </p>
<p>using <a href="http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html" rel="nofollow">Prosthaphaeresis Formula<... |
27,965 | <p>I'm looking at <a href="https://math.stackexchange.com/questions/2669893/calculating-the-sums-of-series">this question</a>. I gave the answer that was accepted. Please bear in mind that, when I answered this question, it was a different edit. In particular, there were more parts to the question.</p>
<p>The reason I... | Community | -1 | <blockquote>
<p>I figured that all the asker needed was one good, fully justified worked example, and they could do the rest on their own.</p>
</blockquote>
<p>Doing the OP's exercises isn't the only way to achieve this goal.</p>
<p>Instead, you could find (or create) a well-posed reference question that has the de... |
191,210 | <p>Let $R$ be the smallest $\sigma$-algebra containing all compact sets in $\mathbb R^n$.
I know that based on definition the minimal $\sigma$-algebra containing the closed (or open) sets is the Borel $\sigma$-algebra. But how can I prove that $R$ is actually the Borel $\sigma$-algebra?</p>
| William | 13,579 | <p>Let $\mathcal{B}$ denote the $\sigma$-algebra of Borel sets, i.e. the smallest $\sigma$ algebra containing the closed sets. Let $\mathcal{C}$ is $\sigma$-algebra generated by all the compact subsets of $\mathbb{R}^n$. </p>
<p>As you mentioned in your previous question, <a href="https://math.stackexchange.com/questi... |
24,318 | <p>I have an expression as below:</p>
<pre><code>Equations = 2.0799361919940695` x[1] + 3.3534325557330327` x[1]^2 -
4.335179297091139` x[1] x[2] + 1.1989715511881491` x[2]^2 -
3.766597877399148` x[1] x[3] - 0.33254815073371535` x[2] x[3] +
1.9050048836042945` x[3]^2 + 1.1386715715291826` x[1] x[4] +
2... | rcollyer | 52 | <p>I would simplify your code a bit, merging everything into the <code>Map</code> statement, and move everything into a function, as follows:</p>
<pre><code>process[func_, xvals_] :=
Block[{points},
points = Map[ With[{val = func@#}, UnitStep[val] val]&, xvals];
Transpose[{xvals, points}]
]
</code></pre>
<p... |
1,647,157 | <p>How can I solve this using only 'simple' algebraic tricks and asymptotic equivalences? No l'Hospital.</p>
<p>$$\lim_{x \rightarrow0}
\frac
{\sqrt[3]{1+\arctan{3x}} - \sqrt[3]{1-\arcsin{3x}}}
{\sqrt{1-\arctan{2x}} - \sqrt{1+\arcsin{2x}}}
$$</p>
<p>Rationalizing the numerator and denominator gives</p>
<p>$$
\lim_{x... | Disintegrating By Parts | 112,478 | <p>The orthogonal complement of $Y$ consists of all $g$ such that
$$
0 = (f,g) = \int_{-\pi}^{0}f(t)\overline{g(t)}+\int_{0}^{\pi}f(t)\overline{g(t)}dt \\
= \int_{0}^{\pi}f(t-\pi)\overline{g(t-\pi)}+f(t)\overline{g(t)}dt \\
= \int_{0}^{\pi}f(t)\overline{\{g(t-\pi)+g(t)\}}dt,\;\;\; f \in Y.
... |
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