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 |
|---|---|---|---|---|
1,129,150 | <p>I know the eigenvalues for a matrix. Let's say they are 2 and 1. How can I find the matrix A for them (all members of A are not null) ?</p>
| 2'5 9'2 | 11,123 | <p>$a$ and $a^2$ are either both even or both odd. So you don't change parity when you replace $a^2$ with $a$:
$$\begin{align}
ab(a+b)+bc(b+c)+ca(c+a)
&=a^2b+ab^2+b^2c+bc^2+ac^2+a^2c\\
&\equiv ab+ab+bc+bc+ac+ac\\
&= 2(ab+bc+ac)\\
\end{align}$$
which is even.</p>
|
2,331,657 | <p>On the generalization of a <a href="https://math.stackexchange.com/questions/2329248/to-compute-frac12-pi-i-int-mathcalc-1zz22-dz-where-mathcalc">recent question</a>, I have shown, by analytic and numerical means, that</p>
<p>$$\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2+\cdots+z^{2n}|^2~dz =2n$$</p>
<p>where $\math... | Lukas Betz | 238,388 | <p>For $z\in\mathcal C$, we have $$|1+...+z^{2n}|^2 = (1+z+...+z^{2n})(1+z^{-1}+...+z^{-2n}) = \sum_{j=0}^{2n}\sum_{k=0}^{2n}z^jz^{-k} = \sum_{j=0}^{2n}\sum_{k=0}^{2n}z^{j-k}.$$</p>
<p>Therefore $$\frac{1}{2\pi i}\int_{\mathcal C}|1+z+...+z^{2n}|^2 dz = \sum_{j=0}^{2n}\sum_{k=0}^{2n}\frac{1}{2\pi i}\int_{\mathcal C}z^... |
1,363,902 | <p>Suppose $A$ and $B$ are linear transformations on finite dimensional vector space $V$,s.t. $A,B\neq 0$ and $A^2=B^2=0$. Suppose the dimension of range $A$ and $B$ are equal, can $A$ and $B$ be similar?</p>
| walkar | 98,077 | <p>If $A^2 = 0$, then $A$ satisfies the polynomial $x^2$, and and similarly for $B$. So then the minimal polynomial for $A$ and $B$ both divide $x^2$. But since $A\neq 0$, $B \neq 0$, then $x^2$ must be the minimal polynomial for both $A$ and $B$. </p>
<p>Then the Jordan Canonical Form of $A$ and $B$ will include at l... |
206,390 | <p>I have a list of strings:</p>
<pre><code>lis = {"a","b","c","12","d","q","r","X","s"}
</code></pre>
<p>I'd like to delete list members starting with "X" moving backwards through the list from "X" until a list member that's a digit character is found, to get:</p>
<pre><code>res = {"a","b","c","12","s"}
</code></pr... | kglr | 125 | <pre><code>Replace[lis, {a___, b_String?(StringMatchQ[NumberString]), Shortest[c___], "X", d___} :>
{a, b, d}]
</code></pre>
<blockquote>
<p>{"a", "b", "c", "12", "s"}</p>
</blockquote>
|
3,809,788 | <p>Are we able to obtain the following algebraically?<span class="math-container">$$\widehat{\mathbf{x}}=\underset{\mathbf{x}}{\operatorname{argmin}}\|A-B(I\otimes \mathbf{x})\|_F^2$$</span>where
<span class="math-container">$A\in \mathbb{R}^{m\times n}$</span>, <span class="math-container">$B\in \mathbb{R}^{m\times (m... | Alejandro Sánchez Yalí | 217,786 | <p>I would do the following, define <span class="math-container">$W=I\otimes x$</span> and <span class="math-container">$h(W)=||A-BW||_{F}^{2}$</span>. Now doing <span class="math-container">$f(V)= ||V||_{F}^{2}$</span> and <span class="math-container">$g(W) = A-BW$</span> then <span class="math-container">$h(W)=f \cir... |
3,809,788 | <p>Are we able to obtain the following algebraically?<span class="math-container">$$\widehat{\mathbf{x}}=\underset{\mathbf{x}}{\operatorname{argmin}}\|A-B(I\otimes \mathbf{x})\|_F^2$$</span>where
<span class="math-container">$A\in \mathbb{R}^{m\times n}$</span>, <span class="math-container">$B\in \mathbb{R}^{m\times (m... | greg | 357,854 | <p><a href="https://en.wikipedia.org/wiki/Vectorization_(mathematics)#Compatibility_with_Kronecker_products" rel="nofollow noreferrer">Vectorize</a> the matrices which appear in the Frobenius norm, i.e.
<span class="math-container">$$\eqalign{
&{\rm vec}(A) = a \\
&{\rm vec}\Big(B(I_n\otimes x)\Big)
&= (I_... |
2,243,674 | <p>$\def\d{\mathrm{d}}$How to solve this ODE? (From a real analysis course, existence and uniqueness of ODE)
$$\frac{\d x}{\d t}=(x+t)t. \quad \forall t\in [0,1], \quad x(0)=0$$</p>
<p>My attempt:</p>
<p>$$\dot{x}=\frac{\d x}{\d t}=V(x(t),t)=(x+t)t$$</p>
<p>So we can use $\phi_v(x)$ such that</p>
<p>$$\phi_v^1(x,t)... | doraemonpaul | 30,938 | <p>Let $u=x+t$ ,</p>
<p>Then $x=u-t$</p>
<p>$\dfrac{dx}{dt}=\dfrac{du}{dt}-1$</p>
<p>$\therefore\dfrac{du}{dt}-1=tu$</p>
<p>$\dfrac{du}{dt}-tu=1$</p>
<p>I.F. $=e^{-\int t~dt}=e^{-\frac{t^2}{2}}$</p>
<p>$\therefore\dfrac{d\left(e^{-\frac{t^2}{2}}u\right)}{dt}=e^{-\frac{t^2}{2}}$</p>
<p>$e^{-\frac{t^2}{2}}u=\int_0... |
101,716 | <p><em>Disclaimer</em>: This is homework, however I am not looking for an answer. I'm only trying to understand the actual question.</p>
<p>I'm given four mutually exclusive and exhaustive events: $A$, $B$, $C$, and $D$. I'm also given $P(A)$, $P(B)$, $P(C)$ and $P(D)$. There is also some minor event $M$, for which I ... | André Nicolas | 6,312 | <p>The correct conditional probability formula is
$$P(B|M)=\frac{P(B\cap M)}{P(M)},\qquad\qquad(\ast)$$
and that is what we should start from. You are essentially assuming that $P(B\cap M)=P(B)$. That can hardly ever be true. Typically your $P(B)/P(M)$ will be greater than $1$, so cannot even be a probability. </p>
<... |
74,592 | <p>Let me denote $X_n$ the set of transpositions in $n$ elements. Equivalently, $X_n$ is the set of doubletons in $[1,n]\times[1,n]$. The cardinality of $X_n$ is $N=\frac{n(n-1)}{2}$.</p>
<p>If $f:{\mathbb Z}/N{\mathbb Z}\rightarrow X_n$ is a bijection, let us denote
$$r(f):=\min\{|\ell-m|;\ell\ne m\quad\hbox{and}\qu... | Brendan McKay | 9,025 | <p>Close to $n/2$ is possible. I'll do odd $n$ and leave even $n$ for someone else's pleasure.</p>
<p>Let $m=(n-1)/2$. For $i=0,\ldots,n-1$ and $j=1,\ldots,m$, let $M(i,j)$ be the pair $\{i-j,i+j\}$ (all values taken mod $n$, of course). The solution is
$$ M(0,1).\ldots,M(0,m),M(1,1),\ldots,M(1,m),\ldots,M(n-1,1),\... |
403,165 | <p>Suppose that $H\triangleleft G, K\le G\ $ and $K\nsubseteq H$. How we can prove that $HK=G?$ </p>
<p>Also $(G:H)=p$ where p is prime.</p>
| Josué Tonelli-Cueto | 15,330 | <p>With your modification, I give you a hint. By the index formula
$$p=(G:H)=(G:HK)(HK:H)$$
Since $p$ is prime, what can you say about the possibilities? Which one is impossible under the given assumptions?</p>
|
403,165 | <p>Suppose that $H\triangleleft G, K\le G\ $ and $K\nsubseteq H$. How we can prove that $HK=G?$ </p>
<p>Also $(G:H)=p$ where p is prime.</p>
| Ayman Hourieh | 4,583 | <p>By <a href="https://math.stackexchange.com/questions/315546/use-the-4th-isomorphism-theorem-to-show-the-index-formula">this formula</a>:
$$
[G : H] = [G : HK][HK : H]
$$</p>
<p>Since $[G : H] = p$ is prime, one of $[HK : H]$ and $[H : HK]$ must be $1$. It cannot be $[HK : H]$ because $K \not \subset H$ so $HK \ne H... |
4,276,716 | <p><span class="math-container">$$
2\tan ^{-1}\left(\sqrt{\frac{a}{b}}\tan \frac{x}{2}\right)=\sin ^{-1}\frac{2\sqrt{ab}\sin x}{\left(b+a\right)+\left(b-a\right)\cos x}
$$</span></p>
<p>I know within inverse of trigonometric function we have the value.
How Do I solve this.</p>
<p>my approach for this solution was:<br /... | Rene Schipperus | 149,912 | <p>Let <span class="math-container">$$\tan A=\sqrt{\frac{a}{b}}\tan\frac{x}{2}$$</span> and
<span class="math-container">$$\sin B=\frac{2\sqrt{ab}\sin x}{(b+a)+(b-a)\cos x}$$</span></p>
<p>You wish to show <span class="math-container">$2A=B$</span>. Calculate</p>
<p><span class="math-container">$$\sin 2A=2\sin A\cos A=... |
1,876,639 | <blockquote>
<p><span class="math-container">$\tan^{-1}x, \tan^{-1}y, \tan^{-1}z $</span> are in arithmetic progression, as are <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span>. (We assume <span class="math-container">$y \ne 0,1,-1$</span>.) Show:... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$\tan^{-1}z-\tan^{-1}y=\tan^{-1}y-\tan^{-1}x$$</p>
<p>$$\implies\tan(\tan^{-1}z-\tan^{-1}y)=\tan(\tan^{-1}y-\tan^{-1}x)$$</p>
<p>$$\iff\dfrac{z-y}{1+yz}=\dfrac{y-x}{1+xy}$$</p>
<p>As $x,y,z$ are in A.P.,$y-x=z-y$</p>
<p><strong>Case</strong>$\#1:$If If $y-x=z-y=0$</p>
<p><strong>Case</strong>$\#2... |
940,410 | <p>Using the method showed <a href="https://math.stackexchange.com/questions/939051/what-is-the-closed-form-of-sum-n-1-infty-frac-it-j-0-left-n">here</a> proposed by Olivier Oloa with simplifications proposed by Anastasiya-Romanova, it is possible to prove that</p>
<p>$$\sum _{n=1}^{\infty }{\frac {{{\it J}_{0}\left(\... | David H | 55,051 | <p><strong>(Too long for comment)</strong></p>
<p>The Bessel function of the first kind of order zero, $J_{0}{\left(z\right)}$, has the following integral representation:</p>
<p>$$J_{0}{\left(z\right)}=\frac{2}{\pi}\int_{0}^{1}\mathrm{d}t\,\frac{\cos{\left(zt\right)}}{\sqrt{1-t^2}}.$$</p>
<p>Represent the Bessel fun... |
1,562,082 | <p>Using the definition in Bartle's Introduction to Real Analysis, I am trying to gain an intuitive understanding of limits that tend to infinity.</p>
<p>Given Definition:</p>
<p>Let ($x_n$) be a sequence of real numbers.</p>
<p>(i) We say that ($x_n$) tends to $\infty$, and write $lim(x_n) = +\infty$ , if for every... | porridgemathematics | 270,179 | <p>Essentially what the definition is saying is, if you claim that a sequence ${x_n}$ is truly 'approaching infinity', if I give you a really large number, say $100000000000000000000000$ you should be able to tell me that there is a point in this sequence of numbers $x_n$, where if you take all terms of the sequence af... |
4,508,353 | <blockquote>
<p>Solve the differential equation and find its solution <span class="math-container">$$\frac{d^3y}{dx^3}=0$$</span></p>
</blockquote>
<p>I haven't learnt yet how to solve differential equations of order <span class="math-container">$3$</span>. Moreover, I don't think this can be solved as there is no vari... | Parcly Taxel | 357,390 | <p>As an equation of the form <span class="math-container">$y^{(n)}=f(x)$</span>, directly integrating three times gives the answer as all polynomial functions of degree <span class="math-container">$2$</span> or less: <span class="math-container">$Ax^2+Bx+C$</span> where <span class="math-container">$A,B,C$</span> are... |
1,662,218 | <p>Among the following, which is closest in value to $\sqrt{0.016}$?</p>
<p>A. $0.4$</p>
<p>B. $0.04$</p>
<p>C. $0.2$</p>
<p>D. $0.02$</p>
<p><strong>E. $0.13$</strong></p>
<p><strong>My Approach:</strong></p>
<p>$(\frac{16}{1000})^\frac{1}{2} = (\frac{4}{250})^\frac{1}{2} = \frac{2}{5\cdot\sqrt{10}} = \frac{\sq... | egreg | 62,967 | <p>Just square:</p>
<p>$0.4^2=0.16$</p>
<p>$0.04^2=0.0016$</p>
<p>$0.2^2=0.04$</p>
<p>$0.02^2=0.0004$</p>
<p>$0.13^2=0.0169$</p>
<p>Can you choose?</p>
|
1,197,547 | <p>If $G=A * B$ is the free product of two groups $A$ and $B$ and $g \in G-A$, then prove that $gAg^{-1} \cap A=1$.</p>
<p>We know $A \cap B=1$, so if we write $g=a_1b_1a_2b_2 \ldots a_nb_n$, does not give me sufficient road to go? How should I approach it?</p>
| David R. | 158,279 | <p>It's still possible to make $1024$ work. Do $4102 = 7 \times 586$. Now, $1024 = 4^{9 - (3 + 1)}$ and $4102 = 14 \times 293$. What I did there was rearrange the digits and move a $2$ in there, and replace the various operators with a single $\times$ sign. I doubt this would've won you the Glenfiddich, however.</p>
|
2,077,883 | <blockquote>
<p>$f$ and $g$ are functions of real variables, strictly increasing and strictly decreasing respectively on $\Bbb R$ (both surjections), I'm asked to prove that there exists at most one solution to the equation $f(x)=g(x)$.</p>
</blockquote>
<p><strong>My attempt:</strong> Suppose $f(x)=g(x)=m,$ for som... | Sil | 290,240 | <p>It seems you have already proven the part you intended in
$$f(a)-g(a)\lt f(x)-g(x)\lt f(b)-g(b)$$
just notice that $f(x)-g(x)=0$ (why?). So this way you have $f(a)<g(a)$ for $a<x$ and $f(b)>g(b)$ for $b>x$, which completes the proof (equality cannot occur).</p>
|
55,679 | <p>Let $G$ be a Lie group and let $\xi.\eta$ be left invariant vector fields. We can now construct right invariant vector fields $X_\xi$ and $X_\eta$ by defining $X_\xi(e)=\xi(e)$ and $X_\eta(e)=\eta(e)$. For $GL_n$, it is true that $[X_\xi,X_\eta]=X_{[\eta,\xi]}$. Is it true for any Lie group?</p>
| Claudio Gorodski | 15,155 | <p>For a left-invariant vector field $X$ on $G$, denote by $X^R$ the right-invariant
vector field with the same value at the identity. From $\iota_*X=\iota_*L_{g*}X=R_{g^{-1}*}\iota_*X$, we see that $\iota_*X$ is right-invariant, where $\iota$ is the inversion map $g\mapsto g^{-1}$. Since $\iota_*X(1)=-X(1)$, we get $... |
877,355 | <p>The "modern"(schematic) definition of a projective variety is the following:</p>
<blockquote>
<p>Let $k$ be an algebraically closed field. A <em>projective variety</em> over $k$ is a closed subscheme of $\mathbb P^n_k=\textrm{Proj}(k[T_1,\ldots,T_n])$ (Remember the structure of $k$-scheme).</p>
</blockquote>
<p>... | Keenan Kidwell | 628 | <p>The choice of whether to require varieties in the scheme-theoretic sense to be (geometrically) irreducible and\or reduced seems to depend on the source. Certainly a $k$-variety ($k$ a field) should be of finite type and separated. The all-mighty Stacks Project also requires integrality (reduced+irreducible), but as ... |
2,437,635 | <p>It is well know that $$\left(\sum_{r=1}^n r\right)^2=\sum_{r=1}^n r^3$$
i.e.
$$(1+2+3+\cdots+n)(1+2+3+\cdots+n)=1^3+2^3+3^3+\cdots+n^3$$
and this is usually proven by showing that the closed form for the sum of cubes is $\frac 14 n^2(n+1)^2$ which can be written as $\left(\frac 12 n(n+1)\right)^2$, and then noticin... | Thomas Andrews | 7,933 | <blockquote>
<p>Lemma: If $f(x,y)$ is a binary function, then:
$$\sum_{i=1}^{n} \sum_{j=1}^{n} f(i,j)=\sum_{k=1}^{n} \left(f(k,k)+\sum_{i=1}^{k-1}f(i,k)+\sum_{j=1}^{k-1}f(k,j)\right)$$</p>
</blockquote>
<p>The <a href="https://math.stackexchange.com/a/1305108/7933">graphical proof</a> can be written starting with ... |
194,220 | <p>Im trying to animate multiple points around multiple parametric plots which depicts orbital motion of body around a planet. Initial conditons are;</p>
<pre><code>μ = 3.986004418*10^14
a = {7.92597218162462`*^6, 7.359004757830201`*^6,
6.970300551929753`*^6}
r = {7.388961739897817`*^6, 7.352270990873303`*^6,
6.... | J. M.'s persistent exhaustion | 50 | <p>I thought it might be useful as a reference to demonstrate a side-by-side comparison of two ways to visualize <a href="https://en.wikipedia.org/wiki/Kepler_orbit" rel="nofollow noreferrer">Keplerian orbits</a> from their <a href="https://en.wikipedia.org/wiki/Orbital_elements" rel="nofollow noreferrer">orbital eleme... |
327,201 | <p>I'm reading Behnke's <em>Fundamentals of mathematics</em>:</p>
<blockquote>
<p>If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology.</p>
</blockquote>
<p>I got curious on this: Are there infinite sets of axioms? The only thing I could think about is the possible ex... | André Nicolas | 6,312 | <p>Many important theories, most significantly first-order Peano arithmetic, and ZFC, the most commonly used axiomatic set theory, have an infinite number of axioms. So does the theory of algebraically closed fields. </p>
|
4,351,449 | <p>Show that : <span class="math-container">$1 - |\phi(t)| \ge \frac{1-|\phi(2t)|}{4}$</span> where <span class="math-container">$\phi(t)$</span> is a characteristic function .</p>
<p>I am able to prove another part of the question <span class="math-container">$\Re(1-\phi(t)) \ge \Re(\frac{1-\phi(2t)}{4})$</span> . Ho... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$X$</span> is a random variable with characteristic function <span class="math-container">$\phi$</span> then for a sutable <span class="math-container">$s$</span> we have <span class="math-container">$1-|\phi (t)|=1-\phi (t)e^{is}=E(1-e^{i(tX+s)})\geq E(1-\cos (tX+s)$</span> . Now u... |
2,834,273 | <blockquote>
<p>Solve:
$\tan{4\theta} = \dfrac{\cos{\theta} - \sin{\theta}}{\cos{\theta} + \sin{\theta}}$
for acute angle $\theta$</p>
</blockquote>
<p>I need help solving that problem. I have tried to do both side, but no result yet.</p>
<p>What I have done.
$\dfrac{\sin{4\theta}}{\cos{4\theta}} = \dfrac{\cos{... | Michael Rozenberg | 190,319 | <p>We need to solve
$$\tan4\theta=\frac{\frac{1}{\sqrt2}\cos\theta-\frac{1}{\sqrt2}\sin\theta}{\frac{1}{\sqrt2}\cos\theta+\frac{1}{\sqrt2}\sin\theta}$$ or
$$\tan4\theta=\frac{\sin(45^{\circ}-\theta)}{\cos(45^{\circ}-\theta)}$$ or
$$\tan4\theta=\tan(45^{\circ}-\theta)$$ or
$$4\theta=45^{\circ}-\theta+180^{\circ},k$$ whe... |
606,158 | <p>I'm really sorry for this basic, stupid question. I have been looking for answers online but I can't find any.</p>
<p>I don't understand the following summation:</p>
<p>$$\sum_{i=0}^{n-1}i=\frac{n(n-1)}2$$</p>
<p>I understand that the basic formula for summations of i is the following:</p>
<p>$$\sum_{i=1}^ni=\fr... | Asinomás | 33,907 | <p>$2(1+2+3+4+5...+n)=(1+2+3...+n)+(n+(n-1)+....+2...+1)$. </p>
<p>Now add the first one in the first bracket with the first one of the second bracket, the second of the first with the second of the second and so on to get:</p>
<p>$(1+n)+(2+(n-1)+...(j+(n-j+1))+...+(n+1)=n(n+1)$
since there are $n$ pairs.</p>
<p>so... |
606,158 | <p>I'm really sorry for this basic, stupid question. I have been looking for answers online but I can't find any.</p>
<p>I don't understand the following summation:</p>
<p>$$\sum_{i=0}^{n-1}i=\frac{n(n-1)}2$$</p>
<p>I understand that the basic formula for summations of i is the following:</p>
<p>$$\sum_{i=1}^ni=\fr... | Bill Dubuque | 242 | <p>$\begin{eqnarray}{\bf Hint}\quad f_1 + \cdots + f_n &=&\, -f_0 + (\color{#c00}{f_0\! +\cdots +\! f_{n-1}}) + \color{#0a0}{f_n}\\ \\ &=&\quad 0 \,\ +\ \color{#c00}{\dfrac{n(n\!-\!1)}2}\ +\ \color{#0a0}n\\ &=&\qquad\quad\ \dfrac{n(n\!+\!1)}2\end{eqnarray}$</p>
|
147,379 | <p>I would like to prove by induction the following inequality:</p>
<p>$\frac{4^n}{n+1} < \binom{2n}{n}$, for all natural numbers n > 1.</p>
<p>Any hints?</p>
| Phira | 9,325 | <p>You know that $$\binom{2(n+1)}{n+1} = \frac{2(n+1)(2n+1)}{(n+1)^2}\binom{2n}{n}.$$</p>
<p>The main step of the induction proof is therefore</p>
<p>$$\frac{4^n}{n+1} \frac{2(n+1)(2n+1)}{(n+1)^2} \ge \frac{4^{n+1}}{n+2}.$$</p>
<p>This inequality is easy to show after simplification.</p>
|
20,659 | <p>When memorizing and recalling the times table, I learned to say "six sevens are forty-two", and always wondered what it would be like to learn to say "six times seven equals forty-two" and whether it would be harder. Likewise, of course, with all the other ones e.g. "seven eights are fifty-s... | ryang | 372 | <p><em>Not a proper acceptable answer, just an expansion of my original comment:</em></p>
<p>A literal/direct/mechanistic recitation probably involves a lighter cognitive load than a quirkier sentence-translated recitation. So:</p>
<ul>
<li>one times five equals five</li>
<li>six times seven equals forty-two</li>
<li>e... |
3,496,299 | <p>I used limit of the function at zero, and got that the limit is zero. So I said, while the limit existed and it is finite then the singularity is Removable Singularity. My function is <span class="math-container">$$f(z)=\frac{e^{z^{-1}}}{\sin(z^{-1})}$$</span> I think that I had something wrong while calculating the... | Taladris | 70,123 | <p>Assume that <span class="math-container">$\lim\limits_{z \to 0}\frac{e^{z^{-1}}}{\sin(z^{-1})}$</span> exists. </p>
<p>Then <span class="math-container">$\lim\limits_{t \to \infty}\frac{e^{t}}{\sin(t)}$</span> also exists. But this latter limit does not exist, since <span class="math-container">$f(t)=\frac{e^{t}}{\... |
2,208 | <p>What are some general guidelines for editing a MathOverflow post? Are there some "do"s and "don't"s I should know about? </p>
<p>(The material that was previously in this question box has been moved to an answer box, where it belongs, in response to a pertinent comment by Noam Elkies.) </p>
| Todd Trimble | 2,926 | <p>Most of the time, the edits that I see on MO are respectfully and tactfully performed and small in scope, and gratefully received by the post's author as improvements. Occasionally though I see flare-ups over an edit, to the point where a post will be self-mutilated or self-deleted; (more rarely) there will be an ed... |
89,407 | <p>Can users of <em>Mathematica</em> on Windows 7 or Windows 8/8.1 upgrade their computers to Windows 10 – a process that began today? Does <em>Mathematica</em> work? Are all the settings, licenses, access to files and folders preserved? Is there anything one must be careful about?</p>
| Arnoud Buzing | 105 | <p>I upgraded three machines from Windows 7 to Windows 10, two of which were work machines containing installations of Mathematica 10.x (and earlier). So far I have encountered zero issues related to Mathematica with this Windows upgrade. From a personal perspective, upgrading to Windows 10 was very simple.</p>
<p>Als... |
3,144,546 | <p>As part of my attempt at solving <a href="https://math.stackexchange.com/questions/3140495/integral-involving-the-logarithm-of-trigonometric-function/3142176?fbclid=IwAR2AX0PmeNzOhLxNLSlOgS3hGW8rY42A-rfhU_4ImH-2uJFSxIu6A336QmM#3142176">this</a> integral, I became stuck with resolve the following definite integral:
<... | Claude Leibovici | 82,404 | <p>Based on series expansion, starting from the usual expansion of <span class="math-container">$\frac{\tan(x)} x$</span>, you have (using the generalized binomial expansion)
<span class="math-container">$$\left(\frac{\tan (u)}{u}\right)^a=1+\frac{a }{3}u^2+\left(\frac{a^2}{18}+\frac{7 a}{90}\right)
u^4+\frac{\left(... |
320,589 | <p>The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas... | Thomas Kalinowski | 12,674 | <p><a href="https://en.wikipedia.org/wiki/K%C5%91nig%27s_theorem_(graph_theory)" rel="nofollow noreferrer">Kőnig's theorem</a> and <a href="https://en.wikipedia.org/wiki/Hall%27s_marriage_theorem" rel="nofollow noreferrer">Hall's marriage theorem</a> are famous and follow from LP duality (together with an integrality a... |
1,261,819 | <p>I found lots of examples of ADMM formalization of equality constraint problems (all with single constraint). I am wondering how to generalize it for multiple constraints with mix of equality and inequality constraints.</p>
<p>I have a problem</p>
<blockquote>
<p>Minimize $||A_x||_1 + \lambda ||A_y||_2 $, such th... | Tobias | 222,993 | <p>One way to formulate the problem using ADMM is to let the ADMM-variable $X$ contain $A_x$ and $A_y$, i.e. $X = [A_x; A_y]$ (semi-colon denotes stacking, as in Matlab etc.), and let $Z=[Z_1; Z_2; Z_3; Z_4]$ contain four blocks corresponding to $A_x$, $A_x$, $A_x$ and $A_y$ respectively. (I will write $Q$ for $X-I$, w... |
3,254,331 | <blockquote>
<p>Prove <span class="math-container">$$\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$$</span></p>
</blockquote>
<p>Out of boredom, I decided to play with some integrals and Inverse Symbolic Calculator and accidentally found this to my surprise</p>
<p><span class="math-container">$... | auscrypt | 675,509 | <p>Note that <span class="math-container">$\arctan \frac{1}{x} = \frac{\pi}{2} - \arctan x$</span> (simply draw a triangle with side <span class="math-container">$1$</span> and <span class="math-container">$x$</span> and consider the two angles). We then obtain <span class="math-container">$$I = \int_0^\infty \left( \a... |
3,254,331 | <blockquote>
<p>Prove <span class="math-container">$$\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$$</span></p>
</blockquote>
<p>Out of boredom, I decided to play with some integrals and Inverse Symbolic Calculator and accidentally found this to my surprise</p>
<p><span class="math-container">$... | xpaul | 66,420 | <p>Let
<span class="math-container">$$ I(a,b)=\int_0^\infty\left(\arctan \frac ax\right)\left(\arctan \frac bx\right) \mathrm d x. $$</span>
Then
<span class="math-container">\begin{eqnarray}
\frac{\partial^2I(a,b)}{\partial a\partial b}&=&\int_0^\infty\frac{x^2}{(x^2+a^2)(x^2+b^2)}\mathrm d x\\
&=&\fra... |
604,051 | <p>Let $A \to B$ be a surjective homomorphism between (unital) noetherian commutative rings with the same Krull dimension. Is the kernel of this map nilpotent ?</p>
<p>Thanks to Makoto Kato and Martin Brandenburg, it seems that the answer to the question is trivially false.</p>
<p>Now assume $A$ is a quotient formal ... | Georges Elencwajg | 3,217 | <p>Geometrically, you are asking whether a closed immersion $\operatorname {Spec} B\hookrightarrow \operatorname {Spec} A$ of affine schemes of the same dimension is surjective.<br>
This can be <strong>seen</strong> to be false (algebraic geometry is a visual art!): just inject the $x$-axis of the plane $\mathbb A^2$ ... |
3,291,889 | <p>All rings are commutative ring with unity.</p>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are two <span class="math-container">$R$</span>-algebras and <span class="math-container">$I$</span> and <span class="math-container">$J$</span> are two ideals of <span class="ma... | David K | 139,123 | <p>Given only the few facts you are allowed to use, there is some ambiguity concerning what the value of <span class="math-container">$\cos(\pi/3)$</span> might be. </p>
<p>As far as I understand, you are supposed to consider two functions <span class="math-container">$f$</span> and <span class="math-container">$g$</s... |
764,508 | <p>I can't solve this. Can any one show me detail solution
$$
\sqrt{x^2+x+1}+x^3=\sqrt{2x+2}+x^2+x
$$</p>
| String | 94,971 | <h2>The first order derivatives</h2>
<p>For any $y\in(c,d)$ we have a unique $x=f^{-1}(y)\in(a,b)$ and by assumption we know that $f'(x)=\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}=k$ is well defined and non-zero.</p>
<p>The next part can be phrased more technically via epsilon-delta arguments, but basicall... |
92,743 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/86263/characteristic-of-a-field-is-0-or-prime">Characteristic of a field is $0$ or prime</a> </p>
</blockquote>
<p>Is there any field of characteristic 4? Or any other composite number? </p>
| Dilip Sarwate | 15,941 | <p>This is a small variation on Fredrik Meyer's answer.</p>
<p>$0$ and $1 \neq 0$ are the additive and multiplicative identities in a field
$\mathbb F$. </p>
<ul>
<li><p>If $1$, $1+1$, $1+1+1, \cdots$ are all distinct elements of $\mathbb F$,
then the <em>characteristic</em> of $\mathbb F$ is said to be $0$. Note th... |
4,046,685 | <p>Clearly, the total number of subsets possible is <span class="math-container">$2^5$</span></p>
<p>For two elements to be common, both subsets need to have at least two elements, so we can form quite a lot of cases which satisfy both conditions.</p>
<p>Now there are far too many cases (IMO) for me to manually curate,... | Brian M. Scott | 12,042 | <p>There are <span class="math-container">$\binom52=10$</span> two-element subsets of <span class="math-container">$\{1,2,3,4,5\}$</span>, and every pair <span class="math-container">$\langle A,B\rangle$</span> of subsets of <span class="math-container">$\{1,2,3,4,5\}$</span> whose intersection has exactly <span class=... |
229,915 | <p>Let $A,B\subseteq\mathbb R^d$ with $A$ closed such that $A\subset\overline{B}$. Does there exist $B'\subset B$ such that $A=\overline{B'}$?</p>
| Community | -1 | <p>Sorry, I misread the question initially.</p>
<p>The answer is no in general. For example, take $B$ to be plane minus $x$-axis, and $A$ to be $x$-axis. If $B'$ exists, it must be a subset of both $A$ and $B$, which is empty.</p>
|
3,506,533 | <blockquote>
<p>Let <span class="math-container">$X,Y$</span> be Banach space and <span class="math-container">$T:X\to Y$</span> bounded . If <span class="math-container">$T(X)$</span> has finite dimensional, then <span class="math-container">$T$</span> is compact. </p>
</blockquote>
<p>Assume <span class="math-con... | ncmathsadist | 4,154 | <p>Let <span class="math-container">$B$</span> be the closed unit ball of <span class="math-container">$X$</span>. Then, <span class="math-container">$T(B)$</span> is a bounded subset of a finite-dimensional space and is therefore precompact. Therefore, <span class="math-container">$T$</span> is compact. </p>
|
1,314,346 | <p>I'm learning about generating functions, and one part that I am struggling with is the logic behind rearranging summations (particularly double summations).</p>
<p>I'll illustrate an example:</p>
<p>Using the Lagrangean Inversion theorem, I get that the $z^n$ coefficient is given by:
$$S_n=\frac{1}{n}[u^{n-1}]\le... | Bobby | 10,809 | <p>Let $ABC$ be the triangle, with $a = |BC|$ and $b = |AC|$. The angles opposite these sides are $\alpha =\angle CAB= \angle ABC$. </p>
<p>Now assume the contrary $a < b$. Choose point P on AC such that $|PC|= |BC|= a$ and let $\omega$ be the angle of $CPB$. Thus $\angle CPB = \angle CBP = \omega$. </p>
<p>We kn... |
347,186 | <p>I have this question:</p>
<blockquote>
<p>Let $x, n$ be integers with $n \geq 2$ and $n$ not dividing $x$. Show that the order o($\bar{x}$) of $x \in Z_n$ is
$o(\bar{x})= \frac{n}{HCF(x, n)}$</p>
</blockquote>
<p>I've been thinking about it for ages but I still don't get why. A hint would be appreciated.</p>
| Hanul Jeon | 53,976 | <p>Since
$$\sum_{n=1}^N 2^{1/n}-2^{1/(n+1)} = 2-2^{1/(N+1)}$$
and
$$\lim_{N\to\infty} 2^{1/(N+1)} =1,$$
so this series is convergent and converges to 1.</p>
|
2,363,236 | <p>This may seem like a stupid question but</p>
<p>What method can one use to convert a decimal number, such as $0.672$ into a whole number?</p>
<p>This isn't rounding as $0.999$ should result in $0$, but saying "floor the number" isn't an answer as I'm looking for a mathematical method in which a number, $n$, can be... | Community | -1 | <p>Rational numbers and irrational numbers are <em>real</em> numbers. </p>
<p>For any $x,y\in{\bf R}$ with $x<y$, there exists rational $a$ and irrational $b$ such that $a,b\in(x,y)$ by <em>density</em>:</p>
<p><a href="https://math.stackexchange.com/q/1027970/9464">What does it mean for rational numbers to be &qu... |
4,071,878 | <p><span class="math-container">$|\sin(a)|=\cos(3a)$</span> is an alternative version of an equation <span class="math-container">$\sqrt{1-x^2}=4x^3-3x$</span>, where I made a substitution <span class="math-container">$x=\cos(a)$</span> for <span class="math-container">$x \in [-1, 1]$</span>. Unfortunately, I have no i... | Koro | 266,435 | <p>The problem with your approach is that <span class="math-container">$f$</span> is not differentiable at all points of <span class="math-container">$[-1,1]$</span>.</p>
<p>Since domain is <span class="math-container">$[-1,1]$</span>, let <span class="math-container">$x=\cos \theta$</span>, where <span class="math-con... |
31,085 | <p>This was asked by my maths lecturer a couple of years ago and ive been wracking my brains ever since:</p>
<blockquote>
<p>Find a number that, when multiplied by
99 will give the original number but
with a 1 at the beginning and a 1 at
the end.</p>
<p>For example: 42546254 * 99 would equal
1425462541 ... | Hans Lundmark | 1,242 | <p>A down-to-earth way of finding the answer is to write the equation as $100x - x = 1x1$,
or $1x1 + x = 100 x$. So if $x=abc\dots xyz$, then</p>
<pre><code> 1abc...xyz1
+ abc...xyz
------------
=abc...xyz00
</code></pre>
<p>This means that the last digit $z$ must be 9, so we now have</p>
<pre><code> 1
1a... |
31,085 | <p>This was asked by my maths lecturer a couple of years ago and ive been wracking my brains ever since:</p>
<blockquote>
<p>Find a number that, when multiplied by
99 will give the original number but
with a 1 at the beginning and a 1 at
the end.</p>
<p>For example: 42546254 * 99 would equal
1425462541 ... | Bill Dubuque | 242 | <p>It was shown that $\rm\ x = 112359550561797752809\:.$<br>
Notice that $1/89\ =\ 0.0112359550561797752808988\ldots$</p>
<p><strong>EXERCISE</strong> $\: $ Explain it (this, perhaps, is <em>the point</em> of the OP).</p>
<p><strong>NOTE</strong> $\ $ This is closely connected with fibonacci numbers. Hint: </p>
<p>... |
2,679,173 | <p>The question is:</p>
<p>A sports club has 3 departments, tennis, squash and badminton. We get
given the following information.</p>
<p>• 90 people are members of the tennis department.</p>
<p>• 60 people are members of the squash department.</p>
<p>• 70 people are members of the badminton department.</p>
<p>• 25... | user061703 | 515,578 | <p>The idea of setting the Venn diagram (mentioned before) is very good, but I will modify it a bit:</p>
<p>Set it to $7$ regions like below ($T$ is for tennis, $S$ is for squash, $B$ is for badminton)</p>
<p>Assume the number of people in each region is $a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$. I will state again... |
1,056,105 | <p>I was reading the section of Ravi Vakil's Algebraic Geometry notes where he discusses elliptic curves.</p>
<p>If we let an elliptic curve be $(E,p)$ (Where $p$ is the distinguished point), we have $\mathcal{O}(3p)$, which has $3$ sections, which gives us a closed embedding into $\mathbb{P}^2$. In the next sentence ... | tracing | 200,415 | <p>The degree of the line bundle is $3$. This means that a generic section of $\mathcal O(3p)$ has $3$ zeroes. This means that a generic line in $\mathbb P^2$ intersects the image of the curve in $3$ points, which means the image has degree $3$. </p>
<p>This kind of computation is part of the basic package related ... |
2,253,752 | <p>The ultrafilter lemma says: <em>Every filter $F$ is contained in a ultrafilter.</em></p>
<p>Question: Is this ultrafilter unique? Or can one find a filter $F$ such that there are several ultrafilters that contain $F$?</p>
| Asaf Karagila | 622 | <p>It is generally the case that it is not unique. Consider the following theorem.</p>
<blockquote>
<p><strong>Theorem.</strong> Let $\cal F$ be a filter on a set $X$. The following are equivalent:</p>
<ol>
<li>$\cal F$ has a unique extension to an ultrafilter on $X$.</li>
<li>$\cal F$ is an ultrafilter on ... |
1,311,023 | <p>The question:</p>
<p>Let $\gamma$ be a contour such that $0 \in I(\gamma),$ where $I$ is the interior of the contour. Show that </p>
<p>$$\int_\gamma z^n \, \text{d}z = \begin{cases} 2\pi i & \text{if } n = -1 \\ 0 & \text{otherwise} \end{cases}$$</p>
<p>By taking $\gamma$ as the ellipse</p>
<p>$$\{ (x,... | Chappers | 221,811 | <p>Change variables to $u=-\log{x}$. Then the limits become $\infty$ and $0$, and $x=e^{-u}$, $dx = -e^{-u} du$, so the integral is, swapping the limits,
$$ \int_0^{\infty} u^{-1/2} e^{-u} \, du = \Gamma(1/2) $$</p>
<p>Your original was in fact the form that Euler originally considered for the $\Gamma$-function, by th... |
2,036,533 | <p>I am trying to understand how $\mathrm W$ is an equivalence relation.</p>
<p>Let $A = \{1,2,3,4,5,6,7\}$ and $B = \{1,2,3,4\}$. </p>
<p>Let $\mathrm W$ be the relation on $P(A)$ defined by: \begin{equation}
\forall X, Y \in P(A), X \mathrm R Y \Leftrightarrow |X \cap B| = |Y \cap B|
\end{equation}</p>
| David | 119,775 | <p>(Answering your comment.) Yes, you are missing something. What you are missing is that this is a relation between <strong>sets</strong>, not between individual elements. So the statement
$$X\ {\rm R}\ Y$$
only makes sense (whether true or false) if $X$ and $Y$ are sets. It does not make sense, and is irrelevant ... |
2,337,692 | <p>Question:</p>
<p>$ Prove\ for\ all\ natural\ n:\ \frac{1}{n+1}+\frac{1}{n+2}+\ldots +\ \frac{1}{3n+1}>1 $</p>
<p>I know that the base case holds. I.H: Assume it is true for $n = k$. Now I am not sure how to prove it for $n = k+1$.</p>
| farruhota | 425,072 | <p>For $n=1$, it is $\frac12+\frac13+\frac14>1.$</p>
<p>Assume it is true for $n=k.$</p>
<p>We'll prove for $n=k+1:$</p>
<p>$$\frac{1}{k+2}+\cdots+\frac{1}{3k+1}+\frac{1}{3k+2}+\frac{1}{3k+3}+\frac{1}{3k+4}=$$
$$\underbrace{\frac{1}{k+1}+\cdots+\frac{1}{3k+1}}_{>1}-\frac{1}{k+1}+\frac{1}{3k+2}+\frac{1}{3k+3}+\... |
2,232,779 | <p>So, the concept of an average truly is somewhat abstract. Most statisticians define it as a "measure of central tendency." Others say it is the "center of gravity" for a set of numbers.</p>
<p>I personally prefer a slightly more concrete explanation: A statistic that describes the "typical", or better yet, "represe... | David G. Stork | 210,401 | <p>There are only two integer partitions of $10$ of length 7 formed from the digits $1,2,3$: $\{ 3,2,1,1,1,1,1 \}$ and $\{ 2,2,2,1,1,1,1 \}$. The number of ways you can form numbers from the first set can be computed using a multinomial:</p>
<p>${7! \over 1! 1! 5!} = 42$</p>
<p>and for the second set is</p>
<p>${7... |
2,232,779 | <p>So, the concept of an average truly is somewhat abstract. Most statisticians define it as a "measure of central tendency." Others say it is the "center of gravity" for a set of numbers.</p>
<p>I personally prefer a slightly more concrete explanation: A statistic that describes the "typical", or better yet, "represe... | Vishnu V.S | 397,349 | <p>The number of solutions is the coefficient of $x^{10}$ in $(x+x^2+x^3)^7$ which is the coefficient of $x^3$ in $(1-x^3)^7(1-x)^{-7}$. Only the first two terms : $1$ and $-7x^3$ are of relevance to us since the other terms have higher powers of $x$. If we select $1$, we'd have to select the coefficient of $x^3$ in $(... |
164,043 | <blockquote>
<p>Let <span class="math-container">$f(x)=x^n+5x^{n-1}+3$</span> where <span class="math-container">$n\geq1$</span> is an integer. Prove that <span class="math-container">$f(x)$</span> can't be expressed as the product of two polynomials each of which has all its coefficients integers and degree <span cl... | Yai0Phah | 23,875 | <p>The notation $[x^t]P(x)$ denotes the coefficient of $x^t$ in polynomial $P(x)$.</p>
<p>Suppose that $f(x)=g(x)h(x)$, where $g(x),h(x)$ are monic polynomials, $g(x),h(x)\in\Bbb Z[x]$ and $\deg g>0$, $\deg h>0$, $\deg g+\deg h=n$. Let $g(x)=\sum_k\alpha_kx^k$, $h(x)=\sum_k\beta_kx^k$, where $\alpha_k,\beta_k$ i... |
638,529 | <p>Find, using the power series: $$y(x)=\sum_{k=0}^\infty a_{k}x^k$$ a solution for the following differential equation: $$y'(x) = -x^2y(x),\,\, y(0)=1$$
What's the convergence radius of the constructed power series? Also give a closed formula.</p>
<p>So far I've come up with $\sum_{k=0}^\infty a_{k+1}(k+1)x^{k} = \su... | TonyK | 1,508 | <p>You can compare the coefficients here too, because $\sum_{k=0}^\infty -a_{k}x^{k+2} = \sum_{k=2}^\infty -a_{k-2}x^k$. (If you haven't seen this before, just write out the first few terms of each side to see why.)</p>
<p>So if we set $a_{-2} = a_{-1} = 0$, we get the recurrence $a_{k+1}(k+1) = -a_{k-2}$ for $k \ge 0... |
3,578,788 | <p>I am struggling with finding the limit:</p>
<p><span class="math-container">$$\lim_{x\to 11} \left(\frac{x}{11}\right)^{\frac{(x-13)\cdot (x-12)}{x-11}}$$</span></p>
<p>I've tried countless methods such as turning it into the form of <span class="math-container">$a^x \to e^{\ln(a^x)}$</span> and yet i didn't manag... | vonbrand | 43,946 | <p>Take logarithms, use l'Hôpital:</p>
<p><span class="math-container">$\begin{align*}
\lim_{x \to 11} \ln \left(\frac{x}{11}\right)^{\frac{(x - 13) (x - 12)}{x - 11}}
&= \lim_{x \to 11} \frac{(x - 13) (x - 12)}{x - 11} \ln \frac{x}{11} \\
&= \lim_{x \to 11} (x - 13) (x - 12) \frac{\ln x / 11}{x - 11... |
3,187,756 | <p>I looked for answers on how to do this on this on this site and couldn't find anything answering this question. Is this what a line integral is used for or is that only to find area under a function f(x,y) along a curve C (on xy-plane for example)?</p>
| uniquesolution | 265,735 | <p>The integral <span class="math-container">$\int_1^{\infty}\frac{1}{\sqrt{x^3+x+1}}\,dx$</span> converges, so for every <span class="math-container">$\varepsilon$</span> there exists <span class="math-container">$M>0$</span> such that <span class="math-container">$$\int_M^{\infty}\frac{1}{\sqrt{x^3+x+1}}\,dx<\v... |
711,802 | <p>Can't find any proof in Shannon's 1948 paper. Can you provide one or disproof?</p>
<p>Thank you.</p>
<p>P.S.</p>
<p>$H(x)$(or $H(y)$) is the entropy of messages produced by the discrete source $x$(or $y$).</p>
<p>$H(x,y)$ is the joint entropy.</p>
<p>They are all entities in information theory.</p>
| Johan Wikström | 10,099 | <p>No it doesn't have to.</p>
<p>$H(X,Y) = H(X) + H(Y|X)$</p>
<p>To lower $H(X,Y)$ while keeping $H(X)$ fixed, you need to lower $H(Y|X)$. You can lower $H(Y|X)$ without lowering $H(Y)$ since $0 \leq H(Y|X) \leq H(Y)$ is a measure on how dependent X and Y is. If they are more dependent, there will be less entropy lef... |
244,959 | <p>I'm trying to do this.
it doesn't have to be the same.</p>
<p><a href="https://i.stack.imgur.com/BjB5I.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BjB5I.png" alt="enter image description here" /></a></p>
| kglr | 125 | <p><a href="https://i.stack.imgur.com/2Gxil.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2Gxil.gif" alt="enter image description here" /></a></p>
<p>Use two functions of your choice:</p>
<pre><code>f1[x_] := -x - Piecewise[{{2 Sin[x ], -2 Pi <= x <= 2 Pi}}]
f2[x_] := x + Piecewise[{{2 Sin[2 ... |
720,084 | <p>I read in Gittman, Hamkins, et al, that ZFC without the power set causes the axiom of replacement to fail. Yet I also read (generally, throughout the literature, but mostly in connection with Cantor's theorem) that the power set is generated by <em>replacing</em> one set with another. Is the term replacement in the ... | sidneimv | 134,984 | <p>$$\ln\left|\frac{x}{x+\cos x}\right|$$</p>
<p>Just a minute and I'll bring you the proof.</p>
|
4,291,509 | <p>Let <span class="math-container">$ν$</span> denote counting measure and <span class="math-container">$λ$</span> denote Lebesgue measure. What is
<span class="math-container">$(ν ⊗λ)(\{(x,x)\})_{x∈\mathbb{R}}$</span>?</p>
<p>I am a little fuzzy on the product measure and have tried two ways to do this. Can someone ex... | FirstName LastName | 741,169 | <p>Using geometry (Pythagoras) : a rectangular triangle with hypotenuse 1 and other sides a and b. For sure: a + b > 1 (as is so for <strong>any</strong> 'non flat' triangle).</p>
<p>Agreed: not really proof by contradiction or contrapositive (except in that a rectangular triangle is never 'flat')</p>
<p>discrete ma... |
4,291,509 | <p>Let <span class="math-container">$ν$</span> denote counting measure and <span class="math-container">$λ$</span> denote Lebesgue measure. What is
<span class="math-container">$(ν ⊗λ)(\{(x,x)\})_{x∈\mathbb{R}}$</span>?</p>
<p>I am a little fuzzy on the product measure and have tried two ways to do this. Can someone ex... | Mohammad Riazi-Kermani | 514,496 | <p>If <span class="math-container">$$a+b \le 1$$</span> upon squaring both sides we get, <span class="math-container">$$a^2+b^2+2ab\le 1$$</span>
That implies <span class="math-container">$$1+2ab\le1$$</span> or <span class="math-container">$$2ab\le 0$$</span>
Which is impossible due to <span class="math-container">$0&... |
359,903 | <p>I've a doubt on how do we prove this kind of stuff involving infinite intersections. My point is: the book I'm working with gives the following example to prove that the intersection of infinite sets may not be open: let $a \in \mathbb{R}^n$, given the familly of balls $B(a; 1/k)=\left\{x \in \mathbb{R}^n \mid d(x,a... | Abel | 71,157 | <p>Formally, if we have some collection $\mathcal{C}$ of sets, then
$a\in\bigcap_{S\in\mathcal{C}}S$ if and only if $a\in S$ for every $S\in\mathcal{C}$.</p>
<p>Thus, in your example, clearly $a\in B(a,\frac{1}{k})$ for every $k$, hence $a\in\bigcap_{k=1}^\infty B(a,\frac{1}{k})$. Furthermore, for any $b\neq a$, we ha... |
3,062,765 | <p>i’d like to calculate Fourier coefficients of <span class="math-container">$\cos 2 \pi f_0 t$</span>.
This is what I did : </p>
<p><span class="math-container">$$ c_k = \frac{1}{T_0}\int_{0}^{T} \cos 2 \pi f_0 t \cdot e^{-2i\pi f_0 t}. $$</span></p>
<p>From Euler formulas:</p>
<p><span class="math-container">... | caverac | 384,830 | <p>The problem start in you very first expression</p>
<p><span class="math-container">$$
c_k = \frac{1}{T}\int_0^T \cos 2\pi f_0 t \cdot e^{-2\pi i \color{red}{k}t / T}~{\rm d}t
$$</span></p>
<p>and now do the same trick you did</p>
<p><span class="math-container">\begin{eqnarray}
c_k &=& f_0\int_0^{1/f_0} \... |
1,438,689 | <p>It is well known that you can use forcing to change the truth value of various sentences ($CH$, $\Diamond$, et cetera). However, often when performing such a construction over a model $V$, the action is generally in $V[G]$ (or some inner model thereof - as when we violate $AC$ in $HOD(x)^{V[G]}$); we want to figure ... | hot_queen | 72,316 | <p>Examples using generic ultrapowers:</p>
<p>(1) (Gititk, Shelah) If there is a total extension of Lebesgue measure, then there is a Sierpinski set (non Lebesgue null set of size $\aleph_1$ each of whose null subsets is countable).</p>
<p>Proof sketch: Force with the null ideal of the total extension and let $j: V \... |
1,438,689 | <p>It is well known that you can use forcing to change the truth value of various sentences ($CH$, $\Diamond$, et cetera). However, often when performing such a construction over a model $V$, the action is generally in $V[G]$ (or some inner model thereof - as when we violate $AC$ in $HOD(x)^{V[G]}$); we want to figure ... | Andrés E. Caicedo | 462 | <p>My favorite family of examples come from the partition calculus. The original proof of the Baumgartner-Hajnal theorem $\omega_1\to (\alpha)^2_n$ for all finite $ n $ and all countable $\alpha $ appealed to the absoluteness of well-foundedness. The homogeneous set was found in an extension where $\mathsf{MA} $ holds,... |
2,459,579 | <p>I ran into a difficult question today as I was trying to find the matrix exponential for a matrix that has a determinant of $0$. Here is the matrix: </p>
<p>$$C = \begin{bmatrix}
1 & 1 \\
-1 & -1 \\
\end{bmatrix} $$</p>
<p>I got only one eigenvalue from the characteristic polynomial, that eigenvalu... | carmichael561 | 314,708 | <p>Note that $C$ is nilpotent:
$$ C^2=\begin{bmatrix}1&1\\-1&-1\end{bmatrix}\begin{bmatrix}1&1\\-1&-1\end{bmatrix}=0$$
Therefore $e^C$ can be computed directly from the definition:
$$ e^C=I+\sum_{k=1}^{\infty}\frac{C^k}{k!}=I+C=\begin{bmatrix}2&1\\-1&0\end{bmatrix}$$</p>
|
1,662,646 | <p>I'm having a problem finding the inverse of $y=2x^2-12x+13$.
At the end I get to the following: </p>
<p>$$x=3 \pm \frac{\sqrt{40+8y}}{4}$$</p>
<p>As far as I know the answer is suppose to be $x= 3 \pm \frac{\sqrt{y+5}}{\sqrt{2}}$
but I am unable to get to it.</p>
<p>Please can someone help me with the process of ... | gammatester | 61,216 | <p>$$\frac{\sqrt{40+8y}}{4} = \frac{\sqrt{8}\sqrt{5+y}}{4}=
\frac{\sqrt{8}\sqrt{5+y}}{\sqrt{16}}=\frac{\sqrt{5+y}}{\sqrt{2}}$$</p>
|
959,742 | <p>For a function $f(x) = x\sqrt{2+x^2}$ find out if it's bijective and if so, find its inverse.</p>
<p>The function is surjective because $x^2 > 0:\forall x\in\mathbb{R}$. I'm having difficulties proving that the function is injective.</p>
<p>I tried the following: $$f(x) = f(y) \iff x\sqrt{2+x^2} = y\sqrt{2+y^2}... | egreg | 62,967 | <p>The condition $x^2\ge0$, which implies $2+x^2>0$ just tells you that the function is defined on $\mathbb{R}$, but not that it is surjective. For instance, the function
$$
g(x)=\frac{1}{\sqrt{2+x^2}}
$$
is not surjective if considered as $g\colon\mathbb{R}\to\mathbb{R}$, because $0<g(x)\le\frac{1}{\sqrt{2}}$ fo... |
96,576 | <p>I will call two graphs <span class="math-container">$G$</span> and <span class="math-container">$H$</span>, <span class="math-container">$r$</span>-equidecomposable (in analogy with <a href="https://en.wikipedia.org/wiki/Hilbert%27s_third_problem" rel="nofollow noreferrer">Hilbert's third problem</a>) if they can be... | Gerhard Paseman | 3,402 | <p>If Joseph is going to take this as an opportunity to show some nice and illustrative pictures, I am going to do similarly, but using words instead. Gjergji probably already knows what I am going to say, but others might find the remarks a useful stepping stone to the subject.</p>
<p>My initial thought was using a ... |
1,917,484 | <p>Let $f:X\rightarrow \mathbb{C}$ be an integrable function and $g_n:X\rightarrow \mathbb{C}$ be a sequence of integrable functions so that $\|g_n\|_1\rightarrow 0$ and $|g_n(x)|\leq 1$ for every $n, x$. Show that $\|fg_n\|_1\rightarrow 0$.</p>
<p>I think the $|g_n(x)|\leq 1$ part is only so we can say $g_n \geq g_n^... | H. H. Rugh | 355,946 | <p>The previous answer is very elegant. A more pedestrian approach is to use:
$$ \int |fg_n| \leq \int |f| 1_{|f|>R} + R \int |g_n| $$
And take the limits in the right order.</p>
|
277,060 | <p>For example I have this equation,I want to use c->a/b ,but it can not work
<span class="math-container">$$
\frac{a^2}{b^2}+\frac{b^2}{a^2}+\frac{a}{b}+e^{a/b}+\frac{b}{a}+\log \left(\frac{a}{b}\right)
$$</span></p>
<pre class="lang-mathematica prettyprint-override"><code>rule = {a/b -> c};
eq = 1/(a/b)^2 + (a/... | Michael Seifert | 27,813 | <p>To see why Mathematica does what it does, it can be instructive to write out the <code>FullForm</code> of the expressions in question. This tells you how Mathematica is storing these things internally.</p>
<pre><code>FullForm[rule]
(* List[Rule[Times[a,Power[b,-1]],c]] *)
</code></pre>
<p>So what Mathematica is doi... |
12,098 | <p>It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer curiosity, I clicked on Google, which took me to <a href="http://mathworld.wolfram.com/HeartCurve.html">this page</a>.</p... | Timothy Wagner | 3,431 | <p>This is really about plotting polar plots, parametric plots and implicitly defined functions in Mathematica.</p>
<p>This is the info on how to draw polar plots</p>
<p><a href="http://mathworld.wolfram.com/PolarPlot.html" rel="nofollow">http://mathworld.wolfram.com/PolarPlot.html</a></p>
<p>Parametric plots</p>
<... |
12,098 | <p>It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer curiosity, I clicked on Google, which took me to <a href="http://mathworld.wolfram.com/HeartCurve.html">this page</a>.</p... | Vicfred | 85,162 | <p>You can plot <a href="http://www.wolframalpha.com/entities/surfaces/taubin%27s_heart_surface/cr/rn/yc/" rel="noreferrer">Taubin's heart surface</a> using <code>ContourPlot3D</code>:</p>
<pre><code>ContourPlot3D[(2 x^2 + y^2 + z^2 - 1)^3 - (1/10) x^2 z^3 - y^2 z^3 == 0,
{x, -1.5, 1.5}, {y, -1.5, 1.5}, ... |
12,098 | <p>It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer curiosity, I clicked on Google, which took me to <a href="http://mathworld.wolfram.com/HeartCurve.html">this page</a>.</p... | David Caliri | 124,897 | <p>The following inputs will plot the following 6 hearts in the picture below respectively.</p>
<pre><code>ContourPlot[(x^2 + y^2 - 1)^3 - x^2 y^3 == 0, {x, -1.5, 1.5}, {y, -1.5, 1.5}, MaxRecursion -> 5]
ContourPlot[x^2 + (y - (2 (x^2 + Abs[x] - 6))/(3 (x^2 + Abs[x] + 2)))^2 == 36, {x, -9, 9}, {y, -9, 9}, MaxRecur... |
12,098 | <p>It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer curiosity, I clicked on Google, which took me to <a href="http://mathworld.wolfram.com/HeartCurve.html">this page</a>.</p... | E. Noujeim | 532,824 | <p>A three-dimensional space curve with the shape of a red heart:</p>
<p><img src="https://i.stack.imgur.com/qnYTO.jpg"></p>
<p>The Mathematica code for the image above is:</p>
<pre><code>ParametricPlot3D[{Cos[u]*(4*Sqrt[1 - v^2]*Sin[Abs[u]]^Abs[u]), v,
Sin[u]*(4*Sqrt[1 - v^2]*Sin[Abs[u]]^Abs[u])},
{u, -Pi, P... |
1,405,889 | <blockquote>
<p>Is $\int_{0}^\infty \frac{\sin(nx)}x \,dx$ is equal to $\pi/2$ for positive real $n$?</p>
</blockquote>
<p>I've come to this answer by inverse Fourier transform. But since there is n, I am quite confused that I didn't get n in the answer. Is this answer incorrect?
Thank you</p>
| Paolo Leonetti | 45,736 | <p>Use that $|a+b|\ge ||a|-|b||$, so that
$$
\lim_{n\to \infty} \left||x_{n+1}|-\left|\frac{x_n}{2}\right|\right|=0.
$$
For each $\varepsilon>0$ there exists $n_0$ such that if $n\ge n_0$ then
$$
\left||x_{n+1}|-\left|\frac{x_n}{2}\right|\right|\le \varepsilon \, \implies \, |x_{n+1}| \le \left|\frac{x_n}{2}\right|+... |
1,405,889 | <blockquote>
<p>Is $\int_{0}^\infty \frac{\sin(nx)}x \,dx$ is equal to $\pi/2$ for positive real $n$?</p>
</blockquote>
<p>I've come to this answer by inverse Fourier transform. But since there is n, I am quite confused that I didn't get n in the answer. Is this answer incorrect?
Thank you</p>
| Gabriel Romon | 66,096 | <p>For the record, here the out-of-the-blue proof outlined by the authors of <em>Selected problems in real analysis</em>.</p>
<p>Let <span class="math-container">$\displaystyle y_n=x_n-\frac{x_{n-1}}{2}$</span>.</p>
<p>Note that <span class="math-container">$$\displaystyle x_n-\frac{x_1}{2^{n-1}}=\sum_{1<k\leq n} \f... |
2,216,070 | <p>I'm having problems finding the primitive function to $\int \frac{1}{x \sqrt{8-x^2}} dx$. I've tried to use the substitution $t = x^2-8$, but then I just get stuck with $\int \frac{1}{(8-t)\sqrt{t}} dt$ instead. Using the substitution $t = \sqrt{x^2-8}$ doesn't get me much further either.</p>
<p>Any help is much ap... | StackTD | 159,845 | <p>You can perform a trigonometric substitution ($x=\sqrt{8}\sin u$, as in <a href="https://math.stackexchange.com/a/2216074/159845">Ziad Fakhoury's answer</a>), but unless you can use the anti-derivative of $\tfrac{1}{\sin u}$ (from a table or by heart), this will lead to rationalization as well and you can do that st... |
3,725,958 | <p>When I search implicit differentiation for equation <span class="math-container">$x^2 + y^2 = r^2$</span> I find results of two versions: one using derivative and the other using differential.</p>
<p>Version1: <span class="math-container">$\frac{d }{dx}(x^2 + y^2 = r^2) \Leftrightarrow 2x + 2y\frac{dy}{dx} = 0 $</sp... | K.defaoite | 553,081 | <p>Any two-variable relation can be summarized by the equation
<span class="math-container">$$f(x,y)=0$$</span>
In your example this would be
<span class="math-container">$$x^2+y^2-r^2=0$$</span>
The total derivative of <span class="math-container">$f$</span> with respect to an arbitrary variable <span class="math-cont... |
1,625,621 | <p>I am stuck on this limit and have no idea how to solve it and which trig identity to use. Any help would be appreciated. Thanks!</p>
<p>$\lim\limits_{x \to 0^-} \frac{\sqrt{1+2\sin^2 \frac{x}{2}-\cos^2x}}{\left\lvert x \right\rvert}$</p>
<p>Note: <strong>Without</strong> using L'Hopitals rule.</p>
| Workaholic | 201,168 | <p><strong>Hint:</strong> Using the identity $1-\cos^2x=\sin^2x$ one gets $$\lim\limits_{x \to 0^-} \frac{\sqrt{1+2\sin^2 \frac{x}{2}-\cos^2x}}{\left\lvert x \right\rvert}=\lim\limits_{x \to 0^-} \frac{\sqrt{\sin^2x+2\sin^2\tfrac x2}}{\sqrt{x^2}}=\lim\limits_{x \to 0^-}\sqrt{\dfrac{\sin^2x+2\sin^2\tfrac x2}{x^2}},$$ wh... |
906,318 | <p>How do you prove $f(x,y) = y - x$ is continuous? The domain is $\mathbb{R^{2}}$ and the codomain is $\mathbb{R}$. Is there an easy way to do it using the definition that the preimage of an open set is an open set? I don't have much experience proving multivariable functions are continuous.</p>
| Community | -1 | <p>OK, let's prove it from the definition you stated.</p>
<p>Let $z$ be any point in $\mathbb{R}$, and let $(a,b)$ be any point such that $f(a,b) = z$. For example, $(a,b) = (0,z)$ is one such point.</p>
<p>Let $\epsilon > 0$. Note that if $|x-a| < \epsilon/2$ and $|y-b| < \epsilon/2$, then
$$\begin{align}
|... |
906,318 | <p>How do you prove $f(x,y) = y - x$ is continuous? The domain is $\mathbb{R^{2}}$ and the codomain is $\mathbb{R}$. Is there an easy way to do it using the definition that the preimage of an open set is an open set? I don't have much experience proving multivariable functions are continuous.</p>
| vociferous_rutabaga | 164,345 | <p>This proof does not use the "preimage of open set" technique, but the idea behind it might be useful going forward.</p>
<p>Given $X$ , $Y$ <a href="http://en.wikipedia.org/wiki/Metric_space" rel="nofollow">metric spaces</a> and $f:X \rightarrow Y$ a function, we have the following <a href="https://proofwiki.org/wik... |
2,767,471 | <p>You enter a metro station in a big hurry, and decide to take the first train that arrives. </p>
<p>There are two lines running through this station: one runs every five minutes (line A), the other every three (line B). To be precise, suppose the next arrival of the A train is uniformly distributed on the interval [... | Karn Watcharasupat | 501,685 | <p>Hint:</p>
<p>Consider the waiting time $T$ such that
$$T=\min\{A,B\}$$</p>
<p>So$$P(T>t)=P(A>t,B>t)=P(A>t)\cdot P(B>t)$$
by independence.</p>
|
2,584,044 | <p>I'm learning about how use mathematical induction.
I'm tasked with proving the inequality shown in (1). It is a requirement that I use mathematical induction for the proof.</p>
<p>$(1) \quad P(n):\quad 2n+1 < 2^{n}, \quad n \ge 3$</p>
<p>I would like some feedback regarding whether my proof is valid and if my u... | farruhota | 425,072 | <p>Alternatively, without induction: for $n\ge 3$:
$$2^n=(1+1)^n=1+n+\cdots +n+1>2n+1.$$</p>
|
85,165 | <p>I have two lists </p>
<pre><code>X = {1, 2, 3};
Y = {5, 6, 7, 8};
</code></pre>
<p>I want to apply function <code>g[x,y_,z_]</code> to all pairs from X*Y, so I need to get a list <code>{g[x,1,5],g[x,1,6]…,g[x,3,8]}</code></p>
<p>I came up with this syntax</p>
<pre><code>g[x, ##] &@(Sequence @@ #) & /@ Tu... | Dr. belisarius | 193 | <p>As requested, I'm copying the above comment as an answer:</p>
<pre><code>g[x, ##] & @@@ Tuples[{X, Y}]
</code></pre>
<p>seems an elegant way to me</p>
|
392,608 | <p>I know this is a very basic question but I need some help.</p>
<p>I have to find the second derivative of: </p>
<p>$$\frac{1}{3x^2 + 4}$$</p>
<p>I start by using the Quotient Rule and get the first derivative to be:</p>
<p>$$\frac{-6x}{(3x^2 + 4)^2}$$</p>
<p>This I believe to be correct.
Following that I procee... | DonAntonio | 31,254 | <p>$$\left(-\frac{6x}{(3x^2+4)^2}\right)'=-\frac{6(3x^2+4)^2-72x^2(3x^2+4)}{(3x^2+4)^4}=$$</p>
<p>$$=-\frac{18x^2+24-72x^2}{(3x^2+4)^3}=-\frac{6(-9x^2+4)}{(3x^2+4)^3}$$</p>
|
1,417,404 | <p>The following came up in my solution to <a href="https://math.stackexchange.com/questions/1410565/can-this-congruence-be-simplified/1410579#1410579">this question</a>, but buried in the comments, so maybe it's worth a question of its own. Consider the Diophantine equation
$$ (x+y)(x+y+1) - kxy = 0$$
For $k=5$ and $... | Ghartal | 83,884 | <p>I think it can be proven using Vieta jumping method that in order for</p>
<p>$k=\dfrac{(x+y)(x+y+1)}{xy}$</p>
<p>to be an integer for positive integers $x$ and $y$ the only values $k$ can take are $5, 6$.</p>
<p>Ok, we have </p>
<p>$x^2+y^2+x+y=(k-2)xy$</p>
<p>Vieta jumping is the key of the solving this proble... |
565,135 | <p>I have to prove the following theorem :</p>
<blockquote>
<p>Let $p$ be a prime number and let $n \ge 1$,be any integer, then there exists a field of order $p^n$.</p>
</blockquote>
<p><strong>My attempt</strong></p>
<p>I started off by considering the polynomial $f(x)$=$x^{p^n}-x \in \Bbb Z_p[x]$.</p>
<p>I took... | Adam Hughes | 58,831 | <p>We note that it is sufficient to show there exists an irreducible polynomial, <span class="math-container">$Q(x)$</span> of degree <span class="math-container">$n$</span> over <span class="math-container">$\Bbb F_p$</span>, the field with <span class="math-container">$p$</span> elements (which you denote by <span cl... |
2,942,681 | <p>In an ordinary function like the temperature—one of the properties of the minimum is that if we go away from the minimum in the first order, the deviation of the function from its minimum value is only second order.</p>
<p>At any place else on the curve, if we move a small distance the value of the function changes... | Botond | 281,471 | <p>We say that a function <span class="math-container">$f: \mathbb{R} \rightarrowtail \mathbb{R}$</span> is differentiable at a point <span class="math-container">$a \in \text{Int}(\text{dom}(f))$</span> if there exists <span class="math-container">$A \in \mathbb{R}$</span> and <span class="math-container">$r: \text{do... |
2,078,592 | <p>Let $A:=\{z\in S^1: z^n=1\}$, considering $S^1\subset \mathbb{C}$.</p>
<p>Then, how do I compute $H_k(S^1\times S^1, A\times \{1\}\cup \{1\} \times S^1)$ when $k=1,2$?</p>
<p>Let's consider the long exact sequence of a pair $(S^1\times S^1, A\times \{1\}\cup \{1\} \times S^1)$:</p>
<p>$...\rightarrow H_k(A\times ... | John Hughes | 114,036 | <p>Hint: What's
$$
H_1(A\times \{1\}\cup \{1\} \times S^1)?
$$
Hint: it's $\Bbb Z$, generated by the $S^1$ factor. </p>
<p>What's
$$
H_1(S^1\times S^1),
$$
and what are its (most natural) generators? </p>
<p>Hint 2: What's $H_2$ of a 1-complex? </p>
|
3,624,230 | <p>An algebra, as far as I know, is closely related to a group with a family of functions being closed under addition, scalar multiplication and then the product of any two functions in the family.</p>
<p>Then there is this separate term I came across on Wikipedia called a <a href="https://en.wikipedia.org/wiki/Lie_al... | Mike Pierce | 167,197 | <p>Abstractly, both an <em>algebra</em> and a <em>Lie algebra</em> are vector spaces equipped with a multiplication. The difference is in the kind of multiplication they are equipped with. An <em>algebra</em> is often called by it's full name, an <em>associative algebra</em>, since the multiplication is <em>associative... |
1,658,279 | <p>I'm new here so apologies if I am not clear enough. I am trying to find the zero divisors of the form $ax + b$ in $\mathbb Z_{10}$. Specifically, I need to find the values of $b$. I know that $2,4,5,6$ and $8$ are zero divisors in $\mathbb Z_{10}$ but I am not sure how to translate these into linear divisors.</p>
<... | Ove Ahlman | 222,450 | <p>You are adding more information to the problem, thus changing the probability.</p>
<p>Compare: Rolling a dice you have $1/6$ to get a $5$. However if I add the information that I did not roll a $1$, then the probability of having rolled a $5$ becomes $1/5$.</p>
<p>In the same way if you add information about how m... |
2,024,468 | <p>I need to find the points of intersection of a circle with radius $2$ and centre at $(0,0)$ and a rectangular hyperbola with equation $xy=1$. As per the topic statement is there any way to solve this without the graphical method. I have tried setting the $y$ values equal but I cant solve the resulting equation for $... | Community | -1 | <p>By a change of variable, $$x^2+y^2=4,\\xy=1$$ can be rewritten</p>
<p>$$X+Y=4,\\XY=1$$ provided you keep in mind that $x$ and $y$ have the same sign.</p>
<p>Then this is a classical sum/product problem, solved by</p>
<p>$$(X-Y)^2=(X+Y)^2-4XY=12,$$ then</p>
<p>$$X,Y=\frac{4\pm\sqrt{12}}2=2\pm\sqrt3.$$</p>
<p>Fin... |
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