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 |
|---|---|---|---|---|
2,781,153 | <p>I've a right triangle that is inscribed in a circle with radius $r$ the hypotunese of the triangle is equal to the diameter of the circle and the two other sides of the triangle are equal to eachother.</p>
<blockquote>
<p>Prove that when you divide the area of the circle by the area of the triangle that you will ... | user061703 | 515,578 | <p>The task given "the right triangle" and "two other sides of the triangle are equal to each other". This means this is a right iscoceles triangle or $45-45-90$ triangle.</p>
<p>Assume that this is the $45-45-90$ triangle $ABC$ right and isoceles at $A$, draw the altitude $AH$ (perpendicular to $BC$), then we can pro... |
223,642 | <p>$z\cdot e^{1/z}\cdot e^{-1/z^2}$ at $z=0$.</p>
<p>My answer is removable singularity.
$$
\lim_{z\to0}\left|z\cdot e^{1/z}\cdot e^{-1/z^2}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{z-1}{z^2}}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{-1}{z^2}}\right|=0.
$$
But someone says it is an essential singularity. I don't know... | TTY | 19,412 | <p>for $z\cdot e^{\frac{-1}{z^2}}$, note if you approach to the origin along the imaginary line, say $z=ih$, we will get $ihe^{\frac{-1}{(i)^2h}}=ihe^{\frac{1}{h}}$, this obviously does not tends to zero as $h \to 0$</p>
|
1,137,079 | <p>I'm new to the concept of complex plane. I found this exercise:</p>
<blockquote>
<p>Let $z,z_1,z_2\in\mathbb C$ such that $z=z_1/z_2$. Show that the length of $z$ is the quotient of the length of $z_1$ and $z_2$.</p>
</blockquote>
<p>If $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ then $|z_1|=\sqrt{x_1^2+y_1^2}$ and $|z_2|... | egreg | 62,967 | <p>If you know that $|ab|=|a|\,|b|$, then use $a=z_1/z_2$ and $b=z_2$:
$$
|z_1|=\left|\frac{z_1}{z_2}z_2\right|=|ab|=|a|\,|b|=
\left|\frac{z_1}{z_2}\right|\,|z_2|
$$
Now, easily,
$$
\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}.
$$</p>
<p>Proving that $|ab|=|a|\,|b|$ is much simpler, even with the definition: if $a... |
4,608,805 | <p>Suppose that I have a class of 35 students whose average grade is 90. I randomly picked 5 students whose average came out to be 85. Assume their grades are i.i.d and of normal <span class="math-container">$N(\mu, \sigma^2)$</span>. From the example I have seen, <span class="math-container">$\mu$</span> is usually ca... | William M. | 396,761 | <p>You are assuming that the grade of each student follows a normal distribution <span class="math-container">$\mathsf{Norm}(\mu, \sigma^2)$</span> <strong>and</strong> you assume <span class="math-container">$\mu = 90.$</span> You may rise the question as to whether is sensical to model grades with normal distribution... |
2,087,724 | <p>Let $\Omega $ a smooth domain of $\mathbb R^d$ ($d\geq 2$), $f\in \mathcal C(\overline{\Omega })$. Let $u\in \mathcal C^2(\overline{\Omega })$ solution of $$-\Delta u(x)+f(x)u(x)=0\ \ in\ \ \Omega .$$
Assume that $f(x)\geq 0$ for $x\in \Omega $. Prove that $$\int_{B(x,r)}|\nabla u|^2\leq \frac{C}{r^2}\int_{B(x,2r)}|... | xpaul | 66,420 | <p>Denote $B(x,r)$ by $B_r$ and choose a smooth function $\xi$ to have the following property
$$ \xi=\left\{\begin{array}{lcr}1,&\text{if }x\in B_r\\
\ge 0,&\text{if }x\in B_{2r}\setminus B_r\\
0,&\text{if }x\in\Omega\setminus B_{2r},
\end{array}
\right. $$
satisfying $|\nabla \xi|\le \frac{C}{r}$. Using $u... |
3,013,529 | <p>Suppose that the function <span class="math-container">$f$</span> is:</p>
<p>1) Riemann integrable (not necessarily continuous) function on <span class="math-container">$\big[a,b \big]$</span>;</p>
<p>2) <span class="math-container">$\forall n \geq 0$</span> <span class="math-container">$\int_{a}^{b}{f(x) x^n} = 0... | Disintegrating By Parts | 112,478 | <p>If you know something about Fourier analysis, you can use the Fejer kernel and the following to conclude that <span class="math-container">$f=0$</span> at all points of continuity:
<span class="math-container">$$
\int_{a}^{b}f(x)e^{-isx}dx = \sum_{n=0}^{\infty}\frac{(-is)^n}{n!}\int_{a}^{b} f(x)x^n dx =... |
3,013,529 | <p>Suppose that the function <span class="math-container">$f$</span> is:</p>
<p>1) Riemann integrable (not necessarily continuous) function on <span class="math-container">$\big[a,b \big]$</span>;</p>
<p>2) <span class="math-container">$\forall n \geq 0$</span> <span class="math-container">$\int_{a}^{b}{f(x) x^n} = 0... | zhw. | 228,045 | <p>To keep the notation simple, suppose <span class="math-container">$f$</span> is Riemann integrable on <span class="math-container">$[-1,1],$</span> <span class="math-container">$f$</span> is continuous at <span class="math-container">$0,$</span> and <span class="math-container">$ \int_{-1}^1 p(x)f(x)\, dx =0$</span>... |
1,002,777 | <p>I want to convert this polynomoial to partial fraction.</p>
<p>$$
\frac{x^2-2x+2}{x(x-1)}
$$</p>
<p>I proceed like this:
$$
\frac{x^2-2x+2}{x(x-1)} = \frac{A}{x} + \frac{B}{x-1}
$$
Solving,
$$
A=-2,B=1
$$
But this does not make sense. What is going wrong?</p>
| Timbuc | 118,527 | <p>An idea I didn't see in the other answers:</p>
<p>$$\frac{x^2-2x+2}{x(x-1)}=\frac{(x-1)^2+1}{x(x-1)}=\frac{x-1}x+\frac1{x(x-1)}=1-\frac1x+\frac1{x(x-1)}$$</p>
<p>And now either directly: $\;\frac1{x(x-1)}=\frac1{x-1}-\frac1x\;$ ,or by partial fractions, so that finally</p>
<p>$$\frac{x^2-2x+2}{x(x-1)}=1-\frac2x+\... |
2,136,024 | <p>I am having problems with this linear algebra proof:</p>
<blockquote>
<p>Let $ A $ be a square matrix of order $ n $ that has exactly one nonzero entry
in each row and each column. Let $ D $ be the diagonal matrix whose $ i^{th} $
diagonal entry is the nonzero entry in the $i^{th}$ row of $A$</p>
<p>For ... | Joshua Ruiter | 399,014 | <p>For part one, you have the right idea. Try multiplying $A$ on the right by various permutation matrices, and just see what happens. For example, multiplying on the right by this matrix swaps columns 1 and 3.
\begin{equation}
\begin{bmatrix}
0 & 0 & a_1 & 0 \\
a_2 & 0 & 0 & 0 \\
0 & 0 &... |
3,979,371 | <p>So I have been struggling with this question for a while. Suppose <span class="math-container">$X$</span> is uniformly distributed over an interval <span class="math-container">$(a, b)$</span> and <span class="math-container">$Y$</span> is uniformly distributed over <span class="math-container">$(-\sigma, \sigma)$</... | Acccumulation | 476,070 | <p><span class="math-container">$X|Z$</span> can't, in general, be uniformly distributed over <span class="math-container">$(z-\sigma,z+\sigma)$</span> because there are values of <span class="math-container">$z$</span> such that <span class="math-container">$(z-\sigma,z+\sigma)$</span> is not a subset of <span class="... |
4,309,247 | <p>My question comes from an exercise in Shilov's <em>Linear Algebra</em>. His hint is to use induction, but I'm struggling to get anywhere. I looked through the book and couldn't find any theorem that seemed useful, so I'm guessing there is some sort of manipulation I must be missing? A good first step to take would b... | esoteric-elliptic | 425,395 | <p>As the author suggests, we shall use induction.</p>
<p>The base case, i.e. <span class="math-container">$m = 1$</span> follows from the hypothesis. Suppose for <span class="math-container">$k\in \mathbb N$</span>, <span class="math-container">$$A^kB - BA^k= kA^{k-1}$$</span>
Multiplying throughout by <span class="ma... |
355,296 | <p>How can we evaluate $$\displaystyle\int \frac{x^2 + x+3}{x^2+2x+5} dx$$ </p>
<p>To be honest, I'm embarrassed. I decomposed it and know what the answer should be but<br>
I can't get the right answer. </p>
| lab bhattacharjee | 33,337 | <p>$$I=\int \frac{x^2 + x+3}{x^2+2x+5} dx=\int \frac{x(x+1)+3}{(x+1)^2+2^2} dx$$ </p>
<p>Putting $x+1=2\tan\theta,dx=2\sec^2\theta d\theta,$</p>
<p>$$I= \frac{2\tan\theta(2\tan\theta-1)+3}{4\sec^2\theta d\theta}2\sec^2\theta d\theta$$</p>
<p>$$2\tan\theta(2\tan\theta-1)+3=4\tan^2\theta-2\tan\theta+3=4\sec^2\theta-2\... |
109,423 | <p>Let $f$ be an isometry (<em>i.e</em> a diffeomorphism which preserves the Riemannian metrics) between Riemannian manifolds $(M,g)$ and $(N,h).$ </p>
<p>One can argue that $f$ also preserves the induced metrics $d_1, d_2$ on $M, N$ from $g, h$ resp. that is, $d_1(x,y)=d_2(f(x),f(y))$ for $x,y \in M.$ Then, it's easy... | Yuri Vyatkin | 2,002 | <p>Your calculation looks like an attempt to prove the naturality of the Levi-Civita connection, the fact that @Zhen Lin implicitly points to. In the settings of the question it can be stated as
$$
\nabla^g_X{Y}=f^* \left( \nabla^{(f^{-1})^* g}_{\operatorname{d}f(X)} \operatorname{d}f(Y) \right)
$$</p>
<p>Notice als... |
1,073,681 | <p>Given two half-integers $a,b\in\mathbb Z+\frac 1 2$, the integers nearest to each is given by $N_a=\{a-0.5,\ a+0.5\}$ and $N_b=\{b-0.5,\ b+0.5\}$.</p>
<p>Is there a general method to find, for the two given half-integers, the values $n\in N_a,m\in N_b$ where $\left|nm-ab\right|$ is minimal?</p>
<hr>
<p>The motiva... | vadim123 | 73,324 | <p>We compute $$nm=ab\pm \frac{a}{2} \pm \frac{b}{2} \pm \frac{1}{4}$$
and hence we wish to minimize $$|nm-ab|=|\pm \frac{a}{2} \pm \frac{b}{2} \pm \frac{1}{4}|$$
In this expression, the three $\pm$ are not all freely chosen; there must be an even number of $-$'s chosen. Let's assume without loss that $a\ge b\ge 0$ (i... |
1,073,681 | <p>Given two half-integers $a,b\in\mathbb Z+\frac 1 2$, the integers nearest to each is given by $N_a=\{a-0.5,\ a+0.5\}$ and $N_b=\{b-0.5,\ b+0.5\}$.</p>
<p>Is there a general method to find, for the two given half-integers, the values $n\in N_a,m\in N_b$ where $\left|nm-ab\right|$ is minimal?</p>
<hr>
<p>The motiva... | Heimdall | 191,910 | <p>What you're trying to do, is to find a number</p>
<p>$$(a\pm{1\over2})(b\pm{1\over2})$$</p>
<p>that's the closest to $ab$.</p>
<p>For start, let's assume $a$ and $b$ are both positive.</p>
<p>If they are both $+$ or both $-$ it's easy to see you are further than having one $+$ and one $-$. To see which one shoul... |
4,376,076 | <p>i) the Matrix P has only real elements</p>
<p>ii) 2+i is an eigenvalue of Matrix P</p>
<p>I got that the zeros has to be <span class="math-container">$(x-(2+i))(x-(2-i))$</span> which is equal to <span class="math-container">$(x-2)^2 -i^2$</span> so the characteristical polynom is equal to <span class="math-containe... | Herrpeter | 190,056 | <p>So the zeros are <span class="math-container">$(x-(2+i))(x-(2-i))$</span> =</p>
<p><span class="math-container">$$=((x-2)+i)((x-2)-i)$$</span>
<span class="math-container">$$=(x-2)^2-i^2 = x^2-4x+4+1=x^2-4x+5$$</span></p>
<p>So we get that characteristical polynom has the factors <span class="math-container">$(2-x)(... |
2,265,782 | <p>Number of twenty one digit numbers such that Product of the digits is divisible by $21$</p>
<p>Since product is divisible by $21$ the number should contain the digits $3,6,7,9$ But i am unable to decide how to proceed...can i have any hint</p>
| Bram28 | 256,001 | <p>Here is a method called the 'short truth-table' method. As the name implies, it is not a full truth-table.</p>
<p>The idea is that you try to make the statement false ... and if you find that you cannot do that, then that means that the statement <em>must</em> be true, i.e. it is a tautology.</p>
<p>OK, so let's ... |
8,997 | <p>I have a set of data points in two columns in a spreadsheet (OpenOffice Calc):</p>
<p><img src="https://i.stack.imgur.com/IPNz9.png" alt="enter image description here"></p>
<p>I would like to get these into <em>Mathematica</em> in this format:</p>
<pre><code>data = {{1, 3.3}, {2, 5.6}, {3, 7.1}, {4, 11.4}, {5, 14... | Mr.Wizard | 121 | <p>Copying <a href="https://superuser.com/a/371320/70766">my answer from SuperUser:</a></p>
<p>I am not familiar with the Excel clipboard format, but there is a lovely suite of tools for pasting tabular data in the form of a Palette. See <a href="http://szhorvat.net/pelican/pasting-tabular-data-from-the-web.html" rel... |
481,167 | <p>Let $V$ be a $\mathbb{R}$-vector space.
Let $\Phi:V^n\to\mathbb{R}$ a multilinear symmetric operator.</p>
<p>Is it true and how do we show that for any $v_1,\ldots,v_n\in V$, we have:</p>
<p>$$\Phi[v_1,\ldots,v_n]=\frac{1}{n!} \sum_{k=1}^n \sum_{1\leq j_1<\cdots<j_k\leq n} (-1)^{n-k}\phi (v_{j_1}+\cdots+v_{j... | Ken | 544,921 | <p>(This is just the summary of the answers by Ewan and Anthony, but a lot simpler.)</p>
<p>Let <span class="math-container">$S_n$</span> denote the symmetric group. Also, let us write <span class="math-container">$X=\{1,\dots,n\}$</span>. We compute <span class="math-container">$$\begin{eqnarray}\sum_{k=1}^{n}\sum_{1... |
3,027,450 | <p>I'm learning about group homomorphisms and I'm confused about what the <span class="math-container">$\phi$</span> transformation is exactly. </p>
<p>If we have some group homomorphism <span class="math-container">$\phi : G\rightarrow H$</span> what exactly does <span class="math-container">$\phi(G)$</span> mean? </... | Robert Israel | 8,508 | <p><span class="math-container">$\phi(G)$</span> is the image of <span class="math-container">$G$</span> under the mapping <span class="math-container">$\phi$</span>, i.e. the set of all <span class="math-container">$\phi(x)$</span> for <span class="math-container">$x \in G$</span>.</p>
|
3,027,450 | <p>I'm learning about group homomorphisms and I'm confused about what the <span class="math-container">$\phi$</span> transformation is exactly. </p>
<p>If we have some group homomorphism <span class="math-container">$\phi : G\rightarrow H$</span> what exactly does <span class="math-container">$\phi(G)$</span> mean? </... | Alekos Robotis | 252,284 | <p><span class="math-container">$\phi(G)$</span> denotes the image of the group <span class="math-container">$G$</span>. This is actually a purely set-theoretic definition. Given a map of sets <span class="math-container">$f:X\to Y$</span>, <span class="math-container">$f(X)$</span> denotes the image of <span class="ma... |
1,109,552 | <p>So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) = 2$ and so $a^2+5b^2 = 2$ , however, since $b^2$ and $a^2$ are both positive integers then $b=0$ and $a=\sqrt{2}$ h... | Robert Soupe | 149,436 | <p>There are only two units in $\mathbb{Z}[\sqrt{-5}]$: 1 and $-1$. To obtain the associate of a number, you multiply that number by a unit other than 1, and in this domain there is only one choice: $-1$. So, for example, the associate of 2 is $-2$, the associate of 3 is $-3$, the associate of $1 - \sqrt{-5}$ is $-1 + ... |
3,472,151 | <p>I find two main sources on how to compute the half-derivative of <span class="math-container">$e^x$</span>. Both make sense to me, but they give different answers.</p>
<p>Firstly, people argue, that
<span class="math-container">$$\begin{align}
\frac{\mathrm{d}}{\mathrm{d} x} e^{k x} &= k e^{k x} \\[4pt]
\frac{\... | Conifold | 152,568 | <p>Both are right. Think of it like this: what is "the" square root of <span class="math-container">$i$</span>? Even for <span class="math-container">$1$</span> there are two of them, but we forget because of the habit to take the positive root. For <span class="math-container">$i$</span> there is no habit available. T... |
526,820 | <p>How do I integrate the inner integral on 2nd line? </p>
<p><img src="https://i.stack.imgur.com/uIxQX.png" alt="enter image description here"></p>
<hr>
<p>$$\int^\infty_{-\infty} x \exp\{ -\frac{1}{2(1-\rho^2)} (x-y\rho)^2 \} \, dx$$</p>
<p>I know I can use integration by substitution, let $u = \frac{x-y\rho}{\sq... | Felix Marin | 85,343 | <p>\begin{align}
&\phantom{=\,\,}\int^\infty_{-\infty}x
\exp\left(%
-\,\frac{\left[x - y\rho\right]^{2}}{2\left[1 - \rho^{2}\right]}
\right)\,dx
\tag{1}
\\[3mm]&=
\int^\infty_{-\infty}\left(x + y\rho\right)
\exp\left(-\,\frac{x^{2}}{2\left[1 - \rho^{2}\right]}\right) \, dx
\tag{2}
\\[3mm]&=
y\rho\,\sqrt{2\p... |
136,021 | <p>There is an equivalence relation between inclusion of finite groups coming from the world of <a href="http://en.wikipedia.org/wiki/Subfactor" rel="noreferrer">subfactors</a>:</p>
<p><strong>Definition</strong>: <span class="math-container">$(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$</span> if <span class="ma... | Owen Sizemore | 5,732 | <p>This is somewhere between a comment and an answer. But it is too long for a comment, so I put it here.</p>
<p>To me the natural thing to look at is the action $G\curvearrowright G/H$. $|G/H|$ captures the index, which you surely want to do, and the action should in some sense capture the position of $H$ inside $G$.... |
136,021 | <p>There is an equivalence relation between inclusion of finite groups coming from the world of <a href="http://en.wikipedia.org/wiki/Subfactor" rel="noreferrer">subfactors</a>:</p>
<p><strong>Definition</strong>: <span class="math-container">$(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$</span> if <span class="ma... | Dave Penneys | 351 | <p>For finite groups, the answer was given by Izumi in his paper "Characterization of isomorphic group-subgroup subfactors" (MR1920326). There he looks at the crossed product subfactor, but you can always take duals.</p>
<p>Edit after @Andre's comment:</p>
<p>The actual condition between the two pairs of subgroups is... |
1,285,014 | <p>Let $R,S$ be commutative rings with identity.</p>
<p>Proving that $X \sqcup Y$ is an affine scheme is the same as proving that $Spec(R) \sqcup Spec(S) = Spec(R \times S)$.</p>
<p>I proved that if $R,S$ are rings, then the ideals of $R \times S$ are exactly of the form $P \times Q$, where $P$ is an ideal of $R$ and... | Sam Walls | 181,802 | <p>If you write out the truth table for your formula...</p>
<pre><code>p q r p → ¬(q∨r)
0 0 0 1
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 0
</code></pre>
<p>To get the DNF, look at every result that is True; those lines directly correspond to the following clauses in DNF: $$(\ln... |
1,537,881 | <p>Find the values of $a$ and $b$ if $$ \lim_{x\to0} \dfrac{x(1+a \cos(x))-b \sin(x)}{x^3} = 1 $$
I think i should use L'Hôpital's rule but it did not work.</p>
| Idris Addou | 192,045 | <p>use the equivalent, near $0$
\begin{eqnarray*}
\cos x &\approx &1-\frac{x^{2}}{2} \\
\sin x &\approx &x-\frac{x^{3}}{6}
\end{eqnarray*}
\begin{eqnarray*}
\frac{x(1+a\cos x)-b\sin x}{x^{3}} &\approx &\frac{x(1+a\left( 1-\frac{x^{2}%
}{2}\right) )-b\left( x-\frac{x^{3}}{6}\right) }{x^{3}} \\
&a... |
1,537,881 | <p>Find the values of $a$ and $b$ if $$ \lim_{x\to0} \dfrac{x(1+a \cos(x))-b \sin(x)}{x^3} = 1 $$
I think i should use L'Hôpital's rule but it did not work.</p>
| Bernard | 202,857 | <p>You can, unless using L'Hospital's rule repeatedly, which is not the alpha and omega of limit computations. The simplest way to go is to use <em>Taylor's polynomial</em>:</p>
<p>$$ \cos x=1-\dfrac{x^2}2+o(x^2),\quad \sin x=x-\frac{x^3}6+o(x^3),$$
whence
$$x(1+a\cos x)-b\sin x=(1+a-b)x+\Bigl(\frac b6-\frac a2\Bigr... |
48,989 | <p>How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?</p>
| Jyrki Lahtonen | 11,619 | <p>Hint: Show that rows of $AB$ are linear combinations of rows of $B$. Transpose this hint.</p>
|
48,989 | <p>How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?</p>
| Listing | 3,123 | <p>Surely vectors that are in the kernel of <span class="math-container">$B$</span> are also in the kernel of <span class="math-container">$AB$</span>. Vectors that are in the kernel of <span class="math-container">$A^T$</span> are also in the kernel of <span class="math-container">$(AB)^T=B^TA^T$</span> therefore with... |
1,828,097 | <p>If we contruct two strainght lines as shown:<a href="https://i.stack.imgur.com/8K5Eo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8K5Eo.png" alt="enter image description here"></a></p>
<p>Then join them such that to complete a triangle. <a href="https://i.stack.imgur.com/Uvtnw.png" rel="nofoll... | Qwerty | 290,058 | <p>Take this example. Find <strong>any</strong> point between $(1,\infty)$ , invert it and you will get a point in $(0,1)$. Since there are infinitely many points you can choose , according to you it must mean $\infty=\infty$ i.e. $(0,1)=(1,\infty)$? </p>
|
788,995 | <p>I need to prove that function $\mathbb R × \mathbb R → \mathbb R $ : $f(x,y) = \frac{|x-y|}{1 + |x-y|}$ is a metric on $\mathbb R$. First two axioms are trivial; it's the triangle inequality which is pain. $\frac{|x-y|}{1 + |x-y|}$ + $\frac{|y-z|}{1 + |y-z|} ≥ \frac{|x-z|}{1 + |x-z|} ⇒ \frac{|x-y| + |y-z| + 2|(x-y)... | Hagen von Eitzen | 39,174 | <p>Have a look at $g(x)=\frac x{1+x}$. All we need is that for $u,v\ge 0$ and $w\le u+v$, we have $$\tag1 g(w)\le g(u)+g(v).$$
For if this is the case, we can let $u=|x-y|$, $v=|y-z|$, $w=|x-z|$; then $w\le u+v$ by the ordinary triangle inequality, and then $(1)$ i sthe desired triangle inequality for $f$.</p>
<p>To s... |
612,827 | <p>I'm self studying with Munkres's topology and he uses the uniform metric several times throughout the text. When I looked in Wikipedia I found that there's this concept of a <a href="http://en.wikipedia.org/wiki/Uniform_space" rel="nofollow">uniform space</a>.</p>
<p>I'd like to know what are it's uses (outside poi... | Matt E | 221 | <p>The concept of <em>uniform space</em> is designed to abstract the notions
of <em>uniform continuity</em>, <em>Cauchy sequences</em>, and <em>completeness</em> from
metric space theory. </p>
<p>What these concepts all have in common is that they require us to
be able to talk about a pair of points $x$ and $y$ being ... |
772,391 | <p>The formula for the Chi-Square test statistic is the following:</p>
<p>$\chi^2 = \sum_{i=1}^{n} \frac{(O_i - E_i)^2}{E_i}$</p>
<p>where O - is observed data, and E - is expected.</p>
<p>I'm curious why it depends on the absolute values? For example, if we change the units we're measuring we'll get a different sta... | 2'5 9'2 | 11,123 | <p>In the version of this test that I am familiar with, individual data is <em>categorical</em>, not quantitative like your examples. And the expected and observed values should be <em>frequencies</em> of some category (a count of how many times it occurs), not some individual's quantitative measurement. The numbers th... |
3,444,556 | <blockquote>
<p>Let <span class="math-container">$f:[-1,1] \to \mathbb{R}$</span> be continuous on <span class="math-container">$[-1,1]$</span>.</p>
<p>Assume <span class="math-container">$\displaystyle \int_{-1}^{1}f(x)x^ndx = 0$</span> for <span class="math-container">$n = 0,1,2,...$</span> </p>
<p>Then s... | user284331 | 284,331 | <p><span class="math-container">\begin{align*}
\int_{-1}^{1}|f(x)||f(x)-p_{n}(x)|dx&\leq\sup_{x\in[-1,1]}|f(x)-p_{n}(x)|\int_{-1}^{1}|f(x)|dx\\
&=K\sup_{x\in[-1,1]}|f(x)-p_{n}(x)|,
\end{align*}</span>
where <span class="math-container">$K$</span> is the integral, which is a finite number, now the supremum can b... |
83,336 | <p>Sorry if this is a naive question-- I'm trying to learn this stuff (cross-posted from <a href="https://math.stackexchange.com/questions/89248/induced-representations-of-topological-groups">https://math.stackexchange.com/questions/89248/induced-representations-of-topological-groups</a>)</p>
<p>If $G$ is a group with... | Marc Palm | 10,400 | <p>In fact the right relation is
$$ Hom_G ( - , Ind_H^G - ) = Hom_H( Res - , -).$$
For compact group, it does not really matter by the Peter Weyl theorem, but it is essential as soon as you omit compactness. </p>
<p>I want to add that Frobenius reciprocity usually boils down to tensor-hom adjointness, so essentailly ... |
2,400,336 | <p>My first try was to set the whole expression equal to $a$ and square both sides. $$\sqrt{6-\sqrt{20}}=a \Longleftrightarrow a^2=6-\sqrt{20}=6-\sqrt{4\cdot5}=6-2\sqrt{5}.$$</p>
<p>Multiplying by conjugate I get $$a^2=\frac{(6-2\sqrt{5})(6+2\sqrt{5})}{6+2\sqrt{5}}=\frac{16}{2+\sqrt{5}}.$$</p>
<p>But I still end up w... | Stefan4024 | 67,746 | <p>Here's a useful formula for this kind of problems:</p>
<p>$$\sqrt{a \pm \sqrt{b}} = \sqrt{\frac{a + \sqrt{a^2 - b}}{2}} \pm \sqrt{\frac{a - \sqrt{a^2 - b}}{2}}$$</p>
<p>where we have $a,b \ge 0$ and $a^2 > b$</p>
|
3,552,219 | <p>I come across an explanation of recursion complexity. This screenshot is in question:</p>
<p><a href="https://i.stack.imgur.com/ySKdo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ySKdo.png" alt="a"></a></p>
<p>How do you get this?</p>
<pre><code>T(n) = 3T(n/4) + n
</code></pre>
<p>The <spa... | Karl | 279,914 | <p>In this example, we're considering an algorithm that, given a problem of size <span class="math-container">$n$</span>, does 1 unit of "local" work and divides the remaining work into 3 smaller problems of the same kind, where each of these subproblems has size <span class="math-container">$n/4$</span>. (Instead of 3... |
3,210,791 | <p>The question is as follows:</p>
<p>Consider the following partial differntial equation (PDE)</p>
<p><span class="math-container">$2\frac{\partial^2u}{\partial x^2}+2\frac{\partial^2u}{\partial y^2} = u$</span></p>
<p>where <span class="math-container">$u=u(x,y)$</span> is the unknown function.</p>
<p>Define the ... | Community | -1 | <ul>
<li><p><span class="math-container">$2\cdot0+2\cdot2x=xy^2$</span> ?</p></li>
<li><p><span class="math-container">$2(-y^2\sin(xy))+2(-x^2\sin(xy))=\sin(xy)$</span> ?</p></li>
<li><p><span class="math-container">$2\dfrac{e^{(x-y)/3}}9+2\dfrac{e^{(x-y)/3}}9=e^{(x-y)/3}$</span> ?</p></li>
</ul>
|
68,438 | <p>I recall reading somewhere that if a conformal class contains an Einstein metric then that metric is the unique metric with constant scalar curvature in its conformal class, with the exception of the case of the round sphere. Does this sound right? If it is true: where can I find the proof of this result? </p>
| Renato G. Bettiol | 15,743 | <p>The first proof of the statement "Einstein metrics are the unique metrics with constant scalar curvature in their conformal class, except for round spheres" is due to Obata in 1971, see <a href="http://www.ams.org/mathscinet-getitem?mr=0303464" rel="noreferrer">MR0303464 M. Obata, The conjectures on conformal transf... |
1,281,507 | <blockquote>
<p>$$x*y = 3xy - 3x - 3y + 4$$</p>
<p>We know that $*$ is associative and has neutral element, $e$.</p>
<p>Find $$\frac{1}{1017}*\frac{2}{1017}*\cdots *\frac{2014}{1017}.$$</p>
</blockquote>
<p>I did find that $e=\frac{4}{3}$, and, indeed, $x*y = 3(x-1)(y-1)+1$. Also,it is easy to check that t... | Travis Willse | 155,629 | <p><strong>Hint</strong> What is $1 \ast y$?</p>
<blockquote class="spoiler">
<p><strong>Additional hint</strong> Note that $1$ occurs in the $2014$-fold product, namely as $\frac{1017}{1017}$.</p>
</blockquote>
|
132,238 | <p>I'm trying to solve a maximization problem that apparently is too complicated (it's a convex function) and NMaximize just runs endlessly.</p>
<p>I'd like to have an approximate result, though. How can I tell <code>NMaximize</code> to just give up after $n$ seconds and give me the best it has found so far?</p>
| Lukas | 21,606 | <p>This is how I usually deal with this kind of problem. Keywords to the solution are <code>TimeConstrained</code>, <code>AbortProtect</code>, <code>Throw</code> and <code>Catch</code>.</p>
<p>Consider the two target functions: </p>
<pre><code>fun[x_] := (Pause[1]; x^4 - 3 x^2 - x);
fun2[x_, y_] := (Pause[1]; x + y);... |
850,390 | <p>Let $f(x)$ be differentiable function from $\mathbb R$ to $\mathbb R$, If $f(x)$ is even, then $f'(0)=0$. Is it always true?</p>
| Marm | 159,661 | <p>Given: $f(x)=f(-x)$</p>
<p>then we obtain: $f'(x)=-f'(-x) \implies f'(0)=-f'(0) \iff 2f'(0)=0$ , hence $f'(0)=0$</p>
|
104,626 | <p>I encountered the following differential equation when I tried to derive the equation of motion of a simple pendulum:</p>
<p>$\frac{\mathrm d^2 \theta}{\mathrm dt^2}+g\sin\theta=0$</p>
<p>How can I solve the above equation?</p>
| user14717 | 24,355 | <p>Start with
$$
\frac{1}{2}\frac{\mathrm{d}\dot{\theta}^{2}}{\mathrm{d}\theta}
= \dot{\theta}\frac{\mathrm{d}\dot{\theta}}{\mathrm{d}\theta}
= \frac{\mathrm{d}\theta}{\mathrm{d}t}\frac{\mathrm{d}\dot{\theta}}{\mathrm{d}\theta} = \frac{\mathrm{d}\dot{\theta}}{\mathrm{d}t} = \ddot{\theta}
$$
Then your equation become... |
2,766,879 | <p>Show that there are no primitive pythagorean triple $(x,y,z)$ with $z\equiv -1 \pmod 4$. </p>
<p>I once have proven that, for all integers $a,b$, we have that $a^2 + b^2$ is congruent to $0$, or $1$, or $2$ modulo $4$. I feel like it is enough to conclude it by considering $a=x$, $b=y$ and $\gcd(x,y)=1$. But I am n... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: you have to prove that $$\frac{x^2+x+2}{x-1}-\frac{\frac{1}{x^2}+\frac{1}{x}+2}{\frac{1}{x}-1}\geq 8$$ and this is equivalent to $$\frac{(x+1)^3}{x(x-1)}\geq 8$$</p>
|
1,638,051 | <p>$$\int\frac{dx}{(x^{2}-36)^{3/2}}$$</p>
<p>My attempt:</p>
<p>the factor in the denominator implies</p>
<p>$$x^{2}-36=x^{2}-6^{2}$$</p>
<p>substituting $x=6\sec\theta$, noting that $dx=6\tan\theta \sec\theta$ </p>
<p>$$x^{2}-6^{2}=6^{2}\sec^{2}\theta-6^{2}=6^{2}\tan^{2}\theta$$</p>
<p>$$\int\frac{dx}{(x^{2}-36... | egreg | 62,967 | <p>Be careful: with the substitution you have
$$
(x^2-36)^{3/2}=(36\tan^2\theta)^{3/2}=216\tan^3\theta
$$
(at least in an interval where $\tan\theta$ is positive) so your integral becomes
$$
\int\frac{6\tan\theta\sec\theta}{216\tan^3\theta}\,d\theta=
\frac{1}{36}\int\frac{\cos\theta}{\sin^2\theta}\,d\theta=
-\frac{1}{3... |
2,683,326 | <p>I have a function $f(x)$ whose second order Taylor expansion is represented by $f_2(x)$. Is it true that $$f(x)>f_2(x)$$ for all $x$? Any help in this regard will be much appreciated. Thanks in advance.</p>
| Graham Kemp | 135,106 | <p>Rather than breaking a stick of length 1 into six parts with five breaks, break a circle of circumference 1 into six parts with six breaks and then pick one from those breaks as the "original ends". </p>
<p>So, what is the probability that all six breaks are not on the same semicircle?</p>
<p>Well wolog we pick o... |
2,837,683 | <p>I have to solve the integral $$\int_D \sqrt{x^2+y^2} dx dy$$ where $D=\{(x,y)\in\mathbb{R^2}: x^{2/3}+y^{2/3}\le1\}$.</p>
<p>I am not able to find a parameterization that suits the integrand.
I tried with $$\cases{x=(r\cos t)^3\\y=(r\sin t)^3}$$ in order to reduce the domain to a circle but then the integral become... | MR ASSASSINS117 | 546,265 | <p>With Polar Coordinates and with the Jacobian Transformation</p>
<p>$$\int_D\sqrt{x^2+y^2}\ dS\implies\iint_{\mathbb R^2} r\cdot rdS\implies\int_0^1\int_0^{2\pi}r^2dtdr$$</p>
|
1,176,958 | <p>I've been struggling with this for over an hour now and I still have no good results, the question is as follows:</p>
<blockquote>
<p>What's the probability of getting all the numbers from $1$ to $6$ by rolling $10$ dice simultaneously?</p>
</blockquote>
<p>Can you give any hints or solutions? This problem seems... | Victor | 142,550 | <p>The way I see this problem, I'd consider two finite sets, namely the set comprised by the $10$ dice (denoted by $\Theta$) and the set of all the possible outcomes (denoted by $\Omega$), in this particular situation, the six faces of the dice.</p>
<p>Therefore, the apparent ambiguity of the problem is significantly ... |
387,749 | <p>This comes from Guillemin and Pollack's book Differential Topology. The book claims that one cannot parametrize a unit circle by a single map. I thought we could (by a single angle $\theta$). </p>
<p>I think one possible answer might be the fact that if we let $\theta$ in [0, 2$\pi$), when $\theta$ approaches 2$\pi... | GCD | 74,703 | <p>Remember that parametrizations are supposed to be homeomorphisms onto their images.</p>
<p>The angle map $\theta\rightarrow e^{i\theta}$ (to use complex coordinates), as a map from $[0,2\pi)$ onto the circle $S^1$ is continuous and bijective. The issue is that it is not a homeomorphism. You can see this explicitly ... |
2,154,608 | <blockquote>
<p>Let $a$, $b$ and $c$ be non-negative numbers such that $a^3+b^3+c^3=3$. Prove that:
$$a^4b+b^4c+c^4a\leq3$$</p>
</blockquote>
<p>This inequality similar to the following.</p>
<blockquote>
<p>Let $a$, $b$ and $c$ be non-negative numbers such that $a^2+b^2+c^2=3$. Prove that:
$$a^3b+b^3c+c^3a\le... | Parcly Taxel | 357,390 | <p>Numberphile made <a href="https://youtu.be/2s4TqVAbfz4" rel="nofollow noreferrer">a video</a> on exactly this topic a while back. The idea is to consider the dihedral angle between adjacent faces for each of the five Platonic solids – a polychoron will consist of instances of one such solid. For the polychoron... |
1,356,783 | <p>What kind of mathematical object is this substitution(is it a function or what). We assuming set of variables exist.</p>
| dtldarek | 26,306 | <p>Let us construct a toy language with terms defined recursively as</p>
<p>$$\mathcal{T} = x \mid y \mid \mathtt{f(}t_1\mathtt{)} \mid \mathtt{g(}t_2\mathtt{,}t_3\mathtt{)} $$
where $x,y \in \mathcal{V}$ are variables and $t_1, t_2, t_3 \in \mathcal{T}$ are terms.
Now we could define substitution for our toy language... |
2,468,329 | <p>Let F be a field and choose an element $u \in F$. Consider the function $\epsilon_u:F[x]\rightarrow F$ given by $$\epsilon_u(a_nx^n+...+a_0)=a_nu^n+...+a_0$$</p>
<p>I am asked to show that this is surjective but not injective, as well as finding its kernel.</p>
<p>My idea is that this function will just send every... | Community | -1 | <p>Surjective: consider what happens to CONSTANT polynomials (polynomials of degree zero) under your mapping. Not injective: Consider the two polynomials $p(x) = x$ (of degree $1$) and $q(x) = u$ (a polynomial of degree zero, i.e. a constant). Are $p$ and $q$ equal? What are their images under the mapping?</p>
<p>Kern... |
719,055 | <p>I'm trying to show that if solid tori $T_1, T_2; T_i=S^1 \times D^2$ ,are glued by homeomorphisms between their respective boundaries, then the homeomorphism type of the identification space depends on the choice of homeomorphism up to, I think, isotopy ( Please forgive the rambling; I'm trying to put together a l... | Community | -1 | <p>In terms of differentials (in the single-variable case), $\frac{dy}{dx}$ is the unique scalar with the property that $\frac{dy}{dx}dx = dy$.</p>
<p>$\frac{dy}{du} \frac{du}{dx}$ therefore has the property that</p>
<p>$$\frac{dy}{du} \frac{du}{dx} dx = \frac{dy}{du} du = dy $$</p>
<p>therefore $\frac{dy}{du}\frac{... |
719,055 | <p>I'm trying to show that if solid tori $T_1, T_2; T_i=S^1 \times D^2$ ,are glued by homeomorphisms between their respective boundaries, then the homeomorphism type of the identification space depends on the choice of homeomorphism up to, I think, isotopy ( Please forgive the rambling; I'm trying to put together a l... | DanielV | 97,045 | <p>The relation
$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$</p>
<p>requires that $u$ has a nonzero relationship with $x$ and $y$. This doesn't have to be explicitly stated when the chain rule is written as
$$D_x f(g(x)) = f'g(x) \cdot g'(x)$$</p>
<p>Example:</p>
<p>Consider using $g = \text{gravitation... |
340,575 | <p>I got my exam on Thursday, and just got a few questions left. Anyway I would aprreciate help a lot! Can anyone please help me to solve this task? You can see the picture below. The need is to finde the size of the two radius. I thought about working with cords, like the cord AC is the same size like another one. Sti... | Adi Dani | 12,848 | <p>$AM_1M_2B$ is a right angle trapez from figure we can see that
$$\frac{r_1+r_2}{2} AB=\frac {AB+r_1+r_2}{2}r_1=AM_1C+BM_2C+ABC $$
where
$$AB=\sqrt{12^2+9^2}=15$$
$$AM_1C=6\sqrt{r_1^2-6^2}$$
$$BM_2C=9/2\sqrt{r_2^2-(9/2)^2}$$
$$ABC=54$$
so we get the system</p>
<p>$$\frac{r_1+r_2}{2} 15=\frac {15+r_1+r_2}{2}r_1$$
$$\... |
1,116,022 | <p>I've always had this doubt.
It's perfectly reasonable to say that, for example, 9 is bigger than 2.</p>
<p>But does it ever make sense to compare a real number and a complex/imaginary one?</p>
<p>For example, could one say that $5+2i> 3$ because the real part of $5+2i
$ is bigger than the real part of $3$? Or i... | Ross Millikan | 1,827 | <p>You can put (partial) orders on the complex numbers. One choice is to compare the real parts and ignore the complex ones. Another is to use the lexicographic order, comparing the real parts and then comparing the imaginary ones if the real parts are equal. Another is to use the modulus. There are many more. The... |
1,116,022 | <p>I've always had this doubt.
It's perfectly reasonable to say that, for example, 9 is bigger than 2.</p>
<p>But does it ever make sense to compare a real number and a complex/imaginary one?</p>
<p>For example, could one say that $5+2i> 3$ because the real part of $5+2i
$ is bigger than the real part of $3$? Or i... | dustin | 78,317 | <p>Since $\mathbb{R}\subset\mathbb{C}$, every $x\in\mathbb{R}$ can be written as $x + i\cdot 0$. Now if we prescribe the lexicographical (dictionary) ordering, we can compare them. </p>
<p>Let $z,w\in\mathbb{C}$ and $z = x+iy$ and $w=a+bi$.
Then the lexicographical ordering is $z < w$ if $x<a$ or $x=a$ and $y<... |
1,116,022 | <p>I've always had this doubt.
It's perfectly reasonable to say that, for example, 9 is bigger than 2.</p>
<p>But does it ever make sense to compare a real number and a complex/imaginary one?</p>
<p>For example, could one say that $5+2i> 3$ because the real part of $5+2i
$ is bigger than the real part of $3$? Or i... | fleablood | 280,126 | <p>Simple answer. No. The complex numbers can not be an ordered field. [if $a \ge 0$ then $a^2 = a*a \ge 0$. If $a < 0$ then $a^2 = a*a > 0$ so $a^2 \ge 0$ for all $a$ so $1 = 1^2 > 0$ and $-1 < 0$. If $\mathbb C$ were an ordered field, $i^2 > 0$ so $-1 > 0$. Impossible. $\mathbb C$ can not be a... |
1,116,022 | <p>I've always had this doubt.
It's perfectly reasonable to say that, for example, 9 is bigger than 2.</p>
<p>But does it ever make sense to compare a real number and a complex/imaginary one?</p>
<p>For example, could one say that $5+2i> 3$ because the real part of $5+2i
$ is bigger than the real part of $3$? Or i... | Abhishek Choudhary | 452,208 | <p>We can compare complex numbers just like we compare two-digit numbers,</p>
<p>For example, <span class="math-container">$53 > 42$</span> can be seen as <span class="math-container">$5 + 3i > 4 + 2i$</span></p>
<p>So, For <span class="math-container">$Z_{1} = a_{1} + ib_{1}$</span> and <span class="math-contain... |
2,651,394 | <p>I am attempting to create a function in Matlab which turns all matrix elements in a matrix to '0' if the element is not symmetrical. However, the element appears to not be reassigning.</p>
<pre><code>function [output_ting] = maker(a)
[i,j] = size(a);
if i ~= j
disp('improper input!')
else
end
c = 1;
b = a.';
... | Gwopmeat | 532,852 | <p>Your answer is correct lab, but the formal definition of a derivative is the ugliness provided in the picture above, and not the nice and neat cot(x) you described. Both methods, the chain rule and formal definition of a derivative, will always provide the same result - a derivative of a univariate function.</p>
|
4,331,790 | <blockquote>
<p><strong>Question 23:</strong> Which one of following statements holds true if and only if <span class="math-container">$n$</span> is a prime number?
<span class="math-container">$$
\begin{alignat}{2}
&\text{(A)} &\quad n|(n-1)!+1 \\
&\text{(B)} &\quad n|(n-1)!-1 \\
&\text{(C)} &... | Diger | 427,553 | <p>Calculating the generating function for the LHS has already been done following your link. It suffices to calculate the generating function for the RHS i.e.</p>
<p><span class="math-container">$$\sum_{n=0}^\infty \mathcal{J}_n x^n = e\sum_{n=0}^\infty x^n \sum_{k=0}^\infty \frac{(-1)^k}{k!} \binom{n+k}{k} = e \sum_... |
549,299 | <p>I'm searching(I searched this site first) for example of fields $F \subseteq K \subseteq L$ where $L/K$ and $K/F$ are normal but $L/F$ is not normal. Presenting some fields just for $F$ or $L$, instead of all three fields will help me too. Thanks for your attention.</p>
| Community | -1 | <p><strong>Hint</strong>: Consider $$\Bbb{Q} \subseteq \cdots \subseteq \Bbb{Q}[\sqrt[4]{2}]$$</p>
|
3,204,950 | <p>This question arose from Physics, where the force on an object attached on a spring is proportional to the displacement to the equilibrium (that is, the rest position). Also, if the displacement to the equilibrium is positive, the force will be negative, as it tries to pull the object back (i.e. if you pull a string... | John Doe | 399,334 | <p>Yes, proportionality does not tell anything about the sign of the proportionality constant. This is probably done in physics so that the proportionality constant <span class="math-container">$k$</span> can be considered to be always positive. This has a physical interpretation , so it is convenient for it to be posi... |
4,417,901 | <p>In the first chapter of "Differential Equations, Dynamical Systems and an Introduction to Chaos" by Hirch, Smale and Devaney, the authors mention the first-order equation <span class="math-container">$x'(t)=ax(t)$</span> and assert that the only general solution to it is <span class="math-container">$x(t)=... | Dr. Sundar | 1,040,807 | <p>Let us define
<span class="math-container">$$
I_n = \int\limits_{x = 0}^\infty \ x^n \, e^{-\lambda x} \ dx \tag{1}
$$</span></p>
<p><strong>Method 1: Using Gamma Functions</strong></p>
<p>Use the substitution
<span class="math-container">$$
\lambda x = t \ \ \mbox{or} \ \ x = {t \over \lambda} \tag{2}
$$</span></p>... |
33,582 | <p>My code finding <a href="http://en.wikipedia.org/wiki/Narcissistic_number">Narcissistic numbers</a> is not that slow, but it's not in functional style and lacks flexibility: if $n \neq 7$, I have to rewrite my code. Could you give some good advice?</p>
<pre><code>nar = Compile[{$},
Do[
With[{
n = 1000... | chyanog | 2,090 | <p>Dynamically generated <code>Do</code> loops:)</p>
<pre><code>cnar =
With[{n = 7},
With[{var = Array[Unique["x"] &, n]},
With[{n1 = FromDigits@var, n2 = Total[var^n]},
Compile[{Null},
Do[If[n1 == n2, Sow@n1], ##],
RuntimeOptions -> "Speed", CompilationTarget -> "C"
] &... |
4,159,341 | <p>There are <span class="math-container">$4$</span> coins in a box. One is a two-headed coin, there are <span class="math-container">$2$</span> fair coins, and the fourth is a biased coin that comes up <span class="math-container">$H$</span> (heads) with probability <span class="math-container">$3/4$</span>.</p>
<p>If... | Ittay Weiss | 30,953 | <p>Firstly, I'll assume that by "randomly select [and flip] 2 coins" you mean "uniformly select 2 coins". Now, naively there are <span class="math-container">$12$</span> possible pairs of coins to consider. However, there are only three types of coins. The outcome of <span class="math-container">$HH... |
2,148,861 | <p>In one of my junior classes, my Mathematics teacher, while teaching Mensuration, told us that <strong>metres square</strong> and <strong>square metres</strong> have a difference between them and <strong>metres cube</strong> and <strong>cubic metres</strong> too have a difference between them and that we should not m... | Ian Miller | 278,461 | <p>They have the same meaning.</p>
<p>What is sometimes confused is saying "four metres squared" versus "four square metres". The first is squaring the quantity four metres so you get an answer of $16m^2$ while the second is just a way of saying $4m^2$.</p>
|
82,770 | <p>I know of two places where $K_{*}(\mathbb{Z}\pi_{1}(X))$ (the algebraic $K$-theory of the group ring of the fundamental group) makes an appearance in algebraic topology. </p>
<p>The first is the Wall finiteness obstruction. We say that a space $X$ is finitely dominated if $id_{X}$ is homotopic to a map $X \righta... | Tim Porter | 3,502 | <p>First a slight quibble: the Whitehead group is the <i>origin</i> of algebraic K-theory, and predates the general stuff by quite a time, so your wording is not quite fair to Whitehead! </p>
<p>On your first question, one direction to look is at Waldhausen's K-theory. (The original paper is worth studying, but you s... |
376,600 | <p>$$\lim_{n\to\infty} \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^n}\,dx $$</p>
<p>Mathematica tells me the answer is 0, but how can I go about actually proving it mathematically?</p>
| robjohn | 13,854 | <p><strong>Hint:</strong> Apply <a href="http://en.wikipedia.org/wiki/Dominated_convergence_theorem" rel="nofollow">Dominated Convergence</a>.</p>
<blockquote>
<p><strong>Dominated Convergence:</strong> Suppose $|f_n(x)|\le g(x)$ where $f_n$ is measurable and $g(x)$ is a non-negative integrable function. If $\lim\li... |
149,872 | <p>How would I show that $|\sin(x+iy)|^2=\sin^2x+\sinh^2y$? </p>
<p>Im not sure how to begin, does it involve using $\sinh z=\frac{e^{z}-e^{-z}}{2}$ and $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$?</p>
| user59776 | 59,776 | <p>\begin{align}
\sin(z)^2 &= (\sin x \cos (iy))^2 +(\cos x \sin(iy))^2 \\
&=\sin^2x \cosh^2y+\cos^2x \sinh^2y \\
&= \sin^2x (1+\sinh^2y )+(1-\sin^2x ) \sinh^2y \\
&=\sin^2 x+\sin^2 x \sinh^2y+ \sinh^2y-\sin^2x \sinh^2y \\
&=\sin^2x+\sinh^2y
\end{align}</p>
|
595,280 | <p>Let $V$ be a vector space over $\Bbb F$, and let $x\not=0,y\not=0 $ be two elements in $V$. </p>
<p>I want to show that $x\otimes_{_F} y=y\otimes_{_F} x$ iff $x=ay$ where $a\in \Bbb F$.</p>
<p>I know the second direction, so only want to see the first direction (If case).</p>
| Shuchang | 91,982 | <p>Take a set of basis $e_1,\ldots,e_n$ of $V$ and let $x=\sum_
{i}x^ie_i,~y=\sum_
{j}y^je_j$, then
$$x\otimes y=\sum_
{i,j}(x^ie_i)\otimes(y^je_j)=\sum_
{i,j}x^iy^je_i\otimes e_j$$
$$y\otimes x=\sum_
{i,j}y^jx^ie_j\otimes e_i$$
The symmetry implies
$$x^iy^j=x^jy^i$$
That is,
$$\frac{x^i}{y^i}=\frac{x^j}{y^j}=a$$
for s... |
125,317 | <p>Consider a (locally trivial) fiber bundle $F\to E\overset{\pi}{\to} B$, where $F$ is the fiber, $E$ the total space and $B$ the base space. If $F$ and $B$ are compact, must $E$ be compact? </p>
<p>This certainly holds if the bundle is trivial (i.e. $E\cong B\times F$), as a consequence of Tychonoff's theorem. It al... | Mariano Suárez-Álvarez | 274 | <p>By local triviality, there is a open covering $\mathcal U$ of $B$ such that for each $U\in\mathcal U$ the open subset $\pi^{-1}(U)$ of $E$ is homeomorphic to $U\times F$ in a way compatible with the projection to $U$. It follows that there is a subbase $\mathcal S$ of the topology of $E$ consisting of open sets each... |
3,121,103 | <p>The integral surface of the first order partial differential equation <span class="math-container">$$2y(z-3)\frac{\partial z}{\partial x}+(2x-z)\frac{\partial z}{\partial y} = y(2x-3)$$</span> passing through the curve <span class="math-container">$x^2+y^2=2x, z = 0$</span> is</p>
<ol>
<li><span class="math-contain... | JJacquelin | 108,514 | <p>Four equations are proposed and it is asked to find which one satisfies both properties :</p>
<ul>
<li><p>First : <span class="math-container">$x^2+y^2=2z$</span> at <span class="math-container">$z=0$</span></p></li>
<li><p>Second : Is solution of the PDE.</p></li>
</ul>
<p>HINT :</p>
<p>By inspection, it is obvi... |
2,483,611 | <p>I believe the answer is 13 * $13\choose4$ * $48\choose9$.</p>
<p>There are $13\choose4$ to draw 4 of the same cards, and multiply by 13 for each possible rank (A, 2, 3, ..., K). Then there are $48\choose9$ to choose the remaining cards.</p>
<p>One thing I am not certain of, is whether this accounts for the possibi... | Claude Leibovici | 82,404 | <p>May be, you could consider the function $$P(x)=x^4 -4(m+2)x^2 + m^2$$
$$P'(x)=4x^3 -8(m+2)x$$ $$P''(x)=12x^2 -8(m+2)$$</p>
<p>The first derivative cancels at $$x_1=-\sqrt{4+2m} \qquad x_2=0\qquad x_3=\sqrt{4+2m}$$ For these points
$$P(x_1)=-3 m^2-16 m-16 \qquad P(x_2)=m^2 \qquad P(x_3)=-3 m^2-16 m-16$$ </p>
<p>So,... |
2,025,934 | <p>May $V$ be an $n$ dimensional Vektorspace such that $\dim (V) =: n \ge 2$.</p>
<p>We shall prove, that there are infinitely many $k$-dimensional subspaces of $V$, $\forall k \in \{1, 2, ..., n-1\}$.</p>
<p>So first, I thought about using induction, the base step is not that hard, for $n=2$ we take two vectors, say... | Djura Marinkov | 361,183 | <p>Subspaces:$(a_1\times a_2\times...\times a_{k-1}\times(a_k+ma_{k+1}))$, $m\in N$ </p>
|
1,619,292 | <p>Let $\mathbf C$ be an abelian category containing arbitrary direct sums and let $\{X_i\}_{i\in I}$ be a collection of objects of $\mathbf C$. </p>
<p>Consider a subobject $Y\subseteq \bigoplus_{i\in I}X_i$ and put $Y_i:=p_i(Y)$ where $p_i:\bigoplus_{i\in I}X_i\longrightarrow X$
is the obvious projection. </p>
<p>I... | Martín-Blas Pérez Pinilla | 98,199 | <p>Divulgative:</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/0631232516" rel="noreferrer">Does God Play Dice? The New Mathematics of Chaos</a> by Ian Stewart.</p>
<p>A bit more advanced:</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/0521477476" rel="noreferrer">Explaining Chaos</a> by Peter Sm... |
2,581,135 | <blockquote>
<p>Find: $\displaystyle\lim_{x\to\infty} \dfrac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}.$</p>
</blockquote>
<p>Question from a book on preparation for math contests. All the tricks I know to solve this limit are not working. Wolfram Alpha struggled to find $1$ as the solution, but the solution process pre... | Mr Pie | 477,343 | <p>Let $\Lambda =$ the limit we need to find. Then, $ \ \Box\ \Lambda = 1$.</p>
<hr>
<p><em>Proof</em>: We will begin our proof using the following <em>Lemma</em>. $$\forall a, b\in\mathbb{R}, \ \sqrt{a + \sqrt{b}} = \sqrt{\frac{a + \sqrt{a^2 - b}}{2}} + \sqrt{\frac{a - \sqrt{a^2 - b}}{2}}.\tag1$$
Substitute $a = x$ ... |
2,828,205 | <p><a href="https://i.stack.imgur.com/JJRaZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JJRaZ.png" alt="enter image description here"></a></p>
<p>First we take identity element from set which is Identity matrix so S=I
for which b(σ(x),σ(y))=b(x,y) which is identity transformation in O(V,b)
so k... | Federico Fallucca | 531,470 | <p>We want prove the surjectivity of the map $\Psi$.</p>
<p>We had fix a base $\beta:=\{v_1,...,v_n\}$ on $V$ and the function $\Psi$ maps every $\sigma\in O(V,b)$ to matrix $A_\sigma$ associate with itself in the base $\beta$.</p>
<p>If $A=(a_{ij})\in O(V,B)$ we can define a function</p>
<p>$\sigma’: \beta\to V$</p... |
276,987 | <p>I want to visualize the following set in Maple:</p>
<blockquote>
<p>$\lbrace (x+y,x-y) \vert (x,y)\in (-\frac{1}{2},\frac{1}{2})^{2} \rbrace$ </p>
</blockquote>
<p>Which commands should I use? Is it even possible?</p>
| Brian M. Scott | 12,042 | <p>Here’s a characterization of maximal non-discrete topologies.</p>
<blockquote>
<p><strong>Lemma.</strong> Let $\tau$ be a non-discrete topology on a set $X$, and let $N=\big\{x\in X:\{x\}\notin\tau\big\}\ne\varnothing$. Then $\tau$ is maximal non-discrete iff </p>
<ol>
<li>$\tau$ induces the discrete topol... |
636,730 | <p>Let $G$ be a group of infinite order . Does there exist an element $x$ belonging to $G$ such that $x$ is not equal to $e$ and the order of $x$ is finite?</p>
| Amr | 29,267 | <p>No. The only element of $\mathbb{Z}$ that has finite order is $0$</p>
|
636,730 | <p>Let $G$ be a group of infinite order . Does there exist an element $x$ belonging to $G$ such that $x$ is not equal to $e$ and the order of $x$ is finite?</p>
| fkraiem | 118,488 | <p>Sometimes no, for example in $(\mathbf{Z},+)$. Sometimes yes, for example in $(\mathbf{R}^*,\times)$.</p>
|
1,618,411 | <p>I'm learning the fundamentals of <em>discrete mathematics</em>, and I have been requested to solve this problem:</p>
<p>According to the set of natural numbers</p>
<p>$$
\mathbb{N} = {0, 1, 2, 3, ...}
$$</p>
<p>write a definition for the less than relation.</p>
<p>I wrote this:</p>
<p>$a < b$ if $a + 1 <... | Nephente | 96,393 | <p>A way to think about the natural numbers is in terms of the <a href="https://en.wikipedia.org/wiki/Peano_axioms" rel="nofollow">Peano Axioms</a>.
There exists a "successor" map </p>
<p>$$ S: \mathbb{N}\rightarrow \mathbb{N} $$
such that in particular</p>
<ul>
<li>$S(0) = 1 $</li>
<li>$0\notin S(\mathbb{N}) $</li>
... |
4,474,806 | <p>I use the following method to calculate <span class="math-container">$b$</span>, which is <span class="math-container">$a$</span> <strong>increased</strong> by <span class="math-container">$x$</span> percent:</p>
<p><span class="math-container">$\begin{align}
a = 200
\end{align}$</span></p>
<p><span class="math-cont... | insipidintegrator | 1,062,486 | <p>Notice that what you have done is basically exploit the first-order approximation of the Taylor series of <span class="math-container">$\frac{1}{1-x}$</span>: <span class="math-container">$$\displaystyle\frac{1}{1-x}=1+x+x^2+x^3+… for |x|\lt1$$</span> <span class="math-container">$≈1+x $</span>; for <span class="ma... |
1,046,066 | <p>Is this series $$\sum_{n\geq 1}\left(\prod_{k=1}^{n}k^k\right)^{\!-\frac{4}{n^2}} $$ convergent or divergent?</p>
<p>My attempt was to use the comparison test, but I'm stuck at finding the behaviour of $\displaystyle \prod_1^n k^k$ as $n$ goes to infinity. Thanks in advance.</p>
| user 1591719 | 32,016 | <p>By <strong>Cesaro-Stolz theorem</strong>, we have that $$\lim_{n\to\infty}\frac{1}{n^2 \log(n)}\sum_{k=1}^{n} k\log(k)=\frac{1}{2}$$ and then we see that for $n$ large, we have that $$\frac{1}{n^2}\sum_{k=1}^{n} k\log(k)\approx\frac{\log(n)}{2}\Rightarrow \frac{-4}{n^2}\sum_{k=1}^{n} k\log(k)\approx -2\log(n)$$and... |
3,165,937 | <p>Are orthogonal groups are lie groups? I think parameter space points corresponds to elements with determinant -1 break analytic property of lie groups , what is the general condition to check a group is lie group or not ?</p>
| Dietrich Burde | 83,966 | <p>Orthogonal groups can be defined over arbitrary fields <span class="math-container">$K$</span> as the subgroup of the general linear group <span class="math-container">$GL_n(K)$</span> given by
<span class="math-container">$$
\operatorname {O} (n,K)=\left\{Q\in \operatorname {GL} (n,K)\;\left|\;Q^{\mathsf {T}}Q=QQ^... |
2,658,195 | <p>I have the following problem with which I cannot solve. I have a very large population of birds e.g. 10 000. There are only 8 species of birds in this population. The size of each species is the same.</p>
<p>I would like to calculate how many birds I have to catch, to be sure in 80% that I caught one bird of each s... | farruhota | 425,072 | <p>It is a binomial experiment. Let $X$ is a number of UPs. When die is rolled three times, i.e. $n=3$, then:
$$P(X=0)=\frac18; \ P(X=1)=\frac38; \ P(X=2)=\frac38; \ P(X=3)=\frac18.$$
So selecting one or two UPs have higher chances. </p>
<p>Similarly if $n=4$, the combinations are:
$$C(4,0)=1; \ C(4,1)=4; \ C(4,2)=6; ... |
546,572 | <p><img src="https://i.stack.imgur.com/aJ2t5.jpg" alt="enter image description here"></p>
<p>Could anyone tell me how to solve 9b and 10? I've been thinking for five hours, I really need help.</p>
| bof | 97,206 | <p>$$f(x)=\cos x^2, A=f^{-1}(1),B=f^{-1}(-1)$$</p>
|
2,196,413 | <blockquote>
<p>Let $R$ be a commutative ring. Denote by $R^*$ the group of invertible elements (this is a group w.r.t multiplication.) Suppose $R^*\cong \mathbb{Z}$. I need to show that $1+1=0$ in $R$.</p>
</blockquote>
<p>I have no clue about why such statement should be true. I don't even have an example for a r... | lhf | 589 | <p><em>Hint 1:</em> $-1$ is a unit and so is a power of $u$, where $u$ is a generator of $R^\times$.</p>
<p><em>Hint 2:</em>
$(-1)^2=1$. What are the elements of finite order in $\mathbb Z$ ?</p>
|
2,414,492 | <p>Check the convergence of $$\sum_{k=0}^\infty{2^{-\sqrt{k}}}$$
I have tried all other tests (ratio test, integral test, root test, etc.) but none of them got me anywhere.
Pretty sure the way to do it is to check the convergence by comparison, but not
sure how.</p>
| farruhota | 425,072 | <p>Referring to Miguel's comment on integral test:
$$\begin{align}
\int_{0}^{\infty} \frac{1}{2^{\sqrt{x}}}dx
&=\int_{0}^{\infty} \frac{1}{2^t}\cdot 2tdt\\
&=2\left(t\cdot \frac{-2^{-t}}{\ln{2}}\bigg{|}_{0}^{\infty}+\int_{0}^{\infty} \frac{2^{-t}}{\ln{2}}dt\right)\\
&=2\left(-\frac{2^{-t}}{\ln^2{2}}\bigg{|... |
1,335,950 | <p>I have the following sum ($n\in \Bbb N)$:
$$ \frac {1}{1 \times 4} + \frac {1}{4 \times 7} + \frac {1}{7 \times 10} +...+ \frac {1}{(3n - 2)(3n + 1)} \tag{1} $$
It can be proved that the sum is equal to
$$ \frac{n}{3n + 1} \tag{2}$$
My question is, how do I get the equality? I mean, if I hadn't knew the formula $(2... | please delete me | 249,542 | <p>Write $\frac{1}{(3n-2)(3n+1)}=\frac{1}{3}(\frac{1}{3n-2}-\frac{1}{3n+1})$ before summing.</p>
|
2,732,202 | <p>$Z_1, Z_2, Z_3,...$ are independent and identically distributed R>V.s s.t. $E(Z_i)^- < \infty$ and $E(Z_i)^+ = \infty$. Prove that $$\frac {Z_1+Z_2+Z_3+\cdots+Z_n} n \to \infty$$ almost surely.</p>
<p>What does $E(Z_i)^+$ $E(Z_i)^-$ mean? I believe it is integrating fromnegative infinity to zero and zero to posi... | Davide Giraudo | 9,849 | <p>If $x$ is a real number, $x^+$ denotes the positive part of $x$, that is, $x^+=\max\{x,0\}$ and $x^-=\max\{x,0\}-x$ so that $x=x^+-x^-$. Write $Z_i=Z_i^+-Z_i^-$. An application of the strong law of large number to $\left(Z_i^-\right)_{i\geqslant 1}$ shows that $n^{-1}\sum_{i=1}^nZ_i^-\to \mathbb E\left[Z_1^-\right] ... |
1,047,544 | <p>I'm doing some research and I'm trying to compute a closed form for $ \mathbb{E}[ X \mid X > Y] $ where $X$, $Y$ are independent normal (but not identical) random variables. Is this known?</p>
| bijection | 35,625 | <p>yes, $\mathbb{E}(X|X > y)$ has a closed form as the expectation of a truncated normal. However, integrating that expression times the pdf of $Y$, is difficult. The normal r.v. are arbitrary. Does anyone have a satisfactory answer or is there no known closed form? </p>
|
3,468,336 | <p>Consider the multiplication operator <span class="math-container">$A \colon D(A) \to L^2(\mathbb R)$</span> defined by
<span class="math-container">$$\forall f \in D(A): \quad(Af)(x) = (1+\lvert x \rvert^2)f(x),$$</span>
where <span class="math-container">$$D(A) := \left \{f\in L^2(\mathbb R): (1+\lvert x \rvert^2)... | Clarinetist | 81,560 | <p>Observe <span class="math-container">$$\left(\sum_{i=1}^{n}X_i\right)^2 = \sum_{i=1}^{n}\sum_{j=1}^{n}X_iX_j = \sum_{i = j}X_iX_j + \sum_{i \neq j}X_iX_j = \sum_{i=1}^{n}X_i^2 + \sum_{i\neq j}X_iX_j$$</span>
yielding an expected value of
<span class="math-container">$$n(\lambda^2 + \lambda) + \lambda^2(n^2 - n) = n... |
1,203,922 | <p>Show that it is possible to divide the set of the first twelve cubes $\left(1^3,2^3,\ldots,12^3\right)$ into two sets of size six with equal sums.</p>
<p>Any suggestions on what techniques should be used to start the problem?</p>
<p>Also, when the question is phrased like that, are you to find a general case that ... | barak manos | 131,263 | <p>First, you've noticed that each side of the equation has to sum up to $3042$.</p>
<p>Since $11^3+12^3=3059>3042$, each one of them has to be on a different group.</p>
<p>Since $3042$ is even, each group must contain an even number of odd numbers.</p>
<p>So each group must contain $0$ or $2$ or $4$ or $6$ odd n... |
1,245,651 | <p>In algebra, I learned that if <span class="math-container">$\lambda$</span> is an eigenvalue of a linear operator <span class="math-container">$T$</span>, I can have
<span class="math-container">\begin{equation}
Tx = \lambda x
\tag{1}
\end{equation}</span>
for some <span class="math-container">$x\neq 0$</span>, whic... | Disintegrating By Parts | 112,478 | <p>The multiplication operator <span class="math-container">$(Mf)(x)=xf(x)$</span> on <span class="math-container">$L^{2}[0,1]$</span> is a classical example of an operator with no eigenvalues, but its spectrum is <span class="math-container">$[0,1]$</span>.</p>
<p><span class="math-container">$M$</span> has no eigenva... |
2,911,187 | <p>Lines (same angle space between) radiating outward from a point and intersecting a line:</p>
<p><a href="https://i.stack.imgur.com/52HY4.png" rel="noreferrer"><img src="https://i.stack.imgur.com/52HY4.png" alt="Intersection Point Density Distribution"></a></p>
<p>This is the density distribution of the points on t... | Andreas | 317,854 | <p>The curve is a density function. The idea is the following. From your first picture, assume that the angles of the rays are spaced evenly, the angle between two rays being $\alpha$. I.e. the nth ray has angle $n \cdot \alpha$. The nth ray's intersection point $x$ with the line then follows $\tan (n \cdot \alpha) =... |
3,583,330 | <p>I've approached the problem the following way : </p>
<p>Out of the 7 dice, I select any 6 which will have distinct numbers : 7C6.</p>
<p>In the 6 dice, there can be 6! ways in which distinct numbers appear.</p>
<p>And lastly, the last dice will have 6 possible ways in which it can show a number.</p>
<p>So the re... | Mathsmerizing | 757,478 | <p>We need 2 alike and 5 distinct numbers on the die out of {<span class="math-container">$1,2,3,4,5,6$</span>}, which can be selected in C(6,1).C(5,5) ways. Now 2 alike and 5 distinct numbers can be arranged in <span class="math-container">$\frac{7!}{2!}$</span> ways. Total number of points in the sample space as you ... |
4,279,076 | <p>I have seen in wikipedia that irrational numbers have infinite continued fraction but I also found <span class="math-container">$$1=\frac{2}{3-\frac{2}{3-\ddots}}$$</span> so my question is that does that mean <span class="math-container">$1$</span> is irrational because it can be written as an infinite continued fr... | L0Ludde0 | 744,946 | <p>No it does not, since the logic behind the statement is</p>
<p><span class="math-container">$x$</span> irrational <span class="math-container">$\implies$</span> <span class="math-container">$x$</span> can be written as an infinite continued fraction.</p>
<p>However, this does not necessarily mean that rationals <em>... |
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