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 |
|---|---|---|---|---|
971,997 | <p>How do we find
$$\Re\left[\int_0^{\large\frac{\pi}{2}} e^{\Large e^{i\theta}}d\theta\right]$$</p>
<p>In the shortest and easiest possible manner?</p>
<p>I cannot think of anything good.</p>
| Semiclassical | 137,524 | <p>An expression in Bessel functions for a generalization of this integral can be developed using the Jacobi-Anger identities $$e^{z \cos \theta}=\sum_{n=-\infty}^\infty I_n(z) e^{i n \theta},\quad e^{i z \sin \theta}=\sum_{n=-\infty}^\infty J_n(z)e^{i n \theta}.$$</p>
<p>Then $e^{z e^{i \theta}}=e^{z\cos \theta}e^{z ... |
2,174 | <p>As a teenager I was given this problem which took me a few years to solve. I'd like to know if this hae ever been published. When I presented my solution I was told that it was similar to one of several he had seen.</p>
<p>The problem:</p>
<p>For an <span class="math-container">$n$</span> dimensional space, develo... | Robin Chapman | 226 | <p>I presume you are asking for the number of regions into which $n$-space
is divided into by $k$ hyperplanes "in general position".
See the notes of Richard Stanley on <a href="http://math.mit.edu/~rstan/arrangements/arr.html" rel="nofollow">hyperplane arrangements</a>.
The answer to your problem is Proposition 2.4, ... |
2,610,048 | <p>Ok, so I'm trying to prove statement in the header. I have read the following discussion on it, but I can't seem to follow it all the way through:</p>
<p><a href="https://math.stackexchange.com/questions/1198735/proving-sum-k-1n-k-k-n1-1/1198743#1198743?newreg=abbcf872d9904cbfa76cd161c4fecdd0">Proving $\sum_{k=1}^... | S. Gates | 519,648 | <p>You have 2 pairs of dice (4 dies) and you roll the dice once (1) and 3 times (2) and are looking for snake eyes (two 1’s). What is the probability (P) of rolling snake eyes exclusively (3) and inclusively (4). Note that there will be 4 answers at the end.</p>
<p>Answer: (can this be explained step by step and not ... |
1,255,629 | <p>Show that </p>
<p>$$\sin\left(\frac\pi3(x-2)\right)$$ </p>
<p>is equal to </p>
<p>$$\cos\left(\frac\pi3(x-7/2)\right)$$</p>
<p>I know that $\cos(x + \frac\pi2) = −\sin(x)$ but i'm not sure how i can apply it to this question.</p>
| egreg | 62,967 | <p>The equality $\sin \alpha=\cos\beta$ can be written
$$
\cos(\pi/2-\alpha)=\cos\beta
$$
which is satisfied when either
$$
\beta=\frac{\pi}{2}-\alpha+2k\pi \qquad(k \text{ integer})
$$
or
$$
\beta=\alpha-\frac{\pi}{2}+2k\pi \qquad(k \text{ integer})
$$
These can be rewritten respectively as
$$
\alpha+\beta=\frac{\pi}{... |
83,648 | <p><a href="http://reference.wolfram.com/language/ref/FullGraphics.html" rel="nofollow noreferrer"><code>FullGraphics</code></a> hasn't worked entirely for a long time and the situation appears to be getting worse instead of better. In <em>Mathematica</em> 10.0, 10.1, 11.3, 12.3 up to 13.1 a simple usage throws numero... | Alexey Popkov | 280 | <h2>Basic implementation</h2>
<p>After working on <a href="https://mathematica.stackexchange.com/a/271580/280">this</a> answer, I realized that the undocumented <a href="https://mathematica.stackexchange.com/a/1411/280"><code>FrontEnd`ExportPacket</code></a> with <code>"InputForm"</code> as second argument ma... |
2,927,374 | <p>In a right triangle, relative to a <span class="math-container">$45^\circ$</span> angle, if we have
<span class="math-container">$$\text{adjacent} = 1$$</span>
<span class="math-container">$$\text{opposite} = 1$$</span></p>
<p>then
<span class="math-container">$$\text{hypotenuse}=\sqrt{o^2+a^2}=\sqrt{1^2+1^2}=\sqr... | Lucas | 478,855 | <p>Nothing went wrong.
<span class="math-container">$$ \frac{\sqrt{\frac{1}{2}}}{1} = \sqrt{\frac12}=\frac1{\sqrt2} $$</span></p>
|
710,146 | <p>Conversely, is it true that if every sequence of pointwise equicontinuous functions from $M$ to $\mathbb{R}$ is uniformly equicontinuous, them $M$ is compact?</p>
| Community | -1 | <p>This is not necessary. To see why this is the case, let us construct a counterexample as follows: Consider a family of constant functions $\{f_n|n\in \mathbb{N}\}$. Here, $f_n(x)=n$ for all $x\in M$. This family is uniformly equicontinuous. However, the sequence $(f_n)_n$ doesn't have a convergent sub-sequence in ev... |
1,313,056 | <p>Got this matrix: </p>
<p>\begin{bmatrix} 1 & 2 \\ -2 & 5 \end{bmatrix}</p>
<p>I should determine if the matrix is diagonalizable or not.
I found the eigenvalues ( only one) = 3.
My eigenvector is then \begin{bmatrix} 1 \\ 1 \end{bmatrix}
This matrix is not diagonizable (from my teachers notes) but i do... | Kolibrie | 202,998 | <p>To answer your comment about the eigenvectors for this matrix:
$$ A = \left[
\begin{array}{cc}
1&0\\
0&1
\end{array}
\right] $$
The characteristic polynomial given by $det(A - rI)$ where I is the n by n identity matrix will have roots equal to the eigenvalues of the matrix.<br>
(Note: $r$... |
4,285,189 | <p>I am aware of the following identity</p>
<p><span class="math-container">$\det\begin{bmatrix}A & B\\ C & D\end{bmatrix} = \det(A)\det(D - CA^{-1}B)$</span></p>
<p>When <span class="math-container">$A = D$</span> and <span class="math-container">$B = C$</span> and when <span class="math-container">$AB = BA$</... | achille hui | 59,379 | <p>Let's say <span class="math-container">$A, B$</span> are <span class="math-container">$n \times n$</span> matrices with entries from a field with characteristic <span class="math-container">$\ne 2{}^{\color{blue}{[1]}}$</span>.</p>
<p>Let <span class="math-container">$I$</span> be the <span class="math-container">$n... |
1,237,618 | <p>I really need Your help.</p>
<p>I need to prove that Euclidean norm is strictly convex. I know that a function is strictly convex if $f ''(x)>0$. Can I use it for Euclidean norm and how? $||x||''=\frac{||x||^2-x^2}{||x||^3}$</p>
<p>Thank You!</p>
| Brenton | 226,184 | <p>It is easier to work with the definition that a function f is convex if for any $\lambda \in [0,1]$ and for all $x,y \in (a,b)$:
$$ f((1-\lambda)x + \lambda y) \leq (1-\lambda)f(x) + \lambda f(y)$$</p>
<p>Then:</p>
<p>$$ ||(1-\lambda x) + \lambda y ||^2 = \sum_{i} |(1-\lambda x_i) + \lambda y_i|^2 \leq \sum_{i} (... |
1,237,618 | <p>I really need Your help.</p>
<p>I need to prove that Euclidean norm is strictly convex. I know that a function is strictly convex if $f ''(x)>0$. Can I use it for Euclidean norm and how? $||x||''=\frac{||x||^2-x^2}{||x||^3}$</p>
<p>Thank You!</p>
| Mercy King | 23,304 | <p>The function
$$
\lVert \mbox{ }\rVert:\mathbb{R}^n\to \mathbb{R},\, f(x)=\lVert x\rVert
$$
is certainly convex. In fact, given $t\in [0,1]$ and $x,y\in \mathbb{R}^n$ we have:
$$
\lVert tx+(1-t)y\rVert=\lVert tx+(1-t)y\rVert\le \lVert tx\rVert+\lVert (1-t)y\rVert=t\lVert x\rVert+(1-t)\lVert y\rVert.
$$
But is it <st... |
80,787 | <p>Hello,</p>
<p>I am very new to the field of approximation theory, and since
an extended search on the Internet did not provide answers for
two rather basic questions, I decided to ask them here. </p>
<p>1) From my understanding upper bounds for</p>
<p>$$ \inf_{q} \int_{-1}^{1} |f(x) - q(x)|^{2p} dt $$</p>
<p>wi... | Ben Adcock | 19,011 | <p>A good introductory lookup for 1) (and similar problems) is the book "Spectral Methods: Fundamentals in Single Domains" by Canuto, Hussaini, Quarteroni & Zang. Chapter 5, in particular. Equation (5.4.16) gives a bound for the $L^p$ norm approximation problem in terms of the L^p smoothness of $f$ and its deriv... |
918,049 | <p>Let $f: X \to Y$ be a map of sets. We are given that $X$ is a topological space. We are to show that there is a topology on $Y$ making $f$ continuous, and moreover, determine if this topology is unique. </p>
<p>Should I read it as "$X$ is just a set on which there is given a topology, i.e sets that are open in $X$ ... | mathlove | 78,967 | <p>HINT : For the $n=p+1$ step, all you need is to prove $$\frac{1+\sqrt{4\alpha +1}}{2}\gt \sqrt{\frac{1+\sqrt{4\alpha +1}}{2}}.$$</p>
|
3,215,553 | <p>Recently I have been reading a lot about <span class="math-container">$\mathbb{Z}_2$</span>-actions on topological spaces. Mainly I was focused on surfaces such as the sphere, torus and Klein bottle and here the existence of a nontrivial <span class="math-container">$\mathbb{Z}_2$</span>-action is rather simple. But... | hmakholm left over Monica | 14,366 | <p><span class="math-container">$X=\mathbb R$</span> easily admits a nontrivial <span class="math-container">$\mathbb Z_2$</span>-action, but it cannot be free -- we can always find a fixed point with the intermediate value theorem. </p>
|
3,818,445 | <p>I'm trying to compute <span class="math-container">$\operatorname{Ext}_{\mathbb{Z}}^1(\mathbb{Z}[1/p],\mathbb{Z})\cong \mathbb{Z}_p/\mathbb{Z}$</span>.</p>
<p>Now I have the projective resolution
<span class="math-container">$$0\rightarrow \bigoplus_{i>1}\mathbb{Z}\xrightarrow{\alpha} \mathbb{Z}\oplus \bigoplus_{... | Fabio Lucchini | 54,738 | <p><span class="math-container">$\newcommand\ZZ{\mathbb{Z}}$</span>
<span class="math-container">$\DeclareMathOperator\Hom{Hom}$</span>
<span class="math-container">$\DeclareMathOperator\Ext{Ext}$</span>
Another approach using a free resolution of <span class="math-container">$\ZZ[1/p]$</span>.
First recall that we hav... |
3,969,679 | <p><strong>The letters in the word GUMTREE and KOALA are rearranged to form a 12-letter word where KOALA appears precisely in order but not necessarily together. How many ways can this happen?</strong></p>
<p>So I attmepted it via this method:</p>
<p>Firstly arrange like this since KOALA must be in order but not necess... | Math Lover | 801,574 | <p>Here is the problem with your method of counting -</p>
<p>You are giving each letter of GUMTREE a choice of being in any of the <span class="math-container">$6$</span> places. So far so good.</p>
<p>But <span class="math-container">$6^7$</span> does not include permutations of "GUMTREE". For example, in th... |
758,158 | <p>I am trying to use <span class="math-container">$f(x)=x^3$</span> as a counterexample to the following statement. </p>
<p>If <span class="math-container">$f(x)$</span> is strictly increasing over <span class="math-container">$[a,b]$</span> then for any <span class="math-container">$x\in (a,b), f'(x)>0$</span>. <... | Mariano Suárez-Álvarez | 274 | <p>Well, you know that $f$ is strictly increasing in $[0,+\infty)$ and in $(-\infty,0]$. Morevoer, $f$ is positive in the first interval and negative in the second one!</p>
|
725,241 | <p>Where s=circumcenter, H= orthocenter, and A'= midpoint of one side of triangle.
<img src="https://i.stack.imgur.com/Pg6rv.png" alt="enter image description here"></p>
<p>How can can I determine the location of the three vertices of the triangle? </p>
| user133281 | 133,281 | <p>Rewrite the right-hand side as $$\sum_{i=1}^k \binom{k-1}{i-1} \binom{n}{n-i}.$$</p>
<p>Now imagine we have $n$ men and $k-1$ women, and we want to count the number of ways to choose $n-1$ of these people. The left-hand side is the number of ways to do this, $\binom{n+k-1}{n-1}$ (we choose $n-1$ out of $n+k-1$ peo... |
2,845,427 | <p>I posted concerning this question a little while ago, not asking for the answer but for an understanding of the setup. Well, I thought I would solve it but I seem to be unable to obtain the required answer. If anyone could help I'd be very grateful.</p>
<p>Here's the question again:</p>
<p><a href="https://i.stack... | Cesareo | 397,348 | <p>Assuming the Hooke's law</p>
<p>$$
F = \lambda\left(\frac{l-l_0}{l_0}\right)
$$</p>
<p>Calling</p>
<p>$$
|AC|_0 = a\frac 43\\
|CB|_0 = a\frac 47\\
|AC| = 2a\sin(\frac{\pi}{3})\\
|CB| = 2a\sin(\frac{\pi}{6})\\
F_{AC} = \lambda\left(\frac{|AC|-|AC|_0}{|AC|_0}\right)(-\cos(\frac{\pi}{6}),\sin(\frac{\pi}{6}))\\
F_{CB... |
1,703,169 | <p>At a recent maths competition one of the questions was to find for which $x,y,z$ this equation holds true:</p>
<p>$$\sqrt{x}-\sqrt{z+y}=\sqrt{y}-\sqrt{z+x}=\sqrt{z}-\sqrt{x+y}$$</p>
<p>where $x,y,z \in \mathbb{R} \cup \{0\}$. So how am I supposed to approach this problem?</p>
<p>Also sorry for not explaining my p... | fleablood | 280,126 | <p>$x \ge 0; y \ge 0; z \ge 0$.</p>
<p>Suppose $z = 0$</p>
<p>Then $\sqrt{x} - \sqrt{y} = \sqrt{y} - \sqrt{x} = - \sqrt{x+y} \implies \sqrt{x} = \sqrt{y} \implies x = y; x + y = 0 \implies x = y = z = 0$.</p>
<p>Likewise if $x = 0$ or $y = 0$ then $x = y = z = 0$ by the same argument so either they all equal 0 or no... |
3,758 | <p>When implicitly finding the derivative of: </p>
<blockquote>
<p>$xy^3 - xy^3\sin(x) = 1$</p>
</blockquote>
<p>How do you find the implicit derivative of:</p>
<blockquote>
<p>$xy^3\sin(x)$</p>
</blockquote>
<p>Is it using a <em>triple</em> product rule of sorts?</p>
| Pierre-Yves Gaillard | 660 | <p>Here is a simple way to compute the $n$-th derivative of a product of $k$ functions. To ease the notation, stick to the case $k=3$. First follow your instinct, and then you'll see that the justification presents no difficulty. Write (with all summation indices varying from 0 to $\infty$)
$$\sum_n\frac{(fgh)^{(n)}}{n... |
3,557,398 | <p>Here are two second-order differential equations.</p>
<p><span class="math-container">$$ y''+9y=\sin(2t) \tag 1 $$</span></p>
<p><span class="math-container">$$ y'' +4y =\sin(2t) \tag 2 $$</span></p>
<p>I am told to use undetermine coefficients method to solve.</p>
<p>For 1), I use <span class="math-container">... | mjw | 655,367 | <p>The statement is true, and could be proved by considering each case:</p>
<p><span class="math-container">$$0<x<y<z$$</span></p>
<p><span class="math-container">$$0<x<z<y$$</span></p>
<p><span class="math-container">$$0<z<x<y$$</span></p>
<p><span class="math-container">$$\textit{etc.}$... |
84,897 | <pre><code>Integrate[a/(Sin[t]^2 + a^2), {t, 0, 2 Pi}]
</code></pre>
<p>$$\int_0^{2 \pi } \frac{a}{a^2+\sin ^2(t)} \, dt$$</p>
<p>gives $0$</p>
<p>This cannot be true. What is going on?</p>
<p>If I insert a number into <code>a</code>, it gives a reasonable result:</p>
<pre><code>NIntegrate[2/(Sin[t]^2 + 4), {t, 0,... | cheap_slut95 | 29,881 | <p>I confirmed the result in both <em>Mathematica</em> 9 and 10 on Linux.</p>
<p>Using an assumption seems to work:</p>
<pre><code>Integrate[a/(Sin[t]^2 + a^2), {t, 0, 2 π}, Assumptions -> a > 0]
</code></pre>
<blockquote>
<pre><code>(2 π)/Sqrt[1 + a^2]
</code></pre>
</blockquote>
<p>which agrees with <code>N... |
3,810,856 | <p>I'm having trouble understanding how this form of the principle (<a href="https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow noreferrer">on wiki</a>) results in the form below.</p>
<p>Wiki form:
<a href="https://i.stack.imgur.com/6IXQZ.png" rel="nofollow noreferrer"><img src="https://... | user | 505,767 | <p>We can cancel out both sides a factor <span class="math-container">$\sin \frac{\pi}{x}>0$</span> to obtain</p>
<p><span class="math-container">$$\frac{\pi}{\sin^2 \frac{\pi}{x}} \gt \frac{x}{\tan \frac{\pi}{x}}\iff \frac{\pi}{\sin^2 \frac{\pi}{x}} \gt \frac{x}{\frac{\sin\frac{\pi}{x}}{\cos\frac{\pi}{x}}}$$</span... |
1,614,899 | <p>Construct Context-free Grammar for integers. Integer can begin with + or - and after that we have non-empty string of digits. Integer must not contain unnecessary leading zeros and zero should not be preceded by + or -.
For example: 0; 123; -15; +9999 are correct, but +0; 01; +-3; +09; + are incorrect.</p>
<p>I hav... | egreg | 62,967 | <p>I'll work under the assumption that a <em>cut</em> $\gamma$ is a subset of $\mathbb{Q}$ such that</p>
<ol>
<li>$\gamma\ne\emptyset$;</li>
<li>for all $a,b\in\mathbb{Q}$, if $a<b$ and $b\in\gamma$, then $a\in\gamma$;</li>
<li>for all $a\in\gamma$, there exists $b\in\gamma$ with $a<b$;</li>
<li>there exists $c\... |
312,012 | <p>What is the exact difference between $\arg\max$ and $\max$ of a function?</p>
<p>Is it right to say the following?</p>
<blockquote>
<p>$\arg\max f(x)$ is nothing but the value of $x$ for which the value of the function is maximal. And $\max f(x)$ means value of $f(x)$ for which it is maximum.</p>
</blockquote>
... | Michael Hardy | 11,667 | <p>Suppose $f(x) = 100 - (x-6)^2$.</p>
<p>Then $\max\limits_x f(x) = 100$ and $\operatorname*{argmax}\limits_x f(x) = 6$.</p>
|
2,112,259 | <p>I want to prove or disprove $\min{S_1}=\min{S_2}\Longleftrightarrow\min{S_1}\in S_2\land\min{S_2}\in S_1$ but I don't know where to start.</p>
<p>Maybe take cases like $\min{S_1}<\min{S_2}$, $\min{S_1}>\min{S_2}$ and $\min{S_1}=\min{S_2}$ and see where it is true?</p>
| Erick Wong | 30,402 | <p>You've already done the hard part. Now that you know $p^2 - q^2 \equiv 1 \pmod 5$. But the only non-zero quadratic residues mod $5$ are $\pm 1$, so there are no solutions unless one of $p$ or $q$ is divisible by $5$. There are not many options for primes divisible by $5$.</p>
|
3,838,889 | <p>Exactly one ace given that it contains exactly two queens?</p>
<p>I am struggling how to solve this maths problem. I know that a hand of <span class="math-container">$10$</span> cards dealt from a normal pack of <span class="math-container">$52$</span> is <span class="math-container">$52\choose 10$</span> <span clas... | tommik | 791,458 | <p>You can use the definition of conditional probability.</p>
<p><span class="math-container">$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$</span></p>
<ul>
<li><p><span class="math-container">$P(A \cap B)$</span> is the probability to have 1A and 2Q in 10 cards. So you have</p>
<ul>
<li><span class="math-container">$\binom{4}{1}$... |
501,993 | <p>I have never encounter "proof" questions before in my career, but this question in textbook troubled me and I have totally no clue where to start.</p>
<p>Prove that the limit
$$\lim_{x\rightarrow0} \frac{1-\sqrt{1-x^2}}{x^2} = \frac{1}{2}$$</p>
| MathAdam | 266,049 | <p>L'Hôpital's rule to the rescue: </p>
<p>$$\lim_{x\rightarrow0} \frac{1-\sqrt{1-x^2}}{x^2} =\lim_{x\rightarrow0} \frac{d/dx(1-\sqrt{1-x^2})}{d/dx(x^2)} = \frac{1}{2\sqrt{1-x^2}}$$</p>
<p>Let $x:=0$</p>
<p>$$\frac{1}{2\sqrt{1-0^2}}=\frac{1}{2}$$</p>
|
551,873 | <p>I have to solve the following problem:
Using $\exists$ Introduction prove that PA$\vdash x\leq y \wedge y\leq z \longrightarrow x\leq z$: I used that if $x\leq y$ then $\ \exists \ r\ x+r=y$ and in the same way $\exists t \ y
+t=z$, but in logic term I don't know how to use these equalities and the Peano's axioms t... | CopyPasteIt | 432,081 | <p>See <a href="https://en.wikipedia.org/wiki/Peano_axioms" rel="nofollow noreferrer">Wikipedia Peano Axioms</a>.</p>
<p>The point of this answer is to show that you can define an ordering of the natural numbers before you define addition.</p>
<p>Proposition 1: If $n$ is any number except $0$, there exist an $m$ such... |
1,717,333 | <p>Feeling unsure, want to check if I am thinking correctly.</p>
<p>Exercise sounds: Let n>1, square-free integer. Show that $\mathbb{Z}[\sqrt{n}]/\langle \sqrt{n}\rangle\simeq\mathbb{Z}_n$</p>
<p>My take: Lets make suitable homomorphism $\mathbb{Z}[\sqrt{n}]\rightarrow\mathbb{Z}_n$. Let $x=a+\sqrt{n}b$, which is the... | learner | 228,313 | <p>Your proof's okay.</p>
<p>The map you define is indeed a homomorphism from $(\Bbb Z[\sqrt n],+,*)$ to $(\Bbb Z_n,+_n,*_n)$ since you have,</p>
<p>$$\begin{align}f((a+\sqrt n b)+(c+\sqrt n d))&=f((a+c)+\sqrt n(b+d))\\&=(a+c)~\bmod~n\\&=(a~\bmod~n)+_n(b~\bmod~n)=f(a+\sqrt n b)+_nf(c+\sqrt n d)\end{align}... |
2,862,166 | <p>In a paper I've been studying it says:</p>
<blockquote>
<p>Let $x$ in the cone $\mathbb R_+^n$ of all vectors in $\mathbb R^n$ with nonnnegative components ($n\in\mathbb N$)</p>
</blockquote>
<p>Somebody tell me what does it means, please? $\mathbb R_+^n$ should be $[0,\infty)\times\dots\times [0,\infty)$ ($n$ t... | Kavi Rama Murthy | 142,385 | <p>A cone $C$ is a set in a vector space with two propertes: $x+y \in C$ whenever $x,y \in C$ and $tx \in C$ whenever $x\in C$ and $t \geq 0$. Since $[0,\infty ) \times [0,\infty ) \times ...[0,\infty ) $ has these properties it is a cone. </p>
|
1,637,237 | <p>I'm studying differential calculus, but one of the questions involves solving an inequality:</p>
<p>$$(x-2)e^x < 0$$</p>
<p>I intend to go deeper in solving inequalities later, but I just want to understand how the teacher got the following solution in order to advance in these lectures:
$$x-2 < 0$$
$$x <... | 5xum | 112,884 | <p>For all values of $x$, $e^x>0$ is true. This means that $a\cdot e^x > 0$ is true if and only if $a>0$.</p>
|
4,114,360 | <p><strong>Question</strong> :</p>
<blockquote>
<p>Andy and Beth are playing a game worth $100. They take turns flipping
a penny. The first person to get 10 heads will win.</p>
<p>But they just realized that they have to be in math class right away
and are forced to stop the game.</p>
<p>Andy had four heads and Beth ha... | Mark Saving | 798,694 | <p>Consider the set <span class="math-container">$S = \{x | \forall n \in \mathbb{N}, x \neq n + \frac{1}{2n}\}$</span>. I claim that this set is open.</p>
<p>For consider some <span class="math-container">$x \in S$</span>. Then by the Archimedean property, we can find some <span class="math-container">$n$</span> such ... |
3,452,518 | <p>I will call an <em>average</em> any continuous function <span class="math-container">$f(v_1,\dots,v_n)$</span> of <span class="math-container">$n$</span> arguments such that it lies in the closed interval <span class="math-container">$[\min(v_{1},\dots ,v_{n});\max(v_{1},\dots ,v_{n})]$</span>, is symmetric for all ... | Andrei | 331,661 | <p>Not sure this matches your needs, but <span class="math-container">$f(v_1,...,v_n)=\min(v_1,...,v_n)$</span> might obey all your requirements. It is homogeneous of degree 1, symmetric in all permutation of arguments, and if only one argument goes to infinity, it will not change it's value. Also, it lies in the close... |
256,415 | <p>I need to store values of a function of x and y defined as <code>f[x_, y_] := Sin[Pi*x/3]*Sin[Pi*y/3]</code> at the following points <code>xval = {0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9, 9.5, 10, 10.5, 11, 11.5, 12, 12.5, 13, 13.5, 14, 14.5, 15};</code>, same for y.</p>
<p>I need... | Nasser | 70 | <p>another alternative is to use <code>meshgrid</code></p>
<pre><code>ClearAll[f, x, y];
f[x_, y_] := Sin[Pi*x/3]*Sin[Pi*y/3];
meshgrid[x_List, y_List] := {ConstantArray[x, Length[x]],
Transpose@ConstantArray[y, Length[y]]}
{xx, yy} = meshgrid[Range[0, 15, .5], Range[0, 15, .5]];
u = f[xx, yy];
pts = Flatten[{xx, yy... |
4,500,550 | <p>I have a very basic question about cohomology of sheaves. Suppose <span class="math-container">$\mathcal{F}$</span> is a sheaf of abelian groups over a topological space <span class="math-container">$X$</span>. Then <span class="math-container">$\mathcal{F}$</span> itself is a topological space with a continuous map... | C.D. | 553,510 | <p>I had some fun playing with examples of the espace étalé in different settings. I'll write <span class="math-container">$X_\mathcal{F}$</span> for the espace étalé of <span class="math-container">$\mathcal{F}$</span> on <span class="math-container">$X$</span>.
Summary:</p>
<ol>
<li>In differential/topological se... |
404,659 | <p>The following is copied verbatim from a book (I. Protasov, <em>Combinatorics of numbers</em>, p. 14):</p>
<blockquote>
<p>Suppose that to each point <span class="math-container">$x$</span> of a set <span class="math-container">$X$</span> a collection <span class="math-container">$\mathcal{B}(x)$</span> of subsets of... | kjo | 13,675 | <p>I think I finally got it. All we need to do is replace B3 with</p>
<p>(B3) each collection $\mathcal{B}(x)$ is closed with respect to finite (including empty) intersections;</p>
<p>Then the empty intersection (namely $X$) will belong to every $\mathcal{B}(x)$.</p>
<p>(It could be argued that the original (B3) al... |
2,025,711 | <p>What is the sum of this series: $$\sum_{n>=1}{(-1)^{n+1}*\frac{2n+1}{n(n+1)}}$$
I don't know how to solve it especialy with that ${(-1)^{n+1}}$</p>
| Fred | 380,717 | <p>Hints: </p>
<p>1.$(-1)^{n+1}*\frac{2n+1}{n(n+1)}=(-1)^{n+1}(\frac{1}{n}+\frac{1}{n+1})$ </p>
<p>and </p>
<ol start="2">
<li>$ \sum_{n \ge 1}(-1)^{n+1}\frac{1}{n}= \ln (2)$</li>
</ol>
|
2,699,842 | <p>I've been looking at this question for hours now, but I can't grasp it: </p>
<p>Let A be a symmetric matrix with minimum eigenvalue $\lambda_{\text{min}}$ . Give a bound on the largest element of $A^{−1}$.</p>
<p>I was looking at the spectral decomposition where $A=VDV^T$ with D a diagonal matrix, however I don't ... | mechanodroid | 144,766 | <p>If a matrix $M$ is symmetric then the operator norm of $M$ is given by </p>
<p>$$\|M\| = \max_{\lambda \in \sigma(M)} |\lambda|$$</p>
<p>The $(i,j)$-th entry of a matrix $M$ is given by $\langle Me_j, e_i\rangle$ where $\{e_1, \ldots, e_n\}$ is the canonical basis.</p>
<p>Therefore $$\left|\langle Me_i, e_j\rangl... |
749,171 | <p>Right now I'm taking a 3 part course on probability and statistics using Schverish & Degroot Probability and Statistics and it is just not helpful. For the first part, which was on Probability, I used a First Course in Probability by Sheldon Ross which was extremely helpful unfortunately it doesn't have much sta... | Community | -1 | <p>$$\frac{f(2)-f(0)}{2-0}=\frac{9-1}{2}=4$$ So solve $$f'(c)=6c-2=4$$ to get $$c=1$$</p>
|
215,227 | <p>I'm looking at a past homework solution and there is a part of it I don't understand. Specifically I'm talking about number 2 in <a href="http://www.math.cmu.edu/~af1p/Teaching/Combinatorics/F08/hw5a.pdf" rel="nofollow">this problem set</a>:</p>
<blockquote>
<p>Let $m=\lfloor(8/7)^{n/3}\rfloor$. Show that there e... | Ross Millikan | 1,827 | <p>The sets $A_i$ were constructed by taking each element of $[n]$ with probability $\frac 12$. An element $p$ can render $A_i \cap A_j \subseteq A_k$ false if it is in $A_i$ and $A_j$ but not in $A_k$. This requires three choices to be right, so happens in $\frac 18$ of the cases. To have $A_i \cap A_j \subseteq A_... |
2,864 | <p>Is there a way to detect double click events? I did not find anything on the doc page of <a href="http://reference.wolfram.com/mathematica/ref/EventHandler.html" rel="nofollow"><code>EventHandler</code></a>.</p>
<p>Use case: I want to re-implement the <code>Crop Image...</code> functionality available form the con... | Mr.Wizard | 121 | <p>You could do something like this:</p>
<pre><code>DynamicModule[{col = Green, time = AbsoluteTime[]},
EventHandler[
Style["text", FontColor -> Dynamic[col]],
"MouseClicked" :>
If[AbsoluteTime[] - time > 0.25,
time = AbsoluteTime[],
col = col /. {Red -> Green, Green -&... |
2,864 | <p>Is there a way to detect double click events? I did not find anything on the doc page of <a href="http://reference.wolfram.com/mathematica/ref/EventHandler.html" rel="nofollow"><code>EventHandler</code></a>.</p>
<p>Use case: I want to re-implement the <code>Crop Image...</code> functionality available form the con... | kglr | 125 | <p>You can use a combination of <code>MouseDown</code> and <code>MouseClickCount</code> as in the following examples: </p>
<p>example 1: double-click increments the value of <code>j</code>:</p>
<pre><code>j = 1; EventHandler[Panel[Dynamic[j]],
"MouseDown" :> If[CurrentValue["MouseClickCount"] == 2, ++j]]
</code><... |
71,670 | <pre><code>TableForm[Table[i/j + 4*Boole[j > i] // N, {i, 3}, {j, 4}], TableHeadings -> {{"Row1", "Row2", "Row3"}, {"Col1", "Col2", "Col3", "Col4"}}]
</code></pre>
<p>Produces the following table:</p>
<p><img src="https://i.stack.imgur.com/Dqnmr.png" alt="enter image description here"></p>
<p>How select the ma... | kglr | 125 | <p>Using <code>Position</code> with <code>MapAt</code>:</p>
<pre><code>ClearAll[sF];
sF = With[{pos = Thread[{Range@Length@#, Position[#, Max@#][[1, 1]] & /@ #}]},
MapAt[Style[#, Bold, Red] &, #, pos]] &;
t = Table[i/j + 4*Boole[j > i] // N, {i, 3}, {j, 4}];
h = {{"Row1", "Row2", "Row3"}, {"C... |
4,132,907 | <p>I'm trying to figure out the identity above though I'm having difficulties towards figuring it out and would kindly appreciate your support!</p>
<p><span class="math-container">$n\binom{n-1}{r-1} = r\binom{n}{r}$</span></p>
<p>What I have tried:
Given that</p>
<p><span class="math-container">$$\binom{n}{r}=\binom{n-... | I am a person | 806,777 | <p><span class="math-container">$\binom{n - 1}{r - 1} = \frac{(n - 1)!}{(r - 1)!(n - r)!}$</span>.</p>
<p><span class="math-container">$n*\frac{(n - 1)!}{(r - 1)!(n - r)!} = \frac{n!}{(r - 1)!(n - r)!}.$</span></p>
<p>Dividing the LHS and RHS by <span class="math-container">$r$</span>, we want to prove that <span class... |
4,132,907 | <p>I'm trying to figure out the identity above though I'm having difficulties towards figuring it out and would kindly appreciate your support!</p>
<p><span class="math-container">$n\binom{n-1}{r-1} = r\binom{n}{r}$</span></p>
<p>What I have tried:
Given that</p>
<p><span class="math-container">$$\binom{n}{r}=\binom{n-... | Bernard | 202,857 | <p>From the formula for binomial coefficients, but the other way: what has to be proved is
<span class="math-container">$$ \binom nr=\frac nr \binom{n-1}{r-1}.$$</span>
Now, we have
<span class="math-container">$$\binom nr=\frac{n!}{r!(n-r)!}=\frac nr\frac{(n-1)!}{(r-1)!(n-r)!}=\frac nr\binom{n-1}{r-1}$$</span>
since <... |
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | that one guy | 137,947 | <p>Simply enough:</p>
<p>r(θ) = sec(θ%(π/n')-π/n)</p>
<p>When n' = n/2 and % is the modulus operator.</p>
<p>It would work just as well with cosecant as with secant.</p>
<p>Also the apothem would be 1 and the radius sec(-π/n).</p>
|
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | amd | 265,466 | <p>A cheap trick: Take an equation that represents the union of the extensions of the polygon’s sides and add a domain-limiting term to both sides. </p>
<p>For example, the equation <span class="math-container">$$(x-C)(x+y-\sqrt2C)(y-C)(x-y+\sqrt2C)(x+C)(x+y+\sqrt2C)(y+C)(x-y-\sqrt2C)+\sqrt{1-x^2-y^2}=\sqrt{1-x^2-y^2... |
1,631,851 | <p>Let $I_1, . . . , I_n$ be any open intervals in $\mathbb{R}$ for $n ≥ 2$. How can I show that $(I_1 × · · · × I_n) \setminus \{a\}$ is open and connected for any $a ∈ \mathbb{R}^n$?</p>
| egreg | 62,967 | <p>The set $I_1\times I_2\times\dots\times I_n$ is open by definition of product topology. The set $\{a\}$ is closed because $\mathbb{R}^n$ is Hausdorff. So
$$
(I_1\times I_2\times\dots\times I_n)\setminus\{a\}=
(I_1\times I_2\times\dots\times I_n)\cap(\mathbb{R}^n\setminus\{a\})
$$
is open because it's the intersectio... |
343,993 | <p>Let $I$ be an ideal of $\mathbb{C}[x,y]$ such that its zero set in $\mathbb{C}^2$ has cardinality $n$. Is it true that $\mathbb{C}[x,y]/I$ is an $n$-dimensional $\mathbb{C}$-vector space (and why)?</p>
| Georges Elencwajg | 3,217 | <p>The answer is no, but your very interesting question leads to about the most elementary motivation for the introduction of scheme theory in elementary algebraic geometry. </p>
<p>You see, if the common zero set $X_{\mathrm{classical}}=V_{\mathrm{classical}}(I)$ consists set-theoretically (I would even say <em>ph... |
79,040 | <p>Let <span class="math-container">$X$</span> be a compact Riemann surface, and <span class="math-container">$f$</span> a meromorphic function on X.
There's a theorem telling us that <span class="math-container">$\deg(\mathrm{div}(f)) = 0$</span>.</p>
<p>But is the inverse statement also true? I mean, is it true that:... | Lucas Kaufmann | 12,501 | <p>Rafael answer shows that if every degree zero divisor on <span class="math-container">$X$</span> is principal then <span class="math-container">$X$</span> is isomorphic to the Riemann sphere <span class="math-container">$\mathbb{C}_{\infty}$</span>.</p>
<p>The converse also holds true, i.e., every degree zero diviso... |
1,950,381 | <p>If $H$ is a subgroup of $S_n$ and if H is not contained in $A_n$ prove that precisely one half of the elements of H are even permutations. I know that multiplying two odd permutations gives a even, and that one even and one odd return an odd permutation but i have no idea where to go from there. </p>
| p Groups | 301,282 | <p>Every subgroup of $S_n$ contains an even permutation (which one?). Then the hypothesis says that $H$ contains an odd permutation also. Then consider the map
$$\sigma:H\rightarrow \{1,-1\}, h\mapsto \mbox{sign}(h).$$
Can you complete proof?</p>
|
2,480,493 | <p>I have a problem with what looks like a very easy equation to solve $\left|\frac{1 + z}{1- i\overline z}\right| = 1$ . ($z$ is a complex number, $\overline z$ is a conjugate of $z$) I got stuck at the point when after replacing $\overline z = a-bi $ and $z = a +bi$ and getting rid of absolute value I end up with $... | xpaul | 66,420 | <p>Let $z=x+yi$ and then from
$$ |1+z|=|1-i\bar{z}| $$
it is easy to obtain
$$ (x+1)^2+y^2=(-y+1)^2+x^2, $$
So
$$ x=-y $$
and hence $z=(1-i)x$.</p>
|
3,515,656 | <p>Given <span class="math-container">$V(x) = -5x_1^2 (x_1^2 - 1) - 4x_1x_2$</span>.
Show that there exist <span class="math-container">$x_0 \in \mathbb{R}^2$</span> arbitrarily close to <span class="math-container">$x = 0$</span> such that <span class="math-container">$V(x_0) > 0$</span>.
I have no idea how to pr... | José Carlos Santos | 446,262 | <p>Note that<span class="math-container">$$V(x,x)=-5x^2(x^2-1)-4x^2=x^2(1-5x^2),$$</span>which is greater than <span class="math-container">$0$</span> if <span class="math-container">$x\in\left(0,\sqrt{\frac15}\right)$</span>.</p>
|
207,515 | <p>Consider the upper half space $\mathbb{R}^n_{+} = \{x = (x_1,..,x_n) \in \mathbb{R}^n : x_n \geq 0\}$. Consider the Laplacian on this space with either the Dirichlet boundary condition or the Neumann boundary condition. My question is: can the Laplacian $\Delta$ under such boundary conditions be treated as a pseudod... | Christian Remling | 48,839 | <p>The formula will work just fine (with no <em>pseudo</em> required), provided you interpret $\widehat{f}$ suitably. Since $f$ is only a function on a half space, we have to make an agreement on what exactly we mean by $\widehat{f}$.</p>
<p>To obtain the Dirichlet Laplacian, define $\widehat{f}$ as the Fourier transf... |
2,658,368 | <p>I know that one can use Category Theory to formulate polynomial equations by modeling solutions as limits. For example, the sphere is the equalizer of the functions
\begin{equation}
s,t:\mathbb{R}^3\rightarrow\mathbb{R},\qquad s(x,y,z):=x^2+y^2+z^2,~t(x,y,z)=1.
\label{equalizer}
\end{equation}
One could then fin... | Nicolas Hemelsoet | 491,630 | <p>In another direction, one can use differential equations to solve a categorical problem.</p>
<p>If one wants to deform-quantize Poisson-Lie group, a crucial step is to deform an infinitesimally braided monoidal category into a braided monoidal category, and this boils down to find a Drinfeld associator, that is a p... |
112,889 | <p>I am new in Wolfram Math and I need a help in making a simple program. Let's suppose we have a list:</p>
<pre><code>list = {{1, 11}, {2, 7}, {4, 2}, {7, 9}, {-2, 3}, {-1, 10}};
</code></pre>
<p>Now, I need to collect the first elements of sublists, but not all of them, only those whose second elements are larger t... | Jason B. | 9,490 | <pre><code>list = {{1, 11}, {2, 7}, {4, 2}, {7, 9}, {-2, 3}, {-1, 10}};
Cases[list, {a_, b_} /; b > 8 :> a]
(* {1, 7, -1} *)
</code></pre>
<p>What I'm doing above is to use <a href="http://reference.wolfram.com/language/ref/Cases.html"><code>Cases</code></a> to select only those sublists whose second element is ... |
112,889 | <p>I am new in Wolfram Math and I need a help in making a simple program. Let's suppose we have a list:</p>
<pre><code>list = {{1, 11}, {2, 7}, {4, 2}, {7, 9}, {-2, 3}, {-1, 10}};
</code></pre>
<p>Now, I need to collect the first elements of sublists, but not all of them, only those whose second elements are larger t... | ubpdqn | 1,997 | <p>Mathematica/Wolfram Language provides a lot of flexibility. Some ways you could do this:</p>
<pre><code>Cases[list, {x_, _?(# > 8 &)} :> x]
Select[list, #[[2]] > 8 &][[;; , 1]]
Pick[list, #[[2]] > 8 & /@ list][[;; , 1]]
True /. GroupBy[list, (#[[2]] > 8 &) -> (#[[1]] &)]
Flatte... |
4,547,151 | <p>Say I have a set of ordered points, like in a polygon, I can name a point in this set <span class="math-container">$P_i$</span>.</p>
<p>Say I have a high dimensional point and I want to denote its x coordinate, I can do <span class="math-container">$P_x$</span> and if I want it's ith coorddinate I can do <span class... | Abel Wong | 1,090,313 | <p>Imagine to cut the mug into slice of circle disc perpendicular to z-axis.
So the radius of each circle disc is x. Area is <span class="math-container">$\pi x^2$</span>. The range of z for the mug is from z=0 to z=1. This is the limit you mentioned.</p>
<p>Then you can take integral.</p>
<p>volume = <span class="math... |
2,713,853 | <p>This is the question:
"Let X and Y be independent random variables representing the lifetime (in 100 hours) of Type A and Type B light bulbs, respectively. Both variables have exponential distributions, and the mean of X is 2 and the mean of Y is 3.
What is the variance of the total lifetime of two Type A bulbs and... | Green | 357,732 | <p>Your definition for $Var(x) = σ^2$ is taken from the normal distribution. Note, however, that in this question $X$ and $Y$ are stated to be exponential distributions with means of $2$ and $3$ respectively.</p>
<p>For an exponential distribution, the mean is equal to $β$ and the variance is equal to $β^2$. That's t... |
427,663 | <p>If $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} \in \Bbb R $</p>
<p>i found one link that had a similar problem.
<a href="https://math.stackexchange.com/questions/343982/prove-if-z-1-and-w-1-then-1-zw-neq-0-and-z-w-over-1">Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w}... | Jyrki Lahtonen | 11,619 | <p>Draw vectors on the complex plane from the origin to a point on the unit circle. Notice that the argument of the sum of two such vectors is the average of the arguments of the two vectors (observe that the latter is defined only modulo a multiple of $\pi$). This follows from the properties of a rhombus (=parallelogr... |
2,790,767 | <p>Let $X_i$ a countable collection of independent random variables with symmetric distribution i.e. $P(X_i\in A)=P(X_i \in -A)$ for all $i\geq 1$. If $\lambda\in\mathbb{R}$ is such that $E(e^{\lambda X_i})$ is finite for all $i$ I want to prove the following Kolmogorov like inequality
$$P\left(\max_{1\leq k\leq n} \le... | Wanshan | 530,208 | <p>I think this bound is incorrect, because the right hand side is too small. For example, if we let $X_i$ to be i.i.d. Rademacher variable, i.e., $P(X_i = 1) = P(X_i = -1) = 1/2$, then any $\lambda\in \mathbb{R}$ can make $E(e^{\lambda X_i})<\infty$. Take $t = 1/2$ and $n=1$, then $P(|X_1|>t) = 1$, but the right... |
3,715,674 | <p>Let <span class="math-container">$X$</span> be a scheme and <span class="math-container">$U$</span> an open subset of <span class="math-container">$X$</span>. Let <span class="math-container">$s \in \Gamma(U, O_X)$</span> and <span class="math-container">$x \in U$</span>.<br>
I am getting confused with what the diff... | Henry Lee | 541,220 | <p>An exact differential equation is defined as an equation which has a solution of the form:
<span class="math-container">$$du(x,y)=P(x,y)dx+Q(x,y)dy$$</span> if the DE is defined as:
<span class="math-container">$$P(x,y)dx+Q(x,y)dy=0$$</span> leading to the general solution of:
<span class="math-container">$$u(x,y)=C... |
54,395 | <p>A generic question: </p>
<p>are there any spectral techniques to estimate the genus of a graph? I am interested in complete balance multipartite graph.</p>
| Igor Rivin | 11,142 | <p>Yes, there are techniques. For graphs of fixed genus and $n$ vertices, the second lowest eigenvalue of the laplacian is of order $O(1/\sqrt{n}),$ where the hidden constant depends on the genus (in an explicit way -- this follows from the Cheeger inequality and the separator theorems of Lipton and Tarjan, see eg, the... |
2,755 | <p>As Akhil had great success with his <a href="https://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry">question</a>, I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with algebraic geometry enou... | Chuck Hague | 1,528 | <p>All of these recommendations are very good, and I'd like to add that the book <a href="https://doi.org/10.1007/978-0-8176-4523-6" rel="nofollow noreferrer">D-Modules, Perverse Sheaves, and Representation Theory</a> (which you can download at the provided link if you have institutional access; otherwise you can get i... |
2,925,320 | <p>I was wondering how did Spectral graph theory, this multidisciplinary area between Linear Algebra and Graph theory start?</p>
<p>How did it emerge? </p>
<p>Was there a certain problem (maybe graph isomorphism problem?) that evoked its formation? </p>
| Chris Godsil | 16,143 | <p>The oldest work on spectral graph theory that I am aware of is the work by chemists on Hueckel molecular orbital theory (<a href="https://en.wikipedia.org/wiki/H%C3%BCckel_method" rel="noreferrer">https://en.wikipedia.org/wiki/Hückel_method</a>). One principal goal of this work was to relate chemical properties of a... |
1,856,225 | <p>Right, so as the final step of my project draws near and after having made a bad layout sort of question, I am posting a new one very clear and unambiguous. I need to find this specific definite integral which Mathematica could not solve:</p>
<p>$$ \int_{x=0}^\pi \frac{\cos \frac{x}{2}}{\sqrt{1 + A \sin^2 \frac{x}... | Yuriy S | 269,624 | <p>I highly doubt this integral has any useful analytic form. I suggest you either use numerical integration or simplify your initial model to obtain a solvable integral.</p>
<p>Using @tired 's suggestion, we can make you integral much easier. First, let's make the following substitutions:</p>
<p>$$y=\sin \frac{x}{2}... |
1,856,225 | <p>Right, so as the final step of my project draws near and after having made a bad layout sort of question, I am posting a new one very clear and unambiguous. I need to find this specific definite integral which Mathematica could not solve:</p>
<p>$$ \int_{x=0}^\pi \frac{\cos \frac{x}{2}}{\sqrt{1 + A \sin^2 \frac{x}... | Mariusz Iwaniuk | 276,773 | <p>This is not a full answer.From answer of user <code>You're In My Eye</code> .First, let's make the following substitutions:</p>
<p>$$y=\sin \frac{x}{2}$$</p>
<p>$$B=\frac{1+A}{\sqrt{A}} \omega$$</p>
<p>$$2 \int_0^1 \sin \left( B~ \text{arctanh}~\sqrt{\frac{1-y^2}{1+Ay^2}} \right) \frac{dy}{\sqrt{1+Ay^2}}$$
substi... |
298,066 | <p>Let $A$ be a commutative ring, and consider the category of bimodules over $A$.</p>
<p>An $A$-bimodule $M$ is called symmetric if $a\cdot m = m \cdot a$ for all $a \in A$, $m \in M$.</p>
<p>Is the category of symmetric bimodules over $A$ closed under extensions?</p>
<p>Namely, given an exact sequence of $A$-bimod... | Johannes Hahn | 3,041 | <p>Easier counterexample than the one Mare gave:</p>
<p>Let $A=k[X]$ and $N:=k^2$ where $X$ acts as $id_N$ on the right and as $\begin{pmatrix}1&1\\&1\end{pmatrix}$ on the left. Then $M:=\langle e_1\rangle$ is a symmetric sub-bimodule ($X$ acts as $id$ on both sides) and the quotient $K:=M/N=\langle e_2+M\rang... |
165,791 | <p>I'm looking for a maximum accuracy quadrature formula:</p>
<p>$$
\int_{-1} ^{1} \sqrt{\frac {1-x}{1+x}} f(x)dx = A_1f(x_1)+A_2f(x_2)+R(f)
$$</p>
<p>I don't know exactly if it's Trapezoidal rule which has the degree of accuracy one, or Simpson's rule with three or Gauss (these 3 I have studied deeply, but might be ... | Brendan McKay | 9,025 | <p>Federico's answer gives you the theory. Since you only ask for two abscissae, it is easy to solve this case by first principles. Just put in the abscissae as variables and do the integral. $f(-3^{-1/2})+f(3^{-1/2})$ gives the right answer for cubic polynomials and not for quartic polynomials. I hope this isn't hom... |
3,258,256 | <p>I know that if we have a matrix A defined for a space V with dimension n that has n distinct eigenvalues, we can write it as <span class="math-container">$X^{-1}DX$</span> where D is a diagonal matrix with the eigenvalues of A on its diagonal and X is a matrix of eigenvectors. </p>
<p>It seems that what we are doin... | symplectomorphic | 23,611 | <p>Your question makes little sense: it is not clear what you mean when you say that under your choice of <span class="math-container">$L$</span>, “<span class="math-container">$X$</span> is just the identity matrix.” But perhaps these comments will clarify the situation.</p>
<p>Diagonalization is really about linear ... |
89,666 | <p>As I am trying to get the solution of the simultaneous ordinary differential equations </p>
<pre><code> b1'[z]-1I*beta1*b1[z]-C1*b2[z]==0,
b2'[z]-1I*beta2*b2[z]+C1*b1[z]==0
</code></pre>
<p>with boundary conditions </p>
<pre><code>b1[1.581825567*10^-6]==0.876212, b2[1.581825567*10^-6]==0.481925
</code... | Michael E2 | 4,999 | <p>Your ODE is linear, so whether it can satisfy your constraint <code>|b1|^2+|b2|^2 <= 1</code> depends primarily on the eigenvalues of the coefficient matrix, as well as the initial condition.</p>
<p>The coefficient matrix is the second array returned by <a href="http://reference.wolfram.com/language/ref/Coeffici... |
4,013,630 | <blockquote>
<p>The population of bacteria triples each day on a petri dish. If it takes 20 days for the population of bacteria to fill the entire dish, how many days will it take bacteria to fill half of the petri dish?</p>
</blockquote>
<ul>
<li><p>My doubt 1: if we don't have a initial population can be solve this p... | Community | -1 | <p><span class="math-container">$$f(n):=\begin{cases}\text{even}(n)&\to-3,\\\text{odd}(n)&\to\ \ \ 7\end{cases}$$</span> is a valid answer.</p>
<hr />
<p>If you really want a "formula",</p>
<p><span class="math-container">$$2-5\cos(n\pi)$$</span>
or
<span class="math-container">$$(10n)\bmod20-3.$$</sp... |
2,332,544 | <p>I've been messing around with a quadratic that popped out of a problem I'm working on, and noticed it has a property I haven't seen before. The form is as follows:</p>
<p>$$ ax^2 + bx + c = 0$$
where
$$a = r_{1}^{2} + r_{2}^{2} - r_{3}^{2}$$
$$b = 2\left( r_{1} t_2 + r_{2} t_2 - r_{3} t_3 \right)$$
$$c = t_{1}^... | Goa'uld | 336,195 | <p>Here is another fun proof. Consider the map</p>
<p>$$ f \colon S^{\infty} \to \mathbb{R} $$</p>
<p>$$ (x_0,x_1,\ldots) \mapsto \sum n x_n $$</p>
<p>This map is well defined since any element of $S^{\infty}$ has only a finite number of nonzero $x_i$. And it is continuous because the restriction to each $S^n$ is co... |
1,902,407 | <p>Expand given function $f$ as Taylor series around $c=3$
$$f(x) = \frac{x-3}{(x-1)^2}+\ln{(2x-4)} $$</p>
<p>and find out open interval at what that series is convergent. What is radius of convergence?</p>
<p>This is what i have so far. We know that $\ln{(1+x)} = \sum_{n=1}^{\infty} \frac{x^n}{n}$ and $\frac{1}{1-x}... | barak manos | 131,263 | <p>$3$ is obviously the second prime.</p>
<p>And $11$ is the third, which you can easily prove by induction for $10^{2n}-1$:</p>
<ul>
<li>Base case: $10^2-1=99$ is divisible by $11$</li>
<li>Inductive step: $10^{2(n+1)}-1=101\cdot(10^{2n}-1)$</li>
</ul>
|
1,902,407 | <p>Expand given function $f$ as Taylor series around $c=3$
$$f(x) = \frac{x-3}{(x-1)^2}+\ln{(2x-4)} $$</p>
<p>and find out open interval at what that series is convergent. What is radius of convergence?</p>
<p>This is what i have so far. We know that $\ln{(1+x)} = \sum_{n=1}^{\infty} \frac{x^n}{n}$ and $\frac{1}{1-x}... | ijbalazs | 403,552 | <p>Let us examine 41. We will show that 41 | 10^100 -1. First we consider the 5th power. 10^5 = 2439 * 41 + 1 so that 41 | 10^5 -1 and further 10^5 -1 | (10^5)^20 - 1 = 10^100 - 1 </p>
|
2,202,498 | <p>Test whether the series $\sum_{n=1}^\infty\frac{1+\sin(n)}{2^n}$ is converges or diverges. I used the limit comparison method in order to test its convergence. I chose $a_n = \frac{1+\sin(n)}{2^n}$ and $b_n = \frac{1}{2^n}$.</p>
<p>$$
\lim_{n \rightarrow \infty}\frac{\frac{1+\sin(n)}{2^n}}{\frac{1}{2^n}} = \lim_{n ... | Alex Provost | 59,556 | <p>In $\mathbb{Z}_7$, we have $-2 = 5$ and $-4 = 3$. Therefore, if we let $A = \left(\begin{matrix}
0&2 \\
2 & 2
\end{matrix}\right)$, the equality $A^2 = 4I + 2A$ with $\mathbb{Z}_7$ coefficients is equivalent to $$0 = A^2 - 2A - 4I = A^2 + 5A + 3I = \phi(X^2 + 5X + 3).$$</p>
|
574,220 | <p><strong>Question 1:</strong></p>
<p>let Polynomial <span class="math-container">$f(x)=\displaystyle\sum_{i=0}^{3}a_{i}x^i,$</span> have three real numbers roots,where <span class="math-container">$a_{i}>0,i=1,2,3$</span>.</p>
<blockquote>
<p>show that:
<span class="math-container">$$g(x)=\sum_{i=0}^{3}a^m_{i}x^i$... | mercio | 17,445 | <p>As I explained in my comment, it is enough to do case (2).</p>
<p>Let $x,y \ge 0$. Then $(X+x)(X+x)(X+y) = X^3+(2x+y)X^2 + (x^2+2xy)X + x^2y$.
The discriminant of $X^3+bX^2+cX+d$ is, according to wikipedia, $\Delta(b,c,d) = b^2c^2-4c^3-4b^3d-27d^2+18bcd$.</p>
<p>The derivative of $m \mapsto \Delta(b^m,c^m,d^m)$ at... |
274,885 | <p>I've been doing some research on ranking algorithms, and I've read theresearch with interest. The aspects of the Schulze Algorithm that appeals to me is that respondents do not have to rank all options, the rank just has to be ordinal (rather than in order), and ties can be resolved. However, upon implementation, I'... | Jay | 2,389 | <p>You can see that ties are not avoidable with a simpler example. Consider this election with three candidates and three votes:</p>
<pre><code>A>B>C (1 single vote)
B>C>A (1 single vote)
C>A>B (1 single vote)
</code></pre>
<p>The result will be a cycle, the Smith set contains all candidates, and yo... |
2,877,032 | <p>$\int y^{2}dx+x^{2}dy$ and $C\left( r\right) =\left(a \cos t,b\sin t\right)$ ,$0<t< \pi$</p>
<p>Could someone give me a hint to evaluate this integral?</p>
<p>my efforts:
$C^{'}\left( r\right)=\left(-a \sin t,b\cos t\right)$ then $\left\| C^{'}\left( r\right) \right\|$=$\sqrt {a^{2}\sin ^{2}t+b^{2}\cos ^{2}t... | Nosrati | 108,128 | <p>Let $x=a\cos t$ and $y=b\sin t$ and replace in integral. Therefore
\begin{align}
\int y^{2}dx+x^{2}dy
&= \int_0^\pi (b\sin t)^2(-a\sin t)\ dt + (a\cos t)^2(b\cos t)\ dt \\
&= ab\int_0^\pi (a\cos^3t-b\sin^3t)\ dt \\
&= \color{blue}{-\dfrac43ab^2}
\end{align}</p>
|
2,071,054 | <p>Let $f(x)$ be continuous in $[a,b]$</p>
<p>Let $A$ be the set defined as:
$$A = \{ x \in [a,b] \mid f(x) = f(a) \}$$</p>
<ol>
<li><p>Does $A$ have a maximum? I guess it it is the max value in $[a,b]$ which $f$ sends to $f(a)$ but I don't know how to prove it.</p></li>
<li><p>Would it still have a maximum if $[a,b]... | Zaid Alyafeai | 87,813 | <p>$$\sum\nolimits_{i=0}^{\infty }{\sum\nolimits_{j=0}^{\infty }{\sum\nolimits_{k=0}^{\infty }{{{3}^{-\left( i+j+k \right)}}}}} = \sum\nolimits_{i=0}^{\infty }{3}^{-i}{\sum\nolimits_{j=0}^{\infty }{3}^{-j}{\sum\nolimits_{k=0}^{\infty }{3}^{-k}}}\\ =\left(\sum_{i=0}^\infty \frac{1}{3^i}\right)^3 = \left(\frac{1}{1-1/3}\... |
1,379,878 | <blockquote>
<p>Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.</p>
</blockquote>
<p>My Try:</p>
<p>I proved that $C(M)$ is ... | user24142 | 208,255 | <p>In the case that $M$ has $n$ unique eigenvalues and eigenvectors, there is a fairly straightforward argument. For the eigenvalue $\lambda$, there are vectors $v_L$ and $v_R$ which are the left and right eigenvectors. Construct the $n\times n$ matrix $C_\lambda = v_L v_R^T$: $C_v$ is such that all its rows are scalar... |
1,685,621 | <p>I know that we can define two vectors to be orthogonal only if they are elements of a vector space with an inner product. </p>
<p>So, if $\vec x$ and $\vec y$ are elements of $\mathbb{R}^n$ (as a real vector space), we can say that they are orthogonal iff $\langle \vec x,\vec y\rangle=0$, where $\langle \vec x,\ve... | Quarky Quanta | 320,443 | <p>Let me try to add my personal intuition to this conversation and see if that helps you a bit. If you take the standard basis vectors, and draw them in the Cartesian coordinate system, then you will note that they satisfy your intuition of what we think of perpendicular vectors. What we further note is that any two v... |
416,497 | <p>I was recently trying to find a numerical solution to a thermodynamics problem and the expression <span class="math-container">$x\ln x$</span> appeared in one of the computations. I did not have to find its value very near <span class="math-container">$0$</span>, so the computer managed fine, but it got me thinking ... | Robert Israel | 13,650 | <p>If <span class="math-container">$x$</span> is represented in floating-point as <span class="math-container">$y \times 10^{-d}$</span>, <span class="math-container">$0.1 < y \le 1$</span>, <span class="math-container">$d \in \mathbb N$</span>, note that</p>
<p><span class="math-container">$$ x \ln(x) = y (\ln(y) -... |
416,497 | <p>I was recently trying to find a numerical solution to a thermodynamics problem and the expression <span class="math-container">$x\ln x$</span> appeared in one of the computations. I did not have to find its value very near <span class="math-container">$0$</span>, so the computer managed fine, but it got me thinking ... | Brendan McKay | 9,025 | <p>Modern mathematics libraries should be able to find <span class="math-container">$\log x$</span> precisely for all floating-point numbers, as the algorithms for doing that have long been known and adopted. My experiments on a fairly recent Intel chip with gnu mathematics library and gcc 10 compiler confirm that.</p>... |
1,183,643 | <p>Given a integer $h$</p>
<blockquote>
<p>What is $N(h)$ the number of full binary trees of height less than $h$?</p>
</blockquote>
<p><img src="https://i.stack.imgur.com/XcNVi.jpg" alt="enter image description here"></p>
<p>For example $N(0)=1,N(1)=2,N(2)=5, N(3)=21$(As pointed by <a href="https://math.stackexc... | Marko Riedel | 44,883 | <p>Here is my contribution to this interesting discussion. Introduce $T_{\le n}(z)$ the OGF of full binary trees of height at most $n$ by the number of nodes. Now a tree of height at most $n$ either has height at most $n-1$ or height exactly $n.$ The latter tree has a subtree of height $n-1$ on the left or the right of... |
1,986,007 | <p>What is the intuitive explanation of the Inverse Function Theorem (and generalized to multiple dimensions)?</p>
| Gyu Eun Lee | 52,450 | <p>The moral story of all differential calculus is this:</p>
<blockquote>
<p>Differentiable functions behave locally like their linearizations.</p>
</blockquote>
<p>The degree to which this statement holds can be quantitatively formulated, for example in terms of error estimates for first-order Taylor expansions. I... |
951,524 | <p>How do I prove, that $(\mathbb R×\mathbb R;+,*) $ is a ring, but not a field, where the $+$ and $*$ operations are: $(a,b)+(c,d):=(a+c,b+d)$ and $(a,b)*(c,d):=(ac,bd)$?</p>
<p>For the solution: so I would have to first show, that $(\mathbb R×\mathbb R;+,*) $ is a ring, I have to prove that using the definition o... | robjohn | 13,854 | <p>The function
$$
\frac{180/z}{z^{180}-1}
$$
has residue $1$ at each root of $z^{180}-1$ $\left(\text{i.e. }e^{k\pi i/90}\text{ for }k=0\dots179\right)$ and residue $-180$ at $z=0$.</p>
<p>On $|z|=1$,
$$
\tan(\theta/2)=-i\frac{z-1}{z+1}
$$
Integrating
$$
f(z)=-\left(\frac{z-1}{z+1}\right)^2\frac{180/z}{z^{180}-1}
$$
... |
1,552,501 | <p><a href="https://i.stack.imgur.com/zFz40.jpg" rel="nofollow noreferrer">My Question</a></p>
<p>First I let u = y' and employed the Chain Rule to obtain du/dx = du/dy * u</p>
<p>But I am not sure where to go from there. Any tips, suggestions, or solutions to the problem would be much appreciated!</p>
<p>I suspect ... | amd | 265,466 | <p>They’re not saying that they’re equal to the angle, only that they’re approximately equal. </p>
<p>The Taylor series for $\sin\theta$ around $0$ is $\theta+\frac{\theta^3}3+\dots$, and that for $\tan\theta$ is $\theta+\frac{\theta^3}6+\dots$, so for very small $\theta$ we can drop the higher-order terms as an appr... |
1,552,501 | <p><a href="https://i.stack.imgur.com/zFz40.jpg" rel="nofollow noreferrer">My Question</a></p>
<p>First I let u = y' and employed the Chain Rule to obtain du/dx = du/dy * u</p>
<p>But I am not sure where to go from there. Any tips, suggestions, or solutions to the problem would be much appreciated!</p>
<p>I suspect ... | marty cohen | 13,079 | <p>This is because
of the only
trig limit that matters:</p>
<p>$\lim_{x \to 0} \frac{\sin x}{x}
= 1
$.</p>
<p>This follows from the expansion
$\sin(x)
=x+\frac{x^3}{6}+...
$.</p>
<p>From this you can deduce that
$\lim_{x \to 0} \frac{1-\cos(x)}{x^2}
=\frac12
$
and
$\lim_{x \to 0} \frac{\tan x}{x}
= 1
$.</p>
|
1,918,781 | <p>Suppose that $\phi:\mathbb R^3\to\mathbb R$ is a harmonic function. I am asked to show that for any sphere centered at the origin, the average value of $\phi$ over the sphere is equal to $\phi(0)$. I am also directed to use Green's second identity: for any smooth functions $f,g:\mathbb R^3\to \mathbb R$, and any sph... | Plutoro | 108,709 | <p>I appreciate the other answers, but I came up with my own answer which, in my humble opinion, is a bit simpler.</p>
<p>For posterity's sake, here is the answer: Let <span class="math-container">$g=1/|\mathbf r|$</span> and <span class="math-container">$f=\phi$</span>. Then the following hold: <span class="math-cont... |
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Joey Lakerdas-Gayle | 13,846 | <p>You can do questions like this in your head pretty easily which is a great skill to teach students. Since <span class="math-container">$3.9$</span> is close to <span class="math-container">$4$</span>, and <span class="math-container">$7.5$</span> is easy to add to itself. So <span class="math-container">$3.9\times 7... |
377,720 | <p>Let $u$ be the distribution on $\mathbb{R}$ given by $$u=\delta_0-\delta_1 $$ </p>
<p>(a) show there exists a continuous function $f$ such that $f''=u$ and indicate such one. I thought of doing this with Fourier-transforms: $$F(f'')=s^2F(f)=F(u) = 1-e^{-2\pi i s}$$ </p>
<p>which would imply $f= F^{-1}(s^{-2}(1- e^... | Sangchul Lee | 9,340 | <p><strong>Answer to (a)</strong> : Intuitively, 1-dimensional Dirac delta function is the derivative of a unit step function (also called the Heaviside step function). Thus $f'' = \delta_0 - \delta_1$ means that $f'$ has a step (jump) of size 1 at $x = 0$ and a step of size -1 at $x = 1$ as follows:</p>
<p>$$ f'(x) =... |
112,575 | <p>If we set $\exp(x)=\sum x^k/k!$, then $\exp(x+y)=\exp(x)\cdot \exp(y)$. In terms of coefficients it means that $(x+y)^n=\sum \frac{n!}{k!(n-k)!} x^ky^{n-k}$, i.e. just binomial expansion.</p>
<p>Now consider logarithm. Set $L(x):=\sum_{k>0} x^k/k$, then $L(x)=-\log(1-x)$ in a sense, and hence $$L(u+v-uv)=L(u)+L(... | David E Speyer | 297 | <p>I don't know a combinatorial interpretation, but here is a quick proof. Let $1 \leq a \leq b$. Write
$$\sum_{0 \leq k \leq a} (-1)^k \frac{(a+b-k-1)!}{(a-k)! (b-k)! k!} = \frac{1}{a!} \sum_{0 \leq k \leq a} (-1)^k \frac{(a+b-k-1)!}{(b-k)!} \binom{a}{k}$$
$$= \frac{1}{a!} \sum_{0 \leq k \leq a} (-1)^k f(k) \binom{a}... |
260,240 | <p><img src="https://i.stack.imgur.com/3WeQw.png" alt="enter image description here"></p>
<p>In the last bullet, it says l must be even and provides an explanation. I don't understand the explanation, however. Why does it have to be even?</p>
| A.Schulz | 35,875 | <p>They mean that in order to have $a=\sqrt{2}\sqrt{l}$ an integer, we must have that $l$ is an even number, say $2r$. In this case $a=\sqrt{2}\sqrt{2r}=2\sqrt{r}$. If $l$ is odd, we can not <em>spilt of the $\sqrt{2}$ from $\sqrt{l}$</em>, thus the $\sqrt{2}$ in $a$ remains, and $a$ will be irrational.</p>
|
155,897 | <p>I would like to exclude non-western characters and words from a text file. I do not know how to insert the text file here, but I suppose you can do without it. All your suggestions will be much appreciated.</p>
<p>Update: I have used:</p>
<pre><code>Alphabet["Russian"]
Select[dict21, Not@StringContainsQ[#, Alterna... | C. E. | 731 | <p>It is especially easy to express the character range <code>Alphabet[]</code> that you mention using <code>RegularExpression</code>:</p>
<pre><code>StringDelete["adeфfgh12cа34", RegularExpression["[^a-z]+"]]
</code></pre>
<blockquote>
<p><code>"adefghc"</code></p>
</blockquote>
<p>It will also be faster than oth... |
1,458,464 | <p>Let</p>
<ul>
<li>$(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces</li>
<li>$I\subseteq[0,\infty)$ be at most countable and closed under addition with $0\in I$</li>
<li>$X=(X_t)_{t\in I}$ be a stochastic process on $(\Omega,\mathcal A)$ with values in $(E,\mathcal E)$</li>
<li>$\mathbb F=(\mathcal F_t... | John Dawkins | 189,130 | <p>The red part is equal to the line just above it because (i) the event $\{\tau=s\}$ is both ${\mathcal F}_s$ measurable and ${\mathcal F}_\tau$ measurable, and (ii) those two $\sigma$-algebras coincide on $\{\tau=s\}$, in the sense that if $A$ is an event and $s\in I$ is fixed then $A\cap\{\tau=s\}\in{\mathcal F}_s$ ... |
58,009 | <p>Let $f: X \to Y$ be a morphism of varieties such that its fibres are isomorphic to $\mathbb{A}^n$. Since the definition of a vector bundle stipulates that $f$ be locally the projection $U \times \mathbb{A}^n \to U$, it is likely that there exist morphisms that are not locally of that form, but I can't come up with a... | Angelo | 4,790 | <p>In Jack's example the fiber is not scheme-theoretically $\mathbb A^1$. You can get a counterexample by taking $Y$ to be a nodal curve, $Y'$ its the normalization, with one of the two points in the inverse image of the node removed, and $X = Y' \times \mathbb A^1$.</p>
<p>If we assume that the map is smooth, this be... |
3,427,640 | <p>This is a problem that came up as I was learning Hermitian/skew-Hermitian transformations:</p>
<hr>
<p>Let <span class="math-container">$T: V\rightarrow E$</span> be a linear transformation where <span class="math-container">$V$</span> is a subspace of a complex Euclidean space <span class="math-container">$E$</sp... | Mathew Westfields | 643,848 | <p>Actually you are done by comparing coefficents in the last equation.</p>
|
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