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 |
|---|---|---|---|---|
3,413,837 | <p>Jerry the mouse is hungry and according to some confidential information, there is a tempting piece of cheese at the end of one of the three paths after the junction he just found himself!</p>
<p>Fortunately, Tom is standing right there and Jerry hopes he can get some useful information as to which path he must get... | Caleb Stanford | 68,107 | <p>Here's another way to do it :) <strong>Ask the following question twice:</strong></p>
<blockquote>
<p>Are you telling the truth and the cheese is in the first path, OR are you lying and the cheese is in the second path?</p>
</blockquote>
<ul>
<li><p><strong>If both answers are True,</strong> then we know that in... |
1,736,341 | <p>As stated in the title: Find the eigenvalues and eigenfunctions for $y''+\lambda y=0$, where $y'(1)=0$ and $y'(2)=0$.</p>
<p>So I have already eliminated the cases for $\lambda=0$ and $\lambda<0$ and I'm focused on the case for $\lambda>0$ now.
The characteristic equation for this case is $r^2+\lambda=0$, so... | Dylan | 135,643 | <p>The system of equations can be rewritten as</p>
<p>$$ \alpha \left( \begin{matrix} -\sin \alpha & \cos \alpha \\ -\sin 2\alpha & \cos 2\alpha \end{matrix} \right) \left( \begin{matrix} k_1 \\ k_2 \end{matrix} \right) = 0 $$</p>
<p>Since your solution vector is in the null space, a solution will exist when ... |
249,107 | <p>Im working on my thesis about semidirect products and splitting lemma. I got the following theorems to prove and Im a not sure how to start. I would appreciate any help.</p>
<p>$\\$
1. Let $f:A\to B$ be a map.</p>
<p>Show:</p>
<p>a) if $g:B\to A$ so that $gf=id_{A}$ then $f$ is injective</p>
<p>b) if $g:B\to A$ ... | Dennis Gulko | 6,948 | <p><strong>Hints:</strong><br>
For (1)a), if $f$ is not injective, then there exist $a\neq b\in A$ such that $f(a)=f(b)$. Hence $gf(a)=gf(b)$. On the other hand, $id_A(a)=a$ and $id_A(b)=b$. Is it possible that $gf=id_A$?<br>
For (1)b), if $f$ is not surjective, then there exists $b\in B$ such that for all $a\in A$, $f... |
2,217,338 | <p>I am trying to define a simple function that is first concave and then convex as shown in the picture below. Since the resulting equation have to be explained/used by a non-technical audience, the function should be ideally as simple as possible, but I have been unable to find any simple form that matches the requir... | Claude Leibovici | 82,404 | <p>For sure, Marco Cantarini provided the good and formal solution of the problem.</p>
<p>I have a modest solution based on series. Starting using $$\sin^2(ax)=\frac{1-\cos(2ax)}2=\sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2 n-1} a^{2 n} }{(2 n)!}x^{2 n}$$ we then have $$x \sin^2(ax) e^{-x^2}=\sum_{n=1}^\infty \frac{(-1)^{... |
691,112 | <p>Suppose $A$ and $B$ are finite sets and $f:A\rightarrow B$. Prove that if $|A|>|B|$, then $f$ is not one-to-one.</p>
<p>Scratch work:</p>
<p>Since the goal is in negation, I try to prove it by contradiction and assume that $f$ is one-to-one. Since $A$ has more elements than $B$, it can't be the case that $f$ is... | Dave Clifford | 82,405 | <p>Let $f(A)$ be the set of images. Then $f(A)\subseteq B$ so $|f(A)|\le |B|$.</p>
<p>If $f$ is one-to-one then $|f(A)|=|A|$.</p>
|
466,722 | <blockquote>
<p>Find a function $f$ and a number $a$ such that:
$$
6+\int_{a}^{x}\frac{f(t)}{t^2}\:\mathrm{d}t=2\sqrt{x}
$$
For all $x>0$</p>
</blockquote>
<p>From Fundamental Theorem of Calculus section. Having some trouble with this. Any help?</p>
| what'sup | 87,411 | <p>$$ \int_a^x \frac{f(t)}{t^2} \ dt = 2\sqrt{x} - 6 $$</p>
<p>$$\mathrm{ differentiate \ using \ leibniz \ rule } $$</p>
<p>$$\frac{f(x)}{x^2} = \frac{1}{\sqrt{x} } \Rightarrow f(x) = x\sqrt{x}$$</p>
<p>$$\int_a^x \frac{\sqrt{t}}{t} \ dt = 2\sqrt{x} - 6 $$</p>
<p>$$\int_a^x t^{-\frac{1}{2}} \ dt = \left | 2\sqrt{... |
1,657,664 | <p>Struggling with a homework problem here and can't understand logically which one would be correct (each has different truth tables). I need to express the following statement using quantifiers, variables, and the predicates M(s), C(s), and E(s) </p>
<blockquote>
<p>"No computer science students are engineering st... | Q the Platypus | 264,438 | <p>The first example says "All computer students are not engineering students". People who are not computer students are free to be maths students or not be maths students.</p>
<p>The second one says "All students are not computer students and they have to be engineering students".</p>
|
2,331,191 | <p>Use either direct proof, proof by contrapositive, or proof by contradiction.</p>
<p>Using proof by contradiction method</p>
<blockquote>
<p>Assume n is a perfect square and n+3 is a perfect square (proof by
contradiction)</p>
<p>There exists integers x and y such that <span class="math-container">$n = x^2$</span> an... | David G. Stork | 210,401 | <p>Let $n=4$ (a perfect square). Note that $n+3 = 7$ is not a perfect square.</p>
<p>Is there anything more to this problem than that?</p>
<p>(Of course if $n=1$ (a perfect square) then $n+3 = 4$ happens to be a perfect square.)</p>
|
3,356,341 | <blockquote>
<p>Let <span class="math-container">$a_1$</span>, <span class="math-container">$a_2$</span> ∈ <span class="math-container">$\mathbb Z$</span>, Show:</p>
<p><span class="math-container">$a_1 \mathbb Z · a_2 \mathbb Z$</span> = <span class="math-container">$(a_1 · a_2)$</span> · <span class="math-container">... | Theo Bendit | 248,286 | <p>The proof is lacking a bit of clarity in its presentation. You start by defining an arbitrary element, <span class="math-container">$x \in a_1 \Bbb{Z}$</span>, then an unrelated arbitrary element <span class="math-container">$y \in a_2 \Bbb{Z}$</span>, and finally, a third arbitrary element <span class="math-contain... |
3,356,341 | <blockquote>
<p>Let <span class="math-container">$a_1$</span>, <span class="math-container">$a_2$</span> ∈ <span class="math-container">$\mathbb Z$</span>, Show:</p>
<p><span class="math-container">$a_1 \mathbb Z · a_2 \mathbb Z$</span> = <span class="math-container">$(a_1 · a_2)$</span> · <span class="math-container">... | Henno Brandsma | 4,280 | <p>Show two inclusions in a systematic way:</p>
<p>Let <span class="math-container">$z \in a \Bbb Z \cdot b \Bbb Z$</span>. This means that we can write <span class="math-container">$z=xy$</span> for some <span class="math-container">$x \in a \Bbb Z $</span> and <span class="math-container">$y \in \Bbb Z $</span>. By ... |
51,757 | <p>I'm trying to find closed form for</p>
<p>$$\sum_{k=1}^{n}\sin\frac{1}{k}$$</p>
<p>I typed it in Mathematica 6.0 and WolframAlpha, but no result what i expected.</p>
<p>Any hints will be appreciated, thank you.</p>
| Henry | 6,460 | <p>I doubt you will find a closed form. </p>
<p>Your expression will give a value slightly less than than the <a href="http://en.wikipedia.org/wiki/Harmonic_number" rel="nofollow">harmonic numbers</a> $\sum_{k=1}^{n}\frac{1}{k}$ which do not have a closed form and as $k$ increases the difference will increase towards... |
3,815,640 | <p>what is the most efficient way to calculate the argument of
<span class="math-container">$$
\frac{e^{i5\pi/6}-e^{-i\pi/3}}{e^{i\pi/2}-e^{-i\pi/3}}
$$</span> without calculator ?</p>
<p>i tried to use <span class="math-container">$\arg z_1-\arg z_2$</span> but the argument of <span class="math-container">$e^{i5\pi/6... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>multiply the numerator and the denominator by <span class="math-container">$$e^{i\frac{\pi}{3}}$$</span></p>
<p>and use the fact that</p>
<p><span class="math-container">$$\arg(1-e^{i\theta})=\frac{\theta}{2}-\frac{\pi}{2}$$</span></p>
|
3,974,394 | <p>I am relatively new to isomorphisms and I don't understand how <span class="math-container">$\varphi$</span> is surjective in this proof. I have searched online, but I still don't understand. If anyone could straight up tell me because I feel like I'm being a bit dumb.</p>
<p><a href="https://i.stack.imgur.com/oIB4A... | PrincessEev | 597,568 | <p>You are mapping</p>
<p><span class="math-container">$$g := g_1 g_2 g_3 \cdots g_n \stackrel{\varphi}{\mapsto} (g_1,g_2,g_3,\cdots,g_n)$$</span></p>
<p>where <span class="math-container">$g_i \in G_i$</span> <span class="math-container">$\forall i \in \{1,\cdots,n\}$</span>. Every element of <span class="math-contain... |
2,340,487 | <p>I was trying to compute this limit:
$$\lim_{x \to 0}\lim_{y \to 0} (x+y)\sin{\frac{x}{y}}$$</p>
<p>And this is my solution:
$$\lim_{x \to 0}\lim_{y \to 0}|(x+y)\sin{\frac{x}{y}}|\leq\lim_{x \to 0}\lim_{y \to 0} |(x+y)|=0$$</p>
<p>So I got the limit 0.</p>
<p>The answer was different. I have no idea what is wrong ... | Paramanand Singh | 72,031 | <p>You are intuitively trying to use a generalization of the following result from calculus of single variable : if $|f(x) |\leq g(x) $ as $x\to a$ and $g(x) \to 0$ as $x\to a$ then $f(x) \to 0$ as $x\to a$. This holds for limits of functions of two variables also. But you are dealing with an iterated limit and not a d... |
4,458,863 | <p>Let <span class="math-container">$z_1,\;z_2,\;z_3\;$</span> be complex number such that <span class="math-container">$|z_1|=|z_2|=|z_3|=|z_1+z_2+z_3|=2\;\;$</span>. If <span class="math-container">$|z_1-z_3|=|z_1-z_2|\; \;$</span> and <span class="math-container">$z_2 \neq z_3.\; \; $</span> Then Find value of <span... | nmasanta | 623,924 | <p><span class="math-container">$$I=\int \sqrt{\cosh(x)}~ dx=\int \sqrt{2\cosh^2\left(\frac x2\right)-1} ~dx=\int \sqrt{2\sinh^2\left(\frac x2\right)+1} ~dx$$</span>
Putting, <span class="math-container">$u=\dfrac{ix}{2}\implies du=\dfrac{i}{2}~dx$</span>,
<span class="math-container">$$I=-2i\int \sqrt{1-2\sin^2 u}~du=... |
649,379 | <p>I'm on the final part of my project, where I have to prove the Noether-Lasker Theorem (or copy out the following proof and "fill in the gaps"). I'm looking for an explanation of what's going on at a macro-level. I think I could follow the proof, but I don't understand how it proves what it says it proves. I've alrea... | zcn | 115,654 | <p>If you're looking for another way to see that the radicals of the primary ideals in a primary decomposition are unique, here is a basic fact which should make the connection to associated primes clearer (this is Lemma 4.4 in Atiyah-MacDonald):
$\newcommand{q}{\mathfrak{q}}$
$\newcommand{p}{\mathfrak{p}}$</p>
<p><st... |
367,643 | <p>A Norman window has the shape of a rectangle with a semi circle on top; diameter of the semicircle exactly matches the width of the rectangle. Find the dimensions of the Norman window whose perimeter is 300 in that has maximal area.</p>
<p><a href="https://i.stack.imgur.com/WBPm7.png" rel="nofollow noreferrer"><im... | AnilB | 64,096 | <p>Area of the window
$$A=\frac{\pi }{8}w^2+wh$$
and the perimeter
$$P=2\, h+w+\frac12\pi\, w$$
The problem is
$$max_{w,h}\ A\qquad s.t.\, P=300$$
Using perimeter constraint you can eliminate for h
$$h=\frac{300- w\, (1+\frac12\pi)}{2}$$
The area is then
$$A=\frac{\pi }{8}w^2+150\,w-\frac{1+\frac12\pi}{2}w^2=-\frac{4+\... |
184,219 | <p>I know that it is the standard functionality of <code>Merge</code> to combine the values of the same keys among associations.</p>
<p>Now I would like to deal with a situation in which, in my associations, the keys are strings (English words). And I want to define the sameness as two words having the same result fro... | kglr | 125 | <pre><code> data = {<|"effect" -> {5, {2, 3}}|>, <|"effects" -> {4, {1, 3, 5}}|>};
Merge[KeyMap[WordStem]/@ data, mergeFunc]
</code></pre>
<blockquote>
<p><|"effect" -> {9, {1, 2, 3, 5}}|></p>
</blockquote>
<p>Also</p>
<pre><code> GroupBy[ data, First @* WordStem @* Keys -> First, mergeFun... |
802,014 | <p>For all sets $A$, $B$, $C$, if $A$ is subset of $B$, $B$ is subset of $C$, and $C$ is subset of $A$, then $A = B = C$.</p>
<p>This is a true statement and I need to provide a proof? Thus, when a statement is false I need to provide it with counterexample whereas if it is true then it has to be provided by a proof?<... | user152102 | 152,102 | <p>We can prove it with set size. Let $\#A$ be the size (order) of $A$, so if $\#A=n$ holds, then $A$ has $n$ elements. Then $A$ is a subset of $B$, so you get $\#A\leq\#B$. Then you get $$\#A\leq\#B\leq\#C\leq\#A. $$ So you get $\#A\leq\#A$, and it is trivial that only $\#A=\#A$ is possible. So you get $$\#A=\#B=\#C=... |
13,030 | <p>At work, we were discussing when is it the best time to change to winter tires for bikes and/or cars.</p>
<p>Using <code>WeatherData[]</code> and <code>DateListPlot[]</code>, it was fairly straightforward for me to create the diagram below:</p>
<p><img src="https://i.stack.imgur.com/Y5wNT.png" alt="Mean temperatur... | Murta | 2,266 | <p>Nice question. First I wrote your code in this way. Where we don't need the function <code>yearStrip</code></p>
<pre><code>{date,year,month,day,temp}={1,1,2,3,2}
cityTemp=WeatherData["Stockholm","MeanTemperature",{{1977,1,1},{2011,12,31},"Day"}];
cityTemp[[All,date,year]]=0
iceRiskDays=Select[cityTemp,#[[temp]]<... |
4,072,386 | <p>I assume this is a simple proof but i'm stuck here.</p>
<p>I need to prove that if <span class="math-container">$A^3B-B$</span> is invertible then <span class="math-container">$BA-B$</span> is invertible.</p>
<p>So <span class="math-container">$A^3B-B=(A^3-I)B$</span> and then both <span class="math-container">$(A^... | Robert Lewis | 67,071 | <p>Note that</p>
<p><span class="math-container">$A^3B - B = (A^3 - I)B; \tag 1$</span></p>
<p>thus the invertibility of <span class="math-container">$A^3B - B$</span> implies both <span class="math-container">$A^3 - I$</span> and <span class="math-container">$B$</span> are invertible. Since</p>
<p><span class="math-c... |
126,074 | <p>I would like to know some applications of Anick's resolution in non-commutative algebras.</p>
| Ronnie Brown | 19,949 | <p>In view of Yemon's reference to group cohomology, I would like to mention Graham Ellis' work on <a href="http://hamilton.nuigalway.ie/" rel="nofollow">"Homological Algebra Programming"</a>. The key point is that he constructs free resolutions inductively together with a contracting homotopy: it is the latter that ... |
1,390,382 | <p>I have a problem that comes from absorbing random walks on a connected undirected graph $G$ with two types of nodes, absorbing nodes and free nodes. We randomly pick a node to start, once the random walk reaches an absorbing node, it will never leave the node again. But if we are at a free node, we will pick an outg... | Aiden | 96,554 | <p>Just come up with an argument from another direction.</p>
<p>Let the number of columns in $T_{ff}$ be $k$.
Suppose, to derive a contradiction, that for any given $i$, $0\le i\le k$, the sum of entries of $i$-th row in $T_{ff}^{n}$ is positive when $n$ goes to infinity. And let us agree for the time being that the l... |
501,678 | <p>Let $f(x)=x-\cos(x)$. Find all points on the graph of $y=f(x)$ where the tangent line has slope 1. (In each answer $n$ varies among all integers).</p>
<p>So far I've used the Sum derivative rule for which I have $1+\sin(x)$. So do I put in 1 in for $x$ for sin$(x)$.</p>
<p>Please Help!!</p>
| Community | -1 | <p>Let $y = \dfrac1x$. We then have
$$L = \lim_{x \to 0^+} \left(\sqrt{\dfrac1x+2} - \sqrt{\dfrac1x}\right) = \lim_{y \to \infty} \left(\sqrt{y+2} - \sqrt{y}\right) = \lim_{y \to \infty} \left(\sqrt{y+2} - \sqrt{y}\right) \times \dfrac{\sqrt{y+2} + \sqrt{y}}{\sqrt{y+2} + \sqrt{y}}$$
This gives us
$$L = \lim_{y \to \inf... |
688,782 | <p>$$a_n=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right) \cdots
\left(1-\frac{1}{n^2}\right)
$$</p>
<p>I have proved that this sequence is decreasing. However I am trying to figure out how to find its limit. </p>
| André Nicolas | 6,312 | <p><strong>Hint:</strong> Rewrite each $1-\frac{1}{k^2}$ as $\frac{(k-1)(k+1)}{k^2}$ and observe the mass cancellations. It will be useful to do this explicitly for say the product of the first $5$ terms. </p>
|
688,782 | <p>$$a_n=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right) \cdots
\left(1-\frac{1}{n^2}\right)
$$</p>
<p>I have proved that this sequence is decreasing. However I am trying to figure out how to find its limit. </p>
| Satish Ramanathan | 99,745 | <p>Answer:</p>
<p>$$\left(\frac{3}{2}\frac{4}{3}\frac{5}{4}\cdots\frac{n}{n-1}\frac{n+1}{n}\right)\cdot\left(\frac{1}{2}\frac{2}{3}\frac{3}{4}\cdots\frac{n-2}{n-1}\frac{n-1}{n}\right)$$</p>
<p>After mass cancellations, pull the $$\frac{n+1}{2}\text{ and }\frac{1}{n}$$</p>
<p>$$\frac{n+1}{2}\cdot\frac{1}{n}$$</p>
<p... |
3,360,879 | <p>Given the localised ring <span class="math-container">$\mathbb{Z}_{(2)}=\{\frac{a}{b}:a,b \in \mathbb{Z}, 2 \nmid b \}$</span>, I want to show that this is an integral domain.</p>
<p>We choose some fraction <span class="math-container">$ \frac{a}{b}\in \mathbb{Z}_{(2)}$</span>,where <span class="math-container">$... | Noah Riggenbach | 482,732 | <p>I think that the slickest proof of this is that <span class="math-container">$\mathbb{Z}_{(2)}$</span> embeds into <span class="math-container">$\mathbb{Q}$</span>. Then since <span class="math-container">$\mathbb{Q}$</span> is a field, all subrings must be domains.</p>
|
3,333,924 | <p>I am wondering about solutions to the following differential equation:
<span class="math-container">$f(x)=C_1 \cdot f'(x+C_2) \; \forall x \in \mathbb{R} \; \exists \; C_1, C_2 \in \mathbb{R}$</span>. With <span class="math-container">$C_1, C_2$</span> being constant. Are the solutions uniquely in the family of sin/... | Community | -1 | <p>A polynomial is an expression obtained by combining constants and variables by means of a <em>finite number</em> of additions and multiplications.</p>
<p>E.g. <span class="math-container">$3xy^3+2x-1$</span>.</p>
<p>An <em>algebraic function</em> of a single variable <span class="math-container">$x$</span> is such t... |
1,841,882 | <p>By Cayley's theorem, we know that for any finite group $G$, there exists $N \in \mathbb{N}$ such that $G$ is isomorphic to a subgroup of $S_N$, the symmetric group on $N$ letters. Can we prove that for every finite group $G$ there is some symmetric group $S_N$ such that $G$ is isomorphic to a $normal$ subgroup of $S... | Stefan4024 | 67,746 | <p><strong>HINT:</strong> Try to prove that for $n \ge 5$, $A_n$, the alternating group of $n$ elements is the only proper and nontrivial normal subgroup of $S_n$.</p>
<p><strong>UPDATE:</strong> This has to do something with the fact that $A_n$ is simple for $n \ge 5$. After proving this and checking the cases $n \le... |
1,641,076 | <p>Let's say I have the following decomposition: </p>
<p>$$\{100,10011,00110\}^*$$</p>
<p>How would I determine if the decomposition is ambiguous or unambiguous?</p>
| Brian Tung | 224,454 | <p>One way is to recognize that the intersection must satisfy the equation for both planes, and must therefore satisfy their sum:</p>
<p>$$
(3x-y+z)+(y+z) = 4+2
$$</p>
<p>$$
3x+2z = 6
$$</p>
<p>You can then let $x = t$, and then $3t+2z = 6$, whence we get $z = 3-\frac{3}{2}t$. You can then rewrite your first equati... |
2,521,710 | <p>I am trying to do a proof for convergence. But I am stuck in my proof not getting any further... What is missing to finish that proof?</p>
<p>$$a_n = \frac{1}{(n+1)^2}$$
Show that: $$\lim_{n \to \infty}a_n=0$$</p>
<p>Let $e > 0$ and $\forall n \ge n_0 = \lceil \frac{1}{\sqrt{\epsilon}}\rceil+1 \in \mathbb Z^+... | NotAMathematician | 485,701 | <p>Do you know about squeezing methods? Since $$0<a_n$$
You can show that the limit of the sequence is "trapped" between $0$, which only means it's equal to zero. Now, since
$$n+1>n$$ then
$$\frac{1}{n+1}<
\frac{1}{n}$$
$$\frac{1}{(n+1)^2}<\frac{1}{n^2}$$
Take in count here we're using only natural number... |
118,701 | <p>I have two vectors of 134 elements each ($mu$, and $gt$). $mu$ contains Integers, and $gt$ contains machine precision Reals. I execute the following simple expression multiple times without changing either mu or gt:
$$
(mu/2*gt).gt
$$
I will get one of two different results: $88474.52216839303$ or $88474.52216839301... | rcollyer | 52 | <p>They are not identical computations. With the first form,</p>
<pre><code>(mu/2 gt).gt
</code></pre>
<p>Mathematica can take advantage of vector arithmetic, usually going through specialized routines like LAPACK. The second form, </p>
<pre><code>Sum[(mu[[i]]/2 gt[[i]]) gt[[i]], {i, Length@mu}]
</code></pre>
<p>ho... |
2,146,508 | <blockquote>
<p>Let $K$ be the algebraic closure of a finite field $k$. Prove that $Gal(K/k) \cong \hat{\mathbb{Z}}$.</p>
</blockquote>
<p>From the definition in the book, here is how $\hat{\mathbb{Z}}$ is defined:
Let $D = Cr(\mathbb{Z}_{p} | \; p \; prime)$, let $\delta: \mathbb{Z} \rightarrow D$ be the map taking... | nguyen quang do | 300,700 | <p>For any field $k$ and a fixed algebraic closure $K$ of $k$, practically by definition, $K$ is the inductive (=direct) limit of its subextensions $L/k$ of finite degree. By (infinite) Galois theory, $Gal(K/k)$ is then the projective (=inverse) limit of its subgroups of finite index. If $k$ is a finite field, for a... |
181,855 | <p>In the latest <a href="http://what-if.xkcd.com/113/" rel="noreferrer">what-if</a> Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of the USA:
<img src="https://i.stack.imgur.com/gyfYt.png" al... | Moritz Firsching | 39,495 | <p>Looking at this old question again, I'm now fairly convinced that the easiest route of solving this problem is using similar ideas to the one suggested by <a href="https://mathoverflow.net/users/297/david-e-speyer">David E Speyer</a> in a <a href="https://mathoverflow.net/questions/181855/what-if-xkcd-com-stabbing-s... |
2,955,780 | <p>The midpoint of a chord of length <span class="math-container">$2a$</span> is at a distance <span class="math-container">$d$</span> from the midpoint of the minor arc it cuts out from the circle. Show that the diameter of the circle is <span class="math-container">$\frac{a^2+d^2}{d}$</span> .</p>
<p>I know I have t... | Bernard | 202,857 | <p>Let <span class="math-container">$A$</span>, <span class="math-container">$B$</span> be the end points of the chord (and of the chords cut out on the circle), <span class="math-container">$C$</span> the mid point of the minor arc, <span class="math-container">$D$</span> the midpoint of the major arc, so that CD is a... |
3,184,802 | <p>Are there any <span class="math-container">$C^\infty$</span> real functions except the exponential family and gamma function family which has all the derivatives of same sign on an interval [a,<span class="math-container">$\infty$</span>) with a<span class="math-container">$\gt$</span>0 ?
I speculate the function is... | Kavi Rama Murthy | 142,385 | <p>Your function is necessarily a 'mixture' of exponentials. All you have to do is apply Bernstein's Theorem (<a href="https://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions</a> ) to <span class="math... |
43,743 | <p>An alternative title is: When can I homotope a continuous map to a smooth immersion?</p>
<p>I have a simple topology problem but it's outside my area of expertise and I worry may be rather subtle. Any help would be appreciated.</p>
<p>The set-up is the following:
Let $M$ be some (closed say) $n$ dimensional mani... | Tom Goodwillie | 6,666 | <p>Reading between the lines, I suppose you have oriented manifolds $\Sigma_i$, so that each of them determines an element of the integral homology group $H_k(M)$. In fact the assumption that these two classes are equal does not even imply that there is a compact oriented $(k+1)$-manifold $\Gamma$ with boundary such th... |
304 | <p>Per <a href="http://blog.stackoverflow.com/2010/07/moderator-pro-tempore/">this post on the SO/SE blog</a> (which, curiously, does not include math.SE in its graphic list), it looks like the admins will choose moderators pro tempore at about 7 days into the public beta. In the roughly 24 hours that we've been in pu... | Isaac | 72 | <p><strong>YES</strong> we do need moderators pro tempore now</p>
|
1,830,989 | <p>so while playing around with circles and triangles I found 2-3 limits to calculate the value of $ \pi $ using the <em>sin, cos and tan</em> functions, I am not posting the formula for obvious reasons.<br>
My question is that is there another infinite series or another way to define the trig functions when the value ... | Noble Mushtak | 307,483 | <p>The summation for $\sin$ with degrees requires $\pi$:
$$\sin(x^\circ)=\sum_{i=1}^\infty (-1)^n\frac{\pi^{2n+1}x^{2n+1}}{180^{2n+1}(2n+1)!}$$
Notice how all I did was substitute $\frac{\pi x}{180}$ into the power series for sin with radians since that's the conversion from degrees to radians. There's just no way to a... |
4,567,410 | <p>I know that when calculating the pre-image of a function <span class="math-container">$f:X \rightarrow Y$</span> in a given subset <span class="math-container">$B$</span>, that is, <span class="math-container">$f^{-1}(B)=\{x \in X\mid f(x) \in B\}$</span>,the <span class="math-container">$f^{-1}$</span> simbol is ju... | Átila Correia | 953,679 | <p>The proposed relation does not hold in general.</p>
<p>In order to conclude so, consider the function <span class="math-container">$f := \{(0,1),(1,1)\}$</span> as well as the sets <span class="math-container">$A = \{0\}$</span> and <span class="math-container">$B = \{1\}$</span>. On the one hand, we have that <span... |
4,194,611 | <p>Let <span class="math-container">$A \in \mathbb{R}^{m \times n}$</span>.</p>
<p><span class="math-container">$\forall i \in \mathbb{N} \; [ x_i \in \mathbb{R}^n \; \mbox{and} \; x_i \geq \textbf{0}] $</span></p>
<p>Assume that the sequence <span class="math-container">$Ax_1, Ax_2, ...$</span> converges to <span clas... | Yuval Filmus | 1,277 | <p>Let <span class="math-container">$\ell$</span> be some linear dependence of the columns of <span class="math-container">$A$</span>, and consider some <span class="math-container">$x_t$</span>. Thus <span class="math-container">$A(x_t + \gamma \ell) = Ax_t$</span> for all <span class="math-container">$\gamma$</span>.... |
152,582 | <p>I have tried to use <code>OpenRead</code> for my application as it appears to be less of a burden on the memory.</p>
<p>I <code>OpenRead</code> a .csv file to extract coordinates in the form {x,y} in the following way</p>
<pre><code>ClearAll[f];
f = OpenRead["mathfile.csv"];
g = ReadList[f, String, RecordL... | terrygarcia | 49,372 | <p>If you use <code>OpenRead</code> as described above you'll have to iterate over your records a second time in order to convert strings to pairs of numbers. This is not memory-efficient. Also, if your input stream will no longer be needed after you load the records, it's better to invoke <code>With[{f=OpenRead["mathf... |
2,316,362 | <p>If $G$ is a Lie Group, a representation of $G$ is a pair $(\rho,V)$ where $V$ is a vector space and $\rho : G\to GL(V)$ is a group homomorphism.</p>
<p>Similarly, if $\mathfrak{g}$ is a Lie Algebra, a representation of $\mathfrak{g}$ is a Lie Algebra homomorphism $\rho : \mathfrak{g}\to \mathfrak{gl}(V)$ to the Lie... | Community | -1 | <p>I think the answer on your question is given probably from a geometric point of view. There is a beautiful theorem from <strong>Lie</strong> himself and is usually referred as <strong>Lie's 3rd Theorem</strong>, and states something which nowadays is rephrased as follows (over the complex numbers)</p>
<p><strong>Th... |
84,249 | <p>I have a data set with evenly spaced data points. The plot is frequency vs. intensity. The overall shape of the plot is an upwards curve into a plateau, this cannot be seen in the data as this is an unimportant feature. There is also an oscillation in this curve. This can be seen in the plot.</p>
<p><img src="https... | Mausy5043 | 29,714 | <p>The data looks like it follows an exponential curve y=Ae^(x+t).</p>
<ol>
<li>find a least-squares fit y=f(x) of the data on such a curve. So, find the function f(x) for the curve.</li>
<li>for each point on the graph do y0 = y-f(x). This flattens out the graph.</li>
<li>then do a FFT analysis on y0 to find the freq... |
1,765,946 | <p>$\newcommand{\Sig}{\Sigma}$
Let $\Sig$ be a diagonal matrix with strictly positive entries on the diagonal. Define $V=\{B \in M_n\mid B\Sig +\Sig B^T=\Sig B +B^T \Sig \}$ (where $M_n$ is the vector space of $n \times n$ real matrices).</p>
<p><strong>What is the dimension of $V$? Is there a "nice" basis for it?</st... | joriki | 6,622 | <p>Since $V$ contains the subspace of symmetric matrices, we can write $B$ as the sum of a symmetric matrix and an antisymmetric matrix that's also in $V$. Antisymmetric matrices $A$ in $V$ fulfil $A\Sigma-\Sigma A=\Sigma A-A\Sigma$ and thus $A\Sigma=\Sigma A$, that is, they commute with $\Sigma$. That means an entry $... |
3,716,619 | <p>Evaluating
<span class="math-container">$$\lim_{x\to 0}\left(\frac{\pi ^2}{\sin ^2\pi x}-\frac{1}{x^2}\right)$$</span>
with L'Hospital is so tedious. Does anyone know a way to evaluate the limit without using L'Hospital? I have no idea where to start.</p>
| md2perpe | 168,433 | <p>Using Maclaurin expansion we get:
<span class="math-container">$$
\frac{\sin\pi x}{\pi x}
= \frac{\pi x - \frac16 (\pi x)^3 + O(x^5)}{\pi x}
= 1 - \frac{\pi^2 x^2}{6} + O(x^4)
\\
\frac{\pi x}{\sin\pi x} = 1 + \frac{\pi^2 x^2}{6} + O(x^4)
\\
\left(\frac{\pi x}{\sin\pi x}\right)^2 = 1 + \frac{\pi^2 x^2}{3} + O(x^4)
... |
3,716,619 | <p>Evaluating
<span class="math-container">$$\lim_{x\to 0}\left(\frac{\pi ^2}{\sin ^2\pi x}-\frac{1}{x^2}\right)$$</span>
with L'Hospital is so tedious. Does anyone know a way to evaluate the limit without using L'Hospital? I have no idea where to start.</p>
| J.G. | 56,861 | <p>Yet another approach: as it's an even function, assume <span class="math-container">$x>0$</span>. Cut a sector in a radius-<span class="math-container">$\sqrt{2}$</span> angle subtending <span class="math-container">$\pi x$</span> radians at the centre so <span class="math-container">$\pi x-\sin\pi x$</span> is t... |
4,621,390 | <p>I'm studying Linear Algebra and have come to think of a column vector as an ordered bunch of objects, where each object is the product of a scalar and a basis vector, vis:</p>
<p><span class="math-container">$\begin{bmatrix}
a \\
b \\
\vdots \\
\end{bmatrix} = a \hat{i} + b \hat{j} + \dots$</span></p>
<p>Where <span... | helixer | 1,073,720 | <p>First of all, basis vectors are linearly independent, so they better be unique, or else they're not basis vectors.</p>
<p>To clarify Seeker's answer:</p>
<blockquote>
<p>The entries in a column vector come from a linear combination of the basis vectors.</p>
</blockquote>
<p>This is referring to all entries at once, ... |
1,098,253 | <p>I have got some trouble with proving that for $x\neq 0$:
$$
\frac{\arctan x}{x }< 1
$$
I tried doing something like $x = \tan t$ and playing with this with no success.</p>
| rar | 196,345 | <p>We want to show that $\arctan(x) \leq x$ for all positive x (or vice-versa for negative x). Notice that at $x=0$, we can evaluate $\arctan(x) = 0$, so the functions are equal. Now, the derivative of $\arctan$ is $1/(1+x^2) < x' = 1$, and paired with our former observation, by a well-known theorem from calculus, t... |
1,098,253 | <p>I have got some trouble with proving that for $x\neq 0$:
$$
\frac{\arctan x}{x }< 1
$$
I tried doing something like $x = \tan t$ and playing with this with no success.</p>
| marwalix | 441 | <p>For $0\lt x \lt \frac{\pi}{2}$ we have $x\leq\tan{x}$ This leads to $\frac{\arctan{x}}{x}\leq 1$
Similarly for $0\gt x \gt -\frac{\pi}{2}$ we have $x\geq\tan{x}$ and this leads to the same inequality. </p>
|
2,929,025 | <p>Bill gave exams for the entrance at some specific gymnasium. <span class="math-container">$602$</span> students took part, which were classified, after the exams, in an ascending order, and the first <span class="math-container">$108$</span> students will be taken, which will accept to enter. Every student that has... | karakfa | 14,900 | <p>easier to compute by hand (or calculator) if you re-write it as nested terms</p>
<p><span class="math-container">$$
1-p^n(1+rn(1+r\frac{(n-1)}2(1+r\frac{(n-2)}3(1+r\frac{(n-3)}4))))
$$</span></p>
<p>where
<span class="math-container">$$
p=0.98 \\
n=112 \\
r=(1-p)/p \\
$$</span></p>
|
3,348,178 | <p>I have three vectors in 3d that originate at a point. If I look at them along a line perpendicular to a plane that intersects two of them, how do I find the angles between those two vectors and the third one?</p>
<p>Clarification because this is frickin difficult to explain:</p>
<p><a href="https://i.stack.imgur.c... | mathcounterexamples.net | 187,663 | <p><span class="math-container">$\sqrt{x^2}= \sqrt{y^2}$</span> doesn’t imply <span class="math-container">$x=y$</span>!!!</p>
<p>Take <span class="math-container">$x=2$</span> and <span class="math-container">$y=-2$</span>.</p>
|
3,348,178 | <p>I have three vectors in 3d that originate at a point. If I look at them along a line perpendicular to a plane that intersects two of them, how do I find the angles between those two vectors and the third one?</p>
<p>Clarification because this is frickin difficult to explain:</p>
<p><a href="https://i.stack.imgur.c... | cansomeonehelpmeout | 413,677 | <blockquote>
<p><span class="math-container">$$x=y\iff -x=y\lor x=-y\tag{1}$$</span></p>
</blockquote>
<p>Doesn't makes sense since <span class="math-container">$-x=y\iff x=-y$</span>, that is, they're equivalent. This reduces <span class="math-container">$(1)$</span> to <span class="math-container">$$x=y\iff -x=y\t... |
3,348,178 | <p>I have three vectors in 3d that originate at a point. If I look at them along a line perpendicular to a plane that intersects two of them, how do I find the angles between those two vectors and the third one?</p>
<p>Clarification because this is frickin difficult to explain:</p>
<p><a href="https://i.stack.imgur.c... | Allawonder | 145,126 | <p>You made your very first mistake in the first line. The statements <span class="math-container">$x=y$</span> and <span class="math-container">$\sqrt{x^2}=\sqrt{y^2}$</span> are not equivalent as claimed, since the latter means <span class="math-container">$|x|=|y|.$</span> Thus, if you take <span class="math-contain... |
298,912 | <p>I was reading some basic information from Wiki about category theory and honestly speaking I have a very weak knowledge about it. As it sounds interesting, I will go into the theory to learn more if it is actually useful in practice.</p>
<p>My question is to know if category theory has some applications in practice... | Martin Brandenburg | 1,650 | <p>The blog entry <a href="http://rs.io/why-category-theory-matters/">"Why Category Theory Matters"</a> by Robert Seaton ends with a quite impressive reference list of applications of category theory to the sciences:</p>
<ul>
<li>Category theory has been used to study <a href="http://reyes-reyes.com/1999/06/01/count-n... |
1,452,121 | <p>I very well know that every open ball is an open set. and that every open set need not be an open ball. But illustrate me some counter example.</p>
| layman | 131,740 | <p>Here is a counterexample.</p>
<p>Consider the set $\Bbb R$ of real numbers. Let $B(0, 2)$ represent the ball of radius $2$ around $0$, i.e, the interval $(0-2, 0+2) = (-2,2)$.</p>
<p>Now let $B(5,2)$ represent the ball of radius $2$ around $5$, i.e., the interval $(5-2, 5+2) = (3,7)$.</p>
<p>The union of these t... |
4,942 | <p>I'd like to control more aspects of a <code>DateListPlot</code>, for example: shading for weekend days, and/or indicators for daytime/nighttime areas. </p>
<p>By way of illustration, here's a simple example of a set of time data points (recent questions on mathematica.stackexchange):</p>
<pre><code>questions =
... | Markus Roellig | 135 | <p>Somehow DateListPlot is resistant to many styling options. Without fiddling with the internals, here is a starting point.</p>
<pre><code>WeekendQ[date_] :=
With[{d = DateString[date, "DayName"]},
MatchQ[d, "Saturday" | "Sunday"]]
weekStyle = {Blue};
weekendStyle = {PointSize -> .015, Directive[Red]};
styleL... |
28,892 | <p>I was searching on MathSciNet recently for a certain paper by two mathematicians. As I often do, I just typed in the names of the two authors, figuring that would give me a short enough list. My strategy was rather dramatically unsuccessful in this case: the two mathematicians I listed have written 80 papers toget... | Deane Yang | 613 | <p>Another thing that is surprisingly rare is a long term collaboration within a single math department. Erwin Lutwak, Gaoyong Zhang, and I, all at NYU-Poly, have a long term relationship that has yielded about jointly authored 21 papers so far, some with an additional author.</p>
|
3,760,450 | <p>I have a class in numerical mathematics, and I received several tasks I should answer. I am not a mathematician, and this is a bit out of my mind range, and I would be grateful for answers.
Question is as follows:</p>
<p>Let <span class="math-container">$x, y \in \Bbb{C}^n$</span> be arbitrary vectors. Show what can... | heropup | 118,193 | <p>Your answer is correct if:</p>
<ol>
<li>The area is also bounded by the line <span class="math-container">$x = 3$</span></li>
<li>We are interested in the unsigned (absolute) area enclosed.</li>
</ol>
<p>I think the most challenging part of the question is determining <span class="math-container">$g$</span>. For <s... |
374,619 | <p>In <a href="https://math.stackexchange.com/a/373935/752">this recent answer</a> to <a href="https://math.stackexchange.com/q/373918/752">this question</a> by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$</p... | Micah | 30,836 | <p>Here's a way of finding, at the very least, a large class of rational solutions. It seems plausible to me that these are all the rational solutions, but I don't actually have a proof yet...</p>
<p>Say we want to solve $(p+q\sqrt{3})^{1/3}+(p-q\sqrt{3})^{1/3}=n$ for some fixed $n$. The left-hand side looks an awful ... |
374,619 | <p>In <a href="https://math.stackexchange.com/a/373935/752">this recent answer</a> to <a href="https://math.stackexchange.com/q/373918/752">this question</a> by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$</p... | mercio | 17,445 | <p>This kind of simplification occurs if and only if $p \pm q\sqrt d$ has a cube root of the form $x \pm y\sqrt d$ with rational $x,y$. So, to get all instances of this, start by choosing $x+y\sqrt d$, and cube it to get the values for $p$ and $q$.</p>
<p>Setting up the system $(x + y\sqrt d)^3 = p + q\sqrt d$, we get... |
1,744,698 | <p>How to show that the characteristic polynomials of matrices A and B are $\lambda^{n-1}(\lambda ^2-\lambda -n)=0$ and $\lambda^{n-1}(\lambda^2+\lambda-n)=0$ respectively by applying elementary row or column operations.</p>
<p>$A=\begin{bmatrix}
1 & 1 & 1 & \cdots & 1 \\
1 & 0 & 0 & \cdots... | Solumilkyu | 297,490 | <p>Use induction on $n$, we only consider the matrix $A$ because solving $B$ is similar. Before starting, I think that the characteristic polynomial of $A$ and $B$ should be of the form $\color{red}{(-1)^{n+1}}\lambda^{n-1}(\lambda^2-\lambda-n)$ and
$\color{red}{(-1)^{n+1}}\lambda^{n-1}(\lambda^2+\lambda-n)$, respectiv... |
226,449 | <p>Many counting formulas involving factorials can make sense for the case $n= 0$ if we define $0!=1 $; e.g., Catalan number and the number of trees with a given number of vetrices. Now here is my question:</p>
<blockquote>
<p>If $A$ is an associative and commutative ring, then we can define an
unary operation on ... | kcrisman | 24,113 | <p>As pointed out in one of the answers to <a href="https://math.stackexchange.com/questions/20969/prove-0-1-from-first-principles">this math.SX question</a>, you can get the Gamma function as an extension of factorials, and then this falls out from it (though this isn't a very combinatorial answer).</p>
|
300,253 | <p>I'm interested in invertible matrices that are built out of invertible sub-blocks. For example, four sub-blocks from $GL_n(F)$ (i.e. the group of $n \times n$ invertible matrices over a field $F$) can be assembled into a $2n \times 2n$ matrix, which may or may not be invertible.</p>
<p>Suppose that a $kn \times kn... | Did | 6,179 | <p>Dunno, maybe that $\mathrm{var}(X_t)=1$ for every $t$.</p>
|
542,951 | <p>I am trying show that the function $f:[0,1]\to \mathbb{R}$ defined by $f(x)=\sin \dfrac{1}{x}$ if $x\neq 0$ and $f(0)=0$ possesses IVP. Though it looks easy, but I am not getting any clue how to start with. Any help would be appreciated.</p>
| Michael Hoppe | 93,935 | <p>You want to show -- despite $f$ is not continuos -- that every value between $f(0)=0$ and $f(1)=\sin(1)$ is attached by $f$. That's simple: consider the restriction of $f$ on $[1/\pi,1]$. Can you work from here applying IVT on that restriction?</p>
|
1,356,932 | <blockquote>
<p><strong>Problem.</strong> Let $h\in C(\mathbb{R})$ be a continuous function, and let
$\Phi:\Omega:=[0,1]^{2}\rightarrow\mathbb{R}^{2}$ be the map defined
by \begin{align*}
\Phi(x_{1},x_{2}):=\left(x_{1}+h(x_{1}+x_{2}),x_{2}-h(x_{1}+x_{2})\right) \tag{1}
\end{align*}
What is the (Lebesgue) meas... | Theo Bendit | 248,286 | <p>The domain of "all functions" seems a little too broad here. I think we should restrict ourselves to functions in $L^2[0, 1]$, so at least we may work within a Hilbert space.</p>
<p>Notice that $\phi : f \mapsto \int_0^1 x^2 f(x) \mathrm{d}x = \langle f(x), x^2 \rangle$ is a bounded linear functional, mapping $L^2[... |
2,150,832 | <p>I don't understand this equation $\int_0^t ds \int_0^{t'} ds' \delta(s-s')= \min(t,t')$.
I tried to work with the property of the dirac delta function that $\int_a^b \delta(x-c)dx = 1$ if $c \in [a,b]$, but I can't see how I can obtain the minimum. Can someone help me? </p>
<p>Thank you in advance!</p>
| Qmechanic | 11,127 | <p>Using the notation for <a href="https://en.wikipedia.org/wiki/Positive_and_negative_parts" rel="nofollow noreferrer">positive and negative parts</a>, we calculate</p>
<p>$$I(t,t^{\prime})~:=~ \int_0^t \!\mathrm{d}s \int_0^{t^{\prime}} \! \mathrm{d}s^{\prime} ~\delta(s\!-\!s^{\prime})
~=~{\rm sgn}(t)~{\rm sgn}(t^{\p... |
2,942,879 | <p>Suppose that <span class="math-container">$lim_{n\rightarrow \infty} a_n = L$</span> and <span class="math-container">$L \neq 0$</span>. Prove there is some <span class="math-container">$N$</span> such that <span class="math-container">$a_n \neq 0$</span> for all <span class="math-container">$n \geq N$</span>.</p>
... | James | 549,970 | <p>WLOG assume <span class="math-container">$L > 0$</span>, so in the definition of limit, we may take <span class="math-container">$\epsilon = L$</span> and then an <span class="math-container">$N$</span> can be found so that for all <span class="math-container">$n \geq N$</span>,</p>
<p><span class="math-containe... |
307,545 | <p>If $\gcd(a,b)=1$, how can I find the values that $\gcd(a+b,a^2+b^2)$ can possibly take? I can't find a way to use any of the elemental divisibility and gcd theorems to find them. </p>
| Andreas Caranti | 58,401 | <p>We have $a^2 + b^2 - a(a+b) = b^2 - ab = -b (a-b)$ and $a^2 + b^2 - b(a+b) = a^2 - ba = a (a-b)$. </p>
<p>So if $d$ divides both $a+b$ and $a^2+b^2$, then $d$ divides $$\gcd(a (a-b), b (a-b)) = \gcd(a, b) (a-b) = a - b.$$</p>
<p>So $d$ divides $a+b + a - b = 2a$ and $a+b - (a - b) = 2b$.</p>
<p>So $d$ divides $2\... |
307,545 | <p>If $\gcd(a,b)=1$, how can I find the values that $\gcd(a+b,a^2+b^2)$ can possibly take? I can't find a way to use any of the elemental divisibility and gcd theorems to find them. </p>
| robjohn | 13,854 | <p>Since <span class="math-container">$\gcd(a,b)=1$</span>, <a href="https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity" rel="nofollow noreferrer">Bezout's Identity</a> says we have an <span class="math-container">$x$</span> and <span class="math-container">$y$</span> so that
<span class="math-container">$$
ax+by=1... |
3,910,623 | <p>There is a problem that appears in an interview<span class="math-container">$^\color{red}{\star}$</span> with <a href="https://en.wikipedia.org/wiki/Vladimir_Arnold" rel="nofollow noreferrer">Vladimir Arnol'd</a>.</p>
<blockquote>
<p>You take a spoon of wine from a barrel of wine, and you put it into your cup of tea... | AlvinL | 229,673 | <p>It helps to discretise the problem. E.g red and blue marbles. Suppose there are <span class="math-container">$M$</span> red marbles in container <span class="math-container">$1$</span> and <span class="math-container">$N$</span> blue marbles in container <span class="math-container">$2$</span>. We can assume <span c... |
7,223 | <p>I want to produce a <em>Mathematica</em> Computable Document in which <code>N</code> appears as a variable in my formulae. But <code>N</code> is a reserved word in the <em>Mathematica</em> language. Is there a way round this other than using a different symbol? It seems a severe limitation if you cannot use <em>Math... | Peter Gorry | 8,273 | <p>There is actually an easy way to achieve this (almost) perfectly. Just start the variable name with a none printing character. I use this technique to have variables for atomic orbitals that start with numbers e.g. 3d or 4f - also not allowed by Mathematica. I just write the name as <code>\[Null]3d</code>. Mathemati... |
3,952,702 | <p>The problem is to make the following integral stationary:
<span class="math-container">$$ \int_{x_1}^{x_2} \frac{\sqrt{1+y'^2}}{y^2}dx $$</span>
to simplify the Euler equation, I tried to change the independent variable:
<span class="math-container">$$ \int_{y_1}^{y_2} \frac{\sqrt{1+x'^2}}{y^2}dy, \: \: \: \: \: \: ... | Parcly Taxel | 357,390 | <p>Do it, substitute <span class="math-container">$u=ky^2,\frac{du}{dy}=2\sqrt{ku}$</span> (assuming <span class="math-container">$k>0$</span>):
<span class="math-container">$$x=\frac1{2\sqrt k}\int\sqrt{\frac u{1-u^2}}\,du$$</span>
This is an elliptic integral. Say we're integrating from <span class="math-container... |
35,393 | <p>I have some data that represents small-scale surface topography. The data is planar but not flat because it's impractical to level this microscope to a tolerance of microns. It looks like this:</p>
<p><img src="https://i.stack.imgur.com/nQcLB.jpg" alt="An example of 3D profilometric data with a nonzero planar slope... | ssch | 1,517 | <p>You can use <a href="http://reference.wolfram.com/mathematica/ref/LinearModelFit.html" rel="nofollow noreferrer"><code>LinearModelFit</code></a> to create a linear-fit of the model, and use that to find an appropriate rotation.</p>
<pre><code>model = LinearModelFit[
data,
{x, y},
{x, y}];
(* We have: z = c... |
2,247,900 | <p>Prove the following inequality</p>
<p>$$\ln \frac{\pi + 2}{2} \cdot \frac{2}{\pi} < \int \limits_0^{\pi/2} \frac{\sin\ x}{x^2 + x} < \ln \frac{\pi + 2}{2}$$</p>
<p>I can prove that $\frac{\sin\ x}{x^2 + x} < \frac{1}{x + 1} \ \forall x \in (0, +\infty) \Rightarrow \int \limits_0^{\pi/2} \frac{\sin\ x}{x^2... | Jack D'Aurizio | 44,121 | <p>The given inequality just depends on the convexity inequality
$$ \forall x\in\left[0,\frac{\pi}{2}\right],\qquad \frac{2}{\pi}x\leq \sin(x)\leq x.$$
We may use another convexity inequality
$$ \forall x\in\left[0,\frac{\pi}{2}\right],\qquad x-\frac{1}{6}x^3\leq \sin(x)\leq x-\left(\frac{4}{\pi^2}-\frac{8}{\pi^3}\rig... |
2,306,709 | <p>Say I have a set of buildings $B$ with a single building $b \in B$ and a set of people $P$ with single persons $p \in P$.</p>
<p>What would be the proper notation for a subset of people in $P$, that live in a building $p$ which comprehensibly associates this subset to the respective building? </p>
| Georgios | 311,495 | <p>You could define a function $f \colon P \to B$ that maps every person to the building he or she lives in. Then, $f^{-1} (b)$ would be the set of people that live in building $b$.</p>
|
2,306,709 | <p>Say I have a set of buildings $B$ with a single building $b \in B$ and a set of people $P$ with single persons $p \in P$.</p>
<p>What would be the proper notation for a subset of people in $P$, that live in a building $p$ which comprehensibly associates this subset to the respective building? </p>
| Mark S. | 26,369 | <p>You could write many different notations, depending on your emphasis/context. I would consider something like "For each $b\in B $, define $P_b $ to be the set of people who live in building $b $.".</p>
|
160,169 | <p>Consider the following implementation of the complex square root:</p>
<pre><code>f[z_]:=Sqrt[(z - I)/(z + I)]*(z + I);
</code></pre>
<p>This implementation has branch points at $\lambda=\pm i$ and a (vertical) branch cut connecting them.</p>
<p>Then</p>
<pre><code>g[z_]:=Sinc[f[z]];
</code></pre>
<p>(recalling ... | Daniel Lichtblau | 51 | <p>It is a problematic result to be sure. In the version under development there is modest improvement to the <code>Series</code>, and no change of note for <code>SeriesCoefficient</code>. The underlying issue I think is an inability to deduce a needed simplification.</p>
<pre><code>f[z_] := Sqrt[(z - I)/(z + I)]*(z +... |
1,009,082 | <p>Given is a linear map f from V to W, whereby V has dimension n and W has dimension m.</p>
<p>Now given n > m, can the map be injective,surjective or invertible?
And what about the same questions, given that m > n?</p>
<p>My thoughts so far: </p>
<ul>
<li><p>Invertibility should be possible in the second case, if ... | Seth | 31,659 | <p>The <a href="http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem#Stone.E2.80.93Weierstrass_theorem.2C_real_version" rel="nofollow">Stone-Weierstrass theorem</a> (a natural generalization of the Weierstrass approximation theorem) says that polynomials with all terms of even degree are uniformly dense in co... |
172,139 | <p>I have a data set, I am trying to join all the data by a line. But I am afraid the plot is not doing it properly. </p>
<p>This is the example (or almost) of my problem:</p>
<pre><code>data = {{0, π}, {π/2, π/2}, {π/2,
3 π/2}, {π, 0}, {π,
2 π}, {3 π/2, π/2}, {3 π/2,
3 π/2}, {2 π, π}};
</code></p... | corey979 | 22,013 | <pre><code>ListLinePlot[data[[FindShortestTour[data][[2]]]],
Frame -> True, AspectRatio -> 1, Epilog -> {Red, PointSize[Large], Point[data]}]
</code></pre>
<p><a href="https://i.stack.imgur.com/un934.png" rel="noreferrer"><img src="https://i.stack.imgur.com/un934.png" alt="enter image description here"></a><... |
3,077,312 | <p>The proof given in my book (and I came up with as well) is:</p>
<p><a href="https://i.stack.imgur.com/H6eqf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/H6eqf.png" alt="Proof"></a></p>
<p>However, the part that throws me off is line #3 where they do <span class="math-container">$\Sigma A_{jk}... | b00n heT | 119,285 | <p>You are not missing much, just the fact that by convention we usually write
<span class="math-container">$$(AB)_{ij}=\sum_k A_{ik}B_{kj}$$</span>
and not (although equivalent)
<span class="math-container">$$(AB)_{ij}=\sum_k B_{kj}A_{ik}$$</span>
because in the first case “ikkj becoms ij”</p>
|
3,077,312 | <p>The proof given in my book (and I came up with as well) is:</p>
<p><a href="https://i.stack.imgur.com/H6eqf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/H6eqf.png" alt="Proof"></a></p>
<p>However, the part that throws me off is line #3 where they do <span class="math-container">$\Sigma A_{jk}... | user408858 | 408,858 | <p>What they want to make clear in line 3 is, that when you want to calculate the entry <span class="math-container">$ij$</span> of the transposed matrix, you can sum over the <span class="math-container">$i$</span>-th column of <span class="math-container">$B$</span> and <span class="math-container">$j$</span>-th row ... |
59,932 | <p>So I need to make a plot of some ~2000 data points I have from a spreadsheet. I'm able to import the data just fine, it stores it like so</p>
<pre><code>sundat
</code></pre>
<blockquote>
<p>{{280.,0.082},{280.5,0.099},......{3995.,0.0087},{4000.,0.00868}}</p>
</blockquote>
<p>And I can call individual data sets... | Simon Woods | 862 | <p>You can also use <a href="http://reference.wolfram.com/language/ref/Dot.html" rel="noreferrer"><code>Dot</code></a>:</p>
<pre><code>sundat.{{2., 0}, {0, .5}}
(* {{560., 0.041}, {561., 0.0495}, {7990., 0.00435}, {8000., 0.00434}} *)
</code></pre>
|
76,420 | <p>The following code segment shows what I'd like to do. I'm a procedural programmer trying to learn the Mathematica functional style. Any help on this would be appreciated.</p>
<pre><code>B = {{1, 3}, {1, 5}, {4, 2}, {5, 2}, {5, 5}}
u = SparseArray[{{1, 2} -> 5, {1, 3} -> 9, {1, 4} -> 6,
{1, 5} ->... | C. E. | 731 | <p>This is just a way to rewrite the data into a - for our purposes - better format:</p>
<pre><code>u = {{1, 2} -> 5, {1, 3} -> 9, {1, 4} -> 6, {1, 5} -> Infinity, {3, 2} -> 2, {3, 4} -> 4, {4, 2} -> 9, {5, 2} -> Infinity, {5, 3} -> Infinity, {5, 4} -> Infinity, {5, 5} -> Infinity};
r... |
76,420 | <p>The following code segment shows what I'd like to do. I'm a procedural programmer trying to learn the Mathematica functional style. Any help on this would be appreciated.</p>
<pre><code>B = {{1, 3}, {1, 5}, {4, 2}, {5, 2}, {5, 5}}
u = SparseArray[{{1, 2} -> 5, {1, 3} -> 9, {1, 4} -> 6,
{1, 5} ->... | djp | 25,325 | <p>I don't think this is a good example for learning functional style. Of course it can be done as other answers show, but they are cryptic for two reasons:</p>
<p>(1) Mathematica doesn't accommodate "for {i, j} in B" (though Simon Woods' answer is pretty close)</p>
<p>(2) your code is actually depending on side eff... |
119,696 | <p>Let us suppose i have two graphs for sequences A and B as follows</p>
<pre><code>a1 := {{0.9, 0.086133}, {0.086133, 0.0082432}, {0.0082432,
0.0007889}, {0.0007889, 0.0000755}, {0.0000755,
7.2256*10^-6}, {7.2256*10^-6, 6.9151*10^-7}, {6.9151*10^-7,
6.618*10^-8}, {6.618*10^-8, 6.3336*10^-9}, {6.3336*10... | e.doroskevic | 18,696 | <h2>Example</h2>
<p><strong>Code</strong></p>
<pre><code>ListPlot[{a, b}, Joined -> True, PlotStyle -> {Red, Yellow}, PlotLegends -> Automatic]
</code></pre>
<p><strong>Note:</strong> <code>a</code> and <code>b</code> are your original data lists <code>a1</code> and <code>a2</code> respectively.</p>
<p><st... |
3,607,430 | <blockquote>
<p>Given that <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span> are the angles of a right-angled triangle, prove that:
<span class="math-container">$$\begin{align}
\sin a\sin b\sin(a-b) &+\sin b\sin c\sin(b-c)+\sin c\sin a\sin(... | John Omielan | 602,049 | <p>You have the right idea. Since the lines involved in the question must have a <span class="math-container">$y$</span>-intercept, they can't be vertical and so they have a specific slope. Thus, they can be written in the form</p>
<p><span class="math-container">$$y = mx + b \tag{1}\label{eq1A}$$</span></p>
<p>The <... |
3,607,430 | <blockquote>
<p>Given that <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span> are the angles of a right-angled triangle, prove that:
<span class="math-container">$$\begin{align}
\sin a\sin b\sin(a-b) &+\sin b\sin c\sin(b-c)+\sin c\sin a\sin(... | Peter Szilas | 408,605 | <p>Intercept equation of a line:</p>
<p><span class="math-container">$x/a+y/b=1$</span>;</p>
<p>This line passes through <span class="math-container">$(x_0,y_0):$</span></p>
<p><span class="math-container">$x_0/a+y_0/b=1$</span>;</p>
<p>Given: <span class="math-container">$c=ab$</span>;</p>
<p>Then </p>
<p><span ... |
300,531 | <p>Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$</p>
<p>where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$).</p>
<hr>
<p>This integral was mentioned in <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant">Wikipedia</a> as in <a href="http://mathw... | robjohn | 13,854 | <p>In <a href="https://math.stackexchange.com/a/300574">this answer</a>, it is shown that since $\Gamma$ is log-convex,
$$
\frac{\Gamma'(x)}{\Gamma(x)}=-\gamma+\sum_{k=1}^\infty\left(\frac1k-\frac1{k+x-1}\right)\tag{1}
$$
Setting $x=1$ yields
$$
\Gamma'(1)=-\gamma\tag{2}
$$
The integral definition of $\Gamma$ says
$$
\... |
103,970 | <p>Here's a very bizarre inconsistency I've just struggled with and I'm wondering why it exists or if I'm missing something.</p>
<p>I have some noisy data and I wish to make a framed plot of the data but allow the data to extend outside the vertical limits of the frame (for stylistic reasons). Like so:</p>
<pre><code... | LLlAMnYP | 26,956 | <p>Here's some fake data:</p>
<pre><code>pts = RandomReal[{-.2, .2}, 37] + (Cos[.1 #] & /@ Range[-1, 35]) // Thread[{Range[-1, 35], #}] &
</code></pre>
<p>Make a plot without the frame that clips the data in the x direction but not in the y direction (by having an extended y range):</p>
<pre><code>plot = Lis... |
2,333,857 | <p>I have this problem that I have worked out. Will someone check it for me? I feel like it is not correct. Thank you!</p>
<p>Rotate the graph of the ellipse about the $x$-axis to form an ellipsoid. Calculate the precise surface area of the ellipsoid. </p>
<p>$$\left(\frac{x}{3}\right)^{2}+\left(\frac{y}{2}\right)^{... | MPW | 113,214 | <p>Your second alternative is the correct one.</p>
<p>The value of $x$ is not a <em>set</em> of numbers, rather it is <em>in a set</em> of numbers.</p>
|
347,315 | <p>How can I find </p>
<ol>
<li>The image of the upper half plane $\mathrm{Im}(z)>0$ under the linear fractional transformation $w=\dfrac{3z + i}{-iz + 1}$.</li>
<li>The image of the set {${z∈C-0:\{\mathrm{Im}(z)} = \mathrm{Re}(z)\}$} under $w=z + \dfrac{1}{z}$.</li>
</ol>
<p>For 1.,
I consider $y>0, x \in R$ a... | sebigu | 32,185 | <p>I can answer the question like this:</p>
<p>Looking at the Fratini subgroups using GAP shows that the Fratini subgroup of $SL_2(3)$ is of order $2$, where the one of $PGL_2(3)$ is trivial.</p>
<p>Assuming the algorithms are correct (which I guess, since they have been testet much), the groups are not isomorphic. Y... |
69,961 | <p>I want to determine the set of natural numbers that can be expressed as the sum of some non-negative number of 3s and 5s.</p>
<p>$$S=\{3k+5j∣k,j∈\mathbb{N}∪\{0\}\}$$</p>
<p>I want to check whether that would be:
0,3, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and so on.</p>
<p>Meaning that it would include 0, 3, 5,... | Brian M. Scott | 12,042 | <p>You can write $$S = \{0,3,5\} \cup \{n\in\mathbb{N}:n\ge 8\}.$$ You can also write $$S = (\mathbb{N}\cup\{0\})\setminus \{1,2,4,7\}.$$</p>
<p>Once you know that $8,9,10\in S$, you can prove that every natural number $n\ge 8$ belongs to $S$ by showing that if $n\ge 8$, there is a non-negative integer $k$ such that $... |
3,547,989 | <p><strong>An ordinary deck of cards is dealt randomly to four players so that each player receives 13 cards.
Find the probability that each player is dealt exactly one ace.</strong></p>
<p>I was wondering if I could compare this scenario with the following:
Given 4 indistinguishable balls and 4 distinguishable boxes,... | Robert Lewis | 67,071 | <p>We have</p>
<p><span class="math-container">$(X(X'X)^{-1}X')^2 = (X(X'X)^{-1}X')(X(X'X)^{-1}X') = X(X'X)^{-1}(X'X)(X'X)^{-1}X'=X(X'X)^{-1}X', \tag 1$</span></p>
<p>which shows that <span class="math-container">$X(X'X)^{-1}X'$</span> is idempotent. Now note that for any idempotent <span class="math-container">$P$<... |
3,335,892 | <p>If I have two injective functions <span class="math-container">$f : A \to B$</span> and <span class="math-container">$g : B \to A$</span>, as Schröder-Bernstein (SB) says, then there is a function <span class="math-container">$h : A \to B$</span> which is bijective.</p>
<p>As for a proof, my reasoning goes somethin... | Asaf Karagila | 622 | <p>This is not obvious at all that <span class="math-container">$|A|\leq|B|$</span> and <span class="math-container">$|B|\leq|A|$</span> imply that <span class="math-container">$|A|=|B|$</span>.</p>
<p>Consider a different notion of equivalence defined, say, on linearly ordered sets, with <span class="math-container">... |
1,978,532 | <p>A sequence with a limit that is a real number is called a convergent sequence. </p>
<p>A sequence which does not converge is said to be divergent. </p>
<p>Find two divergent sequences like $\{x_k\},\{y_k\}$ such that $\lim_{k \rightarrow \infty}|y_k-x_k|=0$ . </p>
<p><strong>Notice that we know $\forall k \in... | ziggurism | 16,490 | <p>The rationals in the interval are one class. $\pi$ plus rationals which lie in the interval are another ($\pi-3$, $\pi-22/7$, etc). $\sqrt{2}-1$ represents a class, so does $e-2$. But I can't construct a list of representatives of all equivalence classes without appealing to the axiom of choice. </p>
<p>The easies... |
1,978,532 | <p>A sequence with a limit that is a real number is called a convergent sequence. </p>
<p>A sequence which does not converge is said to be divergent. </p>
<p>Find two divergent sequences like $\{x_k\},\{y_k\}$ such that $\lim_{k \rightarrow \infty}|y_k-x_k|=0$ . </p>
<p><strong>Notice that we know $\forall k \in... | Piquito | 219,998 | <p>For all irrational $I\in [0,1)$ you have by definition
$$\widetilde I=\text { class of } I=\{I+r;\space r\in \Bbb Q\}$$</p>
|
1,657,115 | <p>$$\int \frac{2x + 10}{(x^2 + 5x + 8) ^ 2}dx$$
we can rewrite as:
$\int \frac{2x + 5}{(x^2 + 5x + 8) ^ 2}dx$ + $\int \frac{5}{(x^2 + 5x + 8) ^ 2}dx$</p>
<p>first one is easy. What can we do with the second one?</p>
| Narasimham | 95,860 | <p>Slope of PD =-1. Slope of AB is +1 . Your last equation is quite correct. After finding x, find the y.</p>
|
4,302,855 | <p>I have tried setting up multiple systems of equations using many known volumes but I always seem to come up short. My last attempt was a hollow cylinder but that leaves you with three unknowns in only two sim. equations (for V and S.A). Can anyone help?</p>
| PC1 | 960,197 | <p>We can use <a href="https://en.wikipedia.org/wiki/Pappus%27s_centroid_theorem" rel="nofollow noreferrer">Pappus's centroid theorems</a> for this purpose.</p>
<p>I will take a torus for example, with a centroid at a distance <span class="math-container">$R$</span> from the origin axis. Let's say that my torus radius ... |
3,051,207 | <p>In the very first chapter in my class "mathematical analysis 1" I've seen something called <strong>the triangle inequality</strong>, which is <span class="math-container">$||a| - |b|| \leq |a \pm b| \leq |a| + |b|$</span>. Now the thing is that I do understand why this is true, but i fail to see what this actually h... | Bernard | 202,857 | <p>Suppose <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are complex numbers, so that they correspond in the Argand-Cauchy plane, to two sides of a triangle, the third side corresponding to <span class="math-container">$a-b$</span>. Then these inequalities correspond to well-known ... |
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