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 |
|---|---|---|---|---|
3,720,272 | <p>My textbook employs a brute force method: add the number of committees that could be formed with one woman, two women, and three women in them. Then, the total number of such committees will be:
<span class="math-container">$$\left(\begin{smallmatrix} 8 \\ 1 \end{smallmatrix}\right)\cdot\left(\begin{smallmatrix} 10 ... | Anurag A | 68,092 | <p>Suppose <span class="math-container">$w_1$</span> was the first woman chosen among possible <span class="math-container">$8$</span>, then say <span class="math-container">$m_1,w_3$</span> were chosen. So your committee is <span class="math-container">$w_1,m_1,w_3$</span>.</p>
<p>In your way of counting: what if <spa... |
3,720,272 | <p>My textbook employs a brute force method: add the number of committees that could be formed with one woman, two women, and three women in them. Then, the total number of such committees will be:
<span class="math-container">$$\left(\begin{smallmatrix} 8 \\ 1 \end{smallmatrix}\right)\cdot\left(\begin{smallmatrix} 10 ... | richard gayle | 164,419 | <p>Anurag has put a finger on it. Your way of counting implicitly makes order of selection matter. Here's a check though. The number of committees with at least one woman should be the total number of all committees less the number that contain only men:</p>
<p><span class="math-container">$$C(18,3)-C(10,3)=696$$</s... |
2,831,270 | <p>I am quite fascinated by the formula for the Mellin transform of the Gaussian Hypergeometric Function, which is given by:</p>
<blockquote>
<p><span class="math-container">$$\mathcal M [_2F_1(\alpha,\beta;\gamma;-x)] = \frac {B(s,\alpha-s)B(s,\beta-s)}{B(s,\gamma-s)}$$</span></p>
</blockquote>
<p><em>Source : <a ... | mrtaurho | 537,079 | <p><strong>Additum</strong></p>
<p>Recently I have come across <a href="https://en.wikipedia.org/wiki/Ramanujan%27s_master_theorem" rel="nofollow noreferrer">Ramanujan's Master Theorem</a>. This Theorem provides an elegant way to show the given relation. Therefore lets write the Gaussian Hypergeometric Function as inf... |
2,377,946 | <blockquote>
<p>The integral is:
$$\int_0^a \frac{x^4}{(x^2+a^2)^4}dx$$</p>
</blockquote>
<p>I used an approach that involved substitution of x by $a\tan\theta$. No luck :\ . Help?</p>
| Math Lover | 348,257 | <p>One way to solve this problem is to do integration by parts.<br>
$$\int{x^3 \frac{x}{(x^2+a^2)^4}dx}=-\frac{x^3}{6(x^2+a^2)^3}+\frac{1}{2}\int{x\frac{x}{(x^2+a^2)^3} dx}.$$
Continuing further
$$\int{x\frac{x}{(x^2+a^2)^3} dx} = -\frac{1}{4} \frac{x}{(x^2+a^2)^2}+\frac{1}{4}\int{\frac{1}{(x^2+a^2)^2}dx}.$$
Now you ca... |
512,768 | <p>I am trying to intuitively understand why the solution to the following problem is $-2$. $$\lim_{x\to\infty}\sqrt{x^2-4x}-x$$
$$\lim_{x\to\infty}(\sqrt{x^2-4x}-x)\frac{\sqrt{x^2-4x}+x}{\sqrt{x^2-4x}+x}$$
$$\lim_{x\to\infty}\frac{x^2-4x-x^2}{\sqrt{x^2-4x}+x}$$
$$\lim_{x\to\infty}\frac{-4x}{\sqrt{x^2-4x}+x}$$
$$\lim_{... | Chris Bonnell | 97,819 | <p>Intuitively, the thing you want to look at with your second attempt is that it's infinity minus infinity. Because the highest order terms are on the same order of magnitude and cancel exactly, the lower-order terms are indeed important to determine what the limit will be.</p>
|
3,956,913 | <p>For this equation :</p>
<blockquote>
<p><span class="math-container">${ (x^2 - 7x + 11)}^{x^2 - 13x +42}=1$</span></p>
</blockquote>
<p>The integer solutions of <span class="math-container">$x$</span> found by WolframAlpha using inverse (logarithmic) function are <span class="math-container">$ 2 , 5 , 6 , 7 .$</span... | player3236 | 435,724 | <p>Notice that <span class="math-container">$n^{1/n} > 1$</span> for all <span class="math-container">$n \in \mathbb N$</span>.</p>
<p>Raise both sides of <span class="math-container">$n^{1/n} \stackrel?> (n+1)^{1/(n+1)}$</span> to the <span class="math-container">$n(n+1)$</span>-th power, now we just need to sho... |
1,419,483 | <p>Can anyone please help me in solving this integration problem $\int \frac{e^x}{1+ x^2}dx \, $?</p>
<p>Actually, I am getting stuck at one point while solving this problem via integration by parts.</p>
| Will R | 254,628 | <p>This is an alternative to the other answer I have provided. I have decided to add it as another answer because I think it uses a sufficiently different approach, and nobody else seems to have hinted at it. We begin just as we begun in my other answer to this same question, and continue until we reach $$\int e^{\tan{... |
4,411,247 | <blockquote>
<p>If <span class="math-container">$G$</span> is finite group, how to prove that <span class="math-container">$f(g)=ag$</span>, <span class="math-container">$a \in G$</span>, is a bijection for all <span class="math-container">$g \in G$</span>? Here <span class="math-container">$ag$</span> is <span class="... | Shaun | 104,041 | <p>Fix <span class="math-container">$a\in G$</span>.</p>
<p>Define</p>
<p><span class="math-container">$$\begin{align}
\hat{f}: G&\to G,\\
x&\mapsto a^{-1}x.
\end{align}$$</span></p>
<p>Then for any <span class="math-container">$g\in G$</span>, we have</p>
<p><span class="math-container">$$\begin{align}
(f\circ... |
443,475 | <p>I am reading some geometric algebra notes. They all started from some axioms. But I am still confused on the motivation to add inner product and wedge product together by defining
$$ ab = a\cdot b + a \wedge b$$ Yes, it can be done like complex numbers, but what will we lose if we deal with inner product and wedge ... | user997712 | 31,611 | <p>In addition to invertibility, as mentioned by Joe, geometric operations can be expressed in simple, co-ordinate free expressions using the geometric product.<br>
For instance, Rotation:
$$ R_{i\theta}(A) = e^{-i\theta/2}Ae^{i\theta/2} $$
rotates the blade $ A $ by an angle $ \theta $ in the plane of the bivector $ i... |
2,713,311 | <p>$ \lim_{x \to \infty} [\frac{x^2+1}{x+1}-ax-b]=0 \ $ then show that $ \ a=1, \ b=-1 \ $</p>
<p><strong>Answer:</strong></p>
<p>$ \lim_{x \to \infty} [\frac{x^2+1}{x+1}-ax-b]=0 \\ \Rightarrow \lim_{x \to \infty} [\frac{x^2+1-ax^2-ax-bx-b}{x+1}]=0 \\ \Rightarrow \lim_{x \to \infty} \frac{2x-2ax-a-b}{1}=0 \\ \Righta... | Clayton | 43,239 | <p>What you have seems fine (though maybe overkill); another way you can approach this problem is using long division to show that $$\frac{x^2+1}{x+1}=x-1+\frac{2}{x+1}.$$ The last term goes to $0$ as $x\to\infty$ while the first two terms combine with those in the original problem to get $(1-a)x-(1+b)\to0$ as $x\to\in... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Scott Morrison | 3 | <p>Dror Bar-Natan has begin putting <a href="http://katlas.math.toronto.edu/drorbn/index.php?title=Dbnvp">many of his lectures and talks</a> online in video format. I'm not claiming that these are the 'best' online maths videos, but they're certainly interesting, and in particular he's come up with some neat tricks to ... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | AgCl | 1,229 | <p>There are Stephen Boyd's lecture videos on convex optimization:</p>
<p><a href="http://www.stanford.edu/class/ee364a/videos.html">http://www.stanford.edu/class/ee364a/videos.html</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Somnath Basu | 1,993 | <p>I guess all of John Conway's lectures are great. Some of those can be found here :
<a href="http://www.math.princeton.edu/facultypapers/Conway/">http://www.math.princeton.edu/facultypapers/Conway/</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Daniel Pape | 11,176 | <p>A few talks under the heading "What is ..." (",,," could be "Morse Theory", for example)
given at the Freie Universität Berlin can be found here:</p>
<p><a href="http://www.scivee.tv/user/5216" rel="nofollow">http://www.scivee.tv/user/5216</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Dave Novick | 13,067 | <p>The University of New South Wales in Sydney has an eLearning channel on YouTube that contains lectures on a number of topics, including Algebraic Topology, Calculus, and Linear Algebra. Some computing and engineering topics are covered as well.</p>
<p><a href="http://www.youtube.com/user/UNSWelearning#p/p" rel="nof... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Javier Álvarez-Vizoso | 10,867 | <p>The complete <a href="http://vod.mathnet.or.kr/sub4_1.php?key_s_title=Lectures+on+Basic+Algebraic+Geometry+by+Miles+Reid+(WCU+project)" rel="nofollow noreferrer">introductory course on Algebraic Geometry by Miles Reid</a> is very interesting (28 lectures following and extending his own undergraduate book on the subj... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Austin Mohr | 8,871 | <p>Timothy Gowers' "<a href="https://www.youtube.com/watch?v=BsIJN4YMZZo" rel="nofollow noreferrer">The Importance of Mathematics</a>" never fails to instill a sense of purpose in my work, even when I feel I'm doing "useless" mathematics.</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | joro | 12,481 | <p>Two recent videotaped lectures by Doron Zeilberger.</p>
<p><a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/famous.html" rel="nofollow">The Joy of Dreaming to be Famous (Videotaped lecture), March 1,2012 </a></p>
<p><a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/astrology.htm... |
1,452,425 | <p>From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not <a href="https://math.stackexchange.com/a/11155/171192">$0^0$ is $1$ is a simple matter of definition</a>.</p>
<p><strong>My question is what the definition of a set is?<... | Asaf Karagila | 622 | <p>Formally speaking, sets are atomic in mathematics.<sup>1</sup> They have no definition. They are just "basic objects". You can try and define a set as an object in the universe of a theory designated as "set theory". This reduces the definition as to what <em>we</em> call "set theory", ... |
1,452,425 | <p>From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not <a href="https://math.stackexchange.com/a/11155/171192">$0^0$ is $1$ is a simple matter of definition</a>.</p>
<p><strong>My question is what the definition of a set is?<... | user247327 | 247,327 | <p>One of the reasons mathematics is so useful is that it is applicable to so many different fields. And the reason for that is that its logical structure starts with "undefined terms" that can then be given definitions appropriate to the field of application. "Set" is one of the most basic "undefined terms". Rather... |
1,452,425 | <p>From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not <a href="https://math.stackexchange.com/a/11155/171192">$0^0$ is $1$ is a simple matter of definition</a>.</p>
<p><strong>My question is what the definition of a set is?<... | epi163sqrt | 132,007 | <p><strong>Note:</strong> According to OPs formulation in the bounty text this answer is aimed to give at least a glimpse of some aspects around the definition of set and set theories with focus on the axiom of choice. The top voted answers already contain the essential information.</p>
<p><strong>On the definition of ... |
1,186,517 | <p>I do some ex for preparing discrete mathematics exam, i get stuck in one problem, anyone could help me?</p>
<blockquote>
<p>How many ways we can partition set {1,2,...,9} into subsets of size 2
and 5?</p>
</blockquote>
<p>anyway, some tutorials for solving such a question...</p>
<p>Edit: Like always Scott is ... | Michael E | 221,334 | <p>Since there are an odd number of elements in the set you must have a subset with 5 elements and therefore you must have two subsets of 2 elements.</p>
<p>First Choose the set with five elements. There are $9 \choose 5$ ways to do this. For <em>each</em> of these choices there are then 4 elements left and you have t... |
87,538 | <p>My problem is the following:</p>
<p>I have a finite surjective morphism $f: X\rightarrow Y$ of noetherian schemes and know that $Y$ is a regular scheme.
(Indeed, in my situation, the two schemes are topologically the same and the arrow is topologically the identity.)</p>
<p>I don't know if $f$ is étale or smooth. ... | Sándor Kovács | 10,076 | <p>I may be misunderstanding the question, but it seems rather straightforward to me. </p>
<p>a) As stated, without assuming, that $X$ is, say, reduced, it is certainly false:
Let $X=\mathrm{Spec}k[\varepsilon]=k[x]/(x^2)$, $Y=\mathrm{Spec} k$ and $f:X\to Y$ the structure map of $X$ as a $Y$-scheme. This is obviously ... |
971,160 | <p>So, this is actually 2 questions in 1. I apologize if that is bad practice, but I didn't want to write 2 questions when they're a word different. So, I have</p>
<ol>
<li>Prove or disprove that if $a|(sb+tc), \forall s,t \in\mathbb{Z}$, then $a|b$, and $a|c$.</li>
</ol>
<p>and then,</p>
<ol start="2">
<li>Prove or... | DeepSea | 101,504 | <ol>
<li>True. </li>
</ol>
<p>Choose $s = 0, t = 1$, then $a|c$, and choose $s = 1, t = 0$, then $a|b$</p>
<ol start="2">
<li>False.</li>
</ol>
<p>Let $a = 2, b = 3, c = 5$, and $s = t = 2$.</p>
|
3,215,381 | <p>When we write <span class="math-container">$\mathbb{Z}_3$</span>, does it mean <span class="math-container">$\mathbb{Z}/3\mathbb{Z}$</span>? Also, does <span class="math-container">$3\mathbb{Z}_3$</span> mean <span class="math-container">$0 \pmod 3$</span>?</p>
| Xander Henderson | 468,350 | <p><em>Many</em> notations in mathematics are overloaded. Whenever you are working in mathematics, it is very important to keep track of the <em>context</em> in which you are working. For example, in the expression
<span class="math-container">$$ 3^s = 2 \implies s \in \left\{ \log_{3}(2) + i\frac{2\pi}{\log(3)} \rig... |
3,002,325 | <p>Proof <span class="math-container">$t$</span> is irrational <span class="math-container">$ t = a-bs $</span> , Given <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are rational numbers, <span class="math-container">$b \neq 0$</span> and <span class="math-container">$s$</span> i... | Ethan Bolker | 72,858 | <p>For the first part: in a comment you have solved for <span class="math-container">$s$</span> in terms of <span class="math-container">$a$</span>, <span class="math-container">$b$</span> and <span class="math-container">$t$</span>. You know the first two are rational. What could you say about <span class="math-conta... |
3,009,362 | <p>I need to find
<span class="math-container">$$\lim_{x\rightarrow -5} \frac{2x^2-50}{2x^2+3x-35}$$</span></p>
<p>Looking at the graph, I know the answer should be <span class="math-container">$\frac{20}{17}$</span>, but when I tried solving it, I reached <span class="math-container">$0$</span>.</p>
<p>Here are the... | user | 505,767 | <p>As an alternative by <span class="math-container">$y=x+5 \to 0$</span></p>
<p><span class="math-container">$$\lim_{x\rightarrow -5} \frac{2x^2-50}{2x^2+3x-35}=\lim_{y\rightarrow 0} \frac{2(y-5)^2-50}{2(y-5)^2+3(y-5)-35}=\lim_{y\rightarrow 0} \frac{2y^2-20y}{2y^2-17y}=\lim_{y\rightarrow 0} \frac{2y-20}{2y-17}$$</spa... |
2,138,448 | <p>Survival game: Consider $3$ players, $A, B$ and $ C$, taking turns shooting at each other. Any player can shoot at only one opponent at a time (and each of them has to make a shot whenever it is his/her turn). </p>
<p>Each shot of $A$ is successful with probability $1/3$, each shot of $B$ is successful with probabi... | Anatoly | 90,997 | <p>It has already correctly been noted that the best strategy for $A $ in this game - assuming no possibility of shooting into the air - is to begin by shooting at $B $. I would add some details on this. </p>
<p>The case in which $A $ decides to shoot at $C $ is relatively simple, since the only possibility for him t... |
1,081,383 | <p>Assume that $A$ is an $n\times n$ symmetric positive-definite matrix.</p>
<p>Prove that:</p>
<blockquote>
<p>the element of $A$ with maximum magnitude must lie on the diagonal. </p>
</blockquote>
| Algebraic Pavel | 90,996 | <p>For an $n\times n$ Hermitian positive semidefinite matrix $A=(a_{ij})$ and $1\leq i,j\leq n$, $$|a_{ij}|^2\leq a_{ii}a_{jj}\leq\max\limits_{k}a_{kk}^2 \quad \Rightarrow \quad|a_{ij}|\leq\max_k a_{kk}.$$</p>
|
2,572,802 | <p>We were asked to find the number of five digit numbers $N=d_1d_2d_3d_4d_5$, where $d_i$ is the $i$th digit of the number and $d_1 < d_2 < d_3 < d_4 < d_5 $. The solution was trivial as for a given selection of five random distinct digits, there is only one way to arrange them in strict increasing order. ... | DanielWainfleet | 254,665 | <p>Let $F(n)$ be the number of $n$-digit numbers whose digits are strictly increasing from L to R. The answer to your Q is $$(F(5)+4F(4)+3F(3)+2F(2)+F(1))+(3F(3)+2F(2))=$$ $$=F(5)+4F(4)+6F(3)+4F(2)+F(1).$$ </p>
<p>Explanation: </p>
<p>(i). $F(5)$ ways with $5$ different digits. </p>
<p>(ii). Given $4$ strictly incre... |
2,572,802 | <p>We were asked to find the number of five digit numbers $N=d_1d_2d_3d_4d_5$, where $d_i$ is the $i$th digit of the number and $d_1 < d_2 < d_3 < d_4 < d_5 $. The solution was trivial as for a given selection of five random distinct digits, there is only one way to arrange them in strict increasing order. ... | Christian Blatter | 1,303 | <p>Your approach can maybe streamlined somewhat, but it is the textbook solution for this problem. </p>
<p>I'd put it this way: Look at weakly ascending seven letter words $\ 1\ d_1\ d_2\ d_3\ d_4\ d_5\ 9\ $. There are six slots for an increment $x_k\geq0$, and the sum of the increments should be $9-1=8$. By "stars an... |
1,730,352 | <p>I am really struggling to understand what modular forms are and how I should think of them. Unfortunately I often see others being in the same shoes as me when it comes to modular forms, I imagine because the amount of background knowledge needed to fully appreciate and grasp the constructions and methods is rather ... | Mathmo123 | 154,802 | <p>The definition of a modular form seems extremely unmotivated, and as @AndreaMori has pointed out, whilst the complex analytic approach gives us the quickest route to a definition, it also clouds some of what is really going on.</p>
<p>A good place to start is with the theory of <em>elliptic curves</em>, which have ... |
3,156,570 | <p>I need to evaluate the following limit:
<span class="math-container">$$
\lim_{x\downarrow 0} \dfrac{(1 - e^x)^{-1}}{x^c}
$$</span>
for different values of the constant <span class="math-container">$c$</span>.</p>
<p><em>What I've tried thus far:</em></p>
<p>We have that
<span class="math-container">$$
\lim_{x\down... | little o | 543,867 | <p>If <span class="math-container">$-1< c < 0$</span> then <span class="math-container">$\frac {1} {x^c(1-e^x)} \rightarrow -\infty$</span> as <span class="math-container">$x \downarrow 0.$</span> If <span class="math-container">$c=-1$</span> then <span class="math-container">$\frac {1} {x^c(1-e^x)} \rightarrow ... |
2,398,215 | <p>If $f$ is continuous on $\mathbb{R}$ any of the following conditions are satisfied then $f$ must be a constant.</p>
<p>(1).$f(x)=f(mx),\forall x\in \mathbb{R},|m|≠1,m\in \mathbb{R}$</p>
<p>(2).$f(x)=f(2x+1),\forall x\in \mathbb{R}$</p>
<p>(3).$f(x)=f(x^2),\forall x\in \mathbb{R}$.</p>
<p>Suppose $f$ satisfy (1).... | Tsemo Aristide | 280,301 | <ol>
<li>Suppose $|m|>1$ $f(x)=f(mx)$ implies that $f(x/m)=f(m(m/x))=f(x)$. We deduce that for every integer $n$, $f({x\over m^n})=f(x)$, since $lim_n{x\over m^n}=0$ and $f$ is continue, we deduce that $f(x)=lim_{n\rightarrow +\infty}f({x\over m^n})=f(0)$.</li>
</ol>
<p>The case $m<1$ consider the sequence $f(m^... |
2,185,489 | <blockquote>
<p>Let $\xi \in \mathcal{L}^2(\Omega,P)$ be a random variable with finite variance. Show that $$(E(\xi))^2 \leq E(\xi^2)$$ </p>
</blockquote>
<p>Since $$\operatorname{Var}(\xi) := E(\xi^2) - (E(\xi))^2$$ this boils down to showing $$\operatorname{Var}(\xi) \geq 0$$ which is quite restrictive. Since the ... | Scientifica | 164,983 | <p>Notice that $$\text{Var}(\xi)=E((\xi-E(\xi))^2)$$</p>
<p>is the definition of variance (at least in a number of books).</p>
|
3,931,807 | <p>I need to find max and min of <span class="math-container">$f(x,y)=x^3 + y^3 -3x -3y$</span> with the following restriction: <span class="math-container">$x + 2y = 3$</span>.</p>
<p>I used the multiplier's Lagrange theorem and found <span class="math-container">$(1,1)$</span> is the minima of <span class="math-conta... | Théophile | 26,091 | <p>As others have said, you don't necessarily need to use Lagrange multipliers. But since you've set up the system, we can see what happens:</p>
<p><span class="math-container">$$
3x^2-3=\lambda\\
3y^2-3=2\lambda\\
x+2y-3=0
$$</span></p>
<p>From the first two equations, we have <span class="math-container">$3y^2-3=2(3x... |
287,129 | <p>The standard definition of computability, for a sequence $s\in\{0,1\}^\omega$, is that there is a Turing machine outputting $s[i]$ on input $i$.</p>
<p>I'm looking for strengthenings of this notion; for example, in the above definition it's not decidable whether there is a $1$ in $s$; or, given $i$, whether there i... | Bjørn Kjos-Hanssen | 4,600 | <p>You can look at <a href="http://www.math.cornell.edu/~minnes/LCsurvey.pdf" rel="noreferrer">automatic structures</a>, replacing Turing machines by finite automata. In that setting, the whole first order theory is decidable (in particular you can add existential quantifiers like you describe), by a Theorem of Hodgson... |
2,536,163 | <p>How to integrate using contour integration?
$$\int_1^{\infty}\frac{\sqrt{x-1}}{(1+x)^2}dx$$
I was putting $y = x-1$ then $\frac{dy}{dx}$= $1-0$, ${dy}={dx} $ then i get $$\int_1^{\infty}\frac{\sqrt{y}}{(2+y)^2}dy$$</p>
<p>I don't know how to take it from here. I would appreciate if someone could help me and give... | Aditya Narayan Sharma | 335,483 | <p>I won't prefer a contour here, making the substitution $y=x-1$ we have,
$$ \displaystyle \int_1^\infty \dfrac{\sqrt{x-1}}{(1+x)^2}\; dx = \int_0^\infty \dfrac{\sqrt{y}}{(2+y)^2}\; dy$$
Next substitute $y=u^2$ ,
$$\displaystyle I = 2\int_0^\infty \dfrac{u^2}{(u^2+2)^2}\; du $$
Using partial fractions this will be eq... |
2,536,163 | <p>How to integrate using contour integration?
$$\int_1^{\infty}\frac{\sqrt{x-1}}{(1+x)^2}dx$$
I was putting $y = x-1$ then $\frac{dy}{dx}$= $1-0$, ${dy}={dx} $ then i get $$\int_1^{\infty}\frac{\sqrt{y}}{(2+y)^2}dy$$</p>
<p>I don't know how to take it from here. I would appreciate if someone could help me and give... | Jack D'Aurizio | 44,121 | <p>$$\begin{eqnarray*}\int_{1}^{+\infty}\frac{\sqrt{x-1}}{(x+1)^2}\,dx &\stackrel{x\mapsto z+1}{=}& \int_{0}^{+\infty}\frac{\sqrt{z}}{(2+z)^2}\,dz\stackrel{z\mapsto 2u}{=}\frac{1}{\sqrt{2}}\int_{0}^{+\infty}\frac{\sqrt{u}}{(1+u)^2}\,du\\&\stackrel{u\mapsto v^2}{=}&\frac{1}{\sqrt{2}}\int_{0}^{+\infty}\fr... |
680,319 | <p>Let's restate this question in using mathematical notation. Let $n,k \in \mathbb{N}$. Let $f(n)=\left\lfloor{\frac{n}{k}}\right\rfloor$. Is it possible to rewrite this using the addition, multiplication, and exponentiation operators? I know it's possible for the case where $k=1$. Quite simply, note that $f(n)=\left\... | JJacquelin | 108,514 | <p>The floor function on form of Fourrier series :</p>
<p><img src="https://i.stack.imgur.com/YQ0ag.jpg" alt="enter image description here"></p>
|
3,394,050 | <p>I'm having trouble with this problem.</p>
<blockquote>
<p>Using logical equivalencies prove that <span class="math-container">$(p \land q)\implies (p \lor q)$</span> is a tautology.</p>
</blockquote>
| Mikhail D | 398,197 | <p>A tautology is a statement that is always 'True'.</p>
<p>In your case you have two variables, each may take the value of either 'true' or 'false'.</p>
<p>You may simply plug these four cases in your expression and get 'True' in each case you try. Note that your logical expression is a composition of three operatio... |
4,488,991 | <p>Let <span class="math-container">$M,M'$</span> be oriented connected compact smooth manifolds of the same dimension, let <span class="math-container">$S$</span> be a smooth manifold,
and let <span class="math-container">$\nu : S\times M\rightarrow M'$</span> be some smooth map.
Let <span class="math-container">$\nu_... | Adayah | 149,178 | <p>Here is an abstract way to construct a bunch of rectangular triangles satisfying the property. They don't need to have integer side lengths, though.</p>
<p>Take any rectangular triangle <span class="math-container">$T$</span>. Let <span class="math-container">$x$</span> denote the product of the lengths of the two s... |
2,600,776 | <blockquote>
<p>A continuos random variable $X$ has the density
$$
f(x) = 2\phi(x)\Phi(x), ~x\in\mathbb{R}
$$
then</p>
<p>(<em>A</em>) $E(X) > 0$</p>
<p>(<em>B</em>) $E(X) < 0$</p>
<p>(<em>C</em>) $P(X\leq 0) > 0.5$</p>
<p>(<em>D</em>) $P(X\ge0) < 0.25$</p>
<p>\begin{eqnarray}... | drhab | 75,923 | <p>Hint:</p>
<p>If $x>0$ then $f(-x)<f(x)$</p>
|
303,933 | <p>How can I prove that
$$ \lim_{n\to\infty} \frac{(\ln(n))^a}{n^b} = 0 \;\forall a,b > 0 $$
? Intuitively it is clear to me because of the behavior of the functions. Thanks for all.</p>
<p><strong>Edit</strong> I'm not able to use L'Hopital rule. Sorry.</p>
| Ishan Banerjee | 52,488 | <p>$lim_{n\to\infty} \frac {log(n)}{n^a}=0$ $ a>0$
(This can be proved with L'hopital's)</p>
<p>Use this to prove your result.</p>
|
1,831,243 | <p>Kronecker "delta" function is generally defined as
$\delta(i,j)=1$ if $i$ is equal to $ j$, otherwise $0$.</p>
<p>How about if $j$ is not an integer? I mean let $j$ is a half open interval defined as $j=(0,1]$ and $ i$ has any value on interval $[0,1]$,
then can we use Kronecker delta to find if $i$ belongs to $j$... | Marc van Leeuwen | 18,880 | <p>No, the convention is that the Kronecker delta tests for <em>equality</em> and not for some other relation such as set membership. Also the elements tested for equality are most often numbers, and even integers, but in a stretch you could use it to test equality of objects of other types. So using the Kronecker delt... |
3,293,082 | <p>I know this question was asked on this site, but I didn't understand the answer. Could someone give me the simplest explanation of this? (High school level explanation)</p>
| Justin Benfield | 297,916 | <p><span class="math-container">$\arccos(x)$</span> is the inverse function of <span class="math-container">$\cos(x)$</span> (restricted to the interval <span class="math-container">$[0,\pi]$</span>)</p>
<p><span class="math-container">$\sec(x)$</span> is the reciprocal, <span class="math-container">$\dfrac{1}{\cos(x)... |
3,293,082 | <p>I know this question was asked on this site, but I didn't understand the answer. Could someone give me the simplest explanation of this? (High school level explanation)</p>
| J.G. | 56,861 | <p><span class="math-container">$\arccos x$</span> is a value whose cosine is <span class="math-container">$x$</span>, whereas <span class="math-container">$\sec x=\frac{1}{\cos x}$</span>. The two kinds of "inverse" trigonometric functions can feel a bit esoteric when you're new to them, but let's pretend <span class=... |
2,000,940 | <p>Suppose $A \subseteq \mathbb{R}$ is measurable and $f\colon A \to \mathbb{R}$ is Lipschitz on the set $A$, i.e there is some $K\ge 0$ such that $\lvert f(x)-f(y)\rvert \le K \lvert x-y\rvert$ for $x,y \in A$. </p>
<p>I'm trying to prove that
$$
m^\ast(f(E)) \le K\,m^\ast(E)\textrm{ for every set }E \subseteq A.
$$<... | Daniel Fischer | 83,702 | <blockquote>
<p>I've tried covering the set $E$ by intervals, yet the function $f$ may not have been defined outside the set $A$.</p>
</blockquote>
<p>Good that you noticed that problem.</p>
<p>But that problem can be dealt with, we can extend $f$ to a Lipschitz continuous $F \colon \mathbb{R} \to \mathbb{R}$ with ... |
125,592 | <p>I'm finding in trouble trying to resolve this exercise. I have to calculate the convolution of two signals:</p>
<p>$$y(t)=e^{-kt}u(t)*\frac{\sin\left(\frac{\pi t}{10}\right)}{(\pi t)} $$</p>
<p>where $u(t)$ is Heavside function</p>
<p>well I applied the formula that says that the convolution of this two signal is... | dayar | 24,144 | <p>It's possible to do this integral in a couples of lines using the residue theorem from complex analysis.</p>
<p>Details: The usual trick to do definite integrals going from $0$ to $2\pi$ is to let $\cos x = \dfrac {z^2 + 1} {2z}$ where $z = {\rm e} ^{{\rm i} x}$. This substitution also implies that ${\rm d} x = \df... |
125,592 | <p>I'm finding in trouble trying to resolve this exercise. I have to calculate the convolution of two signals:</p>
<p>$$y(t)=e^{-kt}u(t)*\frac{\sin\left(\frac{\pi t}{10}\right)}{(\pi t)} $$</p>
<p>where $u(t)$ is Heavside function</p>
<p>well I applied the formula that says that the convolution of this two signal is... | Christian Blatter | 1,303 | <p>Qiaochu Yuan's hint seems to be the simplest approach: By the binomial theorem for any $n\geq0$ one has
$$2^n\cos^n x=(e^{ix}+e^{-ix})^n=\sum_{k=0}^n {n\choose k} (e^{ix})^k\ (e^{-ix})^{n-k}=\sum_{k=0}^n {n\choose k} e^{(2k-n)ix}\ .\qquad(*)$$
Since
$$\int_0^{2\pi}e^{i\ell x}\ dx=\cases{2\pi&$\quad(\ell=0)$\cr 0... |
125,592 | <p>I'm finding in trouble trying to resolve this exercise. I have to calculate the convolution of two signals:</p>
<p>$$y(t)=e^{-kt}u(t)*\frac{\sin\left(\frac{\pi t}{10}\right)}{(\pi t)} $$</p>
<p>where $u(t)$ is Heavside function</p>
<p>well I applied the formula that says that the convolution of this two signal is... | dohmatob | 168,758 | <p>Let <span class="math-container">$X$</span> be a standard normal random variable. Then, your integral <span class="math-container">$I$</span> can be computed as</p>
<p><span class="math-container">$$
I = 2\pi\cdot \mathbb E[X^n] = \begin{cases}2\pi\cdot (n-1)!! = \frac{2\pi}{2^n}{n\choose n/2},&\mbox{ if }n\text... |
1,608,645 | <p>Is there supposed to be a fast way to compute recurrences like these?</p>
<p>$T(1) = 1$</p>
<p>$T(n) = 2T(n - 1) + n$</p>
<p>The solution is $T(n) = 2^{n+1} - n - 2$. </p>
<p>I can solve it with:</p>
<ol>
<li><p>Generating functions.</p></li>
<li><p>Subtracting successive terms until it becomes a pure linear re... | Kaynex | 296,320 | <p>Solve the homogeneous equation by removing any non-functional terms. In this case, simply remove the n.</p>
<p>Solve the specific solution by guessing. In this case, it's not difficult. Try plugging in Ax + b into the functional equation, and see if you can solve for the coefficients.</p>
<p>Then, add both solutio... |
2,396,073 | <p>Let $\omega_1$ be the first uncountable ordinal. In some book, the set $\Omega_0:=[1,\omega_1)=[1,\omega_1]\backslash\{\omega_1\}$ is called the set of countable ordinals. Why? It is obvious that it is an uncountable set, because $[1,\omega_1]$ is uncountable. The most possible reason I think is that for any $x\pr... | Noah Schweber | 28,111 | <p>In fact, $\Omega_0=\omega_1$ (EDIT: with $0$ removed): remember that each ordinal is the set of all smaller ordinals. So in particular, since $\omega_1$ is the smallest uncountable ordinal, it is also the <em>set</em> of all <em>countable</em> ordinals (since each countable ordinal is smaller than $\omega_1$).</p>
|
484,117 | <p>What is the simplest $\Bbb{R}\to\Bbb{R}$ function with two peaks and a valley?</p>
<p>I have a set of points in $\Bbb{R^2}$ and I would like to fit a curve to the points, the points approximately lie on a curve like the one depicted in the following figure:</p>
<p><img src="https://i.stack.imgur.com/8tVLu.png" alt... | doraemonpaul | 30,938 | <p>Let $u=\sqrt{f(t)-y}$ ,</p>
<p>Then $y=f(t)-u^2$</p>
<p>$\dfrac{dy}{dt}=\dfrac{df(t)}{dt}-2u\dfrac{du}{dt}$</p>
<p>$\therefore\dfrac{df(t)}{dt}-2u\dfrac{du}{dt}=u$</p>
<p>$2u\dfrac{du}{dt}+u=\dfrac{df(t)}{dt}$</p>
<p>This belongs to an Abel equation of the second kind.</p>
<p>Let $u=-\dfrac{v}{2}$ ,</p>
<p>Th... |
2,706,165 | <p>So if $y=\log(3-x) = \log(-x+3)$ then you reflect $\log(x)$ in the $y$ axis to get $\log(-x)$.</p>
<p>Then because it is $+3$ inside brackets you then shift to the left by $3$ giving an asymptote of $x=-3$ and the graph crossing the $x$ axis at $(-4,0)$. </p>
<p>However this does not work. The answer shows the $+3... | user | 505,767 | <p>Note that</p>
<ul>
<li>$3-x$ put the origin in x=3 (right translation) and reverse axis direction</li>
<li>thus the vertical asympthote is at $x=3$ and the root is at $x=2$</li>
</ul>
|
4,644,904 | <p>Find the greatest and the least values of the function <span class="math-container">$f(x)=\sin x\sin2x$</span> on the interval <span class="math-container">$(-\infty,\infty)$</span>.</p>
<p>The solution presented is as follows:</p>
<blockquote>
<p>Represent the function <span class="math-container">$y=f(x)=\sin x\si... | Community | -1 | <p><strong>Alternative resolution:</strong></p>
<p>By direct differentiation, we have to solve</p>
<p><span class="math-container">$$\sin(2x)\cos(x)+2\sin(x)\cos(2x)=0$$</span></p>
<p>or</p>
<p><span class="math-container">$$2\sin(x)(3\cos^2(x)-1)=0.$$</span></p>
<p>When <span class="math-container">$\sin(x)=0$</span>,... |
35,151 | <p>Many complexity theorists assume that $P\ne NP.$ If this is proved, how would it impact quantum computing and quantum algorithms? Would the proof immediately disallow quantum algorithms from ever solving NP-Complete problems in Quantum Polynomial time?</p>
<p><a href="http://en.wikipedia.org/wiki/QMA" rel="nofollow... | Artem Kaznatcheev | 8,239 | <p>To your second question. It is unlikely that the current under-review proof of P != NP will allow you to seperate BQP and QMA (or BQP and P, or BQP and NP, or even BPP and NP...). Deolalikar's proof uses descriptive complexity, in particular it uses a correspondence between statements expressible in certain logics a... |
259,308 | <p>The output of <code>ListPointPlot3D</code> is shown below:
<a href="https://i.stack.imgur.com/ypt73.png" rel="noreferrer"><img src="https://i.stack.imgur.com/ypt73.png" alt="enter image description here" /></a>
I only want to connect the dots in such a way that it forms a ring-like mesh. However, when I use <code>Li... | Michael E2 | 4,999 | <p>Using <code>Partition</code>:</p>
<pre><code>Graphics3D[
Polygon[
Flatten[
Partition[Partition[dat, 20], {2, 2}, 1, 1][[All, 1 ;; -2]],
{{1, 2}, {3, 4}}][[All, {1, 2, 4, 3}]]
]
]
</code></pre>
<p><a href="https://i.stack.imgur.com/UIgpv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com... |
138,520 | <p>I am attempting to show that the series $y(x)\sum_{n=0}^{\infty} a_{n}x^n$ is a solution to the differential equation $(1-x)^2y''-2y=0$ provided that $(n+2)a_{n+2}-2na_{n+1}+(n-2)a_n=0$</p>
<p>So i have:
$$y=\sum_{n=0}^{\infty} a_{n}x^n$$
$$y'=\sum_{n=0}^{\infty}na_{n}x^{n-1}$$
$$y''=\sum_{n=0}^{\infty}a_{n}n(n-1)x... | agt | 6,752 | <p>You are correct.<br>
Only you need to go on and observe that the lhs of your last equation factorizes as: $$(n+1)[(n+2)a_{n+2}-2n a_{n+1}+(n-2)a_n]$$</p>
|
3,715,715 | <p>Let’s say I have a set <span class="math-container">$X$</span> and a set <span class="math-container">$Y$</span>, and <span class="math-container">$X \subseteq Y$</span>. Is it possible to state that <span class="math-container">$|X| \leq |Y|$</span> (<span class="math-container">$|X|$</span> cardinality of <span cl... | Brian M. Scott | 12,042 | <p>Yes, it is true. The function <span class="math-container">$\iota:X\to Y:x\mapsto x$</span> is an injection from <span class="math-container">$X$</span> to <span class="math-container">$Y$</span>, and the existence of such an injection by definition means that <span class="math-container">$|X|\le|Y|$</span>.</p>
<p... |
1,846,168 | <p>Let <span class="math-container">$R$</span> be a commutative ring, <span class="math-container">$a \in R$</span>, and <span class="math-container">$\forall i = 1, ...,r \ \ f_i(x) \in R[x]$</span>.</p>
<p>Prove the equality of ideals</p>
<p><span class="math-container">$(f_1(x), ..., f_r(x), x-a ) = (f_1(a), ...f_r(... | m.idaya | 337,282 | <p>as the coefficient dominant of $x-a$ is 1 ivertible in R the euclidien division is ok for any polynomial by $x-a$. so $f_i(x)=g(x)+b $ for some constante $b$ the equality with x=a come $f(a)=b$ and then as reauired the ideal $(f_1(x),...,f_r(x),(x-a))$ is the ideal $(f_1(a),...,f_r(x),(x-a))$</p>
|
750,417 | <p><strong>Question:</strong></p>
<blockquote>
<p>Initially we have a list of numbers $1,2,3,\cdots,2013$.an operation is defined that taking two numbers $a, b$ out from the list, but add $a+b$ into it instead, what is the minimum number of operations required that the sums of any number of numbers in the list can n... | user139388 | 139,388 | <p><strong>EDIT:</strong></p>
<p>I originally had the interpretation that after taking two pieces of paper with numbers written on them, we add the two together and put a piece of paper with that new number on it into the bag. This allows for duplicates (e.g. two copies of "$3$"). The following solution is for the sit... |
679,904 | <p>The question is let $a \in \mathbb{R} $ does not contain 0. Prove that $|a+\frac{1}{a}| \ge 2$. I have no idea how to start this problem and any help on it would be greatly appreciated.</p>
| Community | -1 | <p>We have</p>
<p>$$\left|a+\frac 1 a\right|\ge2\iff a^2+1\ge2|a|\iff (|a|-1)^2\ge0\;\text{which's true}$$</p>
|
1,158,642 | <p>Let $E$ be a vector bundle of rank $r$ and let $\phi:E\rightarrow \mathbb C_p$ non vanishing map to the skyscraper sheaf.
consider the kernel $F$ of this sheaf which is a sub-bundle of $E$, every fiber of $F$ has a rank $r$, just that over $p$ which has rank $r-1$.
So why we say that $F$ has a rank $r$??
thanks <... | Roland | 113,969 | <p>If you know that $F$ is a vector bundle, its rank may only be the rank of $F_{|U}$ where $U$ is the open set $U=X\setminus p$. Hence it has rank $r$, even at $p$ !! Here is an example of this situation.</p>
<p>Consider $X=\mathbb{P}^1$ and let $p$ be any point. Let $E=X\times\mathbb{C}$ be the trivial vector bundle... |
792,356 | <p>There is a dark night and there is a very old bridge above a canyon. The bridge is very weak and only 2 men can stand on it at the same time. Also they need an oil lamp to see holes in the bridge to avoid falling into the canyon.</p>
<p>Six man try to go through that bridge. They need 1,3,4,6,8,9(first man, second ... | Soumyadipto | 149,724 | <p>The correct answer is 31 minutes. This is a very common puzzle question which has been included in various books on brain-teasers, and a few mini games are also based on this puzzle.</p>
<p>The basic trick here is that 1, being the fastest, should ideally cross the bridge with each of the others, so that bringing b... |
2,497,216 | <p>I have this exercise, but I feel like something is wrong. As far as I know, if $m$ is a maximal ideal, then $m \subsetneq A$. But with this hypothesis, I think to take $I = \lbrace 1_A \rbrace$, so the only ideal that contains $I$ is $A$, so $I$ is not maximal. </p>
<p>EDIT:</p>
<p>I wrote something wrong. I didn'... | Andres Mejia | 297,998 | <p>The ideal $(1)$ generates the whole ring, so we usually do not consider it an ideal. A different approach is via Zorn's lemma. If you take the collection of ideals that contain $I$, and any chain in it, we just have to show that the chain has an upper bound (as in an ideal that contains $I$.)</p>
<p>This amounts to... |
582,478 | <p>Please simplify this logic expression for me with helping boolean algebra :</p>
<p>A'C'D + A'BD + BCD + ABC + ACD'</p>
<p>I know that must use consensus theorem .</p>
<p>my solve :</p>
<p>STEP 1 : Terms 1 & 3 ---eliminate---> Term 2</p>
<p>STEP 2 : Terms 3 & 5 ---eliminate---> Term 4</p>
<p>STEP 3 : Te... | sai kiran grandhi | 93,701 | <p>I think you are confusing during the application of minimization by using Boolean algebra formula</p>
<p>First you apply consensus theorem for terms 1,2,3 by taking D as a common factor. This helps in removing the term 2. So the minimized expression is A'C'D + BCD + ABC + ACD'</p>
<p>Now combining the last three t... |
221,351 | <p>I asked the following question (<a href="https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con">https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con</a>) on math.stackexchange.com and ... | John Binder | 30,726 | <p>I think this should work: let $S$ be the set of primes occurring in the denominators of the entries of the elements of $G$, and let $\mathbb{Z}[S^{-1}]$ be the subring of $\mathbb{Q}$ generated by adjoining $1/p$ for $p\in S$.</p>
<p>Let $\mathbb{Q}_S = \prod_{p\in S} \mathbb{Q}_p$ and let $\mathbb{Z}_S = \prod_{p\... |
2,416,424 | <p>It is known that the collection of finite mixtures of Gaussian Distributions over $\mathbb{R}$ is dense in $\mathcal{P}(\mathbb{R})$ (the space of probability distributions) under convergence in distribution metric.</p>
<p>I'm interested to know the following:</p>
<p>Let $P_X$ be a random variable with finite $p$ ... | Dr. Sonnhard Graubner | 175,066 | <p>HINT: write $$\frac{2^x}{2^x-8}-2=\frac{2^x-2^{x+1}+8}{2^x-8}>0$$ and do case work
solving this we get $$3<x<4$$</p>
|
3,449,589 | <p>In Example 1.4 of <em>Lee's Introduction to Smooth Manifolds</em>, which is showing that the <span class="math-container">$n$</span>-sphere, <span class="math-container">$\mathbb{S}^n$</span> is a topological <span class="math-container">$n$</span>-manifold, the following is stated.</p>
<p>In the part where the aut... | Berci | 41,488 | <p>No, <span class="math-container">$g_i$</span> and <span class="math-container">$h_i$</span> <em>ignore</em> the <span class="math-container">$i$</span>th coordinate in the input, and is defined on <span class="math-container">$\Bbb B^n$</span>, just as <span class="math-container">$f$</span>. <br>
The two possible v... |
1,789,373 | <p>I'm trying to figure out why the following is true:</p>
<p>Let $ \kappa $ be an uncountable, regular cardinal. Suppose we turn it into a group (i.e. there are operations $ (\cdot, ^{-1}, e) $ with which $ \kappa $ is a group. My aim is to prove that the set</p>
<p>$$ \{ \alpha \in \kappa : \alpha \text{ is a subgr... | Jack D'Aurizio | 44,121 | <p>By the Cauchy-Schwarz inequality,</p>
<p>$$ (8-e)^2 = (a+b+c+d)^2 \leq 4(a^2+b^2+c^2+d^2) = 4(16-e^2) $$
from which it follows that $e\leq \color{red}{\frac{16}{5}}$. Now it is enough to show that the inequality holds as an equality for some $(a,b,c,d,e)\in\mathbb{R}^5$, pretty easy. In the same way you may also sh... |
1,808,258 | <p>I was reading about orthogonal matricies and noticed that the $2 \times 2$ matrix
$$\begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix} $$
is orthogonal for every value of $\theta$ and that every $2\times 2$ orthogonal matrix can be expressed in this form. I then wonder... | Andrew D. Hwang | 86,418 | <p>First, not every orthogonal matrix is of the stated form, only <em>rotation</em> matrices. There are orthogonal <em>reflection</em> matrices, with determinant $-1$:
$$
\left[
\begin{array}{@{}rr@{}}
\cos\theta & \sin\theta \\
\sin\theta & -\cos\theta
\end{array}\right].
$$</p>
<p>Second, your conjecture is... |
1,808,258 | <p>I was reading about orthogonal matricies and noticed that the $2 \times 2$ matrix
$$\begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix} $$
is orthogonal for every value of $\theta$ and that every $2\times 2$ orthogonal matrix can be expressed in this form. I then wonder... | hmakholm left over Monica | 14,366 | <p>No, that is not the case. Ad you show, the rows in your $A$ will necessarily be orthogonal to each other, and the first row has norm $1$ -- but the norm of the <em>second</em> row is not necessarily $1$.</p>
<p>If you scale each of $x'(t)$ and $y'(t)$ in the second row by $\dfrac{1}{\sqrt{x'(t)^2+y'(t)^2}}$, you do... |
4,506,026 | <p>Consider the set of equations:
<span class="math-container">$$
\begin{cases}
x^2 &= -4y-10\\y^2 &= 6z-6\\z^2 &= 2x+2\\
\end{cases}$$</span></p>
<p>With <span class="math-container">$x,y,z$</span> being real numbers.</p>
<p>By adding the three equations, after simple manipulations, we easily obtain
<span ... | fleablood | 280,126 | <p>A solution to three equations will be a solution to the sum of the equations. But the solution to the sum need not be a solution to all three.</p>
<p>Consider <span class="math-container">$x^2 = 6z-6; y^2 = 2x + 2; z^2 = -4y-10$</span>. Those are <em>different</em> equations and will have different solutions but if... |
1,088,734 | <p>It's possible the integral bellow. What way I must to use for solve it.</p>
<p>$$\int \sin(x)x^2dx$$</p>
| Henry | 130,750 | <p>Using Gauss Lemma (also called Rational Root Theorem), the integer solutions can only be factors of $26$, in this case positive factors. This leaves us with $1$, $2$, $13$ or $26$. Also, since the roots are distinct and by Viete's formula, their product must be $26$, the roots are thus $1$, $2$ and $13$. Thus, agai... |
1,640,217 | <p><em>Use the dot/scalar product to solve the problem</em></p>
<p>Line 1 has vector equation $(2\mathrm{i}-\mathrm{j}) + \lambda(3\mathrm{i} + 2\mathrm{j})$
Find the vector equation of the line perpendicular to Line 1 and passing through the point with position vector $(4\mathrm{i} + 3\mathrm{j})$.</p>
<p>I can solv... | Frentos | 298,557 | <p>Keeping it abstract, you have a line $L_1$ given by $\vec P + \lambda \vec v$, and you want to find a line $L_2$ perpendicular to $L_1$ and passing through an external point $\vec E$.</p>
<p>We want to find the point $\vec C$ on $L_1$ closest to $\vec E$, because then $L_2$ will be the line through $\vec C$ and $\v... |
2,436,419 | <p>So I get the general sense of all of this and calculated all of the partial derivatives but I am unsure what exactly it is asking me for. Any advice or explanation would be greatly appreciated. Thanks!</p>
<p>Let <span class="math-container">$F: R^{3} \to R^{2}$</span> be defined by <span class="math-container">$F(... | David | 136,138 | <p>Calculating the partial derivatives basically tells you the slope of the function in the $x,y,z$ direction. When you calculate $\partial F(1,3,2)$ with respect to $x,y$ and $z$, you get the slope values at the point (1,3,2). What it is asking you is to approximate a point close to the point whose slope you calculate... |
2,436,419 | <p>So I get the general sense of all of this and calculated all of the partial derivatives but I am unsure what exactly it is asking me for. Any advice or explanation would be greatly appreciated. Thanks!</p>
<p>Let <span class="math-container">$F: R^{3} \to R^{2}$</span> be defined by <span class="math-container">$F(... | amd | 265,466 | <p>Go back to the definition of $DF$: it’s the best <em>linear</em> approximation to the change in $F$ at a point. Symbolically, $$F(\mathbf x+\Delta\mathbf x)=F(\mathbf x)+DF(\mathbf x)\Delta\mathbf x+o(\Delta\mathbf x).$$ That is, for a small displacement $\Delta\mathbf x$ from $\mathbf x$, the value of $F(\mathbf x+... |
3,029,446 | <p>If </p>
<ul>
<li><p><span class="math-container">$A$</span> is <span class="math-container">$m \times n$</span> (<span class="math-container">$m<n$</span>), and its rows are independent</p></li>
<li><p><span class="math-container">$B$</span> is <span class="math-container">$n \times p$</span> (<span class="math-... | Robert Israel | 8,508 | <p>Since <span class="math-container">$A$</span> has rank <span class="math-container">$m$</span> and <span class="math-container">$B$</span> has rank <span class="math-container">$p$</span>, <span class="math-container">$AB$</span> has rank at most <span class="math-container">$\min(m,p)$</span>.
<span class="math-con... |
23,953 | <p>I cited the diagonal proof of the uncountability of the reals as an example of a <a href="https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/23708#23708">`common false belief'</a> in mathematics, not because there is anything wrong with the proof but because it is commonly belie... | Harald Hanche-Olsen | 802 | <p><strong>Edit:</strong> A closer look reveals that the proof in the reference below is not the <em>diagonal</em> proof. I am leaving the answer up since the reference might be of interest anyhow.</p>
<p>Cantor had a paper in Crelle's Journal <strong>77</strong> (1874) 258–262. In Christopher P. Grant's translation, ... |
23,953 | <p>I cited the diagonal proof of the uncountability of the reals as an example of a <a href="https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/23708#23708">`common false belief'</a> in mathematics, not because there is anything wrong with the proof but because it is commonly belie... | Hilbert7 | 112,109 | <p>The evidence that you look for can already be found in the second paragraph of <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN237853094&DMDID=DMDLOG_0033" rel="nofollow">Cantor's original paper</a>. There he states "Es läßt sich aber von <em>jenem Satze</em> ein viel einfacherer Beweis liefern, d... |
2,473,780 | <p>So I have the limit $$\lim_{x\rightarrow \infty}\left(\frac{1}{2-\frac{3\ln{x}}{\sqrt{x}}}\right)=\frac{1}2,$$ I now want to motivate why $(3\ln{x}/\sqrt{x})\rightarrow0$ as $x\rightarrow\infty.$ I cam up with two possibilites:</p>
<ol>
<li><p>Algebraically it follows that $$\frac{3\ln{x}}{\sqrt{x}}=\frac{3\ln{x}}{... | marty cohen | 13,079 | <p>For any $a > c > 0, x > 1$,
we have</p>
<p>$\begin{array}\\
\ln(x)
&=\int_1^x \frac{dt}{t}\\
&\lt\int_1^x \frac{dt}{t^{1-c}}\\
&=\int_1^x t^{c-1}dt\\
&=\dfrac{t^c}{c}\big|_1^x\\
&=\dfrac{x^c-1}{c}\\
&<\dfrac{x^c}{c}\\
\text{so}\\
\dfrac{\ln(x)}{x^a}
&<\dfrac1{x^a}\dfrac{x^... |
362,881 | <p>I am going to try to explain this as easily as possible. I am working on a computer program that takes input from a joystick and controls a servo direction and speed. I have the direction working just fine now I am working on speed. To control the speed of rotation on the servo I need to send it so many pulses per s... | A Blumenthal | 54,337 | <p>Hint: $\delta(x)$ is a linear functional on some space of suitably nice (in particular, continuous) functions of $\mathbb{R}$. Given a suitably nice function $f$, what does $\delta(x)[f]$ equal? </p>
<p>When you say $e^x \delta(x) = \delta(x)$, what you're saying is that the left hand side and the right hand side a... |
2,087,235 | <p>I have a question about this question. Find all complex numbers $z$ such that the equation
$$t^2 + [(z+\overline z)-i(z-\overline z)]t + 2z\overline z\ =\ 0$$
has a real solution $t$.</p>
<p><strong>Attempt at a solution</strong></p>
<p>The discriminant is</p>
<p>$[(z+\overline z) - i(z-\overline z)]^2 - 4(2z\ove... | Bernard | 202,857 | <p>Just
set $z=u+iv$. The discriminant becomes
$$\Delta=4[(u+v)^2-2(u^2+v^2)]=-4(u-v)^2.$$
Hence the condition is $\;u=v,\;$ or $\;\arg z\equiv\dfrac\pi4\mod\pi$.</p>
|
148,257 | <p>Let $f:\mathbb R\to\mathbb R$ be continuous. Suppose $(x_n)_n$ and $(y_n)_n$ are sequences in $\mathbb R$ such that the sequence $(x_n-y_n)_n$ converges to $0$. Does this mean that the sequence $(f(x_n)-f(y_n))_n$ converges to $0$?</p>
<p>I feel like it is true, since the definition of continuity states that $f$ pr... | Davide Giraudo | 9,849 | <ul>
<li>The result is true is $f$ is uniformly continuous on $\Bbb R$. </li>
<li>But if it's not the case, we can find $\{x_n\}$ and $\{y_n\}$ which contradict this fact, using the definition of uniform continuity. </li>
</ul>
|
176,488 | <p><strong>Summary:</strong> My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic numbers in the same way that $e$ or $\pi$ behave empirically like random ones. I hope this is not ... | Bjørn Kjos-Hanssen | 4,600 | <p>It sounds like you are talking about what in computability theory and set theory are known as <strong>Cohen generic</strong> reals (the lowest level of which in computability theory is <strong>1-generic</strong>, then 2-generic and so on).</p>
<p>I don't know any really natural example of a 1-generic real, but ther... |
4,206,205 | <p>I came across a theorem in algebraic number theory:</p>
<blockquote>
<p><strong>Theorem</strong> Let <span class="math-container">$A$</span> be a Dedekind ring and <span class="math-container">$M, N$</span> two modules over <span class="math-container">$A$</span>. If <span class="math-container">$M_\mathfrak{p} \sub... | hm2020 | 858,083 | <p><strong>Question:</strong> "My question is not about the theorem per se, but I guess there should be some counterexamples. Chances are there are modules <span class="math-container">$M,N$</span> such that <span class="math-container">$M_p⊂N_p$</span> does not imply <span class="math-container">$M⊂N$</span>.&quo... |
69,050 | <p>The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group,
I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question.
My Question is :</p>
<ol>
<li>Why is the name Qu... | Jack Schmidt | 583 | <blockquote>
<p>Quotient groups $A/B$ count cosets of $B$ inside $A$. The counting even works well with addition.</p>
</blockquote>
<p>The Cartesian plane forms a group A, and a line through the origin is a subgroup B. The cosets of B inside A are all the parallel lines. How many are there?</p>
<p>Suppose <em>B<... |
69,050 | <p>The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group,
I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question.
My Question is :</p>
<ol>
<li>Why is the name Qu... | Beginner | 15,847 | <p>Two of the most basic concepts in mathematics, namely "Sets" and "Relations" are much useful to create new things from old. For example, "integers" can be constructed from natural numbers by putting an equivalence relation on the set $\mathbb{N}\times \mathbb{N}$ (see Concrete Abstract Algebra: Niels Lauritzen), rat... |
69,050 | <p>The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group,
I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question.
My Question is :</p>
<ol>
<li>Why is the name Qu... | Community | -1 | <p>I modify Jack Schmidt's exquisite answer for more details.</p>
<blockquote>
<p>Quotient groups $A/B$ count cosets of $B$ inside $A$. The counting even works well with addition.</p>
</blockquote>
<p>The Cartesian plane forms a group A, and a line through the origin is a subgroup B. The cosets of B inside A are ... |
8,382 | <h3>Context</h3>
<p>I'm writing a function that look something like:</p>
<pre><code>triDiagonalQ[mat_] := MapIndexed[ #1 == 0 || Abs[#2[[1]]-#2[[2]]] <= 1 &, mat, {2}] //
Flatten // And @@ # &
</code></pre>
<p>Now, things like <code>#2[[1]]</code> and <code>#2[[2]]</code> are somewhat hard to read. I... | Mr.Wizard | 121 | <p>My first answer explains that <em>Mathematica</em>'s replacement rules perform destructuring. This answer is intended to complement jVincent's method, which I see is appreciated.</p>
<p>My aim is to provide <code>Attributes</code> for the pattern-based function. This requires that the head evaluate therefore <code... |
17,115 | <p>I am pretty sure that the following statement is true. I would appreciate any references (or a proof if you know one).</p>
<p>Let $f(z)$ be a polynomial in one variable with complex coefficients. Then there is the following dichotomy. Either we can write $f(z)=g(z^k)$ for some other polynomial $g$ and some integer ... | Jonas Meyer | 1,119 | <p>You're right. Quine proved in "<a href="http://www.jstor.org/stable/2039005">On the self-intersections of the image of the unit circle under a polynomial mapping</a>" that if the degree is $n$ and $f(z)\neq g(z^k)$ with $k>1$, then the number of points with at least 2 distinct preimage points is at most $(n-1)^2... |
17,115 | <p>I am pretty sure that the following statement is true. I would appreciate any references (or a proof if you know one).</p>
<p>Let $f(z)$ be a polynomial in one variable with complex coefficients. Then there is the following dichotomy. Either we can write $f(z)=g(z^k)$ for some other polynomial $g$ and some integer ... | Ian Agol | 1,345 | <p>The image of the unit circle is a real-algebraic curve, so the number of self-intersections should be finite. </p>
<p>Addendum:
I'm not sure how to complete the argument, but here's a heuristic (following Speyer's suggestion). The curve $x^2+y^2=1$ is a rational curve (i.e., birational to $\mathbb{CP}^1$, the Riema... |
875,458 | <p>Is it possible to draw a triangle, if the length of its medians $(m_1, m_2, m_3)$ are given only?</p>
<p>Someone asked me this question, but I can not see it. Is it really possible?</p>
<p><strong>UPDATE</strong></p>
<p>Apart from the algebraic solution given by <em>JimmyK4542</em>, can anyone give me a direct co... | Christian Blatter | 1,303 | <p>From JimmyK4542's formulas it follows that
$a={2\over3}\sqrt{2s^2-m_a^2}\>$, where $s:=\sqrt{m_b^2+m_c^2}$. From this one derives the following construction of ${3\over2}a$: </p>
<p><img src="https://i.stack.imgur.com/kskHT.jpg" alt="enter image description here"></p>
<p>Construct $s$ as hypotenuse of a right t... |
1,957,304 | <p>I'm proving the compact-to-Hausdorff lemma (probably not a universal name for it) which is stated as:</p>
<blockquote>
<p>If $X$ is compact, $Y$ Hausdorff, $f:X \rightarrow Y$ a continuous bijection, then $f$ is a homeomorphism.</p>
</blockquote>
<p>However, the following line has popped up in a proof of it:</p>... | carmichael561 | 314,708 | <p>If $y\in f(U)$, then there exists $x\in U$ such that $f(x)=y$. Since $f$ is injective, there cannot be an element $z\in X\setminus U$ such that $f(z)=y$. Therefore $y\not\in f(X\setminus U)$, i.e. $y\in Y\setminus f(X\setminus U)$.</p>
<p>On the other hand, suppose that $y\in Y\setminus f(X\setminus U)$. There exis... |
1,957,304 | <p>I'm proving the compact-to-Hausdorff lemma (probably not a universal name for it) which is stated as:</p>
<blockquote>
<p>If $X$ is compact, $Y$ Hausdorff, $f:X \rightarrow Y$ a continuous bijection, then $f$ is a homeomorphism.</p>
</blockquote>
<p>However, the following line has popped up in a proof of it:</p>... | egreg | 62,967 | <p>The property you want to prove is the same as
$$
Y\setminus f(U)=f(X\setminus U)
$$
Suppose $y\notin f(U)$; then $f^{-1}(y)\notin U$ and therefore $f^{-1}(y)\in X\setminus U$. Therefore $y\in f(X\setminus U)$.</p>
<p>Similarly for the converse inclusion.</p>
<hr>
<p>Note that, in general, when you have a function... |
882,590 | <p><img src="https://i.stack.imgur.com/BMapu.png" alt="enter image description here"></p>
<p>long method: Determine an equation for each and solve using average value formula</p>
<p>alternative methods? </p>
<p>How could you prove the average value to be C over an interval [a,b] if you are given a graph.... looking ... | Varun Iyer | 118,690 | <p>Your equation for this question is:</p>
<p>$$\frac{1}{6}*\int_0^6 h(x) = 2$$</p>
<p>Therefore,</p>
<p>$$\int_0^6 h(x) = 12$$</p>
<p>For answers $c$ and $d$ you can use the area of a triangle,</p>
<p>If you try answer choice $c$,</p>
<p>We get that:</p>
<p>$$A = \frac{1}{2}*4.5*6 - \frac{1}{2}*1.5*(-2) = 12$$<... |
3,802,806 | <p>I and a friend are trying to find all endomorphisms <span class="math-container">$f$</span> of <span class="math-container">$\mathcal{M}_n(\mathbb{R})$</span> such that <span class="math-container">$f({}^t M)={}^t f(M)$</span> for all <span class="math-container">$M$</span>. We believe they are of the form <span cla... | Nick | 27,349 | <p>First, consider a basis of the space <span class="math-container">$\mathcal{M}_n$</span>. For example, the <span class="math-container">$n^2$</span> elementary matrices <span class="math-container">$e_{ij}$</span> which are all zero's except a <span class="math-container">$1$</span> in the <span class="math-containe... |
130,465 | <p>i just started university so im pretty new to all this new math. My problem is to solve this <code>recursive sequence</code>: $a_{n+1} = a_{n}^3$ with: $a_{0} = \frac{1}{2}$ and: $n \in N$</p>
<p>I've to analyse convergence and if its convergent i've to get the limit of this sequence.</p>
<p>I dont know how to sta... | Riemann | 27,899 | <p>Due to $a_{n+1}=a_n^{3}=(a_{n-1}^3)^3=a_{n-1}^{3^2}=……=a_{0}^{3^{n+1}}=(\frac{1}{2})^{3^{n+1}}$. if you know this ,you can do this question.</p>
|
3,753,060 | <blockquote>
<p>If <span class="math-container">$\int f(x)dx =g(x)$</span> then <span class="math-container">$\int f^{-1}(x)dx $</span> is equal to</p>
<p>(1) <span class="math-container">$g^{-1}(x)$</span></p>
<p>(2) <span class="math-container">$xf^{-1}(x)-g(f^{-1}(x))$</span></p>
<p>(3) <span class="math-container">... | Knight wants Loong back | 569,595 | <p>All the other answers have already given you the finest methods of solving the problem. I want to say that as you’re given the options for the integral <span class="math-container">$\int f^{-1} (x) dx$</span>, it would be a nice thing to differentiate the options to see if we get <span class="math-container">$f^{-1}... |
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