qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,027,888 | <h2>Exercise</h2>
<p>If $H$ is the Heaviside function, prove, using the definition below, that $\lim \limits_{t \to 0}{H(t)}$ does not exist.</p>
<hr>
<h2>Definition</h2>
<blockquote>
<p>Let $f$ be a function defined on some open interval that contains the number $a$, except possible $a$ itself. Then we say that ... | David | 651,991 | <p>Main idea: Set <span class="math-container">$\epsilon = 0.1$</span> and see that, for any "candidate to limit" <span class="math-container">$L$</span>, you will always find a point <span class="math-container">$X$</span> as close to <span class="math-container">$0$</span> as you want, such that <span class="math-con... |
163,585 | <p>Which (finite, undirected) graphs have this property?</p>
<p>Every vertex $v$ can be labeled with a positive integer $l(v)$.</p>
<p>Variant 1: For each vertex $v$, $l(v) \geq \Sigma_{[v,w] \in E, w \neq v} l(w)/2$.</p>
<p>Variant 2: For each vertex $v$, $l(v) > \Sigma_{[v,w] \in E, w \neq v} l(w)/2$.</p>
| Generic Human | 26,855 | <p>Here $G$ is a finite undirected graph.</p>
<p>Suppose wlog that $G$ is connected, with at least 2 vertices $1,\dots,n$ and no self-edges. There is no reason to forbid multiple edges. Define $M=A/2$ where $A$ is the adjacency matrix of $G$.</p>
<blockquote>
<p><strong>Theorem</strong>: $G$ satisfies the first con... |
2,903,429 | <p>Show that the equation $$\frac{4}{3}{x^2} + \frac{4}{3}{y^2} + \frac{4}{3}{z^2} + \frac{4}{3}{xy} + \frac{4}{3}{xz} + \frac{4}{3}{yz} = 1$$ represents an ellipsoid. Find the position and lengths of its principal half-axes.</p>
<p>Workings
$$(x+y)^2+(y+z)^2+(z+x)^2=\frac32\,.$$
Common factor $\left(\dfrac32\right)^{... | Claude Leibovici | 82,404 | <p>$$\sum_{j=1}^n \sum_{k=j+1}^n 1=\sum_{j=1}^n (n-j)=n\sum_{j=1}^n 1-\sum_{j=1}^n j=n^2-\frac{1}{2} n (n+1)=\frac{1}{2} (n-1) n$$</p>
|
2,903,429 | <p>Show that the equation $$\frac{4}{3}{x^2} + \frac{4}{3}{y^2} + \frac{4}{3}{z^2} + \frac{4}{3}{xy} + \frac{4}{3}{xz} + \frac{4}{3}{yz} = 1$$ represents an ellipsoid. Find the position and lengths of its principal half-axes.</p>
<p>Workings
$$(x+y)^2+(y+z)^2+(z+x)^2=\frac32\,.$$
Common factor $\left(\dfrac32\right)^{... | JonHales | 413,766 | <p>How I see this is I start with $j = 1$ And so my next sum is $$\Sigma_{k = 2} ^{n} (1) = n-1$$ Now I'll add to this the value for when $j=2$, which will be $$\Sigma_{k = 3}^n (1) = n - 2$$. This pattern continues, so the nested sum you want is really just $$\Sigma_{i = 1}^n (n-i) = \Sigma_{i =1}^n(i) = \frac{1}{2} n... |
3,358,168 | <p><span class="math-container">$x_n$</span> is bounded, <span class="math-container">$\lim_{n \rightarrow \infty}(x_{n+1}-x_n)=0$</span>, <span class="math-container">$\liminf_{n \rightarrow \infty}=l$</span>, <span class="math-container">$\limsup_{n \rightarrow \infty}=L$</span>, show that every point in <span class=... | Matematleta | 138,929 | <p>Suppose <span class="math-container">$l<x<L$</span> but <span class="math-container">$x$</span> is not an accumulation point of <span class="math-container">$(x_n)$</span>. Then there are <span class="math-container">$a,b$</span> such that <span class="math-container">$l<a<x<b<L$</span> and <span c... |
2,006,870 | <p>Is the following inequality true?</p>
<p>$\left( \sum \limits_{i=1}^\infty \sum \limits_{j=1}^\infty \sum \limits_{k=1}^\infty \sum \limits_{l=1}^\infty a_{ij}\,a_{ik}\,a_{jl}\,a_{kl} \right) \leq \left( \sum \limits_{i=1}^\infty \sum \limits_{j=1}^\infty a_{ij}^2 \right)^{1/2}\left( \sum \limits_{i=1}^\infty \sum ... | hmakholm left over Monica | 14,366 | <p>It is easy to find a collision:
$$xy+x-y = (x-1)(y+1) + 1$$
so all we need is to find some number that can be written as a product in more than one way, and then declare the two factors to be $x-1$ or $y+1$, respectively.</p>
<p>For example $12=3\cdot 4=4\cdot 3$, so your function produces $13$ both for inputs $(4,... |
40,532 | <p>I'm having some problems understanding the following paragraph, which I read in a analysis script (hopefully I haven't made any translation errors):</p>
<blockquote>
<p>"A map $f:U \rightarrow Y$, where $U$ is open and $X,Y$ are Banach spaces, is continuous at $x' \in U$ if $$f(x')=\lim_{x\rightarrow x... | Arturo Magidin | 742 | <p>I'll only deal with (1). Like mac, I'm not sure what (3) means here.</p>
<p>(1) When working in polar coordinates, every point in the plane is written in the form $(r,\theta)$, where $r$ denotes the magnitude (distance to the origin) and $\theta$ denotes the argument (direction). Similarly, when we work with comple... |
65,083 | <p>Rotman's book <em>An Introduction to the Theory of Groups</em> (Fourth Edition) asks, on page 22, Exercise 2.8, to show that <span class="math-container">$S(n)$</span> cannot be embedded in <span class="math-container">$A(n+1)$</span>, where <span class="math-container">$S(n)$</span> = the symmetric group on <span c... | Sergei Ivanov | 4,354 | <p>I think the following is sufficiently elementary: a transposition in $S_n$ is an element of order 2 commuting with at least $2(n-2)!$ elements of the group. But $A_{n+1}$ does not have such an element if $n$ is large enough. Indeed, if $\sigma\in A_{n+1}$ is of order 2, then it is a product of $k$ independent transp... |
1,712,481 | <p><span class="math-container">$$I=\displaystyle\int\frac{dx}{(3 + 2\sin x - \cos x)}$$</span></p>
<p>If <span class="math-container">$$\tan\left(\frac{x}{2}\right)=u$$</span></p>
<p>or <span class="math-container">$$x=2\cdot\tan^{-1}(u)$$</span></p>
<p>Then,</p>
<p><span class="math-container">$$\sin{x}=\dfrac{2u}{1+... | Noir | 324,626 | <p>Remember the following trigonometric equations, which are always very helpful to solve integrals:
$$
\begin{matrix}
2\cos(u)^2 = 1 + \cos(2u) & & & \left(\cos(u) + \sin(u)\right)^2 = 1 + \sin(2u) \\
2\sin(u)^2 = 1 - \cos(2u) & & & \left(\cos(u) - \sin(u)\right)^2 = 1 - \sin(2u) \\
\end{matri... |
1,712,481 | <p><span class="math-container">$$I=\displaystyle\int\frac{dx}{(3 + 2\sin x - \cos x)}$$</span></p>
<p>If <span class="math-container">$$\tan\left(\frac{x}{2}\right)=u$$</span></p>
<p>or <span class="math-container">$$x=2\cdot\tan^{-1}(u)$$</span></p>
<p>Then,</p>
<p><span class="math-container">$$\sin{x}=\dfrac{2u}{1+... | Omran Kouba | 140,450 | <p>Generally, consider the real function
$$f(x)=\frac{1}{a+b\cos x+c\sin x}$$
With $a^2>b^2+c^2$ so that $f$ is defined on $\mathbb{R}$.
It is not hard to check that the derivative of
$$F(x)=\frac{x}{d}+\frac{2}{d}\arctan\left(\frac{c\cos x-b\sin x}{d+a+b\cos x+c\sin x}\right)$$
with $d=\sqrt{a^2-b^2-c^2}$, is $f(x... |
131,482 | <p>Noticed strange behaviour of <code>Export[]</code> when saving graphics to EPS. Here is an example figure I am exporting:</p>
<pre><code>fig =
Show[
ListPolarPlot[
Table[{a, .9}, {a, 0, 2 Pi, .001}],
PlotRange -> {{0, 1.1}, {0, 1.1}}, GridLines -> Automatic,
PlotStyle -> PointSize[.005]],
... | Simon Rochester | 8,253 | <p>The default grid lines style uses uses partially transparent grid lines:</p>
<pre><code>InputForm@AbsoluteOptions[fig, GridLinesStyle]
(* {GridLinesStyle -> Directive[GrayLevel[0.5, 0.4]]} *)
</code></pre>
<p>where the second argument of <code>GrayLevel</code> specifies opacity. As pointed out by george2079 in ... |
126,270 | <p>How many seven-digit numbers divisible by 11 have the sum of their digits equal to 59?</p>
<p>I am able to get the seven-digit numbers divisible by 11 </p>
<p>and </p>
<p>I am also able to get the seven-digit numbers whose sum of their digits equal to 59.</p>
<p>But i am not able to get how i can get the count o... | davin | 2,881 | <p>If we look at the number $a_6a_5a_4a_3a_2a_1a_0$:
$$a_6a_5a_4a_3a_2a_1a_0 = a_610^6+a_510^5+ \dots +a_010^0 \\ \equiv a_6(-1)^6 + \dots+a_0(-1)^0 \mod 11 = a_6 -a_5+a_4-a_3+a_2-a_1+a_0 \mod 11$$
So if the number will be divisible by 11 we require
$$a_6 -a_5+a_4-a_3+a_2-a_1+a_0 = 11m$$</p>
<p>We take note that $0 \l... |
307,466 | <p>Is there any simple way to construct an entiere function $f$ such that :
$$\forall p \in {\mathbb N} \quad f(2^p)=(-1)^p$$</p>
| Davide Giraudo | 9,849 | <p>By <a href="http://en.wikipedia.org/wiki/Weierstrass_factorization_theorem" rel="nofollow">Weierstrass products</a>, for each integer $k$ we can find an entire function $f_k$ such that $f_k(2^j)=0$ if $k\neq j$ and $f_k(2^k)=(-1)^k2^k$. Define $f:=\sum_{k=0}^{+\infty}2^{-k}f_k$. Using the relation about elementary f... |
606,656 | <p>So I heard this a long time ago and I recently started thinking about it again. So I was told that the complex function $f(z)=1/z$ maps everything inside a circle to points outside the circle (the remaining part of the complex plane). Why is this?</p>
<p>I realize we can write $$f(z)=\frac{1}{z}=\frac{\overline{z}}... | abiessu | 86,846 | <p>We can also write the numbers $z$ in polar form as $re^{i\theta}$, in which case $$f(z)=f(re^{i\theta})=\frac {e^{-i\theta}}r$$ So the angle is negated and the radius is inverted. This means that $r\lt1\implies \frac 1r\gt 1$, and every complex point is mirrored across the real axis, since $e^{-i\theta}$ preserves... |
3,272,030 | <p>Let's X, Y are random variables (i.e. each one maps elements of a sample space to real numbers). In particular, let's X is such that <span class="math-container">$X:\Omega \rightarrow \mathcal{R}$</span>, where <span class="math-container">$\Omega=\{\omega_1, ..., \omega_n \}$</span> and <span class="math-contain... | Paul Childs | 584,354 | <p>Actually as <span class="math-container">$\{ \omega_1, \omega_2 \} \in \Omega \oplus \Omega$</span> and that the latter is an equally valid sample space, it would <em>no less</em> satisfy the definition of a random variable.</p>
|
3,197,301 | <p>Let <span class="math-container">$OABC$</span> be a tetrahedron such that <span class="math-container">$|OA|=|OB|=|OC|$</span>. Denote by <span class="math-container">$D$</span> and <span class="math-container">$E$</span> the midpoints of segments <span class="math-container">$AB$</span> and <span class="math-contai... | Jack D'Aurizio | 44,121 | <p>Your set is the inverse image of <span class="math-container">$(0,1)$</span> with respect to a continuous function (the distance from the origin). Since <span class="math-container">$(0,1)$</span> is an open set and the mentioned function is continuous, your set is open too.</p>
|
279,238 | <p>I want to create a regular polygon from the initial two points <span class="math-container">$A$</span>, <span class="math-container">$B$</span> and number of vertices <span class="math-container">$n$</span>,<br />
<a href="https://i.stack.imgur.com/dKr9A.png" rel="noreferrer"><img src="https://i.stack.imgur.com/dKr9... | Daniel Huber | 46,318 | <p>Corroborating cvgm interesting work, we may improve a bit. {x1,y1} is not on the polygon and the winding direction is opposite to the positive mathematical winding direction. This may be fixed by:</p>
<pre><code>{x1, y1} = {2, 3};
{x2, y2} = {5, 5};
length = Norm[{x2, y2} - {x1, y1}];
n = 5;
Graphics[{Polygon[
A... |
469,521 | <p>$\theta$, $\phi$ are integrable random variables on a probability space $(\Omega,\mathcal{F},P)$ and $\mathcal{G}$ is $\sigma$-field on $\Omega$ contained in $\mathcal{F}$.
Now we want to prove $E(\theta\mid\mathcal{G})=E(\theta)$ if $\theta$ is independent of $\mathcal{G}$. The proof is, for any $B\in \mathcal{G}$,... | Jacky Zhang | 89,915 | <p>$$\int_{B}E[\theta]dP=E[E[\theta]I_{B}]=E[I_{B}]E[\theta]$$,that's all.</p>
|
3,232,766 | <blockquote>
<p>Is <span class="math-container">$10^{100}$</span> (Googol) bigger than <span class="math-container">$100!$</span>?</p>
<p>If <span class="math-container">$10^{100}$</span> is called as Googol, does <span class="math-container">$100!$</span> have any special name to be called, apart from being called as ... | Paolo Bordignon | 664,304 | <p>With simple ineqalities we have:</p>
<p><span class="math-container">$100!\geq 90^{10}\cdot 80^{10}\cdots 20^{10}\cdot 10^{10}$</span></p>
<p><span class="math-container">$100!\geq (9\cdot 8 \cdots 2 \cdot 1)^{10}\cdot 10^{90}$</span></p>
<p><span class="math-container">$100!\geq (9!)^{10}\cdot 10^{90}>10^{100... |
549,245 | <p>Apparently, it is a common mistake to write $\forall x[R(x) \rightarrow P(x)]$ instead of $\forall x[R(x)\wedge P(x)]$ in some cases, however I can't seem to find the difference between the two. </p>
<p>I did some research in ProofWiki and a Discrete Mathematics book, but I couldn't find this specific difference ex... | Shaun | 104,041 | <p>Assume $[\forall x[Rx\to Px]]\leftrightarrow [\forall x[Rx\wedge Px]]$ and suppose $ \forall x[Rx\to Px] $. Then $ \forall x[Rx\wedge Px] $. Instantiate with $a$ for $x$. Now $Ra\to Pa$ and $Ra$, $Pa$. But then $\neg Ra$ is an option, a contradiction.</p>
<p>You can see these things clearly with tableaux.</p>
<p><... |
1,017,320 | <p>Assume ZFC. Let $B\subseteq\mathbb R$ be a set that is not Borel-measurable. Clearly, $B$ must be uncountable, since countable sets are always Borel being a countable union of measurable singletons.</p>
<p><strong>Question:</strong> can one conclude that $B$ necessarily has the cardinality of the continuum <em>with... | Asaf Karagila | 622 | <p>No, you can't make those inferences. The Borel sets satisfy the continuum hypothesis. Namely any uncountable Borel set has necessarily the cardinality of the continuum.</p>
<p>If $B$ is not Borel measurable, in particular it is not a Borel set. All you can conclude about it is that it is uncountable. If the continu... |
1,037,632 | <p>How to find the last 2 digits of $2014^{2001}$? What about the last 2 digits of $9^{(9^{16})}$?</p>
| Anurag A | 68,092 | <p><strong>Hint:</strong></p>
<p>For finding last two digits you need to reduce this modulo $100$. That is you ne need to find
$$2014^{2001} \equiv ? \pmod{100}.$$
This is the same as asking
$$14^{2001} \equiv ? \pmod{100}.$$</p>
<p>Now in order to facilitate computation, you need to use <a href="http://en.wikipedia... |
4,056,627 | <p>Equation to minimize using Boolean Algebra Laws:
<span class="math-container">$A+\bar{A} B \bar{C}$</span></p>
<p>I have tried doing this but i am unsure of the answer:
<span class="math-container">$$
\begin{array}{l}
\text { Let } K=B \bar{C} \\
A+\bar{A} K=A+K=A+B \bar{C}
\end{array}
$$</span></p>
| kabenyuk | 528,593 | <p>If <span class="math-container">$|G:H|<9$</span>, then since <span class="math-container">$G$</span> is a simple group, it is isomorphic to a subgroup of the symmetric group <span class="math-container">$S_8$</span>. But the group <span class="math-container">$S_8$</span> has a Sylow <span class="math-container"... |
597,610 | <p>can someone explain why when a radical equation is solved some solutions don't work? Is there a rule when a certain transformation is performed on the equation it gives extra solutions? thanks very much</p>
| Cameron Buie | 28,900 | <p>In general, when a transformation is <a href="http://en.wikipedia.org/wiki/One-to-one_function" rel="nofollow"><em>one-to-one</em></a> (or <em>injective</em>), then it is reversible, so results in an equivalent equation. The problem with some radical equations is (for example) that $(-1)^2=1=1^2.$ That is, squaring ... |
120,863 | <blockquote>
<p>A perfect dice is drawn $4$ times.What's the probability to the same number comes out at least $2$ times?</p>
</blockquote>
<p>At first, I applied to binomial law.I made all calculations.</p>
<p>I set up a random variable.The possible values for the variable were $0,1,2,3$ and $4$.
My thought was th... | Michael Hardy | 11,667 | <p>If you find the probability that a $1$ appears at least two times, and similarly a $2$, and a $3$, etc., then you've got the complication that those six events are not mutually exclusive: a $1$ could appear twice <b>and</b> a $2$ twice.</p>
<p>So the simpler way is to find the probability that the event you're look... |
3,270,504 | <blockquote>
<p>Prove that <span class="math-container">$\log|e^z-z|\leq |z|+1$</span> where <span class="math-container">$z\in\mathbb{C}$</span> with <span class="math-container">$|z|\geq e$</span>.</p>
</blockquote>
<p><strong>Background:</strong></p>
<p>This is from a proof that <span class="math-container">$e^z... | Chrystomath | 84,081 | <p><span class="math-container">\begin{align*}
|e^z-z|&\le|e^z|+|z|\\
&\le e^{|z|}+|z|\quad\textrm{from series expansion}\\
&\le e^{|z|+1}\quad\textrm{again from series expansion}
\end{align*}</span></p>
|
3,281,965 | <p>It is known that a way to check whether a number <span class="math-container">$n$</span> is prime, is to check for divisors of <span class="math-container">$n$</span> from <span class="math-container">$2$</span> to <span class="math-container">$\lfloor\sqrt{n}\rfloor$</span>. If we find any divisor, then <span class... | Robert Israel | 8,508 | <p>Suppose <span class="math-container">$n$</span> is a random positive integer from <span class="math-container">$1$</span> to <span class="math-container">$N$</span>. By the Prime Number Theorem, the probability that it is prime is on the order of <span class="math-container">$1/\log N$</span>. On the other hand, f... |
1,647,442 | <p>Prof using the binomial theorem: for all integers $n ≥0$ and for all nonnegative real numbers $x$, $1+nx ≤(1+x)^n$. </p>
<p>Don't have a idea to start this one. I don't know how to use math induction yet, so I need a answer without that.</p>
| Thomas | 26,188 | <p>To take $10\%$ of a quantity $X$, you multiply $X$ by $\frac{1}{10}$. So, for example, $10\%$ of $20$ is $\frac{1}{10}\cdot 20 = 2$.</p>
|
2,859,411 | <p>Let $y(t)$ be a real valued function defined on the real line such that
$y'= y(1 − y)$, with $y(0) \in [0, 1]$. Then $\lim_{t\to\infty} y(t) = 1$.</p>
<p>The solution is given as false .But i have no idea about that. I try some counterexample but it won't work.</p>
<p>How can i find solution with 3 minutes?</p>
| mechanodroid | 144,766 | <p>$$\frac{dy}{dx} = y(1-y) \implies dx = \frac{dy}{y(1-y)}$$</p>
<p>so $$x+C = \ln\frac{y}{1-y} \implies Ce^x = \frac{y}{1-y}$$</p>
<p>and therefore $$y = \frac{Ce^x}{1+Ce^x}$$</p>
<p>The initial condition is</p>
<p>$$y(0) = \frac{C}{1+C} \implies C = \frac{y(0)}{1-y(0)}$$</p>
<p>so $$y = \frac{\left(\frac{y(0)}{... |
161,766 | <p>What is the best way to write a quick function which takes in a vertex (of some undirected graph), and <strong>gives the value of the k-Core's in which that vertex is in?</strong></p>
<p>The obvious modification of <code>kCoreComponents[Graph,k]</code> using <code>Intersection</code> is either taking ages, since it... | Jason B. | 9,490 | <p>I can somewhat reproduce this error. Except for me, if I try to export a graphic with no display set, the kernel just hangs. The answer is to use an <a href="https://en.wikipedia.org/wiki/Xvfb" rel="noreferrer">Xvfb display server</a>, as described <a href="http://elementalselenium.com/tips/38-headless" rel="noref... |
3,173,125 | <p>Let <span class="math-container">$f(x) = x^3 + a x^2 + b x + c$</span> and <span class="math-container">$g(x) = x^3 + b x^2 + c x + a\,$</span> where <span class="math-container">$a, b, c$</span> are integers and <span class="math-container">$c\neq 0\,$</span>. Suppose that the following conditions hold:</p>
<ol>
... | lhf | 589 | <p><em>Hint:</em> Condition 2 can be expressed as <span class="math-container">$f(x)$</span> divides <span class="math-container">$g(x^2)$</span>.</p>
|
1,614,111 | <p>Is there a sequence of positive real numbers $x_1,\ldots,x_n$ for which
$$
\sum_{1\leq i,j\leq n}\left[\frac{\sqrt{ij}}{2}-\min(i,j)\right]x_ix_j> 0?
$$</p>
| Jason | 195,308 | <p>The answer is "yes". Consider the matrix <span class="math-container">$A_n=[a_{ij}]_{1\le i,j\le n}$</span>, defined by</p>
<p><span class="math-container">$$ a_{ij} = \min(i,j) - \frac{\sqrt{ij}}2. $$</span></p>
<p>As mentioned in a comment, this is positive definite for small <span class="math-container">$n$</sp... |
2,880,271 | <p>Let a, b, c and d be real numbers that are not all zero. Let
ax + by = p
cx + dy = q
be a pair of equations in the variables x and y with p, q ∈ R. </p>
<p>Show this system of equations has a unique solution if and only if ab − cd != 0.</p>
<hr>
<p>From Determinant of coefficient matrix, I know (ad -bc) =0 => no ... | mfl | 148,513 | <p>Assume $ad-bc\ne 0.$ Then </p>
<p>$$\begin{array}{l}ax+by=p\\cx+dy=q\end{array}\implies \begin{array}{l}adx+bdy=dp\\bcx+bdy=bq\end{array}\implies x=\dfrac{dp-bq}{ad-bc}.$$ In a similar way you can get $y:$</p>
<p>$$\begin{array}{l}ax+by=p\\cx+dy=q\end{array}\implies \begin{array}{l}acx+bcy=cp\\acx+ady=aq\end{array... |
1,549,506 | <p>To Prove $$\lim_{x \to 0}\frac{x^2+2\cos x-2}{x\sin^3 x}=\frac{1}{12}$$
I tried with L'Hospital rule but in vain.</p>
| Idris Addou | 192,045 | <p>We can use the basic limits
\begin{equation*}
\lim_{x\rightarrow 0}\frac{\sin x}{x}=1,\ \ \ \ \ \ \ \ \
\lim\limits_{x\rightarrow 0}\dfrac{\cos x-1+\frac{1}{2}x^{2}}{x^{4}}=\dfrac{1%
}{24}
\end{equation*}
as follows
\begin{equation*}
\lim\limits_{x\rightarrow 0}\frac{x^{2}+2\cos x-2}{x\sin ^{3}x}%
=\lim\limits_{x\r... |
423,793 | <p>A group of 3141 students gather together. Some of them have 13 friends in this group, some have 33 friends, and the rest has 37 friends. Prove using graph theory that this group does not exist. Assume that if A is friends with B, then B is friends with A.</p>
| Nick Peterson | 81,839 | <p>This seems like homework; so, let me just give you a hint.</p>
<p>Say that you have $a$ people with 13 friends, $b$ with 33, and the remaining $3141-a-b$ have 37. In the graph, this corresponds to having $a$ vertices of degree 13, $b$ vertices of degree $33$, and the rest of degree 37.</p>
<p>So, the total degree... |
229,748 | <p>I have an integral given by <code>intS[zs,c]</code> and I want to plot the expression <code>functionS</code> (which contains <code>intS[zs,c]</code>) with respect to another integral function <code>inta[zs,c]</code> and <code>zs</code>. However, the values of <code>inta[zs,c]</code> are extremely small compared to <... | Sumit | 8,070 | <p>You need to fix <code>BoxRatios</code> for 3d plots like this. I am going to show with a simple example</p>
<pre><code>ParametricPlot3D[{zs, zs^2, zs + c}, {zs, 0, 10}, {c, 0, 10},
PlotStyle -> {LightBlue}, PlotRange -> Full,
LabelStyle -> Directive[Bold, Medium], ImageSize -> 300]
</code></pre>
<p><... |
1,656,963 | <p>If $(a_n) \to 0$, then applying the algebraic limit theorem, what is $ \lim_\limits{n\to\infty} \frac {1+2a_n}{1+3a_n - 4a^2_n}$.</p>
<p>Would just be able to do : $\lim_\limits{n\to\infty} \frac {1+2(0)}{1+3(0) - 4(0)}$. </p>
<p>$\lim \frac {1}{1} = 1$.</p>
| Tsemo Aristide | 280,301 | <p>Hint: The function $f(x) = {{1+2x}\over{1+3x-4x^2}}$ is continuous, so $f(a_n)$ converges towards $f(a)$.</p>
|
3,436,804 | <p>Does the following series converge? If yes, what is its value in simplest form?</p>
<p><span class="math-container">$$\left( \frac{1}{1} \right)^2+\left( \frac{1}{2}+\frac{1}{3} \right)^2+\left( \frac{1}{4}+\frac{1}{5}+\frac{1}{6} \right)^2+\left( \frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10} \right)^2+\dots$$</... | Jam | 161,490 | <h3>Suggestion.</h3>
<p>This is merely a suggestion that is too large to fit in a comment. Euler's integral formula for the harmonic numbers, <span class="math-container">$H_n=\int_0^1\frac{1-x^n}{1-x}\ \mathrm{d}x$</span>, gives us a formula for the series as <span class="math-container">$a\to1^-$</span>:</p>
<p><span... |
1,105,787 | <p>I was given an excercise in my calculus class that i don't really understand, the problem says : Find the area limited by the curves $$ y = \frac{x+4}{x^2+1} ,\space x = -2 ,\space x = 3,\space y = 0 $$</p>
<p>I don't really know what approach to follow here, my guess would be to solve it using riemann sums or mayb... | Tim Raczkowski | 192,581 | <p>Yes, this is a definite integral problem. You can solve it by evaluating</p>
<p>$$\int_{-2}^3{x+4\over x^2+1}\,dx.$$</p>
<p>Hint: Split it into</p>
<p>$$\int_{-2}^3\frac x{x^2+1}\,dx+4\int_{-2}^3\frac1{x^2+1}\,dx,$$</p>
<p>and recall that the derivative of $\tan^{-1}x$ is $\frac1{x^2+1}$.</p>
|
2,265,368 | <p>Let $R$ be a commutative ring with unit and $N$ its nilradical. If $R$ is connected then $R/N$ is also connected; the quotient map induces a homeomorphism between the spectra. I'd like to see a more hands-on proof, but I'm unable to make it work. That is to say, I'd like to show that if $e^2-e\in N$ for some $e\in R... | egreg | 62,967 | <p>See <a href="http://www.math.hawaii.edu/~lee/algebra/idempotent.pdf" rel="nofollow noreferrer">http://www.math.hawaii.edu/~lee/algebra/idempotent.pdf</a>, but I'll repeat the argument here.</p>
<p>Suppose $e-e^2\in N$ and set $f=1-e$. Then $e^kf^k=0$, for some $k\ge1$.</p>
<p>Note that $e^k+N=e+N$ and $f^k+N=f+N$,... |
492,031 | <p>I am currently reading through Hatcher's Algebraic Topology book. I am having some trouble understanding the difference between a deformation retraction and just a retraction. Hatcher defines them as follows:</p>
<p>A <strong>deformation retraction</strong> of a space $X$ onto a subspace $A$ is a family of maps $f_... | Thomas Belulovich | 831 | <p>The difference between a retraction and a deformation retraction does have to do with the "notion of time" as you suggest. </p>
<p>Here's a strong difference between the two:</p>
<p>1) For any $x_0 \in X$, $\{x_0\} \subset X$ has a retract. Choose $r : X \to \{x_0\}$ to be the unique map to the one-point set. Then... |
3,974,768 | <p>Suppose <span class="math-container">$a_n \rightarrow a$</span>, <span class="math-container">$b_n \rightarrow b$</span> and <span class="math-container">$ a < b$</span>.
Is it true that there is <span class="math-container">$ N \in \mathbb{N} $</span> such that for any <span class="math-container">$ m > N $</... | heropup | 118,193 | <p>Use <a href="https://en.wikipedia.org/wiki/Rouch%C3%A9%27s_theorem" rel="nofollow noreferrer">Rouché's theorem</a> with the choice <span class="math-container">$f(z) = z^7 + 1$</span> and <span class="math-container">$g(z) = 7z^4 + 4z$</span>. Then you need to show that for <span class="math-container">$z = 2e^{i\t... |
4,041,595 | <p>I've been having lectures in group theory with Hungerford's book. We were presented with the following theorem:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/pkL0f.png" rel="noreferrer"><img src="https://i.stack.imgur.com/pkL0f.png" alt="enter image description here" /></a></p>
</blockquote>
<p>And then wit... | SSA | 863,540 | <p>Not sure, how much below information can help you. But may help others and I too can get more intuitive insight of this very important theorem, by some experts.</p>
<p>The way I understand the group is a kind of Symmetries of an object. On which one can perform some actions.
Now when it comes to comparing two group... |
2,288,358 | <p>Is there a straightforward way to prove the following inequality:
$$|1 + k\big(\exp(it)-1\big)|\leq 1 $$
where $k\in(0,1)$ and $t \in \mathbb{R}$ (correction, see dxiv answer) with $|t| \leq 1$, other than writing the quantity into its real and imaginary parts and checking that they satisfy the required inequalities... | user407691 | 407,691 | <p>Since the region of integration in u-v coordinates is still a triangle, it can be converted to polar coordinates. We can look at the bounds for r and <span class="math-container">$\theta$</span> separately.</p>
<p>In general with polar coordinates, it's easier when the inner integral is evaluated with respect to r (... |
3,126,080 | <p>Is there any particular equation which doesn't work on the real plane of numbers but works on other planes?</p>
| FormerMath | 174,432 | <p><span class="math-container">$x^2+1=0$</span> doesn't admit roots on <span class="math-container">$\mathbb{R} $</span>, but it does on <span class="math-container">$\mathbb{C}$</span></p>
|
1,827,612 | <p>I have to simplify $\sum_{i,j}\left[{n}\atop{i+j}\right]\binom{i+j}{i}$. I looks like we have $n$ children and we have to answer how many times we can arrange them into circles and color some of circles red.</p>
| Matematleta | 138,929 | <p>Let $\epsilon >0$ and note that there are partitions $P, Q$ such that </p>
<p>$\tag1 \underline{\int_{a}^{b}}{f(x)dx}-\epsilon<L(P,f)$ </p>
<p>$\tag2\underline{\int_{a}^{b}}{g(x)dx}-\epsilon <L(g,Q)$</p>
<p>These inequalities persist if we pass to a common refinement, $T$, so </p>
<p>$\tag 3\underline{\... |
1,307,460 | <p>Need to integrate this function. Need help with my assignment. Thanks</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Use $I=\int_a^bf(x)\ dx=\int_a^bf(a+b-x)\ dx$</p>
<p>$\implies I+I=\int_a^bf(x)\ dx+\int_a^bf(a+b-x)\ dx=\int_a^b[f(x)+f(a+b-x)]\ dx$</p>
<p>Then</p>
<p><strong>Method</strong> $\#1:A\cos x+B\sin x=\sqrt{a^2+b^2}\cos\left(x-\arctan \dfrac BA\right)$</p>
<p><strong>Method</strong> $\#2:$ <a href=... |
4,644,847 | <p>Is there a formal fallacy to describe lack of an observation is not proof that it does not exist, or lack of an occurrence is not proof that it can never happen?</p>
| Prem | 464,087 | <p>In most general terms , we can state it Mathematically / logically / formally like this :</p>
<p>Let X be the Universe (eg Integers or real numbers or .... )<br />
Let Y be the non-empty subset which DOES NOT have Property P.</p>
<p><span class="math-container">$ Y = \{ x \in X : \lnot P(x) \} $</span></p>
<p>Now , ... |
2,651,486 | <p>I have an exercise here that is asking me to write the MacLaurin formula of orders II, III, IV for a multivariable function. Example: $ f(x,y)=\cos x \cos y$</p>
<p>Can anyone tell me what the formula looks like for a multivariable function and maybe guide me through this example? Would be much appreciated!</p>
| Christian Blatter | 1,303 | <p>Your example is special in so far as the function is "separated" into two functions of one variable. In order to prove the formula given by Wikipedia consider for <em>fixed</em> $x$ and $y$ the auxiliary function
$$g(t):= f(t\,x,t\,y)$$
of one variable $t$, and compute its value at $t=1$, using the Taylor expansion ... |
3,019,989 | <p>I'm watching the Limits series on Khan Academy. In many videos Sal repeatedly says that although some people say that functions that tend to infinity have a limit infinity. (For example, in <a href="https://www.khanacademy.org/math/calculus-1/cs1-limits-and-continuity/modal/v/infinite-limits-and-asymptotes" rel="nof... | YiFan | 496,634 | <p>Like Arthur said in the comments, functions being "bounded" and "tending to infinity" mean different things. If <span class="math-container">$\lim_{x\to c}f(x)=\infty$</span>, then it tends to infinity and gets arbitrarily large in the process. Hence it is not bounded, so there does not exist real numbers <span clas... |
3,019,989 | <p>I'm watching the Limits series on Khan Academy. In many videos Sal repeatedly says that although some people say that functions that tend to infinity have a limit infinity. (For example, in <a href="https://www.khanacademy.org/math/calculus-1/cs1-limits-and-continuity/modal/v/infinite-limits-and-asymptotes" rel="nof... | egreg | 62,967 | <p>A <em>necessary</em> condition for a function to have infinite limit is that it is unbounded.</p>
<p>More precisely, if <span class="math-container">$\lim_{x\to c}f(x)=\infty$</span>, then <span class="math-container">$f$</span> is upper unbounded; if <span class="math-container">$\lim_{x\to c}f(x)=-\infty$</span>,... |
3,648,315 | <p><a href="https://i.stack.imgur.com/IKnlD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IKnlD.png" alt="enter image description here"></a></p>
<p>I have an item, let's say sword. </p>
<p>As described above, <br/>
If my sword is at state 1, I can try upgrading it. There are 3 possibilities. <br/... | Ekaveera Gouribhatla | 31,458 | <p>Since our aim is to find <span class="math-container">$x+y$</span>, Let <span class="math-container">$x+y=k$</span>, where <span class="math-container">$k \in \mathbb{Z}$</span></p>
<p>We have <span class="math-container">$$k+2x(k-x)=83$$</span></p>
<p>So</p>
<p><span class="math-container">$$2x^2-2kx+83-k=0 $$<... |
1,639,521 | <p>Let <span class="math-container">$f:[0,\infty)\to\mathbb{R}$</span> differentiable and suppose that <span class="math-container">$$\lim_{x\to\infty}f'(x)=L.$$</span> How can I prove that <span class="math-container">$$\lim_{x\to\infty}\frac{f(x)}{x} = L\;?$$</span></p>
<p>I have solved some similar problems using t... | Joe | 107,639 | <p>Maybe you're searching for a direct computation.</p>
<p>Let us check the case $L\neq0$.</p>
<p>\begin{align*}
L=\lim_{x\to\infty}f'(x)
&=\lim_{x\to\infty}\lim_{y\to x}\frac{f(y)-f(x)}{y-x}\\
&=\lim_{x\to\infty}\lim_{y\to x}\frac{f(x)-f(y)}{x-y}\\
&=\lim_{x\to\infty}\lim_{y\to x}\frac{f(x)\left[1-\frac{... |
440,528 | <p>My question is about group theory:</p>
<blockquote>
<p>How many subgroups does a non-cyclic group contain whose order is 25?</p>
</blockquote>
<p>How can i answer that question?</p>
<p>Can you generalize the answer?</p>
<p>Thanks for your help.</p>
| DonAntonio | 31,254 | <p>As a power of a prime, a non-cyclic group $\;G\;$ of order $\;25\;$ is isomorphic to $\,C_5\times C_5\;$ , with $\,C_5\cong\Bbb Z/5\Bbb Z\;$ is the cyclic group of order $\;5\;$</p>
<p>Now, some hints for you to answer your question:</p>
<p>== A group of the form $\;G=\underbrace{C_p\times C_p\times\ldots\times C_... |
35,375 | <p>Good morning,
today I have read that "number theory is nothing but the study of $\mathrm{Gal}(\mathbb{\bar{Q}}/\mathbb{Q})$", here <a href="http://www.math.uconn.edu/~alozano/elliptic/finding%20points.pdf" rel="nofollow">http://www.math.uconn.edu/~alozano/elliptic/finding%20points.pdf</a>
can anyone give a very naiv... | Keenan Kidwell | 628 | <p>I would say it's not unreasonable to (loosely) describe <em>algebraic</em> number theory as the study of number fields (number fields are the objects you get when you adjoin roots of polynomials like $X^2+1$ to $\mathbb{Q}$). This is admittedly an incredibly vague description and could be taken in any number of ways... |
128,533 | <p>For the description, I have a simplified problem like the following:</p>
<pre><code>MapAt[f[1, #1], {a, b, c, d}, #2] & @@@ {{1, 2}, {3, 4}}
</code></pre>
<p>will give</p>
<blockquote>
<pre><code>{{a, f[1, 1][b], c, d}, {a, b, c, f[1, 3][d]}}
</code></pre>
</blockquote>
<p>But actually <code>{a, f[1, 1][b], ... | Kuba | 5,478 | <pre><code>Fold[
Function[{data, spec}, MapAt[f[1, #1], data, #2] & @@ spec],
{a, b, c, d},
{{1, 2}, {3, 4}}
]
</code></pre>
<blockquote>
<pre><code>{a, f[1, 1][b], c, f[1, 3][d]}
</code></pre>
</blockquote>
|
3,113,850 | <blockquote>
<p><span class="math-container">$$f(x)=\sum_{n=1}^{\infty}{\frac{x^{n-1}}{n}},$$</span></p>
<p>Prove that <span class="math-container">$f(x)+f(1-x)+\log(x)\log(1-x)=\frac{{\pi}^2}{6}$</span></p>
</blockquote>
<p>In my mind though,I think that this is related to Basel problem<span class="math-container">$\l... | Mirko | 248,138 | <p>The matrix</p>
<p><span class="math-container">$$A=\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix} $$</span></p>
<p>satisfies <span class="math-container">$adj(A)=0$</span> and <span class="math-container">$\det(A)=0$</span>. The inverse does not exist if the determinent is zero.</p>
<p>Th... |
2,459,651 | <p>What additional properties must an operation have besides commutativity so that commutativity along with other properties implies associativity?</p>
<p>Where can I read about such structures?</p>
| paul garrett | 12,291 | <p>This is a reasonable question, but a reasonable answer (in my opinion) basically is negative: commutativity and associativity are not reasonably related, in most natural contexts.</p>
<p>The most disturbing-to-me example is that of Jordan algebras, which are commutative but not associative, despite being reasonably... |
1,429,937 | <p>From <a href="http://www.gottfriedville.net/mathprob/comb-subrect.html" rel="nofollow">this</a> page it can be shown that the number of possible rectangles in and m*n grid can be found by first choosing $2$ lines from a possible $m+1$ to form the vertical sides of the rectangle, and then $2$ from the $n+1$ horizonta... | Clive Newstead | 19,542 | <p>If you restrict yourself to the real numbers, then it is not possible in general.</p>
<p>For example, take $a=1$ and $b=2$. Given any $x \in \mathbb{R}$ we have
$$\sin(x+\pi)=-\sin x$$
but since $\sin 2(x+\pi) = \sin 2x$ and $\cos 2(x+\pi) = \cos 2x$, no function of $\sin 2x$ and $\cos 2x$ can have this property.</... |
2,713,555 | <p>I have a question stating</p>
<blockquote>
<p>Let $A= \{ x_1 , x_2 , x_3 , \ldots , x_n \}$ be a set consisting of $n$ distinct elements. How many subsets does it have? How many proper subsets?</p>
</blockquote>
<p>My thought is that there would be subsets with $1$ element, $2$ elements, $3$ element and so on, u... | gt6989b | 16,192 | <p>I disagree. For example, the number of subsets of size 2 is $$ \binom{n}{2} = \frac{n(n-1)}{2} \ne n-1 $$ and in general you are looking at
$$
\sum_{k=0}^n \binom{n}{k} = 2^n,
$$
which is also easy to see from fundamentals by a simple counting argument -- each of the $n$ elements can be either included or excluded, ... |
4,419,360 | <p>In chapter 9 of Spivak's <em>Calculus</em>, on derivatives, he mentions the "Leibnizian Notation" for the derivative of a function <span class="math-container">$f$</span>, <span class="math-container">$\frac{df(x)}{dx}$</span>. In a footnote on page 155, he writes</p>
<blockquote>
<p>Leibniz was led to thi... | José Carlos Santos | 446,262 | <p>Within that axiomatic, there are no infinitesimal numbers, that is, there is no number <span class="math-container">$\mu>0$</span> such that <span class="math-container">$(\forall n\in\Bbb N):\mu<\frac1n$</span>. That's so because the Archimedean property follows from those axioms. And that property states tha... |
301,106 | <p>In the game connect four with a $7 \times 6$ grid like in the image below, how many game situations can occur?</p>
<p><strong>Rules</strong>:</p>
<blockquote>
<p>Connect Four [...] is a two-player game in which the players first
choose a color and then take turns dropping colored discs from the top
into a se... | Damien Merle | 920,944 | <p>So I've worked on it, but I'm not really qualified to do so. First I wanted to calculate every possibility like 7 then 49 and then it was complicated to avoid redundancies and I was facing another problem, the fact that after 6 pieces played, there was a possibility of having a full columns. Too hard for me. So I de... |
283,473 | <p>I'd like to be able to construct polynomials $p$ whose graphs look like this:</p>
<p><img src="https://i.stack.imgur.com/8Rk3a.jpg" alt="enter image description here"></p>
<p>We can assume that the interval of interest is $[-1, 1]$. The requirements on $p$ are:</p>
<p>(1) Equi-oscillation (or roughly equal, anywa... | achille hui | 59,379 | <p>Starting with any Chebyshev polynomial of $1^{st}$ kind $T_n(x), n > 2$,
and any positive root $a$ of it. Compose $T_n(ax)$ with any polynomial $q(x)$
which is monotonic over $[-1,1]$ that satisfies $q(-1) = -1, q(1) = 1$ and
$q'(-1) = q'(1) = 0$ , then $T_n(a q(x))$ is something you want. For example,
$$
T_n\... |
1,691,685 | <p>The answer for it is $$3 + \sum_{k=1}^n (3+k(k-1)2^{k-2})\frac{(-1)^k}{k!} x^k + o(x^n)$$
Well, I've tried to change every $e^x$ to $1 + x + \frac{x}{2!} + ... + o(x^n)$ and got nothing useful. I know how to get Maclaurin series for polynomials but I don't really know what should I do with fractions and how get that... | Nick Peterson | 81,839 | <p><strong>Hint:</strong> I would start by simplifying:
$$
\frac{x^2+3e^x}{e^{2x}}=x^2e^{-2x}+3e^{-x}.
$$</p>
|
2,019,711 | <p>Correction: For what values of the real number $a$, can
$$
a(x_1^2+x_2^2+x_3^2)+2(x_1x_2+x_1x_3+x_2x_3)
$$
be expressed as sum of the form
$$
\alpha x^2+\beta y^2+\gamma z^2
$$
where $\alpha,\beta,\gamma$ are real numbers? </p>
| Community | -1 | <p>The deep reason is that Newton is a "single point" approach which is <em>unaware of the existence of a root</em> ! It can keep iterating even if there is none, and widly wander.</p>
<p>This contrasts with "two points" methods (such as dichotomy, regula falis or the more sophisticated Brent method) that keep track o... |
2,755,785 | <p>I recently came across a problem in which I had to show that a discrete random variable $X$ has two modes $m_1$ and $m_2$. The information given was that $$\frac{P(X=n)}{P(X=n-1)} = \frac{0.9(n-1)}{n-3}$$</p>
<p>I need to show that this distribution has two modes, $m_1$ and $m_2$. My initial thoughts were to try an... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>$$\tan(\tan^{-1}x+\tan^{-1}y+\tan^{-1}z)=\dfrac{x+y+z-xyz}{1-(xy+yz+zx)}$$</p>
<p>Now we have $$x+y+z=xyz=-\dfrac p1$$</p>
<p>and $xy+yz+zx=\dfrac q1$</p>
<p>But $\dfrac{x+y+z-xyz}{1-(xy+yz+zx)}$ will be undefined if $xy+yz+zx=q=1$</p>
|
3,440,171 | <p>How does one write the Taylor expansion of <span class="math-container">$\log\big(\frac{\sin(x)}x\big)$</span>? It is not defined on <span class="math-container">$x=0$</span>.</p>
| user76284 | 76,284 | <p>According to <a href="http://oeis.org/A046989" rel="nofollow noreferrer">OEIS</a>,</p>
<p><span class="math-container">$$ \log \frac{x}{\sin x} = \sum_{n > 0} x^{2n} \frac{2^{2n-1} (-1)^{n+1} B(2n)}{n (2n)!} $$</span></p>
<p>where <span class="math-container">$B$</span> is the <a href="https://en.wikipedia.org/... |
1,046,792 | <p>Given:
$$\sin(x+iy)=\cos\theta+i\sin\theta$$
To prove:
$$x=\arccos (\sqrt{\sin\theta})$$
How I tried:
$$\begin{align*}
\sin x \cosh y &= \cos\theta \\
\cos x \sinh y &= \sin\theta
\end{align*}$$
Then tried to use logarithm of hyperbolic complex number.</p>
<p>Also various trignometric form manipulation but ... | M. Wind | 30,735 | <p>We start off with two real numbers $x$ and $y$, and consider the complex function $\sin(x+iy)$. As has been pointed out by others in this thread, this is a complex number that can written as $z = \sin(x + iy) = \sin(x)\cosh(y)+i\cos(x)\sinh(y)$. </p>
<p>We are then asked to express this result in terms of a new var... |
199,235 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/9505/xy-yx-for-integers-x-and-y">$x^y = y^x$ for integers $x$ and $y$</a> </p>
</blockquote>
<p>Determine the number of solutions of the equation
$n^m = m^n$
where both m and n are integers.</p>
| robjohn | 13,854 | <p><strong>Hint:</strong></p>
<p>Since $m^n=n^m$, take logs and separate the variables:
$$
\frac{\log(m)}{m}=\frac{\log(n)}{n}
$$
This suggests considering the function $f(x)=\frac{\log(x)}{x}$.</p>
<p>$\hspace{2cm}$<img src="https://i.stack.imgur.com/NFeRg.png" alt="enter image description here"></p>
<p><strong>Ano... |
1,567,473 | <p>$-\frac12·\frac{iz-2z}{z^2}+\frac{-1-i}{2i-2z}=
\frac{\frac{-3}4-\frac12i}{z}$</p>
<p>if $z=a+bi$,
How to find $a$ and $b$?
Thank you.</p>
| Martin R | 42,969 | <p>In your case $x_0 = y_0 = 0$. For any $a > 0, b > 0$, the function
$f(x, y) = y^2 + \cos^2 x$ is defined and Lipschitz continous on
the rectangle $R=\{|x|\leq a, |y|\leq b\}$.</p>
<p>On this rectangle, $$|f(x, y)| \le |y|^2 + |\cos x|^2 \le b^2 + 1$$
with equality for $x = 0$ and $y = b$, so
the supremum is $... |
2,334,305 | <blockquote>
<p>Let $F$ be the distribution function of the continuous-type random variable $X$, and assume that $F(x)=0$ for $x \le 0$ and $0 \lt F(x) \lt1$ for $0 \lt x$. Prove that if $P(X \gt x+y |X \gt x)=P(X \gt y),$ then $F(x)=1-e^{- \lambda x}, 0 \lt x.$</p>
</blockquote>
<p>My attempt: </p>
<p>Let $F(x)=P(... | Jonathan Davidson | 457,443 | <p>Let $F(s) = P(X>s)$ Using conditional probability manipulations, $$F(s) = P(X>s) = P(X > s+t\mid X >t) = \frac{P(X>s+t)}{P(X>t)} = \frac{F(s+t)}{f(t)}$$ yielding the functional equation $$F(s+t) = F(s)F(t)$$ Next we consider the difference quotient $$\frac{F(x+h)-F(x)}{h} = F(x)\left(\frac{F(h)+1}{... |
115,657 | <p>In topology the spheres <span class="math-container">$S^n$</span> are the "simplest" closed manifolds, and they are like "Dirac's delta at <span class="math-container">$n$</span>" for (reduced) cohomology groups. Furthermore they are boundaries of the simplest compact manifolds-with-boundary, i.e... | Piotr Achinger | 3,847 | <p>Probably not, at least treating the question naively, since algebraic geometry focuses on smooth projective varieties, and these have nonzero cohomology (for basically any cohomology theory) in degrees $0, 2, \ldots 2\cdot \dim$: the class of the hyperplane section in $H^2$ has nonzero top cup product with itself. <... |
4,587,608 | <p>The following is the definition of a change-of-coordinate matrix from a textbook I'm using:</p>
<p>Let <span class="math-container">$\beta$</span> and <span class="math-container">$\beta'$</span> be two ordered bases for a finite-dimensional vector space V, and let <span class="math-container">$Q=[I_v]_{\beta}^{\bet... | azif00 | 680,927 | <p>Let <span class="math-container">$u = t$</span> and <span class="math-container">$dv = \frac{-t}{\sqrt{1-t^2}}dt$</span>. Then <span class="math-container">$du=dt$</span> and <span class="math-container">$v = \sqrt{1-t^2}$</span>; so
<span class="math-container">\begin{align}
\int \frac{-t^2}{\sqrt{1-t^2}}dt = \int ... |
696,370 | <p>Is it possible that $R/I$ is a field when $R$ is non-commutative ring with unit and $I$ is a maximal left ideal of $R$? If it is not, can anyone give an example of such $R$ and $I$? Thanks.</p>
| DonAntonio | 31,254 | <p>By Cauchy's Theorem, the additive group of $\;\Bbb F\;$ has an element of order two, meaning</p>
<p>$$a+a=a(1+1)=0\implies 1+1=0$$</p>
<p>and clearly (why?) this is the minimal possible integer with this characteristic...</p>
|
997,116 | <p>If one root of the equation $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ is common and $a \ne b$ then:</p>
<p>The options are as follows:
$$\begin{array}{ll}
(A)\quad& a + b = 0\\
(B)& a + b = -1\\
(C)& a - b = 1\\
(D)& a + b = 1
\end{array}$$</p>
<p>Idk how to solve this, please help me.</p>
| Macavity | 58,320 | <p>Hint: Substitute the common root and subtract. </p>
|
1,330,078 | <p>Given a quadrilateral $MNPQ$ for which $MN=26$, $NP=30$, $PQ=17$, $QM=25$ and $MP=28$ how do I find the length of $NQ$?</p>
| Community | -1 | <p>Use the cosine law to find the two partial angles at $M$. Then you know the total angle and you can find $NQ$.</p>
<p>$$\mu_0=\arccos\left(\frac{MQ^2+MP^2-PQ^2}{2MQ\cdot MP}\right)\\
\mu_1=\arccos\left(\frac{MP^2+MN^2-NP^2}{2MP\cdot MN}\right)\\
NQ=\sqrt{MQ^2+MN^2-2MQ\cdot MN\cos\left(\mu_0+\mu_1\right)}.$$</p>
|
18,691 | <p>Let's say I have a folowing set of data:</p>
<ul>
<li>k = 1 : list of values </li>
<li>k = 3 : list of values </li>
<li>k = 10 : list of values</li>
</ul>
<p>I know that to make a <code>BoxWhiskerChart</code> I have to give it a list of listen of these values as data and ks as labels. </p>
<p>How do I force the o... | Jens | 245 | <p>Since the <code>BoxWhiskerChart</code> doesn't label its x-axis coordinate, I decided to make use of the functionality that already exists in <code>ErrorListPlot</code>. The latter is part of the <code>"ErrorBarPlots"</code> package which has to be loaded explicitly.</p>
<p>First I generate the data, then I extract... |
2,758,338 | <p>I need to find the laurent series and the residue of the following complex function
$$f(z)=(z+1)^2e^{3/z^2}$$
at $z=0$.</p>
<p>Since $e^z=\sum z^n/n!$, then
$$e^{3/z^2}=\sum_{n=0}^\infty \frac{3^n/n!}{z^{2n}}$$
thus
$$f(z)=(z^2+2z+1)\sum_{n=0}^\infty \frac{3^n/n!}{z^{2n}}=\sum_{n=0}^\infty \frac{3^n/n!}{z^{2(n-1)}... | Ashwin Trisal | 343,481 | <p>No, this isn't true. Consider $S_5$ (not minimal but easy). Then consider the groups $\langle (1\,2)\rangle$ and $\langle (3\, 4)\rangle$. Both are isomorphic to $\mathbb{Z}/2\mathbb{Z}$, but they're not equal.</p>
|
300,154 | <p>I am considering an optimization problem of the form:
\begin{equation}
\begin{split}
f(s) &= \min_{X} \mathrm{tr}(C(s)X) \\
&\;\;\;\;\;\;\;\;\;\;\; X \ge 0, \\
&\;\;\;\;\;\;\;\;\;\;\; \mathrm{tr}(A_iX) = a_i, \;\; 1 \le i \le M,
\end{split}
\end{equation}
where the minimization is over $n\times n$ Her... | DSM | 155,380 | <p>Consider the following argument for a slightly changed problem (with <span class="math-container">$s^2$</span> in <span class="math-container">$C(s)$</span> replaced by <span class="math-container">$-s^2$</span>) . Not sure if this would be of any help, but writing it anyway. Note that due to concavity of <span clas... |
4,337,887 | <p>I am solving an exercise:</p>
<p>Let <span class="math-container">$T: V \rightarrow W$</span> be a linear transformation.
<span class="math-container">$V$</span> and <span class="math-container">$W$</span> are finite-dimensional inner product spaces. Prove <span class="math-container">$T^*T$</span> and <span class="... | paw88789 | 147,810 | <p>The number of divisors function, sometimes denoted <span class="math-container">$\tau$</span> (tau) includes <span class="math-container">$1$</span> and <span class="math-container">$n$</span> as divisors also. So for instance <span class="math-container">$\tau(12)=6$</span> (divisors are <span class="math-container... |
2,006,927 | <p>I want to prove that $\cap_{i=1}^{\infty}\left(A_i \cap \left(\cup_{j=1}^{\infty}B_j\right)\right)=\cup_{j=1}^{\infty}\left(\left(\cap_{i=1}^{\infty}A_i\right) \cap B_j\right)$ for sets $A_i,B_j$ and natural numbers $i,j$. If an element $x$ belongs to the left hand side, then $x\in A_1$ and $x\in$ some of $B_j$ and ... | DanielWainfleet | 254,665 | <p>For any sets $I,J$ let $A=\cap_{i\in I}A_i,\; B=\cup_{j\in J}B_j,\;C=\cap_{i\in I}(A_i\cap B),\; D=\cup_{j\in J}(A\cap B_j).$</p>
<p>If $I\ne \emptyset$ then for any $x$ we have $$x\in C\iff \forall i\in I\;(x\in A_i\land x\in B)\iff$$ $$\exists j\in J\; (\;x\in B_j\land (\forall i\in I\;(x\in A_i))\;)\iff$$ $$ \e... |
1,000,938 | <p>$\lim_{X \to \infty} \int_0^Xf(x)^2 > 2(\lim_{X \to \infty}\int_0^Xf(x))^2$</p>
| Christian Blatter | 1,303 | <p>Since you just need one example, consider
$$f(t):=e^{-4t}\qquad(t\geq0)\ .$$
Then
$$f(x):=\int_0^x f^2(t)\>dt={1\over8}\bigl(1-e^{-8x}\bigr)$$
and
$$g(x):=2\left(\int_0^x f(t)\>dt\right)^2={1\over 8}\left(1-e^{-4x}\right)^2\ .$$
It follows that
$${f(x)\over g(x)}=\coth(2x)>1\qquad(x>0)\ .$$</p>
|
1,115,793 | <p>Let's consider a number of linear operators, defined on a finite dimensional complex vector space, which two by two commutes with each other. (the amount of them can be infinite). How to prove that that will have a common eigenvector?</p>
<p>The finite case can be done by induction:
1) $n=2$, $AB=BA$, then let $x$ ... | voldemort | 118,052 | <p>Let's say your vector space is $\mathbb{C^n}$. Then $M_n(\mathbb{C})$, i.e. all $n$ by $n$ matrices are the bounded operators on $\mathbb{C^n}$.</p>
<p>Now, $M_n(\mathbb{C})$ is finite dimensional. So, even if you have infinitely many operators, say $A_1,A_2,\cdots$, there will exist $i_1,i_2,\cdots,i_k$ such that ... |
36,272 | <p>Is there a characterisation for which $x\in\mathbb{R}$ the value $\arctan(x)$ is a rational multiple of $\pi$? </p>
<p>Or reformulated: What is the "structure" of the subset $A\subseteq\mathbb{R}$ which fulfils
$$ \arctan(x) \in \pi\mathbb{Q} \Leftrightarrow x\in A$$
for all $x\in\mathbb{R}$?</p>
| Philip Thomas | 114,952 | <p>It is easy to show that for
$$\frac{\pi}{2^{n+1}}=\arctan x,$$
where $n=\left\{0,1,2,3,\ldots\right\}$, the argument $x$ is always an irrational number. Therefore $\arctan x$ cannot be a rational multiple of $\pi$ at $x\in\mathbb{R}$ for this specific case.</p>
|
4,494,199 | <p>This is a problem from a past qualifying exam in complex analysis. I'm working through these to study for my own upcoming qual. For this question, I think my proof is fairly straightforward, but I'd like to know whether or not it is correct and complete. I'm also interested in other ways of answer the question. Than... | Arctic Char | 629,362 | <p>I guess this is a typo, they meant to say</p>
<p><span class="math-container">$$ 1-a \ge b-a,$$</span></p>
<p>which is true from</p>
<p><span class="math-container">$$ (1-a)(1+a) \ge b(b-a)$$</span>
since <span class="math-container">$1+a \ge 1 \ge b$</span>.</p>
<p>(Of course, <span class="math-container">$1-a\ge b... |
158,978 | <p><a href="https://i.stack.imgur.com/5nKVV.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5nKVV.jpg" alt="Explore the Fourier Series for a square wave"></a></p>
<pre><code>f (t) = (4/pi) Sum[(1/n) sin (2 pi (f (t))), {n, 1, Infinity}]
</code></pre>
| m_goldberg | 3,066 | <p>First you need to learn the proper syntax for writing Mathematica expressions. When the sum is written properly as</p>
<pre><code>f[t_] = (4/Pi) Sum[(1/n) Sin[2 Pi n t], {n, 1, ∞, 2}]
</code></pre>
<blockquote>
<p><code>(2 I (ArcTanh[E^(-2 I π t)] - ArcTanh[E^(2 I π t)]))/π</code></p>
</blockquote>
<p>then the ... |
414,400 | <p>Let $ \mathbb{Z}[i]$ denote the ring of the Gaussian intergers. For which of the following value of n is the quotient ring $ \mathbb{Z}[i]/n\mathbb{Z}[i]$ an integral domain?</p>
<p>$ a. 2$</p>
<p>$ b. 13$</p>
<p>$ c. 19$</p>
<p>$ d. 7$</p>
<p>I'm doubtful with the following attempt I made.</p>
<ul>
<li><p><st... | zacarias | 35,464 | <p>The following theorem is well known (see a book on algebraic number theory)</p>
<p>Theorem: Let $p$ be a rational prime (that is, a prime in $\mathbb Z$). Then, $p$ is a prime in the Gaussian integers $\mathbb Z[i]$ if and only if $p\equiv 3 \pmod 4$.</p>
<p>Since $\mathbb Z[i]/n \mathbb Z[i]$ is an integral domai... |
874,607 | <blockquote>
<p>There were 10 questions on a test. A student gets 5 points for every correct answer and 3 points for every partially correct answer. If the student got 19 points, how many correct and partial answers did they have?</p>
</blockquote>
<p>To solve the problem I express the total number of points as the ... | DeepSea | 101,504 | <p>$5x + 3y = 19$. So $5x \leq 19$, thus $x \leq 3$. Check $x = 2$, and $y = 3$ is the only integer values solution. </p>
|
318,299 | <blockquote>
<p>Let <span class="math-container">$U$</span> be an open set in <span class="math-container">$\mathbb R$</span>. Then <span class="math-container">$U$</span> is a countable union of disjoint intervals. </p>
</blockquote>
<p>This question has probably been asked. However, I am not interested in just get... | Guldam | 37,753 | <p>$\mathbb{R}$ with standard topology is second-countable space.</p>
<p>For a second-countable space with a (not necessarily countable) base, any open set can be written as a countable union of basic open set.</p>
<p><a href="https://math.stackexchange.com/questions/1395714/open-set-in-a-second-countable-space-which... |
318,299 | <blockquote>
<p>Let <span class="math-container">$U$</span> be an open set in <span class="math-container">$\mathbb R$</span>. Then <span class="math-container">$U$</span> is a countable union of disjoint intervals. </p>
</blockquote>
<p>This question has probably been asked. However, I am not interested in just get... | Asinomás | 33,907 | <p>The balls with radii $\frac{1}{n}$ and center at a rational number form a basis for the euclidean topology. This family is countable as it is a countable union of countable sets. We have found a countable basis, so we are done.</p>
|
318,299 | <blockquote>
<p>Let <span class="math-container">$U$</span> be an open set in <span class="math-container">$\mathbb R$</span>. Then <span class="math-container">$U$</span> is a countable union of disjoint intervals. </p>
</blockquote>
<p>This question has probably been asked. However, I am not interested in just get... | CopyPasteIt | 432,081 | <p>The balls with radii $\frac{1}{n}$ and center at a rational number form a basis for the euclidean topology. This family is countable since $\mathbb N \times \mathbb Q \equiv \mathbb N$ and we have a countable basis $(B_\lambda)_{ \, \lambda \in \mathbb N \times \mathbb Q}$ of open intervals for $\mathbb R$.</p>
<p... |
183,768 | <p>Prove convergence\divergence of the series:
$$\sum_{n=1}^{\infty}\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$$</p>
<p>Here is what I have at the moment:</p>
<p><strong>Method I</strong></p>
<p>My first way uses a result that is related to <strong><a href="http://en.wikipedia.org/wiki/Wallis_product" rel=... | robjohn | 13,854 | <p>$$
\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}=\frac{(2n)!}{2^{2n}n!^2}\tag{1}
$$
Using Stirling's Formula, we get that
$$
\frac{(2n)!}{2^{2n}n!^2}\sim\frac1{\sqrt{\pi n}}\tag{2}
$$
By the $p$-test,
$$
\sum_{n=1}^\infty \frac1{n^p}\tag{3}
$$
diverges for $p\le1$,
$$
\sum_{n=1}^\infty\frac{1\cdot3\cdot5... |
2,056,209 | <p>We know that the union of countably many countable sets is countable.
What can se say about the union of infinitely many countable sets and its cardinality?
Thanks :)</p>
| BrianO | 277,043 | <p>The cardinality of a union of an infinite collection $\mathscr{S}$ of countably infinite sets is infinite, and is at most equal to the cardinality $\lvert\mathscr{S}\rvert$ of that collection. </p>
<p>That is: </p>
<blockquote>
<p>Suppose $\mathscr{S}$ is an infinite set of countably infinite sets. Then
$$
\al... |
768,925 | <p><img src="https://i.stack.imgur.com/SHRqU.png" alt="enter image description here"></p>
<p>This is quite a tricky question for me, but this is how far I got:<img src="https://i.stack.imgur.com/UV0ou.jpg" alt="enter image description here"></p>
<p>My drawing may not be precise, but I do know the points of tangency. ... | Justin Lindberg | 212,486 | <p>The figure is self-similar; it contains a smaller version of itself. The ratio of the radius of middle circle to that of the largest circle is the same as ratio of the radius of the smallest circle to that of the middle circle. That is, if we let $r$ be the radius of the middle circle, then
$$\frac r{99}=\frac{19}... |
1,824,638 | <p>The figure shows a piece of string tied to a circle with a radius of one unit. The string is just long enough to reach the opposite side of the circle. Find the area of the region, not including the circle itself that is traced out when the string is unwound counterclockwise and continues counterclockwise until it r... | John Molokach | 90,422 | <p>I tried graphing the equations given in the accepted answer by Doug M, but they do not seem to trace out the path of the end of the string as it is being unwound (although his method does gives the correct answer). The way I approached the problem was to copy the equations (5) and (6) from the page <a href="http://... |
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | Paolo Aceto | 5,001 | <p>Rolfsen - Knots and Links</p>
|
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | Spencer | 4,281 | <p>Leon Simon - Lectures on Geometric Measure Theory</p>
|
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | hce | 2,781 | <p>Adams - Lectures on Lie Groups</p>
|
2,031,842 | <p>If 3 is the remainder when dividing $P(x)$ with $(x-3)$, and $5$ is the remainder when dividing $P(x)$ with $(x-4)$, what is the remainder when dividing $P(x)$ with $(x-3)(x-4)$?</p>
<p>I'm completely puzzled by this, I'm not sure where to start...</p>
<p>Any hint would be much appreciated. </p>
| Bill Dubuque | 242 | <p>Notice $\ p\, = 3+(x\!-\!3)q\,\ $ by $\,\ p(3) = 3$</p>
<p>$5 = p(4) = 3+q(4)\,\Rightarrow\, {\color{#c00}2}\, =\, q(\color{#0a0}4) \, $ </p>
<p>Hence $\, p = 3 + (x\!-\!3)(\underbrace{\color{#c00}2+(x\!-\!\color{#0a0}4)r}_{\Large q}\!)\, =\, 2x\!-\!3\, +\, (x\!-\!3)(x\!-\!4)r$</p>
|
1,314,992 | <p>What operations do I need to perform the following conversion?</p>
<p>$$
\frac{\partial ^2y}{\partial x\partial z} \mapsto \frac{\partial ^2y}{\partial x\partial t}
$$</p>
| Intrepid Traveller | 244,431 | <p>Hint: Use the chain rule by changing your z variable to t</p>
|
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