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 |
|---|---|---|---|---|
80,078 | <p>Given $f$:</p>
<p>$$
f(x) = \begin{cases}
\frac1{x} - \frac1{e^x-1} & \text{if } x \neq 0 \\
\frac1{2} & \text{if } x = 0
\end{cases}
$$</p>
<p>I have to find $f'(0)$ using the definition of derivative (i.e., limits). I already know how to differentiate and stuff, b... | Tapu | 17,142 | <p>Please note the formula $(e^h.f)'=e^h(f+f')$.</p>
<p>Now your limit </p>
<p>$$\lim_{h\to0}\frac{e^h(2-h)-h-2}{2(e^h.h^2-h^2)}\quad(=\frac{0}{0})$$$$\lim_{h\to0}\frac{1}{2}\frac{e^h(1-h)-1}{e^h.(h^2+2h)-2h}\quad(=\frac{0}{0})$$$$\lim_{h\to0}\frac{1}{2}\frac{e^h(-h)}{e^h.(h^2+4h+2)-2}\quad(=\frac{0}{0})$$$$\... |
3,845,602 | <p>I was reading Dummit and Foote and encountered the following statement: any two elements in <span class="math-container">$S_n$</span> are conjugate if and only if they have the same cycle types.</p>
<p>However, I am able to produce a counter example:</p>
<p>Let <span class="math-container">$(1 2 3)$</span> and <span... | Shaun | 104,041 | <p>You have conjugated <span class="math-container">$(456)(78)$</span> <em>by</em> <span class="math-container">$(123)$</span>, not shown that they are conjugate <em>with</em> each other.</p>
<p>For example, the conjugate of <span class="math-container">$(456)(78)$</span> <em>by</em> <span class="math-container">$(45)$... |
152,626 | <p>Is there any simple way of computing the following sum?</p>
<p>$$\sum_{k=1}^\infty \frac1{k\space k!}$$</p>
| Ayman Hourieh | 4,583 | <p>The <a href="http://en.wikipedia.org/wiki/Exponential_integral">exponential integral</a> function can be written as:</p>
<p>$$
\mathrm{Ei}(x) = \gamma + \log|x| + \sum_{k=1}^{\infty} \frac{x^k}{k\; k!}
$$</p>
<p>Plug $x = 1$ to get:</p>
<p>$$
\sum_{k=1}^{\infty} \frac{1}{k\; k!} = \mathrm{Ei}(1) - \gamma
$$</p>
... |
3,172,149 | <p>Let <span class="math-container">$B^H(t)$</span> be a fractional Brownian motion with Hurst parameter <span class="math-container">$H\in (0,1)$</span>. We define fractional Gaussian noise as <span class="math-container">$X(t)=B^H(t+1)-B^H(t)$</span>. We know the fBm has covariance <span class="math-container">$R(s,t... | steven | 422,148 | <p>Hint:
<span class="math-container">$\mathbb{E}\left[\left(B_{t}^{H}-B_{s}^{H}\right)\left(B_{u}^{H}-B_{v}^{H}\right)\right]=\frac{1}{2} \mathbb{E}\left[\left(B_{t}^{H}-B_{v}^{H}\right)^{2}+\left(B_{s}^{H}-B_{u}^{H}\right)^{2}-\left(B_{t}^{H}-B_{u}^{H}\right)^{2}-\left(B_{s}^{H}-B_{v}^{H}\right)^{2}\right]=\frac{1}{2... |
29,143 | <p>In what context should I use $=$ and $\equiv$?</p>
<p>What is the precise difference?</p>
<p>Thanks!</p>
<p>(I wasn't sure what to tag this with, any suggestions?)</p>
| Bill Dubuque | 242 | <p>Consider Fermat's little theorem: $\rm\ a^p\ \equiv\ a\ \ (mod\ p)\ $ for all $\rm\ a,\ p\in \mathbb Z\:,\:\ p\:$ prime. This <em>congruence</em> can also be written as an <em>equality</em> in the ring $\rm\:\mathbb Z/p\:,\: $ e.g. $\:$ as $\rm\ \bar a^{\:p}\ =\ \bar a\ $ in $\rm\:\mathbb Z/p\:,\:$ where $\rm\:\bar ... |
27,126 | <p>$$e^{\pi i} + 1 = 0$$</p>
<p>I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us? </p>
<p>Best that I can figure out is that it just emphasizes that the various definitions mathematicians have provided for non-intuitive operations (com... | S. Carnahan | 121 | <p>The answers so far give interpretations of the exponential as a limit of discrete approximations. An alternative interpretation is that any continuous map that takes addition to multiplication on the complex line and takes reals to reals has a purely imaginary kernel isomorphic to the integers. The constant $e$ ar... |
1,733,439 | <p>Trying to prove some uncorrelated things, I came across the following identity:
$$\binom{m}{n}=\sum_{k=0}^{\lfloor n/2 \rfloor} 2^{1-\delta_{k,n-k}} \binom{m/2}{k} \binom{m/2}{n-k}, $$
where $\delta_{i,j}$ is the Kronecker delta, equal to 1 if $i=j$ and vanishing otherwise.
This identity seems to hold for every $m$ ... | Patrick Stevens | 259,262 | <p>It's true. The number of ways we can pick $n$ things from $m$ is:</p>
<p>Divide the $m$ things into two half-sized chunks. Then we need to do one of the following:</p>
<ul>
<li>pick at most $n/2$ things out of the first chunk (say we pick $k$ from the first chunk), and the remaining $n-k$ things out of the second ... |
1,733,439 | <p>Trying to prove some uncorrelated things, I came across the following identity:
$$\binom{m}{n}=\sum_{k=0}^{\lfloor n/2 \rfloor} 2^{1-\delta_{k,n-k}} \binom{m/2}{k} \binom{m/2}{n-k}, $$
where $\delta_{i,j}$ is the Kronecker delta, equal to 1 if $i=j$ and vanishing otherwise.
This identity seems to hold for every $m$ ... | Hypergeometricx | 168,053 | <p>Vandermonde Identity gives
$$\sum_{k=0}^nu_k=\sum_{k=0}^n\binom {m/2}k\binom {m/2}{n-k}=\binom mn\qquad\qquad (1) $$
<strong>If $n$ is even</strong>:<br>
then number of terms is odd, with the middle term being the $n/2$-th, with symmetrical terms on both sides of this, as $k, n-k$ are symmetrical about $n/2$, i.e.
... |
19,399 | <p>I am to give the following for an interview:</p>
<p>"a short 7–10-minute teaching demonstration on logarithms. Please consider this as your first 10 minutes of introducing logarithms as if you have not previously mentioned the word to this class. Treat this as closely to what you would do during an in-person cl... | Carser | 219 | <p>I introduce logarithms as answering the question "to what power do I raise this base to get this result?" Ideally I would review exponents beforehand, but 7 minutes is no time at all. Presumably these students will be familiar with exponents, so you can pose problems like
<span class="math-container">$$2^... |
19,399 | <p>I am to give the following for an interview:</p>
<p>"a short 7–10-minute teaching demonstration on logarithms. Please consider this as your first 10 minutes of introducing logarithms as if you have not previously mentioned the word to this class. Treat this as closely to what you would do during an in-person cl... | SlyPuppy | 625 | <p>I introduce logarithms via a number guessing game. I wrote a run-of-the-mill Python program to do this. It asks for an integer, <span class="math-container">$n>1$</span>, and the class has to guess the random integer between 1 and <span class="math-container">$n$</span>. Each time they do not get the answer right... |
4,226,480 | <p>Assume <span class="math-container">$ u_{0}:\mathbb{R}\to\mathbb{R} $</span> is continuous and such that <span class="math-container">$ \lim_{t\to-\infty}u_{0}\left(t\right)=\lim_{t\to\infty}u\left(t\right)=L $</span>.</p>
<p>Prove that there exists unique <span class="math-container">$ u:\left\{ Im\left(z\right)\ge... | humanStampedist | 474,469 | <p>Let me add to Martin Rs answer: Your <span class="math-container">$g$</span> is actually a bit better. It is a Moebius transform, hence it preserves harmonicity, see e.g. <a href="https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.010/DallAcqua/DaSwChile.PDF" rel="nofollow noreferrer">https://www.uni-ulm.de/... |
786,596 | <p>So I'm trying to solve this practice exam question, </p>
<blockquote>
<p>Let $G$ be a planar graph with at least two edges and does not contain $K_{3}$ as a subgraph. Prove that $|E|\leq 2|V|-4$.</p>
</blockquote>
<p>Now I started doing this by induction, but it seems to me like the base-case is a counter-examp... | Geoff Robinson | 13,147 | <p>Hint: $11^{n+1} - 2^{n+1} = 11(11^{n}-2^{n}) +2^{n}(11-2).$</p>
|
786,596 | <p>So I'm trying to solve this practice exam question, </p>
<blockquote>
<p>Let $G$ be a planar graph with at least two edges and does not contain $K_{3}$ as a subgraph. Prove that $|E|\leq 2|V|-4$.</p>
</blockquote>
<p>Now I started doing this by induction, but it seems to me like the base-case is a counter-examp... | Adi Dani | 12,848 | <p>Hint:
$$11^{n+1}-2^{n+1}=11\cdot11^n-2\cdot2^n=$$
$$=9\cdot11^n+2\cdot11^n-2\cdot2^n=2(11^{n}-2^{n})+9\cdot11^n$$
First part is true from assumption and second part has 9 as a factor</p>
|
1,258,198 | <p>Suppose $a > 1$. I want to compare
$$\int_0^{\infty} \frac{e^{-ax}}{1+x^2}\,\,dx$$ and $$\int_0^{\infty} \frac{e^{-2ax}}{1+x^2}\,\,dx$$</p>
<p>My instinct suggests that after a certain value of $a$, $$\int_0^{\infty} \frac{e^{-2ax}}{1+x^2}\,\,dx < e^{-a}\int_0^{\infty} \frac{e^{-ax}}{1+x^2}\,\,dx$$</p>
<p>bu... | xpaul | 66,420 | <p>I think this is not true. For example, let $a=2$. Then
$$ \int_0^\infty\frac{e^{-4x}}{1+x^2}dx\approx 0.229193> e^{-2}\int_0^\infty\frac{e^{-2x}}{1+x^2}dx\approx 0.0540016. $$</p>
|
1,559,946 | <p>Can anyone solve this?</p>
<p>Find the sum of the series $1 + \frac{1}{2} +\frac{1}{3} + \frac{1}{4} + \frac{1}{5}+ \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \cdots,$ where the denominators are of the form $(2^i) (3^j)(5^k)$?</p>
<p>The test came with the next answer choices:</p>
<p>a) $\frac{7}{2}$</p>
<p>b) $... | Henno Brandsma | 4,280 | <p>Consider $$(1 + \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2^n} + \ldots)(1 + \frac{1}{3} + \frac{1}{9} + \ldots + \frac{1}{3^n} + \ldots)(1 + \frac{1}{5} + \frac{1}{25} + \ldots + \frac{1}{5^n} + \ldots)$$</p>
|
78,617 | <p>I'm sorry I'm French so the subject may not be properly translated, but here's my try:</p>
<p>A goat lives in a rectangular place. She's tied to the point P. The length of the row is 8 meters. The problem is that she can eat flowers: it's the shaded area. The farmer doesn't want the goat to eat the flowers, so he h... | N. S. | 9,176 | <p>Wouldn't be extending the lower edge of the hut by $1+\epsilon$ be enough?</p>
|
3,793,268 | <p>How to prove that <span class="math-container">$x(t) = \cos{(\frac{\pi}{8}\cdot t^2)}$</span> aperiodic?</p>
<p>My process was as follows:</p>
<p><span class="math-container">$x(t+T)= \cos{(\frac{\pi(t+T)^2}{8})}$</span>.</p>
<p>So, <span class="math-container">$T^2 + 2tT -16=0$</span> which seems periodic to me...<... | J.G. | 56,861 | <p>Another proof by contradiction: since <span class="math-container">$\cos\frac{\pi T^2}{8}=1$</span>, some <span class="math-container">$n\in\Bbb N$</span> satisfies <span class="math-container">$T=4\sqrt{n}$</span>, whence<span class="math-container">$$-1=\cos\frac{\pi(T+\pi)^2}{8}=\cos\frac{\pi(16n+8\pi\sqrt{n}+\pi... |
1,537,761 | <blockquote>
<p>Let $n$ be positive number, if $a \equiv b \pmod{2n}$, prove that
$a^2 \equiv b^2 \pmod{4n}$.</p>
</blockquote>
<p>By the congruence in hypothesis, we have $a-b = 2nk$ where $k$ is an integer.
Then $a = b+2nk$ and $a^2 = b^2+4n^2k^2+4knb$. From this we get $a^2-b^2 = 4kn(kn+b)$.</p>
<p>Now I have ... | Mankind | 207,432 | <p>Here's an alternative proof. Assume that $2n|(a-b)$. Then $a$ and $b$ are either both even or both odd, so $2|(a+b)$. Since $a^2-b^2 = (a-b)(a+b)$, we have that $4n|(a^2-b^2)$.</p>
|
195,176 | <p>Find the values of the real constants $c$ and $d$ such that</p>
<p>$$\lim_{x\to 0}\frac{\sqrt{c+dx}-\sqrt{3}}{x}=\sqrt{3}$$</p>
<p>I really have no clue how to even get started.</p>
| F'x | 3,406 | <p>I don't know a way Mathematica is going to solve that directly for you, but it can help you understand what happens. First, let's ask it what the general expression for the limit is:</p>
<pre><code>Limit[(Sqrt[c + d*x] - Sqrt[3])/x, x -> 0]
</code></pre>
<p>The answer is:</p>
<pre><code>DirectedInfinity[-Sqrt[... |
121,645 | <p>I have a (presumably simple) Laplace Transform problem which I'm having trouble with:</p>
<p>$$\mathcal L\big\{t \sinh(4t)\big\} = ?$$</p>
<p>How would I go about solving this? Would you please show working if possible, or alternatively point me in the right direction regarding how to go about solving this?</p>
<... | Rudy the Reindeer | 5,798 | <p><a href="http://www.maths.manchester.ac.uk/~kd/ma2m1/laplace.pdf" rel="nofollow">These notes here</a> look good.</p>
<p>The definition of the Laplace transform is $\mathcal{L}(f)(s) := \int_0^\infty e^{-st} f(t) dt$ and for $\sinh$ the following holds: $\sinh x = \frac12 (e^x -e^{-x})$. Now you put these two things... |
1,724,881 | <p>I'm having a bit of trouble with this problem:</p>
<p>Let $β=\{e^{2x}, xe^{2x}, e^{x}\}$ and define $V=\mbox{span}(\beta)$. Let $T=D-2$ where $D=d/dx$. Show that $\beta$ is a Jordan basis for $T$.</p>
<p>How do I show that $\{e^{2x}, xe^{2x}, e^{x}\}$ is a Jordan basis for $D-2$?</p>
| quid | 85,306 | <p>As said in a comment what needs to be decide is if this basis is formed by a chain of generalized eigen-vectors. </p>
<p>To this end we check the effect of $T$ on the vectors. </p>
<ul>
<li><p>$T(e^{2x}) = 0$, this is does an eigenvactor to the eigenvalue $0$. </p></li>
<li><p>$T(xe^{2x}) = e^{2x}$ and thus $T^2... |
1,722,587 | <p>Those who know golden ratio $\phi$ (phi) constant, know for sure that it is an interesting constant. It is roughly $\phi=1.618034...$ . It is present almost everywhere in nature and it has many very interesting properties.</p>
<p>One of the properties of $\phi$ is: $$\phi^2=\phi+1$$ Is there a constant like $\phi$ ... | Anixx | 2,513 | <p>If you consider <a href="https://en.wikipedia.org/wiki/Split-complex_number" rel="nofollow noreferrer">split-complex numbers</a> or tessarines, then there are four solutions:</p>
<p><span class="math-container">$$\frac{1+\sqrt{5}}2, \frac{1-\sqrt{5}}2, \frac{1+j\sqrt{5}}2, \frac{1-j\sqrt{5}}2$$</span></p>
<p>The las... |
2,041,610 | <p>Let's say I have,</p>
<p><img src="https://i.stack.imgur.com/jRw96.jpg" alt="enter image description here"></p>
<p>Now I have to find the angle CBA. Given that we know just 26 given above.</p>
| Junkai Dong | 366,631 | <p>Hint: $\angle ACB$ is 90 degrees because AB is the diameter of the circle.</p>
|
1,305,151 | <p>I want to prove this without using any of the properties about the field of algebraic numbers (specifically that it is one). Essentially I just want to find a polynomial for which $\cos\frac{2\pi}{n}$ is a root.</p>
<p>I know roots of unity and De Moivre's theorem is clearly going to be important here but I just ca... | robjohn | 13,854 | <p><a href="http://en.wikipedia.org/wiki/Euler%27s_formula" rel="noreferrer">Euler's Formula</a> implies
$$
\left[\cos\left(\frac{2\pi}n\right)+i\sin\left(\frac{2\pi}n\right)\right]^n=1\tag{1}
$$
The <a href="http://en.wikipedia.org/wiki/Binomial_theorem" rel="noreferrer">Binomial Theorem</a> says
$$
\begin{align}
1
&a... |
58,306 | <p>Let $X$ be a topological space, and let $\mathscr{F}, \mathscr{G}$ be sheaves of sets on $X$. It is well-known that a morphism $\varphi : \mathscr{F} \to \mathscr{G}$ is epic (in the category of sheaves on $X$) if and only if the induced map of stalks $\varphi_P : \mathscr{F}_P \to \mathscr{G}_P$ is surjective for e... | Akhil Mathew | 536 | <p>Let $M$ be a smooth manifold, and consider the sheaf of closed 1-forms. There is a surjection from the sheaf of smooth functions to the sheaf of closed 1-forms (namely, the exterior derivative, $f \mapsto df$), which is surjective in the category of sheaves (by the Poincare lemma), but which is in general not surjec... |
58,306 | <p>Let $X$ be a topological space, and let $\mathscr{F}, \mathscr{G}$ be sheaves of sets on $X$. It is well-known that a morphism $\varphi : \mathscr{F} \to \mathscr{G}$ is epic (in the category of sheaves on $X$) if and only if the induced map of stalks $\varphi_P : \mathscr{F}_P \to \mathscr{G}_P$ is surjective for e... | Tomo | 62,940 | <p>Here is a simple 'hands-on' example. Let $X=\{A,B,C\}$ be the three-point space with open sets $\{A,B,C\}$, $\{A,B\}$, $\{B,C\}$, $\{B\}$, and $\emptyset$, and let $\mathscr F$ be the sheaf of abelian groups on $X$ 'generated' by the $\mathbf Z$-valued functions $f\in\mathscr F(\{A,B\})$ and $g\in\mathscr F(\{B,C\})... |
1,234,500 | <p>I am a student in 12th grade and am fond of mathematics. I enjoy reading mathematics but when it comes to problems I just get completely stuck. Its not that I don't understand the problem but often don't know how to go about tackling it. When I see the solution, often I understand it perfectly but arriving at that s... | MCT | 92,774 | <p>The parabola $x^2 + 4x + 3 = (x+3)(x+1)$ is upward facing and has roots at $x = -3, x = -1$, so it is negative in $(-3, -1)$. Break it into cases. One of these cases has two solutions, the other has exactly one (i.e. discriminant = 0).</p>
<p>Case 1: $-3 \leq x \leq -1$. The equation is $-x^2 - 4x - 3 - mx + 2m = 0... |
2,631,733 | <blockquote>
<p>Let $E = \mathcal{C}^0([a,b],\mathbb{R})$, provided with the $||\cdot ||_{\infty}$ norm. Let $\phi: \mathbb{R} \rightarrow \mathbb{R}$ that is $\mathcal{C}^1$. Show that the function given by $\Psi:E \rightarrow \mathbb{R}$:
$$ \Psi(f) = \int^{b}_{a}\phi(f(x))dx $$ is differentiable.</p>
</blockquo... | copper.hat | 27,978 | <p>Since $\phi$ is $C^1$ we can write
$\phi(t+h) -\phi(t)-\phi'(t)h = \int_0^1 (\phi'(t+sh)-\phi'(t)) h ds$. Since
$\phi'$ is uniformly continuous, for any $\epsilon >0$ we can find some $\delta>0$ such that
if $|h| < \delta$, then
$|\phi(t+h) -\phi(t)-\phi'(t)h| \le \epsilon |h|$ for all $x$.</p>
<p>In parti... |
2,803,069 | <p>Prove:
$$|z_1+z_2|\ge\frac{1}{2}(|z_1|+|z_2|)*|\frac{z_1}{|z_1|}+\frac{z_2}{|z_2|}|$$</p>
<p>Except inserting $(a+bi)$ instead of $z$ (which I think will lead me to a dead end), I really don't have a good idea how to confront this exercise, any tips or hints?</p>
<p>I'm not student yet and I just started learning ... | T. J. McIntee | 744,738 | <p>Both are general impossibility theorems that can be presented in the form "No voting system meets all of X criteria." They are very closely related to each other.</p>
<p>If you read Satterthwaite's 1975 paper, you'll see the degree to which the two theorems are closely related: Satterthwaite proves specifically tha... |
2,298,971 | <p>Using Wolframalpha, the limit is $\sqrt{e}$. Now because we already know $\lim\limits_{n \rightarrow \infty}(1+\frac{1}{n})^n = e$ </p>
<p>the term has to be ~$(1+\frac{1}{n})^{\frac{n}{2}}$. My attempt to transform the equation fails though, because im not sure how to get rid of the $-\frac{1}{2}$:</p>
<p>$(1+\fr... | Michael Rozenberg | 190,319 | <p>$$\lim\limits_{n\rightarrow\infty}\left(1+\frac{1}{2n+1}\right)^n=\lim\limits_{n\rightarrow\infty}\left(\left(1+\frac{1}{2n+1}\right)^{2n+1}\right)^{\frac{n}{2n+1}}=e^{\frac{1}{2}}$$</p>
|
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | Joseph O'Rourke | 511 | <p>It may help to compare several exponential curves,
e.g., $y=5^x$, $y=e^x$, and $y=2^x$,
<hr />
<a href="https://i.stack.imgur.com/oXjIj.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/oXjIj.jpg" alt="ThreeExps"></a>
<hr />
and calc... |
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | Daniel Hast | 3,505 | <p>Here's an approach that can be illustrated with a simple picture, and also gives an actual bound on the derivative of $e^x$ as being, at the very least, <em>fairly close</em> to $e^x$ (which makes it much more plausible that they're in fact equal).</p>
<p>Compare the slope of secant line through $(x - 1, e^{x-1})$ ... |
2,571,843 | <p>Let $A = \mathbb{R}\setminus\mathbb{Q}$. Then it can be shown that $A + A = \mathbb{R}$, for example by using the fact that $A$ is $G_{\delta}$. Let $q\in\mathbb{Q}$. This means that $q = r_1+r_2$ where $r_1,r_2$ are irrational numbers. But this is not too surprising, as every rational can be written $q = \left(\fra... | Mohit | 238,205 | <p>Yes, the set of solutions to your equations are:</p>
<p>\begin{align}
q_2 = q - q_1 \\
r = r_1 - q_1
\end{align}</p>
<p>where $q_1$ is a free variable allowed to be any rational number. </p>
|
1,786,514 | <p>Let $S$ be a set containing $n$ elements and we select two subsets: $A$ and $B$ at random then the probability that $A \cup B$ = S and $A \cap B = \varnothing $ is?</p>
<p>My attempt</p>
<p>Total number of cases= $3^n$ as each element in set $S$ has three option: Go to $A$ or $B$ or to neither of $A$ or $B$</p>
<... | copper.hat | 27,978 | <p>Let $\Sigma = \{(A,B) | A,B \subset S \}$, we see that $|\Sigma| = 2^n 2^n = 4^n$. I am assuming that each pair $(A,B)$ is equipropable.</p>
<p>Let $C_1 = \{(A,B)| A \cup B = S \}$.</p>
<p>The sets $E_A = \{(A,B)| A \cup B = S \} = \{(A,B)| A^c \subset B \}$
form a partition of $C_1$ and we see that $|E_A| = 2^{|A... |
1,786,514 | <p>Let $S$ be a set containing $n$ elements and we select two subsets: $A$ and $B$ at random then the probability that $A \cup B$ = S and $A \cap B = \varnothing $ is?</p>
<p>My attempt</p>
<p>Total number of cases= $3^n$ as each element in set $S$ has three option: Go to $A$ or $B$ or to neither of $A$ or $B$</p>
<... | samerivertwice | 334,732 | <p>Right sorry I have a moment free again... The number of "successful" outcomes is the number of ways of composing two disjoint sets whose union is exhaustive over $S$. The probability is that divided by the total number of ways of distributing the elements to A, B, A & B, or neither - which gives $4^n$ total pos... |
3,706,190 | <p>Well it is pretty wierd for me to see this question, the function already is power series isn't it?
Am I missing the purpose of the excersize?</p>
| reuns | 276,986 | <p><span class="math-container">$1$</span> is a root of <span class="math-container">$x^n-1$</span> thus it is irreducible iff <span class="math-container">$n=1$</span>.</p>
<p><span class="math-container">$x^n+1$</span> is irreducible iff <span class="math-container">$f=[\Bbb{F}_p(\zeta_{2n}):\Bbb{F}_p]=n$</span> iff... |
1,273,353 | <p>Suppose the Roulette table has 37 numbers (European Roulette table). During 37 spins, I always do the same bet: 35 numbers straight (35 chips in 35 different numbers).
Then:</p>
<ol>
<li>the probability of winning the 37 consecutive spins is $(\frac{35}{37})^{37}\approx 0.1279$,</li>
<li>the probability of losing ... | Vincenzo Oliva | 170,489 | <p>Let $L=\lim\limits_{n\to\infty}p_n.$ If $b\ge a+1$ the following inequalities hold: $$\require\cancel 0<p_n\le \frac{a\cancel{(a+1)\cdots(a+n)}}{\cancel{(a+1)(a+2)\cdots(a+n)}(a+1+n)}=\frac{a}{a+1+n}\to0,$$ and thus $L=0$ by the squeeze theorem.</p>
<p>For the case $b<a+1$, note that since for any positive $m... |
617,598 | <p>Does anyone know any examples of $f$'s for which $-\triangle u(x) = k f(u(x))$ has an explicit solution (i.e. a formula for the solution, not a numerical approximation scheme) in terms of $k$?</p>
<p>I am interested in examples where $f\geq 0$ is neither constant nor linear. Optimally I would be interested in a smo... | Salech Alhasov | 25,654 | <p>$$\frac{x^2+16y^2}{x} \cdot \frac{x^2+4xy}{x-4y}=\frac{x^2+16y^2}{x} \cdot \frac{x(x+4y)}{x-4y}=(x^2+16y^2)\cdot \frac{(x+4y)}{x-4y}$$</p>
<p>$$=(x+i4y)(x-i4y) \frac{x+4y}{x-4y}$$</p>
|
2,520,301 | <p>So there was an example in my textbook that explained how to show one spanning set is equal to another. However, while I do understand the algebra, I'm not sure why they are allowed to do certain things, or why they do them. </p>
<p>The example:</p>
<p>If x and y are in $R^{n}$ show that span{x, y} = span{x+y, x-y... | Riley | 320,172 | <p>Since we can express $x$ and $y$ in terms of $x+y$ and $x-y$, we can represent any linear combination of $x$ and $y$ as a linear combination of $x+y$ and $x-y$. Therefore, every element of the span of $\{x,y\}$ is in the span of $\{x+y,x-y\}$.</p>
<p>For example: suppose a vector $v\in\mathrm{span}\{x,y\}$. Then $v... |
217,483 | <p><strong>Prove that if $I \subset \mathbb{R}$ is an open interval and $f: I \to \mathbb{R}$ differentiable, and $f$ has only one critical point $x_0$ and this critical point is a local minimum, then $x_0$ is also the absolute min of $f$, using Rolle's and IVT.</strong></p>
<p>To me this seems "obvious", because if $... | Brian M. Scott | 12,042 | <p>Fix $\alpha\in\Bbb N$; the result will follow almost immediately if you can prove that if $\alpha<\beta$, then $S\alpha\le\beta$. </p>
<p>Let $A=\{\gamma\in\Bbb N:\gamma=0\text{ or }S\alpha\le\alpha+\gamma\}$; clearly $0\in A$. Suppose that $\gamma\in A$. If $\gamma=0$, then $S\gamma=1$, and $S\alpha=\alpha+1=\a... |
3,767,935 | <p>Prove or disprove that if <span class="math-container">$$\prod\limits_{x=2}^{\infty} f(x)=0$$</span> and <span class="math-container">$f(x)\neq0$</span> for any <span class="math-container">$x\geq0$</span> then <span class="math-container">$$\prod\limits_{x=2}^{\infty} f(x\varphi)=0$$</span> for any constant <span c... | Peter Foreman | 631,494 | <p>This is not true. Consider any continuous function defined over <span class="math-container">$\mathbb{N}$</span> by
<span class="math-container">$$f(n)=\cases{\frac12&$n$ odd\\1&$n$ even}$$</span>
An explicit example is given by
<span class="math-container">$$f(x)=\frac{3+\cos{(\pi x)}}4$$</span>
Then we hav... |
1,579,579 | <p>I am struggling with showing that for algebraic number $\alpha$, the ring generated by $\mathbb{Q}[\alpha]$ is a field. I understand that to do this, I will have to show that any $r+s\alpha, r,s\in \mathbb{Q}$ has an inverse in $\mathbb{Q}[\alpha]$. I'm lost on how to go about doing this, though. Help? </p>
| Clément Guérin | 224,918 | <ol>
<li><p>From the very definition of <span class="math-container">$\mathbb{Q}[\alpha]$</span> (look in your course if you don't understand), this is a ring (<span class="math-container">$\alpha$</span> could be transcendental here).</p>
</li>
<li><p>From point <span class="math-container">$1$</span>, the only thi... |
372,045 | <p>I'm sure everyone already thought about this at least one time.
Why matrix multiplication is not defined the way showed below?</p>
<p>$$\left( \begin{array}{ccc}
a_{11} & a_{12} & \ldots \\
a_{21} & a_{22} & \ldots \\
\vdots & \vdots & \ddots
\end{array} \right) \cdot
\left( \begin{array}{cc... | Andreas Blass | 48,510 | <p>Suppose we used your proposed product, along with the usual addition and subtraction of matrices. Then all the algebra of $m\times n$ matrices would be the same as if we just used vectors of length $mn$. (In more detail, you can convert any matrix to a vector by just writing the rows of the matrix, one after the o... |
1,372,985 | <p>On <a href="https://en.wikipedia.org/wiki/Annihilator_(ring_theory)#Properties" rel="nofollow">this Wikipedia article</a>, it says that you can define an $R$-module $M$ as an $R/Ann_R(M)$-module using the action $\overline{r}m:=rm.$ </p>
<p>What does that action actually mean? What is $\overline{r}$?</p>
| Math1000 | 38,584 | <p>Since $P$ is symmetric, we know from the spectral theorem that $P$ has real eigenvalues and is diagonalizable. Since $P$ is a doubly stochastic matrix, the stationary distribution is uniform, i.e. $$\pi=\left(\frac16,\frac16,\frac16,\frac16,\frac16,\frac16\right).$$
Since $P$ has a stationary distribution, we know t... |
4,087,134 | <p>I have the polynomial <span class="math-container">$ p(x) = ax^3 + bx^2 + cx + d $</span>. I have to show that:</p>
<p><span class="math-container">$ \int_{-1}^{1} p(x) = p(- \frac{1}{\sqrt{3}}) + p (\frac{1}{\sqrt{3}}) $</span></p>
<p>I'm kind of stuck. My idea so far is to use "proof by symmetry" and the... | Troposphere | 907,303 | <p>There's nothing paradoxical about concluding <span class="math-container">$$ \text{Var}[X] + E[X]^2 = \text{Var}[|X|] + E[|X|]^2. $$</span></p>
<p>Intuitively, <span class="math-container">$\text{Var}[|X|]$</span> is (potentially) smaller than <span class="math-container">$\text{Var}[X]$</span> because the absolute ... |
1,184,501 | <p>I need help putting this in $0/0$ or $\infty/\infty$:</p>
<p>$$\lim_{x\to 0}{1\over xe^{x^{-2}}}$$</p>
<p>I've tried every possible combination, and I don't get what I'm missing. Using a graphic calculator, you easily see that the $\lim_{x\to 0}$ of this function is $0$.</p>
| user3932000 | 196,169 | <p>L'Hôpital's rule is not needed here.</p>
<p>Since $\lim_{x\to 0}\frac{1}{e^{x^2}}=\frac{1}{e^0}=\frac{1}{1}=1$, we only need to consider the $\lim_{x\to 0}\frac{1}{x}$ part. And clearly, this is $\pm\infty$.</p>
|
261,361 | <ol>
<li>In a group of 200 people, number of people having at least primary education (assuming - <em>Category I</em>): number of people having at least middle school education (<em>Category II</em>): number of people having at least high school education (<em>Category III</em>) are in the ratio 7 : 3 : 1</li>
<li>Out ... | Brian Rushton | 51,970 | <p>For problems like this, I would draw a Venn-diagram for the sports and split each of the four pieces of the Venn diagram into four smaller pieces representing education. Some pieces can be filled in immediately (four instance, we mark down five people in the hockey-only diagram in the category-three area, as well as... |
261,361 | <ol>
<li>In a group of 200 people, number of people having at least primary education (assuming - <em>Category I</em>): number of people having at least middle school education (<em>Category II</em>): number of people having at least high school education (<em>Category III</em>) are in the ratio 7 : 3 : 1</li>
<li>Out ... | august | 53,851 | <p>This seems a bit more complicated than just drawing a Venn diagram. I would try to diagram each item out algebraically instead. For example,</p>
<ol>
<li>Group of 200 people, 4 total categories, 3 of them in a ratio of 7:3:1. We can write
$$
w + x + y + z = 200
$$
where $w = $ number of people in Category IV, $x = ... |
3,680,124 | <p>I'm trying to integrate the following:</p>
<p><span class="math-container">$$\int \frac {dx}{x\sqrt{x^2-49}}\,$$</span></p>
<p>using the substitution <span class="math-container">$x=7\cosh(t)$</span></p>
<p>This is as far as I've gotten:</p>
<p><span class="math-container">$\int \frac {dx}{x\sqrt{x^2-49}}\,$</sp... | Quanto | 686,284 | <p>Note</p>
<p><span class="math-container">$$\frac{1}{7} \arctan(\sinh t)
=\frac{1}{7} \arctan\sqrt{\cosh^2t -1}\\
=\frac{1}{7} \arctan\sqrt{\frac{x^2}{49}-1}
=\frac{1}{7} \arctan\frac{\sqrt{x^2-49}}7\\
= \frac{1}{7} \text{arccot } \frac7{\sqrt{x^2-49}}
= \frac{1}{7}(\frac\pi2- \text{arctan} \frac7{\sqrt{x^2-49}})\... |
3,680,124 | <p>I'm trying to integrate the following:</p>
<p><span class="math-container">$$\int \frac {dx}{x\sqrt{x^2-49}}\,$$</span></p>
<p>using the substitution <span class="math-container">$x=7\cosh(t)$</span></p>
<p>This is as far as I've gotten:</p>
<p><span class="math-container">$\int \frac {dx}{x\sqrt{x^2-49}}\,$</sp... | P. Lawrence | 545,558 | <p>My advice is to make the substitution <span class="math-container">$x=7 \sec u$</span> in the original quesion. Then all that remains is to integrate <span class="math-container">$\frac{1}{7}.$</span></p>
|
361,212 | <blockquote>
<p>Suppose $X_1$ is a standard normal random variable. Define
$$X_2=\begin{cases} -X_1, &\text{if} \,\, |X_1|<1 \\ \,\,\,\,X_1, & \text{otherwise}\end{cases}$$ Obtain the cumulative distribution function of $X_1+X_2$ in terms of the cumulative distribution function of a standard normal rand... | Lost1 | 44,877 | <p>what you have written is pretty correct so far. what you need to do. so the CDF is calculated as </p>
<p>$P(Y\leq y)$. You have to work this out for the 3 regions $X\leq -1$, $-1<X<1$,and $X\geq 1$.</p>
<p>The first one is related to that a normal distribution right? because the lower tail has not really ch... |
749,090 | <p>Prove $\ a_{n}<2^{n} $ for every natural number n, where $\ a_{n} $ is defined recursively by $$ a_{1}=1, a_{2}=2, a_{3}=3, a_{n}=a_{n-3}+a_{n-2}+a_{n-1},\ for\ n>=4$$
Once I get the explicit equation, proving this would be easy with induction, however I'm having trouble finding it. I can't find the connecti... | gar | 138,850 | <p>Drawing a diagram is okay for two variables, but in general we can use integration.</p>
<p>For the problem:</p>
<p>$\displaystyle
\mathbb{P}(y\ge x+1)=\dfrac{\displaystyle\int_0^2 \int_{x+1}^3 \, dy\, dx}{\displaystyle\int_0^3 \int_{0}^3 \, dy\, dx}=\dfrac{2}{9}$</p>
|
771,607 | <p>Let $G$ be a finite $p$-group and $K$ be a normal subgroup. I want to show that there exists a normal subgroup $N$ of $G$ such that $N \leq K$ and $[K:N]=p$. I tried in this way: from Sylow's theorem, there exists a normal series $G=G_0 \rhd G_1 \rhd \cdots \rhd G_a=\{e\}$ such that $|G_i/G_{i+1}|=p$, then $K=K \cap... | Mikko Korhonen | 17,384 | <p>Let $K$ be a nontrivial normal subgroup of $G$. If $K$ has order $p$, we can pick $N = 1$, so assume that $K$ has order $\geq p^2$.</p>
<p>Now $K \cap Z(G)$ is nontrivial, so it contains a subgroup $P$ of order $p$. Then $P$ is normal in $G$. If $K$ has order $p^2$ we may pick $N = P$, otherwise repeat this argumen... |
3,462,507 | <p>Before diving into the Sherman-Morrison formula, Meyer in <em>Matrix Analysis and Applied Linear Algebra</em>, pg. 124, starts with</p>
<p><span class="math-container">$$
(I+cd^T)^{-1} = I - \frac{cd^T}{1+dc^T}
$$</span></p>
<p>where <span class="math-container">$c,d$</span> are vectors, and says "it's straightfor... | TZakrevskiy | 77,314 | <p>The important concept here is that you can "use" matrices as arguments in analytic functions (provided that eigenvalues of these matrices lie in the domain of analyticity). To further simplify, if you have a function representable by a power series, you can plug a square matrix instead of a complex variable.</p>
<p... |
924,551 | <p>$$\displaystyle \lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}}$$ </p>
<p>I tried rationalizing the numerator: </p>
<p>$$\lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}} \times \dfrac{(8-\sqrt{x})}{(8-\sqrt{x})}$$ </p>
<p>$$\lim_{x \to \infty}\dfrac{64-16\sqrt{x}+x}{64-x}$$</p>
<p>Is this correct? how do I p... | MCT | 92,774 | <p>We can rewrite the expression as $$\lim \limits_{x \to \infty} (-1 + \frac{16}{8 + \sqrt x})$$</p>
<p>The second term goes to zero, thus the limit is $-1$.</p>
|
519,516 | <p>I have been reading about uniform spaces and topological groups. There does not look to be a lot of literature on the topic, much less accesible literature, and the books that I have been reading do not mention any examples of uniform spaces, other than metric spaces and topological groups. There is another which I ... | Aloginame | 99,524 | <p>The substitution in integrals are the reverse of the chain rule in derivatives.</p>
<p>The following thread gives a fairly good explanation on why does the substitution works</p>
<p><a href="https://math.stackexchange.com/questions/77306/why-does-substitution-work-in-antiderivatives?rq=1">Why does substitution wor... |
1,107,013 | <p>Suppose that $f$ is a differentiable real function in an open set $E \subset \mathbb{R^n}$, and that $f$ has a local maximum at a point $x \in E$. Prove that $f'(x)=0$</p>
| Community | -1 | <p>$$1<\frac{x+1}{\sqrt{x^2+1}}<\sqrt2,$$</p>
<p>because, after squaring
$$x^2+1< x^2+2x+1<2x^2+2,$$
or
$$0<2x< x^2+1,$$
or
$$0< x\land 0<(x-1)^2$$
(except for $x=1$).</p>
<p>Increasing $x$ to infinity, $2\cos x$ goes from $1$ to $\sqrt2$ infinitely many times, and both functions are continuo... |
15,784 | <p>When I edit a cell in the notebook and re-evaluate it, it "overwrites" the input and the output of the previous edit. I want it to copy my edited version to the bottom or something and give me a separate output, not overwriting the old results or old input.</p>
| Sjoerd C. de Vries | 57 | <p>Mike's method seems a fine answer to me, but if you just <em>occasionally</em> want to keep an existing output, for instance as a reference to see whether a new evaluation causes a change or not, I usually place the insert beam between input and output cells and press enter, creating an empty cell in-between. This p... |
1,955,225 | <p>In working with a particular gene for fruit flies, geneticists classify an individual fruit fly as $\small \text{dominant, hybrid or recessive}$. In running an experiment, an individual fruit fly is crossed with a hybrid, then the offspring is crossed with a hybrid and so forth. The offspring in each generation are ... | Mathily | 375,170 | <p><strong>Answer edited to fix the multiplication order error noted by Michael.</strong></p>
<p>If I understand your notation correctly, the answers you are considering are $P^2_{(1,3)}$ and $P^2_{(3,1)}$. You are correct. As we begin with the first generation offspring, a recessive fly, the initial state vector is... |
229,127 | <p>Are $K^{MW}_*(\mathbb{F_q})$ and $K^{MW}_n(\mathbb{F_q})$ already known? Where can I read about it?</p>
| Matthias Wendt | 50,846 | <p>The Milnor-Witt K-theory for finite fields can be put together using
$$
K^{MW}_n\cong K^M_n\times_{I^n/I^{n+1}}I^n
$$
using the known results for Milnor K-theory and the Witt ring. A treatment of the Witt ring dealing with all characteristics can be found in Elman-Karpenko-Merkurjev "The algebraic and geometric the... |
51,898 | <p>I owe the idea of asking this question to Max Muller and
<a href="https://mathoverflow.net/questions/26035/">his curiosity</a>.</p>
<p><em>What is the set of $\alpha$ in the interval $0\le\alpha < 1$ for which
the alternating sum</em>
$$
\sum_{n=1}^\infty\frac{(-1)^{n+[n^\alpha]}}n
$$
<em>converges</em>? Here $[... | Leandro | 2,386 | <p>This is not an answer, but it is too long for a comment.</p>
<p>Hi Wadim, nice problem. I was trying to obtain a partial answer for it based on the following </p>
<p><b>Proposition.</b>
Let be $\xi_1,\xi_2,\ldots$ a sequence of independent Bernoulli random variables with
$\mathbb{P}(\xi_n=+1)=\mathbb{P}(\xi_n=-1)... |
51,898 | <p>I owe the idea of asking this question to Max Muller and
<a href="https://mathoverflow.net/questions/26035/">his curiosity</a>.</p>
<p><em>What is the set of $\alpha$ in the interval $0\le\alpha < 1$ for which
the alternating sum</em>
$$
\sum_{n=1}^\infty\frac{(-1)^{n+[n^\alpha]}}n
$$
<em>converges</em>? Here $[... | Anthony Quas | 11,054 | <p>It converges for all $0\le\alpha<1$. Define the $k$-<em>block</em> to be the set of $n$ such that $[n^\alpha]=k$ (it ranges from $\lceil k^{1/\alpha}\rceil$ to $\lceil (k+1)^{1/\alpha}\rceil-1$). </p>
<p>The absolute value of the contribution to the sum from the $k$-block is at most the reciprocal of its left en... |
1,639,156 | <p>Use logical quantifiers to write:
"Everybody loves somebody sometimes" (Where U=all people)
I came up with this but not sure how to type symbols in here.</p>
<p>$$\forall x \in U\,: \exists y\in U: x \text{ loves } y.$$</p>
<p>So... upside down A="For all"
Backwards E for "there exists"
curly little e for "belongs... | Jack's wasted life | 117,135 | <p>You're almost correct except the first : or 'such that' isn't needed.
$$
\forall\; x\in U\;\exists \;y\in U : x \text{ loves } y
$$
another way to write this is
$x\in U\implies \;\exists y\in U: x$ loves $y$.</p>
|
1,365,268 | <p>Part A is in the title, Part B is here:
Is it true that $(k, n+k)= d$ if and only if $(k, n)=d$?</p>
<p>I am still working on the Part A. </p>
<p>What I have so far:</p>
<p>if $(k, n)= 1$ then $1|k$, $1|n$ and $1|(n-k)$</p>
<p>if $(k, n+k)=1$ then $1|k$, $1|n+k$ and $1|((n+k)- k) \to 1|n$</p>
<p>I was under the... | Claude Leibovici | 82,404 | <p>Consider the function $$f(n,x)=\left(1+\frac{x}{n}\right)^{-n}-2^{-x}$$ What you can notice is that $$f(n,0)=0$$ $$f(n,n)=0$$ Computing the derivative $$\frac{df(n,x)}{dx}=2^{-x} \log (2)-\left(1+\frac{x}{n}\right)^{-n-1}$$ you also find $$\frac{df(n,0)}{dx}=\log (2)-1 <0$$ $$\frac{df(n,n)}{dx}=2^{-n} \log (2)-2^... |
1,365,268 | <p>Part A is in the title, Part B is here:
Is it true that $(k, n+k)= d$ if and only if $(k, n)=d$?</p>
<p>I am still working on the Part A. </p>
<p>What I have so far:</p>
<p>if $(k, n)= 1$ then $1|k$, $1|n$ and $1|(n-k)$</p>
<p>if $(k, n+k)=1$ then $1|k$, $1|n+k$ and $1|((n+k)- k) \to 1|n$</p>
<p>I was under the... | Vlad | 229,317 | <h3>I propose to see how do the limits of this expression compare to each other.</h3>
<p><strong>Recall</strong> the well-known limit:
$$
\lim_{k\to \infty} \left( 1+\frac{1}{k}\right)^{k} =
\lim_{k \to 0} \big( 1+k\big)^{\frac{1}{k}} =
e.
$$</p>
<p>Fix $x$ and let $n \to \infty$.
Denote $m = \dfrac{n}{x}$, so that ... |
2,591,621 | <p>Can this equation $x^3-12x=c$ have $2$ different solutions in $[-2,2]$? In $(-\infty,-2]$? In $[2,+\infty)$?</p>
<p>I said:
Let the equation have 2 different solutions, one in $[-2,x_1]$ and one in $[x_1,2]$ and let $f(x)=x^3-12x-c,f(-2)<0$. According to Bolzano's theorem, $f(-2) \cdot f(x_1)<0 \implies f(x_1... | BallBoy | 512,865 | <p>Picture a graph of $y = x^3 - 12x$; we need to know whether and where a horizontal line (at $y = c$) can intersect the graph twice. See if you can get a handle on what the graph looks like. Where is $y = x^3 - 12x$ increasing? Where is it decreasing?</p>
|
2,591,621 | <p>Can this equation $x^3-12x=c$ have $2$ different solutions in $[-2,2]$? In $(-\infty,-2]$? In $[2,+\infty)$?</p>
<p>I said:
Let the equation have 2 different solutions, one in $[-2,x_1]$ and one in $[x_1,2]$ and let $f(x)=x^3-12x-c,f(-2)<0$. According to Bolzano's theorem, $f(-2) \cdot f(x_1)<0 \implies f(x_1... | abr | 482,023 | <p>Assuming you want the solutions in $\mathbb{R}$ I believe you can answer this question using a divide a conquer approach, using different intervals and different values of c. </p>
<p>I also assume your question is in reality asking for the different values of which your equation will have solutions based on $c$. In... |
392,580 | <p>How to evaluate the following
$$\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)} dx $$
Given hints says to construct a rectangle $0\to R\to R+i\to i \to 0$ and consider $\displaystyle f(z):=\frac{e^{iaz}}{e^{2\pi z}-1} $ and evaluate around it but that does not help.</p>
<p><strong>ADDED::</strong> I need ... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{... |
468,344 | <p>Suppose $0\leq \alpha, \beta, \gamma\leq \pi$ and $\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1$, then what is the maximum and minimum of $\alpha+\beta+\gamma$.</p>
| Calvin Lin | 54,563 | <p>Observe that the second derivative of $\cos^2 x $ is $- 2 \cos 2x$. It is negative on $[0, \frac{\pi}{2} ]$ and positive on $[\frac{\pi}{2}, 0 ]$</p>
<p>To find the maximum, notice that if $\alpha, \beta, \gamma < \frac{\pi}{2} $, then you can replace the corresponding term by $\pi - \alpha, \pi - \beta, \pi - \... |
2,164,465 | <p>Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.</p>
<p>Where would a counting concept like nCr and nPr fall into this mix?</p>
| Sentinel135 | 417,752 | <p>$_nC_r$ and $_nP_r$ are more like counting functions that map $\mathbb N^2\to \mathbb N$. What this means is they aren't operations more or less. they have an equation related to them $P(n,r):= \frac{n!}{(n-r)!}$ and $C(n,r):= \frac{n!}{r!(n-r)!}$. </p>
<p>So in a sense they use multiplication, since $n!= 1*2*3*\do... |
3,381,219 | <p><a href="https://i.stack.imgur.com/mM0OF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mM0OF.png" alt="3B) in the picture"></a>What is an example of an infinite intersection of infinite sets is infinite?</p>
<p>I know that the intersection of infinite sets does not need to be infinite. However,... | Kavi Rama Murthy | 142,385 | <p>Let <span class="math-container">$(B_n)$</span> be a sequence of sets with empty intersection and <span class="math-container">$B_{n+1} \subset B_n$</span>, say <span class="math-container">$B_n =\{n,n+1,...\}$</span>. Now take <span class="math-container">$A_n =B_n \cup E$</span>. Then intersection of <span class=... |
3,381,219 | <p><a href="https://i.stack.imgur.com/mM0OF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mM0OF.png" alt="3B) in the picture"></a>What is an example of an infinite intersection of infinite sets is infinite?</p>
<p>I know that the intersection of infinite sets does not need to be infinite. However,... | Vincent Fourmond | 308,495 | <p>If <span class="math-container">$A_n$</span> is <span class="math-container">$\mathcal{N}$</span> without the first <span class="math-container">$n$</span> prime numbers, then the <span class="math-container">$A_n$</span> are nested subsets with an infinite intersection, the non-prime integers.</p>
|
1,866,639 | <p>Let $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ be a function which is Gâteaux differentiable and let $J_f\in\mathbb{R}^{M\times N}$ be its Jacobian matrix.</p>
<p>Is it true that the Gâteaux derivative of $f$ along a direction $v\in\mathbb{R}^N$ is equal to the matrix-vector product $J_f \cdot v$?</p>
| bartgol | 33,868 | <p>The answer in general is no. The classic counter example is given by the function</p>
<p>$$
f(x,y)=\left\{
\begin{array}{cc}
\frac{x^2y}{x^4+y^2} & (x,y)\neq (0,0),\\
0 & (x,y)=(0,0)
\end{array}
\right.
$$</p>
<p>You can check that at $(0,0)$ this function has all the directional derivatives. However, it i... |
2,317,929 | <p>Let $X=\mathrm{Spec}A$ and $U$ be an affine open subscheme of $X$.
Is there some $a\in A$ such that $D(a)=U$?</p>
| Tsemo Aristide | 280,301 | <p>This would imply that the complementary space of $U$ is $V(I)=V(a)$ and $I$ is principal. This is not always true.</p>
|
979,299 | <p>Assuming that I have $\{x_1,\ldots, x_N\}$ - an iid (independent identically distributed) sample size $N$ of observations of random variable $\xi$ with unknown mean $m_1$, variance (second central moment) $m_{c_2}$ and second raw moment $m_2$. I try to use sample mean $\overline{x}=\frac{1}{N}\sum_{i=1}^Nx_i$ as an ... | Varun Iyer | 118,690 | <p>Using improper integrals,</p>
<p>$$\int_{0}^{\infty} \frac{1}{x^2} = \int_{0}^{3} \frac{1}{x^2} + \int_{3}^{\infty} \frac{1}{x^2}= \lim_{b\to0^{+}} \int_{b}^{3} \frac{1}{x^2} + \lim_{b\to\infty} \int_{3}^{b} \frac{1}{x^2}$$</p>
<p>$$= \lim_{b\to0^{+}} \left[-\frac{1}{x}\right]_{b}^{3} + \lim_{b\to\infty} \left[-\f... |
443,099 | <p>I remember hearing someone say "almost infinite" on one of the science-esque youtube channels. I can't remember which video exactly, but if I do, I'll include it for reference.</p>
<p>As someone who hasn't studied very much math, "almost infinite" sounds like nonsense. Either something ends or it doesn't, there rea... | Asaf Karagila | 622 | <p>Based on the comments suggesting that this was used to describe the number of species of a certain family of insects, or something similar, I would say that this is a perfectly correct <strong>prosaic</strong> use of the term infinite.</p>
<p>This misuse of the word "infinite" alludes to the fact that there are man... |
2,546,274 | <p>Is there a way to solve this system of equations?</p>
<p>$16s-9t=ts$</p>
<p>$107s-100t=ts$</p>
<p>for t and s? Thanks in advance.</p>
| adfriedman | 153,126 | <p>The limit is zero.</p>
<p><strong>Stream Lined Proof</strong>
Fix $m \geq 1$, then splitting the summation:
\begin{align}
0&\leq e^{-\sigma_A-\sigma_B} \sum_{n=0}^{m-1} \frac{(\sigma_A\sigma_B)^n}{(n!)^2} + e^{-\sigma_A-\sigma_B} \sum_{n=m}^{\infty} \frac{(\sigma_A\sigma_B)^n}{(n!)^2}\\
%
&<m \frac{(\sig... |
2,546,274 | <p>Is there a way to solve this system of equations?</p>
<p>$16s-9t=ts$</p>
<p>$107s-100t=ts$</p>
<p>for t and s? Thanks in advance.</p>
| Sangchul Lee | 9,340 | <p><strong>EDITED.</strong></p>
<hr>
<p><strong>1. Probabilistic heuristics.</strong> Let $N_A \sim \operatorname{Poisson}(\sigma_A)$ and $N_B \sim \operatorname{Poisson}(\sigma_B)$ be independent Poisson random variables. As you are already aware of, the quantity in question is</p>
<p>$$\mathbb{P}( N_A = N_B)$$</p>... |
2,105,653 | <p>Show that $\left|Im(2+ z^{c} -4z^2) \right| \leq 9.5$ When $ \left| z \right| \leq \frac {3}{2}$</p>
<p>$z^c$= compliment of z</p>
<p>$\left|Im(2+ z^{c} -4z^2) \right| \leq \left| 2+z^{c} - 4z^2 \right|$</p>
<p>I have tried to split it up directly i have tried to force complete the square i always get a weird va... | mbe | 100,502 | <p>You're trying to prove the unprovable. Let $z=re^{it}$, such that
$$Im(2+ z^{c} -4z^2) =Im(z^c)-4Im(z^2)=-r\sin (t)-4r^2sin(2t).$$
For $r=3/2$, this expression is $\approx-10.076$ at $t\approx 0.814$.</p>
|
78,414 | <p>why is $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I am studying for a differential equation test and I seem to always forget \this, and i am just wondering if there is some intuition i'm missing, or is it just one of those things to memorize? and i know this is not very differential equation related, just one of those things ... | Adrián Barquero | 900 | <p>Well if you find trouble remembering them maybe you can use the <a href="http://en.wikipedia.org/wiki/Euler%27s_formula#Relationship_to_trigonometry" rel="nofollow">formulas</a> $$\cos{x} = \frac{e^{ix} + e^{-ix}}{2} \quad \sin{x} = \frac{e^{ix} - e^{-ix}}{2i}$$ </p>
<p>which you can get from <a href="http://en.wik... |
78,414 | <p>why is $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I am studying for a differential equation test and I seem to always forget \this, and i am just wondering if there is some intuition i'm missing, or is it just one of those things to memorize? and i know this is not very differential equation related, just one of those things ... | lhf | 589 | <p>Here is a geometric interpretation that is easy to remember: the unit circle is parametrized by $(\cos t, \sin t)$ and hence its tangent vector is orthogonal to the position vector. Rotating the position vector by 90 degrees gives you $(-\sin t, \cos t)$ and so $\cos'=-\sin$ and $\sin'=\cos$.</p>
<p>This ar... |
78,414 | <p>why is $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I am studying for a differential equation test and I seem to always forget \this, and i am just wondering if there is some intuition i'm missing, or is it just one of those things to memorize? and i know this is not very differential equation related, just one of those things ... | Scott Carter | 722 | <p>Pause before you answer the question. The derivative of sine is (....) cosine. The integral of sine is (....) minus the cosine. The derivative of cosine is (....) minus the sign. The integral of cosine is (...) sine. Never say the answer without the pause. Think of your best reason for the s-i-g-n (several given abo... |
156,585 | <p>I am struggling to evaluate the following integral:<br>
$$\int \frac{1}{(1-x^2)^{3/2}} dx$$<br>
I tried a lot to factorize the expression but I didn't reach the solution.
Please someone help me.</p>
| Valentin | 31,877 | <p>$$\int \frac{dx}{\left(1-x^2\right)^{3/2}}=[x=\sin t]=\int\frac{\cos t dt}{\cos^3 t}=\int\frac{dt}{\cos^2 t}=\tan t$$</p>
|
87,902 | <pre><code>eq1 := Abs[-3.533147671810^-6] ==
A1 Exp[-(-0.53326099689) ((μ1))^2]
eq2 := 7.2716492165 10^-4 == A2 Exp[-(0.53326099689) ((μ2))^2]
eq3 := Abs[-4.0740049497 10^-10] ==
A3 Exp[-(-8.8857611784 10 ⁻²) ((μ3))^2]
eq4 := -3.1704480355 10^-6 ==
2 (-0.53470532215)... | Dr. belisarius | 193 | <pre><code>eq1 = 3.5331476718 10^-6 == A1 Exp[-(-0.53326099689) m1^2];
eq2 = 7.2716492165 10^-4 == A2 Exp[-(0.53326099689) m2^2];
eq3 = 4.0740049497 10^-10 == A3 Exp[-(-8.885761178410 ^-2) m3^2];
eq4 = -3.1704480355 10^-6 == 2 (-0.53470532215) m1 A... |
1,710,083 | <p>No doubt a similar question has been answered before, but I make my ideal textbook specific. </p>
<p>Does anyone know of an Algebraic Topology textbook with the following properties. </p>
<p>-Accessible (Nothing Hardcore Please, I would consider myself a very average student)</p>
<p>-Solutions (They need not be w... | Captain Lama | 318,467 | <p>I guess the Fulton (Algebraic Topology, a first course) would be a good choice. He stays quite elementary throughout the book, and there are hints for most exercices at the end.</p>
|
1,710,083 | <p>No doubt a similar question has been answered before, but I make my ideal textbook specific. </p>
<p>Does anyone know of an Algebraic Topology textbook with the following properties. </p>
<p>-Accessible (Nothing Hardcore Please, I would consider myself a very average student)</p>
<p>-Solutions (They need not be w... | Wintermute | 67,388 | <p>Check out From Calculus to Cohomology by Madsen. Here is the link</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/0521589568" rel="nofollow">http://www.amazon.com/From-Calculus-Cohomology-Characteristic-Classes/dp/0521589568</a></p>
<p>It is at a lower level than Munkres and has some good simple examples.... |
1,710,083 | <p>No doubt a similar question has been answered before, but I make my ideal textbook specific. </p>
<p>Does anyone know of an Algebraic Topology textbook with the following properties. </p>
<p>-Accessible (Nothing Hardcore Please, I would consider myself a very average student)</p>
<p>-Solutions (They need not be w... | Arteom.k | 174,209 | <p>Elementary Topology Problem Textbook by Viro, Harlamov, etc. as an introduction
It covers only part of a subject, but it has solutions. </p>
|
1,789,077 | <blockquote>
<p>There is a square <span class="math-container">$Q$</span> consisting of <span class="math-container">$(0,0), (2,0), (0,2), (2,2)$</span>.</p>
<p>A point <span class="math-container">$P$</span> satisfies following condition:</p>
<p>The straight line passing through <span class="math-container">$P$</span>... | lEm | 319,071 | <p>$$\sin(2x)+\sin(4x)=2\sin(3x)\cos(x)$$</p>
<p>So</p>
<p>$$\sin(3x) (\cos(x)-1)=0$$</p>
|
3,251,589 | <p><span class="math-container">$$ \lim_{x \to 1} \frac{x^3-1}{x-1}=3
$$</span>
How to prove it using precise definition of limits? While solving it, I get stuck at |(x-1)(x+2)|<ε. I don't know how to take out the inequality for x only as this inequality contains quadratic form.</p>
| Kavi Rama Murthy | 142,385 | <p>Let <span class="math-container">$0 <\epsilon<1$</span>. If <span class="math-container">$|x-1| <\epsilon /4$</span> then <span class="math-container">$|\frac {x^{3}-1} {x-1} -3|=|x^{2}+x+1-3|=|(x-1)(x+1)+(x-1)|<|x-1|(|x-1|+2)+|x-1|<(\epsilon/4) (3)+\epsilon/4 <\epsilon$</span>.</p>
|
3,251,589 | <p><span class="math-container">$$ \lim_{x \to 1} \frac{x^3-1}{x-1}=3
$$</span>
How to prove it using precise definition of limits? While solving it, I get stuck at |(x-1)(x+2)|<ε. I don't know how to take out the inequality for x only as this inequality contains quadratic form.</p>
| Richard Jensen | 658,583 | <p>As you've probably noticed, when <span class="math-container">$x \ne 0$</span>, \frac{x^3-1}{x-1} = x^2 + x + 1$, so you just need to show that |x^2 + x + 1 - 3| = |x^2 + x -2| gets arbitrarily close to 0 (in epsilon delta terms).</p>
|
3,251,589 | <p><span class="math-container">$$ \lim_{x \to 1} \frac{x^3-1}{x-1}=3
$$</span>
How to prove it using precise definition of limits? While solving it, I get stuck at |(x-1)(x+2)|<ε. I don't know how to take out the inequality for x only as this inequality contains quadratic form.</p>
| Arthur | 15,500 | <p>The standard trick to deal with this issue is to declare that you will never pick an <span class="math-container">$x$</span> outside of <span class="math-container">$(0,2)$</span>. In other words, no matter what <span class="math-container">$\varepsilon$</span> is, your <span class="math-container">$\delta$</span> w... |
921,644 | <p>Is this series convergent $1+\dfrac{1}{4}-\dfrac{1}{9}-\dfrac{1}{16}+\dfrac{1}{25}+\dfrac{1}{36}-\dfrac{1}{49}-\dfrac{1}{64}+\cdots\ ?$</p>
<p>Can we write this series as function of $n?$</p>
| Tunk-Fey | 123,277 | <p><strong>Hint :</strong></p>
<p>You may refer to <a href="https://en.wikipedia.org/wiki/Dirichlet_beta_function" rel="nofollow">Dirichlet beta function</a> and <a href="https://en.wikipedia.org/wiki/Dirichlet_eta_function" rel="nofollow">Dirichlet eta function</a> $$\eta(s)=\sum_{n=1}^\infty\frac{(-1)^n}{n^s}=\left(... |
5,119 | <p>If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra?</p>
<p>It seems closed under complement since for all $x$ in the union, $x$ has to belong to some $\mathcal{F}_i$, and so must its complement.</p>
<p>It seems close... | Arturo Magidin | 742 | <p>The problem arises in the countable union; your argument is correct as far as it goes, but from the fact that <span class="math-container">$\cup_{i=1}^n x_i\in \cup_{i=1}^{\infty}F_i$</span> for each <span class="math-container">$n$</span> you cannot conclude that <span class="math-container">$\cup_{i=1}^{\infty} x_... |
4,448,865 | <p>Let the continuous spectrum of a densely defined linear operator <span class="math-container">$L$</span> over a Separable Hilbert space, be defined as the set of all <span class="math-container">$\lambda \in \mathbb C$</span> such that:</p>
<p>(i) <span class="math-container">$L-\lambda$</span> is injective,</p>
<p>... | B. S. Thomson | 281,004 | <p><strong>Commentary:</strong></p>
<p><em>I have the following question about integrable functions. Suppose that <span class="math-container">$\{f_n\}$</span> is a sequence of integrable <span class="math-container">$\dots$</span></em></p>
<p><strong>Hold on there!</strong> If you are deep inside a class lecture or ... |
1,234,820 | <p>I was working on this problem </p>
<blockquote>
<p>Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$.</p>
</blockquote>
<p>My attempt:-
I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should be either zero or the entire ring (since $M_3(\mathbb{R})$ is simple) and in ca... | Crostul | 160,300 | <p>There is no injective ring homomorphism since every matrix of the form $AB-BA$ must be mapped into $0$. To conclude, it is well known that there exist some $AB-BA \neq 0$.</p>
|
2,902,058 | <p>I am solving the following problem:
$$\lim_{R\rightarrow \infty} \int_{C_R} \frac{6z^6 + 5z^5}{z^6 + z^5 + 10}dz,$$
where $C_R=\{z \in \mathbb{C} : |z|=R \}$ for $R>0$.</p>
<p>The only one idea I have is to use the Residue theorem. </p>
<p>But I couldn't apply the theorem to the above problem.</p>
| Mike | 544,150 | <p>You do not have the continuity of $f(x)$, but you DO have, for each $x \in \mathbb{R} \setminus \mathbb{Z}$, continuity of $f(x)$ in a small enough ball $B_x$ around $x$ [I think as far as the exercise it suffices for the student to note this fact?]. That is all you need: For each such $x$ there is an $n_0$ such tha... |
454,622 | <p>I am trying to solve a particular probability question. </p>
<p>I have a fair 10-sides die, whose sides are labelled 1 through 10. I am trying to find the probability of rolling a multiple of 5 or an odd number. </p>
<p>I find the probability as: </p>
<p>P(multiple of 5) OR P(odd number)=P(multiple of 5) + P(odd... | SAAN | 85,982 | <p>Actually when we through a fair die each outcome is identically independently distributed, so there is no need to assume multiple of 5 is independent of rolling an odd number. Two events are said to be dependent if they effect occurrence or non-occurrence of other event. In your case both events are Independent. </... |
454,622 | <p>I am trying to solve a particular probability question. </p>
<p>I have a fair 10-sides die, whose sides are labelled 1 through 10. I am trying to find the probability of rolling a multiple of 5 or an odd number. </p>
<p>I find the probability as: </p>
<p>P(multiple of 5) OR P(odd number)=P(multiple of 5) + P(odd... | Kevin Arlin | 31,228 | <p>No, it's not a fluke, in the sense that $P(A\cap B)=P(A)P(B)$ is the <em>definition</em> of independence of $A$ and $B$, so if it holds, as it does here, $A$ and $B$ are necessarily independent. In another sense, it was a fluke, in that it might not have been obvious that the events <code>multiple of 5</code> and <c... |
1,831,052 | <p><a href="https://i.stack.imgur.com/SxyPL.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SxyPL.jpg" alt="enter image description here"></a></p>
<p>ADE is a straight line .
AE : AD = 3 : 2
Find coordinates of E</p>
<p>My workings </p>
<p>Let E ( X , Y) </p>
<p>Gradient AD = Gradient of AE
$ 1/... | MrYouMath | 262,304 | <p>Hint: $\ln(1-u)=-\sum_{n=1}^{\infty}\frac{u^n}{n}$. Note that $\frac{(3x-1)^n}{6^n}=\left(\frac{x}{2}-\frac{1}{6} \right)^n$</p>
|
1,831,052 | <p><a href="https://i.stack.imgur.com/SxyPL.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SxyPL.jpg" alt="enter image description here"></a></p>
<p>ADE is a straight line .
AE : AD = 3 : 2
Find coordinates of E</p>
<p>My workings </p>
<p>Let E ( X , Y) </p>
<p>Gradient AD = Gradient of AE
$ 1/... | DonAntonio | 31,254 | <p>Hinting: for $\;|z|<1\;$ , we have</p>
<p>$$\frac1{1-z}=\sum_{n=0}^\infty z^n\implies \frac1{z(1-z)}=\sum_{n=0}^\infty z^{n-1}\;\stackrel{\text{Integ.}}\implies\;\log|z|-\log (1-z)=\sum_{n=1}^\infty\frac{z^n}n$$</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.