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 |
|---|---|---|---|---|
2,433,614 | <p>The solution to the initial-value problem $y' = y^{2} + 1$ with $y(0)=1$ is $y = \tan(x + \frac{\pi}{4})$. I would like to show that this is the correct solution in a way that is analogous to my solution to differential equation $y' = y - 12$.</p>
<p><strong>Solution to Differential Equation</strong></p>
<p>If $y'... | John Hughes | 114,036 | <p>Apostol's calculus book has (I believe) a nice approach to sine/cosine: they lay out a few "axioms" that, taken together, define sine and cosine on all points of the form $\frac{n\pi}{2^k}$, where $n$ and $k$ are integers. </p>
<p>Then they show that as a function on this set, sine and cosine are continuous. </p>
... |
927,815 | <p>Find an equation of the plane that passes through the points $A(0, 1, 0)$, $B(1, 0, 0)$ and $C(0, 0, 1)$.</p>
| Paul Sundheim | 88,038 | <p>HINT: The equation of a plane can be expressed as $ax+by+cz=d$ Since the plane goes through $(1,0,0)$, $a=d$. Use the other two points in the same way to get the equation.</p>
|
944,992 | <p>I'm trying to understand each part of this completed proof that my professor did, here is my interpretation in parentheses, please advise as necessary.</p>
<p>Proof: Assume that $2^{1/2}$ is rational, then $2^{1/2}$ = $p/q$ for some integer $p$ and $q$ with $q \neq 0$ and $\gcd(p,q) = 1$</p>
<p>$$(p/q)^2 = 2 = p^2... | Chris Culter | 87,023 | <p>The proof <em>strategy</em> is valid, but it needs some formalization if we want to be confident that it proves exactly what it claims. In this case, the statement is false as written; you need to assume that the polygon is convex!</p>
<p>Here's one way to formalize your proof. In physics, gravity tends to minimize... |
1,918,885 | <p>I would like to show that $\mathbb{Q}$ is the smallest field between $\mathbb{R}$ and $\mathbb{Z}$. In other words, if there is a field $\mathbb{F}$ where $\mathbb{Z} \subset \mathbb{F} \subset \mathbb{R}$, then it must also be that $\mathbb{Q} \subset \mathbb{F}$.</p>
<p>My proof is by contradiction, assume that i... | Rafael | 303,887 | <p>The contradiction is because you are assuming that $\mathbb{Z}\subset\mathbb{F}$. If you assume NOT Q you check NOT P: $\mathbb{Q}\not\subset\mathbb{F}\Rightarrow\mathbb{Z}\not\subset\mathbb{F}$.</p>
|
64,653 | <p>Consider the two (inequivalent) $\mathbb{Z}$-representations $\phi,\psi$ of the symmetric group $S=S_3$ given by</p>
<p>$(1,2)^\phi=\left(\begin{array}{rr}0 &-1\\\ -1 & 0\end{array}\right), \qquad
(1,2,3)^\phi=\left(\begin{array}{rr}0 &1\\\ -1 & -1\end{array}\right);$</p>
<p>$(1,2)^\psi=\left(\begi... | Allen Hatcher | 23,571 | <p>As Tom Goodwillie noted in his comment, $GL(2,\Bbb Z)$ can be identified with $Out(F_2)$, so the question can be rephrased in terms of lifting subgroups of $Out(F_2)$ to $Aut(F_2)$. There is a Realization Theorem for finite subgroups of $Aut(F_n)$ and $Out(F_n)$ which says that such a subgroup can always be realize... |
221,026 | <p>I need to find the value of <span class="math-container">$z$</span> for a particular value of <span class="math-container">$D_c$</span> (eg. <span class="math-container">$500$</span>), but <span class="math-container">$z$</span> is inside an integral, and I'm not able to use <code>Solve</code> since the integral is ... | Bob Hanlon | 9,362 | <pre><code>OmegaM = 0.3111 // Rationalize;
OmegaLambda = 0.6889 // Rationalize;
Dc = 500;
eqn = Integrate[(OmegaM (1 + z1)^3 + OmegaLambda)^(-1/2), {z1, 0, z},
Assumptions -> z > 0]
</code></pre>
<p><a href="https://i.stack.imgur.com/qheZS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qh... |
4,292,091 | <p>I'm reading the definition of <span class="math-container">$inf\emptyset$</span> and <span class="math-container">$sup\emptyset$</span>.</p>
<p>a) I'm wondering why <span class="math-container">$inf\emptyset = \infty$</span> and <span class="math-container">$sup\emptyset = -\infty$</span>. I would have expected both... | José Carlos Santos | 446,262 | <p>Having<span class="math-container">$$\inf\emptyset=\infty\text{ and }\sup\emptyset=-\infty\tag1$$</span>is that only way of defining <span class="math-container">$\inf\emptyset$</span> and <span class="math-container">$\sup\emptyset$</span> so that you always have<span class="math-container">$$A\subset B\implies \in... |
2,707,749 | <p>I know this has to be extremely easy but I'm not going to solve this problem.</p>
<p>The task is to find the angle at point $A$.</p>
<p>Thanks!</p>
<p><img src="https://i.stack.imgur.com/8NTEO.png" alt="Here is the image."></p>
| user | 293,846 | <p>By <a href="http://2000clicks.com/mathhelp/geometrytriangleinscribedanglecircle2.aspx" rel="nofollow noreferrer">extended central angle theorem</a> the angle $A=(175^\circ- 49^\circ)/2=63^\circ $.</p>
|
4,469,136 | <p>I took a number theory course this past semester and I found the idea of there being different primes in different fields. The only fields that we did in detail were the set of reals and the set of Gaussian integers.</p>
<p>What other fields are there, and what numbers are prime in those fields?</p>
| Ethan Bolker | 72,858 | <p>What you are calling a "field" is officially a ring of algebraic integers.</p>
<p>There is lots known about unique factorization and what the primes are in the rings of integers in the fields <span class="math-container">$\mathbb{Q}(\sqrt{d})$</span>.</p>
<p>See the wikipedia page <a href="https://en.wikip... |
1,999,643 | <p>The title states my question: what aspect of closed makes it attractive for optimization?</p>
| the_candyman | 51,370 | <p>Consider these problems:</p>
<p>$$\begin{cases}
\max x \\
\text{s.t.}\\
x \in (0,1)
\end{cases},
\begin{cases}
\max x \\
\text{s.t.}\\
x \in [0,1]
\end{cases}.$$</p>
<p>The first has no solution, while the second has (exactly, $x=1$).</p>
<hr>
<p>Summarizing, it is better to have closed set since sometimes the ... |
925,941 | <p>Solve:</p>
<p>$xy=-30$<br>
$x+y=13$</p>
<p>{15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ?</p>
| Kim Jong Un | 136,641 | <p>Start with
$$
x+y=13 \iff y=13-x
$$
so that
$$
-30=xy=x(13-x)=13x-x^2 \iff x^2-13x-30=0
$$
which solves to give $x=-2$ and $x=15$. The corresponding $y$ values are $15$ and $-2$.</p>
|
925,941 | <p>Solve:</p>
<p>$xy=-30$<br>
$x+y=13$</p>
<p>{15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ?</p>
| Community | -1 | <p>The solutions of the system of equations $xy=p$ and $x+y=s$ are the roots of the quadratic equation:</p>
<p>$$x^2-sx+p=0$$
so in your case we solve
$$x^2-13x-30=0$$
using the discriminant:
$$\Delta=13^2+4\times 30=17^2$$
so the two roots are
$$x_1=\frac{13+17}{2}=15\quad;\quad x_2=\frac{13-17}{2}=-2$$</p>
|
6,990 | <p>The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.</p>
<ul>
<li><p>In how far can this be generalized to other countable sets of functions? For example, if we keep our inner product, can ... | Akhil Mathew | 344 | <p>You need the orthogonality condition to get such an integral representation for the coefficients; otherwise it would probably be more complicated. </p>
<p>The Fourier series of any $L^2$ function converges not only in the norm (which follows from the fact that $\{e^{inx}\}$ is an orthonormal basis) but also almost... |
2,083,475 | <p>This is the conic $$x^2+6xy+y^2+2x+y+\frac{1}{2}=0$$
the matrices associated with the conic are:
$$
A'=\left(\begin{array}{cccc}
\frac{1}{2} & 1 & \frac{1}{2} \\
1 & 1 & 3 \\
\frac{1}{2} & 3 & 1
\end{array}\right),
$$</p>
<p>$$
A=\left(\begin{array}{cccc}
1 & 3 \\
3 & 1
\end{array}\r... | amd | 265,466 | <p>For a hyperbola in standard position, $\tan\alpha=\frac b a$, where $\alpha$ is the angle that the asymptotes make with the transverse axis. If you transform the general equation into standard form, you’ll find that $a^2=-|A'|/\lambda_2|A|$ and $b^2=|A'|/\lambda_1|A|$, where $\lambda_1$ is the positive eigenvalue, s... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| user02138 | 2,720 | <p>\begin{eqnarray}
\zeta(0) = \sum_{n \geq 1} 1 = -\frac{1}{2}
\end{eqnarray}</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Pedro | 23,350 | <p>Considering the main branches</p>
<p>$$i^i = \exp\left(-\frac{\pi}{2}\right)$$</p>
<p>$$\root i \of i = \exp\left(\frac{\pi}{2}\right) $$</p>
<p>And
$$ \frac{4}{\pi } = \displaystyle 1 + \frac{1}{{3 +\displaystyle \frac{{{2^2}}}{{5 + \displaystyle\frac{{{3^2}}}{{7 +\displaystyle \frac{{{4^2}}}{{9 +\displaystyl... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Community | -1 | <p>The Cayley-Hamilton theorem:</p>
<p>If $A \in \mathbb{R}^{n \times n}$ and $I_{n} \in \mathbb{R}^{n \times n}$ is the identity matrix, then the characteristic polynomial of $A$ is $p(\lambda) = \det(\lambda I_n - A)$. Then the Cayley Hamilton theorem can be obtained by "<strong><em>substituting</em></strong>" $\lam... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Diego Silvera | 41,495 | <p>The following number is prime</p>
<blockquote>
<p>$p = 785963102379428822376694789446897396207498568951$</p>
</blockquote>
<p>and $p$ in base 16 is</p>
<blockquote>
<p>$89ABCDEF012345672718281831415926141424F7$</p>
</blockquote>
<p>which includes counting in hexadecimal, and digits of $e$, $\pi$, and $\sqrt{... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| 2'5 9'2 | 11,123 | <p>\begin{align}\frac{64}{16}&=\frac{6\!\!/\,4}{16\!\!/}\\&=\frac41\\&=4\end{align}</p>
<p>For more examples of these <em>weird fractions</em>, see "How Weird Are Weird Fractions?",
Ryan Stuffelbeam, <strong>The College Mathematics Journal</strong>, Vol. 44, No. 3 (May 2013), pp. 202-209.</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Felix Marin | 85,343 | <p>\begin{align}
E &=
\sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}}
=
mc^{2}
+
\left[\sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}} - mc^{2}\right]
\\[3mm]&=
mc^{2}
+
{\left(pc\right)^{2}
\over
\sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}} + mc^{2}}
=
mc^{2}
+
{p^{2}/2m
\over
1 + {\sqrt{... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Shivam Patel | 95,509 | <p>Heres a interesting one again<br>
$3435=3^3+4^4+3^3+5^5%$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Gerry Myerson | 8,269 | <p>$$\lim_{\omega\to\infty}3=8$$ The "proof" is by rotation through $\pi/2$. More of a joke than an identity, I suppose. </p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Neves | 1,747 | <p><span class="math-container">$$
\dfrac{1}{2}=\dfrac{\dfrac{1}{2}}{\dfrac{1}{2}+\dfrac{\dfrac{1}{2}}{\dfrac{1}{2}+\dfrac{\dfrac{1}{2}}{\dfrac{1}{2}+\dfrac{\dfrac{1}{2}}{\dfrac{1}{2}+\dfrac{\dfrac{1}{2}}{\dfrac{1}{2}+\dfrac{\dfrac{1}{2}}{\dfrac{1}{2}+\cdots}}}}}}
$$</span></p>
<p>and more generally we have
<span class... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Neves | 1,747 | <p>$$
\frac{\pi}{2}=1+2\sum_{k=1}^{\infty}\frac{\eta(2k)}{2^{2k}}
$$
$$
\frac{\pi}{3}=1+2\sum_{k=1}^{\infty}\frac{\eta(2k)}{6^{2k}}
$$
where
$
\eta(n)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{n}}
$</p>
|
2,101,059 | <p>Let $d,n~$ be positive integer.</p>
<p>$\int{y(1-y^d)^n}dy$</p>
<p>Is it possible to solve it? If you know the method, please teach me.</p>
| Eff | 112,061 | <p><strong>Hint.</strong> Expand the term $(1-y^d)^n$ using the binomial theorem, and multiply by $y$. Then you will simply have a polynomial, and polynomials are easy to integrate.</p>
|
3,687,484 | <p>Here, <span class="math-container">$H^1(\mathbb{R}^2)$</span> is the standard Sobolev spaces for <span class="math-container">$L^2(\mathbb{R}^2)$</span> functions whose weak derivative belongs to <span class="math-container">$L^2(\mathbb{R}^2).$</span></p>
<p>My question in the title comes from calculus of variatio... | Claudio Moneo | 454,365 | <p>It does not hold for general functions <span class="math-container">$u \in H^{1}(\mathbb{R}^{2})$</span>, even if they are assumed to be smooth.
The reason lies in the following lemma:</p>
<p>Let <span class="math-container">$N \ge 2$</span>. Then for any <span class="math-container">$x_{k} \in \mathbb{R}^{N}, \; \e... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Minhyong Kim | 1,826 | <p>Dear Alex,</p>
<p>It seems to me that the general question in the background of your query on algebra really is the better one to focus on, in that we can forget about irrelevant details. That is, as you've mentioned, one could be asking the question about motivation and decision in any kind of mathematics, or mayb... |
356,940 | <p>According to Wolfie:</p>
<p>$2^{-1} \bmod 5 = 3$</p>
<p><a href="http://www.wolframalpha.com/input/?i=2%5E-1+mod+5" rel="nofollow">http://www.wolframalpha.com/input/?i=2%5E-1+mod+5</a></p>
<p>Why is that?</p>
| André Nicolas | 6,312 | <p>The result says that the (multiplicative) inverse of $2$ modulo $5$ is $3$. This is another way of saying that $3\cdot 2= 1\bmod{5}$.</p>
<p>In general, if $p$ is a prime, and $a$ is a number between $1$ and $p-1$, we can say that $a*{-1}\bmod{p}$ is the number $b$, between $1$ and $p$, such that $ba=1\bmod{p}$.</p... |
1,105,352 | <p>Here are a few definition from my workbook:</p>
<p>Let $a\in\Bbb{R}\cup\{+\infty, -\infty\}$, $D\subset\Bbb{R}$. Then point $a$ is a limit point of $D$ if and only if there exists a sequence $\{x_n\}_{n \geq n_0}$ with terms in set $D - \{a\}$ such that $x_n \rightarrow a$.</p>
<p>And now we have a limit of a func... | hmakholm left over Monica | 14,366 | <p>Depending on what situation you're in, it can either be convenient to say that there is a limit in such cases, or to insist that only the one-sided limit exists. Neither choice is inherently wrong, but of course it pays to be consistent in our use of words. (But beware that textbooks are not always consistent in thi... |
2,378,717 | <p>I wondered many times how the editors of mathematical journals know that the content of a paper or article that was submited by an author is original, this is, that the main content of such calculations and reasonings is unpublished.</p>
<blockquote>
<p><strong>Question.</strong> How do editors of mathematical jo... | thetha | 283,262 | <p>Usually they have payed staff, some Prof. who are working in this field. So they send your paper to them, they review it. Based on their review it will be published or not. It depends on the quality of the journal, how well they check your work. </p>
|
61,659 | <p>How to find: $$\lim_{x \to \frac{\pi}{2}} \frac{\tan{2x}}{x - \frac{\pi}{2}}$$ I know that $\tan(2\theta)=\frac{2\tan\theta}{1-\tan^{2}\theta}$ but don't know how to apply it here.</p>
| Beni Bogosel | 7,327 | <p><a href="http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule" rel="nofollow">l'Hospital Rule</a> works here. The indetermination is $0/0$. </p>
|
3,276,264 | <h1>My attempt</h1>
<p>Based on the sine rule and the graph of <span class="math-container">$\sin A = k a$</span> (where <span class="math-container">$k$</span> is a constant) in interval <span class="math-container">$(0,\pi)$</span>,
increasing <span class="math-container">$a$</span> up to <span class="math-containe... | Thomas Andrews | 7,933 | <p>If <span class="math-container">$a^2+b^2=c^2,$</span> with <span class="math-container">$a,b,c$</span> positive, you'd have: <span class="math-container">$c>a,b>0$</span> and thus <span class="math-container">$ca^2>a^3$</span> and <span class="math-container">$cb^2>b^3.$</span> So you get:</p>
<p><span ... |
225,481 | <p>I would like to write a program that gives me the Klein Gordon Equations given a metric. I will explain.</p>
<p>My code is in the following:</p>
<p><strong>I) Standard Quantities</strong></p>
<p>I have no doubt here the first part is <a href="http://web.physics.ucsb.edu/%7Egravitybook/mathematica.html" rel="nofollow... | Alex Trounev | 58,388 | <p>We can use this code for Klein-Gordon:</p>
<pre><code>Clear[coord, metric, inversemetric, affine, riemann, ricci, scalar, \
einstein, t, r, θ, ϕ]
n = 4;
coord = {t, r, θ, ϕ};
g = {{-(1 - ((2*m)/(r))), 0, 0, 0}, {0, 1/(1 - ((2*m)/(r))), 0,
0}, {0, 0, r^2, 0}, {0, 0, 0, r^2*(Sin[θ]*Sin[θ])}};
g1 = Simplify[Inv... |
2,703,559 | <p>My assignment asks me to prove that the only automorphism of order 2 of $\mathbb{Z}_q$ is $m \mapsto -m$ for $q = 3$ and $q = 5$ and $q = 7$. I have been stuck for ages, and now I wonder if it is true. I need some help to get started. This is my attempt: Assume $k$ and $q$ are relative prime (this is necessary other... | Dietrich Burde | 83,966 | <p>For $q$ prime, $\mathbb{Z}_q$ is a field, so $k^2-1=0$ there has only two solutions, $k=1$ or $k=-1$.</p>
|
2,941,579 | <p>In Taylor's series, to determine the number of terms needed to obtain the desired accuracy, sometimes one needs to solve inequalities of the form <span class="math-container">$$\frac{a^n}{n!}<b,$$</span>
where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are fixed positive n... | Siong Thye Goh | 306,553 | <p><span class="math-container">$$|4x-12|=4|x-3|$$</span></p>
<p>we have <span class="math-container">$|x-3| < \delta$</span>, hence </p>
<p><span class="math-container">$$|4x-12|=4|x-3|< 4 \delta$$</span></p>
|
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | Denis | 66,241 | <p>These interview problems are sometimes weird, where notations are bad, rules are arbitrary, and they expect only one answer where several could fit.</p>
<p>Here is one, which could be the expected one, but probably not:</p>
<p>To compute $a \times b$, take the numerator of $\dfrac{ab^2}{6}$ after simplification of... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | maddog2k | 164,410 | <p>56 Did the question explicitly say there was a pattern to be found or is it just like you've presented it here? The symbols for multiplication(x) and equality(=) have well defined mathematical meaning and therefore 7 x 8 = 56 regardless of what misleading noise was written before. It may just be a test of the abil... |
63,641 | <p>Here is the problem: Two mathematicians meet at a bar. They like each other and tend to collaborate. But it is not so clear what problems could be of common interest to both of them. Of course, the traditional way is they keep describing they work or their field in general so that hopefully they catch something at t... | Qiaochu Yuan | 290 | <p>Here's a comment which might as well be written down. If $f$ is required to be an inner automorphism, then for $G$ finite this question can be understood using the character table of $G$:</p>
<blockquote>
<p>$x$ is conjugate to its inverse if and only if $\chi(x)$ is real for all characters $\chi$.</p>
</blockquo... |
642,497 | <p>How do we prove by vector method that "if the diagonals of a trapezium have equal length then the non-parallel sides of the trapezium have equal length." ? </p>
<p>(taking $ABCD$ to be the trapezium with
$AD$ || $BC$ and $O$ the intersection point of diagonals , if it can be shown by vector method that
$OB=OC$ , ... | Claude Leibovici | 82,404 | <p>I think that an easy way is to build a Taylor expansion around $x=-2$. As a result, you should have
$$9 + 32 (2 + x) - 62 (2 + x)^2 + 38 (2 + x)^3 - 10 (2 + x)^4 + (2 + x)^5$$</p>
|
642,497 | <p>How do we prove by vector method that "if the diagonals of a trapezium have equal length then the non-parallel sides of the trapezium have equal length." ? </p>
<p>(taking $ABCD$ to be the trapezium with
$AD$ || $BC$ and $O$ the intersection point of diagonals , if it can be shown by vector method that
$OB=OC$ , ... | Arthur | 15,500 | <p>Alternative solution: You know you need $1\cdot (x + 2)^5$, since $1$ is the coefficient in front of $x^5$. Calculate $$x^5 - 2x^3 + 6x^2 + 1 - 1\cdot(x + 2)^5$$ and deduce how many $(x + 2)^4$ you need. Rinse and repeat.</p>
|
758,181 | <p><img src="https://i.stack.imgur.com/erzie.png" alt="enter image description here"></p>
<p>Above is just an example I'm trying to work from as I have the solutions. I've seen lots of definitions of what inversions are but they use signs like sigma, and it doesn't really explain what the sigma is. So if anyone could ... | Fly by Night | 38,495 | <p>If $\vec{u}\cdot \vec{w} = 2\times 1 + 1 \times 3$ then that implies that $\vec{u} = (2,1)$ and $\vec{w}=(1,3)$.</p>
<p>If that is the case then $\|\vec{w}\| = \sqrt{1^2+3^1} = \sqrt{10}$ and not $\sqrt{45}$ as you seem to think.</p>
<p>From your calculation that $\|\vec{w}\| = \sqrt{3^2+(-6)^2}$, I assume that $\... |
758,181 | <p><img src="https://i.stack.imgur.com/erzie.png" alt="enter image description here"></p>
<p>Above is just an example I'm trying to work from as I have the solutions. I've seen lots of definitions of what inversions are but they use signs like sigma, and it doesn't really explain what the sigma is. So if anyone could ... | Matias Morant | 140,918 | <p>$$|W|=\sqrt{3^2+1^2}=\sqrt{10}$$
$$\cos ^{-1}\left(\frac{5}{\sqrt{10} \sqrt{5}}\right)=45 {}^{\circ}$$</p>
|
398,002 | <p>There is a <a href="https://en.wikipedia.org/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" rel="noreferrer">classical correspondence</a> between Lie algebras (over <span class="math-container">$\mathbb{R}$</span> or <span class="math-container">$\mathbb{C}$</span>) and Lie groups in the finite dimensional case:... | Alexander Schmeding | 46,510 | <p>Another source discussing the (non-)integrability of infinite dimensional Lie algebras is
T. Robart: Around the exponential mapping, in: Banyaga et al (Eds) Infinite dimensional Lie groups in geometry and representation theory, 2002</p>
<p>There a detailed account of "the exponential catastrophe" can be fo... |
1,201,352 | <p>I have a very simple question which is bugging me.</p>
<p>We are 3 roommates and our total electricity bill is $61 this month,
I was home for the whole month,
Friend X was home for 15 days,
Friend Y was home for 20 days</p>
<p>now the easy question, how much would each person would pay?</p>
<p>My calculation is <... | Jimmy R. | 128,037 | <p>One way to do it is to think that this month had $$30+15+20=65$$ separate days (or say: electricity days). The electricity bill is $61, and it should be splitted as follows</p>
<ol>
<li>you should pay $$\$61\cdot \frac{30}{65}=\$28.15$$</li>
<li>friend X should pay $$\$61\cdot \frac{15}{65}=\$14.08$$ and</li>
<li>f... |
3,534,896 | <p>If <span class="math-container">${\sqrt 3} - {\sqrt 2}, 4- {\sqrt 6}, p{\sqrt 3} - q {\sqrt 2}$</span> form a geometric progression, find the values of p and q.</p>
<p>So I take the second term <span class="math-container">$4-{\sqrt 6} =( {\sqrt 3} - {\sqrt 2}) (r)$</span> , where r is the common ratio.</p>
<p><... | Parcly Taxel | 357,390 | <p>You've made a computation error, perhaps?
<span class="math-container">$$(4-\sqrt6)(2\sqrt3+\sqrt2)=8\sqrt3+4\sqrt2-6\sqrt2-2\sqrt3=6\sqrt3-2\sqrt2$$</span></p>
|
859,493 | <p>Given that any fixed integer $n>0$, let $S=\{1,2,3,4,...,n\}$. Now a Red-Blue subset of $S$ is called $T$. Every element of $T$ is given a colour (either red or blue). For instance $\{17 (\text{red})\}, \{1 (\text{red}), 5 (\text{red})\}$ and $\{1(\text{red}),5(\text{blue})\}$ are $3$ different subsets of $S$.</p... | agha | 118,032 | <p>$$(2x^2+x-3)^8=(2x^2+x-3)\cdot(2x^2+x-3) \cdots (2x^2+x-3)$$</p>
<p>Note that something like $ax$ you get only in one way: if you choose $x$ from one of eight parentheses and $-3$ from the other, so this coefficient is $\displaystyle 8 \cdot (-3)^7 $.</p>
|
2,670,258 | <p>Just to explain the motivation behind some other thing, I shall ask first this question:</p>
<blockquote>
<p>Is $\frac{1}{x}$ continuosly differentiable (or we can examine
$\frac{1}{x^3}$ also) on the set $$E = (- \infty, 0) \cup (0,
> \infty)$$</p>
</blockquote>
<p>The reason why I'm asking this is that w... | 5xum | 112,884 | <p>The funciton $f(x)=\frac1x$ is an element of $C^1(\mathbb R\setminus\{0\})$.</p>
<p>To comment on your question about the initial value problems, I would need more information about what is unclear to you.</p>
|
2,706,061 | <p>This can be rewritten as $A\vec{x}=0$, where $A=(1, -2, 3), \vec{x}=(x, y, z)^T$.
I understand that one can find vectors perpendicular to this place by finding the basis for the null space, as the null space is perpendicular to the row space.</p>
<p>This basis is $(2, 1, 0), (-3, 0, 1)$. I believe that the space de... | José Carlos Santos | 446,262 | <p>Because we're working in a $3$-dimensional vector space and we have found $2$ linearly independent solutions $s_1$ ans $s_2$. Therefore, the space $S$ of <em>all</em> solutions must have dimension <em>at least</em> $2$. But it can't be $3$, since that would meant that $S=\mathbb{R}^3$, but it isn't ($(1\ \ -2\ \ 3)^... |
4,453,356 | <p>I had been having trouble understanding a proof of the irrational nature of √2.</p>
<p>I found this proof in the first page of the foreward to <a href="https://www.cs.ru.nl/%7Efreek/comparison/comparison.pdf" rel="nofollow noreferrer"><em>17 theorem provers of the world</em></a> where a 'geometrical proof' (is that ... | Aaron Montgomery | 485,314 | <p>This is an example of <strong>argument by contradiction</strong>. The key idea is to take a fact that you hope to eventually show is false, and instead assume that it is <em>true</em>, but that the assumption of truth leads you to a contradiction.</p>
<p>The classic example of this style of argument is the proof tha... |
2,003,660 | <p>I got this exercise from the textbook Book of Proof, CH4 E12. I've tackle this problem in the following manner:</p>
<p>Suppose x is a real and $0 < x < 4$, it follows that,</p>
<p>\begin{align*}
&\Rightarrow 0 - 2 < x - 2 < 4 - 2 \\
&\Rightarrow 4 < (x - 2)^2 < 4\\
&\Rightarrow ... | Guy Fsone | 385,707 | <p>Hint See that: $0\le x\le 4 $ implies $0\le 4-x\le 4$ and then,
$$0\le x(4-x) =-(x-2)^2+4\le4$$</p>
<p>Thus directly implies your inequality.$$\dfrac{4}{x(4 - x)} \geq 1.$$</p>
|
1,465,755 | <p>How do I find the angles $\alpha ,\beta ,\theta $ between the vector $(1,0,-1)$ and the unit vectors $i,j,k$ along the axes?</p>
<p>This question is not making sense to me. I know that in order to find the angle between any two nonzero vectors, I just have to take their dot product and divide it by the product of t... | Ben Grossmann | 81,360 | <p>For example, we have
$$
(1,0,-1)\cdot \hat i = (1,0,-1) \cdot (1,0,0) =
(1)(1) + (0)(0) + (-1)(0) = 1
$$</p>
|
2,375,714 | <p>I know this is a really easy question but for some reason I'm having trouble with it.</p>
<p>If $M$ is an object in an additive category $\mathcal C$, and $\text{Hom}_{\mathcal C}(M,M) = 0$, then $M = 0$.</p>
<p>I know that this implies $\text{id}_M =0$ but I'm having trouble showing that $M$ satisfies the conditi... | Pece | 73,610 | <p>You always have that $0 \to M \to 0$ equals $\mathrm{id}_0$, and you just showed that $M\to 0 \to M$ equals $\mathrm{id}_M$. Hence $0\to M$ is an iso (with inverse $M \to 0$).</p>
<p>Remark: the additive structure is not really used here, this is rather a property of pointed categories.</p>
|
160,765 | <p>I have a functor $F:C \to D$ between poset-enriched categories, and I'd like to show that the induced map on classifying spaces is a homotopy-equivalence. To this end, I am trying to establish the presence of initial objects in all the fibers $d\setminus F$ and use the 2-categorical version of Quillen's Theorem A du... | Zhen Lin | 11,640 | <p>There are several possible definitions of initial object in a 2-category <span class="math-container">$\mathfrak{K}$</span>; which one is appropriate depends on your applications.</p>
<ol>
<li><p>A 2-category has an underlying ordinary category, so we may just reuse the standard definition of initial object.</p>
</l... |
457,427 | <p>Find the derivative of the functions:<br>
$$\int_{x^2}^{\sin(x)}\sqrt{1+t^4}dt$$<br></p>
<p>In class we had the following solution:<br>
By the fundamental theorem of calculus we know that <br>
$$\left(\int_a^xf(t)dt\right)'=f(x)$$ So<br>
$$\int_{x^2}^0\sqrt{1+t^4}dt+\int_0^{\sin(x)}\sqrt{1+t^4}dt=$$<br>
$$\int_0^{\... | Eric Auld | 76,333 | <blockquote>
<p>However, if we have that $\left(\int_a^xf(t)dt \right)'=f(x)$ wouldn't the answer just be $$\sqrt{1+\sin(x)^4}-\sqrt{1+x^8}?$$</p>
</blockquote>
<p>Note that hidden in the first formula you mention is multiplication by $\frac {d}{dx} (x)$. The reason you don't see it is because it is equal to one! It... |
4,552,098 | <p>The rules: Two people rolls a single dice. If the dice rolls 1,2,3 or 4, person A gets a point. For the rolls 5 and 6, person B gets a point. One person needs a 2 point lead to win the game.</p>
<p>This is a question taken from my math book. The answer says the probability is <span class="math-container">$\frac{4}{5... | N. F. Taussig | 173,070 | <p>As you showed, person <span class="math-container">$A$</span> can win the first round of two rolls with probability <span class="math-container">$4/9$</span>. However, as Fishbane pointed out in the comments, person <span class="math-container">$A$</span> can also win if both person <span class="math-container">$A$... |
2,081,001 | <p>The number $\frac{22}{7}$ is irrational in our base-$10$ system, but in, say, base-$14$, it is rational (it comes out to $3.2$ in that system).</p>
<p>It's easy for fractions that are irrational as decimals, as you can just represent them in a base that's double the denominator of the fraction. However, what if I h... | FuzzyPixelz | 376,394 | <p>Always refer to definitions, a number $x$ is called <em>irrational</em> iff $\forall p,q \in \Bbb Z : x\neq\frac pq$, that is when you can't express it as ratio of two integers, not based on how it <em>looks</em> using a different number system. <strong>EDIT:</strong> This means that you can never find two integers ... |
2,038,617 | <p>The question asks to show that $$\sum_{k=0}^{3n}(-3)^k \binom{6n}{2k}=2^{6n}$$ by considering the binomial expansion </p>
<p>I thought about the use of $$(1+z)^n=\sum_{k=0}^{n}\binom{n}{k}z^k$$
with suitable complex number $z$, as the formula shows the $(-3)^k$ term might suggest the use of complex number, which ta... | Robert Z | 299,698 | <p>Consider the real part of the binomial expansion of $(1+i\sqrt{3})^{6n}=(2e^{i\pi/3})^{6n}=2^{6n}$:
$$2^{6n}=\mbox{Re}\left((1+i\sqrt{3})^{6n}\right)=
\mbox{Re}\left(\sum_{j = 0}^{6n}{6n \choose j}(i\sqrt{3})^{j} \right)
=\sum_{k = 0}^{3n}{6n \choose 2k}(-3)^k.$$</p>
|
2,737,144 | <p>I tried to prove it by contradiction. </p>
<p>Suppose it is not true that $1\ge\frac{3}{x(x-2)}$, so $1\lt\frac{3}{x(x-2)}$. Then $\frac{3}{x(x-2)}-1\gt0$. Multiply both sides of $\frac{3}{x(x-2)}-1\gt0$ by ${x(x-2)}$.</p>
<p>$(\frac{3}{x(x-2)}-1\gt0)({x(x-2)}\gt0(x(x-2)$</p>
<p>${3-(x(x-2)\gt0}$</p>
<p>${... | Dr. Sonnhard Graubner | 175,066 | <p>From $$x-3\geq 0$$ follows:
$$x-1\geq 2$$
$$(x-1)^2\geq 4$$
$$x^2-2x+1\geq 4$$
$$x^2-2x\geq 3$$
$$x(x-2)\geq 3$$ so $$1\geq \frac{3}{x(x-2)}$$</p>
|
1,679,772 | <p>A binary relation R on $N×N$ is defined as follows$: (a,b)R(c,d)$ if $a≤c$ or $b≤d$. Consider the following
propositions:</p>
<p>$P: R$ is reflexive</p>
<p>$Q: R$ is transitive</p>
<p>Which one of the following statements is TRUE?</p>
<ol>
<li>Both $P$ and $Q$ are true. </li>
<li>$P$ is true and $Q$ is false. </... | MooS | 211,913 | <p>We can also directly use the definition of compactness: For the sake of notation, let $K_1$ be the ambient topological space.</p>
<p>Let the intersection be $\emptyset$, then we have an open cover $K_1 = \bigcup_{i=2}^\infty K_i^c$. By definition we find a finite subcover, i.e. $K_1 = \bigcup_{i=1}^N K_i^c$ for som... |
14,765 | <p>I like to make the "dominoes" analogy when I teach my students induction.</p>
<p>I recently came across the following video:</p>
<p><a href="https://www.youtube.com/watch?v=-BTWiZ7CYoI" rel="noreferrer">https://www.youtube.com/watch?v=-BTWiZ7CYoI</a></p>
<p>In this video, a sequence of concrete block wall caps ar... | Acccumulation | 8,989 | <p>The square root symbol <span class="math-container">$\sqrt {\cdot}$</span> means "the principal square root". For positive numbers, it's understood that it means the positive square root. But your students clearly haven't advanced enough for this to be implicit. Insist that if they want to represent the positive squ... |
14,765 | <p>I like to make the "dominoes" analogy when I teach my students induction.</p>
<p>I recently came across the following video:</p>
<p><a href="https://www.youtube.com/watch?v=-BTWiZ7CYoI" rel="noreferrer">https://www.youtube.com/watch?v=-BTWiZ7CYoI</a></p>
<p>In this video, a sequence of concrete block wall caps ar... | WeCanLearnAnything | 7,151 | <p>Telling and explaining is really not enough. Working exclusively in <em>square</em> roots of numbers all but guarantees blocking vs interleaving problems and encourages shallow robotic thinking and generalizing involving roots.</p>
<p>A few ideas:</p>
<ol>
<li>Error analysis. Have them look at a <em>variety</em> o... |
14,765 | <p>I like to make the "dominoes" analogy when I teach my students induction.</p>
<p>I recently came across the following video:</p>
<p><a href="https://www.youtube.com/watch?v=-BTWiZ7CYoI" rel="noreferrer">https://www.youtube.com/watch?v=-BTWiZ7CYoI</a></p>
<p>In this video, a sequence of concrete block wall caps ar... | Daniel R. Collins | 5,563 | <p>We are all dealing with this, of course. In a theoretical/philosophical sense, you can't "make" students learn anything (<a href="https://www.phrases.org.uk/meanings/you-can-lead-a-horse-to-water.html" rel="nofollow noreferrer">"You can lead a horse to water..."</a>). In a practical/punitive sense, this is what cour... |
14,765 | <p>I like to make the "dominoes" analogy when I teach my students induction.</p>
<p>I recently came across the following video:</p>
<p><a href="https://www.youtube.com/watch?v=-BTWiZ7CYoI" rel="noreferrer">https://www.youtube.com/watch?v=-BTWiZ7CYoI</a></p>
<p>In this video, a sequence of concrete block wall caps ar... | Dominique | 8,190 | <p>When you just know the basics of complex number theory, then you know that <span class="math-container">$x^n = a$</span> has <span class="math-container">$n$</span> solutions, so for <span class="math-container">$n=2$</span>, <span class="math-container">$x^n = a$</span> has two solutions.</p>
<p>What's the big dea... |
197,667 | <p>I have a table with 10 columns, obtained as import from file. I would like to use 4 columns as <code>{x, y, z, u}</code> params for <code>BubbleChart3d</code>, and an additional column to color the points. I'm unable to make this work. The code I have is like this:</p>
<p><code>t</code> is table from a file import.... | Michael E2 | 4,999 | <blockquote>
<p>Does this mean I can be 100% confident that the expression <span class="math-container">$f$</span>
does not converge at <span class="math-container">$x=1/a$</span>?</p>
</blockquote>
<p><strong><em>No:</em></strong></p>
<pre><code>f = ((E + x)^2 - E^2 - x^2 - 2 E x)/(a x - 1);
f /. x -> 1/a
(... |
1,574,290 | <p>How do I prove this? </p>
<p>For the Fibonacci numbers defined by $f_1=1$, $f_2=1$, and $f_n = f_{n-1} + f_{n-2}$ for $n ≥ 3$, prove that $f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n$ for all $n≥ 1$.</p>
| Alexander Heyes | 286,131 | <p>Subscripts and superscripts are a real nuisance in maths - mathematicians tend to privilege intense efficiency and minimalism of notation over readability - it makes sense in the long run, but for a learner it is quite confusing. </p>
<p>Try reading any subscript as "the first $f$, the second $f$" etc., and any sup... |
2,138,963 | <p>finding $\displaystyle \int^{\pi}_{-\frac{\pi}{3}}\bigg[\cot^{-1}\bigg(\frac{1}{2\cos x-1}\bigg)+\cot^{-1}\bigg(\cos x - \frac{1}{2}\bigg)\bigg]dx$</p>
<p>Attempt:</p>
<p>\begin{align}
& \int^{\frac{\pi}{3}}_{-\frac{\pi}{3}}\bigg[\cot^{-1}\bigg(\frac{1}{2\cos x-1}\bigg)+\cot^{-1}\bigg(\cos x - \frac{1}{2}\bigg... | Claude Leibovici | 82,404 | <p><em>Assuming that the solution requires numerical integration.</em></p>
<p>Considering $$f(x)=\cot^{-1}\bigg(\frac{1}{2\cos x-1}\bigg)+\cot^{-1}\bigg(\cos x - \frac{1}{2}\bigg)$$ let us compute $$I_1(k)=\int_{-\frac \pi 3}^{\frac \pi 3-10^{-k}}f(x)\,dx\qquad I_2(k)=\int_{\frac \pi 3+10^{-k}}^\pi f(x)\,dx\qquad I(k)... |
2,423,086 | <p>Show, by listing the elements or how we can list the elements, that </p>
<p>$\mathbb{N}^3$ = $\mathbb{N×N×N}$ is a countable set.</p>
<p><strong>Attempt at a solution:</strong></p>
<p>I was thinking about making use of Cantor's diagonal argument. If this were $\mathbb{N×N}$ I could just do:</p>
<p>(1,1), (1,2), ... | Eric Towers | 123,905 | <p>To go your way, $(x,y,z) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N}$ with $x+y+z = c$ is a finite set. Viewed in $\mathbb{R}^3$ it is a triangle. So explain how to snake across each of these triangles (which will look remarkably like what you are doing in $\mathbb{N}^2$).</p>
<p>Alternatively, let $x = \c... |
11,652 | <p>Last August, so-called "review audits" <a href="https://math.meta.stackexchange.com/q/10740/43351">were introduced</a> on MSE.</p>
<p>These are fake review samples added to the review queues, designed to catch those users who (in a dash to get a badge) practice mindless button mashing instead of careful considerati... | Lord_Farin | 43,351 | <p>Yes.</p>
<p>Please disable the review audits.</p>
|
1,489,078 | <p>Let $P(S)$ denotes the power set of set $S$. Which of the following is always true$?$</p>
<ol>
<li>$P(P(S)) = P(S)$</li>
<li>$P(S) ∩ P(P(S)) = \{ Ø \}$</li>
<li>$P(S) ∩ S = P(S)$</li>
<li>$S ∉ P(S)$</li>
</ol>
<hr>
<p>I try to explain $:$
If $S$ is the set $\{x, y, z \},$ then the subsets of S are:</p>
<p>${}$ (... | layman | 131,740 | <p>No, $\{\}$ is the empty set, and $\{ \{ \} \}$ is the set <em>containing</em> the empty set. They aren't the same -- $\{\}$ has <strong>no elements</strong>, and $\{ \{ \} \}$ has <strong>one element</strong>.</p>
<p>By the way, in $P(P(S))$, you forgot to add the element $\{ \}$, since the empty set is a subset o... |
317,666 | <p>What is the difference between $\mathbb{H}$ and $Q_8$? Both are called <em>quaternions.</em></p>
| Chris Brooks | 7,676 | <p>A good analogy here is the difference between the complex numbers $\mathbb{C}$ and the cyclic group with 4 elements, which can be realized as the group $\{1, i, -1, -i\}\subset\mathbb{C}$ with complex multiplication.</p>
<p>$\mathbb{C}$ is a 2-dimensional vector space over $\mathbb{R}$, and there is a group inside ... |
102,932 | <p>This is a naive question but I hope that the answers will be educational. When is it the case that a finitely presented group $G$ admits a faithful $2$-dimensional complex representation, e.g. an embedding into $\text{GL}_2(\mathbb{C})$? (I am mostly interested in sufficient conditions.) </p>
<p>I think I can figur... | HJRW | 1,463 | <p>This is a very nice question which, as Agol says, is probably out of reach at the moment. To say that there is a classification of the fp subgroups of $GL_2(\mathbb{C})$ would be to say that the set of presentations of those subgroups is recursively enumerable and has solvable Isomorphism Problem. It's hard to gue... |
102,932 | <p>This is a naive question but I hope that the answers will be educational. When is it the case that a finitely presented group $G$ admits a faithful $2$-dimensional complex representation, e.g. an embedding into $\text{GL}_2(\mathbb{C})$? (I am mostly interested in sufficient conditions.) </p>
<p>I think I can figur... | YCor | 14,094 | <p>For a finitely generated group $G$, a necessary and sufficient condition to be embeddable into $\text{GL}_2(\mathbf{C})$ is the following strengthening of being residually finite: for every finite subset $F$ of $G$ and any $m$ there exists a finite field $K$ of characteristic $>m$ and a homomorphism from $G$ to $... |
411,549 | <p>Here is an modular equation</p>
<p>$$5x \equiv 6 \bmod 4$$</p>
<p>And I can solve it, $x = 2$.</p>
<p>But what if each side of the above equation times <strong>8</strong>, which looks like this</p>
<p>$$40x \equiv 48 \bmod 4$$</p>
<p>Apparently now, $x = 0$. Why is that? Am I not solving the modular equation in... | Key Ideas | 78,535 | <p>Hint: mod $\,4\!:\,\ 40\equiv 0\equiv 48,\ $ so $\,\ 40\cdot x\equiv 48\iff 0\cdot x\equiv 0,\ $ true for all $\,x.$</p>
<p>Generally, scaling an equation by a noninvertible factor may increase the solution set.</p>
|
702,506 | <p>We have an exam in $3$ hours and I need help how to solve such trigonometric equations for intervals.</p>
<p>How to solve</p>
<p>$$\sin x - \cos x = -1$$</p>
<p>for the interval $(0, 2\pi)$.</p>
| Michael Hoppe | 93,935 | <p>Since $\cos(\pi/4)=\sin(\pi/4)=\sqrt{2}/2$ we have using the angle sum theorem
$$\sin(x)\sin(\pi/4)-\cos(x)\cos(\pi/4)=-\sqrt{2}/2\iff\cos(x+\pi/4)=\sqrt{2}/2$$
$$\iff
x+\pi/4=\pi/4\lor x+\pi/4=7\pi/4\iff x=3\pi/2$$
as $x\in(0,2\pi)$.</p>
|
716,767 | <p>I feel like an idiot for asking this but i can't get my formula to work with negative numbers</p>
<p>assume you want to know the percentage of an increase/decrease between numbers</p>
<pre><code>2.39 1.79 =100-(1.79/2.39*100)=> which is 25.1% decrease
</code></pre>
<p>but how would i change this fo... | ismael | 55,205 | <h2>Conventional Formula</h2>
<p>The conventional formula for computing the relative growth between two values <span class="math-container">$a$</span> and <span class="math-container">$b$</span> is <span class="math-container">$\displaystyle \frac{b-a}{a}$</span>.</p>
<p>For example, if <span class="math-container">$a ... |
2,721,372 | <p>A sequence is defined by $a_1=2$ and $a_n=3a_{n-1}+1 $ .Find the sum $a_1+a_2+\cdots+a_n$</p>
<p>how to find sum $a_1=2,a_2=7,\ldots$</p>
<p>Also i found the value of $a_n=\frac{5}{6}\cdot3^n-\frac{1}{2}$</p>
| Mauro ALLEGRANZA | 108,274 | <p>The proof relies on <a href="https://en.wikipedia.org/wiki/Principle_of_explosion#Proof" rel="nofollow noreferrer">Ex falso</a> :</p>
<blockquote>
<p>$\vdash \lnot P \to (P \to Q)$.</p>
</blockquote>
<p>We have to apply it in the form :</p>
<blockquote>
<p>$\lnot (x \in \emptyset) \to (x \in \emptyset \to x \... |
2,380,880 | <p>The following question was asked in an exam:<br>
Consider the problem:<br>
Maximize $2y_1+3y_2+5y_3+4y_4$<br>
subject to<br>
$y_1+y_2\leq 1,$ $y_2+y_3\leq 1,$<br>
$y_4+y_1\leq 1,$ $y_3+y_4\leq 1$ and $y_i\geq 0$ for i=1,2,3,4.<br>
Then the optimum value is<br>
1. equal to 8<br>
2. between 8 and 9<br>
3. greater tha... | farruhota | 425,072 | <p>The answer is $D$.</p>
<p>Express it as:
$$2y_1+3y_2+5y_3+4y_4=2(\underbrace{y_1+y_2}_{\le 1})+(\underbrace{y_2+y_3}_{\le 1})+4(\underbrace{y_3+y_4}_{\le 1})\le7.$$
Equality occurs when $(y_1,y_2,y_3,y_4)=(1,0,1,0); (\frac12,\frac12,\frac12,\frac12);(0,1,0,1); etc.$</p>
|
15,316 | <blockquote>
<p>What is the length $f(n)$ of the shortest nontrivial group word $w_n$ in $x_1,\ldots,x_n$ that collapses to $1$ when we substitute $x_i=1$ for any $i$?</p>
</blockquote>
<p>For example, $f(2)=4$, with the commutator $[x_1,x_2]=x_1 x_2 x_1^{-1} x_2^{-1}$ attaining the bound. </p>
<p>For any $m,n \ge... | Erik Demaine | 4,015 | <p>In an unpublished manuscript "Picture-Hanging Puzzles" by Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, and Joseph S. B. Mitchell, we prove the $O(n^2)$ upper bound that comes from iterated commutator with a balanced split, same as sequence A073121. (Indeed, the manuscript cites that sequence.) We conjecture... |
4,107,656 | <p>Let <span class="math-container">$G$</span> be a finite abelian group with a neutral element <span class="math-container">$e$</span>. Prove that for any element <span class="math-container">$g$</span> of <span class="math-container">$G$</span>: <span class="math-container">$g^{|G|}=e$</span>. <span class="math-conta... | Community | -1 | <p>Slight variation on the proof. It's straightforward to show that multiplication by an element of <span class="math-container">$G$</span> defines a bijection. Now that implies that any coset <span class="math-container">$gH$</span> has order <span class="math-container">$|H|$</span>, for <span class="math-container... |
18,530 | <p>Sorry about the title, I have no idea how to describe these types of problems.</p>
<p>Problem statement:</p>
<p>$A(S)$ is the set of 1-1 mappings of $S$ onto itself. Let $S \supset T$ and consider the subset $U(T) = $ { $f \in A(S)$ | $f(t) \in T$ for every $t \in T$ }. $S$ has $n$ elements and $T$ has $m$ eleme... | NebulousReveal | 2,548 | <p><a href="https://math.stackexchange.com/questions/16342/point-of-logarithms">This</a> question might be of relevance to you. It asks about the purpose of logarithms. <a href="http://www.math.umt.edu/tmme/vol5no2and3/TMME_vol5nos2and3_a14_pp.337_344.pdf" rel="nofollow noreferrer">This</a> site gives a brief history o... |
18,530 | <p>Sorry about the title, I have no idea how to describe these types of problems.</p>
<p>Problem statement:</p>
<p>$A(S)$ is the set of 1-1 mappings of $S$ onto itself. Let $S \supset T$ and consider the subset $U(T) = $ { $f \in A(S)$ | $f(t) \in T$ for every $t \in T$ }. $S$ has $n$ elements and $T$ has $m$ eleme... | Travelling Salesman | 141,167 | <p>As Yuan says above, logarithms are numbers describing numbers, intended to give more information about something but in less space or time. </p>
<p>If you want a very simple everyday analogy you could consider some physical cases like steepness, density, or percentages for price increases. All of these are like cru... |
655,005 | <p>Show that if $k \in \mathbb{Z}$, then the integers $6k-1$, $6k+1$, $6k+2$, $6k+3$, and $6k+5$ are pairwise relatively prime.
I am still new and uncomfortable with proofs. Any help would be great. </p>
| MJD | 25,554 | <p>The short answer to the issue raised your question is this: $K^\ast$ is the set of <em>finite</em> sequences of elements of $K$. If $K$ is a finite set, then $K^\ast$ is countable, and Cantor's theorem tells us that the real numbers are not countable.</p>
<p>The mapping may be injective, but the real question is w... |
4,473,264 | <p>I have part of a circle described by three two dimensional vectors.</p>
<ul>
<li>start point <code>s1</code></li>
<li>center point <code>c1</code></li>
<li>end point <code>e</code></li>
</ul>
<p>I move the start point <code>s1</code> by <code>m1</code>, which is a <strong>known</strong> two dimensional vector. My qu... | Lee Mosher | 26,501 | <p>One thing which has not been mentioned explicitly in the comments or the other answer is the following fact:</p>
<blockquote>
<p>Your function <span class="math-container">$f(x)=\sqrt{1+\dfrac{1}{x^4}}$</span> <em>does</em> have an indefinite integral, i.e. an antiderivative, on any interval where it is defined.</p>... |
24,361 | <p>Let $X$ be a topological space and let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over $X$.</p>
<p>Of course, if one has a morphism $f : \mathcal{F} \to \mathcal{G}$ such that for all $x\in X$, $f_x : \mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, then it is known that $f$ itself is an isomorphism.</p>
<p... | Martin Brandenburg | 2,841 | <p>Definitely not. If $X$ is a ringed space, then a $\mathcal{O}_X$-module $F$ is called locally free of rank $1$, if $X$ is covered by open subsets $U_i$ such that $F|_{U_i}$ is free of rank $1$ over $\mathcal{O}_{U_i}$. The correspond to line bundles on $X$. Line bundles form a group, called the Picard group of $X$ a... |
24,361 | <p>Let $X$ be a topological space and let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over $X$.</p>
<p>Of course, if one has a morphism $f : \mathcal{F} \to \mathcal{G}$ such that for all $x\in X$, $f_x : \mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, then it is known that $f$ itself is an isomorphism.</p>
<p... | Robin Chapman | 4,213 | <p>Since there are locally free sheaves that are not globally free, the answer is
clearly no. Indeed the existence of the term "locally free" is a meta-proof
of this. For a concrete example, consider the circle embedded as the central circle
in a smooth Mobius strip. Its normal bundle is locally isomorphic to the
const... |
2,783,129 | <p>How many different value of x from 0° to 180° for the equation $(2\sin x-1)(\cos x+1) = 0$?</p>
<p>The solution shows that one of these is true:</p>
<p>$\sin x = \frac12$ and thus $x = 30^\circ$ or $120^\circ$ </p>
<p>$\cos x = -1$ and thus $x = 180^\circ$</p>
<p><strong>Question:</strong> Inserting the $\arcsin... | mallan | 557,176 | <p>$0\leq \sin(x)\leq1$ in quadrants I <em>and</em> II. That is to say $\sin x$ (and $\cos x$) are stuck in a circle which is periodic. There are an infinite amount of $x$ that satisfy for example $\sin(x)=\frac{1}{2}$, but the simplest answer is enough to suffice</p>
<p>Specifically, $\sin^{-1}x$ isn't a function unl... |
138,866 | <p>I have data in a csv file. The first row has labels, and the first column, too.</p>
<pre><code>Datos = Import["C:\\Users\\jodom\\Desktop\\Data.csv"]
</code></pre>
<p>Tha data in the csv file is that:</p>
<pre><code>{{"No", "Vol", "Vel"}, {1, 500, 45}, {2, 700, 67}, {3, 350, 87}, {4,
123, 23}, {5, 587, 45}, {6, 4... | gwr | 764 | <p>What you can do is this:</p>
<pre><code>association = AssociationThread[ First @ data, ## ]& /@ data; (* data = Datos *)
dataset = Association /* Dataset @@ (
association // Query[ All, Lookup[##, "No"] -> Part[##, 2 ;; 3] & ]
)
</code></pre>
<p><img src="https://i.stack.imgur.com/nPuXK.png" alt="M... |
509,856 | <p>Let $K_n$ be a complete $n$ graph with a color set $c$ with $c=\{\text{Red}, \text{Blue}\}$. Every edge of the complete $n$ graph is colored either $\text{Red}$ or $\text{Blue}$. Since $R(3, 3)=6$, the $K_6$ graph must contain at least one monochromatic $K_3$ graph. How can I prove that this graph must contain anoth... | Sean Eberhard | 23,805 | <p>Since $R(3,3)=6$ there is a monochromatic triangle $\Delta$. Let's say it's blue. Look at the other three vertices. If there is no red edge between them then we've found a second blue triangle, so suppose we have found a red edge $xy$, $x,y\notin\Delta$. If there are two blue edges from $x$ to $\Delta$ then we've fo... |
701,176 | <p>A function $f$ is differentiable over its domain and has the following properties:</p>
<ol>
<li><p>$\displaystyle f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$</p></li>
<li><p>$\lim_{h \to 0} f(h) = 0$</p></li>
<li><p>$\lim_{h \to 0} f(h)/h = 1$</p></li>
</ol>
<p>i) Show that $f(0)=0$</p>
<p>ii) show that $f'(x)=1+[f(x)]^... | Lord Soth | 70,323 | <p>Suppose $n^3-6n^2 = 2$ for some integer $n$. It is easy to check that $n$ cannot be odd. Hence $n = 2k$ for some integer $k$. We have $8k^3-24k^2 = 2$, or equivalently, $k^3-3k^2 = \frac{1}{4}$, which is a contradiction since $k$ was supposed to be an integer.</p>
|
3,310,812 | <p>Solve the equation :
<span class="math-container">$$p^k=kl+1, $$</span> with <span class="math-container">$p$</span> a prime number and <span class="math-container">$k,l\ge 1$</span> two integers.</p>
<p>I know that <span class="math-container">$(p,k,l)=(3,1,2)$</span> is a solution, but can we find all solutions t... | Rory Daulton | 161,807 | <p>This problem has two degrees of freedom: there are three column widths, but the sum of those widths is known so the degrees of freedom are reduced to only two. Therefore this will be easier to solve if we use two variables.</p>
<p>Let's use <span class="math-container">$x$</span> for the width of the first column a... |
738,934 | <p>Write an equation for the line tangent to the graph of x=y^2+4 at the point (5,1).</p>
<p>I have found the derivative y'=1/2y but I do not know what to do next.</p>
| user137794 | 137,794 | <p>Find $y'$ at the point $(5, 1)$. That is the slope of the line. Then you can use point-slope formula.</p>
|
738,934 | <p>Write an equation for the line tangent to the graph of x=y^2+4 at the point (5,1).</p>
<p>I have found the derivative y'=1/2y but I do not know what to do next.</p>
| Sam Ye | 139,806 | <p>Since $y^{\prime}=\frac{1}{2y}$, find $y^{\prime}$ by substitute (5, 1) into the equation which is $y^{\prime} = \frac{1}{2}$ = gradient of the equation.</p>
<p>Equation of a line: $$\begin{align}y-y_1=m(x-x_1)\end{align}$$</p>
<p>Thus, the equation should be $$\begin{align}y-1=\frac{1}{2}(x-5)\end{align}$$</p>
<... |
1,335,446 | <p>Ordinary sphere in $\mathbb{R}^3$ is two-dimensional object (2-sphere), i.e. it requires at least two coordinates to define point on a surface. As I notice, however, there is a catch. <br/>
If we use spherical coordinates,$(\phi,\theta)$, then there are two points where $\phi$ is not defined (polar regions). If we ... | Emilio Novati | 187,568 | <p>A spherical surface in $\mathbb{R}^3$ is a two dimension <a href="https://en.wikipedia.org/wiki/Manifold" rel="nofollow">manifold</a> that is not globally <a href="https://en.wikipedia.org/wiki/Diffeomorphism" rel="nofollow">diffeomorphic</a> to a plane. But we can establish a diffeomorphism locally. This means that... |
1,335,446 | <p>Ordinary sphere in $\mathbb{R}^3$ is two-dimensional object (2-sphere), i.e. it requires at least two coordinates to define point on a surface. As I notice, however, there is a catch. <br/>
If we use spherical coordinates,$(\phi,\theta)$, then there are two points where $\phi$ is not defined (polar regions). If we ... | Community | -1 | <blockquote>
<p>some dimension function that is equal three for 2-sphere (but equal two for 2-sphere with two points excluded)? </p>
</blockquote>
<p>Yes: the smallest number $n$ such that the space can be embedded into $\mathbb{R}^n$, in the sense of being diffeomorphic to a submanifold of $\mathbb{R}^n$. This is s... |
18,224 | <p>I need to generate a very large sparse block matrix, with blocks consisting only of ones along the diagonal. I have tried several ways of doing this, but I seem to always run out of memory.</p>
<p>The fastest way of doing this that I've come up with so far is as follows:</p>
<p>(typically, I will need <code>n</cod... | Carl Woll | 45,431 | <p>I think <a href="http://reference.wolfram.com/language/ref/KroneckerProduct" rel="nofollow noreferrer"><code>KroneckerProduct</code></a> is the right tool for this question:</p>
<pre><code>kp[n_,m_] := KroneckerProduct[
SparseArray[IdentityMatrix[m]],
SparseArray[SparseArray[{}, {n,n}, 1], Automatic, 0]
]
<... |
1,460,561 | <p>Let $\lambda =1$ is the eigenvalue corresponding to the single Jordan block $J$. Prove $J^m \sim J$ with an arbitrary positive integer $m$.</p>
<p>My try: Because $\lambda = 1$ is eigenvalue, $(J-I)^m =0$. After that $(J-I)^{m-1} J = (J-I)^{m-1}$. At this point, I do not know how to continue.</p>
| Nyfiken | 284,069 | <p>I would do it this way: $J=I+N$ where $N$ is the $n\times n$ nilpotent upper triangular matrix. Taking the $m$-th power we obtain
$$
J^m=(I+N)^m=\sum_{k=1}^m\binom{m}{k}N^k=I+mN+(\text{super-super-diagonals}),
$$
from what we get $J^m-I=mN+(\text{super-super-diagonals})$ and, therefore,
$$
\text{rank}(J^m-I)=\text{r... |
1,953,628 | <p>0 choices for the 1st person.
17 choices for the 2nd person (must exclude 1st and his/her two neighbours) </p>
<p>For 2 of these choices of 2nd person, there is one shared neighbour, so 15 remaining choices. (e.g. if they are numbered 1 to 20 in a circle, 1st person is #1, 2nd is #3, then people 20,1,2,3,4 are exc... | N. F. Taussig | 173,070 | <p>Assuming the seating assignments are fixed, the problem is tantamount to arranging $17$ blue chairs and $3$ green chairs in a circle so that no two of the green chairs are adjacent. We will solve the problem for a line, then adjust our answer to account for the fact that the chairs are arranged in a circle.</p>
<p... |
754,603 | <p>I have two equations:</p>
<ol>
<li>$x = 2^n (p+i)^{3n}$ </li>
<li>$x = 14^n p^{3n} $</li>
</ol>
<p>Here $n$, $p$, and $i$ are all integers $\geq0$.</p>
<p>I worked out (using a spreadsheet) that if $i > 1$ then the value of x in expression 1 is larger than the value of x in expression 2.</p>
<p>How can I show... | Git Gud | 55,235 | <p>Given two matrices $M_{m\times n},N_{m\times p}$, there are two ways to interpret the entity $\begin{bmatrix} M & N \end{bmatrix}$.</p>
<p>One is the $m\times (n+p)$ matrix whose $(i,j)$ entry is $\begin{cases} (M)_{(i,j)}, &\text{if }j\leq n\\ (N)_{(i, j-n)}, &\text{if }j\ge n+1\end{cases}$.</p>
<p>In... |
3,727,772 | <p><span class="math-container">$Y_1,Y_2,\ldots,Y_{n+1}$</span> be non-empty subsets of <span class="math-container">$\{1,2,3\ldots,n\}$</span>. Prove that there exists non-empty disjoint subsets <span class="math-container">$A_1$</span> and <span class="math-container">$A_2$</span> of <span class="math-container">$\{1... | Daniel | 391,594 | <p>Previously this was a hint. However, as @mindlack pointed out it does not seem to be straightforward to deduce from here there are two disjoint subsets.</p>
<p><strong>Attempt.</strong> Define <span class="math-container">$[n] = \{1, 2, \ldots, n\}$</span> and let <span class="math-container">$\mathcal{P}([n])$</spa... |
806,779 | <p>Prove, without expanding, that
\begin{vmatrix}
1 &a &a^2-bc \\
1 &b &b^2-ca \\
1 &c &c^2-ab
\end{vmatrix} vanishes.</p>
<p>Any hints ?</p>
| adfriedman | 153,126 | <p>Simply add a multiple of the first column to the last
$$\begin{vmatrix} 1 & a & a^2-bc\\1&b&b^2-ac\\1&c&c^2-ab\end{vmatrix}
=\begin{vmatrix} 1 & a & a^2-bc+(1)(ab+ac+bc)\\1&b&b^2-ac+(1)(ab+ac+bc)\\1&c&c^2-ab+(1)(ab+ac+bc)\end{vmatrix}
= \begin{vmatrix} 1 & a & ... |
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