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 |
|---|---|---|---|---|
21,238 | <p>Would someone be able to point me to a good resource explaining step by step the process for solving inhomogenous recurrence relations? (ie something of the form $ a_n = \sum{{b_i}{a_{n-i}}} + f(n)$ )</p>
| VA. | 1,784 | <p>There is no general criterion, as far as I know, it is all try and see.</p>
<p>Any projective birational morphism $f:X\to Y$ between varieties is the blowup of <em>some</em> sheaf of ideals $I$ on $Y$, so you can see that anything can happen.</p>
|
5,231 | <p>I have coordinates for 4 vertices/points that define a plane and the normal/perpendicular.
The plane has an arbitrary rotation applied to it.</p>
<p>How can I 'un-rotate'/translate the points so that the plane has rotation 0 on x,y,z ?</p>
<p>I've tried to get the plane rotation from the plane's normal:</p>
<pre>... | Community | -1 | <p>From your comments, what I understand of your problem is that you have the coordinates of an arbitrarily oriented rectangle centred on the origin, and you want to find the rotation that will bring it to an axis-aligned rectangle on the $xy$ plane.</p>
<p>Let $u$ and $v$ be the unit vectors that should be mapped to ... |
2,945,913 | <p>I have a quick question about simplifying these exponents and then comparing them:</p>
<p><span class="math-container">$8^{\log_2 n}, 2^{3log_2(log_2n)}$</span> and <span class="math-container">$2^{(log_2(n))^2} $</span></p>
<p>I know the third one evaluates to <span class="math-container">$n^{log_2(n)}$</span>, b... | xbh | 514,490 | <p>Seems fine, but you got a typo at the last part. It should be <span class="math-container">$I(b) = \pi/4$</span> not <span class="math-container">$\pi/2$</span>. Also we could do this quicker:
<span class="math-container">$$
I = \int_0^{\pi/2} \frac {\mathrm dx} {1+\tan(x)^\pi} = \int_0^{\pi/2} \frac {\mathrm dx} {1... |
872,017 | <p>$$\int_0^1 xe^{\sqrt{x}} dx = ? $$</p>
<p>All I can think of is the integration by parts rule, where
$ u = x $ and $ dv= e^{\sqrt(x)} $ $ \Rightarrow du = 1$ and $ v= e^{\sqrt(x)} $ </p>
<p>The answer I get is $e^{\sqrt(x)}(x-1)$ , which is wrong.</p>
<p>Can anyone please explain in detail?</p>
| Mathsource | 12,624 | <p>In fact,
$$
\int xe^{\sqrt{x}}dx = 2\int x\sqrt{x}\dfrac{e^{\sqrt{x}}}{2\sqrt{x}}dx = 2\int x\sqrt{x} e^{\sqrt{x}}d(\sqrt{x}) = 2\int u^{3}e^u du = \dfrac{2}{D}(u^3e^u)
$$
$$
= 2e^u \dfrac{1}{1 + D}u^3 = 2e^u(1 - D + D^2 - D^3)u^3 = 2e^u[u^3 - 3u^2 + 6u - 6]
$$
$$
=2e^{\sqrt{x}}[x^{3/2} - 3x + 6\sqrt{x} - 6] + C
$$<... |
791,719 | <p>I have this inequation:
$$5-3|x-6|\leq 3x -7$$</p>
<p>i solved this this way: </p>
<p>i said, for $x\geq6$ is the modulus positive, so I made 2 cases in which the modulus gives + or - : </p>
<p>1) for $x\geq6$ (positive): </p>
<p>$5-3x+6\leq 3x -7\\
6x\geq30\\
x\geq5$</p>
<p>2) for $x<6$ (negative): </... | pointer | 121,270 | <p>$$arctg(a)+arcctg(a)=\pi/2$$
and
$$tg(\alpha)=1/ctg(\alpha)$$</p>
|
3,766,042 | <p>I was doing the problem</p>
<blockquote>
<p>Find all real solutions for <span class="math-container">$x$</span> in:</p>
<p><span class="math-container">$$ 2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2$$</span></p>
</blockquote>
<p>There was a hint, to prove that <span class="math-container">$2^{x} - 1$</span> has the sa... | OlympusHero | 811,375 | <p>Looks like you're in my class since this is the week that just ended. (on AoPS) If you got an extension, here's a hint as to my solution with some sign work.</p>
<p>Dividing both sides by <span class="math-container">$2$</span>, we get <span class="math-container">$x^2(2^x-1)+x(2^{x^2-1}-1)=2^x-1$</span>. Subtractin... |
362 | <p>What do you guys think of <a href="https://math.stackexchange.com/questions/1187/whats-the-most-effective-ways-of-teaching-kids-times-tables">maths education questions</a>? I wouldn't want the site to be overwhelmed, but they are related to maths. By its nature, many education questions will be subjective and diffic... | BBischof | 16 | <p>I think they have a place, but should follow standards that we can all agree one. One math ed question that I am particularly proud of is <a href="https://mathoverflow.net/questions/8258/whats-a-nice-argument-that-shows-the-volume-of-the-unit-n-ball-in-rn-approaches">https://mathoverflow.net/questions/8258/whats-a-n... |
79,292 | <p>I recently realized that I don't know any non-linear diffeomorphisms of the plane (or $\mathbb{R}^n$ in general) except for linear ones, so I want to ask rather broad questions hoping to be pointed to the appropriate literature.</p>
<p><strike>1) Are there simple ways of constructing autodiffeomorphisms of $\mathbb... | Sanand | 70,278 | <p>Consider $\mathbb{R}^2$ first. Let $f$ be a smooth function on $\mathbb{R}^2$. If we consider a matrix
$$ \begin{bmatrix}
\cos(f(x,y)) & -\sin(f(x,y))\\
\sin(f(x,y)) & \cos(f(x,y))
\end{bmatrix}$$ This is non linear map and it gives a diffeomorphism on $\mathbb{R}^2$. Using these $2\times2$ blocks we can ... |
2,717,007 | <p>the curves are $x^2 = 4y$ and $x^2=4y-4$
these are just the same parabolas but the other one is shifted up by one unit.</p>
<p>I have been thinking of 3 possibilities that might be the answer.</p>
<ol>
<li><p>The area is equal to infinite sq. units</p></li>
<li><p>The area is equal zero</p></li>
<li><p>The area is... | Andrew Li | 344,419 | <p>You can rewrite this as an improper integrals:</p>
<p>$$\int_{-\infty}^\infty \left({x^2 + 4\over 4} - {x^2\over 4}\right) \,\mathrm dx = \int_{-\infty}^\infty 1\,\mathrm dx$$</p>
<p>It becomes obvious this does not converge thus the area is not finite:</p>
<p>$$\lim_{b\to\infty} x \,\Big|^b_{-b} \rightarrow \inf... |
3,971,025 | <p>I need to find the number of conjugated to the permutation (12)(34) in the symmetric group <span class="math-container">$S_6$</span> of rank 6</p>
<p>My answer is 6! = 720</p>
<p>Is this correct?</p>
<p>I concluded that (12)(34)=(12)(34)(5)(6) and the number of combinations for <span class="math-container">$S_6$</sp... | HallaSurvivor | 655,547 | <p>Here <span class="math-container">$L^+$</span> is the set of functions which takes a prime number <span class="math-container">$p$</span> and outputs some natural number <span class="math-container">$n$</span> with the bonus property that <span class="math-container">$f(p) = 0$</span> for all but finitely many prime... |
383,194 | <p>What are the free objects in the category of $G$-sets for a group $G$? </p>
<p>After considerable deliberation (I'm not very bright), I'm pretty sure they are the $G$-sets $X$ on which $G$ acts freely, that is in such a way that only $e$ fixes any elements in $X$. I can prove it -- almost.</p>
<p>Suppose $X$ is a ... | Martin Brandenburg | 1,650 | <p>Sorry I've only read the first line of your question.</p>
<p>If $X$ is a set, the free $G$-set on $X$ is $G \times X$ with the $G$-action $g(h,x)=(gh,x)$. This is because for every $G$-set $Y$ every map $\alpha : X \to Y$ of sets extends uniquely to a $G$-map $G \times X \to Y$ via $(g,x) \mapsto g\, \alpha(x)$.</p... |
187,459 | <p>What are all 4-regular graphs such that every edge in the graph lies in a unique-4 cycle?</p>
<p>Among all such graphs, if we impose a further restriction that any two 4-cycles in the graph have at most one vertex in common, then can we characterize them in some way?</p>
<p>When is it possible to draw such a graph... | Flo Pfender | 12,487 | <p>Here is one more construction which covers a lot of graphs: start with a $4$-regular graph with girth at least $5$, take the line graph and delete a perfect matching in each resulting $K_4$...</p>
<p>This may get us almost all answers to the second question:
If we start with an answer to the second question, and "f... |
332,772 | <p>I am looking for a reference or a proof of the following fact:</p>
<p>Let <span class="math-container">$X_{1}\subset X_{2}\subset\dots $</span> be a sequence of (hausdorff) topological spaces indexed by natural numbers such that each <span class="math-container">$X_{i}\subset X_{i+1}$</span> is a closed subspace ... | David White | 11,540 | <p>This is a Theorem on page 115 of Peter May's book <a href="http://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf" rel="nofollow noreferrer">A concise course in algebraic topology</a>.</p>
<p>A discussion can be found <a href="https://math.stackexchange.com/questions/1341832/when-will-homology-and-direct-limit-co... |
222,639 | <p>It seems to be true that multiply transitive permutation groups have been classified completely (using CFSG), but I am having trouble finding a reference where this classification is actually stated. Is there a canonical reference?</p>
| Derek Holt | 35,840 | <p>Here is a list of the finite $3$-transitive groups, derived by looking through the list of $2$-transitive groups in Section 7.7 of Dixon and Mortimer and identifying those that are $3$-transitive.</p>
<p>Let's first recall the structure of $G := P{\Gamma}L(2,q)$ with $q=p^e$, $p$ prime. Let $S={\rm PSL}(2,q)$. For ... |
762,472 | <p>Let $C$ be a circle of radius $r$ in the plane. Let $p$ be a point in the plane that lies outside of $C$. Show that there are exactly two lines through $p$ that are tangent to $C$.</p>
<hr>
<p>It is one of those questions that seem very intuitive but very hard to prove for me. How do I show that there are "exactly... | Kaj Hansen | 138,538 | <p><strong>Method Using Calculus</strong>: </p>
<p>Say we are given any circle and any point outside of that circle. WLOG, we can translate the circle to be centered at the origin, and rotate our system so that the point is situated along the $y$-axis. </p>
<p>From here, we claim that we can hit that point with ex... |
993,767 | <p>Suppose $V$ is an inner product space over $\mathbb F$ and $u$,$v$ ∈ $V$ and
$\|u\| ≤ \|u + av\|$
for all $a$ ∈ $\mathbb{F}$.Then I want to show that $u$ and $v$ are orthogonal.I want to prove it geometrically.Somebody please give me some hint.</p>
| Did | 6,179 | <p>A different idea, for a more explicit example: $$a_{2^n+k}=2^{-n}\,(2k-2^n)\, n\qquad 0\leqslant k\leqslant2^n-1$$</p>
|
34,247 | <p>We will have an election soon to elect 2 new moderators. Continuing the tradition from past elections (<a href="https://math.meta.stackexchange.com/questions/17598/2014-nominations-for-moderator-on-math-se">2014</a>, <a href="https://math.meta.stackexchange.com/questions/27078/2017-moderator-election-nominating-anot... | Sarvesh Ravichandran Iyer | 316,409 | <p>In the last election, <a href="https://math.meta.stackexchange.com/users/72031/paramanand-singh"><strong>Paramanand Singh</strong></a> missed out in a closely fought race with Xander Henderson. For sure, if they are available, they should be contesting this time.</p>
<p>That's because one can see their responses in ... |
3,756,970 | <p>I know which step is wrong in the following argument, but would like to have contributors' explanations of <em>why</em> it is wrong.</p>
<p>We assume below that weather forecasts always predict whether or not it is going to rain, so <em>not forecast to rain</em> means the same as <em>forecast not to rain</em>. We sh... | Moe Sarah | 787,944 | <p>Every set can become a metric space.</p>
<p>Let <span class="math-container">$X$</span> be a set. Define <span class="math-container">$d: X\times X\rightarrow \mathbb{R}$</span> by <span class="math-container">$d(x,y)=0$</span> if <span class="math-container">$x=y$</span> and <span class="math-container">$1$</span> ... |
45,973 | <p>Let $B,C,D \geq 1$ be positive integers and $(b_n)_{n\geq 0}$ be a sequence with $b_0 = 1, b_n = B b_{n-1} + C B^n + D$ for $n \geq 1$.</p>
<p>Prove that </p>
<p>(a) $\sum_{n\geq 0}^\infty b_n t^n$ ist a rational function</p>
<p>(b) identify a formula for $b_n$</p>
<hr>
<p>Hi!</p>
<p>(a)</p>
<p>As I know I ne... | André Nicolas | 6,312 | <p>I do not know how much help you need. Both Robert Israel and Steve have given an almost complete solution of the problem, and one of theirs should be chosen as the solution to "accept."</p>
<p>You want to find a simple expression for $F(t)$, where
$$F(t)=\sum_0^\infty b_nt^n$$
Substituting in the recurrence for $b... |
2,847,419 | <p>I know that <br/>
$\sigma , \delta$ be 2 function then <br/>
$1)$ $\sigma \circ \delta$ is onto or one-one if both $\sigma $ and $\delta$ is onto or one one.<br/>
I can prove this fact .
I wanted to find the counterexample for both cases if the converse is not true.
<br/> Any Help will be appreciated </p>
| mengdie1982 | 560,634 | <h1>Solution</h1>
<p>Notice <span class="math-container">$$f'(x)=k(x+e^x)^{k-1}(1+e^x).$$</span></p>
<p>Let <span class="math-container">$f'(x)=0$</span>. Then we obtain <span class="math-container">$$x+e^x=0$$</span></p>
<p>This equation has no closed-form solution. But by graphing <span class="math-container">$y=-x$<... |
3,520,722 | <p>I have a few challenges setting up the bounds of integration for the region
<span class="math-container">$$U = \{(x,y) | -1 \leq x-y \leq 1 , \quad 1 \leq xy \leq 2 \}$$</span>
My ultimate goal is to solve <span class="math-container">$$\iint_U x^2y + xy^2 dxdy = \iint_U f(x,y) dxdy$$</span></p>
<p>Here is a plot ... | Claude Leibovici | 82,404 | <p>Starting from
<span class="math-container">$$4\sum_{k=0}^{\infty} \frac{2^k k! \left(k+2\right)!}{\left(2k+4\right)!} $$</span>Consider
<span class="math-container">$$4\sum_{k=0}^{\infty}\frac{k! (k+2)!}{ (2 k+4)!}(2t)^{2k}$$</span> and, now, the trick is to recognize (not so obvious) that this is
<span class="math... |
1,822,160 | <p>Why is the "column space" on the vertical in a matrix? In my mind the column space is that space that the vectors in the matrix have created.
I mean, for example take the equations:</p>
<pre><code>3x + 4y = 5
2x + 8y = 6
</code></pre>
<p>Then the matrix will be:</p>
<p>\begin{pmatrix}
3 & 4 \\
2 & 8
\en... | levap | 32,262 | <p>There is also a space defined by the rows of the matrix and it is called (unsurprisingly) the "row space". When you construct a matrix from a linear system of equations, you are indeed constructing a matrix "row by row" and not "column by column" in the sense that each equation defines a row and not a column and so ... |
1,136,060 | <p>What identity would I need to use to solve for $\theta$? </p>
<p>$5 + \cos(\theta) = 7\sin(\theta)$ </p>
<p>By plugging this into a calculator, I was able to get $\theta \approx 53.13^\circ$. </p>
| Khosrotash | 104,171 | <p>$$a sinx +bcos x=\sqrt{a^2+b^2}sin(x+\alpha)\\tan \alpha =\frac{b}{a}\\$$see an example :$$\sqrt{3}sinx+cosx=\sqrt{3+1}sin(x+arctan(\frac{1}{\sqrt{3}}))=2sin(x+30)$$</p>
|
3,867,197 | <p>Let <span class="math-container">$A$</span> be the following matrix</p>
<p><span class="math-container">$$\left(
\begin{array}{ccc}
1 & 0 & x \\
0 & 1 & y \\
x & y & 1
\end{array}
\right)$$</span></p>
<p>I have to prove that if, at least <span class="math-container">$x+y>\frac{3}{2}$</spa... | Martingalo | 127,445 | <p>By direct definition:
A matrix <span class="math-container">$A$</span> is said to be positive definite if the scalar <span class="math-container">$u^TAu$</span> is strictly positive for every non-zero vectors <span class="math-container">$u$</span>. Let <span class="math-container">$u=(u_1,u_2,u_3)$</span>, <span cl... |
1,704,555 | <blockquote>
<p>If $I\subseteq J$ are ideals in a polynomial ring of $n$ variables, how do I prove that $I = J$ if $\operatorname{in}_{\lt}(I)=\operatorname{in}_{\lt}(J)$, where $\lt$ is any monomial ordering?</p>
</blockquote>
<p>Obviously it suffices to prove that $J \subseteq I$. I'm stuck with how to go forward ... | DonAntonio | 31,254 | <p>Take subspaces of the plane, for example:</p>
<p>$$\begin{align*}&x+2y+z=0\\{}\\&(1,0,-1)+t(-1,1,-1)\;t\in\Bbb R\\{}\\&(1,0,-1)+t(0,1,-2)\;,\;\;t\in\Bbb R\;\;,\;\;\;etc.\end{align*}$$</p>
|
1,704,555 | <blockquote>
<p>If $I\subseteq J$ are ideals in a polynomial ring of $n$ variables, how do I prove that $I = J$ if $\operatorname{in}_{\lt}(I)=\operatorname{in}_{\lt}(J)$, where $\lt$ is any monomial ordering?</p>
</blockquote>
<p>Obviously it suffices to prove that $J \subseteq I$. I'm stuck with how to go forward ... | copper.hat | 27,978 | <p>Let $P$ be the plane, then we must have $Tx = x$ for $x \in P$. Hence any subspace of $P$ is $T$ invariant. Since $\dim P = 2$, you can choose any line
in $P$
through the origin.</p>
<p>For example, note that $(2,-1,0)$ and $(0,5,-2)$ are in $P$. Then
$v_{\alpha,\beta}=\alpha (2,-1,0) + \beta (0,5,-2) \in P$ and so... |
1,221,639 | <p>Consider two random variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>. If X and Y are independent random
variables, then it can be shown that:
<span class="math-container">$$E(XY) = E(X)E(Y).$$</span></p>
<p>Let <span class="math-container">$X$</span> be the random variabl... | Math1000 | 38,584 | <p>Let $M_X(t)$ and $M_Y(t)$ be the moment generating functions of $X$ and $Y$, respectively. Then</p>
<p>$$
\begin{align*}
M_X(t) &= \mathbb E[e^{tX}]\\
&= e^{-t}\mathbb P(X=-1) + \mathbb P(X=0) + e^t\mathbb P(X=1)\\
&= \frac13 e^{-t} + \frac13 + \frac 13 e^t\\
&= \frac13(e^{-t} + 1 + e^t)
\end{align*... |
1,221,639 | <p>Consider two random variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>. If X and Y are independent random
variables, then it can be shown that:
<span class="math-container">$$E(XY) = E(X)E(Y).$$</span></p>
<p>Let <span class="math-container">$X$</span> be the random variabl... | rightskewed | 171,836 | <p>$P(X=-1) = P(X=0) = P(X=1) =\frac{1}{3}$</p>
<p>$Y = X^2$ so $P(Y=1) = \frac{2}{3}$ and $P(Y=0) = \frac{1}{3}$ . $Y$ equals zero iff $X$ equals 0. But $Y$ equals 1 if $X$ is $1$ or $-1$.</p>
<p>$$E[X] = -1.\frac{1}{3} + 0.\frac{1}{3} + 1.\frac{1}{3} = 0$$
$$E[Y] = 1.\frac{2}{3} + 0.\frac{1}{3} = \frac{2}{3}$$</p>... |
1,879,673 | <p>I have woven the below incomplete proof of the following claim:</p>
<blockquote>
<p><em>Claim</em>. If $X$ is completely regular and $Y$ is a compactification of $X$,
then there is a unique, continuous, surjective, closed map
$g:\beta\left(X\right)\to Y$ which is the identity on
$X$.</p>
</blockquote>
<p><... | DanielWainfleet | 254,665 | <p>You proved the existence of a continuous closed surjection $g:\beta X\to Y$ with $g|_X=id_X.$ You did not explicitly prove the uniqueness of such a $g.$ </p>
<p>If $f_1:A\to B$ and $f_2:A\to B$ are continuous and $B$ is Hausdorff then $\{x\in A: f_1(x)=f_2(x)\}$ is closed in $A.$ </p>
<p>In particular if $g_1:\b... |
1,043,734 | <p>I found the following on Wikipedia, <a href="http://en.wikipedia.org/w/index.php?title=Elementary_algebra&oldid=633621020#Properties_of_inequality" rel="nofollow">on the page for Inequalities</a>:</p>
<blockquote>
<p>If $a<b$ and $c<d$ then $a+c < b+d$.</p>
</blockquote>
<p>It references <a href="ht... | Milo Brandt | 174,927 | <p>This holds in any ordered field (or more generally, partially ordered group); the only property we need to take advantage of is translation invariance and transitivity. That is, the properties that
$$a<b\Leftrightarrow a+c<b+c$$
$$a<b \text{ and } b<c\Rightarrow a<c$$</p>
<p>Starting with
$$a<b$$
... |
145,950 | <p>how can I show that any finite CW-space can embedded into an euclidean space of some dimension? Any help or reference would be greatly appreciated.</p>
| Igor Rivin | 11,142 | <p>Well, any simplicial complex can be realized as a subset of the simplex in $\mathbb{R}^V$ (where $V$ is the number of vertices). But a CW complex can only be embedded up to homotopy, it seems (see the answer to your duplicate question on math.stackexchange)</p>
|
699,383 | <p>I am a non-mathematician who knows some elemententary calculus ans I want to prove that the sequence $(x_n)$ given by</p>
<p>$$
x_n=-\sqrt{n} + n\ln\Big(1+\frac{1}{\sqrt{n}}\Big)
$$</p>
<p>is decreasing. Is there an elegant way to show this?</p>
| Artem | 48,057 | <ol>
<li><p>as @user130512 said, you could prove that $x_{n+1}$ is greater than $x_n$. Sometimes, it can help to look at the ratio between the two, that is $x_{n+1}/x_n$. If it is greater than $0$, $x_{n+1}$ is greater than $x_n$.</p></li>
<li><p>I think you also should be able to look at the derivative of $x_n$ to ded... |
699,383 | <p>I am a non-mathematician who knows some elemententary calculus ans I want to prove that the sequence $(x_n)$ given by</p>
<p>$$
x_n=-\sqrt{n} + n\ln\Big(1+\frac{1}{\sqrt{n}}\Big)
$$</p>
<p>is decreasing. Is there an elegant way to show this?</p>
| WimC | 25,313 | <p>This does not answer your question but might be helpful. Let $x>0$ be a real number. Then it follows from $$\log(x) = \int_1^x \frac{dt}{t}$$ and the estimates $$2-t <\frac{1}{t}<1-\frac{(t-1)x}{x+1}$$ for $t\in(1,1+1/x)$ that</p>
<p>$$
x-\frac{1}{2} < x^2\log\left(1+\frac{1}{x}\right)<x-\frac{1}{2}... |
831,618 | <p>Please help me to prove that this integral converges.</p>
<p>$$\int_{0}^1 \frac{1}{\sqrt[3]{1-x^3}}\ dx $$</p>
<p>No ideas. Tried to find function which is bigger and converges, equivalent fun-s, but no result still.</p>
| Lucian | 93,448 | <p><strong>Hint:</strong> Use the fact that $\sqrt{1-x^2}<\sqrt[3]{1-x^3}<1$ for $x\in(0,1)$.</p>
|
147,363 | <blockquote>
<p>If $\alpha$ is an algebraic element of $\mathbb{C}$, then there is a unique non-zero polynomial $f \in \mathbb{Q}[x]$ with leading coefficient $1$ such that $f(\alpha) = 0$, and $f$ is irreducible. </p>
</blockquote>
<p>The first part of this proof would be proving that $f$ is not a unit, but what do... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\rm\:\ f\:\!$ unit (i.e. invertible) $\rm\: \Rightarrow\: 1 = f(x)g(x)\:\Rightarrow\: 1 = f(\alpha)g(\alpha) = 0\cdot g(\alpha) = 0$ </p>
<p>The special case $\rm\:f(x) = x\:$ is one of my <a href="https://math.stackexchange.com/a/2523/242">most popular posts</a> (due to its <em>universal</e... |
375,094 | <p>A metric space <span class="math-container">$(M,d)$</span> is <em>doubling</em> if there exists <span class="math-container">$n$</span> such that every ball of radius <span class="math-container">$r$</span> can be covered by <span class="math-container">$n$</span> balls of radius <span class="math-container">$r/2$</... | YCor | 14,094 | <p>Yes: a f.g. discrete, and more generally compactly generated locally compact group is doubling iff it has polynomial growth.</p>
<p>For f.g. groups, you mentioned <span class="math-container">$\Rightarrow$</span>, and asked <span class="math-container">$\Leftarrow$</span>, which I justify below.</p>
<p>Define <span ... |
3,029,778 | <p>I asked a similar question in <a href="https://math.stackexchange.com/questions/3029766/positive-definite-matrix-implies-the-infimum-of-eigenvalues-are-positive">here</a>, but actually what I want to ask is more difficult as described below:</p>
<p>Suppose <span class="math-container">$P(x): \mathbb{R} \to \mathbb{... | Michael Burr | 86,421 | <p>Originally, the text-part of the question asked whether we knew that, over <span class="math-container">$\Omega$</span>, the determinant of <span class="math-container">$P$</span> is always positive. This is different from the case that the infimum of the determinant is positive (the infimum condition implies that ... |
3,248,569 | <p>I have the following two parametric equations of lines:</p>
<p><span class="math-container">$$\begin{cases} x = -t + 1 \\ y = t + 3 \\ z = -6t \end{cases} \quad \land \quad \begin{cases} x = 2s + 4 \\ y = -s \\ z = 2s + 1 \end{cases}$$</span></p>
<p>I want to examinate their mutual position, that is, I want to fin... | Cesareo | 397,348 | <p>Hint.</p>
<p>Calling <span class="math-container">$\lambda_n = \frac{y_n}{x_n}$</span> we have</p>
<p><span class="math-container">$$
\lambda_n = \frac{\lambda_{n-1}+3}{\lambda_{n-1}+1}
$$</span></p>
<p>giving a sequence <span class="math-container">$\lambda_n$</span> with limit at</p>
<p><span class="math-conta... |
13,882 | <p>Background: When Ueno builds the fully faithful functor from Var/k to Sch/k he mentions that the variety $V$ can be identified with the rational points of $t(V)$ over $k$. I know how to prove this on affine everything and will work out the general case at some future time.</p>
<p>The question that this got me think... | Wanderer | 1,107 | <p>It is certainly true for schemes of finite type over $k$ (algebraically closed) that the closed points are exactly the $k$-points. To see this, notice that if $x \in X$ is any point, then the closure $\overline{\{x\}}$, equipped with its reduced subscheme structure, is integral and has dimension equal to the transce... |
1,836,325 | <p>I am trying to check whether </p>
<p>$f: \mathbb{R} \to \mathbb{R}^\omega$ $f(t) = (t, 2t, 3t, \ldots)$ is continuous or not in the product and box topology. </p>
<p>But I have a feeling I don't have the necessary correct concepts to do these proofs. </p>
<p>I know the product topology:</p>
<ol>
<li><p>Is the co... | syzygy | 349,357 | <p>This is clear: the components of $f$ are continuous. Now use the universal property of product spaces.</p>
|
3,625,233 | <p>We roll a dice until we get 6. Knowing that we have rolled 10 times, evaluate the probability that in the next 20 rolls there will be no 6.
So in this question are we supposed to use binomial distribution or geometric distribution?</p>
| Martin Argerami | 22,857 | <p>Because <span class="math-container">$e^{-x}$</span> and <span class="math-container">$xe^{-x}$</span> are already solutions of the homogeneous part. So if you were to take <span class="math-container">$ae^{-x}+bxe^{-x}$</span>, what you get is a solution of the homogeneous part, and you can never get <span class="m... |
3,625,233 | <p>We roll a dice until we get 6. Knowing that we have rolled 10 times, evaluate the probability that in the next 20 rolls there will be no 6.
So in this question are we supposed to use binomial distribution or geometric distribution?</p>
| user577215664 | 475,762 | <p><span class="math-container">$$y'' +2y' +y = e^{-x}+e^x$$</span>
Rewrite it as :
<span class="math-container">$$(e^xy)''=1+e^{2x}$$</span>
Integrate twice.</p>
<hr>
<p>With undetermined coefficients method the guess should be:
<span class="math-container">$$y=Ae^x+Bx^2e^{-x}$$</span>
Since <span class="math-contai... |
2,878,206 | <p>Let $a_n = \frac{9^n}{n + 5^n}$.</p>
<p>At large $n$ value, $a_n$ is expected to behave like $\frac{9^n}{5^n}$, therefore it diverges.</p>
<p>Using the direct comparison test, how can I find $b_n$ (has to be smaller than $a_n$ to prove that $a_n$ diverges)?</p>
| user | 505,767 | <p>We have that eventually $6^n \ge n+5^n$ therefore</p>
<p>$$a_n = \frac{9^n}{n + 5^n}\ge \frac{9^n}{6^n}=\left(\frac32\right)^n\to \infty$$</p>
<p>indeed by induction </p>
<ul>
<li><p>$n=1\implies 6\ge 1+5$</p></li>
<li><p>assuming $6^n \ge n+5^n$ true we have</p></li>
</ul>
<p>$$6^{n+1}=6\cdot 6^n\ge 6n+6\cdot 5... |
2,057,857 | <p>I am trying to complete my homework based on equivalence relation and I don't seem to understand it properly so I need help !</p>
<p>My question is that do all the elements in my set must satisfy all the three conditions then I can say there is an equivalence relation or I can say there is an equivalence relation o... | Akerbeltz | 351,735 | <p>Let <span class="math-container">$A$</span> be an element of the set of all the finite subsets of <span class="math-container">$\mathbb{R}$</span>, which we shall denote simply by <span class="math-container">$\mathcal{P}_{<\omega}(\mathbb{R})$</span></p>
<p><strong>Assertion</strong>: <span class="math-containe... |
2,885,918 | <p>We consider the following random variable $X$: We have a uniform distribution of the numbers of the unit interval $[0,1]$. After a number $x$ from $[0,1]$ is chosen, numbers from $[0,1]$ are chosen until a number $y$ with $x\leq y$ pops up.</p>
<p>The random variable $X$ counts the number of trials to obtain $y$. H... | uniquesolution | 265,735 | <p>The setting is naturally of independent draws of numbers.</p>
<p>Given $x\in[0,1]$, the probability that $y<x$ is just $x$, so the probability that $n-1$ numbers were chosen and all smaller than $x$ is $x^{n-1}$, and you want the next number to be at least $x$, so the probability for that is $(1-x)$. Thus:</p>
... |
2,607,090 | <p>I have a function for which I know:</p>
<p>$f(2) = 2x -3y \\
f(3) = 5x - 6y \\
f(4) = 9x - 10 y \\
f(5) = 14x - 15y$</p>
<p>Assuming that $f$ is a polynomial, how do I find the general expression for $f$? After many minutes of fiddling I eventually found that this general expression works:</p>
<p>$f(N) = \frac{N(... | filtercoffee | 455,938 | <p>You know :
$f(2) = 2x -3y \\
f(3) = 5x - 6y \\
f(4) = 9x - 10 y \\
f(5) = 14x - 15y$</p>
<p>Which means you know the function is in the form of </p>
<p>$f(N) = Ax - By$</p>
<p>where A and B are expressions involving N.</p>
<p>You can find polynomial expressions for A and B by using <a href="https://en.wikipedia.... |
3,290,199 | <p>If I throw a fair dice <span class="math-container">$12$</span> times, the expected number of <span class="math-container">$6$</span> is <span class="math-container">$2$</span> i.e <span class="math-container">$6$</span> is expected to appear <span class="math-container">$2$</span> times when the dice is thrown... | mlchristians | 681,917 | <p>Continuity over the given interval is all that is required for the integral (term synonymous with <em>antiderivative</em>) to exist over that interval.</p>
<p>When our interval is <span class="math-container">$[a, b]$</span> we may invoke the Fundamental Theorem of Calculus which provides the guarantee. </p>
<p>On... |
122,293 | <p>Let's consider all possible permutations of N numbers. Suppose for each permutation we calculate the sum of absolute differences between consecutive elements. Thus, for (1,2,3) one would have abs(1-2)+abs(2-3)=2. Is it possible to obtain a distribution of such sums for given N? For instance, for N=3 one would have 3... | Andrew D. King | 4,580 | <p>Here is a histogram for length 100. It seems to be normally distributed around $100^2 / 3$, which should put you on the scent, in spite of a complete absence of proof (see edit below).</p>
<p>This is not at all surprising, since if $x$ and $y$ are drawn uniformly at random from the interval $[0,1]$, the expected v... |
635,195 | <p>I'm trying to calculate the following limit: </p>
<p>$$\mathop {\lim }\limits_{x \to {0^ + }} {\left( {\frac{{\sin x}}{x}} \right)^{\frac{1}{x}}}$$</p>
<p>What I did is writing it as: </p>
<p>$${e^{\frac{1}{x}\ln \left( {\frac{{\sin x}}{x}} \right)}}$$</p>
<p>Therefore, we need to calculate: </p>
<p>$$\matho... | Stephen Dedalus | 108,592 | <p>Elementary proof using well known limits and inequalities:
$$\frac{{\ln \left( {\frac{{\sin x}}{x}} \right)}}{x} = \frac{{\ln \left( \left( \frac{{\sin x}}{x}-1 \right) + 1 \right)}}{\frac{{\sin x}}{x}-1 } \cdot \frac{\frac{{\sin x}}{x}-1 }{x} =\frac{{\ln \left( \left( \frac{{\sin x}}{x}-1 \right) + 1 \right)}}{\fr... |
237,838 | <p>The data are for the model $T(t) = T_{s} - (T_{s}-T_{0})e^{-\alpha t}$,
where $T_0$ is the temperature measured at time 0, and $T_{s}$ is the temperature at time $t=\infty$, or the environment temperature. $T_{s}$ and $\alpha$ are parameters to be determined.</p>
<p>How can I fit my data against this model? I'm try... | dantopa | 206,581 | <p>The model is nonlinear, but one of the parameters, $T_{s}$ is linear, which means we can 'remove' it.</p>
<p>Start with a crisp set of definitions: a set of $m$ measurements $\left\{ t_{k}, T_{k} \right\}_{k=1}^{m}.$ The trial function, as pointed out by @Yves Daust, is
$$
T(t) = T_{s} \left( 1 - e^{-\alpha t}\rig... |
301,264 | <p>Note: There is another question of the same title, but it is different and asks for group theory prerequisites in algebraic topology, while i want the topology prerequisites. </p>
<p>I am a physics undergrad, and I wish to take up a course on Introduction to Algebraic Topology for the next sem, which basically teac... | Gaston Burrull | 31,167 | <p>I prefer Munkres over all topology books.</p>
<p>You might starting with Munkres chapter 2, then read chapters 3, 4, 7 (without " * " sections), but if you have enought time is not bad idea reading all of the first part: Chapters 1-8 (long but fun). </p>
<p>I think that chapter 1 is good for you, is an intuitive a... |
114,733 | <p>Say you have the half-plane $\{z\in\mathbb{C}:\Re(z)>0\}$. Is there a rigorous explanation why the transformation $w=\dfrac{z-1}{z+1}$ maps the half plane onto $|w|<1$?</p>
| Adam | 44,654 | <p>( This is more comment then answer but I can't add images in comment )
<img src="https://i.stack.imgur.com/8LsvT.png" alt="Image of 2 maps"></p>
<p>Her is <a href="http://commons.wikimedia.org/wiki/File:Conformal_mapping_from_right_half_plane_to_unit_circle.svg" rel="noreferrer">image and src code</a> </p>
|
3,522,736 | <p>I've been messing around with trying to negate this statement using DeMorgans laws and I keep ending up with incorrect answers such as (~p or ~q) and ~r. If someone could help me with the negation of compound statements. </p>
<p>Thank you.</p>
| Tsemo Aristide | 280,301 | <p><span class="math-container">$f'(x)={{-1}\over{(x+1)^2}}$</span> is never <span class="math-container">$0$</span> we deduce that <span class="math-container">$f$</span> cannot have an extremum since the derivative is zero at an extremum.</p>
|
70,146 | <p>I'm trying to use an image as a <code>ChartLabel</code> and I'm getting strange results.</p>
<p>Here is a bar chart, with labels, that looks ok:</p>
<p><img src="https://i.stack.imgur.com/YmRQp.png" alt="chart ok"></p>
<p>But when I try to replace the "A" label with an image, the output is confusing:</p>
<p><img... | kglr | 125 | <p><strong>Update:</strong> The option <a href="https://reference.wolfram.com/language/ref/LabelingSize.html" rel="nofollow noreferrer"><code>LabelingSize</code></a> provides a convenient way to size the images.</p>
<pre><code>images = ExampleData[{"TestImage", #}] & /@ {"Lena", "Elaine", "Mandrill"};
is = 60;
bs... |
1,610,700 | <blockquote>
<p>$$\int \frac{x-3}{\sqrt{1-x^2}} \mathrm dx$$</p>
</blockquote>
<p>I know that $\int \frac{1}{1-x^2}\mathrm dx=\arcsin(\frac{x}{1})$ but how can I continue from here? </p>
| Harish Chandra Rajpoot | 210,295 | <p>Notice, $$\int \frac{x-3}{\sqrt {1-x^2}}\ dx$$$$=\int \frac{x}{\sqrt {1-x^2}}\ dx-\int \frac{3}{\sqrt {1-x^2}}\ dx$$
$$=-\frac{1}{2}\int \frac{(-2x)}{\sqrt {1-x^2}}\ dx-3\int \frac{1}{\sqrt {1-x^2}}\ dx$$
$$=-\frac{1}{2}\int (1-x^2)^{-1/2}\ d(1-x^2)-3\int \frac{1}{\sqrt {1-x^2}}\ dx$$
$$=-\frac{1}{2}\frac{(1-x^2)^{1... |
864,920 | <p>The matrices $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\2&1\end{pmatrix}$ are well-known to (freely) generate a free group. Some years ago, I read a paper that, if memory serves, sort of summarised what was known about generalisations to pairs of the form $\begin{pmatrix}1&\... | Andreas Caranti | 58,401 | <p>There's a <strong><em>1969</em></strong> paper by Lyndon and Ullman that appears to be it, plots and all.</p>
<blockquote>
<p>R. C. Lyndon and J. L. Ullman, <em>Groups generated by two parabolic linear fractional transformations.</em> Canad. J. Math. <strong>21</strong> (1969) 1388-1403</p>
</blockquote>
<p>The ... |
864,920 | <p>The matrices $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\2&1\end{pmatrix}$ are well-known to (freely) generate a free group. Some years ago, I read a paper that, if memory serves, sort of summarised what was known about generalisations to pairs of the form $\begin{pmatrix}1&\... | i. m. soloveichik | 32,940 | <p>FREE AND NONFREE SUBGROUPS OF $PSL_2(\mathbf{C})$ GENERATED BY TWO PARABOLIC ELEMENTS
by Ju A Ignatov</p>
<p>1979 Math. USSR Sb. 35 49. doi:10.1070/SM1979v035n01ABEH001449</p>
|
2,753,548 | <p>Let $F$ be a field with $7^5$ elements.
$$X=\{a^7-b^7 \mid a,b \in F\}$$
I have no idea how to solve.
Please help me.</p>
| Dietrich Burde | 83,966 | <p>Since $a^7-b^7=(a-b)^7$ in $F$, and $x\mapsto x^7$ is an isomorphism, we see that $X=F$</p>
|
3,279,544 | <blockquote>
<p><span class="math-container">$50$</span> fighters are standing around in a circle. Every fighter may choose
to fight with the person on its left or its right with equal
probability. One person may be chosen by two different persons and
another may not be chosen at all. Can you expect how many fi... | drhab | 75,923 | <p><strong>Guide</strong>.</p>
<p>Number the fighters and let <span class="math-container">$X_i$</span> take value <span class="math-container">$1$</span> if fighter <span class="math-container">$i$</span> is not chosen and <span class="math-container">$0$</span> if he was chosen. </p>
<p>If <span class="math-contain... |
135,936 | <p>I need this one result to do a problem correctly.</p>
<p>I want to show that for any $b \in \mathbb{C}$ and $z$ a complex variable:</p>
<p>$$ |z^2 + b^2| \geq |z|^{2} - |b|^{2}$$ </p>
<p>My attempts have only led me to conclude that </p>
<p>$$ |z^2 + b^2| > \frac{|z|^{2} + |b|^{2}}{2}$$ </p>
| agt | 6,752 | <p>The searched inequality is an instance of the Lipshitz inequality for the distance.</p>
<p>In the concrete case $$||z_1|-|z_2||\leq|z_1-z_2|$$
It is obtained by the triangle inequality $$|z_1|\leq|z_1-z_2|+|z_2|$$ and its symmetric under the exchange of $z_1$ and $z_2$</p>
|
3,419,620 | <p>I'm trying to solve the equation
<span class="math-container">$$(z+i)^2=(\sqrt3+i)^3$$</span> but I don't know how to extract
the roots
<span class="math-container">$$(z+i)^2=(\sqrt3+i)^3 \rightarrow (z+i)^2=8i \rightarrow z^2+(2i)z-(8i+1)=0$$</span></p>
<p><span class="math-container">$z_{1,2}=-i \pm \sqrt{8i}$</s... | user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$(z+i)^2=(\sqrt3+i)^3=8i=8e^{i\left(\frac \pi 2+2k\pi\right)}\implies z+i=2\sqrt 2e^{i\left(\frac \pi 4+k\pi\right)}$$</span></p>
<p>for <span class="math-container">$k=0,1$</span> that is</p>
<ul>
<li><span class="math-container">$z+i=2\sqrt 2e^{i\frac \pi 4}=2+2... |
3,419,620 | <p>I'm trying to solve the equation
<span class="math-container">$$(z+i)^2=(\sqrt3+i)^3$$</span> but I don't know how to extract
the roots
<span class="math-container">$$(z+i)^2=(\sqrt3+i)^3 \rightarrow (z+i)^2=8i \rightarrow z^2+(2i)z-(8i+1)=0$$</span></p>
<p><span class="math-container">$z_{1,2}=-i \pm \sqrt{8i}$</s... | J. W. Tanner | 615,567 | <p>If you note <span class="math-container">$(2+2i)^2=8i$</span>, you'll see your answer matches the book's.</p>
|
3,419,620 | <p>I'm trying to solve the equation
<span class="math-container">$$(z+i)^2=(\sqrt3+i)^3$$</span> but I don't know how to extract
the roots
<span class="math-container">$$(z+i)^2=(\sqrt3+i)^3 \rightarrow (z+i)^2=8i \rightarrow z^2+(2i)z-(8i+1)=0$$</span></p>
<p><span class="math-container">$z_{1,2}=-i \pm \sqrt{8i}$</s... | fleablood | 280,126 | <p>You have <span class="math-container">$z_{1,2} = -i + K$</span> where <span class="math-container">$K^2 = 8i$</span>.</p>
<p>So find <span class="math-container">$K$</span> where <span class="math-container">$K^2 = 8i$</span>.....</p>
<p>Let <span class="math-container">$K = a+bi$</span> so <span class="math-conta... |
351,850 | <blockquote>
<p>A subset $E$ contained in $\mathbb{R}^n$ is such that the function $x \mapsto \left\Vert x\right\Vert^2$ is uniformly continuous on $E$. For $r > 0$, let $E_r$ denote the union of all open balls of radius $r$ contained in $E$. Prove that $E_r$ is bounded for all $r > 0$. Find an example showing ... | Jyrki Lahtonen | 11,619 | <p>The function $f$ is uniformly continuous on the subset $E_r$ as well. </p>
<p>If contrariwise $E_r$ is unbounded for some $r>0$, then there is a sequence of vectors $x_n$ such that $\Vert x_n\Vert\to\infty$, and $B(x_n,r)\subseteq E$. For all $\delta\in(0,r)$ both $x_n$ and $x'_n(\delta)=x_n(1+\delta/(2\Vert x_n... |
14,385 | <p>I have always taught my students that the <span class="math-container">$y$</span>-intercept of a line is the <span class="math-container">$y$</span>-coordinate of the point of intersection of a line with the <span class="math-container">$y$</span>-axis, that is, for the line given by the equation <span class="math-c... | Chickenmancer | 10,192 | <p>It comes down to convention, ultimately, since specifying a unique point $y_0$ associated to $x_0$ is equivalent to specifying the ordered pair $(x_0,y_0)$ on the graph of $f(x)=y.$ That is, since a function is well defined by definition, then there is no ambiguity if one requires that the intercept "$y_0$" be assoc... |
3,465,945 | <p>Prove that <span class="math-container">$\inf f(A) \leq f( \inf A)$</span> if <span class="math-container">$f: [-\infty, + \infty] \to \mathbb{R}$</span> is continuous and <span class="math-container">$A \neq \emptyset$</span> is a subset of <span class="math-container">$\mathbb{R}$</span>.</p>
<p>Attempt;</p>
<p>... | copper.hat | 27,978 | <p>You have <span class="math-container">$\inf_x f(x) \le f(y)$</span> for all <span class="math-container">$y$</span> by definition. Ley <span class="math-container">$y = \inf A$</span> to finish.</p>
|
1,645,232 | <blockquote>
<p>Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}$.</p>
</blockquote>
<p>I know that vectors that are orthogonal will have a dot product of 0. So here's what I was thinking:
\begin{align*}
ax + by &= 0\\
yb &= -ax\\
y &= -ax/b
\end{align*}</p... | Travis Willse | 155,629 | <p>Note that your method implicitly assume $b \neq 0$. Here's a more conceptual way to approach that problem, which avoids that kind of assumption, and which leads directly to the answer in the text:</p>
<p><strong>Hint</strong> The map ${\bf u}^{\flat} : \Bbb R^2 \to \Bbb R$ defined by $${\bf u}^{\flat} : {\bf v} \ma... |
3,191,345 | <p>Evaluate the following definite integral :
<span class="math-container">$$\int_0^{\pi/2}\cfrac{\cos x}{\sqrt{1-\sin x}} \qquad \qquad \qquad (1)$$</span> </p>
<p><span class="math-container">\begin{align}
& = \int_0^{\pi/2}\cfrac{\cos x}{\sqrt{1-\sin x}} \ \ I \ used \ u=1-\sin x \ and \ dx= \cfrac{-du}{cosx... | Jesús Álvarez Lobo | 632,320 | <p>What you submit is correct. The error is in Symbolab; I do not know the cause. Check carefully how you enter the data in Symbolab. It could also come from a bug in the program. (?)</p>
|
3,191,345 | <p>Evaluate the following definite integral :
<span class="math-container">$$\int_0^{\pi/2}\cfrac{\cos x}{\sqrt{1-\sin x}} \qquad \qquad \qquad (1)$$</span> </p>
<p><span class="math-container">\begin{align}
& = \int_0^{\pi/2}\cfrac{\cos x}{\sqrt{1-\sin x}} \ \ I \ used \ u=1-\sin x \ and \ dx= \cfrac{-du}{cosx... | Community | -1 | <p>I put it into symbolab and I got the correct answer!</p>
<p><a href="https://i.stack.imgur.com/hR7RV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hR7RV.png" alt="enter image description here"></a></p>
|
3,191,345 | <p>Evaluate the following definite integral :
<span class="math-container">$$\int_0^{\pi/2}\cfrac{\cos x}{\sqrt{1-\sin x}} \qquad \qquad \qquad (1)$$</span> </p>
<p><span class="math-container">\begin{align}
& = \int_0^{\pi/2}\cfrac{\cos x}{\sqrt{1-\sin x}} \ \ I \ used \ u=1-\sin x \ and \ dx= \cfrac{-du}{cosx... | user0102 | 322,814 | <p>Make the substitution <span class="math-container">$w = 1 - \sin(x)$</span>. Then you get <span class="math-container">$\mathrm{d}w = -\cos(x)\mathrm{d}x$</span>. Thus we have
<span class="math-container">\begin{align*}
\int_{0}^{\pi/2}\frac{\cos(x)}{\sqrt{1-\sin(x)}}\mathrm{d}x = -\int_{1}^{0}\frac{\mathrm{d}w}{\sq... |
672,736 | <p>Let $A = \begin{bmatrix}1&2&1\\0&1&0\\1&3&1\end{bmatrix}$. Find the eigenvalues of $A$.</p>
<p>I think I got a pretty steady ground on how I approached this, I just have some difficulty getting the right answer.</p>
<p>What I have done so far:</p>
<p>$P(\lambda) = det(A - \lambda I)$</p>
... | gt6989b | 16,192 | <p>You did the determinant wrong, the second term is incorrect, as @FH93 hinted.</p>
<p>Another way is to expand where there are most zeros, saves you work. Let's go by 2nd row:
$$
\det \begin{bmatrix}1-\lambda&2&1\\0&1-\lambda&0\\1&3&1-\lambda\end{bmatrix}
= -(1-\lambda)\left[(1-\lambda)^2-1\... |
33,215 | <p>There is a huge debate on the internet on the value of <span class="math-container">$48\div2(9+3)$</span>.</p>
<p>I believe the answer <span class="math-container">$2$</span> as I believe it is part of the bracket operation in BEDMAS. <a href="https://www.mathway.com" rel="nofollow noreferrer">Mathway</a> yields the... | Michael Burge | 5,468 | <p>I would say it isn't even well-defined. In Group Theory or such, you usually pass by a statement that says "associativity means that <span class="math-container">$(1 + 2) + 3$</span> is the same as <span class="math-container">$1 + (2+3)$</span>, so we can write <span class="math-container">$1 + 2 + 3$</span> w... |
33,215 | <p>There is a huge debate on the internet on the value of <span class="math-container">$48\div2(9+3)$</span>.</p>
<p>I believe the answer <span class="math-container">$2$</span> as I believe it is part of the bracket operation in BEDMAS. <a href="https://www.mathway.com" rel="nofollow noreferrer">Mathway</a> yields the... | Gerry Myerson | 8,269 | <p>There is no Supreme Court for mathematical notation; there were no commandments handed down on Sinai concerning operational precedence; all there is, is convention, and different people are free to adhere to different conventions. Wise people will stick in enough parentheses to make it impossible for anyone to mista... |
272,846 | <p>Suppose I have a List of numbers:</p>
<pre><code>num = Range[5]
</code></pre>
<p>I want to combine the second and the third element into a sublist to get the result as {1,{2,3},4,5}.<br />
I tried using this:</p>
<pre><code>MapAt[List, num, {{2}, {3}}]
</code></pre>
<p>which is not giving me the desired result. What... | Nasser | 70 | <p>One possibility is to use <code>ReplacePart</code></p>
<h2>Example 1</h2>
<pre><code>lst = Range[5]
idx = {2, 3};
remove = (# -> Nothing) & /@ idx[[2 ;; -1]]
rep = First@idx -> lst[[First@idx ;; Last@idx]];
ReplacePart[lst, {rep, Sequence @@ remove}]
</code></pre>
<p><img src="https://i.stack.img... |
177,574 | <p>Fix $k \in \mathbb{N}$, $k \geq 1$. Let $p \in [0,1]$ and $x = (x_0, \ldots, x_k)$ be a $(k+1)$-dimensional <em>real</em> vector, and define
$$S(p,x) = -x_0^2 + \sum_{i=0}^k {k \choose i} p^i (1 - p)^{k - i} \cdot (x_i - p)^2.$$
Experiments show that for small values of $k$
$$\exists x \in \mathbb{R}^{k+1} \,.\, \fo... | John Mount | 56,665 | <p>Having trouble formatting. Here is a [line of attack][1] . Also a [proof of the problem mapping][2]. Apparently I am both user 56-something and "John Mount" but have lost control of at least one of those accounts.</p>
<hr>
<p>[1] <a href="http://winvector.github.io/freq/explicitSolution.html" rel="nofollow">htt... |
475,005 | <p>I want to check how many integral numbers in $\big[1,10^6\big]$ include the numbers $1,2,3,4,5$ and how many only them.<br>
how should I check it? this is a problem of inclusion-exclusion? <br>
I would like to get some advice!<br>
Thanks!</p>
| Alraxite | 61,039 | <p>Exclude $10^6$, and consider $[0,999999]$.</p>
<p>For your first question:</p>
<p>There are $10^6$ numbers in the interval $[0,999999]$. Each number in this interval can be thought of as having $6$ digits (So, $27$ would be $000027$). The numbers that do not satisfy the given property are entirely composed of $0,6... |
871,412 | <p>$$I=\int_a^b \sin(\alpha-\beta x^2)\cos(x)\, dx.$$</p>
<p>Can anybody tell me, how to solve this integral ?
I know that this is related to <a href="http://www.it.uom.gr/teaching/linearalgebra/NumericalRecipiesInC/c6-9.pdf" rel="nofollow">Fresnel Integral</a> if the $\cos(x)$ term is absent. </p>
| Claude Leibovici | 82,404 | <p>In almost the same spirit as user71352's answer, the antiderivative of $$I=\int \sin(\alpha-\beta x^2)\cos(x)\, dx$$ can be written as $$2 \sqrt{\frac{2\beta}{\pi}} I=$$ $$\sin \left(\alpha +\frac{1}{4 \beta }\right) \left(C\left(\frac{2 x \beta -1}{\sqrt{2
\pi\beta} }\right)+C\left(\frac{2 x \beta +1}{\sqrt{2 \p... |
2,507,328 | <p>A bit of a beginner question, but I've been told that between 2 x-intercepts, for any polynomial of degree 2 or higher. That is true. But the controversy here is that apparently, it has to be exactly in the middle between the 2. I'm not talking about quadratics; the only turning point is at $x = -b/2a$. I am talking... | supinf | 168,859 | <p>First, it is easy to see that $(-\infty,q)$ is in the sigma algebra for rational $q$.
Since the rational numbers are countable, you can do something like
$$
(-\infty,x)=\bigcup_{q<x, q\in\mathbb Q} (-\infty,q)
$$</p>
|
1,107,317 | <p>I've got this hypergeometric series</p>
<p>$_2F_1 \left[ \begin{array}{ll}
a &-n \\
-a-n+1 &
\end{array} ; 1\right]$</p>
<p>where $a,n>0$ and $a,n\in \mathbb{N}$</p>
<p>The problem is that $-a-n+1$ is negative in this case. So when I try to use Gauss's identity</p>
<p>$_2F_1 \left[ \begin{array}{ll}... | David H | 55,051 | <p>Using <a href="http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/03/06/05/0016/" rel="nofollow">this identity</a> to express the hypergeometric function as a Gegenbauer function, and <a href="http://functions.wolfram.com/HypergeometricFunctions/GegenbauerC3General/03/01/01/0002/" rel="nofollow">... |
309,380 | <p>Let me sum up my - hopefully correct - understanding of the <a href="https://en.wikipedia.org/wiki/Travelling_salesman_problem" rel="nofollow noreferrer">travelling salesman problem</a> and <a href="https://en.wikipedia.org/wiki/Complexity_class" rel="nofollow noreferrer">complexity classes</a>. It's about <a href="... | Timothy Chow | 3,106 | <p>I think you may need to formulate your question more precisely.</p>
<p>Consider the following "nonconstructive proof": If you do blah-blah-blah and the result is 1 then YES, but if the result is 0 then NO, where "blah-blah-blah," upon closer inspection, amounts to running a Turing machine that exhaustively tries al... |
3,430,136 | <p>I would like to prove that the map <span class="math-container">$f: S^n \times S^m \to 2S^{m+n+1}: ((x_1,..,x_{n+1}), (y_1,...,y_{m+1})) \to (x_1,...,x_{n+1},y_1,...,y_{m+1})$</span>
is an imersion. Here <span class="math-container">$2S^{m+n+1}$</span> is the <span class="math-container">$m+n+1$</span> dimensional ... | Ted Shifrin | 71,348 | <p>Working in charts on spheres is (almost?) always painful. So, as another answer suggested, let's make it easier by looking at the map on Euclidean space. We can consider <span class="math-container">$F\colon\Bbb R^{n+1}\times\Bbb R^{m+1} \to \Bbb R^{n+m+2}$</span> given by <span class="math-container">$F(x,y) = (x,0... |
2,144,520 | <p>I don't really understand Cantor's diagonal argument, so this proof is pretty hard for me. I know this question has been asked multiple times on here and i've gone through several of them and some of them don't use Cantor's diagonal argument and I don't really understand the ones that use it. I know i'm supposed to ... | A. Salguero-Alarcón | 405,514 | <p>Suppose $A$ is countable. Then, we will have a bijection:</p>
<p>$1 \to \phi(1)=a= \{a_1, a_2, a_3, a_4,...\}$</p>
<p>$2 \to \phi(2)=b=\{b_1, b_2, b_3, b_4,...\}$</p>
<p>$3 \to \phi(3)=c=\{c_1,c_2,c_3,c_4...\}$</p>
<p>$4 \to \phi(4)=d=\{d_1,d_2,d_3,d_4...\}$</p>
<p>and so on. Now, we're going to build a sequenc... |
2,548,942 | <p>What would be the best approach to calculate the following limits </p>
<p>$$ \lim_{x \rightarrow 0} \left (1+\frac {1} {\arctan x} \right)^{\sin x}, \qquad \lim_{x \rightarrow 0} \frac {\tan ^7 x} {\ln (7x+1)} $$
in a basic way, using some special limits, without L'Hospital's rule? </p>
| Community | -1 | <p>The second is:
$$ \lim_{x\to0}\frac{\sin^7x}{\cos^7x}\frac{1}{\log(7x+1)}= \lim_{x\to0}\frac{\sin^7x}{\cos^7x}\frac{1}{\log(7x+1)}\frac{x^7}{x^7}\frac{7x}{7x}=0$$</p>
<p>For the first use the substitution method.</p>
|
117,500 | <p>How would you go about finding the conjugacy classes of the nonabelian group of order 21, $G:=\left\langle x,y | x^7=e=y^3, y^{-1}xy=x^2\right\rangle$?</p>
| awllower | 6,792 | <p>We can in fact generalize this situation as follows: <a href="https://math.stackexchange.com/questions/153381/estimates-on-conjugacy-classes-of-a-finite-group">Please see this question for reference</a></p>
<blockquote>
<p>Theorem:<br>
Let A be a normal subgroup of G such that A is the centralizer of every non... |
1,986,402 | <blockquote>
<p>How can I simplify $\prod \limits_{l=1}^{a} \frac{1}{4^a} \cdot 16^l$?</p>
</blockquote>
<p>I've tried looking at the terms and finding something in there to conlcude what it might be and also took the $n^{th}$ term of $16^l$ into one fraction but that does rather the opposite of simplification.</p>
| Jack D'Aurizio | 44,121 | <p>$$\prod_{l=1}^{a}\frac{16^l}{4^a} = \frac{1}{4^{a^2}}\prod_{l=1}^{a}16^l = 4^{-a^2} 16^{\sum_{l=1}^{a}l} = 4^{-a^2} 4^{a(a+1)}=\color{red}{4^a}.$$</p>
|
4,247,637 | <p>There are several tea cups in the kitchen, some with handle and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly <span class="math-container">$1200$</span>. What is the maximum possible number of cups in the kitchen?<br>
Here's what I did:<br>
I... | Random Variable | 16,033 | <p>Using the principal branch of the logarithm, let's integrate the function <span class="math-container">$$\frac{e^{i \xi z}}{2 \sqrt{1+z^{2}}}, \quad \xi >0,$$</span> around a semicircular contour in the upper half of the complex plane that is deformed around the branch cut on <span class="math-container">$[i, i \... |
170,967 | <p>Cog $A$ is at position: $Ax$, $Ay$, rotation: $Ar$ and number of teeth: $At$</p>
<p>Cog $B$ is at position: $Bx$, $By$ and number of teeth $Bt$. What is Cog $B$'s rotation such that teeth between Cog $A$ and Cog $B$ line up. There will be the same number of answers as there are teeth, but a 'base angle' is desired.... | sq2 | 35,825 | <p>The solution I have found, which may be @joriki's solution expressed in a different format:</p>
<p>Angle α is of course required, see @joriki's solution.</p>
<p>Br = At / Bt * -Ar + α * (At + Bt) / Bt</p>
<p>and if Bt is even, add π / Bt to Br</p>
|
514,338 | <p>Okay so my algebra knowledge is pretty guff..</p>
<p>I am taking a control systems class and pretty much all the questions I am expected to revise, are about doing this algebraic manipulation and I don't know what steps the tutor is taking to do it..</p>
<p>Okay here goes..</p>
<p>If the transfer function of a sy... | Community | -1 | <p>You can usually resort to doing everything straightforwardly if you don't see how the "tricks" work.</p>
<p>In this case, the straightforward thing is to "compute" the sum:</p>
<p>$$ \frac{3}{20s + 1} + 1 = \frac{20s + 4}{20s+1} $$</p>
<p>There are lots of ways to see how to do this. You can try to find the least... |
3,047,241 | <blockquote>
<p>Let <span class="math-container">$X_1, X_2, \cdots, X_n$</span> be i.i.d. <span class="math-container">$\sim \text{Bernoulli}(p)$</span>. Then <span class="math-container">$\bar{x}$</span> is an unbiased estimator of <span class="math-container">$p$</span>.</p>
</blockquote>
<p>How should I approach ... | Ahmad Bazzi | 310,385 | <p>If you define <span class="math-container">$\bar{x}$</span> as the sample mean, i.e.
<span class="math-container">$$\bar{x} = \frac{1}{n}\sum_{i=1}^n X_i$$</span>
Take expectations on both sides
<span class="math-container">$$E \bar{x} = E \frac{1}{n}\sum_{i=1}^n X_i$$</span>
Since expectation is a linear operator, ... |
83,965 | <p>When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y by evaluating the surface integral of the curl of the vector field over the surface X" or vice versa. The trouble is t... | Igor Rivin | 11,142 | <p>One could argue that complex function theory (the fact that analytic functions integrate to zero around contours) is an application, and a nice one.</p>
|
83,965 | <p>When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y by evaluating the surface integral of the curl of the vector field over the surface X" or vice versa. The trouble is t... | Nilima Nigam | 14,740 | <p>A student may also learn about the content from Stokes theorem from instances where it <em>failed</em> to hold as expected. For example, one has to exercise care when trying to use the theorem on domains with holes. Turn this around: the failure of Stokes to hold <em>as expected</em> tells you about the cohomology o... |
73,559 | <p>I have the following problem. I'd like to add a legend to <code>MatrixPlot</code>. Each colour should have a legend entry. I used <code>PlotLegends</code>, which in principle works. However, if I use more than five colours, this doesn't work anymore.</p>
<pre><code>a = RandomInteger[{1, 6}, {50}];
MatrixPlot[{a}, C... | C. E. | 731 | <p>When <code>Automatic</code> fails the same thing can, luckily, be done manually:</p>
<pre><code>rules = {1 -> Red, 2 -> Blue, 3 -> Green, 4 -> Gray, 5 -> Yellow, 6 -> Orange};
a = RandomInteger[{1, 6}, {50}];
MatrixPlot[
{a},
ColorRules -> rules,
PlotLegends -> SwatchLegend[rules[[All, 2... |
134,205 | <blockquote>
<p>Find the expectation of a Geometric distribution using $\mathbb{E}(X)= \sum_{k=1}^\infty P(X \ge k)$. </p>
</blockquote>
<p>Okay I know how to find the expectation using the definition of the geometric distribution $$P(X=k)= p \cdot(1-p)^{k-1}$$ and I figured that $P(X \ge k)=(1-p)^{k-1}$ but I don't... | David Mitra | 18,986 | <p>For $|r|<1$, the sum of the geometric series $\sum\limits_{k=1}^\infty r^k$ is ${ r\over 1-r}$.
So, write
$$\sum\limits_{k=1}^\infty P[X\ge k]= \sum\limits_{k=1}^\infty (1-p)^{k-1}
= {1\over 1-p}\sum\limits_{k=1}^\infty (1-p)^{k },$$
and apply the formula with $r=1-p$.</p>
|
3,812,087 | <p>Let <span class="math-container">$(M,d)$</span> be a metric space. A set <span class="math-container">$A \subset M$</span> is said to be compact if every open cover of <span class="math-container">$A$</span> has a finite subcover.</p>
<p>Why do we use this definition, rather than the other "definition" whi... | freakish | 340,986 | <p>The question "why we define something as something?" is a really tricky question. There is no some universal reason not to define "compact" as "closed and bounded" or as "finite" or as "empty" or as "green grass". It's just a definition, a label, nothing mo... |
3,812,087 | <p>Let <span class="math-container">$(M,d)$</span> be a metric space. A set <span class="math-container">$A \subset M$</span> is said to be compact if every open cover of <span class="math-container">$A$</span> has a finite subcover.</p>
<p>Why do we use this definition, rather than the other "definition" whi... | Michael Hardy | 11,667 | <p>In some commonplace metric spaces such as <span class="math-container">$\ell^2,$</span> there are sets that are closed and bounded but NOT compact. In particular, the standard orthonormal basis of <span class="math-container">$\ell^2$</span> is an example of such a set. And the closed interval from <span class="math... |
1,201,002 | <p>I´m trying to find a vector $\vec{c} = $ , which is orthogonal to vector $\vec{a}$ and $\vec{b}$:</p>
<p>As far I understood, I have to show that:</p>
<p>$$\langle a,c\rangle=0 $$
$$\langle b,c\rangle=0 $$ </p>
<p>So if I would like to determine an orthogonal vector regarding: \begin{bmatrix}-1\\1\end{bmatrix}
... | David K | 139,123 | <p>Given <span class="math-container">$m$</span> orthogonal vectors <span class="math-container">$v_1, v_2, \ldots, v_m$</span> in <span class="math-container">$\mathbb R^n$</span>, a vector orthogonal to them is any vector <span class="math-container">$x$</span> that solves the matrix equation</p>
<p><span class="mat... |
1,649,194 | <p>Let $S^n\subset\mathbb{R}^{n+1}$ denote the standard unit sphere with normal bundle $\nu$, let $N=(0,\dots,0,1)$ and $S=(0,\dots,0,-1)$. Then there are two sterographic projections
$$\sigma_+\colon S^n-S\to\mathbb{R}^n $$
and
$$\sigma_-\colon S^n-N\to\mathbb{R}^n$$
Both of these maps are homeomorphisms and they for... | William | 14,816 | <p>It looks like, in all dimensions, $\sigma_+$ preserves orientation and $\sigma_-$ reverses it. Here is the case of $S^2$ explicitly worked out:</p>
<p>For a vector space $V$ of dimension $n$, an orientation will be an element of
$$ \{(b_1,\dots,b_n)\ |\ V=\langle b_1,\dots, b_n\rangle \}/GL^+(V) \cong \mathbb{Z}/2... |
2,634,701 | <p>Let $ f: {{\mathbb{R^n}} \rightarrow {{\mathbb{R}} }}$ be continuous and let $a$ and $b$ be points in $ {{\mathbb{R} }} $
Let the function $g: {\mathbb{R}} \rightarrow {\mathbb{R}}$ be defined as:
$$ g(t) = f(ta+(1-t)b) $$
Show that $g$ is continuous .</p>
<p>If I define a function $ h(t)=ta+(1-t)b$, then I have ... | Peter Szilas | 408,605 | <p>$h: \mathbb{R} \rightarrow \mathbb{R^n}$</p>
<p>Denote norm in $\mathbb{R}$ by $|\cdot |$, in $\mathbb{R^n}$ by $||\cdot||.$</p>
<p>Let $a \not= b,$ $ a,b,$ and $t$ be real.</p>
<p>Show $h(t)$ is cont. at $t=t_0.$</p>
<p>Let $\epsilon \gt 0$ be given.</p>
<p>$|h(t)-h(t_0)|= ||(t-t_0)(a-b)|| =|t-t_0| ||a-b||$<... |
1,727,339 | <p>What am I doing wrong?</p>
<p>I've been learning how to put matrices into Jordan canonical form and it was going fine until I encountered this $4 \times 4$ matrix:</p>
<p>$A=\begin{bmatrix}
2 & 2 & 0 & -1 \\
0 & 0 & 0 & 1 \\
1 & 5 & 2 & -1 \\
0 & -4 & 0 & 4 \\
\end... | Disintegrating By Parts | 112,478 | <p>Take a look at
$$
(A-2I)^2 = \begin{bmatrix}
0 & 2 & 0 & -1 \\
0 & -2 & 0 & 1 \\
1 & 5 & 0 & -1 \\
0 & -4 & 0 & 2 \\
\end{bmatrix}^2
$$
The $a_{1,2}$ entry is non-zero. What does that tell you?</p>
|
2,360,268 | <p>Draw a triangle given $A-B=90$(degree) and length of $AC,BC$.</p>
<p><strong>My attempt</strong>:I thought It would be a good idea to draw a right angle so I made the picture below:</p>
<p><a href="https://i.stack.imgur.com/yikB7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/yikB7.png" alt="en... | Michael Rozenberg | 190,319 | <p>By Law of sines for $\Delta ABC$ we obtain
$$\frac{b}{\sin\beta}=\frac{a}{\sin(90^{\circ}+\beta)}$$ or
$$\tan\beta=\frac{b}{a}$$
and the angle with this measure easy to construct.</p>
|
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