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 |
|---|---|---|---|---|
4,064,084 | <p>Does there exists a countable family of infinite sets <span class="math-container">$\{A_n:n\in\mathbb N\}\subset\mathcal P(\mathbb N)$</span> satisfying the following property:
<span class="math-container">$$\text{For every infinite set }I\in\mathcal P(\mathbb N),\text{ there is }n\in\mathbb N\text{ such that }A_n\s... | hgmath | 886,804 | <p>As requested I'm posting the above comment as an answer here.</p>
<p>In order to show that every <span class="math-container">$n\geq 8$</span> can be written as sum of five cubes with absolute value less than <span class="math-container">$n$</span>:</p>
<ul>
<li>we can write for odd numbers <span class="math-contain... |
3,465,018 | <p>Compute <span class="math-container">$\pi_{2}(S^2 \vee S^2).$</span></p>
<p><strong>Hint:</strong>
Use universal covering thm. and use Van Kampen to show it is simply connected.</p>
<p>Still I am unable to solve it, could anyone give me more detailed hint and the general idea of the solution.</p>
| Grisha Taroyan | 402,997 | <p>Following <a href="https://en.wikipedia.org/wiki/Homotopy_excision_theorem" rel="noreferrer">homotopy excision theorem</a> and using an exact sequence of a pair <span class="math-container">$(S^2\times S^2, S^2\vee S^2)$</span> you can write down an exact sequence
<span class="math-container">$$
0 \to \pi_3(S^2\w... |
4,565,728 | <p>Given a collection of topological spaces <span class="math-container">$X_i$</span> indexed by the elements <span class="math-container">$i$</span> of a set <span class="math-container">$I$</span>, we consider the set product <span class="math-container">$P = \prod_{i \in I} X_i$</span> with projections <span class="... | Kritiker der Elche | 908,786 | <p>It is obvious that the box topology on <span class="math-container">$P$</span> makes all projections <span class="math-container">$p_i :P \to X_i$</span> <em>open maps</em>.</p>
<p>In a comment it was conjectured that the box topology is the finest topology on <span class="math-container">$P$</span> such that all p... |
271,844 | <p>I've installed the KnotTheory package, following the instructions <a href="http://katlas.org/wiki/Setup" rel="nofollow noreferrer">here</a>. But when I try to use it I get this error:</p>
<p><code>$CharacterEncoding: The byte sequence {139} could not be interpreted as a character in the UTF-8 character encoding.</co... | Henrik Schumacher | 38,178 | <p>I recall that Claus Ernst once asked me about that. It's time that we make the fix public and searchable:</p>
<p>Go into the file <code>init.m</code> on the <code>KnotTheory</code> package directory.
Then replace the lines</p>
<pre><code>KnotTheoryDirectory[] = (
File /. Flatten[FileInformation[ToFileName[#,"... |
3,115,830 | <p>So my logic to this up until now has been that for any <span class="math-container">$x$</span> the function <span class="math-container">$\left\lfloor\frac{\lceil x\rceil}{2}\right\rfloor$</span> will return an integer that is an element of <span class="math-container">$\mathbb Z$</span>. Thus since you can map any ... | Travis Willse | 155,629 | <p><strong>Hint</strong> Differentiating the relation
<span class="math-container">$$f = a {\bf T} + b {\bf N} + c {\bf B}$$</span>
and using the Frenet-Serret Identities gives
<span class="math-container">\begin{align}
{\bf T}
&= a {\bf T}' + b {\bf N}' + c' {\bf B} + c {\bf B}' \\
&= a (\kappa {\bf N}) + b (-... |
1,721,565 | <p>I'm having trouble with what I have done wrong with the chain rule below. I have tried to show my working as much as possible for you to better understand my issue here.</p>
<p>So:</p>
<p>Find $dy/dx$ for $y=(x^2-x)^3$
<br> So power to the front will equal = $3(x^2-x)^2 * (2x-1)$</p>
<p>Where did the $-1$ come f... | Matan L | 111,632 | <p>Lets denote the following:</p>
<p>$y(x) = (x^2-x)^3$</p>
<p>$f(x)=x^2-x$</p>
<p>$g(x)=x^3$</p>
<p>than $y(x)=g(f(x))$
so by the chain rule you get that $y_x = g_x (f(x)) f_x (x)$</p>
<p>thus in your case $y_x = 3f(x)^2f_x(x)=2(x^2-x)^2(2x-1)$</p>
|
622,278 | <p>What is the relation between the convergence of $\sum a_{n}$ and $\prod (1+a_{n})$ where $a_{n} \in \mathbb{C} \ \forall n$ ?</p>
<p>Where can I find some references about this topic ?</p>
| Sourav D | 114,457 | <p>This is definitely true when $a_n$ are positive numbers as demonstrated <a href="http://cornellmath.wordpress.com/2008/01/26/convergence-of-infinite-products/" rel="nofollow">here</a>.</p>
<p>However, when $a_n \in \mathbb{C}$, then convergence of $\sum|a_n|$ is just a necessary condition for convergence of the inf... |
715,361 | <p>Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that
$$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ \|q\|_{H^1(\Omega)}\leq1,\ n\in\mathbb{N}.\quad (1) $$
Is it then possible to conclude that
$$ \sup_{n\in\mathbb{N}}\|f_n\|_{L^2(\Omega)}... | Héctor Asencio L | 135,194 | <p>If you use $ x=1 $ you get $ 1-1=0 $ on the denominator, which gives you a division by zero. And since that's the only value that gets an equality on both sides of the equation, it shows that there's no solution.</p>
|
2,774,923 | <blockquote>
<p>$ABC$ is a triangle where $AE$ and $EB$ are angle bisectors, $|EC| = 5$, $|DE| = 3$, $|AB| = 9$. Find the perimeter of the triangle $ABC$.
<a href="https://i.stack.imgur.com/4nsiM.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4nsiM.jpg" alt="enter image description here"></a></p... | Community | -1 | <p>Well this is a pretty straightforward question.</p>
<p>Now, According to the angle bisector theorem:</p>
<blockquote>
<p>Consider a triangle ABC. Let the angle bisector of angle A intersect side BC at a point D between B and C. The angle bisector theorem states that the ratio of the length of the line segment BD... |
1,192,357 | <blockquote>
<p>What is the limit of $f(a,b) =\frac{a^\beta}{a^2 + b^2}$ as $(a,b) \to (0,0)$?</p>
</blockquote>
<p>Clearly the answer depends on the value of $\beta$. For $\beta > 0$, we can deduce via inequalities that $\lim_{(a,b) \to (0,0)} f(a,b) = 0$.</p>
<p>However, for $\beta < 0$, the answer is less ... | Eclipse Sun | 119,490 | <p>I will discuss the cases where $\beta$ is not an integer. To make it clearly, let's consider the limit$$\lim_{(a,b)\to(0,0),a>0}\frac{a^\beta}{a^2+b^2}.$$<br>
For $\beta>2$, since $\left|\dfrac{a^\beta}{a^2+b^2}\right|=\left|\dfrac{a^2}{a^2+b^2}\right||a^{\beta-2}|<|a^{\beta-2}|$ , the limit is $0$.<br>
For... |
262,173 | <p>Consider $x^2 + y^2 = r^2$. Then take the square of this to give $(x^2 + y^2)^2 = r^4$. Clearly, from this $r^4 \neq x^4 + y^4$. </p>
<p>But consider: let $x=a^2, y = b^2 $and$\,\,r = c^2$. Sub this into the first eqn to get $(a^2)^2 + (b^2)^2 = (c^2)^2$. $x = a^2 => a = |x|,$ and similarly for $b.$</p>
<p>Now ... | Hagen von Eitzen | 39,174 | <p>Note that $(x^2+y^2)^2=r^4$ does not imply that $r^4\ne x^4+y^4$.</p>
<p>In fact, you show that $$a^4+b^4=c^4$$
provided $x^2+y^2=r^2, x=a^2, y=b^2, z=c^2$. So what?</p>
|
1,001,320 | <p>I was wondering how to do an inequality problem involving QM-AM-GM-HM.</p>
<p>Question: For positive $a$, $b$, $c$ such that $\frac{a}{2}+b+2c=3$, find the maximum of $\min\left\{ \frac{1}{2}ab, ac, 2bc \right\}$.</p>
<p>I was thinking maybe apply AM-GM, however, I'm not sure what to plug in. Any help would be app... | Macavity | 58,320 | <p>Another way: If possible, let the optimum occur when one among $\frac12ab, ac, 2bc$ is lesser than the others. Then note that this smaller term determines the maximum and can be increased at the expense of the variable which is not involved in it. Thus at optimum we must have $\frac12ab = ac = 2bc$, which with th... |
1,042,715 | <p>Given a separable space $X$, if $A$ is discrete subspace of $X$, then $|A|\leq 2^{\aleph_0}$.</p>
<p>Some ideas?. It's similar to "jone's lemma", but without $A$ being closed. Whit what addiotional conditions we can assure the statement becomes true?</p>
| Robert Israel | 8,508 | <p>I nominate Halmos. All of his writing is good, but you might look in particular at
"Finite dimensional vector spaces".</p>
|
1,042,715 | <p>Given a separable space $X$, if $A$ is discrete subspace of $X$, then $|A|\leq 2^{\aleph_0}$.</p>
<p>Some ideas?. It's similar to "jone's lemma", but without $A$ being closed. Whit what addiotional conditions we can assure the statement becomes true?</p>
| JohnD | 52,893 | <p>Pick up an issue of SIAM Review. I admit that I haven't read every article in every issue of late, but I cannot recall an article with less than "very good" writing, and most are excellent.</p>
|
1,042,715 | <p>Given a separable space $X$, if $A$ is discrete subspace of $X$, then $|A|\leq 2^{\aleph_0}$.</p>
<p>Some ideas?. It's similar to "jone's lemma", but without $A$ being closed. Whit what addiotional conditions we can assure the statement becomes true?</p>
| Stiofán Fordham | 28,745 | <p>There is a book by Bott and Tu called '<a href="http://www.springer.com/mathematics/geometry/book/978-0-387-90613-3" rel="nofollow">Differential forms in algebraic topology</a>' which I think satisfies your criteria. In particular, you say it should be engaging and that is what I remember most about when I first rea... |
3,193,696 | <p>Could someone explain what are (at least the four first) moments ? (normalized moment to be more precise) Let <span class="math-container">$X$</span> a r.v. </p>
<ul>
<li><p>So the first moment is the expectation. This will correspond to <span class="math-container">$\mathbb E[X]$</span> and is going to be the "bar... | J.G. | 56,861 | <p>The first moment is the (arithmetic) <em>mean</em>, <span class="math-container">$\mu:=\Bbb E[X]$</span>. If this is finite, any <span class="math-container">$a$</span> satisfies <span class="math-container">$\Bbb E[X-a]=\mu-a$</span>. So if we want to measure spread around <span class="math-container">$\mu$</span>,... |
1,022,380 | <p>in below link, (formula (34)-(40)) there are some definition of Dirac delta function in terms of other functions such as Airy function, Bessel function of the first kind, Laguerre polynomial,....</p>
<p><a href="http://mathworld.wolfram.com/DeltaFunction.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/D... | Ian | 83,396 | <p>Whenever $f$ is a nonnegative function with integral 1, one has approximation by convolution. Specifically, define $f_c(x) = f(x/c)/c$. Then $\lim_{c \to 0^+} g * f_c = g$ for mildly restricted $g$. (For example, $L^p$ for $1 \leq p < \infty$ is enough.) In particular you can take $f(x)=\text{sech}(x)/\pi$.</p>
|
2,826,850 | <p>$x,y$ and $z$ are consecutive integers, such that $\frac {1}{x}+ \frac {1}{y}+ \frac {1}{z} \gt \frac {1}{45} $, what is the biggest value of $x+y+z$ ?.</p>
<p>I assumed that $x$ was the smallest number so that I could express the other numbers as $x+1$ and $x+2$ and in the end I got to a cubic function but I didn'... | Sameer Kailasa | 117,021 | <p>Write $x = y-1$ and $z = y+1$. Then we have $$\frac{1}{y-1} + \frac{1}{y} + \frac{1}{y+1} = \frac{2y}{y^2 - 1} + \frac{1}{y} = \frac{3y^2 - 1}{y^3 - y} > \frac{1}{45}$$ Since $3y^2 - 1< 3y^2$, this implies $$\frac{3y}{y^2 - 1} = \frac{3y^2}{y^3 - y} > \frac{1}{45} \iff 135y > y^2- 1 \iff y-\frac{1}{y} &l... |
4,172,964 | <p>EDIT: Agreed, this isn't a well formed question. But responses below have at least given me a different way to think about it.</p>
<p>EDIT2: Thanks the answers I discovered the Google S2 library (<a href="http://s2geometry.io/devguide/s2cell_hierarchy" rel="nofollow noreferrer">http://s2geometry.io/devguide/s2cell_h... | Cameron Williams | 22,551 | <p>Hint: try defining a new variable <span class="math-container">$w = z^4$</span> so that the equation in terms of <span class="math-container">$w$</span> reads (correcting the <span class="math-container">$z^8$</span> term to <span class="math-container">$|z|^8$</span>)</p>
<p><span class="math-container">$$ |w|^2 + ... |
271 | <p>Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it?</p>
<p>I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers.</p>
<p>The intended u... | Casebash | 123 | <p>Try <a href="http://www.wolframalpha.com/input/?i=1.644934">Wolfram Alpha</a>. It actually does sequences as well.</p>
|
271 | <p>Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it?</p>
<p>I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers.</p>
<p>The intended u... | Douglas S. Stones | 139 | <p>Sometimes the decimal digits of numbers will appear in Sloane's On-Line Encyclopedia of Integer Sequences <a href="http://oeis.org/" rel="nofollow noreferrer">OIES</a>. </p>
<p>E.g. <a href="http://oeis.org/A000796" rel="nofollow noreferrer">here</a> is the decimal expansion of pi.</p>
|
85,841 | <p><a href="http://reference.wolfram.com/language/ref/Binomial.html" rel="nofollow"><code>Binomial[n, k]</code></a> is converted to a polynomial only for <code>k</code> less than 6.</p>
<pre><code>Table[Binomial[n, k], {k, 1, 8}]
(* {n,
1/2 (-1 + n) n,
1/6 (-2 + n) (-1 + n) n,
1/24 (-3 + n) (-2 + n) (-1 + n) n,
1/120 ... | ubpdqn | 1,997 | <p>You could also define your own binomial coefficient function, e.g.</p>
<pre><code>bn[n_, k_] := Fold[(n - #2 + 1) #1/#2 &, 1, Range[k]]
</code></pre>
<p>so, </p>
<pre><code>Grid[{HoldForm[Binomial[n, #]], bn[n, #]} & /@ Range[0, 8],
Frame -> All]
</code></pre>
<p><img src="https://i.stack.imgur.com/... |
1,985,905 | <p>I was wondering if the cardinality of a set is a well defined function, more specifically, does it have a well defined domain and range?</p>
<p>One would say you could assign a number to every finite set, and a cardinality for an infinite set. So the range would be clear, the set of cardinal numbers. But what about... | Mees de Vries | 75,429 | <p>The collection of all sets does not form a set in ZF(-style) set theory, indeed. Note that the same is true for the collection of all cardinals: there is no set containing all cardinals, because then its union would be a set as well, and it would be a greater cardinal than any of its elements.</p>
<p>So the functio... |
1,594,130 | <p>Does there exist a vector field $\vec F$ such that curl of $\vec F$ is $x \vec i+y\vec j+z \vec k$ ? </p>
<p>UPDATE : I did $div(curl \vec F)=0$ as the answers did ; but that assumes a lot i.e. it assumes that components of $F$ have second partial derivatives and continuous mixed partial derivatives ; whereas for c... | Chappers | 221,811 | <p>No: a vector field $F$ can only be the curl of something if $\operatorname{div}{F}=0$, because
$$\operatorname{div}\operatorname{curl} G=0$$
for any twice-differentiable $G$ by antisymmetry. The divergence of your vector field, on the other hand, is $3$.</p>
|
2,574,221 | <p>Does divergence of $\sum a_k$ imply divergence of $\sum \frac{a_k}{1+a_k}$?</p>
<p>Note: $a_k > 0 $</p>
<p>I understand that looking at the contrapositive statement, we can say that the convergence of the latter sum implies $\frac{a_k}{1+a_k}\rightarrow 0$ but from here is it possible to deduce that $a_k\right... | Rigel | 11,776 | <p>If $a_k\geq 0$ for every $k$, then you can reason as follows.</p>
<p>If $(a_k)$ does not converge to $0$, then there exists a subsequence $(a_{n_k})$ with limit $l\in (0,+\infty]$.</p>
<p>It is then easy to verify that
$$
\frac{a_{n_k}}{1+a_{n_k}} \to
\begin{cases}
1, & \text{if}\ l=+\infty,\\
\frac{l}{1+l}\ne... |
2,574,221 | <p>Does divergence of $\sum a_k$ imply divergence of $\sum \frac{a_k}{1+a_k}$?</p>
<p>Note: $a_k > 0 $</p>
<p>I understand that looking at the contrapositive statement, we can say that the convergence of the latter sum implies $\frac{a_k}{1+a_k}\rightarrow 0$ but from here is it possible to deduce that $a_k\right... | Gabriel Romon | 66,096 | <p>If $\sum_n \frac{a_n}{1+a_n}$ converges, $\frac{a_n}{1+a_n}$ goes to $0$ as $n$ goes to $\infty$, hence $a_n$ goes to $0$ as $n\to \infty$.</p>
<p>But since $\lim_{n\to \infty} \frac{a_n}{\frac{a_n}{1+a_n}} = 1$, for large enough $n$, $\left|\frac{a_n}{\frac{a_n}{1+a_n}} - 1 \right|\leq \frac 12$, hence $a_n\leq \f... |
2,985,917 | <p>Would it be possible to calculate which function in the Schwarz class of infinitely differentiable functions with compact support is closest to triangle wave?</p>
<p>Let us measure closeness as <span class="math-container">$$<f-g,f-g>_{L_2}^2 = \int_{-\infty}^{\infty}(f(x)-g(x))^2dx$$</span></p>
<p>I don't e... | User8128 | 307,205 | <p>In general, there is no "closest" Schwarz function to a given <span class="math-container">$L^2$</span> function. Since the Schwarz class is dense in <span class="math-container">$L^2$</span>, we can find Schwarz functions arbitrarily close to any <span class="math-container">$L^2$</span> function. Indeed, let <span... |
3,793,581 | <p>I can solve this integral in a certain way but I'd like to know of other, simpler, techniques to attack it:</p>
<p><span class="math-container">\begin{align*}
\int _0^{\frac{\pi }{2}}\frac{\ln \left(\sin \left(x\right)\right)\ln \left(\cos \left(x\right)\right)}{\tan \left(x\right)}\:\mathrm{d}x&\overset{ t=\sin... | Quanto | 686,284 | <p>Substitute <span class="math-container">$t= \sin^2x$</span></p>
<p><span class="math-container">\begin{align}
&\int _0^{\frac{\pi }{2}}\frac{\ln (\sin x)\ln (\cos x)}{\tan x}\>{d}x \\= &\frac18\int _0^{1}\frac{\ln t\ln (1-t)}{t}\>dt \overset{IBP}= \frac1{16}\int_0^1 \frac{\ln^2 t}{1-t}dt= \frac1{16}\cd... |
1,611,390 | <p>How to show that the following function is an injective function?</p>
<p>$ \varphi : \mathbb{N}\times \mathbb{N} \rightarrow \mathbb{N} \\
\varphi(\langle n, k\rangle) = \frac{1}{2}(n+k+1)(n+k)+n$</p>
<p>I'm starting with $ \frac{1}{2}(a+b+1)(a+b)+a = \frac{1}{2}(c+d+1)(c+d)+c$, but how am I supposed to show from ... | David | 119,775 | <p>Here are two arguments. The first should make the statement "visually obvious" while the second is more formal.</p>
<p>Construct a grid displaying the value of $\phi(n,k)$ for each value of $n$ and $k$:
$$\begin{matrix}
\color{red}{\phantom{k={}}4}&14\cr
\color{red}{\phantom{k={}}3}&9&13\cr
\colo... |
894,159 | <p>I was assigned the following problem: find the value of $$\sum_{k=1}^{n} k \binom {n} {k}$$ by using the derivative of $(1+x)^n$, but I'm basically clueless. Can anyone give me a hint?</p>
| Tulip | 55,386 | <p>Hint : set $$f(x)=(1+x)^n=\sum_{k=0}^n \binom{n}{k}x^k$$ What you get when you evaluate $f'(1)$ ? </p>
<p>Note that you sum can begin from $0$ since the term with $k=0$ is $0$.</p>
|
512,590 | <p>According to the definition my professor gave us its okay for a matrix in echelon form to have a zero row, but a system of equations in echelon form cannot have an equation with no leading variable.</p>
<p>Why is this? Aren't they supposed to represent the same thing?</p>
| Robert Lewis | 67,071 | <p>Using de Moivre's formula $(\cos \alpha + i \sin \alpha)^n = \cos n\alpha + i \sin n\alpha$, we see that setting $z = r(\cos \alpha + i\sin \alpha)$, we have $z^{-1} = r^{-1}(\cos \alpha - i \sin \alpha)$; thus</p>
<p>$z + z^{-1} = r(\cos \alpha + i \sin \alpha) + r^{-1}(\cos \alpha - i \sin \alpha) = 2 \cos \alpha... |
2,303,106 | <p>I was looking at this question posted here some time ago.
<a href="https://math.stackexchange.com/questions/1353893/how-to-prove-plancherels-formula">How to Prove Plancherel's Formula?</a></p>
<p>I get it until in the third line he practically says that $\int _{- \infty}^{+\infty} e^{i(\omega - \omega')t} dt= ... | reuns | 276,986 | <p>As you noted $\int _{- \infty}^{+\infty} e^{i(\omega - \omega')t} dt= 2 \pi \delta(\omega - \omega')$ is of course not true. This is an abuse of notation, what it really means is that the Fourier transform of the (tempered) distribution $f(\omega) = e^{i \omega' t}$ is the (tempered) distribution $\hat{f}(\omega) =... |
2,809,686 | <p>Let S={1,2,3,...,20}. Find the probability of choosing a subset of three numbers from the set S so that no two consecutive numbers are selected in the set.
"I am getting problem in forming the required number of sets."</p>
| Community | -1 | <p><strong><em>Inclusion-Exclusion Principle:-</em></strong><br>
Suppose $A_i$ denote the event that both the numbers $i,$ and $i+1$ ($i=1,2,...,19$) are included in the set.<br>
$n(\bigcup\limits_{i=1}^{19} A_i)$ will denote the total number of cases where we can find at least two consecutive integers. </p>
<p>Her... |
677,241 | <p>Let $A$ be a list of $n$ numbers in range $[1,100]$ (numbers can repeat).
I'm looking for the number of permutations of $A$ which start with a non-decreasing part, where this part ends with the first instance of the highest number, call this "index $i$" (1 based)from the left. After $i$, the remaining permutation is... | frabala | 53,208 | <p>Consider a finite sequence $s$ of length $n$ on integers that range from 0 to 100. Since repetition of numbers is allowed, let's denote with $t^a$ the number of occurrences of the number $a$ within $s$. We also denote with $l$ the largest number within $s$ (I guess it won't always be 100).</p>
<p>Now, assume a give... |
1,274,816 | <p>It seems known that there are infinitely many numbers that can be expressed as a sum of two positive cubes in at least two different ways (per the answer to this post: <a href="https://math.stackexchange.com/questions/1192338/number-theory-taxicab-number">Number Theory Taxicab Number</a>).</p>
<p>We know that</p>
... | Tito Piezas III | 4,781 | <p>In the paper <em><a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Broughan/broughan25.pdf" rel="noreferrer">Characterizing the Sum of Two Cubes</a></em>, Kevin Broughan gives the relevant theorem,</p>
<p><strong><em>Theorem:</em></strong> Let $N$ be a positive integer. Then the equation $N = x^3 + y^3$ has a solu... |
3,792,954 | <p>For vector space <span class="math-container">$V$</span> and <span class="math-container">$v \in V$</span>, there is a natural identification <span class="math-container">$T_vV \cong V$</span> where <span class="math-container">$T_vV$</span> is the tangent space of <span class="math-container">$V$</span> at <span cl... | Patrick Stevens | 259,262 | <p>Differentiate wrt (wlog) <span class="math-container">$x$</span> to show that it's increasing in <span class="math-container">$x$</span> when <span class="math-container">$|y| < 1$</span>; then evaluate at <span class="math-container">$x=-1$</span> and at <span class="math-container">$x=1$</span> to demonstrate t... |
2,022,566 | <p>How can I calculate the below limit?
$$
\lim\limits_{x\to \infty} \left( \mathrm{e}^{\sqrt{x+1}} - \mathrm{e}^{\sqrt{x}} \right)
$$
In fact I know should use the L’Hospital’s Rule, but I do not how to use it.</p>
| Olivier Oloa | 118,798 | <p>One may write, as $x \to \infty$,
$$
\begin{align}
e^{\sqrt{x+1}} - e^{\sqrt{x}}&=e^{\sqrt{x}}\left(e^{\sqrt{x+1}-\sqrt{x}} - 1\right)
\\\\&=e^{\sqrt{x}}\left(e^{\frac1{\sqrt{x+1}+\sqrt{x}}} - 1\right)
\\\\&=e^{\sqrt{x}}\left(1+\frac1{\sqrt{x+1}+\sqrt{x}}+O\left(\frac1x\right) - 1\right)
\\\\& \sim \... |
64,925 | <p>Suppose $G$ is a group and $V$ an irreducible representation of $G$. One has that $V\otimes V\cong \Lambda^2(V)\oplus Sym^2(V)$. It is well-known that if the trivial representation appears as a subrepresentation of $\Lambda^2(V)$ then $V$ is of quaternionic type; while if the trivial representation appears as a subr... | Andreas Blass | 6,794 | <p>This is essentially what Darij wrote, but without mentioning the bilinear forms. (I had written it out before reading far enough into Darij's post to see that he was really doing the same thing, after the part about bilinear forms.) Think of $V\otimes V$ as $\text{Hom}(V^*,V)$. An occurrence of the trivial repres... |
959,201 | <p>I am confused about the following.</p>
<p>Could you explain me why if $A=\varnothing$,then $\cap A$ is the set of all sets?</p>
<p>Definition of $\cap A$:</p>
<p>For $A \neq \varnothing$:</p>
<p>$$x \in \cap A \leftrightarrow (\forall b \in A )x \in b$$</p>
<p><strong>EDIT</strong>:</p>
<p>I want to prove that... | John Hughes | 114,036 | <p>I believe that the statement you want us to prove is wrong. For one thing, there is (in the versions of set theory that I know) no set $R$ of all sets, for if there were, you could form
$$
Q = \{X \in R : X \notin X\}
$$
the set of all sets that are not elements of themselves. The statement $Q \in Q$ then becomes n... |
476,095 | <p>I am attempting to learn about mathematical proofs on my own and this is where I've started. I think I can prove this by induction. Something like:</p>
<p>$n = 2k+1$ is odd by definition</p>
<p>$n = 2k+1 + 2$ (this is where I'm stuck, how do I show that this is odd?)</p>
<p>$n = 2(k+1) + 1$ (if I can show that it... | walcher | 89,844 | <p>An integer $n$ is odd if and only if it is not divisible by $2$ or again if and only if it is of the form $2k+1$ for some integer $k$. If $n=2k+1$ is odd, then $n+2=2k+1+2=2k+2+1=2(k+1)+1$ is obviously odd as well.</p>
|
3,144,813 | <blockquote>
<p>Let <span class="math-container">$X : \mathbb{R} \to \mathbb{R}^n$</span> be a <span class="math-container">$C^1$</span> function. Let <span class="math-container">$\| .\|$</span> be the norm : <span class="math-container">$\| v \| = \max_{1 \leq i \leq N} \mid v_i \mid$</span>. Then is it true that :... | GReyes | 633,848 | <p><span class="math-container">$x=s$</span>, <span class="math-container">$y=0$</span> is just the <span class="math-container">$X$</span>-axis. On that axis you have an initial datum <span class="math-container">$f(x)$</span>. Thus your characteristics will be issued from each point of the form <span class="math-cont... |
2,330,514 | <p>Prove if n is a perfect square, n+2 is not a perfect square</p>
<blockquote>
<p>Assume n is a perfect square and n+2 is a perfect square (proof by
contradiction)</p>
<p>There exists positive integers a and b such that $n = a^2$ and $n + 2= b^2$</p>
<p>Then $a^2 + 2 = b^2$</p>
<p>Then $2 = b^2-a^2... | Ross Millikan | 1,827 | <p>You have to factor $2$ and the only factorization into two terms is $2 \cdot 1$. Since $a,b$ are positive, $b+a \gt b-a$ so we take $b+a=2, b-a=1$</p>
|
3,372 | <p>Have any questions first proposed on Mathoverflow attracted enough interest from experts in their field that solving them would be considered a significant advance?</p>
<p>I don't want to count problems that are known (or strongly suspected) to be at least as hard as some previously described problem, unless the ve... | Martin Sleziak | 8,250 | <p>As far as I know, there is no such thing. See also this post on Mathematics Meta.
<a href="https://math.meta.stackexchange.com/q/12374">Number of people online</a>. On main meta you can find some feature requests such as <a href="https://meta.stackexchange.com/q/2631">View approximate number of users online</a> or <... |
784,258 | <p>I understand that this is an induction question. </p>
<p>I start with the base case (n=1):</p>
<p>$$1 < 2 \tag{That works!}$$</p>
<p>Induction step: Assume the statement works for all $n = k$, Prove for all $n = k+1$</p>
<p>Assume $1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}}+ ... +\frac{1}{\sqrt{k+1}}\le 2\s... | Pedro | 23,350 | <p><em>Hint</em> We have that $(2 x^{1/2})'=x^{-1/2}$. Now, think about $$\int_1^n x^{-1/2}dx$$</p>
|
25,363 | <p>In what way and with what utility is the law of excluded middle usually disposed of in intuitionistic type theory and its descendants? I am thinking here of topos theory and its ilk, namely synthetic differential geometry and the use of topoi in algebraic geometry (this is a more palatable restructuring, perhaps), w... | Wouter Stekelenburg | 3,603 | <p>In a topos, the question is not whether a sentence is true or false, but <em>where</em> it is true, because toposes--at least the geometric ones you're talking about--are generalized spaces. There can be limit points where this question cannot be decided for all sentences. The problem is then that there are other po... |
25,363 | <p>In what way and with what utility is the law of excluded middle usually disposed of in intuitionistic type theory and its descendants? I am thinking here of topos theory and its ilk, namely synthetic differential geometry and the use of topoi in algebraic geometry (this is a more palatable restructuring, perhaps), w... | Frank Quinn | 124,645 | <p>A version of Russell's paradox provides a failure of "law of excluded middle". I don't know if this directly answers your question, but it illustrates the constraints on getting this "law" to work.</p>
<p>Define a "logical function" to be one that returns 'yes' or 'no' when given an input. "law of excluded middle" ... |
101,098 | <p>I apologize in advance because I don't know how to enter code to format equations, and I apologize for how elementary this question is. I am trying to teach myself some differential geometry, and it is helpful to apply it to a simple case, but that is where I am running into a wall.</p>
<p>Consider $M=\mathbb{R}^2$... | Pierre-Yves Gaillard | 660 | <p>Here is, for what it's worth, how I see this. </p>
<p>In differential geometry, it is sometimes convenient to denote the canonical basis of $\mathbb R^n$ by
$$
\left(\frac{\partial}{\partial x_i}\right)_{i=1,\dots,n}\quad,
$$
and the dual basis (which is a basis of $(\mathbb R^n)^*$) by
$$
(dx_i)_{i=1,\dots,n}... |
1,076,974 | <p>Does anyone know how I can determine the equation of the 3D object below? (Maybe there's a program that can do it?) I am looking for a formula to define this 3D object, but am having trouble finding one. </p>
<p>(If you can imagine the 2D object you see revolved about the x-axis, that is the 3D object I'm referring... | coffeemath | 30,316 | <p>Suppose the top of the sketched curve has equation $y=f(x)$ for $0 \le x \le a$ and as in the diagram $f(x) \ge 0$ for $x$ in $[0,a].$ The distance from a 3-d point $(x,y,z)$ to the point $(x,0,0),$ in the plane having constant first coordinate $x,$ is $\sqrt{y^2+z^2},$ and in the revolved figure you want this dista... |
1,600,597 | <p>I'm currently going through Spivak's calculus, and after a lot of effort, i still can't seem to be able to figure this one out.</p>
<p>The problem states that you need to prove that $x = y$ or $x = -y$ if $x^n = y^n$</p>
<p>I tried to use the formula derived earlier for $x^n - y^n$ but that leaves either $(x-y) = ... | fleablood | 280,126 | <p>I'm assuming you are assuming we are dealing with real numbers. (The result is not true in complex numbers as $i^4 = 1^4 = (-1)^4 = (-i)^4$).</p>
<p>Now has it been shown to your satisfaction that every $b > 0$ has a unique positive $n-th$ root? Well... okay, let's do some basics</p>
<p>Okay, suppose $0 < ... |
3,852,952 | <p>Given a projective space <span class="math-container">$\mathbb{P}^n(\mathbb{C})$</span>, I can consider the Grasmannian of lines <span class="math-container">$G(2,n+1)$</span>, which has a structure of projective variety inside <span class="math-container">$\mathbb{P}^N$</span>, where <span class="math-container">$... | xxxxxxxxx | 252,194 | <p>You have that lines in <span class="math-container">$\mathbb{P}^{n}(\mathbb{C})$</span> correspond to points of the Grassmannian <span class="math-container">$G(2,n+1)$</span>. Now when two lines <span class="math-container">$\ell_{1}$</span>, <span class="math-container">$\ell_{2}$</span> of the projective space in... |
926,804 | <p>Is there a word for the quality of a number to be either positive or negative? Consider this question:</p>
<p><em>What's the ... (sign/positivity/negativity, but a word that could describe either) of number <strong>x</strong>?</em></p>
<p>Also, is there an all-encompassing word for the sign put in front of a numbe... | Piquito | 219,998 | <p>If you want to have a "less-simple" answer than the correct one given by @k170, the words positive and negative occur in all totally ordered commutative group with neutral element noted $0$. The positive elements $x$ are those for which $0<x$ and the negative ones are those such that $x<0$. The usual conventi... |
643,560 | <p>Let $\{x_n\}_{n=1}^{\infty}$ and $\{y_n\}_{n=1}^{\infty}$ be sequences of real numbers. Does the following hold:</p>
<p>$$
\limsup x_n +\liminf y_n \le \limsup\,(x_n+y_n).
$$ </p>
<p>This is what I have tried but I am not quite sure if it is correct.
$\text{Fix } K>1. \text{ Let }L=\inf_{1\le i \le k}y_i$. Now... | nmasanta | 623,924 | <p>Let <span class="math-container">$~\limsup x_n=\overline x~$</span> and <span class="math-container">$~\liminf y_n=\underline y~,$</span> then <span class="math-container">$~\overline x~,~\underline y~\in \mathbb R.$</span> <br>
Let <span class="math-container">$~\epsilon>0~$</span> be given.<br>
Then <span class... |
3,450,713 | <p>Let <span class="math-container">$a+3b=7$</span> and <span class="math-container">$c=3$</span>. Then value of <span class="math-container">$a+3(b+c)$</span> is</p>
<p>A) <span class="math-container">$10$</span></p>
<p>B) <span class="math-container">$16$</span> </p>
<p>C) <span class="math-container">$21$</span><... | JMoravitz | 179,297 | <p>We are told that <span class="math-container">$\color{blue}{a+3b} = 7$</span> and that <span class="math-container">$\color{red}{c}=3$</span></p>
<p>We are asked to find <span class="math-container">$a+3(b+c)$</span></p>
<p><span class="math-container">$$\begin{array}{rll}a+3(b+c) &=a+(3b+3c)&\text{distrib... |
186,146 | <p>On a finite dimensional vector space, the answer is yes (because surjective linear map must be an isomorphism). Does this extend to infinite dimensional vector space? In other words, for any linear surjection $T:V\rightarrow V$, AC guarantees the existence of right inverse $R:V\rightarrow V$. Must $R$ be linear?</p>... | Qiaochu Yuan | 232 | <p>No. Let $V = \text{span}(e_1, e_2, ...)$ and let $T : V \to V$ be given by $T e_1 = 0, T e_i = e_{i-1}$. A right inverse $S$ for $T$ necessarily sends $v = \sum c_i e_i$ to $\sum c_i e_{i+1} + c_v e_1$ but $c_v$ may be an arbitrary function of $v$. </p>
|
2,217,454 | <p>Let $\{ x_i : i \in I \}$ be a family of numbers $x_i \in \mathbb R$ with $I$ an arbitrary index set. We say that this family is summable with value $s$ (and write $s = \sum_{i \in I} x_i$ then) if for every $\varepsilon > 0$ there exists some finite set $I_{\varepsilon}$ such that for every finite superset $J \s... | Roberto Rastapopoulos | 388,061 | <p>Assume that
$$ \sum_{i\in I} |x_i| < \infty, $$
and define $J_n = \{i \in I: |x_i| \geq 1/n$}, for $n > 0$, and $J_{\infty} = \{i\in I: x_i \neq 0\}$. By assumption, $J_n$ must be a finite set for all $n$, so
$J_\infty = \bigcup_{n = 1}^{\infty}J_n$ is countable.</p>
|
1,478,314 | <p>In this particular case, I am trying to <strong>find all points $(x,y)$ on the graph of $f(x)=x^2$ with tangent lines passing through the point $(3,8)$</strong>. </p>
<p>Now then, I know the <a href="http://www.meta-calculator.com/online/?panel-102-graph&data-bounds-xMin=-10&data-bounds-xMax=10&data-bo... | Bernard | 202,857 | <p>Either you write the equation of a tangent at $(x_0,x_0^)$2 and check under which condition this equation is satisfied by the $(3,8)$.</p>
<p>Or you write the equation of a line with slope $t$ passing through the point $(3,8)$: $$y-8=t(x-3),$$
and find under which condition this line has a double intersection with ... |
2,114,276 | <p>How to show that $(x^{1/4}-y^{1/4})(x^{3/4}+x^{1/2}y^{1/4}+x^{1/4}y^{1/2}+y^{3/4})=x-y$</p>
<p>Can anyone explain how to solve this question for me? Thanks in advance. </p>
| Sarvesh Ravichandran Iyer | 316,409 | <p>Let <span class="math-container">$x = u^4$</span> and <span class="math-container">$y = v^4$</span>, then the left hand side of the equation simplifies to:
<span class="math-container">$$
(u-v)(u^3 + u^2v + uv^2 + v^3)
$$</span></p>
<p>Now, you can choose to multiply directly:
<span class="math-container">$$
u^4 + ... |
191,210 | <p>Let $R$ be the smallest $\sigma$-algebra containing all compact sets in $\mathbb R^n$.
I know that based on definition the minimal $\sigma$-algebra containing the closed (or open) sets is the Borel $\sigma$-algebra. But how can I prove that $R$ is actually the Borel $\sigma$-algebra?</p>
| Michael Greinecker | 21,674 | <p>It is enough to show that every closed set is in the $\sigma$-algebra. So let $C$ be closed, $x$ be an arbitrary point and $K_n$ the closed ball with center $x$ and radius $n$. Then $K_n\cap C$ is compact for all $n$ and $\bigcup_n (K_n\cap C)=C$.</p>
|
1,647,157 | <p>How can I solve this using only 'simple' algebraic tricks and asymptotic equivalences? No l'Hospital.</p>
<p>$$\lim_{x \rightarrow0}
\frac
{\sqrt[3]{1+\arctan{3x}} - \sqrt[3]{1-\arcsin{3x}}}
{\sqrt{1-\arctan{2x}} - \sqrt{1+\arcsin{2x}}}
$$</p>
<p>Rationalizing the numerator and denominator gives</p>
<p>$$
\lim_{x... | runaround | 310,548 | <p>Using your $g(t) = t - \frac{\pi|{2}, for\, t > 0$, and $g(t) = t +\frac{\pi}{2}\,for\, t< 0$,</p>
<p>We have $$g(t) - t = -\frac{\pi}{2}\, for\, t > 0 $$ and
$$g(t) - t = \frac{\pi}{2} \, for\, t < 0$$
So (g(t) - t) is odd function.</p>
<p>so $$\int_{_pi}^{\pi} s(t)(g(t) - t)dt = 0 $$, for any $s \in ... |
1,927,394 | <blockquote>
<p>Number of all positive continuous function <span class="math-container">$f(x)$</span> in <span class="math-container">$\left[0,1\right]$</span> which satisfy <span class="math-container">$\displaystyle \int^{1}_{0}f(x)dx=1$</span> and <span class="math-container">$\displaystyle \int^{1}_{0}xf(x)dx=\alp... | Thomas Andrews | 7,933 | <p>One way is to think of $f(x)$ as a probability measure. Then you are seeking a continuous random variable $Y$ such that $E(Y)^2=E(Y^2)$. Since $E(Y^2)-E(Y)^2$ is the variance of $Y$, this value is zero if and only if $Y$ is a constant, and thus $f$ would be a delta function, not a continuous real-valued function.</p... |
90,480 | <p>Given two simplicial topological spaces $X_{\bullet}$ and $Y_{\bullet}$ (i.e. a simplicial object in Top) and a continuous map between their geometric realizations $f \colon \lvert X_{\bullet} \rvert \to \lvert Y_{\bullet} \rvert$. Is $f$ homotopic to $\lvert \varphi_{\bullet} \rvert$ for a map $\varphi_{\bullet}$ o... | Ilias A. | 21,369 | <p>I'm just reformulating your question in simplicial case.
If you consider the category of simplicial sets $\mathbf{sSet}$ you can formulate your question as follows:
Is the diagonal functor $diag: [\mathbf{\Delta}^{op},\mathbf{sSet}]\rightarrow \mathbf{sSet}$ from bisimplicial sets to simplicial sets homotopicaly f... |
36,568 | <p>To do Algebraic K-theory, we need a technical condition that a ring $R$ satisfies $R^m=R^n$ if and only if $m=n$. I know some counterexamples for a ring $R$ satisfies $R=R^2$. </p>
<p>Are there any some example that $R\neq R^3$ but $R^2 = R^4$ or something like that?</p>
<p>(c.f. if $R^2=R^4$, then we need that $R... | Simon Wadsley | 345 | <p>Yes. I think you are looking for the Leavitt algebras. I don't know much about them but you could start here: <a href="http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.3827v1.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.3827v1.pdf</a></p>
<p>The idea is that of the Leavitt algebras $R=L(1,n)$ is t... |
774,209 | <p>I got stuck to find a fair formula to calculate the average ranking of the items that I found after consecutive searches, look:</p>
<p><img src="https://i.stack.imgur.com/5LazY.jpg" alt="enter image description here" /></p>
<p>If I calculate the simple average of the item2 for example I get 1,33 as a result, not eve... | Ross Millikan | 1,827 | <p>What are the values in the table? Are they rankings under the three searches? Then you can just average (or sum-gets the same thing and removes the division) the numbers, rank them, and report the result. $1.33333$ for item $2$ is not bad, actually. It will probably be worst (if low numbers are bad), which is exa... |
2,822,355 | <p>this is a problem from one of the former exams from ordinary differential equations.</p>
<p>Find a solution to this equation:</p>
<p>$$x''''+6x''+25x=t\sinh t\cdot \cos(2t)$$</p>
<p>of course the only problem will be to find a particular solution, since the linear part is very simple to solve. My question is how ... | Lutz Lehmann | 115,115 | <p>$$
0=x^4+6x^2+25=(x^2+5)^2-4x^2=(x^2-2x+5)(x^2+2x+5)
$$
has obviously the solutions $x=\pm1\pm2i$ that are all simple and all in resonance with the right side. Thus the method of undetermined coefficients would give you the general form
$$
y=t\Bigl((A_0+A_1t)\cosh(t)\cos(2t)+(B_0+B_1t)\cosh(t)\sin(2t)\\~~~~+(C_0+C_1... |
1,076,292 | <p>I wish to use two points say $(x_1$,$y_1)$ and $(x_2$,$y_2)$ and obtain the coefficients of the line in the following form: $$ Ax + By + C = 0$$</p>
<p>Is there any direct formula to compute.</p>
| Claude Leibovici | 82,404 | <p>You can also remark that $$Ax + By + C = 0=\alpha x+ y + \gamma$$ (with $\alpha=\frac AB$,$\gamma=\frac CB$) and then, applying the conditions, $$\alpha x_1+ y_1 + \gamma=0$$ $$\alpha x_2+ y_2 + \gamma=0$$ and you have to solve for $\alpha $ and $\gamma$ two linear equations. Using the classical methods, you get $$\... |
2,339,974 | <p>I know that there is a theorem that states that If $(G, *)$ and $(H, •)$ are groups, $e_G$ (identity of $G$) and $e_H$ (identity of $H$). Let $f: G\to H$ be a homomorphism. Then </p>
<ol>
<li>$f(e_G) = e_H$.</li>
</ol>
<p>I don't know how to use this, or begin my proof or should I use kernel for this problem?</p>... | user458670 | 458,670 | <p>Note $f(G) < L$. By the first isomorphism theorem $f(G) \approx G/\ker f$. Therefore $|f(G)| = |G|/\lvert\ker f\rvert$. But the order of $f(G)$ divides the order of $L$ which only happens when $|G|=\lvert\ker f\rvert$ which implies $f$ is the trivial map as desired.</p>
|
13,109 | <p>I posted a question half a hour ago. But I think I found the answer myself now.
I understand that answering your own question is appreciated (instead of deleting it).
But I don't know if I should give a hint or a full solution.</p>
<p>It feels a little bit strange to give a hint to my <em>own</em> question, I don't... | robjohn | 13,854 | <p>Generally, hints are given when it seems better that the OP understand how to do the problem rather than being given the answer. Usually, this is when the question is tagged homework, or it seems very likely that the question is from a homework assignment.</p>
<p>In this case, the OP (you) understands the problem, ... |
1,591,863 | <p>I'm studying for the sat, and one question was presented as follows:</p>
<p>If $n$ is a positive integer such that the units (ones) digit of $n^2+4n$ is $7$ and the units digit of n is not $7$, what is the units digit of $n+3$?</p>
<p>So I'm trying to find $n$ such that:</p>
<p>$$(n^2+4n) \mod10=7$$</p>
<p>I kn... | Paul Sinclair | 258,282 | <blockquote>
<p>In this question I am unable to get why we have a terminology of implication here.</p>
</blockquote>
<p>Because that is the problem you've been given. The idea is clearly to test your understanding of implications and of partial orders. Having separate concepts mixed together in problems is quite com... |
253,584 | <p>Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=J_h^\mathtt{T}(x)J_h(x)\in\mathbb{R}^{n\times n}$ and $B(x):=J_h(x)J_h(x)^\mathtt{T}\in\mathbb{R}^{m\times m}$.</p>
... | Peter Michor | 26,935 | <p>Suppose $h$ is real analytic. Then $A(x)$ and $B(x)$ are real analytic in $x\in \mathbb R^n$.
Part L of the main theorem of </p>
<ul>
<li>Andreas Kriegl, Peter W. Michor, Armin Rainer: Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equations and Operator Theory 71,3 (20... |
2,051,555 | <p>I have the following limit to solve.</p>
<p>$$\lim_{x \rightarrow 0}(1-\cos x)^{\tan x}$$</p>
<p>I am normally supposed to solve it without using l'Hôpital, but I failed to do so even with l'Hôpital. I don't see how I can solve it without applying l'Hôpital a couple of times, which doesn't seem practical, nor how ... | lab bhattacharjee | 33,337 | <p>Let $y=\lim_{x\to0}(1-\cos x)^{\tan x}$</p>
<p>$$\ln y=\lim_{x\to0}\dfrac{\ln(1-\cos x)}{\cot x}$$ which is of the form $\dfrac\infty\infty$</p>
<p>So applying L'Hospital $$\ln y=-\lim_{x\to0}\dfrac{\sin x}{\csc^2x(1-\cos x)} =-\lim_{x\to0}\dfrac{\sin x(1+\cos x)(1-\cos x)}{(1-\cos x)}=?$$ </p>
|
92,296 | <p>I trying to review for calculus and I can't figure out how to do $\sqrt{200} - \sqrt{32}$ </p>
| Asaf Karagila | 622 | <p>$$\sqrt{200}-\sqrt{32} = \sqrt{2\cdot 100}-\sqrt{2\cdot16} = \sqrt{2}\sqrt{100}-\sqrt{2}\sqrt{16} = \sqrt{2}(10-4) = 6\sqrt{2} = \sqrt{2\cdot36}=\sqrt{72}$$</p>
|
1,238,783 | <p>I am currently in high school where we are learning about present value. </p>
<p>I struggle with task like these: Say you get 6% interest each year, how much interest would that be each month?</p>
| Claude Leibovici | 82,404 | <p>Almost as Alberto Debernardi answered, let us note $y$ the interest per year and $m$ the interest per month. So, the equation which relates them is $$(1+m)^{12}=1+y$$ So $$m=(1+y)^{1/12}-1$$ To approximate it, you can use the binomial expansion and get $$m\approx \frac{1}{12}y-\frac{11 }{288}y^2=\frac{1}{288} (24-11... |
3,588,053 | <p>Is this a valid proof that the harmonic series diverges?</p>
<ol>
<li>Assume the series converges to a value, S:</li>
</ol>
<p><span class="math-container">$$S=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...$$</span></p>
<ol start="2">
<li>Split the series into two, with alternating even and odd denominator... | Dr. Michael W. Ecker | 438,184 | <p>Rearrangement requires knowledge of absolute convergence. When I wrote up my proof and it was published 23 years ago, that was the comment added. Other than that, your proof is absolutely identical to mine. Here is the reference:</p>
<p>Michael W. Ecker, <em>Divergence Of The Harmonic Series By Rearrangement</em>, ... |
500,323 | <p>As a relative beginner trying to understand math more deeply, I'm trying to learn more about the mathematical laws (the laws of the operations $+, -, \times, \div$)</p>
<p>For example, I know the basic laws (the ones that are just taken to be true) -- the commutative, associative, and distributive laws. What area o... | Don Larynx | 91,377 | <p>Generally, one would need a good experience with number systems to understand the properties of numbers thoroughly well. In order to begin understanding what makes up those laws, just read about them and see if you can make something of it. </p>
<p><a href="http://www.purplemath.com/modules/numbprop.htm" rel="nofol... |
3,704,633 | <p>Evaluate: <span class="math-container">$$\lim_{n \to \infty} \sqrt[n]{\frac{n!}{\sum_{m=1}^n m^m}}$$</span>
In case it's hard to read, that is the n-th root. I don't know how to evaluate this limit or know what the first step is... I believe that: <span class="math-container">$$\sum_{m=1}^n m^m$$</span> doesn't ha... | Integrand | 207,050 | <p>By Stirling's Formula,
<span class="math-container">$$
n! \approx \left(\frac{n}{e}\right)^n\cdot \sqrt{2\pi n}
$$</span>
In the denominator, we have
<span class="math-container">$$
\sum_{m=1}^{n}m^m = n^n + (n-1)^{n-1}+\ldots +2^2+1
$$</span>So,
<span class="math-container">$$
\lim_{n\to\infty}\left(\frac{n!}{\sum_... |
536,128 | <p>I was trying to make sense of a problem when I stumbled upon this on yahoo answers. I was just wondering if it was correct. If it is, can you please maybe explain why?</p>
<p>${\bf r}'(t) = \langle -5 \cos t, -5 \cos t, -4 \sin t \rangle$</p>
<p>${\bf r}''(t) = \langle 5 \sin t, 5 \sin t, -4 \cos t \rangle$. </p>
... | Trevor Wilson | 39,378 | <p>The question is getting rather broad, but let's see if I can answer a part of it.</p>
<p>One answer to the question of why the area under a continuous function is a limit of Riemann sums is that we <em>define</em> it that way. But for that to make sense, we need the see that the limit <em>exists</em>, and also tha... |
2,483,794 | <p>I'm trying to figure out the equality $$\frac{1}{y(1-y)}=\frac{1}{y-1}-\frac{1}{y}$$</p>
<p>I have tried but keep ending up with RHS $\frac{1}{y(y-1)}$.</p>
<p>Any help would be appreciated.</p>
| Mark Bennet | 2,906 | <p>Set $y=2$ - your original formula gives $$\frac 1{-2}=1-\frac 12$$The left-hand side is negative and the right-hand side is positive, so the original statement in your question cannot be true. You seem to have proved that $$\frac 1{y(y-1)}=\frac 1{y-1}-\frac 1y$$ and this is indeed true where the fractions are defin... |
1,285,273 | <p>Looking for hints to find the orthnormal basis for the null space/range of the following matrix</p>
<p>$A = \frac{1}{3}\left( \begin{array}{ccc}
2 & -1 & -1 \\
-1 & 2 & -1 \\
-1 & -1 & 2 \end{array} \right)$</p>
| Alex R. | 22,064 | <p>It sounds like you're interested in Berry Esseen bounds. The simplest result says that if </p>
<p>$Z_n:=\frac{X_1+\cdots+X_n}{\sigma \sqrt{n}},$</p>
<p>and $F_n(z)$ is the cdf of $Z_n$, then </p>
<p>$$|F_n(z)-\Phi(z)|\leq \frac{C \rho}{\sigma^3\sqrt{n}},$$</p>
<p>where $\rho=E[|X_1|^3]$, the third moment (notice... |
3,821,049 | <blockquote>
<p>Find all complex solutions of <span class="math-container">$$e^{-iz}=\frac{-i+\sqrt 2+1}{-i-\sqrt 2-1}$$</span>
If a solution is <span class="math-container">$z=x+iy$</span> we set <span class="math-container">$\mathfrak{Re}(z)=x$</span> and <span class="math-container">$\mathfrak{Im}(z)=y$</span>.</p>
... | vonbrand | 43,946 | <p>Express <span class="math-container">$e^{a + b i} = e^a (\cos b + i \, \sin b)$</span>, and go from there.</p>
|
3,576,979 | <p>Been working on this for some time now but have no idea if it's correct! Any hints are appreciated.</p>
<p>Recall the Fibonacci sequence: <span class="math-container">$f_1 = 1$</span>, <span class="math-container">$f_2 = 1$</span>, and for <span class="math-container">$n \geq 1$</span>, <span class="math-container"... | Deepak | 151,732 | <p>You can use strong induction.</p>
<p>Start with <span class="math-container">$f_k = f_1 + f_2 + ... + f_{k-1} = 2 + ... + f_{k-1}$</span></p>
<p>Assume the proposition holds true for every <span class="math-container">$f_m$</span> where <span class="math-container">$m \leq k-1$</span>.</p>
<p>Then we can bound <s... |
342,306 | <p>An elementary embedding is an injection $f:M\rightarrow N$ between two models $M,N$ of a theory $T$ such that for any formula $\phi$ of the theory, we have $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ where $a$ is a list of elements of $M$.</p>
<p>A critical point of such an embedding is the least ordinal $\alpha$ s... | Miha Habič | 9,440 | <p>The existence of embedding characterizations of many large cardinal notions is something of a happy surprise really. There seems to be no reason, a priori, for the large cardinals of a more combinatorial nature to have such a description, but here we are.</p>
<p>Still, if you require the domain of your embedding to... |
2,017,818 | <p>Find three distinct triples (a, b, c) consisting of rational numbers that satisfy $a^2+b^2+c^2 =1$ and $a+b+c= \pm 1$.</p>
<p>By distinct it means that $(1, 0, 0)$ is a solution, but $(0, \pm 1, 0)$ counts as the same solution.</p>
<p>I can only seem to find two; namely $(1, 0, 0)$ and $( \frac{-1}{3}, \frac{2}{3}... | Tito Piezas III | 4,781 | <p>There are infinitely many. The <em>complete</em> rational solution to
$$a^2+b^2+c^2=1$$
is given by
$$\left(\frac{p^2-q^2-r^2}s\right)^k+\left(\frac{2pq}s\right)^k+\left(\frac{2pr}s\right)^k=1\tag1$$
where $s=p^2+q^2+r^2$ and $k=2$. But eq $(1)$ is also satisfied for $k=1$ if
$$p=\frac{q^2+r^2}{q+r}$$
For example, ... |
3,537,843 | <p>Find values of x such that <span class="math-container">$x^n=n^x$</span>
Here, n <span class="math-container">$\in$</span> I. </p>
<p>One solution will remain <strong>x=n</strong>
But i want to find if any more solutions can exist</p>
<p><span class="math-container">$$x^n=n^x$$</span></p>
| Claude Leibovici | 82,404 | <p><em>Welcome to the wonderful world of <a href="https://en.wikipedia.org/wiki/Lambert_W_function" rel="nofollow noreferrer">Lambert function</a> !</em></p>
<p>If, on the search bar, you just type <em>Lambert</em>, you will find almost 3000 entries.</p>
<p>This function <span class="math-container">$W(z)$</span> can... |
136,340 | <p>I defined the following functions</p>
<pre><code>CreatorQ[_] := False;
AnnihilatorQ[_] := False;
CreatorQ[q] := True;
AnnihilatorQ[p] := True;
CreatorQ[J[n_]] /; n < 0 := True;
AnnihilatorQ[J[n_]] /; n > 0 := True;
</code></pre>
<p>and when I ask for</p>
<pre><code>Assuming[r < 0, CreatorQ[J[r]]]
</code... | Mr.Wizard | 121 | <p>This is what you need:</p>
<pre><code>CreatorQ[_] := False;
AnnihilatorQ[_] := False;
CreatorQ[q] := True;
AnnihilatorQ[p] := True;
CreatorQ[J[n_]] /; Simplify[n < 0] := True;
AnnihilatorQ[J[n_]] /; Simplify[n > 0] := True;
Assuming[r < 0, CreatorQ[J[r]]]
</code></pre>
<blockquote>
<pre><code>True
</cod... |
1,462,908 | <p>Is it possible to have a set of infinite cardinality as a subset of a set with a finite cardinality? It sounds counter-intuitive, but there are things in math that just are so. Can one definitely prove this using only basic axioms? <br />
The main reason I asked this question is because the book <em>Inverted World</... | Sleepy Gary | 276,342 | <p>By definition a set $B$ is a subset of $A$ iff every element of $B$ in $A$. So, the largest subset of a finite set $A$ has exactly as many elements as $A$, but no more.</p>
|
1,462,908 | <p>Is it possible to have a set of infinite cardinality as a subset of a set with a finite cardinality? It sounds counter-intuitive, but there are things in math that just are so. Can one definitely prove this using only basic axioms? <br />
The main reason I asked this question is because the book <em>Inverted World</... | Michael Hardy | 11,667 | <p>"Length" is an inappropriate word here, partly because it's confusing and potentially ambiguous. One can say that the set of all numbers between $0$ and $1$ has finite "length", but it has infinitely many members. It is <b>infinite</b> in the sense usually used when talking about <b>sets</b>, i.e. it has infinitely... |
844,420 | <p>Given a set containing N numbers, minimize the average where you can take out any string of consecutive numbers in the set.
|N|<=100000</p>
<p>Ex. {5, 1, 7, 8, 2}</p>
<p>You can take out {1,7}, etc. but the way to minimize in this case is just to take out {7,8} which will give a minimum average of (5+2+1)/3=2.6... | barak manos | 131,263 | <p>Build a $2$-dimensional table, with each cell $[i][j]$ indicating the sum of elements $i,i+1,\dots,j$:</p>
<pre><code>n = array.length
table[0][0] = array[0]
for j = 1 to n-1:
table[0][j] = table[0][j-1]+array[j]
for i = 1 to n-1:
for j = i to n-1:
table[i][j] = table[i-1][j-1]-array[i]+array[j]
</c... |
1,235,639 | <p>Let $\mathcal{R}$ be the hyperfinite type $II_{1}$ factor and let $\mathcal{U}$ be a free ultrafilter on $\mathbb{N}$.</p>
<p>Is it true that $\mathcal{R}^{\mathcal{U}}$ is never hyperfinite ? How can I see this ?</p>
<p>Thanks</p>
<p><em>I know that under Continuum Hypothesis, every $\mathcal{R}^{\mathcal{U}}$ i... | Martin Argerami | 22,857 | <p>As you say, an ultrapower of $R $ contains $L (\mathbb F_2) $, which is not hyperfinite, while every subfactor of the hyperfinite II $_1$ is hyperfinite. </p>
<p>Another way to see it is that $R $ is separable, while its ultrapowers aren't. </p>
|
1,342,069 | <p>In the <a href="https://en.wikipedia.org/wiki/Forgetful_functor" rel="nofollow">forgetful functor Wikipedia article</a> I read that </p>
<blockquote>
<p>"[Forgetful] Functors that forget the extra sets need not be faithful; distinct morphisms respecting the structure of those extra sets may be indistinguishable ... | Alex G. | 130,309 | <p>One example is the forgetful functor from Schemes to Sets. Given two fields $k_1, k_2$, there may be many different field homomorphisms $k_1 \to k_2$ which give rise to many different morphisms $\text{Spec }k_2 \to \text{Spec }k_1$. However, the underlying sets of both of these schemes are single points, so there is... |
733,280 | <p>I cannot understand why $\log_{49}(\sqrt{ 7})= \frac{1}{4}$. If I take the $4$th root of $49$, I don't get $7$.</p>
<p>What I am not comprehending? </p>
| John Joy | 140,156 | <p>To get used to logarithm rules, try to relate them to exponent rules.</p>
<p>For example, think of a logarithm as the answer to the question "<i>a</i> to what power equals <i>b</i>". So the 2 following statements are equivalent to each other.</p>
<p>$\log a^?=b \iff ?=\log_ab$</p>
<p>Then try to convert logarithm... |
1,779,088 | <blockquote>
<p>Prove
$$\sum_{i=1}^n i^{k+1}=(n+1)\sum_{i=1}^n i^k-\sum_{p=1}^n\sum_{i=1}^p i^k \tag1$$
for every integer $k\ge0$. </p>
</blockquote>
<p>By principle of induction,</p>
<p>$$\sum_{i=1}^n i = n(n+1)- \sum_{p=1}^n p$$
$$2\sum_{i=1}^n i = n(n+1)$$
$$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$
$\implies$$(... | Brian M. Scott | 12,042 | <p>You can also simply reverse the order of summation in the double sum:</p>
<p>$$\begin{align*}
(n+1)\sum_{i=1}^ni^k-\sum_{p=1}^n\sum_{i=1}^pi^k&=(n+1)\sum_{i=1}^ni^k-\sum_{i=1}^n\sum_{p=i}^ni^k\\
&=\sum_{i=1}^n(n+1)i^k-\sum_{i=1}^n(n-i+1)i^k\\
&=\sum_{i=1}^n\big((n+1)-(n+1-i)\big)i^k\\
&=\sum_{i=1}^n... |
18,686 | <p>Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$:</p>
<p>$\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$,</p>
<p>where $\sim$ denotes "homeomorphic to". Obviously, $0 \leq \dim(B) \leq k$.</p>
<p>I have three questions: Given a $B \sub... | HenrikRüping | 3,969 | <p>Of course the point has the desired property, but I guess, this is not the space you are looking for. As François said, $C=\{0;1\}^\omega$ and so we get $C^2\cong C$.</p>
|
4,543,350 | <p>I am having a hard time figuring out this proof.</p>
<p>Let <span class="math-container">$\{x_n\}$</span> be a sequence and <span class="math-container">$x\in\mathbb{R}$</span>. Suppose for every <span class="math-container">$\epsilon>0$</span>, there is an M such that <span class="math-container">$|x_n-x|\leq \e... | FShrike | 815,585 | <p>A more high-level view. Convergence and limit properties can be defined with “open” inequalities <span class="math-container">$…<\varepsilon$</span> or “closed” ones: <span class="math-container">$…\le\varepsilon$</span> in any metric space. The closed and open balls are both local bases for the topology, so it d... |
977,956 | <p>Can you help me solve this problem?</p>
<blockquote>
<p>Simplify: $\sin \dfrac{2\pi}{n} +\sin \dfrac{4\pi}{n} +\ldots +\sin \dfrac{2\pi(n-1)}{n}$.</p>
</blockquote>
| John | 105,625 | <p>$\sin k\alpha=\frac{\cos (k-1)\alpha-\cos (k+1)\alpha}{2\sin\alpha}$ by using compound angle formula given $\sin\alpha \neq0$. Then take summation the numerator will cancel. In your case $\alpha=\frac{2\pi}{n}, k=1\cdots n-1$. So you also need to discuss the case $n=1,2$ separately.</p>
|
134,673 | <p>I need to show that an automorphism of $S_n$ which takes transpositions to transpositions is an inner automorphism.</p>
<p>I thought it could be done by showing that such automorphisms form a subgroups $H\le Aut(S_n)$, that $Inn(S_n)\subset H$ and that they have the same number of elements. The number of inner auto... | Jyrki Lahtonen | 11,619 | <p>Kannappan Sampath's suggestion can be completed as follows.</p>
<p>Let $f$ be an automorphism of $S_n$ with the property that it maps all transpositions to transpositions. So $f(1k)=(a_kb_k)$ for all $k=2,3,\ldots,n$, where $a_k\neq b_k$ are elements of the set $\{1,2,\ldots,n\}$. As the permutations $(12)$ and $(1... |
2,572,304 | <p>Cauchy's Inequality states that, $$ \forall a, b \in R^{n}, |a \cdot b| \leq |a||b| $$. However, the dot product is $$ x \cdot y = x_{1}y_{1}+...+x_{n}y_{n}$$ while the norm of x is $$ |x| = \sqrt[2]{x_{1}^{2} +...+x_{n}^{2}} = \sqrt[2]{x \cdot x}$$. Therefore, $$ |a \cdot b| = \sqrt[2]{(a \cdot b) \cdot (a \cdot b)... | Stephen Meskin | 465,208 | <p>$|x|$ for $x \in R^n$ with $n \ge 2$ is a generalization of $|x|$ for $x \in R^1$</p>
|
2,778,031 | <p>A ball is thrown vertically upward with u velocity. There is air resistance and the air resistance is directly proportional to square of ball's velocity,u. Find the height which the ball can reach.
I started with,
$$a=-g-{{k\over m}u^2},~~~~~~~~~
{d^2x\over dt^2} = -g - {k\over m}\left({dx\over dt}\right)^2$$</p>
... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Yes your set up is correct indeed we have that</p>
<ul>
<li>$a=-g-\frac k m \cdot v^2 \implies \frac{dv}{dt}=-g-\frac k m v^2 \implies \frac{v'(t)}{-g-\frac k m v^2}dt=dt$</li>
</ul>
<p>which can be integrated (see <a href="https://philosophicalmath.wordpress.com/2017/10/21/terminal-v... |
2,778,031 | <p>A ball is thrown vertically upward with u velocity. There is air resistance and the air resistance is directly proportional to square of ball's velocity,u. Find the height which the ball can reach.
I started with,
$$a=-g-{{k\over m}u^2},~~~~~~~~~
{d^2x\over dt^2} = -g - {k\over m}\left({dx\over dt}\right)^2$$</p>
... | David Quinn | 187,299 | <p>HINT...since you are looking for the height not the time, write $$a=v\frac{dv}{dx}=-(g+\frac kmv^2)$$</p>
|
3,317,728 | <p>Suppose that the moment generating function <span class="math-container">$M_X$$(t)$</span> of a random variable <span class="math-container">$X$</span> is given by </p>
<p><span class="math-container">$$ M_X(t)=\frac{e^t+e^{-t}}{6} + \frac 23 $$</span></p>
<p>I need to find the distribution function <span class="... | Feng | 624,428 | <p>Hint: From the moment generating function we can determine the distribution of <span class="math-container">$X$</span>, which is <span class="math-container">$P(X=1)=P(X=-1)=\frac16$</span>, <span class="math-container">$P(X=0)=\frac23$</span>. I believe that you can move on now. </p>
|
2,981,745 | <p>Wolfram Alpha shows that <span class="math-container">$$\int_{0}^{2\pi}x^2\ln (1-\cos x)dx = -\frac{8}{3} \pi (\pi^2 \ln(2) + 3 \zeta(3))$$</span>
I tried to use the Fourier series
<span class="math-container">$$\ln (1-\cos x)=-\sum_{n=1}^{\infty} \frac{\cos^nx}{n}.$$</span>
I am not sure how to continue from this ... | mrtaurho | 537,079 | <p><strong>HINT</strong></p>
<p>By using the basic trigonometric identity</p>
<p><span class="math-container">$$1-\cos(x)=2\sin^2\left(\frac x2\right)$$</span></p>
<p>yor given integral becomes</p>
<p><span class="math-container">$$\begin{align}
\int_{0}^{2\pi}x^2\ln (1-\cos x)~dx &= \int_{0}^{2\pi}x^2 \ln\left... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.