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 |
|---|---|---|---|---|
596,710 | <p>Help me? please How to solve this integral?
$$\int\frac{1+x^2}{1+x^4}\,dx$$</p>
| robjohn | 13,854 | <p>$$
\begin{align}
\int\frac{1+x^2}{1+x^4}\,\mathrm{d}x
&=\frac12\int\frac{\mathrm{d}x}{1-\sqrt2x+x^2}+\frac12\int\frac{\mathrm{d}x}{1+\sqrt2x+x^2}\tag{1}\\
&=\int\frac{\mathrm{d}x}{\left(\sqrt2x-1\right)^2+1}+\int\frac{\mathrm{d}x}{\left(\sqrt2x+1\right)^2+1}\tag{2}\\
&=\frac1{\sqrt2}\left(\tan^{-1}(\sqrt... |
4,174,404 | <p>So I'm trying to find conditions on <span class="math-container">$p$</span> and <span class="math-container">$k$</span> under which <span class="math-container">$1+p^k$</span> is a power of <span class="math-container">$2$</span>, i.e.,
<span class="math-container">$$1+p^k=2^\ell.$$</span></p>
<p>Let's first note so... | arbashn | 930,909 | <p>There's an elementary argument showing there are no solutions for <span class="math-container">$k \gt 1$</span> without invoking <a href="https://en.m.wikipedia.org/wiki/Catalan%27s_conjecture" rel="nofollow noreferrer">Mihăilescu's theorem</a> (previously Catalan's conjecture).</p>
<p>Assume <span class="math-conta... |
187,241 | <p>Let $B$ be a compact manifold, and $\hat{B}\to B$ be the maximal abelian covering of $B$; i.e. $\hat{B}$ is the quotient of the universal cover with respect to the commutator subgroup of $\pi_1(B)$. Given $H\subset H_1(B)$, we can further quotient $\hat{B}$ with respect to $H$ to get a covering $X\to B$ with group o... | Aleksey | 27,140 | <p>I am not too comfortable with spectral sequences and in particular I do not see the obstruction statement in Matthias's answer. It seems there is a direct construction of an invariant cocycle for the total space of the fibration Matthias suggested. I would guess it is a translation of Matthias's answer into less tec... |
2,771,404 | <p>This question is a trigonometric one from what I can tell. I understand that you need to differentiate the given function and the unknown variable on top ($dx$) will correspond to the $x$ unknowns, for example $5x^2$ will give us $10x\frac{dx}{dt}$.</p>
<p>With this specific question I can only get to $-\frac{\sec^... | Rhys Hughes | 487,658 | <p>$$\frac{d(\tan(x))}{dt}=\sec^2x \frac{dx}{dt}$$
$$\frac{d(4t^{-1})}{dt}=-4t^{-2}$$
$$\frac{dx}{dt}=\frac{-4t^{-2}}{\sec^2x}=-\frac{4\cos^2x}{t^2}\because \cos^2x=\frac{1}{\sec^2x}$$</p>
|
695,015 | <p>If $A$ and $B$ are subspaces of $\mathbb R^n$. Is it possible to find a basis for $\mathbb R^n$ that contains a basis for $A$ and $B$?</p>
<p>It has been suggested to me that we define a basis for $A\cap B$ and then use that to define basises $A$ and $B$. I would like to understand why this approach is taken and ... | egreg | 62,967 | <p>Let $\{v_1,\dots,v_k\}$ be a basis of $A\cap B$; you can find vectors $a_1,\dots,a_r\in A$ such that $\{v_1,\dots,v_k,a_1,\dots,a_r\}$ is a basis for $A$; similarly, there are $b_1,\dots,b_s\in B$ such that $\{v_1,\dots,v_k,b_1,\dots,b_s\}$ is a basis for $B$.</p>
<p>Your task is to prove that
$$
\{v_1,\dots,v_k,a_... |
4,365,119 | <blockquote>
<p>If <span class="math-container">$\alpha,\beta,\gamma$</span> are the roots of the equation <span class="math-container">$x^3+x^2-x+1=0$</span>, find the value of <span class="math-container">$\prod\left(\frac1{\alpha^3}+\frac1{\beta^3}-\frac1{\gamma^3}\right)$</span></p>
</blockquote>
<p><strong>My Atte... | JetfiRex | 1,014,890 | <p>Here is the method with least calculation I can think of.</p>
<p>I write <span class="math-container">$a,b,c$</span> for <span class="math-container">$\alpha,\beta,\gamma$</span> instead.</p>
<p>Let <span class="math-container">$p=a+b+c=-1$</span>, <span class="math-container">$q=ab+bc+ca=-1$</span>, <span class="ma... |
150,856 | <p>From the result discussed in <a href="https://mathoverflow.net/questions/75049/does-the-inverse-function-theorem-hold-for-everywhere-differentiable-maps">Does the inverse function theorem hold for everywhere differentiable maps?</a> (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obta... | Peter Michor | 26,935 | <p>If I remember correctly, strict differentiability at one point $x_0$ (and of course invertibility of $df(x_0)$)
is sufficient for the implicit function theorem to hold.
This means something like
$$
df(x_0)(h) = lim_{t\to 0, x\to x_0}\frac1t f(x+th).
$$
I am traveling and I do not have access to the books now. This... |
67,656 | <p>Let $X$ be a projective surface and let $x\in X$ be a smooth point. Consider the blow up $Bl_{x}X$ of $X$ in $x$, and let $E$ be the exceptional divisor. Suppose we know that $E$ is the only (-1)-curve in $Bl_{x}X$, so that $\alpha(E) = E$ for any $\alpha\in Aut(Bl_{x}X)$. Then any automorphisms $\alpha\in Aut(Bl_{x... | Sasha | 4,428 | <p>Another way to explain why $\sigma$ extends to an automorphism of the blow-up is the following. Note that the blowup of a point is the relative Proj of the sheaf of graded algebras $\oplus_{k=0}^\infty I^k$, where $I$ is the ideal of the point $x$. Now if $\sigma$ preserves the point $x$ then it also preserves the i... |
159,654 | <p>Is there a closed form expression for the expectation of $\log(x+a)$ (with $a>0$, the case $a=0$ is obvious) when X follows a beta distribution?</p>
| user64494 | 35,959 | <p>Here is the answer done with Maple. The Maple code $$ with(Statistics):
X := RandomVariable(Beta(alpha, beta)):$$
$$Z := log(X+a)\,\,assuming \,\,a > 0:
simplify(Mean(Z));$$ outputs $$\left(\Gamma \left( -1+\alpha \right) \Gamma \left( \beta \right) {\alpha}
^{2}\ln \left( {a}^{-1} \right) \sin \left( \alpha\,... |
72,588 | <p>Imagine I have a point-like Brownian particle, with diffusion constant $D$, and I place it at some initial coordinate in a cage of known geometry. Assuming the volume $V$ of the cage is "everywhere" accessible, how long does it take the particle to lose its "memory" of the initialization coordinate and achieve a un... | Andrew | 14,551 | <p>For the initial distribution being delta function $\delta(y)$ the result will be the Green function of the Neumann problem for the heat equation. If the domain is regular enough the function can be written as
$$
G(x,y,t)=\sum_{n=1}^\infty \varphi_n(x)\varphi_n(y)e^{-\lambda_n t}
$$
where $\lambda_n$ and $\varphi_n$... |
1,967,312 | <p>Say we have disjoint sets $A$ and $B$ where $A = \{1\}$ and $B = \{2\}$. </p>
<p>Thus the power set of $A$ is $\{\emptyset,1\}$ and the power set of $B$ is $\{\emptyset,2\}$. </p>
<p>I know the intersection set of $A$ and $B$ is $\emptyset$ </p>
<p>My question is this: Given the above information, does that mean ... | celtschk | 34,930 | <p>You write:</p>
<blockquote>
<p>Thus the power set of $A$ is $\{\emptyset,1\}$ and the power set of $B$ is $\{\emptyset,2\}$.</p>
</blockquote>
<p>No, it isn't. In particular, $1\nsubseteq A$ and $2\nsubseteq B$. However since $1\in A$, $\{1\}\subseteq A$ and therefore $\{1\}\in\mathcal P(A)$ (where $\mathcal P(A... |
3,000,236 | <p>I've been bothered with the standard definition of the limit in analysis for a while:
'A sequence <span class="math-container">$(x_n)$</span> has <span class="math-container">$x$</span> as a limit if <span class="math-container">$\forall \epsilon>0\ \exists N\in\mathbb{N}$</span> s.t. <span class="math-container... | Mark S. | 26,369 | <p>The set of limit points/cluster points/accumulation points of a sequence will usually/always be the set you are trying to describe. Definitions vary depending on the source, but "limits of subsequences" probably captures it unambiguously when you're looking at sequences in the reals (or <span class="math-container">... |
3,000,236 | <p>I've been bothered with the standard definition of the limit in analysis for a while:
'A sequence <span class="math-container">$(x_n)$</span> has <span class="math-container">$x$</span> as a limit if <span class="math-container">$\forall \epsilon>0\ \exists N\in\mathbb{N}$</span> s.t. <span class="math-container... | Ittay Weiss | 30,953 | <p>Any particular definition of limit is trying to serve a purpose. The limit definition of a sequence that you mention is the correct one for the purposes of calculus, at least in the sense that it does the job it is designed for. Now, you seem to want a more refined notion of limit of a sequence that will tell you mo... |
3,615,335 | <p>E.g. 2/3 and 3/2 is 66% and 150%. Why is there a 16% difference between the 2 (34% and and 50%) and why does the direction of the percentage change (positive or negative change things). I can do the calculation but don't get the logic. This is probably Mathematical logic I believe?</p>
| Community | -1 | <p>If <span class="math-container">$ v_1,v_2$</span> is a eigenvector of T for the eigenvalue <span class="math-container">$\lambda$</span> then <span class="math-container">$Tv_1=\lambda v_1 $</span> and same for <span class="math-container">$v_2$</span> then
<span class="math-container">$T(v_1+v_2)= T(v_1)+T(v_2)=\l... |
3,893,117 | <p>Claim: <span class="math-container">$2^{n+1} >n^{2}$</span> for any positive integer n.</p>
<p>My proof:</p>
<p><span class="math-container">$n = 1 \longrightarrow$</span></p>
<p><span class="math-container">$2^{1+1} > 1^{2} \iff 4 > 1$</span></p>
<p>Check.</p>
<p><span class="math-container">$n=2\longright... | Donald Splutterwit | 404,247 | <p>We have <span class="math-container">$n$</span> people, One of them will be the Captain & all the others will be one of <span class="math-container">$3$</span> types (A,B or C).</p>
<p>Choose <span class="math-container">$n-k$</span> to be of type <span class="math-container">$A$</span>, now choose <span class="... |
2,874,316 | <p>I started off with:</p>
<p>$$((A\lor B)\land ((B\leftrightarrow A)\rightarrow C))\lor(C\rightarrow \lnot A)$$</p>
<p>And I've managed to get to:</p>
<p>$$((A\lor B) \land (((B\land \lnot A) \lor (A \land \lnot B)) \lor C)) \lor (\lnot C \lor \lnot A)$$</p>
<p>How do I finish and get this into CNF? </p>
| Community | -1 | <p>Resorting to Lambert's $W$ is a little overkill.</p>
<p>With $$x=y\ln y,$$ we have</p>
<p>$$\frac x{\log x}=\frac{y\ln y}{\ln(y\ln y)}=\frac{y\ln y}{\ln y+\ln\ln y}\approx y.$$</p>
<p>Given that the distribution is anyway approximate, this can be a sufficient approximation of the inverse. Or you can use it as a s... |
3,002,520 | <p>I'm stuck on a problem that maybe most of you will be able to solve instantly. Here is the problem:</p>
<p>A publisher needs to bind <span class="math-container">$4500$</span> books. One print shop can bind these books in <span class="math-container">$30$</span> days, another shop can do it in <span class="math-con... | hmakholm left over Monica | 14,366 | <p>You have somehow gotten <span class="math-container">$350$</span> from adding <span class="math-container">$100$</span> and <span class="math-container">$150$</span>. It should be <span class="math-container">$250$</span> only.</p>
|
2,838,874 | <p>Consider $$φ(x, y , z) = x^2 + 2y^2 + 4z^2 −xy −2xz −3yz$$
find the coordinate transformation (translation or rotation) to eliminate $xy$, $xz$ and $yz$.
In $\mathbb R²$, with conic sections, I would do this with
$$\begin{pmatrix}x \\ y\end{pmatrix}=\begin{pmatrix}\cos\alpha &-\sin\alpha\\\sin\alpha & \cos\... | Stefan4024 | 67,746 | <p><strong>HINT:</strong> If $f(x)$ is irreducible, so is $x^nf(1/x)$, where $n = \deg f$</p>
|
1,008,170 | <p>We know that if $x_i^2 = -1 \mod n$ we are able to find more solutions of the form, $x^2 = -y^2 \mod n$ </p>
<blockquote>
<p><strong>Simple Proof:</strong></p>
<p>Let $x_i$ be the initial solution to $x_i^2+1 \equiv 0 \mod n$</p>
<p>$y^2x_i^2+y^2 \equiv 0 \mod n$</p>
<p>$x^2 = -y^2 \mod n$, $x = x_... | Lucian | 93,448 | <p>Let $~\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=a.~$ Cubing both sides, using ${\underbrace{(x+y)}_a}^3=x^3+y^3+3xy~\underbrace{(x+y)}_a$, and simplifying, we have $a^3=4-3a$, whose only real solution, $a=1$, can be found quite easily via the <a href="http://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow... |
1,925,176 | <p>For the question "The corners of a triangle are the points $P(4,7)$, $Q(-2,5)$ and $R(3, -10)$. Find the length of each side of $\triangle PQR$, giving your answers in terms of surds".
I have the answer to the first part as : $|PQ| = 2\sqrt{10}$, $|PR| = \sqrt{290}$, $|RQ| = 5\sqrt{10}$.</p>
<p>The second part asks... | Jason C | 90,044 | <p>Well, it's an application of Pythagoras' Theorem. If the square of your largest edge length (call it $c$) is equal to the sum of the squares of the other two (call them $a$ and $b$), then you've got a right triangle, because the edge lengths satisfy $a^2+b^2=c^2$.</p>
<p>Another way is to check if the dot product o... |
426,661 | <p>Can someone please explain these answers? I have reviewed the slides and read about properties of equality but I still don't understand how to apply it to these sets.</p>
<p>For each the following relations on the set of integers list all that apply
(Reflexive, Symmetric, Antisymmetric, Transitive, or none): </p>
... | Cameron Buie | 28,900 | <p>Let me just look at a few, to show you what's going on.</p>
<hr>
<p>Consider $R_1$.</p>
<p>It is symmetric, because whenever $a\ne b$, then $b\ne a$--so whenever $\langle a,b\rangle \in R_1,$ then $\langle b,a\rangle \in R_1$.</p>
<p>In order for $R_1$ to be reflexive, we'd need to know that $\langle a,a\rangle ... |
3,357,778 | <p>I am having issues solving a problem using the probability approach (do not want to use the combinatorics approach!). Suppose there are four balls in a jar of colors: 1 Red, 1 Blue and 2 Yellow. 2 Balls are to be selected. What is the probability that only one of the two selections will be Yellow?</p>
<p>Should the... | PM. | 416,252 | <p>Your combinatoric approach is at fault, not your probability approach.</p>
<p><span class="math-container">$$
\text{'4 choose 2'} = 4C_2= \binom{4}{2} =\frac{4!}{2!2!}
$$</span></p>
<p>Another (equivalent) approach is to draw the tree of possible outcomes, distinguishing the 2 yellows.</p>
<p>There will be outcom... |
1,652,107 | <p>Since the limit $\frac{\sin(x)}{x}=1$ for $x \rightarrow 0$, I wondered about the infinite product:</p>
<p>$$\prod^{\infty}_{n=1} n \sin \left( \frac{1}{n} \right)=\sin(1) \cdot 2 \sin\left( \frac{1}{2} \right) \cdot 3 \sin\left( \frac{1}{3} \right) \dots$$</p>
<p>By numerical experiment in Mathematica it seems to... | Claude Leibovici | 82,404 | <p>It seems to be true. Consider the Taylor series $$\cos(x)=1-\frac{x^2}{2}+\frac{x^4}{24}+O\left(x^6\right)$$ $$\cos(2x)=1-2 x^2+\frac{2 x^4}{3}+O\left(x^6\right)$$ $$\sqrt{\cos(2x)}=1-x^2-\frac{x^4}{6}+O\left(x^6\right)$$ So, $$1-\cos (x) \sqrt{\cos (2x)}=\frac{3 x^2}{2}-\frac{3 x^4}{8}+O\left(x^6\right)$$</p>
<p>I... |
1,652,107 | <p>Since the limit $\frac{\sin(x)}{x}=1$ for $x \rightarrow 0$, I wondered about the infinite product:</p>
<p>$$\prod^{\infty}_{n=1} n \sin \left( \frac{1}{n} \right)=\sin(1) \cdot 2 \sin\left( \frac{1}{2} \right) \cdot 3 \sin\left( \frac{1}{3} \right) \dots$$</p>
<p>By numerical experiment in Mathematica it seems to... | egreg | 62,967 | <p>Write
$$
\frac{1-\cos x\sqrt{\cos2x}}{x^2}=
\frac{1-\cos x+\cos x-\cos x\sqrt{\cos2x}}{x^2}=
\frac{1-\cos x}{x^2}+\cos x\frac{1-\sqrt{\cos2x}}{x^2}
$$
Since
$$
\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac{1}{2}
$$
and $\lim\limits_{x\to0}\cos x=1$, we just have to compute
$$
\lim_{x\to0}\frac{1-\sqrt{\cos2x}}{x^2}=
\lim_{... |
2,283,981 | <p>When presented with an equation, say, $y=5x^3+7x^2+4x+9$, you can write on the second line, $\frac{dy}{dx}=15x^2+14x+4$. Similarly, $f(x)=5x^3+7x^2+4x+9$ and $f'(x)=15x^2+14x+4$. But is there a way to write "the derivative of $5x^3+7x^2+4x+9$ is $15x^2+14x+4$" in just one line? </p>
<p>What should l write, $\frac{d... | Franklin Pezzuti Dyer | 438,055 | <p>You would denote the derivative of $5x^3+7x^2+4x+9$ as
$$\frac{d}{dx}(5x^3+7x^2+4x+9)$$
That is the only notation I've ever seen unless the expression is expressed as a function.</p>
|
1,839,121 | <p>I want to prove that this inequality holds or not?
The inequality is $\lambda(G) \leq \chi(G) $
where $\lambda(G)$ is the minimum number of edges whose deletion from a graph $G$ disconnects $G$, also called the line connectivity and $\chi(G)$ is the chromatic number of graph.</p>
<p>I would be greatly thankful i... | N. S. | 9,176 | <p>If $\lambda$ denotes the edge connectivity, then what is
$$\lambda(K_{m,n}) , \chi(K_{m,n})?$$</p>
|
117,187 | <p>I have N lines in R^3 (that are very likely skew). I'm looking for an efficient method of computing the radius (and position) of the smallest sphere that intersects every line.</p>
<p>I suspect that there's a solution related to the 'smallest sphere enclosing points' problem, but I failed to find it.</p>
<p>(The p... | David Carchedi | 4,528 | <p>The category can be drastically different. For example, suppose your morphisms are precisely local homeomorphisms. Call this category $Top^{et}$. This category is locally a topos, which is something certainly not the case for the category $Top$ of all continuous maps. Moreover, $Top^{et}$ lacks a terminal object (w... |
117,187 | <p>I have N lines in R^3 (that are very likely skew). I'm looking for an efficient method of computing the radius (and position) of the smallest sphere that intersects every line.</p>
<p>I suspect that there's a solution related to the 'smallest sphere enclosing points' problem, but I failed to find it.</p>
<p>(The p... | Todd Trimble | 2,926 | <p>This seems difficult to answer in any kind of <em>uniform</em> way, i.e., it seems we have to work on a case-by-case basis. Let's start with $\mathrm{Top}_{\mathrm{open}}$, the category of topological spaces and open continuous maps. </p>
<p>A useful heuristic is to let oneself be guided by locales instead of topol... |
508,522 | <p>Take a list of the first $n$ primes $P_n=\{2,3,5,7,11,\ldots\}$ and convert the sequence into a binary string</p>
<p>$$S_n = 101110111\ldots$$</p>
<p>Compress the string with your favorite compression algorithm (the details don't matter) and let the size of the string after compression be $C(S_n)$. Compare this to... | vitaly-t | 689,092 | <p>The first <code>23,163,298</code> primes can be considered compression-friendly. It is the maximum number of primes for which every gap between them is <= <code>255</code>, i.e. fits into a single byte. So for those primes, you can use just a single byte per prime, storing the gap, rather than the actual prime as... |
3,380,002 | <p>we need to find a mapping of those points onto <span class="math-container">$\mathbf{R}^2$</span> so that the two colors classes are can be separated by a line?</p>
<p>So I don't know the radius of those circles. But if I knew, I would take their norm. However that would be a mapping to <span class="math-container"... | Angina Seng | 436,618 | <p>How about the additive group of the polynomial ring <span class="math-container">$k[X]$</span> for <span class="math-container">$k=\Bbb Z/2\Bbb Z$</span>?</p>
|
897,061 | <p>This is just a standard problem from my high school's calculus text, but my proof seems sort of off. This is it:</p>
<p>Let $f(x) = e^x$. The tangent line of $f(x)$ at $x=0$ is $g(x)=x+1$. Since $f''(x_0) \gt 0$ for all $x_0 \in \mathbb R$,the tangent line $g(x) \le f(x)$ for all $x$. Q.E.D.</p>
<p>I didn't show a... | rehband | 154,416 | <p>I don't know how you have defined the exponential function, but you could use the fact that it has the following series expansion $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +\cdots$$</p>
|
19,124 | <p>To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and "identify opposite sides". We can imagine gluing together one pair of opposite sides to get a cylinder, and then gluing together opposite ends of the cylinder to get a torus. (Pro... | WaterMolecule | 12,740 | <h2>East to west on a world map</h2>
<p>The world map is usually what I use to explain toroidal periodic boundary conditions. <strong>The analogy isn't perfect because the Earth isn't a torus, but it gets the important ideas across in a way everyone can understand.</strong> Unfortunately, it doesn't work north-to-south... |
4,117,365 | <p>I try to understand the meaning of E in this formula. The formula is taken from <a href="https://arxiv.org/pdf/1710.09412.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/1710.09412.pdf</a> and it is on the 3rd page. The paper is about minimising vicinal risk loss. <span class="math-container">$\lambda$</span> i... | Eric | 909,833 | <p>The only random variable is <span class="math-container">$\lambda$</span> and the E means we’re taking the expected value over that random variable.</p>
<p>The entire thing is the vicinal distribution.</p>
<p>Basically, they’re saying that the standard method of training is that your input data is a discrete uniform... |
3,908,915 | <p>I was reading Lecture 7 of Afternotes on Numerical Analysis by G. W. Stewart, and he mentions without proof the following result:</p>
<p>If <span class="math-container">$n \epsilon_M \leq 0.1$</span> and <span class="math-container">$\epsilon_i \leq \epsilon_M (i = 1,2,...,n)$</span>, then</p>
<p><span class="math-c... | Ahmad | 411,780 | <p><span class="math-container">$\sum \limits_{k=1}^{n} \frac{(n-1)!}{k! (n-k)! (10 n)^{k-1}} = \frac{1}{n} \sum \limits_{k=1}^{n} \frac{(n)!}{ k! (n-k)! } (\frac{1}{10n})^{k-1} = 10\sum \limits_{k=1}^{n} \binom{n}{k}(\frac{1}{10n})^{k} = 10((1+\frac{1}{10n})^n-1) \to 10 (e^{\frac{1}{10}}-1) $</span> as <span class="ma... |
3,315,113 | <p>Are there some basic criteria by which to check whether a polytope <span class="math-container">$P\subset\Bbb R^n$</span> is a <span class="math-container">$01$</span>-polytope, that is, can be rotated and scaled to have vertices in <span class="math-container">$\{0,1\}^n$</span>, maybe even by using higher-dimensio... | Ethan Bolker | 72,858 | <p>One necessary condition is that the set of distances between vertices (in particular, edge lengths) must scale to a subset of
<span class="math-container">$$
\{ 1, \sqrt{2}, \ldots \}
$$</span>
since those are the distances between vertices of the unit cube.</p>
|
1,609,758 | <p>Is the answer of $\ \ 2415^n \ mod \ \ 2556 \ \ $ is only $711$ and $1989$ for $n\in \mathbb{Z}^{+} >1 \ \ ?$ </p>
<p>If it's true, how to prove that ?</p>
<p>I try to expand $(2556 - 141)^n $ for any $n\in \mathbb{Z}^{+} >1$ , but it doesn't make sense.</p>
<p>According to <a href="http://www.wolfr... | user159517 | 159,517 | <p>I'm assuming that the functions $\varphi_j$ are real-valued. The Cauchy-Schwartz inequality on $\ell^2(\mathbb{N})$ gives</p>
<p>\begin{align*}\sum_{j=1}^{\infty} |\lambda_j \varphi_{j}(x) \varphi_{j}(y)| &= \sum_{j=1}^{\infty} |\sqrt{\lambda_j} \varphi_{j}(x) \sqrt{\lambda_j}\varphi_{j}(y)| \leq \sum_{j=1}^{\... |
1,056,194 | <p>With the regular trig functions, if I ever end up with something like $\operatorname{trig}_1(\operatorname{arctrig}_2(f(x))$, where $\text{trig}_1$ and $\text{trig}_2$ are two arbitrary trigonometric functions, I can draw a right triangle to find a formula for this that doesn't involve any trigonmetric functions.</p... | Community | -1 | <p>You can draw a triangle with an imaginary side !</p>
<p>Indeed, $\cos(x)=\cosh(y)$ and $\sin(x)=i\sinh(y)$, where $y=ix$. Then all the rules known for the trigonometric functions follow for the hyperbolic ones.</p>
<p>For example</p>
<p>$$\cos^2(x)+\sin^2(x)=1\leftrightarrow\cosh^2(y)-\sinh^2(y)=1,\\
\arccos(\sin... |
926,754 | <p>Find a polynomial with the maximum 1. degree which best approximates the $f(x)=e^x$ function in terms of minimax approximation in $[0,1]$.</p>
| Maryam Azizi | 1,011,893 | <p>In the equation <span class="math-container">$e^a−(ma+b)=\rho$</span> a minus sign is missed in the right hand side. The correct equation is <span class="math-container">$e^a−(ma+b)=-\rho$</span>.</p>
|
1,590,619 | <p>I must find the sum $$\sum_\limits{n=1}^{\infty}\frac1{n(2n+1)}$$ I have already tried partial fractions, but obtained an indeterminate form of infinity minus infinity. Perhaps completing the square would help?</p>
| Ron Gordon | 53,268 | <p>Note that</p>
<p>$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \log{2} $$</p>
<p>Then the desired sum is equal to</p>
<p>$$2\sum_{n=1}^{\infty} \left (\frac1{2 n}-\frac1{2 n+1} \right ) = 2 \left (\frac12 - \frac13 + \frac14 - \frac15 +\cdots \right ) = 2 (1-\log{2})$$</p>
|
437,960 | <p>I can't find any logic equivalence or inference rules on this. Personally, I feel that $\neg P \implies \neg T$ and $P \implies \neg T$ would mean that it follows that $\neg T$ is true regardless, and I should be able to use that fact as such in my next step. Is this proper reasoning, though?</p>
| Asaf Karagila | 622 | <p>Note that $P\rightarrow T$ is equivalent to $\lnot P\lor T$. Therefore we have equivalence between $(P\rightarrow\lnot T)\land(\lnot P\rightarrow\lnot T)$ and $(\lnot P\lor\lnot T)\land(P\lor\lnot T)$.</p>
<p>By distributivity this can be written as $(\lnot P\land P)\lor\lnot T$, which is equivalent to $\lnot T$.</... |
209,672 | <blockquote>
<p>Show that <span class="math-container">$n$</span> lines separate the plane into <span class="math-container">$\frac{n^2+n+2}{2}$</span> regions if no two of these lines are parallel and no three pass through a common point.</p>
</blockquote>
<p>I know we start with the base case, where, if we call t... | robjohn | 13,854 | <p>Here is the way I usually think about this (it sort of uses induction).</p>
<p>With $0$ lines, there is $1$ region and no intersections of lines.</p>
<p>Each time a line is added and it crosses $k$ other lines it adds $k+1$ regions and $k$ intersections. Another way of looking at this is that for each line and $k$... |
2,156,709 | <ul>
<li>Prove-($\bigcup\limits_{i}A_i$) $\bigcap $($\bigcup\limits_{j}B_j$)=$\bigcup\limits_{i,j}(A_i \bigcap B_j)$</li>
<li>Prove-($\bigcap\limits_{i}A_i$) $\bigcup $($\bigcap\limits_{j}B_j$)=$\bigcap\limits_{i,j}(A_i \bigcup B_j)$</li>
<li>If {$I_j$} is a family of sets with domain J,say;$\Bbb K$=$\bigcup\limits_{... | lulu | 252,071 | <p>The usual trick is to pass a running total around:</p>
<p>First person passes the total of $S_1+R_1$ to person two where $S_1$ is her salary and $R_1$ is a random number known only to her. Person two then adds $S_2+R_2$ and passes $S_1+S_2+R_1+R_2$ to person three and person three adds $S_3+R_3$ and passes the tot... |
287,516 | <p>I need to calculate value of $1000^{th}$ derivate of the following function at $0$:</p>
<p>$$
f(x) = \frac{x+1}{(x-1)(x-2)}
$$</p>
<p>I've done similar problems before (e.g. $f(x)= \dfrac{x}{e^{x}}$) but the approach I've used would not work in this case and I believe I should expand this function into a power ser... | Julien | 38,053 | <p>As suggested by Argha and David Mitra, begin with the decomposition into partial fractions:
$$
f(x)=\frac{3}{x-2}-\frac{2}{x-1}.
$$
Now if $g(x)=\frac{1}{x-a}$ for some constant $a$, then a proof by induction quickly shows that
$$
g^{(n)}(x)=\frac{(-1)^nn!}{(x-a)^{n+1}}.
$$
It follows that
$$
f^{(1000)}(x)=\frac{3... |
2,797,106 | <p>If $C$ is a fixed non-empty set, which one of the following options imply $A=B$?</p>
<p>1) $A - C = B - C$</p>
<p>2) $A \cup C = B \cup C$</p>
<p>3) $A \cap C = B \cap C$</p>
<p>4) $A \Delta C = B \Delta C$</p>
<p>I'm not sure why all of them are not valid option.</p>
<p>The problem I'm having is with premise.... | fleablood | 280,126 | <blockquote>
<p>The problem I'm having is with premise. If A=B then it would imply that all of the options are true.</p>
</blockquote>
<p>Which is the exact <em>reverse</em> of what is being asked.</p>
<p>The question isn't what can we conclude from $A=B$. It is what can we use to conclude $A=B$.</p>
<p>If the qu... |
3,923,706 | <p>Take two spaces <span class="math-container">$X_1$</span> <span class="math-container">$X_2$</span> and consider <span class="math-container">$\mu\in\mathcal{P}(X_1\times X_2)$</span> i.e the space of Borel probability measures. Let <span class="math-container">$\mu_1,\mu_2$</span> be Borel probability measures on <... | Sangchul Lee | 9,340 | <p>Since <span class="math-container">$\mu_i$</span> is a Dirac mass for each <span class="math-container">$i = 1, 2$</span>, we can find <span class="math-container">$x_i \in X_i$</span> such that <span class="math-container">$\mu_i = \delta_{x_i}$</span>. Now if we write</p>
<p><span class="math-container">$$ E_1 = (... |
842,911 | <p>What is the following product in terms of $n$?</p>
<p>$$\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \cdot \cdots \cdot \frac{2n-1}{2n}$$</p>
<p>Thank you.</p>
| David H | 55,051 | <p>As an alternative answer that has its uses, this product of fractions can be written as a definite integral:</p>
<p>$$\begin{align}
\frac12\frac34\frac56\frac78\dots\frac{2n-1}{2n}&=\frac{(2n-1)!!}{(2n)!!}\\
&=\frac{2}{\pi}\int_{0}^{\pi/2}\sin^{2n}{x}\,\mathrm{d}x
\end{align}$$</p>
|
1,174,463 | <p><span class="math-container">$$\int x^3 \cos(2x^2-5)dx$$</span></p>
<p>I tried using the substitution method but to no avail.</p>
<p>I also tried integrating by parts but that didn't work either.</p>
| Ian | 83,396 | <p>Hint: you don't like the $2x^2-5$. So take $u=2x^2-5$. Then $du=4xdx$, $dx=du/4x$, and $x^2=\frac{u+5}{2}$. So your integral is</p>
<p>$$\frac{1}{4} \int \frac{u+5}{2} \cos(u) du.$$</p>
<p>You can split this into two integrals, one of which is easy and the other of which requires straightforward integration by par... |
495,812 | <p>Let $A$ be an $m \times n$ matrix, and let $A'$ be the result of a sequence of elementary row operations on $A$. Prove that the rows of $A$ span the same space as the rows of $A'$.</p>
<p>I know how to prove that A and A' have both the same rank (rank (A)= dim(Vect(A)) ), but I am no sure how to prove that they bot... | user84413 | 84,413 | <p>It is enough to show this for a single elementary row operation,
so let $r_1,\cdots,r_m$ be the rows of A.</p>
<p>1) If $A^{\prime}$ is formed by interchanging two rows of A, then A and $A^{\prime}$ have the same rows and therefore the same row space.</p>
<p>2) If $A^{\prime}$ is formed by changing $r_i$ to $cr_i$... |
406,255 | <p>$ \int_ {-\pi}^{\pi}\ f(x) \ dx $ is equivalent to integrating $f(x)$ over $-\pi < x < \pi$ (strictly less than) and also $-\pi \le x \le \pi$ (less than or equal). So I just need to reassure myself that the bounds of integration don't change when considering either less than or less than or equal. A quick co... | JLA | 30,952 | <p>Yes that is true (assuming a proper riemann integral, otherwise you must take a limit). Think of it in terms of the area under a curve, ie. you want to calculate the area under the curve f(x) where the end points are $-\pi$, and $\pi$. Clearly the end point don't contribute a nonzero amount to the area.</p>
|
2,643,778 | <p>I can intuitively realize why this is true. All the points that are on the line $y=-x$ are in the set. How can I prove this with sequences?</p>
| Prasun Biswas | 215,900 | <p><strong>Hints:</strong></p>
<ul>
<li><p>The proposed set is the preimage of $[0,\infty)$ under the continuous map $f(x,y)=x+y$</p></li>
<li><p>$[0,\infty)$ is a closed set.</p></li>
<li><p>If $F\subset\Bbb R$ is closed and $f$ is a continuous map $\Bbb R^2\to\Bbb R$, then $f^{-1}(F)$ is closed (try to prove this on... |
2,643,778 | <p>I can intuitively realize why this is true. All the points that are on the line $y=-x$ are in the set. How can I prove this with sequences?</p>
| Marios Gretsas | 359,315 | <blockquote>
<p>We know that $z_n=(x_n,y_n) \to (x,y) \in \Bbb{R}^2 \Leftrightarrow x_n \to^{d_2} x$ and $y_n \to y$ where $d_2$ is the euclideian distance.</p>
</blockquote>
<p>Thus let $z_n \in \{(x,y):x+y \geq 0\}$ and $z_n \to (x_0,y_0)$ </p>
<p>then $z_n=(x_n,y_n)$ where $x_n \to x_0$ and $y_n \to y_0$ and ... |
4,188,560 | <p><a href="https://i.stack.imgur.com/tlruT.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tlruT.png" alt="Polynomials Problem" /></a></p>
<p><strong>OBSERVATIONS</strong></p>
<p>Let <span class="math-container">$$f(x)=1-x+x^2...+x^{16}-x^{17}$$</span>
If we try to replace <span class="math-containe... | Anurag A | 68,092 | <p>Note that
<span class="math-container">$$f(x)=1-x+x^2...+x^{16}-x^{17}=\frac{1-x^{18}}{x+1}.$$</span>
So,
<span class="math-container">$$f(y-1)=\frac{1-(y-1)^{18}}{y}.$$</span>
Look for the coefficient of <span class="math-container">$y^3$</span> in the numerator and that would be <span class="math-container">$\bino... |
304,646 | <p>I am asked to solve the following system of equations using Cramer's Rule</p>
<p>$$
A^2 + B = x\\
AB = 1-x^2
$$</p>
<p>where $A$ and $B$ are 2 unknowns that are to be solved for some given $x$. ie I'm expecting the solution to be $A = a(x)$ and $B = b(x)$ where $a(x)$ and $b(x)$ is some function of $x$.</p>
<p>I... | Community | -1 | <p>You could write that system of equations as</p>
<p>$$
\left( \begin{matrix}A & 1 \\ 0 & A \end{matrix} \right)
\left( \begin{matrix}A \\ B \end{matrix} \right) =
\left( \begin{matrix}x \\ 1-x^2 \end{matrix} \right)
$$</p>
<p>and then solve, but then you get a solution for $A$ and $B$ in terms of $x$ and $... |
1,380,178 | <blockquote>
<p>Find the equation of a circle that intersects the $y$-axis at the origin and at the point $(0,6)$ and also touches the $x$ axis.</p>
</blockquote>
<p>Okay, so I wasn't sure how to do this so I looked at the answer at the back of the book and decided to graph it. The book says the answer is $x^2+y^2-6... | Rory Daulton | 161,807 | <p>Here is a diagram, from Geogebra, that shows an answer. Your question is still ambiguous on the exact placement of the "plane", so I made the initial placement symmetric with the camera. All linear units here are reduced by a factor of $100$, which will not change the final angle. The view of this diagram is vertica... |
30,983 | <p>Firstly, are there always nontrivial real solutions to the sysytem
of equations, $A_{1}x^{5}+B_{1}y^{5}+C_{1}z^{5}=0$ and $A_{2}x+B_{2}y+C_{2}z=0$,
for real numbers $A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, $B_{2}$,
and $C_{2}$? [Answered]</p>
<p>Secondly, are there always nontrivial real solutions to the sysytem
of equa... | Joseph O'Rourke | 6,094 | <p>I did not look at your references, so apologies if this is irrelevant.
Carl Lee defines a <em>regular subdivision</em> of a set of points $V$ (in "Subdivisions and triangulations of polytopes,"
<em><a href="http://www.crcpress.com/ecommerce_product/product_detail.jsf?catno=C3014&isbn=0000000000000&parent_id=... |
98,605 | <p>The <a href="http://en.wikipedia.org/wiki/Characteristic_%28algebra%29"><em>characteristic</em> of a ring</a> (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. <em>Otherwise, we define it to be $0$.</em> </p>
<p>Bu... | lhf | 589 | <p>Given a ring $R$ there is a unique ring homomorphism $\varphi:\mathbb Z\to R$. The characteristic of $R$ is the (canonical, non-negative) generator of $\ker \varphi$.</p>
|
98,605 | <p>The <a href="http://en.wikipedia.org/wiki/Characteristic_%28algebra%29"><em>characteristic</em> of a ring</a> (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. <em>Otherwise, we define it to be $0$.</em> </p>
<p>Bu... | Arturo Magidin | 742 | <ol>
<li><p>Consider the following statement:</p>
<blockquote>
<p>Let $n\geq 0$. The characteristic of $R$ is $n$ if and only if ($ka=0$ for all $a\in R$ $\iff$ $n|k$).</p>
</blockquote>
<p>The statement holds for positive characteristic, but it also holds for characteristic $0$, since $0$ is the only multiple of $... |
98,605 | <p>The <a href="http://en.wikipedia.org/wiki/Characteristic_%28algebra%29"><em>characteristic</em> of a ring</a> (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. <em>Otherwise, we define it to be $0$.</em> </p>
<p>Bu... | apio | 694,806 | <p>It has something to do with the classification of cyclic groups. If <span class="math-container">$(R,+)$</span> is an infinite cyclic group generated by <span class="math-container">$1,$</span> then it is isomorphic to <span class="math-container">$(\mathbb Z,+)$</span> (additive group of integers). If it is a finit... |
1,503,256 | <p>There are 8 employees at a small company. The company have a Christmas tradition where each employee has to bring a small gift. The gifts are then put in a bag and later distributed by a random draw by each of the employees. </p>
<p>a) What is the probability that at least one of the employees don't not get their ... | dineshdileep | 41,541 | <p>Couple of hints:</p>
<p>1) Try expanding the product. Note that for matrices $AB\neq BA$ in general. </p>
<p>2) Try multiplying with $A$ on both sides.</p>
<p>3) What if $A=-I2$?</p>
<p>4) $(AB)^n=B^nA^n$</p>
<p>5) See hint above.</p>
<p>6) $\det(AB)=\det(A)\det(B)$. A matrix is invertible if its determinant i... |
722,890 | <p>Ok, so I know that since the numerator has a higher power that long division is needed. So after doing that, the main fraction is $\frac{-6x-36}{x^2 + 12x + 36}$. I think that's right. But my problem is that after you factor the denominator they're both equal to $(x+6)$. So how would you use partial fraction decompo... | Steven Alexis Gregory | 75,410 | <p><span class="math-container">\begin{array}{rcrrrrrrr}
& & x & -12 \\
& & -- & --- & --- & --- & \\
x^2+12x+36 & ) & x^3 \\
& & x^3 & 12x^2 & 36x \\
& & -- & --- & --- \\... |
598,041 | <p>Prove the following inequality for each $x$:</p>
<p>$$|\sin x+7\cos x| \leq \sqrt{50}$$</p>
<p>My remark: It seems to me that the solution uses the fact that $1^2+7^2=50$.</p>
<p>Thanks in advance!</p>
| DeepSea | 101,504 | <p>$$(1\sin x + 7\cos x)^2 < (1^2 + 7^2)[(\sin x)^2 + (\cos x)^2] = 50$$ </p>
<p>and take square root both sides to get the answer. </p>
|
1,009,996 | <p>Show that if $a,b$ are rational numbers and $x$ is a positive real number then $x^a$$x^b$ $=$ $x^{(a+b)}$</p>
<p>I honestly have no idea how to even do this. Anyone have any hints or a good explanation?
Thanks in advance!</p>
| Charlie Frohman | 90,375 | <p>The proof depends on the definition of $x^a$ you are using. The easiest approach to defining $x^a$ is to define, $\log{y}=\int_1^y\frac{dt}{t}$ and use the inverse function theorem to define $e^x$ to be the inverse function. Finally $x^a=e^{a\log{x}}$. You can now derive the rules of exponents from the rules of lo... |
2,089,980 | <p>Those are the mechanics:</p>
<ul>
<li>You always draw from 2 to 10 cards.</li>
<li>You only use from A to 10. So 40 cards.</li>
</ul>
<p>What I want to obtain:</p>
<ul>
<li>The % of times you gonna obtain a number repeated.</li>
</ul>
<p>I want to make a graphic with how much % of at least one repeat you obtain ... | Marc van Leeuwen | 18,880 | <p>The antonym of "distinct" is "equal", so strictly speaking "$m$ non-distinct boxes" means just one box. This is not what is meant though in this context. It means that among all configurations possible, we should consider equivalent those configurations that can be transformed into one another by permuting the boxes... |
1,990,274 | <p>The problem asks:</p>
<blockquote>
<p>A silo (base not included) is to be constructed in the form of a
cylinder surmounted by a hemisphere. The cost of construction per
square unit of surface area is $7$ times as great for the hemisphere
as it is for the cylindrical sidewall. Determine the dimensions to b... | Anirban Bose | 980,887 | <p>These lecture videos are very good.</p>
<p><a href="https://youtube.com/playlist?list=PLbMVogVj5nJRR7zYZifYopb52zjoScx1d" rel="nofollow noreferrer">https://youtube.com/playlist?list=PLbMVogVj5nJRR7zYZifYopb52zjoScx1d</a></p>
|
3,056,793 | <p>See <a href="https://qr.ae/TUn74K" rel="nofollow noreferrer">What is directional derivative?</a></p>
<blockquote>
<p>It is just the usual derivative i.e. the ONE VARIABLE/ONE DIMENSIONAL
Derivative obtained from breaking down a VECTOR DERIVATIVE.</p>
<p>This example will make it clear. Consider a humming bird moving... | Henry Lee | 541,220 | <p>From the way in which this paragraph is worded it makes most sense to think of a function in more than two dimensions, e.g.
<span class="math-container">$$z=f(x,y)$$</span>
in this context we know that we can take derivatives in two different dimensions, to give:
<span class="math-container">$$\frac{\partial z}{\par... |
179,442 | <p>Recently I've encountered the following equation in a book (don't remember the source, I just took some notes down on my scrap paper) </p>
<p>$$\dot x=t-x^2,\quad x(1)=1.$$ Then I've recognized that this is a Riccati type equation, but not explicitly solvable. The exercise proposed there was however to make a quali... | Severin Schraven | 331,816 | <p>I'd just like to tell you my approach to the first point. Let $u(t)=1/t$ and $v(t)=t^2$. Those are the solutions of</p>
<p>$$ \begin{cases} u'(t)= -u(t)^2 \\ u(1)=1
\end{cases} $$</p>
<p>and</p>
<p>$$ \begin{cases} v'(t)= t \\ v(1)=1
\end{cases}.$$</p>
<p>Hence, we have for $t\geq 1$</p>
<p>$$ 1/t=u(t) \leq f(t... |
2,913,425 | <p>In </p>
<p>E. Swenson, A cut point theorem for CAT(0) groups, J. Differential Geom. 53 (1999), no. 2, 327–358. </p>
<p>the author shows (Theorem 11) that if a group $G$ acts geometrically (i.e. properly discontinuously) and cocompactly by isometries on a CAT(0) space, then $G$ has an element of infinite order. How... | Community | -1 | <p>The author is implicitly assuming that the CAT(0) space is unbounded in the first sentence of the proof of theorem 11 when they say "choose a geodesic ray". In the first paragraph of the section they say that rays are parametrized on $[0,\infty)$. In particular $G$ can not act geometrically on the space if it is fin... |
98,971 | <p>I'm working on an algorithm which should check if two polygons, described by their vertex coordinates, are: one inside the other, are intersecting or are separated</p>
<p>image below describe this three cases:
<a href="https://i.stack.imgur.com/hl6zL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.co... | eldo | 14,254 | <pre><code>R1 = Polygon[{{1, 0}, {1, 1}, {0, 0}}];
R2 = Polygon[{{0, 1}, {1/3, 1/2}, {0, 1/2}}];
Graphics[{R1, Red, R2}, Frame -> True]
</code></pre>
<p><a href="https://i.stack.imgur.com/cD31r.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cD31r.jpg" alt="enter image description here"></a></p>... |
4,521,067 | <blockquote>
<p>If <span class="math-container">$$f(x)=\prod_{k=1}^\infty\left(1-\frac{x^3}{k^3\pi^3}\right)$$</span>
and if <span class="math-container">$$f(x)=\sum_{n=0}^\infty a_n x^n$$</span>
Then find <span class="math-container">$a_n$</span> or convert the above infinite product into an infinite sum.</p>
</block... | Claude Leibovici | 82,404 | <p>Let <span class="math-container">$a=\frac x \pi$</span></p>
<p><span class="math-container">$$f(a)=\prod_{k=1}^\infty\left(1-\frac{a^3}{k^3}\right)=\frac{\prod_{k=1}^\infty(k-a)(k-b)(k-c)}{\prod_{k=1}^\infty k^3}=\frac{\prod_{k=1}^\infty(k-a)}{\prod_{k=1}^\infty k}\frac{\prod_{k=1}^\infty(k-b)}{\prod_{k=1}^\infty k}... |
1,885,438 | <blockquote>
<p>Let $X=\mathbb{R}\times \mathbb{Z}$ and $\tau$ the product topology $\tau_u \times \tau_{CF}$ where $\tau_u$ is the usual topology on $\mathbb{R}$ and $\tau_{CF}$ the cofinite topology on $\mathbb{Z}$. Study the contiuity of $f : (X, \tau)\to (\mathbb{R},\tau_u)$ defined as
$$(t,k)\mapsto t-k$$</p>
... | Brian M. Scott | 12,042 | <p>You have the right general idea, but you’ve not executed it quite right. In order to show that $f$ is not continuous at $\langle 0,0\rangle$, it’s enough to find one open nbhd $U$ of $0$ such that $f^{-1}[U]$ is not open; let’s try $U=\left(-\frac12,\frac12\right)$. Clearly</p>
<p>$$\begin{align*}
f^{-1}[U]&=\l... |
3,514,934 | <blockquote>
<p>If <span class="math-container">$x = 2 + \sqrt{3}$</span> then find <span class="math-container">$x^3 + \dfrac1{x^3}$</span>. </p>
</blockquote>
<p>My attempt: we know that the sum of a number and its reciprocal is always greater than or equal to <span class="math-container">$2$</span> but I don't kn... | J. W. Tanner | 615,567 | <p><span class="math-container">$\left(x+\frac1x\right)^3=x^3+\frac1{x^3}+3\left(x+\frac1x\right),$</span> so <span class="math-container">$x^3+\frac1{x^3}=(x+\frac1x)^3-3(x+\frac1x)$</span>.</p>
<p>When <span class="math-container">$x=2+\sqrt3, $</span> <span class="math-container">$\frac1x=\frac1{2+\sqrt3}\frac{2-\s... |
61,030 | <p><a href="http://reference.wolfram.com/language/ref/Dataset.html"><code>Dataset</code></a> rather conveniently truncates the display of a dataset that has many rows. However, occasionally, I would like to show more of a dataset without having to convert to e.g. a <a href="http://reference.wolfram.com/language/ref/Tab... | Taliesin Beynon | 7,140 | <p>If you are willing to use an undocumented and future-incompatible hack you can set the global variable <code>Dataset`FormattingCompound`PackagePrivate`$Limit</code> to larger than its default value of 64 to see more of the dataset. For example, <code>Dataset[RandomInteger[10, {1000, 5}]]</code> shows 16 rows by defa... |
543,531 | <p>Suppose $X$ and $Z$ are standard normal random variables and define $$Y = X + Z.$$
Then $Y$ is also normal, has zero mean and its variance is two (<a href="http://en.wikipedia.org/wiki/Normal_distribution#Related_distributions" rel="nofollow">Wikipedia</a>). It should also be fair to assert that $Y|(X=x)$ has a norm... | Ian | 103,922 | <p>After some thought, I can answer my own question. The key is using both definitions of $X$, which for clarity are
$$
\begin{align}
X &= Y - Z \\
X &= W,
\end{align}
$$
where $W$ is another standard normal random variable (this is just the original definition of $X$). Even given $Y=y$, the two equations are ... |
288,097 | <p>For $x \in \mathbb{R}^n$ let us denote $x_{[i]}$ the $i$th largest component of $x$ s.t
$$
x_{[1]} \geq x_{[2]} \geq x_{[3]}\ge\cdots
$$
The function
$$
f(x)= \sum_{i=1}^r x_{[i]}
$$
is the sum of $r$ largest element of $x$. Here I want to show that for each $x \in \mathbb{R}^n$
$$
y^Tx \leq f(x)
$$
for $y \in \mat... | robjohn | 13,854 | <p>$[i]$ is a permutation of $(1,2,3,\dots,n)$ so that $x_{[i]}\ge x_{[i+1]}$. Therefore,
$$
\begin{align}
f(x)-y^Tx
&=\sum_{i=1}^rx_{[i]}-\sum_{i=1}^ny_{[i]}x_{[i]}\\
&=\sum_{i=1}^r\left(x_{[i]}-y_{[i]}x_{[i]}\right)
+\sum_{i=r+1}^n\left(0-y_{[i]}x_{[i]}\right)\\
&=\sum_{i=1}^rx_{[i]}\left(1-y_{[i]}\right)... |
1,632,879 | <p>$$\lim_{n\to \infty}\sum_{r=1}^n \tan^{-1} \dfrac{2r+1}{r^4+2r^3+r^2+1}$$</p>
<p>How am I supposed to do it? One thing I see here is </p>
<p>$$\lim_{n\to \infty}\sum_{r=1}^n \tan^{-1} \dfrac{2r+1}{(r^2+r)^2+1}$$</p>
<p>Here derivative of $r^2+r$ is $2r+1$. (if it helps)</p>
<p>It's final answer is</p>
<blockquo... | Saikat | 230,452 | <p>You aren't too far from the answer. Credit on discovering what $(r+1)^2$ was.
Write $r^2+r$ as $r(r+1)$. Squaring that would give you $r^2$ and $(r+1)^2$. And the difference of those two terms is the numerator. Now, it's in the form of $\tan(A-B)$. </p>
<p>Now, telescope this sum which becomes
$\arctan(r)- \arcta... |
3,527,470 | <p>Compute <span class="math-container">$$\lim\limits_{n\to \infty} \int\limits_0^1 x^{2019} \{nx\} dx,$$</span> where <span class="math-container">$\{a\}$</span> denotes the fractional part of the real number <span class="math-container">$a$</span>.<br>
I firstly tried to apply the substitution <span class="math-conta... | marty cohen | 13,079 | <p>Here's a proof that
<span class="math-container">$\lim_{n \to \infty} \int\limits_0^1 f(x) \{nx\} dx
=\dfrac12 \int_0^1 f(x) dx
$</span>.</p>
<p>If <span class="math-container">$f(x) = x^m$</span>,
then
<span class="math-container">$\lim_{n \to \infty} \int\limits_0^1 f(x) \{nx\} dx
=\dfrac12 \int_0^1 x^m dx
=\dfrac... |
3,527,470 | <p>Compute <span class="math-container">$$\lim\limits_{n\to \infty} \int\limits_0^1 x^{2019} \{nx\} dx,$$</span> where <span class="math-container">$\{a\}$</span> denotes the fractional part of the real number <span class="math-container">$a$</span>.<br>
I firstly tried to apply the substitution <span class="math-conta... | robjohn | 13,854 | <p><span class="math-container">$$
\begin{align}
&\int_0^1x^{2019}\{nx\}\,\mathrm{d}x\\
&=\frac1{n^{2020}}\int_0^nx^{2019}\{x\}\,\mathrm{d}x\tag1\\
&=\frac1{n^{2020}}\sum_{k=0}^{n-1}\int_0^1(k+x)^{2019}((k+x)-k)\,\mathrm{d}x\tag2\\
&=\frac1{n^{2020}}\sum_{k=0}^{n-1}\left(\frac{(k+1)^{2021}-k^{2021}}{202... |
27,968 | <p>What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think</p>
<p>$$ \mathbb R^+ $$</p>
<p>but I believe that that is usually used to denote "all real numbers including infinity".</p>
<p>So is there a standard way to denote the set</p>
<p>$$ \{x \in \mathbb R : x \geq 0\} \; ?$$ ... | user264405 | 264,405 | <p>I've learned in elementary school that $\mathbb{R}_{*}$ means the set without the zero, so $\mathbb{R}^{+}=[0,\infty)$ and $\mathbb{R}^{+}_{*}=(0,\infty)$.</p>
|
450,115 | <p>The Wikipedia page of <a href="http://en.wikipedia.org/wiki/Ackermann_function">Ackermann function</a> states that Ackermann function is "roughly comparable" to $f_\omega$ in <a href="http://en.wikipedia.org/wiki/Fast-growing_hierarchy">fast-growing hierarchy</a>. </p>
<p>Is there some standard way to make the "rou... | Andrés E. Caicedo | 462 | <p>I discuss this in some detail in my paper </p>
<blockquote>
<p>Andres E. Caicedo, <a href="http://andrescaicedo.files.wordpress.com/2010/04/regressivepairs.pdf" rel="noreferrer">Regressive functions on pairs</a>, European Journal of Combinatorics, <strong>31 (3)</strong>, (2010), 803–812. <a href="http://www.ams.... |
2,053,714 | <p>what I tried was , I observed the cubes from 1-100.</p>
<p>$1^3=1$ </p>
<p>$2^3=8$ </p>
<p>$3^3=27$ </p>
<p>$4^3=64$ </p>
<p>$5^3=125$ </p>
<p>$6^3=216$ </p>
<p>$7^3=343$ </p>
<p>$8^3=512$ </p>
<p>$9^3=81$ </p>
<p>$10^3=1000$ </p>
<p>$11^3=1331$ </p>
<p>$12^3=1729$ </p>
<p>$13^3=2197$ </p>
<p>$14^3=274... | Shraddheya Shendre | 384,307 | <p><strong>EDIT:</strong><br>
The answer assumes knowledge of modular arithmetic. I saw the OP's comment later. </p>
<p>Let such a number be $n=100x+y$ with $ 1\le x\le 9$ and $0\le y\le 99$. </p>
<p>Then $$n^3 = 10^6x^3 + (3\times 10^4)x^2y + (3\times 10^2)xy^2+y^3$$
Now to get the last two digits we take$\pmod{10... |
1,963,335 | <p><strong>So when I run newton method for certain values in matlab it gives the following result. Question is how do you tell this is linear or quadratic? Book says Notice how the convergence rate actually looks linear for the first 20 iterations and then converges quadratically? How?</strong> </p>
<p>Problem descrip... | Evan Rosica | 209,923 | <p>When you plug both answers into the original differential equation for $f(x)$, you can see that $K=0$, as if $K \neq0$, the given differential equation will not hold. We can see this at a glance by observing that if $K \neq 0$, the left hand side of the original diff eq will have constant term K, but the right side ... |
1,963,335 | <p><strong>So when I run newton method for certain values in matlab it gives the following result. Question is how do you tell this is linear or quadratic? Book says Notice how the convergence rate actually looks linear for the first 20 iterations and then converges quadratically? How?</strong> </p>
<p>Problem descrip... | Reese Johnston | 351,805 | <p>When you differentiated both sides, you lost some information - for example, any constants that might have been there were lost. What that means is that the conclusion you got, that $\frac{dy}{dx} = \frac{-3x^2}{2} + C$, is a <em>necessary</em> condition of a solution, but not necessarily <em>sufficient</em>. To tak... |
795,991 | <p>What I'm really wondering is, does Lagrange polynomial interpolation have an answer for every question of "what's the next integer in this sequence"? Does it define an infinite integer sequence to extend every finite integer sequence?</p>
<p>Let $S$ be a finite set of points, at least two of them, all having consec... | evil999man | 102,285 | <p><strong>Hint :</strong></p>
<p>What is <span class="math-container">$d(x^3y^2)$</span>?</p>
<p>What is <span class="math-container">$d(xy^3)$</span>?</p>
<p>Spoilers below. Hover to see.</p>
<blockquote class="spoiler">
<p> <span class="math-container">$d(x^3y^2)-d(xy^2)+d(x^2)+dy=0$</span></p>
</blockquote>
|
3,372,755 | <p><span class="math-container">$w$</span> and <span class="math-container">$z$</span> are given in polar form as <span class="math-container">$w=Re^{i\Phi}$</span>, <span class="math-container">$z=re^{i\theta}$</span>, <span class="math-container">$R>r>=0$</span>;</p>
<p>I can prove this using decomposition int... | Angina Seng | 436,618 | <p><span class="math-container">\begin{align}
2\Re\left(\frac{z+w}{z-w}\right)&=\frac{z+w}{z-w}+\overline{\left(\frac{z+w}{z-w}
\right)}=\frac{z+w}{z-w}+\frac{\overline{z+w}}{\overline{z-w}}\\
&=\frac{(z+w)(\overline z-\overline w)+(z-w)(\overline z+\overline w)
}{(z-w)\overline{(z-w)}}\\
&=\frac{2z\overlin... |
36,778 | <p>I have a pre-processed image, from which I have to get coordinates of some components.</p>
<p><img src="https://i.stack.imgur.com/xg4lA.png" alt="sample image"></p>
<p>There are 7 green components (pigs actually) that I have already isolated from an angry bird screen shot. So what I want is to locate those six com... | Dr. belisarius | 193 | <pre><code>Show[i, Graphics@{PointSize[Large],
Point@ComponentMeasurements[
MorphologicalComponents[DeleteSmallComponents[
ChanVeseBinarize[i, TargetColor -> Green], 10]],
"Centroid"][[All, 2]]}]
</code>... |
2,354,218 | <p>On page 5 of Royden, Fitzpatrick's real analysis, the book defined cardinality of the set in the following way:</p>
<blockquote>
<p>Given a set $X$, the relation of equipotence (invertible map) is an equivalence relation on the collection of $2^X$ of all subsets of $X$. The equivalence class of a set with respect... | N. S. | 9,176 | <p>You are right, the constant function $f(z)=0$ is a counterexample for this problem.</p>
<p>Now, if $f$ is not the trivial function, it is easy to show that $f(z)$ entire + $f(z)f(-z)=0$ implies that $f(z)=0$ for all $z$. (<strong>hint</strong> Identity theorem).</p>
|
440,806 | <p>I've been trying to solve this problem and I always get 1, but the answer is $1 - 2x$.</p>
<p>If $x<\frac12$ then what is $\left|x-\sqrt{(x-1)^2}\right|$ ?</p>
| Hagen von Eitzen | 39,174 | <p>Note that $\sqrt {t^2}=|t|$ for all real numbers $t$.
By the condition $x<\frac12$, it is clear that $x-1$ is negative, hence $\sqrt{(x-1)^2}=|x-1|=1-x$ and the final result is $|x-(1-x)|=|2x-1|=1-2x$ (because $2x-1<0$).</p>
|
3,664,875 | <p>Is the ring <span class="math-container">$\mathbb{Z[x]}/(2x^2 + 5)$</span> isomorphic to any subring of <span class="math-container">$\mathbb{Z[x]}/(x^3+x+1)$</span>? I believe that the answer is No, but I am not 100 % sure. Here is my reasoning: </p>
<p><span class="math-container">$\mathbb{Z[x]}/(2x^2 + 5) = \{b ... | Dietrich Burde | 83,966 | <p>The argument that "the other ring cannot have cubic terms" is not clear to me. An isomorphic subring could have "cubic terms".</p>
<p>Perhaps the following remark is useful:</p>
<p>Let <span class="math-container">$R=\Bbb Z[x]/(2x^2+5)$</span> and <span class="math-container">$S=\Bbb Z[x]/(x^3+x+1)$</span>.
Then ... |
721,044 | <p>If you have a concatenation operation that takes A and B and makes {A,B}, and sets themselves can be the input of this concatenation operation (i.e. {A,{B,C}}), is it mathematically possible to create a set whose members are A and the set that already includes A? Thanks in advance!</p>
| Asaf Karagila | 622 | <p>Do you mean like $\{A,\{A\}\}$? Just input $A$ and $A$ to have $\{A\}$ and then $A$ and $\{A\}$ to have $\{A,\{A\}\}$.</p>
<p>There's no particular name for this operation. It's just is.</p>
|
721,044 | <p>If you have a concatenation operation that takes A and B and makes {A,B}, and sets themselves can be the input of this concatenation operation (i.e. {A,{B,C}}), is it mathematically possible to create a set whose members are A and the set that already includes A? Thanks in advance!</p>
| Schroedinger's Cat | 136,998 | <p>Now I should mention that I am a bloody maths rookie but I'll nevertheless have a go at it. Others can correct me ....</p>
<p>So I believe you are looking for the notion of a collection perhaps. A collection can contain sets which in turn contain single elements, singletons if I'm not mistaken they are called.</p>
... |
4,050,395 | <p><strong>Problem</strong></p>
<p>A survey of boy-girl couples suggests that in 20% of couples both the boy and the girl love pizza, in 5% of couples the girl loves pizza, but the boy doesn't, in 10% couples the boy loves pizza, but the girl doesn't.</p>
<p>If a boy is chosen randomly and he loves pizza, what's the pr... | J.G. | 56,861 | <p>Note <span class="math-container">$P(A)=P(AB)+P(A\bar{B})=0.2+0.1=0.3$</span>. You can do the rest.</p>
|
2,828,070 | <p>If $$\log_{3x+5} (ax^2+8x+3)>2$$ find the interval in which x lies.</p>
<p>Answer is $\left(-\dfrac43 , -\dfrac{23}{22}\right)$. Now hint me how to solve the problem please.</p>
| sai nithika | 571,498 | <p>Hello Girish Kumar chandora that's a good try but you just missed the point close
What you did is correct to an extent but you have taken
$P(A∩B)$as common but think about this
If it is given that $P(A∩B)/P(B)=P(B∩A)/P(A)$
Then the $P(A∩B)$ would be cancelled on both L.H.S and R.H.S as they both are same (basic r... |
1,471,174 | <p>Let $\psi: \mathbb R \to \Bbb R$ be a given analytic function.</p>
<p>How can I construct a smooth (that is, infinitely differentiable) function $\Phi:\Bbb R\to \Bbb R$ such that $\psi|_D=\Phi|_D$ for some interval $D$, and $\Phi$ is bounded above and below?</p>
<p>E: I've been reminded in the comments that such ... | DanielWainfleet | 254,665 | <p>It still may be impossible.For example if $$f(0)=0,$$ $$f(x)=e^{(-1/x^2)} , x\neq 0$$ then $f$ cannot be extended to an analytic function on any open domain $D\subset C$ for which $0\in D$.</p>
|
317,357 | <p><strong>The Question</strong></p>
<p>Given Hessian manifold <span class="math-container">$M$</span>, there is a natural Kahler structures on <span class="math-container">$TM$</span>. Is it possible to write the Kahler potential of these in terms of the Hessian potential?</p>
<p><strong>Background</strong></p>
<p>... | anna abasheva | 88,385 | <p>My answer deals only with the case when <span class="math-container">$\nabla$</span> is the Levi-Civita connection.</p>
<p>As you seem to know the bundle <span class="math-container">$TTM$</span> naturally decomposes as a direct sum of the vertical subbundle <span class="math-container">$V$</span> and the horizonta... |
2,321,285 | <p>For my abstract algebra course I have to decide whether
$$(*)\qquad X^4 + a^2, \; a \in \mathbb{Z}\;\text{odd}$$ is irreducible over $\mathbb{Q}[X]$ and $\mathbb{Z}[X]$. </p>
<p>Since the degree of the polynomial is 4 and thus even I should be able to split it into two polynomials of degree two $if$ the coefficie... | Trevor Gunn | 437,127 | <p>Hint: the roots of $x^4 + a^2$ are</p>
<p>$$ \pm\sqrt{a}\frac{1 \pm i}{\sqrt{2}}. $$</p>
<p>Separate these into conjugate pairs to find the factorization of $x^4 + a^2$ over $\mathbf{R}[x]$.</p>
<p>If you want to save time, or perhaps check your work, the answer is</p>
<p>$$ x^4 + a^2 = (x^2 - \sqrt{2a} \cdot x ... |
1,479,177 | <p>If there exists $r \in \mathbb{R}$ with $r < 1, K \in \mathbb{N}$, such that $|X_n|^{1/n} < r$ for all $n > K$, then the series $X_n$ (Summation of $X_n$ from $n$ to $\infty$) is absolutely convergent.</p>
<p>We know that $|X_n| < r^n$ for all $n > K$ so sum of all terms would be bounded by sum of f... | user26857 | 121,097 | <p>Since $V$ is a subring of $\mathbb Z$ it is also an ideal of $\mathbb Z$ (why?), so $V=n\mathbb Z$ for some integer $n$. From $17\mathbb Z\subseteq V$ you get $n\mid 17$. Now you can finish the proof.</p>
|
2,468,458 | <p>Given 80 people, how do you arrange so that each person:</p>
<ul>
<li>listens to exactly 12 other people</li>
<li>is listened to by 12 other people</li>
</ul>
<p>This is an interesting question my friend posed and neither of us could find eh solution to. I feel like the answer is easy but still can't manage to fig... | Austin Weaver | 480,825 | <p>Assuming you can listen to and be listened to by the same person, this would just be an undirected graph with exactly $12$ edges connected to each one of $80$ nodes.</p>
<p>For utility, label each person $1$-$80$ and arrange them in a circle.</p>
<p>First, connect each person to the person on their left and right ... |
1,886,332 | <p>Let $f$ be a continuous function from the reals $\mathbb{R}$ onto $I=[0,1]$ with usual topology. Prove that if $C$ is a subset of $I$ and the preimage of $C$ is closed in $\mathbb{R}$ then $C$ is closed in $I$.</p>
<p>My attempt is to use normal space properties but it does not help. </p>
| Tsemo Aristide | 280,301 | <p>Let $D$ be the complementary space of $C$. We are going to show that $D$ is open. Suppose that $D$ is not open. There exists $y\in D$ such that for every open interval $y\in J$, $J\cap[0,1]\cap C$ is not empty. Since $f$ is continuous, $f^{-1}(J\cap [0,1])$ is open. Let $x\in f^{-1}(J\cap [0,1])$ such that $f(x)=y$... |
1,384,455 | <p>Let $X$ be a topological space of infinite cardinality. Is it possible for any $X$ to be homeomorphic to $X\times X$ $?$</p>
<p>For example, $\mathbb R$ is not homeomorphic to $\mathbb R^{2}$, and $S^{1}$ is not homeomorphic to $S^{1} \times S^{1}$ . What other topological spaces might... | Travis Willse | 155,629 | <p>Yes, consider $X := \Bbb Z$ endowed with the discrete topology.</p>
<p>For any topological manifold $M$, $\dim (M \times M) = \dim M + \dim M = 2 \dim M$. Since the dimension of a nonempty topological manifold is well-defined, there is no positive-dimensional topological manifold $M$ for which $M \cong M \times M$,... |
1,384,455 | <p>Let $X$ be a topological space of infinite cardinality. Is it possible for any $X$ to be homeomorphic to $X\times X$ $?$</p>
<p>For example, $\mathbb R$ is not homeomorphic to $\mathbb R^{2}$, and $S^{1}$ is not homeomorphic to $S^{1} \times S^{1}$ . What other topological spaces might... | Alon Amit | 308 | <p>At this level of generality you can make $X=X \times X$ happen quite easily. Take a discrete space of any infinite cardinality, for instance. Or topologize $X=A^B$ by whatever means and compare $X \times X = A^{B \sqcup B}$; under various mild assumptions on $B$ those spaces would be homeomorphic. </p>
|
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