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 |
|---|---|---|---|---|
3,540,956 | <p>Let G be a finite group, H a maximal proper subgroup of G and K a subgroup of H. Is the normalizer of K in G, <span class="math-container">$N_GK$</span>, a subgroup of H.
Now <span class="math-container">$N_GK$</span> is certainly contained in some maximal subgroup, maybe more than one, but why is it contained in H... | EuxhenH | 281,807 | <p>Split the <span class="math-container">$400$</span> students into <span class="math-container">$4$</span> groups of <span class="math-container">$100$</span>. By the pigeonhole principle, at least one of them must have a total age of at least <span class="math-container">$8000/4 = 2000$</span>. But the oldest <span ... |
1,394,490 | <p>I stumbled upon "the God proof" which goes:</p>
<p>$0 = 0 + 0 + 0...$</p>
<p>$ = (1-1) + (1-1) + (1-1) + ...$</p>
<p>$= 1 - 1 + 1 - 1 + 1 - 1 + ...$ </p>
<p>$= 1 + (-1+1) + (-1+1) + (-1+1) + ...$ </p>
<p>$= 1$</p>
<p>Even though this result is obviously wrong, I can't quite pinpoint exactly what the 'illegal'... | Gregory Grant | 217,398 | <p>Infinite sums are not "associative" like that. You can only rearrange the order of summation if the sum converges "absolutely".</p>
|
5,890 | <p>Is there a way to <em>temporarily</em> suppress certain messages, so that I could write for example (with made-up syntax for that feature):</p>
<pre><code>WithOff[Pattern::patv, rule = (f[x_Integer|{x__Integer}] :> g[x])];
rule2 = x_[x__] :> x;
</code></pre>
<p>and get no <code>Pattern::patv</code> message f... | Ajasja | 745 | <p>I agree completely with J.M., <code>Quiet</code> is the answer.</p>
<p>Implementing <code>WithOff</code> using <code>Quiet</code> is (as I'm sure you know) trivial. Here it is, just for fun:</p>
<pre><code>ClearAll[WithOff]
SetAttributes[WithOff, HoldAll];
WithOff[msg_, expr_] := Quiet[expr, {msg}];
WithOff[Patter... |
2,745,623 | <blockquote>
<p>Maximize the generic bivariate quadratic form constrained to the unit circle.</p>
<p>$$\begin{array}{ll} \text{maximize} & f(x_1, x_2) := ax_1^2 + 2bx_1 x_2 + cx_2^2\\ \text{subject to} & g(x_1, x_2) := x_1^2 + x_2^2 - 1 = 0\end{array}$$</p>
</blockquote>
<p>Using the standard Lagrange M... | Fred | 380,717 | <p>Let $x=(x_1,x_2)$ such that $x_1^2+x_2^2=1$ and </p>
<p>$f(x)=\max \{f(u,v):u^2+v^2=1\}$. By Lagrange there is $ \lambda \in \mathbb R$ such that $Ax=\lambda x$.Hence</p>
<p>$\lambda= \lambda x^Tx=x^T(Ax)=f(x)$.</p>
|
1,768,142 | <p>Calculate the line integral
$$
\rm I=\int_{C}\mathbf{v}\cdot d\mathbf{r}
\tag{01}
$$
where
$$
\mathbf{v}\left(x,y\right)=y\mathbf{i}+\left(-x\right)\mathbf{j}
\tag{02}
$$</p>
<p>and $C$ is the semicircle of radius $2$ centred at the origin from $(0,2)$ to $(0,-2)$ to the negative x axis (left half-plane).</p>
<blo... | DonAntonio | 31,254 | <p>The integral becomes, with your parametrization:</p>
<p>$$\int_0^\pi(2\sin t,-2\cos t)\cdot(-2\sin t,2\cos t)dt=\int_0^\pi-4\;dt=-4\pi$$</p>
<p>so yes: I'd say it seems to be you got it right!</p>
|
2,929,238 | <p>Recently I have come arcross the following fraction</p>
<blockquote>
<p><span class="math-container">$$\dfrac{z^2}{(2z^2+3)((s^2+1)z^2+1)}$$</span></p>
</blockquote>
<p>Hence I have encountered this fraction within a task of integration I want to do a partial decomposition. First of all I rewrote it as following... | Yanko | 426,577 | <p>The relation works like that: The zero vector is only in relation with itself. All the other vectors are in relation with each other</p>
<p>Proof: if <span class="math-container">$v=0$</span> then <span class="math-container">$Av=0$</span> for every matrix and therefore <span class="math-container">$0\sim w$</span>... |
1,921,914 | <p>How do you find the sum of $\sum \limits_{i=0}^{n-1}(1+i) $ ?</p>
<p>Actually, I am especially confused because of of the n-1. Usually, I'd start with stuff like:
$$\sum \limits_{i=0}^{0}(1+i) = ?$$
$$\sum \limits_{i=0}^{1}(1+i) = ?$$
$$\sum \limits_{i=0}^{2}(1+i) = ?$$</p>
<p>But I don't know what to do with the... | Brian M. Scott | 12,042 | <p>Change the index of summation: let $k=i+1$. Then as $i$ runs from $0$ to $n-1$, $k$ runs from $0+1=1$ to $(n-1)+1=n$, and you have</p>
<p>$$\sum_{i=0}^{n-1}(1+i)=\sum_{k=1}^nk\;,$$</p>
<p>a summation that you’ve probably already encountered.</p>
<p>In this simple problem you don’t actually have to do that: you ca... |
2,080,460 | <p>The set of uniqueness for $H^2$ is defined to be a set $E\subseteq \mathbb{D}$ such that if $f\in H^2$ and $f|_E =0$ then $f\equiv 0$. Let $$ k_\lambda(z) =\frac{1}{1-z\ \overline{\lambda}}$$ be the reproducing kernel for $H^2$. I want to prove a statement that is claimed in a text and the statement is : If $\{\lamb... | Alex Macedo | 400,433 | <p>Consider a smooth projective curve $C$ of genus $\geq 1$ over a number field $K$. Since we have an exact sequence</p>
<p>$$ 0 \rightarrow \text{Pic}^0(C) \rightarrow \text{Pic}(C) \rightarrow \text{NS}(C) \rightarrow 0,$$</p>
<p>it suffices to show that $\text{Pic}^0(C)$ and $\text{NS}(C)$ are finitely generated.<... |
2,181,989 | <p>In all the question I never took $\sin x =t$ and was able to solve most of them but I got stack in this question and I saw the solution , I found that they took $\sin x =t$ and treated $\sin x$ as just a variable $t$. I found it a bit weird, now many question how would I had known that I had to take $\sin x$ as $t$ ... | Ross Millikan | 1,827 | <p>It will work as long as the first order terms do not cancel out and will often simplify your computations. The problem comes if we ask $$\lim_{x \to 0} \frac {x-\sin x}{x^3}$$ If you blindly substitute in $\sin x=x$ you will get $0$ for the limit because the function is then identically $0$. In truth the limit is... |
42,913 | <p>The following algorithm decides if a number $n>0$ is a totient or a nontotient:</p>
<pre><code>if n = 1
return true
if n is odd
return false
for k in n..n^2
if φ(k) = n
return true
return false
</code></pre>
<p>This is very slow; even using a sieve it takes $n^2$ steps to decide that $n$ is nontotient... | Kimball | 11,323 | <p>As I recall, the finite subgroups of $PGL_N(\mathbb C)$ are classified for $N \le 7$. See Miller--Blichfeldt--Dickson's "Theory of finite groups" (1916) for $N=3$ or Blitchfeldt's "Finite collineation groups" (1917) for $N=3, 4$. (Beware: the terminology is quite old---for instance, isomorphic only means something... |
2,956,158 | <blockquote>
<p>Given <span class="math-container">$z = \cos (\theta) + i \sin (\theta)$</span>,
prove <span class="math-container">$\dfrac{z^{2}-1}{z^{2}+1} = i \tan(\theta)$</span></p>
</blockquote>
<p>I know <span class="math-container">$|z|=1$</span> so its locus is a circle of radius <span class="math-contai... | Martin R | 42,969 | <p>For <span class="math-container">$z = \cos \theta + i \sin \theta$</span> you have
<span class="math-container">$$
\frac{z^2-1}{z^2+1} = \frac{ \cos^2 \theta - \sin^2 \theta + 2i \cos\theta \sin \theta -1}{\cos^2 \theta - \sin^2 \theta + 2i \cos\theta \sin \theta +1 } \, .
$$</span>
Now substitute <span class="math... |
1,563,105 | <p>I have a equation of motion for a forced pendulum show below
$$
{d^2\theta\over dt^2} = -{g\over L}\sin\theta + C\cos\theta\sin(Dt)
$$
$L=10$ cm, $C=2\ \hbox{s}^{-2}$ and $D=5\ \hbox{s}^{-1}$.</p>
<p>I am trying to make this equation dimensionless by setting the follow equations</p>
<p>$$\omega^2 = g/L,\quad ... | Andreas Blass | 48,510 | <p>Suppose, toward a contradiction, that we had a 2-CNF formula $\phi$, using the variables $x_1,x_2,x_3$, and some auxiliary variables $y_1,\dots,y_n$ such that, whenever we assign truth values to the $x$'s, if at least one of the three is true then we can assign values to the $y$'s to make $\phi$ true, but if all thr... |
1,243,159 | <p>I've recently been studying matrices and have encountered a rather intriguing question which has quite frankly stumped me. </p>
<p>Find the $3\times3$ matrix which represents a rotation clockwise through $43°$ about the point $(\frac{1}{2},1+\frac{8}{10})$</p>
<p>For example: if the rotation angle is $66°$ then th... | Emilio Novati | 187,568 | <p>The center of rotation is the point $C=(c_1,c_2)=(1/2,9/5)$ and the angle is $\theta= -43°$.
You can represent this transformation whith a $3 \times 3$ matrix using <a href="http://en.wikipedia.org/wiki/Homogeneous_coordinates" rel="nofollow">homogeneous coordinates</a> in the affine plane.</p>
<p>Note that you ca... |
1,911,037 | <p>So I realized that I have to prove it with the fact that $(x-y)^2+2xy=x^2+y^2$ </p>
<p>So $\frac{(x+y)^2}{xy}+2=\frac{x}{y}+\frac{y}{x}$ $\Leftrightarrow$ $\frac{(x+y)^2}{xy}=\frac{x}{y}+\frac{y}{x}-2$ </p>
<p>Due to the fact that $(x+y)^2$ is a square, it will be positive </p>
<p>$x>0$ and $y>0$ so $xy>... | wythagoras | 236,048 | <p>You made a few mistakes here. </p>
<p>First, actually $(x+y)^2=x^2+2xy+y^2$, what you need is $(x-y)^2+2xy=x^2+y^2$. Now, you get $xy>0$ but $(x-y)^2 \geq 0$, so $\frac{(x-y)^2}{xy} \geq 0$, and this gives you $$\frac{x^2-2xy+y^2}{xy} \geq 0$$</p>
<p>which gives $\frac{x}{y}+\frac{y}{x} \geq 2$, which is the de... |
2,033,832 | <p>I am a math enthusiast in electrical engineering and I am planning on learning Differential Geometry for applications in Control Theory. I want to teach myself this beautiful branch of mathematics in a rigorous way.</p>
<p>I am currently going through Chapman Pugh's Real Analysis, I am then planning on studying Mun... | Pait | 10,746 | <p>Arnold, do Carmo, and Spivak are very good books. Do stay away from Boothby. </p>
<p>Guillemin and Pollack's very readable, very friendly introduction to topology is great, also Milnor's "Topology from the Differentiable Viewpoint". It will be useful to read them before or while you study the geometry part. </p>
<... |
3,087,207 | <p>In "Relational Algebra by Way of Adjunctions," found at <a href="http://www.cs.ox.ac.uk/jeremy.gibbons/" rel="nofollow noreferrer">author's page</a> (<a href="http://dx.doi.org/10.1145/3236781" rel="nofollow noreferrer">doi</a>), section 2.4, an adjunction is described using the signature:</p>
<p><span class="math-... | Gnumbertester | 628,028 | <p>The length of a line segment tangential to a circle around a central angle is the tangent of that central angle. Look at this graph of the tangent function. <a href="https://i.stack.imgur.com/7lgdD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7lgdD.png" alt="enter image description here"></a> <... |
3,087,207 | <p>In "Relational Algebra by Way of Adjunctions," found at <a href="http://www.cs.ox.ac.uk/jeremy.gibbons/" rel="nofollow noreferrer">author's page</a> (<a href="http://dx.doi.org/10.1145/3236781" rel="nofollow noreferrer">doi</a>), section 2.4, an adjunction is described using the signature:</p>
<p><span class="math-... | David K | 139,123 | <blockquote>
<p>Trigonometric ratios all apply only to a right angled triangle</p>
</blockquote>
<p>This statement is not historically accurate.
Historically, trigonometric ratios came from a circle.
The "triangle" part was a later idea.
See <a href="https://math.stackexchange.com/a/2420718">this answer</a> for more... |
2,368,453 | <p>I am trying to derive a proof of the associative property of addition of complex numbers using only the properties of real numbers.</p>
<p>I found the following answer but was hoping someone can explain why it is correct, since I am not satisfied with it (From <a href="https://math.stackexchange.com/questions/10786... | Bill Dubuque | 242 | <p>Below I've annotated the proof with justifications for each equality.</p>
<p><span class="math-container">$$\begin{aligned}
&\quad\ \ z_1 + (z_2 + z_3)\\[.2em]
&= (a + bi) + [(c+di) + (e +fi)]\quad \:\!\text{by definition of a complex number} \\[.2em]
&= (a+bi)+[(c+e)+(d+f)i]\quad\,\ \text{by definitio... |
692,944 | <p>Let $\beta $ be an ordinal such that for all $\gamma $ $2^{\aleph_{\gamma}}$ =
$\aleph_{\gamma + \beta}$. Does $\beta $ have to be infinity?. Under the continuum hypothesis, it is true, let 0= $\gamma $ and then we have an absurdity. But in ZFC, without CH, what happens? I tried to see that for if $\gamma $ is a n... | Unwisdom | 124,220 | <p>The generalized continuum hypothesis (GCH) states that for all $\gamma$:
$$2^{\aleph_{\gamma}}=\aleph_{\gamma+1}.$$
Thus $\beta$ clearly need not be infinite! </p>
<p>Since you can construct a model of ZFC where $2^{\aleph_{0}}=\aleph_{1}$ but $2^{\aleph_{1}}=\aleph_{3}$, it can also be the case that such a $\beta$... |
3,484,483 | <p>I would please like your guidance to find if the series <span class="math-container">$$\sum_{n=1}^{\infty}\left(\frac{1}{n^{2/3}}-\frac{7}{n^{3/2}}\right)$$</span> converges or diverges?</p>
<p>I noticed that We have two separate <span class="math-container">$p$</span>-series and <span class="math-container">$p<... | Simply Beautiful Art | 272,831 | <p>Since <span class="math-container">$\sum\frac7{n^{3/2}}$</span> converges absolutely, it is possible to remove it from the series. This leaves the diverging series <span class="math-container">$\sum\frac1{n^{2/3}}$</span>, so your series diverges.</p>
<p>Note however that it is not possible to split diverging serie... |
720,924 | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| Warbo | 121,746 | <p>It is similar to why SI units are used for scientific calculations.</p>
<p>Many people will notice some quantity which they want to measure, for example length or angle. They will come up with repeatable ways of measuring these, designed to be easy to perform. For example the cubit and the foot are based on body pa... |
720,924 | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| orion | 137,195 | <p>Another perspective, in my opinion the most "proper" one:</p>
<p>degree is just a named numerical constant that equals ${}^\circ=\frac{\pi}{180}$. So when you read $180^\circ$ you are actually multiplying by that constant! In a similar fashion, conversion into degrees just means that you divide & multiply by a ... |
3,498,985 | <blockquote>
<p>Find total number of distinct <span class="math-container">$x\in[0,1]$</span> for which <span class="math-container">$$\int_{0}^{x}\frac{t^2}{1+t^4}dt=2x-1$$</span></p>
</blockquote>
<p>My multiple attempts are as follows:-</p>
<p><strong>Attempt <span class="math-container">$1$</span>:</strong></p>... | Andronicus | 528,171 | <p><strong>Hint</strong>: Notice, that:</p>
<p><span class="math-container">$$\int\frac{x^2dx}{x^4+1}=
\frac{1}{2}\int\frac{2x}{x^4+1}xdx$$</span></p>
<p>Now knowing, that <span class="math-container">$\frac{d}{dx}x=1$</span> and <span class="math-container">$\frac{d}{dx}\arctan{x^2}=\frac{2x}{x^4+1}$</span>, we can ... |
3,498,985 | <blockquote>
<p>Find total number of distinct <span class="math-container">$x\in[0,1]$</span> for which <span class="math-container">$$\int_{0}^{x}\frac{t^2}{1+t^4}dt=2x-1$$</span></p>
</blockquote>
<p>My multiple attempts are as follows:-</p>
<p><strong>Attempt <span class="math-container">$1$</span>:</strong></p>... | Peter Szilas | 408,605 | <p><span class="math-container">$x \in [0,1]$</span>;</p>
<p><span class="math-container">$F(x):=\displaystyle{\int_{0}^{x}}\dfrac{t^2}{1+t^4}dt -2x+1$</span>;</p>
<p>1) <span class="math-container">$F'(x)=\dfrac{x^2}{1+x^4} -2 < x^2-2<$</span></p>
<p><span class="math-container">$1-2<0$</span>;</p>
<p>Sin... |
3,024,496 | <p>I have this determinant which looks like a Vandermonde matrix</p>
<p><span class="math-container">$$D=\begin{vmatrix}1& a_1 & \cdots & a_1^{n-2}& a_1^n\\
1& a_2 & \cdots & a_2^{n-2}& a_2^n\\
\vdots &\vdots & \ddots & \vdots & \vdots\\
1& a_n & \cdots & a_n... | darij grinberg | 586 | <p>Two proofs:</p>
<ol>
<li><p>If I flip the order of the columns of your matrix, I obtain the matrix
<span class="math-container">\begin{equation}
\begin{pmatrix}
a_1^n & a_1^{n-2} & \cdots & a_1 & 1 \\
a_2^n & a_2^{n-2} & \cdots & a_2 & 1 \\
\vdots & \vdots & \ddots & \vdo... |
2,766,332 | <blockquote>
<p>If $a,b$ are elements of a group and $a^2=e, b^6=e, ab=b^4a$, then find the order of $ab$ and express ${(ab)}^{-1}$ in terms of $a^mb^n$ and $b^ma^n$ </p>
</blockquote>
<p>I could find the order of $ab$ to be 6 but struggling to find ${(ab)}^{-1}$ in terms of $a^mb^n$ and $b^ma^n$.</p>
<p>Please hel... | Barry Cipra | 86,747 | <p>As indicated in hints beneath the OP,</p>
<p>$$(ab)^{-1}=b^{-1}a^{-1}=b^5a$$</p>
<p>which gives the inverse in the form $b^ma^n$. To get it in the form $a^mb^n$, use the equation $ab=b^4a$ to obtain</p>
<p>$$(ab)^{-1}=(b^4a)^{-1}=a^{-1}b^{-4}=ab^2$$</p>
<p><strong>Added later</strong>: It's worth noting that $(... |
1,048,045 | <blockquote>
<p>$$\int_0^{\frac{\pi}{2}}\arctan\left(\sin x\right)dx$$</p>
</blockquote>
<p>I try to solve it, but failed. Who can help me to find it?</p>
<p>I encountered this integral when trying to solve $\displaystyle{\int_0^\pi\frac{x\cos(x)}{1+\sin^2(x)}\,dx}$.</p>
| robjohn | 13,854 | <p>Integrating by parts, we get
$$
\int_0^{\pi/2}\sin^{2k+1}(x)\,\mathrm{d}x
=\frac{2k}{2k+1}\int_0^{\pi/2}\sin^{2k-1}(x)\,\mathrm{d}x\tag{1}
$$
Therefore, by induction, we have
$$
\int_0^{\pi/2}\sin^{2k+1}(x)\,\mathrm{d}x
=\frac{2^k\,k!}{(2k+1)!!}\tag{2}
$$
Thus,
$$
\begin{align}
\int_0^{\pi/2}\arctan(\sin(x))\,\mathr... |
236,055 | <p>I would like to extract data of surface properties and grain boundary properties for Cu polymorph #0 from the website: <a href="http://crystalium.materialsvirtuallab.org" rel="nofollow noreferrer">http://crystalium.materialsvirtuallab.org</a> using Mathematica.</p>
<p>So far I tried with <code>Import</code>, but it ... | Jean-Pierre | 72,509 | <p>This url contains raw data:</p>
<pre><code>Import["http://crystalium.materialsvirtuallab.org/crystallium/data/Cu"]
</code></pre>
<p>To access the html of the new page, here is a variation of the method provided in a previous answer, targeting polymorph #0 info. Note that I have added a pause to allow time ... |
249,623 | <p>This problem comes from the response of the author of papers. </p>
<p>Consider two convex bodies $A$ and $B$: </p>
<p>$$A= \{X\in \mathcal{S}^4 : \operatorname{tr}(X) = 1, X\succeq 0 \}$$<br>
$$B = \operatorname{conv} SO(3)$$ </p>
<ol>
<li>$\mathcal{S}^4$ is the set of symmetric $4\times 4$ matrices. </l... | Fedor Petrov | 4,312 | <p>Consider a 3d-rotation with respect to the axis generated by the unit vector $(a,b,c)$ to the angle $\theta$. Its matrix is
$$
M=\pmatrix{\cos \theta+a^2(1-\cos\theta)&ab(1-\cos\theta)-c\sin\theta&
ac(1-\cos\theta)+b\sin\theta
\\ab(1-\cos\theta)+c\sin\theta&\cos \theta+b^2(1-\cos\theta)&bc(1-\cos\the... |
3,832,067 | <p>I recently encountered this question:</p>
<p><strong>The probability that the bulb will work longer than 800 hours is <span class="math-container">$0.2$</span>. We have three bulbs in the hallway. What is the probability that after 800 hours of service at least one of them will still work ?</strong></p>
<p>I underst... | Community | -1 | <p>You should be familiar with binomal distribution.</p>
<p>Let X be the number of lightbulbs still work after 800 hours.</p>
<p><span class="math-container">$P(X \geq 2) = 1 - P(X = 0) - P(X = 1)$</span></p>
<p>So what is the probability that no light bulb works after 800 hours? Easy. It's <span class="math-container"... |
916,120 | <p>What is negation of <strong>All birds can fly.</strong></p>
<p>The question seems bit funny but i don't know which of the following two sentences is correct:</p>
<ol>
<li>Some birds can not fly</li>
<li>There is at least one bird which can not fly.</li>
</ol>
<p>Both the sentence seems almost logically same. But ... | Alice Ryhl | 132,791 | <p>I would say only <strong>2</strong> is the negation of All birds can fly.</p>
<p>All birds can fly is only true if <strong>all</strong> birds can do it.</p>
<p>So if <strong>not all</strong> birds can fly it would be a negation.</p>
<p>There is at least one bird which can not fly, is equal to not all birds can fl... |
916,120 | <p>What is negation of <strong>All birds can fly.</strong></p>
<p>The question seems bit funny but i don't know which of the following two sentences is correct:</p>
<ol>
<li>Some birds can not fly</li>
<li>There is at least one bird which can not fly.</li>
</ol>
<p>Both the sentence seems almost logically same. But ... | Robert Israel | 8,508 | <p>If "some birds" is synonymous with "at least one bird", the two are equivalent. Whether the plural form implies "at least two" rather than "at least one" may be debatable, but this is a question of English rather than mathematics. "There is at least one bird which can not fly" is unambiguous.</p>
|
394,101 | <p>I have an idea for a website that could improve some well-known difficulties around peer review system and "hidden knowledge" in mathematics. It seems like a low hanging fruit that many people must've thought about before. My question is two-fold:</p>
<p><em>Has someone already tried this? If not, who in t... | Carlo Beenakker | 11,260 | <ul>
<li><A HREF="https://selectedpapers.net" rel="noreferrer">SelectedPapers</A> was created "as a space where academics could read, share, and give feedback on articles and papers related to their field.". It is no longer in operation.</li>
<li><A HREF="https://scirate.com" rel="noreferrer">Scirate</A> is a... |
394,101 | <p>I have an idea for a website that could improve some well-known difficulties around peer review system and "hidden knowledge" in mathematics. It seems like a low hanging fruit that many people must've thought about before. My question is two-fold:</p>
<p><em>Has someone already tried this? If not, who in t... | Matthieu Latapy | 158,328 | <p>It seems to me that <a href="https://papers-gamma.link/" rel="noreferrer">Papers<span class="math-container">$^\gamma$</span></a> is close to what you describe: people can upload a paper, inform its authors, and publicly (anonymously or not) discuss it.</p>
<p>I would also like to mention <a href="https://peercommun... |
1,971,645 | <p>I tried doing this problem two ways. I am unable to get the solutions to match each other. Is one of them incorrect?</p>
<p><a href="https://i.stack.imgur.com/5vHwDxx.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5vHwDxx.jpg" alt="enter image description here"></a></p>
| marty cohen | 13,079 | <p>$\begin{array}\\
y+\sqrt{xy}=1\\
\sqrt{xy}=1-y\\
xy = (1-y)^2\\
xy'+y = -2y'(1-y)\\
y = -y'(2-2y+x)\\
y' = \dfrac{-y}{2-2y+x)}\\
\end{array}
$</p>
<p>To get $y$ explicitly,
$xy = (1-y)^2
= 1-2y+y^2
$
so
$y^2-(2+x)y+1 = 0$.</p>
<p>Solving
$y
=\dfrac{2+x\pm\sqrt{x^2+4x+4-4}}{2}
=\dfrac{2+x\pm\sqrt{x^2+4x}}{2}
$.</p>... |
347,494 | <p>I have a question regarding differential forms.</p>
<p>Let $\omega = dx_1\wedge dx_2$. What would $d\omega$ equal? Would it be 0?</p>
| Jim Belk | 1,726 | <p>Yes. The same holds true for any differential form whose coefficients are constant functions. For example, if $\omega = 3(dx\land dy) + 5(dx\land dz) + 7 (dy\land dz)$, then $d\omega = 0$.</p>
<p><strong>Edit:</strong> In general, the exterior derivative is defined by
$$
d\bigl(f\, dx_{i_1}\land\cdots\land dx_{i... |
2,529,616 | <p>$ M= \left[ {
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 1\\
0 & 1 & 1 & 1 & 0\\
0 & 1 & 1 & 1 & 0\\
0 & 1 & 1 & 1 & 0\\
1 & 0 & 0 & 0 & 1\\
\end{array} } \right] $</p>
<p>What is the product of positive eigen values for the above matrix... | Guy Fsone | 385,707 | <blockquote>
<p>Obviously $M$ has rank two. Therefore M has only two non-negative eigenvalues(<strong>count with multplicity</strong>). </p>
</blockquote>
<p>It is easy to check that $v_2 =(1,0,0,0,1)^T$ is an eigenvector associate to the eigenvalue 2. and $v_3 =(0,1,1,1,0)^T$ is an eigenvector associate to the eig... |
3,764,846 | <blockquote>
<p>A martingale <span class="math-container">$\{X_n\}$</span> is bounded in <span class="math-container">$L^2$</span> by definition if <span class="math-container">$\sup\limits_nEX_n^2<\infty$</span>. Show that a martingale <span class="math-container">$\{X_n\}$</span> is bounded in <span class="math-co... | triple_sec | 87,778 | <p><strong>Claim:</strong> Suppose that <span class="math-container">$(X_n)_{n\in\mathbb N}$</span> is a square-integrable martingale adapted to the <span class="math-container">$\sigma$</span>-algebras <span class="math-container">$(\mathscr F_n)_{n\in\mathbb N}$</span>. Then, for any <span class="math-container">$n\i... |
278,528 | <p>Let's suppose we have the following structure of data</p>
<pre><code>data = {{1}, {0.0109, -12.7758, -0.00980164, 0.00032368},
{1.0218, -12.7764, -0.00948724, 0.00064337},
{2.0327, -12.7772, -0.00905215, 0.00095516},
{2}, {0.0109, -12.7758, -0.00980164, 0.00032368},
... | E. Chan-López | 53,427 | <p>Using <code>Table</code>, <code>If</code> and <code>Or</code>:</p>
<pre><code>Table[If[Length[data[[i]]] == 1, data[[i]], Or[If[Length[data[[i]]] == 4
&& First[data[[i]]] < 2, Rest@data[[i]], Nothing]]], {i, 1, Length[data]}]
</code></pre>
<p>Using <code>Table</code>and <code>Which</code>:</p>
<pre><code... |
2,197,576 | <p>$(p\land q)\rightarrow r$ and $(p\rightarrow r)\lor (q\rightarrow r)$</p>
<p>Have to try prove if they are logically equivalent or not using the laws listed below and also if need to use negation and implication laws. I was going to use associative law and then distributive but I wasn't sure how to get rid of the "... | Bram28 | 256,001 | <p>The two statements are not equivalent.</p>
<p>Obviously you cannot use equivalence principles to demonstrate non-equivalence, so let's use a counterexample:</p>
<p>Let $p=True$, $q =False$, and $r=False$</p>
<p>then $(p \land q) \rightarrow r = (T\land F) \rightarrow F = F \rightarrow F = T$</p>
<p>But $(p \righ... |
2,197,576 | <p>$(p\land q)\rightarrow r$ and $(p\rightarrow r)\lor (q\rightarrow r)$</p>
<p>Have to try prove if they are logically equivalent or not using the laws listed below and also if need to use negation and implication laws. I was going to use associative law and then distributive but I wasn't sure how to get rid of the "... | BrunoR | 590,790 | <p>You are missing a equivalence:</p>
<pre><code>1. p → q ≡ ¬p v q
</code></pre>
<p>I find it easier to work on the right side:</p>
<pre><code> (p ∧ q)→r ≡ (p → r) ∨ (q → r)
</code></pre>
<p>Taking only the right side:</p>
<pre><code>(p → r) ∨ (q → r) (Using 1.)
(¬p v r) ∨ (¬q v r) (Commutative laws on the ri... |
3,249,064 | <p>I've read the question: "Why does the derivative of sine only work for radians?" and I can follow the derivation for the derivative of sine when measured in degrees, but the result confuses me. </p>
<p>Does this mean the derivative of the sine changes values when measured in different units? </p>
<p>For example, w... | Ethan Bolker | 72,858 | <p>Several other answers are correct, and appropriate for first year calculus. There is another way to see why from a more advanced perspective.</p>
<p>One way to think about the number <span class="math-container">$e$</span> is to consider the exponential functions
<span class="math-container">$$
f(x) = a^x
$$</span... |
3,603,281 | <blockquote>
<p>A cubic equation <span class="math-container">$x^{3}+ax^2+bx+c$</span> has all negative real roots and <span class="math-container">$a, b, c\in R$</span> with <span class="math-container">$a<3.$</span></p>
<p>Prove that <span class="math-container">$b+c<4.$</span></p>
</blockquote>
<p>My attempt :... | nonuser | 463,553 | <p>Let <span class="math-container">$-p,-r,-q$</span> be it roots, so <span class="math-container">$p,q,r>0$</span>. Now by Vitea formulas we have <span class="math-container">$$p+r+q =a < 3$$</span> and <span class="math-container">$$b+c = pr+pq+qr +qpr$$</span></p>
<p>So we need to check if <span class="math-... |
1,639,241 | <p>I'm using gradient descent with mean squared error as error function to do linear regression. Take a look at the equations first.
<a href="https://i.stack.imgur.com/GN90y.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GN90y.png" alt="enter image description here"></a>
As you can see in eq.1, the... | Clement C. | 75,808 | <p>The $Q$ is a parameter, and $q$ is a variable ranging from $0$ to $Q$: basically, you have $Q+1$ parameters $\textrm{ceps}_0,\dots, \textrm{ceps}_Q$; or, in programming terms, you have an array $\textrm{ceps}[0\dots Q]$.</p>
<p>Similarly, the LPC coefficients are a list of $p$ values $a_1,\dots, a_p$ (i.e., $a_q$ f... |
991,038 | <p>Consider a probability space $\left(\Omega,\mathcal{F},P\right)
$, and a sub-sigma-algebra $\mathcal{G}\subseteq\mathcal{F}
$. As usual, let $L^{2}\left(\Omega,\mathcal{F},P\right)
$ be the space of $\mathcal{F}
$-measurable, square integrable random variable and $L^{\infty}\left(\Omega,\mathcal{G},P\right)
$ b... | Community | -1 | <p>In a ring $(R,+,\times)$ we have $(R,+)$ is an abelian group where one of his axioms is: each element $a$ in this group has a symmetric element denoted by $-a$ such that
$$a+(-a)=-a+a=0$$
hence </p>
<p>$$a+b=a+c\implies -a+a+b=-a+a+c\implies b=c$$</p>
|
991,038 | <p>Consider a probability space $\left(\Omega,\mathcal{F},P\right)
$, and a sub-sigma-algebra $\mathcal{G}\subseteq\mathcal{F}
$. As usual, let $L^{2}\left(\Omega,\mathcal{F},P\right)
$ be the space of $\mathcal{F}
$-measurable, square integrable random variable and $L^{\infty}\left(\Omega,\mathcal{G},P\right)
$ b... | rschwieb | 29,335 | <blockquote>
<p>"why can we add the same thing on both side of the equation it still holds?"</p>
</blockquote>
<p>Because addition is a <strong>function</strong> from $R\times R\to R$.</p>
<p>In particular the restriction to $\{x\}\times R\to R$ Is a function, and actually you can view this as a function from $R\to... |
3,378,754 | <p>if <span class="math-container">$A=1$</span>, <span class="math-container">$y \sim N(1,\sigma^2)$</span></p>
<p>if <span class="math-container">$A=2$</span>, <span class="math-container">$y \sim N(2,\sigma^2)$</span></p>
<p><span class="math-container">$Pr(A=1)=0.5$</span></p>
<p><span class="math-container">$Pr(... | fGDu94 | 658,818 | <p>The disproof is correct and the statement is false.</p>
<p>In fact, <span class="math-container">$f(A) + f(B) = f(A+B)$</span> would hold and im sure you could immediately see this</p>
|
292,594 | <p>Today I have encounter an integral:</p>
<p>$$\int_0^{\infty}\left[\frac{1}{3}\frac{\sin x}{x}+\cdots+\frac{1\times4\times\cdots\times(3n-2)}{3^nn!}\left(\frac{\sin x}{x}\right)^n+\cdots\right]\text{d}x$$</p>
<p>since $$\int_0^{\infty}\frac{\sin x}{x}=\frac{\pi}{2}$$</p>
<p>so I want to estimate $$\sum_{n=1}^{\inf... | Community | -1 | <p>$$a_n = \prod_{k=1}^n \left(1 - \dfrac2{3k}\right) = \dfrac{1 \times 4 \times 7 \times 10 \times \cdots \times (3n-2)}{3 \times 6 \times 9 \times 12 \times \cdots \times 3n} = \dfrac{\Gamma(n+1/3)}{\Gamma(1/3) \Gamma(n+1)}$$
Further the asymptotic for $\Gamma(z)$ is given by
$$\Gamma(z+1) \sim \sqrt{2 \pi z} \left(\... |
292,594 | <p>Today I have encounter an integral:</p>
<p>$$\int_0^{\infty}\left[\frac{1}{3}\frac{\sin x}{x}+\cdots+\frac{1\times4\times\cdots\times(3n-2)}{3^nn!}\left(\frac{\sin x}{x}\right)^n+\cdots\right]\text{d}x$$</p>
<p>since $$\int_0^{\infty}\frac{\sin x}{x}=\frac{\pi}{2}$$</p>
<p>so I want to estimate $$\sum_{n=1}^{\inf... | Did | 6,179 | <p><a href="https://math.stackexchange.com/a/292602">N.S.'s idea</a> can be adapted to yield the exact asymptotics. To wit,
$$
\prod_{k=1}^n\left(1-\frac{2}{3k}\right)=\frac13\prod_{k=2}^n\left(1-\frac{2}{3k}\right)\geqslant\frac13\prod_{k=2}^n\left(\frac{k-1}k\right)^{2/3}=\frac1{3n^{2/3}}.
$$
This uses the fact that,... |
4,016,289 | <p>Let's consider the parametric integral:</p>
<p><span class="math-container">$F:\mathbb{R}\to\mathbb{R}$</span>, where <span class="math-container">$F(x):=\int\limits_0^1 \frac{1}{x}\left(e^{-x^2(1+t)}(1+t)^{-1}-(1+t)^{-1}\right)dt$</span>.</p>
<p>What is the value of the limit: <span class="math-container">$$\lim\li... | Community | -1 | <p>In your second form, although equivalent, you are in fact dividing by <span class="math-container">$0$</span> which you absolutely do not want to do! So you cannot accurately compute the limit as <span class="math-container">$x\rightarrow 0$</span> of <span class="math-container">$\frac{k}{x}$</span>, as you have do... |
4,016,289 | <p>Let's consider the parametric integral:</p>
<p><span class="math-container">$F:\mathbb{R}\to\mathbb{R}$</span>, where <span class="math-container">$F(x):=\int\limits_0^1 \frac{1}{x}\left(e^{-x^2(1+t)}(1+t)^{-1}-(1+t)^{-1}\right)dt$</span>.</p>
<p>What is the value of the limit: <span class="math-container">$$\lim\li... | Rishabh | 883,872 | <p>You have made an error of not converting the limit value when changing the function. Have a look at the equation below to get a better understanding of the flaw in your logic.</p>
<p><span class="math-container">$ Y = \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{x^2+x+1}{2x^2-3} = \lim_{x \to 0} \frac{1+1/x+1/x^2}{2-3... |
1,192,434 | <p>I have a problem with integrating of fraction
$$
\int \frac{x}{x^2 + 7 + \sqrt{x^2 + 7}}
$$
I have tried to rewrite it as $\int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{x^4 + 13x^2 + 42} = \int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{(x^2 + 6)(x^2 + 7)}$ and then find some partial fractions, but it wasn't succesful.</p>
| Tim Raczkowski | 192,581 | <p>Hint: Each term of $A_n$ can be expressed in terms of geometric series.</p>
|
514,702 | <p>I know that this is true and is used to prove that $\mathbb{Q}$ is not a discrete metric space, but I can't figure out, why is it true ?</p>
| Michael Albanese | 39,599 | <p>As you are viewing $\mathbb{Q}$ as a metric space, I assume with the metric $d(x, y) = |x-y|$, $A \subseteq \mathbb{Q}$ is open if for every $a \in A$, there is $r > 0$ such that $B(a, r) \subseteq A$. So $\{a\}$ is open if there is $r > 0$ such that $B(a, r) \subseteq \{a\}$; actually, as $\{a\} \subseteq B(a... |
33,369 | <p>I've been trying to get to grips with the various semantics commonly discussed in formal logic. Specifically, the nature and role of interpretations of first and higher-order logics is slightly unclear.</p>
<p>Here are a few of the specific questions that have occurred to me:</p>
<ul>
<li>Propositional logic only ... | Stefan Geschke | 7,743 | <p>I just say something about your fourth question. In many-sorted logic you have a partition of the domain of your structure into different sorts. An example would be the disjoint union of a vector space over a field $K$ and the field $K$ itself. Both sorts (vectors and scalars)
carry their usual structure (abelia... |
315,697 | <p>Let <span class="math-container">$X$</span> be an irreducible normal projective scheme over <span class="math-container">$\mathbb{C}$</span>. Let <span class="math-container">$U$</span> be the open subscheme of smooth points of <span class="math-container">$X$</span>. Consider the closed subscheme <span class="math-... | Francesco Polizzi | 7,460 | <p>Let me expand my comment into an answer. </p>
<p>Take as <span class="math-container">$X$</span> the cone of vertex <span class="math-container">$v$</span> over an elliptic curve <span class="math-container">$E$</span>. Then <span class="math-container">$X$</span> is simply connected (this is a general property of ... |
2,008,656 | <p>$\lim_{x \to \infty }\sqrt[x]{a^{x}+b^{x}+c^{x}} $;</p>
<p>$a,b,c\in \mathbb{R}$.</p>
<p>I need to find and prove a limit of this sequence. </p>
<p>I know that for example the limit of $a^{x}$ is $\infty$ for $a>1$. And the limit of $\sqrt[x]{a^{x}+b^{x}+c^{x}}$ should be equal $\sqrt[x]{\lim_{x \to \infty }{... | egreg | 62,967 | <p>You can't pull the root outside the limit. And I don't think you can use arbitrary reals; if one of them is negative, even with $x$ running on the the positive integers you'll have big problems to give a meaning to the sequence.</p>
<p>On the other hand, when $x$ is used it is usually understood we're dealing with ... |
2,948,862 | <p>So, I took one intro course in Tensor calculus and this problem reminds of that, except I can't quite recall how derivatives work with respect to components, or what those derivatives produce. Consider the following example: </p>
<p><a href="https://i.stack.imgur.com/KhN1F.png" rel="nofollow noreferrer"><img src="h... | operatorerror | 210,391 | <p>You are right about what the kernel is, however, is it not finite? What's wrong with the basis consisting of the constant function <span class="math-container">$f(x)=1$</span>?</p>
<p>Finding the eigenvalues works the same way. The ODE
<span class="math-container">$$
Df=\lambda f\iff f'=\lambda f
$$</span>
might b... |
65,923 | <p>The <a href="http://en.wikipedia.org/wiki/Parity_of_a_permutation">sign of a permutation</a> $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula</p>
<p>$${\rm sgn}(\sigma) = (-1)^m$$</p>
<p>where $m$ is the number ... | Geoffrey Irving | 38,218 | <p>It's worth mentioning the quadratic time algorithm, since it can be faster for small permutations:</p>
<p>$$\textrm{sgn}(\sigma) = (-1)^{\sum_{0 \le i<j<n} (\sigma_i>\sigma_j)}$$</p>
<p>I.e., the sign is negative iff there are an odd number of misordered pairs of indices. This algorithm also works if we'... |
65,923 | <p>The <a href="http://en.wikipedia.org/wiki/Parity_of_a_permutation">sign of a permutation</a> $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula</p>
<p>$${\rm sgn}(\sigma) = (-1)^m$$</p>
<p>where $m$ is the number ... | FUZxxl | 5,282 | <p>Use the Fisher-Yates-Shuffle in its <a href="https://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle#The_.22inside-out.22_algorithm" rel="nofollow noreferrer">inside-out version</a> to generate a permutation of the numbers $1\dots16$. You can find the parity of this permutation by counting the number of transposi... |
3,194,984 | <p>I got something rather new and I just wanted to make sure my way of thinking in this field is fine. Suppose <span class="math-container">$$X\sim Bin(25,0.61)$$</span> and we are asked to find: <span class="math-container">$E[X^2]$</span>.
So basically I treat this binomial variable as a sum of 25 Bernoulli variables... | user231 | 506,022 | <p>We cannot simply ignore the square, but you have a part of the answer. If we have <span class="math-container">$X = \sum_{i=1}^{25}X_{i}$</span> where <span class="math-container">$X_{i} \sim Ber(0.61)$</span>, then</p>
<p><span class="math-container">$$X^2 = (\sum_{i=1}^{25}X_{i})^2 = \sum_{i=1}^{25}X_{i}^2 + 2\s... |
4,253,330 | <p>Show that the sequence of functions <span class="math-container">$\langle f_n \rangle$</span> defined by <span class="math-container">$$f_n(x) = \frac{nx}{nx+1} \, ; \, n \in \mathbb{N}$$</span> fails to converge uniformly on <span class="math-container">$[0, \infty) .$</span></p>
<hr />
<p>The name of the game here... | MXXZ | 966,405 | <p>Another approach with the <a href="https://www.wikiwand.com/en/Uniform_limit_theorem" rel="nofollow noreferrer">uniform limit theorem</a>:</p>
<p>Assume that <span class="math-container">$f_n$</span> does converge uniformly against <span class="math-container">$f$</span>. It is obvious that each <span class="math-co... |
3,790,726 | <p><span class="math-container">$\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$</span></p>
<hr />
<p>By maths calculator it results 1.
I calculate and results <span class="math-container">$\sqrt{-\frac{1}{2}}$</span>.</p>
<p><span class="math-container">$\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$</span>... | Novice | 702,478 | <p><span class="math-container">$\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}=\sqrt{\frac{3}{4}+{\frac14}}=\sqrt{1}=1.$</span> <br />
Your problem was with not powering the minus in the first expression in numerator.<br />
See the difference between <span class="math-container">$$(-\sqrt3)^2=3$$</span> and <spa... |
395,118 | <p>In a city of over $1000000$ residents, $14\%$ of the residents are senior citizens. In a simple random sample of $1200$ residents, there is about a $95\%$ chance that the percent of senior citizens is in the interval [pick the best option; even if you can provide a sharper answer than you see in the choices, please ... | Abel | 71,157 | <p>In case you don't like the $r=-1$ approach, here is a direct proof.</p>
<p>First write $\displaystyle\ln\frac{1}{y} = \int_1^\frac{1}{y}\frac{1}{t}\,\mathrm{d}t = -\int_\frac{1}{y}^1\frac{1}{t}\,\mathrm{d}t$. Now we change variables to $s = yt$. Then $\mathrm{d}s = y\mathrm{d}t$ and thus $\displaystyle\ln\frac{1}{y... |
2,735,007 | <p>The Jenkins–Traub algorithm for polynomial zeros is a fast globally convergent iterative method published in 1970 by Michael A. Jenkins and Joseph F. Traub. </p>
<p>The published algorithm, e.g. <a href="https://dl.acm.org/citation.cfm?id=355643" rel="nofollow noreferrer">https://dl.acm.org/citation.cfm?id=355643</... | Jens Schwaiger | 532,419 | <p>If one allows complex solutions also, you could take all but one of the variables constant and use one of the root finding algorithms for polynomial in a single variable over the field of complex numbers.</p>
|
920,732 | <p>I have been commanded on homework to find a non-bijective isomorphism in a category whose objects are sets, whose morphisms are set maps, and composition is the usual function composition. So our category is a subcategory of the category of <strong>Set</strong>. </p>
<p>But, I think that such an isomorphism is in f... | Andreas Blass | 48,510 | <p>Given the hint that the identity morphisms can be weird in ((sets)), we can proceed as follows. Let ((sets)) consist of just the one set $\{0,1\}$ and the one map, from $\{0,1\}$ to itself that sends both 0 and 1 to 0. This one map is then the identity map, composition is as usual, and this identity map is a non-b... |
1,794,459 | <p>We have set $S=\{ (a,b) | a,b \in \mathbb{Q}\}$</p>
<p>And we know that $(a,b)\sim \mathbb{R}$ , so $k((a,b))=c$. And $\mathbb Q \sim \mathbb N$, so $k(\mathbb Q)=\aleph_0$.</p>
<p>I don't know how to put all informations together.</p>
<p>I was also thinking that $S \subset \mathbb R$ and than we know $k(S) \le k... | Laurent Duval | 257,503 | <p>Let us start with a simple example. The set of intervals $\{(0,1),(1,2),(2,3)\}$ contains only three intervals. You can easily define a bijection with the set of three integers: $\{0,1,2\}$, with cardinal $3$:
$$(0,1) \leftrightarrow 0$$
$$(1,2) \leftrightarrow 1$$
$$(2,3) \leftrightarrow 2$$
No matter if $(0,1)$ is... |
2,148,389 | <p>Consider the equation $x^2 - 5x + 6 = 0$. By factorising I get $(x-3)(x-2) = 0$. Which means it represents a pair of straight lines, namely $x-2 =0 $ and $x- 3 = 0$, but when I plot $x^2 - 5x + 6 = 0$, I get a parabola, not a pair of straight lines. Why?</p>
<p>Plotting: <code>x^2 - 5x + 6 = 0</code> </p>
<p>at <... | mvw | 86,776 | <blockquote>
<p>I get $(x-3)(x-2) = 0$. which means it represent a pair of striaght
lines namely $x-2 =0 $ and $x- 3 = 0$</p>
</blockquote>
<p>When we just write
$$
(x-3)(x-2) = 0
$$
and ask for $x$ we usually mean the set of solutions
$$
S = \{ x \mid (x-3)(x-2) = 0 \}
$$</p>
<p>Two vertical lines in two dimensi... |
2,148,389 | <p>Consider the equation $x^2 - 5x + 6 = 0$. By factorising I get $(x-3)(x-2) = 0$. Which means it represents a pair of straight lines, namely $x-2 =0 $ and $x- 3 = 0$, but when I plot $x^2 - 5x + 6 = 0$, I get a parabola, not a pair of straight lines. Why?</p>
<p>Plotting: <code>x^2 - 5x + 6 = 0</code> </p>
<p>at <... | Jeppe Stig Nielsen | 70,134 | <p>To get from the equation
$$x^2 - 5x + 6 = 0$$
to a <em>set</em> of solutions, one must do some interpretation. Namely, in what universe or "base set" do we search the solutions?</p>
<p>If this is to be interpreted as $\{ (x,y)\in\mathbb{R}^2 \mid x^2 - 5x + 6 = 0 \}$, then the solution set is indeed two vertical li... |
4,479,797 | <p>Let's say I have an urn with 10 unique objects, and I choose 3 objects from it (each choice is made without replacement). Then the probability of choosing any one object is 3/10. I calculated this probability by summing the probability the object is chosen on the first pick + probability chosen on second pick + prob... | Vedant Chourey | 638,765 | <p>Another simplification can be done for the term
<span class="math-container">$$ \int \frac{y^2}{1+y^4} dy$$</span>
As
Divde by <span class="math-container">$y^2$</span>
<span class="math-container">$$\int \frac{dy}{y^2 + \frac{1}{y^2}}$$</span>
Following by
<span class="math-container">$$ \frac{1}{2}\int \frac{2+\fr... |
3,516,385 | <p>I need to apply a convergency test to</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{n^4}{n!}=\sum_{n=1}^\infty \frac{n^3}{(n-1)!}$$</span></p>
<p>I can't seem to figure if any comparison test apply; those that I tried gave no useful information. Any ideas?</p>
| José Carlos Santos | 446,262 | <p>It is much more natural to use the quotient test here:<span class="math-container">$$\frac{\frac{(n+1)^4}{(n+1)!}}{\frac{n^4}{n!}}=\frac{(n+1)^4}{(n+1)n^4}=\frac{(n+1)^3}{n^4}\to0.$$</span></p>
|
3,516,385 | <p>I need to apply a convergency test to</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{n^4}{n!}=\sum_{n=1}^\infty \frac{n^3}{(n-1)!}$$</span></p>
<p>I can't seem to figure if any comparison test apply; those that I tried gave no useful information. Any ideas?</p>
| Community | -1 | <p><strong>Hint:</strong></p>
<p>You can get rid of the numerator with <span class="math-container">$$n^3=(n-1)(n-2)(n-3)+6(n-1)(n-2)+7(n-1)+1$$</span></p>
<p>and your sum has the general term </p>
<p><span class="math-container">$$\frac1{(n-4)!}+6\frac1{(n-3)!}+7\frac1{(n-2)!}+\frac1{(n-1)!}.$$</span></p>
<p>You s... |
3,847,358 | <p>If <span class="math-container">$\lim \limits_{n \to \infty} x_n + x_{n+1} =0 $</span> is <span class="math-container">$\lim \limits_{n \to \infty} \frac{x_n}{n}=0$</span>?</p>
<p>If <span class="math-container">$\lim \limits_{n \to \infty} x_n - x_{n+1} =0 $</span> then I would say <span class="math-container">$x_n... | johnnyb | 298,360 | <p>I'm not exactly sure of your question, but you can indeed differentiate a differential. It is a slightly different notation, in that the typical notation for a second derivative doesn't allow for this, but if you adopt a different notation for the second derivative, it works perfectly fine. The typical notation fo... |
2,217,630 | <p>I need to find how many <em>real</em> roots this polynomial has and prove there existence. I was wondering if my logic and thought process was correct.</p>
<blockquote>
<p>Determine the number of <em>real</em> roots and prove it for $x^3 - 3x + 2$</p>
</blockquote>
<p>First, note that $f'(x) = 3x^2 - 3$ and so <... | StackTD | 159,845 | <blockquote>
<p>However, $f(1) = 1 - 3 + 2 = 0$ is clearly a root. And by factorizing the polynomial we get $f(x) = (x+2)(x-1)^2$. Indeed, $1$ is a root with a multiplicity of two. </p>
</blockquote>
<p>All the work you did before this becomes unnecessary; after factoring, the roots (and hence the <em>number of</em>... |
2,281,161 | <p>Munkres say it is of the form $\{x\}\times(a,b) $, but for me it just the intervals of the form $(a,b)×(c,d)$. Can anyone explain?</p>
| William Elliot | 426,203 | <p>Sets of the form $\{x\} \times (a,b)$ are a base.<br>
The open sets are any union of base sets.</p>
|
130,804 | <p>I have this question just out of curiosity.</p>
<p>If X is a scheme, then a morphism $f: X \rightarrow X$ can be the identity on the underlying topological space of X, but not the identity on the structure sheaf. For example, f can be the Frobenius morphism.</p>
<p>Does someone know an example of such a morphism w... | Daniele Zuddas | 23,193 | <p>What about "set with piecewise smooth boundary"?</p>
|
49,679 | <p>I'd like to know whether the following statement is true or not.</p>
<p>Let $T_1, T_2\in \mathbb{C}^{n\times n}$ be upper triangular matrices. If there exists a nonsingular matrix $P$ such that $T_1=PT_2P^{-1}$, then there is a nonsingular uppper triangular matrix $T$ such that $T_1=TT_2T^{-1}$. </p>
| Petya | 2,823 | <p>It is not true. </p>
<p>Let upper triangular matrix $T$ be also a differential, i.e. $T^2=0$. Then, acting by upper triangular conjugation one can get a matrix with at most one $1$ in each row and each column, all other elements equal to zero. Moreover, such a matrix (normal form) is unique in the upper triangular... |
98,698 | <p>I have the time domain signal
$$
u_o(t) = u(t)e^{-t/\tau}\eta(t) + \sigma(t)
$$
where $\tau$ is known, $\eta$ is non-Gaussian noise, and $\sigma$ is Gaussian noise. The distribution of $\eta(t)$ is known, but only numerically. I also have prior knowledge that $u(t)$ is a sum of a small number of sinusoids. How can I... | Joseph O'Rourke | 6,094 | <p>I believe you are looking for the radius of a largest empty ball among your point set, a quantity which goes under the name of <em>dispersion</em>.
This plays a role in robotics algorithms, e.g., <a href="http://planning.cs.uiuc.edu/node203.html" rel="nofollow">LaValle's book</a>.
Here is a survey which might lead t... |
1,102,216 | <p>Let $\mathcal{F}\left[f(t)\right](x)$ be the Fourier Transform of $f$, defined regularly as</p>
<p>$$\mathcal{F}\left[f(t)\right](x)=\int_{-\infty}^{\infty}f(t)e^{-itx}\,dt$$</p>
<p>And let $\mathcal{F}^{-1}\left[g(x)\right](t)$ be the Inverse Fourier Transform of $g$, defined regularly as $$ \mathcal{F}^{-1}\left... | Stella Biderman | 123,230 | <p>Hint: Try composing the two formally and noting that $e^{ixt}e^{-ixt}=1$</p>
|
1,517,557 | <p>How should I prove this theorem? What method of proof should I use?</p>
<p>$ 4 \nmid (n-2)^2 \ \Rightarrow \ 6 \nmid n \ \ \ ,n \in \Bbb Z $</p>
| MadMonty | 145,364 | <p>Hint: Try proving the contrapositive, i.e.</p>
<p>If $6 \mid n$, then $4 \mid (n-2)^2$</p>
|
225,953 | <p>I would like to calculate the maximum number of polynomial terms given a certain number of variables and a certain degree. eg. given that the number of variables is 2 and the degree is 3, the maximum number of terms is 9:
<span class="math-container">$$x_1^3 + x_1^2 x_2 + x_1 x_2^2 + x_2^3+ x_1^2 +x_1 x_2 + x_2^2 + ... | Daniel Porumbel | 76,377 | <p>I found another combinatorial proof that does not use complex sums or the well-known stars and bars technique.</p>
<p>I consider $n$ variables and maximum degree $d$. I will show that the number of monomials of degree less than or equal to $d$ (including the monomial
1 that you excluded from your example) is $${n+... |
10,935 | <p>I'm about to give a first-semester calculus lecture covering the mean value theorem for integrals:</p>
<p>If $f$ is continuous on $[a,b]$, then there is some $c\in(a,b)$ such that $(b-a)f(c)=\int_a^b f(x)\,dx$.</p>
<p>In past semesters, I've shown examples in which I confirm that this theorem holds for some specif... | Dan Fox | 672 | <p>The construction of the delta distribution as a limit of box functions. Consider
$$h_{c}(t) = \begin{cases} 0 & t > |c|,\\ \frac{1}{2c} & t \leq |c|.\end{cases}$$
One wants to motivate defining the delta distribuion as the limit (of the linear functionals given by integrating against) $h_{c}(t)$ when $c \... |
3,434,463 | <p>Euler's identity <span class="math-container">$e^{i\pi}+1=0$</span> has always fascinated me, and at the same time freaks me out a bit. Like, they are two <em>very</em> fundamental constants which seem to have absolutely nothing in common, but still there mysteriously is an immediate mathematical connection between ... | J.G. | 56,861 | <p>The Taylor series of <span class="math-container">$e^x,\,\cos x,\,\sin x$</span> for real <span class="math-container">$x$</span> provide a natural identification of <span class="math-container">$e^{ix}=\cos x+i\sin x$</span>, i.e. the unit complex number of argument <span class="math-container">$x$</span>. So <span... |
3,434,463 | <p>Euler's identity <span class="math-container">$e^{i\pi}+1=0$</span> has always fascinated me, and at the same time freaks me out a bit. Like, they are two <em>very</em> fundamental constants which seem to have absolutely nothing in common, but still there mysteriously is an immediate mathematical connection between ... | David | 791,958 | <p>This is going to be a very soft answer, and I will refer to the bible which will intrigue some and infuriate others, so take what you will.</p>
<p>Consider <span class="math-container">$ e ^ {i\theta} $</span>. The formula suggests a circle since as <span class="math-container">$\theta$</span> varies it traces out a... |
2,243,598 | <p>Consider this integral $(1)$</p>
<blockquote>
<p>$$\int_{0}^{\infty}\color{red}{{\gamma+\ln x\over e^x}}\cdot{1-\cos x\over x}\,\mathrm dx={1\over 2}\cdot{\pi-\ln 4\over 4}\cdot{\pi+\ln 4\over 4}\tag1$$</p>
</blockquote>
<p>Recall a well-known integral for $\gamma$:</p>
<p>$$\int_{0}^{\infty}e^{-x}\ln x\,\mathr... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
112,333 | <p>Consider a set of rules, e.g.</p>
<pre><code>{a -> aa, b -> bb, c -> cc, d -> dd, e -> ee}
</code></pre>
<p>I want to remove from this list all patterns of the form e->_ and to do so, I would like to form the Complement of matching patterns in my original set.</p>
<p>However if I use Cases, the des... | Quantum_Oli | 6,588 | <p>Under the Possible Issues tab of the <code>Cases</code> documentation. </p>
<blockquote>
<p>Use <code>HoldPattern</code> to treat the rule itself as a pattern:</p>
</blockquote>
<pre><code>Cases[{a -> aa, b -> bb, c -> cc, d -> dd, e -> ee}, HoldPattern[e -> ee]]
(*{e -> ee}*)
</code></pre>
|
1,168,862 | <p>In a case such as $x^x=27$ we can solve by inspection that $x=3$.</p>
<p>But how can we solve this algorithmically in general, given that $N\in Q$ where $Q\subset\Bbb Z$ and $Q=[1,10^{10}]$ and $x\in\Bbb R$?</p>
| sav | 70,541 | <p>Newton's method has the fastest convergence that I know of.</p>
<p>For solving 0 = f(x)</p>
<p>The idea is to </p>
<p>1.) guess a solution $x_0$ </p>
<p>2.) put it into the function you want to find the zero solution of (ie: calculate $f(x_0)$ ) </p>
<p>3.) calculate the derivative at that point $f'(x_0)$</p>
... |
496,544 | <p>The question asks:</p>
<p>Find the smallest value (for real $x$ and $y$) of: $$x^4+2x^2+y^4-2y^2+3$$</p>
<p>I don't think I understand this question, it is in a completing the square exercise and I don't really know where to start. I can factorise parts and mess around with it but it does not help at all. Any help... | Sid | 91,603 | <p>$x^4+2x^2+y^4-2y^2+3 = (x^2+1)^2 - 1 + (y^2 -1)^2 - 1 + 3 = (x^2+1)^2 + (y^2 - 1)^2 +1$ which must be minimal when the terms $(x^2+1)^2$ and $(y^2 - 1)^2$ are minimal, you should be able to take it from here.</p>
|
2,540,954 | <p>Let $T$ be a (countable) partition of $X$ and let $\sigma(T)$ be the generated $\sigma$-algebra of our interest.</p>
<p>I'm trying to figure out whether $\sigma(T)$ is a complete lattice?</p>
<p><strong>Def.</strong> a <em>complete lattice</em> is a partially ordered set in which all subsets have both a supremum (... | JayTuma | 506,755 | <p>Since the partition is countable, let's say $X = \bigcup_{n \in \mathbb{N}} S_n$ you can define
$$ \varphi : \sigma(T) \to 2^\mathbb{N} \quad
\varphi(S_n) = \{n\} \quad
\varphi \left(\bigcup_{i \in \mathbb{N}} X_i \right) = \bigcup_{i \in \mathbb{N}} \varphi(X_i) $$</p>
<p>It is easy to see that this functio... |
3,093,099 | <p>If <span class="math-container">$x$</span> is positive, what is the maximum value of this expression:</p>
<p><span class="math-container">$$\frac{x^{100}}{1+x+x^2+\ldots+x^{200}}$$</span></p>
<p>This question is from a book of problems on sequence and series under the section on <strong>AM-GM-HM inequality</strong... | jmerry | 619,637 | <p>The denominator (in the original form) is a multiple of an arithmetic mean - a sum of <span class="math-container">$201$</span> terms is <span class="math-container">$201$</span> times their average. So then, depending on taste, you can either apply AM-GM to the denominator or GM-HM to the whole thing.</p>
|
3,093,099 | <p>If <span class="math-container">$x$</span> is positive, what is the maximum value of this expression:</p>
<p><span class="math-container">$$\frac{x^{100}}{1+x+x^2+\ldots+x^{200}}$$</span></p>
<p>This question is from a book of problems on sequence and series under the section on <strong>AM-GM-HM inequality</strong... | Mark Viola | 218,419 | <p>For <span class="math-container">$x>0$</span>, we have from the AM-GM inequality</p>
<p><span class="math-container">$$\begin{align}
\sum_{n=0}^{200}x^n&\ge 201 \sqrt[201]{\prod_{n=0}^{200}x^n}\\\\
&=201 \sqrt[201]{x^{20100}}\\\\
&=201x^{100}
\end{align}$$</span></p>
<p>Hence, we see that </p>
<p><... |
3,093,099 | <p>If <span class="math-container">$x$</span> is positive, what is the maximum value of this expression:</p>
<p><span class="math-container">$$\frac{x^{100}}{1+x+x^2+\ldots+x^{200}}$$</span></p>
<p>This question is from a book of problems on sequence and series under the section on <strong>AM-GM-HM inequality</strong... | Angina Seng | 436,618 | <p>You can instead minimise the reciprocal of your quantity, viz.,
<span class="math-container">$$\frac{1+x+x^2+\cdots+x^{200}}{x^{100}}=x^{-100}+x^{-99}+\cdots+x^{99}+x^{100}.$$</span>
One only needs the two-variable AM/GM inequality to do this, just in the
form <span class="math-container">$y+y^{-1}\ge2$</span> for <... |
1,793,039 | <p>Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb N}\vert$= quantity of natural numbers in $A$, </p>
<p>how can I show that two functions $f,g:\Bbb R \to \mathbb R$ coincide... | joseabp91 | 340,789 | <p>Taking $n \in \mathbb{N}$, with $n \geq 1$, you have
$$
a_n = \frac{{(-1)}^n n}{n + 1} = \frac{{(-1)}^n n}{n \left(1 + \frac{1}{n}\right)} = \frac{{(-1)}^n}{1 + \frac{1}{n}}
$$
and given $\varepsilon > 0$, exists $n_0 \in \mathbb{N}$ such that for all $n \geq n_0$, then
$$
|a_n - {(-1)}^n| < \varepsilon\mbox{.... |
132,980 | <p>Toric varieties and convex polyhedra are intimately connected. Some of this can be found in standard text books (the connection between divisors and mixed volumes seems to be a popular example).</p>
<p>One of the most important objects that are associated to an algebraic variety is its derived category. So I'm wond... | Yochay Jerby | 112,792 | <p>That's a great question! </p>
<p>In the last few years there is plenty of research on the subject - and a few (interesting) open questions. </p>
<p>Before Kawamta's result - note that toric manifolds satisfy $rk(K(X))= \chi(X)$ (this is a result on the fan), so conceptually they are a good candidates to have a ful... |
452,011 | <p>Consider the set $A = \{\theta(f) \mid f : \mathbb{R} \rightarrow \mathbb{R},\;f\,\text{non-decreasing}\}$ where $\theta(f)$ denotes the set of functions which are asymptotically within a constant factor of $f$. Define an order on $A$ by setting $\theta(f) \le \theta(g)$ if $f \in O(g)$.</p>
<p>What is the order st... | Amzoti | 38,839 | <p>This is a very strange system indeed, note I am assuming$\left(Y_1 = \dfrac{dy_1}{dt},~ Y_2 = \dfrac{dy_2}{dt}\right)$.</p>
<p>If we look at a phase portrait, lets see if it sheds any light on matters.</p>
<p><img src="https://i.stack.imgur.com/cFsfl.png" alt="enter image description here"></p>
<p>Well, it looks ... |
452,011 | <p>Consider the set $A = \{\theta(f) \mid f : \mathbb{R} \rightarrow \mathbb{R},\;f\,\text{non-decreasing}\}$ where $\theta(f)$ denotes the set of functions which are asymptotically within a constant factor of $f$. Define an order on $A$ by setting $\theta(f) \le \theta(g)$ if $f \in O(g)$.</p>
<p>What is the order st... | 40 votes | 85,506 | <p>This is a nice example of what a nonlinear term can do to a stable, but not asymptotically stable, equilibrium. It helps to introduce the polar radius $\rho=\sqrt{y_1^2+y_2^2}$, because this function satisfies the ODE
$$\frac{d\rho }{dt} = \frac{y_1}{\rho}\frac{dy_1}{dt}+\frac{y_2}{\rho}\frac{dy_2}{dt} = \rho^3\sin... |
1,900,333 | <p>If $\frac ab$ rounded to the nearest trillionth is $0.008012018027$, where $a$ and $b$ are positive integers, what is the smallest possible value of $a+b$?</p>
<p>I don't see any strategies here for solving this problem, any help? Thanks in advance!</p>
| hmakholm left over Monica | 14,366 | <p>The continued fraction representations of the limits of the interval are
$$ 0.0080120180265 = [0; 124, 1, 4, 2, 1, 463872, 1, 1, 12, 1, 1, 41] \\
0.0080120180275 = [0; 124, 1, 4, 3, 545777, 2, 13, 1, 1, 1, 1, 2] $$</p>
<p>The simplest continued fraction (and therefore also the simplest ordinary fraction!) in that i... |
2,536,791 | <p>I am taking a basic complex analysis course and I'm trying to understand the differences between different forms of convergence.</p>
<p>Specifically, I am trying to distinguish normal convergence from pointwise convergence. I searched around for a similar question, but I was only able to find a comparison between n... | user1992 | 501,968 | <p>$$\int_{0}^{p}\beta e^{-\beta x}dx=-e^{-\beta x}|_{0}^{p}=e^{-\beta x}|_{0}^{p}=1-e^{-\beta p}$$</p>
|
2,372,925 | <p>A book has 500 pages on which typographical errors could occur. Suppose that there are exactly 10 such errors randomly located on those pages. </p>
<p>$a)$ Find the probability that a random selection of 50 pages will contain no errors. </p>
<p>$b)$ Find the probability that 50 randomly selected pages will contain... | Mark Viola | 218,419 | <p>The boundary <span class="math-container">$x=0$</span>, <span class="math-container">$y\ge 0$</span> in the <span class="math-container">$x-y$</span> plane maps to the boundary <span class="math-container">$u=-v$</span>, with <span class="math-container">$u\in [0,\infty)$</span> and <span class="math-container">$v\i... |
2,372,925 | <p>A book has 500 pages on which typographical errors could occur. Suppose that there are exactly 10 such errors randomly located on those pages. </p>
<p>$a)$ Find the probability that a random selection of 50 pages will contain no errors. </p>
<p>$b)$ Find the probability that 50 randomly selected pages will contain... | user5713492 | 316,404 | <p>We have the integral
$$\int\int_Re^{-4u^2-16v^2}2du\,dv$$
The $x$-axis becomes $y=u-v=0$. Since $u=(x+y)/2\ge0$ this boundary becomes the half-line $v=u$ for $u\ge0$.<br>
The $y$-axis becomes $x=u+v=0$, so this boundary becomes the half-line $v=-u$ for $u\ge0$.<br>
Transform to polar coordinates $u=r\cos\theta$, $v=... |
616,393 | <p>Let $z$ be a complex number and $\mathrm{Re}$ denote the real part.</p>
<p>Does there exist a nonconstant entire function $f(z)$ such that $f(z)$ is bounded for $\mathrm{Re}(z)^2 > 1$ ?</p>
| Jean-Claude Arbaut | 43,608 | <p>Look at the function f defined in section 12.2 <a href="http://books.google.com/books?id=MKPe1Q9k-nAC&pg=PA164&lpg=PA164" rel="nofollow">here</a>, in Complex Analysis by Joseph Bak and Donald J. Newman.</p>
<p>It defines an entire function $f$ bounded outside the strip $|\mathrm{Im}(z)|\leq \pi$, and by a s... |
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