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 |
|---|---|---|---|---|
3,314,561 | <p>Consider the triangle <span class="math-container">$PAT$</span>, with angle <span class="math-container">$P = 36$</span> degres, angle <span class="math-container">$A = 56$</span> degrees and <span class="math-container">$PA=10$</span>. The points <span class="math-container">$U$</span> and <span class="math-contain... | Community | -1 | <p>People can maybe talk more generally but I have a really simple example (but helpful in my opinion):</p>
<blockquote>
<p>Not all waves are differentiable. We want all waves to satisfy the wave equation (in some sense). That sense is weak. </p>
</blockquote>
|
3,314,561 | <p>Consider the triangle <span class="math-container">$PAT$</span>, with angle <span class="math-container">$P = 36$</span> degres, angle <span class="math-container">$A = 56$</span> degrees and <span class="math-container">$PA=10$</span>. The points <span class="math-container">$U$</span> and <span class="math-contain... | Jonas Lenz | 450,140 | <p>Let's have a look at the Dirichlet problem on some (say smoothly) bounded domain <span class="math-container">$\Omega$</span>, i.e.
<span class="math-container">$$
-\Delta u=f \text{ in } \Omega\\
u=0~ \text{ on } \partial \Omega
$$</span>
for <span class="math-container">$f \in \text{C}^0(\overline{\Omega})$</spa... |
3,854,785 | <p>Considering <span class="math-container">$$2\sin^2(x) = 1 - \cos(2x)$$</span>
to show <span class="math-container">$$8\sin^4(x) = 3 - 4\cos(2x) +\cos(4x)$$</span>
Assuming I did not how to initially do this proof properly, how would I be able to set up a proof that is still valid to show that <span class="math-conta... | Michael Rozenberg | 190,319 | <p><span class="math-container">$$8\sin^4x=2(1-\cos2x)^2=2-4\cos2x+2\cos^22x=$$</span>
<span class="math-container">$$=2-4\cos2x+1+\cos4x=3-4\cos2x+\cos4x.$$</span>
If really want to use <span class="math-container">$2\sin^2x=1-\cos2x$</span> only, we obtain:
<span class="math-container">$$2-4\cos2x+2\cos^22x=2-4\cos2x... |
3,854,785 | <p>Considering <span class="math-container">$$2\sin^2(x) = 1 - \cos(2x)$$</span>
to show <span class="math-container">$$8\sin^4(x) = 3 - 4\cos(2x) +\cos(4x)$$</span>
Assuming I did not how to initially do this proof properly, how would I be able to set up a proof that is still valid to show that <span class="math-conta... | robjohn | 13,854 | <p><span class="math-container">$$
\begin{align}
4\sin^4(x)
&=1-2\cos(2x)+\cos^2(2x)\tag1\\[6pt]
&=1-2\cos(2x)+1-\sin^2(2x)\tag2\\
&=1-2\cos(2x)+1-\frac12+\frac12\cos(4x)\tag3\\
8\sin^4(x)
&=3-4\cos(2x)+\cos(4x)\tag4
\end{align}
$$</span>
Explanation:<br />
<span class="math-container">$(1)$</span>: squ... |
255,827 | <p>I've had trouble coming up with one.</p>
<p>I know that if I could find </p>
<p>an irreducible poly $p(x)$ over $\mathbb{Q}$
which has roots $\alpha, \beta, \gamma\in Q(\alpha)$,</p>
<p>then $|\mathbb{Q}(\alpha) : \mathbb{Q}| $ = 3 and would be a normal extension,
as $\mathbb{Q}(\alpha)=\mathbb{Q}(\alpha,\beta,\g... | Gerry Myerson | 8,269 | <p>You may know that the Galois group of $x^n-1$ over the rationals is cyclic of order $\phi(n)$ (that's the Euler phi-function). If $\phi(n)$ is a multiple of $3$ (and it's not hard to find such $n$), then you can find a normal extension of the rationals of degree $3$ as a subfield of the splitting field of $x^n-1$. <... |
1,026,506 | <p>If $I_n=\int _0^{\pi }\:sin^{2n}\theta \:d\theta $, show that
$I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$, and hence $I_n=\frac{\left(2n\right)!}{\left(2^nn!\right)2}\pi $</p>
<p>Hence calculate $\int _0^{\pi }\:\:sin^4tcos^6t\:dt$</p>
<p>I knew how to prove that $I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$ ,, but I am... | Idris Addou | 192,045 | <p>Here is the last part of the problem you post.
You can verify the value of the definite integral in the wolfram, at </p>
<p><a href="http://www.wolframalpha.com/input/?i=integral_0" rel="nofollow noreferrer">http://www.wolframalpha.com/input/?i=integral_0</a>^%28pi%29+sin^4%28x%29cos^6%28x%29dx</p>
<p><img src="ht... |
2,524,890 | <p>I know that if matrix $a$ is similar to matrix $b$ then $\operatorname{trace} a=\operatorname{trace} b$.</p>
<p>Does it go to the other side?</p>
<p>Thanks.</p>
| Erik T. | 501,005 | <p>You could just use the common second degree equation formula.</p>
<p>So:</p>
<p>$$ b^2-8b+16a=0\implies b = \frac{8 \pm \sqrt{64-64a}}{2}$$</p>
<p>From this we can conclude that there will only exist a solution over $\mathbb{R}$ when $\sqrt{64-64a}$ exist. Is easy to check that the root only exists when the insid... |
2,600,776 | <blockquote>
<p>A continuos random variable $X$ has the density
$$
f(x) = 2\phi(x)\Phi(x), ~x\in\mathbb{R}
$$
then</p>
<p>(<em>A</em>) $E(X) > 0$</p>
<p>(<em>B</em>) $E(X) < 0$</p>
<p>(<em>C</em>) $P(X\leq 0) > 0.5$</p>
<p>(<em>D</em>) $P(X\ge0) < 0.25$</p>
<p>\begin{eqnarray}... | Matt | 9,666 | <p>This question requires no calculations.
You should not integrate anything to answer it.</p>
<p><strong>The Key</strong><br>
If
$\phi(x)$ is the density function of a distribution $D$,<br>
and
$\Phi(x)$ is the cumulative distribution function of $D$,<br>
then
the density $f(x) = 2\phi(x)\Phi(x)$
corresponds to the d... |
1,921,101 | <p>$∃x.P(x) \Rightarrow ∀x.P(x) $</p>
<p>How can I read this in simple English? I translated it as: There exists an element x for which P(x) implies that for all elements x, P(x) is true - but I feel like this doesn't make much sense. What am I doing wrong here?</p>
| Hayden | 27,496 | <p>$\exists x P(x)$ is read as </p>
<blockquote>
<p>there exists $x$ such that $P(x)$ holds</p>
</blockquote>
<p>Likewise $\forall x P(x)$ is read as </p>
<blockquote>
<p>for every $x$, $P(x)$ holds.</p>
</blockquote>
<p>$A \implies B$ is read as</p>
<blockquote>
<p>If $A$ then $B$.</p>
</blockquote>
<p>Thu... |
19,596 | <p>I am trying to rearrange and manipulate some vector differential equations in <em>Mathematica</em>. As far as I understand you have to tell <em>Mathematica</em> that a variable is a vector by specifying the components of the vector. For example</p>
<pre><code>r = {x, y, z};
</code></pre>
<p>If I want to define vec... | Jens | 245 | <p>The answer depends a lot on what you mean by "doing" vector calculus. You want results to be displayed without using component notation, and that's in general a difficult thing to achieve.</p>
<p>A prerequisite about doing <strong>completely symbolic</strong> vector calculus is to define the simplification rules. B... |
66,671 | <p>$$\text{ABC- triangle:} A(4,2); B(-2,1);C(3,-2)$$<br>
Find a D point so this equality is true:</p>
<p>$$5\vec{AD}=2\vec{AB}-3\vec{AC}$$</p>
| Peđa | 15,660 | <p>So,let's observe picture below.first of all you will need to find point $E$...use that $E$ lies on $p(A,B)$ and that $\left\vert AB \right\vert = \left\vert BE \right\vert $. Since $ p(A,C)\left\vert \right\vert p(F,E)$ we may write next equation: $\frac{y_C-y_A}{x_C-x_A}=\frac{y_E-y_F}{x_E-x_F}$ and $\left\vert EF... |
66,671 | <p>$$\text{ABC- triangle:} A(4,2); B(-2,1);C(3,-2)$$<br>
Find a D point so this equality is true:</p>
<p>$$5\vec{AD}=2\vec{AB}-3\vec{AC}$$</p>
| robjohn | 13,854 | <p>Recall that the vector $\overrightarrow{PQ}$ is the difference of two points $Q{-}P$. In this way,
$$
5\overrightarrow{AD}=2\overrightarrow{AB}-3\overrightarrow{AC}
$$
becomes
$$
5(D-A)=2(B-A)-3(C-A)
$$
All that is left is to solve for $D$.</p>
|
3,786,654 | <blockquote>
<p>Let <span class="math-container">$x_1, x_2, x_3 \in \Bbb R$</span>, satisfy <span class="math-container">$0 \leq x_1 \leq x_2 \leq x_3 \leq 4$</span>. If their squares form an arithmetic progression with common difference <span class="math-container">$2$</span>, determine the minimum possible value of <... | dezdichado | 152,744 | <p>Notice that:</p>
<p><span class="math-container">$$x_3 - x_1 = \dfrac{x_3^2-x_1^2}{x_3+x_1} = \dfrac{4}{\sqrt{x_1^2+4}+x_1}$$</span>
and this is obviously minimized at the largest possible value of <span class="math-container">$x_1.$</span> That value is obtained by observing:
<span class="math-container">$$16\geq x... |
4,068,314 | <p>I do know that double negation and LEM are equivalent, but can we prove
<span class="math-container">$$\vdash \neg \neg (p \vee \neg p)$$</span>
without using either of them, in a Fitch-style proof?</p>
| Marc van Leeuwen | 18,880 | <p>What you may use is the fact that in intuitionistic logic one can derive from <span class="math-container">$\lnot(p\lor q)$</span> that <span class="math-container">$\lnot p\land\lnot q$</span> (and vice versa: the two ways to interpret <span class="math-container">$p$</span> NOR <span class="math-container">$q$</sp... |
804,283 | <p>I have the equation $ t\sin (t^2) = 0.22984$. I solved this with a graphing calculator, but is there any way to solve for $ t$ without graphing? </p>
<p>Thanks!</p>
| Community | -1 | <p>As is said, there is no closed form solution to this equation. No formula if you prefer.</p>
<p>In such cases, numerical methods are used, which means that different values for $t$ are tried, using specific strategies to get closer and closer to the solution.</p>
<p>It is useful to carry out the study of the funct... |
804,283 | <p>I have the equation $ t\sin (t^2) = 0.22984$. I solved this with a graphing calculator, but is there any way to solve for $ t$ without graphing? </p>
<p>Thanks!</p>
| Richard | 14,493 | <p>Using a <a href="https://en.wikipedia.org/wiki/Taylor_series" rel="nofollow">Taylor series</a>, $\sin(x)$ can be written as</p>
<p>$\sin(x)\approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$</p>
<p>Replacing $x$ for $t^2$ gives:</p>
<p>$\sin(t^2)\approx t^2 - \frac{t^6}{3!} + \frac{t^{10}}{5!} ... |
3,128,862 | <p>I'm really stuck in this fairly simple example of conditional probability, I don't understand the book reasoning:</p>
<blockquote>
<p>An ordinary deck of 52 playing cards is randomly divided into 4 piles of 13 cards each.
Compute the probability that each pile has exactly 1 ace. </p>
<p><strong>Solution.</... | Servaes | 30,382 | <p>The parallel lines <span class="math-container">$L$</span> and <span class="math-container">$L'$</span> are contained in a plane <span class="math-container">$V$</span>. That <span class="math-container">$L^{\phi}$</span> and <span class="math-container">$L'^{\phi}$</span> are contained in <span class="math-containe... |
14,612 | <p>For finding counter examples. That does not sound convincing enough, at least not always. Why as a object in its own right the study of Cantor Set has merit ? </p>
| Adam | 4,791 | <p>I think that you may be selling short the value of a counterexample! They are quite useful for making sure that you have not <em>proven</em> <em>too</em> <em>much</em>. i.e. When you have a plausible but only semi-formal argument, how do you tell if it is worth the effort in making it rigorous? Checking against coun... |
2,396,073 | <p>Let $\omega_1$ be the first uncountable ordinal. In some book, the set $\Omega_0:=[1,\omega_1)=[1,\omega_1]\backslash\{\omega_1\}$ is called the set of countable ordinals. Why? It is obvious that it is an uncountable set, because $[1,\omega_1]$ is uncountable. The most possible reason I think is that for any $x\pr... | bof | 111,012 | <p>The set $\Omega=[0,\omega_1)$ is a set of countable ordinals because every element of $\Omega$ is a countable ordinal. To see this, suppose that $x\in\Omega$ is <em>not</em> a countable ordinal. Since $x$ is an ordinal, it follows that $x$ is an <em>uncountable</em> ordinal; but $x\lt\omega_1,$ contradicting the fac... |
295,773 | <p>What would be a good Riemannian Geometry (or Differential Geometry) book that would go well with a General Relativity class (offered by a physics department)? I'm in one right now, but I'd like a pure math perspective on the math that's introduced as I can imagine, inevitably some things would be swept under the rug... | Will Jagy | 10,400 | <p><a href="http://www.math.harvard.edu/~shlomo/docs/semi_riemannian_geometry.pdf" rel="nofollow">STERNBERG_PDF</a> and <a href="http://rads.stackoverflow.com/amzn/click/0125267401" rel="nofollow">O'NEILL</a> ${}{}{}{}{}{}{}$</p>
|
295,773 | <p>What would be a good Riemannian Geometry (or Differential Geometry) book that would go well with a General Relativity class (offered by a physics department)? I'm in one right now, but I'd like a pure math perspective on the math that's introduced as I can imagine, inevitably some things would be swept under the rug... | Javier Álvarez-Vizoso | 4,058 | <p>The new book by Sternberg (a freely available version is linked in the answer by Will Jagy) is very affordable and focused on just what you may need:</p>
<ul>
<li><strong>Sternberg</strong> - <a href="http://amzn.com/0486478556" rel="nofollow noreferrer"><em>Curvature in Mathematics and Physics</em></a>; Dover 2012... |
259,308 | <p>The output of <code>ListPointPlot3D</code> is shown below:
<a href="https://i.stack.imgur.com/ypt73.png" rel="noreferrer"><img src="https://i.stack.imgur.com/ypt73.png" alt="enter image description here" /></a>
I only want to connect the dots in such a way that it forms a ring-like mesh. However, when I use <code>Li... | Daniel Huber | 46,318 | <p>A solution with a hole:</p>
<pre><code>d1 = Select[pts, #[[3]] > -0.1 &];
d2 = Select[pts, #[[3]] < 0 &];
ListPlot3D[{d1, d2}, BoxRatios -> Automatic]
</code></pre>
<p><a href="https://i.stack.imgur.com/uUUBr.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/uUUBr.png" alt="enter im... |
1,854,823 | <p>How do I express the hyperplane $x+y=1$ as the span of two vectors or more?</p>
<p>P. S. We have a 3D space.</p>
| Chee Han | 242,589 | <p>Note that it cannot be expressed as span of some vectors since the set of vectors satisfying the hyplerplane equation does not form a subspace. Nevertheless, let $u=\begin{bmatrix} x \\ y \end{bmatrix}$ be any vector satisfying the equation $x+y=1$. Rearranging gives $y=1-x$ and substituting this into $u$ gives
\beg... |
138,520 | <p>I am attempting to show that the series $y(x)\sum_{n=0}^{\infty} a_{n}x^n$ is a solution to the differential equation $(1-x)^2y''-2y=0$ provided that $(n+2)a_{n+2}-2na_{n+1}+(n-2)a_n=0$</p>
<p>So i have:
$$y=\sum_{n=0}^{\infty} a_{n}x^n$$
$$y'=\sum_{n=0}^{\infty}na_{n}x^{n-1}$$
$$y''=\sum_{n=0}^{\infty}a_{n}n(n-1)x... | Alex R. | 22,064 | <p>Everything seems correct. Before you expand out powers of $n$, notice that equating like powers of $x^n$ in your last sum gives:</p>
<p>$(n+2)(n+1)a_{n+2}-2n(n+1)a_{n+1}+[n(n-1)-2]a_n=0$</p>
<p>Notice that $[n(n-1)-2]=n^2-n-2=(n+1)(n-2)$, so since $n\geq 0$, you can divide the recurrence equation by $(n+1)$ to get... |
4,089,114 | <p>I'm a newbie for mathematics and now I'm learning PDE and stuck on that. Could anyone help me out to understand this elimination from PDE. The equation is similar to solve <span class="math-container">$$(D^2 -6DD'+9D'^2)u = y\cos x$$</span></p>
| jacopoburelli | 530,398 | <p>By <a href="https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem" rel="nofollow noreferrer">Rank-nullity theorem</a> it's sufficient (finite-dimensional case) to prove that the sequence <span class="math-container">$b_k := \dim(B_k) - \dim(B_{k-1})$</span> are non decreasing.</p>
<p>Let's denote with <span cla... |
2,792,751 | <p>Prove that $p_2(n) = \left \lfloor{\frac{n}{2}}\right \rfloor+1$ using the identity $$\frac{1}{(1-x)(1-x^2)}=\frac{1}{2}\left(\frac{1}{(1-x)^2}\right)+\frac{1}{2}\left(\frac{1}{1-x^2}\right)$$</p>
<p>where $p_k(n)$ is the number of partitions of an integer $n$ into a most $k$ parts. The generating function $P_k(x)$... | Václav Mordvinov | 499,176 | <p>Note that this is a <a href="https://en.wikipedia.org/wiki/Bipartite_graph" rel="nofollow noreferrer">bipartite graph</a> $G=(V,E)$. You can find a partition of $V$ into two subsets $V_1,V_2\subset V$, such that $u,w\in V_1\implies \{u,w\}\not\in E$ and similarly $u,w\in V_2\implies \{u,w\}\not\in E$. Now we see tha... |
3,715,715 | <p>Let’s say I have a set <span class="math-container">$X$</span> and a set <span class="math-container">$Y$</span>, and <span class="math-container">$X \subseteq Y$</span>. Is it possible to state that <span class="math-container">$|X| \leq |Y|$</span> (<span class="math-container">$|X|$</span> cardinality of <span cl... | Community | -1 | <p>From Wikipedia: "<span class="math-container">$A$</span> has cardinality less than or equal to the cardinality of <span class="math-container">$B$</span> if there exists an injective function from <span class="math-container">$A$</span> into <span class="math-container">$B$</span>." (That is a definition.)</p>
<p>A... |
2,539,693 | <p>A number theory textbook asked us to compare $\tan^{-1}(\frac{1}{2})$ and $\sqrt{5}$. In fact, these are rather close:</p>
<p>\begin{eqnarray*}
\tan^{-1} \frac{1}{2} &=& 0.46364 \\ \\
\frac{1}{\sqrt{5}} &=& 0.44721
\end{eqnarray*}</p>
<p>So at least numerically I think we have the answer that the ... | Amos Quito | 290,715 | <p>Suppose $\tan(x)=\frac{1}{2}$. Then in particular you know $\frac{\sin(x)}{\cos(x)}=\frac{1}{2}$. You also know $\sin^2(x)+\cos^2(x)=1$. Using the last two equalities you find that $\sin(x)=\frac{1}{\sqrt{5}}$. Since $x\geq\sin(x)$, you're done.</p>
|
2,539,693 | <p>A number theory textbook asked us to compare $\tan^{-1}(\frac{1}{2})$ and $\sqrt{5}$. In fact, these are rather close:</p>
<p>\begin{eqnarray*}
\tan^{-1} \frac{1}{2} &=& 0.46364 \\ \\
\frac{1}{\sqrt{5}} &=& 0.44721
\end{eqnarray*}</p>
<p>So at least numerically I think we have the answer that the ... | Jack D'Aurizio | 44,121 | <p>By <a href="https://arxiv.org/abs/1304.0753" rel="nofollow noreferrer">the Shafer-Fink inequality</a>$^{(*)}$:
$$ \arctan\left(\tfrac{1}{2}\right)> \frac{3\cdot\frac{1}{2}}{1+2\sqrt{1+\left(\frac{1}{2}\right)^2}}=\tfrac{3}{8}\left(\sqrt{5}-1\right) $$
and $\frac{3}{8}\left(\sqrt{5}-1\right)\geq \frac{1}{\sqrt{5}}... |
26,152 | <p>In my textbook, they said:</p>
<p>$$2x^{3} + 7x - 4 \equiv 0 \pmod{5}$$</p>
<p>The solution of this equation are the integers with $x \equiv 1 \pmod{5}$, as can be seen by testing $x = 0, 1, 2, 3,$ and $4.$</p>
<p>And I have no clue how do they had $x \equiv 1 \pmod{5}$. I tested as they suggested:</p>
<p>Let $y... | Community | -1 | <p>What your textbook means is $$\left(2x^{3} + 7x - 4 \equiv 0 \pmod{5} \right) \iff \left(x \equiv 1 \pmod{5} \right)$$</p>
<p>This is what you checked by plugging in the different cases for $x$.</p>
<p>By taking $x \equiv 1 \pmod{5}$, you proved that $2x^{3} + 7x - 4 \equiv 0 \pmod{5}$ and hence $$ \left(x \equiv ... |
758,274 | <p>reference: <a href="https://math.stackexchange.com/questions/428313/what-is-the-orbit-of-a-permutation">What is the orbit of a permutation?</a></p>
<p>To be honest, i don't understand the answer in the link.</p>
<p>The orbit of a group action is defined as follows:</p>
<blockquote>
<p>Let <span class="math-container... | Hagen von Eitzen | 39,174 | <p>If $S_A$ is the group of permutations of $A$, then the very definition of $S_A$ gives you an action of the group on the set $A$. But that is not meant here! Instead, each (fixed) $\sigma\in S_A$ also gives us an action of the group $\mathbb Z$ on the set $A$, namely $\mathbb Z\times A\to A$, $(n,a)\mapsto \sigma^n... |
758,274 | <p>reference: <a href="https://math.stackexchange.com/questions/428313/what-is-the-orbit-of-a-permutation">What is the orbit of a permutation?</a></p>
<p>To be honest, i don't understand the answer in the link.</p>
<p>The orbit of a group action is defined as follows:</p>
<blockquote>
<p>Let <span class="math-container... | blue | 34,139 | <p>If $G$ is a group acting on a set $X$ and $H\le G$ is a subgroup, then in particular $H$ also acts on $X$.</p>
<p>If $g\in G$ and $x\in X$ then the orbit of $x$ under $g$ means the orbit of $x$ under the action of the cyclic group $\langle g\rangle$, whose action on $X$ is determined by $G$ since $\langle g\rangle\... |
4,336,659 | <p>For a beta distribution with parameters <span class="math-container">$a$</span> and <span class="math-container">$b$</span>, we can interpret it as the distribution of the probability of heads for a coin we tossed <span class="math-container">$a+b$</span> times and saw <span class="math-container">$a$</span> heads a... | Rohit Pandey | 155,881 | <p>I'm putting in a partial answer in the hope that either I or someone else will be able to fill in the missing pieces. At a high level, we can condition on the <span class="math-container">$k$</span>-th order statistic. Then, all of the other samples of the uniform can lie either before it or after it. This becomes l... |
3,335,081 | <p>Is a temperature change in Celsius larger than a temperature change in Fahrenheit?</p>
<p><strong>The teacher offers this second way of thinking about the question.</strong></p>
<blockquote>
<p>If the temperature increases by 1 degree Celsius, does it also increase by 1 degree Fahrenheit? Or is one temperature c... | Dilip Sarwate | 15,941 | <blockquote>
<p>Is a temperature change in Celsius larger than a temperature change in Fahrenheit?</p>
</blockquote>
<p>is a very poorly-phrased question for which there can be two diametrically-different interpretations. </p>
<ol>
<li><p>A temperature change of <em>one degree Celsius</em> is a larger temperature ... |
316,866 | <p>Suppose $(a_n)$ is a real sequence and $A:=\{a_n \mid n\in \Bbb N \}$ has an infinite linearly independent subset (with respect to field $\Bbb Q$). Is $A$ dense in $\Bbb R?$</p>
| Jonas Meyer | 1,424 | <p>If $A$ is a linearly independent subset of $\mathbb R$, for each $a\in A$ there is a positive integer $n(a)$ such that $n(a)>|a|$. The set $\left\{\dfrac{a}{n(a)}:a\in A\right\}$ is a linearly independent set with the same cardinality and span as $A$, but it is a subset of $(-1,1)$.</p>
|
118,540 | <p>Let $X$ be a projective surface defined over a field $k$ of characteristic $0$, and let $G$ be a finite group acting biregularly on $X$.</p>
<p>Assuming that $X$ is rational over $k$, is the quotient $X/G$ always rational?</p>
<p>If $k=\mathbb{C}$, we can use Castelnuovo's theorem and see that $X/G$ is unirational... | Christian Liedtke | 16,751 | <p>Just to round out the picture: if the characteristic of $k$ is positive and $G$ is a finite, but non-reduced group scheme (for example, the infinitesimal group scheme $\mu_p$ of $p$.the roots of unity), then the quotient $X/G$ need not even have Kodaira dimension $-\infty$ after desingularization. Moreover, if $G$ i... |
800,363 | <p>What is </p>
<blockquote>
<p>$$\lim_{x\to 0}\left(\frac{x}{e^{-x}+x-1}\right)^x$$</p>
</blockquote>
<p>Using the expansion of <a href="http://en.wikipedia.org/wiki/Exponential_function" rel="nofollow">$e^x$</a>, I get that the function</p>
<blockquote>
<p>$$y=\left(\frac{x}{e^{-x}+x-1}\right)^x$$</p>
</blockq... | Community | -1 | <p>We have using the Taylor series</p>
<p>$$e^{-x}+x-1\sim_0\frac{x^2}{2}$$
hence
$$\frac{x}{e^{-x}+x-1}\sim_0\frac2x$$
and then
$$\left(\frac{x}{e^{-x}+x-1}\right)^x=\exp\left(x\log \left(\frac{x}{e^{-x}+x-1}\right)\right)\sim_0\exp\left(x\log\left(\frac2x\right)\right)\xrightarrow{x\to0}e^0=1$$</p>
|
119,636 | <p>I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived.</p>
<p>For example, when r = 2, the formula is given by:
$\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$
according to <a href="http://www.wolframalpha.com/input/?i=partial+sum+of+n+2%5En" rel="noreferrer">http://www.wolfr... | Julián Aguirre | 4,791 | <p>I suppose you are familiar with the sum of an geometric progression:
<span class="math-container">$$
1+x+x^2+\dots+x^m=\frac{x^{m+1}-1}{x-1}.
$$</span>
Take derivatives an multiply by <span class="math-container">$x$</span>.</p>
|
621,742 | <p>How do you get from$$\int^\infty_0\int^\infty_0e^{-(x+y)^2} dx\ dy$$to
$$\frac{1}{2}\int^\infty_0\int^u_{-u}e^{-u^2} dv\ du?$$ I have tried using a change of variables formula but to no avail.<br>
Edit: Ok as suggested I set $u=x+y$ and $v=x-y$, so I can see this gives $dx dy=\frac{1}{2}dudv$ but I still can't see h... | Robert Israel | 8,508 | <p>Hint: Try $u = x+y$, $v = x-y$</p>
|
3,180,914 | <p>Let <span class="math-container">$G$</span> be a cyclic group of order <span class="math-container">$n$</span>. Let <span class="math-container">$G_k$</span> the subgroup
<span class="math-container">$$G_k=\left\{x^k: x\in G\right\}.$$</span>
Is it true that <span class="math-container">$[G:G_k]\in\{1,k\}$</span>?</... | lhf | 589 | <p>No: For <span class="math-container">$G=C_{12}$</span> we have <span class="math-container">$[G:G_8]=3$</span>.</p>
<p>In general, for <span class="math-container">$G=C_{n}$</span> we have <span class="math-container">$[G:G_k]=\dfrac{n}{\gcd(n,k)}$</span>.</p>
|
3,547,529 | <p>I did the following: I set <span class="math-container">$3^m+3^n+1=x^2$</span> where <span class="math-container">$x\in\Bbb{N}$</span> and assumed it was true for positive integer exponents and for all whole numbers x so that I can later on prove it's invalidity with contradiction. Since <span class="math-container"... | Piquito | 219,998 | <p>COMMENT.-Let <span class="math-container">$F_n(x)$</span> defined as in your problem
<span class="math-container">$$F_n(x)=ax^{2n}+bx^{2n-1}+cx^{2n-2}+dx^{2n-3}\ldots+px+q$$</span> Assuming <span class="math-container">$c\ne0$</span> one has
<span class="math-container">$$F_n(x)=ax^{2n}+bx^{2n-1}\pm F_{n-1}(x)\hspa... |
3,449,589 | <p>In Example 1.4 of <em>Lee's Introduction to Smooth Manifolds</em>, which is showing that the <span class="math-container">$n$</span>-sphere, <span class="math-container">$\mathbb{S}^n$</span> is a topological <span class="math-container">$n$</span>-manifold, the following is stated.</p>
<p>In the part where the aut... | Lee Mosher | 26,501 | <p>You might be misunderstanding the meaning of the notation for the input parameters of <span class="math-container">$g_i$</span> and <span class="math-container">$h_i$</span>. For example, consider
<span class="math-container">$$g_i = f(x^1, \dots, \widehat{x^i}, \dots, x^{n+1})
$$</span>
What this notation means is ... |
1,789,373 | <p>I'm trying to figure out why the following is true:</p>
<p>Let $ \kappa $ be an uncountable, regular cardinal. Suppose we turn it into a group (i.e. there are operations $ (\cdot, ^{-1}, e) $ with which $ \kappa $ is a group. My aim is to prove that the set</p>
<p>$$ \{ \alpha \in \kappa : \alpha \text{ is a subgr... | Batominovski | 72,152 | <p><strong>Physics (Classical Mechanics) Solution:</strong></p>
<p>Consider a $1$-dimensional elastic collision of a particle $X$ of mass $4$ moving at velocity $2$ into a particle $E$ of mass $1$, initially at rest. Due to this collision, $X$ breaks into $4$ smaller particles $A$, $B$, $C$, and $D$ (of course, we ar... |
446,326 | <blockquote>
<p>Let $Q$ be a $3\times3$ special orthogonal matrix. Show that $Q(u\times v)=Q(u)\times Q(v)$ for any vectors $u, v\in\mathbb R^3$.</p>
</blockquote>
<p>I have no idea how to start. I'm not sure if $Q(u)\cdot Q(V)=Q(u\cdot v)$ would helps. Please give me some help. Thanks.</p>
| user81327 | 81,327 | <p>Maybe it will be easiest to show this explicitly for the basis vectors $ \{ (1,0,0) \cdots \} $ , and then the general case follows from linearity of all things involved. It will be useful to note that if $ \vec{Q_1}, \vec{Q_2}, \vec{Q_3} $ are the column vectors of $ Q $, then the fact that $ \det(Q) = 1 = \vec{Q1}... |
4,000,576 | <blockquote>
<p>What is the value of the following integral:
<span class="math-container">$$\int_0^{2\pi}\frac{1}{4\cos^2(t)+9\sin^2(t)}\mathrm{d}t$$</span>
<span class="math-container">$\frac\pi9$</span> ; <span class="math-container">$\frac\pi6$</span> ; <span class="math-container">$\frac\pi3$</span> ; <span class="... | Community | -1 | <p>Here is a solution using differentiation under integral sign. Consider a more general integral, namely:</p>
<p><span class="math-container">\begin{align*}
I_1(\alpha,\beta) & = \int_{0}^{2\pi}\frac{dx}{\alpha \cos^2x+\beta \sin^2x} \\
& = \int_{0}^{2\pi} \frac{\sec^2x}{\alpha + \beta \tan^2... |
35,688 | <p>I'm looking for a fun (not too many tedious calculations) calculus one problem that uses the concept that, after subsitution, you have two integrals of diffent functions with different limits, but equal area. For example:</p>
<p><a href="http://www.wolframalpha.com/input/?i=int%20%28sin%28%28pi%5E2%29/x%29%29/%28x%... | Michael Lugo | 173 | <p>In general there are two ways to compute $E(X^2)$ where $X$ is a random variable with density $f(x)$; for simplicity I'll say $X$ always takes values between $0$ and $a$, so $f(x) = 0$ when $x < 0$ or $x > a$. One is to simply take $\int_0^a x^2 f(x) \: dx$. The other is to find the density $g$ of the random v... |
35,688 | <p>I'm looking for a fun (not too many tedious calculations) calculus one problem that uses the concept that, after subsitution, you have two integrals of diffent functions with different limits, but equal area. For example:</p>
<p><a href="http://www.wolframalpha.com/input/?i=int%20%28sin%28%28pi%5E2%29/x%29%29/%28x%... | Américo Tavares | 752 | <p>If we make the substitution $t=\frac{u}{1+u}$ in the beta function defined by the following first integral, we get the second:</p>
<p>$$B(p,q)=\int_{0}^{1}t^{p-1}(1-t)^{q-1}\;\mathrm{d}t=\int_{0}^{\infty }\frac{%
u^{p-1}}{(1+u)^{p+q}}\;\mathrm{d}u.$$</p>
<p>Since the relation $B(p,q)=\frac{\Gamma (p)\Gamma (q)}{\... |
3,613,235 | <p>I know such integral: <span class="math-container">$\int_0^{\infty}\frac{\ln x}{e^x}\,dx=-\gamma$</span>. What about the integral <span class="math-container">$\int_0^{\infty}\frac{\ln x}{e^x+1}\,dx$</span>? </p>
<p>The answer seems very nice: <span class="math-container">$-\frac{1}{2}{\ln}^22$</span> but how it co... | CHAMSI | 758,100 | <p>Let <span class="math-container">$ m \in\mathbb{N}^{*} : $</span></p>
<p><span class="math-container">\begin{aligned}\sum_{n=1}^{m}{\frac{2}{\sqrt{n}+\sqrt{n+2}}}&=\sum_{n=1}^{m}{\left(\sqrt{n+2}-\sqrt{n}\right)}\\ &=\sum_{n=1}^{m}{\left(\sqrt{n+2}-\sqrt{n+1}\right)}+\sum_{n=1}^{m}{\left(\sqrt{n+1}-\sqrt{n}... |
323,665 | <p>Given the base case <span class="math-container">$a_0 = 1$</span>, does <span class="math-container">$a_n = a_{n-1} + \frac{1}{\left\lfloor{a_{n-1}}\right \rfloor}$</span> have a closed form solution? The sequence itself is divergent and simply goes {<span class="math-container">$1, 2, 2+\frac{1}{2}, 3, 3+\frac{1}{3... | Carlo Beenakker | 11,260 | <p>The sequence <span class="math-container">$a_n$</span> for <span class="math-container">$n\geq 1$</span> has the following formula:
<span class="math-container">$$a_n=\left\lfloor \sqrt{2n}+\tfrac{1}{2}\right\rfloor +\frac{\left\lfloor \frac{1}{2} \left(\sqrt{8 n-7}+1\right)\right\rfloor-\left\lfloor \frac{1}{2} \le... |
323,665 | <p>Given the base case <span class="math-container">$a_0 = 1$</span>, does <span class="math-container">$a_n = a_{n-1} + \frac{1}{\left\lfloor{a_{n-1}}\right \rfloor}$</span> have a closed form solution? The sequence itself is divergent and simply goes {<span class="math-container">$1, 2, 2+\frac{1}{2}, 3, 3+\frac{1}{3... | Stuart LaForge | 23,508 | <p>I have decided to call this sequence <span class="math-container">$\Theta_n$</span> for the triangular-harmonic number sequence because it clearly has properties related to both triangular and harmonic numbers.</p>
<p>I had been studying it by using the recursive definition <span class="math-container">$$\Theta_0 =... |
1,649,907 | <p>Please kindly forgive me if my question is too naive, i'm just a <em>prospective</em> undergraduate who is simply and deeply fascinated by the world of numbers.</p>
<p>My question is: Suppose we want to prove that $f(x) > \frac{1}{a}$, and we <em>know</em> that $g(x) > a$, where $f,g$ and $a$ are all positive... | Win Vineeth | 311,216 | <p>No, it wouldn't imply that. Because, if $g(x)>a$ , Let's take $g(x) = na ; n>1$
$f(x)g(x)=naf(x)$ <br> If you prove $f(x)g(x)>1$ , You get $naf(x) > 1$ $\implies$ $f(x) > $$1\over na$ ; $n>1$ Which is not what you wanted. <br> $f(x)$ can lie between $1\over na$ and $1\over a$ !!</p>
|
3,394,277 | <p>I am new to this site and not familiar with how to type out math notation so I will do my best. I have a problem I am working on regarding the volume of a circle wrapped around a cylinder of variable radius. For the first part of the problem I had no issue creating a function to represent the cross sectional area. U... | Karthik Kannan | 245,965 | <p>As pointed out in the comments it is sufficient to prove that <span class="math-container">$\sigma(\{(-\infty, F(x)]\cap(0, 1):x\in\mathbb{R}\})\subseteq \mathcal{B}(0, 1)$</span>. If <span class="math-container">$F(x)<1$</span> then clearly <span class="math-container">$(-\infty, F(x)]\cap(0, 1) = (0, F(x)]\in\m... |
2,087,235 | <p>I have a question about this question. Find all complex numbers $z$ such that the equation
$$t^2 + [(z+\overline z)-i(z-\overline z)]t + 2z\overline z\ =\ 0$$
has a real solution $t$.</p>
<p><strong>Attempt at a solution</strong></p>
<p>The discriminant is</p>
<p>$[(z+\overline z) - i(z-\overline z)]^2 - 4(2z\ove... | George Law | 141,584 | <p>Hint: Letting $z=a+ib$ reduces the quadratic equation to one with only real coefficients.</p>
|
1,519,952 | <p>Show that
$$S(n,k) = \sum_{m = k-1}^{n-1} {n-1 \choose m} S(m,k-1) $$</p>
<p>-I was having trouble with this proof in class and my professor suggested to look at it as another proof of the following theorem which states:</p>
<p>-For all $n\ge1$
$$B(n) = \sum_{k=0}^{n-1} {n-1 \choose k} B(k) $$
-Unfortunately I s... | Marko Riedel | 44,883 | <p>It may interest the reader to see how this can be done using generating functions.</p>
<p>Fixing the parameter $k$ we seek to show that
$${n\brace k} = \sum_{m=0}^{n-1} {n-1\choose m} {m\brace k-1}.$$</p>
<p>Here we have extended the summation back to zero because the second Stirling number produces zero for those... |
2,163,067 | <p>Prove that $\mathbb{Z}_5[x]$ is a unique factorization domain.</p>
<p>My approach is to prove that $\mathbb{Z}_5[x]$ is a PID, which implies that it is a UFD.</p>
<p>Proof:</p>
<p>Suppose there exists an ideal $I$ in $\mathbb{Z}_5[x]$ such that it is generated by two or more elements of $\mathbb{Z}_5[x]$. That is... | Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>You have to use that, in a polynomial ring over a field, you can perform Euclidean divisions, and consider a non-zero polynomial of least degree in the ideal.</p>
|
8,878 | <p>We can restrict the <strong>movement</strong> of locators in a <code>LocatorPane</code> as follows:</p>
<p><img src="https://i.stack.imgur.com/Is904.png" alt="locator movement"></p>
<p>In the following example, the first locator's movement is confined to the x-axis and the second locator's movement is confined to ... | rm -rf | 5 | <p>That is the default (and expected) behaviour of <code>LocatorPane</code>. This is useful in implementing things like colour pickers, for example, where it is convenient to simply click on any point to select that colour and have the locator move there automatically to indicate selection. </p>
<p>To create locators ... |
8,878 | <p>We can restrict the <strong>movement</strong> of locators in a <code>LocatorPane</code> as follows:</p>
<p><img src="https://i.stack.imgur.com/Is904.png" alt="locator movement"></p>
<p>In the following example, the first locator's movement is confined to the x-axis and the second locator's movement is confined to ... | Vitaliy Kaurov | 13 | <p>This is the bare bone implementation with direct control of mouse events:</p>
<pre><code>DynamicModule[{p = {0, 0}}, EventHandler[Framed@Dynamic[Style[
Graphics[{Red, Disk[p, 0.2]}, PlotRange -> 2], Selectable -> False]],
{"MouseDragged" :> (p = MousePosition["Graphics"])}]]
</c... |
3,784,471 | <p>To solve this exercise,</p>
<p><span class="math-container">$$|\arccos(\cos(x))|<\pi/4$$</span></p>
<p>I have thought to apply this condition,
<span class="math-container">$$|f(x)|<k, \quad k\in \Bbb R^+, \iff -k<f(x)<k$$</span></p>
<p>Hence,</p>
<p><span class="math-container">$$-\frac \pi4<\arccos(\... | Michael Kinyon | 444,012 | <p>Nonassociative division rings with unity are known as <em>semifields</em>. They come up in the coordinatization of projective planes. Their study in the finite case started with</p>
<p>Donald Knuth, Finite semifields and projective planes. <em>J. Algebra</em> <strong>2</strong> (1965), 182-217.</p>
<p>This published... |
479,697 | <p>I have some quetion on essential singularity.</p>
<p>Let $f(z)$, $g(z)$ have the same essential singularity at $z=z_0$.</p>
<p>Then, if $\frac{f(z)}{g(z)}$ is not a constant function on some neighborhood of $z_0$, then $ \frac{f(z)}{g(z)}$ also has essential singularity at $z=z_0$?</p>
<p>If not, could you give m... | Dennis Fleming | 92,318 | <p>I'm going to assume that by essential singularity you mean is 0 at the given point and that the singularity occurs when the function is in the denominator. The classic example for this is the sinc function, sin(x)/x. This is the Fourier transform for a rectangular window. Clearly both functions approach 0 at x = ... |
479,697 | <p>I have some quetion on essential singularity.</p>
<p>Let $f(z)$, $g(z)$ have the same essential singularity at $z=z_0$.</p>
<p>Then, if $\frac{f(z)}{g(z)}$ is not a constant function on some neighborhood of $z_0$, then $ \frac{f(z)}{g(z)}$ also has essential singularity at $z=z_0$?</p>
<p>If not, could you give m... | Silvia Ghinassi | 258,310 | <p>Dominic Michaelis provided the following counterexample in the comments section: if $f$ has an essential singularity at $z=z_0$, then also $g(z)=zf(z)$ has an essential singularity at $z=z_0$ but $\frac{g(z)}{f(z)}=\frac{zf(z)}{f(z)}=z$ which does not have an essential singularity at $z=z_0$.</p>
|
775,265 | <p>Please help me get the answer to this question.</p>
<p>Prove $f(x)=\sqrt{2x-6}$ is continuous at $x=4$ by using precise definition. ($\epsilon-\delta$ definition of limits.)</p>
| re0 | 144,190 | <p>We have $|x-4|< \delta$.</p>
<p>$|f(x)-f(4)|=|\sqrt {2x-6}-\sqrt {2}|=|\sqrt{2}(\sqrt{x-3}-1)|$</p>
<p>Now, multiplying the numerator and denominator by $(\sqrt{x-3}+1)$,</p>
<p>$=\sqrt{2} |\frac{x-4}{\sqrt{x-3}+1}$|</p>
<p>Now, ${\sqrt{x-3}+1}$ is always greater than or equal to $1$. Thus, $\frac{x-4}{\sqrt{... |
1,662,876 | <p>Now we have some examples of what I mean $$\int_0^{2\pi} \sin x~dx=0$$
$$\int_0^{8\pi} \cos 4x~dx=0$$</p>
<p>$$\int_{\pi}^{2\pi} \sin^3 10x~dx=0$$</p>
<p>Looking at the graph of $f(x)=\sin (x)$ for example it makes some sense to me that $$\int_0^{2\pi} \sin x~dx=0$$ because the region below the $x$ axis will "canc... | Mathstudent | 307,000 | <p>Hint : for the first one, use the change of variable $y = x-\pi$ on $[\pi,2\pi]$, the fact that the function $sin$ is an odd function and $$\int_0^{2\pi} \sin x~dx=\int_0^{\pi} \sin x~dx+\int_{\pi}^{2\pi} \sin x~dx$$</p>
|
2,003,916 | <p>Probably a very simple question:</p>
<p>Suppose a hospital orders defibrillators from a manufacturer. It is well known that defibrillations are often not effective, even when the defibrillators themselves are working properly. Suppose research shows that only 15% percent of defibrillations are effective. Over the n... | Thomas Andrews | 7,933 | <p>$ab^p$ and $ba^p$ are both odd if $a,b$ are both odd, and both even otherwise, so $ab^p-ba^p$ is divisible by $2.$</p>
<p>$b^p-b$ is divisible by $p$ for any $b$, so $ab^p-ab$ is divisible by $p$. Similarly, $ba^p-ba$ is divisible by $p$. So $ab^p-ba^p=(ab^p-ab)-(ba^p-ba)$ is divisible by $p$.</p>
<p>The last thin... |
1,033,383 | <p>$ABCD$ is a rectangle and the lines ending at $E$, $F$ and $G$ are all parallel to $AB$ as shown. </p>
<p>If $AD = 12$, then calculate the length of $AG$.<img src="https://i.stack.imgur.com/OUQZ8.png" alt="enter image description here"></p>
<p>Ok, I started by setting up a system of axes where $A$ is the origin an... | Anatoly | 90,997 | <p>I would suggest you to set the origin in $B$. Setting $AB=DC=d$, line $AC$ has equation $y=-\frac{12}{d}x-12$, and point $E$ has coordinates $(-d,-6)$.</p>
<p>Since line $EB$ has equation $y=\frac{6}{d}x$, the $x$-coordinate of its intersection with $AC$ is given by the solution of</p>
<p>$$\frac{6}{d}x=-\frac{12... |
2,747,753 | <p>Let $x\in\mathbb{R}$. Demonstrate that if the numbers $a = x^3–x$ and $b = x^2 +1$ are rational, then $x$ is rational.</p>
| José Carlos Santos | 446,262 | <p><strong>Hint:</strong> $x^3-x=x\bigl((x^2+1)-2\bigr)$</p>
|
8,382 | <h3>Context</h3>
<p>I'm writing a function that look something like:</p>
<pre><code>triDiagonalQ[mat_] := MapIndexed[ #1 == 0 || Abs[#2[[1]]-#2[[2]]] <= 1 &, mat, {2}] //
Flatten // And @@ # &
</code></pre>
<p>Now, things like <code>#2[[1]]</code> and <code>#2[[2]]</code> are somewhat hard to read. I... | M.R. | 403 | <p>Perhaps you could just use With in a Function?</p>
<pre><code>lst = ConstantArray[0, {3, 3}];
MapIndexed[
Function[{value, pos},
With[{i = pos[[1]], j = pos[[2]]},
{value, i, j}
]
],
lst, {2}]
</code></pre>
|
8,382 | <h3>Context</h3>
<p>I'm writing a function that look something like:</p>
<pre><code>triDiagonalQ[mat_] := MapIndexed[ #1 == 0 || Abs[#2[[1]]-#2[[2]]] <= 1 &, mat, {2}] //
Flatten // And @@ # &
</code></pre>
<p>Now, things like <code>#2[[1]]</code> and <code>#2[[2]]</code> are somewhat hard to read. I... | Mr.Wizard | 121 | <p>Leonid provides a nice method for doing this within "pure functions" but I think it should be pointed out that the common method for doing this is pattern matching.</p>
<p><strong>I argue that destructuring is the <em>foundational use</em> of pattern matching in <em>Mathematica</em>.</strong></p>
<p>Every replacem... |
8,382 | <h3>Context</h3>
<p>I'm writing a function that look something like:</p>
<pre><code>triDiagonalQ[mat_] := MapIndexed[ #1 == 0 || Abs[#2[[1]]-#2[[2]]] <= 1 &, mat, {2}] //
Flatten // And @@ # &
</code></pre>
<p>Now, things like <code>#2[[1]]</code> and <code>#2[[2]]</code> are somewhat hard to read. I... | jVincent | 1,194 | <p>I think Mr. Wizard provided a very thorough answer to the question. I would however like to add a slight example of wrapping this up nicely in a format similar to <code>Function[]</code> but using destructuring:</p>
<pre><code>SetAttributes[dFunction, HoldAll]
dFunction[pattern_, body_][arg___] /;MatchQ[{arg}, patt... |
1,794,072 | <p>My attempt :</p>
<p>If $n$ is odd, then the square must be 2 (mod 3), which is not possible.</p>
<p>Hence $n =2m$</p>
<p>$2^{2m}+3^{2m}=(2^m+a)^2$</p>
<p>$a^2+2^{m+1}a=3^{2m}$</p>
<p>$a (a+2^{m+1})=3^{2m} $</p>
<p>By fundamental theorem of arithmetic, </p>
<p>$a=3^x $</p>
<p>$3^x +2^{m+1}=3^y $</p>
<p>$2^{... | Alijah Ahmed | 124,032 | <p>Yet another approach. </p>
<p>For the odd case of $n$, as you mentioned the result is $2\mod 3$ which cannot be a square number.</p>
<p>As for the even case $n=2m$, we have $2^{2m}+3^{2m}\mod 10$ being equal to $2$ (for when $m=0$) or either $3$ (for $n=2+4k,k\geq0$) or $7$ (for $n=4k,k\geq1$), which clearly cann... |
105,750 | <p>Given a <code>ContourPlot</code> with a set of contours, say, this:</p>
<p><a href="https://i.stack.imgur.com/cKoyo.jpg"><img src="https://i.stack.imgur.com/cKoyo.jpg" alt="enter image description here"></a></p>
<p>is it possible to get the contours separating domains with the different colors in the form of lists... | Basheer Algohi | 13,548 | <pre><code>p = ContourPlot[x*Exp[-x^2 - y^2], {x, 0, 3}, {y, -3, 3},
PlotRange -> {0, 0.5}, ColorFunction -> "Rainbow"];
colors = Cases[p, _RGBColor, -1];
poly = Cases[Cases[Normal@p, {__, colors[[2]], __}, -1],Polygon[__], -1];
r = RegionUnion[poly];
lines = Cases[Normal@RegionPlot[r], Line[__], -1];
Graphic... |
881,282 | <p>Same as above, how to simplify it. I am to calculate its $n$th derivative w.r.t x where t is const, but I can't simplify it. Any help would be appreciated. Thank you.</p>
| DeepSea | 101,504 | <p>Let $u = \sqrt{3}x$, and $v = 4 - \sqrt{3}x$, then:</p>
<p>$u + v = 4$, and $\dfrac{3}{u^2} + \dfrac{1}{v^2} = 1 \to 3v^2 + u^2 = u^2v^2 \to 3v^2 = u^2(v^2 - 1) \to 3v^2 = (4-v)^2(v^2 - 1)$. Observe that $v=2$ is a root to the above equation. From this we can use synthetic division to factor the polynomial and fini... |
2,872,807 | <p>I was browsing through facebook and came across this image: <a href="https://i.stack.imgur.com/jozSg.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jozSg.png" alt="enter image description here"></a></p>
<p>I was wondering if we can find more examples where this happens?</p>
<p>I guess this redu... | mengdie1982 | 560,634 | <p>$$\frac{a^3+b^3}{a^3+c^3}=\frac{(a+b)(a^2-ab+b^2)}{(a+c)(a^2-ac+c^2)}=\frac{a+b}{a+c}$$</p>
<p>If $a+c \neq 0$ and $a+b \neq 0,$ then $$a^2-ab+b^2=a^2-ac+c^2,$$namely $$(b+c-a)(b-c)=0.$$</p>
<p>If $b=c$, the case is trivial. If $b \neq c$, then $$b+c=a.$$</p>
|
2,783,423 | <p>If I have a line formed by points A and B, how can I find the distance of another point from that line. Also, whether that line is clockwise or CCW from point A.
<a href="https://i.stack.imgur.com/dpazD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dpazD.png" alt="enter image description here"><... | G Cab | 317,234 | <p><a href="https://i.stack.imgur.com/Ou2nG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ou2nG.png" alt="punto_linea_1"></a></p>
<p>Take the vectors
$$
{\bf v} = \mathop {BA}\limits^ \to \quad \;{\bf z} = \mathop {BZ}\limits^ \to
$$</p>
<p>Compute the unitary vector $\bf t$ parallel to $\bf v... |
2,783,423 | <p>If I have a line formed by points A and B, how can I find the distance of another point from that line. Also, whether that line is clockwise or CCW from point A.
<a href="https://i.stack.imgur.com/dpazD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dpazD.png" alt="enter image description here"><... | Toby Mak | 285,313 | <p>This might be a better method:</p>
<p><img src="https://i.stack.imgur.com/sPY41.png" alt="enter link description here"></p>
<p>Let $A$ be at $(0,2)$, $B$ be at $(3, -2)$ , and $C$ be at $(7, -3)$. </p>
<p>The distance from $A$ to $B$ is $\sqrt{(-2-2)^2+(3-0)^2} = 5$. Let $AD$ be $x$ and $BD$ be $5-x$. You can the... |
3,596,566 | <p>I've been going through <a href="https://math.stackexchange.com/a/27177/288450">this proof</a>.</p>
<p>And I'm wondering what allows me to change the order of the integral and the infinite sum.</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \left( \sum_{n \ge 0} \frac{2^n t^n x^n}{n!} \right) e^{-x^... | zhw. | 228,045 | <p>Let <span class="math-container">$\mu$</span> be a positive measure on <span class="math-container">$X.$</span> If <span class="math-container">$S_N(x)\to S(x)$</span> pointwise <span class="math-container">$\mu$</span> a.e. on <span class="math-container">$X,$</span> and there exists <span class="math-container">$f... |
3,970,641 | <p>I have 927 unique sequences of the numbers 1, 2 and 3, all of which sum to 12 and represent every possible one-octave scale on the piano, with the numbers representing the intervals between notes in half-steps (i.e., adjacent keys). For example, the <a href="https://en.wikipedia.org/wiki/Major_scale" rel="nofollow n... | obscurans | 619,038 | <p>You're starting to think of <a href="https://en.wikipedia.org/wiki/Kolmogorov_complexity" rel="nofollow noreferrer">Kolmogorov complexity</a>, which is a (almost uncomputable) measure of "how hard it is to describe" the sequence. It is completely dependent on "what is allowed to be used to describe&qu... |
3,294,082 | <p>The exercise is to prove that the minimum value between <span class="math-container">$a^{1/b}$</span> and <span class="math-container">$b^{1/a}$</span> is no greater than <span class="math-container">$3^{1/3}$</span>, where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are posit... | Hw Chu | 507,264 | <p>Suppose <span class="math-container">$a \leq b$</span>. Then <span class="math-container">$1/a \geq 1/b$</span> and <span class="math-container">$a^{1/b} \leq b^{1/a}$</span>. Seeing that <span class="math-container">$a^{1/b} \leq a^{1/a}$</span>, It suffices to prove <span class="math-container">$a^{1/a} \leq 3^{1/... |
66,463 | <p>Hi,</p>
<p>Let $\Gamma$ be a free subgroup of rank 2 in $\mathbb{G}_m^2(\mathbb{Q})$. For all but finitely many primes p we can reduce $\Gamma$ modulo p. Let $S$ be the of primes for which $\Gamma$ does not reduce modulo p, and for any $p$ not in $S$, let $\gamma_p$ be the size of $\Gamma \mod p$. My question is wh... | Joe Silverman | 11,926 | <p>Presumably "exceptional" means primes where either one of the generators of $\Gamma$ is 0 or $\infty$ mod p, or where $\Gamma$ mod $p$ has rank smaller than $2$. The following reference is possibly relevant to your question, although we consider a somewhat different sum. We give an upper bound (that should be fairly... |
352,983 | <p>How to find this expression $(1000!\mod 3^{300})$?</p>
| Clive Newstead | 19,542 | <p>$3$ goes into $1000!$ at least $300$ times, since it divides $3, 6, 9, \dots, 900$, and hence $3^{300} \mid 1000!$.</p>
|
99,237 | <p>If we have a directed graph $G = (V,E)$ and we want to find if there is such node $s \in V$ that we can reach all other nodes of $G$</p>
<p>What is a good algorithm to solve this problem and what is its execution time?</p>
| hmakholm left over Monica | 14,366 | <p>Run Tarjan's linear-time <a href="http://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm" rel="nofollow">algorithm for finding strongly connected components</a>.</p>
<p>If there is more than one component with <em>no incoming edges</em>, then there can be no node that can reach everywhere.<... |
3,753,060 | <blockquote>
<p>If <span class="math-container">$\int f(x)dx =g(x)$</span> then <span class="math-container">$\int f^{-1}(x)dx $</span> is equal to</p>
<p>(1) <span class="math-container">$g^{-1}(x)$</span></p>
<p>(2) <span class="math-container">$xf^{-1}(x)-g(f^{-1}(x))$</span></p>
<p>(3) <span class="math-container">... | Rivers McForge | 774,222 | <p>There's a nice visual computation of the antiderivative of an inverse function: <span class="math-container">$$F(x) := \int_0^x f^{-1}(t) dt$$</span> is an antiderivative for <span class="math-container">$f^{-1}(x)$</span>, and for <span class="math-container">$x = a$</span>, <span class="math-container">$F(a)$</spa... |
265,189 | <p>Integrate, $$\int_{0}^{\frac{\pi}{2}}\sin (\tan\theta) \mathrm{d\theta}$$</p>
| Sasha | 11,069 | <p>Making a change of variables $u=\tan(\theta)$:
$$
\int_0^{\pi/2} \sin\left(\tan \theta\right) \mathrm{d} \theta = \int_0^\infty \frac{\sin(u)}{1+u^2} \mathrm{d} u \tag{1}
$$
In order to evaluate this we use the technique of <a href="http://en.wikipedia.org/wiki/Mellin_transform">Mellin transform</a>.</p>
<ol>
<l... |
2,820,796 | <p>In How many ways can a 25 Identical books can be placed in 5 identical boxes. </p>
<p>I know the process by counting but that is too lengthy .
I want different approach by which I can easily calculate required number in Exam hall in few minutes. </p>
<p>Process of Counting :
This problem can be taken partitions of... | Jaroslaw Matlak | 389,592 | <p>You could use a recurrence.</p>
<p>For example: put $a=0,1,2,...,n$ books into the first box and calculate, in how many ways you can put $n-a$ books in $m-1$ boxes. To prevent repetitions, assume, that in the next box you will put no less books, than to the previous one.</p>
<p>Of course:</p>
<ul>
<li>if $n<a... |
2,555,399 | <p>The question is to find out the coefficient of $x^3$ in the expansion of $(1-2x+3x^2-4x^3)^{1/2}$</p>
<p>I tried using multinomial theorem but here the exponent is a fraction and I couldn't get how to proceed.Any ideas?</p>
| lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>The coefficient of $x^3$ in the expansion of $(1-2x+3x^2-4x^3)^{1/2}$</p>
<p>$=$</p>
<p>the coefficient of $x^3$ in the expansion of $(1-2x+3x^2-4x^3+\cdots)^{1/2}$</p>
<p>Using <a href="https://math.stackexchange.com/questions/746388/calculating-1-frac13-frac1-cdot33-cdot6-frac1-cdot3-cdot53-cdot... |
3,864,729 | <p>I am studying for an exam and I am almost grasping compactness. However, some examples are still unclear. E.g. for <span class="math-container">$A = (a, b)$</span> with <span class="math-container">$a < b$</span> and <span class="math-container">$a, b \in \mathbb{R}$</span>.
How can <span class="math-container">... | Darsen | 637,522 | <p>If <span class="math-container">$x\in(a,b)$</span> then <span class="math-container">$a<x<b$</span>. Consider <span class="math-container">$b-x>0$</span>. There is some <span class="math-container">$n_0\in\Bbb N$</span> such that <span class="math-container">$\frac{1}{n_0}<b-x$</span>, so <span class="ma... |
1,748,719 | <p>So there are a few basic formulas I'd like to start with, $W=\int_0^bFdx$, $F=ma$, and $a=\frac{d^2}{dt^2}x$.</p>
<p>In words, Work $(W)$ is defined as the area under a Force versus Displacement $(F/x)$ graph, Force is defined mass times acceleration $(m\cdot a)$, and acceleration is defined as the second derivativ... | Brian | 315,119 | <p>When work is done by a variable force, one would calculate the area under the Force $(F(x))$ vs. Position $(x)$ graph.</p>
<p>So, the work integral would be</p>
<p>$$W=\int_{a}^b F(x) dx=\int_a^b m\cdot a(x) dx$$</p>
<p>Since $a(x)=\frac{dv(x)}{dt}=\frac{dv(x)}{dt}\frac{dx}{dt}=\frac{dv(x)}{dx}v(x)$, then $a(x)dx... |
1,748,719 | <p>So there are a few basic formulas I'd like to start with, $W=\int_0^bFdx$, $F=ma$, and $a=\frac{d^2}{dt^2}x$.</p>
<p>In words, Work $(W)$ is defined as the area under a Force versus Displacement $(F/x)$ graph, Force is defined mass times acceleration $(m\cdot a)$, and acceleration is defined as the second derivativ... | amd | 265,466 | <p>Even though you've accepted a previous answer, I wanted to expand a bit on vnd's comments to the question because they lead to a useful physical result. </p>
<p>Following his and Mackenzie's suggestions, let $v=\frac{da}{dt}$. The integral then becomes, per vnd's comments, $$W = m\int_{x_1}^{x_2}\left(\frac{d^2}{d... |
1,435,269 | <p>Let a sequence $x_n$ be defined inductively by $x_{n+1}=F(x_n)$. Suppose that $x_n\to x$ as $n\to \infty$ and $F'(x)=0$. Show that $x_{n+2}-x_{n+1}=o(x_{n+1}-x_n)$.</p>
<p>I'm not sure how to do this. Any solutions are greatly appreciated. I think The Mean-Value Theorem will be useful and we can assume that $F$ is ... | user402543 | 402,543 | <p>Assuming that $F'\in C^1$, we have:</p>
<p>$\lim_{n\to \infty}\frac{x_{n+2}-x_{n+1}}{x_{n+1}-x_n}=\lim_{n\to \infty}\frac{F(x_{n+1})-F(x_n)}{x_{n+1}-x_n}=F'(x)=0$</p>
<p>Where the last equality has ben obtained by the fact that the sequence is a cauchy sequence (since converges to $x$), and thus:</p>
<p>$\forall ... |
45,911 | <p>I've been wondering for some time now about the difference between a point and a vector. In high school, it was very important to distinguish them from each other, and we used the notation $(x,y,z)$ for points and $[x,y,z]$ for vectors. We always had to translate the point $P=(a,b,c)$ to the vector $\overrightarrow{... | Hans Lundmark | 1,242 | <p>I would say it's a good habit to distinguish points from vectors (in the context that I think you're referring to), even at university!</p>
<p>Geometrically, any point looks just the same as any other point, whereas not all vectors are equal; two vectors can have different lengths, for example, and there is one ver... |
4,539,167 | <p><span class="math-container">$$
g(x) =\min_y f(x, y) =\min_y x^TAx + 2x^TBy + y^TCy
$$</span>
where <span class="math-container">$x\in \mathbb R^{n\times 1}$</span>, <span class="math-container">$y\in \mathbb R^{m\times 1}$</span>, <span class="math-container">$A\in \mathbb R^{n\times n}$</span>, <span class="math-c... | wxydx00 | 410,047 | <p>The <a href="https://en.wikipedia.org/wiki/Envelope_theorem" rel="nofollow noreferrer">envelope theorem</a> should be helpful.</p>
|
1,986,249 | <blockquote>
<p>Let q be a positive integer such that $q \geq 2$ and such that for any
integers a and b, if $q|ab$, then $q|a$ or $q|b$. Show that $\sqrt{q}$
is irrational.</p>
</blockquote>
<p>Proof;</p>
<p>Let assume $\sqrt{q}$ is a rational number, where $n \neq 0$ and $\gcd (m,n)=1$, meaning $\sqrt{q} = \fr... | Jack D'Aurizio | 44,121 | <p>We just have to show that for any prime number $p$, $\sqrt{p}\not\in\mathbb{Q}$.<br>
If we assume $\sqrt{p}=\frac{a}{b}$ with $a,b\in\mathbb{Z}^+$ we get the identity $pb^2=a^2$.<br>
For any $n\in\mathbb{Z}^+$, let $\nu_p(n)=\max\{m\in\mathbb{N}: p^m\mid n\}$. The identity $pb^2=a^2$ implies
$$ \nu_p(pb^2) = \nu_p(a... |
867,209 | <p>I tried to do the implication part. Please, see what I need to do to fix it.</p>
<p>claim: $n|a – b → n|a^2 – b^2$.</p>
<p>claim: $nk = a – b$ for some $k \in \mathbb Z \to nk' = a^2 – b^2$ for some $k' \in Z$.</p>
<p>$(a + 1)^2 – (b +1)^2$</p>
<p>$= a^2 + 2a + 1 -(b^2 +2b + 1)$</p>
<p>$= a^2 + 2a -b^2 -2b$</... | hexaflexagonal | 162,141 | <p>A relation on $A$ is simply a subset of the Cartesian product $A \times A$. For $A=\{a,b,c,d\}$, both $(b,c)$ and $(b,d)$ are contained in $A \times A$; therefore, $\{(b,c),(b,d)\}$ is a relation on $A$. </p>
|
3,598,476 | <p>I have proven that if <span class="math-container">$|x|<\varepsilon,\forall\varepsilon>0$</span>, then <span class="math-container">$x=0$</span>. Further I have proven that ,<span class="math-container">$L=\displaystyle\lim_{n\to\infty}\frac{1}{n} = 0$</span> so that by definition <span class="math-container... | Julio Ignacio Quijas Aceves | 764,383 | <p>You are not wrong nor you are arriving to contradiction. The deal with the first statement, is that <span class="math-container">$x$</span> is fixed, not matter which epsilon you take <span class="math-container">$|x|<\epsilon$</span>. But in the other case, you are taking an <span class="math-container">$\epsilo... |
2,054,676 | <p>I know that m is even and m/2 is odd, but I don't know where/how I can use this. Also, 3y^2 is odd and the sum is odd when x^2 is even. I'm trying to prove that its always odd, but I'm stuck.
Can someone please help?
Thanks</p>
| Peter | 82,961 | <p>Suppose : $m=x^2+3y^2$ is even. </p>
<p>Case $1$ : $x$ is even. Then $y$ must be even as well. So, $m$ is divisble by $4$.</p>
<p>Case $2$ : $x$ is odd. Then, $\ x^2\equiv 1 \mod 4\ $. Since $y$ must be odd as well, we also have $\ y^2\equiv 1\mod 4\ $. Hence, $x^2+3y^2\equiv 0\mod 4$. </p>
<p>So, again, $m$ is d... |
1,379,283 | <p>In my notes it shows how to calculate by using the unit circle. But I do not know why the value of sin is the y coordinate and the value of cos is the x coordinate.</p>
| 3SAT | 203,577 | <p>It is very easy to remember the graph of $\cos(x)$ </p>
<p>[<img src="https://i.stack.imgur.com/iTr0H.png" alt="enter image description here[1"></p>
|
151,425 | <p>I've considered the following spectral problems for a long time, I did not kow how to tackle them. Maybe they needs some skills with inequalities.</p>
<p>For the first, suppose $T:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by
$$Tf(x)=\int_{0}^{x} \! f(t) \, dt$$</p>
<p>How can I calculate:</p>
<ul>
<li>the rad... | Community | -1 | <p>I'm not sure if this can be done with Fourier series, although that's interesting to think about. Anyway, the usual way to see these things is by applying the spectral theorems for compact operators as is done in what follows.</p>
<p>For $f \in L^2[0,1]$ with $\|f\|_2 \le 1$, the Cauchy-Schwarz Inequality shows tha... |
4,187,238 | <p>How many points are common to the graphs of the two equations <span class="math-container">$(x-y+2)(3x+y-4)=0$</span> and <span class="math-container">$(x+y-2)(2x-5y+7)=0$</span>?</p>
<p><span class="math-container">\begin{align*}
(x-y+2)(3x+y-4) &= 0\tag{1}\\
(x+y-2)(2x-5y+7) &= 0\tag{2}
\end{align*}</s... | Rhys Hughes | 487,658 | <p>Compare equations indivually, and work case-by-case.</p>
<p><span class="math-container">$$x-y+2=0\implies y=x+2, x+y-2=0\implies y=2-x$$</span>
<span class="math-container">$$x+2=2-x\implies x=0\to y=2$$</span></p>
<p>So <span class="math-container">$(0,2)$</span> is a solution.</p>
<p><span class="math-container">... |
392,442 | <p>What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or disprove) the primality of every N-th number. I know this is a very large and subjective answer, however, I would li... | Charles | 1,778 | <p>There are dozens, probably hundreds, of formulas for prime numbers. It's a very well-studied problem. Guy's <em>Unsolved Problems in Number Theory</em> has a section devoted to this, and Ribenboim's books cover this in some depth. Many formulas have been published in mathematical journals, and I've seen at least one... |
2,802,959 | <p>If I write
$$
x\in [0,1] \tag 1
$$
does it mean $x$ could be ANY number between $0$ and $1$?</p>
<p>Is it correct to call $[0,1]$ a set? Or should I instead write $\{[0,1]\}$? </p>
<p>Q2:</p>
<p>If I instead have
$$
x\in \{0,1\} \tag 2
$$
does it mean $x$ could be only $0$ OR $1$?</p>
| poyea | 498,637 | <blockquote>
<p>Is it correct to call $[0,1]$ a set?</p>
</blockquote>
<p>Yes, although it doesn't sound natural to me if you <em>"call $[0,1]$ a set"</em>. I'd rather call it the closed interval. You can also write as $\{x|x\in[0,1]\}$ (trivial). </p>
<p>$x\in \{0,1\}$ means $x=1$ or $x=0$.</p>
|
28,811 | <p>There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.</p>
<p>What other conjectures have a large number of proven consequences?</p>
| Charles Matthews | 6,153 | <p>The standard conjectures (<a href="http://en.wikipedia.org/wiki/Standard_conjectures_on_algebraic_cycles">http://en.wikipedia.org/wiki/Standard_conjectures_on_algebraic_cycles</a>) were pretty much designed to be used in this way (and then proved); but proofs are lacking, and some of the results now have non-conditi... |
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