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 |
|---|---|---|---|---|
2,886,544 | <p>For some set $V \subset [a,b]^d$, define the convex hull of $V$ as the set</p>
<p>$$\{\lambda_1v_1 + ... + \lambda_kv_k: \ \lambda_i \ge 0, \ v_i \in V, \ \sum_{i=1}^k \lambda_i = 1, k = 1, 2, 3, ...\}.$$</p>
<p>I don't understand why exactly these vectors form the convex hull of $V$. Why wouldn't I be able to cho... | Jack M | 30,481 | <p>In the following picture, the convex hull of the set of black points is the region inside of the red line.</p>
<p><a href="https://i.stack.imgur.com/gQeDw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gQeDw.png" alt="enter image description here"></a></p>
<p>So yes, the convex hull of the $v_i... |
866,847 | <p><strong>Question:</strong><br/>
Show that $$\sum_{\{a_1, a_2, \dots, a_k\}\subseteq\{1, 2, \dots, n\}}\frac{1}{a_1*a_2*\dots*a_k} = n$$
(Here the sum is over all non-empty subsets of the set of the $n$ smallest positive integers.)</p>
<blockquote>
<p>I made an attempt and then encountered an inconsistency with th... | Thanos Darkadakis | 105,049 | <p>You write:</p>
<p>$\cos (\beta -\alpha) = \cos (-\alpha) \cos\beta + \sin(-\alpha) \sin\beta$.</p>
<p>This is not correct. You should either use the sum of $(\beta)$ and $(-\alpha)$. Or you should use the difference of $(\beta)$ and $(\alpha)$.</p>
<p>1st case:
$\cos (\beta +(-\alpha)) = \cos (-\alpha) \cos(\bet... |
3,599,893 | <p>I had this idea to build a model of Earth in Minecraft. In this game, everything is built on a 2D plane of infinite length and width. But, I wanted to make a world such that someone exploring it could think that they could possibly be walking on a very large sphere. (Stretching or shrinking of different places is OK... | Adayah | 149,178 | <h1>A mathematical model</h1>
<p>Assume you managed to trick the player into thinking they are on a sphere while they are really walking on an infinite plane. What would the world have to look like?</p>
<p>First of all, whenever the player is standing at some point <span class="math-container">$x$</span> on the flat ... |
1,617,239 | <blockquote>
<p><strong>Problem</strong></p>
<p>Find the area of the cone <span class="math-container">$z=\sqrt{2x^2+2y^2}$</span> inscribed in the sphere <span class="math-container">$x^2+y^2+z^2=12^2$</span>.</p>
</blockquote>
<p>I think I have to solve this via the surface integral</p>
<p><span class="math-container... | Jack's wasted life | 117,135 | <p>At the intersection
$$
{z^2\over2}+z^2=12^2\implies z=4\sqrt6
$$
So we have to calculate the surface area of the cone from $z=0$ to $4\sqrt6$. We can use cylindrical polar to parametrize the cone :
$$
\vec r={z\over\sqrt2}\cos\phi\hat i+{z\over\sqrt2}\sin\phi\hat j+z\hat k,\qquad (z,\phi)\in(0,4\sqrt6)\times[0,2\pi... |
18,659 | <p>This is more of a philosophy/foundation question.</p>
<p>I usually come across things like "the set of all men", or for example sets of symbols, i.e. sets of non-mathematical objects.</p>
<p>This confuses me, because as I understand it, the only kind of objects that exists in set theory are sets. It doesn't make s... | Sergei Ivanov | 4,354 | <p>You don't have to <em>define</em> your objects as sets, in fact, you should avoid such unnatural definitions. I don't think a number theorist would be happy to see a proof referring to elements of a natural number or using the identity $1=\{0\}$. Such proofs are not acceptable because they won't survive even the sli... |
4,084,624 | <p>A cool problem I was trying to solve today but I got stuck on:</p>
<p>Find the maximum possible value of <span class="math-container">$x + y + z$</span> in the following system of equations:</p>
<p><span class="math-container">$$\begin{align}
x^2 – (y– z)x – yz &= 0 \tag1 \\[4pt]
y^2 – \left(\frac8{z^2}– x\right... | Eric Towers | 123,905 | <p>From (1):
<span class="math-container">\begin{align*}
0 &= x^2 -(y-z)x-yz \\
&= (x+y+z)(x-y) -y(x-y) \text{,}
\end{align*}</span>
so either
<span class="math-container">$$ x+y+z = y \qquad \text{ or } \qquad x-y = 0 \text{.} $$</span>
We conclude either <span class="math-container">$x = -z$</span... |
3,881,390 | <p>I tried multiplying both sided by 4a
which leads to <span class="math-container">$(6x+4)^2=40 \pmod{372}$</span>
now I'm stuck with how to find the square root of a modulo.</p>
| David Cheng | 452,655 | <p>Completing the square, we have:
<span class="math-container">$$5x^2\equiv 2(x-1)^2$$</span>
Since <span class="math-container">$36\equiv 31+5\equiv 5$</span>, we have:
<span class="math-container">$$36x^2\equiv2(x-1)^2$$</span>
<span class="math-container">$$18x^2\equiv(x-1)^2$$</span>
Adding <span class="math-conta... |
3,881,390 | <p>I tried multiplying both sided by 4a
which leads to <span class="math-container">$(6x+4)^2=40 \pmod{372}$</span>
now I'm stuck with how to find the square root of a modulo.</p>
| Michael Rozenberg | 190,319 | <p><span class="math-container">$$3x^2+4x-2\equiv0\operatorname{mod}31$$</span>it's<span class="math-container">$$
21(3x^2+4x-2)\equiv0\operatorname{mod}31$$</span> or<span class="math-container">$$x^2-9x+20\equiv0\operatorname{mod}31$$</span> or<span class="math-container">$$(x-4)(x-5)\equiv0\operatorname{mod}31,$$</s... |
1,828,729 | <p>I am trying to solve this summation problem .
$$\sum\limits_{k = 0}^\infty {\left( {\begin{array}{*{20}{l}}
{n + k}\\
{2k}
\end{array}} \right)} \left( {\begin{array}{*{20}{l}}
{2k}\\
k
\end{array}} \right)\frac{{{{( - 1)}^k}}}{{k + 1}}$$
It will be grateful if someone could help me !!</p>
| mercio | 17,445 | <p>The line $y=ax+b$ has $2$ tangent points to the curve $y = x^4-2x^2-x$ if and only if $( x^4-2x^2-x)- (ax+b)$ has two (real) double roots $x_1,x_2$, so this polynomial has to be a perfect square $((x-x_1)(x-x_2))^2$</p>
<p>Can you complete the square $(x^4-2x^2+?x+?) = (x^2+?x+?)^2$ and then find its roots $x_1,x_2... |
56,082 | <p>Suppose I have a nested list such as,</p>
<pre><code>{{{A, B}, {A, D}}, {{C, D}, {A, A}, {H, A}}, {{A, H}}}
</code></pre>
<p>Where the elements of interest are,</p>
<blockquote>
<pre><code>{{A, B}, {A, D}}
{{C, D}, {A, A}, {H, A}}
{{A, H}}
</code></pre>
</blockquote>
<p>How would I use select to pick up only eleme... | Mr.Wizard | 121 | <p>I chose a slightly different formulation:</p>
<pre><code>expr = {{{A, B}, {A, D}}, {{C, D}, {A, A}, {H, A}}, {{A, H}}};
Select[expr, Count[#, {A, _}] > 1 &]
</code></pre>
<blockquote>
<pre><code>{{{A, B}, {A, D}}}
</code></pre>
</blockquote>
<p>I will note that this form is faster all four in the Accepted... |
1,509,340 | <p>I'm just wondering, what are the advantages of using either the Newton form of polynomial interpolation or the Lagrange form over the other?
It seems to me, that the computational cost of the two are equal, and seeing as the interpolated polynomial is unique, why ever use one over the other?</p>
<p>I get that they ... | Ian | 83,396 | <p>Frankly, Lagrange interpolation is mostly just useful for theory. Actually computing with it requires huge numbers and catastrophic cancellations. In floating point arithmetic this is very bad. It does have some small advantages: for instance, the Lagrange approach amounts to diagonalizing the problem of finding the... |
1,509,340 | <p>I'm just wondering, what are the advantages of using either the Newton form of polynomial interpolation or the Lagrange form over the other?
It seems to me, that the computational cost of the two are equal, and seeing as the interpolated polynomial is unique, why ever use one over the other?</p>
<p>I get that they ... | Gil | 60,928 | <p>Lagrange method is mostly a theoretical tool used for proving theorems. Not only it is not very efficient when a new point is added (which requires computing the polynomial again, from scratch), it is also numerically unstable.</p>
<p>Therefore, Newton's method is usually used. However, there is a variation of the L... |
1,509,340 | <p>I'm just wondering, what are the advantages of using either the Newton form of polynomial interpolation or the Lagrange form over the other?
It seems to me, that the computational cost of the two are equal, and seeing as the interpolated polynomial is unique, why ever use one over the other?</p>
<p>I get that they ... | Qiaochu Yuan | 232 | <p>Here is an example of a problem that is much easier using Newton interpolation than Lagrange interpolation. Let $p(x)$ be the unique polynomial of degree $n$ such that</p>
<p>$$p(k) = 3^k, 0 \le k \le n.$$</p>
<p>What is $p(n + 1)$? </p>
|
3,242,363 | <blockquote>
<p>Why does this function, <span class="math-container">$$\tan\left(x ^ {1/x}\right)$$</span>
have a maximum value at <span class="math-container">$x=e$</span>?</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/pqE0Q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pqE0Q.png" ... | pancini | 252,495 | <p>You know <span class="math-container">$x$</span>, right? If so, note that
<span class="math-container">\begin{align}
a&=0.029(x+b)+0.3\\
&=0.029(x+0.015(x+a))+0.3\\
&=(0.029+0.029\cdot 0.015)x+0.029\cdot0.015 a+0.3
\end{align}</span></p>
<p>Thus
<span class="math-container">$$(1-0.029\cdot0.015)a=(0.029... |
446,197 | <blockquote>
<p>Dudley Do-Right is riding his horse at his top speed of $10m/s$ toward the bank, and is $100m$ away when the bank robber begins to accelerate away from the bank going in the same direction as Dudley Do-Right. The robber's distance, $d$, in metres away from the bank after $t$ seconds can be modelled by... | Blue | 409 | <p>You mention converting the $\cot$ to $\csc$ ---presumably via the identity $\cot^2 + 1 = \csc^2$--- but perhaps you got off track.</p>
<p>$$\begin{align}
3\left(\cot^2\theta + 1 \right) - \csc^2\theta - 1 &= 3\csc^2\theta - \csc^2\theta - 1 \\
&=2\csc^2\theta - 1 \\[6pt]
&=\csc^2\theta + \left( \csc^2\t... |
2,828,487 | <p>If $\mathcal{R}$ is a von Neumann algebra acting on Hilbert space $H$, and $v \in H$ is a cyclical and separating vector for $\mathcal{R}$ (hence also for its commutant $\mathcal{R}'$), and $P \in \mathcal{R}, Q \in \mathcal{R}'$ are nonzero projections, can we have $PQv = 0$?</p>
<p>[note i had briefly edited this... | Dave L. Renfro | 13,130 | <p>In what follows I’ve restricted myself to those books I actually used at the time (nothing from after about 1991 or 1992), so this does not include anything that appeared afterwards.</p>
<p>I had a one-semester course out of Royden (covered most of the book --- it was a fairly fast-paced course), a one-semester cou... |
930,611 | <blockquote>
<p>Find the maximal value of the function for $a=24.3$, $b=41.5$:
$$f(x,y)=xy\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}$$</p>
</blockquote>
<p>Using the second derivative test for partial derivatives, I find the critical point in terms of $a$ and $b$ by taking partial derivatives of $x$ and $y$ and equa... | marco trevi | 170,887 | <p>I think you are missing some solutions: for instance, if $y=0$ also the points $(a,0)$ and $(-a,0)$ are solutions. Maybe during the calculations you divided by $y$ and/or $x$ without checking what happens when they vanish.</p>
<p>A thing that often helps me "seeing" the problem before actually calculating the solut... |
4,575,771 | <p>I need to show that <span class="math-container">$\int_0^1 (1+t^2)^{\frac 7 2} dt < \frac 7 2 $</span>. I've checked numerically that this is true, but I haven't been able to prove it.</p>
<p>I've tried trigonometric substitutions. Let <span class="math-container">$\tan u= t:$</span></p>
<p><span class="math-cont... | Adam Rubinson | 29,156 | <p>Since <span class="math-container">$t^2 \in [0,1],\ $</span> we may use the Binomial expansion,</p>
<p><span class="math-container">$$ \left( 1+ t^2 \right)^{7/2} = 1 + \frac{7}{2}t^2 + \frac{35}{8} t^4 + \frac{35}{16} t^6 + \frac{35}{128} t^8 + (\text{ alternating sequence of decreasing terms with negative leading ... |
210,110 | <p>A good approximation of $(1+x)^n$ is $1+xn$ when $|x|n << 1$. Does this approximation have a name? Any leads on estimating the error of the approximation?</p>
| Jean-Sébastien | 31,493 | <p>I would say it comes from the Bernoulli inequality. You can read it up on <a href="http://en.wikipedia.org/wiki/Bernoulli%27s_inequality" rel="nofollow">Wiki</a></p>
|
210,110 | <p>A good approximation of $(1+x)^n$ is $1+xn$ when $|x|n << 1$. Does this approximation have a name? Any leads on estimating the error of the approximation?</p>
| EuYu | 9,246 | <p>I would just call it the first order truncation of the <a href="http://en.wikipedia.org/wiki/Binomial_series">Binomial series</a>. If you want more terms of the series, then it's given by
$$(1+x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \mathcal{O}(x^4)$$
for the full series, you can visit the ... |
2,232,060 | <p>$f(x) = \sqrt[3]{1+ \sqrt[3]x}$ </p>
<p>I have to derive in 1st order and 2nd order</p>
<p>$f'(x) = \frac{1}{9x^\frac 23(1+x^\frac 13)^\frac 23}$ Is what I get after the first derivation </p>
<p>Now the teachers assistant is making $some$ $magic$ by showing that </p>
<p>$f(u) = \frac{1}{U^\frac 23}$</p>
<p>$u=... | Ivan Neretin | 269,518 | <p>Let's see. Apparently, $n$ is divisible by 2 and 3, otherwise it couldn't enforce divisibility by 12. Now, the product $ab$ is divisible by 3, which means that at least one of the numbers $a$ and $b$ is divisible by 3, and so is their sum, hence so is the other number. By similar reasoning, both are divisible by 2. ... |
1,936,043 | <p>I would like to prove that the sequence $n^{(-1)^{n}}$ is divergent. </p>
<p>My thoughts: I know $(-1)^n$ is divergent, so $n$ to the power of a divergent sequence is still divergent? I am not sure how to give a proper proof, pls help!</p>
| Ethan Bolker | 72,858 | <p>There are lots of correct answers. Here's a suggestion for how to attack a problem like this.</p>
<p>Before you try to invoke abstract principles like</p>
<blockquote>
<p>$n$ to the power of a divergent sequence is still divergent</p>
</blockquote>
<p>which you rightly wonder about (hence your "?") try <em>writ... |
506,394 | <p>Let $A=\{g\in C([0,1]):\int_{0}^{1}|g(x)|dx<1\}$. If $p\in [0,\infty]$, is $A$ an open set of $(C([0,1]), \left\|{\cdot}\right\|_p)$?</p>
<p>Is it obvious that if $p=1$ then $A$ is open in $(C([0,1]), \left\|{\cdot}\right\|_1)$, because $A=B(0,1)$.</p>
<p>I think $A$ is not open if $p>1$. Any hint to show th... | André Caldas | 17,092 | <p>Notice that from <a href="https://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality" rel="nofollow">Hölder's inequality</a>, valid for $p \geq 1$,
$$
\|f\|_1 = \|f \cdot 1\|_1 \leq \|f\|_p \|1\|_q = \|f\|_p,
$$
where $\frac{1}{p} + \frac{1}{q} = 1$,
since $\|1\|_q = 1$.</p>
<p>But this implies that the identity
$$... |
1,441,905 | <blockquote>
<p>Find the range of values of $p$ for which the line $ y=-4-px$ does not intersect the curve $y=x^{2}+2x+2p$</p>
</blockquote>
<p>I think I probably have to find the discriminant of the curve but I don't get how that would help.</p>
| Narasimham | 95,860 | <p>Equating slope to the required line slope for finding tangency point which demarcates between intersection and non-intersection. </p>
<p>$ 2 x + 2 = - p$</p>
<p>or</p>
<p>$ x = -(1 + p/2) $</p>
<p>and</p>
<p>$ y = p\, ( p/2 +1) -4 $</p>
<p>EDIT 1:</p>
<p>The parabola </p>
<p>$ y = 2 x ( x+1) -4 $</p>
<p>is ... |
1,915,450 | <p>Can anyone help me to prove this? This is given as a fact, but I don't understand why it is true.</p>
<blockquote>
<p>For an integer $n$ greater than 1, let the prime factorization of $n$ be $$n=p_1^ap_2^bp_3^cp_4^d...p_k^m$$
Where a, b, c, d, ... and m are nonegative integers, $p_1, p_2, ..., p_k$ are prime nu... | GoodDeeds | 307,825 | <p>This is solved using combinatorics. Any divisor <span class="math-container">$x$</span> of <span class="math-container">$n$</span> will be of the form
<span class="math-container">$$x=p_1^{n_1}p_2^{n_2}\cdots p_k^{n_k}$$</span>
where <span class="math-container">$0\le n_1\le a$</span>, <span class="math-container">$... |
2,130,836 | <p>My question is really simple: </p>
<p>Let $E$ be a vector space and $A_r(E)$ be the vector space of the alternating $r$-linear maps $\varphi:E\times\ldots \times E\to \mathbb R$. If $v_1,\ldots,v_r$ are linearly independent vectors. Can we get $\omega\in A_r(E)$ such that $\omega(v_1\ldots,v_r)\neq 0$? Is the conve... | B. S. | 231,386 | <p>When <span class="math-container">$2^n-1$</span> is a Mersenne prime,this can be resolved ( although this isn't very helpful, because we only know of 49 Mersenne primes and we don't know if they are finitely many.However, it sure is nice to know that <span class="math-container">$ 2^{74,207,281} − 1$</span> does not... |
4,074,630 | <p>Let <span class="math-container">$f: [a,b] \to [0,\infty)$</span> and <span class="math-container">$f$</span> is Riemann Integrable on every subinterval <span class="math-container">$[a + \epsilon,b]$</span> for <span class="math-container">$\epsilon > 0$</span>. Suppose that the improper Riemann integral exists.... | Eric Towers | 123,905 | <p>Consider the Riemann integral
<span class="math-container">$$ \int_0^1 x^{-2} \,\mathrm{d}x $$</span>
This is an improper Riemann integral due to the unbounded behaviour at the left endpoint of the interval of integration. It's value is defined to be (if this limit exists)
<span class="math-container">$$ \lim_{\... |
2,011,181 | <blockquote>
<p><strong>Question:</strong> Find the area of the shaded region given $EB=2,CD=3,BC=10$ and $\angle EBC=\angle BCD=90^{\circ}$.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/BFf2h.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BFf2h.jpg" alt="Diagram"></a></p>
<p>I first ... | yurnero | 178,464 | <p><strong>Hint</strong>: From the given info, you can compute the sum of the areas of triangles $\triangle EBC$ and $\triangle BDC$:
$$
\frac{1}{2}(2\cdot 10)+\frac{1}{2}(3\cdot 10)=25.
$$
With a quick observation, you can also compute the sum of the areas of triangles $\triangle EBA$ and $\triangle ACD$:
$$
\frac{1}... |
19,356 | <p>So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acqui... | Gerry Myerson | 3,684 | <p>I think one way to answer this question would be to get hold of the qualifying exams from University X from 50-60 years ago and compare them to the exams at the same university today. </p>
|
19,356 | <p>So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acqui... | Dieter | 30,457 | <p>I arrived at this question through my frustration that, despite my master degree, I could not come up with the proof of pi's irrationality just like that. So I studied it and wondered, why was this not on the list of things we learnt at university.</p>
<p>The question is different for an active professional mathema... |
2,322,678 | <p>We have $n$ different elements $(a_1,...,a_n)$ that are all the elements of $K$ and $\in$ finite field $K$.
I want to prove, that $\prod_{i=1}^{n} (X - a_i) + 1 \in K[X]$ doesn't have roots</p>
<p>I know, that if $a_i$ is a root of polynomial $p \in K[X]$ , then exists $f \in K[X]$ such that $p = (x - a_i)f$</p>... | Doug M | 317,162 | <p>$\int_0^\infty \frac {sin x}{x} dx = \int_0^\pi \frac {sin x}{x} dx + \int_\pi^{2\pi} \frac {sin x}{x} dx +\cdots$</p>
<p>or $\sum_\limits{i=0}^{\infty} \int_{i\pi}^{(i+1)\pi} \frac {sin x}{x} dx$</p>
<p>$\frac {\sin x}{x}\le 1$ for all $x$</p>
<p>$\int_0^\pi \frac {sin x}{x} dx < \pi$</p>
<p>$|\int_{i\pi}^{(... |
79,041 | <p>Let <span class="math-container">$\mathfrak{g}$</span> be the Lie algebra of a Lie group <span class="math-container">$G$</span> which acts on a manifold <span class="math-container">$M$</span>.
It is quite standard that the basic forms in <span class="math-container">$\Omega^*(M) \otimes W(\mathfrak{g}^*)$</span> f... | SGP | 11,786 | <p>see the very nice book of <a href="http://books.google.com/books?id=zYMp0GWLFiAC&lpg=PA248&ots=Bx2FxpUDmI&dq=guillemin%2520sternberg%2520supersymmetry&pg=PA182#v=onepage&q&f=false" rel="noreferrer">Guillemin-Sternberg (Supersymmetry and ...)</a>; it also has a reprint of Cartan's paper.</p>
|
1,210,018 | <p>$$ \begin{bmatrix}
1 & 1 \\
1 & 1 \\
\end{bmatrix}
\begin{Bmatrix}
v_1 \\
v_2 \\
\end{Bmatrix}=
\begin{Bmatrix}
0 \\
0 \\
\end{Bmatrix}$$</p>
<p>How can i solve this ?</p>
<p>I found it $$v_1+v_2=0$$ $$v_1+v_2=0$$ .</p>
<p>... | abel | 9,252 | <p>because you have only two variables $v_1, v_2$ you can think of the equation $v_1 + v_2 = 0$ as a line through the origin with slope $-1.$ you have one line in the plane because the second equation does not anything more to this line. any point on this line is a solution. all solutions are given by $v_1 = t, v_2 = ... |
188,158 | <p>I am interested in a function such that <code>f[m, i] = n</code> where <code>m, n</code> are positive integers and <code>n</code> is the <code>i</code>-th number relatively prime with <code>m</code>.</p>
<p>Getting a sample of the possible outputs of <code>f</code> is straightforward. For example, let <code>m = 30<... | Eric Towers | 16,237 | <pre><code>f[m_, i_] := (
m*#[[1]] +
Select[Range[m], GCD[#, m] == 1 &][[ #[[2]] ]]
)& [ QuotientRemainder[i, EulerPhi[m]] ]
RepeatedTiming[f[223 227, 4021987]]
(* {0.058, 4057980} *)
</code></pre>
<p>As long as <code>m</code> is not too big and you repeat <code>m</code>s, you can trade some m... |
2,937,671 | <p>Definition <span class="math-container">$\{A_i\}_{i\in I}$</span> be an indexed family of classes; Let
<span class="math-container">$$A=\bigcup_{i\in I} A_i.$$</span></p>
<p>The <span class="math-container">$product$</span> of the classes <span class="math-container">$A_i$</span> is defined to be the class
<span cl... | Porky | 807,523 | <p>This question appears in Pinter, Exercises 2.5, 6 a). I've been mulling over if for a while. I believe it's not correct as you state above, but as it appears in a textbook I've been trying to work out a way to prove it (perhaps by taking some liberty in notation or defintion!).</p>
<p>If <span class="math-container... |
2,534,999 | <p>I tried to solve $z^3=(iz+1)^3$. I noticed that $(iz+1)^3=i(z-1)^3$ so $(\frac{z-1}{z})^3=i$. How to finish it?</p>
| José Carlos Santos | 446,262 | <p>Let $\omega=\cos\left(\frac{2\pi}3\right)+\sin\left(\frac{2\pi}3\right)i=-\frac12+\frac{\sqrt3}2i$. Then $\omega^3=1$ and$$z^3=(iz+1)^3\iff z=iz+1\vee\omega z=iz+1\vee\omega^2z=iz+1.$$</p>
|
3,779,785 | <p>So I have this problem, <span class="math-container">$W=3^n -n -1$</span>. How to find all <span class="math-container">$n$</span> so <span class="math-container">$W$</span> can be divided by <span class="math-container">$5$</span>.</p>
<p><em>what I tried:</em>
I found all the remainders of <span class="math-contai... | Àlex Rodríguez | 813,535 | <p>First of all, notice that <span class="math-container">$3^4$</span> is congruent 1 mod 5 (by fermat's little theorem). Then, as you said, these residues are 1, 3, 4, 2, so consider next 4 cases:</p>
<ol>
<li>n is congruent 0 mod 4. now, let's see it modulo 5: <span class="math-container">$1-n-1=0 (mod 5)$</span>. T... |
3,773,856 | <p>I'm having trouble with part of a question on Cardano's method for solving cubic polynomial equations. This is a multi-part question, and I have been able to answer most of it. But I am having trouble with the last part. I think I'll just post here the part of the question that I'm having trouble with.</p>
<p>We ha... | José Carlos Santos | 446,262 | <p>Suppose that <span class="math-container">$u$</span> and <span class="math-container">$v$</span> are such that <span class="math-container">$u^3+v^3=-q$</span> and that <span class="math-container">$3uv=-p$</span>. You already know that then <span class="math-container">$u+v$</span> is a root of the depressed equati... |
2,972,355 | <p>How to convert sentence that contains “no more than 3” into predicate logic sentence?</p>
<p>For example: "No more than three <span class="math-container">$x$</span> satisfy <span class="math-container">$R(x)$</span>"
using predicate logic. </p>
<p>This is what I have for "exactly one <span class="math-container"... | Bram28 | 256,001 | <p>Interpreting 'no more than three' as 'at most three' (i.e. it could be three, two, one, or maybe just none at all), you can do:</p>
<p><span class="math-container">$$\exists x \exists y \exists z \forall u (R(u) \rightarrow (u = x \lor u = y \lor u = z))$$</span></p>
|
4,252,431 | <p>Let the constant <span class="math-container">$\alpha > 0$</span> be the problem
<span class="math-container">$$\left\{\begin{array}{cll}
u_t + \alpha u_x & = & f(x,t); \ \ 0 < x < L; \ t > 0\\
u(0,t) & = & 0; \ \ t > 0;\\
u(x,0) & = & 0; \ \ 0 < x < L.
\end{array}\right.... | herb steinberg | 501,262 | <p>Proof by contradiction:</p>
<p>Take the difference of the two equations and divide out common factors to get <span class="math-container">$y^3-y=x^3-x$</span>. This is a cubic in either variable in terms of the other, giving three solutions in each case, possible duplicates (x=y will appear in both sets). Use syn... |
4,527,429 | <p>I am confused as to how we open the abs value, do we get <span class="math-container">$e=0$</span> and <span class="math-container">$e=2x$</span>, or does the identity not exist?</p>
<p>Thanks.</p>
| Anne Bauval | 386,889 | <p>The identity <span class="math-container">$e$</span> does not exist because it should satisfy e.g. <span class="math-container">$-3=-3*e=|-3-e|\ge0.$</span></p>
|
2,412,783 | <p>I'm very new to linear algebra, and I have a homework problem that hasn't been covered in the book or by the professor. It seems like I have a fundamental misunderstanding of what matrices represent, but I can't find a good article or answer.</p>
<blockquote>
<p>Do the three lines $x_1 - 4x_2 = 1$, $2x_1 - x_2 = ... | Sam Mills | 399,086 | <p>Your solution is correct (I assume you have row-reduced properly) but the bottom row gives you no information - does $0x + 0y + 0z = 0$ tell you anything about $x, y,$ or $z$? You can safely scrub out a row of zeroes from a matrix and solve the system from there.</p>
|
3,012,090 | <p>Let <span class="math-container">$x>0$</span>. I have to prove that</p>
<p><span class="math-container">$$
\int_{0}^{\infty}\frac{\cos x}{x^p}dx=\frac{\pi}{2\Gamma(p)\cos(p\frac{\pi}{2})}\tag{1}
$$</span></p>
<p>by converting the integral on the left side to a double integral using the expression below:</p>
<p... | Jack D'Aurizio | 44,121 | <p>The Laplace transform of <span class="math-container">$\cos x$</span> is <span class="math-container">$\frac{s}{1+s^2}$</span> and the inverse Laplace transform of <span class="math-container">$\frac{1}{x^p}$</span> is <span class="math-container">$\frac{s^{p-1}}{\Gamma(p)}$</span>, hence
<span class="math-container... |
2,248,550 | <p>Will be the value in the form of $\frac{"0"}{"0"}$? Do I have to use the L'Hopital rule? Or can I say, that the limit doesn't exist?</p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One may observe that, as $x \to 0^+$, $y\to 1^-$,
$$
x+y-1=x-(1-y)=\left(\sqrt{x}-\sqrt{1-y}\right)\left(\sqrt{x}+\sqrt{1-y}\right).
$$</p>
|
1,778,098 | <p>Let $f,g: M^{k} \to N$ ($M$ and $N$ with out boundary ) such that they are homotopic then for $\omega$ a $k$-form on $N$ do we have that </p>
<p>$$ \int_M f^{\ast} \omega = \int_M g^{\ast} \omega$$ </p>
<p>as conclusion? I can't figure out a proof so I am starting to think that it is not true. I can't use Homolog... | user134824 | 134,824 | <p>Take $M=N=\mathbb R$ and take $\omega$ to be some nonvanishing $1$-form. Any two functions $\mathbb R\to\mathbb R$ are homotopic to each other, hence all are homotopic to zero. If your claim were true it would imply that the integral of any $1$-form is zero, which of course is not the case!</p>
<p>What you might be... |
165,582 | <p>The three lines intersect in the point $(1, 1, 1)$: $(1 - t, 1 + 2t, 1 + t)$, $(u, 2u - 1, 3u - 2)$, and $(v - 1, 2v - 3, 3 - v)$. How can I find three planes which also intersect in the point $(1, 1, 1)$ such that each plane contains one and only one of the three lines?</p>
<p>Using the equation for a plane
$$a_i ... | Cameron Buie | 28,900 | <p>Why does $k+3$ need to be even for the gcd of $k(k+5)/2$ and $k+3$ to divide $2$? For instance, consider the $k=2$ case.</p>
<p>Now, if $2$ had to divide the gcd of those numbers, then we could conclude that $k+3$ must be even.</p>
|
2,375,298 | <p>My question is as follows:</p>
<p>I have four different die and I'm trying to figure out how many possible combinations there are of (6,6,6,3)</p>
<p>My intuition tells me that there are 24 combinations. I'm imagining we have 4 spots:</p>
<hr>
<p>For the first spot there are 4 options (6,6,6,3)
For the second sp... | Aweygan | 234,668 | <p>Here's a proof that the space of weakly Cauchy sequences is closed in $\ell_\infty(X)$:</p>
<p>Let $WC(X)$ denote the subspace of $\ell_\infty(X)$ composed of weakly Cauchy sequences. Let $(x_n)$ be a sequence in $WC(X)$, with $x_n=(x_{nm})$, convergent to some $y=(y_n)\in\ell_\infty(X)$. Fix $f\in X^*$ and $\vare... |
3,865,607 | <p>Given <span class="math-container">$B\subseteq X$</span> with both <span class="math-container">$B$</span> and <span class="math-container">$X$</span> contractible. How would you prove that the inclusion map <span class="math-container">$i:B \to X$</span> is a homotopy equivalence?</p>
<p>Thank you</p>
| Tsemo Aristide | 280,301 | <p>Let <span class="math-container">$H_t$</span> be the homotopy between <span class="math-container">$Id_B$</span> and the constant map <span class="math-container">$f_B(x)=b$</span> and <span class="math-container">$G_t$</span> the homotopy between the identity of <span class="math-container">$X$</span> and the const... |
23,268 | <p>I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects like limits, colimits, and adjoint functors, I was able to do them, although I am sure my proofs were far longer and m... | Martin Brandenburg | 2,841 | <p>I pick up your remarks about sheaves. Indeed, the sheaf condition is a very good example to get a geometric idea of a limit.</p>
<p>Assume that $X$ is a set and $X_i$ are subsets of $X$ whose union is $X$. Then it is clear how to characterize functions on $X$: These are simply functions on the $X_i$ which agree on ... |
23,268 | <p>I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects like limits, colimits, and adjoint functors, I was able to do them, although I am sure my proofs were far longer and m... | Spice the Bird | 14,167 | <p>This answer is sort of an analogy, I am not quite sure how to make it precise. Further, It addresses that part of the question about a fiber product being anything from an intersection to a product (so this is perhaps a narrow answer). I am also not quite sure if this is "geometric". All of this said, a fiber produc... |
67,460 | <p>Denote the system in $GF(2)$ as $Ax=b$, where:
$$
\begin{align}
A=&(A_{ij})_{m\times m}\\
A_{ij}=&
\begin{cases}
(1)_{n\times n}&\text{if }i=j\quad\text{(a matrix where entries are all 1's)}\\
I_n&\text{if }i\ne j\quad\text{(the identity matrix)}
\end{cases}
\end{align}
$$
that is, $A$ i... | hmakholm left over Monica | 14,366 | <p>Some observations, too long for a comment:</p>
<p>If $n$ is odd, your matrix is not invertible, and so there is no solution for arbitrary $b$ (and a solution will not be unique if it exists). First, do some row operations to rewrite the constituent blocks to $$\pmatrix{1&1&1&1\\0&0&0&0\\0&am... |
487,171 | <p>Now I tried tackling this question from different sizes and perspectives (and already asked a couple of questions here and there), but perhaps only now can I formulate it well and ask you (since I have no good ideas).</p>
<p>Let there be $k, n \in\mathbb{Z_+}$. These are fixed.</p>
<p>Consider a set of $k$ integer... | coffeemath | 30,316 | <p>Each sorted string consists of $n_0$ copies of $0$, followed by $n_1$ copies of $1$, etc., ending with $n_{k-1}$ copies of $k-1$. The restriction on the $n_j$ are that they be nonnegative and sum to $n$. The number of solutions to that has a known expression via binomial coefficients, in this case it is $\binom{n+k-... |
3,243,406 | <p>I know that the function <span class="math-container">$f(x)=|x(x-1)^3|$</span> is not derivable in <span class="math-container">$x=0$</span>, but why is it derivable in <span class="math-container">$x=1$</span>?</p>
| XYSquared | 648,514 | <p>The function <span class="math-container">$f(x) = |x|\;|(x-1)^3|$</span>. <span class="math-container">$|x|$</span> certainly has a derivative 1 at <span class="math-container">$x=1$</span>, and <span class="math-container">$h(x)=|(x-1)^3|$</span> indeed has a derivative at <span class="math-container">$x=1$</span> ... |
295,597 | <p>I'm trying to solve this simple integral:</p>
<p>$$\frac12 \int \frac{x^2}{\sqrt{x + 1}} dx$$</p>
<p>Here's what I have done so far:</p>
<ol>
<li><p>$\displaystyle t = \sqrt{x + 1} \Leftrightarrow x = t^2 - 1 \Rightarrow dx = 2t dt$</p></li>
<li><p>$\displaystyle \frac12 \int \frac{x^2}{\sqrt{x + 1}} dx = \int \f... | Alex | 38,873 | <p>There is a (slightly) more obvious way of solving it: rewrite the numerator as $x^2+1-1$ and then the whole integral as a sum of two integrals:
$$
\int \frac{(x^2-1)dx}{\sqrt{x+1}} + \int \frac{dx}{\sqrt{x+1}}
$$
The second integral is easy, the first one is
$$
\int \frac{(x^2-1)dx}{\sqrt{x+1}} =\int \frac{(x+1)(x... |
3,909,005 | <p>I would like to ask what are the derivative values (first and second) of a function "log star": <span class="math-container">$f(n) = \log^*(n)$</span>?</p>
<p>I want to calculate some limit and use the De'l Hospital property, so that's why I need the derivative of "log star":
<span class="math-co... | Shaun | 732,537 | <p>You might try to use the definition of derivative to find your solutions
<span class="math-container">$$
\lim_{\Delta x \rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}
$$</span>
and evaluate at the different intervals that are valid for the function.</p>
|
3,998,098 | <p>I was asked to determine the locus of the equation
<span class="math-container">$$b^2-2x^2=2xy+y^2$$</span></p>
<p>This is my work:</p>
<blockquote>
<p>Add <span class="math-container">$x^2$</span> to both sides:
<span class="math-container">$$\begin{align}
b^2-x^2 &=2xy+y^2+x^2\\
b^2-x^2 &=\left(x+y\right... | Amanuel Getachew | 669,545 | <p>The equation is clearly of a conic section. However,since the coefficient of <span class="math-container">$xy$</span> is non-zero, the conic section is tilted by some angle <span class="math-container">$\theta$</span>. The value of <span class="math-container">$\theta$</span> can be determined as:
<span class="math-... |
56,134 | <p>As mentioned,
I wish to read the first line of a file, and if needed, overwrite it with a new string.
The aim is to have a CSV with a list of possible elements.
Example:</p>
<pre><code>Adding the elements: 5 A, 6 B, and 7 C to a blank CSV:
A B C
5 6 7
Adding 4 A, 9 D:
A B C D
5 6 7
4 0 0 9
Adding 2 B, 7 E
A B C D... | george2079 | 2,079 | <p>Being fairly annoyed that mathematica can't do this straightforward thing..here is a solution using an external python script:</p>
<pre><code> Export["test.txt",
Join[{StringJoin[Join[{"*"}, ConstantArray[" ", {80}], {"*"}]]},
RandomInteger[100, {3, 20}], {"end of file\n"}], "Table"]
FilePrint["t... |
963,503 | <p>Vectors $a$, $b$ and $c$ all have length one. $a + b + c = 0$. Show that
$$
|a-c| = |a-b| = |b-c|
$$
I am not sure how to get started, as writing out the norms didn't help and there is no way to manipulate
$$
|a-c| \le |a-b| + |b-c|
$$
to get an equality. I just need an idea of where to start.</p>
| please delete me | 168,166 | <p>You can try to prove the contrapositive of that implication. Start by assuming that $Z$ is not contained in $X$.</p>
|
4,280,424 | <p>The PDE:
<span class="math-container">$$\frac1D C_t-Q=\frac2rC_r+C_{rr}$$</span></p>
<p>on the domain <span class="math-container">$r \in [0,\bar{R}]$</span> and <span class="math-container">$t \in [0,+\infty]$</span> and where <span class="math-container">$D$</span> and <span class="math-container">$Q$</span> are R... | José Carlos Santos | 446,262 | <p>Yes, <span class="math-container">$\overline A=\Bbb R$</span>, since <span class="math-container">$A$</span> is not closed, from which it follows that the only closed subset of <span class="math-container">$\Bbb R$</span> which contains <span class="math-container">$A$</span> is <span class="math-container">$\Bbb R$... |
4,393,193 | <p>I am looking at the function
<span class="math-container">$$f(x) = \begin{cases}
\dfrac{x^2-1}{x^2-x} & x \ne 0,1\\
0 & x=0\\
2 &x=1
\end{cases}$$</span>
and am trying to show that <span class="math-container">$\lim_{x \to 0} f(x)$</span> DNE. This makes sense to me because <span class="math-container">... | David G. Stork | 210,401 | <p>The general equation of an ellipse (centered on the origin) with principal axes of length <span class="math-container">$a$</span> and <span class="math-container">$b$</span> rotated by angle <span class="math-container">$\theta$</span> is:</p>
<p><span class="math-container">$$\frac{(x \cos \theta + y \sin \theta)^2... |
2,329,730 | <p>Let $G$, an algebraic group, act morphically on the affine variety $X$.</p>
<p>Then we can also have $G$ act on the affine algebra $K[X]$ as follows:
$$\tau_x(f(y))=f(x^{-1}\cdot y),\qquad (x\in G, y\in X)$$</p>
<p>Then $\tau:G\to GL(K[X]),\quad \tau:x\mapsto \tau_x$.</p>
<p>Humphreys says that the reason that th... | Dr. Sonnhard Graubner | 175,066 | <p>if i'm right understand we get
$$x=\frac{b}{27}$$ and $$\frac{b^2}{27}=1024$$ is that what you meant?</p>
|
2,329,730 | <p>Let $G$, an algebraic group, act morphically on the affine variety $X$.</p>
<p>Then we can also have $G$ act on the affine algebra $K[X]$ as follows:
$$\tau_x(f(y))=f(x^{-1}\cdot y),\qquad (x\in G, y\in X)$$</p>
<p>Then $\tau:G\to GL(K[X]),\quad \tau:x\mapsto \tau_x$.</p>
<p>Humphreys says that the reason that th... | TStancek | 109,322 | <p>Since $ax=b$, then $bx=(ax)\cdot x$, so you get quadratic equation $ax^2=c$, or $ax^2-c=0$. Solve it and you will find your solutions.</p>
|
3,965,834 | <p>Does this sum converge or diverge?</p>
<p><span class="math-container">$$ \sum_{n=0}^{\infty}\frac{\sin(n)\cdot(n^2+3)}{2^n} $$</span></p>
<p>To solve this I would use <span class="math-container">$$ \sin(z) = \sum \limits_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{(2n+1)!} $$</span></p>
<p>and make it to <span class="math... | Community | -1 | <p>Clearly,</p>
<p><span class="math-container">$$\left|\sum_{n=0}^\infty\frac{\sin(n)(n^2+3)}{2^n}\right|<\sum_{n=0}^\infty\frac{(n^2+3)}{2^n}.$$</span></p>
<p>Then by the ratio test, for <span class="math-container">$n\ge2$</span>,</p>
<p><span class="math-container">$$\frac12\frac{(n+1)^2+3}{n^2+3}\le\frac67$$</s... |
31,158 | <p>To generate 3D mesh <a href="http://reference.wolfram.com/mathematica/TetGenLink/tutorial/UsingTetGenLink.html#167310445" rel="nofollow noreferrer">TetGen</a> can be easily used. Are there similar functions (or a way to use TetGen) to generate 2d mesh? I know that such functionality can be <a href="https://mathemati... | Misery | 742 | <p>Since Mathematica 10.3 ToElementMesh[] function can be used, along with FEM solver. For details see <a href="https://reference.wolfram.com/language/FEMDocumentation/tutorial/ElementMeshCreation.html" rel="noreferrer">this link</a></p>
|
1,130,487 | <p>Jessica is playing a game where there are 4 blue markers and 6 red markers in a box. She is going to pick 3 markers without replacement.
If she picks all 3 red markers, she will win a total of 500 dollars. If the first marker she picks is red but not all 3 markers are red, she will win a total of 100 dollars. Under ... | mrf | 19,440 | <p>$\ln(2x+2) = \ln 2 + \ln(x+1)$ (assuming $x > -1$). Antiderivatives are only determined up to an additive constant.</p>
|
405,783 | <p>I saw the following in my lecture notes, and I am having difficulties
verifying the steps taken.</p>
<p>The question is:</p>
<blockquote>
<p>Assuming $0<\epsilon\ll1$ find all the roots of the polynomial
$$\epsilon^{2}x^{3}+x+1$$ which are $O(1)$ up to a precision of
$O(\epsilon^{2})$</p>
</blockquote>
<... | Hagen von Eitzen | 39,174 | <p>Much simpler: As $x(\epsilon)\in O(1)$, we have immediately from rewriting the cubic that
$$x(\epsilon)=-1-\epsilon^2x(\epsilon)^3\in -1+O(\epsilon^2).$$</p>
|
269,655 | <p>I am trying to find a nonlinear model from the data.</p>
<p><a href="https://i.stack.imgur.com/W6JEI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/W6JEI.png" alt="enter image description here" /></a></p>
<p>My code is below:</p>
<pre><code>data = {{0.0, 0.0}, {0.05, 0.87}, {0.1, 0.99}, {0.15, 0.... | Michael Seifert | 27,813 | <p>The problem seems to stem from the first and last data points, with <span class="math-container">$x = 0$</span> and <span class="math-container">$x = 1$</span>. My guess is that it has to do with <span class="math-container">$\partial f/\partial x$</span> being singular at these points. In addition, <span class="m... |
386,172 | <p>The expression was simplified in the answer to <a href="https://math.stackexchange.com/questions/384592/finding-markov-chain-transition-matrix-using-mathematical-induction">this question</a>. I'm trying to simplify it but I got stuck. Multiplying all the factors and regrouping didn't work, but maybe I'm doing the wr... | xisk | 73,012 | <p>Start with: $(p)(\frac{1}{2}(2p-1)^n) + (1-p)(-\frac{1}{2}(2p-1)^n) = x$<br>
(I'm setting it equal to $x$ so it's easier to follow.) </p>
<p>$\therefore (p)(2p-1)^{n} + (1-p)(-1)(2p-1)^{n} = 2x$<br>
Factor the $(2p-1)^{n}$:<br>
$\therefore (2p-1)^{n} \cdot (p + (1-p)(-1)) = 2x$<br>
$\therefore (2p-1)^{n} \cdot (p... |
3,652,102 | <p>Let, <span class="math-container">$(P,\le)$</span> be the poset.<br>
I have begun to solve this in the following way-
Note that, <span class="math-container">$rs-r\le rs-s\iff r\ge s$</span><br>
So, without loss of generality assume that <span class="math-container">$r\ge s$</span>, then <span class="math-container"... | Brian M. Scott | 12,042 | <p>HINT: Use <a href="https://en.wikipedia.org/wiki/Dilworth%27s_theorem" rel="nofollow noreferrer">Dilworth’s decomposition theorem</a>. I’ve finished the argument in the spoiler-protected block below.</p>
<blockquote class="spoiler">
<p> Let <span class="math-container">$A$</span> be an antichain in <span class="m... |
802,848 | <p>I am reading this book, <em>Gödel's Proof</em>, by James R. Newman, at location 117 (Kindle), it says,</p>
<blockquote>
<p>For <strong>various reasons</strong>, this axiom, (through a point outside a given line only one parallel to the line can be drawn), did not appear "self-evident" to the ancients.</p>
</block... | mau | 89 | <p>Actually the standard definition of the Fifth Postulate does not involve (of course) infinite: Euclid says that that two lines which cross a third one will eventually meet on the side where the angles made with the third one add to less than two right angles.</p>
<p>Ancient Greeks were uneasy with it because they t... |
4,400,261 | <p>If <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are <a href="https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Standard_normal_random_vector" rel="nofollow noreferrer">bivariate normal PDFs</a> having correlation coefficients <span class="math-container">$ρ_f$</spa... | Leander Tilsted Kristensen | 631,468 | <p>I don't think that you can determine the correlation coefficient without also knowing the variances, but you should be able to determine the new covariance matrix given the covariance matrices for <span class="math-container">$f$</span> and <span class="math-container">$g$</span>. It can be shown that the distributi... |
591,765 | <blockquote>
<p>What is the way to convince myself that $\left\langle(1,2),\ (1,2,3,4)\right\rangle=S_4$ but $\left\langle(1,3),\ (1,2,3,4)\right\rangle\ne S_4$?</p>
</blockquote>
<p>Let $\sigma$ be any transposition and $\tau$ be any $p-$cycle, where $p$ is a prime.
Then show that $S_p=\langle\sigma,\tau\rangle$.</... | Betty Mock | 89,003 | <p>Not sure how to "convince" you, but I suspect you are worried because some of the 2-cycles do the job and others do not. The trick is in picking the right 4-cycle to go with your 2-cycle or vice versa. (1,2) works with (1,2,3,4) because 1 and 2 are adjacent in (1,2,3,4) whereas 1,3 is not. But (1,3) will work wit... |
1,749,128 | <p>How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$? It should be associated with the geometric series. Setting $t=x-3,\ x=t+3$, then I don't know how to continue, could someone clarify the procedure?</p>
| BrianO | 277,043 | <p>Ultimately your question, as expressed in your <strong>Edit</strong>, is a philosophical one, not a mathematical one. If you're a <em>formalist</em>, then no, the objects of consideration have no independent existence, and, visualize what we may, only the formal systems, marks on paper and screens, really exist. If ... |
3,423,225 | <p>I know that the angle <span class="math-container">$\theta$</span> of a right-angled triangle, centered at the origin, is defined as the radian measure of its intersection point with the unit circle, and that <span class="math-container">$\cos(\theta)$</span> and <span class="math-container">$\sin(\theta)$</span> ar... | suhbell | 592,879 | <p>Triangles that have the same angle measurements are similar. Since Both of these triangles are right triangles, and both of these triangles share a common angle, then these triangles have 2 angles that are the same. Since a triangle only has 3 angles, and if 2 angles are the same, then the third angle must be the sa... |
2,882,696 | <p>$a,b,x$ are elements of a group .</p>
<p>$x$ is the inverse of $a$.</p>
<p>Here is my attempt to prove it :-</p>
<p>$a\cdot b = e$</p>
<p>$x\cdot (a\cdot b) = x\cdot e$</p>
<p>$(x\cdot a)\cdot b = x$</p>
<p>$e\cdot b = x$</p>
<p>$b = x$</p>
<p>Are my steps correct?
What I wanted to prove is that if $ab = e$,... | Cornman | 439,383 | <p>No, it does not imply that $x=a$. It implies that $x=b$. Maybe a typo?</p>
<p>We have $ab=e$ since $x=a^{-1}$ we get after multiplying both sides with $a^{-1}$:</p>
<p>$a^{-1}ab=a^{-1}e\Leftrightarrow eb=a^{-1}\Leftrightarrow b=a^{-1}=x$</p>
|
2,882,696 | <p>$a,b,x$ are elements of a group .</p>
<p>$x$ is the inverse of $a$.</p>
<p>Here is my attempt to prove it :-</p>
<p>$a\cdot b = e$</p>
<p>$x\cdot (a\cdot b) = x\cdot e$</p>
<p>$(x\cdot a)\cdot b = x$</p>
<p>$e\cdot b = x$</p>
<p>$b = x$</p>
<p>Are my steps correct?
What I wanted to prove is that if $ab = e$,... | Robert Lewis | 67,071 | <p>This is pretty standard, basic and elementary stuff; the kind of stuff one usually sees in the first few pages of a textbook on group theory; but essential stuff nevertheless.</p>
<p>Our OP neraj's proof that $x = b$ is, of course, unarguably flawless. Lauds.</p>
<p>If we wish to see that</p>
<p>$ab = e \Longrig... |
3,056,121 | <p>I'm trying to find a function with infinitely many local minimum points where x <span class="math-container">$\in$</span> [0,1] and f has only 1 root. No interval should exist where the function is constant.</p>
| Jack D'Aurizio | 44,121 | <p><span class="math-container">$$\int_{0}^{+\infty}\frac{x^{2+\alpha}}{(1+x^2)^3}\,dx \stackrel{(*)}{=}\frac{\pi(1-\alpha^2)}{16\cos\frac{\pi \alpha}{2}} $$</span>
<span class="math-container">$(*)$</span>: we use the substitution <span class="math-container">$\frac{1}{1+x^2}=u$</span>, the Beta function and the refle... |
3,056,121 | <p>I'm trying to find a function with infinitely many local minimum points where x <span class="math-container">$\in$</span> [0,1] and f has only 1 root. No interval should exist where the function is constant.</p>
| Aleksas Domarkas | 562,074 | <p><span class="math-container">$$\int_0^\infty \frac{x^2 \ln(x)}{(1+x^2)^3} {\rm d}x=
\int_0^1 \frac{x^2 \ln(x)}{(1+x^2)^3} {\rm d}x+
\int_1^\infty \frac{x^2 \ln(x)}{(1+x^2)^3} {\rm d}x
$$</span>
Then change in first integral <span class="math-container">$x=\frac{1}{t}$</span>.</p>
|
1,476,946 | <p>So, I'm just starting to peruse "Categories for the Working Mathematician", and there's one thing I'm uncertain on. Lets say I have three objects, $X,Y,Z$ and two arrows $f,g$ such that $X\overset {f} {\to}Y\overset {g} {\to}Z$. Does this necessitate the composition arrow exist so the diagram commutes, i.e mus... | Ben Sheller | 250,221 | <p>If you have arrows $f:X\to Y$ and $g:Y\to Z$, it means that $h=g\circ f:X\to Z$ exists.</p>
<p>It is definitely NOT "If an arrow $h:X\to Z$ exists...", since this would imply that it would have to be true for every arrow $h:X\to Z$.</p>
|
251,466 | <p>Let $A$, $B$ and $C$ be three points in a disk,
does $f\left(A,B,C\right)=\mbox{Area}\left(\mbox{triangle}\,ABC\right)/\mbox{Perimeter}\left(\mbox{triangle}\,ABC\right)$
have maximum on
the boundary? </p>
| coffeemath | 30,316 | <p>First note that if a triangle is subjected to a homothety by factor $r>1$ then the area multiplies by $r^2$ and the perimeter by $r$, so that area/perimeter gets multiplied by $r$. This means for the triangle $ABC$ with longest side say $AB$, that we may expand and move the triangle until vertices $A,B$ are on th... |
68,386 | <p>I'm looking for a theorem of the form </p>
<blockquote>
<p>If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.</p>
</blockquote>
<p>My attempts to do this purely algebraically are not working, so I started looking into methods fr... | Fernando Muro | 12,166 | <p>Thinking geometrically, take the disjoint union of a plane and a point in the 3-dimensional affine space. This has dimension 2. If you remove the plane by inverting its equation, you obtain the the point, which is 0-dimensional. </p>
<p>Algebraically, let $k$ be a field, $R = k[x,y,z]/(x^2-x,xy,xz)$, $v=x$. Then $R... |
2,083,347 | <p>Let's consider a linear operator
$$
Lu = -\frac{1}{w(x)}\Big(\frac{d}{dx}\Big[p(x)\frac{du}{dx}\Big] + q(x)u\Big)
$$
So the Sturm-Liouville equation can be written as
$$
Lu = \lambda u
$$
Why the proper setting for this problem is the weighted Hilbert space $L^2([a,b], w(x)dx)$?</p>
| Varun Iyer | 118,690 | <p>So good job on finding your inverses:</p>
<p>$$
-\frac{x-2}{4}
$$
$$
-\frac{x-6}{5}
$$</p>
<p>Recall that the definition of a function maps $x$ values to $y$ values, or $k(x)$. Therefore, the inverse of a function $k(x)$ maps the $y$ values to the $x$ values.</p>
<p>So we must find our range of $x$ values:</p>
<... |
2,785,698 | <p>Please help me go over this problem; I am a bit confused.</p>
<p>Find ${\displaystyle \frac{\mathrm d}{\mathrm dt} \int_2^{x^2}e^{x^3}\mathrm dx}$.</p>
| ThatEvilChickenNextDoor | 562,857 | <p>First off, I'm assuming that you want to evaluate this:
$$
\frac{d}{dx} \int_2^{x^2}e^{t^3}dt
$$
You want to find the derivative of the integral, so we replace the $x$ inside the integral with a dummy variable to avoid confusion. Now, let's set $f(x)=e^{x^3}$ and say that $F(x)$ is the antiderivative, so $F'(x)=f(x)... |
4,328,117 | <p>I have this lambda expression</p>
<p><span class="math-container">$$(\lambda xyz.xy(zx)) \;1\; 2\; 3$$</span></p>
<p>or</p>
<p><span class="math-container">$$(\lambda x. (\lambda y. (\lambda z.xy(zx))))\;1\;2\;3$$</span>
<span class="math-container">$$(\lambda y. (\lambda z.1y(z1))))\;2\;3$$</span>
<span class="math... | Couchy | 87,768 | <p>First of all, I should point out that there are two ways to understand the lambda calculus, it can be typed or untyped. Not all lambda terms can be typed.</p>
<hr />
<p>First observe that the term <span class="math-container">$λxyz.xy(zx)$</span> can be typed. Since it takes three arguments, let us say the type is
<... |
463,239 | <p>Integrate $$\int{x^2(8x^3+27)^{2/3}}dx$$</p>
<p>I'm just wondering, what should I make $u$ equal to?</p>
<p>I tried to make $u=8x^3$, but it's not working. </p>
<p>Can I see a detailed answer?</p>
| user71352 | 71,352 | <p>Let $u=8x^{3}+27$ then $du=24x^{2}dx$. So</p>
<p>$\displaystyle\int x^{2}(8x^{3}+27)^{\frac{2}{3}}dx=\frac{1}{24}\int(8x^{3}+27)^{\frac{2}{3}}(24x^{2})dx=\frac 1{24}\int u^{\frac{2}{3}}du$</p>
|
877,646 | <p>Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. ,</p>
<p>Following are the given conditions</p>
<p>a) each $A_i$ is non invertible <strong>except $A_0$</strong> because their determinant is zero.</p>
<p>b) $\sum_{n=0}^\infty A_i$ is invertible and determinant is not zero</p>
<p>c) </p>
<... | David Zhang | 80,762 | <p>The mistake is simple--$\mathrm{Ci}$ has a branch cut across the negative real axis, so $\mathrm{Ci}(-\infty - i)$ should indeed evaluate to $-i \pi$ rather than $i \pi$.</p>
|
1,380,697 | <p>I am currently an undergraduate and thinking about applying to graduate school for math. The problem is that I don't know what field I want to go. Taking graduate classes even more confuse me because the more I learn the less I know what specifically I want to do. My question is to where to find an information about... | David Wheeler | 23,285 | <p>As you finish your undergraduate studies, you should have had at least some introduction or passing acquaintance to the following subjects:</p>
<p>Algebra (the abstract kind)</p>
<p>Discrete mathematics (maybe some number theory)</p>
<p>Linear algebra</p>
<p>Real/complex analysis (post-calculus)</p>
<p>Topology... |
3,931,831 | <p>For the scenario given below, I am confused about if the samples are dependent or independent since the scenario does not mention anything about the samples being paired/related or vice versa.</p>
<p>I am aware if terms such as paired, repeated measurements, within-subject effects, matched pairs, and pretest/posttes... | Math Lover | 801,574 | <p>This is from your working -</p>
<p><span class="math-container">$(3x^2 -3, 3y^2 -3) = \lambda (1,2)$</span></p>
<p><span class="math-container">$3x^2 - 3 = \lambda, 3y^2-3 = 2\lambda$</span></p>
<p>Equating <span class="math-container">$\lambda$</span> from both equations,</p>
<p><span class="math-container">$6x^2-6... |
2,362,477 | <p>Solve the equation $f(x) = 2$.
I reached the stage $\sin(x) = {2\over 3}$ but then (as I remember it was solved) using $x = \sin^{-1}(2/3)$ (sine inverse) I get the answer $x = 41.81$ but the correct answer is $x = 0.730$ or $2.41$. Why is this so? Sorry it might be a silly question but it had been long since I stud... | Riccardo.Alestra | 24,089 | <p>The correct answer is:$$x=\arcsin(2/3)=0.7297276563$$</p>
|
4,622,956 | <p>I think <span class="math-container">$\,9\!\cdot\!10^n+4\,$</span> can be a perfect square, since it is <span class="math-container">$0 \pmod 4$</span> (a quadratic residue modulo <span class="math-container">$4$</span>), and <span class="math-container">$1 \pmod 3$</span> (also a quadratic residue modulo <span cla... | Umesh Shankar | 816,291 | <p>Note that if <span class="math-container">$$9\!\cdot\!10^n+4=m^2\implies (m+2)(m-2)=9\!\cdot\!10^n$$</span></p>
<p>Note that <span class="math-container">$5^n$</span> must divide either <span class="math-container">$m+2$</span> or <span class="math-container">$m-2$</span>.
If that happens, the rest of the factors ar... |
3,545,548 | <p><span class="math-container">$\def\LIM{\operatorname{LIM}}$</span>
Let <span class="math-container">$(X,d)$</span> be a metric space and given any cauchy sequence <span class="math-container">$(x_n)_{n=1}^{\infty}$</span> in <span class="math-container">$X$</span> we introduce the formal limit <span class="math-cont... | Paweł Czyż | 551,592 | <p>Let's prove a bit stronger statement.</p>
<p><strong>Propostion</strong> Let <span class="math-container">$x_1, \dots, x_n\in \mathbb R$</span> and <span class="math-container">$0 < k \neq 1$</span> be another real number. Then numbers
<span class="math-container">$$ k^{x_1}, \dots, k^{x_n}$$</span>
are successi... |
3,989,878 | <p>I can't solve this problem. I tried to find <span class="math-container">$\tan x$</span> directly by solving cubic equations but I failed.</p>
<p>The problem is to find <span class="math-container">$\tan x\cot 2x$</span> given that
<span class="math-container">$$\tan x+ \tan 2x=\frac{2}{\sqrt{3}}, \>\>\>\&g... | Quanto | 686,284 | <p>Denote <span class="math-container">$y = \tan x \cot 2x = \frac{1-\tan^2x}2$</span> and express <span class="math-container">$\tan x+ \tan 2x=\frac{2}{\sqrt{3}}$</span> as a system of equations in <span class="math-container">$x,y$</span></p>
<p><span class="math-container">$$\tan x+\frac{2\tan x}{1-\tan^2x}=\left(1... |
2,400,654 | <p>I am told that the statement "any closed set has a point on its boundary" is false, yet I don't know how to disprove it. In fact, I think it is true. </p>
<p>Suppose we have [a,b], a closed set. Then, the boundary would be {a,b}, both of which are the elements of the set. So, there we have a closed set that has poi... | kimchi lover | 457,779 | <p>Of course it's just a convention, so open to change <em>in principle</em>.</p>
<p>One reason for the current convention is that when working with such functions one often makes changes of variables of the form $x = cy$, with attendant $dx = c\,dy$ substitutions. When working this way it is often handy to know that... |
2,791,068 | <p>The Laplace transform of a measure $\mu$ on the real line is defined by
$$f_{\mu}(s)= \int_{\mathbb{R}}e^{-st}d\mu(t), \hspace{1cm} \forall s \geqslant 0.$$
My question is ----</p>
<p>1)Does the Laplace transform of a measure (finite or infinite) always exists?</p>
<p>2)If not, can it be said that the Laplace tran... | Arrhenius Impostor | 563,282 | <p>Your definition of the Laplace transform will not converge in general, because if <span class="math-container">$\mu$</span> has support on the whole real line then the exponential term can blow up, depending on the exact form of <span class="math-container">$\mu$</span>.</p>
<p>The common definition of the Laplace t... |
66,068 | <p>I have a list like this. </p>
<pre><code>cdatalist = {{1., 0.898785, Failed, Failed, 50., 25., "serial"}, {1., 1.31175,1., Failed, 50., 25., "serial"}, {1., 18.8025, Failed, 0.490235, 50., 25., "serial"}, {1., 19.6628, 0.990079, Failed, 50., 25., "serial"}, {1., 39.547, Failed, Failed, 50., 25., "serial"}, {1., 39.... | user2895279 | 11,600 | <pre><code>cdatalist2 =
Cases[cdatalist[[All, 1 ;; 3]], {_?NumericQ, _?NumericQ, _?NumericQ}]
</code></pre>
|
998,769 | <p>A random variable $X$ distributed over the interval $[0, 2\pi]$</p>
<p>a) the pdf of $X$</p>
<p>b) the cdf of $X$</p>
<p>c) $P(\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$</p>
<p>d) $P(-\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$</p>
<p>my answers:</p>
<p>a) pdf of $X$ is $f(x) = \begin{cases}\frac{1}{2\pi},& 0... | Rey | 73,712 | <p>For proving $h$ is injective, you want to show:
$$h(x)=h(y) \iff x=y $$
Which can be proved with something like this: </p>
<p>From $h(x)=h(y)$ we write:
$$\frac{x^3}{|x|} = \frac{y^3}{|y|} $$
$$ \Rightarrow x^2\frac{x}{|x|} = y^2\frac{y}{|y|} $$
$$ \Rightarrow x^2 sign(x) = y^2 sign(y) $$
$$ \Rightarrow \fr... |
3,166,999 | <p>I'm reading Kechris' book "Classical Descriptive Set Theory" and the author gives the following definition (pp. <span class="math-container">$49$</span>, row <span class="math-container">$3$</span>):</p>
<blockquote>
<p>A <strong>weak basis</strong> of a topological space <span class="math-container">$X$</span> ... | Cameron Buie | 28,900 | <p>While it is true that for every <span class="math-container">$x,$</span> any neighborhood <span class="math-container">$U(x)$</span> contains an element of the weak basis, say <span class="math-container">$V(x),$</span> we <em>don't</em> know that <span class="math-container">$V(x)$</span> is a neighborhood of <span... |
4,325,373 | <blockquote>
<p>Using Algebraic approach, test the convexity of the set <span class="math-container">$$S=\{(x_1,x_2):x_2^2\geq8x_1\}$$</span></p>
</blockquote>
<p>Definition of convexity: <span class="math-container">$S \in \mathbb R^2$</span> is a convex set if <span class="math-container">$\forall \alpha \in \mathbb ... | TravorLZH | 748,964 | <p>To verify this, I recommend using Dirichlet series as it is a more powerful device to study Dirichlet convolution. Suppose we define the symbol</p>
<p><span class="math-container">$$
D(s;f)=\sum_{n\ge1}{f(n)\over n^s}
$$</span></p>
<p>It is not difficult to show that <span class="math-container">$D(s;f)D(s;g)=D(s;f*... |
140,819 | <p>Everybody loves the good old quadratic Mandelbrot set. As you probably know, both it and the corresponding quadratic Julia sets are defined by the iteration $f(z) = z^2 + c$.</p>
<p>You might expect, however, that $f(z) = az^2 + bz + c$ would give you more possibilities. However, all the books on the subject assert... | lhf | 589 | <p>For the first question, the key word is <em>conjugation</em>: there is an affine change of coordinates $\phi$ such that $f(z) = az^2 + bz + c$ becomes $g(z)=z^2+c$ in the sense that $f\circ\phi=\phi\circ g$. Since, $g=\phi^{-1} \circ f\circ\phi$, the iterates of $f$ are conjugated to the iterates of $g$ by the same ... |
3,788,298 | <p>Let <span class="math-container">$f(x)$</span> be an integrable function on <span class="math-container">$[0,1]$</span> that obeys the property <span class="math-container">$f(x)=x, x=\frac{n}{2^m}$</span> where <span class="math-container">$n$</span> is an odd positive integer and m is a positive integer. Calculat... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$f$</span> is Riemann integbrable it is continuous almost everywhere. This shows that <span class="math-container">$f(x)=x$</span> almost everywhere (since the equation holds on a dense set). Hence the integral is <span class="math-container">$\int_0^{1} xdx =\frac 1 2 $</span>.</p>
|
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