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 |
|---|---|---|---|---|
629,550 | <p>Investigate the convergence of $\sum a_n$ where $a_n = \displaystyle\int_0^1 \dfrac{x^n}{1-x}\sin(\pi x) \,dx$.</p>
<p>We have thought about using the dominated convergence theorem to find $\lim a_n$, but that would result in something like $\lim a_n = \lim \displaystyle\int_0^1 \dfrac{x^n}{1-x}\sin(\pi x) \,dx = \... | Sasha | 11,069 | <p>Using $\sin(\pi x) \geqslant \pi x(1-x)$ for $0\leqslant x \leqslant 1$
$$
a_n = \int_0^1 \frac{x^n}{1-x} \sin(\pi x) \mathrm{d}x > \int_0^1 \pi x^{n+1} \mathrm{d}x = \frac{\pi}{n+2}
$$
it follows that $\sum_{n=0}^\infty a_n$ diverges.</p>
|
629,550 | <p>Investigate the convergence of $\sum a_n$ where $a_n = \displaystyle\int_0^1 \dfrac{x^n}{1-x}\sin(\pi x) \,dx$.</p>
<p>We have thought about using the dominated convergence theorem to find $\lim a_n$, but that would result in something like $\lim a_n = \lim \displaystyle\int_0^1 \dfrac{x^n}{1-x}\sin(\pi x) \,dx = \... | Jonathan Y. | 89,121 | <p>Though Sasha's solution is perfect, I'd like to suggest another approach (suitable to the [measure-theory] tag):</p>
<p>We note that
$$S_N = \sum_{n=0}^N a_n = \int_0^1 \frac{1-x^{N+1}}{1-x}\frac{\sin(\pi x)}{1-x}dx$$
Now, $f_N := \frac{1-x^{N+1}}{1-x}\frac{\sin(\pi x)}{1-x}\to\frac{\sin(\pi x)}{(1-x)^2}$ pointwise... |
211,920 | <blockquote>
<p>Prove that if $S = S^T$ is symmetric and non-singular, then $S^2$ is
positive definite.</p>
</blockquote>
<p>My attempt:</p>
<p>Suppose $S$ is an $m\times n$ symmetric matrix with linearly independent columns, and suppose $q(x) > 0$, then the matrix $q(x) = \mathbf{x}^\mathrm{T}S\mathbf{x}$ is ... | EuYu | 9,246 | <p>Notice that your ordered basis is in fact an orthogonal basis under the standard inner product. In particular, that means we have
$$v = \frac{\langle v,\ f_1\rangle}{2} f_1 + \frac{\langle v,\ f_2\rangle}{2} f_2 + \frac{\langle v,\ f_3\rangle}{2} f_3 + \frac{\langle v,\ f_4\rangle}{2} f_4$$
This gives you
$$v = f_1... |
3,508,730 | <p>I was trying to prove that <span class="math-container">$(N+\sqrt{N^2-1})^k$</span>, where k is a positive integer, differs from the integer nearest to it by less than <span class="math-container">$(2N-\frac{1}{2})^{-k}$</span>. Note: N is an integer greater than 1. </p>
<p>So, I tried to look for the answer of the... | Hagen von Eitzen | 39,174 | <p>By induction on <span class="math-container">$k$</span>, we have
<span class="math-container">$$ (N\pm \sqrt{N^2-1})^k=a\pm b\sqrt{N^2-1}$$</span>
with <span class="math-container">$a,b\in\Bbb Z$</span>.
And of course
<span class="math-container">$(N-\frac12)^2=N^2-N+\frac14<N^2-1 $</span> implies
<span class="ma... |
127,643 | <p>Hello everybody,</p>
<p>I'm a math student who has just got his first degree, and I am studying algebraic geometry since a few months. Something I have noticed is the (to my eyes) huge amount of commutative algebra one needs to push himself some deeper than the elementary subjects. This can be seen just counting th... | Sándor Kovács | 10,076 | <p>I think this is a very good question, because studying commutative algebra on its own is hard, it is much better to do it with some idea of what all that means geometrically.</p>
<p>In my opinion the best entry to commutative algebra is provided by Miles Reid's <a href="http://rads.stackoverflow.com/amzn/click/052... |
127,643 | <p>Hello everybody,</p>
<p>I'm a math student who has just got his first degree, and I am studying algebraic geometry since a few months. Something I have noticed is the (to my eyes) huge amount of commutative algebra one needs to push himself some deeper than the elementary subjects. This can be seen just counting th... | roy smith | 9,449 | <p>Knowing Sandor, I will heartily second what he said. On second mention, I will say that Mumford's red book on algebraic geometry begins with 5-10 pages (depending on your edition) called "some algebra". This consists of the following subset of Sandor's list: noether's normalization lemma, and cohen - seidenberg's ... |
127,643 | <p>Hello everybody,</p>
<p>I'm a math student who has just got his first degree, and I am studying algebraic geometry since a few months. Something I have noticed is the (to my eyes) huge amount of commutative algebra one needs to push himself some deeper than the elementary subjects. This can be seen just counting th... | Douglas Lind | 8,112 | <p>I am by no means an algebraic geometer, but have used many of its ideas and results for studying algebraic dynamics. I found that the book "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea was a wonderful way to learn both the algebra and geometry, all in the context of concrete computations using Groeb... |
2,498,424 | <p>Consider double sequences $a_{n,m}\in\mathbb R$ where $n,m\in\mathbb Z,$ satisfying</p>
<ol>
<li>$a_{n,m}=a_{n-1,m}+a_{n,m-1}$ for all $n,m\in\mathbb Z,$ and</li>
<li>$\sup_\limits{m\in\mathbb Z}|a_{n,m}|<\infty$ for all $n\in\mathbb Z.$</li>
</ol>
<p>An example solution is $a_{n,m}=(-1)^m2^{-n}.$ A more genera... | charmd | 332,790 | <p>I think that there are solutions which are not of the form (x). I only use the following lemma (see <a href="https://math.stackexchange.com/questions/2640722/sequence-with-bounded-sums"><strong>Sequence with bounded sums</strong></a> for an explicit construction).</p>
<blockquote>
<p><strong>Lemma :</strong> Let ... |
422,196 | <p>After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.</p>
<p>I will therefore move towards a master's degree in engineering and probably work in industry. However, I'm still passionate abo... | Nik Weaver | 23,141 | <p>I think the real viability question here is whether you can maintain your passion for math outside of an academic context. It's certainly possible, but you really have to be motivated to make this work.</p>
<p>If you're doing good work, your other concerns aren't serious obstacles.</p>
|
456,106 | <p>Was solving some exercise of Number theory, and used this theorem
$$m=[a,b]\Longleftrightarrow \left(\frac{m}{a},\frac{m}{b}\right)=1$$Remembered that the teacher showed it in class, but I do not remember how, and I think I may be in the evaluation, and I am also curious how to proof? $\;\;\;$ Please do not use modu... | marty cohen | 13,079 | <p>Let
$a = \prod_p p^{a_i}$
and
$b = \prod_p p^{b_i}$
where the product is over the primes,
all the $a_i$
and $b_i$ are non-negative integers,
and
only a finite number of them
are positive.</p>
<p>Then
$m = \prod_p p^{\max(a_i, b_i)}
$,
so
$m/a = \prod_p p^{\max(a_i, b_i)-a_i}
$
and
$m/b = \prod_p p^{\max(a_i, b_i)-b... |
1,053,640 | <p>"Find all the roots of the polynomial $f(x)=x^2+(3i-2)x-2(1+i)$. Why does the answer not violate the $Conjugate \space Roots \space Theorem \space (CJRT)$"</p>
<p>I tried using the quadratic formula and got to
$$x = \frac{-(3i-2) \pm \sqrt{3-4i}}{2}$$
but I'm having trouble with the $\sqrt{3-4i}$. <br><br> If I u... | Olórin | 187,521 | <p>The answer does not violate the "conjugate roots theorem" (!) because the "theorem" is only valid for polynomial with real coefficients.</p>
<p>For your problem, to find the two values of $\sqrt{3 - 4 i}$, or of a general $\sqrt{x + i y}$, you have to solve the equation $(X+iY)^2 = x+iy$. This is equivalent to $X^2... |
1,053,640 | <p>"Find all the roots of the polynomial $f(x)=x^2+(3i-2)x-2(1+i)$. Why does the answer not violate the $Conjugate \space Roots \space Theorem \space (CJRT)$"</p>
<p>I tried using the quadratic formula and got to
$$x = \frac{-(3i-2) \pm \sqrt{3-4i}}{2}$$
but I'm having trouble with the $\sqrt{3-4i}$. <br><br> If I u... | anomaly | 156,999 | <p>The point of the "conjugate roots theorem" is that if $P$ is a polynomial with real coefficients, then $P(\overline{z}) = \overline{P(z)}$; thus if $P(z_0) = 0$, then $P(\overline{z_0}) = \overline{P(z_0)} = 0$. Your given polynomial $f$ does not have real coefficients and do not satisfy $\overline{f(z)} = f(\overli... |
384,501 | <p>$A= \left[ \begin{array}{ccc}
3 & -1 & 2 \\
-6 & 2 & 4 \\
-3 & 1 & 2 \end{array} \right]$</p>
<p>Applying, $R_{3}-\frac{1}{2}R_{2}$</p>
<p>~ $A= \left[ \begin{array}{ccc}
3 & -1 & 2 \\
-6 & 2 & 4 \\
0 & 0 & 0 \end{array} \right]$</p>
<p>Applying, $R_{2}+2R_{1}$</p>
... | ncmathsadist | 4,154 | <p>Row reduce the matrix and count the nonzero rows that remain.</p>
|
2,432,128 | <p>If there are $5,000,000$ couples in a city, and the probability that a couple matches a specific description is $1\over 12,000,000$, what are the chances that there are two couples that match the specific description given that there is at least one couple that matches the description?</p>
<p>I guess I'm supposed t... | neonpokharkar | 477,567 | <p>$$\newcommand{\C}[2]{^{#1}C_{#2}}$$
Use </p>
<blockquote>
<p>Let p be the probability an event will happen,
$$$$q be it not happening,
$$$$Given n trials have taken place
$$$$Probability that event occurred r times is
$$P(n,r)=\C{n}{r} (p)^r (q)^{n-r}$$</p>
</blockquote>
<p>So </p>
<blockquote>
<p>pro... |
3,473,911 | <blockquote>
<p>What is the derivative of <span class="math-container">$f: \mathbb C \to \mathbb R$</span> where <span class="math-container">$f(z)=z\bar z$</span>?</p>
</blockquote>
<p>Not sure how to go about differentiating this function.</p>
<p>Is it just <span class="math-container">$f'(z)=\bar z$</span>? Not ... | Arthur | 15,500 | <p>Yes, between any two roots of <span class="math-container">$f$</span>, there is a root of <span class="math-container">$f'$</span>. However, just because <span class="math-container">$f'$</span> has a root, that doesn't mean that <span class="math-container">$f$</span> has a root on either side. Consider <span class... |
3,473,911 | <blockquote>
<p>What is the derivative of <span class="math-container">$f: \mathbb C \to \mathbb R$</span> where <span class="math-container">$f(z)=z\bar z$</span>?</p>
</blockquote>
<p>Not sure how to go about differentiating this function.</p>
<p>Is it just <span class="math-container">$f'(z)=\bar z$</span>? Not ... | GEdgar | 442 | <p><strong>Where am I messing up?</strong> Just look at a graph where it fails.</p>
<p><span class="math-container">$$
x^5-5 x -5
$$</span>
<a href="https://i.stack.imgur.com/1JJir.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1JJir.jpg" alt="a"></a></p>
|
2,148,484 | <p>Find the value of the series $$\sum_{n=0}^\infty \frac{n^{2}}{2^{n}}.$$ I tried the problem but not getting the sum. Please help.</p>
| Nosrati | 108,128 | <p><strong>Hint:</strong> Let $$f(x)=\sum_02^{-nx}~~~~~;~~x>0$$ and consider $f''(1)$.</p>
<p><strong>Edit.</strong> Let's know:
\begin{eqnarray}
f(x)&=&\sum_0(2^{-x})^n\\
&=&\frac{1}{1-2^{-x}}\\
f''(x)=\sum_0(-n\ln2)^22^{-nx}&=&2^{-x}(\ln2)^2\frac{1-2^{-x}+2^{1-x}}{(1-2^{-x})^3}\\
f''(1)=(\... |
2,831,968 | <p>$\DeclareMathOperator{\var}{var}$It is just a general question I could not get my mind around.</p>
<p>Assume that $E[X]= 20$ and $\var[X]= 5$, then$$
E[1.2X]= 1.2·E[X]= 1.2×20= 24= 20 + 4 = E[X] + E[0.2X],\\
\var[1.2X]= 1.44·\var[X]= 1.44×5= 7.2.
$$
For$$
\var[1.2X]= \var[X + 0.2X]= \var[X] + \mathord{??} = \var[0.... | Graham Kemp | 135,106 | <blockquote>
<p>how do i split up the variance here if i want to write it as $Var[X]+Var[aX]$
?</p>
</blockquote>
<p>$\begin{split}\mathsf {Var}(1.2X) &=1.2^2\mathsf{Var}(X) &~&=(1+0.2)^2\mathsf{Var}(X) \\ &=\mathsf {Var}(X)+(1.2^2-1)\mathsf{Var}(X) &&=\mathsf{Var}(X)+0.4\mathsf{Var}(X)+0.2... |
2,906,314 | <p>If I have vectors a and b sharing a common point of intersection then I know how to calculate angle between them by using the formula for dot product. But whether b lies to the right or left of a if I am moving along a can not be gotten from this. </p>
<p>What would be the easiest way to find out whether b lies lef... | amd | 265,466 | <p>As David K and others pointed out, in order to distinguish “left” and “right” you need some reference. In a comment you describe consistently looking “down” onto the plane in which the vectors lie. Mathematically, you can specify this by choosing a fixed vector $\mathbf u$ that specifies the “up” direction relative ... |
2,490,654 | <p>$$\int_0^1\sqrt\frac x{1-x}\,dx$$
I saw in my book that the solution is $x=\cos^2u$ and $dx=-2\cos u\sin u\ du$.<br>
I would like to see different approaches, can you provide them?</p>
| 19aksh | 668,124 | <p>Using King's rule, we have,</p>
<p><span class="math-container">$$I= \int_0^1\sqrt{\frac{x}{1-x}}dx =\int_0^1\sqrt{\frac{1-x}{x}}dx$$</span>
<span class="math-container">$$2I = \int_0^1\left(\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}\right)dx = \int_0^1\frac{1}{\sqrt{x-x^2}}dx=\int_0^1\frac{1}{\sqrt{\frac{1}{4}-(x-... |
4,052,683 | <p>If <span class="math-container">$\gcd(a,b,c) = 1$</span> and <span class="math-container">$c = {ab\over a-b}$</span>, then prove that <span class="math-container">$a-b$</span> is a square. <span class="math-container">$\\$</span><br />
Well I tried expressing <span class="math-container">$a=p_1^{a_1}.p_2^{a_2} \cdot... | fleablood | 280,126 | <p>Well <span class="math-container">$\frac {ab}{a-b}$</span> being an integer seems unlikely and specific.</p>
<p>In particular if <span class="math-container">$p$</span> is a prime divisor of <span class="math-container">$c$</span> then <span class="math-container">$p|\frac {ab}{a-b}$</span> so <span class="math-cont... |
128,656 | <p><img src="https://i.stack.imgur.com/AyYxe.jpg" alt=""Put the alphabet in math..."" /></p>
<p><strong>variable</strong>: A symbol used to represent one or more numbers.</p>
<p>Or alternatively: A symbol used to represent any member of a given set.</p>
<p>High school students are justifiably confused by the... | Christian Blatter | 1,303 | <p>It's a philosophical question. So far I have never met a mathematically hard definition of the notion of "variable", as it is used in analysis ("free" and "bound" variables in logic are another matter). Here are some musings on the theme:</p>
<p>Given any set $S$ and any symbol or letter $x$ one can declare $x$ as ... |
411,261 | <p>Question: Use double integral to find the volume of the solid enclosed by the spheres $x^2+y^2+z^2=1$ and $x^2+y^2+(z-1)^2=1$</p>
<p>Alright so I tried to doing this by myself and I'm not sure if this is right. Could someone check over my work?</p>
<p>Curve of intersection:
\begin{align*}
x^2 + y^2 + z^2 &= x^... | Jared | 65,034 | <p>Wolfram|Alpha says:$$\sum_{k=3}^n\binom{n}{k}\binom{k-1}{2}=2^{n-3}n^2-5\cdot2^{n-3}n+2^n-1$$</p>
|
411,261 | <p>Question: Use double integral to find the volume of the solid enclosed by the spheres $x^2+y^2+z^2=1$ and $x^2+y^2+(z-1)^2=1$</p>
<p>Alright so I tried to doing this by myself and I'm not sure if this is right. Could someone check over my work?</p>
<p>Curve of intersection:
\begin{align*}
x^2 + y^2 + z^2 &= x^... | André Nicolas | 6,312 | <p>Use the Binomial Theorem to write down the expansion of
$$\frac{(1+x)^n-1}{x}.$$
Then differentiate twice, and set $x=1$. You will get a very close relative of your sum. </p>
|
411,261 | <p>Question: Use double integral to find the volume of the solid enclosed by the spheres $x^2+y^2+z^2=1$ and $x^2+y^2+(z-1)^2=1$</p>
<p>Alright so I tried to doing this by myself and I'm not sure if this is right. Could someone check over my work?</p>
<p>Curve of intersection:
\begin{align*}
x^2 + y^2 + z^2 &= x^... | Brian M. Scott | 12,042 | <p>Note that</p>
<p>$$\binom{n}k\binom{k-1}2=\frac12\binom{n}k(k-1)(k-2)\;,$$</p>
<p>where the $(k-1)(k-2)$ looks like the coefficient of the second derivative of $x^{k-1}$. That suggests looking at something like</p>
<p>$$g(x)=\sum_{k=3}^n\binom{n}kx^{k-1}$$</p>
<p>and differentiating twice with respect to $x$ to ... |
3,352,919 | <p>Im taking a course in functional analysis and im trying to prove that in infinite dimensions there is no compact unit ball.
I've read some results following the Riesz Lemma but i seem to not quiet understand.
Can someone show another approach to the problem or try explain me the Riesz Lemma approach? i would really ... | Kwin van der Veen | 76,466 | <p>In order to conclude that for certain initial conditions it holds that <span class="math-container">$R_e\to I$</span> one can use the fact that <span class="math-container">$d/dt\,J^{SS}\leq0$</span> and by substituting the asymptotic result of <span class="math-container">$\omega_e\to0$</span> into the dynamics of ... |
317,756 | <p>Let $L>1$. I am looking for the value, or the leading asymptotics for $L\to\infty$, of
$$\int_1^L\int_1^L\int_1^L\int_1^L \dfrac{\mathrm dx_1~\mathrm dx_2 ~ \mathrm dx_3 ~ \mathrm dx_4}{(x_1+x_2)(x_2+x_3)(x_3+x_4)(x_4+x_1)}$$
More generally, I'd like to know the leading asymptotics of an expression like this wit... | David H | 55,051 | <hr />
<p>Define the function <span class="math-container">$\mathcal{I}:\left(1,\infty\right)\rightarrow\mathbb{R}_{>0}$</span> via the quadruple integral</p>
<p><span class="math-container">$$\mathcal{I}{\left(a\right)}:=\int_{\left[1,a\right]^{4}}\mathrm{d}t\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}w\,\frac{1}{\left(t... |
1,492,477 | <p>The sequence $\frac{1}{n}$ is convergent under euclidean metric. But not convergent with discrete metric.</p>
<p>Is there a non-constant convergent sequence with discrete metric ?</p>
| Hosein Rahnama | 267,844 | <p>This is another solution which I believe is simpler than the other one. I suggest this one as it is more compact. Consider the following</p>
<p>$$\begin{array}{l}
Q = \sum\limits_{i = 1}^P {{{\left( {{y_i} - \bar y} \right)}^2}} = \sum\limits_{i = 1}^P {\left( {y_i^2 - 2{y_i}\bar y + {{\bar y}^2}} \right)} = \sum... |
3,504,777 | <p>Here's what I initially started with:</p>
<blockquote>
<blockquote>
<p>Find a 2x2 non zero matrix <span class="math-container">$A$</span>, satisfying <span class="math-container">$A^2=A$</span>, and <span class="math-container">$A\neq I$</span>.</p>
</blockquote>
</blockquote>
<p>I understand that this is ... | Community | -1 | <p><span class="math-container">$A\in M_n(\mathbb{R})$</span> is a projector (eventually non-orthogonal). The projectors are classified by their trace. Assume that <span class="math-container">$rank(A)=trace(A)=r\in (0,n)$</span>.</p>
<p><span class="math-container">$A$</span> is associated to a (unique) decomposition... |
2,618,728 | <p>Please Help me in the following Problem</p>
<blockquote>
<p>What is the Number Of Natural Numbers ,$n\le30$ for which $\sqrt{n+\sqrt{n+\sqrt{n+\cdots}}}$ is also a prime number.</p>
</blockquote>
<p>The only way I am able to find to solve this is calculate each and every term once but it will be extremely length... | Community | -1 | <p>Let the limit be $x$. Then $\sqrt{n+x}=x$. Thus $x^2-x-n=0$... So $x=\frac{1\pm\sqrt{1+4n}}2$. $x$ is positive, so $x=\frac{1+\sqrt{1+4n}}2$. </p>
<p>So let's see:
$n=6 \implies x=3$ </p>
<p>Say $x$ is a prime, $p$. Then $p^2-p-n=0$, so $n=p(p-1)$... </p>
<p>It looks like infinitely many...
(as there are ... |
1,714 | <p>Say I have a function $f(x)$ that is given explicitly in its functional form, and I want to find its Fourier transform[1]. If $f$ is too complicated to have an analytic expression for $\hat f(k)$, how do I obtain it numerically?</p>
<p>The naive and stupid way, which I currently use, is evaluating the integral for ... | Daniel Lichtblau | 51 | <p>This is borrowed from comp.soft-sys.math.mathematica posts, primarily by Szesi Mukasa. The outside factor I cribbed from a text book.</p>
<pre><code>fft[ll_] :=
Exp[I*Pi*(Range[Length[ll]] + Boole[OddQ[Length[ll]]])]*
RotateRight[Fourier[ll, FourierParameters -> {0, 1}],
Quotient[Length[ll], 2]]
</code><... |
3,633,495 | <p>Quadratic functions are of the form:
<span class="math-container">$$f(x)=b^Tx+\frac{1}{2}x^TCx$$</span>
where <span class="math-container">$C$</span> is assumed without loss of generality to be symmetric.
Tried to prove it myself using contradiction but I didn't find a valid argument.</p>
| matiasdata | 127,936 | <p>If <span class="math-container">$f$</span> is strictly convex then for <span class="math-container">$x\neq y$</span> we have <span class="math-container">$f(y)> f(x) + \nabla f(x)(y-x)$</span>. But by 2nd order Taylor we have that <span class="math-container">$f(y)=f(x) + \nabla f(x)(y-x) + (y-x)^T \nabla^2 f(x+\... |
3,800,728 | <blockquote>
<p>If <span class="math-container">$y''-15xy'+8y=e^{x}$</span> and <span class="math-container">$y(0)=y'(0)=-2$</span>. Then find first <span class="math-container">$5$</span> non zero terms of series solution of that equation about <span class="math-container">$x=0$</span></p>
</blockquote>
<p>What i try... | TheSilverDoe | 594,484 | <p>Suppose that you have a series
<span class="math-container">$$y(x) = \sum_{n=0}^{+\infty} b_nx^n$$</span>
with a positive radius of convergence, which is solution of the equation. Then in the open disc of convergence, you can differentiate term by term, so you get
<span class="math-container">$$y''(x) - 15xy'(x) + 8... |
3,800,728 | <blockquote>
<p>If <span class="math-container">$y''-15xy'+8y=e^{x}$</span> and <span class="math-container">$y(0)=y'(0)=-2$</span>. Then find first <span class="math-container">$5$</span> non zero terms of series solution of that equation about <span class="math-container">$x=0$</span></p>
</blockquote>
<p>What i try... | G Cab | 317,234 | <p>You started well and reached to
<span class="math-container">$$
\sum\limits_{0\, \le \,n} {n\left( {n - 1} \right)b_{\,n} \,x^{\,n - 2} } - 15\sum\limits_{0\, \le \,n} {n\,b_{\,n} \,x^{\,n} } + 8\sum\limits_{0\, \le \,n} {\,b_{\,n} \,x^{\,n} } = e^{\,x}
$$</span>
note that you have better and keep the sum to sta... |
3,800,728 | <blockquote>
<p>If <span class="math-container">$y''-15xy'+8y=e^{x}$</span> and <span class="math-container">$y(0)=y'(0)=-2$</span>. Then find first <span class="math-container">$5$</span> non zero terms of series solution of that equation about <span class="math-container">$x=0$</span></p>
</blockquote>
<p>What i try... | Cesareo | 397,348 | <p>This is a pure mechanical procedure so I will let a script in order to obtain the desired results.</p>
<p>First we know that <span class="math-container">$e^x = \sum_{k=0}^{\infty}\frac{x^k}{k!}$</span> so defining <span class="math-container">$Y = \sum_{k=0}^n a_k x^k$</span> we develop the approximate relationship... |
181,797 | <p>I have a table with desired inputs of a given function <code>efvc[n,m,u]</code>. I want to calculate for different values of <code>n</code>, <code>m</code> and <code>u</code>. I imported a Excel table with the values I want. Is there a way to call each row of the table as <code>n</code>,<code>m</code>,<code>u</code>... | kglr | 125 | <pre><code>data = {{{0., 0., 6.}, {0.104716, 0.0000884874, 6.}, {0.206578,
0.000825568, 6.}, {0.301571, 0.000365564, 6.}, {0.410605,
0.0000384834, 6.}, {0.506678, 0.0000494736, 6.}, {0.610588,
0.000617136, 6.}, {0.707025, 0.0000728179, 6.}, {0.80244,
0.0000829668, 6.}, {0.905712, 0.00125702, 6.}... |
1,991,822 | <p>Suppose $f:[0,1]\rightarrow\mathbb{R}$. defined by $f(x)=(-1)^n n $ when $x\in(1/(n+1),1/n]$ and $f(0)=0$. Show that the improper Riemann integral $$\int_{0}^{1} f(x) dx $$ is real number. </p>
<p>First note that $f$ has infinite points when $f$ is discontinuous. I have trouble to compute the integral. I have s... | Surb | 154,545 | <p><strong>Hint</strong></p>
<p>Look at
$$\lim_{n\to \infty }\sum_{k=1}^n\int_{1/(k+1)}^{1/k}(-1)^n n\mathrm d x.$$</p>
|
3,589,178 | <p>I don't know how to do this limit.
<span class="math-container">$$\lim_{x\to\infty}\left(\frac1{x^2\sin^2\frac 1x}\right)^\frac 1{x\sin\frac 1x-1}$$</span>
And here it is as an image, with bigger font:
<a href="https://i.stack.imgur.com/uKwCN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/uKwCN.p... | Mark Viola | 218,419 | <p>Your idea of making the substitution <span class="math-container">$x\mapsto1/x$</span> is fine. We then wish to evaluate the limit as <span class="math-container">$x\to 0$</span> of <span class="math-container">$f(x)$</span> where <span class="math-container">$f(x)$</span> is given by</p>
<p><span class="math-cont... |
2,204,642 | <p>I have a hard time believing that there can exist a bijection $f:\Bbb R^2\to \Bbb R$.</p>
<p>I just cannot get around my intuition that a one-to-one map of a one-dimensional space (or interval) must also be one-dimensional.</p>
<p>I am not a mathematician. I am interested in the topic since it has potential releva... | Jacob Wakem | 117,290 | <p>It is not a mapping of the spaces but a mapping of the sets. A 1-1 function from R^2 to R would be simply to go back and forth choosing digits from each number. Ie: f(.111111...,.22222...)=.12121212121212...</p>
|
2,204,642 | <p>I have a hard time believing that there can exist a bijection $f:\Bbb R^2\to \Bbb R$.</p>
<p>I just cannot get around my intuition that a one-to-one map of a one-dimensional space (or interval) must also be one-dimensional.</p>
<p>I am not a mathematician. I am interested in the topic since it has potential releva... | Jack D'Aurizio | 44,121 | <p>There is a bijection between $\mathbb{R}$ and $(0,1)$ and a bijection between $(0,1)$ and a Cantor set $K$.<br>
The first map is given by $x\mapsto \frac{1}{2}+\frac{1}{\pi}\arctan(x)$ and the second map takes the canonical (i.e. without an infinite tail of digits $1$) base-$2$ representation of $x\in(0,1)$ and does... |
3,709,040 | <p>I'm reading Goldblatt's Topoi and trying to practice categorical reasoning, generalizing the example of <span class="math-container">$\mathbf{Set}^\rightarrow$</span> being a topos.</p>
<p>So, let <span class="math-container">$\mathcal{C}$</span> be a category with a subobject classifier <span class="math-container... | fosco | 685 | <p><span class="math-container">$\require{AMScd}$</span>If <span class="math-container">$\cal E$</span> is an elementary topos, then so is the functor category <span class="math-container">${\cal E}^C$</span> for every small category <span class="math-container">$C$</span>; the classifying monomorphism is just the imag... |
2,894,637 | <p>As stated the condition is:</p>
<ol>
<li>$\int_0^\infty f(x) dx=0$ </li>
<li>$f(x)$ continuous on $x\in[0,\infty)$</li>
</ol>
<p>What I would like to prove is $\lim_{x\to\infty}f(x)=0$</p>
<p>It is easy to prove that if $\lim_{x\to\infty}f(x)$ exists, or $\lim_{x\to\infty}f(x)=\infty$.</p>
<p>But I would like to... | Torsten Schoeneberg | 96,384 | <p>There is a general lemma that says that if $\mathfrak{g}$ is a semisimple Lie subalgebra of some $\mathfrak{gl}_n(K)$ (where $K$ is any field of characteristic $0$), then an element $X \in \mathfrak{g}$ is semisimple / nilpotent as element of $End(K^n)$ if and only if $ad_\mathfrak{g}(X)$ is nilpotent/semisimple. Se... |
1,263,887 | <p>I'm working on a proof right now, and the question asks about an invertible skew-symmetric matrix. How is that possible? Isn't the diagonal of a skew-symmetric matrix always $0$, making the determinant $0$ and therefore the matrix <em>is not</em> invertible?</p>
| Sloan | 217,391 | <p>No, the diagonal being zero does not mean the matrix must be non-invertible. Consider $\begin{pmatrix} 0 & 1 \\ -1 & 0 \\ \end{pmatrix}$. This matrix is skew-symmetric with determinant $1$. <br> Edit: as a brilliant comment pointed out, it <i>is</i> the case that if the matrix is of odd order, then skew-symm... |
2,424,649 | <blockquote>
<p>Show that $\sqrt{2} + \sqrt[3]{5}$ is algebraic of degree $6$ over
$\mathbb{Q}$</p>
</blockquote>
<p>What is the degree of a root? Is it the smallest polynomial that gives this thing as root?</p>
<p>What I tried:</p>
<p>$x = \sqrt{2} + \sqrt[3]{5} \implies x^2 = 2 + 2\sqrt{2}\sqrt[3]{5} + \sqrt[3... | Cornman | 439,383 | <p>Hint: Use the formula for the degree:</p>
<p>$[\mathbb{Q}(\sqrt{2}+\sqrt[3]{5}):\mathbb{Q}]= [\mathbb{Q}(\sqrt{2}+\sqrt[3]{5}):\mathbb{Q}(\sqrt{2})]\cdot [\mathbb{Q}(\sqrt{2}):\mathbb{Q}]$</p>
|
2,424,649 | <blockquote>
<p>Show that $\sqrt{2} + \sqrt[3]{5}$ is algebraic of degree $6$ over
$\mathbb{Q}$</p>
</blockquote>
<p>What is the degree of a root? Is it the smallest polynomial that gives this thing as root?</p>
<p>What I tried:</p>
<p>$x = \sqrt{2} + \sqrt[3]{5} \implies x^2 = 2 + 2\sqrt{2}\sqrt[3]{5} + \sqrt[3... | Community | -1 | <p>$$\sqrt[3]5+\sqrt2=q\implies$$
$$q-\sqrt2=\sqrt[3]5\implies$$
$$q^3-3\sqrt2q^2+6q-2\sqrt2=5\implies$$
$$q^3+6q+5=(3q^2+2)\sqrt2.$$
Square the last identity.</p>
|
21,290 | <p>Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$? </p>
<p>I am looking for elementary example (so there should be a proof of transcendence that does not use any big machinery).</p>
| ACL | 10,696 | <p>Eisenstein proved (actually, stated) in 1852 that if $f=\sum a_n z^n$ is an algebraic power series with rational coefficients, there exist positive integers $A$ and $B$ such that $A a_n B^n$ are integers for all $n$. In particular, as Eisenstein himself remarks, only finitely many prime numbers appear in the denomin... |
21,290 | <p>Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$? </p>
<p>I am looking for elementary example (so there should be a proof of transcendence that does not use any big machinery).</p>
| camilo | 26,139 | <p>over the rationals every power serie with integer coefficents not periodic is tracendent, over Fp a power serie is algebraic iff the secuence of coeficient is p automatic there is a article od jp allouch tracendence of formal series with it information.</p>
|
1,238,430 | <p>I'm working on a project that involves that set $P = \{\{n_1, \ldots, n_k\} \mid k \in \mathbb{N}, n_i \in \mathbb{N} \text{ and } n_1 + \cdots +n_k = n\}$ of all integer partitions of a number $n$. Is there a standard notation for this?</p>
<p>I know that $p(n)$ is commonly used to denote the <em>number</em> of i... | JimT | 409,742 | <p>Often it is simply denoted as $P(n)$ or $\mathcal{P}(n)$. However, I have seen other letters used in books and articles. I would simply go with $\mathcal{P}(n)$ but you should still formally define it in the beginning of your text.</p>
|
2,134,121 | <p>I can't figure out an algebraic proof for the following identity, (and I don't know if I can use the binomial theorem for this one):
<span class="math-container">$$\sum_{k=0}^m(-1)^{m-k}{{n \choose {k}}}= {{n-1}\choose m}$$</span></p>
<p>Thank you, for your help in advance.</p>
| Alexander Burstein | 499,816 | <p>Notice that <span class="math-container">$$(-1)^{m-k}=\binom{-1}{m-k},$$</span> so you want to prove
<span class="math-container">$$\sum_{k=0}^m\binom{-1}{m-k}{\binom{n}{k}}= {\binom{n-1}{m}}.$$</span> But that's just a special case of the Vandermonde identity.</p>
<p>Using generating functions, the identity follow... |
2,776,763 | <p>Consider two sequences $\{b_n\}_{n\in \mathbb{N}}$ and $\{a_n\}_{n\in \mathbb{N}}$. Suppose that
$$
(*) \hspace{1cm} \lim_{n\rightarrow \infty} (b_n+a_n)=L<\infty
$$</p>
<p>Does this imply
$$
\exists \lim_{n\rightarrow \infty} b_n \text{ and it is finite}
$$
$$
\exists \lim_{n\rightarrow \infty} a_n \text{ and i... | CyclotomicField | 464,974 | <p>We begin by treating the vectors as matrices with a single column and use familiar properties of matrix arithmetic to prove this is the case.</p>
<p>First note that $y^Tx$ can be interpreted as a dot product of $x$ and $y$ with $T$ denoting the transpose (assuming the base field is a subfield of $\mathbb{R}$). In p... |
1,872,234 | <p>If we have</p>
<p>$$f(x) = \sum_{n=0}^\infty a_n x^n$$</p>
<p>The $k$th derivative is </p>
<p>$$f^{(k)}(x) = \sum_{n=0}^{\infty} a_{n+k} \frac{(n+k)!}{n!} x^n$$</p>
<p>Which also means that</p>
<p>$$f^{(k)}(0) = k! a_k$$</p>
<p>Implying</p>
<p>$$a_k = \frac{f^{(k)}(0)}{k!}$$</p>
<p>So we can substitute this ... | avs | 353,141 | <p>$f(x) = \sum_{k \geq 0} a_{k} (x-a)^{k}$</p>
<p>$f'(x) = \sum_{k \geq 1} k a_{k} (x-a)^{k-1}$, so $f'(a) = a_{1}$</p>
<p>$f''(x) = \sum_{k \geq 2} (k-1) k a_{k} (x-a)^{k-2}$, so $f''(a) = 2! \, a_{2}$</p>
<p>$f'''(x) = \sum_{k \geq 3} (k-2) (k-1) k a_{k} (x-a)^{k-2}$, so $f'''(a) = 3! \, a_{3}$.</p>
<p>So on.</p... |
39,802 | <p>Suppose $f:X \to Y$ is a function of sets. Then we can take the quotient $X/\text{~}$ by identifying $x \text{~} y$ if and only if $f(x)=f(y)$. Now suppose instead that $f:X \to Y$ is a map of simplicial sets. I want to emulate this homotopically, by adding a 1-simplex between $x$ and $y$ if there is a 1-simplex fro... | Ronnie Brown | 19,949 | <p>You might like to look at the 1978 thesis of Nick Ashley on "Simplicial $T$-complexes and crossed complexes: a
nonabelian version of a theorem of Dold and Kan." available from Esquisses Math. 1978 at <a href="http://ehres.pagesperso-orange.fr/Cahiers/Ctgdc.htm" rel="nofollow">http://ehres.pagesperso-orange.fr/Cah... |
2,816,184 | <p>In the first chapter of Gouvea's intro to $p$-adics, there's a heuristic argument that</p>
<p>$$ \frac{2}{1}+\frac{2^2}{2}+\frac{2^3}{3}+\frac{2^4}{4}+\cdots=0 \tag{$\ast$}$$</p>
<p>as $2$-adic numbers, since it's the Mercator series for $\ln(-1)$ and $2\ln(-1)=\ln(-1)^2=\ln1=0$.</p>
<p>(Like I said, heuristic.)<... | ViHdzP | 718,671 | <p>Since <span class="math-container">$\lim_{n\to\infty}\nu_2\left(\frac{2^n}n\right)=\infty$</span>, it suffices to prove that there are partial sums with arbitrarily high <span class="math-container">$\nu_2$</span>. We adapt the first solution from <a href="http://www.maths.usyd.edu.au/u/SUMSCOMP/sols2002.pdf" rel="n... |
31,562 | <p>How to evaluate the number of ordered partitions of the positive integer <span class="math-container">$ 5 $</span>?</p>
<p>Thanks!</p>
| QuentinUK | 67,690 | <p>Counting in binary the groups of 1s or 0s form the partitions. Half are the same so there are 2^(n-1). As to be expected this gives the same results as the gaps method, but in a different order.</p>
<p>Groups</p>
<pre><code>0000 4
0001 3,1
0010 2,1,1
0011 2,2
0100 1,1,2
0101 1,1,1,1
0110 1,2,1
0111 1,3
</co... |
2,405,709 | <p><strong>Problem</strong></p>
<p>A math book wants me to prove that given two natural numbers $m, \ n$ are not divisible by $5$, then the difference $m^4 - n^4$ <em>is</em> divisible by $5$.</p>
<p><strong>Thoughts</strong></p>
<p>The only method I can think of now, is to go through all the possible ways of writin... | Mark Bennet | 2,906 | <p>$$m^4-n^4=(m-n)(m+n)(m^2+n^2)$$</p>
<p>Now modulo $5$ we have $m^2+n^2\equiv m^2+n^2-5n^2=(m+2n)(m-2n)$.</p>
<p>If $n$ is not divisible by $5$, the numbers $\pm n, \pm 2n$ cover all the non-zero residues modulo $5$, and one of the factors $m\mp n, m\mp 2n$ is therefore divisible by $5$.</p>
|
63,315 | <p>I am trying to identify a sequence related to the von Mangoldt function matrix. Since I believe/conjecture that the columns in the matrix have period lengths as in this sequence b:</p>
<pre><code>b = Table[Product[Exp[MangoldtLambda[n]], {n, 1, k}], {k, 1, nn}];
</code></pre>
<p>and since "b" grows rather fast, th... | Mats Granvik | 328 | <p>I realized now that I included unnecessary many terms of the Dirichlet inverse of the Euler totient.</p>
<p>Therefore a better program is:</p>
<pre><code>ii = 13
aa = Range[ii]*0;
Monitor[Do[
Clear[A, a, b, n, k];
b = Table[Product[Exp[MangoldtLambda[n]], {n, 1, k}], {k, 1, nn}];
a = Table[
If[n == 1, 1,... |
10,948 | <p>This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges absolutely because it behaves roughly like $\sum_{i>0}i^{-2}$.</p>
<p>Some quick facts:</p>
<ul>
<li>Pretty much by c... | Noam D. Elkies | 14,830 | <p>For the record, one <em>can</em> prove the product formula for the sine without complex analysis (and without the Gamma function), from which (as <strong>David Speyer</strong> noted) one can recover $\sum_{i \in \bf Z} (x+i)^{-2}$ as the second logarithmic derivative. See <a href="https://math.stackexchange.com/que... |
3,271,419 | <p>What are the last 2 digits of <span class="math-container">$2017^{2017}$</span>?</p>
<p>Notice that
<span class="math-container">$$2017 (2017) = 2017 ( 2000 + 10 + 7) = (....000) + (....70) + (2017 \times 7)$$</span>
so the last two digits of <span class="math-container">$2017^{2}$</span> are the last two digits o... | Bill Dubuque | 242 | <p>Nice observation! You noticed <span class="math-container">$\!\bmod 100\,$</span> the powers of <span class="math-container">$\,17\,$</span> have the following structure</p>
<p><span class="math-container">$$\begin{array}{r|r r}
n & 17^{\large n}\!\! & \\
\hline
0 &\ \ 01 & 17 & 89 & 13\\
4&... |
3,271,419 | <p>What are the last 2 digits of <span class="math-container">$2017^{2017}$</span>?</p>
<p>Notice that
<span class="math-container">$$2017 (2017) = 2017 ( 2000 + 10 + 7) = (....000) + (....70) + (2017 \times 7)$$</span>
so the last two digits of <span class="math-container">$2017^{2}$</span> are the last two digits o... | Samuel Bowditch | 684,402 | <p>It suffices to evaluate <span class="math-container">$ (2017)^{2017} \pmod {100}: $</span></p>
<p><span class="math-container">$$ \equiv (17)^{2017} \equiv (17^{2})(17) \equiv (-11)^{1008}(17) $$</span></p>
<p><span class="math-container">$$ \equiv (21)^{504}(17) \equiv (21^{5})^{100}(21^{4})(17) \equiv (1)^{100}(... |
3,004,210 | <p>A smilar question has been asked before <a href="https://math.stackexchange.com/questions/23503/create-unique-number-from-2-numbers">Create unique number from 2 numbers</a>.</p>
<blockquote>
<p>is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 an... | hmakholm left over Monica | 14,366 | <p>In the comments you say you want to use your function in a Java program where it looks like you want it to have a signature more or less like</p>
<pre><code>static int pair(int a, int b) {
...
}
</code></pre>
<p>However, then you're out of luck: It is <strong>impossible</strong> for a pure function with this ... |
56,337 | <p>Suppose I have the following list:</p>
<pre><code>list = {a, b, c, d}
</code></pre>
<p>I want to generate this result:</p>
<pre><code>{{f[a, a], f[a, b], f[a, c], f[a, d]}, {f[b, b], f[b, c],
f[b, d]}, {f[c, c], f[c, d]}, {f[d, d]}}
</code></pre>
<p>What could be the shortest way?</p>
<p>The list elements can... | Öskå | 1,356 | <p>There is probably something neater but the following works:</p>
<pre><code>l = {a, b, c, d};
s = SplitBy[Tuples[l, {2}], First];
list = Take[s[[#]], #2] & @@@ Thread@{Range@Length@l, Range[-Length@l, -1]}
Map[f[Sequence @@ #] &, list, {-2}]
</code></pre>
<blockquote>
<pre><code>{{f[a, a], f[a, b], f[a, c],... |
56,337 | <p>Suppose I have the following list:</p>
<pre><code>list = {a, b, c, d}
</code></pre>
<p>I want to generate this result:</p>
<pre><code>{{f[a, a], f[a, b], f[a, c], f[a, d]}, {f[b, b], f[b, c],
f[b, d]}, {f[c, c], f[c, d]}, {f[d, d]}}
</code></pre>
<p>What could be the shortest way?</p>
<p>The list elements can... | C. E. | 731 | <h1>Solutions</h1>
<hr />
<pre><code>Pick[
Outer[f, list, list],
UpperTriangularize@ConstantArray[True, {#, #} &@Length@list]
]
</code></pre>
<p>Using the new <code>Composition</code> shorthand:</p>
<pre><code>Thread@*f @@@ MapIndexed[{#, list[[First@#2 ;;]]} &, list]
</code></pre>
<h1>Timings</h1>
<hr... |
56,337 | <p>Suppose I have the following list:</p>
<pre><code>list = {a, b, c, d}
</code></pre>
<p>I want to generate this result:</p>
<pre><code>{{f[a, a], f[a, b], f[a, c], f[a, d]}, {f[b, b], f[b, c],
f[b, d]}, {f[c, c], f[c, d]}, {f[d, d]}}
</code></pre>
<p>What could be the shortest way?</p>
<p>The list elements can... | RunnyKine | 5,709 | <p>This is not the shortest, but faster than all except Pickett's (almost just as fast)</p>
<pre><code>f4 = Thread@f[#[[1]], #] & /@ Partition[#, Length@#, 1, {1, 1}, {}] &
</code></pre>
<p>OR</p>
<pre><code> dP = Developer`PartitionMap;
</code></pre>
<p>Then:</p>
<pre><code>f5 = dP[Thread@f[#[[1]], #] &am... |
126,251 | <p>Suppose one has a finite number of distances $d_1,\ldots,d_k$ on the Euclidean plane all of which metricize the usual Euclidean topology.</p>
<p>Define for each pair of points $x$ and $y$ in the plane
$$d(x,y) = \inf\left\lbrace d_{i_1}(x_0,x_1) + \cdots d_{i_l}(x_{l-1},x_l) \right\rbrace$$
where the infimum is tak... | Pablo Lessa | 7,631 | <p>The infimum can be zero as pointed out by Anton Petrunin. Here's a construction on the interval <span class="math-container">$[0,1]$</span>.</p>
<p>Consider a sequence of piecewise linear functions <span class="math-container">$f_n:[0,1] \to [0,1]$</span> each of which is strictly increasing defined by (see figure... |
119,981 | <p>Let $C/\mathbb Q$ be a smooth projective curve of genus $g\geq 2$ or a smooth affine curve of genus $g \geq 1$. The exact sequence</p>
<p>$1 \to \pi_1^{et}(C \otimes_\mathbb Q \bar{\mathbb Q}) \to \pi_1^{et}(C) \to \operatorname{Gal}(\bar{\mathbb Q}|\mathbb Q) \to 1$</p>
<p>gives a homomorphism from $\operatorname... | Greg Kuperberg | 1,450 | <p>Thanks to the comments and answers from Scott Carnahan and Michael Greinecker, I think that I understand it better now. I'm going to write this as a CW summary answer and also accept one of the other answers.</p>
<p>People often talk as if all currency is borrowed from the central bank, but that is not really true... |
2,224,980 | <p>Apologies if this kind of question is not allowed here - if so please delete it.</p>
<p>I was just wondering if anyone could recommend a book on mathematical analysis that is interesting enough to sit down and read for enjoyment alone? Something not written in the style of a textbook?</p>
<p>All the best.</p>
| Paul | 202,111 | <p>Counterexamples in Analysis is great for recreational reading.</p>
|
2,344,259 | <blockquote>
<p>$$\int_0^\infty \frac{x^2}{x^4+1} \; dx $$</p>
</blockquote>
<p>All I know this integral must be solved with beta function, but how do I come to the form $$\beta (x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}\;dt \text{ ?}$$</p>
| jgsmath | 455,126 | <p><strong>Hint</strong>: If you <em>have</em> to use Beta function, let $x^2 = \tan \theta$ and use the definition of <a href="https://en.wikipedia.org/wiki/Beta_function" rel="nofollow noreferrer">Beta function</a> in terms of sines and cosines.</p>
|
2,847,359 | <p>So say you had $5^x=25$ where $x$ is obviously $2$, how would you work $x$ out if the question wasn't obvious?</p>
<p><strong>edit:</strong> What if the question was something like $a^x=-1$ (where $a$ is any number).</p>
<p>PS to all the maths elitists out there: Feel free to down vote, I just want to know how to ... | user | 505,767 | <p>We have for $a\neq0,1$ and $b>0$</p>
<p>$$a^x=b\iff \log a^x=\log b\iff x\log a=\log b\iff x=\frac{\log a}{\log b}$$</p>
<p>where $\log$ can be in any positive base $\neq 1$.</p>
|
4,008,196 | <p>Wikipedia contains the following figure (to be found, e.g. <a href="https://en.wikipedia.org/wiki/Monoid" rel="nofollow noreferrer">here</a>) in order to visualize the relations between several algebraic structures. I highlighted a part that I find especially interesting.</p>
<img src="https://i.stack.imgur.com/18HU... | Daniel Hast | 41,415 | <p>Having division is a stronger property than having cancellation. A magma <span class="math-container">$M$</span> (that is, a set with a binary operation, which we write multiplicatively) has division (and is called a <em>quasigroup</em>) if for all <span class="math-container">$a, b \in M$</span>, there exist unique... |
1,517,189 | <p>My first question here..sorry if I'm not very specific but I try to be.</p>
<p>A T-tetromino has three connected blocks in a line and another one above the middle block. How many ways can one be painted on the grid if orientation matters? What about if it doesn't?</p>
| Fabrice NEYRET | 277,841 | <p>As usual: study the difference (from its derivative).</p>
<p>To compare which of $f(t)$ and $g(t)$ is greatest, you just have to study the sign of $h = f-g$. Now if $h(t)$ is monotonous (i.e. going always up, or always down) at least after some $t_0$, then you can predict the sign at infinity.</p>
|
657,162 | <p>Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$</p>
<p>I know that $\prod_{i=3}^k (n-i) < \prod_{i=3}^k n = n^{k-2}$</p>
<p>Also a tighter upper bound is appreciated.</p>
| Nick Peterson | 81,839 | <p>You can start by writing
$$
\prod_{i=3}^{k}(n-i)=n^{k-2}\prod_{i=3}^{k}\left(1-\frac{i}{n}\right)=n^{k-2}\exp\left[\sum_{i=3}^{k}\log\left(1-\frac{i}{n}\right)\right].
$$
Now, you've said that $k\ll n$; thus $\frac{i}{n}\to0$ uniformly over $3\leq i\leq k$. </p>
<p>How you proceed from here really depends on a more... |
3,097,590 | <p>I am reading my textbook and I see this:</p>
<p><a href="https://i.stack.imgur.com/Na9oQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Na9oQ.png" alt="enter image description here"></a></p>
<p>So why are we allowed to just restrict the domain like this? I can see that we're trying to make it s... | Ted | 15,012 | <p>The reason to make this restriction is so that there is a unique <span class="math-container">$\theta$</span> such that <span class="math-container">$x = a \sec \theta$</span>. Effectively, we want an inverse function of <span class="math-container">$\sec$</span>, but (like all trig functions) <span class="math-cont... |
39,315 | <p>This is "escalated" from <a href="https://math.stackexchange.com/questions/4498/classification-of-small-complete-groups">stackexchange</a>. </p>
<p>So, $S_n$ for $n \ne 2,6$, $\text{Aut}(G)$ for $G$ non-cyclic simple, $\text{Hol}(C_p)$ for $p$ odd prime are well-known classes of complete group. </p>
<p>What's the ... | Marty Isaacs | 9,694 | <p>Here's an infinite family of complete groups. Let U be cyclic of prime order p > 2 should and let G be the holomorph of U, so U is normal in G (and in fact is characteristic) and G/U is cyclic of order p-1. It is easy to see that Z(G) = 1. Also U is complemented in G and there are exactly p complements, and they ar... |
4,593,584 | <p>A R.V. <span class="math-container">$X$</span> has a pdf of the form <span class="math-container">$f_{X}(x) = e^{-x} u(x)$</span> and an independent R.V. <span class="math-container">$Y$</span> has a pdf of <span class="math-container">$f_{Y}(y) = 3e^{-3Y} u(y)$</span> using characteristic functions, find the pdf of... | SpectreDNZ | 1,120,328 | <p>To address Alberto Takase's interesting question, there is <a href="https://arxiv.org/abs/1904.09193v2" rel="nofollow noreferrer">a paper by Pradic and Brown</a> that answers it in the affirmative, that Cantor-Bernstein does indeed necessitate excluded middle. As excluded middle is independent from IZF, where the au... |
2,283,705 | <p>In <a href="https://arxiv.org/pdf/0910.5004.pdf" rel="nofollow noreferrer">F.M.S. Lima's paper</a> on Riemann zeta-type functions, he conjectures the following formula:
<span class="math-container">$$\sum_{n=1}^\infty{\frac{\zeta(2n)}{2n(2n+1)\dots(2n+N)}\frac{1}{4^n}}=\frac{1}{2}\left[\frac{\ln{\pi}}{N!}-\frac{H_N}... | James Arathoon | 448,397 | <p><strong>Hint</strong>: I think the analogous conjecture to F.M.S. Lima's linked to above, but instead for even $N\ge 2$ is</p>
<p>$$\sum_{n=1}^\infty{\frac{\zeta(2n)}{2n(2n+1)\dots(2n+N)}\frac{1}{4^n}}=\frac{1}{2}\left[\frac{\ln{\pi}}{N!}-\frac{H_N}{N!}+\sum_{m=1}^{(N-2)/2}(-1)^{m+1}\frac{\zeta(2m+1)}{\pi^{2m}(N-2m... |
2,027,108 | <p>This is <a href="https://books.google.com/books?id=ImCSX_gm40oC&pg=PA116" rel="nofollow noreferrer">exercise 10 - Problems II</a> - of the book <em>"Introduction to Mathematical Reasoning"</em> by Peter J. Eccles.</p>
<blockquote>
<p>We define half-infinite intervals as follows:
\begin{align}
(a, \infty) &a... | Tsemo Aristide | 280,301 | <p>Partition of unity are used to use the well-known definition of the integration in open subset of $R^n$, for every $U\in O$, $U$ is a chart which is diffeomorphic to an open subset of $R^n$.</p>
<p>If $U,V\in O$, $\int_{U\cap V}f=\int_{U\cap V}\sum_{\phi\in \Phi}f.\phi=\int_{U\cap V}f$, since $\sum_{\phi\in \Phi}\p... |
2,027,108 | <p>This is <a href="https://books.google.com/books?id=ImCSX_gm40oC&pg=PA116" rel="nofollow noreferrer">exercise 10 - Problems II</a> - of the book <em>"Introduction to Mathematical Reasoning"</em> by Peter J. Eccles.</p>
<blockquote>
<p>We define half-infinite intervals as follows:
\begin{align}
(a, \infty) &a... | tomasz | 30,222 | <p>Since $\sum_{\phi\in\Phi} \phi=1$, you have by linearity of integration
$$
\sum_\phi \int_A\phi\cdot f \,\textrm{d}x=\int_A\sum_\phi \phi\cdot f \,\textrm{d}x=\int_A 1 \cdot f \,\textrm{d}x=\int_A f\,\textrm{d}x.
$$
There is a technical subtlety in the first equality, because linearity of integration only allows fin... |
442,043 | <p>Assume I have a non-empty finite set $S$ with $x=|S|$. I want to divide the set $S$ into subsets $S_1, S_2, .., S_n$ (<em>Edit:</em> Yes, $S = \cup S_i$, and I'm embarrassed that I forgot to include that) such that: </p>
<ul>
<li>$ |S_i| = y, \forall 1 \le i \le n$ (The cardinality of each subset is fixed) </li>... | ErnestScribbler | 632,197 | <p>Adding to @hardmath's aside. An interesting family of solutions in addition to the finite projective spaces are the dual finite affine planes (2-d vector spaces) over finite fields: <span class="math-container">$GF(q)^2$</span> . In those, it holds that</p>
<p>(a) two distinct points share exactly one line.</p>
<p... |
1,227,759 | <p>$$x \mod 1000 \mod 5$$</p>
<p>I would have thought that it was $x \mod 5000$ except that it doesn't hold true for $x = 5005$ since you'll get zero, but $5005 \mod 5000 = 5$.</p>
| JB King | 8,950 | <p>Since 5 is a divisor of 1000, I believe the expression would simplify to "x mod 5" as adding multiples of 1000 wouldn't change the remainder when divided by 5.</p>
|
269,552 | <p>I have been working on the following problem:</p>
<p>"Let $\sim$ be the equivalence relation on the unit circle $S^1$ defined by $x \sim -x$, $x \in S^1$. Show that $S^1/\sim$ is homeomorphic to $S^1$ and interpret geometrically."</p>
<p>I have applied the following two theorems:</p>
<p>"Let $X$ and $Y$ be space... | Brian M. Scott | 12,042 | <p><img src="https://i.stack.imgur.com/sF6th.png" alt="enter image description here"></p>
<p>Start with a circle and fold it into a figure eight as shown above. Now fold it along the vertical centreline so that $A$ and $A'$ coincide, as do $B$ and $B'$, and $C$ and $C'$. You’ve now identified each point on the origina... |
1,037,972 | <p>The morse code is made up of marks called dots and dashes."Q", for example is (--,--).Is it possible to make up such a code so that every letter of the alphabet is represented by at most three marks?</p>
<p>i have tried this question as follows
with 3 marks we can form 2+4+8=14 letters
my answer is coming correct ... | Suzu Hirose | 190,784 | <p>If we have two marks plus an "end of symbol" then the total number of messages we can make from a maximum of three of them is $2^3+2^2+2=14$ for three-symbol, two-symbol, and one-symbol codes, so it is not possible to represent all 26 letters of the alphabet with at most three marks.</p>
<p>It is worth noticing tha... |
1,037,972 | <p>The morse code is made up of marks called dots and dashes."Q", for example is (--,--).Is it possible to make up such a code so that every letter of the alphabet is represented by at most three marks?</p>
<p>i have tried this question as follows
with 3 marks we can form 2+4+8=14 letters
my answer is coming correct ... | 5xum | 112,884 | <p>There are exactly $2^n$ combinations of marks with length $n$. This means that there are two combinations of marks with length $1$, four of them for length $2$ and $8$ of them for length $3$.</p>
<p>In general, if you want the number of letters you can make with length at least $n$, it is equal to </p>
<p>$$2 + 4 ... |
570,467 | <p>$ E_{n}=2E_{n-1}+ 2^{n-1} $</p>
<p>Can anyone help me to solve this recurrence? Is there a general way to think about recurrence?</p>
| Carsten S | 90,962 | <p>\begin{align*}
E_1&=2 E_0 + 2^0\\
E_2&=2 E_1 + 2^1=4 E_0+ 2^1+2^1\\
E_3&=2 E_2 + 2^2=8 E_0+ 2^2+2^2+2^2
\end{align*}</p>
|
2,101,241 | <p>Find the remainder when $$140^{67}+153^{51}$$ is divided by $17$.</p>
<p>$$140\equiv 4 \pmod {17}$$
$$67\equiv 16 \pmod{17}$$
$$153 \equiv 0 \pmod{17}$$
$$51\equiv 0 \pmod{17}$$</p>
<p>$$\Rightarrow 140^{67}+153^{51}\equiv 4^{16}+0 \equiv 1\pmod{17}$$</p>
<p>Solution should be $13$. What's wrong?</p>
| marwalix | 441 | <p>One has $140\equiv 4\pmod{17}$. Now $4^4=256\equiv 1\pmod{17}$ and $67=16\cdot 4+3$ so</p>
<p>$$140^{67}\equiv (4^4)^{16}\cdot 4^3\equiv 4^3\equiv13 \pmod{17}$$</p>
<p>Similarly one has $153\equiv 0\pmod{17}$ and so $153^{51}\equiv 0\pmod{17}$.</p>
<p>So the answer is indeed $13$</p>
|
2,101,241 | <p>Find the remainder when $$140^{67}+153^{51}$$ is divided by $17$.</p>
<p>$$140\equiv 4 \pmod {17}$$
$$67\equiv 16 \pmod{17}$$
$$153 \equiv 0 \pmod{17}$$
$$51\equiv 0 \pmod{17}$$</p>
<p>$$\Rightarrow 140^{67}+153^{51}\equiv 4^{16}+0 \equiv 1\pmod{17}$$</p>
<p>Solution should be $13$. What's wrong?</p>
| Joffan | 206,402 | <p>You're right to discard the $153$ term as divisible by $17$, so not affecting the remainder, and also right to reduce the $140\bmod 17$, changing the question to just find $4^{67}\bmod 17$. However ${67}$ is an operation count, not a value that can be reduced by the same modulus (and nor could the $51$, but it becam... |
7,787 | <p>I'm trying to expand the following polynomial </p>
<pre><code> Expand[ (A1 a1 + A2 a2 + A3 a3 + A4 a4 + A5 a5 + A6 a6 + A7 a7 + A8 a8)
(D1 a1 + D2 a2 + D3 a3 + D4 a4 + D5 a5 + D6 a6 + D7 a7 + D8 a8)
+ (H1 a1 + H2 a2 + H3 a3 + H4 a4 + H5 a5 + H6 a6 + H7 a7 + H8 a8)
(E1 a1 + E2 a2 + ... | Mr.Wizard | 121 | <p>As R.M observed the border region is numbered <code>1</code>. We can therefore do:</p>
<pre><code>ArrayPlot[regions /. 1 -> Red]
</code></pre>
<p><img src="https://i.stack.imgur.com/KwZzQ.png" alt="Mathematica graphics"></p>
<pre><code>MatrixPlot[regions /. 1 -> Black, Frame -> False]
</code></pre>
<p>... |
7,787 | <p>I'm trying to expand the following polynomial </p>
<pre><code> Expand[ (A1 a1 + A2 a2 + A3 a3 + A4 a4 + A5 a5 + A6 a6 + A7 a7 + A8 a8)
(D1 a1 + D2 a2 + D3 a3 + D4 a4 + D5 a5 + D6 a6 + D7 a7 + D8 a8)
+ (H1 a1 + H2 a2 + H3 a3 + H4 a4 + H5 a5 + H6 a6 + H7 a7 + H8 a8)
(E1 a1 + E2 a2 + ... | Heike | 46 | <p>The function <code>ComponentMeasurements</code> has the option <code>"BorderComponents"</code> which when set to <code>False</code> will ignore components that are connected to the border. You could use this to filter for the internal components only. For example</p>
<pre><code>internal = ComponentMeasurements[regi... |
971,457 | <p>It is asked that I find a function such that $$10-f(x)=2\int_0^xf(t)dt.$$ I tried giving a new function F(x) such that ${dF(x)\over dx}=f(x)$, but all I got was a new equation $$F(x)=10x-2\int_0^xF(t)dt.$$ So how do we find such function. Thanks in advance! (I am new to differential equations, so I do not know much ... | Community | -1 | <p>Differentiate the equation with respect to $t$ and use the Fundamental theorem of calculus: We obtain : </p>
<p>$$ - \frac{ d f }{dt } = 2 f(t) $$</p>
<p>Next, write this equation as follows:</p>
<p>$$ \frac{df}{f} = - 2 dt \implies \int \frac{df}{f} = - 2 \int dt \implies \ln f = -2t + C \implies f(t) = e^{-2t ... |
971,457 | <p>It is asked that I find a function such that $$10-f(x)=2\int_0^xf(t)dt.$$ I tried giving a new function F(x) such that ${dF(x)\over dx}=f(x)$, but all I got was a new equation $$F(x)=10x-2\int_0^xF(t)dt.$$ So how do we find such function. Thanks in advance! (I am new to differential equations, so I do not know much ... | ajotatxe | 132,456 | <p>Try computing the derivative of each member. You have to use the Fundamental Theorem of Calculus:</p>
<p>$$-f'(x)=2f(x)$$</p>
<p>Now, divide by $f$:</p>
<p>$$-\dfrac{f'(x)}{f(x)}=2$$</p>
<p>Note that the first member is the derivative of $-\ln f(x)$. Can you continue?</p>
|
1,725,150 | <p>I am trying to answer the following question:</p>
<p>Let $M_a := \{ (x^1,\ldots,x^n,x^{n+1}) \in \mathbb{R}^{n+1} : (x^1)^2 + \cdots +(x^n)^2 - (x^{n+1})^2 = a\}$. For which values of $a$, $M_a$ is a submanifold of $\mathbb{R}^{n+1}?$</p>
<p>I have to use the regular value theorem, but I think I don't know in fact... | Llohann | 310,768 | <p>If $f$ has constant rank, you have a submersion and $a$ does not matter. But it is not the case (as noticed by Tsemo Aristide):
$$\nabla f(x^1,...,x^{n+1})=(2x^1,...,2x^n,-2x^{n+1}).$$
Therefore, $(0,...,0)$ is not a regular point.</p>
<p>In the theorem, the dependence of the point $a$ appears when you need to chec... |
382,549 | <p>Let $A = diag \left (\lambda_1, ..., \lambda_n \right ) \in \mathbb{R}^{n \times n}$, with $\lambda_1 < \lambda_2 < ... < \lambda_n$.<br>
Let $u = \left (u_1, ..., u_n \right ) ^T \in \mathbb{R}^n$, with $u_i \neq 0 \ \ \forall i$.<br>
How can be shown that: </p>
<ol>
<li>For any $\alpha \in \mathbb{R}, \... | Martin Argerami | 22,857 | <p>Suppose that $(A-\lambda_j I+\alpha\,uu^T)x=0$. We can write this as $(A-\lambda_j)x=-\alpha\,uu^Tx$. If we look at the $j^{\rm th}$ coordinate, the left-hand side is $0$, and the right-hand side is
$$
-\alpha(u^Tx) u_j,
$$
so (using that all coordinates of $u$ are nonzero) $u^Tx=0$. But then the right-hand side is ... |
1,585,323 | <p>Statement :- Let $A$ denote an event whose probability of occurrence in a single trial is $p$. If $k$ denotes the number of occurrences of $A$ in $n$ independent trials, then </p>
<p>$$P\left(\left|\frac kn - p\right|> \epsilon\right) \lt \frac{pq}{n \epsilon ^2}$$</p>
<p>Someone please help me understanding th... | Satish Ramanathan | 99,745 | <p>Chebychev's inequality says the upper bound is $\frac{1}{\epsilon^2}$ for $\epsilon$ standard deviations in $P((|k-\mu|)\ge \epsilon\sigma) = P(|\frac{k}{n}-p|\ge\epsilon\sqrt{\frac{pq}{n}})$. The standard deviation here is $\sqrt{npq}$ and $\mu = np$. For the following expression $\left(\epsilon\sqrt{\frac{pq}{n}... |
1,208,269 | <p>let $x,y,z>0$ such that $x^2+y^2+z^2=1$. Find the minimum of $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$</p>
<p>Is the answer $3\sqrt{3}$ by any chance?</p>
| DeepSea | 101,504 | <p>Multiply the $1^{st}$ equation by $b$, and the $2^{nd}$ equation by $2$ and add them to eliminate $y$: $(ab+2)x = 0 \Rightarrow ab+2 =0$ will yield non-trivial solution. For example, $a= 2, b = -1$ will do.</p>
|
1,208,269 | <p>let $x,y,z>0$ such that $x^2+y^2+z^2=1$. Find the minimum of $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$</p>
<p>Is the answer $3\sqrt{3}$ by any chance?</p>
| Nathan C. | 226,630 | <p>Hint:</p>
<p>$\begin{bmatrix}
a&2|0\\
1&-b|0\\
\end{bmatrix}\to \begin{bmatrix}
1&\frac{2}{a}|0\\
0&\frac{2}{a}+b|0\\
\end{bmatrix}$</p>
<p>We know that if $\frac{2}{a}+b=0$, there are infinitely many solutions, and if it is nonzero we can multiply this number by the additive inverse and continue t... |
1,108,019 | <p>Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show that $T(Y)=\frac{1}{n}\sum_iY_i$ is complete:
$$
h\text{ being a function s.t. }E_\theta(h(T))=0 \forall \theta\in\Th... | DDB | 240,853 | <p>...or, consider your integral being the squared integral of (exp(-r^2)dr, from 0 to Infinity) in polar coordinates which can be solved easely. The result is sqrt{1/2*sqrt(pi)*erf (Infinity)}. erf(infinity) is a fancy way of saying "1".</p>
|
1,355,502 | <p>We inductively define $a^1=a, a^{n+1}=a^n a$. I want to show that $a^{n+m}=a^n a^m$. </p>
<p>By definition, this is true if $m=1$. Now for $m=2$, we have
$$
\begin{align}
a^{n+2} =& a^{(n+1)+1}\\
=& a^{n+1}a \\
=& \left(a^{n}a\right)a \\
=& a^{n}\left(aa\right) \\
=& a^{n}a^2
\end{align}
$$
Ho... | Sepideh Abadpour | 93,266 | <p>assume $m=k\Rightarrow a^{n+k}=a^na^k$<br>
then we should prove that $m=k+1\Rightarrow a^{n+(k+1)}=a^na^{k+1}$<br>
$$a^{n+(k+1)}=a^{(n+k)+1}=a^{n+k}a\qquad\qquad because\;a^{n+1}=a^na$$
$$\Rightarrow a^{n+(k+1)}=a^{n+k}a=a^na^ka=a^na^{k+1}\qquad\qquad because\;a^{n+1}=a^na$$</p>
|
2,895,648 | <blockquote>
<p>Suppose $f: \mathbb{R} \to \mathbb{R}$ is continuous such that for any
real $x$, </p>
<p>$|f(x) - f(f(x))| \leq \frac{1}{2} |f(x) -x|$.</p>
<p>Must $f$ have a fixed point?</p>
</blockquote>
<p>The question seems to invite an eventual application of the standard contraction mapping theorem... | Marco | 582,590 | <p>Let $x_i$ be the sequence defined by $x_0=0$ and $x_{i+1}=f(x_i)$. Then the inequality implies that
$$|x_{i+1}-x_{i+2}|<\frac{1}{2}|x_{i+1}-x_i|.$$
Therefore $|x_{i+1}-x_i|<2^{-i}|x_1-x_0| \rightarrow 0$. For all $m>n>N$, one has
$$|x_m-x_n|\leq |x_m-{x_{m-1}}|+\ldots +|x_{n+1}-x_n|<(2^{-m+1}+\ldots+2... |
3,091,943 | <p>Here is the solution:</p>
<p><a href="https://i.stack.imgur.com/HbkbJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HbkbJ.png" alt="enter image description here"></a></p>
<p>But I could not understand how the last term in the fourth line came from the line before it, could anyone explain this ... | robjohn | 13,854 | <p>Assuming <span class="math-container">$0\le k_j\le n_j$</span> for all <span class="math-container">$1\le j\le m$</span>,
<span class="math-container">$$
\begin{align}
\binom{n_1+n_2}{k_1+k_2}
&=\sum_{j=0}^{k_1+k_2}\binom{n_1}{j}\binom{n_2}{k_1+k_2-j}\tag1\\
&\ge\binom{n_1}{k_1}\binom{n_2}{k_2}\tag2
\end{ali... |
3,882,308 | <p>I am starting to learn about rings and ideals in abstract algebra. I came across a textbook problem that I am having a lot of trouble solving:</p>
<blockquote>
<p>Prove that for any positive integer <span class="math-container">$n$</span> ending in <span class="math-container">$7$</span>, the ideal generated by <spa... | ben huni | 618,745 | <p>Assume <span class="math-container">$n$</span> is positive. Then we have that <span class="math-container">$n=10k+7$</span> for some <span class="math-container">$k \in \mathbb{Z}$</span> and <span class="math-container">$(1-\sqrt{11})(1+\sqrt{11})=-10$</span>.
Thus it follows that <span class="math-container">$\gcd... |
3,882,308 | <p>I am starting to learn about rings and ideals in abstract algebra. I came across a textbook problem that I am having a lot of trouble solving:</p>
<blockquote>
<p>Prove that for any positive integer <span class="math-container">$n$</span> ending in <span class="math-container">$7$</span>, the ideal generated by <spa... | Servaes | 30,382 | <p><strong>Hint:</strong> Note that the ideal also contains the element
<span class="math-container">$$(1+\sqrt{11})(1-\sqrt{11})=1-11^2=-10.$$</span></p>
|
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