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 |
|---|---|---|---|---|
172,080 | <p>Here is a fun integral I am trying to evaluate:</p>
<p>$$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$</p>
<p>I thought about integrating by parts $2n$ times and then using the binomial theorem for $\sin(x)$, that is, using $\dfrac{e^{ix}-e^{-ix}}{2i}$ form in the binomial se... | Graham Hesketh | 66,912 | <p>One more just for luck... </p>
<p>Use the evenness of the integrand, the binomial expansion of $\sin(x)^{2n}$ in terms of exponentials, and the Fourier transform representation of the <a href="http://en.wikipedia.org/wiki/Rectangular_function">rectangular function</a> and you have:</p>
<p>\begin{aligned}
\frac{1}... |
159,761 | <p>I have two lists:</p>
<pre><code>list1 = {"a", "b"};
list2 = {{{1, 2}, {3, 4}}, {{1, 2}}};
</code></pre>
<p>My goal is to create a new list which would be:</p>
<pre><code>{"a u 1:2","a u 2:3","b u 1:2"}
</code></pre>
<p>In other words first element in <code>list1</code> would be distributed before each subelemen... | user1066 | 106 | <p>For the second part of your question, <code>ArrayFlatten</code> and <code>Thread</code> may be combined:</p>
<pre><code>ArrayFlatten[{#}] & /@ Thread[{list1, list2}]
</code></pre>
<blockquote>
<p>{{{a, 1, 2}, {a, 3, 4}}, {{b, 1, 2}}}</p>
</blockquote>
<p>But perhaps more useful is simply the following: </p>... |
1,896,024 | <p><span class="math-container">$f(n) = 2n^2 + n$</span></p>
<p><span class="math-container">$g(n) = O(n^2)$</span></p>
<p>The question is to find the mistake in the following process:</p>
<blockquote>
<p><span class="math-container">$f(n) = O(n^2) + O(n)$</span></p>
<p><span class="math-container">$f(n) - g(n) = O(n^... | Mithlesh Upadhyay | 234,055 | <p>According to definition of big-O notation:</p>
<p>$f(x)=O(g(x)){\text{ as }}$$x\to a$\,
if and only if</p>
<p>${\displaystyle \limsup _{x\to a}\left|{\frac {f(x)}{g(x)}}\right|<\infty }$</p>
<p>$\lim_{x→\infty}\left|{\cfrac {f(n)-g(n)}{(n^2)}}\right|=0$ </p>
<p>If $g(x)$ is nonzero, or at least becomes nonzer... |
1,077,594 | <p>Let $C[a,b]$ be the space of continuous functions on $[a,b]$ with the norm
$$
\left\Vert{f}\right\Vert=\max_{a \leq t \leq b}\left| f(t)\right|
$$</p>
<p>Then $C[a,b]$ is a Banach space. </p>
<p>Let's view $C^1[a,b]$ as a subspace of it. My question is, is this $C^1[a,b]$ a Banach space?</p>
<p>I think it is, sin... | sabachir | 201,840 | <p>we use $$f\left( z \right) = \frac{{e^{iz} }}{{1 + z^2 }}$$</p>
|
1,743,935 | <p>Not sure if I have done this correctly, seems too straight forward, any help is very appreciated. </p>
<blockquote>
<p>QUESTION:<br>
Find the real and imaginary parts of $f(z) = \cos(z)$.</p>
</blockquote>
<p>ATTEMPT:<br>
$\cos(z) = \cos(x+iy) = \cos x\cos(iy) − \sin x\sin(iy) =
\cos x\cosh y − i\sin x\sinh y... | egreg | 62,967 | <p>By definition,
$$
\cos z=\frac{e^{iz}+e^{-iz}}{2},\qquad
\sin z=\frac{e^{iz}-e^{-iz}}{2i}
$$
In particular, for real $y$,
$$
\cos(iy)=\frac{e^{-y}+e^{y}}{2}=\cosh y
$$
and
$$
\sin(iy)=\frac{e^{-y}-e^{y}}{2i}=i\frac{e^{y}-e^{-y}}{2}=i\sinh y
$$</p>
<p>So, yes, you're correct.</p>
|
1,743,935 | <p>Not sure if I have done this correctly, seems too straight forward, any help is very appreciated. </p>
<blockquote>
<p>QUESTION:<br>
Find the real and imaginary parts of $f(z) = \cos(z)$.</p>
</blockquote>
<p>ATTEMPT:<br>
$\cos(z) = \cos(x+iy) = \cos x\cos(iy) − \sin x\sin(iy) =
\cos x\cosh y − i\sin x\sinh y... | Community | -1 | <p>Using the exponential definition of the cosine,</p>
<p>$$2\cos(z)=e^{iz}+e^{-iz}=e^{-y+ix}+e^{y-ix}\\
=e^{-y}(\cos(x)+i\sin(x))+e^{y}(\cos(x)-i\sin(x))\\
=(e^y+e^{-y})\cos(x)-i(e^y-e^{-y})\sin(x)).$$</p>
|
1,219,129 | <p>For any vector space $V$ over $\mathbb{C}$, let $X$ be a set whose cardinality is the dimension of $V$. Then $V \cong \bigoplus\limits_{i \in X} \mathbb{C}$ as vector spaces.</p>
<p>Is there a similar description of arbitrary Hilbert spaces? Is there something they all "look" like?</p>
| Tomasz Kania | 17,929 | <p>Every Hilbert space is isometrically isomorphic to $\ell_2(\Gamma)$ for some set $\Gamma$. This follows directly from <a href="http://en.wikipedia.org/wiki/Parseval%27s_identity" rel="nofollow">Parseval's identity</a>.</p>
|
3,073,832 | <p>I need to understand the meaning of this mathematical concept: "undecided/undecidable". </p>
<p>I know what it means in the English dictionary. But, I don't know what it means mathematically.</p>
<p>If You answer this question with possible mathematical examples, it will be very helpful to understand this issue.<... | hunter | 108,129 | <p>Given a set of axioms, a statement is undecidable if neither it nor its negation follow from the axioms.</p>
<p>Example:
If your only axiom is:
<span class="math-container">$$
\forall z \forall x \forall y \ (y=x) \vee(y=z)\vee (x=z)
$$</span>
(in English, "for any three things, two of them are equal")</p>
<p>the... |
3,073,832 | <p>I need to understand the meaning of this mathematical concept: "undecided/undecidable". </p>
<p>I know what it means in the English dictionary. But, I don't know what it means mathematically.</p>
<p>If You answer this question with possible mathematical examples, it will be very helpful to understand this issue.<... | user3482749 | 226,174 | <p>A statement <span class="math-container">$P$</span> is undecidable in an theory <span class="math-container">$T$</span> if <span class="math-container">$t \cup \{P\}$</span> and <span class="math-container">$t \cup \{\neg P\}$</span> are both consistent. In practice, it's usually used more broadly: <span class="math... |
1,954,411 | <p>Let $N>0$ be a large integer, and $n<N$, then how to simply the following sum
$$\sum\limits_{k=1}^n\frac{N-n+k}{(N-k+1)(N-k+1)(N-k)}.$$
Thank you very much, guys.</p>
<p>Actually for another similar sum $\sum\limits_{k=1}^n\frac{1}{(N-k+1)(N-k)}=\sum\limits_{k=1}^n\frac{1}{N-k}-\frac{1}{N-k+1}=\frac{1}{N-n}-\... | Claude Leibovici | 82,404 | <p><em>I am not sure that you will like it.</em></p>
<p>$$S_n=\sum\limits_{k=1}^n\frac{N-n+k}{(N-k+1)^2(N-k)}$$ $$S_n=\frac{n (n-2 N)}{N (n-N)}+(n-2 N-1)\, \big(\psi ^{(1)}(-N)-\psi ^{(1)}(n-N)\big)$$ where appears the first derivative of <a href="https://en.wikipedia.org/wiki/Digamma_function" rel="nofollow">the diga... |
1,954,411 | <p>Let $N>0$ be a large integer, and $n<N$, then how to simply the following sum
$$\sum\limits_{k=1}^n\frac{N-n+k}{(N-k+1)(N-k+1)(N-k)}.$$
Thank you very much, guys.</p>
<p>Actually for another similar sum $\sum\limits_{k=1}^n\frac{1}{(N-k+1)(N-k)}=\sum\limits_{k=1}^n\frac{1}{N-k}-\frac{1}{N-k+1}=\frac{1}{N-n}-\... | user90369 | 332,823 | <p>With the same method which you have used above you get </p>
<p>$$\sum\limits_{k=1}^n \frac{N-n+k}{(N-k+1)^2(N-k)}=\frac{n(2N-n)}{N(N-n)}-(2N+1-n)\sum\limits_{k=1}^n \frac{1}{( N-k+1)^2}$$</p>
<p><em>Hints</em>:</p>
<p>$\enspace N-n+k=(2N+1-n)-(N-k+1)$</p>
<p>$\enspace \displaystyle \frac{1}{(N-k+1)^2(N-k)}=\frac... |
4,307,016 | <p>Explore convergence of <span class="math-container">$\sum_{n=3}^{\infty}\frac{1}{n\ln n(\ln \ln n)^\alpha}$</span></p>
<p>Tried to use Cauchy integral test,so we need to find</p>
<p><span class="math-container">$$\int_{3}^\infty\frac{dx}{x\ln x(\ln \ln x)^\alpha}=\int_{\ln 3}^{\infty}\frac{dz}{z(\ln z)^\alpha}= \in... | Botnakov N. | 452,350 | <p>You already almost solved the problem.</p>
<p>If <span class="math-container">$\alpha \ne 1$</span> we have
<span class="math-container">$$\int_{\ln (\ln 3)}^{\infty}\frac{du}{(u^\alpha)} = \int_{\ln (\ln 3)}^{\infty} u^{-\alpha} du = \frac{u^{-\alpha+1}}{-\alpha+1} \bigg|_{\ln (\ln 3)}^{\infty} $$</span>
It's fini... |
117,608 | <p>We know that if $G$ is a simple group with $p+1$ Sylow $p$-subgroups, then $G$ is 2-transitive. Now let $G$ be almost simple group with $p+1$ Sylow $p$-subgroups. Is $G$ 2-transitive group?</p>
| Geoff Robinson | 14,450 | <p>I think there is a direct argument. Let $M$ be the unique minimal normal subgroup of $G,$
which is non-Abelian simple. Then $M$ must act faithfully by conjugation
on the $(p+1)$ Sylow $p$-subgroups of $G$- otherwise, $M$
has a normal Sylow $p$-subgroup, which must then be trivial.
But even then, $M$ must normalize, ... |
1,134,854 | <blockquote>
<p>In complex analysis, let $a, b>0$ in $\mathbb R$, $f(s)=\int^{b}_{a}1/t^s dt$, then $f$ is holomorphic for $Re(s)>0$.</p>
</blockquote>
<p>If $s\neq 1$, then $f(s)=\frac{a^{1-s}}{(1-s)}-\frac{b^{1-s}}{(1-s)}$, but if $s=1$, then $f(s)=\ln\big(\frac{b}{a}\big)$, they seems quite different in th... | Martín-Blas Pérez Pinilla | 98,199 | <p>For problems like this, you can use <a href="http://en.wikipedia.org/wiki/Morera%27s_theorem" rel="nofollow">Morera's theorem</a> (and <a href="http://en.wikipedia.org/wiki/Fubini%27s_theorem" rel="nofollow">Fubini</a>).</p>
<p>For your particular case, check that the discontinuity of $\frac{a^{1-s}}{(1-s)}-\frac{b... |
416,153 | <p>Show that $x^2+y^2=p$ has a solution in $\mathbb{Z}$ if and only if $ p≡1 \mod 4$. Thnx, if someone can help</p>
| DonAntonio | 31,254 | <p>Hints: </p>
<p>$$\begin{align*}\bullet&\;\;\;\Bbb Z_p:=\Bbb Z/p\Bbb Z\;\;\text{is a field whenever $\;p\;$ is a prime}\\
\bullet&\;\;\;\text{Doing arithmetic modulo $\;p\;$ :}\;x^2+y^2=p=0\;\wedge\;\;xy\neq 0\iff\left(\frac xy\right)^2=-1\\
\bullet&\;\;\;\left|\;\Bbb Z_p^*\;\right|=p-1\\{}\\
\bullet&... |
187,974 | <p>If $ \cot a + \frac 1 {\cot a} = 1 $, then what is $ \cot^2 a + \frac 1{\cot^2 a}$? </p>
<p>the answer is given as $-1$ in my book, but how do you arrive at this conclusion?</p>
| N. S. | 9,176 | <p><strong>Hint</strong> What do you get if you square $\cot(a)+\frac{1}{\cot(a)}$?</p>
|
914,936 | <p>Does anyone know where I can find the posthumously published (I think) chapter 8 of Gauss's Disquisitiones Arithmaticae?</p>
| Community | -1 | <p>Maser's 1889 German translation has on the title page: "the last third of the text contains Gauss's published papers on The Theory of Numbers, followed by his posthumous writings on that subject". It's about 230 pp., tacked onto books I-VII of Disquisitiones. It's available as a reprint (mine is Chelsea press). I do... |
3,393,244 | <p>My homework is to transform this formula </p>
<p><span class="math-container">$$(A \wedge \neg B) \wedge (A \vee \neg C)$$</span> into this equivalent form: <span class="math-container">$A \wedge \neg B$</span>. Do you have any ideas?</p>
| J.G. | 56,861 | <p>Note that <span class="math-container">$A\land\neg B\to A$</span> and <span class="math-container">$A\to A\lor\neg C$</span>, so your original statement is equivalent to <span class="math-container">$A\land\neg B$</span> by repeated use of <span class="math-container">$(p\to q)\to((p\land q)\equiv p)$</span>.</p>
|
1,943,328 | <p>I know about $S_n$, $D_n$ and $A_n$. And from my limited understanding there seem to be many more. I would like to know whether there is some kind of relation that links a small set of non Abelian groups to create the other ones. Something like with the Abelian groups and the Fundamental Theorem of Abelian Groups.</... | Mees de Vries | 75,429 | <p>The "official" answer is the <a href="https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups">classification of simple finite groups</a>. In some sense, <a href="https://en.wikipedia.org/wiki/Composition_series">all finite groups are built from simple finite groups</a>, so understanding those is a grea... |
557,543 | <p>Does there exists a positive decreasing sequence $\{a_i\}$ with $\sum_{i\in\mathbb{N}} a_i$ convergent, such that $\forall I\subset\mathbb{N},\sum_{i\in I}a_i$ is an irrational number?</p>
<p>Such an example would give rise to a <strong>closed perfect set containing no rationals</strong>. I can only do it for infin... | André Nicolas | 6,312 | <p>Let
$$a_n=\frac{\sqrt{2}}{10^{n!}}.$$
The sum of any finite (non-zero!) number of the $a_i$ is irrational. The sum of an infinite number of the $a_i$ is transcendental, since $\sum_{i=1}^\infty \frac{1}{10^{n_i!}}$ is a <a href="http://en.wikipedia.org/wiki/Liouville_number">Liouville number.</a> </p>
|
793,693 | <p>Since I was interested in maths, I have a question. Is infinity a real or complex quantity? Or it isn't real or complex?</p>
| Jack M | 30,481 | <p>The question is a bit meaningless. "The infinite" is a philosophical concept. There are a <em>wide</em> variety of very different mathematical objects that are used to represent "the infinite", and now that we're in the realm of mathematics and not philosophy, I can make the concrete mathematical claim that <em>no</... |
1,946,824 | <p>In his book "Analysis 1", Terence Tao writes:</p>
<blockquote>
<p>A logical argument should not contain any ill-formed
statements, thus for instance if an argument uses a statement such
as x/y = z, it needs to first ensure that y is not equal to zero.
Many purported proofs of “0=1” or other false statements... | Andreas Blass | 48,510 | <p>The essential fact needed here is that a recursive definition like that of $F_n$ can be replaced by an equivalent explicit definition. Specifically, the Fibonacci sequence $W=\{\langle k,F_k\rangle:k\in\mathbb N\}$ can be defined as the set of all those ordered pairs $\langle k,x\rangle$ such that there exists a fu... |
777,535 | <p>I need to find the full Taylor expansion of $$f(x)=\frac{1+x}{1-2x-x^2}$$</p>
<p>Any help would be appreciated. I'd prefer hints/advice before a full answer is given. I have tried to do partial fractions\reductions. I separated the two in hopes of finding a known geometric sum but I could not.</p>
<p>Edit: I guess... | Community | -1 | <p>The proof assumes that the dot product is linear, which is not trivial to prove without the standard algebraic definition.</p>
<p>The more straightforward proof would be as follows: Create a triangle with the two vectors $a$ and $b$ so that the third side is $a-b$. Define the dot product as $a\cdot b=a_1b_1+a_2b_2$... |
1,756,685 | <p>For natural numbers—that is, integers greater than or equal to 1—prove that: <br/>
$n^{2n+1}\ge(n+1)^{n+1}(n-1)^{n}$ <br/></p>
<p>Equivalently, show that $(1-1/n)^n$ is strictly increasing.</p>
| Jack D'Aurizio | 44,121 | <p>$$\left(1-\frac{1}{n}\right)^n = \left(1-\frac{1}{n}\right)^n\cdot 1\stackrel{\color{red}{AM-GM}}{\color{red}{<}}\left(\frac{n\cdot\left(1-\frac{1}{n}\right)+1}{n+1}\right)^{n+1}=\left(1-\frac{1}{n+1}\right)^{n+1}.$$</p>
|
2,801,433 | <p>I have made the following conjecture, and I do not know if this is true.</p>
<blockquote>
<blockquote>
<p><strong>Conjecture:</strong></p>
</blockquote>
<p><span class="math-container">\begin{equation*}\sum_{n=1}^k\frac{1}{\pi^{1/n}p_n}\stackrel{k\to\infty}{\longrightarrow}2\verb| such that we denote by | p_n\verb| ... | Robert Z | 299,698 | <p>Recall that
<a href="https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes" rel="nofollow noreferrer">$\sum_{n=1}^{\infty} \frac{1}{p_n}$</a> is a divergent series. Then your series is divergent too because, for any positive number $a$,<br>
$$\lim_{n\to \infty}a^{1/n}=\lim_{n\to \infty... |
2,136,079 | <p>A cone $K$, where $K ⊆\Bbb R^n$ , is pointed; which means that it contains no line (or equivalently, $(x ∈ K~\land~ −x∈K) ~\to~ x=\vec 0$.</p>
| Royi | 33 | <p>It means there are no 2 points inside it which creates a line and the whole line is contained by the cone.</p>
<p>For instance, take $ \mathbb{R}^{2} $ it is clearly a cone yet it is not pointed as any line in $ \mathbb{R}^{2} $ is contained by $ \mathbb{R}^{2} $.</p>
<p>Yet if you take $ \mathbb{R}^{2}_{++} $, na... |
3,424,656 | <p>Assume <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are continuous at <span class="math-container">$x=a$</span>. Prove <span class="math-container">$h=\max\{f,g\}$</span> is continuous at <span class="math-container">$x=a$</span>.</p>
<p>My solution:</p>
<p>When <span class="... | copper.hat | 27,978 | <p>We always have <span class="math-container">$x_k \le y_k + |x_k - y_k| \le y_k +\|x-y\|_\infty$</span>.</p>
<p>Hence <span class="math-container">$x_k \le \max_j y_j +\|x-y\|_\infty$</span> and so <span class="math-container">$\max_j x_k \le \max_j y_j +\|x-y\|_\infty$</span>.</p>
<p>Reversing the roles of <span c... |
3,424,656 | <p>Assume <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are continuous at <span class="math-container">$x=a$</span>. Prove <span class="math-container">$h=\max\{f,g\}$</span> is continuous at <span class="math-container">$x=a$</span>.</p>
<p>My solution:</p>
<p>When <span class="... | BR Pahari | 276,873 | <p>At <span class="math-container">$x=a$</span>, use the fact that </p>
<p><span class="math-container">$$\max{(f,g)}=\frac{1}{2}(f+g+|f-g|).$$</span></p>
|
798,897 | <p>In our lecture we ran out of time, so our prof told us a few properties about measure: He said that a measure is $\sigma$-additive iff it has a right-side continuous function that it creates. And he was not only referring to probability measures.
After going through my lecture notes, I thought that this would imply... | Squirtle | 29,507 | <p>Consider the Dirac measure. </p>
<p>$$\delta_a(E)= 1\text{ if }a\in E, 0\text{ otherwise}$$</p>
<p>Or what about the measure that is zero on the empty set and infinity otherwise. </p>
|
2,958,135 | <p>A "standard" example of Bayes Theorem goes something like the following:</p>
<blockquote>
<p>In any given year, 1% of the population will get disease <em>X</em>. A particular test will detect the disease in 90% of individuals who have the disease but has a 5% false positive rate. If you have a family history of <... | ryang | 21,813 | <p>Further to user856's explanation in the comments, here's a complementary answer.</p>
<p>The way to frame/interpret medical tests in general is to understand them as updating one's level of certainty that the patient has the disease:</p>
<ul>
<li>without a medical-test result, the disease prevalence (a measure of dis... |
737,689 | <p>I have to prove that in a partially ordered set, only one of </p>
<blockquote>
<p>$$x<y,x=y,x>y$$ </p>
</blockquote>
<p>can hold. </p>
<p>My book says if both $x<y$ and $x=y$ hold, then this will imply $x<x$, which is a contradiction (contradicting irreflexivity). </p>
<p>I don't understand how thi... | David | 119,775 | <p><strong>Hint</strong>. Use the binomial theorem: since $a=b+kp$ we have
$$a^p=(b+kp)^p=b^p+\cdots\ ,$$
and with a bit of thought you will be able to see why all the remaining terms are divisible by $p^2$.</p>
|
1,136,278 | <p>Prove that $n(n-1)<3^n$ for all $n≥2$. By induction.
What I did: </p>
<p>Step 1- Base case:
Keep n=2</p>
<p>$2(2-1)<3^2$</p>
<p>$2<9$ Thus it holds.</p>
<p>Step 2- Hypothesis: </p>
<p>Assume: $k(k-1)<3^k$</p>
<p>Step 3- Induction:
We wish to prove that:</p>
<p>$(k+1)(k)$<$3^k.3^1$</p>
<p>We ... | axiom | 167,868 | <p>We know that $k(k-1)<3^k$ (The induction assumption)</p>
<p>Multiply 3 both sides, and we get:</p>
<p>$3k(k-1)<3^{k + 1}$</p>
<p>Now we will be done if we prove that
$k(k+1)\le3k(k - 1)$.</p>
<p>This can be rearranged as $2k - 4 \ge 0$, which is true since $k \ge 2$.</p>
<p>Hence proved.</p>
|
1,666,396 | <p>I can show the convergence of the following infinite product and some bounds for it:</p>
<p>$$\prod_{k \geq 2}\sqrt[k]{1+\frac{1}{k}}=\sqrt{1+\frac{1}{2}} \sqrt[3]{1+\frac{1}{3}} \sqrt[4]{1+\frac{1}{4}} \cdots<$$</p>
<p>$$<\left(1+\frac{1}{4} \right)\left(1+\frac{1}{9} \right)\left(1+\frac{1}{16} \right)\cdo... | Yuriy S | 269,624 | <p><strong>This is not an answer</strong>, but it's important and I post it separately from the question itself.</p>
<p>I found in <a href="https://math.stackexchange.com/a/1065075/269624">this answer</a> by @RandomVariable the following series:</p>
<p>$$\sum_{k=1}^{\infty} \frac{\ln (k+1)}{k(k+1)}=\frac{\pi^2}{4}-1-... |
1,666,396 | <p>I can show the convergence of the following infinite product and some bounds for it:</p>
<p>$$\prod_{k \geq 2}\sqrt[k]{1+\frac{1}{k}}=\sqrt{1+\frac{1}{2}} \sqrt[3]{1+\frac{1}{3}} \sqrt[4]{1+\frac{1}{4}} \cdots<$$</p>
<p>$$<\left(1+\frac{1}{4} \right)\left(1+\frac{1}{9} \right)\left(1+\frac{1}{16} \right)\cdo... | Jacob | 181,986 | <p>Working off of other people's findings, you can write
$$\sum_{k=2}^{\infty} \frac{(-x)^k \zeta (k)}{k} = x \gamma + \ln (\Gamma(x+1))$$
$$\frac{d}{dx}\sum_{k=2}^{\infty} \frac{(-x)^k \zeta (k)}{k} = -\sum_{k=2}^{\infty} (-x)^{k-1} \zeta (k)=\gamma+\psi(x+1)=H_x$$
$$\sum_{k=2}^{\infty} (-x)^{k-2} \zeta (k)=\frac{H_x}... |
1,752,021 | <blockquote>
<p>Let $G=S_3\times S_3$ where $S_3$ is the symmetric group. Let $p=
\begin{pmatrix}
1 & 2 & 3 \\
2 & 3 & 1 \\
\end{pmatrix}
$, let $L=(p)$, $K=L\times L$ and $H=\{(I_3,I_3),(p,p),(p^2,p^2)\}$. Show that $K\triangleleft G$, $H\triangleleft K$ but $H$ no is ... | gregorygsimon | 148,402 | <p>Using generators, there won't be much computation needed. </p>
<p>$K$ is generated by $(1,p)$ and $(p,1)$. If $H$ is invariant under conjugation by the generators of $K$, then $H$ is normal in $K$.</p>
<p>Lagrange's theorem proves that $S_3$ is generated by $p$ and any transposition, e.g. $t=
\begin{pmatri... |
1,752,021 | <blockquote>
<p>Let $G=S_3\times S_3$ where $S_3$ is the symmetric group. Let $p=
\begin{pmatrix}
1 & 2 & 3 \\
2 & 3 & 1 \\
\end{pmatrix}
$, let $L=(p)$, $K=L\times L$ and $H=\{(I_3,I_3),(p,p),(p^2,p^2)\}$. Show that $K\triangleleft G$, $H\triangleleft K$ but $H$ no is ... | Community | -1 | <p>Observe that $|L| = 3$ and $|S_3| = 6$, so the index of $L$ in $S_3$ is $2$. Therefore, $L \lhd S_3$. Consequently, $K = L \times L \lhd S_3 \times S_3 = G$.</p>
<p>Then, observe that $K$ is abelian, because it is a direct product of abelian groups, so all of its subgroups are normal. Therefore $H \lhd K$.</p>
<p>... |
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | Federico Poloni | 4,930 | <p>First heard it from a former classmate of mine, might be her own invention:</p>
<blockquote>
<p>When the second derivative is positive, the function is happy (i.e., its graph looks like a smile). When the second derivative is negative, the function is sad (i.e., its graph looks like a frown).</p>
</blockquote>
<... |
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | Elle Najt | 8,029 | <p>To remember concave Up (vs concave down), I remember that the U shape is concave up. Similarly, in conVex, the V is convex. </p>
<p>(If you like, v is the only letter in the word which is the graph of a function. Sadly, concave also has a v, but this mneumonic seems to work anyway. You just have to remember which w... |
1,451,745 | <p>Can someone check my logic here. </p>
<p><strong>Question:</strong> How many ways are there to choose a an $k$ person committee from a group of $n$ people? </p>
<p><strong>Answer 1:</strong> there are ${n \choose k}$ ways. </p>
<p><strong>Answer 2:</strong> condition on eligibility. Assume the creator of the comm... | pre-kidney | 34,662 | <p>Did you define what you mean by "eligible"? I don't follow the argument.</p>
<p>Here is another approach: inductively apply Pascal's identity
$$
\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}.
$$</p>
|
4,103,366 | <p>I have <span class="math-container">$A \in \mathbb{R}^{q\times n }, B \in \mathbb{R}^{n \times p} $</span> with <span class="math-container">$\text{rank}(A)=q~$</span> and <span class="math-container">$~\text{rank}(B)=p$</span>.</p>
<p>Additionally there is the condition: <span class="math-container">$n\geq p \geq q... | nachosemu | 846,505 | <p>Take the following example:</p>
<p><span class="math-container">$$
A = \begin{bmatrix}
0 & 0 & 0& 1 & 0 & 0\\
0 & 0 & 0& 0 & 1 & 0 \\
0 & 0 & 0& 0 & 0 & 1 \\
\end{bmatrix}, B = \begin{bmatrix}
1 & 0\\
0 & 1\\
0 & 0\\
0 & 0\\
0 & 0\\
\end... |
4,351,990 | <p>I have just finished my undergrad and while I haven't studied much in representation theory I find it a very fascinating subject. My current interest is in differential equations, and I am wondering is there any ongoing research that combines these two areas?</p>
| A. Thomas Yerger | 112,357 | <p>I'll offer another view: physics, especially quantum mechanics, is essentially about the interplay between representation theory and differential equations. Representations of groups like the unitary groups and the Heisenberg group(s) encode information about symmetries of physical systems, invariances under various... |
270,849 | <p>I am trying to show that </p>
<p>$P(E\mid E\bigcup F) \geq P(E \mid F)$.</p>
<p>This is intuitively clear. But when expanding I get $P(E)\ P(F)\geq P(E\bigcup F)\ P(E \bigcap F)$. How to continue?</p>
| Davide Giraudo | 9,849 | <p><strong>Hint:</strong> the difference between these two terms is $P(E\cap F^c)P(F\cap E^c)$. </p>
|
3,301,115 | <p>I'm currently taking an introductory course in graph theory, and this problem is giving me a bit of a hard time. Where would I even start? Thanks a bunch?</p>
| Michael Rozenberg | 190,319 | <p>By C-S
<span class="math-container">$$\sum_{cyc}\frac{a}{\sqrt{1-bc}}\leq\sqrt{\sum_{cyc}a\sum_{cyc}\frac{a}{1-bc}}.$$</span>
Thus, it's enough to prove that
<span class="math-container">$$\sum_{cyc}\frac{a}{1-bc}\leq\frac{9}{2(a+b+c)},$$</span> which is true by SOS:
<span class="math-container">$$\frac{9}{2(a+b+c)}... |
546,276 | <p>Let $\{s_n\}$ be a sequence in $\mathbb{R}$, and assume that $s_n \rightarrow s$. Prove that $s^k_n\rightarrow s^k$ for every $k \in\mathbb{N}$</p>
<p>Ok, so we need $|s^k_n - s^k| < \varepsilon$. I rewrote this as</p>
<p>$$|s_ns^{k-1}_n - ss^{k-1}|=|(s_n-s)(s^{k-1}_n + s^{k-1}) -s_ns^{k-1}+ss_n^{k-1}|$$</p>
<... | DanielV | 97,045 | <p>$y = \frac 1 4 x^3 + 12x + 6$</p>
<p>$L = -24x - 32$</p>
<p>$P$ is a line that is perpendicular to $L$.</p>
<p>The slope of a perpendicular is the negative of the multiplicative inverse, that is: $$\frac{dP} {dx} = -(\frac {dL} {dx})^{-1} = -\frac{dx} {dL}$$</p>
<p>To solve the problem we want $y$ where:
$$\und... |
2,147,458 | <p>Solve the following integral:
$$
\frac{2}{\pi}\int_{-\pi}^\pi\frac{\sin\frac{9x}{2}}{\sin\frac{x}{2}}dx
$$</p>
| Jack D'Aurizio | 44,121 | <p>Such integral equals:
$$\frac{4}{\pi}\int_{-\pi/2}^{\pi/2}\frac{\sin(9x)}{\sin(x)}\,dx=\frac{4}{\pi}\int_{-\pi/2}^{\pi/2}\frac{e^{9ix}-e^{-9ix}}{e^{ix}-e^{-ix}}\,dx \tag{1}$$
that is:
$$ \frac{4}{\pi}\int_{-\pi/2}^{+\pi/2}\left(e^{8ix}+e^{6ix}+\ldots+1+\ldots+e^{-6ix}+e^{-8ix}\right)\,dx=\frac{4}{\pi}\int_{-\pi/2}^{... |
2,905,022 | <p>I recently stumbled upon the problem $3\sqrt{x-1}+\sqrt{3x+1}=2$, where I am supposed to solve the equation for x. My problem with this equation though, is that I do not know where to start in order to be able to solve it. Could you please give me a hint (or two) on what I should try first in order to solve this equ... | Barry Cipra | 86,747 | <p>Hint: Try the substitution $u=\sqrt{x-1}$.</p>
<p><strong>Added later</strong>: The answer hints so far, including mine (above), have all aimed at squaring away the square root symbols, leaving a quadratic equation that's easy to solve, with the caveat that the solutions to the quadratic are not necessarily soluti... |
213,916 | <p>Let $ D\subset \mathbb{C}$ be open, bounded, connected and with smooth boundary. Let $f$ be a nonconstant holomorphic function in a neighborhood of the closure of $D$ , such that $|f(z)|=c \forall z\in \partial D$, show that $f$ takes on each value $a$, such that $|a| < |c| $ at least once in $D$.</p>
| Christopher A. Wong | 22,059 | <p>The underlying principle in this problem is the open mapping property for holomorphic functions. However, this problem can be cleaned up by using some more specialized results.</p>
<blockquote>
<p><strong>Claim 1</strong>. If $f(z)$ must vanish somewhere on $D$.</p>
</blockquote>
<p><em>Proof</em>: As $f$ is non... |
3,732,571 | <p>I know that there exists a bijection between <span class="math-container">$[0,1]$</span> and <span class="math-container">$[0,1]\cup\{2\}$</span>, but late last night I was not able to come up with a trivial solution. Will be glad if one can provide such example.</p>
| Aryaman Maithani | 427,810 | <p>Define <span class="math-container">$f:[0, 1]\to[0,1]\cup\{2\}$</span> as
<span class="math-container">$$f(x) = \begin{cases}
2 & x = 1\\
\dfrac{1}{n-1} & x\text{ is of the form }\dfrac{1}{n};\;n\ge 2\\
x & \text{otherwise}
\end{cases}$$</span></p>
<p>The idea basically is to take the countable subset <s... |
3,732,571 | <p>I know that there exists a bijection between <span class="math-container">$[0,1]$</span> and <span class="math-container">$[0,1]\cup\{2\}$</span>, but late last night I was not able to come up with a trivial solution. Will be glad if one can provide such example.</p>
| FiMePr | 802,801 | <p>Well, I'm not sure there is a simple formula for this.</p>
<p>The idea is to find an infinite countable subset <span class="math-container">$C$</span> of <span class="math-container">$[0,1]$</span>, add the point <span class="math-container">$2$</span> to it and define a bijection between <span class="math-container... |
380,177 | <p>In mathematics, I want to know what is indeed the difference between a <strong>ring</strong> and an <strong>algebra</strong>?</p>
| mdp | 25,159 | <p>A ring $R$ has operations $+$ and $\times$ <a href="https://en.wikipedia.org/wiki/Ring_(mathematics)#Definition" rel="noreferrer">satisfying certain axioms which I won't repeat here</a>. An (associative) algebra $A$ similarly has operations $+$ and $\times$ satisfying the same axioms (it doesn't need a multiplicativ... |
380,177 | <p>In mathematics, I want to know what is indeed the difference between a <strong>ring</strong> and an <strong>algebra</strong>?</p>
| rschwieb | 29,335 | <p>One thing that complicates answering this question is that rings are almost always assumed to be associative, but algebras are frequently not assumed to be associative. (In other words, my impression is that it's more common to allow 'algebra' to name something nonassociative than it is to use 'ring' to mean somethi... |
1,212,425 | <p>This is a homework problem that I cannot figure out. I have figured out that if $n^2 + 1$ is a perfect square it can be written as such:</p>
<p>$n^2 + 1 = k^2$.</p>
<p>and if $n$ is even it can be written as such:</p>
<p>$n = 2m$</p>
<p>I believe I'm supposed to use the fact that if $n \pmod{4} \equiv 0$ or $1$ ... | Brian M. Scott | 12,042 | <p>You have the useful fact backwards. It’s not true that if $n$ is congruent to $0$ or $1$ mod $4$, then $n$ is a perfect square: $5\equiv1\pmod4$, and $8\equiv0\pmod4$, but neither $5$ nor $8$ is a perfect square. What <em>is</em> true is that if $n$ is a perfect square, then $n\equiv0\pmod 4$ or $n\equiv1\pmod 4$. T... |
3,814,195 | <p>As an applied science student, I've been taught math as a tool. And although I've been studying <strong>a lot</strong> throughout the years, I always felt like I am missing depth. Then I read geodude's answer on this <a href="https://math.stackexchange.com/questions/721364/why-dont-taylor-series-represent-the-enti... | Lawrence Mano | 201,051 | <p>Complex Analysis by Ahlfors is a masterpiece! But whatever book you read, you must read with not only your mind but also with your heart and soul!! You must feel the subject and only then Complex Analysis will stick.</p>
|
2,991,825 | <p>I'm trying to find the general solution to this matrix
<span class="math-container">\begin{bmatrix}1&-2&1&3&0\\2&-4&4&6&4\\ -2&4&-1&-6&2\\1&-2&-3&3&-8\end{bmatrix}</span></p>
<p>Ax=<span class="math-container">$\begin{bmatrix}1&6&0&-7&\... | Doug M | 317,162 | <p>use that <span class="math-container">$|\sin x| < |x|$</span></p>
<p><span class="math-container">$\sin \frac{\pi}{2^{n+1}} < \frac{\pi}{2^{n+1}}$</span></p>
<p><span class="math-container">$0<2^{n}\sin \frac{\pi}{2^{n+1}} < \frac {\pi}{2}$</span></p>
<p>The sequence is bounded... can we show that it ... |
2,208,943 | <p>I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far.</p>
<p>One thing I feel I am lacking in is motivation. That is, the difference in rigour between the... | Stella Biderman | 123,230 | <p>In general, the push for rigor is usually in response to a failure to be able to demonstrate the kinds of results one wishes to. It's usually relatively easy to demonstrate that there exist objects with certain properties, but you need precise definitions to prove that no such object exists. The classic example of t... |
2,208,943 | <p>I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far.</p>
<p>One thing I feel I am lacking in is motivation. That is, the difference in rigour between the... | danny | 429,130 | <p>I list here few excellent texts on Real Analysis,have a look at them. </p>
<p>1)Understanding Analysis by Stephen Abbott</p>
<p>2)Real Mathematical Analysis by Pugh</p>
<p>3)Counterexamples in analysis by Gelbaum</p>
<p>For historically inclined yet mathematical you may try
The Calculus gallery by William Dunha... |
1,393,265 | <p>How to prove that$(n!)^{1/n}$ tends to infinity as limit tends to infinity?
I tried to do this by expanding $n!$ as $n\times (n-1)\times (n-2)\cdots 4\times3\times2\times 1$ and taking out n common from each factor so that I can have $n$ outside the radical sign, But then the last terms would be $(4/n)\times(3/n)\ti... | 3d0 | 217,450 | <p>Using Sterling:</p>
<p>$$\sqrt{2\pi}\ n^{n+1/2}e^{-n} \le n! \le e\ n^{n+1/2}e^{-n} $$</p>
<p>Lets apply $^{\frac{1}{n}}$</p>
<p>$$(\sqrt{2\pi}\ n^{n+1/2}e^{-n})^{\frac{1}{n}} \le (n!)^{\frac{1}{n}} \le (e\ n^{n+1/2}e^{-n})^{\frac{1}{n}} $$</p>
<p>The left side
$$\sqrt[2n]{2\pi}\ n^{1+\frac{1}{2n}}e^{-1} \le (n!... |
275,371 | <p>I was wondering if it is possible to decompose any symmetric matrix into a positive definite and a negative definite component. I can't seem to think of a counterexample if the statement is false.</p>
| Community | -1 | <p>This answer is years late but I like this proof.
The set of positive definite matrices is open in the set of symmetric matrices and must thus contain <span class="math-container">$\frac{n(n+1)}{2}$</span> matrices which are linearly independent. Take that as a basis for the vector space of symmetric matrices. Then ... |
1,407,797 | <p>P is the middle of the median line from vertex A, of ABC triangle. Q is the point of intersection between lines AC and BP.</p>
<p><a href="https://i.stack.imgur.com/ka8E8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ka8E8.png" alt="enter image description here"></a></p>
| suncup224 | 61,149 | <p><strong>Hint:</strong> </p>
<p>Part 1: Apply <a href="https://en.wikipedia.org/wiki/Menelaus'_theorem" rel="nofollow">Menelaus' theorem</a> on triangle $AMC$ and line $BPQ$.</p>
<p>Part 2: Apply Menelaus' theorem on triangle $BCQ$ and line $APM$ (needs the answer form part 1).</p>
<p>I'm giving only hints but... |
364,278 | <p>Let <span class="math-container">$X$</span> be a variety over a number field <span class="math-container">$K$</span>. Then it is known that for any topological covering <span class="math-container">$X' \to X(\mathbb{C})$</span>, the topological space <span class="math-container">$X'$</span> can be given the structur... | S. carmeli | 115,052 | <p>Adding on Will's and Sasha's answers, the condition of having a rational point, or at least a "1-truncated homotopy fixed point" for the action is necessary. For example, let <span class="math-container">$C_2$</span> act on the circle <span class="math-container">$S^1$</span> by half rotation. The covers o... |
45,570 | <p>I'm writing a little package in Mathematica for geology where a particular stone may be approximated as an hemisphere. Anyway this is a rough estimation because a real hemisphere has its height as loong as its radius. Instead, a reservoir stone (for an hydrocarbon) has often a form of a section of an hemisphere, its... | Jens | 245 | <p>The ragged edges were caused by the fact that the parametrization of the surface in terms of height over the equatorial plane is singular at the equator, as you can also see in the increased distance between mesh lines on the plotted surface. So the main part of the solution is to choose a better parametrization, an... |
155,547 | <p>Given $X_1, \ldots, X_n$ from $\mathcal{N} (\mu, \sigma^2)$.</p>
<p>I have to compute the probability:
$$P\left(|\bar{X} - \mu| > S\right)$$
where $\bar{X}$ is the sample mean and $S^2$ is the sample variance.</p>
<p>I tried to expand:
$$P\left(\bar{X}^2 + \mu^2 - \bar{X}\mu > \frac{1}{n}\sum {X_i}^2 + \frac... | Did | 6,179 | <p>The empirical mean and empirical variance of i.i.d. normal samples are independent and follow known distributions, which are respectively normal and chi-squared. This indicates that
$$
\mathrm P\left(|\bar X-\mu|\gt S\right)=\mathrm P\left((n-1)Z_1^2\gt n(Z_2^2+\cdots+Z_n^2)\right),
$$
where $(Z_k)_{1\leqslant k\leq... |
329,513 | <p>$$
\int \frac{\sqrt{\frac{x+1}{x-2}}}{x-2}dx
$$</p>
<p>I tried:
$$
t =x-2
$$
$$
dt = dx
$$
but it didn't work.
Do you have any other ideas?</p>
| littleO | 40,119 | <p>The phenomenon in convex optimization that the dual of the dual problem is (usually) the same as the primal problem is seemingly a total surprise, and it is only rarely explained. But there's a nice, enlightening explanation that I learned from reading Ekeland and Temam. This material can also be found in the book... |
2,631,220 | <p>If $z$ is a variable complex number , and $a$ is a fixed complex number , is it true that if $z$ , $a$ satisfy the following condition </p>
<p>$|z+a| = |z-a|$ </p>
<p>Then the locus of $z$ is the perpendicular bisector of $a$ and $-a$ ?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>let $$z=x+iy$$ then $$|x+a+iy|=|x-a+iy|$$ and we get
$$\sqrt{(x+a)^2+y^2}=\sqrt{(x-a)^2+y^2}$$
Can you proceed?</p>
|
92,020 | <p>Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$.<br>
Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that<br>
$$\limsup_{\omega,t \in\Omega\times[S,T]} \sum_i^m \sum_j^n | g_{ij}(\omega,t)|<\infty$$ and<br>
$$\int_S^T E|G(\omega,t)... | Brenton | 226,184 | <p>This is an excerpt from the book "Stochastic Differential Equations and Applications" by Xuerong Mao. I've frequently used this book as a reference for SDEs. This looks like exactly what you're asking for, but specifically for $p=6$ (Theorem 5.21 they refer to is the Ito Isometry property)</p>
<p><img src="https://... |
1,092,665 | <p>My question is really simple, how can I write symbolically this phrase: </p>
<blockquote>
<p>$x=\sum a_mx^m$ where $m$ range over
$\{1,\ldots,g\}\setminus\{t_1,\ldots,t_u\}$</p>
</blockquote>
<p>Being more specific, I would like to know how to write with mathematical symbols this part: "range over $\{1,\ldots,... | Community | -1 | <p>You could put the limits of the sum behind the sum, making them ore inline instead of the big displaystyle format.</p>
<p>If you are using LaTeX I do wonder why there is no space left to use the slightly better looking alternative Kez gave. (if you are not using LaTeX, why not..?)</p>
<p>$\Large x=\sum_{{m=1\ldots... |
201,236 | <p>how to compute numerically the integral </p>
<pre><code>NIntegrate[6 x/(1 - x), {x, 0, 1}]
</code></pre>
<p>to give a value which is approximately equal to 891.441</p>
| David Keith | 44,700 | <p>The integral does not exist:</p>
<pre><code>Limit[Integrate[6 x/(1 - x), x], x -> 1]
(* \[Infinity] *)
out = Quiet@
Table[{wp,
NIntegrate[6 x/(1 - x), {x, 0, 1}, Exclusions -> 1,
WorkingPrecision -> wp]}, {wp, 10, 500, 5}];
ListPlot[out, Frame -> True,
FrameLabel -> {"Working Pre... |
201,236 | <p>how to compute numerically the integral </p>
<pre><code>NIntegrate[6 x/(1 - x), {x, 0, 1}]
</code></pre>
<p>to give a value which is approximately equal to 891.441</p>
| bill s | 1,783 | <p>You can approach this by finding the integral from 0 to something a little less than 1:</p>
<pre><code>f[eps_] = Integrate[6 x/(1 - x), {x, 0, eps}, Assumptions -> 0 < eps < 1]
-6 (eps + Log[1 - eps])
</code></pre>
<p>Now you can see explicitly that as eps -> 1, the integral diverges to -Infinity.</p>
<p... |
1,512,171 | <p>I want to show that there exists a diffeomorphic $\phi$ such that the following diagram commutes:
$$
\require{AMScd}
\begin{CD}
TS^1 @>{\phi}>> S^1\times\mathbb{R}\\
@V{\pi}VV @V{\pi_1}VV \\
S^1 @>{id_{S^1}}>> S^1
\end{CD}$$
where $\pi$ is the associated projection of $TS^1$, and $\pi_1(x,y)=x$ is ... | C. Falcon | 285,416 | <p>$f$ isn't Riemann-integrable but Lebesgue-integrable and indeed its integral is $1$, because $f=1$ almost everywhere on $[0,1]$, since $\mathbb{Q}$ is countable.</p>
|
1,341,440 | <p>I came across a claim in a paper on branching processes which says that the following is an <em>immediate consequence</em> of the B-C lemmas:</p>
<blockquote>
<p>Let $X, X_1, X_2, \ldots$ be nonnegative iid random variables. Then $\limsup_{n \to \infty} X_n/n = 0$ if $EX<\infty$, and $\limsup_{n \to \infty} X_... | Bhaskar Vashishth | 101,661 | <p>Let $|G|=8$ where $x\in G$ and $|x|=4$. Now $|x^2|=2$ and denote $Z=Z(G)$</p>
<p>Consider canonical homomorphism $\eta :G \to G/Z$.</p>
<p>Case 1- $|Z|=8$, $G$ is abelian, nothing to prove.</p>
<p>Case 2- $|Z|=4$, then $\eta : G \to \Bbb{Z}_2$, so if $x \to \bar{0}$ then so does $x^2$, and if $x \to \bar{1}$, the... |
3,476,022 | <p>I was watching this Mathologer video (<a href="https://youtu.be/YuIIjLr6vUA?t=1652" rel="noreferrer">https://youtu.be/YuIIjLr6vUA?t=1652</a>) and he says at 27:32</p>
<blockquote>
<p>First, suppose that our initial <em>chunk</em> is part of a parabola, or if you like a cubic, or any polynomial. If I then tell you... | ncmathsadist | 4,154 | <p>Polynomials are analytic functions. If two analytic functions agree on a set having a limit point, they must be equal by the <a href="https://en.wikipedia.org/wiki/Identity_theorem" rel="nofollow noreferrer">Identity Theorem.</a></p>
|
173,112 | <blockquote>
<p>Solve for $x$. $12x^3+8x^2-x-1=0$ all solutions are rational and between $\pm 1$</p>
</blockquote>
<p>As mentioned in my previous answers, I'm guessing I have to use the Rational Root Theorem. But I've done my research and I do not understand what to plug in or anything about it at all. Can someone p... | Bill Dubuque | 242 | <p>The reciprocals of the roots are roots of the (negated) reversed polynomial $\rm\:x^3+x^2-8\,x-12.\:$ By the Rational Root Test all its roots are integers. If the roots are $\rm\:a,b,c\:$ then by Vieta's Formulas we have $\rm\:abc = 12,\ a+b+c =-1\:$ so $\rm\:a,b,c = \ldots$</p>
<p><strong>Remark</strong> $\ $ I ch... |
1,175,993 | <p>I want to show $T=d/dx$ is unbounded on $C^1[a,b]$ with $b>1$. Take a sequence $f(x)=x^n$, and $\|T\|=\sup_{x\in[a,b]}\frac{\|Tx\|}{\|x\|}=\frac{\|n\cdot b^{n-1}\|}{\|b\|}$. I want to claim as $n$ goes to infinity, the operator norm goes to infinity, and hence it's unbounded. But the definition of operator norm o... | pabodu | 64,543 | <p>Pedro M., I expect $||f||_1$ in $C^1$ is $||f||_C+||f'||_C$. The operator $T$ is supposed to map functions from $C^1$ into itself. Definitely, its domain is narrower than $C^1$.</p>
|
1,688,762 | <p>$$\int \sqrt{\frac{x}{2-x}}dx$$</p>
<p>can be written as:</p>
<p>$$\int x^{\frac{1}{2}}(2-x)^{\frac{-1}{2}}dx.$$</p>
<p>there is a formula that says that if we have the integral of the following type:</p>
<p>$$\int x^m(a+bx^n)^p dx,$$ </p>
<p>then:</p>
<ul>
<li>If $p \in \mathbb{Z}$ we simply use binomial expa... | 3SAT | 203,577 | <blockquote>
<p>$$\int \sqrt{\frac{x}{2-x}}dx$$</p>
</blockquote>
<p>Set $t=\frac {x} {2-x}$ and $dt=\left(\frac{x}{(2-x)^2}+\frac{1}{2-x}\right)dx$</p>
<p>$$=2\int\frac{\sqrt t}{(t+1)^2}dt$$</p>
<p>Set $\nu=\sqrt t$ and $d\nu=\frac{dt}{2\sqrt t}$</p>
<p>$$=4\int\frac{\nu^2}{(\nu^2+1)^2}d\nu\overset{\text{ partia... |
276,329 | <p>I have a problem, from Gelfand's "Algebra" textbook, that I've been unable to solve, here it is:</p>
<p><strong>Problem 268.</strong> </p>
<p>What is the possible number of solutions of the equation $$ax^6+bx^3+c=0\;?$$</p>
<p>Thanks in advance.</p>
| DonAntonio | 31,254 | <p>Put $\,t:=x^3\,$ , so your equation becomes</p>
<p>$$(*) at^2+bt+c=0\Longrightarrow \Delta= b^2-4ac$$</p>
<p>Now, if $\,\Delta=0\,\,$ then $\,(*)\,$ has one unique solution. $\,x^3=t={-b/2a}\,$ , and if $\,\Delta >0\,$ then there're two solutions for $\,t=x^3\,$.</p>
<p>Since $\,3\,$ is an odd natural we don't... |
2,451,350 | <p>Currently I am reading into functional data analysis. A common assumption is that the expected value of some random function is $0$, i.e. $\mathbb{E}(x) = 0$ where $x \in L^2$, the space of all squared integrable functions with inner product $\langle x,y \rangle = \int x(t)y(t) \text{d}t$. </p>
<p>My question might... | Kenny Lau | 328,173 | <blockquote>
<p>If $Ax = b$ has a solution $x = u$, then $u + v$ is also a solution to $Ax = b$ for all solutions $x = v$ to $Ax = 0$.</p>
</blockquote>
<p>This sentence may be a little bit difficult to understand. Allow me to rephrase it:</p>
<blockquote>
<p>If $Ax = b$ has a solution $x = u$, then let $x=v$ be ... |
4,602,683 | <p>Let <span class="math-container">$\mathbb{F}$</span> be a field, and consider <span class="math-container">$\mathbb{F}^\mathbb{F}$</span> as an algebra over <span class="math-container">$\mathbb{F}$</span> with the standard function multiplication. Let <span class="math-container">$D$</span> be a linear transformati... | Marius S.L. | 760,240 | <p>Not an answer, but a helpful heuristic.</p>
<p>The chain rule and the product rule cannot be compared. A product of functions, which you called standard, requires two functions with the same domain and range: <span class="math-container">$f,g\, : \,D\rightarrow R$</span> since we plug in the same value and multiply ... |
168,053 | <p>If g is a positive, twice differentiable function that is decreasing and has limit zero at infinity, does g have to be convex? I am sure, from drawing a graph of a function which starts off as being concave and then becomes convex from a point on, that g does not have to be convex, but can someone show me an example... | Community | -1 | <p>Since the functions mentioned so far are <strong>eventually</strong> convex, here is one more:
$$
f(x)=e^{-x}(3+2\sin x)
$$
The first derivative $$f\,'(x)=e^{-x}(2\cos x-2\sin x-3)$$ is always negative because $\cos x-\sin x\le \sqrt{2}$ for all $x$. But the second derivative $$f\,''(x)=e^{-x}(3-4\cos x)$$ changes ... |
2,129,830 | <p>I am wondering if this is generally true for any topology. I think there might be counter examples, but I am having trouble generating them. </p>
| Nosrati | 108,128 | <p>In the Sierpinski topology $\{X,\emptyset,\{0\}\}$, the set $\{0\}$ is an open set that isn't the interior of any closed set.</p>
|
50,002 | <p>a general version: connected sums of closed manifold is orientable iff both are orientable.
I think this can be prove by using homology theory, but I don't know how.Thanks.</p>
| Jason DeVito | 331 | <p>If the connect sum is orientable, so are both pieces:</p>
<p>Proof: We'll use the fact that an $n$-manifold is closed and orientable iff $H_n(M) = \mathbb{Z}$. Assume $M_1$ is nonorientable and consider the connect sum $M_1\sharp M_2$.</p>
<p>The pair $(M_1\sharp M_2, M_2-\{p\})$ gives rise to a long exact seque... |
942,470 | <p>I am trying to count how many functions there are from a set $A$ to a set $B$. The answer to this (and many textbook explanations) are readily available and accessible; I am <strong>not</strong> looking for the answer to that question and <strong>please do not post it</strong>. Instead I want to know what fundamen... | adrija | 173,185 | <p>A function $f:A\rightarrow B$ is a rule that assigns to an element of $A$ an $unique$ element of $B$. So, first of all, given $a\in A$, you can't say that it maps to nothing or to a subset of two or more elements. That won't be a function at all from $A$ to $B$, but since with each element of $A$ you are associating... |
2,542,056 | <p>Baire's Category Theorem states that a meager subset of a complete metric space has empty interior. </p>
<p>Are there examples of meager subsets of non-complete metric spaces which do not have empty interior?<br>
In particular, are the rationals numbers as a subset of themselves an example?</p>
| HBHSU | 397,029 | <p>Use $\mathbb{Q}$ as the underlying space. For each $G_n$, use $\mathbb{Q}-\{{q_n\}}$, where $q_n\in\mathbb{Q}$, so that the countably infinite intersection is $\emptyset$.</p>
|
4,203,906 | <p>Does there exist real numbers a and b such that</p>
<p>(i) <span class="math-container">$a+b$</span> is rational and <span class="math-container">$a^
n +b^
n$</span>
is irrational for each natural <span class="math-container">$n ≥ 2$</span>;</p>
<p>(ii) <span class="math-container">$a+b$</span> is irrational and <sp... | AAA | 627,380 | <p>For ii, I claim that if <span class="math-container">$a^n+b^n$</span> is rational for all <span class="math-container">$n\geq 2$</span>, then <span class="math-container">$a+b$</span> is rational.</p>
<p>Proof: Let <span class="math-container">$f_n=a^n+b^n$</span>. Then <span class="math-container">$f_n^2-f_{2n}=2(a... |
3,255,654 | <p>In a multiple choice question, there are five different answers, of which only one is correct. The probability that a student will know the correct answer is 0.6. If a student does not know the answer, he guesses an answer at random.</p>
<p>a) What is the probability that the student gives the correct answer?</p>
... | Henno Brandsma | 4,280 | <p><span class="math-container">$P(B|A_2)$</span> is the probability that the student gives the correct answer, while guessing (i.e. not knowing), so that is <span class="math-container">$\frac{1}{5} = 0.2$</span> </p>
<p>Clearly, by definition almost, <span class="math-container">$P(B|A_1)=1$</span>, if the student k... |
1,480,331 | <blockquote>
<p>Let $A$ be an $m \times n$ matrix with $m < n$ and $\operatorname{rank}(A) = m$. Prove that there exist infinitely many matrices $B$ such that $AB = I$.</p>
</blockquote>
<p>Stumped. How do I begin to prove this?</p>
| Robert Israel | 8,508 | <p>Note that this does exist, because with probability $1$ you will eventually get a $6$ or an odd number. Suppose the first time you get a $6$ or an odd number is on the $n$'th roll. There is one $6$ and there are $3$ odd numbers, so the conditional probability, given that it happens on the $n$'th roll, is $1/4$. A... |
288,499 | <p>Simply stated, I've been trying for a long time to either find in the literature, or derive myself, a notion of path in Cech closure spaces, that specialises to paths in a topological space, and to graph-like paths in so-called "quasi-discrete closure spaces". </p>
<p>Let me recall the definitions:</p>
<p>A closur... | Igor Rivin | 11,142 | <p>I am not sure of the notation, but I assume this can be derived from the Schlafli formula for the volume of a tetrahedron (so this seems to indicate that Gauss knew Schlafli's formula three quarters of a century prior to Schlafli):</p>
<p>$$
dV = -\frac12 \sum_{ij}l_{ij} d \alpha_{ij},$$ where $l$ is the length of... |
3,096,572 | <p>I am trying to find whether the following is stable absolutely using the improved Euler and the Adams-Bashforth 2 scheme,
<span class="math-container">$u'=\begin{bmatrix} -20&0&0\\ 20&-1&0\\0&1&0\end{bmatrix}u=Au$</span>, where the timestep is <span class="math-container">$\frac{1}{2}$</span... | Cesareo | 397,348 | <p>With the improved Euler method we have</p>
<p><span class="math-container">$$
u_{k+1}=\left(I_3+hA + \frac h2 A^2\right)u_k = M u_k
$$</span></p>
<p>this schema is contractive if <span class="math-container">$\max|\text{eigenvalues}(M)| < 1$</span> but calculating we get</p>
<p><span class="math-container">$$
... |
4,507,155 | <p>In <a href="https://math.stackexchange.com/questions/4454551/are-fracp212-and-fracp5np5-12-are-coprime-to-each-other">previous post</a>, I got the answer that <span class="math-container">$\gcd \left(\frac{p^2+1}{2}, \frac{p^5-1}{2} \right)=1$</span>, where <span class="math-container">$p$</span> is prime number.</p... | Keith Backman | 29,783 | <p>For primes of the form <span class="math-container">$6k-1$</span>, and choosing <span class="math-container">$m$</span> odd and <span class="math-container">$n$</span> even, you will obtain <span class="math-container">$p^m=6a-1$</span> and <span class="math-container">$p^n=6b+1$</span>. Hence <span class="math-cont... |
431,236 | <p>I have a cylinder of radius 4 and height 10 that is at a 30 degree angle. I need to find the volume.</p>
<p>I have no clue how to do this, I have spent quite a while on it and went through many ideas but I think my best idea was this.</p>
<p>I know that the radius is 4 so if I cut the cylinder in half from corner ... | apnorton | 23,353 | <p>Picture for reference:<br>
<img src="https://i.stack.imgur.com/VIfvF.jpg" alt="Graphic"></p>
<p>Let's get our terms straight here. $h$ is the height of the cylinder; $\ell$ is the side length, and $r$ is the radius. This cylinder is tilted at $30^\circ$.</p>
<p>The volume of a cylinder like this is given by the ... |
1,998,244 | <p>Given the equation of a damped pendulum:</p>
<p>$$\frac{d^2\theta}{dt^2}+\frac{1}{2}\left(\frac{d\theta}{dt}\right)^2+\sin\theta=0$$</p>
<p>with the pendulum starting with $0$ velocity, apparently we can derive:</p>
<p>$$\frac{dt}{d\theta}=\frac{1}{\sqrt{\sqrt2\left[\cos\left(\frac{\pi}{4}+\theta\right)-e^{-(\the... | Simply Beautiful Art | 272,831 | <p>Out of the 17 who had at least one brother, 11 had no sisters. Combined with the 18 who had at least one sister, this gives us 29 students. Adding in the 5 who had no brothers or sisters, we get 34. So there is exactly 34 students.</p>
<p>$$\text{no sister/brother $+$ no sisters some brothers $+$ sisters, with o... |
649,502 | <p>What do we mean when we talk about a topological <em>space</em> or a metric <em>space</em>? I see some people calling metric topologies metric spaces and I wonder if there is some synonymity between a topology and a space? What is it that the word means, and if there are multiple meanings how can one distinguish t... | ncmathsadist | 4,154 | <p>I think of a "space" as the conceptually smallest place in which a given abstraction makes sense. For example, in a metric space, we have distilled the notion of distance. In a topological space, we are in the minimal setting for continuity. </p>
|
66,570 | <pre><code>tmp = {x, y, z}^{1, 2, 3}
Times @@ tmp
Length[%]
</code></pre>
<p>This gives a length of 3. But I was expecting 1.</p>
<p>What is exactly this "length" of x*y^2*z^3 called?
I would think this as a scalar of length 1?</p>
<p>Thanks!</p>
| Kellen Myers | 9,482 | <p>The <code>Length</code> operator will operate on lists, but if your object is not a list, it is not automatically considered to be length 1. When you apply <code>Times@@</code> to your <code>tmp</code>, it is no longer a list.</p>
<p><code>Length</code> will apply to many other expressions. For example:</p>
<pre><... |
66,570 | <pre><code>tmp = {x, y, z}^{1, 2, 3}
Times @@ tmp
Length[%]
</code></pre>
<p>This gives a length of 3. But I was expecting 1.</p>
<p>What is exactly this "length" of x*y^2*z^3 called?
I would think this as a scalar of length 1?</p>
<p>Thanks!</p>
| Basheer Algohi | 13,548 | <p>According to the documentation:</p>
<pre><code>Length[expr]
</code></pre>
<p>gives the number of elements in expr.</p>
<p>in your case</p>
<pre><code>tmp = {x, y, z}^{1, 2, 3}
r=Times @@ tmp
(*x y^2 z^3*)
</code></pre>
<p>r is an expression with thee elements. to see this you do:</p>
<pre><code>FullForm[r]
(*... |
3,745,273 | <p>I am looking for a way to solve :</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{x\sin(3x)}{x^4+1}\,dx $$</span></p>
<p>without making use of complex integration.</p>
<p>What I tried was making use of integration by parts, but that didn't reach any conclusive result. (i.e. I integrated <span cla... | Claude Leibovici | 82,404 | <p><em>Too long for a comment</em></p>
<p>The more general problem
<span class="math-container">$$I=\int_{-\infty}^\infty \frac {x^m\,\sin(px)}{P_{2n}(x)} \,dx \qquad\text{where}\qquad m <2n\qquad p > 0$$</span> is not too bad if you feel conformatble with the manipulation of complex numbers. For sure, the condit... |
2,994,900 | <p>Prove that <span class="math-container">$$\sum_{d\mid q}\frac{\mu(d)\log d}{d}=-\frac{\phi(q)}{q}\sum_{p\mid q}\frac{\log p}{p-1},$$</span>
where <span class="math-container">$\mu$</span> is Möbius function, <span class="math-container">$\phi$</span> is Euler's totient function, and <span class="math-container">$q$<... | Fabio Lucchini | 54,738 | <p>Let me write <span class="math-container">$n$</span> instead of <span class="math-container">$q$</span>.
We have
<span class="math-container">\begin{align}
\sum_{d|n}\frac{\mu(d)\log(d)}d
&=\sum_{d|n}\frac{\mu(d)}d\sum_{p|d}\log(p)\\
&=\sum_{p|n}\log(p)\sum_{p|d|n}\frac{\mu(d)}d\\
&=\frac 1n\sum_{p|n}\lo... |
50,362 | <p>I have a question about the basic idea of singular homology. My question is best expressed in context, so consider the 1-dimensional homology group of the real line $H_1(\mathbb{R})$. This group is zero because the real line is homotopy equivalent to a point. The chain group $C_1(\mathbb{R})$ contains all finite ... | Dylan Wilson | 423 | <p>Remember that $C_2X$ does not just consist of all maps $\sigma: \Delta^2 \rightarrow X$ but also all formal sums of these. In particular, consider a map of the unit square that realizes a nullhomotopy of $\mu$. Divide the unit square into two triangles, label the vertices, orient the edges properly, and interpret th... |
3,424,687 | <blockquote>
<p>Let <span class="math-container">$n$</span> be a positive integer and a complex number with unit modulus is a solution of the equation <span class="math-container">$z^n+z+1=0$</span>. Prove that <span class="math-container">$n $</span> can't be <span class="math-container">$196$</span>. </p>
</blockqu... | lhf | 589 | <p><span class="math-container">$z^n+z+1=0$</span> implies <span class="math-container">$1=|z|^n=|z^n|=|z+1|$</span>.</p>
<p>If moreover <span class="math-container">$|z|=1$</span>, then <span class="math-container">$z$</span> is a primitive cubic root of <span class="math-container">$1$</span> and so <span class="mat... |
339,142 | <p>I'm trying to understand the difference between the sense, orientation, and direction of a vector. According to
<a href="http://www.eng.auburn.edu/users/marghdb/MECH2110/c1_2110.pdf">this</a>,
sense is specified by two points on a line parallel to a vector. Orientation is specified by the relationship between the ve... | JohnDee | 118,544 | <p>Here is how I think of it, lets construct a vector from scratch using just two points in space A,B. Draw a line segment between the two points A,B. The magnitude of the line segment is the 'length' of the vector. The 'orientation' of the line segment we can define as the angle that the line segment makes with the ho... |
96,468 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/22537/how-many-fixed-points-in-a-permutation">How many fixed points in a permutation</a> </p>
</blockquote>
<p>Suppose we have a collection of n objects, numbered from 1 to n. These objects are placed in ... | André Nicolas | 6,312 | <p>We will count the number of ways to have exactly $p$ objects in the correct places. Then one only needs to divide by $n!$ to find the probability.</p>
<p>Which $p$ objects are in the correct place? These can be chosen in $\binom{n}{p}$ ways. Now for every such choice, we must arrange the remaining $n-p$ objects so ... |
816,088 | <blockquote>
<p>The sum of two variable positive numbers is $200$.
Let $x$ be one of the numbers and let the product of these two numbers be $y$. Find the maximum value of $y$.</p>
</blockquote>
<p><em>NB</em>: I'm currently on the stationary points of the calculus section of a text book. I can work this out in my... | Tunk-Fey | 123,277 | <p>Let the first number be $x$ and the second number be $z$. We have
$$x+z=200\quad\Rightarrow\quad z=200-x.$$
We want to maximize $$y=xz=x(200-x)=200x-x^2.$$
Setting the first derivative equals $0$ yields
\begin{align}
\frac{d}{dx}\left(200x-x^2\right)&=0\\
200-2x&=0\\
2x&=200\\
x&=100.
\end{align}
Che... |
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