qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,161,294 | <p>I was wondering... $1$, $\phi$ and $\frac{1}{\phi}$, they have something in common: they share the same decimal part with their inverse. And here it comes the question:</p>
<p>Are these numbers unique? How many other members are in the set if they exist? If there are more than three elements: is it finite or infin... | Joffan | 206,402 | <p>So we want values of $0<x<1$ such that $x+k= \frac{\large 1}{\large x}$ for positive integer $k$, meaning $x^2+kx-1 =0$. This has a positive solution in the range for every $k$.</p>
|
2,161,294 | <p>I was wondering... $1$, $\phi$ and $\frac{1}{\phi}$, they have something in common: they share the same decimal part with their inverse. And here it comes the question:</p>
<p>Are these numbers unique? How many other members are in the set if they exist? If there are more than three elements: is it finite or infin... | Cye Waldman | 424,641 | <p>The golden ratio is one of only two numbers that share certain properties, such as that the golden ratio is one less than its square. I have a complete description of these <em>morphic</em> numbers in my response to another post here: <a href="https://math.stackexchange.com/questions/394111/what-real-number-is-exact... |
1,735,910 | <p>In <a href="https://www.youtube.com/watch?v=aHU-L3BLd_w">a recent video</a> the legendary Matt Parker claimed he kept flipping a two-sided (fair) coin untill he scored a sequence of ten consecutive 'switch flips', i.e. letting $T$ denote a tail and $H$ a head, then a sequence of ten switch flips is defined to be eit... | joriki | 6,622 | <p>The probability to get $10$ consecutive switch flips can be modelled as a Markov chain, with the states corresponding to the number of consecutive switch flips in the immediate past. The state in which $10$ consecutive switch flips have been encountered is absorbing. The stationary distribution with eigenvalue $1$ h... |
499,476 | <p>Use mathematical induction to prove that the derivative of $f(x)=\sin(ax+b)$ is given by</p>
<p>$f^{(n)}(x)= (-1)^ka^n\sin(ax+b)$ if $n=2k$, and $(-1)^ka^n\cos(ax+b)$ if $n=2k+1$</p>
<p>for a number $k=0,1,2,3,...$</p>
<p>I have done som proofs by induction, but I seem to struggle as soon as trig functions appea... | AlexR | 86,940 | <p>Let $f(x) = \sin(ax+b)$ then we claim for $n\in\mathbb N_0$
$$\begin{align*}
f^{(2n)}(x) & = (-1)^n a^{2n} \sin(ax+b) \\
f^{(2n+1)}(x) & = (-1)^n a^{2n+1} \cos(ax+b)
\end{align*}$$
The induction start for $n=0$ is obvious. Then we show for $k\in\mathbb N_0$:
$$\begin{align*}
f^{(2k)}(x) & = f^{(2k-1)'}(x... |
1,698,039 | <p>Alright, so let's say I have $$\frac{x^{-6}}{-x^{-4}}$$ The answer is $\dfrac{1}{x^2}$, but why isn't it $\dfrac{1}{-x^2}$?</p>
| frog1944 | 215,383 | <p>If the questions is asking; simplify $\frac {x^{-6}}{-x^{-4}}$, then the answer is $\frac {1}{-x^2}$. But if the answer is $\frac {1}{x^2}$, then I believe the question is $\frac {x^{-6}}{(-x)^{-4}}$.</p>
|
2,814,703 | <p>I am reading <a href="https://en.wikipedia.org/wiki/Lower_limit_topology" rel="nofollow noreferrer">lower limit topology</a> on Wikipedia, which states that the lower limit topology </p>
<blockquote>
<p>[...] is the topology generated by the basis of all half-open intervals $[a,b)$, where a and b are real numbers... | Henno Brandsma | 4,280 | <p>$$(a,b) = \bigcup \{[x,b): a < x < b \}$$</p>
<p>Every $[x,b) \subseteq (a,b)$ whenever $a < x < b$ for the right to left inclusion, and on the other hand, if $a < x < b$, $x \in [x,b)$, which shows the left to right inclusion. If you want a countable union at all cost (topologies are closed under... |
2,406,043 | <p>Let the triangle $\triangle ABC$ have sides $a,b,c$ and be inscribed in a circle with radius $R$. If $p=\frac{a+b+c}{2}$ The radius of the circle can be expressed as</p>
<p>a) $$R=\frac{\sqrt{p(p-a)(p-b)(p-c)}}{4abc}$$</p>
<p>b) $$R=\frac{4\sqrt{p(p-a)(p-b)(p-c)}}{abc}$$</p>
<p>c) $$R=\frac{abc}{4\sqrt{p(p-a)(p-b... | Michael Rozenberg | 190,319 | <p>The proof without AM-GM.</p>
<p>Let $x=a^3$, $y=b^3$ and $z=c^3$.</p>
<p>Hence, $abc=1$ and
$$x+y+z-3=a^3+b^3+c^3-3=a^3+b^3+c^3-3abc=$$
$$=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)=$$
$$=\frac{1}{2}(a+b+c)((a-b)^2+(a-c)^2+(b-c)^2)\geq0.$$
Done!</p>
|
2,489,498 | <p>A={a,b,c,d}</p>
<p>R={(a,b),(a,c),(c,b)}</p>
<p>According to the definition for transitive relation, if there is (a,b) and (b,c) there should be (a,c)</p>
<p>In the above relation there is (a,c),(c,b) as well as (a,b). Shouldn't it be transitive?</p>
| zwim | 399,263 | <p>To get rid of $\sqrt{x}$ you could substitute $u=\sqrt{x}$.</p>
<p>Then $\displaystyle I=\int_0^1 \sqrt{x}\ln(x)\,dx=\int_0^1 u\ln(u^2)2u\,du=4\int_0^1 u^2\ln(u)\,du$</p>
<p>Notice that $u\mapsto u^2\ln(u)$ is continuous on any $[\delta,1]$ with $0<\delta\ll 1$ so the part $\displaystyle \int_{\delta}^1 u^2\ln(... |
3,636,667 | <blockquote>
<p>Evaluate
<span class="math-container">$$\lim_{n\to\infty}\frac{1}{n^{p+1}}\cdot \sum_ \limits{i=1}^{n} \frac{(p+i)!}{i!} , p \in N$$</span> </p>
</blockquote>
<p>Now, I found this problem while doing some practice and I am curious on how to solve it . I have no good ideas yet, so I will appreciate... | grand_chat | 215,011 | <p>The limit is <span class="math-container">$\frac1{p+1}$</span>. There is a nice closed form:
<span class="math-container">$$
\sum_ \limits{i=1}^{n} \frac{(p+i)!}{i!} = \frac1{p+1} \underbrace{(n+1)\cdots (n+p+1)}_{\text{$p+1$ factors}} -p!
$$</span>
(even nicer if you absorb <span class="math-container">$p!$</span>... |
4,274,600 | <p><span class="math-container">$(x^3+x+1)^{-1} \mod (x^4+x+1)$</span> over <span class="math-container">$\text{GF}(2)$</span></p>
<p>I understand well how to solve the equation without inverse but don't know how to solve it with inverse.</p>
| Misha Lavrov | 383,078 | <p>For a problem with bigger polynomials, I'd want to try fancier tools. But in this problem, I'd be tempted to compute</p>
<p><span class="math-container">\begin{align}
A &= (x^3+x+1)(1) \bmod x^4+x+1 \\
B &= (x^3+x+1)(x) \bmod x^4+x+1 \\
C &= (x^3+x+1)(x^2) \bmod x^4+x+1 \\
D &= (x^3+x+1)(... |
3,684,917 | <p>Let <span class="math-container">$C_{1}$</span> and <span class="math-container">$C_{2}$</span> be polytopes in <span class="math-container">$\mathbb{R}^{n}$</span> such that
<span class="math-container">$C_{1}=conv\left( V\right) $</span> with <span class="math-container">$V$</span> being a set of vertices. If
<s... | John Hughes | 114,036 | <p>I'm going to write a formula <span class="math-container">$H(a, b, n, p)$</span> for the number of items congruent to <span class="math-container">$n$</span>, modulo <span class="math-container">$p$</span>, in the interval <span class="math-container">$a \le k < b$</span>. If you want to apply it to get the answ... |
909,228 | <p>I'm trying to find a closed form for the following sum
$$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$
where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number.</p>
<p>Could you help me with it?</p>
| Mhenni Benghorbal | 35,472 | <p>You can have instead the equivalent integral representation </p>
<blockquote>
<p>$$ I = \int_{0}^{1}\frac{\ln^2(u)\ln(1-u/2)}{u(u-2)}du \sim .5582373010. $$</p>
</blockquote>
<p>Try to evaluate the above integral. See my <a href="https://math.stackexchange.com/questions/275643/proving-an-alternating-euler-sum-su... |
909,228 | <p>I'm trying to find a closed form for the following sum
$$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$
where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number.</p>
<p>Could you help me with it?</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
909,228 | <p>I'm trying to find a closed form for the following sum
$$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$
where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number.</p>
<p>Could you help me with it?</p>
| Dennis Orton | 761,314 | <p>By first finding the following integral by using the algebraic identity
<span class="math-container">$a^2b=\frac{1}{6}\left(a+b\right)^3-\frac{1}{6}\left(a-b\right)^3-\frac{1}{3}b^3$</span> one can easily prove avoiding Euler sums that:
<span class="math-container">$$\int _0^1\frac{\ln ^2\left(1-x\right)\ln \left(1+... |
1,285,621 | <p>Below is the joint distribution of Boolean random variables X1, X2 and X3. How do I find variance and expectation of X2? I understand that variance is
"average of squares of difference from mean value". But here we have boolean values. I am confused. Can somebody suggest how to do this? I am a newbie to pr... | AlexR | 86,940 | <p>It's no different than an integer variable with values $\{0,1\}$. Naturally, both EV and variance will be in the real interval $[0,1]$.<br>
What you have to do is simply find the distribution of $X_2$ from the table - this will be parameterized by $p\in[0,1]$ with $P(X_2=1) = p$ and $P(X_2=0)=1-p$.<br>
The expected ... |
1,285,621 | <p>Below is the joint distribution of Boolean random variables X1, X2 and X3. How do I find variance and expectation of X2? I understand that variance is
"average of squares of difference from mean value". But here we have boolean values. I am confused. Can somebody suggest how to do this? I am a newbie to pr... | Lawrence | 214,733 | <p>It looks like you're asking why the variance and expected value of a boolean variable isn't itself boolean. The problem goes away if you consider variance to be a measure of the variable rather than the variable itself, and consider 'expected value' to be a form of scaled probability rather than the domain value to ... |
1,285,621 | <p>Below is the joint distribution of Boolean random variables X1, X2 and X3. How do I find variance and expectation of X2? I understand that variance is
"average of squares of difference from mean value". But here we have boolean values. I am confused. Can somebody suggest how to do this? I am a newbie to pr... | BruceET | 221,800 | <p>For reference, I am repeating the PDF (or PMF) table of $X_2.$
Then $E(X_2) = \sum_{i=0}^6 v_i p_i,$ the terms and total of
which are in the third row. (My notation may different from
that of your book, so make sure you connect the table with
whatever formula your book has.)</p>
<pre><code> value 0 0 1 ... |
2,030,116 | <p>How can i prove that $\sqrt[12]{2}$ is irrational number? </p>
<p>I'm trying: </p>
<p>$$\sqrt[12]{2} = \frac{p}{q}$$ where $p$, $q$ are integers</p>
<p>it follows that :</p>
<p>$$p^{12} = 2q^{12} $$</p>
<p>What is argument of irrationality in this case?
From what we know that the right-hand side has an even n... | Jack M | 30,481 | <p>The basic argument that $\sqrt 2$ is irrational is generalized using the fundamental theorem of arithmetic.</p>
<p>Supose $(\frac p q)^{12} = 2$. Then $p^{12} = 2 q^{12}$. The prime factorization on the left hand side contains a number of $2$-s which is a multiple of $12$, but the prime factorization on the right h... |
105,857 | <p>Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Is $\text{SL}_2(\mathcal{O})$ generated by elementary matrices? If it isn't, is there any other natural generating set for it?</p>
<p>The usual argument shows that this is true for $\mathcal{O} = \mathbb{Z}$ (or, more generally, a Euclidean d... | Alex | 57,771 | <p>If $\mathcal O$ be the ring of integers in an algebraic number field, whether $SL_2(\mathcal O)$ is generated by elementary matrices depends on the field $k$:</p>
<ul>
<li><p>If $k = \Bbb Q$, or $k = \Bbb Q(\sqrt{-D}$ for $D\in\{1,2,3,7,11\}$, then $SL_2(\mathcal O)$ is generated by elementary matrices.</p></li>
<l... |
801,081 | <p>I was doing some school work and got bored so I started messing with k-gonal numbers. I started with the triangular numbers, square numbers and looked for patterns. I noticed something.</p>
<p>Let $n^{(k)}$ denote the $n$-th $k$-gonal number. For example, $3^{(3)}$ is the third triangular number, 6.</p>
<p>I f... | Janaka Rodrigo | 1,043,137 | <p>Method (iii) <br/>
Let u(k,n) be the n th k - gonal number <br/>
Consider the regular polygon of k sides which represents u(k,n) <br/>
When straight lines drawn through a selected vertex joining other vertices there will be k-2 triangles and out of which any triangle can be used to represent the triangle number u... |
2,199,222 | <p>I have the feeling of being stuck or missing something trying to prove
$$ \lim_{N\to\infty}\sum_{k=1}^{N} \frac{1}{N+k} =\int_{1}^{2} \frac{1}{x} dx = ln(2)$$</p>
<p>Using Riemann-Sums I have shown that $$\int_{1}^{a} \frac{1}{x} dx=\lim_{N\to\infty}\sum_{k=1}^{N} (a^{1/N}-1)=\lim_{N\to\infty}N(a^{1/N}-1)=\lim_{h\t... | user209663 | 209,663 | <p>I think it's called Skew Hermitian matrix. </p>
<p>Here is the Wikipedia link <a href="https://en.wikipedia.org/wiki/Skew-Hermitian_matrix" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Skew-Hermitian_matrix</a><a href="https://en.wikipedia.org/wiki/Skew-Hermitian_matrix" rel="nofollow noreferrer">1</a><... |
1,749,730 | <p>What is the maximum number of faces of totally convex solid that one can "see" from a point? </p>
<p>...and, more importantly, how can I ask this question better? (I'm a college student with little experience in asking well formed questions, much less answering them.) </p>
<p>By "see" I mean something like this: y... | Ethan Bolker | 72,858 | <p>You can always find a place from which to see at least half the faces.</p>
<p>To see why, start by considering a polyhedron with central symmetry. Imagine
a viewpoint from which you don't see any lines as points or faces as lines (i.e. general position) and far enough away so that you can see all the faces whose n... |
1,749,730 | <p>What is the maximum number of faces of totally convex solid that one can "see" from a point? </p>
<p>...and, more importantly, how can I ask this question better? (I'm a college student with little experience in asking well formed questions, much less answering them.) </p>
<p>By "see" I mean something like this: y... | Joseph Malkevitch | 1,369 | <p>Perhaps this paper which deals with the situation in a rather abstract way, and for higher dimensional polytopes might interest some: <a href="http://www-math.mit.edu/~rstan/transparencies/vis.pdf" rel="nofollow">http://www-math.mit.edu/~rstan/transparencies/vis.pdf</a></p>
|
555,045 | <p>Let $K$ be a quadratic number field.
Let $R$ be an order of $K$, $D$ its discriminant.
By <a href="https://math.stackexchange.com/questions/546281/on-a-certain-basis-of-an-order-of-a-quadratic-number-field">this question</a>, $1, \omega = \frac{(D + \sqrt D)}{2}$ is a basis of $R$ as a $\mathbb{Z}$-module.</p>
<p>L... | Makoto Kato | 28,422 | <p>I realized after I posted this question that the proposition is not correct.
I will prove a corrected version of the proposition.</p>
<p>We need some notation.
Let $\sigma$ be the unique non-identity automorphism of $K/\mathbb{Q}$.
We denote $\sigma(\alpha)$ by $\alpha'$ for $\alpha \in R$.
We denote $\sigma(I)$ by... |
266,832 | <p>Let $p(x) = \sum_{k \geq 0} a_k x^k$ where the $a_k$'s are IID random variables taken from a mean-zero random variable taking finitely many values in $\mathbb{R}$; it clearly converges for $-1<x<1$. Is it a.s. true that the sign of $p(x)$ oscillates infinitely often as $x \rightarrow 1^-$? That is, is it the c... | Robert Israel | 13,650 | <p>We can construct inductively a sequence $t_n \to 1-$ and an increasing sequence $K_n$ of positive integers
such that with probability $> 1 - 1/n^2$, $f(t_n)$ has the same sign as
$V_n = \sum_{k=K_{n-1}+1}^{K_n} a_k t_n^k$. Since $\sum 1/n^2 < \infty$,
almost surely $f(t_n)$ has the same sign as $V_n$ for a... |
4,510,795 | <p>I need to find the % difference between two numbers. One person told me to use <span class="math-container">$\frac{x-y} {x} $</span>, another told me to use <span class="math-container">$\frac x y$</span> <span class="math-container">$- 1$</span> . Who is right?</p>
<p>Example:
Today's price: <span class="math-c... | Samuel Lelièvre | 162,322 | <p>Sage's <code>solve</code> has an optional parameter <code>to_poly_solve</code>.</p>
<p>It helps in this case:</p>
<pre><code>sage: x = var('x')
sage: rad_pol = sqrt(x) + sqrt(1 - x) - 1
sage: solve(rad_pol, x)
[sqrt(x) == -sqrt(-x + 1) + 1]
sage: solve(rad_pol, x, to_poly_solve=True)
[x == 0, x == 1]
</code></pre>
<... |
1,500,827 | <p>A composite number $n$ is a Fermat-pseudoprime to base $a$, if</p>
<p>$$a^{n-1}\equiv\ 1\ (\ mod\ n)$$</p>
<p>If $n-1=2^s\times t$ , $t$ odd , $n$ is a strong a-PRP, if either
$2^t\equiv 1\ (\ mod\ n)$ or there is a number $u$ with $0\le u<s$ and
$\large 2^{2^u\times t}\equiv -1\ (\ mod\ n\ )$</p>
<p>I want t... | Brian M. Scott | 12,042 | <p>Evaluating a few of the differences by hand is a good idea, but in the end you have to do it symbolically rather than just numerically:</p>
<p>$$\begin{align*}
a_{n+1}-a_n&=\left(1+\frac1{2^n}\right)-\left(1+\frac1{2^{n-1}}\right)\\
&=\frac1{2^n}-\frac1{2^{n-1}}\\
&=\frac1{2^n}-\frac2{2^n}\\
&=-\fra... |
2,475,757 | <p>I want to determine if the following integrals converge or diverge.</p>
<ol>
<li>$\int_{0}^\infty \frac{\sqrt{x}}{\sqrt[3]{x^5+1}}dx.$</li>
<li>$\int_{0}^\infty \sin\frac{1}{x^2+1}dx$.</li>
<li>$\int_{\sqrt{2}}^2 \frac{dx}{\sqrt{x^2-2}}dx.$</li>
<li>$\int_{0}^1 \frac{\ln{x}}{x}dx.$</li>
</ol>
<hr>
<p><strong>(1):... | user247327 | 247,327 | <p>Oh, bother! Amanda R just pointed out the balls are numbered 1 to 3, not 0 to 2! The computation is basically the same.</p>
<p>The largest of the two numbers will be 2 for (1, 2), (2 ,1), or (2, 2). That is three out of the total 3*3= 16 possible results drawing two balls. P(2)= 3/9.</p>
<p>(Although you didn't... |
4,302,213 | <blockquote>
<p>Let <span class="math-container">$R,S$</span> be rings and <span class="math-container">$\varphi : R\to S$</span> be a ring homomorphism. Verify that</p>
<ol>
<li><span class="math-container">$\varphi(na) = n\varphi(a)$</span> for all <span class="math-container">$n\in\mathbb Z$</span> and <span class="... | Wuestenfux | 417,848 | <p>For the first property,
use induction to show that <span class="math-container">$\phi(na) = n\phi(a)$</span> for all <span class="math-container">$a$</span> and <span class="math-container">$n\geq 0$</span>.
This is actually what you have done.</p>
<p>For negative <span class="math-container">$n$</span> one must be ... |
3,347,342 | <blockquote>
<p><span class="math-container">$$\frac{2}{5}^{\frac{6-5x}{2+5x}}<\frac{25}{4}$$</span></p>
</blockquote>
<p>I can write this as
<span class="math-container">$$\frac25 ^{\frac{6-5x}{2+5x}} <\frac25 ^{-2}$$</span>
Therefore <span class="math-container">$$\frac{6-5x}{2+5x}<-2$$</span>
Solving it... | Mostafa Ayaz | 518,023 | <p>Note that <span class="math-container">$$a^x>a^y\implies \begin{cases}x>y&,\quad a>1\\x<y&,\quad 0<a<1\end{cases}$$</span></p>
|
2,347,820 | <p>What is the solution to $\log_{10} x -x=2?$</p>
<p>I have tried to solve it but I couldn't. I've got to $x^x =200$.</p>
| Robert Israel | 8,508 | <p>Assuming you're using the principal branch of log,
$$x = -\frac{W(-100 \ln(10))}{\ln(10)} $$
where $W$ is the principal branch of the Lambert W function. Other branches of log would correspond to other branches of Lambert W. Since $-100 \ln(10) < -1/e$, no solutions are real.</p>
|
2,347,820 | <p>What is the solution to $\log_{10} x -x=2?$</p>
<p>I have tried to solve it but I couldn't. I've got to $x^x =200$.</p>
| Jack D'Aurizio | 44,121 | <p>If $x=10^{x+2}$ has a real solution such solution has to be $\geq 0$, since $10^{x+2}\geq 0$ for any $x\in\mathbb{R}$.<br>
But if $x\geq 0$ and $x=10^{x+2}$ then $x\geq 100$ since $10^{x+2}$ is an increasing function.<br>
If $x\geq 100$ and $x=10^{x+2}$ then $x\geq 10^{102}\geq 10^{10^2}$ for the same reason.<br>
A... |
2,811,870 | <p>This is a question from Brilliant.org</p>
<blockquote>
<p>The triangle $ABC$ has $AB = 9$ and $AC:BC = 40:41$. What is the maximum possible area of $ABC$?</p>
</blockquote>
<p>For this question, I considered the equation $A=\frac 12ab\sin\theta$.</p>
<p>Since $\sin\theta\le 1$, then $A$ is maximised when $\sin\... | dan_fulea | 550,003 | <p>As mentioned in a comment, the best way to realize where is the error is to consider the similar problem where $AB=9$ is given and fixed, and where $AC:BC=1$. Then in this case the triangle is isosceles, and we have the option to choose its height as big as we want. The bigger this heigth $h$, (the smaller the angle... |
997,587 | <p>The first sequence given is 3, 7, 16, 41, 77,....
I really am quite stuck on this because I can't seem to find any relationship between one term and the terms prior to it. I first noticed that it seemed like we were adding a perfect square to each one, since 3+4=7, 7+9=16, etc. But we skipped over adding the perfect... | Brian M. Scott | 12,042 | <p>The first one is a mistake: either it was intended to be $3,7,16,32,57,\ldots$, corresponding to the description $s_1=3$, $s_n=s_{n-1}+n^2$ for $n>1$, or there is simply not enough information to allow one to guess how it is intended to differ from that sequence.</p>
<p>For the second one you can try ‘unwinding’... |
3,611,072 | <p>Show that if a prime <span class="math-container">$p ≠ 3$</span> is such that <span class="math-container">$p≡1$</span> (mod 3) then p can be written as <span class="math-container">$a^2-ab+b^2$</span> where a and b are integers. </p>
<p>I have no idea how to approach this question, so any help much appreciated! T... | Tob Ernack | 275,602 | <p>Given that <span class="math-container">$p = a^2 - ab + b^2$</span>, we can factor over <span class="math-container">$\mathbb{Z}[\zeta_3]$</span> (ring of Eisenstein integers) to obtain <span class="math-container">$p = \left(a + \zeta_3b\right)\left(a + \zeta_3^2b\right)$</span>.</p>
<p>Therefore the ideal <span c... |
3,611,072 | <p>Show that if a prime <span class="math-container">$p ≠ 3$</span> is such that <span class="math-container">$p≡1$</span> (mod 3) then p can be written as <span class="math-container">$a^2-ab+b^2$</span> where a and b are integers. </p>
<p>I have no idea how to approach this question, so any help much appreciated! T... | nguyen quang do | 300,700 | <p>This is just a matter of quadratic reciprocity law.</p>
<p>Let <span class="math-container">$\omega$</span> be a primitive cubic root of unity and <span class="math-container">$K=\mathbf Q(\omega)= \mathbf Q(\sqrt {-3})$</span> . It is known that the ring of integers of <span class="math-container">$K$</span> is <s... |
672,744 | <p>Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis.</p>
<p>Having trouble evaluating the integral: </p>
<p>Solved for $x$:</p>
<ul>
<li>$x=0, y=1$</li>
<li>$x=2, y=13$</li>
</ul>
<p>$$\int_1^{13} 2\pi\sqrt\frac{y-1}3 \cdot \sqrt{1+\sqrt\frac{y-1}3'}^2\,dy$$</p>
<p>I... | Semsem | 117,040 | <p>Note that $((\sqrt{\frac{y-1}{3}})')^2=(\frac{1}{2\sqrt{\frac{y-1}{3}}} (1/3))^2=\frac{1}{4(y-1)}$
$$=\int_1^{13} 2\pi\sqrt\frac{y-1}3 \cdot \sqrt{1+\frac{1}{4(y-1)}}dy
\\= \int_1^{13} 2\pi\sqrt{\frac{y-1}3+\frac{1}{12}}dy
\\= \int_1^{13} 2\pi\sqrt{\frac{y}3-\frac{1}{4}}dy
\\= 6\pi\{(\frac{y}3-\frac{1}{4})^\frac{3}... |
1,896,008 | <p>Is the following statement correct: </p>
<blockquote>
<p>If $A$ and $B$ are closed subsets of $[0,\infty)$, then $A+B=\{x+y:x \in A,y \in B\}$ is closed in $[0,\infty)$.</p>
</blockquote>
| Yaddle | 333,729 | <p>Let $(c_n)_{n \in \mathbb N}$ a convergent sequence in $A + B$ with
$$c := \lim_{n \to \infty} c_n.$$
We need to show, that $c \in A+B$. There exist sequences $(a_n)_{n \in \mathbb N}$ in $A$ and $(b_n)_{n \in \mathbb N}$ in $B$ with $c_n = a_n + b_n$. Since $A,B \subseteq [0, \infty)$, we have that $a_n \leq c_n$ ... |
2,153,340 | <p>Let $G$ be a group, and $C$ a set of proper subgroups of $G$.</p>
<p>Each subgroup in $C$ is normal subgroup of $G$.</p>
<p>For $G_1 , G_2\in C$, if $G_1 \ne G_2$ then $G_1\cap G_2=\{e_G\}$</p>
<p>$\bigcup\limits_{H\in C}H= G$.</p>
<p>Need to prove that G is Abelian group, hint someone?</p>
| user300 | 226,798 | <p>Hint: Let $x,y\in G$. Then there exist $G_1,G_2\in C$ such that $x\in G_1,y\in G_2$. Then show $xyx^{-1}y^{-1}\in G_1\cap G_2$</p>
|
3,670,240 | <p>It' not a physics question, just ..coincidence ;) (i'm concerned about mathematical rightness of it)</p>
<p>Let's consider <span class="math-container">$U,T,S,P,V\in\mathbb{R_{>0}}$</span> such that
<span class="math-container">$$dU=TdS-PdV$$</span></p>
<ul>
<li>Based on this, how we can rigorously proof that <... | Giorgio Pastasciutta | 660,461 | <p>Thinking back, i couldn't accept that such a simple question had such a long and complicated proof, so eventually i come up with:</p>
<p>Assume <span class="math-container">$U=U(S,V)$</span>, this implies
<span class="math-container">$$\frac{dU}{dA}\bigg|_{S,\,V}=0\qquad\forall\, A\;\text{not dependent on S,V,U}$$<... |
2,066,455 | <p>I ask this question mainly to resolve (hopefully) and error with the following problem. </p>
<p>The United States Court consists of $3$ women and $6$ men. In how many ways can a $3$-member committee be formed if each committee must have at least one woman?</p>
<p>My approach:
Since each group needs at least one wo... | AlgorithmsX | 355,874 | <p>You have $8$ choices for the person sitting on one end (because there are only $8$ people that can be on either end), you have $7$ choices for the person sitting on the other end (because you already put a person on the end and you still can't put two people on the end), and then you have $8!$ for the rest of the pe... |
4,269,898 | <p>I've a question concerning inverse limits, since I don't usually work with them this extensively.</p>
<p>I'm considering the inverse limit of the following "bi-inverse system" of <span class="math-container">$R$</span>-modules and black arrows <span class="math-container">$f_{\bullet,\bullet}$</span>, and ... | Ittay Weiss | 30,953 | <p>A functor <span class="math-container">$j\colon C\to D$</span> between small categories is initial if pulling back limits along it does not alter limits (in the precise sense that the evident induced morphism is an isomorphism). So, you are asking whether the diagonal inclusion in your case is initial.</p>
<p>If we ... |
88,565 | <p>Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:</p>
<p><span class="math-container">$$x^2\ne x\implies x\ne 1$$</span></p>
<p>I immediately answered true, but for some reason, everyone (including my classmates and ... | P.K. | 34,397 | <p><em>Thing to note.</em> This is called <a href="http://en.wikipedia.org/wiki/Truth_tables#Logical_implication" rel="noreferrer">logical implication</a>.</p>
<blockquote>
<p>$x^2≠x⟹x≠1$: Is this sentence true or false, and why?</p>
</blockquote>
<p>We can always check that using an example. Let us look at this im... |
2,757,870 | <p>I've come to this:
$$f: \mathbb{N} \to \{\ldots, -6,-4,-2,0,2,4,6, \ldots\},\qquad f(n) =
\begin{cases}
2n & \text{ if } n \text{ is odd} \\
-n & \text{ if } n \text{ is even}
\end{cases}$$</p>
<p>I don't know what to do with this though. I never know how to format a proof correctly.</p>
| fleablood | 280,126 | <p>Well, you've found your bijection. So the next step is to prove it is a bijection..... which you can't because it is not.</p>
<p>First prove it is surjective: that for any $e\in E$ that there is an $n\in \mathbb N$ so that $f(n) = e$. </p>
<p>Failure of pf: let $e=2k $. In $k\le 0$ then $f(-e) = e$.
If $k>... |
2,670,082 | <p>I have quite an interesting infinite totient sum. My task is to evaluate</p>
<p>$\sum_{n=1}^{\infty} \frac{\phi(n)}{5^n +1}.$</p>
<p>The problem is that I have no idea how to go from here as I have never seen such a problem before. The usual techique of writing $n$ and $\phi(n)$ in terms of the prime factorization... | David | 119,775 | <p>Basic facts -</p>
<ol>
<li>Using the idea behind Achille Hui's hint and noting that odd numbers can only have odd factors,</li>
</ol>
<blockquote>
<p><span class="math-container">$$\color{red}{\sum_{\textstyle{n=1\atop n\ \rm odd}}^\infty
\frac{\phi(n)q^n}{1-q^{2n}}}
=\sum_{\textstyle{n=1\atop n\ \rm odd}}^\i... |
84,138 | <p>I recently have a paper rejected from a very good (but not the top) journal. The referee report said the result was good and certainly belong there, but he did not think I did enough to back up my claims (it was a rather long and harsh criticism at the exposition). Now I know for sure that my result is good and my p... | rview | 20,137 | <p>yes you should resubmit it</p>
|
3,935,494 | <p>Usually the inverse of a square <span class="math-container">$n \times n$</span> matrix <span class="math-container">$A$</span> is defined as a matrix <span class="math-container">$A'$</span> such that:</p>
<p><span class="math-container">$A \cdot A' = A' \cdot A = E$</span></p>
<p>where <span class="math-container... | Math Lover | 801,574 | <p>In the second problem, please note your set excludes the case where none of the fetus is a baby boy as you know at least one of them is.</p>
<p>So the probability in the second case that all three fetuses are baby boys <span class="math-container">$ \displaystyle = \frac{0.485^3}{1-0.515^3}$</span></p>
|
3,935,494 | <p>Usually the inverse of a square <span class="math-container">$n \times n$</span> matrix <span class="math-container">$A$</span> is defined as a matrix <span class="math-container">$A'$</span> such that:</p>
<p><span class="math-container">$A \cdot A' = A' \cdot A = E$</span></p>
<p>where <span class="math-container... | Mehul | 857,496 | <p>now, I am not sure about an "official" sol but here is mine.</p>
<p>now in these cases, I would use sets and a Venn diagram. let's name the persons 1,2,3
now let A = {events|1 has a boy} and similarly for B and C so the Venn diagram would look like-<a href="https://i.stack.imgur.com/HDVit.jpg" rel="nofollo... |
80,918 | <p>Could anyone please tell me what could be the math function to get the number of zeros in given decimal representation of numbers? I scratched my head on Combination and Permutation but couldn't come up with generic answer. The number length can be up to 1000 digits, so you can represent a number as a String.</p>
<... | Arturo Magidin | 742 | <p>Say you are writing all $d$ digit numbers, writing leading zeros. Then you would write out $d\times 10^d$ digits total, with each digit occurring exactly $\frac{1}{10}$th of the time, so you would write a total of $d\times 10^{d-1}$ zeros.</p>
<p>So, for example, to write all numbers between $0$ and $99$, writing t... |
2,280,666 | <p>Let AD be the altitude corresponding to the hypotenuse BC of the right triangle ABC. The circle of diameter AD intersects AB at M and AC at N shown. Prove $\frac{BM}{CN}$= $\bigg(\frac{AB}{AC}\bigg)^{3}$.</p>
<p>So far I have...</p>
<p>The power of B is $BD^{2}=(BM)(BA)$</p>
<p>The power of C is $CD^{2}=(CN)(CA)$... | CY Aries | 268,334 | <p>As $\triangle ABD\sim\triangle CAD$,</p>
<p>$$\frac{BD}{AD}=\frac{AD}{CD}=\frac{AB}{CA}$$</p>
<p>Therefore,</p>
<p>$$\frac{BD}{CD}=\frac{BD}{AD}\cdot\frac{AD}{CD}=\left(\frac{AB}{CA}\right)^2$$</p>
<p>and thus</p>
<p>$$\frac{BD^2}{CD^2}=\left(\frac{AB}{CA}\right)^4$$</p>
<p>Using power of a point,</p>
<p>$$\f... |
2,280,666 | <p>Let AD be the altitude corresponding to the hypotenuse BC of the right triangle ABC. The circle of diameter AD intersects AB at M and AC at N shown. Prove $\frac{BM}{CN}$= $\bigg(\frac{AB}{AC}\bigg)^{3}$.</p>
<p>So far I have...</p>
<p>The power of B is $BD^{2}=(BM)(BA)$</p>
<p>The power of C is $CD^{2}=(CN)(CA)$... | Peter Szilas | 408,605 | <p>$\angle AMD = 90 °$, so $MD$ || $AC$.</p>
<p>(Thales circle over $AD$.)</p>
<p>Intercept Theorem:</p>
<p>1) $BM/BA$ = $BD/BC$ .</p>
<p>$\angle AND = 90°$, so $ND$ || $AB$.</p>
<p>(Thales circle over $AD$.)</p>
<p>Intercept Theorem:</p>
<p>2) $CN/CA$ = $CD/CB$.</p>
<p>Dividing1) by 2):</p>
<p>$BM/CN$ × $CA/... |
2,251,998 | <p>Here is a question that I am working on:</p>
<blockquote>
<p>If $G$ is a group such that <em>every</em> non-identity element has order $2$, show that $G$ is abelian (commutative).</p>
</blockquote>
<p><strong>My attempt</strong></p>
<p>Suppose that for all $a \in G$, we have $$a^2 = Id$$</p>
<p>My goal is to s... | TMM | 11,176 | <p>See <a href="https://isc.carma.newcastle.edu.au/advanced" rel="nofollow noreferrer">inverse symbolic calculator</a>, which I think was originally coined Plouffe's inverter. It does exactly what you want. If you enter your decimals there, it will give your candidate solution as one match, other close matches for inst... |
137,136 | <p><a href="https://i.stack.imgur.com/tk4kk.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tk4kk.png" alt="Mathcad example of solving coil impedance"></a></p>
<p>I am new to Mathematica.</p>
<p>I am trying to figure out how to do the same thing in Mathematica as the image depicts done in Mathcad.<... | yode | 21,532 | <p>If we change the last <code>Transpose</code> into a <code>f</code>,it will return a uncalculate structure</p>
<pre><code>Dataset[{{0, 10}, {2, 11}, {3, 12}}][Transpose /* Map[MinMax] /* f]
</code></pre>
<blockquote>
<p>f[{{0, 3}, {10, 12}}]</p>
</blockquote>
<p>But the <code>Transpose</code> doesn't work here,s... |
1,550,603 | <p>I got confirmed from a graduate school starting from next year and I will major algebraic geometry.</p>
<p>Until now, I have never thought that I study little things than others with my age. However, I heard that <strong>some</strong> of my colleagues already studied Hartshorne at least once and quite a few of them... | Noah Schweber | 28,111 | <p><em>Your question is very broad, and I'm not sure this fully addresses it; but this is too long for a comment, and hopefully you find it useful nonetheless.</em></p>
<hr>
<p>I think there's a couple false assumptions here.</p>
<p>First, that there is a "better" way to approach studying mathematics. People vary wi... |
211,705 | <p>I am given a table of possible <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> values that can be generated in a casino. In the game, both are generated with each turn.</p>
<p><img src="https://i.stack.imgur.com/G0nLn.jpg" alt="enter image description here" /></p>
<blockquote>... | John Gowers | 26,267 | <p>What you are trying to do is to find the value of </p>
<p>$$\mathbb{E}(Y)=\sum_yy\mathbb{P}(Y=y)$$</p>
<p>for various random variables $Y$ depending on $X_1,X_2$. Notice that this is exactly what you did in the first question: you worked out the possible values for $Y=8X_1$, and then multiplied each value by the ... |
1,146,802 | <p>Often seen similar systems of equations. Usually consider such systems in which decisions no. Such as there. <a href="https://math.stackexchange.com/questions/1146460/is-there-a-b-c-d-in-mathbb-n-so-that-a2b2-c2-b2c2-d2">Is there $a,b,c,d\in \mathbb N$ so that $a^2+b^2=c^2$, $b^2+c^2=d^2$?</a></p>
<p>I think it wo... | Kieren MacMillan | 93,271 | <p>The <span class="math-container">$2.n.m$</span> Diophantine equations have well-known complete solutions (a.k.a. parameterizations) for all <span class="math-container">$n,m \ge 1$</span> (for example, see <a href="https://www.jstor.org/stable/3620159" rel="nofollow noreferrer">Bradley's paper</a> or <a href="https:... |
1,535,731 | <p>I am currently self-learning about matrix exponential and found that determining the matrix of a diagonalizable matrix is pretty straight forward :).</p>
<p>I do not, however, know how to find the exponential matrix of a non-diagonalizable matrix.</p>
<p>For example, consider the matrix
$$\begin{bmatrix}1 & 0... | Ben Grossmann | 81,360 | <p>There are two facts that are usually used for this computation:</p>
<blockquote>
<p><strong>Theorem:</strong> Suppose that $A$ and $B$ commute (i.e. $AB = BA$). Then $\exp(A + B) = \exp(A)\exp(B)$</p>
<p><strong>Theorem:</strong> Any (square) matrix $A$ can be written as $A = D + N$ where $D$ and $N$ are su... |
3,668,101 | <p>I know that if <span class="math-container">$n \bmod k \le k-1$</span> then this sum is converge then it has finite sum, I just guess it's <span class="math-container">$\ln(k)$</span> because when <span class="math-container">$k=1$</span> sum is <span class="math-container">$0=ln(1)$</span>. I really don't know how ... | Tony419 | 786,419 | <p>Also, @xpaul 's computation shows that the initial sum is asymptotically equal to <span class="math-container">$\log k$</span>. Indeed, @xpaul derived
<span class="math-container">\begin{eqnarray}
&&\sum_{n=1}^{\infty }\frac{(n)\mod(k)}{n(n+1)}=\sum_{m=0}^{\infty }\sum_{r=1}^{k-1}\frac{(mk+r)\mod(k)}{(mk+r)... |
255,811 | <p>I'm recalling this question from memory, so I may be messing it up a bit.</p>
<p>Let $a/3+b/2+c=0$. Show that $ax^2+bx+c=0$ has at least one root in $[0,1]$ using the Mean Value Theorem.</p>
<p>Let $f(x)=ax^2+bc+c$. Then $f(0)=c$ and $f(1)=a+b+c$. Also $f'(x)=2ax+b$. So there exists $f(\xi)=[f(1)-f(0)]/1=a+b-c... | user1551 | 1,551 | <p>Apply MVT to $g(x) = \int (ax^2+bx+c) dx$.</p>
|
1,373,170 | <p>How can I solve $e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}=1$ for $x$,</p>
<p>where $N\geq 1, k_1,\ldots,k_N \in \mathbb{R}, k_1,\ldots,k_N < 0, x\in \mathbb{R}$ and $x >0$.</p>
<p>I looked at the basic rules of exponentiation and logarithms and they do not seem to help simplify the equation in this particular ca... | johannesvalks | 155,865 | <p>Write an algorithm to solve the problem...</p>
<p>Let</p>
<blockquote>
<p>$$
Q = \left( \sum_{m=1}^{n} \exp(k_m y) - 1 \right)^2.
$$</p>
</blockquote>
<p>In case we have the solution, we have
$$
Q = 0
$$.</p>
<p>For changes in $Q$ we have</p>
<blockquote>
<p>$$
\delta Q = 2 Q \left( \sum_{m=1}^{n} k_m \exp(... |
1,373,170 | <p>How can I solve $e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}=1$ for $x$,</p>
<p>where $N\geq 1, k_1,\ldots,k_N \in \mathbb{R}, k_1,\ldots,k_N < 0, x\in \mathbb{R}$ and $x >0$.</p>
<p>I looked at the basic rules of exponentiation and logarithms and they do not seem to help simplify the equation in this particular ca... | marty cohen | 13,079 | <p>Let
$f(x)
= e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}-1
=\sum_{i=1}^N e^{k_i/x}-1
$,
and let
$K = \sum_{i=1}^N k_i
$.</p>
<p>The restrictions that
$x > 0$
and
$k_i < 0$
are important in what follows.</p>
<p>$f'(x)
=\sum -\frac{k_i}{x^2}e^{k_i/x}
=-\frac1{x^2}\sum k_ie^{k_i/x}
=\frac1{x^2}\sum |k_i|e^{k_i/x}
$,
so... |
1,951 | <p>In <a href="https://matheducators.stackexchange.com/a/1949/704">this answer</a>, user <a href="https://matheducators.stackexchange.com/users/942/robert-talbert">Robert Talbert</a> stated that</p>
<blockquote>
<p>There are some amazing things you can do pedagogically with clickers.</p>
</blockquote>
<p>I'd like t... | Robert Talbert | 942 | <p>First of all I would highly recommend <a href="http://amzn.com/B002UHTTYO">Derek Bruff's definitive book on this subject</a>. There are more good ideas in that book than any one faculty member can expect to implement. If there's anything I do with clickers in the classroom that works, it's probably appropriated from... |
1,672,847 | <p>A stick of total length $1$ is split at a randomly selected point $X$, i.e. $X$ is uniformly distributed in the interval $[0, 1]$.</p>
<p>Determine the expected length of the piece that contains the point $1/3$.</p>
<p>I've figured out so far that I need to determine a function $f(x)$ so that the length of the pie... | André Nicolas | 6,312 | <p>Given that the length $X$ is $\le 1/3$, the expectation of $X$ is $1/6$, and the expected length of the piece that contains $1/3$ is $5/6$.</p>
<p>Given that $X\gt 1/3$, the expectation of $X$ is $2/3$, so the expected length of the piece that contains $1/3$ is $1/3$.</p>
<p>Thus by the Law of Total Expectation, t... |
2,440,802 | <p>The number of positive integers that $n$ can take in between the range $100$ to $200$.</p>
<p>I tried a lot using the prime factorization method but no use. </p>
| Piquito | 219,998 | <p>What I want is just to find out a solution. I feel the one I have found is the only but I have not stopped to prove this.</p>
<p>The identities $$(8t+2)^2-(8t+2)-2=8(8t^2+3t)\\(27s+24)^2+2(27s+24)-3=27(27s^2+50s+23)$$ establish two parametrizations, respectively for the diophantine equations $$n^2-n-2=8y\\n^2+2n-3=... |
2,136,411 | <p>According to the Theorem 12.7 of the book Analytic Nymber Theory by Apostol, $$\zeta(1-s) = 2(2\pi)^{-s} \Gamma(s) \cos \big(\frac{\pi s}{2}) \zeta(s)$$ which results in (as the book also says) that $\zeta(-2n) =0$ for $n=1,2,3, \dots$, the so-called trival zeros of $\zeta(s)$. </p>
<p>But how on earth $\zeta(-2n) ... | Jan Eerland | 226,665 | <p>Notice, that the Riemann zeta function is only defined when the $\Re\left(\text{s}\right)>1$, so:</p>
<p>$$\zeta\left(\text{s}\right)=\sum_{\text{n}\in\mathbb{N}^+}\frac{1}{\text{n}^\text{s}}\space\space\space\to\space\space\space\zeta\left(-2\text{s}\right)\ne\sum_{\text{n}\in\mathbb{N}^+}\frac{1}{\text{n}^{-2\... |
2,558,870 | <p>Suppose $f:[0,1]\to \mathbb{R}$ is uniformly continuous, and $(p_n)_{n\in\mathbb{N}}$ is a sequence of polynomial functions converging uniformly to $f$.</p>
<p>Does it follow that $\mathcal{F}=\{p_n\mid n\in\mathbb{N}\}\cup \{f\}$ is equicontinuous?</p>
<p>Also, if $C_n$ are the Lipschitz constants of the polynomi... | user | 505,767 | <p>$$(A-pI)v_i=q_iv_i-pv_i\implies (A-pI)^{-1}(A-pI)v_i=(A-pI)^{-1}(q_i-p)v_i\implies (A-pI)^{-1}v_i=(q_i-p)^{-1}v_i \quad \square$$</p>
|
2,970,370 | <p>For <span class="math-container">$f \in C^0([0,1])$</span>, I have the following partial differential equation:</p>
<p><span class="math-container">$$u''(x) = f(x)$$</span> in <span class="math-container">$\Omega = (0,1)$</span>
<span class="math-container">$$u'(0) = u'(1) = 0$$</span></p>
<p>Why is this equation ... | Lutz Lehmann | 115,115 | <p>Set <span class="math-container">$v=u'$</span> then you are trying to solve the problem
<span class="math-container">$$
v'=f(x)\\
v(0)=v(1)=0
$$</span>
But <span class="math-container">$$v(1)-v(0)=\int_0^1v'(s)\,ds=\int_0^1f(s)\,ds$$</span>
so the equality of the boundary values is only possible if <span class="math... |
4,381,145 | <blockquote>
<p>Show that the three vector fields <span class="math-container">$X = y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}, Y = z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}$</span> and <span class="math-container">$Z=x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}$</span> on <s... | Clara B. | 572,257 | <p>By Proposition <span class="math-container">$5.38$</span> in John Lee's book "Smooth manifolds" (2. Edition), <span class="math-container">$T_p\mathbb S^2=\ker df_{|p}$</span>, where <span class="math-container">$f:\mathbb R^3\to\mathbb R$</span>, <span class="math-container">$(x,y,z)\mapsto x^2+y^2+z^2-1$... |
391,333 | <p>It is well known that $\sum_{k = 1}^{n}k^3 =\Big [\sum_{k=1}^{n}k^1\Big]^2$. My question is very simple.</p>
<blockquote>
<p>There are $3$-tuples $(p, q, \alpha) \in
\mathbb{N}\times\mathbb{N}\times\mathbb{N}$, in addition to $(3,1,2)$,
such that $\alpha\geq 2$ and $$\sum_{k = 1}^{n}k^{\,p} =\Big [\sum_{k... | Calvin Lin | 54,563 | <p>[The following claims should be obvious, and will not be proven.]</p>
<p>Claim: The degree of $ \sum_{i=1}^n i^r$ is $r+1$.</p>
<p>Claim: The leading coefficient of $ \sum_{i=1}^n i^r$ is $\frac{1}{r+1} $.</p>
<p>The first claim gives us $p+1 = \alpha (q+1) $. The second claim gives us $\frac{1}{p+1} = \left( \fr... |
2,669,292 | <p>$g_n(x) = \frac{\ln(1+x/n)}{n}$ on $\mathbb{R}$.
Don't they all converge to 0?</p>
| José Carlos Santos | 446,262 | <p>Please note that$$g_n(ne^n)=\frac{\ln(1+e^n)}n>\frac{\ln(e^n)}n=1.$$Therefore, your sequence cannot converge uniformly to $0$. This, in spite of your (correct) statement that for each <em>individual</em> $x$, $\lim_{n\to\infty}g_n(x)=0$. But uniform convergence is not about that; that's the realm of <em>pointwise... |
3,995,492 | <p>I have no clue how to do this, I manage to get I get that <span class="math-container">$11^{36} \equiv 1 \hspace{0.1cm} \text{mod} (13)$</span> but I can't get anywhere from there.</p>
| sirous | 346,566 | <p>Use this relation:</p>
<p><span class="math-container">$6\times 11-5\times 13=1$</span></p>
<p>So we can write:</p>
<p><span class="math-container">$$11^{36}\equiv (6\times 11-5\times 13=1) \ mod (13) \equiv 6 \times 11\ mod (13)$$</span></p>
<p>Therefore:</p>
<p><span class="math-container">$11^{36}\equiv 6\times 1... |
3,078,097 | <blockquote>
<p>Why it is impossible to split the natural numbers into two sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span> such that for distinct elements <span class="math-container">$m, n \in A$</span> we have <span class="math-container">$m + n \in B$</span> and vice-versa?... | bof | 111,012 | <p>It is a famous fact of <a href="https://en.wikipedia.org/wiki/Ramsey%27s_theorem#Example:_R(3,_3)_=_6" rel="nofollow noreferrer">Ramsey theory</a> that, if each edge of <span class="math-container">$K_6$</span> (a complete graph on <span class="math-container">$6$</span> vertices) is colored red or blue, there will ... |
366,249 | <p>$3xy^2dx+2x^3dy$
where is the boundary of the region between the circles $x^2+y^2=25$ and $x^2+y^2=64$ having positive orientation.</p>
<p>Not quite sure how to evaluate this...</p>
| doraemonpaul | 30,938 | <p>Let $y=\sum\limits_{n=0}^\infty a_n(x+1)^{n+r}$ ,</p>
<p>Then $y'=\sum\limits_{n=0}^\infty(n+r)a_n(x+1)^{n+r-1}$</p>
<p>$y''=\sum\limits_{n=0}^\infty(n+r)(n+r-1)a_n(x+1)^{n+r-2}$</p>
<p>$\therefore(x+1)^2\sum\limits_{n=0}^\infty(n+r)(n+r-1)a_n(x+1)^{n+r-2}+(x+1)\sum\limits_{n=0}^\infty(n+r)a_n(x+1)^{n+r-1}-\sum\l... |
134,987 | <blockquote>
<p>$$3x^2 + 2y^4 = z^4$$</p>
</blockquote>
<p><em>How do I solve this??</em> I would like to use so-called "elementary number theory", not abstract algebra (e.g. $\mathbb{Z} ( \sqrt d)$) or elliptic curves.</p>
<p>Note: I'm not asking <em>what</em> the solutions are, but rather <em>how</em> to find the... | Will Jagy | 10,400 | <p>EDIT: it would appear that what you want to know a procedure. So, put all the degree four terms together on one side of the equals sign and re-write that as a quadratic form in substitute variables, as below.</p>
<p>The reason you use 3 is this: write $$ 3 x^2 = u^2 - 2 v^2, $$ where we will eventually put back $u... |
1,626,362 | <p><code>The following is a short extract from the book I am reading:</code> </p>
<blockquote>
<p>If given a Homogeneous ODE:
$$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}+5\frac{\mathrm{d} y}{\mathrm{d}x}+4y=0\tag{1}$$
Letting
$$D=\frac{\mathrm{d}}{\mathrm{d}x}$$ then $(1)$ becomes
$$D^2 y + 5Dy + 4y=(D^2+5D+4)... | Justpassingby | 293,332 | <ol>
<li><p>Assume there are constants $A$ and $B$ such that the function $A\exp(-x)+B\exp(-4x)$, is identically zero. The constants $A$ and $B$ are (over)determined by filling in (for example) $x=\ln1=0,\ln2,\ln3$ so that the only possibility becomes $A=B=0.$</p></li>
<li><p>The solutions of a homogeneous equation or ... |
1,626,362 | <p><code>The following is a short extract from the book I am reading:</code> </p>
<blockquote>
<p>If given a Homogeneous ODE:
$$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}+5\frac{\mathrm{d} y}{\mathrm{d}x}+4y=0\tag{1}$$
Letting
$$D=\frac{\mathrm{d}}{\mathrm{d}x}$$ then $(1)$ becomes
$$D^2 y + 5Dy + 4y=(D^2+5D+4)... | BLAZE | 144,533 | <p>Given the general form of a second order ODE is
$$a\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}+b\frac{\mathrm{d}y}{\mathrm{d}x}+cy=0\tag{1}$$</p>
<p>where $a,b,c$ are known real coefficients. The most important property of $(1)$ is <em>linearity</em>. This means that if we have two solutions $y_1$ and $y_2$, then any lin... |
3,167,261 | <blockquote>
<p>Let <span class="math-container">$\mathcal{O}$</span> be an open subset of the plane <span class="math-container">$\mathbb{R}^{2}$</span> and
let the mapping <span class="math-container">$F : \mathcal{O} \rightarrow \mathbb{R}^{2}$</span> be
represented by <span class="math-container">$F(x, y) = ... | avs | 353,141 | <p>The real and imaginary parts of a holomorphic function are harmonic. Source: <a href="https://books.google.com/books/about/Elementary_Theory_of_Analytic_Functions.html?id=KsGbqTBjyoUC" rel="nofollow noreferrer">Cartan's <em>Elementary Theory of Analytic Functions</em></a>.</p>
|
1,177,782 | <p>I tried to prove the following theorem and was wondering if someone could please tell me if my proof can be fixed somehow...</p>
<p>Theorem: Let $H$ be a Hilbert space and $x_n\in H$ a bounded sequence. Then $x_n$ has a weakly convergent subsequence.</p>
<p>My idea for a proof:</p>
<p>The map $\phi: H \to H^\ast$... | Vincent Boelens | 94,696 | <p>I think this can be done without invoking Banach-Alaoglu or the Axiom of Choice. I will sketch the proof. By the Riesz representation theorem (which as far as I can tell can be proven without Choice), a Hilbert space is reflexive. Furthermore, it is separable iff its dual is.</p>
<p>To show the weak convergence of t... |
2,669,278 | <p>I noticed a strange thing with my calculator.<br>
When I start with any number like 1,2,3 or 1.2, 1.34 .... or even 0.<br>
And repeatedly take the cosine function of this number.<br>
I get the same following number. I don't thing this is a coincidence since it's happening with any number I try. </p>
<pre>0.9998477... | Brian Tung | 224,454 | <p>Your calculator must be operating in degrees. Since $0.9998\ldots$ degrees is very close to $0$ (being less than $1/90$ of the way from $0$ radians to $\pi/2$ radians), its cosine must be very close to $1$. What you are finding is the fixed point of the function $\cos \theta$, where $\theta$ is expressed in degree... |
33,543 | <p>Let $M$ be a filtered module over a filtered algebra $A$, and suppose $gr(M)$ is flat over $gr(A)$, where $gr$ means the associated graded module and algebra, respectively.</p>
<p>What can one say in general about the flatness of $M$ over $A$, or with relevant assumptions (for instance in the above, we should assum... | Mariano Suárez-Álvarez | 1,409 | <p>Let me suppose, as in your examples, that we have a base field $k$.</p>
<p>It is well known that to check that a right $A$-module $M$ is flat it is enough to show that whenever $I\leq_\ell A$ is a left ideal, the map $M\otimes_AI\to M\otimes_A A$ induced by the inclusion $I\to A$ is injective. This condition can be... |
320,348 | <p>I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$</p>
<p>I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively things like $${2n+2\choose n+1}={2n+2\over n+1}{2n+1\choose n}=2\cdot {2n+1\over n+1}{2n\choose n},$$</p>
<p... | Ishan Banerjee | 52,488 | <p>Split the <span class="math-container">$2n$</span> elements into two groups of size <span class="math-container">$n$</span>
Then the no. of ways of choosing <span class="math-container">$n$</span> from the <span class="math-container">$2n$</span> is the no. of ways of choosing <span class="math-container">$i$</span>... |
114,122 | <p>I am trying to figure out the maximum possible combinations of a (HEX) string, with the following rules:</p>
<ul>
<li>All characters in uppercase hex (ABCDEF0123456789)</li>
<li>The output string must be exactly 10 characters long</li>
<li>The string must contain at least 1 letter</li>
<li>The string must contain a... | hmakholm left over Monica | 14,366 | <p>I think the sanest way to split this is by how many <em>different</em> letters and numbers is contained in the string. Call the <em>total</em> number of different symbols $k$. The possibilities are then:</p>
<pre><code>k=5 1+4 2+3 3+2 4+1
k=6 1+5 2+4 3+3 4+2 5+1
k=7 1+6 2+5 3+4 4+3 5+2 6+1
k=8 ... |
3,045,677 | <p>I am trying to create a program for a school project where I need to plot points of a triangle given all 3 side lengths (a=10, b=20, c=30).
I tried the solution from the other topic and it didn't work since the result produced was C(20,0) and that cant be right since one of the sides is already placed on the axis a... | user2661923 | 464,411 | <p>Given sides a,b,and c, use the law of cosines to calculate the angle <span class="math-container">$\theta$</span> between a and b. then (for example), plot b as horizontal, and plot a as a directional vector, with angle <span class="math-container">$\theta.$</span> </p>
|
217,436 | <p>I don't know where to start... It's a multiple-choice question: I can choose from $\sqrt{2}, 0, 2, 1$</p>
<p>Thank you!</p>
| lab bhattacharjee | 33,337 | <p>Using <a href="http://en.wikipedia.org/wiki/Euler%27s_formula" rel="nofollow">this</a> or <a href="http://mathworld.wolfram.com/EulerFormula.html" rel="nofollow">this</a>, $$e^{\frac{i\pi}4}=\cos \frac{\pi}4 +i\sin\frac{\pi}4=\frac{1+i}{\sqrt 2}$$</p>
<p>$$(1-i)\cdot e^{\frac{i\pi}4}=(1-i)\cdot \frac{(1+i)}{\sqrt 2... |
217,436 | <p>I don't know where to start... It's a multiple-choice question: I can choose from $\sqrt{2}, 0, 2, 1$</p>
<p>Thank you!</p>
| dot dot | 21,681 | <p>$$e^{i\pi/4}(1-i)=(1-i)(1+i)\frac{\sqrt{2}}{2}=2\frac{\sqrt{2}}{2}=\sqrt{2}$$</p>
|
2,206,247 | <p><strong>Question:</strong> Consider the following non linear recurrence relation defined for $n \in \mathbb{N}$:</p>
<p>$$a_1=1, \ \ \ a_{n}=na_0+(n-1)a_1+(n-2)a_2+\cdots+2a_{n-2}+a_{n-1}$$</p>
<p>a) Calculate $a_1,a_2,a_3,a_4.$</p>
<p>b) Use induction to prove for all positive integers that:</p>
<p>$$a_n=\dfra... | Χpẘ | 309,642 | <p>Yes it is related to triangle inequality. If the two sides of right triangle have lengths of $\sqrt{a}$ and $\sqrt{b}$, then hypotenuse has length $\sqrt{a+b}$. Therefore $\sqrt{a}+\sqrt{b} > \sqrt{a+b}$ </p>
|
1,238,210 | <p>How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$ ?</p>
<p>P.S: This is my method as I thought:
$\int _0^x\:\:e^{t^2}dt>\int _1^x\:e^tdt=e^x-e$ which is divergent, so all your answers, helped me to think otherwise, maybe my method help something else :D</p>
| Alessio Bocci | 386,931 | <p>It is vary simple:
$$y(x)=\int_0^xe^{t^2}dt=\frac{1}{2} \sqrt{\pi } \text{erfi}(x)$$
So:
$$\lim_{x \rightarrow+\infty}y(x)=+\infty$$</p>
|
1,934,033 | <p>I'm new here. I wish to ask a question regarding predicate logic:</p>
<p>I was given three predicates:</p>
<p><strong>parent(p,q): p is the parent of q.</strong></p>
<p><strong>female(p): p is a female.</strong></p>
<p><strong>p = q: p and q are the same person.</strong></p>
<p>Now, I was tasked with translatin... | Anonymous Coward | 251,953 | <p>No, it is not in this particular case. You conclude that by substitution.</p>
<p>female(q) can have two values : true or false.</p>
<ul>
<li>For female q, female(q)=true you have : parent(Alice,true).</li>
<li>For non-female q, female(q)=false you have : parent(Alice,false).</li>
</ul>
<p>Since false and true are... |
64,643 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/4467/a1-2-is-either-an-integer-or-an-irrational-number">$a^{1/2}$ is either an integer or an irrational number</a> </p>
</blockquote>
<p>I know how to prove $\sqrt 2$ is an irrational number. Who can tell ... | Ross Millikan | 1,827 | <p>If you follow through the usual proof for $\sqrt{2}$ substituting $3$ for $2$, it goes through just fine. Let $\sqrt{3}=\frac{p}{q}, p,q $ relatively prime. $3=\frac{p^2}{q^2}$, so $3$ divides $p$ and so on.</p>
|
12,057 | <p>Let $p,q$ belong to $\mathbb{N}$ and are relatively prime to each other. If $\alpha,\beta$ belong to $\mathbb{N}$, are also relatively prime to each other,then are $(p\beta+q\alpha)$ and $q\beta$ always relatively prime ?</p>
| Bill Dubuque | 242 | <p>Simply repeatedly apply <span class="math-container">$\rm\color{#C00}E = $</span> Euclid's lemma:</p>
<p><span class="math-container">$\begin{align} 1 &= (pb+qa,qb)\\
\overset{\rm\color{#C00}E }\iff\ \ \ \ \ \ \ 1 &= (pb+qa,q) = (pb,q) \overset{\rm\color{#C00}E }= (b,q)\quad\ {\rm via}\quad\ (p,q) = 1\\
{... |
4,249,573 | <p>I'm studying Set Theory in my own, with Goldrei's textbook. The chapter I'm reading is on order-isomorphism and well-ordering. One exercise asks (i) to argue that, in general, a collection of well-ordered sets order-isomorphic to a given well-ordered set is a proper class (rather than a set). The proof, IMHO, is fai... | Lazy | 958,820 | <p>Let’s assume <span class="math-container">$X\neq\emptyset$</span> (else the property is trivially true). First note that you are looking at classes of order-isomorphic ordered sets, so pairs <span class="math-container">$(A,\leq)$</span> so that all for all such pairs exists a bijection that leaves the order intact.... |
28,877 | <p>Since I self-study mathematical analysis without <em>formal</em> teacher, I can only appeal to help from out site most of the time. It's obvious that to grasp the underlying concepts in mathematics, we must roll the sleeves and solve problems.</p>
<p>It's clear that there are actually mistakes and misunderstanding ... | user21820 | 21,820 | <p>Not yet mentioned...</p>
<ol>
<li><p>Learn at least one formal deductive system for first-order logic (you can see my profile for <a href="http://math.stackexchange.com/a/1684204/21820">one variant of Fitch-style natural deduction</a> and <a href="http://math.stackexchange.com/a/1788516/21820">some basic examples</... |
966,278 | <p>Given a recursive relation
$$a_n = \begin{cases}
(1 - 2b_n)a_{n-1} + b_n, & n > 1 \\
\frac{1}{2}, & n =1
\end{cases}
$$, how can I expression $a_n$ in term of $b_i, i \in \{1, 2, \dots n\}$?</p>
| Peter Huxford | 152,620 | <p>Suppose that $a_n=\frac{1}{2}$ for some $n$. Then according to the recurrence:</p>
<p>\begin{align*}
a_{n+1}&=(1-2b_{n+1})a_n+b_{n+1} \\
&= \frac{1}{2}-b_{n+1}+b_{n+1} \\
&= \frac{1}{2}
\end{align*}</p>
<p>Since $a_1=\frac{1}{2}$, by induction this shows that $a_n=\frac{1}{2}$ for all $n$.</p>
|
3,425,369 | <p>I'm taking a linear algebra course and I'm having trouble proving linear (in)dependence of functions. I understand that I have to prove that the <span class="math-container">$a_1f(x) + a_2g(x) = 0$</span> but I don't know how to actually do that. For example given a pair of functions 1 and t, how do you prove linear... | Matthew Leingang | 2,785 | <p>With polynomials, or functions in general, you can evaluate them at certain points and explore the consequences.</p>
<p>For instance, suppose that there exist scalars <span class="math-container">$a_1$</span> and <span class="math-container">$a_2$</span> such that <span class="math-container">$a_1 \cdot 1 + a_2 \cd... |
1,285,774 | <p>I have looked at similar questions under 'Questions that may already have your answer" and unless I have missed it, I cannot find a similar question.</p>
<p>I am trying to answer the following:</p>
<p>Let $A = \left(\begin{matrix}
a & b \\
b & d \\
\end{matrix}\right)$ be a symmetric 2 x 2 matrix. Prove t... | user99163 | 99,163 | <p>Hint: $A$ is positive definite iff $\vec x^{T}A \vec x>0$ for all $\vec x = (x, y) \in\mathbb{R^{2}}\setminus \{(0,0\}$</p>
<p>iff $ax^{2}+2bxy+dy^{2}>0$ for all $x,y\in\mathbb{R}$, $(x, y) \neq (0,0)$</p>
<p>iff $a(x+\frac{by}{a})^{2}+(d-\frac{b^{2}}{a})y^{2}>0$ for all $x,y\in\mathbb{R}$, $(x, y) \neq ... |
2,342,124 | <p><a href="https://i.stack.imgur.com/QdbFG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/QdbFG.png" alt="enter image description here"></a></p>
<p>Well this seems like <span class="math-container">$1-|t|$</span> for <span class="math-container">$|t|<1$</span> and <span class="math-container">... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
1,156,738 | <p>Is it fine to say "Groups $A$ and $B$ are isomorphic." or should one say "Groups $A$ and $B$ are isomorphic to each other."?</p>
| mweiss | 124,095 | <p>Both formulations are common. You could also say "$A$ is isomorphic to $B$."</p>
|
200,322 | <p>Is there a compact topological space $(X,\tau)$ such that for no cardinal $\kappa$ there is a surjective continuous map $e:\{0,1\}^\kappa \to X$? </p>
<p>(We assume that $\{0,1\}$ is endowed with the discrete topology, and $\{0,1\}^\kappa$ has the product topology.)</p>
| Tomasz Kania | 15,129 | <p>To elaborate on Joseph's answer, the class of continuous images of Cantor cubes has a fancy name, they are the so called <a href="http://en.wikipedia.org/wiki/Dyadic_space" rel="nofollow">dyadic spaces</a>. There is a nice result by Haydon: every Dugundji space is dyadic. (A space $X$ is Dugundji if the conclusion o... |
232,436 | <p>How do I solve? I've tried to multiply and divide by the conjugate cannot advance.
$$\lim_{x\rightarrow +\infty} \sqrt{(x-a)(x-b)}-x$$</p>
| André Nicolas | 6,312 | <p>Multiplying and dividing by the conjugate works fine. Let $x$ be positive and larger than $a$ and $b$. We quickly obtain
$$\frac{-ax-bx+ab}{\sqrt{(x-a)(x-b)}+x}.$$
Divide top and bottom by $x$. (That is another commonly useful kind of move.) We get
$$\frac{-a-b+\frac{ab}{x}}{\sqrt{\left(1-\frac{a}{x}\right)\left(1... |
232,436 | <p>How do I solve? I've tried to multiply and divide by the conjugate cannot advance.
$$\lim_{x\rightarrow +\infty} \sqrt{(x-a)(x-b)}-x$$</p>
| juantheron | 14,311 | <p>$\bf{My\; Solution::}$ Given $\displaystyle \lim_{x\rightarrow \infty}\sqrt{(x-a)(x-b)}-x\;,$ Now Using $\bf{A.M\geq G.M}$</p>
<p>Now When $x\rightarrow \infty,$ Then $(x-a)\;,(x-b)\rightarrow \infty$</p>
<p>So $\displaystyle \frac{(x-a)+(x-b)}{2}\geq \sqrt{(x-a)(x-b)}\Rightarrow x-\left(\frac{a+b}{2}\right)\geq \... |
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