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 |
|---|---|---|---|---|
947,290 | <p>In a cyclic group of order 8 show that element has a cube root. So for some $a\in G$ there is an element $x \in G$ with $x^3=a.$</p>
<p>Also show in general that if $g=<a>$ is a cyclic group of order m and $(k,m)=1$ then each element in G has a $k$th root. What element will $a^k$ generate? Use this to expre... | Arpit Kansal | 175,006 | <p>Perhaps simpler approach consider $f:G\to G$ defind as $x\to x^3$.Now by using the fact that G is abelian and does not have any element of order $3$,show that $G$ is automorphism and hence done.Also i think abelian is sufficient condition!</p>
|
136,067 | <p>Assume $f(x)>0$ defined in $[a,b]$, and for a certain $L>0$, $f(x)$ satisfies the Lipschitz condition $|f(x_1)-f(x_2)|\leq L|x_1-x_2|$.</p>
<p>Assume that for $a\leq c\leq d\leq b$,$$\int_c^d \frac{1}{f(x)}dx=\alpha,\int_a^b\frac{1}{f(x)}dx=\beta$$Try to prove$$\int_a^b f(x)dx \leq \frac{e^{2L\beta}-1}{2L\alp... | Yimin | 30,330 | <p>If $f(x)$ is smooth, or $f\in C^1$, assume $\displaystyle h(t)=\int_a^t f(s)\mathrm{d}s$, and $\displaystyle g(t)=\int_a^{t}\frac{1}{f(s)}\mathrm{d}s$,we just can focus on
\begin{equation}
\frac{h(t)}{\exp(2Lg(t)-1)}
\end{equation}</p>
<p>because we know that there is a $\xi\in(c,d)$ s.t.
\begin{equation}
\frac{h... |
317,753 | <p>I am taking real analysis in university. I find that it is difficult to prove some certain questions. What I want to ask is:</p>
<ul>
<li>How do we come out with a proof? Do we use some intuitive idea first and then write it down formally?</li>
<li>What books do you recommended for an undergraduate who is studying ... | oks | 60,529 | <p>To come out with a proof I pretty much always started by
1. imagining a specific example
2. drawing the example as picture if possible
3. persuading myself (by looking at the picture) that the thing we were being asked to prove was actually true, then
4. making up some notation to describe what I was looking at... |
22,101 | <p>The general rule used in LaTeX doesn't work: for example, typing <code>M\"{o}bius</code> and <code>Cram\'{e}r</code> doesn't give the desired outputs.</p>
| mweiss | 124,095 | <p>There are certain system-dependent ways to enter diacriticals and other special characters. For example, on a Macintosh computer running any operating system prior to 10.10 (Yosemite):</p>
<ul>
<li>The keystroke combination <code>option-U + vowel</code> produces the vowel with an umlaut over it.</li>
<li>The keyst... |
1,650,204 | <p>I was given this problem and I can't seem to think of a solution.</p>
<p>Here is a possibly helpful graphic:</p>
<p><a href="https://i.stack.imgur.com/VKZkv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VKZkv.png" alt="Here is a possibly helpful link:" /></a></p>
<blockquote>
<p>Given two parall... | Bobson Dugnutt | 259,085 | <p>Just to supplement @PVanchinathan's excellent answer and because the comment became too long, I'm writing this answer. </p>
<p>The movement of the dots represents the linear transformation on a whole. Some vectors are also shown. For instance, the red ones are all vertical/horizontal in the original representation,... |
718,266 | <p>Is there a simple intuitive graphical explanation of Clifford Algebra for the layman? Since Clifford Algebra is a Geometric Algebra, surely the best way to present those concepts is with graphical figures.</p>
| user48672 | 138,298 | <p>To get some geometrical meaning you can look at some special Clifford algebras:</p>
<p>$\mathcal{Cl}_{0,1}$ is isomorphic to the complex plane.</p>
<p>$\mathcal{Cl}_{2,0}$ is isomorphic to the Euclidean plane.</p>
<p>$\mathcal{Cl}_{3,0}$ is isomorphic to the 3D Euclidean space.</p>
<p>$\mathcal{Cl}_{3,1}$ is iso... |
2,437,983 | <p>What is the chance that at least two people were born on the same day of the week if there are 3 people in the room?</p>
<p>I know how to get the answer which is 19/49 when considering all 3 people <strong>not being born on the same day</strong>. However, when I try to calculate the answer directly I seem to get it... | Community | -1 | <p>Half-open intervals $\mathcal{A}=\{[a,b):a<b\}$ are Borel sets. Thus, $\sigma(\mathcal{A})\subset \mathcal{B}_{\mathbb{R}}$. On the other hand, any open set in $\mathbb{R}$ can be approximated by a countable union of the intervals from $\mathcal{A}$ which implies that $\mathcal{B}_{\mathbb{R}}\subset\sigma(\mathc... |
3,460,595 | <p>I am given the following sequence:</p>
<p><span class="math-container">$$a_n = 1^9 + 2^9 + ... + n^9 - an^{10}$$</span></p>
<p>Where <span class="math-container">$a \in \mathbb{R}$</span>. I have to find the value of <span class="math-container">$a$</span> for which the sequence <span class="math-container">$a_n$<... | Community | -1 | <p>Consider</p>
<p><span class="math-container">$$a_{n+1}=a_n+(n+1)^9+an^{10}-a(n+1)^{10}.$$</span></p>
<p>In this recurrence, the term of degree <span class="math-container">$9$</span> has the coefficient <span class="math-container">$1-10a$</span>. If this coefficient is nonzero, the polynomial grows to infinity. O... |
267,706 | <p>I'm making an animation of a <a href="https://en.wikipedia.org/wiki/Reuleaux_triangle" rel="nofollow noreferrer">Reuleaux triangle</a> rolling on a straight line like this
<a href="https://i.stack.imgur.com/m0IMm.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/m0IMm.gif" alt="rolling Reuleaux tria... | Daniel Huber | 46,318 | <p>With code from the Wolfram Demo project for a Reuleaux triangle from <code>https://demonstrations.wolfram.com/ARotatingReuleauxTriangle/</code> and some small changes:</p>
<pre><code>angle[vec_] :=
Arg[First[vec] + I*Last[vec]] + If[Last[vec] >= 0, 0, 2*Pi]
centerpath[t_] :=
Piecewise[{{{1 + Cos[Mod[t, 2 Pi... |
426,998 | <p>Motivated in the <a href="https://cs.stackexchange.com/questions/12830/can-expected-depth-of-an-element-and-expected-height-differ-significantly">analysis of algorithms</a>, consider the following setup.</p>
<p>Assume we have discrete random variables $X^{(n)}_1, \dots, X^{(n)}_n$ which we can not assume to be iden... | András Salamon | 3,362 | <p>(<em>Note:</em> this answer was to an earlier version of the question. My understanding was that the distribution of $X_j$ was fixed. The current version of the question indicates that the parameters of its distribution depend on $n$ also. The bounds still apply, but may be less directly useful for this scenario.... |
426,998 | <p>Motivated in the <a href="https://cs.stackexchange.com/questions/12830/can-expected-depth-of-an-element-and-expected-height-differ-significantly">analysis of algorithms</a>, consider the following setup.</p>
<p>Assume we have discrete random variables $X^{(n)}_1, \dots, X^{(n)}_n$ which we can not assume to be iden... | Raphael | 3,330 | <p>The following method can yield bounds stronger than the one András cites but requires even more knowledge about the distribution of the <span class="math-container">$X^{(n)}_i$</span>. The idea is to use bounds on the tail probabilities of the <span class="math-container">$X^{(n)}_i$</span> to bound the tail of thei... |
3,660,652 | <p>To which of the seventeen standard quadrics (<a href="https://mathworld.wolfram.com/QuadraticSurface.html" rel="nofollow noreferrer">https://mathworld.wolfram.com/QuadraticSurface.html</a>) do these two equations reduce?
<span class="math-container">\begin{equation}
Q_1^2+3 Q_2 Q_1+\left(3
Q_2+Q_3\right){}^2 = 3 ... | Bernard | 202,857 | <p>Here is how to obtain all solutions with congruences:</p>
<p>This relation means that
<span class="math-container">\begin{align}
781 + 256 (3d-1)\bmod 81&\iff 52+ 13(3d-1)\equiv 0
\iff39(1+d)\equiv 0 \bmod 81\\
\scriptstyle\text{(simplifying by }3)&\iff 13(1+d)\equiv 0 \bmod 27\\
\scriptstyle (13\text{ is a... |
3,833,767 | <p>I am trying to brush up on calculus and picked up Peter Lax's Calculus with Applications and Computing Vol 1 (1976) and I am trying to solve exercise 5.2 a) in the first chapter (page 29):</p>
<blockquote>
<p>How large does <span class="math-container">$n$</span> have to be in order for</p>
<p><span class="math-cont... | saulspatz | 235,128 | <p>If you get an answer like <span class="math-container">$1.29999$</span>, you'll simply have to compute more terms of the series, but in all likelihood, you'll be able to make a definite statement. Try to compute the first four digits after the decimal point. You may be doubtful about the fourth digit, but in all l... |
152,295 | <p>What is the definition of picture changing operation?
What is a standard reference where it is defined - not just used?</p>
| QGravity | 64,606 | <p>This is a physical description of Picture-Changing Operation:</p>
<p>In the RNS formulation of superstring theory, the worldsheet theory has superconformal gauge invariance. One thus needs to fix the gauge. This means that one can (locally) fix the form of metric and the gravitino field. This will introduce ghost a... |
3,583,475 | <p>Write as a single fraction:</p>
<p><span class="math-container">$(4x+2y)/(3x) - (5x+9y)/(6x) + 4$</span></p>
<p>Simplify your answer as much as possible.</p>
<p>The answer that I got from when I did the math was: (27x-5y)/(6x). But I have asked some of my friends who some got a different answer from mine. Plea... | fleablood | 280,126 | <p>Put them over a common denominator:</p>
<p><span class="math-container">$\frac {4x+2y}{3x}\cdot \frac 22 - \frac {5x + 9y}{6x} + 4\cdot\frac {6x}{6x}=$</span></p>
<p><span class="math-container">$\frac {2(4x+2y) - (5x+9y) +4\cdot 6x}{6x}=$</span></p>
<p><span class="math-container">$\frac {(8x+4y) -(5x +9y) + 24x... |
194 | <p>In many parts of the world, the majority of the population is uncomfortable with math. In a few countries this is not the case. We would do well to change our education systems to promote a healthier relationship with math. But in the present situation, how can we help the students who come to our classes, which the... | adamblan | 93 | <p>There are a few strategies that are supported by experimental research which I will share here, but they all have to do with <strong>stereotype threat</strong>. I am sure there are other types of anxiety related to math which would not be helped by these strategies.</p>
<p>First, the <a href="http://en.wikipedia.o... |
194 | <p>In many parts of the world, the majority of the population is uncomfortable with math. In a few countries this is not the case. We would do well to change our education systems to promote a healthier relationship with math. But in the present situation, how can we help the students who come to our classes, which the... | Mandy Jansen | 739 | <p>I agree with the idea that different people might be anxious about mathematics for different reasons...</p>
<p><a href="https://www.npr.org/blogs/health/2012/11/12/164793058/struggle-for-smarts-how-eastern-and-western-cultures-tackle-learning" rel="nofollow noreferrer">Culturally</a>, in the United States, we tend t... |
194 | <p>In many parts of the world, the majority of the population is uncomfortable with math. In a few countries this is not the case. We would do well to change our education systems to promote a healthier relationship with math. But in the present situation, how can we help the students who come to our classes, which the... | Allen Seay | 5,247 | <p>Sue VanHattum, in reference to your question "How can we help students who are very ANXIOUS about math?" I don't know that this will answer your question, but I want to refer you to www.mathmidway.org or midway@mathmuseum.net. Phone is (631) 444-0945. One of the questions in the booklet is "how do you get a 10-year-... |
1,281,967 | <p>This is a dumb question I know.</p>
<p>If I have matrix equation $Ax = b$ where $A$ is a square matrix and $x,b$ are vectors, and I know $A$ and $b$, I am solving for $x$.</p>
<p>But multiplication is not commutative in matrix math. Would it be correct to state that I can solve for $A^{-1}Ax = A^{-1}b \implies x =... | Bradley Morris | 235,142 | <p>If $A$ is invertible then your solution works. Use this little result to determine if $A$ is invertible:</p>
<p>$A^{-1}$ exists $\Leftrightarrow$ det($A$) $\neq 0$. Where det($A$) is the determinant of $A$.</p>
<p>A little reading on determinants: <a href="http://mathworld.wolfram.com/Determinant.html" rel="nofoll... |
2,130,911 | <p>I'm unsure how to compute the following : 3^1000 (mod13)</p>
<p>I tried working through an example below,</p>
<p>ie) Compute $3^{100,000} \bmod 7$
$$
3^{100,000}=3^{(16,666⋅6+4)}=(3^6)^{16,666}*3^4=1^{16,666}*9^2=2^2=4 \pmod 7\\
$$</p>
<p>but I don't understand why they divide 100,000 by 6 to get 16,666. Where di... | Jack D'Aurizio | 44,121 | <p>There is a fast&brutal solution that requires very little knowledge:
$$ 3^{1000} \equiv 3\cdot(3^3)^{333} \equiv 3\cdot 1^{333} \equiv \color{red}{3}\pmod{13}.$$
A similar approach works in the other case, too:
$$ 3^{10000}\equiv 3\cdot(3^3)^{3333} \equiv 3\cdot(-1)^{3333} \equiv -3\equiv \color{red}{4}\pmod{7}.... |
1,447,852 | <p>Compute this sum:</p>
<p><span class="math-container">$$\sum_{k=0}^{n} k \binom{n}{k}.$$</span></p>
<p>I tried but I got stuck.</p>
| MadMonty | 145,364 | <p>A more intuitive way of thinking about this is to ask, "Given n people, how many possible 'teams' of people are there, given that each team has a leader?".</p>
<p>So on one hand, if a team has $k$ people in it, then there are ${n}\choose{k}$ ways to pick those $k$ people, and any of those $k$ people can be leader, ... |
883,972 | <p>Let:</p>
<p>$$f(n) = n(n+1)(n+2)/(n+3)$$</p>
<p>Therefore :</p>
<p>$$f∈O(n^2)$$</p>
<p>However, I don't understand how it could be $n^2$, shouldn't it be $n^3$? If I expand the top we get $$n^3 + 3n^2 + 2n$$ and the biggest is $n^3$ not $n^2$.</p>
| Did | 6,179 | <p>$$n+2\leqslant n+3\implies f(n)\leqslant n(n+1)=n^2+n\leqslant2n^2$$
$$(n+1)(n+2)=n(n+3)+2\geqslant n(n+3)\implies f(n)\geqslant n^2$$</p>
|
883,972 | <p>Let:</p>
<p>$$f(n) = n(n+1)(n+2)/(n+3)$$</p>
<p>Therefore :</p>
<p>$$f∈O(n^2)$$</p>
<p>However, I don't understand how it could be $n^2$, shouldn't it be $n^3$? If I expand the top we get $$n^3 + 3n^2 + 2n$$ and the biggest is $n^3$ not $n^2$.</p>
| IAmNoOne | 117,818 | <p>Because formally, $$\lim_{n \to \infty} \left | \frac{f(x)}{g(x)} \right |= \lim_{n \to \infty} \left | \frac{\frac{n(n+1)(n+2)}{n+3}}{n^2} \right |= \lim_{n \to \infty} \left | \frac{n(n+1)(n+2)}{n^2(n+3)} \right | = 1.$$</p>
<p>So $f\in O(n^2)$ indeed.</p>
|
883,972 | <p>Let:</p>
<p>$$f(n) = n(n+1)(n+2)/(n+3)$$</p>
<p>Therefore :</p>
<p>$$f∈O(n^2)$$</p>
<p>However, I don't understand how it could be $n^2$, shouldn't it be $n^3$? If I expand the top we get $$n^3 + 3n^2 + 2n$$ and the biggest is $n^3$ not $n^2$.</p>
| evinda | 75,843 | <p>$$f(n)=\frac{n(n+1)(n+2)}{n+3}=\frac{(n^2+n)(n+2)}{n+3}=\frac{n^3+2n^2+n^2+2n}{n+3}=\frac{n^3+3n^2+2n}{n+3} \\ =n^2-\frac{6}{n+3}+2$$</p>
<p>Let $f(n)=O(n^2)$.Then, $\exists c>0 \text{ and } n_0 \geq 1 \text{ such that } \forall n \geq n_0: \\ f(n) \leq cn^2 \Rightarrow n^2-\frac{6}{n+3}+2 \leq cn^2 \Rightarrow ... |
1,905,863 | <p>I'm on the section of my book about separable equations, and it asks me to solve this:</p>
<p>$$\frac{dy}{dx} = \frac{ay+b}{cy+d}$$</p>
<p>So I must separate it into something like: $f(y)\frac{dy}{dx} + g(x) = constant$</p>
<p>*note that there are no $g(x)$</p>
<p>but I don't think it's possible. Is there someth... | Leucippus | 148,155 | <p>Consider: $$\frac{dy}{dx} = \frac{ay+b}{cy+d}$$
which can be seen as the following:
\begin{align}
1 &= \frac{c y + d}{a y + b} \, \frac{dy}{dx} \\
&= \frac{c}{a} \, \left[ 1 + \left(\frac{d}{c} - \frac{b}{a} \right) \, \frac{a}{ay + b} \right] \, \frac{dy}{dx}
\end{align}
which becomes
$$\frac{a}{c} \, dx = ... |
1,074,534 | <p>How can I get started on this proof? I was thinking originally:</p>
<p>Let $ n $ be odd. (Proving by contradiction) then I dont know.</p>
| drhab | 75,923 | <p>Let $n$ be the smallest positive number that has $k>1$ divisors and let $n=p_1^{r_1}\times\cdots\times p_s^{r_s}$ be its factorization in primes. If $n$ is odd then $2<p_i$ for $i=1,\dots,s$. Replacing one of the $p_i$ by $2$ results in a smaller number that has the same number of divisors ($k=r_1\times\cdots\... |
187,618 | <p>I am trying to solve the following problem.</p>
<p>The time $T$ required to repair a machine is an exponentially distributed random variable with mean 10 hours.</p>
<p>a) What is the probability that a repair takes at least 15 hours given that its duration
exceeds 12 hours?
b) What is the probability that the comb... | Did | 6,179 | <p>$$\mathrm P(T_1+T_2\gt20)=\mathrm P(T_1\gt20)+\int_0^{20}\mathrm P(T_2\gt20-t)\cdot\lambda\mathrm e^{-\lambda t}\cdot\mathrm dt
$$
$$
\mathrm P(T_1+T_2\gt20)=\mathrm e^{-20\lambda}+\int_0^{20}\mathrm e^{-\lambda (20-t)}\cdot\lambda\mathrm e^{-\lambda t}\cdot\mathrm dt=\ ...$$</p>
|
201,381 | <p>I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis. </p>
<blockquote>
<p>Can you suggest some fundamental papers(or books); so after reading these I can have, hopefully(p... | Desiderius Severus | 43,737 | <p>In an other fashion, you can be interested in how Fourier analysis (series decompositions, Poisson formula) is fundamental in :</p>
<ul>
<li>Trace formulas (kind of generalization of Poisson formula in the non-real-and-commutative case)</li>
<li>Computing functional equations for zêta-functions and reaching Tamagaw... |
4,294,577 | <p>If I have a function with all positive integer for the coefficients, is there a way to have a lower bound? Zero isn't an option, because I've done the rational root theorem and found all possible roots. If you need, I can provide the function and its list of possible roots below:</p>
<p><span class="math-container">... | Alexey Do | 532,569 | <p>Since the Bourbaki style is terrible to track down all the detals, I post this one to help others those who want to read a full proof of this problem; of course, I adapt modern notations.</p>
<p>Fix a base <span class="math-container">$\Delta$</span> for the root system <span class="math-container">$\Sigma$</span>. ... |
2,875,907 | <p>There are set of rods of length <span class="math-container">$1,2,3,4 \dots N$</span>.
Two players take turns to chose 3 rods and compose triangle with non-zero area. After that this particular 3 rods are removed.
If it is not possible to compose triangle then player looses.</p>
<p>Who has winning strategy?</p>
<... | Jaap Scherphuis | 362,967 | <p>This is a nim-like <a href="https://en.wikipedia.org/wiki/Impartial_game" rel="noreferrer">impartial game</a>, so each possible position has a nim-value associated with it.
I wrote a little program to calculate the nim-values, and the values of the starting positions are:</p>
<pre><code>N 4 5 6 7 8 9 10 11 12... |
3,385,420 | <p>The question is from <em>Cambridge Admission Test 1983</em></p>
<blockquote>
<p>A room contains m men and w women. They leave one by one at random until only people of the same sex remain. show by a carefully explained inductive argument, or otherwise, that the expected number of people remaining is <span class="... | drhab | 75,923 | <p>Number the men with <span class="math-container">$1,\dots,m$</span> and the women with <span class="math-container">$1,\dots,w$</span>.</p>
<p>For <span class="math-container">$i=1,\dots,m$</span> let <span class="math-container">$X_{i}$</span> take value <span class="math-container">$1$</span> if man <span class="... |
853,774 | <blockquote>
<p>If $(G,*)$ is a group and $(a * b)^2 = a^2 * b^2$ then $(G, *)$ is abelian for all $a,b \in G$.</p>
</blockquote>
<p>I know that I have to show $G$ is commutative, ie $a * b = b * a$</p>
<p>I have done this by first using $a^{-1}$ on the left, then $b^{-1}$ on the right, and I end up with and expres... | user160992 | 160,992 | <p>There is only one operation defined for the group, namely $*$, so if you want to be pedantic/exact, $a*b$ is a valid statement, while $ab$ is not defined.</p>
<p>However, in practice we shorten the notation, so $a*b$ can be written as $ab$.</p>
<p>So your final expression is equivalently $ab=ba$ or $a*b=b*a$. They... |
1,050,232 | <p>Prove the following proposition</p>
<p>Let $x, y \in \mathbb{ R}>0$. If $x < y$ then $0 < y^{-1 }< x^{-1}.$</p>
<p>So far I've gotten that since $x, y > 0$ then $x^{-1}, y^{-1} > 0$. </p>
| Julián Aguirre | 4,791 | <p>Since $a,b\in(\frac\pi8,\frac\pi4)$, we have $2c\in(\frac\pi4,\frac\pi2)$. Then
$$
\cos\frac\pi2<\cos 2c<\cos\frac\pi4\implies\cos^22c\le\frac{1}{2}
$$
and
$$
\frac{1}{\cos^22c}\ge2.
$$</p>
|
1,050,232 | <p>Prove the following proposition</p>
<p>Let $x, y \in \mathbb{ R}>0$. If $x < y$ then $0 < y^{-1 }< x^{-1}.$</p>
<p>So far I've gotten that since $x, y > 0$ then $x^{-1}, y^{-1} > 0$. </p>
| ir7 | 26,651 | <p>If $c\in(\frac\pi8,\frac\pi4)$, then $2c\in(\frac\pi4,\frac\pi2)$ and:</p>
<p>$$ 0 =\cos^2\left(\frac{\pi}{2}\right) <\cos^2(2c) < \cos^2\left(\frac{\pi}{4}\right) = \frac{1}{2}. $$</p>
<p>Hence:</p>
<p>$$ \frac{2}{\cos^2(2c)} > 4.$$</p>
<p><strong>Edit:</strong> </p>
<p>If $ a= b$, then</p>
<p>$$ |\t... |
11,518 | <p>How to prove that $\mathcal{O}_{\sqrt[3]{3}}$ is an euclidean domain? I heard that one should prove the following but why it is enough?</p>
<p>For any $ a,b,c\in\mathbb{R}$, prove that there are $ x,y,z\in\mathbb{R}$ such that $ x-a,y-b,z-c\in\mathbb{Z}$ and that
$$-1\leq x^3+3y^3+9z^3-9xyz\leq 1.$$</p>
| Alex B. | 3,212 | <p>Note that the ring of integers of $\mathbb{Q}(\sqrt[3]{3})$ is $\mathbb{Z}[\sqrt[3]{3}]$ with basis $1,\sqrt[3]{3},\sqrt[3]{9}$ over $\mathbb{Z}$. You want to show that the norm, defined by
\begin{eqnarray*}
N(a+b\sqrt[3]{3} + c\sqrt[3]{9}) & = & (a+b\sqrt[3]{3} + c\sqrt[3]{9})(a+\zeta_3b\sqrt[3]{3} + \zeta_... |
3,407,368 | <p>Please help me to think through this.</p>
<p>Take Riemann, for example. Finding a non-trivial zero with a real part not equal to <span class="math-container">$\frac{1}{2}$</span> (i.e., a counterexample) would disprove the conjecture, and also so it to be decidable.</p>
<p>How about demonstrating that Riemann is u... | Robert Israel | 8,508 | <p>They do not differ. "Any non-trivial zeros that we can find through brute force checking " is exactly the same as "All non-trivial zeros".
That is, there is a "brute-force" procedure that will enumerate all the zeros. </p>
<p>If the RH is false, it is provably false. So if it happens to be undecidable, it must be... |
1,437,287 | <p>On <a href="https://en.wikipedia.org/wiki/Geometric_series#Geometric_power_series" rel="nofollow">Wikipedia</a> it is stated that by differentiating the following formula holds:</p>
<p>$$ \sum_n n q^n = {1\over (1-q)^2}$$</p>
<p>Does this not require a proof? It seems to me because the series is infinite it is not... | Michael Hardy | 11,667 | <p>It does require proof. That the derivative of a sum of finitely many terms is the sum of the derivatives is proved in first-semester calculus, but it doesn't always work for infinite series. For example, let
$$
g_n(x) = \frac{\sin(nx)} {n^2}.
$$
Then $\displaystyle\sum_{n=1}^\infty g_n(x) \vphantom{\dfrac \sum {\d... |
1,686,568 | <p>I am learning about tensor products of modules, but there is a question which makes me very confused about it! </p>
<p>If $E$ is a right $R$-module and $F$ is a left $R$-module, then suppose we have a balanced map (or bilinear map) $E\times F\to E\otimes F$. If some element $x\otimes y \in E\otimes F$ is $0$, then ... | rschwieb | 29,335 | <p>It is not possible to find a nice characterization of when simple tensors are zero.</p>
<p>To give an example where $a, b$ are nonzero but $a\otimes b=0$, consider $\Bbb Z/6\Bbb Z\otimes_\Bbb Z \Bbb Z$ where $2\otimes 3=2\cdot3\otimes 1=0\otimes 1=0$.</p>
|
446,272 | <p>let
$$\left(\dfrac{x}{1-x^2}+\dfrac{3x^3}{1-x^6}+\dfrac{5x^5}{1-x^{10}}+\dfrac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$$</p>
<p>How find the $a_{2^n}=?$</p>
<p>my idea:let
$$\dfrac{nx^n}{1-x^{2n}}=nx^n(1+x^{2n}+x^{4n}+\cdots+x^{2kn}+\cdots)=n\sum_{i=0}^{\infty}x^{(2k+1)n}$$
Thank you everyone</... | Mariano Suárez-Álvarez | 274 | <p>The square of the sum is $$\sum_{u\geq0}\left[\sum_{\substack{n,m,k,l\geq0\\(2n+1)(2k+1)+(2m+1)(2l+1)=u}}(2n+1)(2m+1)\right]x^u.$$</p>
<p>It is easy to use this formula to compute the first coefficients, and we get (starting from $a_1$)
$$0, 1, 0, 8, 0, 28, 0, 64, 0, 126, 0, 224, 0, 344, 0, 512, 0, 757, 0,
1008, 0... |
299,471 | <p>I know this is just $S^2$. To see it, I use the CW structure of $S^1$ x $S^1$ , consisting of one 0-cell, two 1-cells and a 2-cell. Then since the reduced suspension is the cartesian product identifying the wedge (or smash product) , what just remains is the 0.cell and the 2-cell...This is very theoretical and I don... | S. carmeli | 115,052 | <p>Here is a way to see this. After you collapse one circle, you end up with a sphere with two points attached. Then, the remaining circle (1-cell) is the image of the path connecting this two points. If you first glue the two points and then collapse the whole segmnent between them, is the same as collapsing the segme... |
330,508 | <p>Going from $2\cos(2\theta+\pi/3)$ to $\cos2\theta−\sqrt{3}\sin2\theta$ is simple enough, however I'm stuck on going from $2\cos(2\theta+\pi/3)$ to $−2\sin(2\theta−\pi /6)$. How do i do this?</p>
| Brian M. Scott | 12,042 | <p>Use the identities for the sine and cosine of the sum or difference of two angles:</p>
<p>$$\begin{align*}
\sin\left(2\theta-\frac{\pi}6\right)&=\sin2\theta\cos\frac{\pi}6-\cos2\theta\sin\frac{\pi}6\\
&=\frac{\sqrt3}2\sin2\theta-\frac12\cos2\theta\;,
\end{align*}$$</p>
<p>and</p>
<p>$$\begin{align*}
\cos\... |
330,508 | <p>Going from $2\cos(2\theta+\pi/3)$ to $\cos2\theta−\sqrt{3}\sin2\theta$ is simple enough, however I'm stuck on going from $2\cos(2\theta+\pi/3)$ to $−2\sin(2\theta−\pi /6)$. How do i do this?</p>
| Kang Oedin | 848,913 | <p>First, let's recap:<br />
<span class="math-container">$\sin{\pi\over3}=\frac{\sqrt{3}}{2}$</span>
<br /><span class="math-container">$\cos{\pi\over3}=\frac12$</span>
<br /><span class="math-container">$\sin{\pi\over6}=\frac12$</span>
<br /><span class="math-container">$\cos{\pi\over6}=\frac{\sqrt{3}}{2}$</span>
<br... |
2,941,854 | <p>I want to determine all the <span class="math-container">$x$</span> vectors that belong to <span class="math-container">$\mathbb R^3$</span> which have a projection on the <span class="math-container">$xy$</span> plane of <span class="math-container">$w=(1,1,0)$</span> and so that <span class="math-container">$||x||... | trancelocation | 467,003 | <p>I assume you mean vectors <span class="math-container">$v$</span> which have the same projection <span class="math-container">$w = (1,1,0)$</span> onto the <span class="math-container">$xy$</span>-plane:</p>
<p><span class="math-container">$$P_{xy}v = w = (1,1,0)$$</span></p>
<p>So, you look for all
<span class="... |
2,337,524 | <p>We have $p(x)$ a degree $m$ polynomial and $q(x)$ a degree $k$ polynomial. We also know that $p(x) = q(x)$ has at least $n+1$ solutions. And, $n\geq m\land n\geq k$.</p>
<p>Now, I tried graphing a little to see if I see a pattern </p>
<p>I tried making $$y= x^{2} $$
and
$$y = -x^{2}+5 $$
There were two points of... | Robert Israel | 8,508 | <p>$P(x) - Q(x)$ is a polynomial of degree at most $\max(m,k)$, and therefore has at most $\max(m,k)$ roots unless it is the $0$ polynomial, i.e. unless $P = Q$.</p>
|
2,736,323 | <blockquote>
<p>Given that $Y \sim U(2, 5)$ and $Z = 3Y - 4$, what is the distribution for $Z$?</p>
</blockquote>
<p>I've worked out that for $Y \sim N(2, 5)$, $Z \sim N(2, 45)$ since </p>
<p>$$\mu=3\cdot2 - 4 = 2$$</p>
<p>and </p>
<p>$$\sigma^2=3^2 \cdot 5 = 45$$</p>
<p>I'm wondering how the working differs whe... | Maffred | 279,068 | <p>Call $U$ the CDF of $U(2,5)$, it is $U(t) = \frac{1}{3}t - \frac{2}{3}$.</p>
<p>$A(t) = \mathbb P(Z \leq t) = \mathbb P (3Y-4 \leq t) = \mathbb P (Y\leq \frac{t+4}{3}) = U(\frac{t+4}{3}) = \frac{1}{3}[\frac{t+4}{3}] -\frac{2}{3}$ for $2 \leq \frac{t+4}{3} \leq 5$, $0$ elsewhere, i.e. for $2 \leq t \leq 11$. </p>
<... |
2,736,323 | <blockquote>
<p>Given that $Y \sim U(2, 5)$ and $Z = 3Y - 4$, what is the distribution for $Z$?</p>
</blockquote>
<p>I've worked out that for $Y \sim N(2, 5)$, $Z \sim N(2, 45)$ since </p>
<p>$$\mu=3\cdot2 - 4 = 2$$</p>
<p>and </p>
<p>$$\sigma^2=3^2 \cdot 5 = 45$$</p>
<p>I'm wondering how the working differs whe... | Remy | 325,426 | <p>If $Y\sim \mathsf {Unif}(2,5)$ and $Z=3Y-4$ then $Z\sim \mathsf {Unif}(2,11)$.</p>
<p>The transformation stretches the distribution of $Y$ by a factor of $3$ and then shifts it $4$ units to the left. Recalling that </p>
<p>$$F_Y(y) = \mathsf{P}({Y\leq y})=\frac{y-2}{3}$$ for $2< y <5$ we get that for $2< ... |
462,569 | <blockquote>
<p>Consider the polynomial ring <span class="math-container">$F\left[x\right]$</span> over a field <span class="math-container">$F$</span>. Let <span class="math-container">$d$</span> and <span class="math-container">$n$</span> be two nonnegative integers.</p>
<p>Prove：<span class="math-container">$x^d-1 \... | lhf | 589 | <p>Here is another take.</p>
<ul>
<li><p><span class="math-container">$F=\mathbb Q$</span>: Write <span class="math-container">$x^n-1 = (x^d-1)q(x)$</span> with <span class="math-container">$q(x) \in \mathbb Z[x]$</span>, because <span class="math-container">$x^d-1$</span> is monic. Now take derivatives: <span class="... |
186,726 | <p>Just a soft-question that has been bugging me for a long time:</p>
<p>How does one deal with mental fatigue when studying math?</p>
<p>I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my brain starts to get foggy and my mental condition d... | Sasha | 11,069 | <p>This question partially belongs to the sister SE site: <a href="http://productivity.stackexchange.com">productivity.SE</a></p>
<p>To fight the mental fatigue the following things will help:</p>
<ul>
<li>doing physical exercises, as they improve oxygen supply to the brain (e.g. walking, working out, etc)</li>
<li>g... |
186,726 | <p>Just a soft-question that has been bugging me for a long time:</p>
<p>How does one deal with mental fatigue when studying math?</p>
<p>I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my brain starts to get foggy and my mental condition d... | akkkk | 28,826 | <p>I was thinking about a specific mathematician, and after Thomas mentioned him I thought I'd make a comment:</p>
<p>Coffee.</p>
<p>As the very productive mathematician Paul Erdős did not say (it was actually Alfréd Rényi, according to wikipedia):</p>
<blockquote>
<p>A mathematician is a machine for turning coffe... |
1,928,259 | <p>I have the following problem: </p>
<blockquote>
<p>The function $f(x)$ is odd, its period is $5$ and $f(-8) = 1$. What is $f(18)$?</p>
</blockquote>
<p>So, $f(-8) = f(-8 + 5) = 1$. I also know that you could replace $(-8)$ with $(-3)$ and still get the same result of $1$.</p>
<p>I'm just learning about periods.... | sobol | 369,285 | <p>This notation is pretty common in thermodynamics. It just means that the derivative of $f$ is taken with respect to $y$ keeping $x$ constant i.e., it is just a normal partial derivative of $f$ with respect to $y$. The resulting entity is both a function of $x$ and $y$.</p>
<p>$$f=f(x,y) \\
x=x(y,...)$$</p>
<p>$(\f... |
2,061,547 | <p>I am solving for the zeroes of the function:</p>
<blockquote>
<p>$$\frac{\cos(x)(3\cos^2(x)-1)}{(1+\cos^2(x))^2}$$</p>
</blockquote>
<p>The zeroes of the function I found were done by setting $\cos(x)=0$, and $3\cos^2(x)-1=0$</p>
<p>For the $3\cos^2(x)-1=0$ I solved it and got $x=\cos^{-1}(\frac{\sqrt3}{3})$ bu... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> Around any non-zero value $L$, there exists a neighborhood $(L-\epsilon, L + \epsilon)$ small enough so that it contains at most one element of $S$.</p>
|
176,691 | <p>Let $A'$ denotes the complement of A with respect to $ \mathbb{R}$ and $A,B,T$ are subsets of $\mathbb{R}$. I am trying to prove $A' \cap (A' \cup B') \cap T= A' \cap T$, but I got some problems along the way.</p>
<p>$A' \cap (A' \cup B') \cap T= (A' \cap A') \cup (A' \cap B') \cap T= A' \cup (A \cup B) \cap T =(A'... | Jānis Lazovskis | 30,265 | <p>Well, given arbitrary sets $X,Y$ we always have $X\cap(X\cup Y) = X$ since the intersection of a set with a union of that same set and anything else is just the first set itself (I hope that's not too wordy). Try drawing all possibilites with sets to see this. So $A'\cap (A' \cup B') = A'$ and that gives you your an... |
4,646,773 | <p>I started with an integral <span class="math-container">$ \int_{0}^{2\pi} \sqrt{2[\sin^2(t) + 16\cos^2(t) - 4\sin(t)\cos(t)]} \,dt $</span></p>
<p>And I simplified it to <span class="math-container">$ \int_{0}^{2\pi} \sqrt{17 + 15\cos(2t) - 4\sin(2t)} \, dt$</span></p>
<p>My question: I know this can be simplified w... | Astyx | 377,528 | <p>The <span class="math-container">$m$</span>-th term of the <span class="math-container">$n$</span>-th line of that "triangle" is <span class="math-container">$u_{n,m} = \sum_{k=0}^n (m+k){n\choose k} = 2^{n-1}(2m+n)$</span> (you can check this by induction).</p>
<p>Then all that is left is writing <span cl... |
3,106,696 | <p>I am confused on the notation used when writing down the solution of x and y in quadratic equations.
For example in <span class="math-container">$x^2+2x-15=0$</span>, do I write :</p>
<p><span class="math-container">$x=-5$</span> AND <span class="math-container">$x=3$</span></p>
<p>or is it</p>
<p><span class=... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\ (f_n,f_{n-1}) = (\overbrace{f_n-f_{n-1}}^{\Large 2n},\,f_n) = \overbrace{(2n,n(n\!+\!1)\!\color{#c00}{+\!1})}^{\Large 2,n\ \mid\ n(n+1)\ \ \ \ \ \ \ }=1\ $</span> by Euclid. </p>
|
3,106,696 | <p>I am confused on the notation used when writing down the solution of x and y in quadratic equations.
For example in <span class="math-container">$x^2+2x-15=0$</span>, do I write :</p>
<p><span class="math-container">$x=-5$</span> AND <span class="math-container">$x=3$</span></p>
<p>or is it</p>
<p><span class=... | Stefan Lafon | 582,769 | <p>Suppose <span class="math-container">$d$</span> divides both <span class="math-container">$f(n)$</span> and <span class="math-container">$f(n+1)$</span>. Then it divides their difference
<span class="math-container">$$f(n+1)-f(n)=2(n+1)$$</span>
Because, as you said <span class="math-container">$f(n)$</span> is odd,... |
3,604,388 | <p>Let <span class="math-container">$P_n$</span> be the statement that <span class="math-container">$\dfrac{d^{2n}}{dx^{2n}}(x^2-1)^n = (2n)!$</span> </p>
<p>Base case: n = 0, <span class="math-container">$\dfrac{d^0}{dx^0}(x^2-1)^0 = 1 = 0!$</span></p>
<p>Assume <span class="math-container">$P_m = \dfrac{d^m}{dx^m}... | farruhota | 425,072 | <p>Assume:
<span class="math-container">$$P_m = \dfrac{d^{2m}}{dx^{2m}}(x^2-1)^m = (2m)!$$</span></p>
<p>Then:
<span class="math-container">$$P_{m+1}=\dfrac{d^{2m}}{dx^{2m}}\left(\dfrac{d^2}{dx^2}(x^2-1)^{m+1}\right)=
\dfrac{d^{2m}}{dx^{2m}}\left(\dfrac{d}{dx}\left[2x(m+1)(x^2-1)^m\right]\right)=\\
2(m+1)\dfrac{d^{2m}... |
2,548,469 | <p>Suppose there is a sequence of iid variates from $U(0,1)$, $X_1,X_2,\dots$ If we stop the process when $X_n>X_{n+1}$, what is the expected number of generated variates?</p>
<p>I am just thinking about treating it as a Bernoulli process, so that I can use geometric distribution. Is this the right approach?</p>
| Dylan | 135,643 | <p>You can rewrite the integrand as</p>
<p>$$ \frac{(x-1)(x+1)}{(x+1)^2\sqrt{x^2\left(x + \dfrac{1}{x} + 1\right)}} = \frac{x^2 - 1}{(x^3 + 2x^2 + x)\sqrt{x + \dfrac{1}{x} + 1}} \\
= \frac{1 - \dfrac{1}{x^2}}{\left( x + \dfrac{1}{x} + 2 \right)\sqrt{x + \dfrac{1}{x} + 1}} $$</p>
<p>Then make the substitution $u^2 = ... |
848,229 | <p>Two teams take part at a KO-tournament with n rounds. Assuming, that the teams
win all their games until they are paired together, what is the probability
that they both meet in the final ?</p>
<p>I figured out that the solution is </p>
<p>$P_n=\prod_{j=2}^n (1-\frac{1}{2^j-1})=\frac{2^{n-1}}{2^n-1}$</p>
<p>So,... | Kushashwa Ravi Shrimali | 42,058 | <p>$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \large \textbf{Method -1} $
$$
\begin{align}
I & = \sin (bx) \left( \cfrac{1}{a} e^{ax} \right) - {\int} \left( b\cos (bx) \left(\cfrac{e^{ax}}{a} \right) \right) \\
& = \sin (bx) \left( \cfrac{1}{a} e^{ax} \right) - \cfrac{b}{a} \left[ (\cos (bx)) ... |
848,229 | <p>Two teams take part at a KO-tournament with n rounds. Assuming, that the teams
win all their games until they are paired together, what is the probability
that they both meet in the final ?</p>
<p>I figured out that the solution is </p>
<p>$P_n=\prod_{j=2}^n (1-\frac{1}{2^j-1})=\frac{2^{n-1}}{2^n-1}$</p>
<p>So,... | Kushashwa Ravi Shrimali | 42,058 | <p>$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \large{\textbf{Method - 2}}$</p>
<p>Let : $I = \int e^{ax} \sin (bx) $
And say :
$$y_1 = e^{ax} \sin (bx) \\
y_2 = e^{ax} \cos (bx) $$</p>
<p>And : $$y'_1 = ae^{ax}\sin (bx) + e^{ax}b\cos (bx) \\
y'_2 = -e^{ax} b\sin (bx) + ae^{ax} \cos (bx) $$
Or we can ... |
848,229 | <p>Two teams take part at a KO-tournament with n rounds. Assuming, that the teams
win all their games until they are paired together, what is the probability
that they both meet in the final ?</p>
<p>I figured out that the solution is </p>
<p>$P_n=\prod_{j=2}^n (1-\frac{1}{2^j-1})=\frac{2^{n-1}}{2^n-1}$</p>
<p>So,... | user111187 | 111,187 | <p><strong>Method 3</strong></p>
<p>We have $$\int e^{ax} e^{ibx} dx = \frac{e^{ax} e^{ibx}}{a+ib}.$$
Taking the imaginary part yields
$$ \int e^{ax} \sin{bx} dx = \frac{e^{ax}}{a^2+b^2}(a \sin bx - b \cos bx).$$
By taking the real part we also get for free
$$ \int e^{ax} \cos{bx} dx = \frac{e^{ax}}{a^2+b^2}(a \cos bx... |
77,311 | <p>I am a first-time user pf <em>Mathematica</em> (V10). I know it's easy to install palettes, but uninstalling them drives me crazy. I want to delete one. Who can help me to do that? </p>
| Mike Honeychurch | 77 | <p>Here is a bare bones tool to remove a palette and place it in a new directory. You can modify to delete the file entirely if you wish. You can modify the sources. There is an internal FE command to update the palette menu but I do not have that. You'll have to restart Mathematica.</p>
<pre><code>DynamicModule[{new,... |
211,175 | <p>In Gradshteyn and Ryzhik, (specifically starting with the section 3.13) there are several results involving integrals of polynomials inside square root. These are given in terms of combinations of elliptic integrals. See for instance: <a href="https://i.stack.imgur.com/6Cqyb.jpg" rel="nofollow noreferrer"><img src=... | J. M.'s persistent exhaustion | 50 | <p><strong>Update</strong></p>
<p>The Carlson integrals (e.g. <a href="https://reference.wolfram.com/language/ref/CarlsonRF.html" rel="nofollow noreferrer"><code>CarlsonRF[]</code></a>) are now built-in, as of version 12.3; this supersedes the functionality of my old package.</p>
<hr />
<p>Since TheDoctor mentioned <a ... |
2,650,628 | <p>The equation $\log_e(x) + \log_e(1+x) =0$ can be written as:</p>
<p>a) $x^2+x-e=0$</p>
<p>b) $x^2+x-1=0$</p>
<p>c) $x^2+x+1=0$</p>
<p>d) $x^2+xe-e=0$</p>
<p>I tried differentiating both sides, then it becomes $\frac{1}{x}+\frac{1}{1+x}=0$, but I dont get any of the answers.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>then you will get $$\ln(x)+\ln(1+x)=\ln(1)$$ or $$x(x+1)=1$$
can you finish?</p>
|
3,079,929 | <p>Find all positive triples of positive integers a, b, c so that <span class="math-container">$\frac {a+1}{b}$</span> , <span class="math-container">$\frac {b+1}{c}$</span>, <span class="math-container">$\frac {c+1}{a}$</span> are also integers. </p>
<p>WLOG, let a<span class="math-container">$\leqq b\leqq c$</span>... | Hagen von Eitzen | 39,174 | <p>If any two of <span class="math-container">$a,b,c$</span> are equal, then wlog. <span class="math-container">$a=b$</span>. As <span class="math-container">$\frac{b+1}{a}=1+\frac1a$</span> is an integer, we conclude <span class="math-container">$a=b=1$</span>. The remaining conditions are that <span class="math-conta... |
941,709 | <p><strong>Question:</strong> Let $X$ be any set with at least two elements. Assume that the only open
subsets of $X$ are the empty set $\emptyset$ and $X$ itself.
- Which subsets of $X$ are closed?
- Which subsets of $X$ are compact?</p>
<p><strong>My thoughts:</strong> Thus also $\emptyset$ and $X$ have to be also c... | Mike Earnest | 177,399 | <p>For a set to be $not$ compact, it needs to have an open cover without any finite subcovers. Are there any sets which have open covers which can't be reduced to finite ones? Think about what open covers must look like in this topology.</p>
|
586,112 | <p>Consider following statment $\{\{\varnothing\}\} \subset \{\{\varnothing\},\{\varnothing\}\}$</p>
<p>I think above statement is false as $\{\{\varnothing\}\}$ is subset of $\{\{\varnothing\},\{\varnothing\}\}$ but to be proper subset there must be some element in $\{\{\varnothing\},\{\varnothing\}\}$ which is not i... | BCLC | 140,308 | <p>I can't believe I haven't found a single answer that says this, but:</p>
<p><strong>This is exactly what the fundamental theorem of calculus is about.</strong></p>
<ol>
<li><p>An integral is an area function. An integral of function <span class="math-container">$f:\mathbb R \to \mathbb R$</span> (usually assumed co... |
1,839,693 | <p>I think it must be true. Yet I have no rigorous proof for that. So, what I need to prove is that "group being simple" is invariant under isomorphism.</p>
<p>That if $G \cong H$, then either both are simple groups, or both are not simple.</p>
| Sidharth Ghoshal | 58,294 | <p>Recall a group is simple if it doesn't have a normal subgroup other than itself and the identity group, or put another way "no non trivial normal subgroup".</p>
<p>So a natural proof by contradiction arises.</p>
<p>Since $G \cong H$ we can consider an isomorphism $\pi: H \rightarrow G$.</p>
<p>Suppose now that $H... |
17,455 | <p>Assume $S_1$ and $S_2$ are two $n \times n$ (positive definite if that helps) matrices, $c_1$ and $c_2$ are two variables taking scalar form, and $u_1$ and $u_2$ are two $n \times 1$ vectors. In addition, $c_1+c_2=1$, but in the more general case of $m$ $S$'s, $u$'s, and $c$'s, the $c$'s also sum to 1.</p>
<p>What ... | Jonas Meyer | 1,424 | <p>The condition that $S_1$ and $S_2$ are positive definite is relevant to the existence of the inverse in the definition of the function. I assume that it is taken as given that the inverse exists at the relevant values of $c_1$ and $c_2$. This would be true in particular if $c_1$ and $c_2$ were positive.</p>
<p>By... |
4,385,209 | <p>Let's start by generalizing the concept of a metric space. An <span class="math-container">$S$</span>-metric space is a set <span class="math-container">$X$</span> with a function <span class="math-container">$d : X \times X \to S$</span> such that</p>
<ul>
<li><span class="math-container">$d(x,y) = 0 \iff x = y$</s... | Paul Frost | 349,785 | <p>We understand that <span class="math-container">$\mathbb R$</span> and <span class="math-container">$\mathbb Q$</span> are both endowed with the standard metric <span class="math-container">$d(x,y) = \lvert x - y \rvert$</span>. These metrics induce the usual topologies on <span class="math-container">$\mathbb R$</s... |
259,808 | <p>For example, suppose I wanted to determine which of the following has the fastest asymptotic growth:</p>
<ol>
<li><p>$n^2\log(n)+(\log(n))^2$</p></li>
<li><p>$n^2+\log(2^n)+1$</p></li>
<li><p>$(n+1)^3+(n-1)^3$</p></li>
<li><p>$(n+\log(n))^22^{100}$</p></li>
</ol>
<p>Are there any straightforward methods to tell wh... | Samir | 1,057,918 | <p>'A' does a certain task in a time 'time' n3+1100 where n is the number of element processed algorithnm 'B' does the same task in a 'time' of 60n2+6000 if any would less efficient algorithm excute more quickly than the more efficient algorithm</p>
|
4,076,033 | <p>I know how to check if a vector or a matrix is linearly dependent or independent , but how do I apply it on this problem?</p>
<p>Let V1 , V2 , V3 be vectors
How do I prove that the vector V3 = ( 2, 5, -5) is linearly dependent on V1 = ( 1,-2,3) and V2 = (4,1,1) ?</p>
<p>Will it be enough or correct if I solved the e... | pmun | 468,438 | <p>To solve this you can approach with basic methodology:</p>
<p>Consider the equation <span class="math-container">$c_1(1,-2,3)+c_2(4,1,1)=(2,5,-5).$</span> Then you will have system of linear equations:</p>
<p><span class="math-container">$c_1+4c_2=2, -2c_1+c_2=5, 3c_1+c_2=-5$</span>.</p>
<p>Finding that whether <spa... |
2,172,836 | <p>I'm writing a small java program which calculates all possible knight's tours with the knight starting on a random field on a 5x5 board.</p>
<p>It works well, however, the program doesn't calculate any closed knight's tours which makes me wonder. Is there an error in the code, or are there simply no closed knight's... | TonyK | 1,508 | <p>No closed knight's tour is possible on a board with an odd number of squares, because each move changes the colour of the knight's square. So after an odd number of moves, you can't be back at the starting square, because it's the wrong colour.</p>
|
4,821 | <p>A quick bit of motivation: recently a question I answered quite a while ago ( <a href="https://math.stackexchange.com/questions/22437/combining-two-3d-rotations/178957">Combining Two 3D Rotations</a> ) picked up another (IMHO rather poor) answer. While it was downvoted by someone else and I strongly concur with the... | André Nicolas | 6,312 | <p>I believe that if one has a "competing" answer, then the task of dealing with a conspicuously weak answer should in general be left to others.</p>
<p>If the question is quite old, so that a new very weak answer is unlikely to get scrutiny, I think one should wait a while, and then perhaps leave a gentle comment.</p... |
252,272 | <p>I'm working with trace of matrices. Trace is defined for square matrix and there are some useful rule, i.e. <span class="math-container">$\text{tr}(AB) = \text{tr}(BA)$</span>, with <span class="math-container">$A$</span> and <span class="math-container">$B$</span> square, and more in general trace is invariant unde... | joriki | 6,622 | <p>Yes, the cyclic invariance holds irrespective of the dimensions of the matrices. The trace of a product in either order is simply the sum of all products of corresponding entries.</p>
|
966,835 | <p>I want to find the asymptotic complexity of the function:</p>
<p>$$g(n)=n^6-9n^5 \log^2 n-16-5n^3$$</p>
<p>That's what I have tried:</p>
<p>$$n^6-9n^5 \log^2 n-16-5n^3 \geq n^6-9n^5 \sqrt{n}-16n^5 \sqrt{n}-5 n^5 \sqrt{n}=n^6-30n^5 \sqrt{n}=n^6-30n^{\frac{11}{2}} \geq c_1n^6 \Rightarrow (1-c_1)n^6 \geq 30n^{\frac{... | Exodd | 161,426 | <p>there's a faster way: if </p>
<p>$$
\lim_{x\to \infty}\frac{f(x)}{g(x)}\in\mathbb{R}/\{0\}
$$
then </p>
<p>$$
f(n)=\Theta(g(n))
$$</p>
<p>And this is easy to prove</p>
|
331,859 | <p>I need to find the antiderivative of
$$\int\sin^6x\cos^2x \mathrm{d}x.$$ I tried symbolizing $u$ as squared $\sin$ or $\cos$ but that doesn't work. Also I tried using the identity of $1-\cos^2 x = \sin^2 x$ and again if I symbolize $t = \sin^2 x$ I'm stuck with its derivative in the $dt$.</p>
<p>Can I be given a h... | Mikasa | 8,581 | <p>Just adding a good point, however, you got the answer completely $\ddot\smile$ :</p>
<blockquote>
<p>Consider $$\int\sin^m(x)\cos^n(x)dx$$ where in $m,n\in\mathbb Q$. Whenever $m+n$ is an even integer, you can use $t=\tan(x)$ or $t=\cos(x)$ as a good substitution.</p>
</blockquote>
<p>And here $m+n=8$ is an even... |
3,108,847 | <p>I am trying to prove that
If <span class="math-container">$z\in \mathbb{C}-\mathbb{R}$</span> such that <span class="math-container">$\frac{z^2+z+1}{z^2-z+1}\in \mathbb{R}$</span>. Show that <span class="math-container">$|z|=1$</span>.</p>
<p>1 method , through which I approached this problem is to assume <span cla... | David K | 139,123 | <p>We are given that the imaginary part of <span class="math-container">$\frac{z^2+z+1}{z^2-z+1}$</span> is zero.
Therefore <span class="math-container">$\frac{z^2+z+1}{z^2-z+1}$</span> is equal to its own conjugate:
<span class="math-container">$$\frac{z^2+z+1}{z^2-z+1} = \frac{\bar z^2+\bar z+1}{\bar z^2-\bar z+1}.$$... |
1,274,514 | <p>I want to show that proposition<span class="math-container">$5.33$</span> in introduction to homological algebra Rotman :let <span class="math-container">$I$</span> be a directed set , and let <span class="math-container">$\{A_i,\alpha_j^i\}$</span>, <span class="math-container">$\{B_i,\beta_j^i\}$</span>, and <span... | Daniel Valenzuela | 156,302 | <p><em>Hint:</em> $sr: \{A_i,\alpha^i_j\} \to \{B_i,\beta^i_j\}$ is the zero morphism of direct systems.</p>
|
2,343,993 | <blockquote>
<p>Find the limit -$$\left(\frac{n}{n+5}\right)^n$$</p>
</blockquote>
<p>I set it up all the way to $\dfrac{\left(\dfrac{n+5}{n}\right)}{-\dfrac{1}{n^2}}$ but now I am stuck and do not know what to do.</p>
| bloomers | 432,669 | <p>Hint: $\frac{n}{n+5} = 1 + \frac{-5}{n+5} $</p>
|
2,343,993 | <blockquote>
<p>Find the limit -$$\left(\frac{n}{n+5}\right)^n$$</p>
</blockquote>
<p>I set it up all the way to $\dfrac{\left(\dfrac{n+5}{n}\right)}{-\dfrac{1}{n^2}}$ but now I am stuck and do not know what to do.</p>
| farruhota | 425,072 | <p>Alternatively:
$$\lim_\limits{n\to\infty} \left(\frac{n}{n+5}\right)^n=\lim_\limits{n\to\infty} \frac{1}{\left(1+\frac{5}{n}\right)^n}=$$
$$\lim_\limits{n\to\infty} \frac{1}{\left(\underbrace{\left(1+\frac{1}{\frac{n}{5}}\right)^{\frac{n}{5}}}_{=e}\right)^5}=\frac{1}{e^5}.$$</p>
|
2,343,993 | <blockquote>
<p>Find the limit -$$\left(\frac{n}{n+5}\right)^n$$</p>
</blockquote>
<p>I set it up all the way to $\dfrac{\left(\dfrac{n+5}{n}\right)}{-\dfrac{1}{n^2}}$ but now I am stuck and do not know what to do.</p>
| Paramanand Singh | 72,031 | <p>Let us assume the fundamental limit $$\lim_{n\to\infty} \left(\frac{n+1}{n}\right)^{n}=\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^{n}=e\tag{1} $$ Taking reciprocals we get $$\lim_{n\to\infty} \left(\frac{n} {n+1}\right)^{n}=\frac{1}{e}\tag{2}$$ And note that the above limit holds if $n$ is replaced by $n+k$ where ... |
3,283,724 | <p>Let A, B ⊆ R, and let f : A → B be a bijective function. Show that if <span class="math-container">$f$</span> is strictly increasing on A, then <span class="math-container">$f^{-1}$</span> is strictly increasing on B.</p>
<p>How would I write this proof? I think by contradiction but I don't know where to start.</p... | drhab | 75,923 | <p>Let <span class="math-container">$x,y\in B$</span> with <span class="math-container">$x<y$</span>.</p>
<p>Now find <span class="math-container">$u,v\in A$</span> with <span class="math-container">$x=f(u)$</span> and <span class="math-container">$y=f(v)$</span> (possible because <span class="math-container">$f$</... |
791,372 | <p>Hi I am trying to solve this double integral
$$
I:=\int_0^\infty \int_0^\infty \frac{\log x \log y}{\sqrt {xy}}\cos(x+y)\,dx\,dy=(\gamma+2\log 2)\pi^2.
$$
Thank you.</p>
<p>The constant in the result is given by $\gamma\approx .577$, and is known as the Euler-Mascheroni constant. I was thinking to write
$$
I=\Re \... | Ron Gordon | 53,268 | <p>Consider</p>
<p>$$\int_0^{\infty} dx \, x^{\alpha} e^{i x}$$</p>
<p>We know from Cauchy's theorem that this integral is equal to (when it converges)</p>
<p>$$i \, e^{i \pi \alpha/2} \int_0^{\infty} du \, u^{\alpha} \, e^{-u} = i \, e^{i \pi \alpha/2} \, \Gamma(\alpha+1)$$</p>
<p>Differentiating both sides with r... |
1,823,187 | <blockquote>
<p>There are $n \gt 0$ different cells and $n+2$ different balls. Each cell cannot be empty. How many ways can we put those balls into those cells?</p>
</blockquote>
<p>My solution:</p>
<p>Let's start with putting one different ball to each cell.
for the first cell there are $n+2$ options to choose a b... | Sangchul Lee | 9,340 | <p>Define $f$ by</p>
<p>$$ f(x) = \mathrm{e}^{-x} + \sum_{n=2}^{\infty} \max\{0, n - n^4|x - n|\}. $$</p>
<p>It is easy to check that</p>
<p>$$ \int_{1}^{\infty} f(x) \, \mathrm{d}x = \int_{1}^{\infty} \mathrm{e}^{-x} \, \mathrm{d}x + \sum_{n=2}^{\infty} \frac{1}{n^2} < \infty. $$</p>
<p>On the other hand,</p>
... |
2,426,535 | <p>In the book <em>Simmons, George F.</em>, Introduction to topology and modern analysis, page no- 98, question no- 2, the problem is : <strong><em>Let $X$ be a topological space and a $Y$ be metric space and $f:A\subset X\rightarrow Y$ be a continuous map. Then $f$ can be extended in at most one way to a continuous ... | Francesco Polizzi | 456,212 | <p>The following is a well-known result in point-set topology.</p>
<blockquote>
<p><strong>Proposition.</strong> Two continuous functions $f, g \colon X \to Y$ from a topological space $X$ to a Hausdorff space $Y$, that coincide over a dense subset $D \subseteq X$, necessarly coincide everywhere.</p>
</blockquote>
... |
2,426,535 | <p>In the book <em>Simmons, George F.</em>, Introduction to topology and modern analysis, page no- 98, question no- 2, the problem is : <strong><em>Let $X$ be a topological space and a $Y$ be metric space and $f:A\subset X\rightarrow Y$ be a continuous map. Then $f$ can be extended in at most one way to a continuous ... | MSIS | 678,294 | <p>Some caveats: If A is also a metric space, then f must be uniformly continuous, as this preserves Cauchy sequences , while standard continuity does not.</p>
<p>An example, for <span class="math-container">$A =\mathbb Q \subset \mathbb R$</span> , choose <span class="math-container">$f(x):= \frac {1}{ x-\sqrt 3} $</s... |
296,536 | <p>Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit
$$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$
exists in $[0,\infty]$ for any $f$ (That is, I want the limit to exist, but perhaps not be finite.)</p>
<p>Moreover generally I would like to ... | Iosif Pinelis | 36,721 | <p>$\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}{\operatorname{\mathsf P}}$ </p>
<p>In the excellent answer, Willie Wong offered a construction of a... |
537,021 | <p>Say I divide a number by 6, will a number modulus by 6 always be between 0-5? If so, will a number modulus any number (N) , the result should be between $0$ and $ N - 1$?</p>
| mdp | 25,159 | <p>I will assume that by "number", you mean an integer, although it doesn't matter for everything I say.</p>
<p>The answer to your question depends a little on your point of view. I would say that $x\bmod n$ is usually best interpreted as the set of numbers $[x]_n=\{x+kn:k\in\mathbb{Z}\}$. However, there is a unique e... |
1,742,982 | <p>I was trying to solve the equation using factorial as shown below but now I'm stuck at this level and need help.</p>
<p>$$C(n,3) = 2*C(n,2)$$</p>
<p>$$\frac{n!}{3!(n-3)!} = 2\frac{n!}{2!(n-2)!}$$</p>
<p>$$3! (n - 3)! = (n - 2)!$$</p>
| Nick Matteo | 59,435 | <p>Well, the function is always negative if $x<-5$ or $x>3$, is equal to zero when $x$ is $-5$ or $3$, and grows in magnitude without bound as $x$ increases past $3$, so the range includes $(-∞,0]$.</p>
<p>Between $-5$ and $3$ the function is positive (except at $0$), so it's a question of finding the maximum ac... |
2,249,841 | <p>Let $a_n$ denote the number of those permutations $\sigma$ on $\{1,2,3....,n\}$ such that $\sigma$ is a product of exactly two disjoint cycles. Then </p>
<ol>
<li><p>$a_5 = 50$</p></li>
<li><p>$a_4 = 14$</p></li>
<li><p>$a_5 = 40$</p></li>
<li><p>$a_4 = 11$</p></li>
</ol>
<p>I tried specifically for $a_5$ and $a... | Michael Lugo | 173 | <p>There's a well-known formula for the number of permutations with $p_i$ cycles of length $i$ for each $i$, namely </p>
<p>$$ {n! \over \prod_i i^{p_i} (p_i)!}$$</p>
<p>(see for example <a href="http://blog.plover.com/math/fixpoints.html" rel="nofollow noreferrer">this post by Mark Jason Dominus</a>). In the case w... |
51,096 | <p>Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?</p>
| Shai Covo | 2,810 | <p>Let
$$
A_0 = \lbrace 1,3,5,7,9,\ldots \rbrace
$$
and
$$
A_1 = \lbrace 2^n 1 : n \in \mathbb{N} \rbrace,
$$
$$
A_2 = \lbrace 2^n 3 : n \in \mathbb{N} \rbrace,
$$
$$
A_3 = \lbrace 2^n 5 : n \in \mathbb{N} \rbrace,
$$
$$
A_4 = \lbrace 2^n 7 : n \in \mathbb{N} \rbrace,
$$
$$
A_5 = \lbrace 2^n 9 : n \in \mat... |
184,361 | <p>I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:</p>
<blockquote>
<p><strong>DEFINITION</strong> Given $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x < z+1$$ and we d... | Brian M. Scott | 12,042 | <p>Let $n=\lfloor x\rfloor$, and let $\alpha=x-n$; clearly either $0\le\alpha<\frac12$, or $\frac12\le\alpha<1$. Then </p>
<p>$$\lfloor 2x\rfloor=\lfloor 2n+2\alpha\rfloor=2n+\lfloor 2\alpha\rfloor=\begin{cases}
2n,&\text{if }0\le\alpha<\frac12\\
2n+1,&\text{if }\frac12\le\alpha<1\;,
\end{cases}$$<... |
2,698,553 | <p>Is the natural ring morphism $\mathbb{C}\otimes_{\mathbb{Z}}\mathbb{C}\to\mathbb{C}\otimes_{\mathbb{Q}}\mathbb{C}$ an isomorphism?</p>
<p>In other words, is there a $\mathbb Z$-linear map $f:\mathbb{C}\otimes_{\mathbb{Q}}\mathbb{C}\to\mathbb{C}\otimes_{\mathbb{Z}}\mathbb{C}$ such that
$$
f(z\otimes w)=z\otimes w
$... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p> $$\mathbf C\otimes_{\mathbf Z }\mathbf C\simeq(\mathbf C\otimes_{\mathbf Z}\mathbf Q)\otimes_{\mathbf Q}\mathbf C. $$
Now $\;\mathbf C\otimes_{\mathbf Z}\mathbf Q\simeq \mathbf C$ because of the universal property of rings of fractions.</p>
|
2,698,553 | <p>Is the natural ring morphism $\mathbb{C}\otimes_{\mathbb{Z}}\mathbb{C}\to\mathbb{C}\otimes_{\mathbb{Q}}\mathbb{C}$ an isomorphism?</p>
<p>In other words, is there a $\mathbb Z$-linear map $f:\mathbb{C}\otimes_{\mathbb{Q}}\mathbb{C}\to\mathbb{C}\otimes_{\mathbb{Z}}\mathbb{C}$ such that
$$
f(z\otimes w)=z\otimes w
$... | Pierre-Yves Gaillard | 660 | <p>In this post $U,V,X$ and $Y$ are $\mathbb Q$-vector spaces, $a$ is an integer, $b$ is a nonzero integer, and $u,v$ and $x$ are vectors of $U,V$ and $X$ respectively.</p>
<p>$(\star)$ A $\mathbb Z$-linear map $f:X\to Y$ is automatically $\mathbb Q$-linear.</p>
<p>Proof of $(\star)$: We have
$$
f\left(\frac abx\rig... |
756,735 | <blockquote>
<p>Let $n>0$ be a positive integer. For all $x\not=0$, prove that $f(x) = 1/x^n$ is differentiable at $x$ with $f^\prime(x) = -n/x^{n+1}$ by showing that the limit of the difference quotient exists.</p>
</blockquote>
<p>I am having trouble seeing how I can manipulate the difference quotient in order ... | Angel | 109,318 | <p><span class="math-container">$$\lim_{h\to0}\frac{\frac{1}{(x+h)^n}-\frac{1}{x^n}}{h}=\lim_{h\to0}\frac{\frac{1}{x^n(1+\frac{h}{x})^n}-\frac{1}{x^n}}{h}=\lim_{h\to0}\frac{1}{x^n}\frac{\frac{1}{(1+\frac{h}{x})^n}-1}{h}=\lim_{h\to0}\frac{1}{x^{n+1}}\frac{\frac{1}{(1+\frac{h}{x})^n}-1}{\frac{h}{x}}.$$</span> Now let <sp... |
2,666,425 | <p>I'm working through Vakil's excellent The Rising Sea notes, and in an exercise, the following question is posed:</p>
<p>If $X$ is a topological space, show that the fibered product always exists in the category of open sets of $X$, by describing what a fibered product is. </p>
<p>Now I know intuitively the fibered... | Angina Seng | 436,618 | <p>The category of open sets is a poset; that is there's at most one
morphism between objects; here there's a morphism $U\to V$ iff
$U$ is an open subset of the open subset $V$. In general if one
has arrows $U\to W$ and $V\to W$ in a poset, then they have a pullback
(fibre product) iff $U$ and $V$ have a least upper bo... |
2,688,829 | <p>I was wondering how to find the growth rate of the function defined by the number of ways to partition $2^n$ as powers of 2. After a search through OEIS I came across <a href="https://oeis.org/A002577" rel="nofollow noreferrer">OEIS A002577</a> which is what I'm looking for. I can't seem to find any link to asymptot... | Parcly Taxel | 357,390 | <p>Go a bit deeper. <a href="https://oeis.org/A000123" rel="nofollow noreferrer">OEIS A000123</a> gives the number of ways of partitioning $2n$ (multiplication, not exponentiation) into powers of two. In the formulas section there is this from Philippe Flajolet (typesetting and expansion of abbreviations is mine):</p>
... |
3,337,475 | <p>This is definitely the most difficult integral that I've ever seen.
Of course, I'm not able to solve this.
Could you help me?</p>
<p><span class="math-container">$$\int { \sin { x\cos { x } \cosh { \left( \ln { \sqrt { \frac { 1 }{ 1-\sin { x } } } +\tanh ^{ -1 }{ \left( \sin x \right) +\tanh ^{ -1 }{ \left( \co... | David G. Stork | 210,401 | <p><em>Mathematica</em> gives:</p>
<p><span class="math-container">$$-\frac{\sqrt{\frac{1}{1-\sin (x)}} \sqrt{\sin ^2(2 x)} \csc^2(x) \\ \left(-90 \sin \left(\frac{x}{2}\right)+35 \sin \left(\frac{3 x}{2}\right)-3 \sin \left(\frac{5 x}{2}\right)+15 \cos \left(\frac{3 x}{2}\right)+3 \cos \left(\frac{5 x}{2}\right)+30 \... |
543,938 | <p>Can anyone share a link to proof of this?
$${{p-1}\choose{j}}\equiv(-1)^j(\text{mod}\ p)$$ for prime $p$.</p>
| sanjshakun | 73,938 | <p>Definition of $a \equiv b \pmod{c}$ requires $a,b,c$ to be integers. (See David Burton's Elementary Number Theory for a definition and a similar problem.) Here is a way to do it. $$(p-1)(p-2)\ldots(p-j) \equiv (-1)^j j! \pmod{p}.$$ Therefore, $$\binom{p-1}{j} j! \equiv (-1)^j j! \pmod{p}.$$ Now we can "cancel" $j!$ ... |
524,686 | <p>Let X be a uniform random variable on [0,1], and let $Y=\tan\left (\pi \left(x-\frac{1}{2}\right)\right)$. Calculate E(Y) if it exists. </p>
<p>After doing some research into this problem, I have discovered that Y has a Cauchy distribution (although I do not know how to prove this); therefore, E(Y) does not exist.<... | drhab | 75,923 | <p>$-\pi^{-1}\ln\cos\pi\left(x-\frac{1}{2}\right)$ serves as primitive
of $\tan\pi\left(x-\frac{1}{2}\right)$ on $\left(0,1\right)$</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.