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 |
|---|---|---|---|---|
2,792,770 | <p>I found the following question in a test paper:</p>
<blockquote>
<p>Suppose $G$ is a monoid or a semigroup. $a\in G$ and $a^2=a$. What can we say
about $a$?</p>
</blockquote>
<p>Monoids are associative and have an identity element. Semigroups are just associative. </p>
<p>I'm not sure what we can say about $a... | drhab | 75,923 | <p>Let $\Omega=\Omega'$ and let $\mathcal F$ be a proper subcollection of $\mathcal F'$.</p>
<p>Then the identity function $\Omega\to\Omega'$ is not measurable.</p>
|
3,169,668 | <p>If A and B don't commute are there counterexamples that AB is diagonalizable but BA not?</p>
<p>I read that if AB=BA then both AB and BA are diagonalizable. </p>
| egreg | 62,967 | <p>This is true in the special case when both <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are invertible.</p>
<p>If we denote by <span class="math-container">$E_C(\lambda)$</span> the eigenspace of the matrix <span class="math-container">$C$</span> relative to the eigenvalue <sp... |
2,285,202 | <p>So I want to find a field extension that has the galois group $Z_{3} \times Z_{3} \times Z_{3} $. Now if the 3's where changed to 2's then I guess for example $(x^2-2)(x^2-3)(x^2-5)$ would suffice but I don't see how any clever way to do it with $Z_{3}$. I tried a bit with cyclotomic extensions but came up empty han... | Angina Seng | 436,618 | <p>If we take three primes $p\equiv1\pmod3$ then each cyclotomic
field $\Bbb Q(\zeta_p)$ has a cyclic cubic subextension (generated by
a Gaussian period). Take the compositum of these three fields.</p>
<p>To be specific, take $p=7$, $13$ and $19$. Then we can take $\Bbb Q(\gamma_7,\gamma_{13},\gamma_{19})$ where
$$\ga... |
1,929,698 | <p>Let $f(x)=\chi_{[a,b]}(x)$ be the characteristic function of the interval $[a,b]\subset [-\pi,\pi]$. </p>
<p>Show that if $a\neq -\pi$, or $b\neq \pi$ and $a\neq b$, then the Fourier series does not converge absolutely for any $x$. [Hint: It suffices to prove that for many values of $n$ one has $|\sin n\theta_0|\ge... | fonini | 113,664 | <p>First of all, I like to write the Fourier coefficients of this function as
<span class="math-container">$$\widehat{\chi_{\left[a,b\right]}}\left(n\right)=\frac{b-a}{2\pi}\exp\left(-i\frac{a+b}{2}n\right)\mathrm{sinc}\left(\frac{b-a}{2\pi}n\right)$$</span> where <span class="math-container">$$\mathrm{sinc}\,\eta
= \f... |
2,012,223 | <p>I've got a problem with finding main argument of these complex number. How can i evaluate this two examples?</p>
<p>$$\sin \theta - i\cos \theta$$</p>
<p>$$\frac{(1-i\tan \theta)}{1+\tan \theta}$$</p>
| robjohn | 13,854 | <p>Since $\arg(zw)=\arg(z)+\arg(w)$,
$$
\begin{align}
\arg(\sin(\theta)-i\cos(\theta))
&=\arg(-i)+\arg(\cos(\theta)+i\sin(\theta))\\
&=\theta-\frac\pi2
\end{align}
$$
and
$$
\begin{align}
\arg\left(\frac{1-i\tan(\theta)}{1+\tan{\theta}}\right)
&=\arg(\cos(-\theta)+i\sin(-\theta))-\arg(\sin(\theta)+\cos(\the... |
2,034,523 | <p>I am sure there is a general and simplified way to solve this problem, I am just unable to figure out the generalized formula (if there is one). </p>
<p>Say we have to write a <strong>code with 4 digits</strong>, the digits can range from <strong>0</strong> to <strong>9</strong>. </p>
<p>All digits in the code <... | Robert Z | 299,698 | <p>Hint. Once you choose $4$ distinct digits in $\{0,1,\dots,9\}$ (you can do it in $\binom{10}{4}$ ways) in how many ways can you arrange them in decreasing order? How many in increasing order? Now take them away from the $4!$ possible permutations.</p>
|
2,369,081 | <blockquote>
<p>Evaluate the integral $$\int_0^1\frac{x^7-1}{\log (x)}\,dx $$</p>
</blockquote>
<p>[1]: <a href="https://i.stack.imgur.com/lcK2p.jpgplz" rel="nofollow noreferrer">https://i.stack.imgur.com/lcK2p.jpgplz</a> I'm trying to solve this definite integral since 2 hours. Please, I need help on this.</p>
| hamam_Abdallah | 369,188 | <p><strong>hint for the convergence</strong></p>
<p>$$f : x\mapsto \frac {x^7-1}{\log (x)} $$
is continuous at $(0,1) $ , and is locally integrable.</p>
<p>Near $0$, $f $ is bounded since $\lim_{0^+}f (x)=0$ thus $$\int_0^\frac 12 f (x)dx $$ is convergent.</p>
<p>Near $1$, $$\log (x)\sim x-1$$ and
$$\lim_{1^-}f (x)=... |
2,369,081 | <blockquote>
<p>Evaluate the integral $$\int_0^1\frac{x^7-1}{\log (x)}\,dx $$</p>
</blockquote>
<p>[1]: <a href="https://i.stack.imgur.com/lcK2p.jpgplz" rel="nofollow noreferrer">https://i.stack.imgur.com/lcK2p.jpgplz</a> I'm trying to solve this definite integral since 2 hours. Please, I need help on this.</p>
| robjohn | 13,854 | <p>This can be written as a <a href="http://mathworld.wolfram.com/FrullanisIntegral.html" rel="nofollow noreferrer">Frullani Integral</a>:
$$
\begin{align}
\int_0^1\frac{x^7-1}{\log(x)}\,\mathrm{d}x
&=\int_0^\infty\frac{e^{-u}-e^{-8u}}{u}\,\mathrm{d}x\tag{1}\\
&=\lim_{\epsilon\to0}\int_\epsilon^\infty\frac{e^{-... |
3,453,175 | <p>If <span class="math-container">$y=\dfrac {1}{x^x}$</span> then show that <span class="math-container">$y'' (1)=0$</span></p>
<p>My Attempt:</p>
<p><span class="math-container">$$y=\dfrac {1}{x^x}$$</span>
Taking <span class="math-container">$\ln$</span> on both sides,
<span class="math-container">$$\ln (y)= \ln \... | Heatconomics | 531,927 | <p>Let <span class="math-container">$y=\frac{1}{x^x}$</span>, then <span class="math-container">$ln(y)=-xln(x)$</span>, then <span class="math-container">$y'(x)=-(1+ln(x))y(x)$</span>. Taking a second derivative, we have that <span class="math-container">$y''(x)=-y'(x)(1+ln(x))-\frac{y(x)}{x}$</span>. Evaluating at <sp... |
38,552 | <p>The following Cubics have 3 real roots but the first has Galois group $C_3$ and the second $S_3$</p>
<ul>
<li>$x^3 - 3x + 1$ (red)</li>
<li>$x^3 - 4x + 2$ (green)</li>
</ul>
<p>Is there any geometric way to distinguish between the two cases? Obviously graphing this onto the real line does not help.</p>
<p><img sr... | Qiaochu Yuan | 232 | <p>There's no reason to expect that the set of real points tells you the full story in an arithmetic situation. For example, can you tell that $\pi$ is transcendental but that $\sqrt{10}$ isn't from looking at their relative positions on the number line? </p>
<p>One thing you can do which (depending on your tastes) co... |
38,552 | <p>The following Cubics have 3 real roots but the first has Galois group $C_3$ and the second $S_3$</p>
<ul>
<li>$x^3 - 3x + 1$ (red)</li>
<li>$x^3 - 4x + 2$ (green)</li>
</ul>
<p>Is there any geometric way to distinguish between the two cases? Obviously graphing this onto the real line does not help.</p>
<p><img sr... | Gerry Myerson | 8,269 | <p>Almost all cubics (with integer coefficients and three real roots) have Galois group $S_3$. What exactly is meant by "almost all" is a little technical, but the phrase can be made precise, and the result rigorously proved. One consequence is that if you start with a $C_3$ cubic and perturb the roots the tiniest litt... |
4,166,579 | <p>I'm trying to solve this proof but I'm not completely sure how to start. Discrete has been pretty rough for me so far so any help would be greatly appreciated!</p>
| Peter Szilas | 408,605 | <p>Option.</p>
<p>Assume <span class="math-container">$T \subset S$</span> (strict inclusion).</p>
<p>There is a <span class="math-container">$s \in S$</span> and <span class="math-container">$s \not \in T$</span>.</p>
<p><span class="math-container">$f(s)\not \in f(T)$</span> since <span class="math-container">$f$</sp... |
1,703,120 | <p>So I have a vector <span class="math-container">$a =( 2 ,2 )$</span> and a vector <span class="math-container">$b =( 0, 1 )$</span>.<br />
As my teacher told me, <span class="math-container">$ab = (-2, -1 )$</span>.</p>
<p><span class="math-container">$ab = b-a = ( 0, 1 ) - ( 2, 2 ) = ( 0-2, 1-2 ) = ( -2, -1 )$</sp... | Samuel | 433,229 | <p>( Vector AB ) = ( Vector B ) - ( Vector A )</p>
<p>Think of this logically when you have the equation 10 - 2 you get 8 ( a positve value )
However if you do 2 - 10 you get the same magnitude 8 but opposite direction -8.</p>
<p>Use this to understand the vectors since the point of Vector AB is moving from A to B yo... |
2,187,509 | <p>We're currently implementing the IBM Model 1 in my course on statistical machine translation and I'm struggling with the following appplication of the chain rule.
When applying the model to the data, we need to compute the probabilities of different alignments given a sentence pair in the data. In other words to com... | skyking | 265,767 | <p>There's an ad-hoc solution at least going something like this:</p>
<p>You can use taylor expansion to show that (or the convexity of the function). Consider the function $f(x) = x^2e^x-\ln x - 1$ and you have:</p>
<p>$$f'(x) = (x^2+2x)e^x - 1/x$$
$$f''(x) = (x^2+4x+4)e^x + 1/x^2 = (x+2)^2e^x + 1/x^2$$</p>
<p>We s... |
4,312,890 | <p>I'm working my way through Grimaldi's textbook, and there's one exercise in the supplementary exercises for Chapter 4 that I don't understand how to approach.</p>
<p>Here is the problem: if <span class="math-container">$n \in Z^+$</span>, how many possible values are there for <span class="math-container">$gcd(n,n+... | Lai | 732,917 | <p>Denote the greatest common divisor of x and y be (x,y), then
<span class="math-container">$$(n,n+3000)=(n,3000)$$</span>
Since <span class="math-container">$$
3000=2^{3} \times 3 \times 5^{3}
$$</span> Therefore there are <em><strong>at most</strong></em> <span class="math-container">$$(3+1)\times (1+1)\times (3+1)=... |
2,263,230 | <p>Let's say I wanted to express sqrt(4i) in a + bi form. A cursory glance at WolframAlpha tells me it has not just a solution of 2e^(i<em>Pi/4), which I found, but also 2e^(i</em>(-3Pi/4))</p>
<p>Why do roots of unity exist, and why do they exist in this case? How could I find the second solution? </p>
| Adam Hughes | 58,831 | <p>By definition $\zeta\in\Bbb C$ is a root of unity if there is $n\in\Bbb N$ so that $\zeta^n=1$. Roots of unity exist thanks to $e^{2\pi i}=1$ and the usual fact about exponentials that $(e^a)^b=e^{ab}$ so that $e^{2\pi i/n}$ is always an $n^{th}$ root of unity.</p>
<p>To see how you can get them all just note that ... |
279,994 | <p>Note that S [n] is the sum of the first n terms of the sequence a [n]. It is known that a [1]==1, and the sequence {S [n]/a [n]} is an equal difference sequence with a tolerance of 1/3. Find the general term formula of sequence a [n]</p>
<p>Let b [n]==S [n]/a [n], first work out the general term formula of b [n], an... | csn899 | 68,574 | <p>A friend provided me with the answer:</p>
<pre><code>ClearAll["`*"]
sol = First@RSolve[{b[n + 1] == b[n] + 1/3, b[1] == 1}, b[n], n]
s[n_] = b[n] a[n] /. sol
sola = First@
RSolve[{a[n + 1] == s[n + 1] - s[n], a[1] == 1}, a[n], n]
Sum[1/a[n] /. sola, {n, 1, n0}]
</code></pre>
|
2,643,705 | <p>If $A,B,C,D$ are all matrices and $A=BCD$ (with dimensions such that all matrix multiplications are defined), how does one solve for $C$? </p>
<p>In the particular context I'm working in, $B$ and $D$ are both orthogonal, and $C$ is diagonal. I'm not sure if that's necessary to solve for $C$.</p>
| Siong Thye Goh | 306,553 | <p>Hint:</p>
<ul>
<li>$B^TB=I$</li>
<li><p>$DD^T = I$</p></li>
<li><p>You might like to premultiply and postmultiply the equation by some matrices to isolate $C$.</p></li>
</ul>
|
2,643,705 | <p>If $A,B,C,D$ are all matrices and $A=BCD$ (with dimensions such that all matrix multiplications are defined), how does one solve for $C$? </p>
<p>In the particular context I'm working in, $B$ and $D$ are both orthogonal, and $C$ is diagonal. I'm not sure if that's necessary to solve for $C$.</p>
| idok | 514,894 | <p>Since $B, D$ are orthogonal,</p>
<p>$B^t B = D D^t = I$</p>
<p>multiply your equation with $B^t$ from the left and by $D^t$ from the right to get
$$C = B^t A D^t$$</p>
|
3,353,826 | <p>All the vertices of quadrilateral <span class="math-container">$ABCD$</span> are at the circumference of a circle and its diagonals intersect at point <span class="math-container">$O$</span>. If <span class="math-container">$∠CAB = 40°$</span> and <span class="math-container">$∠DBC = 70°$</span>, <span class="math-c... | polettix | 264,102 | <p>If you are fine going always in one direction for halfway values, you can resort to the programming trick of using <span class="math-container">$\lfloor x + \frac{1}{2} \rfloor$</span> (halfways towards <span class="math-container">$+\infty$</span>) or <span class="math-container">$\lceil x - \frac{1}{2} \rceil$</sp... |
693,640 | <p>Assmue $f(x_{1},x_{2},\cdots,x_{n})$ is a second degree real polynomial with $n(n\ge 2)$ variables.
Let $S$ be such that $f(x_{1},x_{2},\cdots,x_{n})$ is the set of minimum and maximum points.
In other words：
$$S=\{(b_{1},b_{2},\cdots,b_{n})\in R^n| f(x_{1},x_{2},\cdots,x_{n})\le f(b_{1},b_{2},\cdots,b_{n}),\forall... | Christian Blatter | 1,303 | <p>Any real symmetric polynomial $f$ of degree $\leq2$ in the variables $x_1$, $\ldots$, $x_n$ can be written in the form
$$f(x_1,\ldots,x_n)=a_0+ a_1 \sigma_1+a_2\sigma_1^2 + a_3\sigma_2\ ,\tag{1}$$
where $\sigma_1$ and $\sigma_2$ denote the elementary symmetric polynomials of degree $1$ and $2$ in the $x_i$. Since $\... |
3,977,281 | <p>Is it possible for a function to not have a maxima or a minima? (S.t. I can't find the decreasing and increasing interval.) If so, how do we show it mathematically?</p>
<p>I was practicing and found these two functions.</p>
<p><span class="math-container">$a. f(x) = x+\sqrt{x^2-1} $</span> and <span class="math-cont... | José Carlos Santos | 446,262 | <p><em>a.</em> As you wrote, <span class="math-container">$f'(x)=1+\frac x{\sqrt{x^2-1}}$</span>. It turns out that <span class="math-container">$f'(x)<0$</span> if <span class="math-container">$x<-1$</span> and that <span class="math-container">$f'(x)>0$</span> if <span class="math-container">$x>1$</span>.... |
3,423,674 | <p>According to my calculus professor and MIT open coursework, taking the derivative of (x^2+4)^-1 is an application of the chain rule, not the power rule. The answer to the question is -(x^2+4)^-2, which makes sense to me, but I just don't understand why this is considered an application of the chain rule rather than ... | herb steinberg | 501,262 | <p>The derivative is <span class="math-container">$-2x(x^2+4)^{-2}$</span>. Thus the chain rule. You missed <span class="math-container">$2x$</span>.</p>
|
24,810 | <p>The title says it all. Is there a way to take a poll on Maths Stack Exchange? Is a poll an acceptable question?</p>
| Asaf Karagila | 622 | <p>You can post a question, with several answers asking people to vote accordingly.</p>
<p>This is relatively acceptable on meta (although a discussion should be had first). If I would see something like that on the main site, I'd immediately downvote, close and then delete (and maybe flag to get the process done even... |
4,090,416 | <blockquote>
<p>Suppose <span class="math-container">$f(x)$</span> be bounded and differentiable over <span class="math-container">$\mathbb R$</span>, and
<span class="math-container">$|f'(x)|<1$</span> for any <span class="math-container">$x$</span>. Prove there exists <span class="math-container">$M<1$</span> s... | Umberto P. | 67,536 | <p>You can assume without loss of generality that <span class="math-container">$f(0) = 0$</span>.</p>
<p>Suppose to the contrary that no such <span class="math-container">$M$</span> exists. Then for each <span class="math-container">$k \ge 2$</span> there exists a point <span class="math-container">$x_k$</span> satisfy... |
1,027,330 | <p>How does one figure out whether this series:
$$\sum_{n=3}^{\infty}(-1)^{n-1}\frac{1}{\ln\ln n}$$
converges or diverges? And, what is the general approach behind solving for convergence/divergence in a series that seems to "oscillate" (thanks to the -1 in this case)? </p>
<p>I have so far tried to split the functio... | Emanuele Paolini | 59,304 | <p>Check that you can apply <a href="http://en.wikipedia.org/wiki/Alternating_series_test" rel="nofollow">Leibniz alternating series test</a></p>
|
1,027,330 | <p>How does one figure out whether this series:
$$\sum_{n=3}^{\infty}(-1)^{n-1}\frac{1}{\ln\ln n}$$
converges or diverges? And, what is the general approach behind solving for convergence/divergence in a series that seems to "oscillate" (thanks to the -1 in this case)? </p>
<p>I have so far tried to split the functio... | user860374 | 137,485 | <p>Let $b_n = \frac{1}{\ln{(\ln{(n)})}}$</p>
<p>Since $\ln(n)$ is increasing, we know $\ln{(\ln{(n)})}$ also increases, thus we have that:</p>
<p>$b_n = \frac{1}{\ln{(\ln{(n)})}}$ is monotonically decreasing on $[2,\infty)$ and also
$$\lim_{n\to \infty}b_n= \lim_{n\to \infty}\frac{1}{\ln{(\ln{(n)})}} = 0.$$
Thus, fro... |
2,563,303 | <blockquote>
<p><strong><em>Question:</em></strong> If $z_0$ and $z_1$ are real irrational numbers I write $$q=z_0+z_1\sqrt{-1}$$ Surely $q$ is just a complex number. Under what condition will the <a href="https://en.wikipedia.org/wiki/Absolute_value#Complex_numbers" rel="nofollow noreferrer">number</a> $|q|$ be an ... | Fred | 380,717 | <p>If $(A_n)$ is a positive sequence with $\lim_{n \to \infty}\frac{A_{n+1}}{A_n}= 0$, then there is $N \in \mathbb N$ such that $\frac{A_{n+1}}{A_n} <1$ for $n \ge N$. This gives</p>
<p>$A_{n+1}<A_N$ for all $n>N$.</p>
<p>Therefore $\lim_{n \to \infty}A_n = \infty$ can not occure.</p>
|
1,004,303 | <p>Let $S=\{(x,0)\} \cup\{(x,1/x):x>0\}$. Prove that $S$ is not a connected space (the topology on $S$ is the subspace topology)</p>
<p>My thoughts: Now in the first set $x$ is any real number, and I can't see that this set in open in $S$. I can't find a suitable intersection anyhow.</p>
| copper.hat | 27,978 | <p>Let $\phi(x) = {1 \over 2x}$. Let $U = \{ (x,y) | x>0, \ y > \phi(x)\}$,
$V = \{ (x,y) | x>0, \ y < \phi(x)\}$. Then $U,V$ are disjoint, open, $S \cap U \neq \emptyset$, $S \cap V \neq \emptyset$ and $S \subset U \cup V$. Hence $S$ is not connected.</p>
|
1,223,209 | <p>As part of another problem I am working on, I have the following product to work out. </p>
<p>$\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \cdot h $</p>
<p>where $h$ is a scalar. My question is, if I commute the row vector and the scalar then I can just multiply it through. If I think of the $h$ as a $1 \times... | Mick A | 153,109 | <p>Your suspicion is right in that you didn't take the conditional probabilities into account. The numbers of new/old balls in each draw of $3$ balls <em>do</em> have hypergeometric conditional distributions. As you seem to have done, I'll define "success" in this distribution as getting an old ball. If $X_k$ is the nu... |
1,817,367 | <blockquote>
<p>Prove that $\forall k = m^2 + 1. \space m \in \mathbb{Z}^+$, if $k$ is divisible by any prime then that prime is congruent to $1, 2 \pmod 4$.</p>
</blockquote>
<p>I am unable to realize why it can't have $2$ prime factors congruent to $3 \pmod 4$. Can anyone please help me proceed?</p>
<p>Thanks.</p... | Ege Erdil | 326,053 | <p>Another solution: Suppose that $ p $ divides $ m^2 + 1 = (m-i)(m+i) $. Since primes that are 3 mod 4 are inert in $ \mathbb{Z}[i] $, and since $ p $ divides neither of the factors on the right, but divides $ m^2 + 1 $, it cannot be prime in $ \mathbb{Z}[i] $, which means it is 1 or 2 modulo 4. (This is because $ \ma... |
1,817,367 | <blockquote>
<p>Prove that $\forall k = m^2 + 1. \space m \in \mathbb{Z}^+$, if $k$ is divisible by any prime then that prime is congruent to $1, 2 \pmod 4$.</p>
</blockquote>
<p>I am unable to realize why it can't have $2$ prime factors congruent to $3 \pmod 4$. Can anyone please help me proceed?</p>
<p>Thanks.</p... | awllower | 6,792 | <p>Another approach that looks similar to the one by @Starfall:<br>
First it is easy to show that $p\not\equiv0\pmod4,$ as every number divisible by $4$ is not a prime.<br>
Then, as stated in <a href="https://math.stackexchange.com/questions/38431/dedekinds-theorem-on-the-factorisation-of-rational-primes">this question... |
1,305,935 | <p>Let $f(n)$ be non-negative real valued function defined for each natural number $n$.</p>
<p>If $f$ is convex and $lim_{n\to\infty}f(n)$ exists as a finite number, then can we conclude that $f$ is non-increasing?</p>
| m.Just | 1,130,811 | <p>The margin equals the shortest distance between the points of the two hyperplanes. Let <span class="math-container">$\mathbf{x_1}$</span> be a point of one hyperplane, and <span class="math-container">$\mathbf{x}_2$</span> be a point of the other hyperplane.
We want to find the minimal value of <span class="math-con... |
3,573,575 | <p>I'm trying to find the eigenvalues of a matrix <span class="math-container">$$A=\begin{bmatrix}2/3 & -1/4 & -1/4 \\ -1/4 & 2/3 & -1/4 \\ -1/4 & -1/4 & 2/3\end{bmatrix}$$</span></p>
<p>The eigenvalues of this matrix, are the roots <span class="math-container">$\lambda$</span> of the equation ... | user757704 | 757,704 | <p>It's equal to <span class="math-container">$- \frac{1}{4}J + \frac{11}{12} I$</span>, where <span class="math-container">$I$</span> is the identity matrix and <span class="math-container">$J$</span> is the matrix of all <span class="math-container">$1$</span>s. Note that <span class="math-container">$J$</span> has t... |
2,060,891 | <p>Number of solutions of $a^3=e$ in $C_9$</p>
<p>The solution goes: $a^3=e$ if and only if $a$ lies in the unique subgroup of $C_9$ of order $3$ thus there are $3$ solutions.</p>
<p>I'm questioning why? </p>
| Sarvesh Ravichandran Iyer | 316,409 | <p>It is a misprint. The answer is $\frac 16$ because the cases are as you have given i.e. $(4,6),(5,5),(6,4),(5,6),(6,5),(6,6)$, and these are six cases out of $36$, giving the probability of $ \frac 16$.</p>
|
4,326,547 | <p>I've solved linear ODEs before. This however is something completely new to me. I want to solve it without using approximations or anything.</p>
<p><span class="math-container">$s''( t) s( t) =( s'( t))^{2} +B( s( t))^{2} s'( t) -g\cdot s( t) s'( t)$</span></p>
<p>These are the equations I started with</p>
<p><spa... | Cesareo | 397,348 | <p>Making <span class="math-container">$y = \int s dt$</span> we have</p>
<p><span class="math-container">$$
\frac{y'''}{y''}=\frac{y''}{y'}+B y'-g
$$</span></p>
<p>then</p>
<p><span class="math-container">$$
\ln\left(\frac{y''}{y'}\right)=B y - g t+C
$$</span></p>
<p>or</p>
<p><span class="math-container">$$
y''=y'C_1... |
3,853,509 | <blockquote>
<p>prove <span class="math-container">$$\sum_{cyc}\frac{a^2}{a+2b^2}\ge 1$$</span> holds for all positives <span class="math-container">$a,b,c$</span> when <span class="math-container">$\sqrt{a}+\sqrt{b}+\sqrt{c}=3$</span> or <span class="math-container">$ab+bc+ca=3$</span></p>
</blockquote>
<hr />
<p><str... | Fred | 380,717 | <p>Hint: <span class="math-container">$f(x)= \frac{1}{2} \cdot \frac{1}{1-(-\frac{3}{2}x^2)}.$</span></p>
<p>Now geometric seies !</p>
|
1,464,143 | <p>$\lim_{n \to \infty} n\ln\left(1+\frac{1}{n}\right)$ using L'Hòpital rule show that this is $1$. Can you do this since there isn't a division and $n$ will obviously tend to infinity and $\ln\left(1+\frac{1}{n}\right)$ will tend to $0$? So there limits aren't matching?</p>
<p>So I set $u=n $</p>
<p>$du=1$</p>
<p>$... | Mark Viola | 218,419 | <p>$$\lim_{n\to \infty}n\log\left(1+\frac1n\right)=\lim_{n\to \infty}\frac{\log\left(1+\frac1n\right)}{1/n}\lim_{n\to \infty}\frac{\left(\frac{1}{1+1/n}\right)(-1/n^2)}{-1/n^2}=1$$</p>
|
302 | <p>I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first found? And what is the...significance?</p>
| Marko Amnell | 2,594 | <p>The book <em>A Mathematical History of the Golden Number</em> by Roger Herz-Fischler is an exhaustive study of nearly all references to the golden ratio, from the earliest times, and <a href="http://ebookee.org/A-Mathematical-History-of-the-Golden-Number_887455.html" rel="nofollow">is available as a free e-book</a>.... |
34,671 | <p>Going through some old papers, I came up with a simple-looking problem I thought about 5 years ago or so. </p>
<p>MO wants motivation ... Associated to a probability measure on a metric space is something called "quantization dimension" ... this involves defining a function $D \colon (0,\infty) \to (0,\infty)$. E... | Gerald Edgar | 454 | <p>My response to the answer by fedja, Jun 5, 2011. This should be a comment, but won't fit. </p>
<p>It didn't work. Taking values of $A,B,\epsilon$ that satisfy your conditions, then tracing back through using 20-digit arithmetic, I get these values: $s_1=0.34018988053902955186$, $s_2=0.98903555253485545775$, $p_... |
4,122,732 | <p>I was solving exercise 3.125 of Wackerly's Probability book and i did not understand the solution given in the solutions manual.</p>
<p>The problem says:</p>
<p>Customers arrive at a shop following a poisson distribution for an average of 7 customers per hour. What is the probability that exactly two clients arrive ... | Vons | 274,987 | <p>Simple answer is that for both problems, you have a span of two hours, so the calculation is identical for both of them. On average 7 customers arrive in 1 hour so 14 customers arrive in 2 hours. Using <span class="math-container">$X\sim\text{Poisson}(14)$</span> to be the number of customer arrivals in a 2 hour tim... |
3,997,632 | <p>Use the Chain Rule to prove the following.<br />
(a) The derivative of an even function is an odd function.<br />
(b) The derivative of an odd function is an even function.</p>
<p><strong>My attempt:</strong></p>
<p>I can easily prove these using the definition of a derivative, but I'm having trouble showing them us... | peek-a-boo | 568,204 | <p><span class="math-container">$f$</span> being even means for every <span class="math-container">$x\in\Bbb{R}$</span>, <span class="math-container">$f(x)=f(-x)$</span>. Or, if we define the function <span class="math-container">$u:\Bbb{R}\to\Bbb{R}$</span> as <span class="math-container">$u(x)=-x$</span>, then the co... |
3,335,060 | <blockquote>
<p>The numbers of possible continuous <span class="math-container">$f(x)$</span> defiend on <span class="math-container">$[0,1]$</span> for which <span class="math-container">$I_1=\int_0^1 f(x)dx = 1,~I_2=\int_0^1 xf(x)dx = a,~I_3=\int_0^1 x^2f(x)dx = a^2 $</span> is/are</p>
<p><span class="math-container"... | J.G. | 56,861 | <p>If you know your powers of <span class="math-container">$3$</span> well, you know <span class="math-container">$2.7^3=19.683$</span>. Since <span class="math-container">$e>2.718=2.7\left(1+\frac{2}{300}\right)$</span>,<span class="math-container">$$e^3>19.683\left(1+\frac{2}{100}\right)=19.683+0.39366>20.$$... |
2,777,555 | <p>Prove that if $f(0)=0$ and $f'(0)=0$, then $f(x)=0$ for all $x$. </p>
<p>Hint: The idea
is to multiply both sides of the equation $f''(x)+ f(x) = 0$ by something that
makes the left-hand side of the equation into the derivative of something.</p>
<p>I'm not sure how to proceed and don't really understand the hint.<... | Mohammad Riazi-Kermani | 514,496 | <p>$$f''(x)+ f(x) = 0 $$</p>
<p>$$ f'(x) f''(x) +f'(x)f(x) =0 $$</p>
<p>$$ (1/2)(f^2 + f'^2 )' =0$$</p>
<p>$$f^2 + f'^2=C$$</p>
<p>Since $$ (f^2 + f'^2)(0)=0$$</p>
<p>We get $C=0$, that is $f(x)=0$ </p>
|
264,587 | <p><strong>NOTE</strong></p>
<p>I'm sorry, my question was not clear. I want to know all the ways to split a list with a given length simply, <strong>rather than split a cyclic substitution</strong>.
If a given list has length <span class="math-container">$N$</span> and the rule is <span class="math-container">${m, n, ... | kglr | 125 | <pre><code>kSP = ResourceFunction["KSetPartitions"];
partitionLst[a_, p_] := Select[Sort@Map[Length] @ # == Sort @ p &][
DeleteDuplicates @ Sort @ kSP[a, Length @ p]]
partitionLst[{a, b, c, d}, {1, 3}]
</code></pre>
<blockquote>
<pre><code>{{{a}, {b, c, d}}, {{a, b, c}, {d}}, {{a, b, d}, {c}}, {{a,... |
3,981,458 | <p>A star graph <span class="math-container">$S_{k}$</span> is the complete bipartite graph <span class="math-container">$K_{1,k}$</span>. One bipartition contains 1 vertex and the other bipartition contains <span class="math-container">$k$</span> vertices. <a href="https://en.wikipedia.org/wiki/Star_(graph_theory)" re... | ArsenBerk | 505,611 | <p>It is clear that
<span class="math-container">$$\bar{d}(S_k) = \frac{k+k\cdot1}{k+1} = \frac{2k}{k+1} = 2-\frac{2}{k+1}$$</span></p>
<p>Now, in <span class="math-container">$H$</span>, we can consider only two cases:</p>
<p><strong>Case 1:</strong> Center vertex <span class="math-container">$v \notin H$</span> (vert... |
3,981,458 | <p>A star graph <span class="math-container">$S_{k}$</span> is the complete bipartite graph <span class="math-container">$K_{1,k}$</span>. One bipartition contains 1 vertex and the other bipartition contains <span class="math-container">$k$</span> vertices. <a href="https://en.wikipedia.org/wiki/Star_(graph_theory)" re... | Misha Lavrov | 383,078 | <p>All trees are balanced, so in particular stars are balanced.</p>
<p>An <span class="math-container">$n$</span>-vertex tree has average degree <span class="math-container">$2-\frac 2n$</span>. Any subgraph of average degree <span class="math-container">$2$</span> can be reduced to a subgraph of minimum degree <span c... |
838,400 | <p>One question asking if $\mathbb{Z}^*_{21}$ is cyclic.</p>
<p>I know that the cyclic group must have a generator which can generate all of the elements within the group.</p>
<p>But does this kind of question requires me to exhaustively find out a generator? Or is there any more efficient method to quickly determine... | Alex Jordan | 157,500 | <p>I believe this is the same as what KCd said, but I will be more specific. $Z^*_{21}$ contains two subgroups of order 2, namely $<8>$ and $<13>$. However, for $Z^*_{21}$ to be cyclic, it must have only one subgroup of order 2. This fact comes from the fundamental theorem of cyclic groups:</p>
<blockquote... |
341,823 | <p>Let <span class="math-container">$E\subset B_1(0)\subset \mathbb{R}^n$</span> be a compact set s.t. <span class="math-container">$\lambda(E)=0$</span>, where <span class="math-container">$\lambda$</span> is the Lebesgue measure, and <span class="math-container">$B_1(0)$</span> is the Euclidean unit ball centered at ... | Iosif Pinelis | 36,721 | <p>If <span class="math-container">$E\ne\emptyset$</span>, then <span class="math-container">$d(x,E)\le2$</span> for all <span class="math-container">$x\in B_1(0)$</span>. So, your integral is <span class="math-container">$\le\lambda(B_1(0))\ln2<\infty$</span>. </p>
|
90,459 | <p>I want to find the degree of $\mathbb{Q}(\sqrt{3+2\sqrt{2}})$ over $\mathbb{Q}$. I observe that
$3+2\sqrt{2}=2+2\sqrt{2}+1=(\sqrt{2}+1)^2$ so
$$
\mathbb{Q}(\sqrt{3+2\sqrt{2}})=\mathbb{Q}(\sqrt{2}+1)=\mathbb{Q}(\sqrt{2})
$$
so the degree is 2.</p>
<p>Is there a more mechanical way to show this without noticing the... | Sam | 3,208 | <p>We can immediately see that $\sqrt{3+2\sqrt{2}}$ is a root of </p>
<p>$$(X^2 -3)^2 -8 = X^4 -6X^2 + 1$$</p>
<p>So we can ask, whether this polynomial is irreducible or not.</p>
<p>The polynomial has not roots in $\mathbb Q$, since for a root $\frac rs \in \mathbb Q$ with $(r,s) = 1$, we would need to have $r|1$, ... |
363,767 | <p>An ellipse is specified $ x^2 + 4y^2 = 4$, and a line is specified $x + y = 4$. I need to find the max/min distances from the ellipse to the line.</p>
<p>My idea is to find two points $(x_1, y_1)$ and $(x_2,y_2)$ such that the first point is on the ellipse and the second point is on the line. Furthermore, the line ... | lab bhattacharjee | 33,337 | <p>We can solve without using Lagrange Multiplier Method</p>
<p>We have $$\frac{x^2}4+\frac{y^2}1=1$$</p>
<p>So, any point$(P)$ on the ellipse can be represented as $(2\cos t,\sin t)$</p>
<p>SO, the distance from the line : $x+y-4=0$ from $P(2\cos t,\sin t)$</p>
<p>is $$\frac{|2\cos t+\sin t-4|}{\sqrt{1^2+1^2}}=\fr... |
1,278,329 | <p>Solve the recurrence $a_n = 4a_{n−1} − 2 a_{n−2}$</p>
<p>Not sure how to solve this recurrence as I don't know which numbers to input to recursively solve?</p>
| Jared | 138,018 | <p>Assume the solution is $a_n = Ar^n$ then you get:</p>
<p>$$
a_n = Ar^n, a_{n - 1} = A \frac{r^n}{r}, a_{n - 2} = A\frac{r^n}{r^2}
$$</p>
<p>Plug this into your recursion relation:</p>
<p>\begin{align}
a_n = 4a_{n - 1} - 2a_{n - 2}\\
Ar^n = 4A \frac{r^n}{r} - 2A\frac{r^n}{r^2} \\
Ar^n \big(1 - \frac{4}{r} + \frac{... |
1,262,174 | <p>I am currently teaching Physics in an Italian junior high school. Today, while talking about the <a href="http://en.wikipedia.org/wiki/Dipole#/media/File:Dipole_Contour.svg" rel="noreferrer">electric dipole</a> generated by two equal charges in the plane, I was wondering about the following problem:</p>
<blockquote>... | Narasimham | 95,860 | <p>HINT:</p>
<p>Almost sure area can be evaluated between limits $(u_1,u_2),(v_1,v_2)$ for holomorphic functions of complex variables in conformal maps.</p>
<p>The equipotentials and lines of force form an orthogonal net of curvilinear rectangle curves (resemble Cassinian Ovals, but different.. one out of which is s... |
1,262,174 | <p>I am currently teaching Physics in an Italian junior high school. Today, while talking about the <a href="http://en.wikipedia.org/wiki/Dipole#/media/File:Dipole_Contour.svg" rel="noreferrer">electric dipole</a> generated by two equal charges in the plane, I was wondering about the following problem:</p>
<blockquote>... | achille hui | 59,379 | <p>Consider following parametrization of the first quadrant of the $(x,y)$ plane:</p>
<p>$$[1,\infty) \times [0,1] \ni (u,v)
\quad\mapsto\quad (x,y) \in [0,\infty)^2 \quad\text{s.t.}\quad
\begin{cases}
r_1 &= \sqrt{(x+1)^2+y^2} = u+v\\
r_2 &= \sqrt{(x-1)^2+y^2} = u-v
\end{cases}$$
In this parametrization, the... |
4,408,507 | <p>We study the definition of Lebesgue measurable set to be the following:</p>
<p>Let <span class="math-container">$A\subset \mathbb R$</span> be called Lebesgue measurable if <span class="math-container">$\exists$</span> a Borel set <span class="math-container">$B\subset A$</span> such that <span class="math-container... | Esgeriath | 1,021,258 | <p>You probably switched the inequality sign. If we have triangle with angles <span class="math-container">$\alpha, \beta, \gamma$</span>, then inequality <span class="math-container">$$
\sin\left(\frac\alpha 2\right)\sin\left(\frac\beta 2\right)\sin\left(\frac\gamma 2 \right)\geq \frac 1 8
$$</span>
does not need hold... |
268,185 | <p><strong>Question:</strong> </p>
<ol>
<li>Given a PDE, is there a general method to show that it is <em>not solvable</em> using the inverse scattering transform?</li>
<li>Specifically, for the perturbed 1D NLS or the 2D cubic NLS, where was it first shown that these equations can not be solved using <em>any</em> for... | mo-user | 127,891 | <p>Another method for testing integrability is the Painleve test, see e.g. <a href="https://arxiv.org/pdf/solv-int/9804003.pdf" rel="nofollow noreferrer">these</a> lecture notes and references therein. It has some caveats: for example, certain changes of variables do not preserve the Painleve property. Yet another poss... |
2,159,915 | <p>Consider the following system of ODE:</p>
<p>$$\begin{array}{ll}\ddot y + y + \ddot x + x = 0 \\
y+\dot x - x = 0 \end{array}$$</p>
<p><strong>Question</strong>: How many initial conditions are required to determine a unique solution?</p>
<p>A naive reasoning leads to four: $y(0),\dot y(0), x(0)$ and $\dot x(0)$.... | Jean Marie | 305,862 | <p>This problem depends on 3 arbitrary constants. Here is why.</p>
<p>Let $$\tag{1}\begin{cases}u&=&x+y\\v&=&x-y\end{cases}$$</p>
<p>The given differential system, written under the form:</p>
<p>$$\tag{2}\left\{\begin{array}{rclr}\ddot{(x+y)}& = & - (x+y) \ \ \ \ \ & (a) \\
\dot x& =... |
2,901,734 | <p>As title says find the minimum value of $(1+\frac{1}{x})(1+\frac{1}{y})$when given that $x+y=8$ using AGM inequality including Arithmetic Mean, Geometric Mean, and Harmonic Mean.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>Hint: Expanding your term we get
$$1+\frac{x+y}{xy}+\frac{1}{xy}=1+\frac{9}{xy}$$
By AM-GM we get</p>
<p>$$\frac{x+y}{2}\geq \sqrt{xy}$$ from here we get</p>
<p>$$1+\frac{9}{xy}\geq \frac{9}{16}+1$$</p>
|
182,346 | <p>Let's call a polygon $P$ <em>shrinkable</em> if any down-scaled (dilated) version of $P$ can be translated into $P$. For example, the following triangle is shrinkable (the original polygon is green, the dilated polygon is blue):</p>
<p><img src="https://i.stack.imgur.com/M0LOu.png" alt="enter image description here... | Gabriel C. Drummond-Cole | 3,075 | <p>Any <s>simply connected</s> polygon must be star-shaped to be shrinkable. I have made minor edits below to treat the more general case.</p>
<p>Let $D$ be a polygon with convex hull $H$. Assume we are given a non-trivial shrinking of $D$; view this as a map from $H$ to itself. This map must have a fixed point $x$, e... |
3,363,875 | <p>when I read a book,they say this is clear:</p>
<p>let <span class="math-container">$n$</span> be postive integer,then have
<span class="math-container">$$(-1)^n(n+1)\equiv n+1\pmod 4$$</span>
Why don't I feel right?</p>
| Tsemo Aristide | 280,301 | <p>If <span class="math-container">$n$</span> is even <span class="math-container">$(-1)^n=1$</span> and <span class="math-container">$(-1)^n(n+1)=(n+1)$</span></p>
<p>if <span class="math-container">$n=2p+1$</span>, <span class="math-container">$(-1)^n(n+1)=-(2p+2)$</span>.</p>
<p><span class="math-container">$(-1)^... |
1,515,776 | <p>How can I solve something like this?</p>
<p>$$3^x+4^x=7^x$$</p>
<p>I know that $x=1$, but I don't know how to find it. Thank you!</p>
| Paolo Leonetti | 45,736 | <p>Defining $f(x):=\mathrm{log}_7\left(3^x+4^x\right)$, you want to search for fixed points of $f$. But
$$
f^\prime(x)=\frac{1}{\ln 7}\cdot \frac{3^x\ln 3+4^x \ln 4}{3^x+4^x}>0
$$
and
$$
f^{\prime\prime}(x)=\frac{1}{\ln 7}\cdot \frac{3^x4^x}{(3^x+4^x)^2} \cdot ((\ln 4)^2+(\ln 3)^2-2\ln 3 \ln 4)
$$
which is positive ... |
1,515,776 | <p>How can I solve something like this?</p>
<p>$$3^x+4^x=7^x$$</p>
<p>I know that $x=1$, but I don't know how to find it. Thank you!</p>
| juantheron | 14,311 | <p>Here $\displaystyle 3^x+4^x = 7^x\Rightarrow \bf{\underbrace{\left(\frac{3}{6}\right)^x+\left(\frac{4}{6}\right)^x}_{Strictly\ decreasing\; function}} = \underbrace{\left(\frac{7}{6}\right)^x}_{Strictly\; increasing\; function}$</p>
<p>So these two curve Intersect each other exactly one Point.</p>
<p>So we can eas... |
3,944,628 | <p>I'm reading a book and, in its section on the definition of a stopping time(continuous), the author declares at the start that for the whole section every filtration will be complete and right-continuous.</p>
<p>So, in the definition of a Stopping Time, how important are these conditions? Why would they matter?</p>
| lab bhattacharjee | 33,337 | <p>Where you have stopped,
let <span class="math-container">$$z=-2\sin^2x+1+2\sin x\cos y$$</span></p>
<p><span class="math-container">$$\iff2\sin^2x-2\sin x\cos y+z-1=0$$</span></p>
<p>As <span class="math-container">$\sin x$</span> is real, the discriminant must be <span class="math-container">$\ge0$</span></p>
<p><s... |
4,090,408 | <p>Show that <span class="math-container">$A$</span> is a whole number: <span class="math-container">$$A=\sqrt{\left|40\sqrt2-57\right|}-\sqrt{\left|40\sqrt2+57\right|}.$$</span>
I don't know if this is necessary, but we can compare <span class="math-container">$40\sqrt{2}$</span> and <span class="math-container">$57$<... | egreg | 62,967 | <p>The check you're doing is indeed necessary in order to remove the absolute value.</p>
<p>However, when you arrive at <span class="math-container">$3200\mathrel{\Diamond}3249$</span> you can realize that the difference is a square, which is precisely the condition for a radical of the form
<span class="math-container... |
85,126 | <blockquote>
<p>Show that any sequence of positive numbers $(a_n)$ satisfying $$0< \frac{a_{n+1}}{a_n} \leq 1+ \frac{1}{n^2}$$
must converge.</p>
</blockquote>
<p>I have tried taking the limit of the inequality which yields that $0 \leq \lim \frac{a_{n+1}}{a_n} \leq 1$. If $\lim \frac{a_{n+1}}{a_n} \lt 1$, the... | Gerry Myerson | 8,269 | <p>I agree with the suggestions in the comments that for most values of $a,b$ there will be no algebraic solution. As for a numerical algorithm, are you familiar with <a href="http://en.wikipedia.org/wiki/Newton%27s_method" rel="nofollow">Newton's Method</a> (often called Newton-Raphson)? </p>
|
716,859 | <p>Define the mean of order $p$ of $a$ and $b$ as $s_p(a,b)$ $=$ $({a^p + b^p\over 2})^{1/p}$.</p>
<p>I have to find the limit of the sequence $s_n(a,b)$. I already know this sequence is bounded above by $b$ (from a previous question) and if I assume the limit exists I can show it is $b$. What I cannot show is that th... | Martín-Blas Pérez Pinilla | 98,199 | <p>Use the squeeze theorem. Supposing $b>a\ge 0$:
$$\frac b{2^{1/p}}=\left(\frac{b^p}2\right)^{1/p}\le\left(\frac{a^p + b^p}2\right)^{1/p}
\le\left(\frac{b^p + b^p}2\right)^{1/p}=b
$$
ant take $\lim_{p\to\infty}$.</p>
|
165,900 | <p>Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) \in k p$. My question: is $G$ some well known algebraic group? </p>
| Alexander Premet | 24,386 | <p>I think so. This can be deduced from the classification of cubic forms in 3 varibles; see
Tadayuki Abiko, Classification of cubic forms with three variables, Hokkaido Math. J. Vol. 10, 1981, 239-248. Google is the quickest way to find this paper which is freely available via Project Euclid.</p>
|
3,722,407 | <p>I am struggling with this problem:</p>
<blockquote>
<p>Let n be an even number, and denote <span class="math-container">$[n]=\{1,2,...,n\}$</span>. A sequence of
sets <span class="math-container">$S_1 , S_2 , \cdots , S_m \subseteq [n]$</span> is considered <em>graceful</em>
if:</p>
<ol>
<li><span class="math-contai... | Community | -1 | <p><strong>If this can help:</strong></p>
<p>A first approximation is <span class="math-container">$x_n=\dfrac1n$</span>. A better approximation can be found in the form <span class="math-container">$\dfrac{1+t}n$</span>. We write</p>
<p><span class="math-container">$$\left(\frac{1+t}n\right)^n-n\left(\frac{1+t}n\right... |
3,722,407 | <p>I am struggling with this problem:</p>
<blockquote>
<p>Let n be an even number, and denote <span class="math-container">$[n]=\{1,2,...,n\}$</span>. A sequence of
sets <span class="math-container">$S_1 , S_2 , \cdots , S_m \subseteq [n]$</span> is considered <em>graceful</em>
if:</p>
<ol>
<li><span class="math-contai... | N. S. | 9,176 | <p>Hopefully I didnt do any mistake:</p>
<p>Note that</p>
<p><span class="math-container">$$P_n(\frac{1}{n}+\frac{1}{n^{n+1}})= (\frac{1}{n}+\frac{1}{n^{n+1}})^n - \frac{1}{n^n}>0$$</span></p>
<p>Now, let <span class="math-container">$\alpha >1$</span>. We have
<span class="math-container">$$P_n(\frac{1}{n}+\frac... |
79,726 | <p>Let $R$ be a commutative ring with unity. Let $M$ be a free (unital) $R$-module.</p>
<p>Define a <em>basis</em> of $M$ as a generating, linearly independent set.</p>
<p>Define the <em>rank</em> of $M$ as the cardinality of a basis of $M$ (as we know commutative rings have IBN, so this is well defined).</p>
<p>A <... | Pierre-Yves Gaillard | 660 | <p>The statement in the Edit of Georges's answer also holds in the non-commutative case:</p>
<blockquote>
<p>Let $R$ be an associative ring with $1$. If $M$ is an $R$-module, if $B$ is an infinite basis of $M$, and if $S\subset M$ is a generating subset, then we have $$|S|\ge|B|,$$ where, for any set $X$, the symbol... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Victor Protsak | 5,740 | <p><a href="https://en.wikipedia.org/wiki/Feit%E2%80%93Thompson_theorem">Feit-Thompson theorem</a> $ $</p>
<hr>
<p><strong>Edit</strong> (GK): This would also be my first answer, let me add a few details. The <strong>Feit-Thompson theorem</strong> asserts that every finite group of odd order is solvable. An... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Allen Knutson | 391 | <p><a href="http://terrytao.wordpress.com/2012/03/23/some-ingredients-in-szemeredis-proof-of-szemeredis-theorem/" rel="nofollow noreferrer">Szemerédi’s theorem</a>, that inside a positive-density set of naturals there are arbitrarily long arithmetic progressions. To quote Terry Tao, "...the pieces of Szemerédi’s proof ... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Stanley Yao Xiao | 10,898 | <p>The proof of the <a href="https://en.wikipedia.org/wiki/Thue%E2%80%93Siegel%E2%80%93Roth_theorem" rel="nofollow">Thue-Siegel-Roth theorem</a> is still very difficult, as no substantial improvement to Roth's original argument is known.</p>
<p>The Thue-Siegel-Roth Theorem states that for any non-rational algebraic nu... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Gil Kalai | 1,532 | <p><strong>The decomposition theorem for intersection homology</strong></p>
<p>The decomposition theorem for (middle perversity) intersection homology (for algebraic varieties) was proved in 1982 by Beilinson-Bernstein-Deligne-Gabber. I don't understand it well enough to describe it (but please replace this sentence w... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Dylan Thurston | 5,010 | <p>I'm surprised no one has mentioned <a href="https://en.wikipedia.org/wiki/Michael_Freedman">Freedman's</a> Theorem from 1982 yet. Technically this theoerem says that <a href="https://en.wikipedia.org/wiki/Casson_handle">Casson handles</a> are standard, but more broadly this completes the classification of topologica... |
2,115,199 | <p>Let $A \in \text{End}(V)$ be an endomorphism, and $\mathbb Q[A]$ a subalgebra in $\text{End}(V)$ generated by $A$.</p>
<p>Is $\mathbb Q[A]$ always at most dim$V$-dimensional? How to prove it</p>
| Adam Hughes | 58,831 | <p>Recall that the minimal polynomial $m_A(x)$ divides the characteristic polynomial $p_A(x)$ which has degree $\dim V$ and that</p>
<blockquote>
<p>$$\Bbb Q[A]\cong\Bbb Q[x]/(m_A(x))$$</p>
</blockquote>
<p>and this is a $\deg m_A(x)\le \deg p_A(x)=\dim V$ algebra.</p>
|
2,292,656 | <p>Let $L/K$ be a degree $n$ extension of fields, where $K$ has discrete valuation $v$, which can be prolonged to the discrete valuations $w_i$ on $L$. We can therefore define the completion of $K$ w.r.t. $v$ to be $\hat K$, and the completion of $L$ w.r.t. $w_i$ to be $\hat L_i$, then in Theorem II.3.1 of Serre's <em>... | Dominic Wynter | 275,050 | <p>In order to define the canonical morphism $\varphi:L\otimes_K\hat K\to\prod_i\hat L_i$, we will make use of the completion functor $\mathrm{comp}:\mathbf{DVF}\to\mathbf{DVF}$ acting on the category of discretely valued fields, with continuous field embeddings (that is to say, field extensions that prolong the discre... |
192,072 | <p>Bonjour!<br>
I'm trying this number-theory problem, but i don't have any idea how to solve it.<br>
Can you give me some hints ?</p>
<p>We have got any $\mathbb{Z_+}$ number. Let it be $n$.<br>
Then we must proof that $2 \nmid \sigma(n) \implies n = k^2 \vee n = 2k^2$.<br>
Thanks for any help </p>
| Hagen von Eitzen | 39,174 | <p>If $n$ is odd, then $\sigma(n)$ is the sum of $\tau(n)$ odd divisors. For this sum to be odd, $\tau(n)$ must be odd. But that means that it is not possible to pair off divisors as pairs $(d, \frac n d)$, i.e. there is one divisor $d$ with $d=\frac nd$ and hence $n=d^2$.</p>
<p>If $n=2^rm$ with $m$ odd and $r>0$ ... |
2,437,026 | <p>Suppose that:</p>
<p>$$
X \sim Bern(p)
$$</p>
<p>Then, intuitively $X^2 = X \sim Bern(p)$. However, when I try to think of it logically, it doesn't make any sense. </p>
<p>As an example, $X$ is $1$ with probability $p$ and $0$ with probability $1-p$. Then, $X^2 = X\cdot X$ is $1$ only when both $X$'s are $1$, whi... | Siong Thye Goh | 306,553 | <blockquote>
<p>$X^2=X.X$ is $1$ only when both $X$'s are $1$, which occurs with probability $\color{blue}p$. </p>
</blockquote>
<p>Notice that $X$ and $X$ are identical, and dependent. The two $X$'s refer to the same thing. </p>
<p>Since $X$ takes binary value, we indeed have $X^2=X$.</p>
|
3,111,489 | <p>For which <span class="math-container">$p,q$</span> does the <span class="math-container">$\int_0^{\infty} \frac{x^p}{\mid{1-x}\mid^q}dx$</span> exist ?</p>
<p>Can you help me, I have been siting hours on this question .</p>
<p>I got that for <span class="math-container">$ q<1$</span> and <span class="math-cont... | gt6989b | 16,192 | <p>I would try to get rid of the absolute value to simplify:
<span class="math-container">$$
\int_0^\infty \frac{x^pdx}{|1-x|^q}
= \int_0^1 \frac{x^p dx}{(1-x)^q}
+ \int_1^\infty \frac{x^p dx}{(x-1)^q}
$$</span>
Now the first integral has a problem at <span class="math-container">$x \to 1^-$</span>, and the second as... |
1,463,567 | <p>We have the following theorem </p>
<p>If |G| = 60 and G has more than one Sylow-5 subgroup, then G is simple.</p>
<p>Since order of the rigid motion of the dodecahedron group is 60, so all we have to do is to show that it has more than one sylow-5 subgroup, but I don't know how to do this as I don't know the eleme... | Najib Idrissi | 10,014 | <p>Yes, the assumption that the set is open is necessary. The plane $\mathbb{R}^2$ is locally path-connected, and the "topologist's sine curve" $$S = \operatorname{closure} \bigl\{(x,\sin(1/x) \mid x > 0 \bigr\}$$ is connected (because it's the closure of a connected set), but it's not path-connected (a classical ex... |
238,076 | <p>I was thinking about this when flying on the plane which was approaching and slowing down.</p>
<p>Assume an object is approaching its target which is at a certain initial distance d at time t0.</p>
<p>It starts at a speed that will allow it to reach the target in exactly one hour (e.g. d=100km, it starts at 100 km... | ebsddd | 36,669 | <p>Split the problem into two parts: the time it takes to get from 100 to 10 km, and from 10 to 0 km. Second part is trivial, so I'll focus on the first.</p>
<p>For the first part, let $x(t) = d(t) - 10$, so that when you're at the 10 km mark, $x = 0$. Then, since $v(t)=-d(t)$, $v(t)=-x(t)-10$. As a differential equat... |
2,296,256 | <p>I need help how to mathematically interpret an ODE (Newton's second law). I used to the ODE in this form:
$$
m\ddot x(t)=F(t)\tag{1}
$$</p>
<p>However, in another book they wrote:
$$
m\ddot x=F(x,\dot x) \tag{2}
$$
where $F: \mathbb{R}^n \times \mathbb{R}^n\rightarrow \mathbb{R}^n$.</p>
<p><strong>Questions:</stro... | marty cohen | 13,079 | <p>If f is a function, then, by definition, i=j implies f(i)=f(j).</p>
<p>Nothing else is needed.</p>
|
2,296,256 | <p>I need help how to mathematically interpret an ODE (Newton's second law). I used to the ODE in this form:
$$
m\ddot x(t)=F(t)\tag{1}
$$</p>
<p>However, in another book they wrote:
$$
m\ddot x=F(x,\dot x) \tag{2}
$$
where $F: \mathbb{R}^n \times \mathbb{R}^n\rightarrow \mathbb{R}^n$.</p>
<p><strong>Questions:</stro... | Mr. Xcoder | 435,694 | <p>$$\forall\:i,j: i>j \implies f(i) > g(j)\tag1$$
$$\forall\:i,j: i<j \implies f(i) < g(j)\tag2$$</p>
<p>Bypassing the redundancy, if you simply suppose that $i\neq j,\: f(i)=g(j)$, there are two cases here we need to handle:</p>
<ul>
<li><p>$i>j,\text{ and (1)}\implies \underbrace{f(i) > g(j)}_{\... |
1,624,221 | <p>For the former one, I am aware that if let $F(x)=\int_a^x f(t)dt$, then it also equals $\int_0^x f(t)dt-\int_0^a f(t)dt$, so $F'(x)= f(x)-0=f(x)$. But who can tell me why $\int_0^a f(t)dt$ is $0$?</p>
| Hugh | 94,681 | <p>Use partial fraction decomposition to represent the sum and notice that it is a telescoping series.</p>
|
1,380,348 | <p>100-sided dice was rolled 98 times,
Numbers form 1 to 50 were rolled exactly once, except number 25, which wasn't rolled yet.
Number 75 was rolled 49 times
You can only bet if the next roll result will be below 51 or above 49.</p>
<p>How do you choose ?, how to calculate which bet is better ?
ELI5 please</p>
<p>A... | user2566092 | 87,313 | <p>If the die is fair, then the probability the roll will be above $51$ is $49/100$ and the probability the roll is below $49$ will be $48/100$. The previous rolls do not change the fact that the die is fair. So technically, since it's slightly more likely, you should bet that the roll is above $51$.</p>
<p>If you do ... |
1,380,348 | <p>100-sided dice was rolled 98 times,
Numbers form 1 to 50 were rolled exactly once, except number 25, which wasn't rolled yet.
Number 75 was rolled 49 times
You can only bet if the next roll result will be below 51 or above 49.</p>
<p>How do you choose ?, how to calculate which bet is better ?
ELI5 please</p>
<p>A... | hvedrung | 245,555 | <p>Bet on 75. Fair dice can't fall on one outcome in 49 rolls out of 98. The probability is $ 10^{-96}$. It is greater than the number of atoms in the universe.</p>
<p>PS. If this is study task than bets are equal. Next outcome has absolutely no dependance from previous ones.</p>
|
3,219,635 | <p>Suppose <span class="math-container">$f$</span> is continuous on <span class="math-container">$\Bbb R$</span>, define <span class="math-container">$F(x)=\int_a^bf(x+t)\cos t\,dt,x\in [a,b]$</span>.</p>
<p>How to show <span class="math-container">$F(x)$</span> is differentiable on <span class="math-container">$[a,b]... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$F(x)=\int_{a+x}^{b+x} f(s) [\cos \,s \cos \,x+\sin \,s \sin \,x]ds$</span>. Split this into two terms and pull out <span class="math-container">$\cos\, x,\sin\,x$</span> from the integral. Use the fact indefinite integrals of continuous functions are differentiable. </p>
|
4,044,654 | <p>We say that a continuous function <span class="math-container">$u:\mathbb{R}^d\to \mathbb{R}$</span> is subharmonic if it satisfies the mean value property
<span class="math-container">$$u(x)\leq \frac{1}{|\partial
B_r(x)|}\int_{\partial B_r(x)}u(y)\,\mathrm{d}y \qquad (\star)$$</span> for
any ball <span class="ma... | Peter Morfe | 711,689 | <p>Martin gives a very reasonable answer extending the question's second argument; here's how to extend the first one.</p>
<p>If <span class="math-container">$u: \mathbb{R}^{d} \to \mathbb{R}$</span> is continuous and convex, then mollification of <span class="math-container">$u$</span> gives a family of smooth functio... |
2,268,299 | <p>Let's say I have a series of real values $y_0,y_1,y_2\cdots$.
My question is if it's always possible to find (at least one) $C^\infty$ real function such that
\begin{equation}
f^{(n)}(0)=y_n
\end{equation}
and in the affirmative case, how.
It is a kind of "reverse taylor" problem... any hints?</p>
| Community | -1 | <p>Yes, it is possible to find a function that satisfies the condition
$f^{\left(n\right)}\left(0\right)=y_n$
when the sequence $\left<y_n\right>$ is given, though infinite such functions exist.</p>
<p>One way of finding a possible function would be to a polynomial of degree $n$. Consider
$$p\left(x\right)=a_0+a... |
208,744 | <p>I was asked to show that $\frac{d}{dx}\arccos(\cos{x}), x \in R$ is equal to $\frac{\sin{x}}{|\sin{x}|}$. </p>
<p>What I was able to show is the following:</p>
<p>$\frac{d}{dx}\arccos(\cos(x)) = \frac{\sin(x)}{\sqrt{1 - \cos^2{x}}}$</p>
<p>What justifies equating $\sqrt{1 - \cos^2{x}}$ to $|\sin{x}|$?</p>
<p>I ... | Mikasa | 8,581 | <p>If $y=\arccos(x)$ then $x=\cos(y)$ and so $$\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}=\frac{-1}{\sin(y)}$$ if $\sin(y)\neq0$. Here, $$0<y=\arccos(x)<\pi, \sin(y)>0$$ and $$\sin(y)=\sqrt{1-\cos^2(y)}=\sqrt{1-x^2}$$ It means that $$\frac{dy}{dx}=\frac{d}{dx}\arccos(x)=\frac{-1}{\sqrt{1-x^2}}$$ where in $x\in (-1,... |
10,600 | <p>As mentioned in <a href="https://matheducators.stackexchange.com/questions/1538/counterintuitive-consequences-of-standard-definitions">this question</a> students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \t... | Mikhail Katz | 1,385 | <p>There is something to be learned from the history of the concept. The modern concept of continuity was introduced by Cauchy in 1821. This concept is probably best explained in two stages:</p>
<p>(1) Cauchy's definition of continuity as infinitesimal $x$-increment always producing an infinitesimal change in $y$;</p... |
10,600 | <p>As mentioned in <a href="https://matheducators.stackexchange.com/questions/1538/counterintuitive-consequences-of-standard-definitions">this question</a> students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \t... | Peter Saveliev | 10,168 | <p>Property (3), i.e., the $\varepsilon-\delta$ definition of continuity, has numerous motivations/interpretations. For example, continuity can be interpreted as <em>accuracy</em>. Suppose we are shooting a cannon located at the top of a hill. Even when the mathematics is perfect, our limited knowledge of the many para... |
186,890 | <p>Working with other software called SolidWorks I was able to get a plot with a curve very close to my data points:</p>
<p><a href="https://i.stack.imgur.com/DooKo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DooKo.png" alt="enter image description here"></a></p>
<p>I tried to get a plot as acc... | kglr | 125 | <p>Using halirutan an J.M.'s <code>IPCUMonotonicInterpolation</code> from <a href="https://mathematica.stackexchange.com/q/14662/125">this q/a</a>:</p>
<pre><code>Plot[IPCUMonotonicInterpolation[dados]@t, {t, 0, 5},
Epilog -> {Red, PointSize[Large], Point@dados},
AspectRatio -> 1/GoldenRatio, GridLines ->... |
3,628,374 | <p>We have that <span class="math-container">$W \in \mathbb{R}^{n \times m}$</span> and we want to find <span class="math-container">$$\text{prox}(W) = \arg\min_Z\Big[\frac{1}{2} \langle W-Z, W-Z \rangle+\lambda ||Z||_* \Big]$$</span></p>
<p>Here, <span class="math-container">$||Z||_*$</span> represents the trace nor... | lhf | 589 | <p>Here is a solution using linear algebra over <span class="math-container">$\mathbb F_p$</span>.</p>
<p><span class="math-container">$g$</span> satisfies <span class="math-container">$0=g^p-I=(g-I)^p$</span>.
Therefore, <span class="math-container">$1$</span> is the only eigenvalue of <span class="math-container">$g... |
3,628,374 | <p>We have that <span class="math-container">$W \in \mathbb{R}^{n \times m}$</span> and we want to find <span class="math-container">$$\text{prox}(W) = \arg\min_Z\Big[\frac{1}{2} \langle W-Z, W-Z \rangle+\lambda ||Z||_* \Big]$$</span></p>
<p>Here, <span class="math-container">$||Z||_*$</span> represents the trace nor... | Jyrki Lahtonen | 11,619 | <p>It sounds like you want a mixture of group actions and linear algebra.</p>
<hr>
<p>Let <span class="math-container">$g\in G=GL_2(\Bbb{F}_p)$</span> be an element of order <span class="math-container">$p$</span>. Let's denote <span class="math-container">$H=\langle g\rangle$</span>, and <span class="math-container"... |
3,671,223 | <p>First and foremost, I have already gone through the following posts:</p>
<p><a href="https://math.stackexchange.com/questions/2463561/prove-that-for-all-positive-integers-x-and-y-sqrt-xy-leq-fracx-y">Prove that, for all positive integers $x$ and $y$, $\sqrt{ xy} \leq \frac{x + y}{2}$</a></p>
<p><a href="https://ma... | saulspatz | 235,128 | <p>The well-known fact your teacher is talking about is that the square of any really number is nonnegative. That is a theorem. As for why you should start with this particular instance of that theorem, that takes a little insight. </p>
<p>If you were trying to prove the theorem yourself, you might start by seeing... |
2,512,424 | <p>It is an easy exercise to show that all finite groups with at least three elements have at least one non-trivial automorphism; in other words, there are - up to isomorphism - only finitely many finite groups $G$ such that $Aut(G)=1$ (to be exact, just two: $1$ and $C_2$).</p>
<p>Is an analogous statement true for a... | Mikko Korhonen | 17,384 | <p>Ledermann and B.H.Neumann ("On the Order of the Automorphism Group of a Finite Group. I", Proc. Royal Soc. A, 1956) have shown the following:</p>
<blockquote>
<p><strong>Theorem.</strong> Let $n > 0$. There exists a bound $f(n)$ such that if $G$ is a finite group with $|G| \geq f(n)$, then $|\operatorname{Aut}... |
2,512,424 | <p>It is an easy exercise to show that all finite groups with at least three elements have at least one non-trivial automorphism; in other words, there are - up to isomorphism - only finitely many finite groups $G$ such that $Aut(G)=1$ (to be exact, just two: $1$ and $C_2$).</p>
<p>Is an analogous statement true for a... | Brauer Suzuki | 960,602 | <p>A more accessible account on the theorem of Ledermann-Neumann mentioned in the accepted answer can be found here: <a href="https://www.tandfonline.com/doi/full/10.1080/00029890.2020.1803625" rel="nofollow noreferrer">Math. Monthly</a> or <a href="https://arxiv.org/abs/1909.13220" rel="nofollow noreferrer">arxiv</a><... |
81,209 | <p>I feel a bit ashamed to ask the following question here. </p>
<blockquote>
<p>What is (actually, is there) Galois
theory for polynomials in
$n$-variables for $n\geq2$?</p>
</blockquote>
<p>I am preparing a large audience talk on Lie theory, and decided to start talking about symmetries and take Galois theory... | Tim Porter | 3,502 | <p>This will not answer the question but is more than a comment in addition it may be very naive! (This is a hard question not a soft question!!!)</p>
<p>I wonder if given the Galois group <-> étale fundamental group link works for dimension 1, should there not be a link '2-Galois thingie'<->étale 2-type, and he... |
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