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 |
|---|---|---|---|---|
151,192 | <p>By Cantor's normal form theorem, any ordinal $r$ can be expressed as:
$r=\omega^{k_1}a_1 + \omega^{k_2}a_2 + \ldots$. ($k_1>k_2>\ldots$)</p>
<p>I want to know whether the class of all $a_i$'s is countable or not.</p>
<p>If it is not countable how do i prove a problem such as:
$\omega^{k_1}a_1 + \omega^{k_2}a... | g.castro | 25,312 | <p>Each $a_i$ in the Cantor normal form $\omega^{k_1}\cdot a_1 + \cdots + \omega^{k_n}\cdot a_n$ is a positive natural number. </p>
<p>So the "class of all $a_i$'s" is the set of all positive natural numbers, which is countable. </p>
<p>If you meant "the class of all tuples $(a_1,\ldots, a_n)$", then this is the set ... |
3,429,020 | <p>This is the complex function: </p>
<p><span class="math-container">$$f(z) = 6\bar z^2-2\bar z - 4i |z|^2$$</span></p>
<p>which is also problem number 7.4, b on page 46 of this book: <a href="https://nnquan.files.wordpress.com/2013/01/giao-trinh-ham-phuc.pdf" rel="nofollow noreferrer">https://nnquan.files.wordpress... | Math1000 | 38,584 | <p>If <span class="math-container">$x=0$</span> then <span class="math-container">$f_n(x)=0$</span> and trivially <span class="math-container">$\lim_{n\to\infty}f_n(x)=0$</span> For <span class="math-container">$x\in (0,1]$</span>, we have <span class="math-container">$f_n(x) = nxe^{-nx^2}\leqslant ne^{-nx^2}=ne^{-n}e^... |
1,234,820 | <p>I was working on this problem </p>
<blockquote>
<p>Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$.</p>
</blockquote>
<p>My attempt:-
I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should be either zero or the entire ring (since $M_3(\mathbb{R})$ is simple) and in ca... | Robert Israel | 8,508 | <p>A ring homomorphism from any matrix ring containing $\mathbb R I$ to $\mathbb R$ is also an algebra homomorphism: </p>
<ol>
<li>If $\phi$ is not identically $0$, $\phi(I) = 1$ because $\phi(x) = \phi(I x) = \phi(I) \phi(x)$. </li>
<li>If $n \in \mathbb Z$, we then get $\phi(nI) = n \phi(I) = n$.</li>
<li>If $r$ is... |
1,234,820 | <p>I was working on this problem </p>
<blockquote>
<p>Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$.</p>
</blockquote>
<p>My attempt:-
I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should be either zero or the entire ring (since $M_3(\mathbb{R})$ is simple) and in ca... | Dietrich Burde | 83,966 | <p>It is known that every injective ring homomorphism $\phi:M_n(K)\rightarrow M_m(K)$ must satisfy $n\le m$, see for example Lemma $3.2$ <a href="http://arxiv.org/pdf/math/0509188.pdf" rel="nofollow">here</a>. For $m=1$ and $n=3$ this is a contradiction. One could also use the Amitsur-Levitzki theorem.</p>
|
818,015 | <p>I try to get the variables for this equation:</p>
<p>$$\begin{cases}
6x_1 + 4x_2 + 8x_3 + 17x_4 &= -20\\
3x_1 + 2x_2 + 5x_3 + 8x_4 &= -8\\
3x_1 + 2x_2 + 7x_3 + 7x_4 &= -4\\
0x_1 + 0x_2 + 2x_3 -1x_4 &= 4
\end{cases}$$</p>
<p>So i started with:</p>
<p>$$ \begin{pmatrix}
6 & 4 &am... | Alec | 114,173 | <p><a href="http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule" rel="nofollow">L'hopital's rule</a> is useful in this case, since the limits are of indeterminate form $0/0$. Then, taking the derivative of the numerator and denominator:
$$
\lim_{x\to 0}\frac{1−\cos x}{x^2} = \lim_{x\to 0}\frac{\sin x}{2x} = \frac{1}... |
1,182,383 | <p>So I'm revising definitions of algebra for my exam and I'm wondering what an Ideal actually is?</p>
<p>I believe the definition is:</p>
<p>$I$ is an ideal of $R$ if $xr,rx\in I$ where $r\in R$ and $x\in I$</p>
<p>However, after looking about the web, I'm finding that unity has a lot to do with it. I'm struggling ... | Mathmo123 | 154,802 | <p>Whenever we study a new algebraic object like a group or a ring, one of the most useful things to study is the maps between those objects. In group theory, we are interested in homomorphisms between groups. In ring theory, we are interested in homomorphisms between rings. Knowing about the possible homomorphisms fro... |
1,182,383 | <p>So I'm revising definitions of algebra for my exam and I'm wondering what an Ideal actually is?</p>
<p>I believe the definition is:</p>
<p>$I$ is an ideal of $R$ if $xr,rx\in I$ where $r\in R$ and $x\in I$</p>
<p>However, after looking about the web, I'm finding that unity has a lot to do with it. I'm struggling ... | Colin McLarty | 99,616 | <p>For Emmy Noether an ideal was basically a modulus, as in solving an equation modulo 5. She gets this from Dedekind but makes it more far-reaching. </p>
<p>For her arithmetic modulo 5 was properly done as algebra in the ring $\mathbb{Z}/(5)$ where $(5)$ is the ideal generated by $5\in\mathbb{Z}$. Of course she k... |
2,959,026 | <blockquote>
<p>I have the matrix</p>
<p><span class="math-container">$$Q=\begin{bmatrix}-1&2\\0&1\\\end{bmatrix}$$</span></p>
<p>and want to find the invariant points.</p>
</blockquote>
<p>To do this, I solve the equation:</p>
<p><span class="math-container">$$\begin{bmatrix}-1&2\\0&1\\\end{bmatrix}\be... | Siong Thye Goh | 306,553 | <p>The non-zero invariant points are eigenvectors with eigenvalue <span class="math-container">$1$</span>. If the problem has full rank, then the only solution would be the zero vector.</p>
<p>As you have shown, the only condition is <span class="math-container">$x=y$</span>. That is, the invariant points are multiple... |
1,353,892 | <p>Just a small thought popped up in my mind; and now I'm stuck on it. Any idea on how to find the value of $\tan 20^\circ$? I tried doing it by using the multiple angle formulas, but I didn't get an answer... How do I proceed? </p>
| user21820 | 21,820 | <p>$\cos(3t) = 4 \cos(t)^3 - 3 \cos(t)$</p>
<p>$\sin(3t) = 3 \sin(t) - 4 \sin(t)^3$.</p>
<p>Substituting $t = 20^\circ$ gives cubics that can be solved in closed form if you allow the operation of taking cube roots.</p>
|
129,294 | <p>I found the answer in this <a href="https://mathematica.stackexchange.com/questions/67306/insert-at-specific-resulting-positions/67309#67309">post</a> very interesting to do what I need, but I would like something where I could provide a <code>list to be modified</code>, a <code>list with values</code> that will be ... | m_goldberg | 3,066 | <p>Here is somewhat obscure one-liner.</p>
<pre><code>data = {0, 1, 2, 3, 4, 5, 6, 7};
new = {a, b, c, d, e};
where = {2, 6, 7, 8, 13};
(RightComposition @@ Thread[Insert[new, where]]) @ data
</code></pre>
<blockquote>
<p><code>{0, a, 1, 2, 3, b, c, d, 4, 5, 6, 7, e}</code></p>
</blockquote>
|
3,630,967 | <p>More precisely, define <span class="math-container">$\phi(x) = \frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$</span>. </p>
<p>Does there exists a constant <span class="math-container">$L$</span> such that <span class="math-container">$$|\phi(x)-\phi(y)|\le L|x-y|,$$</span> for all <span class="math-container">$x,y\in \math... | dohmatob | 168,758 | <p>In fact, one generalize the accepted answer to show that every 1D gaussian density is <span class="math-container">$1/4$</span>-Lipschitz, irrespective of the mean and standard deviation.</p>
<hr />
<p>Indeed, consider a 1D Gaussian density centered at <span class="math-container">$\mu \in \mathbb R$</span>, with st... |
119,443 | <p>My buddy and I are arguing over something that cropped up in this past weekend's Texas Hold'em tournament.</p>
<p>A player got "knocked out" (lost all their chips) early on in the game. The person hosting the tournament told them they could buy back in, and my buddy got upset. He argued that it gave the player buyi... | Community | -1 | <p>One of the biggest pitfalls of using math to prove a point is accidentally proving something that isn't quite the issue at hand. (or even unrelated)</p>
<p>While the <em>specific</em> point you have made is true, I suspect you have fallen into this trap: that you are completely neglecting two very important possibi... |
2,155,755 | <p>What is the difference (or connection) between the dimension of a vector space and the dimension in terms of bases?</p>
<p>For instance, when we talk about the vector space $\mathbb{R}^3$, we are talking about a 3-dimensional vector space. This vector space contains vectors with three elements: $(x_1, x_2, x_3)$.</... | David K | 139,123 | <p>Every $\mathbb R^n$ is a vector space (or at least <em>can</em> be a vector space),
but not every vector space is $\mathbb R^n.$</p>
<p>In your example with the three vectors $\{(1, 1, -2, 0, -1), (0, 1, 2, -4, 2), (0, 0, 1, 1, 0) \}$, the vectors belong to $\mathbb R^5,$
which we can consider to be a $5$-dimension... |
2,980,366 | <p>Euler's theorem was expanded to encompass polyhedrons homeomorphic to not only spheres but also <span class="math-container">$g$</span>-holed toruses. I've tried to understand proofs about how <span class="math-container">$2-2g$</span> is a topological invariant but have always had trouble with the use of planar gra... | Liviu Nicolaescu | 111,906 | <p>Very simple examples of <span class="math-container">$k$</span>-planes in <span class="math-container">$\mathbb{C}^n$</span> are the graphs linear operators <span class="math-container">$\mathbb{C}^k\to\mathbb{C}^{n-k}$</span>. These linear operators form a complex vector space of dimension <span class="math-co... |
17,423 | <p>In most of books on elementary algebra, intermediate algebra and college algebra, the degree of the non-zero polynomial <span class="math-container">$$f(x)=a_nx^n+\cdots a_1x+a_0$$</span> with <span class="math-container">$a_n\neq 0$</span> is defined to be <span class="math-container">$n$</span>. </p>
<p>But I am ... | Vassilis Markos | 12,829 | <p>In general, I agree with @Henry Towsner on the fact that the proofs should not always be presented in an elementary course.</p>
<p>However, I have to disagree on the implicit "well-definition property" of any definition. Such a definition would require that some sort of uniqueness property has been proved, which ca... |
1,402,887 | <p>I was going through some sample papers of math, and I found this question which I cant solve:</p>
<p>If $abc=1$, find $1/(1+a+b^{-1})+1/(1+b+c^{-1})+1/(1+c+a^{-1})$.</p>
<p>Please help me with this.... I have spent almost 3 hours on this question...</p>
<p>Thanks for the help.</p>
| Barry Cipra | 86,747 | <p>A somewhat plodding, but nonetheless effective, approach is simply to eliminate one of variables, say letting $c=(ab)^{-1}$, and then simplify the fractions:</p>
<p>$$\begin{align}
{1\over1+a+b^{-1}}+{1\over1+b+c^{-1}}+{1\over1+c+a^{-1}}
&={1\over1+a+b^{-1}}+{1\over1+b+ab}+{1\over1+(ab)^{-1}+a^{-1}}\\\\
&={... |
363,335 | <p>Consider the element $w=x^2yx^{-1}y^{-1}x^{-1}yxy^{-1}x^{-1}$ of the free group $F_2=\langle x,y\rangle$. By considering the image of this element under the abelianization map (equivalently, by adding exponents), we know that $w$ is in the commutator subgroup $(F_2)_2 = [F_2,F_2]$. In fact I can algorithmically deco... | Sean Eberhard | 23,805 | <p>Not long after I posted my question I found the standard answer to my question. I'll include it here for the benefit of anybody else with a similar question. A key search-word is "basic commutators", and the following can be found in full detail in the classic group theory text of M. Hall, who first defined basic co... |
3,634,864 | <p>Say we have a proposition If P then Q, and let Q = A <span class="math-container">$\land$</span> B</p>
<p>If we do a proof by contradiction, we assume the negation of Q: <span class="math-container">$\lnot (A \land B) = \lnot A \lor \lnot B$</span></p>
<p>From here, can we continue our contradiction proof by star... | Graham Kemp | 135,106 | <p>You may prove <span class="math-container">$(\neg A\vee\neg B)\to\neg P$</span> using a Proof by Cases. Assume <span class="math-container">$\neg A$</span> and derive <span class="math-container">$\neg P$</span>, <strong>and</strong> assume <span class="math-container">$\neg B$</span> and derive <span class="math-c... |
1,303,263 | <p>Given 2 vectors,$u=(3,5)$,$v=(s,s^2)$,in what situations do u and v parallel?$(s≠0)$</p>
<p>In order to be parallel,$u$ must be proportional to $v$,vice verse.Let $k$ be a scalar $neq 0$,then $ku=(3k,5k)=v=(s,s^2)$,which gives $3k=s$;$5k=s^2 \rightarrow 5k=9k^2 (k \neq 0) \rightarrow 5=9k→k=\frac{5}{9}$,plug $k=\fr... | nullUser | 17,459 | <p>Eyebrow raising indeed, though the pattern does not continue as you suggest. I get
$$
0, 1, 1, 2, 3, 5, 7, 10, 16, 23, 37, 55, 84, 125, 198
$$</p>
<p>Remember that the the number of primes has a well known growth rate (<a href="https://en.wikipedia.org/wiki/Prime_number_theorem">https://en.wikipedia.org/wiki/Prime_... |
957,302 | <p>Let <span class="math-container">$k$</span> a field. Let <span class="math-container">$f$</span> be the ring injective homomorphism</p>
<p><span class="math-container">$$ f:k[x] \rightarrow k[x,y]/(xy-1)$$</span></p>
<p>obtained as the composition of the inclusion <span class="math-container">$k[x] \subset k[x,y]$</... | orangeskid | 168,051 | <p>HINT:</p>
<p>$k[x,y]/(xy-1)$ is naturally isomorphic to the ring of fractions $k[x][\frac{1}{x}] = S^{-1}\ k[x]$, where $S= \{1,x,x^2, \ldots\}$. </p>
|
957,302 | <p>Let <span class="math-container">$k$</span> a field. Let <span class="math-container">$f$</span> be the ring injective homomorphism</p>
<p><span class="math-container">$$ f:k[x] \rightarrow k[x,y]/(xy-1)$$</span></p>
<p>obtained as the composition of the inclusion <span class="math-container">$k[x] \subset k[x,y]$</... | user26857 | 121,097 | <p>Let $P$ be a prime ideal in $k[x,y]$.</p>
<p>If $xy-1,x\in P$ then $1\in P$, contradiction.</p>
<p>If $xy-1,x-1\in P$ then $y-1\in P$, so we can take $P=(x-1,y-1)$. (Note that $xy-1\in(x-1,y-1)$.)</p>
|
724,012 | <p>not sure how to approach the following $\frac{\sin(a)}{\sin(b)}<\frac{a}{b}<\frac{\tan(a)}{\tan(b)}$ for $0<b<a<\pi/2$. Hints would be appreciated!</p>
| Robert Israel | 8,508 | <p>Maple does this using EllipticK:</p>
<p>> int(1/sqrt(3-cos(t)),t=0..Pi);
$$ \text{EllipticK}\left(\dfrac{1}{2}\sqrt{2}\right)$$</p>
<p>> select(has,[FunctionAdvisor(specialize,%)],GAMMA);
$$
[[{\rm EllipticK} \left( \dfrac12\sqrt {2} \right) =\dfrac12{\frac {{\pi }^{
3/2}}{ \Gamma \left( 3/4 \right) ^{2}}},\m... |
2,331,657 | <p>On the generalization of a <a href="https://math.stackexchange.com/questions/2329248/to-compute-frac12-pi-i-int-mathcalc-1zz22-dz-where-mathcalc">recent question</a>, I have shown, by analytic and numerical means, that</p>
<p>$$\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2+\cdots+z^{2n}|^2~dz =2n$$</p>
<p>where $\math... | Count Iblis | 155,436 | <p>You can simplify things as follows. We have:</p>
<p>$$f(z) = \sum_{k=0}^{2n}z^k = \frac{z^{2n+1}-1}{z-1}$$</p>
<p>Then on the unit circle, we have:</p>
<p>$$\left|f(z)\right|^2 = f(z)f^*(z) = f(z)f(z^*) = f(z)f\left(z^{-1}\right) = \frac{\left(z^{2n+1}-1\right)^2}{z^{2n}(z-1)^2}$$</p>
<p>The integral over the ... |
2,331,657 | <p>On the generalization of a <a href="https://math.stackexchange.com/questions/2329248/to-compute-frac12-pi-i-int-mathcalc-1zz22-dz-where-mathcalc">recent question</a>, I have shown, by analytic and numerical means, that</p>
<p>$$\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2+\cdots+z^{2n}|^2~dz =2n$$</p>
<p>where $\math... | Enredanrestos | 106,433 | <p>$$
\frac{1}{2\pi i}\int_0^{2\pi}\sum_{l=0}^{2n}e^{il\theta}\sum_{m=0}^{2n}e^{-im\theta}ie^{i\theta}d\theta=\frac{1}{2\pi}\sum_{l,m=0}^{2n}\int_0^{2\pi}\exp(i\theta(l-m+1))d\theta
$$</p>
<p>I think the last integral is zero unless $l-m+1=0$, which happens $2n$ times when $l$ and $m$ go between 0 and 2n. </p>
|
970,654 | <p>I'm interested in the growth of $$f(n):=\sum_{x=1}^{n-1} \left\lceil n-\sqrt{n^{2}-x^{2} } \right\rceil \quad \text{for}\quad n\rightarrow\infty $$</p>
<h3>Progress</h3>
<p>(From comments) I've got
$$\frac{f(n)}{n^2} \ge 1-n^{-1} (1+\sum\limits_{x=1}^{n-1} \sqrt{1-\frac{x^2}{n^2}} )$$ and $$\frac{f(n)}{n^2}\le... | hjhjhj57 | 150,361 | <p>You're almost there. The key idea is the definition you give:
$$
\times_{i \in I} G_i:= \lbrace x:I \to \bigcup_{i \in I} G_i \mid x(i) \in G_i , \forall i \in I\rbrace
$$
So, this new space you're defining is a set of <strong>functions</strong> (you can call one this functions $x$ or $(x)_{i\in I}$ as long as you k... |
604,459 | <p>The complex <em>solid spherical harmonics</em> can be defined as</p>
<p>$$
U_n^m(\boldsymbol{r}) = r^n P_n^m(\cos{\theta}) e^{im\phi},
$$</p>
<p>where $r,\theta,\phi$ are the usual spherical coordinates of $\boldsymbol{r}=(x,y,z)$. Note that $U_n^m(\boldsymbol{r})$ is a homogeneous polynomial in $x$, $y$, and $z$ ... | Andrew D. Hwang | 86,418 | <p>Here are details of the strategy outlined in the comments for recursively calculating the coefficients of $U_n^m(\mathbf{r})$. This doesn't fully answer the question, but perhaps it will be useful nonetheless.</p>
<p>To fix notation, let $(r, \phi, \theta)$ denote spherical coordinates with the physicists' conventi... |
238,702 | <p>I am a highschool freshman, and I really like to have goals for my life, one of the big ones is my career of choice. Previously, I have always wanted to be a programmer, and I have written a lot of code. But it seems to me that programming can get slightly bland, whereas math never disappoints me. So my question ... | Community | -1 | <p>There's a lot of options and it depends on what specifically in math you want to be doing. If you truly want to, as you said above, be sitting next to a whiteboard doing math all day it sounds like you would like to become a professor. I believe professors tend to be content and fulfilled people, though their sala... |
2,049,525 | <p>Show that the expression
$$(x^2-yz)^3+(y^2-zx)^3+(z^2-xy)^3-3(x^2-yz)(y^2-zx)(z^2-xy)$$
is a perfect square and Find its square root.</p>
| Robert Z | 299,698 | <p>The given expression can be factored as
$$[(x+y+z)(x^2+y^2+z^2-xy-yz-zx)]^2.$$</p>
|
2,285,276 | <p>A wheel is spun with the numbers $1, 2$, and $3$ appearing with equal probability of $1\over 3$ each. If the number $1$ appears, the player gets a score of $1.0$; if the number $2$ appears, the player gets a score of $2.0$, if the number $3$ appears, the player gets a score of $X$, where $X$ is a normal random varia... | Masacroso | 173,262 | <p>Just for the record I will add my own solution to complete the path of my question (based in the previous answers).</p>
<hr>
<p>Any possible functional $f_{c,k}$ over $x^k$ can be written as</p>
<p>$$f_{c,k}(x^k)=c\tag{3}$$</p>
<p>for any $c\in\Bbb R$. But observe that if we define $$f_{c,k}(x^k):=c B_k(x^k)$$</... |
1,756,593 | <p>I have to prove that the series $\displaystyle\sum_{n=1}^\infty {\ln(n^2) \over n^2 }$ converges. The ratio test is inconclusive, so I should use the comparison test, but which series should I compare it with? I tried ${1\over n}$, ${1 \over n^2 }$, but I need a bigger series which converges to prove that this one c... | Aloizio Macedo | 59,234 | <p>Consider $a_n:=\frac{\ln (n^2)}{n^2}$, $b_n:=\frac{1}{n^{3/2}}$. We have that</p>
<p>$$\lim\limits_{n \to \infty}\frac{a_n}{b_n}=\lim\limits_{n \to \infty}\frac{\ln (n^2)}{n^{1/2}}=\lim\limits_{n \to \infty}\big(2\cdot \frac{\ln (n)}{n^{1/2}}\big)=0.$$</p>
<p>Therefore, there exists $N$ such that $n>N$ implies ... |
1,662,218 | <p>Among the following, which is closest in value to $\sqrt{0.016}$?</p>
<p>A. $0.4$</p>
<p>B. $0.04$</p>
<p>C. $0.2$</p>
<p>D. $0.02$</p>
<p><strong>E. $0.13$</strong></p>
<p><strong>My Approach:</strong></p>
<p>$(\frac{16}{1000})^\frac{1}{2} = (\frac{4}{250})^\frac{1}{2} = \frac{2}{5\cdot\sqrt{10}} = \frac{\sq... | parsiad | 64,601 | <p>Though squaring each and checking works equally as well, here's another way to figure it out:
$$
\sqrt{0.016}=\frac{\sqrt{1000}}{\sqrt{1000}}\sqrt{0.016}=\frac{\sqrt{16}}{\sqrt{1000}}=\frac{4}{\sqrt{1000}}\approx\frac{4}{\sqrt{1024}}=\frac{4}{\sqrt{2^{10}}}=\frac{4}{2^{5}}=\frac{4}{32}=\frac{1}{8}=0.125
$$</p>
|
1,662,218 | <p>Among the following, which is closest in value to $\sqrt{0.016}$?</p>
<p>A. $0.4$</p>
<p>B. $0.04$</p>
<p>C. $0.2$</p>
<p>D. $0.02$</p>
<p><strong>E. $0.13$</strong></p>
<p><strong>My Approach:</strong></p>
<p>$(\frac{16}{1000})^\frac{1}{2} = (\frac{4}{250})^\frac{1}{2} = \frac{2}{5\cdot\sqrt{10}} = \frac{\sq... | colormegone | 71,645 | <p>If you rationalize the denominator of your ratio to get $ \ \frac{ \sqrt{10} }{25} \ $ , you can use the fact that $ \ \sqrt{10} \ $ is a little bigger than 3 to estimate that the number in question is a bit larger than $ \ \frac{3}{25} \ = \ 0.12 \ $ . No other choice but (E) is close to that.</p>
|
4,419,897 | <p>Let <span class="math-container">$(X,\| \cdot \|_X)$</span>, <span class="math-container">$(Y,\| \cdot \|_Y)$</span> be normed linear spaces and <span class="math-container">$T: X \rightarrow Y$</span> be a surjective linear operator. Show that the following are equivalent:</p>
<p>(1) <span class="math-container">$T... | Kavi Rama Murthy | 142,385 | <p>(1) implies (2): Take <span class="math-container">$U$</span> to be the open unit ball. Then <span class="math-container">$T(U)$</span> is an open set containing <span class="math-container">$T0=0$</span>. So there exists <span class="math-container">$r >0$</span> such that <span class="math-container">$B(0,r) \s... |
388,292 | <p>Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment.</p>
<p>Are there any books which gives a good account of associative algebras, and the representation theory of ass... | Julian Kuelshammer | 15,416 | <p>In addition to Mariano's list:</p>
<ul>
<li>Etingof et al.: Introduction to representation theory (which is also available <a href="http://www-math.mit.edu/~etingof/replect.pdf" rel="nofollow">online</a>).</li>
</ul>
<p>Addendum: And one should also mention (although as a starter its speed is too high)</p>
<ul>
<... |
2,838,938 | <p>Given:</p>
<ul>
<li>$\theta$ (a negative angle)</li>
<li>$v_0$ (initial velocity)</li>
<li>$y_0$ (initial height)</li>
<li>$g$ (acceleration of gravity)</li>
</ul>
<p>I want to find the range of a projectile (ignoring wind resistance)</p>
<p>Hours of searching have given no useful results. Those that I thought we... | David K | 139,123 | <p>Assuming a projectile is launched at angle $\theta$ from horizontal from a height $y_0$ above the ground, where the ground is assumed to be represented by the line $y = 0.$
The measurement of the angle is such that $\theta = 0$ if the projectile
initially is moving horizontally to the right, and otherwise the angle ... |
3,738,622 | <p>I need change the summation order in the double sum
<span class="math-container">$$
S_{m,n}=\sum_{j=0}^m \sum_{k=0}^n a_{j,k} x^{j-k} B_{m+n-j-k},
$$</span>
to separate <span class="math-container">$B$</span> and get somethink like to
<span class="math-container">$$
S_{m,n}=\sum_{s=0}^{m+n} \left( \sum_{k=0}^* *... | epi163sqrt | 132,007 | <p>Here is an answer in the same line as the answer of @YvesDaoust. In order to change the summation to the wanted form <span class="math-container">$S_{m,n}=\sum_{s=0}^{m+n} \left( \sum_{k=0}^* **** \right) B_{s}$</span> we sum up the terms corresponding to the example graphic below:</p>
<p> ... |
278,694 | <p>How to plot a function line with Markers:</p>
<pre><code>f[x_, m_] = m Sin[x]
g[x_, m_] = 2 m Cos[x]
Manipulate[
Plot[{f[x,m], g[x,m]}, {x, 0, 10},
PlotLegends -> Automatic], {m, 0, 5}]
</code></pre>
<p><a href="https://i.stack.imgur.com/wjVEs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com... | bmf | 85,558 | <p>Edit 3: please see below</p>
<pre><code>f[x_, m_] := m Sin[x]
g[x_, m_] := 2 m Cos[x]
r0 = Table[{x, f[x, m]}, {x, 0, 10}];
r1 = Table[{x, g[x, m]}, {x, 0, 10}];
Manipulate[
ListPlot[{r0, r1} /. m -> mm,
PlotMarkers -> {"\[FilledSquare]", "\[FilledCircle]"},
Joined -> True, PlotRa... |
877,355 | <p>The "modern"(schematic) definition of a projective variety is the following:</p>
<blockquote>
<p>Let $k$ be an algebraically closed field. A <em>projective variety</em> over $k$ is a closed subscheme of $\mathbb P^n_k=\textrm{Proj}(k[T_1,\ldots,T_n])$ (Remember the structure of $k$-scheme).</p>
</blockquote>
<p>... | user160609 | 160,609 | <p>I don't think many people would refer to an arbitrary closed subscheme of projective space as a <em>variety</em>. It is fairly standard to require varieties to be
(geometrically) reduced and irreducible. </p>
<p>One reason is that they then admit a field of rational functions, which records information about th... |
2,437,635 | <p>It is well know that $$\left(\sum_{r=1}^n r\right)^2=\sum_{r=1}^n r^3$$
i.e.
$$(1+2+3+\cdots+n)(1+2+3+\cdots+n)=1^3+2^3+3^3+\cdots+n^3$$
and this is usually proven by showing that the closed form for the sum of cubes is $\frac 14 n^2(n+1)^2$ which can be written as $\left(\frac 12 n(n+1)\right)^2$, and then noticin... | Bernard | 202,857 | <p>By <em>induction</em>:</p>
<p>You have to know the high-school formula $\;\displaystyle\sum_{r=1}^n r=\frac{n(n+1)}2$.</p>
<p>Suppose $\;\Bigl(\displaystyle\sum_{r=1}^n r\Bigr)^2=\sum_{r=1}^n r^3 $ for some $n\ge 1$. Then
\begin{align}
\Bigl(\mkern1mu\sum_{r=1}^{n+1} r\Bigr)^2&=\Bigl(\displaystyle\sum_{r=1}^n ... |
3,936,782 | <p>I am interested in the following problem and its generalizations. Say we are on the real axis, and we have <span class="math-container">$n$</span> points <span class="math-container">$x_1, \ldots, x_n$</span> on this line. I would like to minimize/maximize the length someone would walk to go through all of them in a... | Alex Ravsky | 71,850 | <p>Enumerate the points providing <span class="math-container">$x_1<x_2<\dots<x_n$</span>.</p>
<blockquote>
<p>to minimize, we can pass by all of them from the leftmost one to the rightmost one</p>
</blockquote>
<p>Yes, any path going through the points has a fragment connecting <span class="math-container">$... |
1,821,426 | <p>I would like to know if a homogeneous linear differential equation, with variable coefficients which are periodic, is stable.</p>
<p>So the differential equation can be written as,</p>
<p>$$
\dot{y}(t)=A(t)y(t), \tag{1}
$$</p>
<p>$$
A(t+T)=A(t). \tag{2}
$$</p>
<p>I would suspect that the solution could be of the... | Kwin van der Veen | 76,466 | <p>The exponential of a matrix can be defined as</p>
<p>$$
e^{F(t)} = \sum_{n=0}^\infty \frac{\left(F(t)\right)^n}{n!} = I + F(t) + \frac{1}{2}\left(F(t)\right)^2 + \frac{1}{6}\left(F(t)\right)^3 + \cdots.
$$</p>
<p>When taking the derivative of this infinite sum you get,</p>
<p>$$
\frac{d}{dt}e^{F(t)} = \dot{F}(t) ... |
327,201 | <p>I'm reading Behnke's <em>Fundamentals of mathematics</em>:</p>
<blockquote>
<p>If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology.</p>
</blockquote>
<p>I got curious on this: Are there infinite sets of axioms? The only thing I could think about is the possible ex... | Peter Smith | 35,151 | <p>As others have noted, first order PA has an infinite number of axioms. But it is worth stressing that this is <em>not</em> just a superficial feature of the usual mode of presenting the induction axioms via a schema or template. It is provable that there is no finitely axiomatized theory in the language of PA whose ... |
327,201 | <p>I'm reading Behnke's <em>Fundamentals of mathematics</em>:</p>
<blockquote>
<p>If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology.</p>
</blockquote>
<p>I got curious on this: Are there infinite sets of axioms? The only thing I could think about is the possible ex... | Petr | 37,490 | <p>Perhaps surprisingly even the classical (Łukasiewicz's) axiomatization of propositional logic has an infinite number of axioms. The axioms are all <a href="https://en.wikipedia.org/wiki/Substitution_instance">substitution instances</a> of</p>
<ul>
<li>$(p \to (q \to p))$</li>
<li>$((p \to (q \to r)) \to ((p \to q) ... |
1,797,788 | <p>Looking at a continuous projection $B\times F\rightarrow B$, are slices $\left\{b\right\}\times F\subset B\times F$ homeomorphic to $F$?</p>
| DanielWainfleet | 254,665 | <p>(1). Let $T_B$ be the topology on $B$ and $T_F$ be the topology on $F.$ The Tychonoff product topology $T_{B\times F}$ on $B\times F$ is defined to be the weakest topology such that the projections $\pi_B:B\times F$ and $\pi_F:B\times F$ are continuous.</p>
<p>(2). So $c\times f=$ $(c\times F)\cap (B\times f)=$ $(... |
1,636,807 | <p>Ok, here is what I think. Please correct me if I am wrong.
$$\sqrt{9} \neq 3$$
and also
$$\sqrt{9} \neq -3$$</p>
<p>Now let's assume, that above statements are false, then we have $-3 = \sqrt{9} = 3$ and since $3 \neq -3$ the assumption must be wrong.
Ok, square root must be equal to 3 and -3 at the same time. As o... | AmpleMimic | 235,107 | <p>It's not a mistake to plot the negative branch of the graph $\sqrt{x}$. Rather it arises from the convention that the sign $\sqrt{x}$ means <em>the positive root of x</em>, whilst $-\sqrt{x}$ means <em>the negative root of x</em>. If we wish our graph to represent a <em><a href="https://en.wikipedia.org/wiki/Functio... |
3,743,691 | <p>I am trying to describe this quotient group <span class="math-container">$\mathbb{Z}\times\mathbb{Z}/\mathbb{3Z}\times\mathbb{Z}$</span> Let's denote with <span class="math-container">$A$</span> and <span class="math-container">$B$</span> respectively <span class="math-container">$\mathbb{Z} \times \mathbb{Z} $</spa... | ancient mathematician | 414,424 | <p>@lhf has given the right solution, but your comment suggests you don't understand it.</p>
<p>Here's a more elementary way.</p>
<p>We want to give a list of the distinct cosets <span class="math-container">$a+B$</span> - I use your notation.</p>
<p>When does <span class="math-container">$a+B=x+B$</span>?. Answer, if ... |
121,522 | <p>I have 2 lists. For example,</p>
<pre><code>list1 = {{a,b}, {c, d}, {e, f}, {g, h}}
list2 = {{1}, {2}, {3}, {4}}
</code></pre>
<p>I want to merge them such that I get the result:</p>
<pre><code>list3 = {{1,a,b}, {2,c, d}, {3,e, f}, {4,g, h}}
</code></pre>
<p>and the method generalizes to a large number of subli... | nadlr | 29,902 | <p>One way could be</p>
<pre><code>list3 = Partition[Flatten@Riffle[list2,list1], 3]
</code></pre>
<p>There a few things to watch out for though. <code>Riffle</code> truncates lists if the two arguments are not of the same length. Also, a more general way to generate your <code>list2</code> might be to use <code>Rang... |
3,254,331 | <blockquote>
<p>Prove <span class="math-container">$$\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$$</span></p>
</blockquote>
<p>Out of boredom, I decided to play with some integrals and Inverse Symbolic Calculator and accidentally found this to my surprise</p>
<p><span class="math-container">$... | Lai | 732,917 | <p>Letting <span class="math-container">$y=\arctan \frac{1}{x} $</span> yields
<span class="math-container">$$
\begin{aligned}
I=& \int_0^{\frac{\pi}{2}} y^2 \csc ^2 y d y \\
&=-\int_0^{\frac{\pi}{2}} y^2 d(\cot y) \\
&=-\left[y^2 \cot y\right]_0^{\frac{\pi}{2}}+2 \int_0^{\frac{\pi}{2} } y \cot yd y \\
&... |
203,033 | <p>I want to plot an amplitude vs frequency with this formula:
<a href="https://i.stack.imgur.com/YQFJV.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YQFJV.jpg" alt="enter image description here"></a></p>
<p>By using this data:</p>
<p><a href="https://i.stack.imgur.com/B0VX3.jpg" rel="nofollow no... | Αλέξανδρος Ζεγγ | 12,924 | <p>The input data is arranged as</p>
<pre><code>data = {{4.19, 3.93, 3.7, 3.49, 3.31, 3.31, 3.14, 2.99, 1.75}, {0.18, 0.35, 0.45, 1.06, 1.13, 0.77, 0.57, 0.34, 0.14}}\[Transpose];
</code></pre>
<p>Then fit a nonlinear model</p>
<pre><code>model = (a ω^2) / Sqrt[(ω^2 - ωe^2)^2 + 4 γ^2 ωe^2];
fittedparameters = FindFi... |
3,291,889 | <p>All rings are commutative ring with unity.</p>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are two <span class="math-container">$R$</span>-algebras and <span class="math-container">$I$</span> and <span class="math-container">$J$</span> are two ideals of <span class="ma... | TonyK | 1,508 | <p>You can use <span class="math-container">$\cos 3x=4\cos^3x-3\cos x$</span>, along with <span class="math-container">$\cos\pi=-1$</span>. If we let <span class="math-container">$t=\cos\pi/3$</span>, we have
<span class="math-container">$$4t^3-3t=-1$$</span>
<span class="math-container">$4t^3-3t+1$</span> factorises a... |
374 | <p>Should <a href="https://mathematica.stackexchange.com/questions/tagged/strings" class="post-tag" title="show questions tagged 'strings'" rel="tag">strings</a> be a synonym of <a href="https://mathematica.stackexchange.com/questions/tagged/text" class="post-tag" title="show questions tagged 'text'" re... | István Zachar | 89 | <p>Following Sjoerd's argument, I would sugges the removal of <a href="https://mathematica.stackexchange.com/questions/tagged/text" class="post-tag" title="show questions tagged 'text'" rel="tag">text</a>, as if it is understood as a style, the question should be tagged as <a href="https://mathematica.stackexch... |
1,525,340 | <p>I have a fairly simply question which I am not sure about. A 3 digits number is being chosen by random (100-999). What is the probability of getting a number with two identical digits ? (like 101). Thank you !</p>
| fleablood | 280,126 | <p>There are ten numbers of the form aa*. 9 of them are not aaa. There are 9 possible chooses for a. So there are 81 numbers where the first two digits are the same.</p>
<p>By exact same reasoning there are 81 numbers where where the first and last digit are the same. </p>
<p>Counting the numbers where the last tw... |
229,915 | <p>Let $A,B\subseteq\mathbb R^d$ with $A$ closed such that $A\subset\overline{B}$. Does there exist $B'\subset B$ such that $A=\overline{B'}$?</p>
| Austin Mohr | 11,245 | <p>Here is a counterexample that is essentially the same as Sanchez's answer, but a dimension simpler. Let</p>
<ul>
<li>$A = \{0 \}$ and</li>
<li>$B = (0,1)$,</li>
</ul>
<p>so that $A \subseteq \overline{B}$. The only set whose closure is a singleton is the set itself, but that is not a valid choice in our instance s... |
419,091 | <blockquote>
<p><span class="math-container">$G$</span> is an infinite group.</p>
<ol>
<li><p>Is it necessary true that there exists a subgroup <span class="math-container">$H$</span> of <span class="math-container">$G$</span> and <span class="math-container">$H$</span> is maximal ?</p>
</li>
<li><p>Is it possible that... | Jared | 65,034 | <p>As Prism states in the comments, the Prüfer group is an example of a group with no maximal subgroup. Define $\mathbb{Z}(p^{\infty})$ to be the set of all $p^n$-th roots of unity as $n$ ranges over the natural numbers. The operation is multiplication.</p>
<p>It can be shown that any subgroup of $\mathbb{Z}(p^{\inf... |
419,091 | <blockquote>
<p><span class="math-container">$G$</span> is an infinite group.</p>
<ol>
<li><p>Is it necessary true that there exists a subgroup <span class="math-container">$H$</span> of <span class="math-container">$G$</span> and <span class="math-container">$H$</span> is maximal ?</p>
</li>
<li><p>Is it possible that... | Matteo Vannacci | 397,291 | <p>If the group <span class="math-container">$G$</span> is infinite and finitely generated then the answer is YES. Fix <span class="math-container">$g\in G$</span>, then by Zorn's lemma there exists a (proper) maximal subgroup that contains <span class="math-container">$g$</span>.</p>
<p>Here is a proof of the above. L... |
907,336 | <p>show that:
$$\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}-\left(\dfrac{1}{a+b+c+1}+\dfrac{1}{b+c+d+1}+\dfrac{1}{c+d+a+1}+\dfrac{1}{d+a+b+1}\right)\ge 0$$
where $abcd=1,a,b,c,d>0$</p>
<p>I have show three variable inequality</p>
<p>Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$.... | RE60K | 67,609 | <p>Let's use Lagrange Multipliers:
Let use take $f$ to be:
$$f(a,b,c,d)=\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}-\left(\dfrac{1}{a+b+c+1}+\dfrac{1}{b+c+d+1}+\dfrac{1}{c+d+a+1}+\dfrac{1}{d+a+b+1}\right)$$
subject to constraint $abcd=1$:
$$g(a,b,c,d)=abcd=1$$
Here are the equations we need to solve:
$$... |
2,147,807 | <p>Essentially what the title asks. For an argument $x$, how can I analytically acquire values for the function:
$$
f(x)=\sum_{k=0}^{\infty}\frac{x^{2k+1}}{(2k+1)(k!)}
$$
Again, it is important that I know how to do this <strong>analytically</strong>, as there are other series comparable to this one that I also wish to... | Jack D'Aurizio | 44,121 | <p>$$f(x)=\int_{0}^{x}e^{t^2}\,dt =\int_{0}^{x}\exp\left(x^2+t^2-2tx\right)\,dt=xe^{x^2}\int_{0}^{1}e^{-t^2 x^2}\,dt = \frac{\sqrt{\pi}}{2} e^{x^2}\,\text{Erf}(x)$$
has the following <a href="http://functions.wolfram.com/GammaBetaErf/Erf/10/01/" rel="nofollow noreferrer">continued fraction representation</a>:</p>
<p>$... |
1,893,356 | <p>This is what I came up with so far:</p>
<p>Inductive step: assume $2^n > n^4$.
Need to prove $2^{n+1} > (n+1)^4$
$$
2^{n+1} = 2 \cdot 2^n > 2 \cdot n^4\\
(2 \cdot n^4)^{1/4} = (2)^{1/4} \cdot n > n+1 \implies 2n^4 > (n+1)^4 \implies 2^n > (n+1)^4
$$</p>
<p>Is there a better way to solve this prob... | Daniel W. Farlow | 191,378 | <p>As others have noted, your induction proof ultimately suffers from establishing what your base case is. Your deduction that
$$
2^{1/4}(n)>n+1
$$
requires $n\geq6$, as Barry notes, but your inequality is not even true until $n\geq17$, as lulu notes. This means that your base case should be, at minimum, for when $n... |
347,186 | <p>I have this question:</p>
<blockquote>
<p>Let $x, n$ be integers with $n \geq 2$ and $n$ not dividing $x$. Show that the order o($\bar{x}$) of $x \in Z_n$ is
$o(\bar{x})= \frac{n}{HCF(x, n)}$</p>
</blockquote>
<p>I've been thinking about it for ages but I still don't get why. A hint would be appreciated.</p>
| Michiel | 53,881 | <p><strong>Show convergence</strong></p>
<p>You can use the <a href="http://en.wikipedia.org/wiki/Integral_test_for_convergence" rel="nofollow">integral tests for convergence</a> on this problem, because the integral of this function is fairly easy to determine. The integral test states that your series of the form:
$... |
113,295 | <p>Suppose $x_1, x_2$ are IDD random variables uniformly distributed on the interval $(0,1)$. What is the pdf of the quotient $x_2 / x_1$?</p>
| Antoni Parellada | 152,225 | <p>Late to the party, but I want to have one post on SE that addresses this question in its raw formulation (no secondary issues or finer points in many similar questions), and taking advantage of the comment to the accepted answer about using triangles.</p>
<p>Renaming the rv's, <span class="math-container">$X$</span>... |
1,744,160 | <p>I want to evaluate </p>
<p>$$\lim _{x\to \pi }\left(\frac{\sin x}{\pi ^2-x^2}\right)$$</p>
<p>without using L'Hopital's rule or Taylor series. My thinking process was something like this: in order to get rid of the undefined state, I need to go from $\sin x$ to $\cos x$. I tried this substitution: $t = \frac{\pi}{... | DeepSea | 101,504 | <p><strong>hint</strong>: $\sin x = \sin(\pi - x)$, and $\pi^2 - x^2 = (\pi -x)(\pi + x)$. </p>
<p>Also use $\dfrac{\sin(\pi - x)}{\pi - x} \to 1$ when $x \to \pi$.</p>
|
246,445 | <p>I often see the sentence "let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with a certain distribution". But given a random variable $X$ on a probability space $\Omega$, how do I know that there is a sequence of INDEPENDENT random variables of the same distribution on $\Omega$?</p>
| S.D. | 20,309 | <p>The easiest way to show the existence in our case is to construct such a probability space. The intuition is that it is easy to define the joint probability measure on some "simple" sets using the iid property and then one leverages the celebrated <a href="http://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_extension... |
2,904,878 | <p>I want to prove that
let S be the set on which group G operates. Let H ={ g∈G | g.s=s for all s∈S} prove that H is normal subgroup of G.
This group action gives an homomorphism whose kernal is H.
Then it follows directly from statement that H is normal.</p>
<p><strong>How to prove that if the subgroup is the kerne... | random | 513,275 | <p>For the modified question the answer is yes.</p>
<p>Let m and M be the minimum and maximum of $f$ on $[a,b]$ and assume, without loss of generality, that $g(x)\gt 0$. Then $m\int_{a}^{b} g(x)\,dx=\int_{a}^{b} m\,g(x)\,dx\le \int_{a}^{b} f(x)g(x)\,dx\le\int_{a}^{b} M\,g(x)\,dx=M\int_{a}^{b} g(x)\,dx$, so
$$m\le{\int... |
1,440,988 | <p>Show that the set $ \left\{\dfrac{1}{x^2-1}\mid x\in(0,1)\right\} $ is not bounded.</p>
<p>We should assume that it is bounded, then try to prove the opposite, but I don't know where to start. </p>
| Arthur | 15,500 | <p>This is one way to do it:</p>
<ol>
<li>Let $M< 0$ be a candidate for a bound</li>
<li>Solve $\frac{1}{x^2-1}=M-1$ (be careful with multiple solutions, make certain that one of them is in $(0,1)$)</li>
<li>Conclude $M$ is not a bound after all</li>
</ol>
|
1,703,116 | <p><em>Suppose you repeatedly roll a six-sided dice. What's the probability you'll see a "1" before you see a "6"?</em> </p>
<p>The Pr of rolling a "1" is the same as the probability of rolling a '6'.
So should the probability be $$1 - [(1-p)^{k-1}p]?$$ </p>
| Masacroso | 173,262 | <p>The probability to see a $1$ before a $6$ is the sum of probabilities for every possible number of throws i.e. </p>
<p>$$\Pr[1\text{ before a }6]=\sum_{n\ge 0}\left(\frac16\right)\cdot\left(\frac46\right)^n=\frac16\cdot\frac1{1-\frac46}=\frac12$$</p>
|
1,703,116 | <p><em>Suppose you repeatedly roll a six-sided dice. What's the probability you'll see a "1" before you see a "6"?</em> </p>
<p>The Pr of rolling a "1" is the same as the probability of rolling a '6'.
So should the probability be $$1 - [(1-p)^{k-1}p]?$$ </p>
| Win Vineeth | 311,216 | <p>Let's see it this way- </p>
<p>Both $1$ and $6$ are not possible together.</p>
<p>So, Probability of $1$ before $6 +$ Probability of $6$ before $1$ = $1$.</p>
<p>For a fair die, both are equal. So, each is $\frac 12$</p>
|
172,432 | <p>Jyrki Lahtonen has suggested I write a blog post relating binary quadratic forms to quadratic field class numbers, <a href="https://math.stackexchange.com/questions/209512/binary-quadratic-forms-over-z-and-class-numbers-of-quadratic-%EF%AC%81elds/209543#comment1727526_209543">https://math.stackexchange.com/questions... | Will Jagy | 3,324 | <p>Alright, i have gotten the binary forms part of the story. I may or may not ever know the field side of things, that's life.</p>
<p>From experiments with the website <a href="http://www.numbertheory.org/php/classnopos.html" rel="nofollow">http://www.numbertheory.org/php/classnopos.html</a> and comparison with my ow... |
4,226,997 | <p>If I perform vector multiplication as below and then want to notate summation along the last axis of the resulting matrix, how do I show that in summation notation?</p>
<p>Example:
<span class="math-container">$$\pmatrix{ 2 \\ 3 } \pmatrix{1 & 1 & 1 \\ 2 & 2 & 2} = \pmatrix{2 & 2 & 2 \\ 6 &am... | user | 505,767 | <p>It should be</p>
<p><span class="math-container">$$\pmatrix{ 2 & 0\\0 & 3 } \pmatrix{1 & 1 & 1 \\ 2 & 2 & 2} = \pmatrix{2 & 2 & 2 \\ 6 & 6 & 6} = \mathbf{R}\in \mathbb{R}^{2 \times 3}$$</span></p>
<p>for the matrix product and for the vector</p>
<p><span class="math-container"... |
621,210 | <p>As it is clear from the title, what is the cardinality of the set $\{ (x,y) \in \Bbb{R}^2 \; | \; y > x > 0 , x^x = y^y \}$?</p>
| Michael Albanese | 39,599 | <p>Let $f(x) = x^x$. Then $f'(x) = x^x(\ln x - 1)$. </p>
<p>For $0 < x < e^{-1}$, $f'(x) < 0$ so $f$ is strictly decreasing on $(0, e^{-1})$. For $e^{-1} < x < \infty$, $f'(x) > 0$ so $f$ is strictly increasing on $(e^{-1}, \infty)$. We also have a critical point for $f(x)$ at $x = e^{-1}$. In fact, ... |
4,121,836 | <p>Let <span class="math-container">$R,R',S$</span> be finite (unital, associative) rings. Assume that <span class="math-container">$R \times S \cong R' \times S$</span> as rings. Does it follow that <span class="math-container">$R \cong R'$</span> as rings ? What if we assume <span class="math-container">$R,R',S$</spa... | Eric Wofsey | 86,856 | <p>Let <span class="math-container">$R$</span> be a ring with finitely many central idempotents. The set of central idempotents of <span class="math-container">$R$</span> then forms a finite Boolean algebra, which is isomorphic to the power set of a finite set. Under this isomorphism, the singleton sets correspond to... |
4,121,836 | <p>Let <span class="math-container">$R,R',S$</span> be finite (unital, associative) rings. Assume that <span class="math-container">$R \times S \cong R' \times S$</span> as rings. Does it follow that <span class="math-container">$R \cong R'$</span> as rings ? What if we assume <span class="math-container">$R,R',S$</spa... | Martin Brandenburg | 1,650 | <p>Here is an alternative solution, based on the following "Fake Yoneda Lemma". It does not just hold for rings, but for all algebraic objects of a given type (with the same proof), see my answer <a href="https://math.stackexchange.com/questions/614496">here</a>.</p>
<blockquote>
<p>Let <span class="math-con... |
4,361,279 | <p>In my book's table for antiderivatives of some functions, I came across the following,</p>
<p><span class="math-container">$$\int{e^{ax}dx=\frac{1}{a}e^{ax}} + C, \qquad a \neq0\tag{1}$$</span></p>
<p>I can't understand the reasoning behind the condition <span class="math-container">$a\neq0$</span>. Also,</p>
<p><sp... | Angel | 109,318 | <p>The conundrum here is that <span class="math-container">$$\int_0^te^{ax}\,\mathrm{d}x=\begin{cases}\frac{e^{at}-1}{a}&a\neq0\\t&a=0\end{cases}.$$</span> This is because for <span class="math-container">$a\neq0,$</span> it is true that <span class="math-container">$$\frac{\mathrm{d}}{\mathrm{d}t}\frac1{a}e^{a... |
2,017,133 | <p>$$\sum_{i = 1}^n (2i+3) = n(n+4)$$
for all n >= 1.</p>
<p>Was a homework problem that was given no solution. Was told last lines weren't correctly written.
My attempt:
Let P(n) = n(n+4) for all n >= 1
Basis Step: P(2) = 2(6) = 12 >= 1
Inductive Step:
$$\sum_{(i=1}^{k+1} (k+1)(k+5)$$</p>
<p>= k(k+4) + (k+1)
= $$k... | John | 7,163 | <p>Assume the statement holds for $n=k$:</p>
<p>$$\sum_{i = 1}^k (2i+3) = k(k+4).$$</p>
<p>Write out the left side for $n=k+1$ and split off the last term:</p>
<p>$$\sum_{i = 1}^{k+1} (2i+3) = \left[\sum_{i = 1}^{k} (2i+3)\right] + 2(k+1) + 3.$$</p>
<p>The manipulation above is just algebra, but we now have a piece... |
789,420 | <p>If $M$ is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is properly a manifold).</p>
| Moishe Kohan | 84,907 | <p>This is an addendum to Brad's answer (including references as requested). </p>
<p>Every topological manifold (with or without boundary) is locally contractible (this is immediate from the definition). In particular, it is ANR (absolute neighborhood retract) - this requires a bit of a proof, which can be found in Bo... |
1,113,516 | <p>I learnt that $(\mathbb{R},\times) < (\mathbb{C},\times)$, Which means the first is a subgroup of the second one. But in the first group inequality is defined, while it's not in the latter. This got me thinking, is there anything about a group which tells you which symbols ($=,\neq,<,\geq,\cdots$) are defined ... | mathcounterexamples.net | 187,663 | <p>You should find a way to write the problem as a matrix problem and then to reduce the matrix to a simpler form.</p>
|
1,665,714 | <p>I am looking for the following sets for all $z \in \mathbb{C}$</p>
<p>$$\{z: \cos(z)=0\} \text{ and } \{z: \sin(z)=0\}$$</p>
<p>I believe the best way to do this is consider the exponential form so</p>
<p>$$\cos(z)=\frac{e^{iz}+e^{-iz}}{2} \text{ and } \sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$$</p>
<p>So for $\cos(z)=0... | Raymond Manzoni | 21,783 | <p>Let's examine the most significant terms of the product for $n\gg 1$ :
\begin{align}
\tag{1}P_n&:=\prod_{k=1}^{\infty}\zeta (2kn)\\
&=\zeta (2n)\;\zeta (4n)\cdots\\
\tag{*}&=(1+2^{-2n}+3^{-2n}+4^{-2n}+o(4^{-2n}))\;(1+2^{-4n}+o(4^{-2n}))\;(1+o(4^{-2n}))\\
\tag{2}P_n&=1+2^{-2n}+3^{-2n}+2\cdot 4^{-2n}+o... |
595,864 | <blockquote>
<p><em>Theorem</em>: For every positive integer n greater than $2$, then $\phi(n)$ is an even integer.</p>
</blockquote>
<p>I know this theorem and the same is used much, but I was curious how it would be to demonstrate it, show me anyone know how or where to find it?</p>
| Tobias Kildetoft | 2,538 | <p>One way to prove it is the following: When $n\geq 3$ we have that $-1\neq 1$ in $\mathbb{Z}/n\mathbb{Z}$, so $-1$ is an element of order $2$ in $(\mathbb{Z}/n\mathbb{Z})^*$ and the result follows by Lagrange since $\varphi(n)$ is the order of that group.</p>
|
595,864 | <blockquote>
<p><em>Theorem</em>: For every positive integer n greater than $2$, then $\phi(n)$ is an even integer.</p>
</blockquote>
<p>I know this theorem and the same is used much, but I was curious how it would be to demonstrate it, show me anyone know how or where to find it?</p>
| CODE | 75,231 | <p><strong>Lemma</strong>: If $\gcd(n,k)=1$ then $\gcd(n,n-k)=1$.<br><br> Proof: Suppose $d|n,d|n-k$ so $d|n-(n-k)$ and $d|k$. So we have $d|k$ and $d|n$ and according to the first part, $d=1$. Thus, $\gcd(n,n-k)=1$.<br> This means you count each $k$ that $\gcd(n,k)=1$ twice.</p>
|
347,523 | <p>MacLane and Moerdijk's <a href="http://books.google.co.uk/books?id=SGwwDerbEowC&dq=mac+lane+moerdijk&source=gbs_navlinks_s" rel="nofollow">Sheaves in Geometry and Logic</a> has a section on Continuous Group Actions (Sec. III.9). On page 152, there is an isomorphism displayed:</p>
<p>$$Hom_G(G/U, X) \cong X... | Qiaochu Yuan | 232 | <p>$G/U$ has a distinguished element, namely the coset of the identity. A $G$-morphism $G/U \to X$ is completely determined by where it sends this coset, and the possible points in $X$ it can be sent to are precisely points fixed by the action of $U$. (This should make sense on the point-set level, and then one only ha... |
41,725 | <p>Asked this question in a different formulation in cstheory, got some pointers, but no definitive answer ... maybe someone here knows.</p>
<p>Suppose I need to compute the factorization of a block of consecutive numbers N, N+1, ... N+n. </p>
<p>As far as I understand, there are two extreme cases. On one hand, if n ... | tdnoe | 2,360 | <p>Using a sieve of Eratosthenes approach, it is easy to create a list $S$ of the smallest prime factor of every number less than $N$. Then to factor any $n < N$, we just recursively look up the factors: let $n_1=n$, $f_i = S(n_i)$, and $n_{i+1} = n_i/f_i$. With 4 GB of memory, numbers less than ${10}^9$ are quickl... |
2,152,485 | <blockquote>
<p>Use the formula for the angle between two vectors $v$ and $u$ to show $$|v \times(u \times v)|=\sqrt{((u.u)(v.v)^2-(v.v)(u.v)^2}$$</p>
</blockquote>
<p>I have endlessly used $u.v = |u||v|\cos(\theta)$ but to no avail:</p>
<p>This was my square root after simplifying</p>
<p>$$ \sqrt{\cos^3(\theta)|u... | Ángel Mario Gallegos | 67,622 | <p>Let $\theta$ the angle between $u$ and $v$. Since $u\times v$ is a vector orthogonal to both of $u$ and $v$ we have
\begin{align*}
|v\times (u\times v)| &=|v||u\times v|\\
&=|v||u||v||\sin \theta|\\
&=|v||u||v|\sqrt{1-\cos^2\theta}\\
&=|v|\sqrt{|u|^2|v|^2-(u\cdot v)^2}\\
&=\sqrt{|u|... |
122,986 | <p>I've read the axioms of a field. To understand the generality of the axioms, could you give me an example of a field which is not (isomorphic to) a subset of complex number (with or without modulus operations).</p>
| marlu | 26,204 | <p>$\mathbb Z/p\mathbb Z$ is a (finite) field for every prime $p$. It is not isomorphic to a subfield of $\mathbb C$ since it has characteristic $p$, i.e. in $\mathbb Z/p\mathbb Z$ we have $1+\ldots+1 = 0$. In $\mathbb C$, however, such an equation is not true.</p>
|
711,802 | <p>Can't find any proof in Shannon's 1948 paper. Can you provide one or disproof?</p>
<p>Thank you.</p>
<p>P.S.</p>
<p>$H(x)$(or $H(y)$) is the entropy of messages produced by the discrete source $x$(or $y$).</p>
<p>$H(x,y)$ is the joint entropy.</p>
<p>They are all entities in information theory.</p>
| leonbloy | 312 | <p>Both cases are possible.</p>
<pre><code> H(X) [=========================]
H(Y) [===============] (original)
H(X,Y) [==================================]
H(X) [=========================] constant
H(Y) [=======] less than original
H... |
3,241,924 | <p>Let <span class="math-container">$T: V\to V$</span> and <span class="math-container">$V$</span> be a vector space of finite dimension.</p>
<p>Let <span class="math-container">$M$</span> be the minimal polynomial of <span class="math-container">$T$</span> and write it as <span class="math-container">$M = M_1 \cdot M_... | Berci | 41,488 | <p>One inclusion is immediate: <span class="math-container">$U\cap W_i\subseteq U$</span> for all <span class="math-container">$i$</span>, hence <span class="math-container">$\bigoplus_i U\cap W_i\subseteq U$</span>.</p>
<p>For the converse, we have to assume that the polynomials <span class="math-container">$M_i$</sp... |
97,922 | <p><code>MyDataSet[Select[#point[Fsr] == 1 &]]</code> returns the row I want to delete, now I am trying what looks to me the most logic command to remove these rows, which is
<code>MyDataSet[Delete[#point[Fsr] == 1 &]]</code>
<code>MyDataSet[DeleteCases[#point[Fsr] == 1 &]]</code>, but neither works .... a... | alancalvitti | 801 | <p>Here's one way, not necessarily the most efficient:</p>
<p><code>MyDataSet[Select[#point[Fsr] == 1 & /* Not]]</code></p>
|
1,118,338 | <p>Let $\{b_n\}$ be a sequence with limit $\beta$. Show that if $B$ is an upper bound for $\{b_n\}$, then $\beta \leq B$.</p>
<p>What I have:</p>
<p>Assume that $\beta>B$, so $\beta-B>0$. Since $\{b_n\}$ converges to $\beta$ we know that there exists $N\in \mathbb{B}$ such that $|b_n-\beta|<\beta -B$ for al... | Bernard | 202,857 | <p>Remember $ \lvert b_n-\beta\rvert=\max\{b_n-\beta,\beta-b_n\} $ so the inequality $\lvert b_n-\beta\rvert <\beta-B$ implies $\beta-b_n<\beta-B$, which implies $b_n>B$, so that $B$ wouldn't be an upper bound for $b_n$.</p>
|
48,227 | <p>I am stuck with this problem,</p>
<blockquote>
<p>A function $f(x)$ is defined as $f(x)
= \sinh(x)$. Another function $g(x)$ is such that $f(g(x)) = x$.</p>
<p>Find the value of $\large
g(\frac{e^{2012}-1}{2e^{1006}})$</p>
</blockquote>
<p>I tried representing $f(x) =\large \frac{e^{2x}-1}{2e^{x}}$,and th... | jspecter | 11,844 | <p>The idea is to notice $\mathrm{sinh}:\mathbb{R} \rightarrow \mathbb{R}$ is a bijection. Hence, any right inverse is a left inverse. It follows </p>
<p>$$g(\frac{e^{2012}-1}{2e^{1006}}) = g\circ f(1006) = 1006$$ </p>
|
2,452,466 | <p>This is the problem: </p>
<p>$$\int \frac{x+2}{x^2+x}$$</p>
<p>I am supposed to write $(y+2)$ as $\frac{1}{2}(2x+1)+\frac{3}{2}$</p>
<p>$$\frac{1}{2}\int \frac{2x+1}{x^2+x}+\frac{3}{2}\int \frac{3}{x^2+x}$$</p>
<p>Fine, is there supposed to be some way for me to know which fractions to use here or am I supposed ... | Error 404 | 206,726 | <p>You can do this : $$\frac {(x+1)+1}{x(x+1)}=\frac 1x + \frac 1{x(x+1)}=\frac 1x + \frac {(x+1-x)}{x(x+1)}=\frac 1x + \left(\frac 1x - \frac 1{x+1}\right)=\frac 2x - \frac {1}{x+1}.$$</p>
|
28,027 | <p>Let $L/K$ be a finite Galois extension with Galois group $G$ and $V$ a $L$-vector space, on which $G$ acts by $K$-automorphisms satisfying $g(\lambda v)=g(\lambda) g(v)$. It is known that the canonical map</p>
<p>$V^G \otimes_K L \to V$</p>
<p>is an isomorphism. However, I can't find any short and nice proof for t... | Dustin Clausen | 3,931 | <p>I dunno about the explicit inverse, but there are two simple ways I know of showing the map is an isomorpism. The first is just to apply Grothendieck's faithfully flat descent theory to L/K -- one identifies the descent data on an L-vector space as exactly the kind of Galois action you describe. The other, maybe m... |
28,027 | <p>Let $L/K$ be a finite Galois extension with Galois group $G$ and $V$ a $L$-vector space, on which $G$ acts by $K$-automorphisms satisfying $g(\lambda v)=g(\lambda) g(v)$. It is known that the canonical map</p>
<p>$V^G \otimes_K L \to V$</p>
<p>is an isomorphism. However, I can't find any short and nice proof for t... | KConrad | 3,272 | <p>Martin, <a href="http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/galoisdescent.pdf" rel="nofollow">http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/galoisdescent.pdf</a>
is a handout on this kind of stuff
and Theorem 2.14 there gives a proof of the bijection between $K$-forms of $V$ and $G$-structures... |
720,084 | <p>I read in Gittman, Hamkins, et al, that ZFC without the power set causes the axiom of replacement to fail. Yet I also read (generally, throughout the literature, but mostly in connection with Cantor's theorem) that the power set is generated by <em>replacing</em> one set with another. Is the term replacement in the ... | r9m | 129,017 | <p>$\displaystyle\int \dfrac{\cos x + x \sin x}{x(x + \cos x)}dx$</p>
<p>$\displaystyle = \int \dfrac{x+\cos x}{x(x+\cos x)}dx + \int \dfrac{x\sin x - x}{x(x+\cos x)}dx$</p>
<p>$\displaystyle= \ln|x| - \int \frac{d(x + \cos x)}{x+\cos x}dx = \ln \left|\frac x{x+\cos x}\right| + c$.</p>
|
1,620,900 | <p>Let $\mathbb F_p(t)$ the field of rational functions with coefficients in $\mathbb F_p$.</p>
<p>Is it true or not that every finite extension $K$ of $\mathbb F_p(t)$ is $K\cong\mathbb F_{p^m}(t)$ for some $m\ge 1$?</p>
| Jyrki Lahtonen | 11,619 | <p>No. Other such extensions are fields of functions on some curve. For that to make sense you need to know a few basics from algebraic geometry. The idea is that the field of rational functions corresponds to the line, but more complicated curves have more complicated function fields.</p>
<p>For example, if $p>3$,... |
4,291,509 | <p>Let <span class="math-container">$ν$</span> denote counting measure and <span class="math-container">$λ$</span> denote Lebesgue measure. What is
<span class="math-container">$(ν ⊗λ)(\{(x,x)\})_{x∈\mathbb{R}}$</span>?</p>
<p>I am a little fuzzy on the product measure and have tried two ways to do this. Can someone ex... | PC1 | 960,197 | <p>If you start with <span class="math-container">$a+b\leq1$</span> then <span class="math-container">$b\leq1-a$</span>. We also have by hypothesis that <span class="math-container">$a^2+b^2 = 1$</span>, which means that:
<span class="math-container">$$a^2 + b^2 = 1$$</span>
<span class="math-container">$$a^2 + (1-a)^2... |
2,000,268 | <p>We usually tend to say the "Average" is whether "Mean", "Median" or "Mode" and in colloquial usage "Average" is always equivalent to "Mean".</p>
<blockquote>
<p>But my <strong>question</strong> is: Is there any precise rigorous definition of "Average of a statistical population" in statistics (regardless of our k... | staydirty | 533,557 | <p>For continuously distributed $x$, $\langle x\rangle=\int xp(x)dx$ (appropriate limits), where $\langle x\rangle$ is the "average" or "most expected" or "expectation value" of $x$, and $p(x)$ is the distribution function. For a discrete population, $\langle x\rangle=\sum_i x_ip(x_i)$. </p>
|
808,964 | <blockquote>
<p>Let $a_n$ be a positive sequence such that $S_n = \sum\limits_{k=1}^n a_k$ diverges.
I'm trying to prove $\sum\limits_{k=1}^n \frac{a_n}{S_n}$ diverges.</p>
</blockquote>
<p>I tried summation by parts, limit comparison and Stolz theorem in many different combinations and am still stuck with nothing... | Did | 6,179 | <p>Note that, for every $M\geqslant N$,
$$\sum_{n=N+1}^M\frac{a_n}{S_n}\geqslant\sum_{n=N+1}^M\frac{a_n}{S_M}=1-\frac{S_N}{S_M}$$
hence, for every $N$, considering that $S_M\to+\infty$ when $M\to\infty$,
$$
\sum_{n=N+1}^\infty\frac{a_n}{S_n}\geqslant1.
$$
This shows that the series diverges since <em>the rest of a summ... |
1,668,792 | <p>This is a question given in our weekly test.</p>
<p>$$f = \lim_{x\to 0^+}\{[(1+x)^{1/x}]/e\}^{1/x}.$$</p>
<p>Find the value of $f$. I tried to use 1^ infinity form but I didn't get it. So anybody please help me.</p>
| egreg | 62,967 | <p>Your limit $f$ exists if and only if its logarithm exists:
\begin{align}
\log f
&=\lim_{x\to0^+}\log\bigl(\bigl((1+x)^{1/x})/e\bigr)^{1/x}\bigr)
\\[6px]
&=\lim_{x\to0^+}\frac{\dfrac{1}{x}\log(1+x)-1}{x}
\\[6px]
&=\lim_{x\to0^+}\frac{\log(1+x)-x}{x^2}
\\[6px]
&=\lim_{x\to0^+}\frac{x-x^2/2+o(x^2)-x}{x^... |
2,459,579 | <p>I ran into a difficult question today as I was trying to find the matrix exponential for a matrix that has a determinant of $0$. Here is the matrix: </p>
<p>$$C = \begin{bmatrix}
1 & 1 \\
-1 & -1 \\
\end{bmatrix} $$</p>
<p>I got only one eigenvalue from the characteristic polynomial, that eigenvalu... | R.W | 253,359 | <p>Note the answer given by @carmichael561 is the same as the one given by <a href="http://www.wolframalpha.com/input/?i=matrixexp(t+%7B%7B1,1%7D,%7B-1,-1%7D%7D)" rel="nofollow noreferrer">wolfram</a> but using a $tC$ matrix to get</p>
<p>$$e^{tC} = I + tC = \begin{bmatrix}1+t&t\\-t&1-t\end{bmatrix}$$</p>
<p... |
2,459,579 | <p>I ran into a difficult question today as I was trying to find the matrix exponential for a matrix that has a determinant of $0$. Here is the matrix: </p>
<p>$$C = \begin{bmatrix}
1 & 1 \\
-1 & -1 \\
\end{bmatrix} $$</p>
<p>I got only one eigenvalue from the characteristic polynomial, that eigenvalu... | Will Jagy | 10,400 | <p>Alright, you called your matrix $C.$ Once we get the Jordan normal form, call it $J,$ using
$$ A^{-1} C A = J, $$ so that
$$ A J A^{-1} = C, $$
we get
$$ e^C = e^{A J A^{-1}} = A e^J A^{-1}. $$
Note that AJA was <a href="https://en.wikipedia.org/wiki/Aja_%28album%29" rel="nofollow noreferrer">a platinum selling al... |
713,135 | <p>I feel like that question's got an obvious answer, but I somehow missed it during my probability class. There are random variables, which distributions can be expressed if a form of functions - like Gaussian, uniform, binomial etc. If I'm going to take a ruler and measure the length of my laptop again and again, all... | Marc | 132,141 | <p>I think what you describe is a stochastic process $\{X_n\}_{n\in\mathbb{N}}$ in which each variable $X_n$ can have a different distribution. Unfortunately, if there is no structure in these distributions, not many interesting things can be concluded from such a process. A type of stochastic process which is studied ... |
1,662,646 | <p>I'm having a problem finding the inverse of $y=2x^2-12x+13$.
At the end I get to the following: </p>
<p>$$x=3 \pm \frac{\sqrt{40+8y}}{4}$$</p>
<p>As far as I know the answer is suppose to be $x= 3 \pm \frac{\sqrt{y+5}}{\sqrt{2}}$
but I am unable to get to it.</p>
<p>Please can someone help me with the process of ... | Martín-Blas Pérez Pinilla | 98,199 | <p>$$\sqrt{40+8y} = \sqrt{8}\sqrt{5+y}$$</p>
|
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