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 |
|---|---|---|---|---|
3,506,091 | <p>Solve <span class="math-container">$2^m=7n^2+1$</span> with <span class="math-container">$(m,n)\in \mathbb{N}^2$</span></p>
<p>Here is what I did:
First try, I have seen first that the obvious solutions are <span class="math-container">$n=1$</span> and <span class="math-container">$m=3$</span> , and <span class="ma... | Mastrem | 253,433 | <p><strong>tl;dr</strong> The only solutions are <span class="math-container">$(m,n)=(6,3)$</span> and <span class="math-container">$(m,n)=(3,1)$</span>.</p>
<p>This answer is rather long and has been split in two parts. In part one, we show that, with the exception of <span class="math-container">$(6,3)$</span> we mus... |
3,506,091 | <p>Solve <span class="math-container">$2^m=7n^2+1$</span> with <span class="math-container">$(m,n)\in \mathbb{N}^2$</span></p>
<p>Here is what I did:
First try, I have seen first that the obvious solutions are <span class="math-container">$n=1$</span> and <span class="math-container">$m=3$</span> , and <span class="ma... | Community | -1 | <p>HINT</p>
<p><span class="math-container">$2^m\equiv 1$</span> mod <span class="math-container">$7$</span> and so <span class="math-container">$m=3k$</span>. For <span class="math-container">$n>0$</span>, we now have
<span class="math-container">$$2^k-1=au^2, 2^{2k}+2^k+1=bv^2$$</span>
where either <span class="m... |
885,450 | <blockquote>
<p>After covering a distance of 30Km with a uniform speed, there got some
defect in train engine and therefore its speed is reduced to 4/5 of
its original speed. Consequently, the train reaches its destination 45
minutes late. If it had happened after covering 18Km of distance, the
train would h... | Start wearing purple | 73,025 | <p>The logarithm of the expression under the limit can be rewritten as
$$\sum_{k=0}^{2^{2^n-1}}\ln\left(1+\frac{1}{2^{2^n}+2k}\right)=\sum_{k=0}^{2^{2^n-1}}\frac{1}{2^{2^n}+2k}+O\left(2^{-2^n}\right).$$
Denoting $N=2^{2^n-1}$, it is easy to see that the limit of the logarithm can be computed as the limit of a Riemann s... |
105,040 | <p>This question in stackExchange remained unanswered. </p>
<p>Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M_n(\mathbb F)$ what is the cardinality of $C_{M_n(\mathbb F)} (A)$ , the centralizer of $A$ in $M_n(\mathbb F)$? There a... | Alireza Abdollahi | 19,075 | <p>Let me add some cases in which one has a clear answer:</p>
<p>[R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991., Corollary 4.4.18]. Let $F$ be a field and $n$ is a natural number. If
$A\in M_n(F)$ is a cyclic matrix, then $C_{M_n(F)}(A)$ is the set of all matrices wh... |
105,040 | <p>This question in stackExchange remained unanswered. </p>
<p>Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M_n(\mathbb F)$ what is the cardinality of $C_{M_n(\mathbb F)} (A)$ , the centralizer of $A$ in $M_n(\mathbb F)$? There a... | Amritanshu Prasad | 9,672 | <p>Treat $F^n$ as an $F[t]$-module $M^A$, where $t$ acts by the matrix $A$. Then the centralizer can be thought of as $\mathrm{End}_{F[t]} M^A$. Now, $M^A$ has a primary decomposition</p>
<p>$ M^A = \bigoplus_{p \in \mathrm{Irr}(F[t])} M_p$</p>
<p>where $M_p$ consists of vectors in $M^A$ which are annihilated by some... |
105,040 | <p>This question in stackExchange remained unanswered. </p>
<p>Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M_n(\mathbb F)$ what is the cardinality of $C_{M_n(\mathbb F)} (A)$ , the centralizer of $A$ in $M_n(\mathbb F)$? There a... | Victor Miller | 2,784 | <p>I've recently come across this question about the size of the centralizer (I assume that you mean the subgroup of $\text{GL}_n(F)$ which commute with the matrix). It seems to be hard to find. I wonder why it's been omitted from every Algebra book (even Bourbaki!) that I've consulted. There is a derivation of it i... |
300,745 | <p>If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded?</p>
<p>($a$ and $b$ being finite numbers).</p>
<p>I tried proving and disproving it. Couldn't find an example for a non-bounded image. </p>
<p>Is there any basic proof or counter example for any of the cases?</p>
<p>Thanks a mi... | N. S. | 9,176 | <p>Let $\delta$ be so that $|x-y|< \delta \Rightarrow |f(x)-f(y)|<1$. Now prove that there exists a finite set $F$ so that </p>
<p>$$(a,b) \subset \cup_{z \in F} (z-\frac{\delta}{2}, z+\frac{\delta}{2} ) \,.$$</p>
<p>Now, what can you say about $|f(x)|$ and $\max\{f(z)|z \in F \}+1\,.$?</p>
<p>P.S. The existen... |
741,436 | <p>I get stuck at the following question:</p>
<p>Consider the matrix<br>
$$A=\begin{bmatrix}
0 & 2 & 0 \\
1 & 1 & -1 \\
-1 & 1 & 1\\
\end{bmatrix}$$</p>
<p>Find $A^{1000}$ by using the Cayley-Hamilton theorem.</p>
<p>I find the characteristic polynomial by $P(A) = -A^{3} + 2A^2 = 0$ (by Cayle... | Potato | 18,240 | <p>Your formula tells you, after you multiply through by $A^{997}$, that
$$A^{1000}=2A^{999}.$$
Similarly,
$$2A^{999}=4A^{998}.$$</p>
<p>This process can be repeated to find $A^{1000}$ in terms of $A^2$, which you can then compute. </p>
|
741,436 | <p>I get stuck at the following question:</p>
<p>Consider the matrix<br>
$$A=\begin{bmatrix}
0 & 2 & 0 \\
1 & 1 & -1 \\
-1 & 1 & 1\\
\end{bmatrix}$$</p>
<p>Find $A^{1000}$ by using the Cayley-Hamilton theorem.</p>
<p>I find the characteristic polynomial by $P(A) = -A^{3} + 2A^2 = 0$ (by Cayle... | Oria Gruber | 76,802 | <p>There is another way of approaching this.</p>
<p>You could divide $x^{1000}$ by the characteristic polynomial:</p>
<p>$x^{1000} = (-x^3+2x^2)Q+R$ where $R$ is a polynomial of degree less than 3 with unknown coefficients.</p>
<p>write down $R=ax^2+bx+c$ and evaluate $R$ at the roots of the characteristic polynomia... |
741,436 | <p>I get stuck at the following question:</p>
<p>Consider the matrix<br>
$$A=\begin{bmatrix}
0 & 2 & 0 \\
1 & 1 & -1 \\
-1 & 1 & 1\\
\end{bmatrix}$$</p>
<p>Find $A^{1000}$ by using the Cayley-Hamilton theorem.</p>
<p>I find the characteristic polynomial by $P(A) = -A^{3} + 2A^2 = 0$ (by Cayle... | Guy Fsone | 385,707 | <p>Set $X_n=A^n$ since, $A^3 =2A^2$
then, we have $$X_{n+3} = A^{n+3} = 2A^2A^n = 2X_{n+2}$$
Hence, $(X_n)_n$ is geometric and
$$X_n =2^{n-2}X_2\Longleftrightarrow A^n =2^{n-2} A^2$$
that is for $n=1000$ we get
$$A^{1000} = 2^{998}A^2$$</p>
|
1,250,020 | <blockquote>
<p>Find $\int \frac{1+\sin x \cos x}{1-5\sin^2 x}dx$</p>
</blockquote>
<p>I used a bit of trig identities to get: $\int \frac {2+\sin (2x)}{-4+\cos(2x)}dx$ and using the substitution: $t= \tan (2x)$ I got to a long partial fractions calculation which doesn't seem right.</p>
<p>Any hints on how to do it... | Américo Tavares | 752 | <p>The denominator of your second integral should be $-3+5\cos 2x$, because
from the identities $\sin x\cos x=\frac{\sin 2x}{2}$ and $\sin ^{2}x=\frac{
1-\cos 2x}{2}$ we obtain</p>
<p>\begin{equation*}
\frac{1+\sin x\cos x}{1-5\sin ^{2}x}=\frac{2+\sin 2x}{-3+5\cos 2x}.
\end{equation*}</p>
<p>To evaluate </p>
<p>\beg... |
131,179 | <p>Consider the assumptions</p>
<pre><code>$Assumptions = {Element[a,Reals], Element[z,Complexes]}
</code></pre>
<p>I'm looking for a test, to be applied on <code>a</code> and <code>z</code>, that gives <code>True</code> if the argument is a complex number such as <code>a</code> and <code>False</code> if it's real su... | Edmund | 19,542 | <p>The following uses <a href="http://reference.wolfram.com/language/ref/$Assumptions.html" rel="nofollow noreferrer"><code>$Assumptions</code></a> to <em>loosely</em> check if a symbol has a definition using a particular domain.</p>
<pre><code>ClearAll[symbolDomainQ];
SetAttributes[symbolDomainQ, HoldFirst];
symbol... |
11,244 | <p>In order to evaluate new educational material the contentment of students with this material is often measured. However, just because a student is contented doesn't mean that he/she has actually learned something. Is there any research investigating the correlation between students contentment and the educational qu... | Michael Hardy | 205 | <p>Vast numbers of students are trained from early childhood to engage in a ruthless competition for grades. When such students are required to take math courses, if they're not interested in understanding math, the course degenerates into one in which getting good grades depends only on believing and obeying and work... |
159,026 | <p>It is known (and easy) to prove that if $T: H\longrightarrow H $ is compact, where $H$ is a Hilbert space, then for any orthonormal basis $ e_n $ we have $||Te_n||\longrightarrow 0$.</p>
<p>My question is the following:
Let $P$ be a orthogonal projection on $H$. Let $ e_n $ be a fixed orthonormal basis such that $$... | Bill Johnson | 2,554 | <p>No. Consider $H = (\sum_n \ell_2^{2^n})_2$ with the unit vector basis. In $\ell_2^{2^n}$, let $x_n$ be the sum of the unit vector basis. For the subspace take the closed linear span of $(x_n)$.</p>
|
159,026 | <p>It is known (and easy) to prove that if $T: H\longrightarrow H $ is compact, where $H$ is a Hilbert space, then for any orthonormal basis $ e_n $ we have $||Te_n||\longrightarrow 0$.</p>
<p>My question is the following:
Let $P$ be a orthogonal projection on $H$. Let $ e_n $ be a fixed orthonormal basis such that $$... | Alex Degtyarev | 44,953 | <p>No. Let the basis be $e_{1,1}, e_{2,1}, e_{2,2},\ldots e_{n,1},\ldots,e_{n,n},\ldots$, in this order, i.e., split it into groups of growing size. Then the projector to the subspace generated by $e_{1,1},\frac12(e_{2,1}+e_{2,2}),\ldots,\frac1n(e_{n,1}+\ldots+e_{n,n}),\ldots$ is a counterexample.</p>
|
1,227,609 | <p>Let $X,Y,Z$ be finite sets, and consider probability distributions $p$ over $X\times Y\times Z$. If we know the marginals of $p$ over all the pairs $X\times Y$, $X\times Z$ and $Y\times Z$, is that enough to pin down $p$ uniquely?</p>
| snar | 24,723 | <p>No, marginals never give you the joint distribution. For a simpler case, take $p$ uniformly distributed on $[0,1]^2$ and $q$ uniformly distributed on the diagonal line segment $\{(x,y) : x = y, 0 \leq x \leq 1\} \subset \mathbb{R}^2$. Both have both marginals being uniform on $[0,1]$ but $p$ and $q$ are distinct.</p... |
419,176 | <p>Let $f:\left\{x\in\mathbb{R}^n\vert\parallel x\parallel<1\right\}\rightarrow\mathbb{R}$ be an one-to-one bounded continuous function.<br>
I want to construct such $f$ which is not uniformly continuous.<br><br>
In this case, I thought I can construct $f$ with a restriction $n=2$.<br>
But I'm confused because $f$ i... | Mher | 80,548 | <p>You can't construct such function. You can prove that $f$ has the limit on the boundary of the closed ball and $f$ will be uniformly continuous.</p>
<p>Let $U_n$ be the open unit ball in $\mathbb{R}^n$. Consider $f$ on the interval $[0,x)\subset U_n$ that connects the center of the ball and the point $x$ from the e... |
419,176 | <p>Let $f:\left\{x\in\mathbb{R}^n\vert\parallel x\parallel<1\right\}\rightarrow\mathbb{R}$ be an one-to-one bounded continuous function.<br>
I want to construct such $f$ which is not uniformly continuous.<br><br>
In this case, I thought I can construct $f$ with a restriction $n=2$.<br>
But I'm confused because $f$ i... | Etienne | 80,469 | <p>For $n=1$, as pointed out by Vishal, there is no such function $f$.</p>
<p>For $n\geq 2$, such an $f$ doesn't exist either, but for a "trivial" reason: there is no continuous and one-to-one function at all from the open unit ball $B\subset\mathbb R^n$ into $\mathbb R$. One can see this as follows. Assume that $f:B\... |
2,634,277 | <p>I am working on some development formulas for surfaces and as a byproduct of abstract theory i get that:
$$\int_{-\frac{\pi}{2}}^\frac{\pi}{2}\frac{1+\sin^2\theta}{(\cos^4\theta+(\gamma\cos^2\theta-\sin\theta)^2)^\frac{3}{4}}d\theta$$
is independent on the parameter $\gamma\in\mathbb{R}$. I thought that there was so... | JanG | 266,041 | <p>Put
\begin{equation*}
I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dfrac{1+\sin^2\theta}{(\cos^4\theta +(\gamma\cos^2\theta-\sin \theta)^2)^{\frac{3}{4}}}\, d\theta
\end{equation*}
If $x = \dfrac{\sin\theta}{\cos^2\theta}$, $\, y = \gamma-x$ and $y = \sqrt{z}$ then
\begin{equation*}
dx = \dfrac{\cos^2\theta+2\sin^2\th... |
520,046 | <blockquote>
<p>Find the smallest natural number that leaves residues $5,4,3,$ and $2$ when divided respectively by the numbers $6,5,4,$ and $3$.</p>
</blockquote>
<p>I tried
$$x\equiv5\pmod6\\x\equiv4\pmod5\\x\equiv3\pmod4\\x\equiv2\pmod3$$What $x$ value?</p>
| wendy.krieger | 78,024 | <p>The number which leaves (5, 4, 3, 2) mod (6, 5, 4, 3) is one less than the one that leaves a residue of (0, 0, 0, 0), mod (6, 5, 4, 3). So one finds $n=\operatorname{lcm}(6,5,4,3)-1$ to get 59, which is the desired answer.</p>
|
2,594,669 | <p>Given the Pythagoras Theorem: <strong>a² + b² = c²</strong></p>
<p>Is there a way to get the value of <strong>b</strong> when we only have a value for <strong>a</strong> and the angle <strong>α</strong>?</p>
<p>To be frank, I have no clue about that, what I want isn't the angle of <strong>β</strong> but the length... | ArsenBerk | 505,611 | <p>Pythagoras Theorem is a theorem about right-angle triangles so if we know one of the angles (except $90^{\circ}$, which is not the case in here), then we know all of the angles and knowing one of the sides of the triangle is enough for finding other two.</p>
|
3,471,292 | <p>I need to find the value of the series <span class="math-container">$\sum_{n=0}^{\infty}\frac{(n+1)x^n}{n!}$</span>.I've computed its radius of convergence which comes out to be zero.</p>
<p>I'm not getting how to make adjustments in the general terms of the series to get the desired result...</p>
| lab bhattacharjee | 33,337 | <p>Hint</p>
<p><span class="math-container">$$\dfrac{(n+1)x^n}{n!}=x\cdot\dfrac{x^{n-1}}{(n-1)!}+\dfrac{x^n}{n!}$$</span></p>
<p>Now <span class="math-container">$$e^y=\sum_{r=0}^\infty\dfrac{y^r}{r!}$$</span></p>
|
3,471,292 | <p>I need to find the value of the series <span class="math-container">$\sum_{n=0}^{\infty}\frac{(n+1)x^n}{n!}$</span>.I've computed its radius of convergence which comes out to be zero.</p>
<p>I'm not getting how to make adjustments in the general terms of the series to get the desired result...</p>
| Marios Gretsas | 359,315 | <p>Let <span class="math-container">$a_n$</span> the sequence in the series.</p>
<p>Now <span class="math-container">$a_n=\frac{z^n}{(n-1)!}+\frac{z^n}{n!}$</span></p>
<p>Now <span class="math-container">$\sum_{n=0}^{\infty}\frac{nz^n}{n!}=\sum_{n=1}^{\infty}\frac{z^n}{(n-1)!}=z\sum_{k=0}^{\infty}\frac{z^k}{k!}=ze^z$... |
228,889 | <p><em>[Attention! This question requires some reading and it's answer probably is in form of a "soft-answer", i.e. it can't be translated into a hard mathematical proposition. (I hope I haven't scared away all readers with this.)]</em></p>
<p>Consider the following three examples:</p>
<p>1) <em>[If this example seem... | user642796 | 8,348 | <p>Take a <a href="http://en.wikipedia.org/wiki/Vitali_set">Vitali set</a> in the unit interval $[0,1)$, and call it $V$. (Define an equivalence relation on $[0,1)$ by $x \sim y$ iff $x - y \in \mathbb{Q}$, and let $V$ be obtained from the Axiom of Choice by choosing exactly one element from every $\sim$-equivalence c... |
2,361,920 | <p>The question is as follows:
<a href="https://i.stack.imgur.com/RHPwG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RHPwG.png" alt="enter image description here"></a> </p>
<p>I can not solve this question so I am asking what exactly in the probability theory that I must revise so I could solve i... | Graham Kemp | 135,106 | <p>There are three colours. You are to draw two socks, and may as well place them in front of you. For each of the colours, what is the probability that the sock on your left is that colour and, <em>given that</em>, that the sock on the right is so too?</p>
<p>$$\def\blank{{\underline{\quad}}} \blank\tim... |
22 | <p>By matrix-defined, I mean</p>
<p>$$\left<a,b,c\right>\times\left<d,e,f\right> = \left|
\begin{array}{ccc}
i & j & k\\
a & b & c\\
d & e & f
\end{array}
\right|$$</p>
<p>...instead of the definition of the product of the magnitudes multiplied by the sign of their angle... | Isaac | 72 | <p>Note that if you replace $i$, $j$, and $k$ with $m$, $n$, and $p$, the determinant becomes the dot-product of the vector $(m, n, p)$ with the cross-product of the two original vectors. If $(m, n, p) = (a, b, c)$ or $(m, n, p) = (d, e, f)$, the determinant is zero (any matrix with two identical rows has determinant ... |
2,076 | <p>I'm attempting for the first time to create a map within <em>Mathematica</em>. In particular, I would like to take an output of points and plot them according to their lat/long values over a geographic map. I have a series of latitude/longitude values like so:</p>
<pre><code> {{32.6123, -117.041}, {40.6973, -111.9}... | Stefan | 2,448 | <p>I am a big fan of the <code>Texture</code> functionality...
(I'd like to point out, that this is all inspired by graphics/visualisation genius <a href="https://mathematica.stackexchange.com/users/820/yu-sung-chang">Yu-Sung Chang</a>)</p>
<pre><code>nightEarth = SphericalPlot3D[1, {u, 0, Pi}, {v, 0, 2 Pi},
PlotPo... |
93,499 | <p>Let $P$ be a normal sylow $p$-subgroup of a finite group $G$.</p>
<p>Since $P$ is normal it is the unique sylow $p$-subgroup.</p>
<p>I would like to say if $\phi$ is an automorphism then $\phi(P)$ is also a sylow $p$-subgroup. Then uniqueness would finish the proof. But is that true?</p>
<p>Does an automorphism o... | Santropedro | 83,420 | <p>Take any automorphism $f$, then $\#f(P)=\#P$ (because in particular $f$ is a bijection), and since also the image of subgroups (in this case $P$) is a subgroup, you conclude $f(P)$ is a p-sylow, and since you remarked:</p>
<blockquote>
<p>Since $P$ is normal it is the unique Sylow $p$ subgroup.</p>
</blockquote>
... |
2,806,432 | <p>Let $(\mathbb{R}^N,\tau)$ a topological space, where $\tau$ is the usual topology.
Let $A\subset\mathbb{R}^N$ a compact. If $(A_n)_n$ is a family of open such that
\begin{equation}
\bigcup_nA_n\supset A,
\end{equation}
then, from compact definition
\begin{equation}
\bigcup_{i=1}^{k}A_i\supset A
\end{equation}
Now, ... | drhab | 75,923 | <p>The values $f(x)$ where $f$ denotes a PDF cannot be interpreted as probabilities (as you seem to think). Note for instance that they can take values that exceed $1$.</p>
<p>That is the mistake you made.</p>
<p>We have the equalities $H+T=50$ and $H+8T=225$ here leading to $H=25=T$, so the question can be rephrased... |
1,369,409 | <p>I have a bit of an advanced combination problem that has left me stumped for a few days. Essentially my question is if you have n sets of items, and you can select a different number of items from each set, how do you compute the combinations without first creating new sets.</p>
<p>An example in pictures:
I have th... | Mark | 169,199 | <p>This seems like a good approach. Combinations from A are {Dog, Cat}, {Dog, Rhino}, and {Cat Rhino}. There are 3 combinations for set B, and 6 combinations for set C. Thus you will have $3*3*6=54$ different sets that meet your criteria.</p>
|
1,421,740 | <p>Let $90^a=2$ and $90^b=5$, Evaluate </p>
<h1>$45^\frac {1-a-b}{2-2a}$ </h1>
<p>I know that the answer is 3 when I used logarithm, but I need to show to a student how to evaluate this without involving logarithm. Also, no calculators.</p>
| lab bhattacharjee | 33,337 | <p>$$2=90^a=(2\cdot5\cdot3^2)^a\iff5^a3^{2a}=2^{1-a}$$</p>
<p>and similarly, $$5=90^b\iff5^{1-b}3^{-2b}=2^b$$</p>
<p>Equating the powers of $2,$
$$\implies(5^a3^{2a})^b=(5^{1-b}3^{-2b})^{1-a}\iff3^{2b}=5^{1-a-b}$$</p>
<p>$$45^{1-a-b}=(3^2\cdot5)^{1-a-b}=3^{2-2(a+b)}\cdot5^{1-a-b}=3^{2-2(a+b)}\cdot3^{2b}$$
$$\implies... |
1,044,009 | <p>I have two numbers $N$ and $M$.
I efficiently want to calculate how many pairs of $a$,$b$ are there such that $1 \leq a \leq N$ and $1 \leq b \leq M$ and $ab$ is a perfect square.</p>
<p>I know the obvious $N*M$ algorithm to compute this. But i want something better than that. I think it can be done in a better tim... | Mike Pierce | 167,197 | <p><strong>Step 1:</strong></p>
<p>Write a function $f$ that takes a perfect square $x$ and returns a list of all (or just the number of) possible pairs $(a,b)$ such that $ab = x$.</p>
<p><strong>Step 2:</strong></p>
<p>Have the function $f$ iterate over the squares of the integers $1$ to $n$, where
$n = \lfloor\sqr... |
1,736,098 | <p>Wrote some Python code to verify if my Vectors are parallel and/or orthogonal. Parallel seems to be alright, orthogonal however misses out in one case. I thought that if the dotproduct of two vectors == 0, the vectors are orthogonal? Can someone tell me what's wrong with my code?</p>
<pre><code>def isParallel(self,... | paulmelnikow | 640,314 | <p>If you're working with 3D vectors, you can do this concisely using the toolbelt <a href="https://github.com/lace/vg" rel="nofollow noreferrer">vg</a>. It's a light layer on top of numpy and it supports single values and stacked vectors.</p>
<pre><code>import numpy as np
import vg
v1 = np.array([1.0, 2.0, 3.0])
v2 ... |
381,566 | <p>I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange.</p>
<p>I just wanted to ask if there is a notion of fractional derivative that is linear and satisfy the following property <span class="math-container">$D^u((f)... | Tom Copeland | 12,178 | <p>The generalized Leibniz formula applicable to the classic fractional integroderivative is</p>
<p><span class="math-container">$$ D^{\omega}\; f(x)g(x) = \sum_{n \geq 0} \binom{\omega}{n} [D^{\omega-n}f(x)]D^ng(x)=(D_L+D_R)^{\omega} g(x)f(x),$$</span></p>
<p>where <span class="math-container">$D_L$</span> acts on the... |
876,209 | <p>I am confused on how to write a formal proof for sum notations. How would I write a formal proof for this example?</p>
<p>Prove that $$\sum\limits_{k = 0}^\infty\frac{2}{3^k} = 3.$$ Prove that for any $\alpha \in \{0, 2\}^\mathbb{N}$ that $$0 \le \sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \le 3.$$</p>
| Joel | 85,072 | <p>It all really depends on how many theorems you have at your disposal. Since you have $\sum_{k=0}^\infty \frac{2}{3^k} = 3$ already, you know by the comparison theorem that $$\sum_{k=0}^\infty \frac{\alpha(k)}{3^k} \le \sum_{k=0}^\infty \frac{2}{3^k} = 3$$ and it is positive since all of the terms are positive.</p>
... |
3,525,488 | <p>So I have the polar curve </p>
<p><span class="math-container">$r=\sqrt{|\sin(n\theta)|}$</span></p>
<p>Which I am trying to evaluate between <span class="math-container">$0$</span> and <span class="math-container">$2\pi$</span>. By smashing it into wolfram it returns a constant value 4 for any <span class="math-c... | Claude Leibovici | 82,404 | <p>The constant value of <span class="math-container">$4$</span> is normal
<span class="math-container">$$\int_{0}^{2\pi}|\sin(t)|\,dt=\int_{0}^{\pi}\sin(t)\,dt-\int_{\pi}^{2\pi}\sin(t)\,dt=2+2$$</span>
<span class="math-container">$$\int_{0}^{2\pi}|\sin(2t)|\,dt=\int_{0}^{\frac\pi 2}\sin(2t)\,dt-\int_{\frac\pi 2}^{\pi... |
1,781,117 | <h1>The question</h1>
<p>Prove that:
$$\prod_{n=2}^∞ \left( 1 - \frac{1}{n^4} \right) = \frac{e^π - e^{-π}}{8π}$$</p>
<hr>
<h2>What I've tried</h2>
<p>Knowing that:
$$\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right)$$
evaluating at $z=i$ gives
$$ \frac{e^π - e^{-π}}{2i} = \sin(πi) = πi \prod_{n=1}^∞ \... | Jorge Martín Pérez | 308,745 | <p>I'll reproduce the answer @C.Dubussy have just deleted:
$$ \prod_{n=2} \left( 1 - \frac{1}{n^4} \right) = \prod_{n=2}^∞ \left( 1 + \frac{1}{n^2}\right) \prod_{n=2}^∞ \left( 1 - \frac{1}{n^2}\right) = \frac{\sin{iπ}}{iπ} \prod_{n=2}^∞ \frac{n-1}{n} \prod_{n=2}^∞ \frac{n+1}{n} $$</p>
<p>And because the last product ... |
1,781,117 | <h1>The question</h1>
<p>Prove that:
$$\prod_{n=2}^∞ \left( 1 - \frac{1}{n^4} \right) = \frac{e^π - e^{-π}}{8π}$$</p>
<hr>
<h2>What I've tried</h2>
<p>Knowing that:
$$\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right)$$
evaluating at $z=i$ gives
$$ \frac{e^π - e^{-π}}{2i} = \sin(πi) = πi \prod_{n=1}^∞ \... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
2,077,694 | <p>How to find $A = M^{A}_{B}$ in linear transformation $F = \mathbb{P_{2}} \rightarrow \mathbb{R^{2}} $, where $ F(p(t)) = \begin{pmatrix} p(0) \\ P(1) \end{pmatrix},$
$ A = \{1,t,t^{2}\},$
$B=\left \{ \begin{pmatrix} 1\\ 0\end{pmatrix},\begin{pmatrix} 0\\ 1\end{pmatrix} \right \}$?</p>
| tomasz | 30,222 | <p><strong>Hint</strong>: If $d$ is a proper divisor of $n$, then $dk$ is a proper divisor of $nk$.</p>
|
9,484 | <p>Let <span class="math-container">$F(k,n)$</span> be the number of permutations of an n-element set that fix exactly <span class="math-container">$k$</span> elements.</p>
<p>We know:</p>
<ol>
<li><p><span class="math-container">$F(n,n) = 1$</span></p>
</li>
<li><p><span class="math-container">$F(n-1,n) = 0$</span></p... | Reid Barton | 126,667 | <p>A permutation of {1, ..., n} with k fixed points is determined by choosing which k elements of {1, ..., n} it fixes and choosing a derangement of the remaining n-k elements. So,</p>
<p>$F(k, n) = {n \choose k} F(0, n-k)$.</p>
<p>(This formula is also on the page Michael Lugo linked to.) You have already given on... |
1,821,849 | <p>Let $K/F$ be a field extension and $L_1,L_2$ subfields of $K$ such that $L_1$ and $L_2$ have finite degree over $F$. </p>
<p>Does $L_1 \cong L_2$ imply $[L_1 : F ]=[L_2 : F]$? Obviously, if the isomorphism fixes $K$ (which isn't always necessarily true) the result holds. The result even holds if $F$ is of finite de... | Jyrki Lahtonen | 11,619 | <p>No.</p>
<p>A simple counterexample is $K=\Bbb{Q}(\pi)$, $F=\Bbb{Q}(\pi^6)$,
$L_1=\Bbb{Q}(\pi^2)$ and $L_2=\Bbb{Q}(\pi^3)$. Then</p>
<ol>
<li>All the fields $K,F,L_1,L_2$ are simple transcendental extensions of $\Bbb{Q}$ and thus they are all isomorphic to each other.</li>
<li>$[K:F]=6$, $[L_1:F]=3$, $[L_2:F]=2$.</... |
346,950 | <p>The equation $$3\sin^2 x - 3\cos x -6\sin x + 2\sin 2x + 3=0$$ has a solution $x = 0$. That is mean it has a factor $\cos x - 1$. I tried write the given equation has the form
$$(\cos x - 1)P(x)=0.$$ I am looking for the factor $P(x)$. How to do that?</p>
| Ron Gordon | 53,268 | <p>It looks like the simplest way to factor out the zero-ing function is to use a half angle. For example, I rewrite your equation as</p>
<p>$$\sin{\frac{x}{2}} \left [ 2 \cos{\frac{x}{2}} ( 3 \sin{x} + 4 \cos{x} - 6) + 6 \sin{\frac{x}{2}} \right ] = 0$$</p>
<p>Note that </p>
<p>$$\sin{\frac{x}{2}} = \sqrt{\frac{1-... |
346,950 | <p>The equation $$3\sin^2 x - 3\cos x -6\sin x + 2\sin 2x + 3=0$$ has a solution $x = 0$. That is mean it has a factor $\cos x - 1$. I tried write the given equation has the form
$$(\cos x - 1)P(x)=0.$$ I am looking for the factor $P(x)$. How to do that?</p>
| lab bhattacharjee | 33,337 | <p>$$\implies 3\left(\sin^2x-2\sin x+1\right)+2\sin2x-3\cos x=0$$</p>
<p>$$\implies 3\left(1-\sin x\right)^2+\cos x(4\sin x-3)=0$$</p>
<p>$$\text{We know,}\sin x=\frac{2t}{1+t^2},\cos2x=\frac{1-t^2}{1+t^2}\text{ where }t=\tan\frac x2$$</p>
<p>So, $1-\sin x=\frac{(1-t)^2}{1+t^2}, 4\sin x-3=4\frac{2t}{1+t^2}-3=-\frac{... |
138,658 | <p>Suppose $X$ is a topological space, and $\mu$ is a Borel measure on $X$. Also suppose we have an $n$-dimensional vector bundle $E \to X$, with an inner product $\langle \cdot,\cdot \rangle_x$ on the fibre $E_x$ for all $x \in X$, in such a way that each $E_x$ is complete and such that there exists a vector bundle tr... | TaQ | 12,643 | <p>At least if $X$ is $\sigma$-compact and if compact sets have finite measure and if the local vector bundle trivializations are homeomorphisms, I would expect the representation of $(L^p(E))^*$ as $L^{p'}(E)$ to hold. The proof would proceed by first dividing $X$ into countably many disjoint relatively compact sets $... |
1,784,679 | <p>if $p,q,r$ are three positive integers prove that</p>
<p>$$LCM(p,q,r)=\frac{pqr \times HCF(p,q,r)}{HCF(p,q) \times HCF(q,r) \times HCF(r,p)}$$</p>
<p>I tried in this way;</p>
<p>Let $HCF(p,q)=x$ hence $p=xm$ and $q=xn$ where $m$ and $n$ are relatively prime.</p>
<p>similarly let $HCF(q,r)=y$ hence $q=ym_1$ and $... | sTertooy | 336,630 | <p>I decided to write my comment as an answer. Rather than start with naming $HCF(p,q)$, $HCF(q,r)$ and $HCF(r,p)$, start with $HCF(p,q,r)$. So let's call $HCF(p,q,r) = h$.</p>
<p>Next, write $HCF(p,q) = xh$, $HCF(q,r) = yh$ and $HCF(r,p) = zh$. It should be clear why we can assume the factor $h$ appears in all three,... |
4,377,932 | <p><span class="math-container">$$3000 = ( p6300 + (1-p)2200 ) / 1.06
\\
3000 = p6300 + 2200 - p2200 / 1.06
\\
3000 = p4100 + 2200 / 1.06
\\
800 = p4100 / 1.06
\\
p4100 = 800\times1.06
\\
p4100 = 848
\\
p = 0.2068$$</span></p>
<p>The correct answer is <span class="math-container">$p=0.239.$</span>
What mistake(s) have ... | Magdiragdag | 35,584 | <p>As written, the first line says that <span class="math-container">$(1 - p) 2200$</span> has to be divided by <span class="math-container">$1.06$</span>. However, you're messing that up in several ways in the subsequent steps.</p>
<p>Edit: in the comments the OP said the correct answer is 0.239. And, also mentioned i... |
4,377,932 | <p><span class="math-container">$$3000 = ( p6300 + (1-p)2200 ) / 1.06
\\
3000 = p6300 + 2200 - p2200 / 1.06
\\
3000 = p4100 + 2200 / 1.06
\\
800 = p4100 / 1.06
\\
p4100 = 800\times1.06
\\
p4100 = 848
\\
p = 0.2068$$</span></p>
<p>The correct answer is <span class="math-container">$p=0.239.$</span>
What mistake(s) have ... | ryang | 21,813 | <blockquote>
<p>I'm not good at solving an equation with brackets. Could you show it in a simple way, with steps, with the equation and values from my initial post?</p>
</blockquote>
<p>Click <a href="https://www.symbolab.com/solver/step-by-step/3000%20%3D%20%5Cleft(p6300%20%2B%20%5Cleft(1-p%5Cright)2200%5Cright)%20%2F... |
3,313,603 | <p>I am assigned with a question which states the rate of a microbial growth is exponential at a rate of (15/100) per hour. where y(0)=500, how many will there be in 15 hours?</p>
<p>I know this question is generally modelled as: </p>
<p><span class="math-container">$y=y_0*e^{kt}$</span></p>
<p>However, my solution ... | Abdulrahman Albishri | 1,024,527 | <p>The general equation can be modeled in the following way:</p>
<p>If we call the bacteria <span class="math-container">$B$</span>, and time (obviously) <span class="math-container">$t$</span>, then <span class="math-container">$B$</span> is clearly a function of time <span class="math-container">$B(t)$</span>. Notice... |
3,405,622 | <p>Let <span class="math-container">$\int_{0}^{1}fg \text{ }d\mathbb{P}=0$</span>, for all <span class="math-container">$f$</span> <span class="math-container">$\in$</span> <span class="math-container">$L^{\infty}([0,1],\mathbb{P})$</span> and <span class="math-container">$g$</span> be a fixed function in <span class="... | Kavi Rama Murthy | 142,385 | <p>Yes. Take <span class="math-container">$f=I_{g>0}$</span> to get <span class="math-container">$\int_{g>0} gdP=0$</span>. Similarly <span class="math-container">$\int_{g<0} gdP=0$</span>. These give <span class="math-container">$\int |g|dP=0$</span> so <span class="math-container">$g=0$</span> almost everyw... |
2,857,769 | <blockquote>
<p>Find $t$ such that $$\lim_{n\to\infty} \frac {\left(\sum_{r=1}^n r^4\right)\cdot\left(\sum_{r=1}^n r^5\right)}{\left(\sum_{r=1}^n r^t\right)\cdot\left(\sum_{r=1}^n r^{9-t}\right)}=\frac 45.$$</p>
</blockquote>
<p>At first sight this question scared the hell out of me. I tried using the general known... | Oleg567 | 47,993 | <p>The skeleton:</p>
<p>First thought is to apply integrals:</p>
<p>$$
\dfrac{\int\limits_1^n x^4 dx \int\limits_1^n x^5 dx}{\int\limits_1^n x^t dx \int\limits_1^n x^{9-t} dx} \sim
\dfrac{\dfrac{n^5}{5} \cdot \dfrac{n^6}{6}}{\dfrac{n^{t+1}}{t+1} \cdot \dfrac{n^{9-t+1}}{9-t+1}}
$$</p>
<p>Then we'll have fraction
$$
... |
2,512,556 | <p>What would be the solution of $ y''+y=\cos (ax) \ $ if $ \ a \to 1 \ $. </p>
<p><strong>Answer:</strong></p>
<p>I have found the complementary function $ y_c \ $ </p>
<p>$ y_c(x)=A \cos x+B \sin x \ $</p>
<p>But How can I find the particular integral if $ a \to 1 \ $ </p>
| Ian | 83,396 | <p>For simplicity let's just solve the complex exponential problem $z''+z=e^{iax}$ (your solution is the real part of this). When $a \neq 1$, the particular solution to this is of the form $Ae^{iax}$ (where $A$ is in general complex). When $a=1$ this is no longer a particular solution, because it is actually a homogene... |
3,500,799 | <p>what is the dimension of the vector space spanned by the set of vectors <span class="math-container">$(a,b,c) $</span>where <span class="math-container">$a^2+b^2=c$</span>?</p>
| Lukas Rollier | 737,665 | <p>It is a subspace of <span class="math-container">$\mathbb{R}^3$</span> containing the linearly independent vectors <span class="math-container">$(1,0,1)$</span>, <span class="math-container">$(0,1,1)$</span>, and <span class="math-container">$(1,2,5)$</span>. Hence, it must be <span class="math-container">$\mathbb{R... |
12,102 | <p>For local field, the reciprocity map establishes almost an isomorphism from the multiplicative group to the Abelian Absolute Galois group. (In global case the relationship is almost as nice). It is tempted to think that there can be no such nice accident. </p>
<p>Do we know any explanation which suggest that there ... | Johnson Jia | 3,247 | <p>The reciprocity map in the local case can be motivated by considering the unramified case. Let me try to explain this. For simplicity, let us consider the case of a finite abelian unramified extension $K_n/\bf Q_p$ of degree $n$. Denote by $k_n$ the residue field of $K_n$. The unramified condition gives rise to the ... |
3,995,046 | <p>Refer to <a href="https://oeis.org/A340800" rel="nofollow noreferrer">https://oeis.org/A340800</a> to notice that the number of primes between two primes having the same last digit is increasing as the primes themselves increase. Is there an explanation for this? How can the size of primes have any influence on th... | Rhys Hughes | 487,658 | <p>Let <span class="math-container">$$\mathfrak P(x,n)=\Bbb P(\exists m\leq n \in \Bbb N : x+10m \text{ is prime} | x \text{ is prime})$$</span></p>
<p>The relative density of primes decreases as numbers get larger. Because of this, <span class="math-container">$\mathfrak P$</span> is very high for small <span class="m... |
3,995,046 | <p>Refer to <a href="https://oeis.org/A340800" rel="nofollow noreferrer">https://oeis.org/A340800</a> to notice that the number of primes between two primes having the same last digit is increasing as the primes themselves increase. Is there an explanation for this? How can the size of primes have any influence on th... | Robert Israel | 8,508 | <p>Of course the general trend of A340800 must be increasing (although it is not monotonic). For any <span class="math-container">$M$</span>, the maximum of <span class="math-container">$A072971(k)$</span> for <span class="math-container">$k \le M$</span> is some finite
value <span class="math-container">$N$</span>, a... |
68,803 | <p>I am trying to understand how all the players in the title relate, but with all the grading shifts,and difficult isomorphisms involved in the subject I am having a hard time being sure that I have the picture right. I am going to write what I think is true, and if someone would confirm or deny it, that would be real... | Tim Perutz | 2,356 | <p><b>Some blah on symplectic homology vs. cohomology.</b> There's an invariant $SH(M)$ of Liouville domains $M$ which some people call symplectic homology and some symplectic cohomology. This is the direct limit of Hamiltonian Floer groups associated with functions of increasing eventual slope. The dual theory has two... |
2,804,716 | <p>Given this Maclaurin series:</p>
<p>$$f(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}$$</p>
<p>And the following Catenary curve, assuming that $a=1$:</p>
<p>$$g(x)=\frac{a(e^\frac{x}{a}+e^{-\frac{x}{a}})}{2}$$</p>
<p>Why does $f(x)=g(x)$ seem to hold true (at least when graphed)?</p>
<p>I'm looking for a purely al... | cansomeonehelpmeout | 413,677 | <p>Remember that $$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$</p>
<p>Thus $$\frac{1}{2}(e^\frac{x}{a}+e^\frac{-x}{a})=\frac{1}{2}\sum_{n=0}^{\infty}\left [\left (\frac{x}{a}\right )^n\frac{1}{n!}+\left (\frac{-x}{a}\right )^n\frac{1}{n!}\right ]$$</p>
<p>This can be simplified to $$\frac{1}{2}\sum_{n=0}^\infty \left (\f... |
1,641,255 | <p>I am having difficulty find the centroid of the region that is bound by the surfaces $x^2+y^2+z^2-2az=0$ and $3x^2+3y^2-z^2=0$ (lying above $xy$ plane, consider the inner region). I know the first surface is a sphere, while the second is an infinite cone.</p>
<p>I just dont know how to even approach it as I dont kn... | user84413 | 84,413 | <p>By symmetry, you have that $\overline{x}=\overline{y}=0$; so you just need to find $\overline{z}$.</p>
<p>If you use spherical coordinates, you have </p>
<p>$\hspace{.5 in}\displaystyle\overline{z}=\frac{\int z\;dV}{\int 1 \;dV}$ where</p>
<p>$\;\;\displaystyle\int z\;dV=\int_0^{2\pi}\int_0^{\frac{\pi}{6}}\int_0^... |
1,641,255 | <p>I am having difficulty find the centroid of the region that is bound by the surfaces $x^2+y^2+z^2-2az=0$ and $3x^2+3y^2-z^2=0$ (lying above $xy$ plane, consider the inner region). I know the first surface is a sphere, while the second is an infinite cone.</p>
<p>I just dont know how to even approach it as I dont kn... | Community | -1 | <p>We can solve for $a=1$ and in the end multiply by $a$.</p>
<p>As said, in cylindrical coordinates</p>
<p>$$r^2+(z-1)^2\le 1,\\
3r^2\le z^2.$$</p>
<p>The intersection of the two surfaces is a circle in the plane $z=\frac32$ (by eliminating $r^2$).</p>
<p>We can decompose the domain in horizontal slices, which are... |
2,963,587 | <p>I'm working on a relatively low-level math project, and for one part of it I need to find to a function that returns how many many configurations are reachable within n moves. from the solved state.</p>
<p>Because there are 18 moves ( using the double moves metric ), one form of the function could be <span class="... | Mark S | 447,928 | <p><strong>Too long for a comment, making CW</strong></p>
<p>Alexander Chervov recently asked a very similar <a href="https://mathoverflow.net/questions/322877/number-of-positions-of-rubiks-cube-grows-with-multiplier-13-with-the-distance">question</a> on MO. He noted that the <a href="http://cube20.org" rel="nofollow... |
1,783,458 | <blockquote>
<p>Prove that the equation
$$z^n + z + 1=0 \ z \in \mathbb{C}, n \in \mathbb{N} \tag1$$
has a solution $z$ with $|z|=1$ iff $n=3k +2, k \in \mathbb{N} $.</p>
</blockquote>
<hr>
<p>One implication is simple: if there is $z \in \mathbb{C}, |z|=1$ solution for (1) then $z=cos \alpha + i \cdot sin\al... | David | 119,775 | <p>You have to prove that if $n=3k+2$, then the equation has a solution $z$ with $|z|=1$.</p>
<p>So, suppose $n=3k+2$ and take $z=\frac12(-1\pm i\sqrt3)$, which is the solution you found for the other direction. Certainly it is true that $|z|=1$, and by substituting and simplifying you can check that $z^n+z+1=0$.</p>... |
366,096 | <p>Let's consider $J\subset \mathbb R^2$ such that J is convex and such that it's boundary it's a curve $\gamma$. Let's suppose that $\gamma$ is anti-clockwise oriented, let's consider it signed curvature $k_s$. I want to prove the intuitive following fact:</p>
<p>$$
\int\limits_\alpha {k_s } \left( s \right)ds \geqs... | Robert Israel | 8,508 | <p>If $s$ is arc-length, $T(s)$ is the unit tangent vector and $N(s)$ the counterclockwise unit normal, $\dfrac{d}{ds} T(s) = k(s) N(s)$. It's convenient to consider the plane as the complex plane, so $T(s) = e^{i\theta(s)}$ and $N(s) = i e^{i \theta(s)}$. Then we have
$\dfrac{d\theta}{ds} = k(s)$. Now you want to s... |
2,663,303 | <blockquote>
<p>Let $G$ be finite. Suppose that $\left\vert \{x\in G\mid x^n =1\}\right\vert \le n$ for all $n\in \mathbb{N}$. Then $G$ is cyclic.</p>
</blockquote>
<p>What I have attempted was the fact that every element is contained in a maximal subgroup following that <a href="https://groupprops.subwiki.org/wiki/... | Ri-Li | 152,715 | <p>I have another idea, please have a look.</p>
<p>For each prime $p\mid |G|$ we have $H$, a Sylow $p$-subgroup(uniqueness comes from the given condition) s.t. $|H|=p^n$(say). Then $G=H_1 \oplus \cdots \oplus H_n$. Now each $H_i$ has at least one normal subgroup of order $p^a$ s.t. $a\mid n_i$ for all $0 \leq a \leq n... |
445 | <p>Under what circumstances should a question be made community wiki?</p>
<p>Probably any question asking for a list of something (e.g. <a href="https://math.stackexchange.com/questions/81/list-of-interesting-math-blogs">1</a>) must be CW. What else? What about questions asking for a list of applications of something ... | Larry Wang | 73 | <p>The only useful effect I see in making a question community wiki is to force answers to be community wiki: that is, answers should be more easily editable, or should not grant reputation. So when do we want this? </p>
<p>Polls or lists, <a href="https://math.stackexchange.com/questions/329/best-ever-book-on-number... |
990,340 | <p>I have an arguement with my friends on a probability question.</p>
<p><strong>Question</strong>: There are lots of stone balls in a big barrel A, where 60% are black and 40% are white, black and white ones are identical, except the color.</p>
<p>First, John, blindfolded, takes 110 balls into a bowl B; afterwards, ... | G Tony Jacobs | 92,129 | <p>The exponents are all multiples of 17. Thus, we can order the following:</p>
<p>$7<2^3<3^2$</p>
<p>Raising each to the $17$th power, we obtain:</p>
<p>$7^{17} < 2^{51} < 3^{34}$.</p>
<p>The inequality is preserved, because $x\mapsto x^{17}$ is a strictly increasing function.</p>
|
990,340 | <p>I have an arguement with my friends on a probability question.</p>
<p><strong>Question</strong>: There are lots of stone balls in a big barrel A, where 60% are black and 40% are white, black and white ones are identical, except the color.</p>
<p>First, John, blindfolded, takes 110 balls into a bowl B; afterwards, ... | Yola | 31,462 | <p>You have $7^{17}, (2^3)^{17}, (3^2)^{17}$</p>
|
94,525 | <p>I am trying to solve the equation
$$z^n = 1.$$</p>
<p>Taking $\log$ on both sides I get $n\log(z) = \log(1) = 0$.</p>
<p>$\implies$ $n = 0$ or $\log(z) = 0$</p>
<p>$\implies$ $n = 0$ or $z = 1$.</p>
<p>But I clearly missed out $(-1)^{\text{even numbers}}$ which is equal to $1$.</p>
<p>How do I solve this equati... | Gigili | 181,853 | <p>To solve it algebraically, I'd say:</p>
<ul>
<li>For even values of $n$, $z=1$ or $z=-1$ are the solutions.</li>
<li>For odd values of $n$, $z=1$ is the answer.</li>
<li>And if $n=0$, $\forall z \in \mathbb{R}$ would be valid as an answer.</li>
</ul>
|
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Andrew Stacey | 45 | <p>Any vector bundle is a groupoid: you can add and subtract vectors only if they are in the same fibre. Similarly, if you take a vector bundle E → M (or some other fibre bundle) then consider the automorphism bundle Aut(E) → M where a point in Aut(E) above p ∈ M is an automorphism of E<sub>p</sub>. Th... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| javier | 914 | <p>Beyond all the categorical and bundle-like examples already given, you can easily understand groupoids as generalizations of groups in a purely geometrical sense.</p>
<p>If you think of groups as the sets of <em>symmetries</em> of certain geometrical objects, then groupoids are <strong>local</strong> symmetries of ... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Sean Rostami | 2,047 | <p>Lots of things are groupoids, but many are not groups. There is a theory of groupoids, and if you don't acknowledge groupoids, they won't let you use their theory =)</p>
<p>The fundamental group(oid) example is really good. What happens if you want to do Van Kampen's theorem on a pair of sets whose intersection is ... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Maik Köster | 41,401 | <p>One reason why people get existed about groupoids which has not been mentioned so far: The category of reduced smooth orbifolds is isomorphic to a category of proper effective etale Lie groupoids. For some investigations on orbifolds, working with their local charts can become very clumpsy. Sometimes, the correspon... |
3,845,968 | <p><a href="https://math.stackexchange.com/q/29666/717872">There</a>
<a href="https://math.stackexchange.com/q/11/717872">are</a>
<a href="https://math.stackexchange.com/q/363977/717872">tens</a>
<a href="https://math.stackexchange.com/q/3339682/717872">of</a>
<a href="https://math.stackexchange.com/q/2005492/717872">p... | Ennar | 122,131 | <p>What you wrote is simply
<span class="math-container">$$\frac 19 = 0.\bar 1 \implies 1 = \frac 99 = 0.\bar 9$$</span>
which is correct.</p>
<p>However, you didn't write anything to prove that <span class="math-container">$\frac 19 = 0.\bar 1$</span>, so I wouldn't count this as a proof.</p>
<p>Any proof really needs... |
2,204,812 | <p>The solution of the differential equation $\frac{dy}{dx}-xtan(y-x)=1$ will be?</p>
<p>For solving such problems first we should see if the equation is in variable seperable form or not. Obviously here it is not. So I tried to see if it can be made to variable seperable by substitution, but substituting $y-x=z$ woul... | projectilemotion | 323,432 | <p>Your decision to substitute $z=y-x$ is a good one.</p>
<p>Differentiating with respect to $x$ gives:
$$\frac{dz}{dx}=1\cdot \frac{dy}{dx}-1$$
Therefore, rearranging gives:
$$\frac{dy}{dx}=\frac{dz}{dx}+1$$
Substituting gives a separable ODE:
$$\frac{dz}{dx}+1-x\tan{z}=1$$
$$\frac{dz}{dx}=x\tan{z}$$
Can you continue... |
3,932,803 | <p>I need to prove <span class="math-container">$$ \lim_{x\rightarrow\ 0}\frac{x^2-8}{{x-8}} =1 $$</span> using epsilon-delta definition.
I know I need to show that for every <span class="math-container">$\epsilon >0$</span> there exist a <span class="math-container">$\delta >0$</span> such that if <span class="m... | 5xum | 112,884 | <p><strong>Hint</strong>:</p>
<p><span class="math-container">$$\left|\frac{x^2-x}{x-8}\right| = |x|\cdot \left|\frac{x-1}{x-8}\right|.$$</span></p>
<p>Now, if <span class="math-container">$\delta<1$</span>, then you can further estimate that:</p>
<ol>
<li><span class="math-container">$|x-1| < 3$</span></li>
<li>... |
3,932,803 | <p>I need to prove <span class="math-container">$$ \lim_{x\rightarrow\ 0}\frac{x^2-8}{{x-8}} =1 $$</span> using epsilon-delta definition.
I know I need to show that for every <span class="math-container">$\epsilon >0$</span> there exist a <span class="math-container">$\delta >0$</span> such that if <span class="m... | Neat Math | 843,178 | <p>There are many ways to get an upper bound for <span class="math-container">$\left| \frac{x-1}{x-8} \right|$</span> when <span class="math-container">$x$</span> is sufficiently small. A couple of things you can do to make this easier: 1) Use the triangle inequality, as one of the answerers did; 2) separate a constant... |
2,002,385 | <blockquote>
<p>Prove $|e^{i\theta_1}-e^{i\theta_2}|\geq|e^{i\theta_1/2}-e^{i\theta_2/2}|$
where $ \theta_1, \theta_2 \in (0,\pi]$.</p>
</blockquote>
<p>Even though geometrically it is an obvious fact, somehow I couldn't prove it elegant way (it's really frustrating), and I'm sure some of you guys know how to prov... | Jan Eerland | 226,665 | <p>When $\theta_1\space\wedge\space\theta_2\in\mathbb{R}$:</p>
<ol>
<li>$$e^{\theta_1i}-e^{\theta_2i}=\cos\left(\theta_1\right)+\sin\left(\theta_1\right)i-\left(\cos\left(\theta_2\right)+\sin\left(\theta_2\right)i\right)=$$
$$\cos\left(\theta_1\right)-\cos\left(\theta_2\right)+\left(\sin\left(\theta_1\right)-\sin\left... |
780,611 | <p>The differential equation in question is a FODE,</p>
<p>$$
\frac{dy}{dt} = -\frac{a^2\sqrt{2g}}{\sqrt{(R+y)(R-y)}}
$$</p>
<p>Upon first inspection, this is separable, but I don't know how to proceed from there.</p>
<p>Thanks.</p>
| Chinny84 | 92,628 | <p>$$
\frac{dy}{dt} = -\frac{a^2\sqrt{2g}}{\sqrt{(R+y)(R-y)}}
$$
is indeed seperable
$$
\int\sqrt{R^{2}-y^{2}}dy = -a^{2}\sqrt{2g}t + \lambda
$$
using $y=R\cos\theta$ leads to
$$
\int R\sin\theta d\left(R\cos\theta\right) = -\int R^2\sin^{2}\theta d\theta = -a^{2}\sqrt{2g}t + \lambda
$$
Do you need this to be completed... |
2,930,292 | <p>I'm currently learning the unit circle definition of trigonometry. I have seen a graphical representation of all the trig functions at <a href="https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/unit-circle-definition-of-trig-functions/a/trig-unit-circle-review" rel="nofollow noreferrer">khan academ... | Mohammad Riazi-Kermani | 514,496 | <p>It would be very helpful if you connected the center of the unit circle to the point of tangency.</p>
<p>Notice that this segment has length one and is perpendicular to the tangent line.</p>
<p>Now there are many similar right triangles to be found and the proprtionality relations gives you the trig functions as i... |
265,949 | <p>Consider the set of all $n \times n$ matrices with real entries as the space $ \mathbb{R^{n^2}}$ .
Which of the following sets are compact?</p>
<ol>
<li>The set of all orthogonal matrices.</li>
<li>The set of all matrices with determinant equal to unity.</li>
<li><p>The set of all invertible matrices.</p>
<p>I am ... | froggie | 23,685 | <p>To check that a set of $n\times n$ matrices $\subset \mathbb{R}^{n^2}$ is compact, it is probably easiest to check that it is closed and bounded. </p>
<p>I'm going to give an example that is not any of the three you have in the problem, but will hopefully help you with them. Let $S$ be the collection of $n\times n$... |
514,922 | <p>I need to prove the following affirmation: If $ \lim x_{2n} = a $ and $ \lim x_{2n-1} = a $, prove that $\lim x_n = a $ (in $ \mathbb{R} $ )</p>
<p>It is a simple proof but I am having problems how to write it. I'm not sure it is the right way to write, for example, that the limit of $(x_{2n})$ converges to a:</p>
... | Hagen von Eitzen | 39,174 | <p>You can do so. As a matter of fact,you consider the sequence $(y_n)$ given by $y_n=x_{2n}$ and could write - as you may be more accustomed to - that $|y_n-a|<\epsilon$ for all $n>n_1$. But as $y_n=x_{2n}$ you really get back what you (correctly) wrote: $|x_{2n}-a|<\epsilon$.</p>
|
2,918,091 | <p>Suppose I want to find the locus of the point $z$ satisfying $|z+1| = |z-1|$</p>
<p>Let $z = x+iy$</p>
<p>$\Rightarrow \sqrt{(x+1)^2 + y^2} = \sqrt{(x-1)^2 + y^2}$ <br/>
$\Rightarrow (x+1)^2 = (x-1)^2$ <br/>
$\Rightarrow x+1 = x-1$ <br/>
$\Rightarrow 1= -1$ <br/>
$\Rightarrow$ Loucus does not exist</p>
<p>Is my a... | Fabio Lucchini | 54,738 | <p>When you removed squares, you have also to consider $x+1=1-x $ which gives $x=0$.</p>
<p>Alternatively, this is the locus of the points equidistant from $\pm 1$, that's the axis of the segment $[-1,1] $.</p>
|
140,134 | <p>Does anyone know whether there is any geometric applications of the Iwaniec's conjecture on $ l^p $ bound of Beurling Alfhors transform (or the complex Hilbert transform). One application could have been was Behrling's conjecture (solved). Is there is any thing else that might be of geometric significance.
For p=2, ... | Fabrice Baudoin | 48,356 | <p>Iwaniec conjecture is closely connected to the important Morrey's conjecture on the relationship between rank one convexity and quasiconvexity.</p>
<p>You will find a nice discussion in the Section 5 of the following survey by Banuelos</p>
<p><a href="http://arxiv.org/pdf/1012.4850v2.pdf" rel="nofollow">http://arx... |
2,125,206 | <ol>
<li><p>Let $W$ be the region bounded by the planes
$x = 0$, $y = 0$, $z = 0$, $x + y = 1$, and $z = x + y$.</p></li>
<li><p>$(x^2 + y^2 + z^2)\, \mathrm dx\, \mathrm dy\, \mathrm dz$; $W$ is the region bounded by $x + y + z = a$ (where $a > 0$), $x = 0$, $y = 0$, and $z = 0$.</p></li>
</ol>
<p>$x,y,z$ being $0... | Doug M | 317,162 | <p>1) give you nothing to integrate. But we can still find the limits</p>
<p>$z = 0$ lower limit for $z$</p>
<p>$z = x+y$ upper limit for $z$</p>
<p>$y = 0$ lower limit for $y$</p>
<p>$y = 1-x$ upper limit for $y$</p>
<p>$x = 0$ lower limit for $x$</p>
<p>$x = 1$ upper limit for $x$</p>
<p>For the last one, bec... |
447,484 | <p>I am just learning about differential forms, and I had a question about employing Green's theorem to calculate area. Generalized Stokes' theorem says that $\int_{\partial D}\omega=\int_D d\omega$. Let's say $D$ is a region in $\mathbb{R}^2$. The familiar formula to calculate area is $\iint_D 1 dxdy = \frac{1}{2}\int... | Steven Gubkin | 34,287 | <p>I would like to point out the integrating $xdy$ to get area has a natural geometric interpretation: you are summing the areas of small horizontal rectangles. The sign of these areas is determined by whether you are moving up or down, and the sign of x. Draw a picture of a wild blob, intersect it with a horizontal... |
108,372 | <p>Given a map $\psi: S\rightarrow S,$ for $S$ a closed surface, is there any algorithm to compute its translation distance in the curve complex? I should say that I mostly care about checking that the translation distance is/is not very small. That is, if the algorithm can pick among the possibilities: translation dis... | Ian Agol | 1,345 | <p>I don't know an algorithm, but here's a possible approach. As Richard and Lee have observed, one may assume that <span class="math-container">$\psi$</span> is pseudo-Anosov. In that case, the mapping torus <span class="math-container">$T_\psi$</span> is a hyperbolic 3-manifold fibering over <span class="math-contain... |
108,372 | <p>Given a map $\psi: S\rightarrow S,$ for $S$ a closed surface, is there any algorithm to compute its translation distance in the curve complex? I should say that I mostly care about checking that the translation distance is/is not very small. That is, if the algorithm can pick among the possibilities: translation dis... | Mark Bell | 3,121 | <p>Recently, <a href="https://arxiv.org/abs/1609.09392" rel="nofollow noreferrer">Richard Webb and myself gave a polynomial-time algorithm</a> for computing the asymptotic translation length of a mapping class
$$ \ell(h) = \lim_{n \to \infty} d(x, h^n(x)). $$
This appears as Algorithm 6 of the paper and relies on being... |
141,823 | <p>I am thinking about the simplest version of Hensel's lemma. Fix a prime $p$. Let $f(x)\in \mathbf{Z}[x]$ be a polynomial. Assume there exists $a_0\in \mathbf{F}_p$ such that $f(a_0)=0\mod p$, and $f'(a_0)\neq 0\mod p$. Then there exists a unique lift $a_n\in \mathbf{Z}/p^{n+1}\mathbf{Z}$ for every $n$. I know there ... | Daniel Litt | 6,950 | <p>There is indeed a proof along these lines. Suppose one has a polynomial $f$ over $\mathbb{Z}_p$; I'll use $f$ to refer to its reductions mod $p^k$ as well. You have a diagram:</p>
<p>$$\text{Spec}((\mathbb{Z}/p^k)[t]/f(t))\to~\text{Spec}((\mathbb{Z}/p^{k+1})[t]/f(t))~~~~~~~~~~~~~$$
$$\downarrow\uparrow~~~~~~~~~~~... |
1,168,968 | <p>So I'm doing some cryptography assignment and I'm dealing with a modular arithmetic in hexadecimal. Basically I have the values for $n$ and the remainder $x$, but I need to find the original number $m$, e.g.</p>
<p>$$m \mod 0x6e678181e5be3ef34ca7 = 0x3a22341b02ad1d53117b.$$</p>
<p>I just need a formula to calculat... | kryomaxim | 212,743 | <p>Derivative of the numerator: $(5+ \sqrt{x^2+5})'= (\sqrt{x^2+5})'= \frac{(x^2+5)'}{2}\sqrt{x^2+5}^{-1} = \frac{2x}{2 \sqrt{x^2+5}} = \frac{1}{\sqrt{1+\frac{5}{x^2}}}$.</p>
<p>Derivative of the denominator: $(x-6)'=1$. </p>
<p>L'Hospital's rule is possible.</p>
|
2,377,946 | <blockquote>
<p>The integral is:
$$\int_0^a \frac{x^4}{(x^2+a^2)^4}dx$$</p>
</blockquote>
<p>I used an approach that involved substitution of x by $a\tan\theta$. No luck :\ . Help?</p>
| Robert Z | 299,698 | <p>Let $x=at$ and by <a href="https://en.wikipedia.org/wiki/Partial_fraction_decomposition" rel="nofollow noreferrer">Partial-Fraction Decomposition</a> we get
\begin{align*}\int_0^a \frac{x^4}{(x^2+a^2)^4}dx&=\frac{1}{a^3}\int_0^1 \frac{t^4}{(t^2+1)^4}dt\\
&=\frac{1}{a^3}\int_0^1\left(\frac{1}{(t^2+1)^2}-\fra... |
1,956,897 | <p>I have the question "Find the centre and radius of the circle whose equation is $x^2+y^2+x+3y-2=0$"</p>
<p>So I've worked out the centre to be $(-1/2, -3/2)$ which when I checked the solutions is correct however I got $9/2$ for the radius and in the solutions the radius should be $\frac{3\sqrt{2}}{2}$.</p>
<p>Coul... | Theorem | 346,898 | <p>You got $R^2$! you need $R$. So just take the square root.</p>
|
1,956,897 | <p>I have the question "Find the centre and radius of the circle whose equation is $x^2+y^2+x+3y-2=0$"</p>
<p>So I've worked out the centre to be $(-1/2, -3/2)$ which when I checked the solutions is correct however I got $9/2$ for the radius and in the solutions the radius should be $\frac{3\sqrt{2}}{2}$.</p>
<p>Coul... | Stefan | 375,579 | <p>$x^2+y^2+x+3y-2=0 \Leftrightarrow (x+0.5)^2+(y+1.5)^2=4.5$</p>
<p>i suppose this is what you did.
The common formula for circles is $(x-a)^2+(y-b)^2=r^2$ so you just need to take the root of $\frac{9}{2}$.</p>
<p>$\sqrt{\frac{9}{2}}=\frac{3}{\sqrt{2}}=3\frac{\sqrt{2}}{2}$</p>
|
798,307 | <p>I have a linear functional from the space of nxn matrices over a field F. The functional satisfies $f(A) = f(PAP^{-1})$ for all invertible $P$ and $A$ an nxn matrix. I'm trying to show that $f(A) = \lambda tr(A)$ for some constant $\lambda$.</p>
<p>So far I have:</p>
<ul>
<li>The linear functionals have basis $f_{... | EPS | 133,563 | <p>From the given condition it is clear that $f$ satisfies $f(AB)=f(BA)$ for any two square matrices $A$ and $B$. Next recall that the kernel of the trace function on $M_n(\mathbb{R})$ is $n^2-1$ dimensional. Thus any square matrix $X$ can be written in the form $X=Z+cA$ where $Z$ is a matrix with zero trace and $A$ is... |
3,306,089 | <p>I came across this meme today:</p>
<p><a href="https://i.stack.imgur.com/RfJoJ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RfJoJ.jpg" alt="enter image description here"></a></p>
<p>The counterproof is very trivial, but I see no one disproves it. Some even say that the meme might be true. Wel... | Ross Millikan | 1,827 | <p>I am sure it has <span class="math-container">$31415$</span> again in the decimal expansion, but why should it continue <span class="math-container">$926535$</span> after that? Sometimes it will, but it will eventually diverge from the decimals at the start. You have not made any argument that when you see <span c... |
2,401,281 | <blockquote>
<p>Show that for $\{a,b,c\}\subset\Bbb Z$ if $a+b+c=0$ then $2(a^4 + b^4+ c^4)$ is a perfect square. </p>
</blockquote>
<p>This question is from a math olympiad contest. </p>
<p>I started developing the expression $(a^2+b^2+c^2)^2=a^4+b^4+c^4+2a^2b^2+2a^2c^2+2b^2c^2$ but was not able to find any useful... | ajotatxe | 132,456 | <p>$$c^4=(-a-b)^4=(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$$</p>
<p>Therefore,</p>
<p>$$2(a^4+b^4+c^4)=4(a^4+2a^3b+3a^2b^2+2ab^3+b^4)$$</p>
<p>Now compute $(a^2+ab+b^2)^2$.</p>
|
2,401,281 | <blockquote>
<p>Show that for $\{a,b,c\}\subset\Bbb Z$ if $a+b+c=0$ then $2(a^4 + b^4+ c^4)$ is a perfect square. </p>
</blockquote>
<p>This question is from a math olympiad contest. </p>
<p>I started developing the expression $(a^2+b^2+c^2)^2=a^4+b^4+c^4+2a^2b^2+2a^2c^2+2b^2c^2$ but was not able to find any useful... | lhf | 589 | <p>We can simplify the algebra by noticing that the expressions are homogenous:</p>
<p>$
2(a^4+b^4+c^4)
\\\quad=2(a^4+b^4+(a+b)^4)
\\\quad=2b^4(x^4+1+(x+1)^4),
\quad x=a/b
$</p>
<p>$
2(x^4+1+(x+1)^4)
\\\quad=4 x^4 + 8 x^3 + 12 x^2 + 8 x + 4
\\\quad=4 (x^2 + x)^2 + 8 (x^2 + x) + 4
\\\quad=4(y^2+2y+1),
\quad y=x^2+x
\\... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Matt Noonan | 2,510 | <p>Along the sphere eversion lines, there is also the energy-minimizing sphere eversion constructed by Rob Kusner. I think there is a video of it at <a href="https://www.youtube.com/watch?v=I6cgca4Mmcc" rel="nofollow noreferrer">https://www.youtube.com/watch?v=I6cgca4Mmcc</a>, though it isn't labelled as such.</p>
<p>... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Matt | 4,265 | <p>MIT's OpenCourseWare has a few math courses up:</p>
<p><a href="http://ocw.mit.edu/OcwWeb/web/courses/av/index.htm#Mathematics">http://ocw.mit.edu/OcwWeb/web/courses/av/index.htm#Mathematics</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Jan Weidner | 2,837 | <p>A nice introduction to representation theory of compact lie groups, sl2(R) and other topics:
<a href="http://www.math.utah.edu/vigre/minicourses/sl2/schedule.html" rel="nofollow">http://www.math.utah.edu/vigre/minicourses/sl2/schedule.html</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Anweshi | 2,938 | <p>I am surprised that nobody mentioned the four-week <a href="http://www.uni-math.gwdg.de/aufzeichnungen/SummerSchool/">workshop at Göttingen</a> on arithmetic geometry in 2006 summer. Almost all of the videos are still available. Wonderful videos.</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | T.B. | 12,312 | <p>Ken Ribet's introductory lecture on Serre's modularity conjecture. Useful and quite easy to follow and understand. <a href="http://fora.tv/2007/10/25/Kenneth_Ribet_Serre_s_Modularity_Conjecture" rel="nofollow">http://fora.tv/2007/10/25/Kenneth_Ribet_Serre_s_Modularity_Conjecture</a></p>
|
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