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 |
|---|---|---|---|---|
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Ian Agol | 1,345 | <p>There's an identity for chromatic polynomials of planar triangulations called the "golden identity" found by Tutte, giving a quadratic relation between the values of the chromatic polynomial at <span class="math-container">$\phi+1$</span> and <span class="math-container">$\phi+2$</span>. In a fairly <a hre... |
373,951 | <p>Suppose we have a linearly ordered group over $\mathbb Z^n$ where the ordering goes left-to-right, i.e. when deciding if $(x_1,x_2,\dots)<(y_1,y_2,\dots)$ we first check if $x_1< y_1$, if it is then $X< Y$. If they are the same, we compare $x_2< y_2$ and so on.</p>
<p>I believe there is an isomorphism t... | Cameron Buie | 28,900 | <p>No. Note that an order-isomorphism will be an injective continuous function when both are considered in the order topology. Consider then $\Bbb R^2$ ordered as you describe, and in particular consider the images of the connected sets $\{x\}\times\Bbb R$ of $\Bbb R^2$ under any continuous function. The continuous ima... |
373,951 | <p>Suppose we have a linearly ordered group over $\mathbb Z^n$ where the ordering goes left-to-right, i.e. when deciding if $(x_1,x_2,\dots)<(y_1,y_2,\dots)$ we first check if $x_1< y_1$, if it is then $X< Y$. If they are the same, we compare $x_2< y_2$ and so on.</p>
<p>I believe there is an isomorphism t... | Pete L. Clark | 299 | <p>An ordered abelian group <span class="math-container">$(G,+,<)$</span> is <strong>Archimedean</strong> if for all <span class="math-container">$x,y \in G$</span> with <span class="math-container">$x > 0$</span>,
there is a positive integer <span class="math-container">$n$</span> with <span class="math-containe... |
2,255,192 | <p>I'm going through the exercises in Georgi E Shilov's Linear Algebra book and am on chapter 1 problem 2: "Write down all the terms appearing in the determinant of order four which have a minus sign and contain $ a_{23}$"</p>
<p>the answers I have arrived at are: </p>
<p>$a_{11}$$a_{23}$$a_{32}$$a_{44}$</p>
<p>$a_{... | Jed | 441,147 | <p>Every iteration the green area is reduced by 25%.
As the iteration number tends toward infinity, the green area will approach 0.
Since the white area is everything else, its area will tend toward the original area (1).</p>
|
101,972 | <p>I have a grammatically computable function $f$, which means that a grammar $G = (V,\Sigma,P,S)$ exists, so that</p>
<p>$SwS \rightarrow v \iff v = f(w)$.</p>
<p>Now I have to show that, given a grammatically computable function $f$, a Turing Machine $M$ can be constructed, so that $M$ computes $f$.</p>
<p>Here is... | Marc van Leeuwen | 18,880 | <p><a href="http://en.wikipedia.org/wiki/Polynomial_long_division" rel="nofollow">Polynomial long division</a> is the way to go. Especially over a finite field where you don't have to worry about fractional coefficients (working over for instance the rational numbers these can get extremely unwieldy surprisingly soon).... |
101,972 | <p>I have a grammatically computable function $f$, which means that a grammar $G = (V,\Sigma,P,S)$ exists, so that</p>
<p>$SwS \rightarrow v \iff v = f(w)$.</p>
<p>Now I have to show that, given a grammatically computable function $f$, a Turing Machine $M$ can be constructed, so that $M$ computes $f$.</p>
<p>Here is... | ego | 21,356 | <p>$f$ corresponds to the binary number $1011$ and $g$ to $110$ if you identify $x$ with $2$. Appending a $0$ (rsp. multiplication by $2$) corresponds to multiplying with $x$ and $\oplus$ (exclusive or) is addition.</p>
<pre><code>1011:110 = 11, i.e., the quotient is $x+1$
110
---
111
110
---
1, i.e., the rema... |
3,494,470 | <p>Will all units in <span class="math-container">$\mathbb{Z}_{72}$</span> be also units (modulo <span class="math-container">$8$</span> and <span class="math-container">$9$</span>) of <span class="math-container">$\mathbb{Z}_8$</span> and <span class="math-container">$\mathbb{Z}_9$</span>? </p>
<p>I think yes, becaus... | Bill Dubuque | 242 | <p>Yes, that is correct. More generally if <span class="math-container">$\,f(x)\,$</span> is polynomial with integer coef's and it has a root <span class="math-container">$\,x\equiv r\pmod{\!mn}\,$</span> then that root <em>persists</em> <span class="math-container">$\bmod m\,$</span> & <span class="math-container"... |
3,887,156 | <p>I understand that the vertical shift is <span class="math-container">$0$</span> that is why the graph starts at <span class="math-container">$(0,0)$</span>. Also I understand that the amplitude is <span class="math-container">$3$</span> because the maximum y value is <span class="math-container">$3$</span> and the m... | Zhooo | 819,254 | <p>We have <span class="math-container">$V=span\{(1,1,0),(0,0,1)\}$</span> now we know that
<span class="math-container">$$
\dim\frac{\mathbb{R}^3}{V}=3-\dim V=1
$$</span></p>
<p>so, if <span class="math-container">$v$</span> it's a generator of the quocient, the map <span class="math-container">$v\mapsto 1$</span> ext... |
1,903,520 | <p>By generalizing the approach in <a href="https://math.stackexchange.com/questions/1903152/integral-involving-a-dilogarithm-versus-an-euler-sum">Integral involving a dilogarithm versus an Euler sum.</a> meaning by using the integral representation of the harmonic numbers and by computing a three dimensional integral... | Przemo | 99,778 | <p>By using the functional equations for the trilogarithm we simplified the result as follows:
\begin{eqnarray}
&&S^{(3)}(x)= \\
&&\frac{ \text{Li}_3(x)}{(1-x)}+
3\frac{\text{Li}_3(1-x)-\zeta (3)}{(1-x)}+
\log(1-x)\frac{ \left(-2 \log ^2(1-x)+3 \log (x) \log(1-x)-\pi ^2 \right)}{2 (1-x)}
\end{eqnarray}<... |
417,181 | <p>We have to prove that if the difference between two prime numbers greater than two is another prime,the prime is $2$.
It can be proved in the following way.</p>
<p>1)$Odd -odd =even$. </p>
<p>Therefore the difference will always even.</p>
<p>2)The only even prime number is $2$.Therefore the difference will be $2$... | mrf | 19,440 | <p>Assume that $\sum_{n=1}^\infty b_n$ is a divergent positive series. Then you can always find another divergent positive series $\sum_{n=1}^\infty a_n$ with the property that
$$
\lim_{n\to\infty} \frac{a_n}{b_n} = 0.
$$
(In your case, $b_n = \dfrac{1}{n\log n}$.)</p>
<p>One way to see this is to use a theorem of Abe... |
69,658 | <p>Given a fiber bundle $f: E\rightarrow M$ with connected fibers we call the image $f^*(\Omega^k(M))\subset \Omega^k(E)$ the subspace of basic forms. Clearly, for any vertical vector field $X$ on $E$ we have that the interior product $i_X(f^*\omega)$ and the Lie derivative $L_X(f^*\omega)$ vanish for all $\omega \in \... | Jason DeVito | 331 | <p>First, this can be checked locally, so we may as well assume $E = F\times M$.</p>
<p>Use coordinates $x_i$ on $F$ and $y_j$ on $M$. Then a $k-$form is given by $\alpha =\sum f_{IJ} dx^I\wedge dy^J$. Here, $I = \{i_1,...,i_s\}$ and $dx^I$ means $dx^{i_1}\wedge...\wedge dx^{i_s}$ and we have $|I|+|J|=k$.</p>
<p>Th... |
2,525,498 | <p>I'm an undergrad, and I've been presented with the following problem:</p>
<blockquote>
<p>Fundamental Theorem of Arithmetic: Let $\mathbb{N}_{>0}$ be the monoid of positive
integers with binary operation given by ordinary multiplication, let $P$ be
the set of primes in $\mathbb{N}$, let $M$ be a commutativ... | Burrrrb | 322,248 | <p>The function has to be monotone, assume it's increasing, now consider $f^{-1}(1)$ and observe that $1$ is not in $(0,1)$. </p>
|
2,225,150 | <p>I am seek for a rigorous proof for the following identity</p>
<p>$\sum_{i = 0}^{T} x_i \sum_{j = 0}^{i} y_j = \sum_{i = 0}^{T}y_i\sum_{j = i}^{T} x_j$. </p>
<p>By setting some small $T$ and expand the formulas, it is then clear to see the result. I am asking for help to give a formal proof of this identity, by reo... | Leox | 97,339 | <p>Direct way
\begin{gather*}
\sum_{i = 0}^{T} x_i \sum_{j = 0}^{i} y_j=x_0 y_0+x_1(y_0+y_1)+x_2(y_0+y_1+y_2)+\cdots+x_k(y_0+y_1+\cdots+y_k)+\cdots+x_T(y_0+y_1+\cdots+y_T)=y_0(x_0+x_1+\cdots+x_T)+y_1(x_1+x_2+\cdots+x_T)+\cdots+x_T y_T=\sum_{i=0}^T y_i \sum_{j=i}^T x_j.
\end{gather*}</p>
|
2,225,150 | <p>I am seek for a rigorous proof for the following identity</p>
<p>$\sum_{i = 0}^{T} x_i \sum_{j = 0}^{i} y_j = \sum_{i = 0}^{T}y_i\sum_{j = i}^{T} x_j$. </p>
<p>By setting some small $T$ and expand the formulas, it is then clear to see the result. I am asking for help to give a formal proof of this identity, by reo... | Andrew D. Hwang | 86,418 | <p>The formal approach for all such formulas is mathematical induction. Fix sequences $(x_{i})_{i=0}^{\infty}$ and $(y_{j})_{j=0}^{\infty}$ of summands from some commutative ring (e.g., the field of real numbers). (If you're only interested in your "change of index" formula up to some fixed finite number of summands, t... |
4,027,071 | <p>Let <span class="math-container">$\mathbf{C}$</span> be a category with products, and let <span class="math-container">$A, B, C \in \mathbf{C}$</span>. I wish to show that there exists a morphism <span class="math-container">$h: (A \times B) \times C \to A \times (B \times C)$</span> which is an isomorphism.</p>
<p>... | fosco | 685 | <p>See 1.5 "Diagram Chasing" here <a href="https://compose.ioc.ee/categoryTheory2020/week3/week3.pdf" rel="nofollow noreferrer">https://compose.ioc.ee/categoryTheory2020/week3/week3.pdf</a></p>
|
4,027,071 | <p>Let <span class="math-container">$\mathbf{C}$</span> be a category with products, and let <span class="math-container">$A, B, C \in \mathbf{C}$</span>. I wish to show that there exists a morphism <span class="math-container">$h: (A \times B) \times C \to A \times (B \times C)$</span> which is an isomorphism.</p>
<p>... | Berci | 41,488 | <p><strong>Hint:</strong> As in the comments, prove that both <span class="math-container">$(A\times B)\times C$</span> and <span class="math-container">$A\times (B\times C)$</span> are limits for the discrete 3-point diagram <span class="math-container">$A,B,C$</span> (ternary product).</p>
|
1,292,490 | <blockquote>
<p>Let $(a_{ij})$ be a real $n \times n$ matrix satisfying,</p>
<ol>
<li>$a_{ii} > 0 \space (1 \leq i \leq n) ,$</li>
<li>$a_{ij} \leq 0 \space (i \ne j, 1 \leq i,j \leq n) ,$</li>
<li>$\sum_{i=1}^ {i=n} \space a_{ij} > 0 (1 \leq j \leq n).$ </li>
</ol>
<p>Then $\det (A) > 0... | Pegah | 242,413 | <p>You can also use Gaussian Elimination Method.
Show that in each step of elimination, elements of main diagonal stay positive.
So, you will have an upper triangular matrix in which all the elements of main diagonal are positive. Now use the fact that in an upper triangular matrix, determinant is equal to the product ... |
2,332,419 | <p>What's the angle between the two pointers of the clock when time is 15:15? The answer I heard was 7.5 and i really cannot understand it. Can someone help? Is it true, and why?</p>
| Archis Welankar | 275,884 | <p>At $15$ ie $3$ o' clock minute hand will be at exact $3$. Now for every minute the hour hand moves $0.5$ degrees . Calculation we have $360^o=12$hrs ie $360=12×60$minutes thus $0.5deg/min $ hence the answer is $15\times 0.5=7.5$degrees.</p>
|
3,745,551 | <p>I often see people say that if you have 2 IID gaussian RVs, say <span class="math-container">$X \sim \mathcal{N}(\mu_x, \sigma_x^2)$</span> and <span class="math-container">$Y \sim \mathcal{N}(\mu_y, \sigma_y^2)$</span>, then the distribution of their sum is <span class="math-container">$\mathcal{N}(\mu_x + \mu_y, \... | User5678 | 632,875 | <p>I think what you are looking for is a Multivariate Gaussian — assign each unit to a different dimension and then you have a random Gaussian vector. If there is no correlation between the two dimensions this reduces to each coming from its own Gaussian.</p>
|
393,467 | <p>I am looking for a proof that:</p>
<p>if <span class="math-container">$A_{11}A_{12}...A_{1n}$</span>; <span class="math-container">$A_{21}A_{22}...A_{2n}$</span>; <span class="math-container">$\cdots$</span>; <span class="math-container">$A_{i1}A_{i2}...A_{in}$</span>; <span class="math-container">$\cdots$</span>; <... | Iiro Ullin | 219,013 | <p>Such inequality is impossible: consider <span class="math-container">$p(x)=1$</span>, <span class="math-container">$q(x)=1/(2\sqrt{x})$</span>, as probability densities on <span class="math-container">$(0,1)$</span>. Then <span class="math-container">$D_{KL}(p\parallel q)$</span> is finite, while <span class="math-c... |
393,467 | <p>I am looking for a proof that:</p>
<p>if <span class="math-container">$A_{11}A_{12}...A_{1n}$</span>; <span class="math-container">$A_{21}A_{22}...A_{2n}$</span>; <span class="math-container">$\cdots$</span>; <span class="math-container">$A_{i1}A_{i2}...A_{in}$</span>; <span class="math-container">$\cdots$</span>; <... | Ze-Nan Li | 235,487 | <p>Now, I am trying to answer this question.</p>
<p><strong>Proposition</strong>. <em>If <span class="math-container">$p$</span> and <span class="math-container">$q$</span> are two probability densities, and (upper) bounded by <span class="math-container">$\tau_1$</span> and <span class="math-container">$\tau_2$</span>... |
1,824,966 | <p>Ok, I was asked this strange question that I can't seem to grasp the concept of..</p>
<blockquote>
<p>Let $T$ be a linear transformation such that:
$$T \langle1,-1\rangle = \langle 0,3\rangle \\
T \langle2, 3\rangle = \langle 5,1\rangle $$
Find $T$.</p>
</blockquote>
<p>Is there suppose to be a funct... | user247327 | 247,327 | <p>"Given T find T" makes no sense! But if you are given "$T$" (standard font) and asked to find "$\mathbf{T}$" (bold face) where the bold face has <strong>already</strong> been defined to be the "matrix associated with the linear transformation, then, yes.</p>
<p>You should also understand that the matrix representi... |
37,052 | <p>This is my first question with mathOverflow so I hope my etiquette is up to par here.</p>
<p>My question is regarding a <span class="math-container">$3\times3$</span> magic square constructed using the la Loubère method (see <a href="http://en.wikipedia.org/wiki/Magic_square#Method_for_constructing_a_magic_square_of... | José Figueroa-O'Farrill | 394 | <p>The original reference for Wick's theorem is, not surprisingly, Wick's original 1950 paper: <a href="https://doi.org/10.1103/PhysRev.80.268" rel="nofollow noreferrer"><em>The Evaluation of the Collision Matrix</em></a> published in the Physical Review <strong>80</strong> (2) pp. 268-272. He also shows how to comput... |
3,896,345 | <p>I've been studying a paper in which the author says:</p>
<p>Fix <span class="math-container">$n$</span> such that <span class="math-container">$m^n \prod_{j=1}^n \frac{j}{j+\delta} > 1$</span>, where <span class="math-container">$1<m<\infty$</span>, and <span class="math-container">$\delta >0$</span>.</p... | Mathick | 846,353 | <p>I think I solved this, or at least I managed to convince myself that this is true. Approximately, we have that</p>
<p><span class="math-container">$\prod_{j=1}^{n} \frac{j}{j+\delta} = \prod_{j=1}^{n} \left(1 - \frac{\delta}{j+\delta}\right) \geq c_0 \exp\left[\sum_{j=1}^{n} \log\left(1 - \frac{\delta}{j+\delta}\rig... |
3,669,937 | <p>I had this problem where i had the application <span class="math-container">$\varphi: \mathbb Z[i] \Rightarrow \mathbb Z/(2)$</span> where <span class="math-container">$\varphi(a+bi)=\bar{a}+\bar{b}$</span>. I had to find the kernel and prove that is a factor ideal. I proofed that the kernel is formed by all the com... | J. W. Tanner | 615,567 | <p>A generator of the ideal is <span class="math-container">$1+i$</span>.</p>
<p>Let <span class="math-container">$a+bi$</span> be a multiple of <span class="math-container">$1+i.$</span> </p>
<p>Then <span class="math-container">$a+bi= (x+yi)(1+i)=(x-y)+(x+y)i,$</span> so <span class="math-container">$a+b=2x$</span... |
167,262 | <p>I make a circle with radius as below</p>
<pre><code>Ctest = Table[{0.05*Cos[Theta*Degree], 0.05*Sin[Theta*Degree]}, {Theta, 1, 360}] // N;
</code></pre>
<p>And herewith is my list of data points</p>
<pre><code>pts = {{0., 0.}, {0.00493604, -0.00994539}, {0.00987001, -0.0198918}, {0.0148019, -0.0298392}, {0.019731... | bill s | 1,783 | <p>Another way to specify the condition on a variable that it be either zero or 1 is to observe that the square of the variable must equal itself, e.g. x^2==x has only solutions x=0 and x=1. So we could add these conditions to the solution. Here is a small example:</p>
<pre><code>Solve[Flatten[{2 x[1] + 3 x[2] - 2 x[3... |
250,074 | <p>How can one generate a random vector <span class="math-container">$v=[v_1, v_2, v_3]^T$</span> satisfying <span class="math-container">$\sqrt{v_1v_1^* + v_2 v_2^* + v_3 v_3^*} = 1$</span>, where <span class="math-container">$T$</span> and <span class="math-container">$*$</span> denote the transpose and complex-conju... | mikado | 36,788 | <p>This is a very simple 1-liner giving a list of <code>n</code> such random vectors</p>
<pre><code>sphericalrandom[n_] := Normalize /@ RandomVariate[NormalDistribution[0, 1], {n, 3}]
</code></pre>
<p>Note that these are uniformly distributed on the sphere, since the multivariate normal distribution is invariant under ... |
2,618,746 | <p>The distance between two stations $X$ and $Y$ is 220 km.</p>
<p>Trains $P$ and $Q$ leave station $X$ at 7 am and 8:15 am respectively at the speed of 25 km/hr and 20 km/hr respectively for journey towards $Y$.</p>
<p>Train $R$ leaves station $Y$ at 11:30 am at a speed of 30 km/hr for journey towards $X$. </p>
<p>... | lab bhattacharjee | 33,337 | <p>Let the time of equidistance be $t$ hours after $7$ AM</p>
<p>So, $Q$ will travel $20(t-5/4)=20t-25$ km</p>
<p>$P$ will travel $25t$ km</p>
<p>and $R$ will travel $30(t-9/2)$ km hence is $220- 15(2t-9)=355-30t$ km away from $X$</p>
<p>We need $$20t-25+355-30t=2\cdot25t$$</p>
|
3,928,937 | <p>Determine all the solutions of the congruence<br />
<span class="math-container">$x^{85} ≡ 25 \pmod{31}$</span><br />
using index function in base <span class="math-container">$3$</span> module <span class="math-container">$31$</span>.<br />
It is clear to me that <span class="math-container">$3$</span> is primitive... | lab bhattacharjee | 33,337 | <p>Using <a href="https://mathworld.wolfram.com/DiscreteLogarithm.html" rel="nofollow noreferrer">Discrete logarithm</a> with respect to base <span class="math-container">$3$</span>,</p>
<p><span class="math-container">$85\cdot$</span>ind<span class="math-container">$_3x\equiv2\cdot$</span>ind<span class="math-contain... |
604,824 | <p>So the puzzle is like this:</p>
<blockquote>
<p>An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to pour and washes away all its scent trail. This ant has the strength of traveling 280 ft more then it ... | Will Nelson | 62,773 | <p>Rescale so that radius $R=1$. Assume the starting point (the center of the circle) is at $(0,0)$. Unless I've made a calculation error (I don't think so), the minimum is achieved with the following path:</p>
<ul>
<li><p>Straight line from $(0,0)$ to $\left(\frac{\sqrt{3}}{3},-1\right)$.</p></li>
<li><p>Straight lin... |
604,824 | <p>So the puzzle is like this:</p>
<blockquote>
<p>An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to pour and washes away all its scent trail. This ant has the strength of traveling 280 ft more then it ... | Constructor | 114,355 | <h3>Historical summary</h3>
<p>It is the famous problem invented by R. Bellman in 1956 [1]. It is known as <a href="http://mathworld.wolfram.com/LostinaForestProblem.html" rel="noreferrer" title="Page on 'Wolfram MathWorld' site">'Lost in a Forest Problem'</a>:</p>
<blockquote>
<p>What is the best path to f... |
959,322 | <p>Solve $$ \sum_{k = 1}^{ \infty} \frac{\sin 2k}{k}$$</p>
<p>I first tried to use Eulers formula</p>
<p>$$ \frac{1}{2i} \sum_{k = 1}^{ \infty} \frac{1}{k} \left( e^{2ik} - e^{-2ik} \right)$$</p>
<p>However to use the geometric formula here, I must subtract the $k=0$ term and that term is undefinted since $1/k$. I... | Leucippus | 148,155 | <p>The summation is as follows:
\begin{align}
S &= \sum_{n=1}^{\infty} \frac{\sin(2an)}{n} = \frac{1}{2i} \, \sum_{n=1}^{\infty} \frac{ e^{2ai n} - e^{- 2ai n}}{n} \\
&= - \frac{1}{2i} \left( \ln(1 - e^{2ai}) - \ln(1 - e^{-2ai}) \right) \\
&= - \frac{1}{2i} \ln\left( \frac{1 - e^{2ai}}{1 - e^{- 2ai}} \right... |
379,194 | <p>Let $X$ be a topological space and $X^*$ be its supspace. It is stated in my textbook that if $c(A)$ represents the closure of set $A$ in $X$, then $c(A) \bigcap X^*$ is closed in $X^*$. </p>
<p>A closed set is one which contains all its limit points, and a limit point of a set is a point such that every open set c... | Stefan Hansen | 25,632 | <p><strong>Hints</strong>: If $f$ and $g$ are differentiable at $x$, then
$$
(f+g)'(x)=f'(x)+g'(x),
$$
and if $a$ is a constant, then
$$
(af)'(x)=a\cdot f'(x)
$$
If $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$ then
$$
(f\circ g)'(x)=f'(g(x))\cdot g'(x).
$$
And lastly if $f(x)=x^n$ for $n\in\mathbb{... |
3,623,924 | <p>Trying to solve the following problem:</p>
<p>Let <span class="math-container">$f(x)$</span> be a continuous real-valued function on <span class="math-container">$[0,3]$</span>. Given any <span class="math-container">$\varepsilon>0$</span> prove there exists a polynomial, <span class="math-container">$p(x)$</spa... | user27182 | 22,020 | <p>SW says that for any <span class="math-container">$\epsilon > 0$</span> there is a polynomial <span class="math-container">$p(x)$</span> such that <span class="math-container">$\forall x \in [0, 3]$</span> we have <span class="math-container">$|f(x) - p(x)| < \epsilon / 4$</span>. </p>
<p>Then
<span class="m... |
2,311,979 | <p>Let $A = (a_{i,j})_{n\times n}$ and $B = (b_{i,j})_{n\times n}$</p>
<p>$(AB) = (c_{i,j})_{n\times n}$, where $c_{i,j} = \sum_{k=1}^n a_{i,k} b_{k,j}$, so</p>
<p>$(AB)^T = (c_{j,i})$, where $c_{j,i} = \sum_{k=1}^n a_{j,k}b_{k,i} $, and
$B^T = b_{j,i}$ and $A^T = a_{j,i}$, so </p>
<p>$B^T A^T = d_{j,i}$ where $d_... | Sangchul Lee | 9,340 | <p>You seem to know that $(i,j)$-entry of $B^T$ is $b_{j,i}$, that is probably why you are writing $B^T = (b_{j,i})$. The issue is, this notation is confusing as it is not telling you which index denotes the row and which denotes the column. I guess this is where you get confused.</p>
<p>To make things clear, let us u... |
1,463,881 | <blockquote>
<p>By considering $\sum_{r=1}^n z^{2r-1}$ where z= $\cos\theta + i\sin\theta$, show that if $\sin\theta$ $\neq$ 0, $$\sum_{r=1}^n \sin(2r-1)\theta=\frac{\sin^2n\theta}{\sin\theta}$$</p>
</blockquote>
<p>I couldn't solve this at first but with some hints some of you gave, I was able to come up with my ow... | K_P | 181,185 | <p>The problem should be equivalent to $x_1 +x_2+x_3+x_4+x_5=5$ with $0≤x_1≤3$ and $0≤x_2<3$ and $x_3 \ge 0$.The formula should be $n-r+1\choose {r-1}$ wich gives ${9 \choose 4}=126 $ solutions without the restrictions. Applying the restrictions we need to get rid of the instances of $x_1=4$ (there are 4 of them) an... |
1,001,839 | <p>$$\frac{\pi x y^2}{4}$$</p>
<p>Is this function continuous? I really haven't worked with continuity with multivariable funtions before, so I am a little stumped. How would one answer such a question? </p>
<p>I'm reading a bit ahead of my level, and I'm seeing all these epsilon delta things... is that what I am sup... | Race Bannon | 188,877 | <p>You are asking if the function $f(x,y) = \frac{\pi xy^{2}}{4}$ is continuous? If you had a functions of x or y alone, then it'd be easy to see that it's continuous, right? Together, they should still be continuous... I hope this agrees with your intuition. Proving that the function is continuous, you just need the d... |
189,380 | <p>How can I solve this ODE:</p>
<p>$y(x)+Ay'(x)+Bxy'(x)+Cy''(x)+Dx^{2}y''(x)=0$</p>
<p>Can you please also show the derivation.</p>
| Robert Israel | 8,508 | <p>I don't think you'll find an "elementary" solution in general. Maple finds a rather complicated solution involving hypergeometric functions:
$$\displaystyle S\, := \,y \left( x \right) ={\it \_C1}\,{\mbox{$_2$F$_1$}(1/2\,{\frac {-d+B+ \sqrt{{d}^{2}+ \left( -2\,B-4 \right) d+{B}^{2}}}{d}},1/2\,{\frac {-d+B- \sqrt{{d... |
86,202 | <p>Let $\mathcal{L},\mathcal{U}$ be invertible sheaves over a
noetherian scheme $X$, where $X$ is of finite type over a noetherian
ring $A$. If $\mathcal{L}$ is very ample, and $\mathcal{U}$ is
generated by global sections, then $\mathcal{L} \otimes \mathcal{U}$
is very ample.</p>
<p>Since $\mathcal{L}$ is very ample,... | red_trumpet | 312,406 | <p>It is true that if <span class="math-container">$i: X \to Y$</span> is an immersion, and <span class="math-container">$j:X \to Z$</span> is <em>any</em> morphism (all over <span class="math-container">$S$</span>), then <span class="math-container">$(i, j): X \to Y \times_S Z$</span> is an immersion. See <a href="htt... |
2,623,560 | <blockquote>
<p>Decide if $\mathbb Z[i]/\langle i\rangle$ and $\mathbb Z$ are isomorphic, if $\mathbb Z[i]/\langle i+1\rangle$ and $\mathbb Z_2$ are isomorphic</p>
</blockquote>
<p>I know that in the first case if there exist such homomorphism then $f(i)=0$ (and in the second case $f(i+1)=0$), but I don't know exa... | Pedro | 178,668 | <p>To show the first isomorphism you can use one of the <strong>isomorphism theorems</strong>:</p>
<p>$$(A+I)/I \cong A/(A\cap I) $$</p>
<p>In this case $A=\mathbb{Z}\subseteq \mathbb{Z}[i]$ and $I=(i)$.</p>
<p>In the second case, note that $(i+1)^{2}=(2)$, so we have $\mathbb{Z}\cap (i+1)^{2}=(2)$. Then apply again... |
2,623,560 | <blockquote>
<p>Decide if $\mathbb Z[i]/\langle i\rangle$ and $\mathbb Z$ are isomorphic, if $\mathbb Z[i]/\langle i+1\rangle$ and $\mathbb Z_2$ are isomorphic</p>
</blockquote>
<p>I know that in the first case if there exist such homomorphism then $f(i)=0$ (and in the second case $f(i+1)=0$), but I don't know exa... | lhf | 589 | <p>Here is an answer for the second question:</p>
<p>Since $2=(1+i)(1-i)$, we have $2 \in (1+i)$.</p>
<p>Thus, $a+bi \equiv (a\bmod 2)+(b\bmod 2)\,i \equiv (1+i)$.</p>
<p>Therefore, the classes of $\mathbb Z[i]$ mod $(1+i)$ are reduced to the classes of $0,1,i,1+i$.</p>
<p>Now $0 \equiv 1+i$ and $1 \equiv i$ and so... |
425,981 | <p>Let $F$ be an infinite field such that $F^*$ is a torsion group. We know that $F^*$ is an Abelian group. So every subgroup of $F^*$ is a normal subgroup.</p>
<p>My question:</p>
<p>Does $F^*$ have a proper subgroup with finite index?</p>
| Jyrki Lahtonen | 11,619 | <p>Let $F$ be the algebraic closure of a finite field. Each element of $F^*$ belongs to a finite field, so is a torsion element. On the other hand $F^*$ cannot have a subgroup of index $n>1$. For if $A$ is such a subgroup, then $x^n\in A$ for all $x\in F^*$. But if $z\in F^*\setminus A$, then $z$ has an $n$th root i... |
425,981 | <p>Let $F$ be an infinite field such that $F^*$ is a torsion group. We know that $F^*$ is an Abelian group. So every subgroup of $F^*$ is a normal subgroup.</p>
<p>My question:</p>
<p>Does $F^*$ have a proper subgroup with finite index?</p>
| DonAntonio | 31,254 | <p><strong>Note:</strong> This uses the very same reasoning as Jyrki's answer but with a different, perhaps slightly more groupwise, approach:</p>
<p>(1) An abelian group $\,A\,$ (with multiplicative operation, to fit within our problem) is <em>divisible</em> if</p>
<p>$$\forall\,g\in G\;\wedge\;\forall n\in\Bbb N\;... |
1,554 | <p>Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold structure extend across the point singularity?</p>
<p>(Penny Smith and I wrote a paper on this many years ago, but we... | Igor Belegradek | 1,573 | <p>Here is what seems to be a counterexample. Let (M,g) be a simply-connected closed Riemannian manifold. Then M times (0,infinity) with the warped product metric dr^2 + r^2 g has bounded curvature and the completion at r=0 is a point. If the metric is smooth, then M is diffeomorphic to a sphere, so any other M gives a... |
1,554 | <p>Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold structure extend across the point singularity?</p>
<p>(Penny Smith and I wrote a paper on this many years ago, but we... | Rafe Mazzeo | 888 | <p>Take a cone over a finite quotient S^{2n-1}/\Gamma. The curvature is 0, but the manifold
structure does not even extend. (More generally, you can take the cone over any compact
Einstein manifold of dimension n-1 with Einstein constant n-2.) </p>
|
1,554 | <p>Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold structure extend across the point singularity?</p>
<p>(Penny Smith and I wrote a paper on this many years ago, but we... | valeri | 1,988 | <p>What if you try a family of triangles, parallel to some two-direction s.t. their union contains singularity? (Like tetraedr for n=3)? Then their geometry (angles, sides, etc) are controlled from "outside" the singularity, so they all have uniformly bounded cirvature - including that one which contains singularity. L... |
1,930,401 | <p>Are there any non-linear real polynomials $p(x)$ such that $e^{p(x)}$ has a closed form antiderivative? If not, is the value of $\int_{0}^{\infty}e^{p(x)}dx$ known for any $p$ with negative leading term other than $-x$ and $-x^2$?</p>
| Hrhm | 332,390 | <p>This proof was given <a href="http://www.drking.org.uk/hexagons/misc/polymax.html" rel="nofollow noreferrer">here</a>.</p>
<p>Take the polygon and arrange it so that all the vertices lie on a circle:
<a href="https://i.stack.imgur.com/U4u4q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/U4u4q.pn... |
3,084,934 | <p>I want to prove or disprove that the Fourier transform <span class="math-container">$\mathcal F \colon (\mathcal S(\mathbb R^d), \lVert \cdot \rVert_1) \to L^1(\mathbb R^d)$</span> is unbounded, where <span class="math-container">$\lVert\cdot \rVert_1$</span> denotes the <span class="math-container">$L^1(\mathbb R^d... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>We have to show that for each <span class="math-container">$\epsilon > 0 \ \exists \delta > 0$</span> such that whenever <span class="math-container">$|x-a| < \delta$</span> you have
<span class="math-container">$$
\epsilon
> \left|\frac{1}{f(x)} - 0 \right|
= \left|\frac... |
456,583 | <p>I was searching for a Latex symbol that indicates $A \Rightarrow B$ and $A \not\Leftarrow B$ ($B$ if not only if $A$, $B$ ifnf $A$). I thought of using $A \Leftrightarrow B$ with the left arrow tick <code><</code> crossed out. Since I did not find such a symbol:</p>
<p>Is there a Latex symbol for this?</p>
<p>H... | Andreas Blass | 48,510 | <p>When people assert implications, they often implicitly involve universal quantification. For example, "if $n$ is a prime number greater than $2$, then $n$ is odd" really means "for all integers $n$, if $\dots$." When one denies an implication, one includes the universal quantifier in the denial, so it becomes an e... |
34,487 | <p>A few years ago Lance Fortnow listed his favorite theorems in complexity theory:
<a href="http://blog.computationalcomplexity.org/2005/12/favorite-theorems-first-decade-recap.html" rel="nofollow">(1965-1974)</a>
<a href="http://blog.computationalcomplexity.org/2006/12/favorite-theorems-second-decade-recap.html" rel=... | Kevin H. Lin | 83 | <p>Well I guess after Cook, Karp's paper "Reducibility among combinatorial problems" is the second most obligatory and canonical thing to mention. This paper was the first to demonstrate to the world the diversity and ubiquity of NP-complete problems.</p>
|
3,702,094 | <p>For a school project for chemistry I use systems of ODEs to calculate the concentrations of specific chemicals over time. Now I am wondering if </p>
<p><span class="math-container">$$ \frac{dX}{dt} =X(t) $$</span></p>
<p>the same is as </p>
<p><span class="math-container">$$ X(t)=e^t . $$</span> </p>
<p>As far a... | Community | -1 | <p>It is true that <span class="math-container">$$X(t)=e^t$$</span> is a solution of the differential equation <span class="math-container">$X'(t)=X(t)$</span>. But we know from the theory that the solution must be a family of functions, depending on an arbitrary constant.</p>
<p>The usual way to solve this separable ... |
2,547,488 | <p>We have always been taught that a function assigns to every element in the domain a single <em>unique</em> element in the range. If a rule of assignment assigns to one element in the domain more than one element in the range then it isn't a function.</p>
<p>Now in Munkres' <em>Topology</em>, on page 107, it says:</... | sti9111 | 233,046 | <p>$f^{-1}(U)=\{x\in X | f(x)\in U\}$, here we do not need that the function $f^{-1}$
there exist, we only use definition of the pre-image of set by $f,$ and off course
any constant function is continuous because given any pen set $U$ in $Y$
we have two possibilites, $y_0\in U$ in this case $f^{-1}(U)=X$ which is open ... |
1,783,200 | <p>Prove or disprove the following statement:</p>
<p><strong>Statement.</strong> <em>Continuous for each variables, when other variables are fixed, implies continuous?</em> More clearly, prove or disprove the following problem:</p>
<p>Let $\displaystyle f:\left[ a,b \right]\times \left[ c,d \right]\to \mathbb{R}$ for... | Quản Bá Hồng Nguyễn | 286,880 | <p>I have built a counter-example for this statement. Because the building process is quite complicated (it has images too). So I post my <em>counter example</em> in the following link:
<a href="https://hongnguyenquanba.wordpress.com/2016/05/12/problem-6/" rel="nofollow">https://hongnguyenquanba.wordpress.com/2016/05/1... |
69,476 | <p>Hello everybody !</p>
<p>I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer scientist each day) that if the result is exact this may not be the computationally fastest way possible to do it... | paperclip optimizer | 472,708 | <p>If I understood your question correctly, you are willing to do an arbitrarily large amount of precomputing on your polynomial in order to make the evaluation process at run time as fast as possible.</p>
<p>In other words, you want to find a some way of making your polynomial evaluation "sparse" in some sen... |
1,336,937 | <p>I think: <em>A function $f$, as long as it is measurable, though Lebesgue integrable or not, always has Lebesgue integral on any domain $E$.</em></p>
<p>However Royden & Fitzpatrick’s book "Real Analysis" (4th ed) seems to say implicitly that “a function could be integrable without being Lebesgue measurable”. I... | Tyr Curtis | 209,743 | <p>Let $A\subset [0,1]$ be a nonmeasurable set. Then consider the function $f(x)=1$, if $x\in A$ and $f(x)=-1$, if $x \in [0,1]\setminus A$ and zero everywhere else. Then clearly $f$ is not measurable, but $|f(x)|=\chi_{[0,1]}(x)$, so it's integrable.</p>
|
856,334 | <p>the problem is based on this picture. <img src="https://i.stack.imgur.com/51fmQ.jpg" alt="enter image description here"></p>
<p>at beginning or we say $t=0$, $P$ is a circle of which the center is at the point $(0,r)$, $r_0=1$ is the initial radius of this circle. $AB$ is a vector which has an angle $\theta$ from t... | MvG | 35,416 | <h1>Some analysis</h1>
<p>I choose $x$ as a free parameter, and $y(x)$ as a function to describe the movement. You want $l=2r$ which you can rewrite as</p>
<p>\begin{align*}
x^2+(y-r)^2&=(2r)^2\\
x^2+y^2-2yr+r^2&=4r^2\\
3r^2+2yr-(x^2+y^2)&=0\\
r&=\frac{-y+\sqrt{3x^2+4y^2}}{3}
\end{align*}</p>
<p>The ... |
1,640,733 | <p>I think it is true that any power of a logarithm, no matter how big, will eventually grow slower than a linear function with positive slope.</p>
<p>Is it true that for any exponent $m>0$ (no matter how big we make $m$), the function $f(x)$ $$f(x)=(\ln x)^m$$</p>
<p>will eventually always be less than $g(x) = x$... | DanielWainfleet | 254,665 | <p>No calculus required.Taking logs to any base $b=1+r$ with $r>0,$ then for $x>(1+r)^2$ we have $\log_b x>2.$ So for $x>(1+r)^2$ let $n_x$ be the positive integer such that $$n_x\leq \log_bx<n_x+1.$$ We have then $b^{n_x}\leq x<b^{n_x+1}$. And since $n_x\geq 2$ and $r>0$, we have, by the Binomial ... |
88,122 | <p>For the easiest case, assume that $L/E$ is Galois and $E/K$ is Galois. Under what conditions can we conclude that $L/K$ is Galois? I guess the general case can be a bit tricky, but are there some "sufficiently general" cases that are interesting and for which the question can be answered?</p>
<p>EDIT: Since Jyrki's... | Lior B-S | 26,713 | <p>If every $K$-automorphism of $E$ can be extended to a $K$-automorphism of $L$, then $L/K$ is Galois. </p>
|
191,175 | <p>How to calculate the limit of $(n+1)^{\frac{1}{n}}$ as $n\to\infty$?</p>
<p>I know how to prove that $n^{\frac{1}{n}}\to 1$ and $n^{\frac{1}{n}}<(n+1)^{\frac{1}{n}}$. What is the other inequality that might solve the problem?</p>
| Jonathan | 37,832 | <p>With
$$y=\lim_{n\to\infty} (n+1)^{1/n},$$
consider, using continuity of $\ln$,
$$\ln y=\lim_{n\to\infty} \frac{1}{n}\ln(n+1)=0.$$
This tells you that your limit is $1$.</p>
<p>Alternately,
$$n^{1/n}<n^{1/n}\left(1+\frac{1}{n}\right)^{1/n}<n^{1/n}\left(1+\frac{1}{n}\right),$$
where the middle guy is your expre... |
191,175 | <p>How to calculate the limit of $(n+1)^{\frac{1}{n}}$ as $n\to\infty$?</p>
<p>I know how to prove that $n^{\frac{1}{n}}\to 1$ and $n^{\frac{1}{n}}<(n+1)^{\frac{1}{n}}$. What is the other inequality that might solve the problem?</p>
| Martin Argerami | 22,857 | <p>For the other inequality, you could use
$$
(n+1)^{\frac1n}\leq (2n)^{\frac1n}=2^{\frac1n}\,n^{\frac1n}.
$$</p>
|
3,780,575 | <p>We know that if <span class="math-container">$f$</span> is continuous on [a,b] and <span class="math-container">$f:[a,b] \to \mathbb{R}$</span>, then there exists <span class="math-container">$c \in [a,b]$</span> with <span class="math-container">$f(c)(a-b) = \int_a^bf(x)dx$</span></p>
<p>If we change ''f is continu... | user69608 | 793,349 | <p>Found an another solution:</p>
<p>we have <span class="math-container">$a^2=b(c+b)$</span></p>
<p>A triangle of smallest perimeter means <span class="math-container">$gcd(a,b,c)=1$</span></p>
<p>In fact <span class="math-container">$gcd(b,c)=1$</span> since any common factor of <span class="math-container">$b,c$</sp... |
1,169,336 | <p>Using the formal definition of convergence, Prove that $\lim\limits_{n \to \infty} \frac{3n^2+5n}{4n^2 +2} = \frac{3}{4}$.</p>
<p>Workings:</p>
<p>If $n$ is large enough, $3n^2 + 5n$ behaves like $3n^2$</p>
<p>If $n$ is large enough $4n^2 + 2$ behaves like $4n^2$</p>
<p>More formally we can find $a,b$ such that... | AAkash | 219,951 | <p>It simple</p>
<p>$$ \lim_{x \to \infty} \dfrac{3 + \dfrac{5}{n}}{4 + \dfrac{2}{n^2}}$$</p>
<p>$$ = \dfrac{3}{4}$$</p>
|
1,169,336 | <p>Using the formal definition of convergence, Prove that $\lim\limits_{n \to \infty} \frac{3n^2+5n}{4n^2 +2} = \frac{3}{4}$.</p>
<p>Workings:</p>
<p>If $n$ is large enough, $3n^2 + 5n$ behaves like $3n^2$</p>
<p>If $n$ is large enough $4n^2 + 2$ behaves like $4n^2$</p>
<p>More formally we can find $a,b$ such that... | Timbuc | 118,527 | <p>Perhaps simpler:</p>
<p>With the Squeeze Theorem:</p>
<p>$$\frac34\xleftarrow[x\to\infty]{}\frac{3n^2}{4n^2}\le\frac{3n^2+5n}{4n^2+2}\le\frac{3n^2+5n}{4n^2}=\frac34+\frac54\frac1{n}\xrightarrow[n\to\infty]{}\frac34+0=\frac34$$</p>
<p>With arithmetic of limits:</p>
<p>$$\frac{3n^2+5n}{4n^2+2}=\frac{3n^2+5n}{4n^2+... |
2,280,243 | <blockquote>
<p>A tribonacci sequence is a sequence of numbers such that each term from the fourth onward is the sum of the previous three terms. The first three terms in a tribonacci sequence are called its <em>seeds</em> For example, if the three seeds of a tribonacci sequence are $1,2$,and $3$, it's 4th terms is $... | Stefan Gruenwald | 149,416 | <p>Here is a little python program for your particular tribonacci sequence. You first seed your list with the start values (in your case 6,19,22). The you load your variables with them (a,b,c) and you define how the variables will be updated: a will be the sum of a+b+c in the next iteration. b will be set to the old va... |
52,299 | <p>Hello everybody.</p>
<p>I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1-} f(re^{i\phi})=0$.</p>
<p>Does anyone know such an example.</p>
<p>Best
CJ</p>
| David Hansen | 1,464 | <p>According to a footnote in the famous Hardy-Ramanujan paper "Asymptotic formulae in combinatory analysis", the function $f(q)=\prod_{n=1}^{\infty}\frac{1}{1-q^n}$ vanishes like $f(re^{i\theta})=o((1-r)^{1/4-\varepsilon})$ for almost all $\theta$. No proof is given, though I can't imagine Hardy would have made a stat... |
201,163 | <p>I have a data set that contains data of the form (x0, y0, f1, f2, i1, i2, i3). The (x0, y0) are the coordinates, while the values f1 and f2 are real numbers (i1, i2, i3 correspond to some integers which are used as indices). The data can be downloaded <a href="http://www.mediafire.com/file/0xtxn8rggjorhdj/basins_%25... | kickert | 54,320 | <p>The <code>Predict</code> function can provide you the information you need.</p>
<p>Start by importing your data into Mathematica. For me, it was easiest to change the file extensions to <code>.txt</code> and use <code>SemanticImport</code>.</p>
<pre><code>rawdata = SemanticImport["basins_(L4).txt"] // Normal;
mis... |
4,056,273 | <h2><a href="https://gateoverflow.in/357468/gate-cse-2021-set-1-ga-question-9" rel="nofollow noreferrer">GATE CSE 2021 Set 1 | GA Question: 9</a></h2>
<hr />
<p>Given below are two statements 1 and 2, and two conclusions I and II</p>
<ul>
<li><strong>Statement 1:</strong> All bacteria are microorganisms.</li>
<li><stro... | JJacquelin | 108,514 | <p><span class="math-container">$$\frac{d \beta}{dt}=\frac{c}{R(\beta(t))+c_0}\quad;\quad
R(\beta)=\frac{1}{\sqrt{1-\big(e \cos(\beta)\big)^2}}$$</span></p>
<p><span class="math-container">$$\left(\frac{1}{\sqrt{1-\big(e \cos(\beta)\big)^2}}+c_0\right)\frac{d \beta}{dt}=c$$</span></p>
<p><span class="math-container">$$... |
4,056,273 | <h2><a href="https://gateoverflow.in/357468/gate-cse-2021-set-1-ga-question-9" rel="nofollow noreferrer">GATE CSE 2021 Set 1 | GA Question: 9</a></h2>
<hr />
<p>Given below are two statements 1 and 2, and two conclusions I and II</p>
<ul>
<li><strong>Statement 1:</strong> All bacteria are microorganisms.</li>
<li><stro... | Eugene | 726,796 | <p><span class="math-container">$$
\begin{aligned}
\dot{\beta}{(\theta)} &= \frac{c}{R(\beta{(\theta)}) + c_0} \Leftrightarrow \\
&\Leftrightarrow \left(R(\beta{(\theta)}) + c_0\right)\dot{\beta}{(\theta)} = c \Leftrightarrow \\
&\Leftrightarrow \int\limits_0^\theta\left(R(\beta{(\tau)}) + c_0\right)\dot{\b... |
939,868 | <p>There are a lot math journals with title "acta" includes, for instance, Acta Mathematica, acta arithmetica, etc. Would you explain what "acta" means?</p>
| Lucian | 93,448 | <p>Acta is the plural of actum, a Latin word refering to $($official$)$ papers or documents.</p>
|
113,963 | <p>While the common approach to algebraic groups is via representable functors, it seems that there is no such for differential algebraic groups (defined by differential polynomials). Neither the book by E. Kolchin, nor the texts by Ph. J. Cassidy contain anything like this — they work only with the groups of points ov... | anonymous | 21,885 | <p>A functorial-schematic approach to differential/difference algebraic groups is surely possible. Why this is not to be found in the literature is probably due to historic reasons. The major bulk of results in differential and difference algebra was obtained in a time where algebraic geometry in the style of Weil's "F... |
2,354,004 | <p>I'm struggling with the following sum:</p>
<p>$$\sum_{n=0}^\infty \frac{n!}{(2n)!}$$</p>
<p>I know that the final result will use the error function, but will not use any other non-elementary functions. I'm fairly sure that it doesn't telescope, and I'm not even sure how to get $\operatorname {erf}$ out of that.</... | J.G. | 56,861 | <p>Here's a hint. Write the fraction as $\int_0^\infty\frac{x^n}{(2n)!}e^{-x}dx$. The sum is then $\int_0^\infty\cosh\sqrt{x}e^{-x} \, dx=\int_0^\infty 2y\cosh y e^{-y^2} \, dy$. Can you take it from there?</p>
|
150,180 | <p>I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points:</p>
<p>Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered pairs $(E,E^{'})$ of elliptic curves together with cyclic isogeny $E\rightarrow E^{'}$ of degree $N$. Gross u... | Anton Fonarev | 10,941 | <p>Under the Noetherian hypothesis, the category of coherent sheaves is the smallest abelian subcategory (say, in the category of $\mathcal{O}_X$-modules), containing all line bundles. I don't have a reference in mind, but any standard algebraic geometry text should work (Harstsorne, Liu, Vakil's notes, Stacks project ... |
3,599,893 | <p>I had this idea to build a model of Earth in Minecraft. In this game, everything is built on a 2D plane of infinite length and width. But, I wanted to make a world such that someone exploring it could think that they could possibly be walking on a very large sphere. (Stretching or shrinking of different places is OK... | Daniel R. Collins | 266,243 | <p>Not really a full answer, but elaboration on the OP's "I was creating a hole in the world" observation: It's been known since antiquity that one can make a <a href="https://en.wikipedia.org/wiki/Stereographic_projection" rel="nofollow noreferrer">stereographic projection</a> of a plane onto sphere, missing one singl... |
3,599,893 | <p>I had this idea to build a model of Earth in Minecraft. In this game, everything is built on a 2D plane of infinite length and width. But, I wanted to make a world such that someone exploring it could think that they could possibly be walking on a very large sphere. (Stretching or shrinking of different places is OK... | Tanner Swett | 13,524 | <p>I'd like to add another visual which complements <a href="https://math.stackexchange.com/a/3600741/13524">James K's answer</a>.</p>
<p>What you need is this:</p>
<p><a href="https://i.stack.imgur.com/5MyN4.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5MyN4.png" alt="A plain white square pillo... |
1,617,239 | <blockquote>
<p><strong>Problem</strong></p>
<p>Find the area of the cone <span class="math-container">$z=\sqrt{2x^2+2y^2}$</span> inscribed in the sphere <span class="math-container">$x^2+y^2+z^2=12^2$</span>.</p>
</blockquote>
<p>I think I have to solve this via the surface integral</p>
<p><span class="math-container... | Thomas Rasberry | 265,575 | <p>You could try to project the (double) cone and the sphere onto the $xz$-plane via substituting $y=0$, leaving you with the question of simply rotating the line $z=\sqrt{2}x$ around the $z$ axis to generate the (double) cone (where would this intersect with the circle the sphere would project on the $xz$-plane?). You... |
636,246 | <blockquote>
<p>Let $g(x)=x^2\sin(1/x)$, if $x \neq 0$ and $g(0)=0$. If $\{r_i\}$ is the numeration of all rational numbers in $[0,1]$, define
$$
f(x)=\sum_{n=1}^\infty \frac{g(x-r_n)}{n^2}
$$
Show that $f:[0,1] \rightarrow R$ is differentiable in each point over [0,1] but $f'(x)$ is discontinuous over each $r_n$... | David Mitra | 18,986 | <p>A standard result from analysis is: </p>
<p>Let $I$ be a bounded interval of $\Bbb R$ and let $(f_n)$ be a sequence of functions on $I$ to $\Bbb R$. Suppose there exists $x_0\in I$ such that $(f_n(x_0))$ converges, and that the sequence $(f_n')$ of derivatives exists on $I$ and converges uniformly on $I$ to a func... |
10,942 | <p>I heard about it sometime somewhere and want to read about it now, but I can't recall what the name is:</p>
<p>Start with $a_1 = \ldots =a_n=1$. Choose a number between 1 and $n$ with probability $a_i/(a_1+ \ldots + a_n)$ to choose $i$. If $i_0$ is the number chosen, increase $a_i$ by 1 and now choose another numbe... | Wok | 2,380 | <p>Thanks for the pointer, Shai Covo. The name of this specific process is <a href="http://en.wikipedia.org/wiki/Preferential_attachment" rel="nofollow">preferential attachment</a>.</p>
|
302,061 | <p>Can you say how to find number of non-abelian groups of order n?</p>
<p>Suppose n is 24 ,then from structure theorem of finite abelian group we know that there are 3 abelian groups.But what can you say about the number of non-abelian groups of order 24?</p>
<p>The following link is a list of number of groups of or... | DonAntonio | 31,254 | <p>Can be pretty messy to do this, but using semidirect products you can get quite some answers.</p>
<p>In you example, $\,24=2^3\cdot 3\,$ and from Sylow Theorems and some element counting either the Sylow $\,2-$ subgroup or the $\,3-$subgroup <em>must</em> be normal. From here you obtaing some action by automorphism... |
152,620 | <p>The following question is from Golan's linear algebra book. I have posted a solution in the answers. </p>
<p><strong>Problem:</strong> Let $F$ ba field and let $V$ be a vector subspace of $F[x]$ consisting of all polynomials of degree at most 2. Let $\alpha:V\rightarrow F[x]$ be a linear transformation satisfying</... | André Nicolas | 6,312 | <p>Your calculation looks fine. Equivalently, but with rather less work, note that
$$x^2-x=(x^2+x+1)-2(x+1)+1,$$
so we can compute $\alpha(x^2-x)$ directly by linearity, without computing $\alpha$ at the most common basis elements.</p>
|
56,082 | <p>Suppose I have a nested list such as,</p>
<pre><code>{{{A, B}, {A, D}}, {{C, D}, {A, A}, {H, A}}, {{A, H}}}
</code></pre>
<p>Where the elements of interest are,</p>
<blockquote>
<pre><code>{{A, B}, {A, D}}
{{C, D}, {A, A}, {H, A}}
{{A, H}}
</code></pre>
</blockquote>
<p>How would I use select to pick up only eleme... | Chris Degnen | 363 | <pre><code>list = {{{A, B}, {A, D}}, {{C, D}, {A, A}, {H, A}}, {{A, H}}};
If[Count[First /@ #, A] >= 2, #, ## &[]] & /@ list
</code></pre>
<blockquote>
<p>{{{A, B}, {A, D}}}</p>
</blockquote>
|
2,994,970 | <p>As far that i have known, i understand the notion "a function on the circle" by each one of the followings (both equivalent):</p>
<ol>
<li>A function is defined on <span class="math-container">$\mathbb{R}$</span> that is <span class="math-container">$2\pi-$</span>periodic.</li>
<li>A function that is defined on <sp... | Hans Lundmark | 1,242 | <p>Actually the two conditions are equivalent (unless you require the functions to be continuous).</p>
<p>In the second case, you could equally well say that <span class="math-container">$f$</span> should be defined on <span class="math-container">$[a,b)$</span> to begin with, and forget about the requirement <span cl... |
3,070,258 | <p>In a (Partial Differential Equations / Laplace Equation) , I try to solving a problem of Laplace eq. by using separation of variables method.</p>
<p>I usually using the rule : if <span class="math-container">$e^{2 \sqrt{k} b} = 1$</span>, then I have: <span class="math-container">$2\sqrt{k} b = 2ni\pi$</span>. </p>... | user3482749 | 226,174 | <p>Yes, indeed: the result that you quote (after correcting the missing/surplus <span class="math-container">$2$</span>) is true for all complex values of <span class="math-container">$b$</span>, and so, if <span class="math-container">$e^{2\sqrt{k}\pi} = 1$</span>, then you have <span class="math-container">$2\sqrt{k}... |
231,479 | <p>Is there a function that can create hexagonal grid?</p>
<p>We have square grid graph, where we can specify <code>m*n</code> dimensions:</p>
<pre><code>GridGraph[{m, n}]
</code></pre>
<p>We have triangular grid graph (which works only for argument <code>n</code> up to 10 - for unknown reason):</p>
<pre><code>GraphDat... | azerbajdzan | 53,172 | <p>Here is my generalization of the code from link provided by @LouisB:</p>
<pre><code>HexagonalGridGraph2[{wide1_Integer?Positive, wide2_Integer?Positive,
wide3_Integer?Positive}, opts : OptionsPattern[Graph]] :=
Module[{cells, edges, vertices},
cells =
Flatten[Table[
CirclePoints[{Sqrt[3] (1 j + k -... |
231,479 | <p>Is there a function that can create hexagonal grid?</p>
<p>We have square grid graph, where we can specify <code>m*n</code> dimensions:</p>
<pre><code>GridGraph[{m, n}]
</code></pre>
<p>We have triangular grid graph (which works only for argument <code>n</code> up to 10 - for unknown reason):</p>
<pre><code>GraphDat... | Anton Antonov | 34,008 | <p>There is a resource function that makes hexagonal graphs: <a href="https://resources.wolframcloud.com/FunctionRepository/resources/HexagonalGridGraph" rel="nofollow noreferrer"><code>HexagonalGridGraph</code></a>. (Contributed by WRI.)</p>
<p><a href="https://i.stack.imgur.com/gGunX.png" rel="nofollow noreferrer"><i... |
4,575,771 | <p>I need to show that <span class="math-container">$\int_0^1 (1+t^2)^{\frac 7 2} dt < \frac 7 2 $</span>. I've checked numerically that this is true, but I haven't been able to prove it.</p>
<p>I've tried trigonometric substitutions. Let <span class="math-container">$\tan u= t:$</span></p>
<p><span class="math-cont... | River Li | 584,414 | <p><em>Alternative proof</em>:</p>
<p>Clearly, we have
<span class="math-container">$\sqrt{1 + t^2} \le 1 + t^2/2$</span>. Thus, we have
<span class="math-container">$$(1 + t^2)^{7/2} \le (1 + t^2)^3 (1 + t^2/2) = \frac12t^8 + \frac52 t^6 + \frac92 t^4 + \frac72 t^2 + 1.$$</span></p>
<p>Thus, we have
<span class="math-... |
4,237,342 | <p>I am a researcher and encountered the following challenging function in my work:</p>
<p><span class="math-container">$$f(S)=\sum_{k=1}^{S-1}(\ln (S)-\ln (k))^2 \bigg [ \frac{1}{(S-k)^2}+\frac{1}{(S+k)^2} \bigg ]$$</span></p>
<p>And I am only interested in the first term of the Taylor expansion of this function when ... | user247327 | 247,327 | <p>Evaluate the first term- when k'= 1? That should be easy:
<span class="math-container">$(ln(S)- ln(1))^2\left[\frac{1}{(S-1)^2}- \frac{1}{(S+1)^2}\right]$</span>
<span class="math-container">$= (ln(S)- 0)^2\left[\frac{(S+1)^2- (S-1)^2}{(S+1)^2(S-1)^2}\right]$</span>
<span class="math-container">$= (ln(S))^2\left[\f... |
2,232,060 | <p>$f(x) = \sqrt[3]{1+ \sqrt[3]x}$ </p>
<p>I have to derive in 1st order and 2nd order</p>
<p>$f'(x) = \frac{1}{9x^\frac 23(1+x^\frac 13)^\frac 23}$ Is what I get after the first derivation </p>
<p>Now the teachers assistant is making $some$ $magic$ by showing that </p>
<p>$f(u) = \frac{1}{U^\frac 23}$</p>
<p>$u=... | Fabian Schn. | 432,082 | <p>If I get you right, we have to find $n,a,b$ so that the conditions </p>
<ul>
<li>$12 \mid a+b $</li>
<li>$n \mid ab $</li>
<li>$12 \mid a $</li>
<li>$12 \mid b $</li>
</ul>
<p>hold true.</p>
<p>Well, if you are searching for the smallest possible values, than you might be able to simply check the first numbers wi... |
4,062,987 | <p>I was reading this question here: <a href="https://math.stackexchange.com/questions/1230688/what-are-the-semisimple-mathbbz-modules">What are the semisimple $\mathbb{Z}$-modules?</a> and I understood everything except why we need <span class="math-container">$\alpha_p$</span> copies here <span class="math-container... | PierreCarre | 639,238 | <p>You can start by noting that the sequence is decreasing. In fact,
<span class="math-container">$$
\dfrac{a_{n+1}}{a_n} = \dfrac{2n+1}{2n+2} < 1 \Rightarrow a_{n+1}< a_n.
$$</span></p>
<p>Also, the sequence is clearly bounded. Since it is decreasing and positive, we have that <span class="math-container">$0 <... |
1,936,043 | <p>I would like to prove that the sequence $n^{(-1)^{n}}$ is divergent. </p>
<p>My thoughts: I know $(-1)^n$ is divergent, so $n$ to the power of a divergent sequence is still divergent? I am not sure how to give a proper proof, pls help!</p>
| Dr. Sonnhard Graubner | 175,066 | <p>note that $$(-1)^{n}=-1$$ if $n$ is odd and $$(-1)^{n}=1$$ if $n$ is even. </p>
|
2,130,836 | <p>My question is really simple: </p>
<p>Let $E$ be a vector space and $A_r(E)$ be the vector space of the alternating $r$-linear maps $\varphi:E\times\ldots \times E\to \mathbb R$. If $v_1,\ldots,v_r$ are linearly independent vectors. Can we get $\omega\in A_r(E)$ such that $\omega(v_1\ldots,v_r)\neq 0$? Is the conve... | WafflesTasty | 70,877 | <p>I was looking for this as well, and eventually figured it out myself. So here's my solution for future reference. The short answer is, <span class="math-container">$2^n - 1$</span> never divides <span class="math-container">$3^n - 1$</span>. Here's the proof, making use of the Jacobi symbol.</p>
<p>Assume <span cla... |
63,671 | <p>The problem is:</p>
<p>For infinite independent Bernoulli trials, prove that the total number of successful trials $N$ have the following property:</p>
<p>$$ [N < \infty] = \bigcup\limits_{n=1}^{\infty}\,[N \le n] $$</p>
<p>Actually this is just part of bigger problem in a book, and the equation is given as an... | Pablo | 15,860 | <p>Maybe we don't have to do thermal average of n and we need :
$$n = \frac{1}{{z^{ - 1} e^{\varepsilon /kT} - 1}} = \sum\limits_{n = 1}^\infty {\left( {ze^{ - \varepsilon /kT} } \right)^l }
$$
so:
$$\sum\limits_{} {'... = \sum\limits_{i,j,k = 1}^\infty {\sum\limits_{} ' } } z^{ijk} \frac{{e^{ - \frac{{\hb... |
63,671 | <p>The problem is:</p>
<p>For infinite independent Bernoulli trials, prove that the total number of successful trials $N$ have the following property:</p>
<p>$$ [N < \infty] = \bigcup\limits_{n=1}^{\infty}\,[N \le n] $$</p>
<p>Actually this is just part of bigger problem in a book, and the equation is given as an... | Pablo | 15,860 | <p>I have tried to solve this integral:
$$F\left( {a,b,c,d} \right) = \int {\frac{{e^{ - \frac{{\hbar ^2 }}{{2m}}\left( {ak_a^2 + bk_b^2 + ck_\gamma ^2 + dk_\lambda ^2 } \right)} }}{{\left( {k_a^2 + k_b^2 - k_\gamma ^2 - k_\lambda ^2 } \right)}}k_a k_b k_\gamma k_\lambda dk_a dk_b dk_\gamma dk_\lambda }
$$
... |
2,868,047 | <p>My question is in relation to a problem I am trying to solve <a href="https://math.stackexchange.com/questions/2867002/finding-mathbbpygx">here</a>. If $g(.)$ is a monotonically increasing function and $a <b$, is it always true that $a<g(a)<g(b)<b$? Why or why not?</p>
| Jaroslaw Matlak | 389,592 | <p><strong>Hint</strong></p>
<p>You can compute the probability of not getting triple. The number $f_n$ of possibilities of not getting triples in $n$ throws is:
$$f_1 = \binom{6}{1}\\
f_2 = \binom{6}{2}2!+\binom{6}{1}\\
f_3=\binom{6}{3}3!+\binom{6}{1}\binom{5}{1}\frac{3!}{2!}\\
f_4=\binom{6}{4}4!+\binom{6}{1}\binom{5... |
451,063 | <p>Alright so I am having the following issue: I want to figure out how to find the fourier coefficients of the following function:
$$D(X)=\frac {a'(x)} {1+a'(x)^2}$$</p>
<p>Where $a(x)$ is an arbitrary function. I already have a model for finding the fourier coefficients for $a(x)$ and $a'(x)$:</p>
<pre><code>fc =... | AlexR | 86,940 | <p>You can try to approximate $D$ using the <code>ifft</code> of $a'$, additionally your formula seems to be
$$ \hat{f}_m = \frac{1}{N} \sum_{j=0}^{N-1} f_j \omega_N^{mj} $$
where $\omega_N = \exp(\frac{2\pi i}{N})$.</p>
|
1,603,323 | <p>If <span class="math-container">$A$</span> is a positive definite matrix can it be concluded that the kernel of <span class="math-container">$A$</span> is <span class="math-container">$\{0\}$</span>? </p>
<p>pf: R.T.P <span class="math-container">$\ker A = 0$</span>.
Suppose not, i.e., there exists some <span class... | Hanno | 316,749 | <p><span class="math-container">$\color{blue}{\text{user}1551}$</span>'s comment as answer.</p>
<p>If <span class="math-container">$x\in\ker A$</span>, then <span class="math-container">$Ax=0\,$</span> and in turn <span class="math-container">$\,x^TAx=0$</span>. As <span class="math-container">$A$</span> is positive d... |
19,356 | <p>So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acqui... | Deane Yang | 613 | <p>I advise against using MathOverflow as a guide to what most young mathematicians do or ought or learn. The last time I saw such a strong bias towards "abstract nonsense" was when I was a graduate student at Harvard (in the early 80's), where if you wanted to do differential geometry rather than derived categories, y... |
19,356 | <p>So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acqui... | Gerald Edgar | 454 | <p>In Littlewood's <em>Miscellany</em> there is an essay "A Mathematical Education" where he describes the situation before 1907.</p>
|
2,835,474 | <p>What is linear about a linear combination of things?. In linear algebra, the "things" we are dealing with are usually vectors and the linear combination gives the span of the vectors. Or it could be a linear combination of variables and functions. But why not just call it combination. Why is the term "linear" includ... | wjmccann | 426,335 | <p>When we want to talk about things being linear we are restricting ourselves to only two potential operations</p>
<ol>
<li>We can add things together</li>
<li>We can scale things</li>
</ol>
<p>We call these operations linear because, well, they operate on a line! When you scale a vector, the new vector is a "furthe... |
2,520,044 | <p>$$\lim_{x\to2}{\frac{\sqrt{3x-2}-\sqrt{5x-6}}{\sqrt{2x-1}-\sqrt{x+1}}}$$</p>
<p>Evaluate the limit.</p>
<p>Thanks for any help</p>
| Michael Rozenberg | 190,319 | <p>$$\lim_{x\to2}{\frac{\sqrt{3x-2}-\sqrt{5x-6}}{\sqrt{2x-1}-\sqrt{x+1}}}=\lim_{x\to2}{\frac{(3x-2-(5x-6))(\sqrt{2x-1}+\sqrt{x+1})}{(2x-1-(x+1))(\sqrt{3x-2}+\sqrt{5x-6})}}=$$
$$=-2\cdot\frac{\sqrt3+\sqrt3}{2+2}=-\sqrt3$$</p>
|
1,159 | <p>A lot of times I see theorems stated for local rings, but usually they are also true for "graded local rings", i.e., graded rings with a unique homogeneous maximal ideal (like the polynomial ring). For example, the Hilbert syzygy theorem, the Auslander-Buchsbaum formula, statements related to local cohomology, etc.<... | Jan Weidner | 2,837 | <p>I will try to provide some geometric intuition, why there should be an analogy between local rings and graded rings with unique homogeneous maximal ideal. Maybe this also helps to guess whether a statement true for local rings should still hold in the graded case.
A graded k-Algebra can be thought of as an affine sp... |
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