qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,454,317 | <p>Assume that $x$ and $y$ are both differentiable functions of t and find the required values of $\frac{dy}{dt}$ and $\frac{dx}{dt}$.</p>
<p>$$xy = 8$$</p>
<p>(a) Find $\frac{dy}{dt}$, given $x = 4$ and $\frac{dx}{dt} = 13$. </p>
<p>$$y=\frac{8}{x}\\
y'=-\frac{8}{x^2}\\
dy=-8dx $$</p>
<p>$dt=1$?</p>
<p>Can someon... | Gaurang Tandon | 89,548 | <p><strong>Quick and easy:</strong><br>
1. Taking the differential of the given equation, we get:
$xdy+ydx=0$<br>
2. Since $x=4$, use the original equation to get $y (=2)$<br>
3. Putting it back in our new equation, and dividing by $dt$, we get: $$4\frac{dy}{dt}+2\frac{dx}{dt}=0$$
4. Put the given value ($\frac{dx}{dt... |
1,449,845 | <p>So the biquadratic equation is $x^4+(2-\sqrt3)x^2+2+\sqrt3=0$. Let $a_1,a_2,a_3,a_4$ be its roots. So we have to find the value of $(1-a_1)(1-a_2)(1-a_3)(1-a_4)$ . <br>
<strong>My attempt:</strong> <br>
So of we put $x^2=t$, and let the roots of the new quadratic equation be $a_1,a_2$. So we get that $a_1=-a_3;a_2=-... | N. S. | 9,176 | <p>Here is your mistake:</p>
<p>As the new quadratic is a new equation, to avoid confusion I will use $b_1,b_2$ for the roots</p>
<p>$$1-(a_1^2+a_2^2)+(a_1a_2)^2=1-(b_1+b_2)+b_1b_2=1+(2-\sqrt{3})+2+\sqrt{3}
$$</p>
<p>You used $a_1,a_2$ to denote both the original roots and their squares!!!!</p>
|
1,822,362 | <p><a href="https://i.stack.imgur.com/jOOyV.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jOOyV.jpg" alt="enter image description here"></a></p>
<p>I'm not sure if I understand this question and was wondering if anyone could provide any insight to an answer. The only thing I can think of adding is... | parag | 957,295 | <p>This is also a correct formulation.
<span class="math-container">$$
4x_1+4(y_1+3y_2+4y_3+10y_4)\ge 10\\
y_1+y_2+y_3+y_4\le 1\\
y_1,y_2,y_3,y_4 \in \{0,1\}\\
x_1 \ge 0
$$</span></p>
<p><span class="math-container">$y_1+y_2+y_3+y_4 = 1 $</span> ( avoiding <span class="math-container">$x_1 \ge 0 $</span> ) also works ... |
2,921,981 | <p>I know that the cube is the only 3 d shape which falls in polyhedrons but still is composed of squares, exclusively although its a even sided shape.
I have noticed that after square there is no single polyhedron which is exclusively made out of a polygon with even number of sides. For example the hexagon, which does... | Dr. Richard Klitzing | 518,676 | <p>Regular polyhedra are the tetrahedron (3 triangles per vertex, 4 triangles in total), the octahedron (4 triangles per vertex, 8 triangles in total), the icosahedron (5 triangles per vertex, 20 triangles in total), the cube (3 squares per vertex, 6 i squares in total), and the dodecahedron (3 pentagons per vertex, 12... |
143,696 | <p>Does anyone know if there is a way to scale a graph from the command Periodogram?
More specifically in terms of the y axis. I am wondering if there is a way that I can make the highest value on the y axis of these graphs to be "1" while still keeping the ratio in comparison to the highest value of all the other valu... | Igor Rivin | 11,539 | <p>If you are doing <code>Periodogram[list]</code>, then dividing <code>list</code> by <code>Max[Abs[Fourier[list]]</code> should do the trick, I should think.</p>
|
4,493,427 | <p>Suppose you have a compact set <span class="math-container">$\Omega \subset \mathbb{R}^n$</span> partitioned as <span class="math-container">$\Omega = \bigcup_{j=1}^n \Omega_j$</span> with that <span class="math-container">$\Omega_i \cap \Omega_j = \emptyset$</span> if <span class="math-container">$i \neq j$</span>... | Leonid | 679,193 | <p>You can approximate <span class="math-container">$s$</span> arbitrarily by a continuous piecewise linear function. for instance, let the function be <span class="math-container">$y_j$</span> on <span class="math-container">$\Omega_j$</span> except on an arbitrarily small subset of it so that it can jump linearly int... |
4,357,325 | <p>I came across this question under summation:</p>
<p>Find the sum: <span class="math-container">$$\sum_{r=0}^n \left[\frac{r}{n} - \alpha \right]^2 {n \choose r}x^r(1-x)^{n-r} $$</span></p>
<p>To start with this, I wrote it as <span class="math-container">$\sum_{r=0}^n \left[\frac{r-n\alpha}{n} \right]^2 {n \choose ... | RobPratt | 683,666 | <p>Correct, and here's an alternative approach that uses the <a href="https://en.wikipedia.org/wiki/Binomial_distribution" rel="nofollow noreferrer">binomial distribution</a>. Let <span class="math-container">$X$</span> be a <span class="math-container">$\text{Bin}(n,x)$</span> random variable. Then <span class="math... |
4,601,103 | <p>Let <span class="math-container">$X_i$</span> be <span class="math-container">$i.i.d$</span> integrable random variables with bounded variance and <span class="math-container">$f$</span> be a continuous function with compact support. I want to prove <span class="math-container">$$
\lim_{N\to\infty} \frac{1}{N} \sum_... | geetha290krm | 1,064,504 | <p>This follows immediately from SLLN's since <span class="math-container">$(f(X_i))$</span> is also i.i.d. Measurability and boundedness is enough for this; continuity is not needed. [Convergence holds in the almost sure sense].</p>
|
69,760 | <p>The classical Schauder estimates (see the link)
<a href="http://en.wikipedia.org/wiki/Schauder_estimates" rel="nofollow">http://en.wikipedia.org/wiki/Schauder_estimates</a></p>
<p>Requires $f\in C^\alpha$ in order to get a solution $u\in C^{2+\alpha}$ of the equation</p>
<p>$$\Delta u=f$$</p>
<p>In fact, we can c... | timur | 824 | <p>Merely $f\in C$ cannot guarantee $u\in C^2$, as there are examples of $u\not\in C^2$ with $\Delta u\in C$, cf. an exercise in Gilbarg-Trudinger. This failure can be formulated in terms of non-closedness of the range of the Laplacian, or unboundedness of the inverse between certain spaces. The spaces $C^k$ (as well a... |
2,765,102 | <p>I am given the equation of a surface:
$$x^3+y^3+z^3-3xyz =0$$
And I need to find the equation of the plane tangent to this surface at $(1,1,1).$ <br>
At first, this task did not look easy for me as we are not given an explicit equation of a surface, but I tried using implicit differentiation assuming that $z$ depend... | Cedron Dawg | 556,138 | <p>The surface can be considered a level surface of:</p>
<p>$$ F(x,y,z) = x^3+y^3+z^3-3xyz $$</p>
<p>The gradient is always orthogonal to the level surface, and thus will serve to define the normal to the tangent plane. When the gradient is zero, this doesn't apply.</p>
<p>The gradient of $F$ is:</p>
<p>$$ \nabla ... |
1,601,095 | <p>Let
$$f_n(x) = \begin{cases}
\sin nx & 0 \leq x \leq \frac\pi n\\
0 & x \geq \frac\pi n
\end{cases}$$</p>
<p>Then my book says that $f_n \to f \equiv 0$ on the interval $[0, +\infty)$.<br>
I don't understand why $0$ is included. I think that the interval of convergence should be $(0, +\infty)$ because $f(0)... | Jendrik Stelzner | 300,783 | <p>Because $f_n(0) = \sin(n \cdot 0) = \sin(0) = 0$ for all $n$ we have $f_n(0) \to 0$. Suppose that $x > 0$. Then there exists some $N \geq 1$ with $x \geq \frac{\pi}{N}$. We then have $x \geq \frac{\pi}{n}$ for all $n \geq N$. So for all $n \geq N$ we have $f_n(x) = 0$. Therefore $f_n(x) \to 0$.</p>
|
1,407,053 | <p>Show using logarithms that if $y^k = (1-k)zx^k(a)^{-1}$ then $y = (1-k)^{(1/k)}z^{(1/k)}x(a)^{(-1/k)}$.</p>
| Vlad | 229,317 | <p>Take logarithm of both sides of the equation $\,y^k = \left(1-k\right)\,z\,x^k\,\left(a\right)^{-1},\,$ get</p>
<p>$$
\begin{aligned}
y^k = \left(1-k\right)\,z\,x^k\,a^{-1}
&\implies
\ln\left( y^k \right) = \ln\left(\left(1-k\right)\,z\,x^k\,\left(a\right)^{-1} \right)
\\
&\implies
k\ln y = \ln\left(1-... |
98,839 | <pre><code>I want to visualize Ricci flow solution on the following sphere
</code></pre>
<p>Let $r> 0$ </p>
<p>$L = \{ (x cos \theta, x sin \theta, x) | r < x < R \}$</p>
<p>$S$ : $(z-\sqrt{2} r)^2 + x^2 + y^2 = r^2$ </p>
<p>$T$ : $ (z- \sqrt{2} R)^2 + x^2 + y^2 = R^2$ </p>
<p>If $R$ is sufficiently la... | Community | -1 | <p>I believe that the Ricci flow on orbifolds was first studied by Hamilton in the paper "Three-Orbifolds with Positive Ricci Curvature", answering a question of Thurston. This paper, originally written in the early eighties, is published in the "Collected Papers on Ricci Flow" book edited by H.-D. Cao, etal (2003). It... |
1,998,810 | <p>There exists $x \in \mathbb{R}$ such that the number $f(x)=x^2 +5x +4$ is prime.</p>
<p>I can't understand where to start. </p>
<p>This is what I have so far: </p>
<p>Let P(x) be the statement "$x^2 + 5x +4$ is prime".
Then we have $\exists x \in \mathbb{R}, P(x)$.</p>
<p>I built a table and I suspect that this ... | Jacob Wakem | 117,290 | <p>The function goes to infinity as x goes to infinity. It is also continuous. Thus it assumes all sufficiently large real numbers. In particular it assumes all sufficiently large prime numbers.</p>
|
24,321 | <p>I'm trying to figure out the probability of a 3rd failure occurring on the 5th attempt of doing something. Let's just call the probability of success of failure P(S) or P(F), I won't put numbers as I want to actually learn.</p>
| Qiaochu Yuan | 232 | <p>The line through $(a_0 : ... : a_n)$ and $(b_0 : ... : b_n)$ is uniquely parameterized by $(a_0 X + b_0 Y : a_1 X + b_1 Y : ... a_n X + b_n Y)$. Note that this precisely describes a morphism $\mathbb{P}^1 \to \mathbb{P}^n$ which is an isomorphism onto its image. I am surprised this is not given somewhere in Fulton. ... |
638,048 | <p>I am looking at an exercise,where given that $a_{n}=\sqrt{2+\sqrt{2+...+\sqrt{2}}}$,I have to show that $\lim_{n \to \infty}a_{n}=2$.
We find that $a_{0}=0,a_{1}=\sqrt{2},a_{2}=\sqrt{2+a_{1}} \text{ and so on and we conclude that }a_{n+1}=\sqrt{2+a_{n}}$.Then we take the function $\varphi(x)=\sqrt{2+x}$,in order to ... | Yiorgos S. Smyrlis | 57,021 | <p>(1) Show inductively that $a_n$ is increasing.</p>
<p>(2) Show inductively that $a_n<2$.</p>
<p>Then, (1)+(2) imply that $a_n$ is convergent. Say $a_n\to x$. But then $a_{n+1}=\sqrt{2+a_n}\to x$.
At the same time $\sqrt{2+a_n}\to\sqrt{2+x}$. Hence $x=\sqrt{2+x}$, thus $x^2-x-2=0$. This implies that $x=2$ or $x=... |
638,048 | <p>I am looking at an exercise,where given that $a_{n}=\sqrt{2+\sqrt{2+...+\sqrt{2}}}$,I have to show that $\lim_{n \to \infty}a_{n}=2$.
We find that $a_{0}=0,a_{1}=\sqrt{2},a_{2}=\sqrt{2+a_{1}} \text{ and so on and we conclude that }a_{n+1}=\sqrt{2+a_{n}}$.Then we take the function $\varphi(x)=\sqrt{2+x}$,in order to ... | kmitov | 84,067 | <p>$a_1=\sqrt{2}$, $a_{n+1}=\sqrt{2+a_n}, n=1,2,\cdots$</p>
<p>If we take the sequense $b_1=2$ and $b_{n+1}=\sqrt{2+b_n}, n =1,2, \cdots$</p>
<p>we can see that for every $n$ $a_n \le b_n=2$. Therefore, the sequence $a_n$ is bounded from above. On the other hand $a_1=\sqrt{2} \le a_2=\sqrt{2+\sqrt{2}}$
and by induct... |
3,382,305 | <p>I know the two definitions for continuity, (sequential and epsilon-delta)</p>
<p>Given <span class="math-container">$x_0 \in D, \forall \epsilon > 0, \exists \delta > 0, |x - x_0| < \delta \rightarrow |f(x) - f(x_0)| < \epsilon $</span> </p>
<p>and</p>
<p>f is continuous if <span class="math-container... | amsmath | 487,169 | <p>You are given that
<span class="math-container">$$
\forall\epsilon>0\,\exists\delta>0 : |x-x_0|<\delta\,\Longrightarrow|f(x)-L|<\epsilon.
$$</span>
For any <span class="math-container">$\epsilon>0$</span> you can thus choose <span class="math-container">$x=x_0$</span> (because <span class="math-contai... |
69,680 | <p>Evaluating the following expression results in a non-zero output</p>
<pre><code>FullSimplify[Integrate[2 f[x], {x,0,1}] - 2 Integrate[f[x], {x,0,1}]]
</code></pre>
<p>I think the output should be zero, but do not know how to simplify this expression. Any suggestions ?</p>
| Nasser | 70 | <p>Use the Jens trick:</p>
<pre><code>f /: Integrate[f[x_], x_] := if[x];
SetAttributes[if, {NumericFunction}];
</code></pre>
<p>And now</p>
<pre><code>FullSimplify[Integrate[2 f[x], {x, 0, 1}] - 2 Integrate[f[x], {x, 0, 1}]]
(* 0 *)
</code></pre>
<p>You can read more about this in <a href="https://mathematica.sta... |
1,026,725 | <p>How many even 3 digit numbers contain at least one 7.
I got 126, but it was not an answer choice for the problem. Can anyone help?</p>
| Adhvaitha | 191,728 | <p>Count the number of $3$ digit even numbers that do not contain $7$ and subtract from the total number of $3$ digit even numbers.</p>
<blockquote class="spoiler">
<p>Let $xyz$ be the $3$ digit even number. Total number of $3$ digit even numbers is $450$, since $x$ has $9$ options, $y$ has $10$ options and $z$ has ... |
1,026,725 | <p>How many even 3 digit numbers contain at least one 7.
I got 126, but it was not an answer choice for the problem. Can anyone help?</p>
| Mark Bennet | 2,906 | <p>The final (units) digit is even, and there are five choices for that independent of whatever else happens - so the answer must be divisible by five.</p>
<p>There are $450$ even numbers with three digits. If there is no seven the hundreds digit is one of eight (no zero or seven), and the tens digit is one of nine (n... |
2,340,231 | <p>I have the following ODE for complex $z$:</p>
<p>$$\dot{z}=i|z|^2z$$</p>
<p>and I would like to find the most general exact solution possible. It is easy to see that $z=0$ and $z=e^{it}$ are two solutions, but I am hoping for a way to see if these are the only ones or if more may be found.</p>
<p>Thanks</p>
| user254433 | 254,433 | <p>The complex conjugate of your equation (which we denote by (1)) is
$$
\overline z'(t)=-i|z|^2\overline z(t),\qquad (2).
$$
Let's add $\overline z\cdot(1)$ to $z\cdot (2)$:
$$
\frac{d}{dt}\left(z(t)\overline z(t)\right)=0.
$$
This implies $|z(t)|=|z(0)|=:r$. Then (1) reduces to
$$
z'(t)=ir^2 z(t)
$$
This means $z(t)... |
2,050,385 | <p>Solve $15x$ "congruent to" $20\mod 88$</p>
<p>So far I think I know $15\mod 88$ is $-41$ or if positive $47$`</p>
| Sarvesh Ravichandran Iyer | 316,409 | <p>A trick to remember.</p>
<p>You are trying to solve the equation $15x - 20 = 88y$ for some $x,y$. Factorize five out, and you get that $5(3x-4) = 88y$. Now, since $5$ is prime and $5$ doesn
't divide $88$, it divides $y$. </p>
<p>Now, it is a simple question of trial and error with multiples of $5$ (you don't have... |
666,806 | <p>The question I have is mostly on stability analysis but the problem is:<br><br><br>
Consider a nonlinear pendulum. Using a linearized stability analysis, show that the inverted position is unstable. What is the exponential behavior of the angle in the neighborhood of this unstable equilibrium position.<br><br></p>
... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\down}{\downarrow}... |
3,310,683 | <p>Could you help me please with a question about integrals? Can an integral have more than one answer? For example with this integral:
<span class="math-container">$$\int\sqrt{1+\sqrt{1-x^2}}dx$$</span>
Doing by replacing u=<span class="math-container">$\sqrt{1-x^2}$</span>, I have this solution:
<span class="math-c... | azif00 | 680,927 | <p>Yes, an integral can be have more than one "answer", hence the importance of the constant that is added when finished integrating. More correctly, if two functions <span class="math-container">$f$</span> and <span class="math-container">$g$</span> they have the same derivative, so we can say that these differ by a c... |
142,481 | <ol>
<li><p>Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?</p></li>
<li><p>I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) and the functor $Ho(M)\to Ho(Pro-M)$ to be an exact full embedding. Which restrictions on M ... | Felix Goldberg | 22,051 | <ol>
<li><p>You might want to look up <a href="http://www.sciencedirect.com/science/article/pii/S0024379510004775" rel="nofollow">this paper</a> where reduced graphs are studied, in the sense of Servatius (or very similar to it), from the viewpoint of spectral graph theory.</p></li>
<li><p>People have studied twin-free... |
142,481 | <ol>
<li><p>Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?</p></li>
<li><p>I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) and the functor $Ho(M)\to Ho(Pro-M)$ to be an exact full embedding. Which restrictions on M ... | Ira Gessel | 10,744 | <p>A graph in which no two vertices have the same neighborhood is called a <em>point-determining graph</em> or <em>mating graph</em> or <em>mating-type graph</em>. See <a href="https://oeis.org/A006024" rel="nofollow">A006024</a> for references.</p>
|
1,597,186 | <blockquote>
<p>In $a^n \equiv b^n \pmod m$, does $n$ have to be an integer?</p>
</blockquote>
<p>I just came upon this rule and am wondering its limits. Thank you</p>
| Lubin | 17,760 | <p>What’s necessary is that exponentiation, $a^n$, must be well-defined for the values of $a$ and $n$, each in a specified domain; and the relation $a\equiv_m b$ must be well-defined within the range of values of the exponentiation. Here’s a relatively obscure example:</p>
<p>If $a$ is a principal unit in a $p$-adic d... |
2,469,745 | <p>I have seen this question on this site so I know this is a duplicate. I do not understand all the explanations on the other questions and they are years old.</p>
<blockquote>
<p>Let $G$ and $H$ be groups. Suppose $J$ is a normal subgroup of $G$ and $K$ is a normal subgroup of $H$. Show that $f(x,y)=(Jx,Ky)$ is a ... | Community | -1 | <p>Take the base-2 logarithm of both members, and you get</p>
<p>$$2+4+6+\cdots+2x=(-2)(-36)$$</p>
<p>or</p>
<p>$$1+2+3+\cdots+x=36.$$</p>
<p>$36$ is the eighth triangular number.</p>
<hr>
<p>Even though this is irrelevant to the given problem, you convert a power of $2$ to a power of $3$ by writing</p>
<p>$$2^a... |
266,283 | <p>Recall that a space $X$ is a locally compact if for every $x\in X$ there exists a neighbourhood $U$ of the point of $x$ such that $cl U$ is compact subspace of $X$. How we can show that compact $cl U$ is $T_1$?</p>
| Rustyn | 53,783 | <p>$\mathbb{Z}$ with the digital line topology is a counterexample to this claim. Denote $\mathbb{Z}$ with the digital line topology as: $\mathbb{Z}_{\text{DL}}$ Note that every integer $m$ has a neighborhood contained in a compact subset of $\mathbb{Z}_{\text{DL}}$. If $m$ is odd take the neighborhood to be $\{m\}$ wh... |
2,522,125 | <blockquote>
<p>Is it true that if $A$ and $B$ are $2\times 2$ matrices and $AB=0$ then $A=0$ or $B=0$. Prove it, or prove the contrary. </p>
</blockquote>
<p>I tried saying that if:
$$A= \begin{pmatrix}
0 & 0 \\
0 & 0 \\
\end{pmatrix}\quad\text{and}\quad B= \begin{pmatrix}
... | Robert Z | 299,698 | <p>For example you may try with
$$A=B=\begin{bmatrix}
0 & 1 \\
0 & 0 \\
\end{bmatrix}.$$</p>
|
2,522,125 | <blockquote>
<p>Is it true that if $A$ and $B$ are $2\times 2$ matrices and $AB=0$ then $A=0$ or $B=0$. Prove it, or prove the contrary. </p>
</blockquote>
<p>I tried saying that if:
$$A= \begin{pmatrix}
0 & 0 \\
0 & 0 \\
\end{pmatrix}\quad\text{and}\quad B= \begin{pmatrix}
... | symplectomorphic | 23,611 | <p>Note that your own calculation shows that if $a=g=1$ but all other variables vanish, then the product $AB$ will vanish, too. (There are other possibilities. Robert's solution is $b=f=1$.)</p>
<hr>
<p>You may be interested to know that the failure of $AB=0$ to imply $A=0$ or $B=0$ when $A$ and $B$ are matrices mean... |
368,129 | <p>30 years ago, Yves Colin de Verdière introduced the algebraic graph invariant <span class="math-container">$\mu(G)$</span> for any undirected graph <span class="math-container">$G$</span>, see [1]. It was motivated by the study of the second eigenvalue of certain Schrödinger operators [2,3]. It is defined in purely ... | M. Winter | 108,884 | <p>The following embeddability characterization for <span class="math-container">$\mu(G)\le 5$</span> is merely a conjecture, but I think it fits the question:</p>
<blockquote>
<p><strong>Conjecture</strong> (van der Holst, [1])<strong>:</strong> <span class="math-container">$\mu(G)\le 5$</span> if and only if <span cl... |
3,973,885 | <p>To add some context, let <span class="math-container">$(\Omega,\mathcal{F},\mu)$</span> be a measure space. The statement of the Extended Dominated Convergence Theorem(EDCT) is:</p>
<blockquote>
<p><span class="math-container">$\lbrace f_{n}\rbrace_{n\geq 1}$</span> be a sequence of functions from <span class="math-... | Oliver Kayende | 704,766 | <p>Note <span class="math-container">$c\cdot v_1=(c,0,c)\neq(1,1,2)$</span> and thus <span class="math-container">$v_1,v_2$</span> are linearly independent because <span class="math-container">$v_2\notin\text{Span}(\{v_1\})$</span>. Therefore <span class="math-container">$\mathcal B:=\{v_1,v_2\}$</span> is a Hamel basi... |
142,220 | <p>Fermat proved that <span class="math-container">$x^3-y^2=2$</span> has only one solution <span class="math-container">$(x,y)=(3,5)$</span>.</p>
<p>After some search, I only found proofs using factorization over the ring <span class="math-container">$Z[\sqrt{-2}]$</span>.</p>
<p>My question is:</p>
<p>Is this Fermat'... | Carlo Beenakker | 11,260 | <p>Fermat never gave a proof, only announced he had one (sounds familiar?). Euler did give a proof, which was flawed, see Franz Lemmermeyer's <A HREF="http://www.fen.bilkent.edu.tr/~franz/ant/ant01.pdf" rel="noreferrer">lecture notes,</A> or see page 4 of David Cox's <A HREF="http://math.stanford.edu/~lekheng/flt/cox.p... |
142,220 | <p>Fermat proved that <span class="math-container">$x^3-y^2=2$</span> has only one solution <span class="math-container">$(x,y)=(3,5)$</span>.</p>
<p>After some search, I only found proofs using factorization over the ring <span class="math-container">$Z[\sqrt{-2}]$</span>.</p>
<p>My question is:</p>
<p>Is this Fermat'... | Kieren MacMillan | 19,844 | <p><strong>Lemma.</strong>
Let <span class="math-container">$a$</span> and <span class="math-container">$b$</span> be coprime integers, and let <span class="math-container">$m$</span> and <span class="math-container">$n$</span> be positive integers such that <span class="math-container">$a^2+2b^2=mn$</span>. Then there... |
29,619 | <p>From the book,<br>
Suppose $p \equiv 1 \pmod{4}$, then by law of quadratic reciprocity, we have:
$$\left(\frac{3}{p}\right) = \left(\frac{p}{3}\right) $$
Next, if $p \equiv 2 \pmod{3}$, then $p \equiv 5 \pmod{12}$
Hence, $$\left(\frac{3}{p}\right) = \left(\frac{p}{3}\right) = -1$$</p>
<p>How do they get those Leg... | Mark | 7,667 | <p>Personally I find that a huge step in understanding things is knowing motivation behind the theorem. Douglas and yunone have both probably explained the meaning behind those equations pretty well but take a look at this Wikipedia article for more historical context: <a href="http://en.wikipedia.org/wiki/Quadratic_re... |
29,619 | <p>From the book,<br>
Suppose $p \equiv 1 \pmod{4}$, then by law of quadratic reciprocity, we have:
$$\left(\frac{3}{p}\right) = \left(\frac{p}{3}\right) $$
Next, if $p \equiv 2 \pmod{3}$, then $p \equiv 5 \pmod{12}$
Hence, $$\left(\frac{3}{p}\right) = \left(\frac{p}{3}\right) = -1$$</p>
<p>How do they get those Leg... | Bill Dubuque | 242 | <p>If $\rm\:q\:$ and $\rm\:p = 4\:k+1\:$ are distinct odd primes then by the law of quadratic reciprocity we have</p>
<p>$\displaystyle\rm\quad\quad\quad\quad\ \ \frac{p-1}{2}\ =\ 2\:k\ \ \Rightarrow\ \ \left(\frac{q}{p}\right)\ \times\ \left(\frac{p}{q}\right)\ =\ (-1)^{\frac{p-1}{2}\ \frac{q-1}{2}}\: =\ 1$</p>
<p>T... |
470,427 | <p>I am trying to understand the topology on $\{0,1\}^X$, where $X$ is uncountable. The topology on $\{0,1\}$ is the discrete and I am using the product topology on $\{0,1\}^X$.
My question is, who are the basic open sets? From my understanding of the definition of product topology, basic sets should either contain fin... | Alexey | 48,558 | <p>You may start by defining a <em>sub-basic open set</em> by fixing an $i\in X$ and an open $U\subset\{0, 1\}$ (or $U$ can be taken from a topology basis of $\{0, 1\}$, or from a sub-basis; practically, $U = \{0\}$ or $U = \{1\}$), and setting</p>
<p>$$
O_{i, U} = \{ f\in\{0, 1\}^X\mid f(i)\in U\}.
$$</p>
<p>A basic... |
278,357 | <p>How do I prove that</p>
<p>$$ \{\alpha\in\mathsf{On}\,|\,\alpha\prec A\}\in V,$$</p>
<p>assuming $A\in V$? I know that if AC is assumed, this set is equal to $\mbox{card}(A):=\mu_\alpha(\alpha\approx A)$, and hence it exists. But in the absence of the AC, we allow the possibility of sets such that $\forall\alpha\i... | Asaf Karagila | 622 | <p>First note that every bounded class of ordinals is a set. This is trivial using the subset schema applied to any upper bound of the class. Here is a nice proof as for why such classes classes of ordinals cannot be a proper class:</p>
<hr>
<p>Assume by contradiction that $A$ is a set such that $\{\alpha\in\mathsf{O... |
1,379,849 | <blockquote>
<p>Find the number of seven digit whole numbers in which only $2$ and $3$ are present as digits if no two $2$'s are consecutive in any number?</p>
</blockquote>
<p><strong>My Approach</strong>:
We can make numbers and see like: $2323232$, $2333333$, $2332332$, etc. Please suggest alternate solution of t... | Michael Galuza | 240,002 | <p>From Fermat's Little Theorem
$$
2^6\equiv 1\pmod 7,\enspace 2^7\equiv 2\pmod 7
$$
$128=2^7$; so,
$$
128^{128^{128}}\equiv 2^{128^{128}}\pmod 7.
$$
Now we should find remainder of $128^{128}$:
$$
128^{128}\equiv ? \pmod 6
$$
(because $2^6\equiv 1\pmod 7$). Ok (by $\pmod 6$),
$$
128^{128}= 2^{128} = 2\cdot2^{127} \pmo... |
1,379,849 | <blockquote>
<p>Find the number of seven digit whole numbers in which only $2$ and $3$ are present as digits if no two $2$'s are consecutive in any number?</p>
</blockquote>
<p><strong>My Approach</strong>:
We can make numbers and see like: $2323232$, $2333333$, $2332332$, etc. Please suggest alternate solution of t... | Taylor | 246,759 | <p>$128 \equiv 2 \: mod \: 7$ like you said. Observe that $2^3\equiv 1\:mod\:7$ so that $2^k\equiv 2^{(k\:mod\:3)}\:mod\:7$. Thus, we need to compute $128^{128}\equiv 2^{128}\:mod\:3$. We now apply similar reasoning as before and observe that $2^2 \equiv 1 \: mod \: 3$ so that $2^k\equiv 2^{(k \: mod \: 2)} \: mod\: 3$... |
2,409,744 | <p>Given,
$|\frac{1}{x}-2|<4$, I can solve this via the theorem approach $|x-a|<b\Rightarrow-b<x-a<b$..... but in the above question, there comes a possibility in $-4<\frac{1}{x}-2$ where the solution for it is less than $-\frac{1}{2}$ but it can change if I assume $\frac{1}{x}$ to be a negative value. H... | nonuser | 463,553 | <p>You have $-2<{1\over x} <6$ thus:</p>
<p>If $x>0$ we have $-2x<1<6x$ so $x>1/6$.</p>
<p>If $x<0$ we have $-2x>1>6x$ so $x<-1/2$. </p>
<p>So $x\in (-\infty, -{1\over 2})\cup ({1\over 6},\infty)$.</p>
|
1,234,217 | <p>Why do we choose Natural number to describe whether a set is countable or not? How can we say that Natural Number is countable?</p>
| Hagen von Eitzen | 39,174 | <p>The special role of the naturals is that their cardinality is the smallest that is not finite. That this cardinality bears the special name "countable" is of course owed to the fact that we use the naturals for counting :)</p>
|
2,780,281 | <p>Hey this is my first time using this website so please fix my formatting if it is bad.</p>
<p>Can someone please help me compute this$$ \prod_{n=1}^\infty\bigg(1+\frac{(-1)^n}{n+1}\bigg) $$</p>
| Masacroso | 173,262 | <p>HINT: an infinity product as $\prod_{k=1}^\infty a_k$ is defined as the limit of the sequence $(b_n)_{n\in\Bbb N}$ defined by the partial products $b_n:=\prod_{k=1}^n a_k$. Now write some $b_n$ and see what the sequence seems to be.</p>
<blockquote class="spoiler">
<p> Note that $b_{2n+1}=\frac12$ and $b_{2n}=\fr... |
288,106 | <p>Let $W$ be a cobordism between $n$-manifolds $M$ and $N$, and $f\colon W \mapsto X$ be a map to some manifold $X$. </p>
<p>Does anybody know of any nice examples of general relationships between the images of the maps $g_*\colon H_*(M)\mapsto H_*(X)$ and $h_*\colon H_*(N)\mapsto H_*(X)$ induced by the restrictions ... | Danny Ruberman | 3,460 | <p>You can say a few things using Poincaré duality and the exact sequence of the pair $(W,\partial W)$. For instance, $M$ is $2k$-dimensional (and everything is oriented), then the signatures of $M$ and $N$ are the same. The standard proof of this says among other things that the rank of the kernel of the inclusion map... |
288,106 | <p>Let $W$ be a cobordism between $n$-manifolds $M$ and $N$, and $f\colon W \mapsto X$ be a map to some manifold $X$. </p>
<p>Does anybody know of any nice examples of general relationships between the images of the maps $g_*\colon H_*(M)\mapsto H_*(X)$ and $h_*\colon H_*(N)\mapsto H_*(X)$ induced by the restrictions ... | Mark Grant | 8,103 | <p>You may be able to glean some information from thinking about the Stiefel-Whitney and Pontrjagin numbers of your maps $g$ and $h$. This is explained in Section 17 of Conner and Floyd's "Differentiable Periodic Maps".</p>
<p>Here is the situation in the non-oriented case for mod 2 homology. Since your maps $g:M\to X... |
288,106 | <p>Let $W$ be a cobordism between $n$-manifolds $M$ and $N$, and $f\colon W \mapsto X$ be a map to some manifold $X$. </p>
<p>Does anybody know of any nice examples of general relationships between the images of the maps $g_*\colon H_*(M)\mapsto H_*(X)$ and $h_*\colon H_*(N)\mapsto H_*(X)$ induced by the restrictions ... | David C | 27,816 | <p>If we consider cohomology with $\mathbb{Z}/2$-coefficients, if $M$ is a manifold $H^*(M;\mathbb{Z}/2)$ it satisfies Poincaré duality, and is an unstable algebra over the Steenrod algebra. Brown and Peterson in their paper "Algebraic bordism groups" (Annals of maths, 1964) had the wonderful idea to give an algebraic ... |
4,549,427 | <p>I would like to know if <span class="math-container">$\forall xx=x$</span> is an axiom in axiomatic set theory like in other first order languages, or a theorem? If it is a theorem, how to prove it?</p>
<p><strong>Update:</strong>
In the first order language materials I read,the equality is one of the logical symbol... | Just a user | 977,740 | <p>It can go either way. By <a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory" rel="nofollow noreferrer">wikipedia</a>, "The converse of this axiom (Axiom of extensionality) follows from the substitution property of equality. If the background logic does not include equality <span class="m... |
3,900,569 | <p><strong>Question:</strong></p>
<p>Let <span class="math-container">$\{x_{n}\}$</span> be a recursively defined sequence defined as:
<span class="math-container">$$x_{1} = \frac{1}{2}$$</span>
<span class="math-container">$$x_{n+1} = \frac{1-x_{n}}{4}$$</span></p>
<p>I want to show first that <span class="math-contai... | player3236 | 435,724 | <p>You have found <span class="math-container">$|x_{n+1}-x_n| =\frac14|-x_n+x_{n-1}| = \frac14|x_n-x_{n-1}|$</span>, which satisfies the inequality. If you insist, write <span class="math-container">$|x_{n+1}-x_n| \le |x_{n+1}-x_n| = \frac14|x_n-x_{n-1}|$</span> and now you have the inequality.</p>
<p>Using this we see... |
1,050,725 | <p>How might I go about finding the basis of a vector spaced spanned by a set of three vectors? For example, if given the set of vectors <code>{(1,2,3),(4,5,6),(7,8,9)}</code> how would I find the basis?</p>
<p>I've tried looking <a href="http://mathworld.wolfram.com/VectorSpaceBasis.html" rel="nofollow">here</a> but ... | mfl | 148,513 | <p>Since </p>
<p>$$(7,8,9)=2\cdot(4,5,6)-(1,2,3)$$ means $(7,8,9)$ is linear combination of $(1,2,3)$ and $(4,5,6).$ Since $(1,2,3)$ and $(4,5,6)$ are linearly independent, what is a possible basis?</p>
|
1,366,462 | <p>The line $x + y − 1 = 0$ intersects the circle $x^2 + y^2 = 13$ at $A(\alpha_1, \alpha_2)$ and $B(\beta_1, \beta_2)$.
Without finding the coordinates of A and B, find the length of the chord AB. </p>
<p><strong>Hint:</strong> Form a quadratic equation in $x$ and evaluate $|\alpha_1 − \beta_1 |$, and similarly find ... | Emilio Novati | 187,568 | <p>Hint:
the difference of the roots of a quadratic equation $ax^2+bx+c=0$ is
$$
|x_1-x_2|= \dfrac{\sqrt{\Delta}}{|a|}
$$
as you can easely prove by the solution formula.</p>
<blockquote class="spoiler">
<p> In your case you find:$\sqrt{\Delta}=5 \,,\,a=1 \Rightarrow |x_1-x_2|=|\alpha_1-\beta_1|=5$</p>
</blockquote>... |
3,621,259 | <blockquote>
<p>How many <span class="math-container">$5$</span>-digit numbers can be made out of the digits <span class="math-container">$1, 2, 2, 2, 3$</span> ?</p>
</blockquote>
<p>I know it might be a basic question, but I do not remember how I used to solve these kinds of questions.</p>
| PrincessEev | 597,568 | <p>You cannot, since you're basically assuming</p>
<p><span class="math-container">$$\frac{ \sum_i f(i) }{ \sum_i g(i) } = \sum_i \frac{f(i)}{g(i)}$$</span></p>
<p>This is not true in general. You're assuming this because each integral is basically its own, individual (limit of a) sum. For instance, it is generally n... |
3,621,259 | <blockquote>
<p>How many <span class="math-container">$5$</span>-digit numbers can be made out of the digits <span class="math-container">$1, 2, 2, 2, 3$</span> ?</p>
</blockquote>
<p>I know it might be a basic question, but I do not remember how I used to solve these kinds of questions.</p>
| Community | -1 | <p>No, and for several reasons:</p>
<ol>
<li><p><span class="math-container">$\int_0^1 f(x)$</span> is not a thing. The operation is <span class="math-container">$\int_a^b(\bullet)\,dx$</span>: it takes functions and it returns numbers.</p></li>
<li><p><span class="math-container">$\frac{\int_0^1 f(x)\,dx}{\int_0^1 g(... |
26,256 | <p>One can define the Euler characteristic χ for a graph as the number of vertices minus the number of edges. Thus an <span class="math-container">$n$</span>-cycle has <span class="math-container">$\chi = 0$</span> and <span class="math-container">$K_4$</span> has <span class="math-container">$\chi=-2$</span>.
Is t... | Tony Huynh | 2,233 | <p>There is indeed a version of Gauss-Bonnet for graphs $G$ embedded on a 2-manifold. Here, the <em>combinatorial curvature</em> at a vertex $x$ of $G$ is</p>
<p>$1-\frac{deg(x)}{2} + \sum_{f \sim v} \frac{1}{size(f)}$,</p>
<p>where $f \sim v$ means that the face $f$ is incident to the vertex $x$. </p>
<p>This <a ... |
139,125 | <p>This is a variation on an earlier question resolved by <em>user35353</em>: <a href="https://mathoverflow.net/questions/139105/can-a-tangle-of-arcs-interlock">Can a tangle of arcs interlock?</a> In that question, the arcs were restricted to circular arcs, and <em>user35353</em>'s proof that one arc can be removed wit... | Cristi Stoica | 10,095 | <p><img src="https://i.stack.imgur.com/xKsHq.gif" alt="Tangled ellipses"></p>
<p>I think that this configuration cannot be untangled.</p>
|
1,575,676 | <p><a href="https://i.stack.imgur.com/xYGNz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xYGNz.png" alt="enter image description here"></a></p>
<p>This is from Spivak Calculus on Manifolds, section 5.3</p>
<p>I have done part a, but I am stuck on part (b) and have been for a day now:</p>
<p>let... | Community | -1 | <p>The correct notation for the sixth derivative is</p>
<p>$$\frac{d^6 y}{dx^6}$$</p>
<p>not $\frac{d^6 y}{d^6x}$. This notation is meant to be suggestive of taking the sixth power of the operator $d/dx$; that is,</p>
<p>$$\frac{d^6 y}{dx^6} = \underbrace{\frac{d}{dx} \cdots \frac{d}{dx}}_{6} y$$</p>
<p>Imagine $dx... |
1,575,676 | <p><a href="https://i.stack.imgur.com/xYGNz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xYGNz.png" alt="enter image description here"></a></p>
<p>This is from Spivak Calculus on Manifolds, section 5.3</p>
<p>I have done part a, but I am stuck on part (b) and have been for a day now:</p>
<p>let... | Community | -1 | <p>For convenience, first transform</p>
<p>$$-2\cos^2(x)=-1-\cos(2x).$$</p>
<p>Then the sixth derivative is $$2^6\cos(2x),$$ because $\cos(x)''=-\cos(x)$ and because of the scaling of the variable .</p>
<p>At $x=\dfrac\pi4$, $$0.$$</p>
|
1,575,676 | <p><a href="https://i.stack.imgur.com/xYGNz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xYGNz.png" alt="enter image description here"></a></p>
<p>This is from Spivak Calculus on Manifolds, section 5.3</p>
<p>I have done part a, but I am stuck on part (b) and have been for a day now:</p>
<p>let... | Kaster | 49,333 | <p>If you recall the definition of the derivative, then you can write
\begin{align}
y'(x) &= \lim_{h \to 0} \frac {y(x+h) - y(h)}h \\
y''(x) &= \lim_{h \to 0} \frac {y'(x+h) - y'(x)}h = \lim_{h \to 0} \frac{\frac {y(x+2h) - y(x+h)}h - \frac {y(x+h) - y(x)}h}h = \lim_{h \to 0} \frac {y(x+2h) - 2y(x+h) + y(x)}{h^... |
13,460 | <p>there're some students, who belive that <span class="math-container">$$\frac10 = \infty $$</span></p>
<p>I need to teach them that this is not true and <span class="math-container">$\frac10 $</span> is undefined, mathematically and give a good picture (for their minds)</p>
<p>what is the proper way to teach them wit... | CosmoVibe | 9,383 | <p>A little late to the party, but I wanted to add my two cents.</p>
<p>The students should understand that they are looking at the value of <span class="math-container">$1/0$</span> as a limit. This is good but it's not the entire picture, and using that value in normal computation will break algebra.</p>
<p>See the... |
3,947,337 | <p>Be <span class="math-container">$E$</span> a normed vector space. If <span class="math-container">$A \subset E$</span> is compact and <span class="math-container">$p \in E$</span>, prove that <span class="math-container">$A_p = \{x+p:x \in A\} $</span> is also compact.</p>
<p>Hello, everyone! All right? I tried to p... | Christian Blatter | 1,303 | <p>Given <span class="math-container">$p$</span>, the translation <span class="math-container">$T: \>x\mapsto x+p$</span> of <span class="math-container">$K$</span> is bijective and continuous, and so is <span class="math-container">$T^{-1}: \>x\mapsto x-p$</span>. It follows that <span class="math-container">$T$... |
125,369 | <p>Probably a simple question, but I can't find an answer anywhere, not even in the suggested questions with similar titles. It might also be that I just don't get the correct terminology. This is not really a field with which I'm familiar.</p>
<p>I have a system of equations of the form $aw+bx+cy+dz=e$ and I want to ... | Dirk | 3,148 | <p>Basically, you need a description of the range of $A$ and check, whether $b$ lies in or out.</p>
<p>A projection onto the range of $A$ can be computed by several means, e.g. the QR decomposition. If you have $P$ as a projection matrix onto the range of $A$, just check of $Pb=b$.</p>
|
2,112,802 | <p>If $A \subseteq \mathbb{R} $
$$
\exists p \in A, \forall q \in A , q \leq p $$</p>
<hr>
<p>Can I just use a specific value for $p$ and arbritary value for $q$ to disprove this?</p>
<p>$p = 3$ and $q = p + 1$, hence $q > p$</p>
<hr>
<p>Also, how would should one go about this one: </p>
<p>If .. $\exists p \... | MPW | 113,214 | <p><strong>Hint:</strong> When trying to form equations from descriptions, "is" usually corresponds to an equal sign. "The distance between a point $(a,b)$ and a point $(c,d)$" is the quantity $\sqrt{(a-c)^2+(b-d)^2}$. Do you know how to find the distance between a point and a vertical line?</p>
|
3,200,995 | <p>I would like to approximate <span class="math-container">$$\frac{\cos^2{x}}{\sin(x) \tan(x)}$$</span> using the small angle approximations. </p>
<p>Throughout I will use <span class="math-container">$\sin(x) \approx x$</span>, <span class="math-container">$\tan(x) \approx x$</span>, <span class="math-container">$\c... | MachineLearner | 647,466 | <p>We have to be more precise when talking about something being constant. A constant is a number or vector.</p>
<p>In optimal control theory, the Hamiltonian <span class="math-container">$\mathcal{H}$</span> can additionally be a function of <span class="math-container">$x(t)$</span>, <span class="math-container">$u(... |
186,336 | <p>I have a notebook with grouped cells, where the input cells are hidden and only output cells open. Is it possible in an easy way to evaluate the whole notebook and preventing opening all groups, preserving the group structure?</p>
| Bill Watts | 53,121 | <p>Based on the comment by theorist</p>
<pre><code>$Assumptions = a > 0 && ko > 0 && B ∈ Reals
int = 2*Pi*B^2*Integrate[BesselK[0, ko*ρ]^2*ρ, ρ]
(*π B^2 ρ^2 (BesselK[0, ko ρ]^2 - BesselK[1, ko ρ]^2)*)
</code></pre>
<p>Check answer</p>
<pre><code>D[int, ρ] // FullSimplify
(* 2 π B^2 ρ BesselK[0... |
1,386,677 | <p>Proving that $$\sin x > \frac{(\pi^{2}-x^{2})x}{\pi^{2}+x^{2}}, \qquad\forall x>\pi$$</p>
| Euler88 ... | 252,332 | <p>Hint: You can make $f(x)=(\pi^2+x^2)\sin x-(\pi^2-x^2)x$. So $f'(x)=2x\sin x+(\pi^2+x^2)\cos x+3x^2-\pi^2$. See that $f'(\pi)=0$, calculate $f''$ and show that $f''(\pi)>0$. Next try to see if $\pi$ is a minimum of f. Since $f(\pi)=0$ the results follows.</p>
|
1,386,677 | <p>Proving that $$\sin x > \frac{(\pi^{2}-x^{2})x}{\pi^{2}+x^{2}}, \qquad\forall x>\pi$$</p>
| Intelligenti pauca | 255,730 | <p>With the substitution: $x=t+\pi$ the inequality can be rewritten as
$$
{\sin t\over t}<1+{\pi t\over 2\pi^2+2\pi t+t^2},\quad\hbox{for $t>0$},
$$
which is obviously true.</p>
|
2,326,259 | <p>I tried the following $$I = \langle X^2,X+1\rangle =\langle X^2,X+1,X^2+2(X+1)\rangle =\langle X^2,X+1,(X+1)^2+1 \rangle$$</p>
<p>Yet no matter how I arrange it, I cannot obtain $1$. Can someone help me out?</p>
| José Carlos Santos | 446,262 | <p>$-(X+1)^2+X^2+2(X+1)=1$. So, your ideal contains $1$.</p>
|
4,019,129 | <p>Let <span class="math-container">$E$</span> be a Banach space, <span class="math-container">$\iota:E\to E''$</span> denote the canonical injection of <span class="math-container">$E$</span> into <span class="math-container">$E''$</span>, i.e. <span class="math-container">$$\iota x:=\langle x,\;\cdot\;\rangle\;\;\;\t... | Peter Morfe | 711,689 | <p>Given a set <span class="math-container">$E$</span>, denote by <span class="math-container">$M(E)$</span> the set of all bounded functions on <span class="math-container">$E$</span> with the supremum norm topology. We will consider <span class="math-container">$M(V_{\delta})$</span>, where <span class="math-contain... |
4,019,129 | <p>Let <span class="math-container">$E$</span> be a Banach space, <span class="math-container">$\iota:E\to E''$</span> denote the canonical injection of <span class="math-container">$E$</span> into <span class="math-container">$E''$</span>, i.e. <span class="math-container">$$\iota x:=\langle x,\;\cdot\;\rangle\;\;\;\t... | 0xbadf00d | 47,771 | <p><a href="https://math.stackexchange.com/users/711689/peter-morfe">Peter Morfe</a>'s <a href="https://math.stackexchange.com/a/4019557/47771">answer</a> answered my question, which is why I've accepted it. However, since the whole stuff with the <span class="math-container">$\sigma_c(E',E)$</span>-topology on <span c... |
3,694,658 | <p>How can I prove that?</p>
<p><span class="math-container">$1^3+2^3+\cdots+(n-1)^3<\frac{n^4}{4}$</span> </p>
| fleablood | 280,126 | <p>Induction:</p>
<p>Base case: <span class="math-container">$n=2$</span> and <span class="math-container">$1^3 =1 < 4=\frac {2^4}4$</span>.</p>
<p>Induction step:</p>
<p>If we assume that <span class="math-container">$1^3 +2^3+.... +(n-1)^3 < \frac {n^4}{4}$</span> then</p>
<p><span class="math-container">$[... |
1,427,970 | <p>Let $G$ act on $\Omega$ transitively, and let $|G| = |\Omega| + 1$ (both sets are assumed to be finite). I want to show from first principles (using maybe arguments like the pigeonhole principle, but not Burnside's lemma) that there exists a non-trivial element having a fixed point. For example let $\Omega = \{\alph... | uniquesolution | 265,735 | <p>Note that $g_{n-1}\neq g_{i-1}$ so there is some element $\omega\in\Omega$ for which $g_{i-1}(\omega)\neq g_{n-1}(\omega)$ and so $g_{i-1}^{-1}g_{n-1}(\omega)\neq\omega$ whence $g_{i-1}^{-1}g_{n-1}$ cannot be the identity. </p>
|
593,438 | <p>I was reading Mathematics for Economists by Simon and Blume.</p>
<blockquote>
<p>The level set $x^2+y^2+z^2=1$ is a two-dimensional sphere of radius $1$.</p>
</blockquote>
<p>How to actually know that it is two dimensional?</p>
| LASV | 89,736 | <p>Every point of the sphere has a neighbourhood that looks like $\mathbb{R}^2$ (If we are talking about dimension as a manifold).</p>
|
29,181 | <p>I ran across this infinite product:</p>
<p>$$\lim_{n\to\infty}\prod_{k=2}^n\left(1-\frac1{\binom{k+1}{2}}\right)$$</p>
<p>I easily found that it converges to 1/3. Using my calculator, I found that</p>
<p>$$1-\frac1{\binom{k+1}{2}}=\frac{(k-1)(k+2)}{k(k+1)}$$</p>
<p>Then, here is my question</p>
<p>$$\prod_{k=2}... | Ross Millikan | 1,827 | <p>$\binom{k+1}{2}=\frac{(k+1)k}{2}$, so $1-\frac{1}{\binom{k+1}{2}}=\frac{k(k+1)-2}{k(k+1)}=\frac{(k-1)(k+2)}{k(k+1)}$. Then $$\prod_{k=2}^{n}\frac{(k-1)(k+2)}{k(k+1)}=\frac{2(n-1)!(n+2)!}{6n!(n+1)!}=\frac{n+2}{3n}$$ where the 2 comes because the $(n+1)!$ in the denominator really starts at 3 and the 6 comes because... |
135,423 | <p>I would like to label a curve inside ListLinePlot. Let's say I have the following list:</p>
<pre><code>Table[{x, x^2}, {x, 0, 10, 0.1}]
</code></pre>
<p>What I expected is a labeled curve. The label should also be placed above the curve and in the middle. It should also be rotated with the curve like the following... | Vitaliy Kaurov | 13 | <p>Use <code>Text</code> or <code>Inset</code> in <code>Epilog</code>:</p>
<pre><code>ListLinePlot[Table[{x, x^2}, {x, 0, 10, 0.1}],
Epilog -> Text[Style[x^2, 15], {4.5, 25}, Automatic, {1, .6}]]
</code></pre>
<p>or</p>
<pre><code>ListLinePlot[Table[{x, x^2}, {x, 0, 10, 0.1}],
Epilog -> Text[Style["x^2", 1... |
135,423 | <p>I would like to label a curve inside ListLinePlot. Let's say I have the following list:</p>
<pre><code>Table[{x, x^2}, {x, 0, 10, 0.1}]
</code></pre>
<p>What I expected is a labeled curve. The label should also be placed above the curve and in the middle. It should also be rotated with the curve like the following... | m_goldberg | 3,066 | <p>To use <code>Labeled</code> to make the kind of label you want, you have to wrap a specific point with <code>Labeled</code>. You also have to use <code>Rotate</code> to get the rotation.</p>
<p>Here is the code:</p>
<pre><code>data = Table[{x, x^2}, {x, 0, 10, 0.1}];
data[[61]] = Labeled[data[[61]], Rotate[Style[x^2... |
385,887 | <p>I'm looking for an example of a mathematical relation that is symmetric but not reflexive. A standard non-mathematical example is siblinghood. </p>
| Federica Maggioni | 49,358 | <p>Let $X=\{a,b\}$ be a set, ($a$ and $b$ distinct). Define the relation $R$ on X by $R=\{(a,a)\}$. Then $R$ is symmetric (and transitive), but not reflexive on $X$ since $(b,b)$ is not in $R$.</p>
|
386 | <p>The Reshetikhin-Turaev construction take as input a Modular Tensor Category (MTC) and spits out a 3D TQFT. I've been told that the other main construction of 3D TQFTs, the Turaev-Viro State sum construction, factors through the RT construction in the sense that for each such TQFT Z there exists a MTC M such that the... | Igor Korepanov | 2,955 | <p>There must be lots of <em>fermionic</em> TQFTs behaving in a very different way compared to RT. One feature of these fermionic (based on <em>Berezin integral</em>) theories is that they are quite straightforwardly generalized to any dimensions, not just 3 - which, by the way, provides you with a powerful and intrigu... |
863,101 | <p>Can anyone help me to find the derivative of this function? I know I have to use the quotient rule: $\dfrac{f(x)}{g(x)}=\dfrac{f′(x)g(x)−f(x)g′(x)}{(g(x))^2}$, but I don't know how I use this when the function is:</p>
<blockquote>
<p>$$f(x,y) = \frac{7y + x^2}{1+y^2}$$ </p>
</blockquote>
<p>$f_x (x,y) = ?$ </p... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Specialize $\, g(x) = 2x+1\,$ below.</p>
<p><strong>Theorem</strong> $\ $ Let $\,g(x)\,$ be a polynomial with integer coeffcients such that $\,\color{#c00}{2\nmid g(n)}\,$ for integers $\,n.$</p>
<p>Then $\ 3\mid g(1)\,\Rightarrow\, 6\mid f(n) = n(n\!+\!1)g(n).\ $ If so, then $\ 4\mid f(... |
189,603 | <p>Let's suppose that we have the following equation</p>
<pre><code>Clear["Global`*"];
m = 1/2;
V = m/Sqrt[(x - m)^2 + (y - m)^2 + (z - m)^2] + m/Sqrt[(x + m)^2 + (y + m)^2 + (z + m)^2]
+ 1/2*(x^2 + y^2);
Vx = D[V, x];
Vy = D[V, y];
Vz = D[V, z];
</code></pre>
<p>Then we can use the <code>FindRoot</code> module fo... | Henrik Schumacher | 38,178 | <p>The iterates of Newton's method for the equation <span class="math-container">$F(x) = 0$</span> with a initial guess <span class="math-container">$x_0$</span> look like
<span class="math-container">$$x_{k+1} = x_k - DF(x_k)^{-1} \, F(x_k).$$</span>
If <span class="math-container">$x_0$</span> is chosen badly, then ... |
1,365,882 | <p>The theorem statement is "if $f$ is continuous on $[a,b]$, $f$ is bounded on $[a,b]$". This is proven in the textbook Calculus by the author Apostol by the "method of successive bisection", which I'm sure many are familiar with. The proof is done by contradiction. </p>
<p>Here is my concern with this proof: we take... | Race Bannon | 188,877 | <p>Take a look at it from another perspective. $f$ being bounded means it is either bounded above or below. Suppose $f$ is not bounded above on $[a,b]$. Then $\exists x_1,\cdots,x_k$ such that $f(x_1) > 1, \cdots, f(x_k) > k, \forall k \in \mathbb{N}$.</p>
<p>Your sequence $(x_n)_{n \in \mathbb{N}}$ is unbounded... |
104,492 | <p>Three people A,B,C attend the following game: from 0~100, the Host will come up a number with Uniform, but he doesn't tell them the number, the attendee will guess a number and the closes one will win. A choose the number first and tell the number, B will tell another different number based on A's number, C choose a... | Greg Martin | 16,078 | <p>To temporarily avoid problems due to the discreteness of the set $\{0,1,\dots,100\}$, let's pretend that the three people are guessing a real number between 0 and 1. If the first two guesses are $a$ and $b$, say $0<a<b<1$, then C will want to guess</p>
<ul>
<li>a tiny bit less than $a$, if $\max\{a,\frac{b... |
1,124 | <p>Recently, Oleksandr kindly showed a <a href="https://mathematica.stackexchange.com/questions/1096/list-of-compilable-functions">list of Mathematica commands that can be compiled</a>.
RandomVariate was part of that list. However, whether this can be compiled depends upon the distribution that is being sampled. </p>
... | Ian Hincks | 1,167 | <p>Another distribution which is missing is <code>MultinormalDistribution[v,S]</code>. However, variates from this distribution are just affine transformations of vectors of iid normal variates. Indeed, if <code>y=RandomVariate[NormalDistribution[],Length@s]</code>, then <code>x=v+A.y</code> has the distribution of int... |
2,488,165 | <p>Find the line integral of $x^2+y^2$ over the polar curve $r=e^{\theta}$</p>
<p>Not really sure on how to find the curve on terms of a parameter in order to evaluate the integral</p>
| Connor Harris | 102,456 | <p>Note that $x^2 + y^2 = r^2$, so the integral you're trying to find is $\int r^2\, ds$, where $ds$ is the arc-length form $$ds = \sqrt{r^2 + \left( \frac{dr}{d\theta}\right)^2}\, d\theta$$
In this case $r = \frac{dr}{d\theta} = e^{\theta}$, so plugging in gives $$\int r^2 ds = \int (e^\theta)^2 \sqrt{2 (e^\theta)^2}\... |
1,715,945 | <p>I need help solving/understanding this question:</p>
<p>L (x,y) : "x loves y".
Translate "there are exactly two people whom Lynn loves".
Its answer includes a variable "z". I do not get that part with the variable "z". How did it come here when it was not introduced in the question? Detailed solution is appreciated... | ralleee | 209,617 | <p>Let $a_1,\dots a_n$ be generators for $F_n$, and $g_1,\dots g_n$ generators for $G$. Note that a homomorphism from $F_n$ is given by the images of $a_i$. So let's take the homomorphism given by $a_i\mapsto g_i$. Any element in $G$ can be written as product of $g_i$-s. Then the product of corresponding $a_i$-s will b... |
2,205,209 | <p>Today I came across a question in which equations of two lines (Which were parallel) were given and it was asked to find their angle bisector.</p>
<p>My answer for this was :</p>
<p>Since there is no point of intersection of Parallel lines, there is no origin of angle bisector. So, answer should be <strong>Doesn't... | McMillenandhiswife | 429,771 | <p>In <a href="https://en.wikipedia.org/wiki/Projective_geometry" rel="nofollow noreferrer">Projective Geometry</a> two parallel lines intersect at the infinity point. If you then define the angle bisector as a line through this intersection point, that has the same angle to both of the other lines, every parallel line... |
42,864 | <blockquote>
<p>Given a monic polynomial <span class="math-container">$f\in\mathbb{Z}[x]$</span>, how can I determine whether there is <span class="math-container">$k\in\mathbb{Z}^+$</span> such that <span class="math-container">$f\mid x^k-1$</span>?</p>
</blockquote>
<p>For example, <span class="math-container">$x^2-x... | Bill Dubuque | 242 | <p>There are various algorithms known for such, based on properties of roots of unity, e.g.<br />
<img src="https://i.stack.imgur.com/1Sb8F.jpg" alt="enter image description here" /><br />
See for example the following papers.</p>
<p>F. Beukers , C. J. Smyth. <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=... |
934,051 | <p>Let $A$ and $B$ be sets. Show that $A = \bigcup C$ for some $C \subset B$ iff for every $a \in A$ there exists $b \in B$ such that $a\in b$ and $b \subset A$ </p>
<p>I am having trouble proving the right-to-left implication, but have proved the left-to-right implication. </p>
<p>Could anyone provide guidance? </p>... | Sheheryar Zaidi | 131,709 | <p>As a matter fact, I fiddled around with the problem and got this: </p>
<p>Let $m>n$ then if $a$ divides $x^m-1$ and $x^n-1$, it divides $(x^m-1)-(x^n-1) = x^m-x^n = x^n(x^{m-n} - 1)$ Now if $a$ divides $x^n$ then clearly they're not coprime. Let's assume it divides $x^{m-n}-1$ instead, then it must divide $(x^{... |
3,420,053 | <p>It is a pigeonhole problem.</p>
<p>I have already known that there are <span class="math-container">$1972$</span> remainders in total and the two numbers which have the same remainder can be subtracted and the difference between the two numbers is divisible by <span class="math-container">$1973$</span>.</p>
<p>BUT... | Arthur | 15,500 | <p>Dividing that difference by <span class="math-container">$10^n$</span> for a suitable value of <span class="math-container">$n$</span> will not change whether it's divisible by <span class="math-container">$1973$</span>. And thus you are done.</p>
|
1,279,829 | <p>I know that other groups like $\mathbb{Z}_4 \times \mathbb{Z}_3 \times \mathbb{Z}_5$ are isomorphic to $\mathbb{Z}_{60}$. </p>
| Bruno Joyal | 12,507 | <p>A funny geometric explanation is that $\text{Spec}(\mathbb Z/2\mathbb Z \times \mathbb Z/30\mathbb Z)$ and $\text{Spec}(\mathbb Z/60\mathbb Z)$ do not have the same number of points. Since $30=2 \times 3 \times 5$, the ring $\mathbb Z/2\mathbb Z \times \mathbb Z/30\mathbb Z$ is a product of $4$ fields hence $\text{S... |
3,652,180 | <blockquote>
<p>Let <span class="math-container">$D$</span> be an integral domain which is not a field and <span class="math-container">$Q=\text{Frac}(D)$</span> the field of fractions of <span class="math-container">$D$</span>. Then <span class="math-container">$Q$</span> as a <span class="math-container">$D$</span>... | Hugo | 787,513 | <p>As a non zero projective module has maximal submodules, if K had a projective cover, then K would have a maximal submodule (The maximal submodules of a projective cover of K are in bijective correspondence with the maximal submodules of K). Then it would be a divisible simple module, which can not exist.</p>
|
1,125,862 | <p>Proof: There exists $a = 0$ (For every $b$, an element of the set of positive numbers, such that: $b > a$)</p>
<p>$$a + b > 0 \implies b > 0 \implies a < b.$$</p>
<p>Thus, we have shown that $0 < b$ for every $b$ that is an element of the set of positive numbers. Now, we want to show that $ab = 0 \i... | Litho | 197,288 | <p>Consider a map $f:M_2(\mathbb{R}) \to \mathbb{R}^3$
$$
\left(\begin{matrix} a &b \\ c& d\end{matrix}\right) \mapsto (a-b,b-c,c-d).
$$
Then $q(m) = ||f(m)||^2$ for any $m\in M_2(\mathbb{R})$, where $||\cdot||$ is the standard Euclidean norm on $\mathbb{R}^3$. So if $m_1,m_2,m_3,m_4$ are pairwise orthogonal el... |
3,110,664 | <p>Given the equation <span class="math-container">$k = (p - 1)/2$</span> where <span class="math-container">$p$</span> is any prime number, what is the chance that a randomly chosen element from the set of all <span class="math-container">$k$</span>s will be divisible by 3? Or rather, how can this probability calculat... | zhw. | 228,045 | <p>Let <span class="math-container">$Z$</span> be the zero set of <span class="math-container">$f.$</span> If <span class="math-container">$Z$</span> were uncountable, it would have a limit point <span class="math-container">$a\in \mathbb R^n.$</span> Thus there would exist a sequence of distinct points <span class="ma... |
2,246,777 | <p>I was curious as to whether $$\lim_{x \to 0}\frac{1}{x^2}=\infty $$
Or the limit does not exist? Because doesn't a limit exist if and only if the limit tends to a finite number? </p>
| Alexander Day | 432,831 | <p>good question. if you have $$\lim_{x \to 0} \frac{1}{x^2}$$</p>
<p>as $x \to 0$, the denominator gets closer to 0, and the number gets bigger. example:</p>
<p>$$\frac {1}{0.01^2}=10000$$
$$\frac {1}{0.001^2}=1000000$$
$$\frac {1}{0.00001^2}=100000000$$</p>
<p>BUT(and this is a big but) you cannot have division b... |
551,994 | <p>I want to prove this statement:</p>
<p>$$(A_1 \cup A_2)^c = {A_1}^c \cup {A_2}^c$$</p>
<p>where the $c$ means the complement.</p>
<p>Any help would be greatly appreciated.</p>
| Cameron Buie | 28,900 | <p>You will have trouble proving that $(A_1\cup A_2)^c=A_1^c\cup A_2^c$, since it is not true, in general. (In fact, it holds precisely when $A_1=A_2$.) However, the following <em>are</em> true in general:</p>
<ul>
<li>$(A_1\cup A_2)^c=A_1^c\cap A_2^c$</li>
<li>$(A_1\cap A_2)^c=A_1^c\cup A_2^c$</li>
</ul>
<hr>
<p><s... |
551,994 | <p>I want to prove this statement:</p>
<p>$$(A_1 \cup A_2)^c = {A_1}^c \cup {A_2}^c$$</p>
<p>where the $c$ means the complement.</p>
<p>Any help would be greatly appreciated.</p>
| msh210 | 4,992 | <p>This is false in the general case. For example, if $A_1$ is the set of integers ($\{\ldots,-2,-1,0,1,2,\ldots\}$) and $A_2$ is the set of positive real numbers, and the universe is the set of all real numbers, then $(A_1 \cup A_2)^c$ doesn't contain $-1$ or $\frac12$, but $A_1^c \cup A_2^c$ contains them.</p>
|
871,526 | <p>Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$</p>
<p>What is $\tau$, I can't figure that part out.</p>
<p>All ideas are welcome.</p>
| Holy cow | 155,188 | <p>I will keep updating, this is where I am now .</p>
<p>We know that $$\textbf{T} = \dfrac{\textbf{r}'}{\left| \textbf{r}'\right|}$$</p>
<p>$$\textbf{r}' = \left| \textbf{r}'\right|\hspace{1mm}\textbf{T}$$</p>
<p>$\color{red}{\text{Note that}}$ $|\textbf{r}'| = s'$</p>
<p>$$\textbf{r}' =s'\hspace{1mm}\textbf{T}$$<... |
871,526 | <p>Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$</p>
<p>What is $\tau$, I can't figure that part out.</p>
<p>All ideas are welcome.</p>
| Vincenzo Tibullo | 6,266 | <p>Take into account that, for a generic function $f$,
$$
f'=\frac{df}{dt}=\frac{df}{ds}\frac{ds}{dt}=s'\frac{df}{ds}
$$
so that
$$
\mathbf{N}'=s'\frac{d\mathbf{N}}{ds}
$$
and
$$
\frac{d\mathbf{N}}{ds}=-\kappa\mathbf{T}+\tau\mathbf{B}
$$
see <a href="http://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas" rel="no... |
125,517 | <p>I'm plotting an ellipse <code>1/(1 - e Cos[x])</code> with <code>e=0.9</code> using <code>PolarPlot</code>. Taking $x\in[0,2\pi)$, i.e. one revolution, everything looks fine. But taking <code>n=500</code> revolutions produces something very vaguely resembling an ellipse:</p>
<pre><code>e = 0.9;
n = 500;
PolarPlot[1... | C. E. | 731 | <p>It's a listable expression, so you can plot it with a high precision fixed step size very quickly:</p>
<pre><code>e = 0.9;
n = 500;
Graphics@Line@With[{x = Range[0, 2 n Pi, 0.01]},
Transpose[{Cos[x]/(1 - e Cos[x]), Sin[x]/(1 - e Cos[x])}]
] // AbsoluteTiming
</code></pre>
<p><img src="https://i.stack.imgur... |
125,517 | <p>I'm plotting an ellipse <code>1/(1 - e Cos[x])</code> with <code>e=0.9</code> using <code>PolarPlot</code>. Taking $x\in[0,2\pi)$, i.e. one revolution, everything looks fine. But taking <code>n=500</code> revolutions produces something very vaguely resembling an ellipse:</p>
<pre><code>e = 0.9;
n = 500;
PolarPlot[1... | Michael E2 | 4,999 | <p>The following tables show that the major part of the time to plot is proportional to the number of function evaluations, which for the OP's function is about a 10 microseconds per evaluation, or about 100 times as slow as C.E.'s method. Recursive subdivision adds a small overhead, which is not tremendously signific... |
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