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 |
|---|---|---|---|---|
2,762,040 | <p>I started to think about this problem and then factored $n^5 - n$ to $(n^2 - 1)(n^2 + 1)(n)$, and later to $(n-1)(n)(n+1)(n^2 + 1)$. I know that $(n-1)(n)(n+1)$ is divisible by $6$, but it is not that case $5$ divides $n^2 + 1$ for any integer $n$, so i can´t use the multiplication property. Can anyone help me finis... | David C. Ullrich | 248,223 | <p>One of the other answers requires that you know something - the other is very clever. A hint for a proof that doesn't require either non-trivial theorems or cleverness: $n=5k+j$, where $j\in\{0,1,2,3,4\}$.</p>
|
16,754 | <p>Let $c$ be an integer, not necessarily positive and not a square. Let $R=\mathbb{Z}[\sqrt{c}]$
denote the set of numbers of the form $$a+b\sqrt{c}, a,b \in \mathbb{Z}.$$
Then $R$ is a subring of $\mathbb{C}$ under the usual addition and multiplication.</p>
<p>My question is: if $R$ is a UFD (unique factorization ... | Matt E | 221 | <p>The answer is yes. The argument is as follows: if $R$ is a UFD, then it is necessarily integrally closed in its fraction field $K = \mathbb Q(\sqrt{c})$, and thus is
equal to the full ring of algebraic integers in $K$. A general fact about such full rings of algebraic integers is that if they are UFDs then they ar... |
4,171,907 | <blockquote>
<p>If <span class="math-container">$3\sin x +5\cos x=5$</span> then prove that <span class="math-container">$5\sin x-3\cos x=3$</span></p>
</blockquote>
<p>What my teacher did in solution was as follows</p>
<p><span class="math-container">$$3\sin x +5\cos x=5 \tag1$$</span></p>
<p><span class="math-contain... | Umesh shankar | 80,910 | <p>Yes you are right,he/she should have broken the problem into two cases.</p>
<p>Case <span class="math-container">$1.$</span> When <span class="math-container">$\sin x\ne 0$</span>, then according to your teacher <span class="math-container">$5\sin x-3\cos x=3$</span>.</p>
<p>Case <span class="math-container">$2.$</s... |
59,046 | <p>Let $A \in M_n(\mathbb R)$ and suppose its minimal polynomial is:
$$M_A(t)=\prod_{i=1}^{k}(t-\lambda_i)^{\textstyle s_i}.$$</p>
<p>When $\lambda _1,\lambda_2,\lambda _3,......,\lambda _k$ are distinct eigenvalues.</p>
<p>We define a new matrix: $B\in M_{2n}(\mathbb R)$ by:
$$\left(\begin{matrix}
A &I_n \\
0... | Yuval Filmus | 1,277 | <p>Prove by induction that for any polynomial $P$,
$$ P(B) = P\left( \begin{bmatrix} A & I \\ I & A \end{bmatrix} \right) =
\frac{1}{2} \begin{bmatrix} P(A+I)+P(A-I) & P(A+I)-P(A-I) \\ P(A+I)-P(A-I) & P(A+I)+P(A-I) \end{bmatrix}. $$</p>
<p>Hence $$P(B)=0 \Leftrightarrow P(A+I) = P(A-I) = 0.$$
You take... |
3,362,916 | <p>I'm trying to graph <span class="math-container">$|x+y|+|x-y|=4$</span>. I rewrote the expression as follows to get a function that resembles the direction of unit vectors at <span class="math-container">$\pi/4$</span> to the horizontal axis (take it to be <span class="math-container">$x$</span>)<span class="math-co... | fGDu94 | 658,818 | <p>Once <span class="math-container">$x$</span> is fixed you can look for solutions in three distinct regions: <span class="math-container">$y>x$</span>, <span class="math-container">$-x<y<x$</span>, <span class="math-container">$y<-x$</span>. By doing this you can simplify the absolute values and find solu... |
3,362,916 | <p>I'm trying to graph <span class="math-container">$|x+y|+|x-y|=4$</span>. I rewrote the expression as follows to get a function that resembles the direction of unit vectors at <span class="math-container">$\pi/4$</span> to the horizontal axis (take it to be <span class="math-container">$x$</span>)<span class="math-co... | ASTRONO | 706,272 | <p>The given graph forms a square with side 4.Simple logic is that when (X+y)+(x-y)=c
When (X+y) approaches c,(x-y) approaches 0.
From the above thing the upper boundary will be 2x=4.
You get X=2,y=2.
Because there is mod if you can solve one boundary it gets mirrored about both X and y axis</p>
|
3,362,916 | <p>I'm trying to graph <span class="math-container">$|x+y|+|x-y|=4$</span>. I rewrote the expression as follows to get a function that resembles the direction of unit vectors at <span class="math-container">$\pi/4$</span> to the horizontal axis (take it to be <span class="math-container">$x$</span>)<span class="math-co... | Doug M | 317,162 | <p>Looks like you have said</p>
<p><span class="math-container">$u = \frac {\sqrt {2}}{2} (x + y)\\
v = \frac {\sqrt {2}}{2} (x - y)$</span></p>
<p>Which rotates coordinate system 45 degrees.</p>
<p>And your expression becomes</p>
<p><span class="math-container">$|\sqrt 2 u| + |\sqrt 2 v| = 4$</span> or <span class... |
3,058,019 | <blockquote>
<p>Two numbers <span class="math-container">$297_B$</span> and <span class="math-container">$792_B$</span>, belong to base <span class="math-container">$B$</span> number system. If the first number is a factor of the second number, then what is the value of <span class="math-container">$B$</span>?</p>
</... | nonuser | 463,553 | <p>Since <span class="math-container">$b+1>0$</span> and <span class="math-container">$$(b+1)(2b+7)\mid (7b+2)(b+1)\implies 2b+7\mid 7b+2$$</span></p>
<p>we have <span class="math-container">$$2b+7\mid (7b+2)-3(2b+7) = b-19$$</span></p>
<p>so if <span class="math-container">$b-19> 0$</span> we have <span class... |
2,109,347 | <p>My statistics note states that the variance of the empirical distribution is
$v= \sum_{i=1}^{n}(x_i-\bar x )^2\frac {1} {n}$ which the author then re-writes as
$v= \sum_{i=1}^{n}x_i^2 (\frac {1} {n}) - \bar x^2$. How is this achieved?</p>
| szw1710 | 130,298 | <p>$$\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x})^2=\frac{1}{n}\left(\sum_{i=1}^n x_i^2-2\sum_{i=1}^n x_i\cdot \bar{x}+\sum_{i=1}^n \bar{x}^2\right)=\frac{1}{n}\left(\sum_{i=1}^nx_i^2-2\bar{x}\sum_{i=1}^nx_i+\bar{x}^2\sum_{i=1}^n 1\right).$$</p>
<p>Hence</p>
<p>$$
\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x})^2=\frac{1}{n}\sum_{i=1}^... |
2,869,442 | <blockquote>
<p>Check whether the series
$$\sum_{n=1}^{\infty}\int_0^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^2}\ dx$$
is convergent.</p>
</blockquote>
<p>I tried to sandwich the function by $\dfrac{1}{1+x^2}$ and $\dfrac{x}{1+x^2}$ , but this did not help at all.
Any other way of approaching?</p>
| Chappers | 221,811 | <p>Since $\sqrt{x}$ is increasing and $1/(1+x^2)$ is decreasing, we have
$$ \int_0^{1/n} \frac{\sqrt{x}}{1+x^2} \, dx < \frac{1}{\sqrt{n}} \int_0^{1/n}\frac{dx}{1+x^2} < \frac{1}{\sqrt{n}} \int_0^{1/n} dx = \frac{1}{n^{3/2}}. $$
And $\sum_n 1/n^{3/2}$ converges.</p>
|
2,291,310 | <p>I'm seeking an alternative proof of this result:</p>
<blockquote>
<p>Given $\triangle ABC$ with right angle at $A$. Point $I$ is the intersection of the three angle lines. (That is, $I$ is the incenter of $\triangle ABC$.) Prove that
$$|CI|^2=\frac12\left(\left(\;|BC|-|AB|\;\right)^2+|AC|^2\right)$$</p>
</bloc... | Jack D'Aurizio | 44,121 | <p>Alternative proof: by Stewart's theorem the length $\ell_c$ of the angle bisector through $C$ is given by
$$ \ell_c^2 = \frac{ab}{(a+b)^2}\left[(a+b)^2-c^2\right] $$
and by Van Obel's theorem and the bisector theorem $\frac{CI}{\ell_c}=\frac{a+b}{a+b+c}$. It follows that, in general:
$$ CI^2 = \frac{ab}{(a+b+c)^2}\l... |
19,285 | <p>Is anyone aware of Mathematica use/implementation of <a href="http://en.wikipedia.org/wiki/Random_forest">Random Forest</a> algorithm?</p>
| Seth Chandler | 5,775 | <p>I'm going to be bold and attempt to edit the Ross code so that it is (a) a little easier to understand and (b) takes the same form of argument as LinearModelFit and other Mathematica prediction creators. I've also added some annotations to the critical code. My variable names are now far longer than the Ross names ... |
519,764 | <p>Question: show that the following three points in 3D space A = <-2,4,0>, B = <1,2,-1> C = <-1,1,2> form the vertices of an equilateral triangle.</p>
<p>How do i approach this problem?</p>
| Community | -1 | <p><strong>Hint</strong>: A triangle is equilateral if and only if all its sidelengths are equal. We compute</p>
<p>$$\|A - B\| = \|\langle 3, 2, 1\rangle\| = \sqrt{3^2 + 2^2 + 1^2} = \sqrt{14}$$</p>
<p>$$\|A - C\| = \|\langle -1, 3, -2\rangle\| = ? $$</p>
<p>$$\|B - C\| = \|\langle 2, 1, -3\rangle\| = ? $$</p>
|
519,764 | <p>Question: show that the following three points in 3D space A = <-2,4,0>, B = <1,2,-1> C = <-1,1,2> form the vertices of an equilateral triangle.</p>
<p>How do i approach this problem?</p>
| rnjai | 42,043 | <p>Find the distance between all the pairs of points
$$|AB|,|BC|,|CA|$$
and check if
$$|AB|=|BC|=|CA|$$
For example:
$$|A B| = \sqrt{(-2-1)^2 + (4-2)^2 + (0-(-1))^2} = \sqrt{14}$$</p>
|
519,764 | <p>Question: show that the following three points in 3D space A = <-2,4,0>, B = <1,2,-1> C = <-1,1,2> form the vertices of an equilateral triangle.</p>
<p>How do i approach this problem?</p>
| David Park | 99,469 | <p>This would be the Grassmann algebra approach. This is a fairly simple problem and the main advantage of Grassmann algebra is that we can set up the objects and calculate with them in a natural manner.</p>
<pre><code><< GrassmannCalculus`
SetPreferences["Grassmann3Space", "Vector"]
</code></pre>
<p>The follow... |
496,011 | <p>Suppose that $ p_n(t) $ is the probability of finding n particle at a time t. And the dynamics of the particle is described by this equation : </p>
<p>$$ \frac{d}{dt} p_n(t) = \lambda \Delta p_n(t) $$</p>
<p>Defining one - dimensional lattice translation operator $ E_m = e^{mk} $ with $ km - mk = 1 $ and $\Delta ... | Cameron Buie | 28,900 | <p>I'm afraid not. You should not have an equation with a "differential factor" on one side and not on the other (that is, $dx=2x$ is nonsense). For more on how to deal with differential factors, you might find the second part of <a href="https://math.stackexchange.com/a/155769/28900">this answer</a> (from "Now, if I w... |
496,011 | <p>Suppose that $ p_n(t) $ is the probability of finding n particle at a time t. And the dynamics of the particle is described by this equation : </p>
<p>$$ \frac{d}{dt} p_n(t) = \lambda \Delta p_n(t) $$</p>
<p>Defining one - dimensional lattice translation operator $ E_m = e^{mk} $ with $ km - mk = 1 $ and $\Delta ... | abiessu | 86,846 | <p>HINT: when completing a $u$ substitution, the typical flow is as follows:</p>
<p>$$\int x^3\sqrt{x^2+1}dx$$</p>
<p>Substitute $u=x^2+1$, then $du=2xdx$ and we have $x^2=u-1$:</p>
<p>$$\int {(u-1)\sqrt u du\over 2}$$</p>
<p>From there, the remaining steps are to integrate, then reverse-substitute to obtain the f... |
14,448 | <p>Here's the most common way that I've seen letter grades assigned in undergrad math courses. At the end of the semester, the professor: 1) computes the raw score (based on homework, quizzes, and tests) for each student; 2) writes down all the raw scores in order; 3) somewhat arbitrarily clusters the scores into group... | Matt Ollis | 10,159 | <p>As @shoover hints at in their comment, the question assumes that you've already answered the question of why you are assigning letter grades at all. Assuming the answer to that is something along the lines of "because I have no choice" or "that's just how it's done", here's how I approach it. (Perhaps useful info:... |
2,121,583 | <p>Question: Let $f,g: X \rightarrow \mathbb{R}$ continous (over $X$, and $X$ is a metric space). If $\overline{Y}\subset X $, and $f(y)=g(y)$ for every $y\in Y $, prove that $\left.f\right|_\overline{Y}= \left.g\right|_\overline{Y}$.</p>
<p>Attempt:</p>
<p>Since $\overline{Y} \subset X$, it follows that $\left.f\ri... | Arthur | 15,500 | <p>Assume there is a point $y_0\in\overline Y$ such that $f(y_0)\neq g(y_0)$. Then, by continuity, there must be an $\epsilon>0$ such that for any $y\in B_{\epsilon}(y_0)$, we have $f(y)\neq g(y)$. But because some of those $y$ necessarily must be in $Y$, we have reached a contradiction.</p>
|
29,861 | <p>The meta question <em><a href="https://math.meta.stackexchange.com/q/29857/290189">Not actually a question, just a rant!</a></em> has inspired me to ask for <em>what</em> an answerer can do in case of self-deletion by the question asker while the answerer is typing the answer.</p>
<p>Since per se site is supposed t... | daniel | 18,124 | <p>Would it not be possible to add a feature that gives notice of imminent self-closure? Like a 5-minute warning? Or just require notice in a comment? I have closed quite a few of my own questions but try to indicate imminent closure in a comment in case someone is working on an answer so they have a chance to object. ... |
4,168,223 | <p>Let <span class="math-container">$R$</span> be the row reduced echelon form of a <span class="math-container">$4 \times 4$</span> real matrix <span class="math-container">$A$</span> and
let the
third column of <span class="math-container">$R$</span> be <span class="math-container">$\left[\begin{array}{l}0 \\ 1 \\ 0 ... | Anurag A | 68,092 | <p>Assuming that you are using the <strong>same</strong> <span class="math-container">$\bf{b}$</span> in <span class="math-container">$[A | \bf{b}]$</span> and <span class="math-container">$[R | \bf{b}]$</span> (i.e. you have not row reduced the right hand side column), then <span class="math-container">$Q$</span> need... |
2,394,716 | <p>Let $A = \{\frac{x}{2} - \lfloor\frac{x+1}{2}\rfloor : x \in \mathbb{R} \}$ </p>
<p>Does <strong>supremum</strong> and <strong>infimum</strong> of $A$ exist ? If the answer is yes then find them .</p>
<p>My try : I rewrite the expression $\frac{x}{2} - (\lfloor 2x \rfloor - \lfloor x \rfloor)$ but it doesn't h... | Dietrich Burde | 83,966 | <p>It really depends on the context. For solving linear equations by substitution, the standard form of $-4x+3y=5$ would be multiplying by $\frac{1}{4}$ to obtain
$x=\frac{3y-5}{4}$ for substitution. So the normal form could be
$$
-4x+3y-5=0,
$$</p>
<p>or
$$
\; x-\frac{3}{4}y+\frac{5}{4}=0.
$$
The last normal form au... |
4,250,292 | <p>I am stuck trying to solve the following integral:</p>
<p><span class="math-container">$$\int_{0}^{3}\frac{1}{3}e^{3t-2}dt$$</span></p>
<p>I understand that I can take out <span class="math-container">$\frac{1}{3}$</span> of the integral, and that the integral of <span class="math-container">$e^{3t-2}$</span> is <sp... | Alessio K | 702,692 | <p>For <span class="math-container">$a\neq 0$</span> we have</p>
<p><span class="math-container">$$\int e^{at}dt=\frac{1}{a}e^{at}+C$$</span></p>
<p>Thus we have</p>
<p><span class="math-container">$$\int \frac{1}{3} e^{3t-2}dt=\frac{1}{3}\int e^{3t}e^{-2}dt$$</span>
<span class="math-container">$$=\frac{1}{3}e^{-2}\in... |
468 | <p>Textbook writers are blessed with only solving problems with neat answers. Numerical coefficients are small integers, many terms cancel, polynomials split into simple factors, angles have trigonometric functions with known values. Pure bliss.</p>
<p>The "real life" is different (as any of us knows).</p>
<p>Giving ... | Markus Klein | 114 | <p><strong>I would encourage to have a significant amount of non-"round" numbers in your homework and also in exams.</strong> Some reasons:</p>
<ul>
<li>Except when you would teach calculations with numbers, normally the way you do things is important, not actual calculations. Even if the students feel that something ... |
468 | <p>Textbook writers are blessed with only solving problems with neat answers. Numerical coefficients are small integers, many terms cancel, polynomials split into simple factors, angles have trigonometric functions with known values. Pure bliss.</p>
<p>The "real life" is different (as any of us knows).</p>
<p>Giving ... | Jyrki Lahtonen | 282 | <p>Like others I more often than not craft the problems to have nice solutions. Just to keep the students on their toes I occasionally insert something not so nice. As we are largely discussing eigenvalue problems, let me propose the following trick I picked up from a senior colleague. </p>
<p>Use a problem, where the... |
1,377,412 | <p>I am brand new to ODE's, and have been having difficulties with this practice problem. Find a 1-parameter solution to the homogenous ODE:$$2xy \, dx+(x^2+y^2) \, dy = 0$$assuming the coefficient of $dy \ne 0$
The textbook would like me to use the subsitution $x = yu$ and $dx=y \, du + u \, dy,\ y \ne 0$
Rewriting t... | Chiranjeev_Kumar | 171,345 | <p>$$2xy\mathrm dx+(x^2+y^2)\mathrm dy = 0$$</p>
<p>$$2xy \mathrm dx=-(x^2+y^2)\mathrm dy$$</p>
<p>$$\frac{dx}{dy}=-\frac{(x^2+y^2)}{2xy}$$</p>
<p>Now put, $x=yu \Rightarrow \mathrm dx=u\mathrm dy+y\mathrm du$, i.e.
$\frac {dx}{dy}=u+y\frac{du}{dy}$</p>
<p>Using these substitution and after simplification a you wi... |
1,851,209 | <p>Let $L:X\to Y$ an linear operator. I saw that $L$ is bounded if $$\|Lu\|_Y\leq C\|u\|_X$$
for a suitable $C>0$. This definition looks really weird to me since such application is in fact not necessary bounded as $f:\mathbb R\to \mathbb R$ defined by $f(x)=x$. So, is there an error in <a href="https://en.wikipedia... | Surb | 154,545 | <p><strong>Answer 1</strong></p>
<p>Let $A\in \Bbb R^{n\times n}$ and $v\in\Bbb R^n$ such that $Av\neq 0$.</p>
<p>Then it is true that $$\lim_{s\to \infty} \|A(sv)\|=\|Av\|\lim_{s\to \infty} s=\infty.$$
However, there exists a $C>0$ such that
$$\frac{\|A(sv)\|}{\|sv\|}=\frac{\|Av\|}{\|v\|}\leq C \qquad \forall s&... |
113,843 | <p>Here comes some sample data</p>
<pre><code>data = {{0, 0.7, 0.4}, {1, 0.831177, 0.51854}, {2, 1.11106, 0.463533},
{3, 1.84226, -0.642571}, {4, 0.677049, -0.327877},
{5, 0.77886, -0.451322}, {6, 0.965874, -0.508772},
{7, 1.34397, -0.202473}, {8, 1.01761, -0.717013},
{9, -0.0507992... | Hubble07 | 7,009 | <pre><code>r3 = AppendTo[Table[{Graphics[{Text[
Which[i == 1, Subscript[P, 0], i == Length[d0], Subscript[P, f],
True, ToString[i - 1]], Offset[{0, 10}, d0[[i]]]]}],
Graphics[{PointSize[Large], Which[i == 1, Red],
Which[i == Length[d0] - 1, {Point[d0[[i]]], Blue,
Point[d0[[i + 1]]]}, True, Point[d... |
3,406,056 | <p>By the definition of matrix exponentiation,</p>
<p><span class="math-container">$$A^k = \begin{cases}
I_n, & \text{if } k=0 \\[1ex]
A^{k-1}A, & \text{if } k\in \mathbb {N}_0 \\
\end{cases}$$</span></p>
<p>In my book, there's an exercise to do <span class="math-container">$D^k$</span>, where <span class="... | Mohammad Riazi-Kermani | 514,496 | <p>Both are correct but they are two different topics.</p>
<p><span class="math-container">$$A^k=\left\{
\begin{array}{c}
I_n, if\, k=0 \\
A^{k-1}A, if\, k\in {\displaystyle \mathbb {N} }_0 \\
\end{array}
\right.$$</span>
is the definition of the matrix power <span class="math-container">$A^k$</span> in general.</p... |
2,614,127 | <p>I have been trying to show the statement below using the $AC$ but I am starting to think that it is not strong enough to do it. </p>
<p><strong>Context:</strong> Let $\Gamma$ be an uncountable linearly ordered set with a smallest element (not necessarily well-ordered). </p>
<p>For each $\alpha\in\Gamma$, let $C_\a... | Eric Wofsey | 86,856 | <p>Let me first ignore your requirements that $\Gamma$ is uncountable and that the containments are strict, since these requirements are wholly artificial and have nothing to do with what's going on, and just make counterexamples a bit messier. Here, then, is the prototypical counterexample. Let $\Gamma=\mathbb{N}$, ... |
17,436 | <p>I have this list:</p>
<pre><code>a = {{{0, 0}, {1, 7}, {2, 0}, {3, 2}, {4, 7}}, {{0, 0}, {1, 0}, {2, 1}, {3, 2}, {4, 7}}}
</code></pre>
<p>and I'd like to transform it to this:</p>
<pre><code>a = {{{0, 0}, {1, 7}, {2, Na}, {3, 2}, {4, 7}}, {{0, 0}, {1, Na}, {2, 1}, {3, 2}, {4, 7}}}
</code></pre>
<p>ie I'd like t... | image_doctor | 776 | <p>This will work if all your first elements of the sub-lists are of the form {0,0}</p>
<pre><code>a = a /. {{x_?Positive, 0} :> {x, Na}};
a
</code></pre>
<blockquote>
<p>{{{0, 0}, {1, 7}, {2, Na}, {3, 2}, {4, 7}}, {{0, 0}, {1, Na}, {2,
1}, {3, 2}, {4, 7}}}</p>
</blockquote>
|
17,436 | <p>I have this list:</p>
<pre><code>a = {{{0, 0}, {1, 7}, {2, 0}, {3, 2}, {4, 7}}, {{0, 0}, {1, 0}, {2, 1}, {3, 2}, {4, 7}}}
</code></pre>
<p>and I'd like to transform it to this:</p>
<pre><code>a = {{{0, 0}, {1, 7}, {2, Na}, {3, 2}, {4, 7}}, {{0, 0}, {1, Na}, {2, 1}, {3, 2}, {4, 7}}}
</code></pre>
<p>ie I'd like t... | Rolf Mertig | 29 | <p>Another quick possibility</p>
<pre><code>a=a/. {n_, r:{_Integer,_Integer}...} :> Join[{n},{ r} /. 0->Na]
</code></pre>
|
747,519 | <p><img src="https://i.stack.imgur.com/jYzfz.png" alt="enter image description here"></p>
<p>I tried this problem on my own, but got 1 out of 5. Now we are supposed to find someone to help us. Here is what I did:</p>
<p>Let $f:[a,b] \rightarrow \mathbb{R}$ be continuous on a closed interval $I$ with $a,b \in I$, $a \... | DanielWainfleet | 254,665 | <p>(1).... Let $J=[a,b]$ with $a\leq b.$ Any sequence $(x_n)_{n\in N}$ of members of $J$ has a convergent subsequence. That is, there is a strictly increasing $g:N\to N$ such that $(x_{g(n)})_{n\in N}$ is a convergent sequence.</p>
<p>Proof: Let $J_1=[a,(a+b)/2]$ if $\{n: x_n\in [a,(a+b)/2]\}$ is an infinite set, oth... |
613,105 | <p>I was observing some nice examples of equalities containing the numbers $1,2,3$ like $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi$ and $\log 1+\log 2+ \log 3=\log (1+2+3)$. I found out this only happens because $1+2+3=1*2*3=6$.<br> I wanted to find other examples in small numbers, but I failed. How can we find all of the s... | N. S. | 9,176 | <p>Without loss of generality $a \leq b \leq c$. Then $a+b+c \leq 3c$ and hence</p>
<p>$$abc=a+b+c \leq 3c$$</p>
<p>Thus, either $c =0$, in which case $a=b=c=0$, or </p>
<p>$$ab \leq 3 \,.$$</p>
<p>This leads to only four possibilities to check: $a=0$ or $(a,b)=(1,1)$ or $(a,b)=(1,2)$ or $(a,b)=(1,3)$.</p>
|
794,389 | <p>Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2$ squares, with sides $1, 2, 3\ldots n^2$ ($n$ odd). Which seems like it would work with ANY odd $n$. It's so simple, surely it's well-known, but I haven't seen it in my (brief) web travels. I ha... | Ross Millikan | 1,827 | <p>It looks like you can tile an arbitrarily large section of the plane, but not the whole thing. As you say, it looks like you cannot extend it toward the southeast. If I challenge you to cover a $1,000,000 \times 1,000,000$ square you can do it, but you need to plan ahead by putting the correct number of small squa... |
102,721 | <p>This is probably a very simple question, but I couldn't find a duplicate.</p>
<p>As everybody knows, <code>{x, y} + v</code> gives <code>{x + v, y + v}</code>. But if I intend <code>v</code> to represent a vector, for example if I am going to substitute <code>v -> {vx, vy}</code> in the future, then the result <... | Ruslan | 5,208 | <p>Another option, which doesn't require activation/replacement-by-identity/manual-indexing/etc. is to create a custom Plus function, which only evaluates for lists:</p>
<pre><code>vectorPlus[a_?ListQ,b_?ListQ]=a+b;
</code></pre>
<p>With it you have:</p>
<pre><code>vectorPlus[x, {q,r,s}]
vectorPlus[{q,r,s}, y]
vecto... |
1,815,662 | <blockquote>
<p>Prove that $X^4+X^3+X^2+X+1$ is irreducible in $\mathbb{Q}[X]$, but it has two different irreducible factors in $\mathbb{R}[X]$.</p>
</blockquote>
<p>I've tried to use the cyclotomic polynomial as:
$$X^5-1=(X-1)(X^4+X^3+X^2+X+1)$$</p>
<p>So I have that my polynomial is
$$\frac{X^5-1}{X-1}$$ and now... | egreg | 62,967 | <p>Finding the complex roots of the polynomial is easy: if $\varphi=2\pi/5$, the roots are
$$
r_1=e^{i\varphi},\quad
r_2=e^{2i\varphi},\quad
r_3=e^{3i\varphi}=\bar{r}_2\quad
r_4=e^{4i\varphi}=\bar{r}_1
$$
and so the factorization over the reals is
$$
(X^2-(r_1+\bar{r}_1)X+1)((X^2-(r_2+\bar{r}_2)X+1).
$$
What you want t... |
167,575 | <p>I have 6 sets of 4D points. Here is an example of one set :</p>
<pre><code>{{30., 5., 111.925, 113.569}, {30., 7.5, 114.7, 158.286}, {30., 10., 115.625, 206.023},
{30., 12.5, 115.625, 257.528}, {30., 15., 117.475, 294.663}, {30., 17.5, 119.325, 328.03},
{30., 20., 121.175, 357.982}, {30., 22.5, 122.1, 393.646}, {... | Carl Woll | 45,431 | <p>Why not use <a href="http://reference.wolfram.com/language/ref/BubbleChart" rel="nofollow noreferrer"><code>BubbleChart</code></a> with style wrappers?:</p>
<pre><code>BubbleChart[
Replace[
{{3,4,3,5},{4,1,4,8}},
{a_, b_, c_, d_} :> Style[{a, b, c}, Lighter[Green, d/10]],
{1}
]
]
... |
65,059 | <p>I have two points ($P_1$ & $P_2$) with their coordinates given in two different frames of reference ($A$ & $B$). Given these, what I'd like to do is derive the transformation to be able to transform any point $P$ ssfrom one to the other.</p>
<p>There is no third point, but there <em>is</em> an extra constra... | Ben Blum-Smith | 13,120 | <p>I think that the two points should be enough information even without the additional information about the $y$-axis of the second frame. <strong>EDIT:</strong> joriki pointed out that this isn't true (see comments). The additional piece of information about the $y$-axis is needed, but then the problem should be sol... |
4,639,966 | <p>I recently saw the expansion <span class="math-container">$(1+ \frac{1}{x})^n = 1 + \frac{n}{x} + \frac{n(n-1)}{2!x^2} + \frac{n(n-1)(n-2)(n-3)}{3!x^3}.... $</span> where <span class="math-container">$n \in \mathbb Q$</span></p>
<p>From what I understood, they have taken the Taylor series of <span class="math-co... | aschepler | 2,236 | <p>This equation is not a Taylor series, but it is correct, at least for <span class="math-container">$x \geq 1$</span>.</p>
<p>The usual Taylor series</p>
<p><span class="math-container">$$ (1+t)^n = 1 + nt + \frac{n(n-1)}{2!} t^2 + \frac{n(n-1)(n-2)}{3!} t^3 + \cdots $$</span></p>
<p>is a true equation for every rati... |
2,124,068 | <p>I came across the following problem in a book I was reading on continuous probability distributions:-</p>
<p>$Q.$ Let $Y$ be uniformly distributed on $(0,1)$. Find a function $\phi$ such that $\phi(Y )$ has the gamma density $\Gamma(\frac12,\frac12)$.</p>
<p>I know that the probability density represented by $\Gam... | m_goldberg | 72,018 | <p>Consider that </p>
<pre><code>54 {Cos[t], Sin[t], t/10 + 12.5/54} /. t -> 0
</code></pre>
<p>gives</p>
<blockquote>
<p><code>{54, 0, 12.5}</code>.</p>
</blockquote>
<p>Since </p>
<pre><code>54 {Cos[t], Sin[t], t/10 + 12.5/54}
</code></pre>
<p>is the parametric expression of a helix with radius 54, it repr... |
2,017,993 | <p>Is $0$ an eigenvalue for a compact normal operator?</p>
<p>Many texts mention that compact normal operators have a complete orthonormal basis of eigenvectors. If they do, then what about the kernel of the operator? The elements in the kernel, may not be eigenvectors.</p>
<p>Where is the mistake in my understanding... | Martin Argerami | 22,857 | <p>The elements of the kernel are precisely the eigenvectors for zero.</p>
<p>So, when $T $ is normal and compact you can form an orthonormal basis by joining orthonormal bases for each eigenspace (including the kernel, if nonzero).</p>
|
422,948 | <p>How could I/is it possible to take a fourier transform of text? i.e. What domain would/does text exist in? Any help would be great.</p>
<p>NOTE: I do not mean text as an image. I understand it's value, but I'm wondering if it is possible to map text to some domain and transform text on the basis of letters. This is... | Bert Newton | 986,909 | <p>I had a similar idea last night when I was trying to explain the concept of FFTs for fundamental analysis and synthesis of sounds to someone, and the analogy that popped into my head was of the prevalence of lowercase letters, uppercase letters, and punctuation in a page of text corresponding to signals that occur ... |
4,106,273 | <p>In how many ways can a committee of four be formed from 10 men (including Richard) <br>
and 12 women (including Isabel and Kathleen) if it is to have two men and two women <br></p>
<p>a) Isabel refuses to serve with Richard,</p>
<p>b) Isabel will serve only if Kathleen does, too</p>
<p>My Thoughts : <br>
a) Total nu... | herb steinberg | 501,262 | <p>Part (b). There are <span class="math-container">$^{20}C_3$</span> ways when Isabel is on and Kathleen is not.</p>
<p>Part (a) is correct.</p>
|
3,999,652 | <p>Let triangle <span class="math-container">$ABC$</span> is an equilateral triangle. Triangle <span class="math-container">$DEF$</span> is also an equilateral triangle and it is inscribed in triangle <span class="math-container">$ABC \left(D\in BC,E\in AC,F\in AB\right)$</span>. Find <span class="math-container">$\cos... | Semiclassical | 137,524 | <p>As others have done, I'll assume <span class="math-container">$AB=8,DF=5$</span> without loss of generality. Then equilateral triangles <span class="math-container">$\triangle ABC$</span> and <span class="math-container">$\triangle DEF$</span> have area <span class="math-container">$(\sqrt{3}/4)AF^2 = 16\sqrt{3}$</s... |
420,294 | <p>While reading Bill Thurston's <a href="http://www.ams.org/publications/journals/notices/201601/rnoti-p31.pdf" rel="noreferrer">obituary</a> in the Notices of the AMS I came across the following fascinating anecdote (pg. 32):</p>
<blockquote>
<p>Bill’s enthusiasm during the early stages of mathematical discovery was ... | Wahome | 122,319 | <p>As Derek Holt suggested in a comment, it seems Thurston was indeed thinking of word acceptors that returned normal forms for elements of automatic groups. From a 1989 research report of his titled <a href="http://timo.jolivet.free.fr/docs/ThurstonLectNotes.pdf" rel="nofollow noreferrer">Groups, tilings and finite st... |
1,382,479 | <p>I would like to know why $a^p \equiv a \pmod p$ is the same as $a^{p-1} \equiv 1 \pmod p$, and also how Fermat's little theorem can be used to derive Euler's theorem, or vice versa. </p>
<p>Please keep in mind that I have little background in math, and I am trying to understand these theorems to understand the math... | hmakholm left over Monica | 14,366 | <p>It is not obvious how to derive Euler's theorem in its full generality from Fermat's little theorem -- if the modulus has a non-trivial square factor, then Fermat's little theorem doesn't seem to provide enough.</p>
<p>Fortunately, for RSA you don't need Euler's theorem in its full generality; it is enough to know:... |
1,382,479 | <p>I would like to know why $a^p \equiv a \pmod p$ is the same as $a^{p-1} \equiv 1 \pmod p$, and also how Fermat's little theorem can be used to derive Euler's theorem, or vice versa. </p>
<p>Please keep in mind that I have little background in math, and I am trying to understand these theorems to understand the math... | Thomas Andrews | 7,933 | <p>(For the first part.)</p>
<p>More generally, if $b\mid ac$ and $b$ and $a$ are relatively prime, then $b\mid c$. In this case, $b=p$ and $c=a^{p-1}-1$.</p>
<p>This more general theorem can be seen by "unique factorization," also known as the fundamental theorem of arithmetic. But it can also be proved first as a l... |
4,487,494 | <blockquote>
<p><strong>Problem:</strong> Let <span class="math-container">$x$</span> and <span class="math-container">$y$</span> be non-zero vectors in <span class="math-container">$\mathbb{R}^n$</span>.<br>
(a) Suppose that <span class="math-container">$\|x+y\|=\|x−y\|$</span>. Show that <span class="math-container">... | Sourav Ghosh | 977,780 | <p><span class="math-container">$a)$</span> Using <a href="https://en.m.wikipedia.org/wiki/Polarization_identity" rel="nofollow noreferrer">polarization identity</a></p>
<p><span class="math-container">$4\langle x, y\rangle =\|x+y\|^2-\|x-y\|^2=0$</span></p>
<p><span class="math-container">$\langle x, y\rangle =x\cdot ... |
4,226,030 | <blockquote>
<p>I want to solve
<span class="math-container">$$C\cos(\sqrt\lambda \theta) + D\sin(\sqrt\lambda \theta) = C\cos(\sqrt\lambda (\theta + 2m\pi)) + D\sin(\sqrt\lambda (\theta + 2m\pi))$$</span>
The solution must be valid for all <span class="math-container">$\theta$</span> in <span class="math-container">$\... | David K | 139,123 | <p>Let <span class="math-container">$f_{C,D,\lambda}(\theta)
= C\cos(\sqrt\lambda \theta) + D\sin(\sqrt\lambda \theta).$</span>
Your want the values of the parameters <span class="math-container">$C,$</span> <span class="math-container">$D,$</span> and <span class="math-container">$\lambda$</span> such that</p>
<p><sp... |
634,929 | <p>How can I evaluate this integral?</p>
<p>$$
\int{x^{3}\,{\rm d}x \over \left(x - 1\right)^{2}\sqrt{x^{2} + 2x + 4}}$$</p>
<p>I would be grateful for any tips.</p>
| lab bhattacharjee | 33,337 | <p>As $x^2+2x+4=(x+1)^2+(\sqrt3)^2,$ using <a href="http://en.wikipedia.org/wiki/Trigonometric_substitution" rel="nofollow">Trigonometric substitution</a></p>
<p>let us set $x+1=\sqrt3\tan\psi$ and assuming $0<\psi<\frac\pi2$ so that $\sec\psi,\tan\psi>0$</p>
<p>$$I=\int\frac{x^3}{(x-1)^2\sqrt{x^2+2x+4}}dx=... |
1,842,365 | <p>I recently got acquainted with a theorem:</p>
<p>If $f(x)$ is a periodic function with period $P$, then $f(ax+b)$ is periodic with period $\dfrac{P}{a}$ , $a>0$.</p>
<p>I am having a difficulty in understanding this theorem. Does this theorem mean that $f(ax+b)=f(ax+b+ \dfrac{P}{a})$? </p>
<p>If the above mean... | Gordon | 169,372 | <p>We need only show that, for any Borel set $A \in \mathbb{R}$,
\begin{align*}
\int_{Z_1+Z_2 \in A} Z_1 dP = \int_{Z_1+Z_2 \in A} Z_2 dP.
\end{align*}
We denote by $F$ the common cumulative distribution function of $Z_1$ and $Z_2$. Then, from the independence assumption,
\begin{align*}
\int_{Z_1+Z_2 \in A} Z_1 dP &am... |
1,504,214 | <p>When Mr. and Mrs. Smith took the airplane, they had together 94 pounds of baggage.
He paid 1.50 and she paid 2.00 for excess weight. If Mr. Smith made the trip by himself with the combined baggage of both of them, he would have to pay $13.50. How many pounds of baggage can one person take along without being charged... | Tavasanis | 39,690 | <p>An exclusively algebraic solution follows. Generally, the cost a passenger should pay if he/she has excess baggage, is $$c=(w_t-w_f)\; q,$$where $w_t$ is the weight the passanger shows over the counter, $w_f$ is the maximum free weight, and $q$ is the amount per pound in excess.</p>
<p>The couple showed 94 pounds. ... |
79,270 | <p>For integer $n\ge 3$, consider the graph on the set of all even vertices of the $n$-dimensional hypercube $\{0,1\}^n$ in which two vertices are adjacent whenever they differ in exactly two coordinates. This is an $(n(n-1)/2)$-regular graph on $2^{n-1}$ vertices. Is there any standard name / notation for this graph? ... | Zack Wolske | 18,086 | <p>Conway & Sloane's "Sphere Packings, Lattices and Groups" references Coxeter's "Regular Polytopes" for the phrase "halfcube", but Coxeter only uses the notation $h\Pi_n$, saying $h$ can be taken to stand for half- or hemi-, for an arbitrary polytope $\Pi_n$ {$p, q, \ldots, w$} with even $p$ (in your case, {$4,3,3... |
1,255,803 | <p>My understanding is that the thesis is essentially a <em>definition</em> of the term "computable" to mean something that is computable on a Turing Machine.</p>
<p>Is this really all there is to it? If so, what makes this definition so important? What makes this definition so significant as to warrant having it's ow... | Christopher | 73,985 | <p>The idea is that we have an informal notion of "computable" - that is, "something that can be computed". (This is explicitly not a precise definition). We also have a formal definition of "computable", that is, "computable by a Turing machine". The Church-Turing thesis is that these two notions coincide, that is, an... |
1,189,814 | <p>Can a finitely generated module $M$ over a commutative ring have $\operatorname{Ann}(x) \neq 0$ for all $x \in M$ while $\operatorname{Ann}(M) = 0$?</p>
<p>It's not difficult to show that there is no such module if the ring is a integral domain. For general, I guess the answer is yes. But I failed to find a desired... | user26857 | 121,097 | <p>Let $R$ be a UFD which is not a PID, e.g. $R=\mathbb Z[X]$, and $M=\bigoplus_{p\text{ prime}} R/(p)$. Note that every non-invertible element of $R$ is a zero-divisor on $M$. Let $I=(p_1,p_2)$ with $p_1,p_2$ primes such that $I\ne R$. Since $I$ does not contain invertible elements, every element of $I$ is a zero-divi... |
1,477,871 | <p>If the space $X$ is banach , then I want to show that any linear map $T:X \to X$ is continuous iff the null space is closed. I could show that if $T$ is continuous then the null space is closed. But I am unable to prove the converse. Any hints are appreciated. Thanks</p>
| gerw | 58,577 | <p>This is not true. See the $T$ in <a href="https://math.stackexchange.com/a/426494/58577">this answer</a>. Since this $T$ is injective, its kernel is $\{0\}$.</p>
<p>Note that a similar assertion is true for linear functionals $T : X \to \mathbb{R}$.</p>
|
1,516,450 | <p>When finding the Pythagorean triple where $a+b+c=1000$,
Wolfram alpha shows me that $a< -500\left(\sqrt{2} - 2\right)$</p>
<p>When I input $a^2+b^2=c^2, a<b<c$ and $a+b+c=1000$</p>
<p>How does wolfram arrive at that inequality:</p>
<p>$a< -500\left(\sqrt{2} - 2\right)$</p>
<p>Here is the link: <a hre... | Brian Tung | 224,454 | <p>For the inequality, we observe that since $b > a$, we must have</p>
<p>$$
c^2 = a^2+b^2 > 2a^2
$$</p>
<p>or</p>
<p>$$
c > a\sqrt{2}
$$</p>
<p>Thus</p>
<p>$$
1000 = a+b+c > a+a+a\sqrt{2} = a(2+\sqrt{2})
$$</p>
<p>or</p>
<p>$$
a < \frac{1000}{2+\sqrt{2}} = \frac{1000(2-\sqrt{2})}{2^2-(\sqrt{2})^2... |
1,516,450 | <p>When finding the Pythagorean triple where $a+b+c=1000$,
Wolfram alpha shows me that $a< -500\left(\sqrt{2} - 2\right)$</p>
<p>When I input $a^2+b^2=c^2, a<b<c$ and $a+b+c=1000$</p>
<p>How does wolfram arrive at that inequality:</p>
<p>$a< -500\left(\sqrt{2} - 2\right)$</p>
<p>Here is the link: <a hre... | poetasis | 546,655 | <p>Let <span class="math-container">$(A+B+C)=P=(m^2-n^2 )+2mn+(m^2+n^2 )=2m^2+2mn$</span></p>
<p><span class="math-container">$$\text{We solve for }\\ n=\frac{P-2m^2}{2m}\text{ where }\biggl\lceil\frac{\sqrt{P}}{2}\space\biggr\rceil\le m\le\biggl\lfloor\sqrt\frac{P}{2}\biggr\rfloor$$</span></p>
<p>Given <span class="m... |
2,859,463 | <blockquote>
<p>Prove or disprove. All four vertices of every regular tetrahedron in $ \mathbb{R}^3$ have at least two irrational coordinates.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/hYrWv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hYrWv.png" alt="enter image description here"... | Ross Millikan | 1,827 | <p>You can place one of the vertices at the origin. A second one can be at $(1,0,0)$</p>
|
3,831,198 | <p>Suppose that <span class="math-container">$100$</span>kg of a radioactive substance decays to <span class="math-container">$80$</span>kg in <span class="math-container">$20$</span> years.</p>
<p>a) Find the half-life of the substance (round to the nearest year).</p>
<p>b)Write down a function <span class="math-conta... | user577215664 | 475,762 | <p><strong>Hint:</strong>
You have the radioactive decay law:
<span class="math-container">$$N(t)=N_0e^{-\lambda t}$$</span>
You have <span class="math-container">$N=80$</span> the time <span class="math-container">$t$</span> and <span class="math-container">$N_0=100$</span> deduce <span class="math-container">$\lambda... |
3,757,222 | <p>Let <span class="math-container">$n_{1}, n_{2}, ... n_{k} $</span> be a sequence of k consecutive odd integers. If <span class="math-container">$n_{1} + n_{2} + n_{3} = p^3$</span> and <span class="math-container">$n_{k} + n_{k-1} + n_{k-2} + n_{k-3} + n_{k-4} = q^4$</span> where both p and q are prime, what is k?<... | fisura filozofica | 356,159 | <p>You got it to <span class="math-container">$3n_1+6=p^3$</span> but how about <span class="math-container">$3(n_1+2)=p p^2$</span> and <span class="math-container">$p$</span> is prime so <span class="math-container">$p=3$</span>. Using this you can find <span class="math-container">$n_1$</span> then <span class="math... |
265,635 | <p>I'm trying to create a user-defined function that computes the equivalent resistance of <span class="math-container">$n$</span> resistors in parallel.</p>
<p>As we know, such formula is:</p>
<p><span class="math-container">$R_\text{eq.p} = \dfrac{1}{\displaystyle\sum_{k=1}^{n} \dfrac{1}{R_k}} = \left( \displaystyle\... | Syed | 81,355 | <pre><code>Clear[Rp]
Rp[rin_List] := Module[{r},
r = DeleteCases[rin, \[Infinity]];
If[Total[r] === 0
, \[Infinity]
, Times @@ r/Total[Times @@@ Subsets[r, {Length@r - 1}]]
]
]
testCases = {{4 k, 4 k}, {Quantity[6, "KiloOhms"],
Quantity[4, "KiloOhms"]}, {1, 2, 3}, {r1, r2}, {r1... |
4,020,464 | <p>So <span class="math-container">$Ax = λx$</span>;</p>
<p><span class="math-container">$A(Ax) = λ(Ax) \to (A^2)x = (λ^2)x$</span></p>
<p>I kind of dont know how to get the <span class="math-container">$1$</span> and <span class="math-container">$I$</span> here...</p>
<p>Any help is appreciated</p>
| Vons | 274,987 | <p>Let's look at the pieces</p>
<p><span class="math-container">$$\begin{split}X&\sim N(\mu_1,\sigma_1^2)\\
Y&\sim N(\mu_2,\sigma_2^2)\end{split}$$</span></p>
<p>Then what the book means by <span class="math-container">$\bar X-\bar Y$</span> is a sum of m+n independent normal random variables is that <span clas... |
1,691,605 | <p>I've been given the following definition:</p>
<blockquote>
<p>For a THMC with one step transition matrix $\mathbf{P}$, the row vector $\mathbf{\pi}$ with elements $(\pi_{i})_{i \in S}$ (where $S$ is the state space) is a stationary distribution iff $\mathbf{\pi \; P} = \mathbf{\pi}$</p>
</blockquote>
<p>However,... | frog | 84,997 | <p>Using l'Hospitals rule you get
$$
\lim_{x\to a}\frac{a^{a^x}a^x(\ln a)^2-a^{x^a}ax^{a-1}\ln a}{a^x\ln a - ax^{a-1}}=a^{a^a}\ln a.
$$
EDIT: If you're looking for method that does not involve taking derivatives, I suggest the following:
$$\lim_{x\to a}\frac{a^{a^x}-a^{x^a}}{a^x-x^a}=\lim_{x\to a}a^{a^x}\frac{1-a^{x^a-... |
1,691,605 | <p>I've been given the following definition:</p>
<blockquote>
<p>For a THMC with one step transition matrix $\mathbf{P}$, the row vector $\mathbf{\pi}$ with elements $(\pi_{i})_{i \in S}$ (where $S$ is the state space) is a stationary distribution iff $\mathbf{\pi \; P} = \mathbf{\pi}$</p>
</blockquote>
<p>However,... | Fnacool | 318,321 | <p>Let $f(t) = a^t=e^{(\ln a) t}$. Now $f'(t) = (\ln a) (a^t)$. </p>
<p>By the mean value theorem, </p>
<p>$$\frac{f(t)-f(s) }{ t-s} = f'(c)$$</p>
<p>for some intermediate $c$.</p>
<p>In our case $t$ and $s$ are functions of $x$, satisfying $\lim_{x\to a} t(x) =\lim_{x\to a} s(x) = a^a$. Therefore, the intermediate... |
546,123 | <p>Let <span class="math-container">$X$</span> be any uncountable set with the cofinite topology. Is this space first countable?</p>
<p>I don't think so because it seems that there must be an uncountable number of neighborhoods for each <span class="math-container">$ x \in X$</span>. But I am not sure if this is true.... | Ari Royce Hidayat | 435,467 | <p><em>(Just a rewrite of an already excellence answer above and its comment.)</em></p>
<p>Cofinite topology is not first countable. To prove it, we show that some point of <span class="math-container">$X$</span> does not have a countable local base.</p>
<p>Let <span class="math-container">$x \in X$</span>, and suppose... |
4,562,451 | <p>I had this maths question:</p>
<blockquote>
<p>Given that <span class="math-container">$$8\sqrt{p} = q\sqrt{80}$$</span> where <span class="math-container">$p$</span> is prime, find the value of <span class="math-container">$p$</span> and the value of <span class="math-container">$q$</span></p>
</blockquote>
<p>I di... | Oscar Lanzi | 248,217 | <p>Clearly <span class="math-container">$4p$</span> has to have a prime factor of <span class="math-container">$5$</span> but <span class="math-container">$4$</span> does not have that factor. So the prime <span class="math-container">$p$</span> has to have a factor of <span class="math-container">$5$</span>, therefore... |
4,562,451 | <p>I had this maths question:</p>
<blockquote>
<p>Given that <span class="math-container">$$8\sqrt{p} = q\sqrt{80}$$</span> where <span class="math-container">$p$</span> is prime, find the value of <span class="math-container">$p$</span> and the value of <span class="math-container">$q$</span></p>
</blockquote>
<p>I di... | Nicolino | 415,177 | <p>The answer is very different depending on whether q is required to be an integer or not. Let's look at both cases, starting from the equation <span class="math-container">$4p = 5q^2$</span> that you derived.</p>
<p>If q is restricted to the integers, then that means <span class="math-container">$5q^2$</span> is divi... |
2,417,542 | <p>$$\sum_{i,j}{n\brack i+j}\binom{i+j}i$$
Does this have a combinatorial interpretation? I don't see how to use Stirling numbers of the first kind in interpretations. I know that the answer is $(n+1)!$ , but the original question didn't provide it.</p>
| Behnam Esmayli | 283,487 | <p>To find the effect of this matrix on unit vector $\vec{I}$ multiply the matrix into $(1,0)^T$. So, $\vec{i}$ goes to $(o,-1)=-\vec{j}$. Thus a rotation of $-\pi /2$ or $3\pi /2$. Which one? Look at effect on $\vec{j}$. Multiply into $(0,1)$... You'll see that $-\pi/2$ is the right rotation whose matrix is the one gi... |
4,035,300 | <p>If someone could just do a very basic walkthrough on how you would go about answering this question it would be greatly appreciated as I'm practising for an exam!</p>
<p>'''</p>
<p>When a company bids for contracts it estimates the probability of winning each contract is <span class="math-container">$0.18$</span>, i... | Neat Math | 843,178 | <p>Let <span class="math-container">$f(x)=x^3-x^2+2x-1, g(x)=x^3+x^2-1$</span>. We have</p>
<p><span class="math-container">$$-g(x)\cdot g(-x)=-(x^3+x^2-1)((-x)^3+(-x)^2-1)\\
=(x^3+x^2-1)(x^3-x^2+1) = x^6 - (x^2-1)^2\\
=x^6-x^4+2x^2-1=f(x^2)$$</span></p>
<p>Now if <span class="math-container">$g(x)=(x-r)(x-s)(x-t)$</sp... |
1,724,419 | <p>I can create a large collection of normalized real valued $n$-dimensional vectors from some random process which I hypothesis should be equidistributed on the unit sphere. I would like to test this hypothesis.</p>
<ul>
<li>What is a good way numerically to test if vectors are equidistributed on the unit sphere? I ... | G Cab | 317,234 | <p>I would proceed on the basis that a (hollow) sphere with $N$ mass = $1$ points uniformly distributed shall have mass-centre (1st moment) =$0$, moment of inertia (2nd moment) = $\rho (n)N$, around any ax.
Where $\rho (n) = 2/3$ in the case $n=3$, while for the n-dimensional sphere in general it shall be calculated .... |
2,373,175 | <p>It is probably an easy solution to this problem but I am either too overwhelmed already or not smart enough. Please help me out with this problem from the last year test</p>
<p><a href="https://i.stack.imgur.com/rzQWv.jpg" rel="nofollow noreferrer">Here is the problem</a></p>
| farruhota | 425,072 | <p>Note:
$$
\begin{array}{c|l}
n & \text{# of perfect squares} \\
\hline
1 & 1 \\
2 & 1 \\
3 & 1 \\
4 & 2 \\
5 & 2 \\
\cdot & \cdot \\
8 & 2 \\
9 & 3 \\
10 & 3 \\
\cdot & \cdot \\
15 & 3 \\
16 & 4 \\
\cdot & \cdot \\
n & [\sqrt{n}]
\end{array}
$$
where $[n]$ ... |
6,831 | <p>I would like for the autocomplete feature to search through contexts, for example if I have a symbol named A`B`C`MyFunction, when I type A` and press "cmd + shift + k" it will complete it.</p>
<p><em>Edit</em></p>
<p>To be clear, I don't want to have to type the path because it's usually very long, and I don't wan... | Brett Champion | 69 | <blockquote>
<p><em><strong>This is obsolete in Mathematica 9, which automatically includes contexts in completions.</strong></em></p>
<p><em><strong>Undocumented function: use at your own risk, subject to change in future versions, etc....</strong></em></p>
</blockquote>
<hr />
<p>The function you're interested in is ... |
1,859,178 | <p>Let parabola $\Gamma_{1}:$$y^2=4x$,and hyperbolic curve $\Gamma_{2}$: $x^2-y^2=1$.it is well known this two is symmetric.so the two point $A$ and $B$ about $x$ axial symmetry,or mean $x_{A}=x_{B}$.see figure,</p>
<p><a href="https://i.stack.imgur.com/wCPAN.png" rel="nofollow noreferrer"><img src="https://i.stack... | Peter G. Chang | 339,525 | <p>The confusion arises from ignoring an implicit condition of the first equation, $y^2 = 4x$. Since $y^2 \geq 0$, we have the condition $x \geq 0$.</p>
<p>Thus of the two solutions for $x$ you have, the positive $x = 2 + \sqrt{5}$ is the only solution.</p>
<p>We see that there are two values of $y$ that satisfy the ... |
1,859,178 | <p>Let parabola $\Gamma_{1}:$$y^2=4x$,and hyperbolic curve $\Gamma_{2}$: $x^2-y^2=1$.it is well known this two is symmetric.so the two point $A$ and $B$ about $x$ axial symmetry,or mean $x_{A}=x_{B}$.see figure,</p>
<p><a href="https://i.stack.imgur.com/wCPAN.png" rel="nofollow noreferrer"><img src="https://i.stack... | John Molokach | 90,422 | <p>$x_B$ is an extraneous solution. Try finding the value of $y$ in $y^2=4x_B$ and you'll find nonreal answers. </p>
|
1,859,178 | <p>Let parabola $\Gamma_{1}:$$y^2=4x$,and hyperbolic curve $\Gamma_{2}$: $x^2-y^2=1$.it is well known this two is symmetric.so the two point $A$ and $B$ about $x$ axial symmetry,or mean $x_{A}=x_{B}$.see figure,</p>
<p><a href="https://i.stack.imgur.com/wCPAN.png" rel="nofollow noreferrer"><img src="https://i.stack... | DRF | 176,997 | <p>Ahh your edit clarified your mistake. While $x_A=x_B$ if you are looking at the $x$-coordinates of the two points you have in your picture these $x_A$ and $x_B$ don't correspond to the two solutions of the quadratic you found.</p>
<p>One of those solutions $x=2-\sqrt{5}$ does not solve the pair of equations since a... |
1,025,321 | <p>$\ln(1+xy) = xy$</p>
<p>When I try to implicitly differentiate this I get</p>
<p>$\frac{1}{1+xy}(y + xy')$ = (y + xy')</p>
<p>At which point the $(y + xy')$ terms cancel out, leaving no $y'$ to solve for.</p>
<p>However, the answer to this is $-\frac{y}{x}$... How do you get this?</p>
| C-star-W-star | 79,762 | <p>Not every smooth function induces a smooth map:
$$\Phi:\mathbb{B}\to\mathbb{R}:\quad\varphi(|x|):=\frac{1}{1-|x|}$$
Just have a careful look at its diagram:</p>
<p><img src="https://i.stack.imgur.com/p4Ck8.gif" alt="Diffeomorphic Ball"></p>
<p><em>(Note that it is not even differentiable at zero!)</em></p>
<p>The... |
1,025,321 | <p>$\ln(1+xy) = xy$</p>
<p>When I try to implicitly differentiate this I get</p>
<p>$\frac{1}{1+xy}(y + xy')$ = (y + xy')</p>
<p>At which point the $(y + xy')$ terms cancel out, leaving no $y'$ to solve for.</p>
<p>However, the answer to this is $-\frac{y}{x}$... How do you get this?</p>
| C-star-W-star | 79,762 | <p><strong>Standard Approach</strong></p>
<p>Consider the identification:
$$\Phi:\mathbb{B}^n\to\mathbb{R}^n:\Phi(x):=\frac{x}{\sqrt{1-\|x\|_E^2}}$$
Its inverse is given explicitely by:
$$\Psi:\mathbb{R}^n\to\mathbb{B}^n:\Psi(y):=\frac{y}{\sqrt{1+\|y\|_E^2}}$$
The argument of the roots never vanish:
$$1-\|x\|_E^2\neq0... |
355,489 | <p>What are suggestions for reducing the transmission rate of the current epidemics?</p>
<p>In summary, my best one so far is (once we are down to the stay home rule) to discretize time, i.e., to introduce the following rule for the general populace not directly involved in necessary services:</p>
<p><em>If members o... | Gil Kalai | 1,532 | <p>Mathematically speaking it looks that if we split the categories <span class="math-container">$C_i$</span> into subcategories with zero (or small) cross transmission edges then this suppress the exponential growth at some scale. (And this is also the case for other geometric limitation on the graph of possible trans... |
808,144 | <p>Here is a fun looking one some may enjoy. </p>
<p>Show that:</p>
<p>$$\int_{0}^{1}\log\left(\frac{x^{2}+2x\cos(a)+1}{x^{2}-2x\cos(a)+1}\right)\cdot \frac{1}{x}dx=\frac{\pi^{2}}{2}-\pi a$$</p>
| Tolaso | 203,035 | <p>I understand that this an old question but I would like to share my 2 cents. First of all we recall the following Fourier identities:</p>
<p><strong>Lemma 1:</strong> Let <span class="math-container">$x \in [-\pi, \pi]$</span> then </p>
<p><span class="math-container">$$\sum_{n=1}^{\infty} \frac{\sin nx}{n} = \fra... |
2,477,107 | <p>Okay so I have to prove this. I can write that if 2 divides n and 7 divides n, then there must be integers k and m such that
$2*k=n$
and
$7*m=n$</p>
<p>So $14*k*m=n^2$</p>
<p>But what to do after that?</p>
<p>If I say that then 14 divides $n^2$, I get bit of a circular argument, but if I write that n divides $14*... | Math Lover | 348,257 | <p>Following from what you have written, $$n = 2k=7m \implies k=\frac{7m}{2}.$$
Since $k$ is an integer and $\gcd(2,7)=1$, $m/2$ must be an integer; i.e., $m/2=r \implies m=2r$, where $r$ is an integer. Therefore,
$$n=7m=7\times 2r = 14 r.$$
Q.E.D.</p>
|
2,875 | <p>I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group <code>G</code> being <code>SL(2, R)</code>, can be completely described and that there is a discrete and continuous part of the spectrum of <code>L^2(G)</code>.</p>
<ol>
<li>How are those ... | David Bar Moshe | 1,059 | <p>I just want to elaborate more on questions 3. and 4. I'll consider the locally isomorphic groups SU(1,1) of SL2(R) and SU(2) of SO(3)</p>
<p>There is an analogy between the discrete series of SU(1,1) and the unitary irreps of SO(3). Both have holomorphic representations on the group's orbit on the flag manifold S^2... |
228,224 | <p>I am faced with the following expression</p>
<p><span class="math-container">$$
-\frac{(1 - a x^{2})^{b/2}}{b} {{}_2F_1} (1, \frac{b}{2}; \frac{c}{2}; 1 - a x^{2}) = - p t
$$</span></p>
<p>where <span class="math-container">$ a, b, c, p $</span> are constant values. Also, <span class="math-container">$ {{}_2F_1} $</... | cvgmt | 72,111 | <p>Since we can solve t ,so we can also use <code>ParametricPlot</code></p>
<pre><code>Clear["`*"];
a = 1;
b = 2;
c = 3;
t = ((1 - a x^2)^(b/2)/b) Hypergeometric2F1[1, b/2, c/2, 1 - a x^2]/p;
ParametricPlot[
Table[{t, x}, {p, {1, 2, 3}}] // Evaluate, {x, -2, 2}, {t, -1, 2},
Axes -> False, FrameLabel -&g... |
3,700,814 | <p>Consider the vector space <span class="math-container">$V=C(\mathbb{R},\mathbb{R})$</span> and <span class="math-container">$V\ni U=\{f\in C(\mathbb{R},\mathbb{R}) |f(0)=0\}$</span>. I want to find a complement of <span class="math-container">$U$</span>, such that <span class="math-container">$V=U\oplus W$</span>. T... | Digitallis | 741,526 | <p>As others have pointed out the <span class="math-container">$W$</span> you defined is not a subspace. It also looks like you think the choice of <span class="math-container">$W$</span> should always be <span class="math-container">$W = U^c \cup \{ f_0\}$</span> that is not the case, precisely because this will not ... |
1,607,190 | <p>Prove by induction that $8^{n} − 1$ for any positive integer $n$ is divisible by $7$. </p>
<p>Hint: It is easy to represent divisibility by $7$ in the following way: $8^{n} − 1 = 7 \cdot k$ where k is a positive integer.</p>
<p>This question confused me because I think the hint isn't true. If $n = 1$ and $k = 2$ f... | Harish Chandra Rajpoot | 210,295 | <p>Notice the following steps of mathematical induction, </p>
<ol>
<li><p>Setting $n=1$, $$8^1-1=7$$ above number is divisible by $7$ for $n=1$</p></li>
<li><p>assume that $8^n-1$ is divisible by $7$ for $n=k$ then $$8^k-1=7m$$
or $$8^k=7m+1\tag 1$$
where, $m$ is some integer </p></li>
<li><p>setting $n=k+1$, $$8^{k+1... |
1,016,884 | <p>Four friends, Andrew, Bob, Chris and David, all have different heights. The sum of their heights is 670 cm.
Andrew is 8cm taller than Chris and Bob is 4cm shorter than David.
The sum of the heights of the tallest and shortest of the friends is 2cm more than the sum of the heights of the other two.
Find the height of... | advocateofnone | 77,146 | <p>Let the heights of andrew bob chris and david be $a,b,c,d$. Then
$$a+b+c+d=670$$
$$a=c+8$$
$$d=b+4$$</p>
<p>This tells us that either $a$ or $d$ is the tallest height and $c$ or $b$ the shortest. So there are in total $4$ options try each of them and see if all the constraints are satisfied. For example if andrew i... |
4,150,776 | <p>Let <span class="math-container">$(a_n)_{n=1}^\infty$</span> Let be a positive, increasing, and unbounded sequence. Prove that the series:</p>
<p><span class="math-container">$$\sum_{n=1}^\infty\left(\frac{1}{a_{2n-1}}-\frac{1}{a_{2n}}\right)$$</span></p>
<p>convergent.</p>
<hr />
<p>We know that since <span class=... | Thomas Andrews | 7,933 | <p>Hint: Let</p>
<p><span class="math-container">$$b_n=\frac1{a_{n-1}}-\frac1{a_{n}}$$</span></p>
<p>Each <span class="math-container">$b_n$</span> is non-negative, since <span class="math-container">$a_n$</span> is increasing.</p>
<p>Then your series is: <span class="math-container">$$\sum_{n=1}^\infty b_{2n}$$</span>... |
4,118,161 | <p>Hello I am solving the following problem and could use some help.</p>
<p>Let (C[0,1],<span class="math-container">$d_\infty$</span>) be the metric space of continuous functions on [0,1] where the distance function is defined by</p>
<p>Let <span class="math-container">$d_\infty(f,g)=\sup_{x∈[0,1]}|f(x)−g(x)|. $</span... | mathcounterexamples.net | 187,663 | <p><strong>For (a)</strong></p>
<p><span class="math-container">$f \in \mathcal C^1([0,1], \mathbb R) =V$</span> is a fixed point if and only if</p>
<p><span class="math-container">$$f(x) = \int_0^x f(t) \ dt$$</span> for all <span class="math-container">$x \in [0,1]$</span>. As <span class="math-container">$f$</span> ... |
4,118,161 | <p>Hello I am solving the following problem and could use some help.</p>
<p>Let (C[0,1],<span class="math-container">$d_\infty$</span>) be the metric space of continuous functions on [0,1] where the distance function is defined by</p>
<p>Let <span class="math-container">$d_\infty(f,g)=\sup_{x∈[0,1]}|f(x)−g(x)|. $</span... | Kavi Rama Murthy | 142,385 | <p>Answer for the second part: <span class="math-container">$|T^{2}f(x)|=|\int_0^{x}\int_u^{x}dt du|=|\int_0^{x}(x-u)f(u)du|\leq \|f\|\int_0^{x}f(u)du=\frac 1 2 \|f\|$</span>.</p>
|
405,648 | <p>Is there a sensible characterization of groups <span class="math-container">$G$</span> with the following property?</p>
<blockquote>
<p>Every extension of groups <span class="math-container">$1\to G\to H\to K\to 1$</span> is split.</p>
</blockquote>
<p>A complete group <span class="math-container">$G$</span> has tha... | YCor | 14,094 | <blockquote>
<p><strong>Proposition.</strong> Given a group <span class="math-container">$G$</span>, this happens (every exact sequence <span class="math-container">$1\to G\to H\to H/G\to 1$</span> splits) iff <span class="math-container">$G$</span> has a trivial center and <span class="math-container">$1\to G\to \math... |
2,889,929 | <p>I have following function:</p>
<p>$$f(x)=x^2\cdot\left({\sin{\frac 1 x}}\right)^2$$</p>
<p>I want to find the limit of the function for $x\rightarrow0^\pm$. First I analyze $\frac 1 x$:</p>
<ul>
<li>$\frac {1}{x}\rightarrow +\infty$ for $x\rightarrow0^+$</li>
</ul>
<p>but the $\sin$ of infinity does not exist. T... | Jan | 254,447 | <p>I am not going to be giving you the answer, but I will hopefully give you the means to find it yourself.</p>
<p>So we have three shooters who can all hit or miss the target. So there are many possible outcomes of the shooting. For example, A can miss, B can hit and C can miss. Clearly this is different from A hitti... |
1,550,923 | <p>I know of two applications of vector spaces over $\mathbb Q$ to problems posed by people not specifically interested in vector spaces over $\mathbb Q$:</p>
<ul>
<li>Hilbert's third problem; and</li>
<li>The Buckingham pi theorem.</li>
</ul>
<p>What others are there?</p>
| user293794 | 293,794 | <p>One nice application is proving the following theorem:</p>
<blockquote>
<p>A rectangle
R
with side lengths
$1$
and
$x$
, where
$x$
is irrational, cannot be “tiled” by finitely many squares (so that the squares
have disjoint interiors and cover all of
R
).</p>
</blockquote>
<p>The proof can ... |
4,576,868 | <p>I was provided the following generating function, and was unsure how to use it. I have never seen an example where the function “involved” itself.
The generating function is
<span class="math-container">$F(z)^8$</span>
Where
<span class="math-container">$$F(z)=z+z^6 F(z)^5+z^{11} F(z)^{10}+z^{16} F(z)^{15}+z^{21} F(... | Leucippus | 148,155 | <p>A method that will take a while but gives the first few terms for the equation
<span class="math-container">$$ F(x) = x + x^6 F(x)^5 + x^{11} F(x)^{10} + x^{16} F(x)^{15} + x^{21} F(x)^{20} $$</span>
is to let <span class="math-container">$F(x) = a_{0} + a_{1} \, x + a_{2} \, x^2 + \cdots$</span>.</p>
<p>First notic... |
3,003,672 | <p>Say I have an infinte 2D grid (ex. a procedurally generated world) and I want to get a unique number for each integer coordinate pair. How would I accomplish this?</p>
<p>My idea is to use a square spiral, but I cant find a way to make a formula for the unique number other than an algorythm that just goes in a squa... | Kamil Maciorowski | 331,040 | <p>Other answers state how to convert integers to naturals, I won't repeat this step. Let's suppose you have two naturals, e.g.:</p>
<p><span class="math-container">$$ 123 $$</span>
<span class="math-container">$$ 98765 $$</span></p>
<p>Add leading zeros to obtain equal number of digits:</p>
<p><span class="math-con... |
3,807,665 | <p>Let <span class="math-container">$ V $</span> a finite dimensional vector space and <span class="math-container">$ V^* $</span> the dual space. Further let be <span class="math-container">$ B=(v_1,...,v_n) $</span> a base of <span class="math-container">$ V $</span> and <span class="math-container">$ B^*=(v_1^*,...,... | Peter Franek | 62,009 | <p>This seems too complicated. Just note that <span class="math-container">$v_i^*(\sum_j a_j v_j) = v_i^*(\sum_j b_j v_j)$</span> implies <span class="math-container">$a_i=b_i$</span>.</p>
|
3,807,665 | <p>Let <span class="math-container">$ V $</span> a finite dimensional vector space and <span class="math-container">$ V^* $</span> the dual space. Further let be <span class="math-container">$ B=(v_1,...,v_n) $</span> a base of <span class="math-container">$ V $</span> and <span class="math-container">$ B^*=(v_1^*,...,... | Kevin Aquino | 804,948 | <p>It can be useful to note that if <span class="math-container">$\mathcal{B}= (v_{1}, \ldots, v_{n})$</span> is a basis for <span class="math-container">$V$</span> and <span class="math-container">$\mathcal{B}^{*}= (\varphi_{1}, \ldots, \varphi_{n})$</span> is the corresponding dual basis, then for any <span class="ma... |
3,807,665 | <p>Let <span class="math-container">$ V $</span> a finite dimensional vector space and <span class="math-container">$ V^* $</span> the dual space. Further let be <span class="math-container">$ B=(v_1,...,v_n) $</span> a base of <span class="math-container">$ V $</span> and <span class="math-container">$ B^*=(v_1^*,...,... | janmarqz | 74,166 | <p>Choose each <span class="math-container">$v^*_i$</span> and apply <span class="math-container">$v^*_i(v)=v^*_i(w)$</span> then <span class="math-container">$a_i=b_i$</span>, so <span class="math-container">$v=w$</span>.</p>
|
96,437 | <p>In Mathematica 9.0, the documentation for the Curl function states that in n-dimensions "the resulting curl is an array with depth n-k-1 of dimensions". Accordingly, if a 2-dimensional array is feeded in the Curl function in 3-D space, it returns a scalar value. </p>
<p>However, it does not agree with the definitio... | Kagaratsch | 5,517 | <p>Assuming that by $e_{ijk}$ you mean the totally anti-symmetric tensor $\epsilon_{ijk}$, the expression you cite only is valid in three dimensions (since only in three dimensions $\epsilon_{ijk}$ has three indices). With the above assumption, the equation you provide can be implemented as follows</p>
<pre><code>twot... |
4,380,274 | <p>Indefinite integral is pretty easy to solve, I did it by substitution and I'm pretty sure it can be done relatively easy via integration by parts.
The problem are boundaries.</p>
<p>After substitution <span class="math-container">$arcsin x=t$</span> we get</p>
<p><span class="math-container">$$\int_0^\frac{\pi}{2} \... | Quanto | 686,284 | <p>You may not split the integral into two divergent integrals. Instead, integrate as a whole as shown below</p>
<p><span class="math-container">\begin{align}\int_0^\frac{\pi}{2} \frac{(\sin t-t)\cos t}{\sin^3t}dt
&=\frac12\int_0^\frac{\pi}{2} (t-\sin t)d\left(\frac{1}{\sin^2t}\right)\\
&= \frac12\frac{t-\sin t... |
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