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 |
|---|---|---|---|---|
2,437,026 | <p>Suppose that:</p>
<p>$$
X \sim Bern(p)
$$</p>
<p>Then, intuitively $X^2 = X \sim Bern(p)$. However, when I try to think of it logically, it doesn't make any sense. </p>
<p>As an example, $X$ is $1$ with probability $p$ and $0$ with probability $1-p$. Then, $X^2 = X\cdot X$ is $1$ only when both $X$'s are $1$, whi... | Graham Kemp | 135,106 | <blockquote>
<p>As an example, $X$ is $1$ with probability $p$ and $0$ with probability $1-p$. Then, $X^2 = X\cdot X$ is $1$ only when both $X$'s are $1$, which occur with probability $p^2$, and so it doesn't seem like $X^2 = X$. Can someone tell me what is wrong here?</p>
</blockquote>
<p>There are not "both $X$'s"... |
1,133,817 | <p>Using Legendre polynomial generating function
\begin{equation}
\sum_{n=0}^\infty P_n (x) t^n = \frac{1}{\sqrt{(1-2xt+t^2)}}
\end{equation}
Or $$ P_n(x)=\frac{1}{2^n n!} \frac{d^n}{dx^n} [(x^2-1)^n] $$ </p>
<p>Show$$ P_{2n}(0)=\frac{(-1)^n (2n)!}{(4)^n (n!)^2} $$
And $$ P_{2n+1}(0)=0$$</p>
<p>I expressed ... | Joe | 107,639 | <p>Assume $\lim_{x\to0}f(x)$ exists.
If by contradiction $\lim_{x\to0}f(x)=a\neq0$ then $\lim_{x\to0}\frac{f(x)}{x}=\infty$.</p>
|
238,076 | <p>I was thinking about this when flying on the plane which was approaching and slowing down.</p>
<p>Assume an object is approaching its target which is at a certain initial distance d at time t0.</p>
<p>It starts at a speed that will allow it to reach the target in exactly one hour (e.g. d=100km, it starts at 100 km... | Ross Millikan | 1,827 | <p>Until the final speed, if $x$ is the distance to the target, we have $\frac {dx}{dt}=-x$. This is solved by $x=x_0e^{-t}$ where $x_0$ is the distance at time $0$. Then we look how long it takes to get to $10$, so we solve $10=100e^{-t}, -t=\ln \frac 1{10}, t=\ln 10$. Then, as you say, it takes an hour to get ther... |
2,510,723 | <p>Does the series $\displaystyle\sum_{n=1}^{\infty} \dfrac{19+(n+5)!}{(n+7)!}$ Converge or diverge</p>
<blockquote>
<p>I tried using the ratio test, but it gave me this which I don't see how you'd be able to solve.
$$\lim_{n\to\infty} |\frac{a_{n+1}}{a_{n}}| = \lim_{n\to\infty}\frac{19+(n+6)!}{19+(n+5)!*(n+8)}$$... | Nosrati | 108,128 | <p>$$\sum_{n=1}^{\infty} \frac{19+(n+5)!}{(n+7)!}=19\sum_{n=1}^{\infty} \frac{1}{(n+7)!}+\sum_{n=1}^{\infty} \frac{1}{(n+7)(n+6)}$$
and
$$ \frac{1}{(n+7)!}<\dfrac{1}{n^2}~~~,~~~\frac{1}{(n+7)(n+6)}<\dfrac{1}{n^2}$$</p>
|
1,624,221 | <p>For the former one, I am aware that if let $F(x)=\int_a^x f(t)dt$, then it also equals $\int_0^x f(t)dt-\int_0^a f(t)dt$, so $F'(x)= f(x)-0=f(x)$. But who can tell me why $\int_0^a f(t)dt$ is $0$?</p>
| Jimmy R. | 128,037 | <p>Since $$\frac{1}{n^2-1}=\frac{1}{(n-1)(n+1)}=\frac12\left(\frac{1}{n-1}-\frac{1}{n+1}\right)$$ as you already have, then write for $N\in \mathbb N$<br>
\begin{align}\sum_{n=2}^{N}\left(\frac{1}{n-1}-\frac{1}{n+1}\right)&=\left(1-\frac13\right)+\left(\frac12-\frac14\right)+\left(\frac13-\frac15\right)+\dots+\left... |
2,125,018 | <blockquote>
<p>You toss a fair coin 3x, events:</p>
<p>A = "first flip H"</p>
<p>B = "second flip T"</p>
<p>C = "all flips H"</p>
<p>D = "at least 2 flips T"</p>
<p><strong>Q:</strong> Which events are independent?</p>
</blockquote>
<p>From the informal def. it is where one doesnt affect th... | L Parker | 410,830 | <p>I retain the relationship $\alpha = \pi - \gamma$ as proposed by JeanMarie; allow me to explain the coordinates of $a$ using 1 variable only: </p>
<p>Let slope of $L_1 = \tan \theta_0$. We have</p>
<p>$$
\begin{align}
a&= (L_1 \cos \theta_0, L_1 \sin\theta_0) \\
b&= (L_3 \cos \theta, L_3 \sin\theta)
\end{a... |
42,881 | <p>In short, my question is the same as my previous one except that everything is now wrapped up in a module.</p>
<p>The relevant code I'm working with is:</p>
<pre><code>getinter[a_, b_, u0_, k_, m_, hbar_, Nu_, Np_, up_] :=
Module[{ekp, ms, LUs, env, eenv, envpart, f, kppart, g, approx,
approx1, papprox, appro... | freddieknets | 631 | <p>You are having a scoping problem here, because inside a <code>Module</code> your variable x will be renamed internally.</p>
<p>Just replace your temporary definition of <code>f</code> with a <code>ReplaceAll</code>. I don't have your definitions for <code>energies</code>, <code>effmass</code>, etc, so I'm not 100% ... |
3,991,572 | <p>I need to solve the following problem:
<span class="math-container">$\lim_{x\to 3}(x-3) \cot{\pi x}$</span>.
Can anyone give me a hint? I have no idea.</p>
| Parcly Taxel | 357,390 | <p>There is no need to cancel terms across both series because of the <span class="math-container">$(3/4)^n$</span>. Evaluating both series individually (using the Maclaurin series of <span class="math-container">$\log(1-x)$</span>) gives
<span class="math-container">$$S=\frac{16}{\log4}-\left(-\frac{33}2+\frac{16}{\lo... |
3,991,572 | <p>I need to solve the following problem:
<span class="math-container">$\lim_{x\to 3}(x-3) \cot{\pi x}$</span>.
Can anyone give me a hint? I have no idea.</p>
| lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>Let <span class="math-container">$\dfrac{7n+32}{n(n+2)}\cdot\left(\dfrac34\right)^n=f(n)-f(n-2)$</span></p>
<p>where <span class="math-container">$f(m)=\dfrac a{m+2}\cdot\left(\dfrac34\right)^m$</span></p>
<p>so that for <span class="math-container">$r\ge4,$</span> <span class="math-container">$$\sum_{n... |
3,812,432 | <p>For <span class="math-container">$a,b,c>0.$</span> Prove<span class="math-container">$:$</span> <span class="math-container">$$4\Big(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2} \Big)+\dfrac{81}{(a+b+c)^2}\geqslant{\dfrac {7(a+b+c)}{abc}}$$</span></p>
<p>My proof is using SOS<span class="math-container">$:$</span... | tkf | 117,974 | <p>Given a string that avoids <span class="math-container">$000,111,222$</span> add <span class="math-container">$i$</span> modulo <span class="math-container">$3$</span> to the <span class="math-container">$i$</span>'th digit to obtain a string that avoids <span class="math-container">$012,120,201$</span>. This is cl... |
179,886 | <p>I have series of values which, by visual inspection, appear to be sums of certain constants, not divisible by each other, with rational weights. I want to convert these sums to vectors of weights for a specific basis vector.</p>
<p>I have an initial solution based on <code>FindInstance</code>, which works reasonabl... | kirma | 3,056 | <p>This is really an extended comment to @DanielLichtblau's answer, touching the subject of <code>WorkingPrecision</code>. I arrived at this completely separately...</p>
<p>Essentially the improvement here is calculating necessary <code>$MaxExtraPrecision</code> on basis of input:</p>
<pre><code>ClearAll[splitSumCoef... |
2,303,795 | <p>So I know that by Euler's homogeneous function theorem $m$ is a positive number, but why is it an integer?
And how to prove that $f$ is polynomial of degree $m$?</p>
| RRL | 148,510 | <p>Note that</p>
<p>$$\int_0^x \sin^2(t +1/n) \, dt = \frac{x}{2} - \frac{1}{4}\sin(2x+2/n) + \frac{1}{4} \sin (2/n),$$</p>
<p>and</p>
<p>$$\int_0^x \sin^2(t) \, dt = \frac{x}{2} - \frac{1}{4}\sin(2x) .$$</p>
<p>Hence,</p>
<p>$$|f_n(x) - f(x)| \leqslant \frac{1}{4}\left|\sin(2x+2/n) - \sin(2x) \right|+ \frac{1}{4}... |
3,131,516 | <p>I would like to know if this differential equation can be transformed into the hypergeometric differential equation</p>
<p><span class="math-container">$ 4 (u-1) u \left((u-1) u \text{$\varphi $1}''(u)+(u-2) \text{$\varphi $1}'(u)\right)+\text{$\varphi $1}(u) \left((u-1) u \omega ^2-u (u+4)+8\right)=0$</span></p>
| Abhinav | 646,522 | <p>Being a high school student I can help you out.
<span class="math-container">$$2x^2+3x-2$$</span>
<span class="math-container">$$=2x^2+4x-x-2$$</span>
<span class="math-container">$$=2x(x+2)-(x+2)$$</span>
<span class="math-container">$$=(2x-1)(x+2)$$</span></p>
|
2,885,754 | <p>I just sat a real analysis exam and this was a question in it that I couldn't answer...</p>
<p>Prove that $\left|e^\frac{-x^2}{2t}-e^\frac{-y^2}{2t}\right| \leq \frac{|x-y|}{t}$ for $x,y \in [-1,1] ,t>0$</p>
<p>I ended up trying to set $f(x,y)=e^\frac{-x^2}{2t}-e^\frac{-y^2}{2t}$, then attempted trying $f(-1,-1... | tmaths | 584,390 | <p>You can write when $|x| \leqslant |y|$ and using triangle inequality for integrals :</p>
<p>\begin{align*}
\left|
e^{-x^2/2t} - e^{-y^2/2t}
\right| = \left| \int_{x}^{y}- \frac{u}{t} e^{-u^2/2t} du\right| &\leqslant \int_{|x|}^{|y|} \frac{|u|}{t} du \\
&\leqslant \int_{|x|}^{|y|} \frac{u}{t} du \\
&= \... |
2,885,754 | <p>I just sat a real analysis exam and this was a question in it that I couldn't answer...</p>
<p>Prove that $\left|e^\frac{-x^2}{2t}-e^\frac{-y^2}{2t}\right| \leq \frac{|x-y|}{t}$ for $x,y \in [-1,1] ,t>0$</p>
<p>I ended up trying to set $f(x,y)=e^\frac{-x^2}{2t}-e^\frac{-y^2}{2t}$, then attempted trying $f(-1,-1... | Rigel | 11,776 | <p>Observe that, by the mean value theorem,
$$
|e^a - e^b| \leq |a-b|,
\qquad \forall a,b \leq 0.
$$
Hence, if $x,y\in [-1,1]$,
$$
\left|
e^{-x^2/2t} - e^{-y^2/2t}
\right| \leq
\left|
{x^2/2t} - {y^2/2t}
\right| \leq
|x+y| \, \frac{|x-y|}{2t} \leq \frac{|x-y|}{t}\,.
$$</p>
|
2,716,585 | <p>Evaluate $\sum_{k=1}^{n-3}\frac{(k+3)!}{(k-1)!}$. </p>
<p>My strategy is defining a generating function,
$$g(x) = \frac{1}{1-x} = 1 + x + x^2...$$
then shifting it so that we get,
$$f(x)=x^4g(x) = \frac{x^4}{1-x}= x^4+x^5+...$$
and then taking the 4th derivative of f(x).
Calculating the fourth derivative is going t... | zwim | 399,263 | <p>If you consider $U_4(k)= k(k+1)(k+2)(k+3)(k+4)$</p>
<p>Then $U_4(k)-U_4(k-1)=k(k+1)(k+2)(k+3)\bigg((k+4)-(k-1)\bigg)=5U_3(k)$</p>
<p>Thus you get a telescoping series and</p>
<p>$$\sum\limits_{k=1}^{n-3}k(k+1)(k+2)(k+3)=\dfrac 15\bigg(U_4(n-3)-U_4(0)\bigg)=\dfrac 15U_4(n-3)\\=\dfrac{(n-3)(n-2)(n-1)(n)(n+1)}5$$</p... |
2,716,585 | <p>Evaluate $\sum_{k=1}^{n-3}\frac{(k+3)!}{(k-1)!}$. </p>
<p>My strategy is defining a generating function,
$$g(x) = \frac{1}{1-x} = 1 + x + x^2...$$
then shifting it so that we get,
$$f(x)=x^4g(x) = \frac{x^4}{1-x}= x^4+x^5+...$$
and then taking the 4th derivative of f(x).
Calculating the fourth derivative is going t... | N. Shales | 259,568 | <p>When you differentiate $g(x)$ $4$ times all powers of $x$ initially less than $4$ disappear anyway. </p>
<p>Then:</p>
<p>$$g^{(4)}(x)=4!(1-x)^{-5}=\sum_{k\ge 0}\frac{(k+4)!}{k!}x^k$$</p>
<p>Operating with $(1-x)^{-1}=1+x+x^2+x^3+\cdots$ on both sides gives a new expansion with terms which are partial sums of the ... |
69,542 | <p>My first question here would fall into the 'ask Johnson' category if there was one (no pressure Bill). I'm interested in constructing a uniformly convex Banach space with conditional structure without using interpolation. The constructions of Ferenczi and Maurey-Rosenthal both use interpolation. </p>
<p>Using ex... | Bill Johnson | 2,554 | <p>Kevin, there are non reflexive spaces with non trivial type--even of type 2. James constructed the first one; his argument is very complicated. Later Pisier-Xu did it much more simply using interpolation between $\ell_1$ and $\ell_\infty$, but using the universal non weakly compact operator instead of the formal i... |
249,074 | <p>I am trying to solve the following problem:</p>
<blockquote>
<p>Show that a unit-speed curve $\gamma$ with nowhere vanishing curvature is a geodesic on the ruled surface $\sigma(u,v)=\gamma(u)+v\delta(u)$, where $\gamma$ is a smooth function of $u$, if and only if $\delta$ is perpendicular to the principal normal... | robjohn | 13,854 | <p>A unit-speed curve $\gamma(u)$ (i.e. parametrized by arc-length) is a geodesic in a surface $S$ iff $\gamma''(u)$ is perpendicular to $S$.</p>
<p>The normal to $\sigma(u,v)=\gamma(u)+v\delta(u)$ is parallel to
$$
\frac{\partial\sigma}{\partial u}\times\frac{\partial\sigma}{\partial v}
=(\gamma'+v\delta')\times\delt... |
1,512,515 | <p>I have tested all the primes up to 50,000,000 and did not find a single prime which satisfies the condition "sum of digits of prime number written in base7 divides by 3". E.g. </p>
<ul>
<li>13 (Base10) = 16 (Base7) --> 7 (sum of digits in base 7)</li>
<li>1021 (Base10) = 2656 (Base7) --> 19</li>
<li>823541 (Base10)... | phuclv | 90,333 | <p>Do you see that in decimal the divisibility of 9 and 3 is checked by <em>divisibility of sum of the digits for 9 or 3</em>? Similarly you can check divisibility of 15, 5 and 3 in hexadecimal, 7 in octal, 3 in base 4 and 2, 3, 6 in base 7 because <strong>this is true for all factors of <span class="math-container">$b... |
772,665 | <p><strong>Question:</strong></p>
<blockquote>
<p>For any $a,b\in \mathbb{N}^{+}$, if $a+b$ is a square number, then $f(a)+f(b)$ is also a square number. Find all such functions.</p>
</blockquote>
<p><strong>My try:</strong> It is clear that the function
$$f(x)=x$$ satisfies the given conditions, since:
$$f(a)+f(... | Community | -1 | <p>It's not a complete answer, but as mentioned in comments, this problem probably missed some restrictions, and so have too many solutions. Thus I decided to answer this question for the case that $f$ have constant value in infinite (or finite by little changes) partition of $\mathbb N$.<br>
I expect another answers f... |
383,063 | <p>I need some help with this exercise:</p>
<p>Suppose $A\subseteq{G}$ is abelian, and $|G:A|$ is a prime power. Show that $G'\lt{G}$</p>
<p>Thank you very much in advance.</p>
| Jack Schmidt | 583 | <p>Hint: Reduce to a simple group and apply theorem 3.9.</p>
<p>Reduction to simple group:</p>
<blockquote class="spoiler">
<p> Suppose by way of contradiction that $G'=G$. If $G$ is not simple, then $G$ has a proper non-identity normal subgroup $N$. If $AN=G$, then $G/N = AN/N \cong A/A\cap N$ is abelian, so $G=G'... |
204,150 | <p>If I had a list of let's say 20 elements, how could I split it into two separate lists that contain every other 5 elements of the initial list?</p>
<p>For example:</p>
<pre><code>list={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
function[list]
(*
{1,2,3,4,5,11,12,13,14,15}
{6,7,8,9,10,16,17,18,19,20}
*)
</... | Alx | 35,574 | <pre><code>Flatten /@ {#[[1 ;; ;; 2]], #[[2 ;; ;; 2]]} &@Partition[list, 5]
</code></pre>
<p>gives desired </p>
<pre><code>{{1, 2, 3, 4, 5, 11, 12, 13, 14, 15}, {6, 7, 8, 9, 10, 16, 17, 18, 19, 20}}
</code></pre>
|
1,761,668 | <p>Wikipedia says about logical consequence:</p>
<blockquote>
<p>A formula φ is a logical consequence of a formula ψ if every interpretation that makes ψ true also makes φ true. In this case one says that φ is logically implied by ψ.</p>
</blockquote>
<p>But if φ and ψ are both true under some interpretations, then... | Martin Argerami | 22,857 | <p>Sure. Let $x\in M$. Put $\delta=1-\|x\|$. If we show that the ball of radius $\delta$ around $x$ is contained in $M,$ this implies that $M$ is open. </p>
<p>The key fact is that $\|z\|=(z_1^2 +\ldots+z_{n+1}^2)^{1/2}$ is a norm. In particular, it satisfies the triangle inequality. </p>
<p>So, if $\|z-x\|<\delta... |
4,481,695 | <p>I tried substituting <span class="math-container">$x+3$</span> to see if I could simplify in any way, but couldn't think of anything. Also tried using <span class="math-container">$\ln$</span> and <span class="math-container">$\exp$</span>, but in the end just got to <span class="math-container">$\ln(0)$</span>. Can... | Frank W | 552,735 | <p>If you prefer a proof without using derivatives: Observe that our limit is also equal to</p>
<p><span class="math-container">$$\lim\limits_{x\to-3}\frac {4^{(x+3)/5}-1}{x+3}=\frac 15\lim\limits_{x\to0}\frac {4^x-1}x$$</span></p>
<p>Where the substitution <span class="math-container">$x\mapsto\tfrac 15(x+3)$</span> w... |
1,801,112 | <p>Find the simplest solution:</p>
<p>$y' + 2y = z' + 2z$ I think proper notation is not sure, y' means first derivate of y. ($\frac{dy}{dt}+ 2y = \frac{dz}{dt} + 2z$)</p>
<p>$y(0)=1$</p>
<p>I got kind of confused, is $y=z=1$ a proper solution here? Or is disqualified because a constant is not reliant on time and so... | Aaron Meyerowitz | 84,560 | <p>The answer you propose certainly can't be faulted.</p>
<p>There is not a unique accepted criterion of "simplest" in mathematics. Certainly we expect people to agree in many specific cases. Many items in high stakes tests are of the form "find the next term in the sequence" but what is the simplest sequence starting... |
4,066,512 | <p>Find <span class="math-container">$\sum_{j=0}^{n}\sum_{i=j}^{n} {n \choose i}{i \choose j}$</span>.</p>
<p>I don't know how to double summations like this very well. Can someone expand this to show how the <span class="math-container">$i=j$</span> thing works?</p>
<p>I tried the following:
<span class="math-containe... | Especially Lime | 341,019 | <p>See robjohn's answer for the problem in your attempt.</p>
<p>However, a good way to start these problems is to stop and think what the sum actually represents. <span class="math-container">$\binom ni\binom ij$</span> is the number of ways to choose a committee of <span class="math-container">$i$</span> people out of... |
4,287,733 | <blockquote>
<p>How can we show <span class="math-container">$$\frac{1-x^n}{1-c^n} + \left(1-\frac{1-x}{1-c}\right)^n \leq 1 $$</span> for all <span class="math-container">$n \in \mathbb{N}$</span>, <span class="math-container">$0 \leq c \leq x
\leq 1, c \neq 1$</span>?</p>
</blockquote>
<p>The context is <a href="ht... | Ivan Kaznacheyeu | 955,514 | <p>If <span class="math-container">$x=c$</span> or <span class="math-container">$n=1$</span> then inequality holds. So consider <span class="math-container">$x > c, n > 1$</span>:</p>
<p><span class="math-container">$$\frac{1-x^n}{1-c^n}+\left(\frac{x-c}{1-c}\right)^n\leq 1\Leftrightarrow
1-x^n+(1-c^n)\left(\frac... |
3,831,387 | <p><span class="math-container">$X,Y\sim N(0,1)$</span> and are independent, consider <span class="math-container">$X+Y$</span> and <span class="math-container">$X-Y$</span>.</p>
<p>I can see why <span class="math-container">$X+Y$</span> and <span class="math-container">$X-Y$</span> are independent based on the fact th... | tommik | 791,458 | <p>To understand a very intuitive brainstorming let's start with <span class="math-container">$X,Y$</span> iid <span class="math-container">$N(\theta;1)$</span> distribution.</p>
<p>You will probabily know that <span class="math-container">$X+Y$</span> is a <em>"complete sufficient statistic"</em> for <span c... |
3,597,172 | <h2>The problem</h2>
<p>Let <span class="math-container">$f: \mathbb{R} \to \mathbb{R}$</span></p>
<p>Determine <span class="math-container">$f(x)$</span> knowing that </p>
<p><span class="math-container">$ 3f(x) + 2 = 2f(\left \lfloor{x}\right \rfloor) + 2f(\{x\}) + 5x $</span>, where <span class="math-container">$... | Nitin Uniyal | 246,221 | <p>For surjectivity of a linear map <span class="math-container">$A: U\rightarrow V$</span>, you must have
<span class="math-container">$rank(A)=dim(V)$</span>.</p>
<p>In your case <span class="math-container">$rank(A)=2=dim(V)$</span> so <span class="math-container">$A$</span> is ....</p>
|
308,117 | <p>I have the matrix
$$A := \begin{bmatrix}6& 9& 15\\-5& -10& -21\\ 2& 5& 11\end{bmatrix}.$$ Can anyone please tell me how to both find the eigenspaces by hand and also by using the Nullspace command on maple? Thanks.</p>
| Mikasa | 8,581 | <p>You know that, the solution sets of homogeneous linear systems $Ax=0$ provide an important source of vector spaces called Nullspace. Here, $det(A)\neq0$ so the above system has only one triple in $\mathbb R^3$ as its solution $$0^*=(0,0,0)$$ so the Nullspace is a trivial vector subspace of $\mathbb R^3$ $$\langle 0^... |
2,903,557 | <p>I am having trouble proving the following identity: </p>
<p>$$\frac{\sinh \tau +\sinh i\sigma }{\cosh \tau +\cosh i\sigma }=-\coth \left(i \frac{\sigma +i\tau }{2}\right)$$</p>
<p>I have tried using identities and the definitions but haven't had much luck. This is a missing step in inverting the bipolar coordinate... | user | 505,767 | <p>We have that</p>
<p>$$T_r=\frac{n-r+1}{r(r+1)(r+2)}=\frac{n+1}{r(r+1)(r+2)}-\frac{r}{r(r+1)(r+2)}=$$$$=(n+1)\left(\frac{1}{2r}+\frac{1}{2(r+2)}-\frac{1}{r+1}\right)+\frac1{r+2}-\frac1{r+1}$$</p>
<p>and by telescoping we can see that the first term gives</p>
<p>$$(n+1)\left(\frac12+\frac14+\frac12\cdot 2\sum_{r=3}... |
1,052,073 | <p><strong>Assume $V$ is a real $n$-dimensional vector space, and $v,w \in V $. Define $ T \in L(V)$ by $ T(u) = u - (u,v)w$. Find a formula for Trace(T)</strong></p>
<p>All I know about this is that trace is sum of the diagonal entries of the matrix. So how do I find the diagonal entries? I don't really know what ste... | Henry | 6,460 | <p>Experimentally it appears that seems that something close to Hagen von Eitzen's comment of the construction using a regular pentagon</p>
<p><img src="https://i.stack.imgur.com/7C0ZP.png" alt="enter image description here"></p>
<p>works for all obtuse triangles. Here is an example, with two isosceles triangles cut... |
1,052,073 | <p><strong>Assume $V$ is a real $n$-dimensional vector space, and $v,w \in V $. Define $ T \in L(V)$ by $ T(u) = u - (u,v)w$. Find a formula for Trace(T)</strong></p>
<p>All I know about this is that trace is sum of the diagonal entries of the matrix. So how do I find the diagonal entries? I don't really know what ste... | Hagen von Eitzen | 39,174 | <p>Let $ABC$ be a triangle with angles $\alpha=\angle BAC<\frac\pi2,\beta=\angle CBA<\frac\pi2,\gamma=\angle ACB$.
Let $F$ be the orthogonal projection of $C$ to $AB$ (which is between $A$ and $B$).
For any point $P$ between $C$ and $F$, let $\ell $ be the line through $P$ parallel to $AB$.
Let $E$ be the interse... |
2,099,516 | <p>For independent Gamma random variables $G_1, G_2 \sim \Gamma(n,1)$, $\frac{G_1}{G_1+G_2}$ is independent of $G_1+G_2$. Does this imply that $G_1+G_2$ is independent of $G_1-G_2$? Thanks!</p>
| mbe | 100,502 | <p>Since a rigorous proof as well as a simulation study have been proportioned already, I thought I'd throw in a slightly $\textbf{heuristic}$ argument on why whey are not independent. </p>
<p>Given a realisation of $G_1-G_2$, set $Y=|G_1-G_2|$. Then we know at least that $G_1\geq Y$ or $G_2\geq Y$, such that necessar... |
2,403,608 | <p>I was asked to solve for the <span class="math-container">$\theta$</span> shown in the figure below.</p>
<p><a href="https://i.stack.imgur.com/3Yxqv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3Yxqv.png" alt="enter image description here" /></a></p>
<p>My work:</p>
<p>The <span class="math-con... | Xander Henderson | 468,350 | <p>First, some general advice</p>
<ul>
<li>Often when you are presented a problem like this, the thing that you are looking for is something that you can't get at directly. <strong>Be prepared to work indirectly.</strong> You have to make some kind of round-about argument, and determine a lot of intermediate unknown... |
1,293,725 | <p>Here I have a question:</p>
<p>Solve for real value of $x$:
$$|x^2 -2x -3| > |x^2 +7x -13|$$</p>
<p>I got the answer as $x = (-\infty, \frac{1}{4}(-5-3\sqrt{17}))$ and $x=(\frac{10}{9},\frac{1}{4}(3\sqrt{17}-5)$</p>
<p>Please verify it if it is correct or not. Thanks</p>
| DeepSea | 101,504 | <p><strong>hint</strong>: Use $|A| > |B| \iff A^2 > B^2 \iff (A-B)(A+B) > 0$. Apply this property to $A = x^2-2x-3, B = x^2+7x-13$</p>
|
3,492,155 | <p>I am looking for the values <span class="math-container">$ x \in R $</span> which satisfy the following equation :</p>
<p><span class="math-container">$ e^{-\alpha x} = \frac{a}{x - c} $</span></p>
<p>Where <span class="math-container">$ \alpha $</span>, <span class="math-container">$ a $</span> and <span class="m... | GEdgar | 442 | <p>Step by step:
<span class="math-container">$$
e^{-\alpha x} = \frac{a}{x-c}
\\
(x-c)e^{-\alpha x} = a
\\
(x-c)e^{-\alpha x} e^{\alpha c} = a e^{\alpha c}
\\
(x-c)e^{-\alpha (x-c)} = a e^{\alpha c}
\\
-\alpha(x-c)e^{-\alpha (x-c)} = -a\alpha e^{\alpha c}
\\
-\alpha(x-c) = W(-a\alpha e^{\alpha c})
\\
x-c = -\frac{W(-a... |
286,930 | <p>I have been assigned this problem for homework:</p>
<blockquote>
<p>Show that, if $a < b + \epsilon$ for every $\epsilon \gt 0$, then $a\le b$.</p>
</blockquote>
<p>I have tried to go about this using Induction, but I don't know what the base case would be. It is obvious to me in my mind, but I don't know ho... | lamb_da_calculus | 54,044 | <p>I'm assuming this is from a first proof-based course, so apologies if this comes off as condescending. </p>
<p>Induction is harder to use in this case for precisely the reason you stated: what is the base case? We don't have any equivalent of some smallest nonzero number 1 with which to start like we do in $\mathbb... |
286,930 | <p>I have been assigned this problem for homework:</p>
<blockquote>
<p>Show that, if $a < b + \epsilon$ for every $\epsilon \gt 0$, then $a\le b$.</p>
</blockquote>
<p>I have tried to go about this using Induction, but I don't know what the base case would be. It is obvious to me in my mind, but I don't know ho... | icurays1 | 49,070 | <p>I'm a snob, so I like using contrapositive instead of contradiction whenever possible. The contrapositive would go like this: "if $a>b$, then there exists an $\epsilon>0$ such that $a\geq b+\epsilon$." Finding this epsilon should be easy with a number line. </p>
<p>Edit: the original poster figured it out in ... |
2,518,305 | <p><a href="http://www.mit.edu/%7Esame/pdf/qualifying_round_2017_answers.pdf" rel="noreferrer">This is a problem from MIT integration bee 2017.</a></p>
<p><span class="math-container">$$\int_0^{\pi/2} \frac 1 {1+\tan^{2017} x} \, dx$$</span></p>
<p>I have tried substitution method, multiplying numerator and denominator... | jonsno | 310,635 | <p>Try using $$\int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx$$</p>
<p>and the fact that $\tan(\frac{\pi}{2}-x) = \cot(x)$ to convert to get the following:
$$I = \int_0^{\pi / 2} \frac 1 {1+ \tan^{2017}(x)} \, dx =
\int_0^{\pi / 2} \frac 1 {1+ \tan^{2017}(\pi/2-x)} \, dx \\
= \int_0^{\pi / 2} \frac 1 {1+ \cot^{2017... |
260,865 | <p>I am fairly new to mathematica and I am trying to plot a 3D curve defined by multiple formulas. I have the curve <span class="math-container">$K$</span> from the point <span class="math-container">$(\frac{1}{2},-\frac{1}{2}\sqrt{3},0)$</span> to <span class="math-container">$(\frac{1}{2},\frac{1}{2}\sqrt{3},2\sqrt{3... | Bob Hanlon | 9,362 | <p>You could also use <a href="https://reference.wolfram.com/language/ref/ParametricPlot3D.html" rel="nofollow noreferrer"><code>ParametricPlot3D</code></a>. The curve is made up of two segments as a function of <code>x</code></p>
<pre><code>ParametricPlot3D[{
{x, Sqrt[1 - x^2], Sqrt[1 - x^2]/x + Sqrt[3]},
{x, -Sqr... |
260,865 | <p>I am fairly new to mathematica and I am trying to plot a 3D curve defined by multiple formulas. I have the curve <span class="math-container">$K$</span> from the point <span class="math-container">$(\frac{1}{2},-\frac{1}{2}\sqrt{3},0)$</span> to <span class="math-container">$(\frac{1}{2},\frac{1}{2}\sqrt{3},2\sqrt{3... | Daniel Huber | 46,318 | <p>Here is an analytical solution:</p>
<pre><code>eq = {x^2 + y^2 == 1 && z == Sqrt[3] + y/x, x > 1/2};
Reduce[eq, z, Reals]
</code></pre>
<p><a href="https://i.stack.imgur.com/rS8Af.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rS8Af.png" alt="enter image description here" /></a></p>
<p... |
260,865 | <p>I am fairly new to mathematica and I am trying to plot a 3D curve defined by multiple formulas. I have the curve <span class="math-container">$K$</span> from the point <span class="math-container">$(\frac{1}{2},-\frac{1}{2}\sqrt{3},0)$</span> to <span class="math-container">$(\frac{1}{2},\frac{1}{2}\sqrt{3},2\sqrt{3... | cvgmt | 72,111 | <p>Use the hint by @J.M.can'tdealwithit</p>
<pre><code>sol = Solve[{x == Cos[t], y == Sin[t], z == y/x + Sqrt[3],
x >= 1/2, -π <= t <= π},{x,y,z}]
plot = ParametricPlot3D[{x, y, z} /. sol[[1]], {t, -π, π},
PlotStyle -> {Thick, Red}]
</code></pre>
<p><a href="https://i.stack.imgur.com/tYzh7.png" rel... |
2,970,234 | <p>So, I'm given the following query:</p>
<p>Write the Taylor series centered at <span class="math-container">$z_0 = 0$</span> for each of the following complex-valued functions:</p>
<p><span class="math-container">$$f(z) = z^2\sin(z),\quad g(z) = z\sin(z^2)$$</span></p>
<p>Then, use these series to help you compute... | Nosrati | 108,128 | <p><span class="math-container">\begin{align}
\lim_{z\to0} \frac{z^2\sin(z)-z\sin(z^2)}{z^5}
&= \lim_{z\to0} \frac{\sum_{n=0}^{\infty} \dfrac{(-1)^nz^{2n+3}}{(2n+1)!}- \sum_{n=0}^{\infty} \dfrac{(-1)^nz^{4n+3}}{(2n+1)!}}{z^5} \\
&= \lim_{z\to0} \frac{\left(z^3-\dfrac{z^5}{3!}+\dfrac{z^7}{5!}-\cdots\right)-\left... |
2,502,711 | <p>If the cross ratio $Z_1, Z_2, Z_3$ and $Z_4$ is real, then</p>
<p>which of the following statement is true? </p>
<p>1)$Z_1, Z_2$ and $Z_3$ are collinear</p>
<p>2)$Z_1, Z_2$ and $Z_3$ are concyclic</p>
<p>3)$Z_1, Z_2$ and $Z_3$ are collinear when atleast one $Z_1, Z_2$ or $Z_3$ is real </p>
<p>My attempt : By ... | doraemonpaul | 30,938 | <p>Hint:</p>
<p>$\dfrac{d^2z}{dx^2}=be^{-az}$</p>
<p>$\dfrac{dz}{dx}\dfrac{d^2z}{dx^2}=be^{-az}\dfrac{dz}{dx}$</p>
<p>$\int\dfrac{dz}{dx}\dfrac{d^2z}{dx^2}~dx=\int be^{-az}\dfrac{dz}{dx}~dx$</p>
<p>$\int\dfrac{dz}{dx}~d\left(\dfrac{dz}{dx}\right)=\int be^{-az}~dz$</p>
<p>$\dfrac{1}{2}\left(\dfrac{dz}{dx}\right)^2=... |
3,965,668 | <p>I started off my proof by of course stating that the different right triangles I would be comparing should have the same area A. I was able to show that what the question is asking is true visually and computationally using the Pythagorean Theorem, and even using the triangle inequality, but I don't really know how ... | Martin R | 42,969 | <p>Yes. For all <span class="math-container">$t \in [0, 1]$</span> is
<span class="math-container">$$
e^{\phi(t) - f(\gamma(t))} = \frac{e^{\phi(t)}}{e^{f(\gamma(t))}} = \frac{\gamma(t)}{\gamma(t)} = 1
$$</span>
because <span class="math-container">$e^{f(w)} = w$</span> holds for all <span class="math-container">$w \i... |
1,408,467 | <p>I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm only trying to find the non-redundant constraints that define a feasible region (i.e. I have no objective function), given a set of possibly redundant inequality constraints.</p>
<p>For instance, if I have:... | Robert Israel | 8,508 | <p>You could try maximizing $x_1 + 0 x_2$ subject to the first $5$ constraints.
The constraint is redundant iff the optimal objective value $\le -6$. </p>
|
32,294 | <p>I sort of asked a version of this question before and it was unclear; try I will now to make an honest attempt to state everything clerly.</p>
<p>I am trying to evaluate the following, namely $\nabla w = \nabla |\vec{a} \times \vec{r}|^n$, where $\vec{a}$ is a constant vector and $\vec{r}$ is the vector $<x_1,x_... | Hans Lundmark | 1,242 | <p>Let $f(r)=|a \times r|$. (I'll be lazy and omit the vector arrows.)</p>
<p>Then, to begin with, $\nabla(f(r)^n) = n f(r)^{n-1} \nabla f(r)$ by the chain rule. So it only remains to compute $\nabla f(r)$, and if you can't think of any clever way, then you just do it by brute force in terms of coordinates: take $a=(a... |
1,238,481 | <p>Let $\vec p, \vec q$ and $\vec r$ are three mutually perpendicular vectors of the same magnitude. If a vector $\vec x$ satisfies the equation $\begin{aligned} \vec p \times ((\vec x - \vec q) \times \vec p)+\vec q \times ((\vec x - \vec r) \times \vec q)+\vec r \times ((\vec x - \vec p) \times \vec r)=0\end{aligned}... | Asaf Karagila | 622 | <p>Let me first clarify the issue with an inaccessible, since it seems to me that my words were somewhat misunderstood.</p>
<ol>
<li><p>I only mentioned inaccessible cardinals, since it is a plausible axiom, and if you believe it to be consistent, then of course there will be a (standard) model of <span class="math-con... |
32,809 | <p>Is it possible to define new graphics directives?</p>
<p>For example, suppose I want to be able to use the following code:</p>
<pre><code>Graphics[{ BigPointSize[0.07], SmallPointSize[0.04],
Red, BigPoint[{1,1}], BigPoint[{1,3}], SmallPoint[{3,1}],
Blue, SmallPoint[{2,2}], SmallPoint[{3,2}], ... | Vahagn Poghosyan | 9,567 | <pre><code>BigPointSizeValue = 1;
BigPointSize[s_] := (BigPointSizeValue = s;)
BigPoint[p_] := {PointSize[BigPointSizeValue], Point[p]}
SmallPointSizeValue = 1;
SmallPointSize[s_]:=(SmallPointSizeValue = s;)
SmallPoint[p_] := {PointSize[SmallPointSizeValue], Point[p]}
</code></pre>
|
32,809 | <p>Is it possible to define new graphics directives?</p>
<p>For example, suppose I want to be able to use the following code:</p>
<pre><code>Graphics[{ BigPointSize[0.07], SmallPointSize[0.04],
Red, BigPoint[{1,1}], BigPoint[{1,3}], SmallPoint[{3,1}],
Blue, SmallPoint[{2,2}], SmallPoint[{3,2}], ... | Vahagn Poghosyan | 9,567 | <pre><code>Protect[BigPointSize, BigPoint, SmallPointSize, SmallPoint];
UserGraphics[gr_, opt : OptionsPattern[Graphics]] := Module[
{
BigPointSizeStack, BigPointSizeExec, BigPointExec,
SmallPointSizeStack, SmallPointSizeExec, SmallPointExec,
TempList, TempListExec,
},
BigPointSizeStack = {0.1}; (*... |
4,020,986 | <blockquote>
<p>For every <span class="math-container">$n \in \mathbb{N}$</span> denote <span class="math-container">$x_n=(n,0) \in \mathbb{R^2}.$</span> Show that the set <span class="math-container">$\mathbb{R^2} \setminus \{x_n \mid n \in \mathbb{N} \}$</span> is an open subset of the plane.</p>
</blockquote>
<p>The... | lhf | 589 | <p>Consider the function <span class="math-container">$\mathbb R^2 \to \mathbb R^3$</span> given by <span class="math-container">$f(x,y)=(\sin(x \pi),x,y)$</span>. Then <span class="math-container">$f$</span> is continuous.</p>
<p>Let <span class="math-container">$C=\{0\} \times [0,\infty) \times \{0\}$</span>. Then <s... |
4,020,986 | <blockquote>
<p>For every <span class="math-container">$n \in \mathbb{N}$</span> denote <span class="math-container">$x_n=(n,0) \in \mathbb{R^2}.$</span> Show that the set <span class="math-container">$\mathbb{R^2} \setminus \{x_n \mid n \in \mathbb{N} \}$</span> is an open subset of the plane.</p>
</blockquote>
<p>The... | Henno Brandsma | 4,280 | <p>As <span class="math-container">$\Bbb N$</span> and <span class="math-container">$\{0\}$</span> are closed in <span class="math-container">$\Bbb R$</span>, <span class="math-container">$\Bbb N \times \{0\}$</span> is closed in <span class="math-container">$\Bbb R^2$</span> and so your set, which is the complement of... |
3,001,747 | <p>I'd need some help evaluating this limit:</p>
<p><span class="math-container">$$\lim_{x \to 0} \frac{\ln\sin mx}{\ln \sin x}$$</span></p>
<p>I know it's supposed to equal 1 but I'm not sure how to get there.</p>
| user | 505,767 | <p><strong>HINT</strong></p>
<p>We need <span class="math-container">$x>0$</span> and <span class="math-container">$m>0$</span> then by standard limits</p>
<p><span class="math-container">$$\frac{\ln\sin mx}{\ln \sin x}=\frac{\ln\frac{\sin mx}{mx}+\ln (mx)}{\ln \frac{\sin x}x+\ln x}=\frac{\ln\frac{\sin mx}{mx}+... |
3,001,747 | <p>I'd need some help evaluating this limit:</p>
<p><span class="math-container">$$\lim_{x \to 0} \frac{\ln\sin mx}{\ln \sin x}$$</span></p>
<p>I know it's supposed to equal 1 but I'm not sure how to get there.</p>
| Mostafa Ayaz | 518,023 | <p>According to L^Hopital's rule we have <span class="math-container">$$\lim_{x \to 0} \frac{\ln\sin mx}{\ln \sin x}=\lim_{x \to 0} \frac{m\cos mx}{\cos x}{\sin x\over \sin mx}=1$$</span></p>
|
347,226 | <p>If $V$ is a vector space over the field $F$ then verify that </p>
<p>$$(\alpha_1+ \alpha_2)+(\alpha_3+\alpha_4)=[\alpha_2+(\alpha_3+\alpha_1)]+\alpha_4$$</p>
<p>for all the vectors $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ in $V$?</p>
<p>i think we will use properties of vector spaces like associative,cummutative,etc... | cthl | 69,820 | <p>This is just the associativity and commutativity of the addition:
In your example you would use the associativity in the following way: $(\alpha_1+\alpha_2)+(\alpha_3+\alpha_4)=((\alpha_1+\alpha_2)+\alpha_3)+\alpha_4=[\alpha_1+(\alpha_2+\alpha_3)]+\alpha_4=(\ast)$. Now we use commutativity and get $(\ast)=[\alpha_1+... |
347,226 | <p>If $V$ is a vector space over the field $F$ then verify that </p>
<p>$$(\alpha_1+ \alpha_2)+(\alpha_3+\alpha_4)=[\alpha_2+(\alpha_3+\alpha_1)]+\alpha_4$$</p>
<p>for all the vectors $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ in $V$?</p>
<p>i think we will use properties of vector spaces like associative,cummutative,etc... | Eleven-Eleven | 61,030 | <p>$(\alpha_1+\alpha_2)+(\alpha_3+\alpha_4)$ = $[(\alpha_1+\alpha_2)+\alpha_3]+\alpha_4$ = $[(\alpha_2+\alpha_1)+\alpha_3]+\alpha_4$ = $[\alpha_2+(\alpha_1+\alpha_3)]+\alpha_4$ = $[\alpha_2+(\alpha_3+\alpha_1)]+\alpha_4$</p>
|
57,213 | <p>Let <span class="math-container">$A \in \mathbb{Q}^{6 \times 6}$</span> be the block matrix below:</p>
<p><span class="math-container">$$A=\left(\begin{array}{rrrr|rr}
-3 &3 &2 &2 & 0 & 0\\
-1 &0 &1 &1 & 0 & 0\\
-1&0 &0 &1 & 0 & 0\\
-4&6 ... | Pierre-Yves Gaillard | 660 | <p><strong>Hint</strong></p>
<p>Let $A$ be an square matrix with coefficients in a field $K$, and let $g$ be its minimal polynomial. </p>
<p>Then the epimorphism $K[X]\to K[A]$, $f\mapsto f(A)$ induces an isomorphism $K[X]/(g)\to K[A]$. </p>
<p>Assume that $g$ splits over $K$. </p>
<p>Then the Chinese Remainder The... |
1,364,430 | <p><strong>Problem</strong></p>
<p>How many of the numbers in $A=\{1!,2!,...,2015!\}$ are square numbers?</p>
<p><strong>My thoughts</strong></p>
<p>I have no idea where to begin. I see no immediate connection between a factorial and a possible square. Much less for such ridiculously high numbers as $2015!$.</p>
<p... | ThePortakal | 137,487 | <p><strong>Hint:</strong> (for example) $13!, 14!, \dots , 25!$ are all nonsquare numbers because all of them are divisible by $13$ only once. (because $13$ is a prime)</p>
<p>Similarly, $17!, 18!, \dots, 33!$ are nonsquare numbers.</p>
<p>Go on like this.</p>
|
117,619 | <p>I need to evaluate the following real convergent improper integral using residue theory (vital that i use residue theory so other methods are not needed here)
I also need to use the following contour (specifically a keyhole contour to exclude the branch cut):</p>
<p><a href="https://i.stack.imgur.com/4wwwj.png" rel... | Amir Alizadeh | 53,185 | <p>Close format for this type of integrals:
$$ \int_0^{\infty} x^{\alpha-1}Q(x)dx =\frac{\pi}{\sin(\alpha \pi)} \sum_{i=1}^{n} \,\text{Res}_i\big((-z)^{\alpha-1}Q(z)\big) $$
$$ I=\int_0^\infty \frac{\sqrt{x}}{x^3+1} dx \rightarrow \alpha-1=\frac{1}{2} \rightarrow \alpha=\frac{3}{2}$$
$$ g(z) =(-z)^{\alpha-1}Q(z) =\fr... |
283,747 | <p>Let $BG$ denote the classifying space of a finite group $G$. For which group cohomology classes $c\in H^2(G;\mathbb{Z}/2)$ does there exist a real vector bundle $E$ over $BG$ such that $w_2(E)=c$?</p>
| Matthias Wendt | 50,846 | <p>I don't know of a general criterion, but here are two relevant references.</p>
<p>1) Several statements concerning the relation between the cohomology ring and the subring of Stiefel-Whitney classes are discussed in: </p>
<ul>
<li>P. Guillot. The computation of Stiefel-Whitney classes.
Annales de l'institut Fouri... |
2,647,123 | <p>I'm asked to to find a $3\times3$ matrix, in which no entry is $0$ but $A^2=0$. </p>
<p>The problem is if I I brute force it, I am left with a system of 6 equations (Not all of which are linear...) and 6 unknowns. Whilst I could in theory solve that, is there more intuitive way of solving this problem or am I going... | Focus | 516,040 | <p>There's an idea. We can try to put $A = \alpha \beta'$, where $\alpha,\beta$ are $3 \times 1$ matrix.</p>
<p>Then, $A^2 = \alpha \beta' \alpha \beta'$, so if $\alpha,\beta \ne 0$(all the components are not $0$),$\beta' \alpha = 0$ things will be solved.</p>
<p>So,we could let</p>
<p>$\alpha = \begin{pmatrix} 1 \... |
2,873,520 | <p>I want to find out how interference of two sine waves can affect the output-phase of the interfered wave. </p>
<p>Consider two waves,</p>
<p>$$ E_1 = \sin(x) \\
E_2 = 2 \sin{(x + \delta)}
$$</p>
<p>First off, I don't know how to prove it, but I can see visually (plotting numerically) that the sum of these waves... | Siong Thye Goh | 306,553 | <p>$$L= a_0(x) + a_1(x)\frac{d}{dx}+ \ldots + a_n(x) \frac{d^n}{dx^n} $$</p>
<p>is known as a linear <a href="https://en.wikipedia.org/wiki/Differential_operator" rel="nofollow noreferrer">differential operator</a>.</p>
<p>We have
$$Ly= a_0(x)y + a_1(x)\frac{dy}{dx}+ \ldots + a_n(x) \frac{d^ny}{dx^n} $$</p>
<p>$$a_0... |
3,239,540 | <p><span class="math-container">$$S=\sum_{k=2}^{n}\frac{k^{2}-2}{k!}, n\geq 2$$</span></p>
<p>I got <span class="math-container">$S=\sum_{k=2}^{n}\frac{1}{(k-2)!}+\frac{1}{(k-1)!}-\frac{1}{k!}-\frac{1}{k!}$</span></p>
<p>I give k values but not all terms are vanishing.I remain with <span class="math-container">$\frac... | P. Quinton | 586,757 | <p>For the sum until the index <span class="math-container">$n$</span>, you can apply the same trick :
<span class="math-container">\begin{align*}
\sum_{k=2}^n \frac{k^2-2}{k!} &= \sum_{k=2}^n \left[\frac{k (k-1)}{k!} + \frac{k}{k!}-\frac{2}{k!} \right]\\
&= \color{blue}{\sum_{k=2}^n \frac{k (k-1)}{k!}} + \colo... |
1,456,444 | <p>How can I go about solving this Pigeonhole Principle problem? </p>
<p>So I think the possible numbers would be: $[3+12], [4+11], [5+10], [6+9], [7+8]$</p>
<p>I am trying to put this in words...</p>
| Yiyuan Lee | 104,919 | <p>The sentence you want is probably something along the lines of : "We need to pick $6$ distinct numbers from $5$ pairs of numbers. By the pigeonhole principle, both numbers in at least one of the pairs must be picked."</p>
|
2,218,716 | <p>Explain why 1/i is − i. </p>
<p>(That is: explain why the multiplicative inverse of i is the complex number − i.)</p>
<p>And then the hint that I was given was, what property defines the multiplicative inverse?</p>
<p>I know how to algebraically prove 1/i = -i, but need help writing the proof.</p>
| Isaac Browne | 429,987 | <p>$i^2=-1$ is the definition of i. Thus,
$$i*-i=--1=1$$
The multiplicative inverse is simply whatever you multiply by to get the identity, thus it is proved!</p>
|
3,249,735 | <p>My question relates to this problem:</p>
<p>Prove by induction that 54 divides <span class="math-container">$2^{2k+1}-9k^2+3k-2$</span>. </p>
<p>My solving so far gives this answer: (after all calculations)</p>
<p><span class="math-container">$2^{2(k+1)+1}-9(k+1)^2+3(k+1)-2= 54 \cdot2^2k+27k^2-27k$</span></p>
<p... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>Note that the first and last terms are even and hence are divisible by 2. The middle two are really
<span class="math-container">$$
3k-9k^2 = 3k(1-3k),
$$</span>
and the factors always have different parity, hence one of them is always even as well.</p>
|
2,627,131 | <blockquote>
<p>let <span class="math-container">$f(x)= 1+\sqrt{x+k+1}-\sqrt{x+k} \ \ k \in \mathbb{R}$</span>
Number of answers :</p>
<p><span class="math-container">$$f(x)=f^{-1}(x) \ \ \ \ :f^{-1}(f(x))=x$$</span></p>
</blockquote>
<p>MY Try :</p>
<p><span class="math-container">$$y=1+\sqrt{x+k+1}-\sqrt{x+k} \\( y... | xpaul | 66,420 | <p>Let $y=f(x)=f^{-1}(x)$ and then
\begin{eqnarray}
y=1+\sqrt{x+k+1}-\sqrt{x+k},\tag{1}\\
x=1+\sqrt{y+k+1}-\sqrt{y+k}.\tag{2}
\end{eqnarray}
Subtracting (1) from (2) gives
\begin{eqnarray}
x-y&=&(\sqrt{y+k+1}-\sqrt{x+k+1})-(\sqrt{y+k}-\sqrt{x+k})\\
&=&\frac{y-x}{\sqrt{y+k+1}+\sqrt{x+k+1}}-\frac{y-x}{\sq... |
2,174,912 | <p>I' just so stumped right now. I want to get $x^{n}$ to equal $x^{2n+1}$. Right now I have that:
$$(\sqrt{x})^{2n} = x^n$$
But I don't know what to do to x to get:
$$x^n = \{\text{something done to $x$}\}^{2n+1}$$</p>
| Community | -1 | <p>$(x^n)=x^{\frac{n}{2n+1}^{2n+1}}$ $\forall n\in\Bbb R/\{-\frac{1}{2}$}</p>
|
2,197,967 | <p>Can someone explain how is the RHS concluded? I did with sample numbers and it is all correct. but I can't figure out how C(12,6) comes to play.
$$
\binom{12}{0} + \binom{12}{1} + \binom{12}{2} + \binom{12}{3} + \binom{12}{4} + \binom{12}{5} = (2^{12} - \binom{12}{6}) / 2
$$</p>
| Mythomorphic | 152,277 | <p>Suppose we toss a fair coin for $12$ times. The number of combinations is $2^{12}$.</p>
<p>Now notice the number of outcomes $N(H>T)$ such that the total number of Head is more than Tail is same as that of Tail more than Head $N(H<T)$, by symmetry.</p>
<p>We know $N(H>T)$ is the total number of choosing l... |
4,548,329 | <p>Find the first derivative of <span class="math-container">$$y=\sqrt[3]{\dfrac{1-x^3}{1+x^3}}$$</span></p>
<p>The given answer is <span class="math-container">$$\dfrac{2x^2}{x^6-1}\sqrt[3]{\dfrac{1-x^3}{1+x^3}}$$</span> It is nice and neat, but I am really struggling to write the result exactly in this form. We have ... | Átila Correia | 953,679 | <p><strong>HINT</strong></p>
<p>I would start with noticing that</p>
<p><span class="math-container">\begin{align*}
y^{3} = \frac{1 - x^{3}}{1 + x^{3}} = -1 + \frac{2}{1 + x^{3}} \Rightarrow 3y^{2}y' = -\frac{6x^{2}}{(1 + x^{3})^{2}}
\end{align*}</span></p>
<p>Can you take it from here?</p>
|
3,888,259 | <p>The special linear group of invertible matrices is defined as the kernel of the determinant of the map:</p>
<p><span class="math-container">$$\det:GL(n,\mathbb{R}) \mapsto \mathbb{R}^*$$</span></p>
<p>In my mind the kernel of a linear map is the set of vectors that are mapped to the zero vector. So the map above wou... | José Carlos Santos | 446,262 | <p>But <span class="math-container">$\det$</span> is <em>not</em> a linear map. It is a group homomorphism. Its kernel is<span class="math-container">$$\left\{M\in GL(n,\Bbb R)\,\middle|\,\det(M)=1\right\},$$</span>since <span class="math-container">$1$</span> is the identity element of <span class="math-container">$\b... |
4,639,011 | <p>I'm interested in finding the asymptotic at <span class="math-container">$n\to\infty$</span> of
<span class="math-container">$$b_n:= \frac{e^{-n}}{(n-1)!}\int_0^\infty\prod_{k=1}^{n-1}(x+k)\,e^{-x}dx=e^{-n}\int_0^\infty\frac{e^{-x}}{x\,B(n;x)}dx$$</span>
Using a consecutive application of Laplace' method, I managed ... | Svyatoslav | 869,237 | <p>This is an attempt to generalize the asymptotic. Namely, to consider a bit more general case:
<span class="math-container">$$b_n(\lambda)=e^{-n}\int_0^\infty\frac{x^\lambda \,e^{-x}}{x\,B(n;x)}dx\,,\,\,\lambda\geqslant0\tag{1}$$</span>
Using the second approach proposed by @Gary, we get
<span class="math-container">... |
119,876 | <pre><code>Module[{x},
f@x_ = x;
p@x_ := x;
{x, x_, x_ -> x, x_ :> x}
]
?f
?p
</code></pre>
<p>gives</p>
<pre><code>{x$17312, x$17312_, x_ -> x, x_ :> x}
f[x_]=x
p[x_]:=x
</code></pre>
<p>but I'd like to get</p>
<pre><code>{x$17312, x$17312_, x$17312_ -> x$17312, x$17312_ :> x$17312}
f[x$17312... | Kuba | 5,478 | <p>While <code>Set</code> isn't a scoping construct (SC), it is considered one by other SCs outer to it. <a href="http://reference.wolfram.com/language/ref/Set.html" rel="nofollow noreferrer"><strong>ref / Set / Details[[-3]]</strong></a> (thanks to <a href="https://mathematica.stackexchange.com/users/280/alexey-popkov... |
402,802 | <p>I have read that $$y=\lvert\sin x\rvert+ \lvert\cos x\rvert $$ is periodic with fundamental period $\frac{\pi}{2}$.</p>
<p>But <a href="http://www.wolframalpha.com/input/?i=y%3D%7Csinx%7C%2B%7Ccosx%7C" rel="nofollow">Wolfram</a> says it is periodic with period $\pi$.</p>
<p>Please tell what is correct.</p>
| WiseStrawberry | 61,705 | <p>If you look at the graphs you can see the difference. what <strong>is</strong> a fundamental period? </p>
|
397,347 | <p>I'm trying to figure out how to evaluate the following:
$$
J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx
$$
I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies J=-I'(1)$, but I couldn't figure out what $I(s)$ was. My other idea was contour integration, but I'm not sure how to deal... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
2,971,143 | <p>Let me choose <span class="math-container">$n=1$</span> for my induction basis: <span class="math-container">$2 > 1$</span>, true.</p>
<p>Induction Step : <span class="math-container">$2^n > n^2 \rightarrow 2^{n+1} > (n+1)^2 $</span></p>
<p><span class="math-container">$2^{n+1} > (n+1)^2 \iff$</span></... | user | 505,767 | <p>You have assumed for the <strong>base case</strong></p>
<ul>
<li><span class="math-container">$n=1 \implies 2>1$</span></li>
</ul>
<p>and it is not wrong, then for the <strong>induction step</strong> <span class="math-container">$P(n) \implies P(n+1)$</span> you have found that it works only for <span class="ma... |
76,778 | <p>I'm reading Yao's unpredictability -> pseudorandomness construction
and Goldreich/levin's pseudorandom permutation -> pseudorandom generator construction.</p>
<p>My question is:</p>
<p>is there a direct way to show that:</p>
<p>given a pseudorandom function, we can construct a pseudorandom permutation out of it?<... | Steve | 18,168 | <p>That would be the celebrated Luby Rackoff result.</p>
|
1,908,844 | <p>The following example is taken from the book "Introduction to Probability Models" of Sheldon M. Ross (Chapter 5, example 5.4).</p>
<blockquote>
<p>The dollar amount of damage involved in an automobile accident is an
exponential random variable with mean 1000. Of this, the insurance
company only pays that amou... | Alex R. | 22,064 | <p>If you want a generating function approach, your sum is equivalent to finding the $x^k$ coefficient of $(1+x)^n(1+x)^m=(1+x)^{m+n}.$ This is easily seen to be $\binom{n+m}{k}$. </p>
|
2,916,246 | <p>I've found this to be difficult to solve:</p>
<p>$$ \frac{d^2 x }{dt^2} + (a x + b) \frac{dx}{dt} = 0 $$</p>
<p>I've done some reading, and I guess I could write this as:</p>
<p>$$ \frac{d^2 x }{dt^2} + b \frac{dx}{dt} + ax \frac{dx}{dt} = 0 $$</p>
<p>If I then treat $v(x) = \frac{dx}{dt}$ as an independent var... | user577215664 | 475,762 | <p>$$\frac{d^2 x }{dt^2} + (a x + b) \frac{dx}{dt} = 0$$
$$x'' + (a x + b) x' = 0$$
$$x'' + a xx' + bx' = 0$$
$$x'' + \frac a 2 (x^2)' + bx' = 0$$
Integrate
$$x'+ \frac a 2 x^2 + bx = K$$
$$(x+\frac ba)'+ \frac a 2 (x^2 + \frac {2b}ax+\frac {b^2}{a^2}) = C$$
Substitute $z=x+\frac ba$
$$z'+ \frac a 2 z^2 = C$$
This last... |
2,111,402 | <p>Simple exercise 6.2 in Hammack's Book of Proof. "Use proof by contradiction to prove"</p>
<p>"Suppose $n$ is an integer. If $n^2$ is odd, then $n$ is odd"</p>
<p>So my approach was:</p>
<p>Suppose instead, IF $n^2$ is odd THEN $n$ is even</p>
<p>Alternatively, then you have the contrapositive, IF $n$ is not even... | Nitin Uniyal | 246,221 | <p>The contrapositive of the original statement i.e. "If $n$ is even the $n^2$ is even" is easy to prove. </p>
<blockquote>
<p>Let $n=2k$, $k\in\mathbb Z$ then $n^2=(2k)^2=2(2k^2)$ is even.</p>
</blockquote>
|
26,083 | <p>I have a data set in form of:(this is just an example)</p>
<pre><code>1324501020
3241030205
4332020134
</code></pre>
<p>the data are stored in a text file (e.g. data.txt) but I need to convert them into a matrix format such that each number be place in a cell like this:</p>
<pre><code>1 3 2 4 5 0 1 0 2 0
3 2 4 1 ... | george2079 | 2,079 | <pre><code>MatrixForm[
Read /@ StringToStream /@ Characters[#] & /@ {"1234" , "5678"}]
</code></pre>
<p><img src="https://i.stack.imgur.com/aCR3S.jpg" alt="enter image description here"> </p>
<p>perhaps a bit cleaner..</p>
<pre><code> IntegerDigits /@ Read /@ StringToStream /@ {"1234", "5678"}
</code></pre... |
26,083 | <p>I have a data set in form of:(this is just an example)</p>
<pre><code>1324501020
3241030205
4332020134
</code></pre>
<p>the data are stored in a text file (e.g. data.txt) but I need to convert them into a matrix format such that each number be place in a cell like this:</p>
<pre><code>1 3 2 4 5 0 1 0 2 0
3 2 4 1 ... | Hans | 7,695 | <p>Try:</p>
<pre><code>TableForm[IntegerDigits[{1324501020, 3241030205, 4332020134}]]
(* ->
1 3 2 4 5 0 1 0 2 0
3 2 4 1 0 3 0 2 0 5
4 3 3 2 0 2 0 1 3 4
*)
</code></pre>
<p>This will make the list look like a matrix table form. <a href="https://mathe... |
3,408,082 | <blockquote>
<p><span class="math-container">$\textbf{Definition}$</span>: We say <span class="math-container">$f:\mathbb{R} \to \mathbb{R}$</span> is <em>intersecting</em> if for every nonempty <span class="math-container">$A \subset \mathbb{R}$</span>, <span class="math-container">$f[A] \cap A \neq \varnothing$</span... | ℋolo | 471,959 | <p>A complete answer for <span class="math-container">$\ell$</span>-intersecting for infinite <span class="math-container">$\ell$</span>:</p>
<p>Let <span class="math-container">$f:Q\to Q$</span>, define <span class="math-container">$D(f) = \{x \in Q: f(x) \neq x\}$</span> and <span class="math-container">$\ell(f)$</s... |
2,781,827 | <p>I need to find a symmetric matrix of real values (not the zero matrix) of any order that is orthogonal to any diagonal matrix of real values.
Any hints?</p>
| Community | -1 | <p><em>Hint:</em> Using the definition of orthogonality of matrices to be $\text { tr}(AB^t)=0$ (see <a href="https://math.stackexchange.com/a/1262311">here</a>), all that matters is what happens on the diagonal of the product matrix $AB^t$.</p>
<p>Think zeros on the diagonal... since $B$ is diagonal, this will res... |
3,082,779 | <p>I love watches, and I had an idea for a weird kind of watch movement (all of the stuff that moves the hands). It is made up of a a central wheel, with one of the hands connected to it (in this case, it will be the hour hand). This hand goes through a pivot, and then displays the time. I attached a video of a 3d mock... | Giuseppe Negro | 8,157 | <blockquote>
<p><strong>SUMMARY</strong>. This is formally wrong, as Ivo already showed. <strong>If the metric is <span class="math-container">$\delta_{\mu \nu}dx^\mu\otimes dx^\nu$</span></strong>, then it is morally correct. Otherwise, it is not; see the example at the end of this post.</p>
</blockquote>
<p>My und... |
2,756,686 | <p>I have a second derivative that I need to use to find inflection points to create a graph. The second derivative is $$f^{\prime\prime}(x)=-4\pi^2\cos(\pi(x-1))$$</p>
<p>So I set the equation to $0$ and solve for $x$</p>
<p>$$-4\pi^2\cos(\pi(x-1))=0$$</p>
<p>I divide by the constant $-4\pi^2$ and get </p>
<block... | Greg | 495,411 | <p>Assuming you're looking within $0 \le t \le 2\pi$.</p>
<p>Where does $\cos t = 0$?<br>
At $\pi/2$ and $3\pi/2$.<br>
Set $\pi(x-1)$ equal to each of these and solve for $x$. You'll get $3/2$ and $1/2$.</p>
|
262,425 | <p>I'm trying to integrate a function that involves a <em>finite</em> sum:</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty}\sum_{j=1}^n (e^{-b t^2}r_j) \,dt$$</span></p>
<p>I think it should be possible to take the exponent <em>outside</em> the sum:</p>
<p><span class="math-container">$$\int_{-\infty}^{\in... | Bob Hanlon | 9,362 | <p>Use a replacement <a href="https://reference.wolfram.com/language/ref/Rule.html" rel="nofollow noreferrer"><code>Rule</code></a> to swap the order when appropriate.</p>
<pre><code>Clear["Global`*"]
swap = Integrate[Sum[f_, iter1_, opts1___], iter2_, opts2___] :>
Sum[Integrate[f, iter2, opts2], iter... |
713,098 | <p>The answer to my question might be obvious to you, but I have difficulty with it. </p>
<p>Which equations are correct:</p>
<p>$\sqrt{9} = 3$</p>
<p>$\sqrt{9} = \pm3$</p>
<p>$\sqrt{x^2} = |x|$</p>
<p>$\sqrt{x^2} = \pm x$</p>
<p>I'm confused. When it's right to take an absolute value? When do we have only one va... | MPW | 113,214 | <p>The confusion about the sign is understandable. The square root symbol applied to a positive number always yields a positive number (disregarding the case of zero for the sake of simplicity here). The problem arises when you don't know ahead of time whether $x$ is positive or not. It is true that one of the numbers ... |
441,374 | <p>Let $K_{\alpha}(z)$ be the <a href="https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_I.CE.B1_.2C_K.CE.B1" rel="nofollow noreferrer">modified Bessel function of the second kind of order $\alpha$</a>.</p>
<p>I need to compute the following integral:</p>
<p>$$\int_0^\infty\;\;K_0\left(\sqrt{a(... | Ron Gordon | 53,268 | <p>OK, I think I have a way to make an approximation that holds up pretty well by using Laplace's method twice. </p>
<p>Begin by using the representation</p>
<p>$$K_0(u) = \int_0^{\infty} dt \, e^{-u\cosh{t}}$$</p>
<p>Then the integral you seek may be written, when we reverse the order of integration, as</p>
<p>$$\... |
598,635 | <p>Prove the two Identities for
$-1 < r < 1$</p>
<p>$$\sum_{n=0}^{\infty} r^n\cos n\theta =\frac{1-r\cos\theta}{1-2r\cos\theta+r^2}$$</p>
<p>$$\sum_{n=0}^{\infty} r^n\sin{n\theta}=\frac{r \sin\theta }{1-2r\cos\theta+r^2}$$</p>
<p>Sorry could not figure out how to format equations</p>
| DonAntonio | 31,254 | <p>Hint:</p>
<p>$$\sum_{n=0}^\infty \left(re^{\theta i}\right)^n=\frac1{1-re^{i\theta }}\;,\;\;\text{as long as}\;\;|r|<1$$</p>
<p>But</p>
<p>$$\frac1{1-re^{i\theta}}=\frac{1-re^{-i\theta}}{|1-re^{i\theta}|^2}$$</p>
<p>and now just take the real and the imaginary parts...and remember, of course, that a complex s... |
2,011,236 | <p>I was reading <a href="https://web.williams.edu/Mathematics/lg5/Hindman.pdf" rel="nofollow noreferrer">this</a> discussion of Hindman's Theorem by Leo Goldmakher, and was tripped up by his introduction of a topology on $U(\mathbb N)$. (He is using $U(\mathbb N)$ to denote the space of ultrafilters on the natural num... | Noah Schweber | 28,111 | <p>The answer to your first question is yes.</p>
<hr>
<p>For your second question, I like the following. Think of a set $A\subseteq\mathbb{N}$ as a question you're asking about some number $N$ that I know and you're trying to figure out (a la "Twenty Questions" but for large values of twenty) - the question is, "Is $... |
2,189,445 | <p>I try to solve this:
$$
\frac{\partial^{2} I}{\partial b \partial a} = I.
$$
I guessed $ I = C e^{a+b} $, but it's not the general solution. So, how to find the last one?</p>
| skyking | 265,767 | <p>The statement (premise) is not a paradox, it's a lie - a false statement. If we let $xSy$ mean $x$ shaves $y$ and $b$ is the barber the statement becomes:</p>
<p>$$\forall x(bSx\Leftrightarrow \neg xSx)$$</p>
<p>this means especially $bSb\Leftrightarrow \neg bSb$ which is formally false or expressed mathematically... |
3,086,878 | <p>Recently I came across this general integral,
<span class="math-container">$$\int \frac {dx}{(x^2-2ax+b)^n}$$</span>
Putting <span class="math-container">$x^2-2ax+b=0$</span> we have,
<span class="math-container">$$x = a±\sqrt {a^2-b} = a±\sqrt {∆}$$</span>
Hence the integrand can be written as,
<span class="math-co... | Aleksas Domarkas | 562,074 | <p>Let <span class="math-container">$b\neq a^2$</span>,
<span class="math-container">$$S(n)=\int \frac {dx}{(x^2-2ax+b)^n}$$</span>
With method of undetermined coefficients we find formula
<span class="math-container">$$S(n)=\frac{Ax+B}{(x^2-2ax+b)^{n-1}}+CS(n-1)$$</span>
We get
<span class="math-container">$$1=-\left... |
3,086,878 | <p>Recently I came across this general integral,
<span class="math-container">$$\int \frac {dx}{(x^2-2ax+b)^n}$$</span>
Putting <span class="math-container">$x^2-2ax+b=0$</span> we have,
<span class="math-container">$$x = a±\sqrt {a^2-b} = a±\sqrt {∆}$$</span>
Hence the integrand can be written as,
<span class="math-co... | Przemo | 99,778 | <p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span> and <span class="math-container">$x$</span> be real and <span class="math-container">$n$</span> be a positive integer. Let <span class="math-container">$\Delta:= a^2-b$</span>.
then the following formula holds:
<span class="math-con... |
737,915 | <p>I'm reading Calculus: Basic Concepts for High School Students and am trying to digest the definition of 'limit of function'. There are two details that I am struggling to fully accept:</p>
<ol>
<li><p>If you are supposed to pick an interval $(a - \delta, a + \delta)$ but $a$ can be an undefined point at the end of ... | Rustyn | 53,783 | <p>Any function that has a jump discontinuity is an example. Take for example the function defined by: $f(x) = x$ if $x\ne 4$, $f(x) = 5$ if $x=4$. Then $$\lim_{x\to 4}f(x) = 4$$ but $f(4) = 5$. </p>
|
4,159,771 | <p>I understand the geometric intuition behind determinants but what is the real life use of it? I'm not looking for answers along the lines of "it helps to find solutions to linear systems" etc, unless this is one of those concepts that is useful because it allows us to do "more math". I'm more int... | umby | 839,300 | <p>Simplifying, in the deformation theory of bodies, the determinant of the deformation tensor gives the ratio between the final volume, in the deformed state, and the initial volume, in the undeformed state. Then, if det > 0 (< 0), you have a deformation causing an expansion (contraction), whether if det = 0 the... |
3,121,361 | <p>Given <span class="math-container">$G$</span> has elements in the interval <span class="math-container">$(-c, c)$</span>. Group operation is defined as:
<span class="math-container">$$x\cdot y = \frac{x + y}{1 + \frac{xy}{c^2}}$$</span></p>
<p>How to prove closure property to prove that G is a group?</p>
| Calum Gilhooley | 213,690 | <p>If <span class="math-container">$x \geqslant y > 0$</span>, and <span class="math-container">$n$</span> is a positive integer, then
<span class="math-container">$$
x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + \cdots + y^{n-1})
\geqslant n(x - y)y^{n-1}.
$$</span>
Therefore, for <span class="math-container">$n > 1$... |
2,555,499 | <p>Let $v_1=(1,1)$ and $v_2=(-1,1)$ vectors in $\mathbb{R}^2$. They are <strong>clearly linearly independent</strong> since each is not an scalar multiple of the other. The following information about a linear transformation $f: \mathbb{R}^2 \to \mathbb{R}^2$ is given: $$f(v_1)=10 \cdot v_1 \text{ and } f(v_2)=4 \cdot ... | Benji Altman | 398,014 | <p>This is where definitions become necessary. How I was taught order of operations was that division and multiplication have the same precedence, and that weather written as $ab$ or $a\cdot b$ or even $a\times b$, it's all the same thing (assuming $a$ and $b$ are numbers and not vectors or something). Based on what I ... |
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