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 |
|---|---|---|---|---|
629,923 | <p>Let $G$ be a finite solvable group and let $x_1,x_2,...,x_n$ be elements of $G$ of pairwise relatively prime orders. If the product $x_1x_2...x_n=1$, prove that each $x_i=1$.</p>
<p>I have no idea. Tell me some hints.</p>
<p>Thanks in advanced.</p>
| achille hui | 59,379 | <p>Let $\displaystyle a_n = \frac{n!}{n^n} e^n$. For any $x > 0$, MVT tell us</p>
<p>$$\log(1+x) = \frac{x}{1+y} < x\quad\text{ for some } y \in (0,x)$$</p>
<p>This implies for any $n \ge 1$, we have</p>
<p>$$\log(1+\frac{1}{n}) < \frac{1}{n}\quad\implies\quad (1+\frac{1}{n})^n < e$$</p>
<p>As a result,... |
2,295,515 | <p>I was doing a problem and I ran into this triangle:</p>
<p><a href="https://i.stack.imgur.com/sFSxM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sFSxM.png" alt="enter image description here"></a></p>
<p>The problem itself is not that important for this question, but rather the explanation of ... | marty cohen | 13,079 | <p>To bisect the triangle,
you have to bisect
the two sides separately.
Only one side
has a bisector of length 6.4.
The other side has
a bisector of length 6.0.</p>
|
3,561,155 | <p>For real numbers <span class="math-container">$a$</span> and <span class="math-container">$b$</span> define</p>
<p><span class="math-container">$$f(x) = \frac1{ax+b}$$</span></p>
<p>For which <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are there three distinct real numbers <... | Quanto | 686,284 | <p>Assume <span class="math-container">$a\ne 0$</span>. Then, <span class="math-container">$$x_1 = \frac1{ax_3+b}= \frac1{a\frac1{ax_2+b}+b}=\frac1{a\frac1{a\frac1{ax_1+b}+b}+b}$$</span></p>
<p>which leads to</p>
<p><span class="math-container">$$(a+b^2)(ax_1^2+bx_1-1) = 0\tag 1$$</span></p>
<p>and similarly, </p>
... |
2,068,499 | <p>Related to <a href="https://math.stackexchange.com/questions/2068466/confusion-of-central-limit-theory">Confusion of central limit theory</a>, and the fact that I just finished my first course in (Master's-level) graduate-level probability, which relates to this material.</p>
<p>The Central Limit Theorem states tha... | Community | -1 | <p>Nice question! If the problem is that the variance vanishes as $n \to \infty$, here's an intuitive explanation. Let's see it from this view point: Why does <em>variance</em> exists at all? Because we are <em>uncertain</em> about the <em>random variable</em> $X$. The variance of $\overline{X_n}$ is a <em>measure of u... |
4,457,260 | <p>I need to translate the distance between two points into an angle from 0 to 359. To do this I use the new position coordinates which is defined by a vector and subtract the original position. This generates a difference which I can then use to define the slopes absolute value (I know it can be done with out the abso... | true blue anil | 22,388 | <p>Not <em>quite</em> a routine stars and bars question.</p>
<p>The first digit has to be <span class="math-container">$\geq 1$</span>, whereas the others can be <span class="math-container">$\geq 0$</span></p>
<p>We can equalize the lower limits by substituting <span class="math-container">$X_1 = x_1-1$</span> as sugg... |
2,118,974 | <p>Prove that:$$\frac {\sec (16A) - 1}{\sec (8A) - 1}=\frac {\tan (16A)}{\tan (4A)}$$.</p>
<p>My Attempt,</p>
<p>$$L.H.S= \frac {\sec (16A)-1}{\sec (8A)-1}$$
$$=\frac {\frac {1}{\cos (16A)} -1}{\frac {1}{\cos (8A)} -1}$$
$$=\frac {(1-\cos (16A)).(\cos (8A)}{(\cos (16A))(1-\cos (8A))}$$.</p>
<p>What should I do next?... | lab bhattacharjee | 33,337 | <p>As $\cos2y=1-2\sin^2y,\sin2y=2\sin y\cos y$</p>
<p>$$\dfrac{1-\sec16A}{\tan16A}=\cdots=\dfrac{\cos16A-1}{\sin16A}=-\dfrac{2\sin^28A}{2\cos8A\sin8A}=-\tan8A$$</p>
<p>Similarly, $$\dfrac{1-\sec8A}{\tan8A}=-\tan4A$$</p>
<p>Can you take it home from here?</p>
|
4,050,395 | <p><strong>Problem</strong></p>
<p>A survey of boy-girl couples suggests that in 20% of couples both the boy and the girl love pizza, in 5% of couples the girl loves pizza, but the boy doesn't, in 10% couples the boy loves pizza, but the girl doesn't.</p>
<p>If a boy is chosen randomly and he loves pizza, what's the pr... | Henry | 6,460 | <p>Hint:</p>
<p>In <span class="math-container">$100$</span> couples:</p>
<ul>
<li>How many couples do you expect to see where both the boy and the girl love pizza?</li>
<li>How many couples do you expect to see where the boy loves pizza but the girl does not?</li>
<li>How many couples do you expect to see where the bo... |
2,615,176 | <p>The nature of the curve $f(x) = x (\log_{2} x)^6$ is baffling me. I expect the curve looks rather like plot of $f(x) = x\log_{2} x$; however probing on Wolfram Alpha says it does not. The shape of the graph looks more like a squashed $f(x) = x^2$ which is not all that surprising but just doesn't seem right.</p>
<p>... | user | 505,767 | <p>The key fact are that </p>
<p>$$(\log_2x)^6\ge0$$ </p>
<p>$$(\log_21)^6 = 0$$ </p>
<p>$$x\to 0^+ \implies x(\log_2 x)^6\to0$$ </p>
<p>thus there exist a maximum for $x\in(0,1)$.</p>
<p>The difference for $x>1$ with $x\log_2x$ can be understood taking a look at the second derivatives of the functions</p>
<p... |
317,357 | <p><strong>The Question</strong></p>
<p>Given Hessian manifold <span class="math-container">$M$</span>, there is a natural Kahler structures on <span class="math-container">$TM$</span>. Is it possible to write the Kahler potential of these in terms of the Hessian potential?</p>
<p><strong>Background</strong></p>
<p>... | Gabe K | 125,275 | <p>It turns out the answer was simpler than I initially thought, so thought I would provide an answer in case others were interested. S.Surace's suggestion in the comments was exactly right, but I didn't understand how to use it when I first asked the question.</p>
<p>Given any set of coordinates <span class="math-con... |
29,435 | <p>Suppose $f$ and $g$ are two embeddings of $S^1$ in $R^3$ or in $S^3$. How do I show whether they form a link or not?</p>
| Douglas Zare | 8,345 | <p>I think the common interpretation of your question is to determine whether you can pull the components apart with an ambient isotopy. This is a harder problem than you might expect, but it is doable. The key is that being able to pull the knots apart is equivalent to whether there is a topological sphere in the comp... |
2,321,285 | <p>For my abstract algebra course I have to decide whether
$$(*)\qquad X^4 + a^2, \; a \in \mathbb{Z}\;\text{odd}$$ is irreducible over $\mathbb{Q}[X]$ and $\mathbb{Z}[X]$. </p>
<p>Since the degree of the polynomial is 4 and thus even I should be able to split it into two polynomials of degree two $if$ the coefficie... | dxiv | 291,201 | <blockquote>
<p>After coefficient comparison I get several conditions</p>
</blockquote>
<p>Hint: it follows that $d=-b\,$, then either $b=0$ (which can be easily discarded) or $e=c=\pm a$ and $b^2 = 2c$. But the square of an integer cannot be twice an odd number, so the latter cannot hold.</p>
|
2,321,285 | <p>For my abstract algebra course I have to decide whether
$$(*)\qquad X^4 + a^2, \; a \in \mathbb{Z}\;\text{odd}$$ is irreducible over $\mathbb{Q}[X]$ and $\mathbb{Z}[X]$. </p>
<p>Since the degree of the polynomial is 4 and thus even I should be able to split it into two polynomials of degree two $if$ the coefficie... | Anurag A | 68,092 | <p>Since $a$ is odd, $a^2+1 \equiv 2 \pmod{4}$. So consider the polynomial $(x+1)^4+a^2$. Irreducibility of this polynomial can be ascertained by Eisenstein's criterion. This will show that your actual polynomial (which is a shift of this one) is also irreducible.</p>
|
2,634,752 | <p>So the question is to solve this differential equation:
$y'' - 4y' + 5y = 1 + t$.</p>
<p>I've got the general solution to $\ C_1e^{2t}\cos(2t) + C_2e^{2t}\sin(2t) $</p>
<p>However, I don't really know how to proceed with the particular solution.</p>
<p>I know that t is a polynomial of the first degree and should... | Dr. Sonnhard Graubner | 175,066 | <p>make for the particular solution the ansatz $$y_p=At^2+Bt+C$$</p>
|
583,544 | <p>Sometimes I see an exercise like this: </p>
<blockquote>
<p>Let $f:[a,b] \to \mathbb R$ be differentiable. (A few more givens here.) Show that $f'$ has such-and-such property.</p>
</blockquote>
<p>What is usually meant by that? Should $f'$ be defined on $[a,b]$ with one-sided derivatives in $a,b$? Or should one ... | Nate | 91,364 | <p>This just begs to be telescoped:</p>
<p>$$\sum_{k=2}^\infty\sum_{n=1}^k \frac{1}{k(k-1)n} = \sum_{k=2}^\infty \frac{1}{k(k-1)} + \sum_{n=2}^\infty \frac{1}{n} \sum_{k=n}^\infty\frac{1}{k(k-1)}$$</p>
<p>$$= \sum_{k=2}^\infty (\frac{1}{k-1} -\frac{1}{k}) + \sum_{n=2}^\infty \frac{1}{n} \sum_{k=n}^\infty(\frac{1}{k-... |
2,676,479 | <p>Say you have $f(x) $ and $g(x)$ and $g(x) = f^{-1}(x) $.</p>
<p>I observed that these two curves need not intersect, for example with $f(x) = e^x$ and $g(x) = \ln x $ never intersecting each other.</p>
<p>I also observed that a function can either have one, two, or three intersections with its inverse, but I was u... | Community | -1 | <p>A function can have over $3$ intersection points with its inverse relation.</p>
<p>For example, take $f(x)=x^4-3x^2+2x-3$.</p>
<p>Finding points of intersection of $f(x)$ and $f^{-1}(x)$:</p>
<p><a href="https://i.stack.imgur.com/kRvcF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kRvcF.png" ... |
2,004,745 | <blockquote>
<p>Find point on the curve $x^2+2y^2=6$ whose distance from the line $x+y-7=0$ is, minimum</p>
</blockquote>
<p>$\bf{My\; Try::}$ Let $(x,y)$ be any point on the curve $x^2+2y^2=6\;,$ Then we have to</p>
<p>minimize $\displaystyle \left|\frac{x+y-7}{\sqrt{2}}\right|$</p>
<p>Using $\bf{Cauchy\; Schwarz... | Jacky Chong | 369,395 | <p>Using Stirling's approximation, we have that
\begin{align}
\frac{(2n)!e^nn^n}{n!(2n)^{2n}} \approx \sqrt{4\pi n}\left(\frac{2n}{e}\right)^{2n} \frac{e^nn^n}{(2n)^{2n}} \frac{1}{\sqrt{2\pi n}}\left( \frac{e}{n}\right)^n = \sqrt{2}.
\end{align}</p>
|
1,078,493 | <p>How can I prove that for each $n\in\Bbb Z^+$ $$\frac{f_{2n}}{f_{2n-1}}\leq\frac{1+\sqrt{5}}{2}$$ where each $f_i$ is a term of the Fibonacci sequence. </p>
<p>Any help is really appreciated</p>
| lab bhattacharjee | 33,337 | <p>$$\frac{a^{2n}-b^{2n}}{a^{2n-1}-b^{2n-1}}-a=\frac{b^{2n-1}(a-b)}{a^{2n-1}-b^{2n-1}}$$</p>
<p>will be $<0$ for real $a-b>0, b<0$</p>
<p>Using <a href="https://www.proofwiki.org/wiki/Euler-Binet_Formula" rel="nofollow">Euler-Binet Formula</a>, $a=\dfrac{1+\sqrt5}2>0,b=\dfrac{1-\sqrt5}2<0\implies a-b&... |
3,086,979 | <p>Attempt:
I've used that <span class="math-container">$\sin(x) = (2t)/(1+t^2)$</span> and <span class="math-container">$\cos(x) = (1-t^2)/(1+t^2)$</span>.
However I don't seem to get a perfect square, instead I get <span class="math-container">$$(2/((1+t^2)^2))(14t^2 +20t + 38)$$</span>.</p>
<p>I'm not sure if there... | Aleksas Domarkas | 562,074 | <p>I get
<span class="math-container">$$\frac{4 {{\left( t+5\right) }^{2}}}{{{\left( {{t}^{2}}+1\right) }^{2}}}$$</span></p>
|
1,684,045 | <p>Let $0<2x<\pi$. If $$\sum_{n=-2}^{\infty}\cos^n x=8,$$ then please find $x.$ </p>
<p>I tried $\sum_{n=-2}^{\infty}\cos^n x=\frac{1}{1-\cos x}+\frac{1}{\cos x}+\frac{1}{\cos^2 x}=\frac{1+\cos x}{\cos^2 x \sin^2 x}=8.$ But I cant find $x.$</p>
| hamid kamali | 208,328 | <p><strong>Hint</strong>: If $-1 \lt \cos(x)\lt 1$ and $\cos(x)\ne 0$, we get that:
$$
\begin{align}
\sum_{n=-2}^{\infty} \cos(x)&=\frac{1}{\cos^2(x)}+\frac{1}{\cos(x)}+\frac{1}{1-\cos(x)}\\
&=\frac{1-\cos(x)+(1-\cos(x))\cos(x)+\cos^2(x)}{(1-\cos(x))\cos^2(x)}\\
&=\frac{1}{(1-\cos(x))\cos^2(x)}
\end{align}
... |
2,824,996 | <p>I'm investigating iterations using Newton's method for $f(x)=x^5-x+1$.</p>
<p>I'm getting unusual results though. I've found that some starting values will result with a "triple oscillation" of results where iterations will loop between values of approximately $1.0003$, $0.7504$ and $-0.0871$. </p>
<p>However the ... | Elias Costa | 19,266 | <p>Use the <a href="https://en.wikipedia.org/wiki/Kantorovich_theorem" rel="nofollow noreferrer">Kantorovich's theorem on Newton's</a>
method in its classical formulation. Set the function <span class="math-container">$F:(-\infty,+\infty)\to \mathbb{R}$</span> by
<span class="math-container">$$
F(x)=e^x-x^{100}.
$$</... |
423,172 | <p>Let us suppose you have a certain function $f(x)$ and you want to find out in which intervals this function is decreasing, constant or increasing. I know you need to follow these steps:</p>
<ol>
<li>Find out $f'(x)$.</li>
<li>Find out the values for which $f'(x)=0$. In other words, we need to find out the zeroes of... | Mark McClure | 21,361 | <p>For the types of functions that you are probably dealing with, you can simply expand your step 2 to examine points where $f'(x)=0$ <em>or</em> $f'(x)$ is undefined. If this set has only isolated points, then these are the only points where the derivative can change sign. In the example that you give, we have
$$f'(... |
2,349,503 | <p>Let $f(x)=\frac{1}{1-x^2}$. </p>
<p>With a $u=x^2$ substitution, $f(x)$ becomes $g(u)=\frac{1}{1-u}$, which being the limit of a geometric series has the form $$g(u)=\sum_{k=0}^\infty u^k$$</p>
<p>Plugging the substitution back in yields a power series representation for $f(x)$</p>
<p>$$\frac{1}{1-x^2}=\sum_{k=0}... | José Carlos Santos | 446,262 | <p>$|x^2|<1\iff-1<x^2<1\iff 0\leqslant x^2<1\text{ (because $x^2\geqslant0$)}\iff x\in(-1,1)$</p>
|
2,349,503 | <p>Let $f(x)=\frac{1}{1-x^2}$. </p>
<p>With a $u=x^2$ substitution, $f(x)$ becomes $g(u)=\frac{1}{1-u}$, which being the limit of a geometric series has the form $$g(u)=\sum_{k=0}^\infty u^k$$</p>
<p>Plugging the substitution back in yields a power series representation for $f(x)$</p>
<p>$$\frac{1}{1-x^2}=\sum_{k=0}... | Trevor Gunn | 437,127 | <p>We have
\begin{align}
\sum_{k \ge 0} \frac{x^k}{1 + x} &= \sum_{k \ge 0} \sum_{\ell \ge k} (-1)^{\ell-k}x^\ell \\
&= \sum_{\ell \ge 0} \sum_{k \le \ell} (-1)^{\ell-k}x^\ell \\
&= \sum_{\ell \ge 0} x^\ell \sum_{k \le \ell} (-1)^{\ell-k}
\end{align}</p>
<p>where</p>
<p>$$ \sum_{k \le \ell} (-1)^{\ell-k} ... |
477,695 | <p>I was asked to compute the 8-th roots of $i$. </p>
<p>Correct me if I am wrong, but am I looking for 8 different solutions to this? My thought was to first consider a root of the form $cos(x) + i sin(x)$. So, we are dealing with $cos(8x) + isin(8x)$. At this point I believe the $cos(8x)$ is $1$. Therefore we have $... | Cameron Buie | 28,900 | <p>You started off okay, in assuming that an eighth root of $i$ has form $\cos x+i\sin x,$ from which we can indeed conclude that its eighth power is $\cos(8x)+i\sin(8x).$ However, its eighth power should be $i$, not $1$, so we need $x$ to satisfy $\cos(8x)=0$ and $\sin(8x)=1.$ We need only solve $\sin(8x)=1$. Do you k... |
253,495 | <p>It used to be that if I typed <code>==</code> or <code>-></code> these would automatically be rendered as <code>\[Equal]</code> and <code>\[Rule]</code>.</p>
<p>Now, for some reason, this feature has disappeared in 12.3.1.</p>
<p>I would like to have it back but I can't find any relevant option neither in the Set... | MannyC | 55,861 | <p>I discovered <a href="https://mathematica.stackexchange.com/a/251013/55861">this answer</a> afterwards.</p>
<p>It turns out that that didn't work for me. I had to set this option instead</p>
<pre><code>CurrentValue[$FrontEnd, {StyleHints, "OperatorRenderings"}] = <|
"|->" -> "\[Fu... |
42,053 | <p>I have this problem to solve:</p>
<blockquote>
<p>Hockey teams 1 and 2 score goals at times of Poisson process with rates 1 and 2. Suppose that $N_1(0)=3$ and $N_2(0)=1$.
What is the probability that $N_1(t)$ will reach 5 before $N_2(t)$ does?</p>
</blockquote>
<p>I've re-worded this to: What is the prob that ... | Shai Covo | 2,810 | <p>To find the probability of team 1 scoring a goal before team 2, you can consider the corresponding waiting times, which are exponentially distributed with means $1$ and $1/2$, respectively (and recall some previous exercise). Or, more simply, consider $\frac{{\lambda _1 }}{{\lambda _1 + \lambda _2 }}$ (and $\frac{{... |
237,496 | <p>I have a complex data set and I want to fit with a mathematical model using <code>NonlinearModelFit</code>, but an error appears. Following is the data I'm using and the code that I have tried.</p>
<pre><code> dataset = {{0.1, 78.2356 - 0.8683 I}, {0.1998, 78.4897 - 0.5794 I}, {0.2996,
78.1675 - 1.0737 I}, {0... | JimB | 19,758 | <p>I suspect that you need to have the same values of the parameters for both the real and imaginary parts. If the variances associated with the real and imaginary regressions are the same and the random errors are independent, then consider using <a href="https://mathematica.stackexchange.com/questions/866/simultaneo... |
1,293,971 | <p>Let $R = \mathbb{C}[x]$, $M = \mathbb{C}^2$. Does there exist a module structure over $M$ as an $R$-module or not?</p>
| Crostul | 160,300 | <p>There exist a lot of ways to show this. I think that the simplest is to say that we can define a $\Bbb{C}[x]$-module structure on $M$ by composing
$$\Bbb{C}[x] \longrightarrow \Bbb{C} \longrightarrow \operatorname{End}(M)$$
where the map $\Bbb{C}[x] \longrightarrow \Bbb{C}$ is any ring morphism (for example $f \maps... |
2,344,414 | <p>Can anyone tell me how I should calculate this problem?</p>
<p>You are on vacation in a foreign country, and you need help. There are 50 people nearby, and you know <strong>25% of the population of this country speaks English</strong>. </p>
<p>If you were to call for help: </p>
<p>(a) <strong>What is the probabil... | G Tony Jacobs | 92,129 | <p>Each person has a $.75$ probability of not speaking English. If we can assume that this probability applies independently to each person in the crowd (a <em>very</em> questionable assumption), then we would calculate the probability that none of them speaks English by raising $.75$ to the $50$th power. That yields a... |
1,552,992 | <p>Let $G$ be abelian. Let $n \in \mathbb{Z}^+$ such that the order of $G$ and $n$ are relatively prime. Show that the function $\varphi : G \rightarrow G$ defined by $\varphi (a) = a^n $ is an automorphism for $a \in G$.</p>
<p>I'm stuck in proving 1-1-ness of the function. Let $a,b$ in $G$ such that $\varphi (x) = \... | Jytug | 122,418 | <p><strong>Hint</strong>: use the fact that $ \gcd(|G|, n ) = 1 $, hence, by Euclid's algorithm (or from the mentioned Bézout's identity), there exist integer $ x,y $ such that
$$ x|G| + yn = 1 $$ So</p>
<p>$$ a = a ^{x|G|}a^{yn} = a^{yn} = b^{yn} = b $$</p>
|
4,315,926 | <p>This question is in the context of a robotics problem. The goal is to track a robot using both its onboard odometry system and a VR system (HTC Vive Pro) using a VR controller mounted to the robot.</p>
<p>What is <strong>known</strong> is the transformation between odometry origin and the robot (measurements <span c... | Quanto | 686,284 | <p>Express the integral as
<span class="math-container">\begin{align}
\int_{0}^{\pi} \frac{d x}{(a-\cos x)^{n}}
=\sum_{k=0}^{[\frac{n-1}2]} \frac{\binom{n-1}{2k}a^{n-2k-1}}{(a^2-1)^{n-1/2}}\int_0^{\pi}\cos^{2k}x \ dx
= \frac{\pi P(a)}{\left(a^{2}-1\right)^{n-{1}/{2}}}
\end{align}</span>
which leads to
<span class="math... |
1,514,028 | <p>It's pretty obvious to me as I see the plot of the two functions but how can I prove it with some algebra?</p>
| More water plz | 282,704 | <p>So here's an answer involving Calculus:</p>
<p>consider $f(x)=x-\sin x $</p>
<p>$f'(x) =1- \cos x > 0$</p>
<p>for $x \in [0,1]$, $f$ is increasing, so $f(x)>f(0)=0$ for $x \in [0,1]$</p>
<p>which means $x>\sin x \implies |x|>|\sin x|$ since $x$ and $\sin x$ is positive here.</p>
<p>for $x \in [-1,0]... |
2,989,607 | <blockquote>
<p>Christina is standing in a rectangular garden. Her distances from the corners of the garden are <span class="math-container">$6$</span> meters, <span class="math-container">$7$</span> meters, <span class="math-container">$9$</span> meters, and <span class="math-container">$d$</span> meters, where <spa... | user | 505,767 | <p><strong>HINT</strong></p>
<p>We have 5 unknown to determine, then we can guess at first <span class="math-container">$6$</span> opposite to <span class="math-container">$d$</span></p>
<p><a href="https://i.stack.imgur.com/2Nxak.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2Nxak.jpg" alt="ente... |
811,490 | <p>Two points A and B are on different sides of a line. Find a point Y on the line such that the absolute value of the difference from Y to A and Y to B is maximal.</p>
<p>My thoughts are as follows. Let's say that point Y is on a line but is not collinear with A and B. Then I could draw a triangle by joining the 3... | user153386 | 153,386 | <p>(0,0) because that would make both lines the maximum size they could be</p>
|
3,738,646 | <p>I think that a constant sequence is a Cauchy sequence because of the definition of Cauchy sequence:</p>
<p>The sequence <span class="math-container">$\{q_n\}$</span> of rationals is a Cauchy sequence if, for every <span class="math-container">$\epsilon>0$</span>, where <span class="math-container">$\epsilon$</spa... | ECL | 786,460 | <p>First call <span class="math-container">$f$</span> the function
<span class="math-container">$$f:x\mapsto \sqrt{2+\sqrt{x}}\,.$$</span>
<span class="math-container">$f$</span> is an increasing function because <span class="math-container">$g:y\mapsto\sqrt{y}$</span> and <span class="math-container">$h:y\mapsto 2+y$... |
3,424,899 | <p>Let be vetors <span class="math-container">$(1, 2, 3,−1)$</span> and <span class="math-container">$(3, 6, 9,−3)$</span> a solutions of some system of linear equations with factors from <span class="math-container">$R$</span>. Prove that the vector <span class="math-container">$(0, 0, 0, 0)$</span> is also a solutio... | John Omielan | 602,049 | <p>Since <span class="math-container">$\frac{1}{1+z^2} = \frac{1}{1-(-z^2)}$</span>, the Taylor expansion around <span class="math-container">$z = 0$</span> (i.e., the Maclaurin series) would be the sum of an infinite <a href="https://en.wikipedia.org/wiki/Taylor_series#Geometric_series" rel="nofollow noreferrer">Geome... |
5,197 | <p>Is it possible to split a list into two lists at a specific position? The main list is for example: </p>
<pre><code>data={{xa,ya},{xb,yb},{xc,yc},...,{xz,yz}}.
</code></pre>
<p>I want to split this list into two new lists: </p>
<pre><code>data1={{xa,ya},{xb,yb},...,{xi,yi}}
</code></pre>
<p>and </p>
<pre><cod... | DavidC | 173 | <p>Following up on J.M.'s suggestion,</p>
<pre><code>m = 3;
data1 = Take[data, m]
data2 = Take[data, -(Length[data] - m)]
</code></pre>
<p>You might also obtain data2 as follows:</p>
<pre><code>data2 = Complement[data, data1]
</code></pre>
<p>I'm uncertain whether the second approach would maintain order invariant ... |
2,934,625 | <p>There is a little thing that I do not understand, about the Fourier transform of a product of functions in <span class="math-container">$L^1$</span> (and only in this space), with the relation <span class="math-container">${\mathcal F}(f g)(\lambda) = \mathcal{F}(f)\star \mathcal{F}(g) (\lambda)$</span> <strong>(and... | René Gy | 130,022 | <p>This is not an answer, but a conjecture which does not fit into a comment. </p>
<p>Let <span class="math-container">$q$</span> be a prime number and <span class="math-container">$n=\prod_q q^{v_q}$</span> be the prime factorization of the integer <span class="math-container">$n >1 $</span>. We have:</p>
<block... |
2,809,243 | <p>I want to show that there doesn't exist any $3$-dimensional manifold for which the fundamental group is isomorphic to the direct sum $\Bbb Z\oplus \Bbb Z\oplus\Bbb Z\oplus\Bbb Z$
I have calculated the fundamental group of $K_{3,3}$ graph and $K_5$ graph and if i have done right it was the direct sum of four copies o... | Community | -1 | <p>Here is a proof for <em>closed manifolds</em> avoiding the sphere theorem. It is unclear to me if something like this can be made workable for manifolds with boundary.</p>
<p>Given a space $X$ with fundamental group $\pi$, the natural map $f: H_2(X) \to H_2(\pi)$, induced by the map $X \to B\pi$ classifying the uni... |
2,152,736 | <p>With respect to a matrix $A\in\mathbb{R}^{n\times n}$, is there any if and only if conditions that make sure the eigen values of $A$ remain real and distinct? $A$ here is not symmetric but is stable and also there exists a $P$ such that $PAP^{-1}$ is symmetric.
For having real eigen values, $A$ needs to symmetric or... | Blazej | 155,834 | <p>As pointed out by Peter, it is sufficient that the characteristic polynomial $\mathrm{det}(A-t I)$ has no double zeros. While this condition might be difficult to check in general, you have rather special form of $A$. You can compute the characteristic polynomial (by hand or using some software like mathematica to m... |
98,451 | <p>I need to do an inverse Laplace transform of a function with essential singularities for a specific problem. I find it is very similar to an equation J. Noolandi worked out in one of his papers in 1977. The paper is at <a href="http://link.aps.org/doi/10.1103/PhysRevB.16.4466" rel="nofollow noreferrer">this page</a>... | Zen Harper | 6,651 | <p>After taking a quick look at the paper, I agree with Robert Israel.</p>
<p>Almost no detailed justification is given (<em>perhaps not surprising for a Physics journal...</em>), and it seems to be not so easy to justify.</p>
<p>Take the special case <span class="math-container">$N=1$</span> for simplicity; so up to c... |
1,884,791 | <p>I'm trying to prove the convergence of </p>
<p>$$\int^{\infty}_{0} \frac{1}{e^x+1}dx$$</p>
<p>Any suggestion or hint for how to deal with it?</p>
<p>Thanks in advance!</p>
| Nate Stemen | 357,182 | <p>What if you compare the integrand to $\mathrm{e}^{-x}$ because we know how to integrate that. We also know that it is always greater than $0$, so use the squeeze theorem.</p>
|
2,483,254 | <p>I have to compute the limit</p>
<p>$$
\lim\limits_{N \to +\infty} \sqrt{N+1} \log \left(1+\frac{x}{N+1}\right)
$$
where $x \ge 0$ is fixed. I tried to see the previous as</p>
<p>$$
\log \lim\limits_{N \to +\infty} \left(1+\frac{x}{N+1}\right)^{\sqrt{N+1}}
$$
and to change variable, but it doesn't work. Intuitively... | Ennar | 122,131 | <p>You can write:</p>
<p>$$\lim_{n\to\infty}(\sqrt{n+1}\ln(1+\frac x{n+1})) = \lim_{n\to\infty}\frac{(n+1)\ln(1+\frac x{n+1})}{\sqrt{n+1}} = \lim_{n\to\infty}\frac{\ln(1+\frac x{n+1})^{n+1}}{\sqrt{n+1}}. $$</p>
<p>Can you finish from here?</p>
|
2,998,342 | <p>Find <span class="math-container">$y$</span> such that </p>
<p><span class="math-container">$$\lfloor y \rfloor + \lfloor 3y \rfloor = 5$$</span></p>
<p>First, I use the properties that <span class="math-container">$$n \leq y <n+1$$</span>
And suppose </p>
<p><span class="math-container">$$\lfloor y\rfloor = 5... | xarles | 39,290 | <p>I try to explain an elementary proof. First, we consider the equation <span class="math-container">$2^n=p^q-1$</span>. We have
<span class="math-container">$$2^n=p^q-1=(p-1)(p^{q-1}+p^{q-2}+\cdots + p+1)$$</span>
Hence <span class="math-container">$p$</span> is odd, <span class="math-container">$p^{q-1}+p^{q-2}+\cd... |
3,151,329 | <p>How to find range of this function, it looks easy but somehow I am not able to do that . <br> What is more important to me is , why is wolframalpha unable to find the range , is it not a simple enough function ?
<img src="https://i.stack.imgur.com/pwGkm.png" alt="enter image description here"></p>
| skewfield | 527,888 | <p>Since the domain is <span class="math-container">$[-1,1]$</span> (otherwise the square root is not defined), we only have to take <span class="math-container">$x$</span> in that domain. Now, the first factor take values in <span class="math-container">$[-1,1]$</span>, instead the second one in <span class="math-cont... |
3,151,329 | <p>How to find range of this function, it looks easy but somehow I am not able to do that . <br> What is more important to me is , why is wolframalpha unable to find the range , is it not a simple enough function ?
<img src="https://i.stack.imgur.com/pwGkm.png" alt="enter image description here"></p>
| lab bhattacharjee | 33,337 | <p>As <span class="math-container">$\sqrt{1-x^2}\ge0,1-x^2\ge0\implies-1\le x\le1$</span></p>
<p><strong>Method</strong> <span class="math-container">$\#1:$</span></p>
<p>Using <a href="https://artofproblemsolving.com/wiki/index.php/Arithmetic_Mean-Geometric_Mean_Inequality" rel="nofollow noreferrer">Arithmetic Mean-... |
1,055,229 | <p>Consider the following exercise from a book I'm reading:</p>
<p>If $n$ is odd show that
$$ f: O(n) \to SO(n) \times \{1,-1\}, A \mapsto (A \operatorname{det}{A}, \operatorname{det}{A})$$</p>
<p>is an isomomorphism.</p>
<p>But why does $n$ have to be odd? I can easily show that it's a group homo., injective and s... | Simon S | 21,495 | <p>If $n$ is even, $n = 2k$, then</p>
<p>$$\operatorname{sgn}(\det(A \det A)) = \operatorname{sgn}(\det A (\det A)^{2k}) = \operatorname{sgn}(\det A) \cdot \operatorname{sgn}(\det A)^{2k} = \operatorname{sgn}(\det A)$$</p>
<p>so $A \det A$ isn't a member of $SO(n)$ unless $A$ was originally.</p>
|
3,237,995 | <p>Let <span class="math-container">$\Bbb L$</span> be a positive integer and define <span class="math-container">$\Bbb Z_L$</span> and <span class="math-container">$\Bbb Z^*_L$</span> as <span class="math-container">$$\Bbb Z_L=\{ 0,1,...,L-1\},\qquad\Bbb Z^*_L=\{i\in \Bbb Z_L\mid \gcd(i,L)=1\}$$</span> where <span cla... | trancelocation | 467,003 | <p>You can conclude the invertibility using the cancelling rule for modular equivalences:</p>
<ul>
<li><span class="math-container">$(\star)$</span>: If <span class="math-container">$(k,L)=1$</span>, then <span class="math-container">$ka \equiv kb \mod L \Leftrightarrow a \equiv b \mod L$</span></li>
</ul>
<p>Now, ta... |
1,527,183 | <p>This is likely a duplicate, but can't find it on MSE.</p>
<p>Let's say I have a normally distributed population with $\mu=2.75$ and $\sigma=0.25$. If $x$ is a value in the population of interest, using the normal probability distribution function I find that</p>
<p>$$P(x=3)\approx 0.96788,$$</p>
<p>and using the... | John Molokach | 90,422 | <p>I will assume upvotes validate the accuracy of the following answer:</p>
<p>The first value is not a probability, but a density value, $D_p$ given by</p>
<p>$$D_p=\frac1{\sigma\sqrt{2\pi}}e^\frac{-(x-\mu)^2}{2\sigma^2}.$$</p>
<p>The density for a value $x=a$ should be used to compare whether a value $x_i$ is more... |
4,447,551 | <p>Suppose we have a sample of <span class="math-container">$X_i = x_i$</span> for <span class="math-container">$i \in [1,n]$</span> where <span class="math-container">$\forall X_i$</span> they are iid with uniform distribution on the interval <span class="math-container">$[a,b]$</span></p>
<p>Now we want to find the M... | Andrei | 331,661 | <p><strong>THIS IS NOT AN ANSWER</strong>, but I can't post picture in the comment, and my clarification request is long.</p>
<p>You are not providing enough information. Look at the black line below <span class="math-container">$y=-x-m$</span>, and the intersections with the red, green, and blue curves. The RGB curves... |
614,045 | <p>I need to find $a\in \Bbb Z, 0\le a\lt10 : f(1 + \frac{a}{10}) = 0$ for a number of different quadratic functions, for example $f(x) = -x^2 + 4x - 3$, by "using a table and a graph". Can someone explain what that actually is, and how one uses the specified methods to solve this sort of problem?</p>
| AJMansfield | 50,951 | <p>Here is how you use a table to solve the problem. Make a list of all the allowed values of $a$, the corresponding values of $x$, and the resulting value for $f(x)$</p>
<pre><code> a │ x │ f(x)
───┼─────┼──────
0 │ 1.0 │ ?.??
1 │ 1.1 │ ?.??
2 │ 1.2 │ ?.??
┊ ┊
9 │ 1.9 │ ?.??
</code></pre>
<p>Now, lookin... |
614,045 | <p>I need to find $a\in \Bbb Z, 0\le a\lt10 : f(1 + \frac{a}{10}) = 0$ for a number of different quadratic functions, for example $f(x) = -x^2 + 4x - 3$, by "using a table and a graph". Can someone explain what that actually is, and how one uses the specified methods to solve this sort of problem?</p>
| Tyler Clark | 3,791 | <p>For the first one, try the quadratic formula (or factoring, but if you cannot see how to factor it, the quadratic formula will give you what you need anyway). You have that $x= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$ where the quadratic you have is of the form $y=ax^2+bx+c$. Therefore, in your case, $a=-1$, $b=4$, and $c... |
518,794 | <p>$f:R^2$ \{y=0} $\Rightarrow R$ , $f:(x,y)\Rightarrow x/y$.<br>
Prove (formally) that $f$ is continuous. </p>
<p>I think what I should show is that any point that belongs to an open ball of radius $\epsilon$ of image, has a pre-image that belongs to an open ball around (x,y), and since image and pre-image are both o... | Elchanan Solomon | 647 | <p>Yes, to show that such a function $f$ is continuous you want to show that the preimage of an open ball of radius $\epsilon$ contains an open ball around $(x,y)$. Here's why you should believe this is true. If $(x,y)$ is close to $(x',y')$, say </p>
<p>$$\sqrt{(x' - x)^2 + (y'-y)^2} <\delta$$</p>
<p>then we know... |
3,051,737 | <ul>
<li>Show that If <span class="math-container">$X$</span> is an any set, then <span class="math-container">$X\subset X\cup\left\{ X\right\}$</span></li>
</ul>
<p>Proof. Let <span class="math-container">$t\in X$</span>. We must show <span class="math-container">$t\in X\cup\left\{ X\right\}$</span>, that is we need ... | William Elliot | 426,203 | <p>Exercise.<br>
Show for all sets A and B, that A <span class="math-container">$\subseteq$</span> A <span class="math-container">$\cup$</span> B. </p>
<p>With that, your problem is just a special case.</p>
|
129,655 | <p>Here is a nice <em>sufficient (but not necessary)</em> condition on whether a quintic is solvable in radicals or not. Given,</p>
<p>$x^5+10cx^3+10dx^2+5ex+f = 0\tag{1}$ </p>
<p>If there is an ordering of its roots such that,</p>
<p>$x_1 x_2 + x_2 x_3 + x_3 x_4 + x_4 x_5 + x_5 x_1 - (x_1 x_3 + x_3 x_5 + x_5 x_2 +... | David Mitra | 18,986 | <p>I imagine it can be construed as the following:</p>
<p>If $\def\i{{\bf i}}\def\j{{\bf j}}\def\vdot{{\bf\cdot}}\def\r{{\bf r}}\def\ts{\textstyle}\def\/{}{\bf F}(x,y)=M(x,y)\i+N(x,y)\j$ gives a force field, the total work done by $\bf F$ on a point moving along the curve $C$ parameterized by ${\bf r}(t), a\le t\le ... |
129,655 | <p>Here is a nice <em>sufficient (but not necessary)</em> condition on whether a quintic is solvable in radicals or not. Given,</p>
<p>$x^5+10cx^3+10dx^2+5ex+f = 0\tag{1}$ </p>
<p>If there is an ordering of its roots such that,</p>
<p>$x_1 x_2 + x_2 x_3 + x_3 x_4 + x_4 x_5 + x_5 x_1 - (x_1 x_3 + x_3 x_5 + x_5 x_2 +... | bgins | 20,321 | <p>Along the line $x+y=1$ we have $dy=-dx$, so
$$\int_{\gamma}y^2dx+x^2dy=\int_1^0 \left((1-x)^2-x^2\right) dx=\int_0^1(2x-1)\,dx=\left[x^2-x\right]_0^1=0.$$</p>
<p>If the curve were closed, we could use <a href="http://en.wikipedia.org/wiki/Green%27s_theorem" rel="nofollow">Green's theorem</a>:
$$\int_{\partial R}Ldx... |
1,201,065 | <p>So we have that $a=1$, $b=0$, $c=1$, so our point reduces to $(t,t^{2}+1)$. Plugging that into the general equation for a line we have $t^{2}+ 1 = mt + b$ where $m$ is the slope and $b$ is the $y$-intercept. So we need this line to intersect $f$ once. Setting the graph of $f$ and the line equal to each other we hav... | Mankind | 207,432 | <p>I would utilize the result that a non-vertical line will intersect the graph of $f(x)=x^2+1$ only once, <em>exactly when the line is tangent to the graph of $f$</em>. You can try to prove this.</p>
<p>It doesn't say so in your question that the point $(t,at^2+bt+c)$ is a point on the graph of $f$, but from what you... |
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | mathreadler | 213,607 | <p>The $k$:th number will be:
$$[1,0,0]{\bf C}^k[1,2,3]^T$$</p>
<p>where ${\bf C}= \left[\begin{array}{ccc}0&1&0\\0&0&1\\1&0&0\end{array}\right]$</p>
<p>Now you can exchange the vector to the right for any ordered set of numbers you want to $\bf C$ycle through. </p>
<hr>
<p>(It is actually a... |
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | David Quinn | 187,299 | <p>As an alternative, here is a purely trigonometric function which generates the required sequence:</p>
<p>$$f(x)=\frac{\sqrt{3}\sin(\frac 23\pi(x-2))}{\cos(\frac 23\pi(x-2))+2}+2$$</p>
<p>Here we have $f(1)=1$, $f(2)=2$, $f(3)=3$, $f(4)=1$, and so on.</p>
|
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | HelloGoodbye | 119,068 | <p>The function</p>
<p>$$f(x)=\left\{
\begin{array}{ll}
x, & \text{if } x \leq n\\
f(x-n), & \text{otherwise}
\end{array}\right.$$</p>
<p>maps the set $\mathbb{Z}^+$ to the sequence $\{1, 2, \ldots, n, 1, 2, \ldots, n, \ldots\}$. Is this way "more systematic" to you?</p>
|
3,615,874 | <p>I'm trying to solve <span class="math-container">$\frac{d}{dx}e^{x}$</span>: I know <span class="math-container">$\ln(e^{x})=x$</span>, so <span class="math-container">$$(A)\frac{d}{dx}\ln(e^{x})=\frac{d}{dx}x=1$$</span>I can also apply the chain rule : <span class="math-container">$$(B)\frac{d}{dx}\ln(e^{x})=\frac{... | Luca Goldoni Ph.D. | 264,269 | <p>We have to prove that
<span class="math-container">$$
\forall \varepsilon > 0\exists \delta \left( \varepsilon \right) > 0:0 < \left| x \right| < \delta \left( \varepsilon \right) \Rightarrow \left| {x^4 \sin ^2 \frac{1}
{x}} \right| < \varepsilon
$$</span>
Since, for every <span class="math-conta... |
83,507 | <p>Is there any shorter / more handy / more flexible way to get directory tree as a nested <code>Rules/Associations</code> list?</p>
<p>I tried to incorporate <code>FileNames</code> + <code>GatherBy</code> and <code>GroupBy</code>, but nothing was as short and handy as old fashioned:</p>
<pre><code>ClearAll @ filesIn... | 2012rcampion | 21,750 | <p>First off, a more compact form of <code>filesInDir</code>:</p>
<pre><code>fileTree[f_, d_] /; DirectoryQ[f] && d > 0 :=
FileNameTake[f] -> (fileTree[#, d - 1] & /@ FileNames[___, f])
fileTree[f_, _] := FileNameTake[f]
</code></pre>
<p>Example usage:</p>
<pre><code>fileTree[$UserBaseDirectory, ... |
65,183 | <p>Let $p$ be an odd prime. I am interested in how many quadratic residues $a$ sre there such that $a+1$ is also a quadratic residue modulo $p$. I am sure that this number is
$$
\frac{p-6+\text{mod}(p,4)}{4},
$$
but I have neither proof nor reference. It is a particular case of the question in the title: if $a$ and $b... | arithboy | 9,944 | <p>There is an elementary argument regarding the last problem. Denote by $N(p)$ the number of pairs of $(a,b)$ such that $a,b,a+b$ are all quadratic residues mod $p$.
Hence we have
$$N(p)=\frac{1}{8}\mathop{\sum\sum}_{\substack{a,b\bmod p\\(ab(a+b),p)=1}}\left(1+\left(\frac{a}{p}\right)\right)\left(1+\left(\frac{b}{p}\... |
1,782,032 | <blockquote>
<p>Find the value of <span class="math-container">$$\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\cdots}}}}$$</span></p>
</blockquote>
<p>I know how to solve when all surds are of the same order, but what if they are different?</p>
<p>Technically, (as some users wanted to know <strong><em>exactly</em></str... | egreg | 62,967 | <p>The text is saying that the <em>restriction</em> of $f$ to each of the intervals that make the domain is a continuous function; the reason is that all these restrictions are constant.</p>
<p>So, yes, (the restriction of) the function can be continuous on $\{0\}$ without the function being continuous at $0$. More p... |
477,559 | <p>I understand there is propositional logic, first-order logic, second-order logic higher-order logic, and type theory, where the latter logics are extensions of the former logics. Can someone explain the differences between these types of logics for someone who knows some basic linear algebra, set theory, calculus, a... | Hunan Rostomyan | 89,895 | <p>I'll speak about their <em>grammatical</em> differences, leaving their proof- and model-theoretic differences for someone more qualified to discuss. Each of these logics has a <em>vocabulary</em> $V$, which is the set of symbols out of which its well-formed formulas (e.g. terms, sentences) are generated. One usuall... |
1,064,534 | <p>$$x^3-3y^2x=-1$$
$$3yx^2 -y^3=1$$</p>
<p>This was the real part and imaginary part on a previous question I asked, instead of the system it was easier to just use polar coordinates to solve, but if this was just a system unrelated to the problem, how would one solve it? I seem to have trouble when the terms are lik... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$x^3+3x^2y-3xy^2-y^3=0$$</p>
<p>$$(x-y)(x^2+xy+y^2)+3xy(x-y)=0\iff(x-y)(x^2+xy+y^2+3xy)=0$$</p>
<p>Check both cases</p>
|
3,104,936 | <p>I was wondering if anyone could explain to me why this holds true?</p>
<p><span class="math-container">$$C\left(n, k\right) = \frac{C\left(n, k-1\right) \cdot C(n - (k - 1), 1)}k$$</span></p>
<p>or</p>
<p><span class="math-container">$$C\left(3, 2\right) = \frac{C\left(3, 1\right) \cdot C(2, 1)}2$$</span></p>
<p... | 유현우 | 641,702 | <p>LHS is the number of ways to choose <span class="math-container">$k$</span> objects from <span class="math-container">$n$</span> objects.
RHS is nothing but, now from <span class="math-container">$n$</span> objects first choose <span class="math-container">$k - 1$</span> and then from remaining <span class="math-co... |
1,896,399 | <p>Im asked to solve find the derivative of: $$ \frac{\ln x}{e^x}$$
<strong>my attempt</strong>
$$D\frac{\ln x}{e^x} = \frac{\frac{1}{x}e^x + \ln (x) e^x}{e^x} = e^x \frac{\frac{1}{x}+\ln x}{e^{2x}} = \frac{\frac{1}{x}+\ln x}{e^x}$$</p>
<p>But this is apparently wrong and the correct answer is:
$$\frac{\frac{1}{x} - ... | Bishnu Basyal | 362,423 | <p>Use correct Quotient rule, D(u/v) = (u' v - u v') / v^2</p>
<pre><code> D( lnx / e^x ) = ( 1/x * e^x - lnx * e^x )/ e^2x
= ( 1/x - lnx)/e^x
</code></pre>
|
3,519,261 | <p>Here is the question.</p>
<blockquote>
<p>If <span class="math-container">$s_1=0$</span>, prove that <span class="math-container">$s_{n+1}=s_n+\frac{1}{2}(x-s_n^2)$</span> is monotone and converges if <span class="math-container">$0<x<1$</span>. </p>
</blockquote>
<p>I tried this question by induction, bu... | angryavian | 43,949 | <p>Here are two approaches</p>
<ul>
<li>reindexing:
<span class="math-container">$$\sum_{i=1}^n (k+i)^2 = \sum_{j=k+1}^{n+k} j^2 = \sum_{j=1}^{n+k}j^2 - \sum_{j=1}^k j^2 = \frac{(n+k)(n+k+1)(2n+2k+1) - k(k+1)(2k+1)}{6}$$</span></li>
<li>expanding:
<span class="math-container">$$\sum_{i=1}^n (k+i)^2 = \sum_{i=1}^n(k^2+... |
3,519,261 | <p>Here is the question.</p>
<blockquote>
<p>If <span class="math-container">$s_1=0$</span>, prove that <span class="math-container">$s_{n+1}=s_n+\frac{1}{2}(x-s_n^2)$</span> is monotone and converges if <span class="math-container">$0<x<1$</span>. </p>
</blockquote>
<p>I tried this question by induction, bu... | J. W. Tanner | 615,567 | <p><span class="math-container">$$\sum_{i=1}^3 {(2+i)^2}=\sum_{i=1}^3 {(4+4i+i^2)}=\sum_{i=1}^3 4+\sum_{i=1}^3 4i+\sum_{i=1}^3 i^2$$</span></p>
<p><span class="math-container">$$=4\times3+4\dfrac{3(3+1)}2+\dfrac{3(3+1)(2\times3+1)}6=12+24+14=50.$$</span></p>
<hr>
<p>Alternatively, <span class="math-container">$$\sum... |
954,658 | <p>How does
$$
\frac{1}{\log(1+\frac{d}{a})}
$$
simplify to $\frac{a}{d}$ in the limit that $d$ is tiny relative to $a$? I was wondering if a Taylor expansion would work though I am not sure how to work it.</p>
| Akiva Weinberger | 166,353 | <p>$\ln(1+\epsilon)=\epsilon+O(\epsilon^2)$. Or, another way of saying it:</p>
<p>$\ln(1+\epsilon)\approx\epsilon$ when $\epsilon$ is small. Since $\frac da$ is small, we have:</p>
<p>$$\frac1{\log(1+\frac da)}\approx\frac1{\frac da}=\frac ad$$</p>
|
682,057 | <p>I'm seeing a physics paper about percolation (<a href="http://arxiv.org/abs/cond-mat/0202259" rel="nofollow">http://arxiv.org/abs/cond-mat/0202259</a>). In the paper the following asymptotic relation is used without derivation.
$$
\sum_{k=0}^{\infty} k P(k) (1 - \epsilon)^{k-1} \sim \left<k\right> - \left<k... | Claude Leibovici | 82,404 | <p>I have not been able to see how is coming the $\epsilon^{\lambda - 2}$ term. However, and, may be, this could be a track $$
\sum_{k=0}^{\infty} k P(k) (1 - \epsilon)^{k-1}=\frac{c \Phi (1-\epsilon,\lambda -1,0)}{1-\epsilon}$$ when $P(k) = c k^{-\lambda}$ ($\Phi$ being the the Hurwitz-Lerch transcendent $\Phi(z,s,a)$... |
975,398 | <p>I want to know about the convergence or divergence of the following series:</p>
<p>$$\sum \sin (a_n) $$</p>
<p>where </p>
<p>$$a_n=\frac{(-1)^n}{n}$$</p>
<p>The tests that I tried were inconclusive. Is it possible to know?</p>
<p>Or, more generally, does this series converge for all conditionally convergent ser... | achille hui | 59,379 | <p>For the first part </p>
<blockquote>
<p>Does $\sum\limits_{n=1}^\infty \sin\frac{(-1)^n}{n}$ converges?</p>
</blockquote>
<p>The answer is YES by
<a href="http://en.wikipedia.org/wiki/Alternating_series_test" rel="nofollow">Alternating series test</a>. </p>
<p>For the second part</p>
<blockquote>
<p>Does $\... |
241,394 | <p>The convergence in the Central Limit Theorem is weak convergence, which is weaker than convergence in probability. I set it as an exercise to find an example that convergence in distribution does not imply convergence in probability:</p>
<blockquote>
<p>Let $(X_j)_{j\geq 1}$ be i.i.d. with $E[X_1]=0$ and $\sigma_... | TKM | 60,361 | <p>Assume $n=2m$</p>
<p>$P(|\frac{S_{n}}{\sqrt{n}}-\frac{S_{m}}{\sqrt{m}}|>\varepsilon)=$</p>
<p>$1-\int_{-\varepsilon \sqrt{nm}}^{\varepsilon \sqrt{nm}}\int\cdots \int f(z-\sqrt{m}\sum x_{i}-(\sqrt{m}-\sqrt{n})\sum x_{i} ) f((\sqrt{m}-\sqrt{n})x_{2})\cdots f((\sqrt{m}-\sqrt{n})x_{m})f(\sqrt{m}x_{m+1})\cdots f(\s... |
3,618,340 | <p>I read a set of notes on projective curves, where it introduced discrete valuations on a field <span class="math-container">$K$</span> as a surjective homomorphism <span class="math-container">$v:K^\times \to \mathbb Z$</span> where <span class="math-container">$v(x+y) \geq \min(v(x), v(y))$</span> for all <span cla... | Viktor Vaughn | 22,912 | <p>Let <span class="math-container">$X$</span> be a smooth projective curve over a field <span class="math-container">$k$</span>, let <span class="math-container">$P$</span> be a point on <span class="math-container">$X$</span>, let <span class="math-container">$\DeclareMathOperator{\O}{\mathcal{O}} \O_P$</span> be the... |
1,604,226 | <p>Let $a,b,c$ be positive real numbers. Prove that </p>
<p>$\frac{bc}{a^2+bc}+\frac{ca}{b^2+ca}+\frac{ab}{c^2+ab}\le{\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}}$.</p>
<p>I have no idea how to solve this question.</p>
| Chad Shin | 302,037 | <p>This proof appears to be a bit complex. </p>
<p>$ \sum _{ cyc }^{ { } }f(a,b,c)=f(a,b,c)+f(b,c,a)+f(c,a,b)$</p>
<p>Note that the $(LHS)\le \sum _{ cyc }^{ { } } \frac { (\frac { b+c }{ 2 } )^{ 2 } }{ a^{ 2 }+(\frac { b+c }{ 2 } )^{ 2 } } (\because AM-GM)$</p>
<p>WLOG $a+b+c=3$</p>
<p>Since $\sum _{ cyc }^{ }{... |
1,931,075 | <p>How many distinct ways can the letters of the word UNDETERMINED be arranged so that all the vowels are in alphabetical order?
Do I need to consider the positions of the vowels?</p>
| Dhruv Reshamwala | 426,496 | <p>For the word UNDETERMINED, vowels must be in the order "EEEIU"</p>
<p>If we ignore the vowels for a minute here and replace them with a <em>dash</em>, we have to arrange the following: -ND-T-RM-N-D. We can do that as we don't care of the rearrangements within the <em>dashes</em> as only one possibility is there ("E... |
4,593,996 | <p>I am looking for someone to explain how the images of basis vectors are calculated. There are a ton of problems like this in my textbook, but there is no explanation on how to do the calculation.</p>
<p><a href="https://i.stack.imgur.com/o7csZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/o7csZ.... | Irving Rabin | 977,423 | <p>The question has given you a linear map <span class="math-container">$T:\mathbb{P}_2 \rightarrow \mathbb{P}_2$</span> and defined it by <span class="math-container">$T(a_0+a_1t+a_2t^2) = a_1+2a_2t$</span>, so it's already given you exactly what this linear map is. You are then given a basis for <span class="math-con... |
3,428,083 | <p>I'm reading a book, and it defines the following definition: </p>
<p>Let <span class="math-container">$f:X \to [-\infty,+\infty]$</span> be a function , where <span class="math-container">$(X,d)$</span> is some metric space.</p>
<p>Define <span class="math-container">$\lim\inf\limits_{y\to x}f(y) :=\inf \{\lim\inf... | Olivier Roche | 649,615 | <p>Let <span class="math-container">$(E,<)$</span> be a well ordered set.</p>
<p>Suppose for contradiction that there is an infinite decreasing sequence <span class="math-container">$(x_n)_{n\in\mathbb{N}}$</span> of elements of <span class="math-container">$E$</span>.<br>
Since <span class="math-container">$E$</sp... |
273,412 | <p>I wrote the following table:</p>
<pre><code>k = 4;
T = Table[{a[n], 0, 1}, {n, 1, k}]
</code></pre>
<p>Which generates:</p>
<pre><code>{{a[1], 0, 1}, {a[2], 0, 1}, {a[3], 0, 1}, {a[4], 0, 1}}
</code></pre>
<p>Now in the I want to use this in the next table:</p>
<pre><code>Table[Table[a[n], {n, 1, 4}], x]
</code></pr... | Bob Hanlon | 9,362 | <pre><code>$Version
(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)
Clear["Global`*"]
k = 4;
T = Table[{a[n], 0, 1}, {n, 1, k}]
(* {{a[1], 0, 1}, {a[2], 0, 1}, {a[3], 0, 1}, {a[4], 0, 1}} *)
Attributes[Table]
(* {HoldAll, Protected, ReadProtected} *)
</code></pre>
<p>Since <code>Table... |
4,183,945 | <p>I have the following question:
Let <span class="math-container">$A$</span> be a symmetric positive definite real matrix. Then we are asked to show that there exits two matrices, <span class="math-container">$C$</span> and <span class="math-container">$D$</span> where <span class="math-container">$C$</span> is lower ... | Ali Shadhar | 432,085 | <p><span class="math-container">$$\int _0^1\frac{\operatorname{Li}_2\left(-x^2\right)}{\sqrt{1-x^2}}\:\mathrm{d}x=\int _0^1\frac{1}{\sqrt{1-x^2}}\left(\int_0^1\frac{x^2\ln(y)}{1+x^2 y}\mathrm{d}y\right)\mathrm{d}x$$</span></p>
<p><span class="math-container">$$=\int _0^1\ln(y)\left(\int_0^1\frac{x^2}{(1+x^2y)\sqrt{1-x^... |
1,562,416 | <p>Suppose $32$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so no two of the three chosen objects are adjacent nor diametrically opposite.
This is again a problem in a math contest in India and this is how I tried it:</p>
<p><strong>Number of ways of... | mrprottolo | 84,266 | <p>Your argument is not completely correct, as you counted more than once some combination. This a way to fix it:</p>
<p>Number of ways to choose $3$ adjacent points : $32$</p>
<p>Number of ways to choose $3$ points such that $\textbf{only}$ two of them are adjacent : $32\cdot28$. Because there are $32$ couples of ad... |
2,200,672 | <p>Find the roots of $f(x)=x^2+x+1$ modulo $7$, and modulo $13$, and modulo $91$ </p>
<p>I think for mod $7$ and $13$, it can be done by trial and error. But how about mod $91$?</p>
| Barry Cipra | 86,747 | <p>For any odd modulus, </p>
<p>$$x^2+x+1\equiv0\iff4x^2+4x+4\equiv0\iff(2x+1)^2\equiv-3$$</p>
<p>For the prime modulus $7$,</p>
<p>$$(2x+1)^2\equiv-3\equiv4\implies2x+1\equiv\pm2\implies4(2x+1)\equiv\pm8\implies x+4\equiv\pm1$$</p>
<p>which implies $x\equiv2$ or $x\equiv4$ mod $7$.</p>
<p>For the prime modulus $1... |
1,449,503 | <p>Could I not represent every odd square number in $\mathbb N$ using the following notation: $(2n-1)^2$ where $n \in \mathbb N$. For every $n= 1,2,3...$ I get the set $1,9,25...$ every odd square number. Why can't I do this? </p>
<p>I ask because then I could prove that if a natural numbers square is odd then the nu... | preferred_anon | 27,150 | <p>What if there is an even number which squares to an odd number? This is actually the key to the proof that you <em>can</em> represent any odd square in this way. Your 'proof' would be circular.</p>
|
4,417,018 | <p>Consider</p>
<p><span class="math-container">$$n \rightarrow ... \rightarrow an+b$$</span></p>
<p>to be a sequence of natural numbers to which we apply <span class="math-container">$3n+1$</span> or <span class="math-container">$\frac{n}{2}$</span> operations.</p>
<p>This sequence is called a cycle, if and only if <s... | Eric Snyder | 585,349 | <p>If we allow ourselves to apply <span class="math-container">$n_{i+1} = 3n_i + 1$</span> even when <span class="math-container">$2 \mid n_i$</span>, then we can find other cycles, for instance:</p>
<p><span class="math-container">$$4 \xrightarrow{*} 13 \rightarrow 40 \rightarrow 20 \rightarrow 10 \rightarrow 5 \right... |
3,403,664 | <p>I have been asked to integrate <span class="math-container">$\int Ae^{at} dt$</span> and I know that the integral is <span class="math-container">$\frac Aae^{at} + c$</span>.</p>
<p>Although this is simple it goes against my understanding that <span class="math-container">$\int a^{n} da = \frac a{n+1}^{n+1} + c$</... | Simon Fraser | 717,270 | <p>It can be shown that the derivative of <span class="math-container">$e^x$</span> is <span class="math-container">$e^x$</span> and so by the chain rule, the derivative of <span class="math-container">$e^{ax}$</span> with respect to <span class="math-container">$x$</span> is <span class="math-container">$ae^{ax}$</spa... |
1,779,314 | <p><strong>harmonic numbers</strong>
$H_n=\sum_{i=1}^{n}\frac{1}{i}$</p>
<p><strong>Euler's constant</strong></p>
<p>$\gamma=\lim_{n\to \infty} [H_n-\ln(n)]$</p>
<p><strong>Factorial</strong></p>
<p>$n!=n(n-1)(n-2)\cdots2\cdot1$; valid for all non-negative integers</p>
<p>Show that,</p>
<p>$$(s+1)\gamma-\int_0^1\... | Laars Helenius | 112,790 | <p>We need to be very careful about what we are assuming is distinguishable and what is not distinguishable. Both @almagest and @windircursed have correct answers with different sets of assumptions: </p>
<p>@almagest assumed that both the baskets and the fruit are indistinguishable while </p>
<p>@windircursed assumed... |
257,368 | <p>I have the following system of ODEs:</p>
<p>dx/dt = a/(1 + z) - Q*z</p>
<p>dy/dt = Qx - qy</p>
<p>dz/dt = qy - cz/(K + z).</p>
<p>Assuming K = 1, Q = q < c, and a = c*(Sqrt[c/Q] - 1), is there a way to use Mathematica to test whether a Hopf bifurcation exists?</p>
<p>Any help would be greatly appreciated!</p>
<p>... | Daniel Huber | 46,318 | <p>You will have a Hopf bifurcation if two Eigenvalues of the Jacobian at a fixpoint have pure imaginary conjugate complex values.</p>
<p>Therefore, we first need the equilibrium or fix points. With the parametervalues from your post we get the fixpoints:</p>
<pre><code>rhs= {a/(1+z)- Q z, Q x- q y, q y - c z/(K+z)} //... |
168,089 | <p>Recently I was stumped by the calculation of the probability
$$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$
where $A_i \sim \text{exp}(\lambda), S_i \sim \text{exp}(\mu), L \sim \text{exp}(\lambda)$ are mutually independent, $\lambda \neq \mu$ are two positive integers,... | Bjørn Kjos-Hanssen | 4,600 | <p>You are looking at <a href="http://en.wikipedia.org/wiki/Gamma_distribution#Summation" rel="nofollow">summation of independent gamma random variables</a> $\sum_i X_i$, in the hard case where the $X_i$ have different scale parameters. It was studied in: </p>
<p>P. G. Moschopoulos (1985) The distribution of the sum o... |
2,832,802 | <blockquote>
<p>Let $F = \mathbb{Q}$ and $K$ be the splitting field of $x^4 - 7$ over $F$. Determine $Gal(K/F)$ and find all the intermediate subfields of $K/F$.</p>
</blockquote>
<p>We have $x^4 - 7 = (x-\sqrt[4]{7})(x + \sqrt[4]{7})(x - i\sqrt[4]{7})(x + i\sqrt[4]{7})$ and so, $K = \mathbb{Q}(\sqrt[4]{7},i)$. Now,... | Nicholas Camacho | 404,283 | <p>Let $G:=\operatorname{Gal}(K/F)$. Since $K/\Bbb Q(i)$ is Galois, there exists $\sigma\in \operatorname{Gal}(K/\Bbb Q(i))\leq G$ with $\sigma(\sqrt[4]{7}) = i\sqrt[4]{7}$. This has order $4$. Similarly, there exists $\tau\in \operatorname{Gal}(K/\Bbb Q(\sqrt[4]{7}))\leq G$ with $\tau(i\sqrt[4]{7})) = -i\sqrt[4]{7}$.... |
1,501,988 | <p>How do I solve the following equation?
$$\lvert{x}\rvert- \lvert{2+x}\rvert= x$$</p>
<p>I was thinking about dividing it into 4 cases: plus plus, plus minus, minus plus and minus minus. What is the best way to solve this?</p>
| Flounderer | 13,203 | <p>Another way is to use $|x| = \sqrt{x^2}$. If $|x| - |2+x| = x$ then
$$\begin{align*}
\sqrt{x^2} - \sqrt{(x+2)^2} &= x\\
x^2 + (x+2)^2 -2\sqrt{x^2}\sqrt{(x+2)^2} &= x^2\\
(x+2)^2 &= 2\sqrt{x^2} \sqrt{(x+2)^2}\\
(x+2)^4 &= 4x^2(x+2)^2
\end{align*}$$
So either $x= -2$, which doesn't satisfy the origina... |
990,638 | <p>Say $X=\{a,b,c,d\}$ and $\tau=\{\emptyset,X,\{a\},\{c\}, \{a,c\}, \{a,b,d\}, \{b,c,d\},
\{b,d\}\}$</p>
<p>This is a topological space as it satisfies the appropriate properties but how do you show $\tau$ is (or is not) path-connected?</p>
| Jeremy Brazas | 88,848 | <p>There is something wrong with your topology and this example certainly isn't path-connected even if you use the topology generated by $\tau$, but here is the idea of how to define paths in finite spaces and show that a <a href="http://en.wikipedia.org/wiki/Finite_topological_space" rel="nofollow">finite space</a> is... |
1,756,667 | <p><a href="https://i.stack.imgur.com/6HG0u.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6HG0u.jpg" alt="enter image description here"></a></p>
<p>I have calculated:</p>
<p>$\det A_1=2$, $\det A_2=3$, $\det A_3=4$, so I was putting some numbers in $\det A_{n+2} = a \det A_{n+1} + b\det A_n$ like... | Solumilkyu | 297,490 | <p>Use induction on $n$. For $n=1$, the result is trivial. Now,
suppose that there is an $N\in\mathbb{N}$ such that $\det A_n=n+1$ for all
$n\le N$. Then for the case $N+1$, we have
\begin{align}
\det A_{N+1}
&=2\det A_N-\det
\begin{pmatrix}
2&1&0&0&&&&&\\
1&2&1&0&&a... |
3,769,681 | <p>What's the name for sequences sitting between geometric and arithmetic?</p>
<p>E.g. let <span class="math-container">$x_{n+1}=ax_n+b$</span></p>
<p>I can't find a general name for these.</p>
<p>These sequences may also be a <a href="https://en.wikipedia.org/wiki/Lucas_sequence" rel="nofollow noreferrer">Lucas Sequen... | John Bentin | 875 | <p>The relations <span class="math-container">$x_{n+1}=ax_n+b\,$</span> (<span class="math-container">$n=0,1,...$</span>), with <span class="math-container">$a$</span> and <span class="math-container">$b$</span> constant, describe a form of iterated affine transformation, and so the generated <span class="math-containe... |
2,244,321 | <p>Let $f, g$ absolutely continuous on $[a,b]$ with $f'(x)=g'(x)$ a.e. on $[a,b]$. Prove if $f=g$ at a single point in $[a,b]$ then $f=g$ everywhere</p>
<p>This is not homework, this is part of a list of problems that it would be advisable to solve before a general exam. Any suggestion is welcome, I have tackled it on... | Mohan | 245,104 | <p>With your definition, it is clear that if $F$ is not $M$-flat, you can find a finitely generated submodule $M'\subset M$ with $F\otimes M'\to F\otimes M$ not injective. Since these generators come from a $M_i$ for some $i$ (being a finite set), you may assume that $M'\subset M_i$. So, you will get a non-zero element... |
1,206,284 | <p>Can someone help me solve $(y')^2=(y/c)^2-1$? WolframAlpha is giving me $\frac 12(c^2 e^{(x/c)-k}+e^{k-(x/c)})$. One book I have is giving me $y=c\cdot \cosh(\frac {x+b}c)$ -- but that one won't work for the HW problem I'm solving because with the conditions $y(0)=y(D)=0$ it gives $y$ as identically $0$ -- so it m... | 3d0 | 217,450 | <p>$$\dfrac{dy}{dx} = \sqrt{\left(\frac{y}{c}\right)^2-1} \implies \int{\dfrac{dy}{\sqrt{\left(\dfrac{y}{c}\right)^2-1}}} =\int1dx$$</p>
<p>we let $u=\dfrac{y}{c}$, so $du = \dfrac{dy}{c}$</p>
<p>$$c\int{\dfrac{du}{\sqrt{u^2-1}}} =x+k$$</p>
<p>With $k$ a constant. Now, we know this is a "classic" integral: arccosh</... |
1,206,284 | <p>Can someone help me solve $(y')^2=(y/c)^2-1$? WolframAlpha is giving me $\frac 12(c^2 e^{(x/c)-k}+e^{k-(x/c)})$. One book I have is giving me $y=c\cdot \cosh(\frac {x+b}c)$ -- but that one won't work for the HW problem I'm solving because with the conditions $y(0)=y(D)=0$ it gives $y$ as identically $0$ -- so it m... | Lutz Lehmann | 115,115 | <h2>"Elementary" solution (The Sh. Holmes type "elementary")</h2>
<p>From the form of the equation we see that it is a hyperbolic relation
$$
1=(y/c)^2-(y')^2
$$
that can be parametrized as $y(x)/c=\cosh(u(x))$ and $y'(x)=\sinh(u(x))$. Consistency is reached when the derivative of the first coincides with the second,... |
111,835 | <p>If $p$ is prime, then $F_{p-\left(\frac{p}{5}\right)}\equiv 0\bmod p$, where $F_j$ is the $j$th Fibonacci number, and $\left(\frac{p}{5}\right)$ is the Jacobi symbol.</p>
<p>Who first proved this? Is there a proof simple enough for an undergraduate number theory course? (We will get to quadratic reciprocity by th... | Gerry Myerson | 8,269 | <p>I recommend Chapter XVII of Volume 1 of Dickson's History of the Theory of Numbers. He cites results of Legendre, Gauss, Dirichlet, and Lagrange, among others, none of them exactly the one you cite, but all of them very closely related, and more general. </p>
<p>Hardy and Wright give two proofs. It's Theorem 180 in... |
2,154,351 | <p>I've recently (yesterday) started to studying complex analysis on my own. So i came across this question at the end of the first chapter
" Let $G$ be an open set in $ \mathbb{C} $ . Is $\{ z : z \in G \lor \bar z \in G \}$ open?</p>
<p>My attempt :</p>
<p>By the definition a set is open if $ A \subseteq \mathbb{C... | Christian Blatter | 1,303 | <p>If $G\subset {\mathbb C}$ is open then the set $\bar G$ obtained by reflecting $G$ in the real axis is open. Your set is just $G\cup\bar G$, hence open. </p>
|
980,875 | <p>I want to minimize the function </p>
<p>$$ \frac{x}{1-x^2} + \frac{y}{1-y^2} + \frac{z}{1-z^2} $$ subject to the constraint $$x^2 + y^2 + z^2 = 1 \space\text{and} \space x,y,z > 0$$</p>
<p>Wolfram Alpha tells me that the minimum occurs at $(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{3\sqrt{... | Macavity | 58,320 | <p>We can write the inequality to prove as
$$\frac{x}{1-x^2}+\frac{y}{1-y^2}+\frac{z}{1-z^2}\ge \frac{3\sqrt3}2 \tag{$\star$}$$
As equality is achieved for $x=y=z=\frac1{\sqrt3}$, if $(\star)$ holds then we have established the minimum.</p>
<p>Consider function $f(t) = \dfrac{t}{1-t^2}-\frac{\sqrt 3}2 -\frac{\sqrt 3}2... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.