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 |
|---|---|---|---|---|
2,672,968 | <p>In a basic probability course, there seems to be an implicit assumption that any continuous random variable has a density function, but I don't really see why this should be. My intuition is that there's some probability space $(\Omega, \Sigma, P)$ and a measurable function $X : \Omega \to \mathbb{R}$, and you'd li... | David C. Ullrich | 248,223 | <p>You're right, a "continuous" random variable need not have a density. The actual story is that if $\mu$ is any Borel measure on $\Bbb R$ with $\mu(\Bbb R)=1$ there is a random variable $X$ such that $$P(X\in E)=\mu(E).$$Now $X$ has a density if and only if $\mu$ is "absolutely continuous".</p>
<p>So the actual stor... |
4,552,917 | <blockquote>
<p>Coin A is a fair coin and coin B is weighted so that the probability of obtaining heads is 0.6. An experiment consists of randomly choosing a [single, fixed] coin and then flipping it 10 times. Given that the coin came up heads exactly 6 times in the 10 flips, what is the probability that you selected... | user170231 | 170,231 | <p>Expand the integral as</p>
<p><span class="math-container">$$\int_1^{1+2\pi} \cos(x) e^{-\sin^2(x)} \, dx = \left\{\int_0^{2\pi} - \int_0^1 + \int_{2\pi}^{1+2\pi} \right\} \cos(x) e^{-\sin^2(x)} \, dx$$</span></p>
<p>For the first integral, we have</p>
<p><span class="math-container">$$\begin{align*}
\int_0^{2\pi} \... |
1,131,959 | <p>Let $G$ be a finite group and $K$ a field with $\mathrm{char}(K) \nmid |G|$, and let $M$ be a cyclic $KG$-module.</p>
<p>For a cyclic generator $v \in M$ we can pick group elements $g_1, \dots, g_d \in G$ such that $\{ g_1v, \dots, g_d v \}$ is a $K$-basis of $M$. Does that automatically mean that $\{ g_1w, \dots, ... | Jeremy Rickard | 88,262 | <p>Let $G$ be the (dihedral of order $8$) subgroup of $\operatorname{GL}(2,\mathbb{R})$ generated by $g:=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$ and $h:=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$.</p>
<p>Then every non-zero element of $\mathbb{R}^2$ generates $\mathbb{R}^2$ as an $\mathbb{R}G$-module. If $... |
4,156,262 | <p>I am working from a textbook that gives us that</p>
<blockquote>
<p>The derivative of the inverse function is <span class="math-container">$$ \frac{dx}{dy} = \large{\frac{1}{\frac{dy}{dx}}}.\qquad (*)$$</span></p>
</blockquote>
<p><span class="math-container">$$$$</span>
Now I am stuck on part <span class="math-con... | Stuck | 545,750 | <p>The derivative of an inverse can be written more clearly as
<span class="math-container">$$(f^{-1})'(x)=\frac{1}{f'(f^{-1}(x))},$$</span>
and part <span class="math-container">$(a)$</span> can be shown straightforwardly using the inverse function theorem (<a href="https://en.wikipedia.org/wiki/Inverse_function_theor... |
1,923,416 | <p>I would like to calculate the exponential map in the n-sphere, however, i don't know how to get started. Someone could give me a tip or bibliographic reference?</p>
| ricvo | 537,923 | <p>A good book with examples on the n-sphere is Absil: Optimization Algorithms on Matrix Manifolds. Geodesics at chapter 5 I believe.</p>
<p>An easy way to get geodesics on the n-sphere is to use the embedding map in R^n+1. The connection on the n-sphere is the projection of the connection in R^n+1 (that is flat!). Us... |
364,321 | <p>Is it possible to nest big-oh notation with omega-notation? I came across this here, while doing calculations on an exercise: </p>
<p>$$
f(x) \in O(\Omega(\log x))
$$</p>
<p>I'm really unsure on how to properly read this. Either it means, that $f(x)$ is unbounded, as $\Omega(\log x)$ is anything growing as fast or... | Aryabhata | 1,102 | <p>Yes, it seems quite strange, and no, it (probably) cannot be omitted.</p>
<p>Typically, when used in a contexts like: $f(x) = x + O(\log x)$, it normally means that there is some element $g$ in the set $O(\log x)$ which appears as $f(x) = x + g(x)$.</p>
<p>In $f \in O(\Omega(\log x))$), If we take it to mean that ... |
46,898 | <p>Given a formal power series $\sum a_n z^n$ and a radius of convergence $R>0$, there are various ways to extend the function to the boundary such as</p>
<ul>
<li>Abel's theorem</li>
<li>Fatou's lemma </li>
<li>$H^\infty$ theorem.</li>
</ul>
<p>What is an example of a function that does (almost) nowhere to the bo... | gary | 6,595 | <p>The sum $f(z) = \sum_{n = 0}^\infty z^{n!}.$ extends nowhere outside of $|z|\lt1$; there are infinitely many points in the set of n-th roots of unity in the boundary each of which blow up to infinity; the set of n-th roots is dense in the boundary $|z|=1$, so the series cannot squeeze out anywhere. </p>
|
232,613 | <p>Let $A$ be a set has $n$ element, my question is how many partial order on it?</p>
<p>For $n=0,1$, $N_P(n)=1$
Case $n=2$, $N_P(n)=3$
Case $n=3$, $N_P(n)=19$</p>
<p>Is there a general formula?</p>
<p>Update:</p>
<p>It seems closed formula have not been found yet. But does any generating function be known current?... | Benjamin Dickman | 37,122 | <p>Instead of moving from the second expression to the third, try to move from the third to the second.</p>
<p>More generally, this technique is called "decomposing into partial fractions." </p>
<p>In this particular case, you might start by writing:
$$\frac{1}{(2n+1)(2n-1)} = \frac{A}{2n+1} + \frac{B}{2n-1}$$</p>
<... |
433,130 | <p>A fact about triangulated categories is that (exact) localisation functors and so-called colocalisation functors come in pairs, making an exact localisation triangle. I've tried to come up with less traditional examples.</p>
<p>Let <span class="math-container">$A$</span> be a commutative ring, and <span class="math-... | user1092847 | 131,945 | <p>The derived tensor product with <span class="math-container">$A/I$</span> is typically not idempotent (because <span class="math-container">${\rm Tor}_{*}^A(A/I, A/I)$</span> is nontrivial), so this won't give you a localization sequence.</p>
<p>One case where it is idempotent is when <span class="math-container">$I... |
3,214,753 | <p>In class, my professor showed that the function:</p>
<p><span class="math-container">$$f(z)=\frac{1}{z}$$</span></p>
<p>does not have a primitive function in the domain <span class="math-container">$\mathbb C$</span> since:</p>
<p><span class="math-container">$$\int_{|z|=R}=2\pi i$$</span></p>
<p>She then said t... | Mark | 470,733 | <p>The difference is that in <span class="math-container">$\mathbb{C}\setminus (-\infty,0]$</span> there is a branch of logarithm. Let's call it <span class="math-container">$L$</span> and this is by definition of a branch a continuous function. Then at any point <span class="math-container">$z_0$</span>:</p>
<p><span... |
359,437 | <p>A (split) reductive linear algebraic group is equivalently described by combinatorial information called a root datum.</p>
<p>A toric variety is described by combinatorial information called a fan. </p>
<p>Both correspondences use the character lattice. </p>
<p>The reference:</p>
<p><a href="http://u.cs.biu.ac.... | skd | 102,390 | <p>Not an answer, but: you can construct a fan from a root system. Let <span class="math-container">$R$</span> be a root system in an Euclidean space, and let <span class="math-container">$\Lambda_R$</span> be the root lattice with dual lattice <span class="math-container">$\Lambda_R^\vee$</span>. The fan <span class="... |
3,526,650 | <p>I was studying <code>Shannon</code>'s <code>Entropy</code> function and for a 35% chance of a particular event, the formula produced the answer 1.5 bits.</p>
<pre><code>log2(0.35) = 1.5 bits(approx.) of information.
</code></pre>
<p>I know it's pretty trivial in context of practical applications, but how can one v... | Doug M | 317,162 | <p>One bit of perfect information will allow a person to double their money. The simplest idea, is if you had a tipster than knew who would win the football game, you could go to a bookie, place a wager and double your money.</p>
<p>Suppose you have a tipster that is right 55% of the time. It is still worth followin... |
381,771 | <p>Let <span class="math-container">$p$</span> be a prime. How many solutions <span class="math-container">$(x, y)$</span> are there to the equation <span class="math-container">$x^2 + 3y^2 \equiv 1 \pmod{p}$</span>? Here <span class="math-container">$x, y \in \{0, 1, \ldots p-1\}$</span>. This paper (<a href="https://... | Fedor Petrov | 4,312 | <p>You may simply rationally parametrize this curve: if <span class="math-container">$y=0$</span> we have 2 solutions, otherwise we may denote <span class="math-container">$x=1+ty$</span>, and <span class="math-container">$x^2+3y^2=1$</span> reads as <span class="math-container">$2t+(3+t^2)y=0$</span>. If <span class="... |
2,431,679 | <p>First I denote $x=\arctan{2}$ and $y=\arcsin{\frac{1}{\sqrt{10}}}$ and then use the addition formula for sine:</p>
<p>$$\sin{(x-y)}=\sin{x}\cos{y}-\cos{x}\sin{y}=\sin{x}\cos{y}-\cos{x}\cdot \frac{1}{\sqrt{10}}.$$</p>
<p>Now I use the fact that $\cos{y}=1-\sin^2{y}$ which gives</p>
<p>$$\cos{y}=1-\sin^2{\left(\arc... | nonuser | 463,553 | <p>Put $\arctan 2 = x$ then $\tan x= 2$, so $\sin x = {2\over \sqrt{5}}$ and $\cos x = {1\over \sqrt{5}}$.</p>
<p>Let $\arcsin {1\over \sqrt{10}}= y$ so $\sin y = {1\over \sqrt{10}}$, thus $\cos y = {3\over \sqrt{10}}$. </p>
<p>Now:</p>
<p>$$\sin{(\arctan{2}-\arcsin{\frac{1}{\sqrt{10}}})} = \sin (x-y) = \sin x \cos... |
2,431,679 | <p>First I denote $x=\arctan{2}$ and $y=\arcsin{\frac{1}{\sqrt{10}}}$ and then use the addition formula for sine:</p>
<p>$$\sin{(x-y)}=\sin{x}\cos{y}-\cos{x}\sin{y}=\sin{x}\cos{y}-\cos{x}\cdot \frac{1}{\sqrt{10}}.$$</p>
<p>Now I use the fact that $\cos{y}=1-\sin^2{y}$ which gives</p>
<p>$$\cos{y}=1-\sin^2{\left(\arc... | Raffaele | 83,382 | <p>$x=\arctan 2$ means $\tan x=2$ thus</p>
<p>$\sin x = \dfrac{\sqrt{\sin^2 x}}{\sqrt{1}}=\dfrac{\sqrt{\sin^2 x}}{\sqrt{\sin^2 x+\cos^2 x}}=\dfrac{\sqrt{\frac{\sin^2 x}{\cos^2 x}}}{\sqrt{\frac{\sin^2 x}{\cos^2 x}+\frac{\cos^2 x}{\cos^2 x}}}=\dfrac{\tan x}{\sqrt{\tan^2 x+1}}$</p>
<p>$\sin x=\dfrac{2}{\sqrt 5}$ and $\c... |
3,032,802 | <p>Does anyone know how to show <span class="math-container">$ \lim_{n\rightarrow\infty} \sum_{k=1}^{\infty}k^{1/2} n^{-1} (1 - \cos(n^2 k^{-2}))=\infty$</span>? My thoughts were that you should you a series approximation of some sort, but I don't know how to get through it.</p>
| Fat ninja | 549,040 | <p>Define <span class="math-container">$$f(x)=
\begin{cases}
c,\;\;x\in [\dfrac{\pi}{2},\dfrac{3\pi}{2}]\\
0,\;\;otherwise
\end{cases}$$</span>
where <span class="math-container">$c\approx 0$</span> and <span class="math-container">$c> 0$</span>. Then <span class="math-container">$1-\cos(x)\geq f(x)$</span> for <spa... |
1,821,448 | <p>$$\iiint \frac{1}{\sqrt{x^2+y^2+(z-2)^2}}$$</p>
<p>for $x^2+y^2+z^2 = 1$</p>
<p>I've used spherical coordinates, like this:
$x=\rho sin\phi cos\theta$; $y=\rho sin\phi sin\theta$; $z=\rho cos\phi$ and $J=\rho^2 sin\phi$</p>
<p>$$\rho^2 sin^2\phi cos^2\theta + \rho^2 sin^2\phi sin^2\theta +\rho^2 cos^2\phi = 1$$
$... | Majid | 254,604 | <p>I think it would be better if you first use $t=z-2$. This way, you have $(x^2+y^2+z^2=1)\rightarrow (x^2+y^2+t^2+4t+3=0)$. Then, use the spherical coordinates. Now, your integral would be really simple, and you need to find the proper boundaries for new integral. </p>
<p>$(x^2+y^2+t^2+4t+3=0) \rightarrow \rho^2+4\r... |
3,440,648 | <blockquote>
<p>Prove that <span class="math-container">$$ \lim_{x \to 0} \frac{ \sqrt{x + \sqrt{ x + \sqrt{x}}} }{\sqrt[8]{x}}= 1$$</span> </p>
</blockquote>
<p>Let <span class="math-container">$x=y^8$</span> with <span class="math-container">$t>0$</span>. Then after we get
<span class="math-container">$$
\lim ... | EuYu | 9,246 | <p>Let <span class="math-container">$f$</span> be the map that takes <span class="math-container">$a/b$</span> to <span class="math-container">$(a+b)/(a+2b)$</span>. We can prove inductively that the <span class="math-container">$n$</span>th iteration of this process gives
<span class="math-container">$$f^n(a/b) = \fra... |
3,440,648 | <blockquote>
<p>Prove that <span class="math-container">$$ \lim_{x \to 0} \frac{ \sqrt{x + \sqrt{ x + \sqrt{x}}} }{\sqrt[8]{x}}= 1$$</span> </p>
</blockquote>
<p>Let <span class="math-container">$x=y^8$</span> with <span class="math-container">$t>0$</span>. Then after we get
<span class="math-container">$$
\lim ... | Arthur | 15,500 | <p>Your numerators and denominators follow the same recursive relationship that defines the Fibonacci sequence. I.e. each time you make a new number (either a new numerator or a new denominator), the new number is equal to the sum of the two most recent previously made numbers.</p>
<p>Any sequence that follows this re... |
313,064 | <p>I have a problem with an exercise:
Find the probability that when throwing three dice, the six will occur on one of them (no matter which one), if on the facets of the remaining two dice, two different numbers will outcome.</p>
<p>The answer given to this exercise is 1/2. However I think it is $\frac{60}{6^3}=\frac... | Ross Millikan | 1,827 | <p>I read the question to ask the chance we get $6ab$, with $a \ne 6, b \ne 6, a \ne b$. It can't be $\frac 12$ as the chance of getting at least one $6$ is $\frac {91}{216} \lt \frac 12$ before we worry about $a$ and $b$</p>
<p>For the specific answer, there are $5$ ways to choose $a$, $4$ ways to choose $b$, and $6... |
238,408 | <p>How can I solve this system of equations in Mathematica?</p>
<pre><code>DCp1total = WA1*DCpA0 + WB1*DCpB0
DCp2total = WA2*DCpA0 + WB2*DCpB0
WA1 + WB1 = 1
WA2 + WB2 = 1
DCp1_total = 0.6
DCp2_total = 0.3
DCpA0 = 0.5
DCpB0 = 0.4
</code></pre>
<p>Unkowns: <code>WA1, WA2, WB1</code> and <code>WB2</code>.</p>
| cvgmt | 72,111 | <pre><code>Solve[{DCp1total == WA1*DCpA0 + WB1*DCpB0,
DCp2total == WA2*DCpA0 + WB2*DCpB0, WA1 + WB1 == 1, WA2 + WB2 == 1,
DCp1total == 0.6, DCp2total == 0.3, DCpA0 == 0.5, DCpB0 == 0.4}]
</code></pre>
<blockquote>
<p><code>{{WA1 -> 2., WA2 -> -1., WB1 -> -1., WB2 -> 2.}}</code></p>
</blockquote>
|
568,930 | <p>Prove or disprove: There exists an element in the quotient group $\mathbb{R}/\mathbb{Z}$ of order 6. By counting formula, since $|\mathbb{R}/\mathbb{Z}|$ is infinite, it seems possible to have an element of order 6. But how to find one? In addition, what is $\mathbb{R}/\mathbb{Z}$, please?</p>
| Benjamin Dickman | 37,122 | <p>The group $\mathbb{R}$ under addition is the set of real numbers along with the standard binary operation that takes input $a, b \in \mathbb{R}$ and gives out $a+b$.</p>
<p>It's a group since it's nonempty, addition is associative, it has an identity (viz., $0$), and every element has an inverse (viz., $r$ has inve... |
151,516 | <p>I am trying to find this by using integration by parts but I am not sure how to do it.</p>
<p>$$\int_0^{\pi/2} (\sin x)^7 (\cos x)^5 \mathrm{d} x$$</p>
<p>I tried rewriting as </p>
<p>$$\int_0^{\pi/2} \sin x\cdot\ (\sin x)^6\cdot\ (\cos x)^5 \mathrm{d} x = \int_0^{\pi/2}\sin x(1-\ (\cos x)^3)\cdot\ (\cos x)^5 \m... | davidlowryduda | 9,754 | <p>Integration by parts is not the method to use here. $u$-substitution is the method to use. This is perhaps not so obvious, except that as you noticed, IBP doesn't seem to get you very far right off.</p>
<p>You have $\displaystyle \int \sin x (1 - \cos^3 x) \cos ^5 x dx = \int \sin x \cos ^5 x dx - \int \sin x \cos ... |
151,516 | <p>I am trying to find this by using integration by parts but I am not sure how to do it.</p>
<p>$$\int_0^{\pi/2} (\sin x)^7 (\cos x)^5 \mathrm{d} x$$</p>
<p>I tried rewriting as </p>
<p>$$\int_0^{\pi/2} \sin x\cdot\ (\sin x)^6\cdot\ (\cos x)^5 \mathrm{d} x = \int_0^{\pi/2}\sin x(1-\ (\cos x)^3)\cdot\ (\cos x)^5 \m... | Pedro | 23,350 | <p>This type of integrals can be solved to give the general result:</p>
<p>$$\int_0^{\pi /2}\sin^m\theta\cos^n\theta d \theta=\begin{cases} \frac{(m-1)!!(n-1)!!}{(m+n)!!} \text{ if any exponent is odd}\cr \frac{(m-1)!!(n-1)!!}{(m+n)!!}\frac{\pi} 2 \text{ both even exponents} \end{cases}$$</p>
<p>We first prove the r... |
612,058 | <blockquote>
<p>Let $g:\mathbb{R}\to \mathbb{R}$ and $f:\mathbb{R}\to \mathbb{R}$ be functions, such that $f(x)\leq g(x)$ for each $x \in \mathbb{R}$. Tell if $\sup(f)\leq \sup(g)$ is true or false. If true, prove it. If false give a counterexample.</p>
</blockquote>
<p>Well, $\sup(f)$ stands for the supremum of the... | copper.hat | 27,978 | <p>If $f(x) \le g(x)$ for all $x$, then you must have $f(x) \le \sup_y g(y)$ for all $x$. If this is true, you must have $\sup_z f(z) \le \sup_y g(y)$, which is the desired result.</p>
|
1,046,515 | <p>Let $$A = \begin{pmatrix}a & b\\c & d\end{pmatrix}.$$</p>
<p>Show that </p>
<p>1) $A$ is diagonalisable if $(a - d)^2 + 4 bc > 0$</p>
<p>2) $A$ is not diagonalisable if $(a - d)^2 + 4 bc < 0$</p>
| Aman | 13,380 | <p>Hint: Look at the characteristic polynomial and see what those conditions imply.</p>
|
1,046,515 | <p>Let $$A = \begin{pmatrix}a & b\\c & d\end{pmatrix}.$$</p>
<p>Show that </p>
<p>1) $A$ is diagonalisable if $(a - d)^2 + 4 bc > 0$</p>
<p>2) $A$ is not diagonalisable if $(a - d)^2 + 4 bc < 0$</p>
| Timbuc | 118,527 | <p>The characteristic polynomial is</p>
<p>$$\det(xI-A)=\begin{vmatrix}x-a&-b\\-c&x-d\end{vmatrix}=x^2-(a+d)x+(ad-bc)$$</p>
<p>The above quadratic's discriminant is</p>
<p>$$\Delta=(a+d)^2-4(ad-bc)=(a-d)^2+4bc$$</p>
<p>Now, what does having positive discriminant imply for the roots of a quadratic? And what ... |
177,181 | <p>I have a object which is similar to the shape of irregular rock like this <img src="https://i.stack.imgur.com/Zz55K.gif" alt="enter image description here"></p>
<p>I would like to find the volume of this. How to do it?</p>
<p>If I have to find the volume, what are the things I would need. eg., If it is cylindrical... | wesley | 36,649 | <p>If you don't want to use Archimedes' solution, and you know the material, you can look up the density and weigh the rock.</p>
|
245,884 | <p>Someone wrote on the <a href="http://en.wikipedia.org/wiki/Laplace_transform" rel="nofollow">Wikipedia article for the Laplace trasform</a> that 'this transformation is essentially bijective for the majority of practical uses.'</p>
<p>Can someone provide a proof or counterexample that shows that the Laplace transfo... | Jean-Sébastien | 31,493 | <p>It depends on you set of definition. If you ask wheter it is bijective from the functions defined on $\mathbb{R}$, compare the laplace transform of
$$
f(x)=\begin{cases}
\sin(x)&x\geq0\\
0&x<0
\end{cases}
$$</p>
<p>and
$$
g(x)=\sin(x), \forall x.
$$</p>
|
245,884 | <p>Someone wrote on the <a href="http://en.wikipedia.org/wiki/Laplace_transform" rel="nofollow">Wikipedia article for the Laplace trasform</a> that 'this transformation is essentially bijective for the majority of practical uses.'</p>
<p>Can someone provide a proof or counterexample that shows that the Laplace transfo... | Robert Israel | 8,508 | <p>Bijective from what space of functions to what space of functions?</p>
<p>For example, by the Paley-Wiener theorem the Laplace transform is a bijection from $L^2(0,\infty)$ to the functions $F$ analytic in the open right half plane whose restrictions to vertical lines in the right half plane have uniformly bounded ... |
245,884 | <p>Someone wrote on the <a href="http://en.wikipedia.org/wiki/Laplace_transform" rel="nofollow">Wikipedia article for the Laplace trasform</a> that 'this transformation is essentially bijective for the majority of practical uses.'</p>
<p>Can someone provide a proof or counterexample that shows that the Laplace transfo... | Christian Blatter | 1,303 | <p>For "the majority of practical uses" it is important that the Laplace transform ${\cal L}$ is <em>injective</em>. This means that when you have determined a function $s\mapsto F(s)$ that suits your needs, there is at most one process $t\mapsto f(t)$ such that $F$ is its Laplace transform. You can then look up this u... |
3,036,724 | <p>I think that this means that <span class="math-container">$|x^3 + 6x^2 + 2^x − 6| \equiv 0 \bmod p$</span>. Ignoring the absolute value first, I get <span class="math-container">$
x(x^2+2)+6(x^2−1) \equiv 0 \mod p \Rightarrow x(x^2+2) \equiv -6(x^2−1) \bmod p$</span>. But I'm not sure how to proceed. </p>
| Dietrich Burde | 83,966 | <p>Hint: Let <span class="math-container">$f(x)=x^3+6x^2+2x-6$</span>. Then we have <span class="math-container">$$3\mid f(x)$$</span> for all <span class="math-container">$x\in \Bbb{Z}$</span>.</p>
|
1,271,119 | <p>Say our world is actually 4-dimensional, and I have a tesseract sitting on my table. To a being only capable of perceiving 3 dimensions, what would it actually look like?</p>
<p>Would it just look like a cube?</p>
<p>I understand that the <em>projection</em> of a tesseract into 3D space looks like the image shown ... | mathlove | 78,967 | <p>Having $a=dk^3,b=dk^2,c=dk$ gives you$$\begin{align}\sqrt{\frac{a^5+b^2c^2+a^3c^2}{b^4c+d^4+b^2cd^2}}&=\sqrt{\frac{d^5k^{15}+d^2k^4\cdot d^2k^2+d^3k^9\cdot d^2k^2}{d^4k^8\cdot dk+d^4+d^2k^4\cdot dk\cdot d^2}}\\&=\sqrt{\frac{dk^{15}+k^6+dk^{11}}{dk^9+1+dk^5}}\\&=\sqrt{\frac{k^6(dk^9+1+dk^5)}{dk^9+1+dk^5}}... |
1,313,670 | <p>The standard birthday question is, how many people have to be in a math class for the probability of two having a shared birthday is > 1/2. With standard assumptions about birthday distributions and independence of birthdays.</p>
<p>A slightly different question.</p>
<p>How many people need to be in a stadium for ... | Ross Millikan | 1,827 | <p>The chance that there are at least two people born on Jan 1 is $1-\left(\frac {364}{365}\right)^N-N\frac {364^{N-1}}{365^N}$. If we pretend that the days are independent, which is not quite fair but will not be too far off, we want $$\left(1-\left(\frac {364}{365}\right)^N-N\frac {364^{N-1}}{365^N}\right)^{365}=\fr... |
585 | <p>I would like to give the built-in function <code>PlusMinus</code>, which is undefined by default, the intuitive meaning: considering all possible combinations of $+$ and $-$.</p>
<p>For simple expressions, this can be implemented as follows:</p>
<pre><code>PlusMinus[a_,b_] := Flatten[{a+b,a-b}]
a \[PlusMinus] b \[... | Rojo | 109 | <p>Here's my shot, haven't tested it much yet, but it's pretty weird.
Flattens out in NumericFunctions, I thought that made sense.</p>
<pre><code>ClearAll[PlusMinus];
Module[{PlusMinusList},
SetAttributes[PlusMinus, {Flat, OneIdentity, NumericFunction}];
SetAttributes[PlusMinusList, {Flat, OneIdentity}];
PlusMinus[... |
2,557,395 | <blockquote>
<p>Let $R$ be a ring and $M,N$ be $R-$modules. Let $\pi: M \to N$ be a
module homomorphism. Let $K$ be a $R$-submodule of $N$. Then
$\pi^{-1}(K)$ is a submodule of $M$</p>
</blockquote>
<p>Surprisingly, I couldn't find this result on google. </p>
<p>Is my proof correct?</p>
<p><strong>Proof:</str... | egreg | 62,967 | <p>Yes, your statement and proof are correct.</p>
<p>This fact is important in the sometimes called “correspondence theorem”: if $f\colon M\to N$ is a homomorphism, the mappings
$H\mapsto f(H)$ and $K\mapsto f^{-1}(K)$ define bijections, inverse of each other, between the set of submodules of $M$ containing $\ker f$ a... |
2,771,404 | <p>This question is a trigonometric one from what I can tell. I understand that you need to differentiate the given function and the unknown variable on top ($dx$) will correspond to the $x$ unknowns, for example $5x^2$ will give us $10x\frac{dx}{dt}$.</p>
<p>With this specific question I can only get to $-\frac{\sec^... | user577215664 | 475,762 | <p>I guess x is a function of t</p>
<p>$$\tan(x)=4t^{-1}$$
$$\frac {d\tan(x)}{dt}=-4t^{-2}$$
$$\frac {x'}{\cos^2(x)}=-4t^{-2}$$
$$ x'= \frac {dx}{dt}=-4{\cos^2(x)}t^{-2}$$</p>
<p>Note that $\sec(x) =\frac 1 {\cos(x)}$</p>
|
67,656 | <p>Let $X$ be a projective surface and let $x\in X$ be a smooth point. Consider the blow up $Bl_{x}X$ of $X$ in $x$, and let $E$ be the exceptional divisor. Suppose we know that $E$ is the only (-1)-curve in $Bl_{x}X$, so that $\alpha(E) = E$ for any $\alpha\in Aut(Bl_{x}X)$. Then any automorphisms $\alpha\in Aut(Bl_{x... | J.C. Ottem | 3,996 | <p>Let $\sigma:X\to X$ be an automorphism fixing a point $x\in X$. Then since the inverse image of $x$ is a Cartier divisor in both $Bl_x X$ and $Bl_x \sigma(X)$, by the universal property of blow-ups, we get an automorphism $\tilde{\sigma}$ of the blow-up $Bl_x X$. Since $\tilde{\sigma}$ restricts to $\sigma$ on the c... |
1,967,312 | <p>Say we have disjoint sets $A$ and $B$ where $A = \{1\}$ and $B = \{2\}$. </p>
<p>Thus the power set of $A$ is $\{\emptyset,1\}$ and the power set of $B$ is $\{\emptyset,2\}$. </p>
<p>I know the intersection set of $A$ and $B$ is $\emptyset$ </p>
<p>My question is this: Given the above information, does that mean ... | PMar | 378,354 | <p>More properly, the powersets are { $\emptyset$, { 1 } } and { $\emptyset$, { 2 } }.</p>
<p>Their intersection is then { $\emptyset$ }, <strong>not</strong> $\emptyset$.</p>
|
1,232,871 | <p>A closed subset of a locally connected space is locally connected?</p>
<p>A closed subset of a locally path-connected space is locally path-connected?</p>
<p>Is there any counter-example?</p>
| Anubhav Mukherjee | 72,329 | <p>consider $\mathbb{R^2}$...if you wish for a connected closed subset which does not follow two properties...then cosider comb space <a href="http://en.wikipedia.org/wiki/Comb_space" rel="nofollow">http://en.wikipedia.org/wiki/Comb_space</a> as a subset of $\mathbb{R^2}$</p>
|
1,701,470 | <p>Could someone please point out resources or provide the solution with detailed steps to calculate the following limits?</p>
<p>If possible, please provide solutions with or without using L'Hopital's rule.</p>
<p>$$
\underset{x\rightarrow\infty}{\lim}\left[\frac{e^{x}}{e^{x^{2}}}\right]
$$</p>
<p>Thanks,</p>
| martini | 15,379 | <p>There is no need using l'Hospital here, for
$$ \frac{e^x}{e^{x^2}} = e^{x-x^2} $$
And now $\lim_{x \to \infty} x-x^2 = -\infty$, that is,
$$ \lim_{x\to \infty} \frac{e^x}{e^{x^2}} = \lim_{y \to -\infty} e^y = 0 $$</p>
<hr>
<p>Addendum: We will show that $\lim_{x\to \infty} (x-x^2) = -\infty$. For $x \ge 1$, we h... |
1,701,470 | <p>Could someone please point out resources or provide the solution with detailed steps to calculate the following limits?</p>
<p>If possible, please provide solutions with or without using L'Hopital's rule.</p>
<p>$$
\underset{x\rightarrow\infty}{\lim}\left[\frac{e^{x}}{e^{x^{2}}}\right]
$$</p>
<p>Thanks,</p>
| P. Appell | 175,495 | <p>Maybe by writting $$\frac{e^{x}}{e^{x^2}} = e^{x-x^2}=e^{-x(x-1)}$$ you get that the limit is $0$ since $-x(x-1)\to -\infty$ as $x \to \infty$.</p>
|
77,306 | <p>I'm not entirely sure what I want to ask here, so please bear with me!</p>
<p>I think the explanation they give us in school for how finding the antiderivative by substitution works is: $$\int f(g(t))g'(t)dt=(Fg)(t)=F(g(t))=F(u)=\int f(u)du$$</p>
<p>But I never really understood the equality $F(u)=\int f(u)du$... | Jan Rothkegel | 154,733 | <p>What you are looking for is an intuitive interpretation of the formula for integration by substitution. I will try to provide it in rather simple words. Consider the example <span class="math-container">$\displaystyle F(x)=\int 2xe^{x^2}\, dx $</span>.</p>
<p>You can easily infer by the inverse chain rule <span cla... |
242,892 | <p>Let's take two independent, random variables $X$ and $Y$ from set $\{1,\ldots,n\}$
probability, that $X =i$ AND $Y = j$ is $\frac{1}{n^2}$.
I want to determine the expected value, so I started:
$$E[Z] = E[|X-Y|] = \sum_i (a_i) \cdot P(|X-Y|=a_i),$$ (sum is to $n$) but I don't now how to count $P(|X-Y|=a_i)$, cause ... | TenaliRaman | 29,755 | <p>$$E[Z] = E[|X - Y|] = \sum_{i = 1}^n \sum_{j = 1}^n |i - j|P(X = i, Y = j) = \frac{1}{n^2}\sum_{i = 1}^n \sum_{j = 1}^n |i - j| = \frac{2}{n^2}\sum_{i = 1}^{n} \sum_{j = 1}^{i - 1} (i - j) = \frac{2}{n^2}\sum_{i = 1}^{n} i(i - 1) - \frac{i(i - 1)}{2} = \frac{2}{n^2}\sum_{i = 1}^{n}\frac{i(i - 1)}{2} = \frac{1}{n^2}\... |
2,393,044 | <p>For two discrete random variables $X,Y$, it is known that their covariance is equal to $\text{Cov}(X,Y) = E(XY) - E(X)E(Y)$. Define a "variant" of the total variation distance in the following manner:
$$ \delta(X,Y) = \sup_{\alpha, \beta} \{| P(X = \alpha \text{ and } Y = \beta) - P(X = \alpha)P(Y = \beta)| .\} $$... | David Cohen | 451,175 | <p>Let $X$ be uniformly distributed in the three element set $\{0,1/2,1\}$, and let $Y=X$. Then $E(XY)-E(X)E(Y)=5/12$ and $\delta(X,Y)=P(X=0 \text{ and }Y=0)-P(X=0)P(Y=0)=2/9$, so $\delta$ is smaller than the covariance.</p>
|
1,158,652 | <p>$$f(x) = \frac{1}{1-2\cos^2x}$$</p>
<p>The result of $f'(x)$ should be equals</p>
<p>$$f'(x) = \frac{-4\cos x\sin x}{(1-2\cos^2x)^2}$$</p>
<p>I'm trying to do it in this way but my result is wrong.</p>
<p>$$f'(x) = \frac {1'(1-2\cos x)-1(1-2\cos^2x)'}{(1-2\cos^2x)^2} =
\frac {1-2\cos^2x-(1-(2\cos^2x)')}{(1-2\c... | Emanuel Camacho | 217,396 | <p>The problem is in this step, from here</p>
<p>$$f'(x) = \frac {1'(1-2\cos x)-1(1-2\cos^2x)'}{(1-2\cos^2x)^2} $$
to here
$$\frac {1-2\cos^2x-(1-(2\cos^2x)')}{(1-2\cos^2x)^2}$$</p>
<p>because $$1'=0.$$</p>
|
993,051 | <p>Just by looking at the function:
$$y(t) = \tan(t)+t$$
I can immediately see that there is a root at $t=0$, though after graphing it I can see many more roots and I can calculate them using computer aided programs; i.e. the next root is at $t=2.0288$.</p>
<p>What I want to know is: what methods I can use to solve t... | robjohn | 13,854 | <p>Newton's method will usually converge more quickly, but the following is conceptually simpler and converges reasonably quickly, too.</p>
<p>If $\tan(t)+t=0$, then $t=-\tan^{-1}(t)$. However, $\tan^{-1}(t)$ does not give all the angles whose tangent is $t$; we also have $t=k\pi-\tan^{-1}(t)$.</p>
<p>Since $\left|\f... |
3,357,778 | <p>I am having issues solving a problem using the probability approach (do not want to use the combinatorics approach!). Suppose there are four balls in a jar of colors: 1 Red, 1 Blue and 2 Yellow. 2 Balls are to be selected. What is the probability that only one of the two selections will be Yellow?</p>
<p>Should the... | Community | -1 | <blockquote>
<p>Since in the comments you seem slightly confused you might find this helpful:-</p>
</blockquote>
<p>Should the solution be this: P(1st pick=Yellow)*P(2nd pick=Not Yellow) + P(1st pick=Not Yellow)*P(2nd pick=Yellow) = ((2/4) (2/3)) + ((2/4) (2/3)) = 2/3?</p>
<blockquote>
<p>Yes, completely correct<... |
2,575,760 | <p>$g(x,y)=f(x^3+3y+2), f:R \rightarrow R$, directional derivative in the direction of the vector $(1,0)$ at the point $(0,0)$</p>
<p>My attempt:</p>
<p>I'm not sure how to prove this is zero?
Using the definition I got to $\lim_{h \rightarrow 0} \frac{f(h^3+2)-f(2)}{h}$ but I dont know how to compute that.
I'm aware... | user | 505,767 | <p>Let's use the definition of <strong>directional derivative</strong></p>
<p>$$\frac{\partial g}{\partial \vec v}=\nabla\cdot\vec v=v_1\frac{\partial g}{\partial x}+v_2\frac{\partial g}{\partial y}$$</p>
<p>In your case since $\vec v=(1,0)$:</p>
<p>$$\frac{\partial g}{\partial \vec v}=\nabla\cdot\vec v=1\cdot\frac{... |
1,063,090 | <p>This does not really make sense to me for several reasons: </p>
<ul>
<li>The integers are usually constructed using a given construction of the natural numbers</li>
<li>Historically natural numbers were conceived of before the integers</li>
<li>The notation $\mathbb{Z}^+$, is more cumbersome to use than $\mathbb{N}... | Graham Kemp | 135,106 | <p>There has always been some ambiguity as to whether $\Bbb N$, the natural numbers, includes zero or not due to inconsistent usage by past authors.</p>
<p>Where as using $\Bbb Z^+$, the positive integers, or $\Bbb Z^\ast$, the nonnegative integers, is unambiguous about the matter.</p>
|
1,063,090 | <p>This does not really make sense to me for several reasons: </p>
<ul>
<li>The integers are usually constructed using a given construction of the natural numbers</li>
<li>Historically natural numbers were conceived of before the integers</li>
<li>The notation $\mathbb{Z}^+$, is more cumbersome to use than $\mathbb{N}... | Robert Soupe | 149,436 | <p>$\mathbb{Z}^+$ means "positive integers."</p>
<p>$\mathbb{N}$ means "natural numbers," which includes zero, no, wait, it doesn't, yeah, it does... Look at this page: <a href="http://oeis.org/wiki/Latin_alphabet" rel="nofollow">http://oeis.org/wiki/Latin_alphabet</a> On the rightmost column of the table, it says:</p... |
1,839,121 | <p>I want to prove that this inequality holds or not?
The inequality is $\lambda(G) \leq \chi(G) $
where $\lambda(G)$ is the minimum number of edges whose deletion from a graph $G$ disconnects $G$, also called the line connectivity and $\chi(G)$ is the chromatic number of graph.</p>
<p>I would be greatly thankful i... | Nitin Uniyal | 246,221 | <p>Here is an example to disaprove the inequality. For a complete bipartite graph $K_{3,4}$, $\lambda (K_{3,4})=3$ and $\chi (K_{3,4})=2$.</p>
|
3,138,158 | <p>How can I show that if the cube power of a matrix is the null matrix, then the matrix itself is singular?</p>
| Daniel Schepler | 337,888 | <p>Suppose that <span class="math-container">$A$</span> is an <span class="math-container">$n \times n$</span> matrix with <span class="math-container">$n \ge 1$</span>, and choose any nonzero vector <span class="math-container">$x \in \mathbb{R}^n$</span>. Then there are three possibilities:</p>
<ul>
<li><span class... |
897,061 | <p>This is just a standard problem from my high school's calculus text, but my proof seems sort of off. This is it:</p>
<p>Let $f(x) = e^x$. The tangent line of $f(x)$ at $x=0$ is $g(x)=x+1$. Since $f''(x_0) \gt 0$ for all $x_0 \in \mathbb R$,the tangent line $g(x) \le f(x)$ for all $x$. Q.E.D.</p>
<p>I didn't show a... | ShakesBeer | 168,631 | <p>My favourite proof is letting $f(x)=e^x-x-1$. $f(0)=0$ and it's easy to show that it has a minimum at $x=0$.
<br>
However you do it, you are somehow going to go through derivatives or calclulus, explicitly or implicitly.</p>
<p>EDIT:
A completely alternative proof: Using the taylor series ($e^x = 1 + x + \frac{x^2... |
897,061 | <p>This is just a standard problem from my high school's calculus text, but my proof seems sort of off. This is it:</p>
<p>Let $f(x) = e^x$. The tangent line of $f(x)$ at $x=0$ is $g(x)=x+1$. Since $f''(x_0) \gt 0$ for all $x_0 \in \mathbb R$,the tangent line $g(x) \le f(x)$ for all $x$. Q.E.D.</p>
<p>I didn't show a... | 5xum | 112,884 | <p>Sure, your proof is pretty short, but it is only short for someone who already did some analysis. For example, you used the following theorems (which is true, of course) in your proof:</p>
<blockquote>
<p>If $f$ is convex on $\mathbb R$, then for any tangent function $g$, we
know that $g(x)<f(x)$ for all $x$... |
2,107,874 | <p>What are the other angles in triangle and what is $\tan(\frac{\beta-\gamma}{2})$, if $\alpha = 60$ and $\frac{b}{c}=2+\sqrt3$?
and
I tried to solve it using law of sines and law of cosines but I couldn't find answer.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>we have $$\frac{b}{c}=\frac{\sin(\beta)}{\sin(\gamma)}=2+\sqrt{3}$$
since we have $$\gamma=\frac{2}{3}\pi-\beta$$ we get
$$\frac{\sin\left(\frac{2}{3}\pi-\beta\right)}{\sin(\beta)}=\frac{1}{2+\sqrt{3}}$$
can you proceed from here?</p>
|
135,863 | <p>There are $3$ production lines in a factory. $2$ of them are new lines and $1$ of them is an old line. All production lines produce at the same rate. The old production line has $8\%$ defective rate while the new production lines have only $3\%$ defective rate. The components are shipped to customers in $100$-unit l... | Beni Bogosel | 7,327 | <p>Suppose $u$ is harmonic, i.e. $\Delta u=0$. Then if $u$ would be necessarily convex it is easy to see that $-u$ is also harmonic, and $u$ would be necessarily concave also. </p>
<p>Therefore if $u$ harmonic implies $u$ convex it follows that every harmonic function is both convex and concave (i.e. the images of $u$... |
2,173,289 | <p>I want to calculate </p>
<p>$$\int_0^\infty \frac{\ln x}{x^2+4}\,dx$$</p>
<p>using the Residue Theorem. The contour I want to use is almost the upper half circle of radius $R$, but going around $0$ following a half-circle of radius $\epsilon$, and using the branch cut at $3\pi/2$. The picture is below. Let's call ... | Mark Viola | 218,419 | <p>After the edit, you should have</p>
<p>$$\begin{align}
\int_{-R}^{-\epsilon} \frac{\log(x)}{x^2+4}\,dx+\int_\epsilon^R\frac{\log(x)}{x^2+4}\,dx&=\int_\epsilon^R\frac{\log(-x)}{x^2+4}\,dx+\int_\epsilon^R\frac{\log(x)}{x^2+4}\,dx\\\\
&=\color{blue}{2\int_\epsilon^R \frac{\log(x)}{x^2+4}\,dx}+\color{red}{i\pi\... |
19,124 | <p>To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and "identify opposite sides". We can imagine gluing together one pair of opposite sides to get a cylinder, and then gluing together opposite ends of the cylinder to get a torus. (Pro... | Xander Henderson | 8,571 | <p>A possibility is to show your students Google's implementation of the game Snake. If you enter the term <a href="https://www.google.com/search?q=snake" rel="nofollow noreferrer"><code>snake</code></a> into Google's search engine, the there is a box at the top showing</p>
<p><a href="https://i.stack.imgur.com/KBL4q.... |
19,124 | <p>To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and "identify opposite sides". We can imagine gluing together one pair of opposite sides to get a cylinder, and then gluing together opposite ends of the cylinder to get a torus. (Pro... | David E Speyer | 51 | <p>One thing that I like about the Pacman analogy is that, if you draw Pacman's eye (or Ms. Pacman's bow, which is my usual choice), then Pacman is not mirror symmetric. This means you can talk about nonorientable surfaces and draw how Pacman goes around a Mobius strip/Klein bottle/projective plane and comes back rever... |
1,330,752 | <p>Please excuse the simplicity of this question, but I am very new to groups and fields. I only seek an simplistic / intuitive expalnation, and confirmation / refutation re whether I am on the right track. I confess I am a little averse to reading about them from the ground up, as my motivation is quickly deteriorated... | Mark Bennet | 2,906 | <p>You have an interesting thought. However, if you know about Normal Subgroups, you know that the kernel of a homomorphism is worth naming and study. An Ideal is useful because it is the Kernel of a homomorphism from one Ring to another.</p>
<p>This means an Ideal has to relate to both the additive and multiplicative... |
372,135 | <p>Question: Let $P = 27+108x+90x^2-80x^3-60x^4+48x^5-8x^6$, find $\sqrt[3]{P}$.</p>
<p>Just wondering if there is a general way for dealing with this sort of question, I was able to figure how to find $\sqrt{P}$, but $\sqrt[3]{P}$ is beyond me. Long division is used in an example, unfortunately it lacks justification... | Community | -1 | <p>The hope is that $\sqrt[3]{P}$ is also a polynomial. If $\sqrt[3]{P}$ were to be a polynomial, then $\sqrt[3]{P}$ should be of the form
$$\sqrt[3]{P} = (-2x^2 + ax + 3)$$ since the coefficient of $x^6$ is $-8$ and the constant term is $27$. Can you take this further?</p>
|
183,123 | <p>By using Cantor's Theorem: if We have a Set A, with n members the Cardinality of the Power Set is $P(A)=2^n$.
There is Formula to calculate n Power Sets $P(...P(P(A))...)$ of a set A with m elements without doing the simple iteration $(((2^2)^2)..)^2^m$ n times?
Many thanks</p>
| Rasmus | 367 | <p>The term $(((2^2)^2)\ldots)^2$ simplifies to $2^{2^n}$.This follows from the general identity $(x^m)^n=x^{m\cdot n}$ and induction.</p>
<p><strong>However</strong>, what you should actually be computing (as S4M points out in the comments) is an expression like
$$
2^{2^{2^{2^{2^n}}}}
$$
which does not simplify any f... |
833,017 | <blockquote>
<p>What is the maximum size of an antichain of $[n]:=\{1,2,3,\dots,n\}$ (say $\mathcal{A}$) such that $\mid A\cap B\mid \ge k$ where
$A,B\in \mathcal{A}$ and $1\le k\le n-1$?</p>
</blockquote>
<p>By antichain, I mean an antichain of the poset $(2^{[n]},\subseteq)$ where $2^{[n]}$ is the power set of $... | Sam Spiro | 209,092 | <p>Answer: ${n \choose \lceil (n+k)/2\rceil}$.</p>
<p>Here's the lower bound. Let $F=\{B_1,\ldots, B_m\}$ be the set of all subsets $B_i\subset [n]$ with $|B_i|=\lceil (n+k)/2\rceil$. It is clear that this is an anti-chain. Moreover, if $i\ne j$ we have $|B_i\cap B_j|=|B_i|+|B_j|-|B_i\cup B_j|\ge 2\lceil (n+k)/2\rc... |
1,110,831 | <p>I have $\pi:4y-3z-4=0 \quad A=(2,4,4) \quad B=(2,-2,-4)$.
I have to calculate the equation of the plane $\sigma$ throw $A$ and $B$ and orthogonal to $\pi$.
What is the solution?
Thanks in advice!</p>
| Don Larynx | 91,377 | <p>$Y > 0$ if $AX > B$ is the only case if you want to avoid trivial solutions</p>
|
1,854,963 | <p>Let $\overline{\mathbb{F}}_2$ be an algebraic closure of $\mathbb{F}_2$, and let $\alpha\in\overline{\mathbb{F}}_2$ be such that $\alpha^2+\alpha+1=0$. Prove that if a polynomial $P$ is irreducible in $\mathbb{F}_2$ but $P$ is reducible in $\mathbb{F}_2[\alpha]$, the deg $P$ is even.</p>
| Batominovski | 72,152 | <blockquote>
<p><strong>Lemma:</strong> Let $q$ be a power of a prime natural number. Write $\bar{\mathbb{F}}_q$ for the algebraic closure of the field $\mathbb{F}_q$ with $q$ elements. Fix $\alpha\in\bar{\mathbb{F}}_q$ such that $\big[\mathbb{F}_q[\alpha]:\mathbb{F}_q\big]=m$ for some positive integer $m$. Suppos... |
1,517,202 | <p>Question: Find $\frac{dy}{dx}$ implicity if $x=\tan(xy)$.</p>
<p>So far, I have used implicit differentiation on both sides of the equation and ended up with two different answers. The first one was a bit more work than the second one. </p>
<p>My first answer is $\frac{dy}{dx}=\frac{\cos^2{xy}(1-y\sec^2{xy})}{x}$.... | Claude Leibovici | 82,404 | <p>Consider $$F=\tan(xy)-x=0$$ So $$F'_x=y \sec ^2(x y)-1$$ $$ F'_y=x \sec ^2(x y)$$ and $$\frac{dy}{dx}=-\frac{F'_x}{F'_y}=\frac{\cos ^2(x y) \left(1-y \sec ^2(x y)\right)}{x}=\frac{\cos ^2(x y)-y}{x}$$ and this is exactly your second answer (before the last simplification).</p>
|
2,887,176 | <p>Can transfinite induction be used to demonstrate that 0.9...=1?</p>
<p>More generally, can it be used to prove limits of sequences?</p>
| Lee Mosher | 26,501 | <p>No, transfinite induction is not designed for this job.</p>
<p>The job that transfinite induction is designed for is to prove universally quantified statements of the form</p>
<blockquote>
<p>For all $x \in \omega$, $P(x)$ is true</p>
</blockquote>
<p>where $\omega$ is a well-ordered set. In ordinary induction ... |
252,257 | <p>I need to show that the DE $y'=y^{\alpha}$, where $\alpha$ is a constant with $0<\alpha<1$, has infinitely many solutions passing through the point $(0,0)$.
Also I need four of such solutions.
Thank you for your help!
Klara</p>
| Thomas | 49,167 | <p>You have
$$ y' = \frac{dy}{dx} = y^{\alpha}, \ y(0)=0, \ 0<\alpha<1$$
From this we can start
\begin{align}
\frac{dy}{dx} &= y^{\alpha}
\\ \Leftrightarrow \frac{1}{ y^{\alpha}} dy &= dx
\\ \Leftrightarrow \int \frac{1}{ y^{\alpha}} dy &= \int 1 dx
\end{align}
Since this really does smell like ho... |
1,400,987 | <blockquote>
<p>Given $x\in \left(0; \frac\pi2\right)$. Prove that $$\sin x>\frac{2x}{\pi}$$</p>
</blockquote>
<p>This is my try:</p>
<p>Let $y=\sin x-\frac{2x}\pi\implies y ' = \cos x - \frac2\pi\implies y ''=-\sin x <0; \forall x\in \left(0; \frac\pi2\right)$
$\implies \left\{\begin{matrix} y '(x)> y '(\... | juantheron | 14,311 | <p><a href="https://i.stack.imgur.com/ZwzBR.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZwzBR.png" alt="enter image description here"></a></p>
<p>Now Slope of $$\displaystyle \bf{OB} = m_{0B} = \frac{\sin x-0}{x-0} = \frac{\sin x}{x}.$$</p>
<p>and Slope of $$\displaystyle \bf{OA} = m_{0A} = \fr... |
1,590,619 | <p>I must find the sum $$\sum_\limits{n=1}^{\infty}\frac1{n(2n+1)}$$ I have already tried partial fractions, but obtained an indeterminate form of infinity minus infinity. Perhaps completing the square would help?</p>
| Jack D'Aurizio | 44,121 | <p>$$\sum_{n=1}^{+\infty}\frac{1}{n(2n+1)}=2\sum_{n=1}^{+\infty}\int_{0}^{1}\left(x^{2n-1}-x^{2n}\right)\,dx = 2\int_{0}^{1}\frac{x}{1+x}\,dx = 2-\log 4.$$</p>
|
437,960 | <p>I can't find any logic equivalence or inference rules on this. Personally, I feel that $\neg P \implies \neg T$ and $P \implies \neg T$ would mean that it follows that $\neg T$ is true regardless, and I should be able to use that fact as such in my next step. Is this proper reasoning, though?</p>
| Brian M. Scott | 12,042 | <p>I don’t know in what system you’re working, but $\neg P\to\neg T$ is equivalent to $T\to P$, and $P\to\neg T$ is equivalent to $T\to\neg P$. $(T\to P)\land(T\to\neg P)$ is equivalent to $T\to(P\land\neg P)$, which is equivalent to $\neg T\lor(P\land\neg P)$, i.e., to $\neg T\lor\bot$, where I use $\bot$ for a contra... |
437,960 | <p>I can't find any logic equivalence or inference rules on this. Personally, I feel that $\neg P \implies \neg T$ and $P \implies \neg T$ would mean that it follows that $\neg T$ is true regardless, and I should be able to use that fact as such in my next step. Is this proper reasoning, though?</p>
| Hagen von Eitzen | 39,174 | <p>The Wikipedia <a href="http://en.wikipedia.org/wiki/List_of_rules_of_inference" rel="nofollow">list of rules of inference</a> lists <em>case analysys</em>
$$\begin{align}&\phi\lor\psi\\&\phi\to\chi\\&\underline{\psi\to\chi}\\&\chi\end{align} $$
We could make use of this with $\phi\equiv P$, $\psi\equ... |
209,672 | <blockquote>
<p>Show that <span class="math-container">$n$</span> lines separate the plane into <span class="math-container">$\frac{n^2+n+2}{2}$</span> regions if no two of these lines are parallel and no three pass through a common point.</p>
</blockquote>
<p>I know we start with the base case, where, if we call t... | shraman shraman | 330,248 | <p>Suppose we draw <span class="math-container">$n$</span> straight lines on the plane so that every pair of
lines intersects (but no <span class="math-container">$3$</span> lines intersect at a common point). Into
how many regions do these <span class="math-container">$n$</span> lines divide the plane?</p>
<p>With <... |
352,733 | <p>Can anyone show me where I can find a proof of the following?
$$H_n \sim \ln{n}+\gamma+\frac{1}{2n}-\sum_{k=1}^\infty \frac{B_{2k}}{2k n^{2k}}=\ln{n}+\gamma+\frac{1}{2n}-\frac{1}{12n^2}+\frac{1}{120n^4}-\cdots$$</p>
| Community | -1 | <p>As Steven Stadnicki points out in the comments, this is just an application of <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula" rel="noreferrer">Euler–Maclaurin formula</a>.</p>
<hr>
<p>Abel's partial summation technique:
\begin{align*}
\sum_{n=1}^{N} a(n) f(n) & = \sum_{n=1}^{N} f(n) (A... |
352,733 | <p>Can anyone show me where I can find a proof of the following?
$$H_n \sim \ln{n}+\gamma+\frac{1}{2n}-\sum_{k=1}^\infty \frac{B_{2k}}{2k n^{2k}}=\ln{n}+\gamma+\frac{1}{2n}-\frac{1}{12n^2}+\frac{1}{120n^4}-\cdots$$</p>
| Richard M. Woodward | 71,227 | <p>You can probably get what you need from Donald E. Knuth's "The Art of Computer Programming," Volume 1 (Fundamental Algorithms), Addison Wesley. In my (Second) Edition the relevant section is 1.2.11.2 (page 108).</p>
<p>I used this back in the mid-1970's to prove that the average rounding error when using computer a... |
287,516 | <p>I need to calculate value of $1000^{th}$ derivate of the following function at $0$:</p>
<p>$$
f(x) = \frac{x+1}{(x-1)(x-2)}
$$</p>
<p>I've done similar problems before (e.g. $f(x)= \dfrac{x}{e^{x}}$) but the approach I've used would not work in this case and I believe I should expand this function into a power ser... | Argha | 35,821 | <p><strong>Hint:</strong> Note that $f(x)=\frac{3}{x-2}-\frac{2}{x-1}$ and if $g(x)=\frac{1}{x-a}$ then $g^n(x)=\frac{(-1)^n n! }{(x-a)^{n+1}}$ where $a$ is a constant and $g^n(x)$ is the $n$-th derivative of $g(x)$.</p>
<p>Here $$\begin{align} \frac{A}{x-2}+\frac{B}{x-1} &=\frac{x+1}{(x-2)(x-1)}\\ \implies A(x-1)... |
80,265 | <p>Let $\alpha$ be a root of the polynomial $X^3-X-1$. The polynomial has discriminant $-23$ which is not a square, so the splitting field must have Galois group $S_3$. I would like to figure out the splitting of primes in $\mathbb{Q}(\alpha)$. The ramified case is trivial, so assume that $p$ is not ramified. By lookin... | Qiaochu Yuan | 232 | <p>Depends on what you mean by "explicit." Prime splitting in this case is governed by the coefficients of the <a href="http://en.wikipedia.org/wiki/Modular_form" rel="nofollow noreferrer">modular form</a></p>
<p>$$f(q) = q \prod_{n=1}^{\infty} (1 - q^n) (1 - q^{23n}).$$</p>
<p>See <a href="https://mathoverflow.net/q... |
668,580 | <p><strong>Theorem</strong> <em>Suppose</em> $$\begin{align}(a)&\sum_{n=0}^\infty a_n\quad\text{converges absolutely}\\(b)&\sum_{n=0}^\infty a_n=A\\(c)&\sum_{n=0}^\infty b_n=B\\(d)&c_n=\sum_{k=0}^n a_k b_{n-k}\quad(n=0,1,2,\dots)\end{align}$$ Then $$\sum_{n=0}^\infty c_n=AB$$</p>
<p><strong>Proof.</str... | Bobby | 10,809 | <p>Assume that $ \overline{(x-1)(x+1)} = \bar{0}$ and by definition of $\mathbb Z_p^*$, we have $p\,|\,(x-1)(x+1)$. Thus, either $p\,|x+1$ or $p\,|\,x-1$ and it implies that $\bar{x} = \bar{1},\overline{p-1}$.</p>
<p>The converse is clear.</p>
<p>Note that $\overline 1=\{\ldots,-p+1,p+1,2p+1\ldots\}$ or $\overline 0... |
1,802,892 | <p>If $ Q(x)= x^2-5x+1 $ , find $ \dfrac {Q(5+h)-Q(5)}{h} $</p>
<p>can someone show the steps to reach the answer of $h+5$?</p>
<p>I got it down to $\frac{h^2+5h-2}{h}$</p>
| Lanier Freeman | 262,519 | <p>$$\frac{Q(5+h)-Q(5)}{h}$$</p>
<p>I get this:</p>
<p>$$\frac{h^2+10h+25-5h-25+1-1}{h}$$</p>
<p>$$\frac{h^2+5h}{h}$$</p>
<p>$$h+5$$</p>
|
1,850,977 | <p>I am creating "children" in script and would like to calculate the number of children before launching the script. I have "width" the number of child nodes and "depth" the depth of the nodes.</p>
<ul>
<li>5 wide and 1 deep = 5</li>
<li>5 wide and 2 deep = 30 = (5+25)</li>
<li>5 wide and 3 deep = 155 = (5+25+125)</l... | C. Falcon | 285,416 | <p>Notice that: $$3^{2n+5}+40n+13=9(3^{2n+3}+40n−27)−320n+256.$$</p>
|
1,850,977 | <p>I am creating "children" in script and would like to calculate the number of children before launching the script. I have "width" the number of child nodes and "depth" the depth of the nodes.</p>
<ul>
<li>5 wide and 1 deep = 5</li>
<li>5 wide and 2 deep = 30 = (5+25)</li>
<li>5 wide and 3 deep = 155 = (5+25+125)</l... | Nate | 91,364 | <p>Let $A_n = 3^{2n+3}+40n-27$, then $A_n = 11A_{n-1} - 19A_{n-2} + 9A_{n-3}$.</p>
<p>From this it's clear that if 64 divides $A_n$ three consecutive values of $n$ then it holds for the next. So by induction it's enough to check it for $n = -1,0,1$, which is easy enough to do by hand. </p>
|
138,857 | <p>At an ambulance stations there are 5 nurses and 4 medics. Find in how many ways can we create a team with three people, where at least one of them is a medic.</p>
<p>I solved the problem this way: there are $4\cdot C_5^2$ ways to form a team with one medic and two nurses and $C_4^2 \cdot 5$ ways to form a team with... | Denmark_Cun | 463,424 | <p>Best way</p>
<p>(Count of those with at least 1 medic) = (Count of all combinations) - (Count those
with no medics)</p>
<p>$${}^9\mathrm C_3-{}^5\mathrm C_3 = 74$$</p>
|
452,570 | <p>I need help with this problem:</p>
<blockquote>
<p>Let $\{u_{n}\}$ and $\{v_{n}\}$, $(n \in \mathbb{N})$ be two different Hilbert bases in a Hilbert space $H$. Define a linear operator $T$ so that, for $x \in H$:
$$T(x) = \sum_{n=1}^{\infty} \lambda(n)u_{n}⟨v_{n} \mid x⟩$$ where $\lambda(n)$ is a fixed sequence... | martini | 15,379 | <p>We have for $x \in H$ $\def\norm#1{\left\|#1\right\|}\def\sp#1{\left<#1\right>}$
\begin{align*}
\norm{Tx}^2 &= \norm{\sum_{n=1}^\infty \lambda_n u_n\sp{v_n, x}}^2\\
&= \sp{\sum_{n=1}^\infty\lambda_n\sp{v_n, x}u_n, \sum_{m=1}^\infty \lambda_m \sp{v_m, x}u_m}\\
&= \sum_{n=1}^\infty\sum_{m=1... |
1,229,488 | <p>I'm not sure where to begin on this problem. Do you begin by solving the integral of each one then applying the conditions to each respective integral? Any help would be appreciated.</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$\frac{d[f''(x)]}{dx}=\sin x\implies d[f''(x)]=\sin x\ dx$$</p>
<p>Integrating both sides wrt $x,$ to find $f''(x)=-\cos x+C$ where is an arbitrary constant </p>
<p>Now $x=0\implies-9=f''(0)=-\cos0+C\iff C=-8$</p>
<p>Continue this way to find $f(x)$</p>
|
4,539,866 | <blockquote>
<p>If <span class="math-container">$a+b+c=1$</span> and <span class="math-container">$a, b, c\geq0$</span>, <br />
what is the maximum value of <span class="math-container">$\frac{1}{a^2-4a+9}+\frac{1}{b^2-4b+9}+\frac{1}{c^2-4c+9}$</span> ?</p>
</blockquote>
<p>I found its answer by using CAS Calculator. ... | River Li | 584,414 | <p>We have, for all <span class="math-container">$x\in [0, 1]$</span>,
<span class="math-container">$$\frac{1}{18}x + \frac{1}{9} - \frac{1}{x^2 - 4x + 9}
= \frac{x(x-1)^2}{18(x^2-4x+9)} \ge 0. \tag{1}$$</span></p>
<p>Using (1), we have
<span class="math-container">$$\frac{1}{a^2 - 4a + 9}
+ \frac{1}{b^2 - 4b + 9} + \f... |
4,182,765 | <p>Claim: <span class="math-container">$3\mid n^4-n^2$</span> for all <span class="math-container">$n \in \mathbb{Z^+}$</span>, <span class="math-container">$n\ge2$</span>.</p>
<p>What I've done so far:</p>
<p>Base case: Let <span class="math-container">$n=2$</span>. Then <span class="math-container">$\exists k \in \m... | Lalit Tolani | 913,165 | <p>Note that claim is</p>
<p><span class="math-container">$3|n^4-n^2$</span></p>
<p><span class="math-container">$\implies 3|n*(n-1)*n*(n+1)$</span></p>
<p>Clearly the last three terms written are three consecutive natural numbers , therefore at most one of them will be a multiple of three and at least one will be even... |
1,967,460 | <p>Given for $\alpha>1$
$$\int^{\pi}_0 \frac{dx}{\alpha-\cos x}=\frac{\pi}{\sqrt{\alpha^2-1}}$$</p>
<p>Prove that $$\int^{\pi}_0 \ln \left(\frac{b-\cos x}{a- \cos x}\right)dx=\pi \left(\frac{b+\sqrt{b^2-1}}{a+\sqrt{a^2-1}}\right)$$
if $a, b>1$</p>
| Mark Viola | 218,419 | <blockquote>
<p>Since one posted solution relied on, and one comment discussed the real analysis methodology of differentiating under the integral, I thought it might be instructive to present a way forward that relies on contour integration. To that end, we proceed.</p>
</blockquote>
<hr>
<p>Let $I(a,b)$ be the i... |
1,937,175 | <p>I have an example of LU factorization and I need to understand somethings..</p>
<p>I need to solve matrix A = $\begin{bmatrix}1&2&1\\2&5&3\\1&4&9\end{bmatrix}$ using LU factorization with partial pivoting
, I know the first step is to factor PA = LU
$$\begin{bmatrix}0&1&0\\1&0&am... | Junsworth | 369,635 | <p>Please tell me if this is wrong but these are my steps for finding L and U.</p>
<p>So starting with the original matrix A:\begin{bmatrix}1&2&1\\2&5&3\\1&4&9\end{bmatrix}
Swap R1 and R2 \begin{bmatrix}2&5&3\\1&2&1\\1&4&9\end{bmatrix}
R2 - 1/2R1 and R3 - 1/2R1 \begin{bm... |
1,937,175 | <p>I have an example of LU factorization and I need to understand somethings..</p>
<p>I need to solve matrix A = $\begin{bmatrix}1&2&1\\2&5&3\\1&4&9\end{bmatrix}$ using LU factorization with partial pivoting
, I know the first step is to factor PA = LU
$$\begin{bmatrix}0&1&0\\1&0&am... | Brian Fitzpatrick | 56,960 | <p>Here are the steps for producing our desired <span class="math-container">$PA=LU$</span> factorization using maximal partial pivoting
<span class="math-container">\begin{align*}
\overset{A}{\left[\begin{array}{rrr}
1 & 2 & 1 \\
2 & 5 & 3 \\
1 & 4 & 9
\end{array}\right]}\xrightarrow{R_1\leftri... |
2,631,937 | <p>I understand the Divergence theorem is as follows, </p>
<p>$\iiint\limits_{v} (\nabla \bullet \textbf{F}) \;dV= \iint\limits_{dv} \textbf{F} \bullet d \textbf{S}$</p>
<p>My query is if i already know $(\nabla \bullet \textbf{F})= 0$ then can i assume the following,</p>
<p>$(\nabla \bullet \textbf{F})\iiint\limits... | kiyomi | 527,262 | <p>The <strong>Divergence Theorem</strong> states that given $V\subseteq\mathbb{R^n}$ which has a smooth boundary $S$, and given a differentiable vector field $\mathbf{F}$ (of class $C^1$) which is continuous in a neighborhood of $V$, then the following is true:</p>
<blockquote>
<p>$$\iiint_V\left(\nabla\cdot\mathb... |
1,976,538 | <p>How do you know your answer to a probability question is correct without checking the solutions?</p>
<p>For example, when you have an equation you can get a graphing calculator, create the graph for the first part of the equation, the graph for the second and then get the intersection to make sure your calculations... | Sarvesh Ravichandran Iyer | 316,409 | <p>I answered the earlier one, I'll answer this one too.</p>
<p>See, the bottom row contains nine lockers. The favourable case consists of choosing three lockers from these nine. So $\binom{9}{3}$ is the number of favourable cases.</p>
<p>The total number of cases is $\binom{27}{3}$. So the answer would be $\dfrac{\... |
1,976,538 | <p>How do you know your answer to a probability question is correct without checking the solutions?</p>
<p>For example, when you have an equation you can get a graphing calculator, create the graph for the first part of the equation, the graph for the second and then get the intersection to make sure your calculations... | Thanassis | 8,153 | <p>Assuring (making sure) a solution is correct is a philosophical problem really, but I know you are not interested in this. You basically want to validate your solution, or in other words increase your confidence that your solution is correct. </p>
<p>A generic way to validate probability problems is to <strong>run ... |
2,778,914 | <p>In the book of <em>Foundations of Geometry</em> by Hilbert, at page 2, it is stated that</p>
<blockquote>
<p>I, 1. Two distinct points A and B always completely determine a
straight line a. We write AB = a or BA = a.</p>
<p>I, 2. Any two distinct points of a straight line completely determine
that line; ... | Eric Wofsey | 86,856 | <p>As far as I can tell, this is best described as an error in the text, possibly one introduced by the translator. These statements appear to be going to some contortions to state everything in terms of the notation "<span class="math-container">$AB=a$</span>", which is not used in the original German (whic... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | მამუკა ჯიბლაძე | 41,291 | <p>Being a shameless platonist, for me the answer is obvious. We only need mathematics since it is the only way to be able to reach certain things transcending ourselves - just as music or poetry is the only way to be able to reach certain other things (or maybe other aspects of the same things). Computers certainly ma... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | Douglas Zare | 2,954 | <p>The particular techniques used to make progress on go do not seem to help much with mathematics. While we might figure out how to get computers to prove deep theorems requiring the introduction of new mathematical ideas later, most of the work has yet to be done. </p>
<p>Neural networks are reasonably good at regre... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | Anixx | 10,059 | <p>Thomas Breuer has proven (<a href="https://homepages.fhv.at/tb/tb/cms/index.html%3Fdownload=tbDISS.pdf" rel="nofollow noreferrer">Classical Observables, Measurement, and Quantum Mechanics</a>, <a href="https://homepages.fhv.at/tb/tb/cms/index.html%3Fdownload=tbPHILSC.pdf" rel="nofollow noreferrer">The Impossibility ... |
1,894,926 | <p>I know no formal system containing arithmetic can prove itself consistent. But is it possible to add some <em>ungrounded</em> axiom in the sense described here <a href="https://en.wikipedia.org/wiki/Liar_paradox#Saul_Kripke" rel="nofollow">https://en.wikipedia.org/wiki/Liar_paradox#Saul_Kripke</a> such that the new ... | user21820 | 21,820 | <p><code>I know no formal system containing arithmetic can prove itself complete.</code>
$
\def\nn{\mathbb{N}}
\def\t{\text}
\def\th{\t{Th}}
\def\con{\t{Con}}
\def\prov{\square}
\def\pa{\t{PA}}
$</p>
<p>Wrong. Any inconsistent formal system that can state completeness of itself proves itself complete. [Edit: In my ori... |
485,755 | <p>Let's call a point $P$ which satisfies the following condition 'a rational point'.</p>
<p><strong>Condition</strong>: Each distance $PA, PB, PC$ from a point $P$ to three vertices $A, B, C$ of an equilateral triangle $ABC$ which has edge-length $1$ is <strong>rational number</strong>.</p>
<p>I'm interested in this... | David E Speyer | 448 | <p>For the record, I will solve the much easier problem of finding infinitely many rational points on the circumcircle. Let's consider the case of a point $P$ on the arc from $A$ to $B$. By <a href="http://en.wikipedia.org/wiki/Ptolemy%27s_theorem" rel="nofollow noreferrer">Ptolemy's theorem</a>, $|PA| + |PB| = |PC|$, ... |
304,646 | <p>I am asked to solve the following system of equations using Cramer's Rule</p>
<p>$$
A^2 + B = x\\
AB = 1-x^2
$$</p>
<p>where $A$ and $B$ are 2 unknowns that are to be solved for some given $x$. ie I'm expecting the solution to be $A = a(x)$ and $B = b(x)$ where $a(x)$ and $b(x)$ is some function of $x$.</p>
<p>I... | ebsddd | 36,669 | <p>Multiply both sides of the first equation to get $A^3+AB=Ax$. Plug in the second equation and rearrange the terms to get $A^3-Ax+(1-x^2)=0$. Solve this cubic either <a href="http://en.wikipedia.org/wiki/Cubic_function#Cardano.27s_method" rel="nofollow">by hand</a> or using <a href="http://www.wolframalpha.com/input/... |
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