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 |
|---|---|---|---|---|
2,403,741 | <blockquote>
<p>Solve for $\theta$ the following equation.
$$\sqrt {3} \cos \theta - 3 \sin \theta = 4 \sin 2\theta \cos 3\theta.$$</p>
</blockquote>
<p>I tried writing sin and cos expansions but it is becoming too long.Please help me.</p>
| Michael Rozenberg | 190,319 | <p>Because $13=6+6+1=2+2+9$. </p>
<p>Now we need to check other sums.</p>
|
244,679 | <p>I have a two variable function <code>z[x,y] = f[x,y] + g[x,y]</code>, such that I know the functional form of <code>f[x,y]</code> but not of <code>g[x,y]</code>. I have to do some symbolic calculations with the function <code>z[x,y]</code>, but I would like to keep only the first order in <code>g[x,y]</code> (treati... | Adam | 74,641 | <p>The following (and my comment) overlooks the fact that <code>u1</code> and <code>f</code> are dependent. Copy code, don't rewrite it -- also don't over edit once copied!</p>
<hr />
<p>When I look at <code>Normal@f</code> for various values of <code>dos</code>, I find the "edge" elements of <code>f</code> ... |
180,647 | <p>Two persons have 2 uniform sticks with equal length which can be cut at any point. Each person will cut the stick into $n$ parts ($n$ is an odd number). And each person's $n$ parts will be permuted randomly, and be compared with the other person's sticks one by one. When one's stick is longer than the other person's... | rschwieb | 29,335 | <p>The previous answers that basically say "No, addition/subtraction is not defined between matrices of different dimensions" are the correct answer to your question.</p>
<p>Actually though, something like this is done formally in <a href="http://en.wikipedia.org/wiki/Clifford_algebra" rel="nofollow">Clifford algebras... |
167,446 | <p>Let $p$ be a prime number, $C_p$-cyclic group of order $p$, and $G$ an elementary p-group of order $p^n$. Let us denote by Cext$(G,C_p)$ the group of all central extensions of $C_p$ by $G$. Is the number of non isomorphic groups in Cext$(G,C_p)$ known as a function of $n$? </p>
| Derek Holt | 35,840 | <p>I voted to close because I was unsure which way around the extension went but, as Yves said, the question is almost trivial if $C_p^n$ is the normal subgroup.</p>
<p>So, suppose that $N \unlhd G$ with $N=C_p$ and $G/N \cong C_p^n$.</p>
<p>Recall that a $p$-group of this form is called <em>extraspecial</em> if $N=Z... |
2,651,054 | <p>I have this expression:
$$(x + y + z’)(x’ + y’ + z)$$ which I am trying to simplify. I decide to multiply it out in order to get, $${\color{red}{(xx')}}+(xy')+(xz)+(yx')+{\color{red}{(yy')}}+(yz)+(z'x')+(z'y')+{\color{red}{(z'z)}}.$$
I know that the $xx', yy'$ and $zz'$ would just be $0$, however, now I am stuck. ... | Mohammad Riazi-Kermani | 514,496 | <p>$$z^3+2=a_0+a_1(z-1)+a_2(z-1)^2+a_3(z-1)^3.$$</p>
<p>$$ z=1 \implies a_0 =3$$ Differentiate, you get $$3z^2 =a_1+2a_2 (z-1)+ 3a_3(z-1)^2$$ </p>
<p>$$ z=1 \implies a_1=3$$</p>
<p>Differentiate, you get</p>
<p>$$ 6z=2a_2+6a_3(z-1)$$</p>
<p>$$ z=1 \implies a_2=3$$</p>
<p>Differentiate, you get</p>
<p>$$ a_3 =1$$... |
2,180,700 | <p>A and B toss a fair coin 10 times. In each toss, if its a head A's score gets incremented by 1, if its a tail B's score gets incremented by 1.</p>
<p>After 10 tosses, the person with the greatest score wins the game.</p>
<p>What is the probability that A wins?</p>
<p>And if B alone gets an extra toss. What is the... | user247327 | 247,327 | <p>It should be obvious, from symmetry, that, in the first case, where the coin is flipped 10 times, that A and B have the same probability of winning. But if there are 5 heads and 5 tails, neither wins. The probability of that is $(1/2)^{10}\begin{pmatrix}10 \\ 5 \end{pmatrix}= \frac{252}{1024}= \frac{63}{251}$ so t... |
957,940 | <p>I'm "walking" through the book "A walk through combinatorics" and stumbled on an example I don't understand. </p>
<blockquote>
<p><strong>Example 3.19.</strong> A medical student has to work in a hospital for five
days in January. However, he is not allowed to work two consecutive
days in the hospital. In how... | Chris Taylor | 4,873 | <p>Let's look at a simpler example - choosing 3 days from 6, such that there are no two consecutive days. Obviously there are four ways to do this - any of $(1,3,5)$, $(1,3,6)$, $(1,4,6)$ or $(2,4,6)$.</p>
<p>Formalizing the problem, you need to choose $(a_1,a_2,a_3)$ such that $1 \leq a_i \leq 6$ and there are no con... |
957,940 | <p>I'm "walking" through the book "A walk through combinatorics" and stumbled on an example I don't understand. </p>
<blockquote>
<p><strong>Example 3.19.</strong> A medical student has to work in a hospital for five
days in January. However, he is not allowed to work two consecutive
days in the hospital. In how... | Florian D'Souza | 176,860 | <p>The number 27 is there because he has to work 5 days in January, and January has 31 days. So, after he picks the 1st day to work, he can only pick from among 30 days for the 2nd day, et cetera until he has 27 days from which he can pick his 5th day.</p>
<p>The subtractions, I believe, are just illustrating that onc... |
957,940 | <p>I'm "walking" through the book "A walk through combinatorics" and stumbled on an example I don't understand. </p>
<blockquote>
<p><strong>Example 3.19.</strong> A medical student has to work in a hospital for five
days in January. However, he is not allowed to work two consecutive
days in the hospital. In how... | najayaz | 169,139 | <p>These kind of problems can be generalized to a single formula in the following manner:</p>
<p>Suppose I have $n$ objects arranged in a row and I pick any $r$ objects such that no two of them are consecutive. I want to calculate in how many ways I can do this. To do that I first randomly select $r$ objects and take ... |
2,942,263 | <p>I am curious whether there is an algebraic verification for <span class="math-container">$y = x + 2\sqrt{x^2 - \sqrt{2}x + 1}$</span> having its minimum value of <span class="math-container">$\sqrt{2 + \sqrt{3}}$</span> at <span class="math-container">$\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{6}}$</span>. I have been to... | Cesareo | 397,348 | <p>Without derivatives,</p>
<p>Considering the functions</p>
<p><span class="math-container">$$
f(x,y)=y-x-2\sqrt{x^2-\sqrt 2 x-1} = 0\\
y = \lambda
$$</span></p>
<p>Their intersection is at the solution for</p>
<p><span class="math-container">$$
f(x,\lambda)=\lambda-x-2\sqrt{x^2-\sqrt 2 x-1} = 0
$$</span></p>
<p>... |
2,942,263 | <p>I am curious whether there is an algebraic verification for <span class="math-container">$y = x + 2\sqrt{x^2 - \sqrt{2}x + 1}$</span> having its minimum value of <span class="math-container">$\sqrt{2 + \sqrt{3}}$</span> at <span class="math-container">$\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{6}}$</span>. I have been to... | roman | 53,017 | <p>One of the approaches may be as follows:</p>
<p>Suppose there exists some <span class="math-container">$a$</span> which is the minimum. Then:</p>
<p><span class="math-container">$$
\begin{align}
x + 2\sqrt{x^2 - \sqrt2 x+1} &= a \\
2\sqrt{x^2 - \sqrt2 x+1} &= a-x
\end{align}
$$</span></p>
<p>Square both s... |
1,002,719 | <p>If we have</p>
<p>$f: \{1, 2, 3\} \to \{1, 2, 3\}$</p>
<p>and</p>
<p>$f \circ f = id_{\{1,2,3\}}$</p>
<p>is the following then always true for every function?</p>
<p>$f = id_{\{1,2,3\}}$</p>
| Milo Brandt | 174,927 | <p>Notice that we could easily show that $f$ is bijective, since if $f\circ f$ is bijective, as the identity is, it must be that $f$ is injective (since were it not, $f\circ f$ could not be either) and that $f$ is surjective (for the same reason).</p>
<p>In particular, this implies that $f$ will be a member of the sym... |
458,088 | <p>I would like to find an approximation when $ n \rightarrow\infty$ of $ \frac{n!}{(n-2x)!}(n-1)^{-2x} $. Using Stirling formula, I obtain $$e^{\frac{-4x^2+x}{n}}. $$ The result doesn't seem right!</p>
<p>Below is how I derive my approximation. I use mainly Stirling Approximation and $e^x =(1+\frac{x}{n})^n $.</p>
... | zyx | 14,120 | <p>Online topology game illustrating a recent theorem</p>
<p><a href="http://www.sci.osaka-cu.ac.jp/math/OCAMI/news/gamehp/etop/gametop.html" rel="nofollow">http://www.sci.osaka-cu.ac.jp/math/OCAMI/news/gamehp/etop/gametop.html</a></p>
|
1,329,398 | <p>So, I've posted a question regarding Wikipedia's quartic page. This was from the first question.</p>
<blockquote>
<p>I'm trying to implement the general quartic solution for use in a ray tracer, but I'm having some trouble. The solvers I've found do cause some strange false negatives leaving holes in the tori I'm t... | asmeurer | 781 | <p>As noted in a comment by Robert Israel, if you are using floating point numbers, it's better to use a root-finding algorithm than the exact formula for quartic roots, as root finding algorithms tend to have much better numerical stability, and avoid issues like this one. They also tend to work much more generally (i... |
49,074 | <p>It might sound silly, but I am always curious whether Hölder's inequality $$\sum_{k=1}^n |x_k\,y_k| \le \biggl( \sum_{k=1}^n |x_k|^p \biggr)^{\!1/p\;} \biggl( \sum_{k=1}^n |y_k|^q \biggr)^{\!1/q}
\text{ for all }(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in\mathbb{R}^n\text{ or }\mathbb{C}^n.$$
can be derived from the ... | Did | 6,179 | <p>Yes it can, assuming nothing more substantial than the fact that midpoint convexity implies convexity. Here are some indications of the proof in the wider context of the integration of functions.</p>
<p>Consider positive $p$ and $q$ such that $1/p+1/q=1$ and positive functions $f$ and $g$ sufficiently integrable wi... |
3,345,063 | <p><a href="https://i.stack.imgur.com/T3Ue9.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/T3Ue9.png" alt="Given a triangle whose apex angle is \theta" /></a></p>
<p><strong>Given a triangle with two circles and apex angle equals <span class="math-container">$\theta$</span>.</strong></p>
<p><em><str... | Henry | 6,460 | <p>The question does not actually mention the base edge</p>
<p>You introduced the tangent where the two circles touch, which necessarily creates a small isosceles triangle. You can then make the base tangent parallel to this, ensuring a similar large triangle which is then isosceles</p>
<p>So there is no scalene case ... |
1,855,650 | <p>Need to solve:</p>
<p>$$2^x+2^{-x} = 2$$</p>
<p>I can't use substitution in this case. Which is the best approach?</p>
<p>Event in this form I do not have any clue:</p>
<p>$$2^x+\frac{1}{2^x} = 2$$</p>
| Zau | 307,565 | <p>Hint:
By AM-GM inequality , $2^x+2^{-x} \geq 2 \times \sqrt{2^x \times 2 ^{-x}} = 2 $
When $2 ^x = 2^{-x}$ the equality holds.</p>
<p>Alternative solution:</p>
<p>Easy to show that $x = 0$, is one possible solution.</p>
<p>$$\frac{d(2^x+2^{-x})}{dx} = ( 2^x -2^{-x}) \log 2$$
which positive over $(0,+\infty)$, n... |
1,839,057 | <p>Where n is an integer, $n\ge1$ and $(A,B)$ just constants </p>
<blockquote>
<p>$$I=\int_{-n}^{n}{x+\tan{x}\over A
+B(x+\tan{x})^{2n}}dx=0$$</p>
</blockquote>
<p>It is obvious that</p>
<p>$$\int_{-n}^{n}x+\tan{x}dx=0$$</p>
<p>Let make a substitution for <em>I</em> $$u=x+\tan{x}\rightarrow du=1+\sec^2{x}dx$$</p>... | Hosein Rahnama | 267,844 | <p>Let us call the integrand as</p>
<p>$$f(x)={x+\tan{x}\over A
+B(x+\tan{x})^{2n}}$$</p>
<p>Then it is quite evident that it is an odd function of $x$</p>
<p>$$f(-x)=-f(x)$$</p>
<p>and hence you can easily conclude</p>
<p>$$\int_{-n}^{n}f(x)dx=0$$</p>
|
1,839,057 | <p>Where n is an integer, $n\ge1$ and $(A,B)$ just constants </p>
<blockquote>
<p>$$I=\int_{-n}^{n}{x+\tan{x}\over A
+B(x+\tan{x})^{2n}}dx=0$$</p>
</blockquote>
<p>It is obvious that</p>
<p>$$\int_{-n}^{n}x+\tan{x}dx=0$$</p>
<p>Let make a substitution for <em>I</em> $$u=x+\tan{x}\rightarrow du=1+\sec^2{x}dx$$</p>... | Mc Cheng | 327,363 | <p>Let:<br>
$$f(x)={x+\tan{x}\over A
+B(x+\tan{x})^{2n}}$$
Set $x \mapsto -u$ </p>
<p>\begin{align}
f(-u)&={-u+\tan{-u}\over A
+B(-u+\tan{-u})^{2n}} \\
&={-u-\tan{u}\over A
+B(-1)^{2n}(u+\tan{u})^{2n}} \\
&=(-1){u+\tan{u}\over A
+B(u+\tan{u})^{2n}} \\
&=-f(u)
\end{align}
Since <em>x</em> and <em>u</em... |
681,608 | <blockquote>
<p>Prop. 6.9: Let $X \to Y$ be a finite morphism of non-singular curves, then for any divisor $D$ on $Y$ we have $\deg f^*D=\deg f\deg D$.</p>
</blockquote>
<p>I can not understand two points in the proof:</p>
<p>(1) (Line 9) Now $A'$ is torsion free, and has rank equal to $r=[K(X):K(Y)]$.</p>
<p>Sinc... | SomeEE | 126,056 | <p>Line 9</p>
<p>$A'$ is a localization of the ring A which is defined as the integral closure of B in K(X). This gives us $Quot(A) = Quot(A') = K(X)$ and so $Quot(A')$ is $r$ dimensional over $Quot(B)$. $A'$ is torsion free and finitely generated over the PID $\mathcal{O}_{Q}$ so $A' = \mathcal{O}_{Q}^{\oplus n}$ f... |
192,883 | <p>Can anyone please give an example of why the following definition of $\displaystyle{\lim_{x \to a} f(x) =L}$ is NOT correct?:</p>
<p>$\forall$ $\delta >0$ $\exists$ $\epsilon>0$ such that if $0<|x-a|<\delta$ then $|f(x)-L|<\epsilon$</p>
<p>I've been trying to solve this for a while, and I think it w... | Charlie Parker | 118,359 | <p>The main issue with the definition that you suggested is that its trivially satisfied as other people pointed out. If you want to get $\delta$ close (really close) to $a$ then your definition says you need to find <strong>any</strong> $\epsilon$ (thats the problem! its too flexible). Trivially set $\epsilon = \infty... |
1,097,134 | <p>this is something that came up when working with one of my students today and it has been bothering me since. It is more of a maths question than a pedagogical question so i figured i would ask here instead of MESE.</p>
<p>Why is $\sqrt{-1} = i$ and not $\sqrt{-1}=\pm i$?</p>
<p>With positive numbers the square r... | Quality | 153,357 | <p>In fact, there are two complex square roots of -1. i and -i. So in fact the answer to your question that it is equal to +i or -i..</p>
|
124,662 | <p>Is topology on $\mathbb{R}/\mathbb{Z}$ compact? If it is, how to prove it? </p>
<p>$\mathbb{R}/\mathbb{Z}$ denotes the set of equivalence classes of the set of real numbers, two real numbers being equivalent if and only if their difference is an integer.</p>
| azarel | 20,998 | <p>$\bf Hint:$ Find a compact subset $K$ of $\mathbb R$ so that the quotient map $\pi: \mathbb R\to \mathbb {R/Z}$ restricted to $K$ is onto.</p>
|
92,660 | <p>Let $X$ be a nonsingular projective variety over $\mathbb{C}$, and let $\widetilde{X}$ be the blow-up of X at a point $p\in X$.
What relationships exist between the degrees of the Chern classes of $X$ (i.e. of the tangent bundle of $X$) and the degrees of the Chern classes of $\widetilde{X}$?</p>
<p>Thanks.</p>
| Liviu Nicolaescu | 20,302 | <p>Assume $X$ is smooth compact of dimension $n$ and $x_0\in X$ is the point where we perform the blowup. Set $ X_* := X \setminus x_0 $, $ \tilde{X}_* := \tilde{X} \setminus E$. Denote by $N$ a tubular neighborhood of $E$ in $\tilde{X}_* $. By Mayer-Vietoris, the Chern classes of $ \tilde{X} $ are determined once ... |
209,856 | <p>Of course, I can use Stirling's approximation, but for me it is quite interesting, that, if we define $k = (n-1)!$, then the left function will be $(nk)!$, and the right one will be $k! k^{n!}$. I don't think that it is a coincidence. It seems, that there should be smarter solution for this, other than Stirling's ap... | cactus314 | 4,997 | <p>For $(nk)!$ your factors are $1,2,3,\dots, k$ then $k+1, \dots, 2k,2k+1 \dots, k!$.</p>
<p>For $k! k^{n!}$ your factors are $1,2,3,\dots, k$ but then constant $k,\dots,k$.</p>
<p>So every factor of <strong>(nk)!</strong> is > or = to each factor of <strong>k!k^(n!)</strong></p>
|
4,096,771 | <p>Given a sequence of iid random variables <span class="math-container">$(Y_i)_{i=1}^\infty$</span> on a probability space <span class="math-container">$(\Omega, \mathcal{F}, \mathbb{P})$</span> such that <span class="math-container">$\mathbb{E}|Y_i| < \infty$</span> and <span class="math-container">$\mathbb{E}Y_i ... | stochastic-conch | 765,353 | <p>Since we are in discrete time, it is enough to show that <span class="math-container">$\mathbb{E}[X_{n+1}\mid\mathcal{F}_{n}]=X_n$</span> a.s. for all <span class="math-container">$n\in\mathbb{N}$</span>. Notice that <span class="math-container">$X_{n+1}=Y_{n+1}+X_n$</span>, so that
<span class="math-container">$$
\... |
424,694 | <p>Let <span class="math-container">$p$</span> be a prime, and consider <span class="math-container">$$S_p(a)=\sum_{\substack{1\le j\le a-1\\(p-1)\mid j}}\binom{a}{j}\;.$$</span>
I have a rather complicated (15 lines) proof that <span class="math-container">$S_p(a)\equiv0\pmod{p}$</span>. This must be
extremely classic... | Ofir Gorodetsky | 31,469 | <p>Let <span class="math-container">$$P(x)=(1+x)^a-1-x^a=\sum_{1 \le j \le a-1} \binom{a}{j}x^j.$$</span> Working in a field <span class="math-container">$F$</span> where <span class="math-container">$|\{\mu \in F: \mu^{p-1}=1\}|=p-1$</span> (roots of unity of order <span class="math-container">$p-1$</span> exist), we ... |
2,916,037 | <p>Some cute results have every digit doubled. </p>
<p>\begin{align}
99225500774400 = {} & \frac{40!}{31!} \\[8pt]
33554433 = {} & 2^{25} +1 \\[8pt]
222277 = {} & -22^{2^2}+77^3 \\[8pt]
8811551199 = {} & 95^5 + 64^5 \\[8pt]
7755660000 = {} & 95^5 + 65^4 \\[8pt]
334444448888 = {} & 6942^3 - 1... | Deepesh Meena | 470,829 | <p>$$\frac{2244}{9999}=0.\overline{2244}$$
$$\frac{22446688}{10^8-1}=0.\overline{22446688}$$
$$\frac{112233445566778899}{10^{18}-1}=0.\overline{112233445566778899}$$
$$\frac{2}{909}=0.\overline{0022}$$
$$\frac{1133}{9999}=0.\overline{1133}$$</p>
<p>$$\frac{1}{11}=0.\overline{09}$$</p>
|
2,055,559 | <blockquote>
<p>Let <span class="math-container">$a,b,c$</span> be the length of sides of a triangle then prove that:</p>
<p><span class="math-container">$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0$</span></p>
</blockquote>
<p>Please help me!!!</p>
| Nat | 401,304 | <p>Label the triangle so that $a>b>c>0$</p>
<p>Then $ab > b^2$ and $cb> c^2$ and $ac> a^2$</p>
<p>So $ab-b^2 > 0$ and $a^2(ab-b^2)>0$</p>
<p>Similarly for the other two terms, so that their sum will naturally be greater than $0$</p>
|
356,530 | <p>I'm a really confused as to how to start this question, would really appreciate any help you guys could give me!</p>
| Brian M. Scott | 12,042 | <p>HINT: There are $2^{17}$ bit strings of length $17$. Count those with exactly $0,1,2,3$, and $4$ ones and subtract from $2^{17}$. For example, there is just one bit string with no ones. To find the number with exactly $2$ ones, observe that there are $\binom{17}2$ ways to choose positions for the ones, and once you’... |
273,499 | <blockquote>
<p>Show that every group $G$ of order 175 is abelian and list all isomorphism types of these groups. [HINT: Look at Sylow $p$-subgroups and use the fact that every group of order $p^2$ for a prime number $p$ is abelian.]</p>
</blockquote>
<p>What I did was this. $|G| = 175$. Splitting 175 gives us $175 ... | rschwieb | 29,335 | <p>This line seems especially mistaken: "I think, by definition of a normal subgroup, they are abelian and so this tells us that G is abelian." Certainly normal subgroups need not be abelian: for an example you can take the alternating subgroup of the symmetric group for any $n>5$.</p>
<p>The Sylow theorems tell y... |
89,188 | <p>Norimatsu's lemma says that on a smooth projective complex variety $X$ of dimension $n$, then we have $H^i(X,\mathcal O_X(-A-E))=0$ for $i < n$ when $A$ is an ample divisor and $E$ is a simple normal crossings (SNC) divisor. Does this statement remain true if $E$ is an effective divisor with SNC support? In ot... | Francesco Polizzi | 7,460 | <p>Here is a counterexample.</p>
<p>Let $X$ be a smooth cubic surface in $\mathbf{P}^3$ and $E$ a line on it; moreover take $A=-K_X$.</p>
<p>Then by Serre duality $$H^1(X, -A-aE)=H^1(X, K_X + A + aE)=H^1(X, aE).$$</p>
<p>On the other hand $h^0(X, aE)=1$ since $E^2=-1$ and $h^2(X, aE)=h^0(X, K_X-aE)=0$ since $K_X$ is... |
2,794,704 | <p>Can the following sum be further simplified? $${1\over 20}\sum_{n=1}^{\infty}\left(n^2+n\right)\left(\frac45\right)^{n-1}$$
(It's part of a probability problem)</p>
| Claude Leibovici | 82,404 | <p>$$f(x)=\sum_{n=1}^{\infty}n(n+1)x^{n-1}=\frac 1x\sum_{n=1}^{\infty}[n(n-1)+2n]x^{n}$$
$$f(x)=\frac {x^2}x\sum_{n=1}^{\infty}n(n-1)x^{n-2}+\frac {2x}x\sum_{n=1}^{\infty}nx^{n-1}=x \left(\sum_{n=1}^{\infty}x^{n}\right)''+2\left(\sum_{n=1}^{\infty}x^{n}\right)'$$ Use now
$$\sum_{n=1}^{\infty}x^{n}=\frac 1{1-x}$$</p>
|
3,141,510 | <p>Calculate sum
<span class="math-container">$$ \sum_{k=2}^{2^{2^n}} \frac{1}{2^{\lfloor \log_2k \rfloor} \cdot 4^{\lfloor \log_2(\log_2k )\rfloor}} $$</span></p>
<p>I hope to solve this in use of Iverson notation:</p>
<h2>my try</h2>
<p><span class="math-container">$$ \sum_{k=2}^{2^{2^n}} \frac{1}{2^{\lfloor \log... | epi163sqrt | 132,007 | <p>Here is an answer following rather closely OP's approach.</p>
<blockquote>
<p>We obtain for <span class="math-container">$n\in\mathbb{Z}, n\geq 0$</span>:
<span class="math-container">\begin{align*}
\color{blue}{\sum_{k=2}^{2^{2^n}}}&\color{blue}{\frac{1}{2^{\left\lfloor\log_2 k\right\rfloor}4^{\left\lfl... |
1,876,708 | <p>Let $m<n$. Why $\mathbb R^m$ is closed in $\mathbb R^n$ ? For example, let us take $\mathbb R^3$ and the subspace $\mathbb R^2$. It looks weird to me that $\mathbb R^2$ is closed in $\mathbb R^3$. To me it looks impossible. It may be open, but not closed. Any explanation is welcome.</p>
| Surb | 154,545 | <p>Let $\boldsymbol x=(x,y,0)\in\mathbb R^2$. We denote $$B(\boldsymbol x,\varepsilon)=\{\boldsymbol u\in \mathbb R^3\mid \|\boldsymbol x-\boldsymbol u\|\leq \varepsilon\}.$$</p>
<p>I think it's very easy to show (even to see), that for all $\varepsilon>0$, $$B(\boldsymbol x,\varepsilon)\cap \mathbb R^2\neq \empty... |
548,776 | <p>I am new in Topos theory. I have actually just started learning. I am reading MacLane-Moerdijk's book, as it was suggested to me as the best introduction to the subject. Unfortunately I can not make sense of the following.</p>
<p>In section 5 of Chapter I, (page 41) they build what is needed to prove that in the pr... | Ittay Weiss | 30,953 | <p>$u:C'\to C$ is a morphism, and $P$ is a presheaf, thus $P(u):P(C)\to P(C')$ is just a function between two sets, which you can apply to $p\in P(C)$ and get an element in $P(C')$, that is simply the element $P(u)(p)$, and the requirement is, as you write, just $P(u)(p)=p'$. </p>
<p>Notice that there is no guess work... |
8,816 | <p>What is the result of multiplying several (or perhaps an infinite number) of binomial distributions together?</p>
<p>To clarify, an example.</p>
<p>Suppose that a bunch of people are playing a game with k (to start) weighted coins, such that heads appears with probability p < 1. When the players play a round, t... | Mariano Suárez-Álvarez | 1,409 | <p>In [Matsusaka, T. Polarized varieties, fields of moduli and generalized Kummer varieties of polarized abelian varieties. Amer. J. Math. 80 1958 45--82.] it is proved that a non-singular projective variety has a maximal algebraic group of automorphisms (that is, every group which acts on the variety by automorphism... |
1,459,334 | <p>How can we show that additive inverse of a real number equals the number multiplied by -1, i.e. how can we show that $(-1)*u = -u$ for all real numbers $u$?</p>
| timber | 275,791 | <p>\begin{align}
&& (1 + (-1)) &= 0 \\
&\implies& (1+(-1))\cdot u &= 0\cdot u = 0\\
&\implies& u + (-1)\cdot u &= 0\\
&\implies& (-u) + (u + (-1)\cdot u) &= -u\\
&\implies& (-u + u) + (-1)\cdot u &= -u\\
&\implies& (-1)\cdot u &= -u
\end{align}</p>... |
1,282,486 | <p>Given the function $f(x) = |8x^3 − 1|$ in the set $A = [0, 1].$
Prove that the function is not differentiable at $x = \frac12.$ </p>
<p>The answer in my book is as follows:</p>
<p>$$\lim_{x \to \frac12-} \dfrac{f(x)-f(1/2)}{x-1/2} = -6$$
$$\lim_{x \to \frac12+} \dfrac{f(x)-f(1/2)}{x-1/2} = 6$$ </p>
<p>Can anyone... | copper.hat | 27,978 | <p>Let $f_+(x)= 8x^3-1$, $f_-(x) = 1-8x^3$, these are both smooth.</p>
<p>In the following $f$ is the function the OP defined. (I am not redefining $f$.)</p>
<p>If $x \ge { 1\over 2}$ then $f(x) = f_+(x)$.</p>
<p>If $x \le { 1\over 2}$ then $f(x) = f_-(x)$.</p>
<p>Hence the one sided derivatives of $f$ will match t... |
422,143 | <p>f differentiable function in R. $f(x)= e^{f'(x)}$
$f(0)=1$</p>
<p>I have proved that $f(x)=1$ for every $x\lt0$. im stuck for $x\gt0
$</p>
| TZakrevskiy | 77,314 | <p>Take $f(x)=1$ and then apply <a href="http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" rel="nofollow">Picard–Lindelöf theorem</a></p>
<p><strong>Edit.</strong></p>
<p>Well, for short, this is the theorem that guarantees local existence and unicity of solutions of Cauchy problems for ODE (i.e. one... |
19,880 | <p>I want to write down $\ln(\cos(x))$ Maclaurin polynomial of degree 6. I'm having trouble understanding what I need to do, let alone explain why it's true rigorously.</p>
<p>The known expansions of $\ln(1+x)$ and $\cos(x)$ gives:</p>
<p>$$\forall x \gt -1,\ \ln(1+x)=\sum_{n=1}^{k} (-1)^{n-1}\frac{x^n}{n} + R_{k}(x... | Chris Card | 1,470 | <p>Try starting from the definition of the MacLaurin series (e.g. as defined here: <a href="http://mathworld.wolfram.com/MaclaurinSeries.html" rel="nofollow">http://mathworld.wolfram.com/MaclaurinSeries.html</a>)?</p>
|
359,212 | <p>I mean, $\Bbb Z_p$ is an instance of $\Bbb F_p$, I wonder if there are other ways to construct a field with characteristic $p$?
Thanks a lot!</p>
| Jim | 56,747 | <p>There are also extensions of $\mathbb F_p$. For example $\mathbb F_2[t]/(t^2 + t + 1)$ is a field with $4$ elements. It has characteristic $2$.</p>
<p>You might try perusing the <a href="http://en.wikipedia.org/wiki/Finite_field">wikipedia page</a> which has more examples.</p>
|
359,212 | <p>I mean, $\Bbb Z_p$ is an instance of $\Bbb F_p$, I wonder if there are other ways to construct a field with characteristic $p$?
Thanks a lot!</p>
| azimut | 61,691 | <p>For each power $p^r$ of $p$, there is a unique finite field of characteristic $p$ and size $p^r$, it is denoted by $\mathbb F_{p^r}$.
For the construction, you take an irreducible polynomial $f\in\mathbb F_p[x]$ of degree $r$. Then $\mathbb F_{p^r} \cong \mathbb F_p[x]/(f)$.
The smallest non-trivial example is $\mat... |
359,212 | <p>I mean, $\Bbb Z_p$ is an instance of $\Bbb F_p$, I wonder if there are other ways to construct a field with characteristic $p$?
Thanks a lot!</p>
| Seirios | 36,434 | <p>As said in the other answers, if $P$ is an irreducible polynomial on $\mathbb{F}_p$ of degree $d$, then $k=\mathbb{F}_p[X]/ (P)$ is a field of cardinality $p^d$ and is of characteristic $p$. For a nice presentation of $k$, you can say that $k=\mathbb{F}_p(A) \subset M_d(\mathbb{F}_p)$ where $A$ is the <a href="http:... |
2,571,031 | <p>I need to solve the quadratic programming problem $$ \text{minimize}\,\, \sum_{j=1}^{n}(x_{j})^{2} \\ \text{subject to}\,\,\, \sum_{j=1}^{n}x_{j}=1,\\ 0 \leq x_{j}\leq u_{j}, \, \, j=1,\cdots , n $$</p>
<p>I know that the first thing I need to do is form the Lagrangian. </p>
<p>Now, for a problem in standard form... | max_zorn | 506,961 | <p>I ran into space limitations, so here is what I wanted as a comment:</p>
<p>Hi @ALannister: </p>
<p>1] if the $u_i$'s are large, then the solution is $\tfrac{1}{n}\mathbf{e}_n$, where $\mathbf{e}_n$ is the vector of all $1$'s in $\mathbb{R}^n$. </p>
<p>2] view your problem as a projection problem: you want to pro... |
1,988,420 | <p>An Ant is on a vertex of a triangle. Each second, it moves randomly to an adjacent vertex. What is the expected number of seconds before it arrives back at the original vertex?</p>
<p>My solution: I dont know how to use markov chains yet, but Im guessing that could be a way to do this. I was wondering if there was ... | Graham Kemp | 135,106 | <p>The ant moves $1$ time, then moves another $N$ times to return to the original vertex. After the first move, the ant either takes one more move back to the original vertex, or it moves to the other non-original vertex, then continues trying from there. Recursively then:</p>
<p>$$\mathsf E(N) ~=~ \tfrac 12+\t... |
512,037 | <p>This is a question from our reviewer for our exam for linear algebra. I just want to have some ideas how to tackle the problem.</p>
<p>If $A$ is an $n\times n$ matrix with integer coefficients, such that the sum of each row's elements is equal to $m$, show that $m$ divides the determinant.</p>
| achille hui | 59,379 | <p>Given any $n \times n$ matrix $A = (a_{ij})$ whose row sums all equal to $m$, i.e.</p>
<p>$$\sum_{j=1}^n a_{ij} = m, \quad\text{ for } 1 \le i \le n \tag{*1}$$</p>
<p>Let $\text{adj}(A) = (b_{ij})$ be its <a href="http://en.wikipedia.org/wiki/Adjugate_matrix" rel="nofollow">adjugate matrix</a>.
We know</p>
<p>$$\... |
897,633 | <p><strong>First question:</strong></p>
<p>Let's say we have a hypothesis test:</p>
<p>${ H }_{ 0 }:u=100$
and ${ H }_{ 1 }:u\neq 100$.</p>
<p>The sample has a size of 10 and gives an average $u=103$ and a p-value = 0.08.
The level of significance is 0.05.</p>
<p>I'm asked the following question (exam):</p>
<p>A) ... | user99680 | 99,680 | <p>Aren't you also given either the population or sample deviation? If any of these are given, you assume the data is distributed a certain way, e.g., normally, with $\mu=100, \sigma=\sigma_0$. Knowing this, i.e., the (assumed) parameters of the population you are sampling from, then allows you to compute the probabil... |
3,193,305 | <p>A random variable <span class="math-container">$x$</span> from the set <span class="math-container">$\{1, 2, ... ,n\}. $</span> Let <span class="math-container">$x$</span> has distribution function <span class="math-container">$f(k) = Y(n) · g^k$</span> where <span class="math-container">$g$</span> is a fixed numbe... | Robert Lewis | 67,071 | <p>Given the equation</p>
<p><span class="math-container">$y'' + a(x)y = 0, \tag 1$</span></p>
<p>with Wronskian</p>
<p><span class="math-container">$W[y_1, y_2] = y_1y_2' - y_1'y_2, \tag 2$</span></p>
<p>we have</p>
<p><span class="math-container">$W' = (y_1y_2' - y_1'y_2)' = y_1'y_2' + y_1y_2'' - y_1''y_2 - y_1'... |
1,464,522 | <blockquote>
<p>Let <span class="math-container">$O_n(\mathbb Z)$</span> be the group of orthogonal matrices (matrices <span class="math-container">$B$</span> s.t. <span class="math-container">$BB^T=I$</span>) with entries in <span class="math-container">$\mathbb Z$</span>.<br>
1) How do I show that <span class="ma... | Empy2 | 81,790 | <p>HINT: Every column vector has length $1$, and all $b_{ij}$ are integers, so exactly one of $b_{i1}$ is $\pm1$ and all the others are zero.</p>
|
1,652,165 | <p>On a empty shelf you have to arrange $3$ cans of soup, $4$ cans of beans, and $5$ cans of tomato sauce. What is the probability that none of the cans of soup are next to each other?</p>
<p>I tried working this out but get very stuck because I'm not sure that I'm including all the possible outcomes. </p>
| Ross Millikan | 1,827 | <p>I would use inclusion/exclusion. There are $12!$ ways to arrange the cans. To have two soup next to each other, group two cans of soup into a pair. There are six ways to do that, then $11!$ ways to order the cans with the pair together. We have counted the ways to have all three together twice, however, which is... |
1,640,285 | <p>A single-celled spherical organism contains $70$% water by volume. If it loses $10$% of its water content, how much would its surface area change by approximately?</p>
<ol>
<li>$3\text{%}$</li>
<li>$5\text{%}$</li>
<li><p>$6\text{%}$</p></li>
<li><p>$7\text{%}$</p></li>
</ol>
| Harish Chandra Rajpoot | 210,295 | <p>Let $r$ be the original radius the original volume of water is $$V_1=\frac{70}{100}\times \frac{4}{3}\pi r^3=\frac{14}{15}\pi r^3$$ & volume of organic material
$$V_2=\frac{30}{100}\times \frac{4}{3}\pi r^3=\frac{2}{5}\pi r^3$$
& surface area $S_0=4\pi r^2$</p>
<p>when it loses $10$% of water then remaini... |
990,796 | <p>I have a homework question in a discrete mathematics class that asks me to determine how many 7-digit id numbers <strong>do not</strong> contain three consecutive sixes. </p>
<p>It seems clear that I should approach this by determining the number that <strong>do</strong> have three consecutive sixes and subtracting... | Tim | 171,636 | <p>This short python script:</p>
<pre><code>for i in range (10**6,(10**7)-1):
if '666' in str(i):
x += 1
</code></pre>
<p>spat out an answer of 42291 at the end!</p>
<p>It works by getting every number from 0 to $7^{10}$ (282475249), not 0 to 9,999,999. I am not sure which you are looking for. This may b... |
1,684,741 | <p>I'm able to show it isn't absolutely convergent as the sequence $\{1^n\}$ clearly doesn't converge to $0$ as it is just an infinite sequence of $1$'s. How do I prove the series isn't conditionally convergent to prove divergence!</p>
| Obinna Nwakwue | 307,490 | <p>As far as I know, here is a way you can do this. Let a set $F$ be equal to: $$F = \{1, 16, 81, 256, 625, 1296, \cdot \cdot \cdot \}$$ which is every positive perfect fourth power. Now, let $d = y^4$ in which $d \in F$. Now, you are going to have to test each number (sorry, but guess and check is the only way I know ... |
2,972,950 | <p>Everything on this question is in complex plane.</p>
<p>As the book describes a property of a winding number, it says that:</p>
<blockquote>
<p>Outside of the [line segment from <span class="math-container">$a$</span> to <span class="math-container">$b$</span>] the function <span class="math-container">$(z-a) / ... | d0SO'N | 605,549 | <p>Sorry, I misread the question at first; thanks to Eric for pointing it out. Also see his answer for the geometric perspective.</p>
<p>Anyways, if the statement were true, then there exists <span class="math-container">$c$</span> such that <span class="math-container">$k(c - b) = c - a$</span> for a real number <spa... |
1,301,116 | <p>We know that if $f \in \mathcal R[a,b]$ and if $a = c_0 < c_1<\cdots<c_m =b$, then the restrictions of $f$ to each subinterval $[c_{i-1},c_i]$ are Riemann integrable.</p>
<p>Is the converse true, i.e if $f ; [a,b] \to \Bbb R$, and $a = c_0 < c_1<\cdots<c_m =b$ and that the restrictions of $f$ to e... | Tim Raczkowski | 192,581 | <p>Must be true. Probably easiest to show that if $f$ is not Riemann integrable over some $[c_{i-1},c_i]$, then it is not integrable over $[a,b]$.</p>
|
25,488 | <p>I have noticed a common pattern followed by many students in crisis:</p>
<ul>
<li>They experience a crisis or setback (injury, illness, tragedy, etc)</li>
<li>This causes them to miss a lot of class.</li>
<li>They may stay away from class longer than they "need to" because of shame: they feel that since t... | Stef | 16,159 | <p>I've personally known two students who went through such a crisis, and didn't flunk. What set them apart from what you describe is that in both cases, they advised the university of their personal issue when it happened, and were able to come up with an acceptable plan together with the university.</p>
<p>This compl... |
25,488 | <p>I have noticed a common pattern followed by many students in crisis:</p>
<ul>
<li>They experience a crisis or setback (injury, illness, tragedy, etc)</li>
<li>This causes them to miss a lot of class.</li>
<li>They may stay away from class longer than they "need to" because of shame: they feel that since t... | Tommi | 2,083 | <p>One obvious solution to the problem is obligatory student presence at contact teaching (maybe 80 %). This will make students concerned about being present and can stop the process where shame at not having been there causes more absence.</p>
<p>In particular, an attendance policy usually carries with it an assumptio... |
3,509,441 | <p>Given a Complex Matrix <span class="math-container">$A$</span> which is <span class="math-container">$n \times n$</span>. How would I go about showing that <span class="math-container">$A^*A$</span> is <span class="math-container">$$\sum_{i=1}^n \sum_{j = 1}^n | a_{ij} |^2$$</span></p>
<p>Here <span class="math-con... | Vsotvep | 176,025 | <p>Here's an alternative proof, using induction.</p>
<p>For <span class="math-container">$n=0$</span> the case is easy: there is only one pair of disjoint subsets of <span class="math-container">$\varnothing$</span>, namely <span class="math-container">$(\varnothing,\varnothing)$</span>.</p>
<p>Suppose that there are... |
438,263 | <p>Is there a concrete example of a <span class="math-container">$4$</span> tensor <span class="math-container">$R_{ijkl}$</span> with the same symmetries as the Riemannian curvature tensor, i.e.
<span class="math-container">\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} = R_{klij}, \\
R... | Ofir Gorodetsky | 31,469 | <p>If you can motivate the problem and make some partial progress on it, you can try and publish it as a paper in a specialized journal, or at the very least upload it to the arXiv.</p>
<p>If you only have empirical evidence, there are journals that are receptive to this kind of this ("Mathematics of Computation&q... |
1,315,805 | <blockquote>
<p>Let the series $$\sum_{n=1}^\infty \frac{2^n \sin^n x}{n^2}$$. For $x\in (-\pi/2, \pi/2)$, when is the series converges?</p>
</blockquote>
<p>By the root-test:</p>
<p>$$\sqrt[n]{a_n} = \sqrt[n]{\frac{2^n\sin^n x}{n^2}} = \frac{2\sin x}{n^{2/n}} \to 2\sin x$$</p>
<p>Thus, the series converges $\iff ... | T. Eskin | 22,446 | <p>It's almost done. Remember that the root test has absolute values inside the root, so you're looking at $\lim_{n\to\infty}|a_{n}|^{\frac{1}{n}}$. As you noticed, this gives us the convergence in $-1<2\sin x<1$. For the boundary value $2\sin x=1$ you have the convergent series $\sum n^{-2}$, and for the boundar... |
36,774 | <p>Do asymmetric random walks also return to the origin infinitely?</p>
| athos | 26,632 | <p>No, it doesn't.</p>
<p>For a random walk, consider point of view $v_k$ as:</p>
<ol>
<li>let $+1$ and $-k$ be the two "fozen points", $p_k$ is the probability hitting $+1$, $1-p_k$ hitting $-k$, or, equivalently</li>
<li>let $-1$ and $+k$ be the two "fozen points", $q_k$ is the probability hitting $-1$, $1-q_k$ hit... |
296,727 | <p><b>Assuming that G is a finite cyclic group, let "a" be the product of all the elements in the group.</b> </p>
<p>i. <b> If G has odd order, then a=e.</b> Is this because there are an even number of non-trivial elements must have their inverses within the non-trivial factors within the product?</p>
<p>ii. <b> If G... | Calvin Lin | 54,563 | <p>Since we know it is a finite cyclic group,</p>
<p><strong>Hint:</strong> $ 1 + 2 + \ldots + n = \frac {n (n+1)}{2}$</p>
<p>This is a multiple of $n$ if $n$ is odd, and not a multiple of $n$ if $n$ is even.</p>
<hr>
<p>Your statements are true, and don't just apply to a finite cyclic group. I would prefer your so... |
3,062,597 | <p>For the statement of Rouché's theorem, I've always seen that both <span class="math-container">$f$</span> and <span class="math-container">$g$</span> have to be holomorphic on and inside a simple closed curve <span class="math-container">$ C $</span>. However, I am solving a problem which seems to suggest that I sho... | Lutz Lehmann | 115,115 | <p>Yes, it is sufficient since by continuity the inequality assumption of Rouché's theorem extends to some neighborhood inside the boundary curve, and thus inside the holomorphic domain. In other words, shift the curve along some inside normal vector field a little bit, which is possible because of the compactness of t... |
4,620,319 | <p>Let's assume that for <span class="math-container">$0<\beta<\alpha<\frac{\pi}{2}$</span>, <span class="math-container">$\sin(\alpha+\beta) = \frac{4}{5}$</span>, and <span class="math-container">$\sin(\alpha-\beta) = \frac{3}{5}$</span>. Then, how could we find <span class="math-container">$\cot(\beta)$</sp... | Lai | 732,917 | <p><span class="math-container">$$
\begin{aligned}
& \sin ^2(\alpha+\beta)+\sin ^2(\alpha-\beta)=\frac{16}{25}+\frac{9}{25}=1 \\
\Rightarrow \quad & \sin ^2(\alpha+\beta)=1-\sin ^2(\alpha-\beta)=\cos ^2(\alpha-\beta)\\ \Rightarrow \quad &
(\sin \alpha \cos \beta+\sin \beta \cos \alpha)^2=(\cos \alpha \cos ... |
234,851 | <p>Find the length of the curve $x=0.5y\sqrt{y^2-1}-0.5\ln(y+\sqrt{y^2-1})$ from y=1 to y=2.</p>
<p>My attempt involves finding $\frac {dy}{dx}$ of that function first, which leaves me with a massive equation.</p>
<p>Next, I used this formula, </p>
<p>$$\int_1^2\sqrt{1+(\frac{dy}{dx})^2}$$</p>
<p>this attempt leave... | mythealias | 31,292 | <p>You should get $$\frac{dx}{dy} = \sqrt{y^2 - 1}$$
So make sure that you are not making a mistake there.</p>
<p>From there use the correct equation for $L$ as mentioned by Vafa.</p>
|
234,851 | <p>Find the length of the curve $x=0.5y\sqrt{y^2-1}-0.5\ln(y+\sqrt{y^2-1})$ from y=1 to y=2.</p>
<p>My attempt involves finding $\frac {dy}{dx}$ of that function first, which leaves me with a massive equation.</p>
<p>Next, I used this formula, </p>
<p>$$\int_1^2\sqrt{1+(\frac{dy}{dx})^2}$$</p>
<p>this attempt leave... | preferred_anon | 27,150 | <p>We have $$2x=y\sqrt{(y^{2}-1)}-\ln(y+\sqrt{y^{2}
-1})$$
Make the substitution $y=\cosh(u)$. Over the interval you are concerned with,
$$2x=\cosh(u)\sinh(u)-u=\frac{1}{2}\sinh(2u)-u$$
Note that $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}=\sinh(u)\frac{du}{dx}$. Differentiate the above with repect to $x$:
$$2=\cosh(2u)\... |
3,140,696 | <p>I am trying to figure out the proper definition of a small circle on a biaxial ellipsoid of revolution. One definition is the intersection of the ellipsoid with a cone emanating from the center of the ellipsoid.</p>
<p>The other way I can imagine to define it is a plane intersecting the ellipsoid in which the plane... | Intelligenti pauca | 255,730 | <p>The intersection with a plane is, in general, an ellipse. The intersection with a cone is a more complicated curve, in general not lying on a plane. If what you want is a planar circle all lying on the ellipsoid, then I'm afraid that is not possible, unless particular conditions are met (e.g. a rotational symmetric ... |
165,489 | <p>I have problem solving this equation, smallest n such that $1355297$ divides $10^{6n+5}-54n-46$. I tried everything using my scientific calculator, but I never got the correct results(!).and finally I gave up!. Could you help me find the first 2 solutions for this equation ? (thanks.)</p>
| Michael E2 | 4,999 | <p>A more <em>Mathematica</em> way to do an exhaustive search:</p>
<pre><code>pp = 1355297;
Pick[#,
Mod[10^5 NestList[Mod[10^6 #, pp] &, 10^6, Length@# - 1] - (54 n + 46 /. n -> #), pp],
0] &@ Range[10^7] // AbsoluteTiming
(* {0.665031, {2331259, 3776127, 5366598, 5505709, 5652052, 7317951, 8306396, 895... |
4,452,885 | <p>Let <span class="math-container">$z = e^{i\theta}, \theta \in \mathbb{R}$</span>. Then, does there exist <span class="math-container">$n \in \mathbb{N}$</span> such that:</p>
<p><span class="math-container">$$1 - z^n = re^{2 \pi i \tau}$$</span></p>
<p>for some <span class="math-container">$\tau \in \mathbb{Q}$</sp... | Conrad | 298,272 | <p><span class="math-container">$\arg (1-z^n)=n\theta/2-\pi/2=2\pi \tau+2k\pi$</span> for some <span class="math-container">$k \in \mathbb Z$</span> so <span class="math-container">$\theta=\frac{4\pi (\tau+k+1/4)}{n}$</span> is a rational multiple of <span class="math-container">$\pi$</span></p>
|
2,716,363 | <p>I understand the core principles of how to prove by induction and how series summations work. However I am struggling to rearrange the equation during the final (induction step).</p>
<p>Prove by induction for all positive integers n,</p>
<p>$$\sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2$$</p>
<p>After both proving fo... | E-A | 499,337 | <p>They factored out the $1/4 (k+1)^2$. So, from the first part of that sum, they got a $k^2$, and second part, they got a $4 (k+1)$. Just multiply it through if you want to see why it holds.</p>
|
696,511 | <p>Find the absolute maximum and minimum values of the function:</p>
<p>$$f(x,y)=2x^3+2xy^2-x-y^2$$</p>
<p>on the unit disk $D=\{(x,y):x^2+y^2\leq 1\}$.</p>
| MGA | 60,273 | <p>We first take the partial derivatives with respect to each variable and set them to zero:</p>
<p>$$\frac{\partial f}{\partial x}=6x^2 + 2y^2 - 1=0$$</p>
<p>$$\frac{\partial f}{\partial y}=4xy - 2y=0$$</p>
<p>From the second equation, we have either: $y=0$ or $x=\frac{1}{2}$. Now we substitute each value into the ... |
5,528 | <p>Let H be a subgroup of G. (We can assume G finite if it helps.) A complement of H in G is a subgroup K of G such that HK = G and |H∩K|=1. Equivalently, a complement is a transversal of H (a set containing one representative from each coset of H) that happens to be a group.</p>
<p>Contrary to my initial naive... | Steve D | 1,446 | <p>Version 2 of "<a href="https://arxiv.org/abs/math/0703471v2" rel="nofollow noreferrer">Factorization problems for finite groups</a>", which discusses bicrossed products (knit products), gives an elementary introduction, classifies some examples, and shows that $A_6$ cannot be written as such a product.</p>
|
508,104 | <p>I want to understand more about this proof from Lang's Algebra:</p>
<p>Let $B$ be a subgroup of a free abelian group $A$ with basis $(x_i)_{i=1...n}$. It has already been shown that $B$ has a basis of cardinality $\leq n$.</p>
<blockquote>
<p>...
We also observe that our proof shows that there exists at leas... | Boris Novikov | 62,565 | <p>1) "$r$-fold" means a direct sum of $r$ exemplars of a group.</p>
<p>2) Every element of $B/pB$ has the order $p$. It is known that such Abelian group is a direct sum of cyclic groups of order $p$.</p>
|
3,165,460 | <p>I am reading a survey on Frankl's Conjecture. It is stated without commentary that the set of complements of a union-closed family is intersection-closed. I need some clearer indication of why this is true, though I guess it is supposed to be obvious. </p>
| Eric Wofsey | 86,856 | <p>There are no uncountable scattered subsets of <span class="math-container">$[0,1]$</span>. This follows from the theory of Cantor-Bendixson rank, for instance. Given any scattered <span class="math-container">$A\subset[0,1]$</span>, there must be some ordinal <span class="math-container">$\alpha$</span> such that ... |
960,865 | <p>How can i prove 2 is a primitive root mod 37, without calculating all powers of 2 mod 37?</p>
| Jyrki Lahtonen | 11,619 | <p>As Peter Taylor explained, it suffices to verify that neither $2^{12}$ nor $2^{18}$ is congruent to $1$ modulo $37$. </p>
<p>If $2^{18}$ were $\equiv1\pmod{37}$, then (because there exists a primitive root) two would have to be a quadratic residue modulo $37$. But $37\equiv 5\pmod8$, so one of the supplements to th... |
1,318,462 | <p>I am struggling with the following problem.Any help will be appreciated.</p>
<p>If the following statement true then please give a proof otherwise give a counterexample.</p>
<ol>
<li><p>If $a^{27} \equiv 1 \pmod{37}$, then $a^9 \equiv 1 \pmod{37}$ </p></li>
<li><p>$a^{9} \equiv 1 \pmod{37}$, then $a^3 \equiv 1 \pm... | Bill Dubuque | 242 | <p>$(1)\ \ $ By Fermat $\ 1 \equiv a^{36}\equiv a^{27} a^9\,$ so $\,a^{27}\equiv 1\,\Rightarrow\,a^9\equiv 1$ </p>
<p>$(3)\ \ $ Similarly $\ a^{36}\equiv 1\equiv a^5\,\Rightarrow\ a = a^{36}/(a^5)^7\equiv 1$</p>
<p>$(2)\ \ $ By Fermat $\,(2^4)^9\equiv 1\,$ but $\,(2^4)^3\equiv 8^4\equiv (-10)^2\equiv -11\not\equiv 1... |
2,155,180 | <p>Let $f,g$ be analytic on some domain $\Omega \subset \mathbb{C}$. By Cauchy's formula, we have
$$
\frac{1}{2\pi i} \oint_{\partial\Omega}
\frac{f(z) \, g(z)}{z - z_0}
\, dz
=
f(z_0) \, g(z_0)
=
-\frac{1}{4\pi^2}
\oint_{\partial\Omega}
\frac{f(u)}{u - z_0}
\, du
\,
\oint_{\partial\Omega}
\frac{g(v)}{v - z_0}
\, d... | Manuel Guillen | 416,103 | <p>\begin{align*}
20x &\equiv 49 &\pmod{23} \\
-3x &\equiv 3 &\pmod{23} \\
x &\equiv -1 &\pmod{23} \\
&\equiv 22 &\pmod{23}
\end{align*}</p>
|
343,281 | <p>Consider the following note written by Gerhard Gentzen in early 1932, on the onset of his work on a consistency proof for arithmetic:</p>
<blockquote>
<p>The axioms of arithmetic are obviously correct, and the principles of proof obviously preserve correctness. Why cannot one simply conclude consistency, i.e., w... | Panu Raatikainen | 102,468 | <p>Gentzen's remark has some bite in the case the standard first-order arithmetic PA, because we plausibly know a bit more arithmetically.
But he apparently did not understand the generality of the incompleteness theorems: they hold for any theory which includes elementary arithmetic and happens to be consistent. In g... |
13,835 | <p>Given that $$X = \{(x,y,z) \in \mathbb{R}^3 |\, x^2 + y^2 + z^2 - 2(xy + xz + yz) = k\}\,,$$ where $k$ is a constant. Also given that a group $G$ is represented by $$\langle g_1,g_2,g_3|\, g_1^2 = g_2^2 = g_3^2 = 1_G\rangle\,.$$ $G$ acts on $X$ such that $$g_1 \cdot (x,y,z) = (2(y+z) - x,y,z)\,,$$ $$g_2 \cdot (x,y,z... | whuber | 91 | <p>Because the image of the group under this (linear) representation is infinite, we will need to limit the orbits. </p>
<h3>Working in the abstract group</h3>
<p>Presuming it may eventually be of interest to depict multiple orbits, let's compute a large number of group elements once and for all. It seems efficient... |
3,039,040 | <p>In the equation <span class="math-container">$3^x=2y^2-1$</span>,
<span class="math-container">$x$</span>, <span class="math-container">$y$</span> are natural numbers.
I found <span class="math-container">$x=1$</span> or <span class="math-container">$2$</span> (mod <span class="math-container">$4$</span>), and <span... | YiFan | 496,634 | <p>Take both sides of the equation modulo <span class="math-container">$3$</span>. If <span class="math-container">$x\geq 1$</span> then <span class="math-container">$2y^2\equiv 1$</span> modulo <span class="math-container">$3$</span>, but if <span class="math-container">$y\equiv 0$</span>, then <span class="math-conta... |
11,973 | <p>I have a list of strings called <code>mylist</code>:</p>
<pre><code>mylist = {"[a]", "a", "a", "[b]", "b", "b", "[ c ]", "c", "c"};
</code></pre>
<p>I would like to split <code>mylist</code> by "section headers." Strings that begin with the character <code>[</code> are section headers in my application. Thus, I ... | faysou | 66 | <p>Here's an answer based on the solution of Murta that parses recursively a list based on different delimiters that can be patterns or string patterns. This can be useful for example to parse a debug output where loops are involved.</p>
<pre><code>splitByPattern[l_List,p_?System`Dump`validStringExpressionQ]:=splitByP... |
11,973 | <p>I have a list of strings called <code>mylist</code>:</p>
<pre><code>mylist = {"[a]", "a", "a", "[b]", "b", "b", "[ c ]", "c", "c"};
</code></pre>
<p>I would like to split <code>mylist</code> by "section headers." Strings that begin with the character <code>[</code> are section headers in my application. Thus, I ... | Lou | 150 | <p>Here's my version based on Position.</p>
<pre><code>mylist = {"[a]", "a", "a", "[b]", "b", "b", "[ c ]", "c", "c"};
split[lst_List, pat_String] := Module[{len, pos},
len = Length[lst];
pos = Partition[Flatten[{Position[lst, _String?(StringMatchQ[#, pat ~~ __] &)],len + 1}], 2, 1];
lst[[#[[1]] ;; #[[2]] -... |
2,083,460 | <p>While trying to answer <a href="https://stackoverflow.com/questions/41464753/generate-random-numbers-from-lognormal-distribution-in-python/41465013#41465013">this SO question</a> I got stuck on a messy bit of algebra: given</p>
<p>$$
\log m = \log n + \frac32 \, \log \biggl( 1 + \frac{v}{m^2} \biggr)
$$</p>
<p>I n... | Jan Eerland | 226,665 | <p>Assuming that all variables are real and positive:</p>
<p>$$\ln\text{m}=\ln\text{n}+\frac{3}{2}\cdot\ln\left(1+\frac{\text{v}}{\text{m}^2}\right)\space\Longleftrightarrow\space\frac{2}{3}\cdot\left(\ln\text{m}-\ln\text{n}\right)=\ln\left(1+\frac{\text{v}}{\text{m}^2}\right)$$</p>
<p>Now, use:</p>
<p>$$\ln\text{a}... |
2,083,460 | <p>While trying to answer <a href="https://stackoverflow.com/questions/41464753/generate-random-numbers-from-lognormal-distribution-in-python/41465013#41465013">this SO question</a> I got stuck on a messy bit of algebra: given</p>
<p>$$
\log m = \log n + \frac32 \, \log \biggl( 1 + \frac{v}{m^2} \biggr)
$$</p>
<p>I n... | projectilemotion | 323,432 | <p>$$\log {m}=\log {n}+\frac{3}{2} \log {\left(1+\frac{v}{m^2}\right)}$$
$$\log {m}=\log {n}+ \log {\left(\left(1+\frac{v}{m^2}\right)^{\frac{3}{2}}\right)}$$
$$\log {m}=\log {\left({n\cdot\left(1+\frac{v}{m^2}\right)^{\frac{3}{2}}}\right)}$$
Exponentiate both sides:
$$m=n\cdot\left(1+\frac{v}{m^2}\right)^{\frac{3}{2}}... |
2,593,627 | <p>I struggle to find the language to express what I am trying to do. So I made a diagram.</p>
<p><a href="https://i.stack.imgur.com/faHgE.png" rel="noreferrer"><img src="https://i.stack.imgur.com/faHgE.png" alt="Graph3parallelLines"></a></p>
<p>So my original line is the red line. From (2.5,2.5) to (7.5,7.5).</p>
<... | Narasimham | 95,860 | <p>You forgot marking the axes.</p>
<p>In the straight line $y=mx+c$, the $c$ or $y$ coordinate part to be added is $\dfrac{1}{\sqrt2}= \sin 45^{\circ}$ if you want shift up parallelly,</p>
<p>and it is $\dfrac{-1}{\sqrt2}$ if you want shift down parallelly. Here slope $m=1$;</p>
<p>So red line</p>
<p>$$ y= m x +0... |
233,238 | <p>I am just practicing making some new designs with Mathematica and I thought of this recently. I want to make a tear drop shape (doesn't matter the orientation) constructed of mini cubes. I am familiar with the preliminary material, I am just having some difficulty getting it to work.</p>
| kglr | 125 | <p>A few additional methods:</p>
<pre><code>Level[#, {-1}] & /@ list
Apply[Sequence, list, {-2}]
Map[Splice, list, {-2}]
Join[{#}, #2] & @@@ list
Prepend[#2, #]& @@@ list
MapAt[Splice, {All, 2}] @ list
ReplacePart[{_, 2, 0} -> Sequence] @ list
Map[Map @ Apply @ Sequence] @ list
Delete[{2, 0}] /@ ... |
1,873,180 | <p>The final result should be $C(n) = \frac{1}{n+1}\binom{2n}{n}$, for reference.</p>
<p>I've worked my way down to this expression in my derivation:</p>
<p>$$C(n) = \frac{(1)(3)(5)(7)...(2n-1)}{(n+1)!} 2^n$$</p>
<p>And I can see that if I multiply the numerator by $2n!$ I can convert that product chain into $(2n)!$... | Gregory Simon | 208,825 | <p>$\int f(x)\, dx$ generally means the set of all anti-derivatives of $f(x)$.</p>
<p>If you have a set $A$ and a set $B$ then in this context $A-B$ should be interpreted as $\{a-b:\ a\in A, \ b\in B\}$.</p>
<p>Therefore, $\int f(x)\, dx - \int f(x)\, dx = \mathbb R$ (assuming we are talking about real integrable fun... |
3,869,990 | <p><span class="math-container">$\newcommand{\C}{\mathcal{C}}$</span>
<span class="math-container">$\newcommand{\D}{\mathcal{D}}$</span>
<span class="math-container">$\newcommand{\A}{\mathcal{A}}$</span>
<span class="math-container">$\newcommand{\S}{\mathcal{S}}$</span>
<span class="math-container">$\newcommand{\Psh}{\... | fosco | 685 | <p>You <span class="math-container">$F_1$</span> is just the left Kan extension along <span class="math-container">$F^\text{op}$</span>; it is going to be fully faithful for a general fact about Kan extensions: extending along a fully faithful functor gives an isomorphism <span class="math-container">$H\cong Lan_F(HF)$... |
3,869,990 | <p><span class="math-container">$\newcommand{\C}{\mathcal{C}}$</span>
<span class="math-container">$\newcommand{\D}{\mathcal{D}}$</span>
<span class="math-container">$\newcommand{\A}{\mathcal{A}}$</span>
<span class="math-container">$\newcommand{\S}{\mathcal{S}}$</span>
<span class="math-container">$\newcommand{\Psh}{\... | Béranger Seguin | 290,401 | <p><span class="math-container">$$ \newcommand{\C}{\mathcal{C}} $$</span>
<span class="math-container">$$ \newcommand{\D}{\mathcal{D}} $$</span>
<span class="math-container">$$ \newcommand{\Hom}{\mathrm{Hom}} $$</span>
<span class="math-container">$$ \newcommand{\Psh}{\mathrm{Psh}} $$</span></p>
<p>Thanks a lot to Fosc... |
1,937,762 | <p>Let's say I have a ratio of polynomials as follows</p>
<p>$P(x)=\frac{a_0x^n+a_1x^{n-2}+a_2x^{n-4}+...}{b_0x^n+b_1x^{n-2}+b_2x^{n-4}+...}$.</p>
<p>The polynomials are finite. Is there a procedure to convert it into a polynomial</p>
<p>$P(x) = A_0 + A_1 f(x) + A_2g(x) + ...$</p>
<p>where $f(x)$ and $g(x)$ are som... | Andrew D. Hwang | 86,418 | <p>Let's re-index the polynomials, writing
$$
a(x) = \sum_{j=0}^{n} a_{j} x^{j},\qquad
b(x) = \sum_{j=0}^{m} b_{j} x^{j},\qquad
P(x) = \frac{a(x)}{b(x)}.
$$
(That is, let the coefficient index <em>match</em> the power of $x$; start with the constant terms.)</p>
<ol>
<li><p>If $b_{0} = b(0) \neq 0$, the rational functi... |
3,858,517 | <p>Is it possible to count exactly the number of binary strings of length <span class="math-container">$n$</span> that contain no two adjacent blocks of 1s of the same length? More precisely, if we represent the string as <span class="math-container">$0^{x_1}1^{y_1}0^{x_2}1^{y_2}\cdots 0^{x_{k-1}}1^{y_{k-1}}0^{x_k}$</s... | G Cab | 317,234 | <p>Consider a binary string with <span class="math-container">$s$</span> ones and <span class="math-container">$m$</span> zeros in total.<br />
Let's put an additional (dummy) fixed zero at the start and at the end of the string.
We individuate as a <em>run</em> the consecutive <span class="math-container">$1$</span>'... |
286,798 | <blockquote>
<p>Find the limit $$\lim_{n \to \infty}\left[\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{n^2}\right)\right]$$</p>
</blockquote>
<p>I take log and get $$\lim_{n \to \infty}\sum_{k=2}^{n} \log\left(1-\frac{1}{k^2}\right)$$</p>
| lab bhattacharjee | 33,337 | <p>The $r$th term is $\frac{r^2-1}{r^2}=\frac{(r-1)}r\frac{(r+1)}r$</p>
<p>So, the product of $n$ terms is $$\frac{3.1}{2^2}\frac{4.2}{3^2}\frac{5.3}{4^2}\cdots \frac{(n-1)(n-3)}{(n-2)^2}\frac{(n-2)n}{(n-1)^2}\frac{(n-1)(n+1)}{n^2}$$
$$=\frac12\frac32\frac23\frac43\cdots\frac{n-2}{n-1}\frac n{n-1}\frac{n-1}n\frac{n+1}... |
2,637,812 | <p>Here is dice game question about probability.</p>
<p>Play a game with $2$ die. What is the probability of getting a sum greater than $7$?</p>
<p>I know how the probability for this one is easy, $\cfrac{1+2+3+4+5}{36}=\cfrac 5{12}$.</p>
<p>I don't know how to solve the follow-up question:</p>
<p>Play a game with ... | Prasun Biswas | 215,900 | <p>$$\frac{x^2-\log(1+x^2)}{x^2\sin^2x}=\frac{\dfrac 1{x^2}-\dfrac{\log(1+x^2)}{x^4}}{\dfrac{\sin^2x}{x^2}}$$</p>
<p>Taking $z=x^2$, note that the numerator becomes $$\dfrac 1z-\dfrac{\log(1+z)}{z^2}=\dfrac{z-\log(1+z)}{z^2}\to\dfrac{1-\frac 1{1+z}}{2z}\to\dfrac{1}{2(z+1)^2}\to\frac 12$$</p>
<p>as $z=x^2\to 0$ as $x\... |
919,699 | <p>I am very bad at problems involving expected return and was hoping some one could help me out.</p>
<p>You are offered a chance to play a game for $48 against 99 other players(100 including you) the game consists of 16 rounds and in each round you have 4 chances to win. The first winner picked in each round gets 30 ... | amcalde | 168,694 | <p>Your expected winnings for a round are:
$$(1/100)\cdot 30 + (1/100)\cdot 50 + (1/100)\cdot 30 + (1/100) \cdot 100 = \frac{21}{10}$$</p>
<p>Do this 16 times. You expect to win
$$16 \cdot \frac{21}{10} \$= \frac{168}{5} \$= 33.6 \$$$</p>
<p>Not worth the money.</p>
<p>Payout is $-48\$ + 33.6\$ = -14.4\$$</p>
|
397,040 | <p>What is the domain for $$\dfrac{1}{x}\leq\dfrac{1}{2}$$</p>
<p>according to the rules of taking the reciprocals, $A\leq B \Leftrightarrow \dfrac{1}{A}\geq \dfrac{1}{B}$, then the domain should be simply $$x\geq2$$</p>
<p>however negative numbers less than $-2$ also satisfy the original inequality. When am I missin... | user49685 | 49,685 | <p>No, you are not missing anything, that rule only works for $A; B > 0$, so $\dfrac{1}{x} \le \dfrac{1}{2} \Leftrightarrow \left[ \begin{array}{l} x < 0 \\ x \ge 2 \end{array} \right.$</p>
<p>Of course it's easy to see that for $x < 0$, we'll always have $\dfrac{1}{x} < 0 < \dfrac{1}{2}$.</p>
|
4,058,600 | <p>Please pardon the elementary question, for some reason I'm not grocking why all possible poker hand combinations are equally probable, as all textbooks and websites say. Just intuitively I would think getting 4 of a number is much more improbably than getting 1 of each number, if I were to draw 4 cards. For example,... | 311411 | 688,046 | <p>Often the books and websites speak a little too loosely. I would say that "getting 4 of a kind" is not a poker hand. It is a set of many different poker hands. A "possible poker hand" is completely specific, e.g. the hand 4 of spades, 4 of hearts, 4 of clubs, 7 of clubs, 8 of diamonds.</p>
|
698,743 | <blockquote>
<p>Let the real coefficient polynomials
$$f(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$$
$$g(x)=b_{m}x^m+b_{m-1}x^{m-1}+\cdots+b_{1}x+b_{0}$$
where $a_{n}b_{m}\neq 0,n\ge 1,m\ge 1$, and let
$$g_{t}(x)=b_{m}x^m+(b_{m-1}+t)x^{m-1}+\cdots+(b_{1}+t^{m-1})x+(b_{0}+t^m).$$
Show that</p>
<p>... | Lutz Lehmann | 115,115 | <p>You just need to check that the resultant $r(t)=Res_x(f(x), g_t(x))$, which is a polynomial in $t$, is not the zero polynomial. Any root of $r(t)$ marks a value of $t$ where both polynomials have a common factor. If $r(t)\ne 0$, the polynomials $f(x)$ and $g_t(X)$ are relatively prime.</p>
<p>And as in the other an... |
536,187 | <p>Let $\mathbb{R}^{\omega}$ be the countable product of $\mathbb{R}$. Make t a topological space using the box topology. Let $\pi_{n}$ denote the usual projection maps. </p>
<p>Fix $N \in \mathbb{Z}_+$ and define $A_N = \{x \in \mathbb{R}^\omega$ $|$ $\pi_{k}(x) = 0$ $\forall k>N\}$. Show that $A_N$ is closed in t... | Stefan Hamcke | 41,672 | <p><strong>Hint:</strong> If $y\notin A_N$, then for some $k>N$ we have $\pi_k(y)=r\ne0$. Can you find a box around $y$ which does not intersect $A_N$. Think about what it means for a box $B$ to be disjoint from $A_N$.</p>
|
516,544 | <p>The following is an <a href="http://placement.freshersworld.com/placement-papers/IBM/Placement-Paper-Whole-Testpaper-37851" rel="nofollow">aptitude problem (question no: 29-32)</a>, I am trying to solve:- </p>
<blockquote>
<p>Questions 29 - 32:</p>
<p>A, B, C, D, E and F are six positive integers such that</... | Prasoon Shrivastav | 106,352 | <p>B + C + D + E = 4A ------1</p>
<p>C + F = 3A ----------------2</p>
<p>C + D + E = 2F -----------3</p>
<p>F = 2D --------------------4</p>
<p>E + F = 2C + 1-----------5</p>
<p>Also A can 13,17 or 19</p>
<p>Eliminating D from all the equations we get</p>
<p>B+2F = 4A ------6</p>
<p>C+F = 3A -------7</p>
<p>C+... |
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